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+{"text":"\\section{Introduction}\n5G mobile networks promise to bring a new era of ultra high-speed communications that surpasses previous generations by several order of magnitudes in communication capacity \\cite{boccardi2014five}. One of the core technologies behind such a spectacular revolution is spatial devision multiple access (SDMA). SDMA enables massive Multi-Input-Multi-Output (MIMO) communication by providing an ability to focus energy on users' devices, empowering pushing the capacity of the network to such a immense boundaries required for 5G communications \\cite{agiwal2016next,hejazi2021dyloc}. Simultaneously, mobile mm-wave communication is enabled through 5G networks, that transform directional communication from a promising aspect of next generation networks, into a must-have feature \\cite{agiwal2016next,rappaport2013millimeter}. Mm-wave communication experiences huge attenuation in the open air, therefore the transmitted energy needs to be directed into narrow rays, to meet sufficient signal-to-noise-ratio (SNR) thresholds required at receivers \\cite{rappaport2019wireless}. In addition to 5G applications, DoA estimation is a required aspect of UAV-to-device and satellite-to-device high frequency and ultra high-speed communication \\cite{agrawal20165g}. Moreover, mm-wave and Terahertz radars used for autonomous driving exploit DoA estimation techniques to estimate angles of the objects around \\cite{dickmann2016automotive}. As directional communication has gained importance in new generation communications, DoA estimation has obtained gravity as an enabler of directional communication. To clarify this necessity, consider that any two devices that exploit directional antennas cannot communicate unless they ascertain in which direction they should send\/receive signals to\/from the other device. Moreover, this knowledge of angle (or position) of the other device should be maintained during the communication period otherwise the link will be disrupted \\cite{nitsche2014ieee}. \n\nThe most common DoA estimation techniques use directional antennas mounted on both the transmitter and the receiver to obtain the initial guess of the relative angle between two devices, this process is also referred as initial access (IA) \\cite{giordani2016initial}. To fulfil this strategy, the first device starts searching for the second device through a beam training protocol, until it finds the other device. Next, the second device repeats the same procedure until the link is established, at this point, they employ tracking techniques to maintain the directional connection between them \\cite{giordani2018tutorial}.\nAlthough such a strategy looks favorable for DoA estimation, it is highly probable that it does not work well when a large number of devices are packed into a specific area or in the presence of a strong multi-path between two device. Moreover, beams are most of the time busy with beam training\/tracking searches instead of transmission\/reception which reduces the communication capacity \\cite{giordani2019standalone}. In other words, using the same antenna for communication and direction-finding, requires using a common resource for two inherently antithetical task in terms of directional antenna requirements. Higher communication capacity requires highly directional antennas to reduce interference and to maximize signal power at the receiver, conversely, as antenna's beams become narrower the beam training\/tracking periods increase and consequently the overhead escalates which eventually reduces the effective communication capacity. To overcome deficiencies of such a strategy we propose to avoid using directional antennas for DoA estimation at both sides of the link, and estimate DoA based on measuring phase difference of arrival (PDoA) of signal between two antennas mounted on the device for multiple frequencies. Meanwhile, we can allocate a directional antenna exclusively for communication purposes. In our proposed strategy, we avoid spatial search to establish the link in the first place, on the other hand, we rely on the received signal in two omni-directional antennas. Subsequently, we amplify the attenuated received signal by a huge processing gain, then estimate DoAs of all of propagation paths between two devices.\nWe will show that exploiting our proposed technique, we can convert spatial search duration to a means to increase DoA estimation precision, and more importantly, we can allocate a specific highly directional antenna for communication, and consequently take advantage of the whole communication capacity such a directionality provides. \n\n\nIn our proposed technique, two antennas are mounted on the device with several mm gap between them, and the PDoA of signal measured through a novel technique named standing wave spectrometry for multiple frequencies. Standnig wave spectrometry is widely used in optical applications to measure phase difference between two rays at multiple frequecies of the optical spectrum \\cite{sabry2015monolithic}\\cite{wolffenbuttel2005mems}\\cite{jovanov2010standing}. To the best of authors knowledge, it is the first time that this technique is introduced for RF mm-wave applications. By applying spectrometry not only we can estimate the DoA of a signal precisely, but also we can estimate multi-path DoAs and the power of each path for a mm-wave propagation environment. Although the proposed approach is inherently a wide-band (WB) technique, it does not require ultra high speed sampling rates essential for must of WB techniques. Consequently, the proposed technique provides us with two main advantages: more data about the DoA of incoming signal, and reduced cost and complexity of the receiver. The first is obtained by discriminating between all incoming propagation paths between the source and the device. The second is secured by greatly reducing the complexity of the DoA estimation through simplification of the receiver by bypassing signal down-conversion and reducing the number of required antennas. Furthermore, we show that the proposed phase difference measurements equals to highly accurate measurement of time difference of arrival (TDoA) of signal between two antennas in the Fisher sense. Moreover, we will prove that the cramer-rao lower bound of error (CRLB) of DoA estimation using the proposed technique equals to a uniform linear array (ULA) that employs multiple antennas, in the Fisher sense. \n\n\n\n\\section{Related Works}\n\nDoA estimation techniques have plethora of applications in Radar, Sonar and Electronic Ware-fare (EW) literature. In these applications, DoA estimation is mainly used to find the relative direction between two objects. Primitive DoA estimation techniques use pencil beam antennas (e.g. dish antennas) along with mechanical actuators for steering the beam and spatial search \\cite{skolnik2001radar},\\cite{barshan1992bat},\\cite{poisel2012electronic},\\cite{hejazi2013lower},\\cite{hejazi2013new},\\cite{khalili2013secant}. More recent techniques, use beamforming techniques over array antennas to obtain narrow beams. In beamforming, input\/output of each antenna of an array, is multiplied by a weight (e.g. a phase shift) to form a desired beam shape. In beamforming, there is no need for mechanical steering, and beams can be steered electronically by changing weights of the antennas. Spatial scanning provided by beamforming proves to be much more faster than the mechanical scanning, moreover, can generate multiple beams simultaneously. Therefore, modern phased array radars can search the environment very fast, and can track and engage with multiple targets concurrently \\cite{mailloux2017phased}.\n\nRecently, DoA estimation also has gained attention as an enabler of ultra-high-speed (Multiple Gbps) directional communications between two devices or a base-station and multiple devices. 5G communication mainly utilizes advanced beamforming capabilities and array antennas for directional communications. 3 different architectures has been introduced for beamforing for 5G applications: 1-Analogue 2-Digital 3-Hybrid \\cite{kutty2015beamforming}. In Analogue beamforming, the beam is shaped via a single RF chain, and so only one beam can be shaped in each time slot. This structure is more power efficient compared to the two other architectures, however, is not as flexible as them in generating multiple beams. Digital beamforing, allocates a specific Rf chain and data-convertor for each antenna and potentially can generate several beams simultaneously. This structure is the most flexible one, however is very power hungry and complicated in comparison to other techniques \\cite{yang2018digital}. Hybrid beaforming scheme assigns multiple RF chains for antennas, while, the number of RF chains is less than the number of antennas. This type of beamforming is the most common scheme for 5G applications, since it can balance a trade-off between complexity, flexibility and power consumption \\cite{sohrabi2016hybrid,molisch2017hybrid}. All directional antennas powered by various beamforming architectures require spatial search to initiate a communication link . Giordani et. al showed that overhead caused by beam-training protocols heavily limits number of array elements at both base stations and user equipments, moreover, several milliseconds is required to establish a link between a base station and user equipment \\cite{lien20175g}. \n\nInterferometric wide-band DoA estimation, has been widely investigated in EW and lightning localization applications \\cite{mardiana2000broadband},\\cite{wu1995direction}, \\cite{hejazikookamari2018novel},\\cite{kookamari2017using},\\cite{hejazi2014sar},\\cite{hejazi2020tensor},\\cite{hejaziwireless},\\cite{joneidi2019large}. In this technique PDoA of signal between two antennas placed more than half-wavelength apart is measured. Since the phase difference is ambiguous and can represent several DoAs, a number of techniques has been introduced to disambiguate the phase. These techniques include: correlative interferometry (CORR), second order difference array (SODA), SODA-Base Inference (SBI) and Common Angle Search (CAS). CORR employs PDoAs between at least two pairs of antennas and compare measurements with a pre-prepaired database of measurements to determine DoA \\cite{kebeli2011extended}. SODA and SBI operate an additional antenna pair with less than half a wavelength gap between antennas to  translate PDoA to an unambigeous DoA. SODA and SBI only works well when input SNR is high enough \\cite{mollai2018compact,mollai2019wideband}. CAS utilizes two or more antenna pairs and introduces the common angle recommanded by all PDoAs as the unambiguous DoA \\cite{searle2017disambiguation}. These techniques can estimate DoA very precisely in a wide-band frequency range, however, none of them can distinguish between DoAs, if two or more signals with differnet DoAs are received simultaneously at the antenna pairs. \n\nHere in section \\ref{Fisher}, we prove that phase interferometry meaurements (PIM) between two antennas equals to highly precise time difference of arrival (TDoA) measurements in the Fisher sense. Moreover, we demonstrate that DoA estimation using PIM between two antennas several wavelength apart equals to DoA estimation using a large ULA in the Fisher sense. Since PIMs represent ambigeous DoAs, we introduce phase spectrometry (PS) to disambiguate PDoAs in section \\ref{PS}. In contrast with Interferometric DoA estimation, we prove that PS can distinguish between multiple concurrent DoAs. Furthurmore, we introduce standing wave receiver (SWR) to extract PDoAs, which is much less complicated than beamforming receivers. We explain how SWR does not need any down-conversion or high sampling rates to extract PDoA. In section \\ref{DoARes} we investigate DoA estimation resolution provided by PS. Then we introduce two approaches to implement PS, one through a long wave-guide, another via employing a frequency code-book in section \\ref{longSW} and \\ref{FreqCB} respectively. In section \\ref{SNR}, we analyse SNR improvement caused by PS. Furthermore, we will show how the whole time required by directional techniques for spatial search can be effectively consumed in PS to improve DoA estimation precision.  We discuss the ability of the proposed technique to identify  DoA of signals from several devices in both uplink and downlink scenarios in section \\ref{scale}. Moreover, we introduce an alternative architecture of the technique that provides us with ultra-fast DoA estimation capability in section \\ref{UFast}. In Section \\ref{sim}, we examine PS performance via various simulations. Finally we conclude the paper in section \\ref{conc}. \n\n\\section{Phase Interferometry Measurements}\n\\label{Fisher}\nConsider 2 antennas with gap $D$ mounted on a device (Figure \\ref{PIMDEFGEO}), referred as phase interferometry array (PIA), both of them are receiving a signal emitted by a source $s(t)$.  The signal is a monotone with carrier frequency $f_c$   \n\\begin{equation}\n    s(t)=a \\: e^{j2\\pi f_c t}\\,,\n    \\label{sigmod}\n\\end{equation}\nwhere $a$ is the amplitude of the signal. Both the first and the second antennas receive the signal, denoted by $s^{(1)}_{R}(t)$ and $s^{(2)}_{R}(t)$ respectively,  with a relative delay $\\Delta (t)$ which results in a phase difference between two signals. We define phase interferometry measurements (PIM) as \n\\begin{equation}\n    \\Delta \\phi = s^{(1)}_{R}(t)s^{*(2)}_{R}(t)=a^2_R \\: e^{j2\\pi f_c \\Delta (t)}+v_n=b e^{j2\\pi f_c \\Delta (t)}+v_n  \\,,\n\\end{equation}\nwhere $a_R$ is the amplitude of the signal received at the PIA and $v_n$ is white noise, we also refer to $e^{j2\\pi f_c \\Delta (t)}$ as PDoA throughout this paper. In the next section we prove that PIM is equivalent to DoA estimation using a ULA in the Fisher sense. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3in,height=2.3in]{PIMDEFGEO.png}\n    \\caption{PIM illustration, two antennas implemented on a device receive a signal ($s(t)$) emitted by a source ($s^{(1)}_{R},s^{(2)}_{R}$). PIM is defind as the interaction of two signals $\\Delta \\phi = s^{(1)}_{R}s^{*(2)}_{R} $}\n    \\label{PIMDEFGEO}\n\\end{figure}\n\\subsection{Fisher Information Matrix of PIM, TDoA \\& DoA}\nGiven noise is Gaussian and independent for each PIM, Fisher information matrix (FIM) of $\\Delta \\phi $ with respect to an arbitrary vector $\\boldsymbol{x}$ , e.g. unknowns to be estimated, can be derived as \\cite{farina1999target}\n\\begin{align}\n\\sum_{\\mathbb{P}} \\frac{1}{\\sigma^2} \\nabla_{\\boldsymbol{x}} \\Delta \\phi^{H} \\nabla_{\\boldsymbol{x}} \\Delta \\phi=&\\sum_{\\mathbb{P}} \\frac{1}{\\sigma^2} (-j2\\pi f_c \\nabla_{\\boldsymbol{x}} (\\Delta (t))^{H} be^{-j2\\pi f_c \\Delta (t)})  (j2\\pi f_c \\nabla_{\\boldsymbol{x}} \\Delta (t) be^{j2\\pi f_c \\Delta (t)}))= \\nonumber \\\\\n&\\sum_{\\mathbb{P}} \\frac{4b^2\\pi^2 f^2_c}{\\sigma^2} \\nabla_{\\boldsymbol{x}} \\Delta (t)^{H} \\nabla_{\\boldsymbol{x}} \\Delta (t) \\,,\n\\end{align}\nwhere $\\mathbb{P}$ is the set of all PIMs, and $\\nabla_{\\boldsymbol{x}}$ is the gradient operator with respect to (w.r.t) $x$. Therefore, FIM of PIM is exactly equals to the following observations, \n\\begin{equation}\n\\delta (t)=b\\Delta (t)+\\frac{v_s}{2\\pi f_c} \\,.\n\\label{delti}\n\\end{equation}\nwhere $\\delta (t)$ is an observation of TDoA of signal between two antennas. Therefore, PIM with additive white noise power $\\sigma^2$ equals to TDoA observations with additive white noise power $\\frac{\\sigma^2}{4\\pi^2 f^2_c}$ of the same PIA in the fisher sense. Assuming far field criteria is fulfilled \\cite{chen2002source}, we have\n\\begin{equation}\n\\delta (t)=b\\frac{D}{c} cos(\\theta_A)+\\frac{v_s}{2\\pi f_c} \\,,\n\\label{delti1}\n\\end{equation}\nwhere $c$ is the speed of light and $\\theta_A$ is DoA of signal and $D$ is the gap between two antennas. CRLB of $\\theta_A$ estimation based on measurements as of (\\ref{delti1}) can be derived as follows\n\\begin{equation}\n\\mathrm{CRLB}_{\\theta_{A}}=\\frac{\\frac{\\sigma^2_s}{b^2}}{(\\frac{D}{c}2\\pi f_c)^2 sin^2(\\theta_A)}=\\frac{\\frac{\\sigma^2_s}{b^2}}{(\\frac{2\\pi D}{\\lambda})^2 sin^2(\\theta_A)}\\,.\n\\label{CRMPIM}\n\\end{equation}\nNow lets take a look at CRLB of DoA estimation using a ULA in which antennas are placed half wavelength apart \\cite{penna2011bounds},\n\\begin{equation}\n\\mathrm{CRLB}_{\\theta_{A}}=\\frac{6 \\frac{\\sigma^2_s}{b^2}}{\\pi^2 m(m^2-1)sin^2(\\theta_A)} \\approx \\frac{6 \\frac{\\sigma^2_s}{b^2}}{\\pi^2 m^3 sin^2(\\theta_A)}\\,,\n\\label{CRBULZ}\n\\end{equation}\nwhere $m$ is the number of array elements. Given the same SNR, DoA estimation using PIM and a ULA array are equivalent in the Fisher sense when, \n\\begin{equation}\nm=(24)^{\\frac{1}{3}}(\\frac{D}{\\lambda})^{\\frac{2}{3}} \\approx 2.8845 (\\frac{D}{\\lambda})^{\\frac{2}{3}} \\,.\n\\label{m\/d}\n\\end{equation}\nFigure \\ref{mdlambda} illustrates \\eqref{m\/d}, as an example, DoA estimation using a PIA with $\\frac{D}{\\lambda}=200$ is equivalent to a ULA with 100 elements in the Fisher sense.  Consequently, DoA estimation using PIM with gap $D$ between two antennas equals to DoA estimation exploiting a ULA with $m$ antennas placed half wavelength apart, in which $m$ obeys (\\ref{m\/d}). This could lead to a huge reduction in complexity of the antenna array required for high precision DoA estimation -that reduces the required number of antennas from $m$ to 2-; if so, why is it not a common DoA estimation technique now? it is because DoA estimation using PIM is ambiguous and there are a number of different DoAs that can be inferred from a specific PIM \\cite{vinci2011novel}; As $D$ increases CRLB decreases, however, ambiguity escalates. Moreover, DoA  estimation using PIM is not capable of detecting and discriminating between multiple concurrent DoAs. In section \\ref{PS}, we propose a solution to estimate DoA using PIMs observed for multiple frequencies, instead of only measuring PIM for only a single frequency. We will see that this approach not only leads to PIM disambiguation, but also provides us with DoA estimation of all signal propagation paths between the source and the device. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.9in,height=3in]{mdlambda.png}\n    \\caption{$m$ versus $\\frac{D}{\\lambda}$, where $m$ is the number of array elements of a ULA that is equivalent to (in the Fisher sense) a phase interferomery array (PIA) with gap $D$ between two antennas}\n    \\label{mdlambda}\n\\end{figure}\n\\subsection{Relationship Between DoA Estimation Precision, Beam-width and Resolution}\n\\label{DoABW}\nIn this section, we explain why DoA estimation precision and antenna beam-width are not necessarily\ncoupled, which further proves that spatial division (SD) and IA can be considered and performed as two completely independent tasks. Referring to (\\ref{CRMPIM}) and (\\ref{CRBULZ}), CRLB of angle estimation precision is directly related to SNR, as SNR increases precision improves; in other words, we can obtain any arbitrary precision if SNR is high enough regardless of $m$ or $D$. Although SNR can be improved by increasing the number of antennas, in a ULA, it can also be improved by integration, which is the time interval we can coherently receive and integrate a signal. Equivalently, angle precision can be improved only by integration, which come at a time cost, regardless of $m$ or $D$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.2in,height=3in]{FigSDMA.pdf}\n    \\caption{Visualisation of spatial division concept. The antenna is able to discriminate between user 1 and user 2,3 because the angular distance between them are more than beam-width. While, it is not able to discriminate between user 2 and user 3, since their angular distance is less than the antenna beam-width.}\n    \\label{SDMA}\n\\end{figure}\n\n\nNow let's take a look at angle resolution concept. Angle resolution help us to measure the capability of a technique to discriminate between multiple incoming signals from different DoAs. We define angle resolution as the minimum angular distance between two incoming signals  that can be discriminated by a technique. Angle resolution is directly coupled with SD capability of a technique. A ULA can discriminate between two DoA if their angular distance is more than it's beam-width. Similarly, in the transmit mode, if the angular distance between two users is more than the beam-width and antenna sends signal to one of them, it causes much less interference for the second device compared to the situation where their angular distance are less than the beam-width (Figure \\ref{SDMA}). In a ULA, beam-width is merely determined by the number of array elements and  equals to $\\frac{2}{m}$. Therefore, the SD capability of a ULA is solely governed by its number of array elements.     \n\nFor ultra-fast mm-wave communication, devices has to be equipped with a highly directional antenna that enables SDMA. On the other hand, for initial access (IA), a good angle estimation is required. As we discussed earlier, an angle estimation with a desired precision can be obtained when SNR is high enough at the receiver. SNR and resolution are not two mutually-coupled aspects of a DoA estimation technique. Especially in the case of array antennas, resolution is governed by number of array elements, while DoA estimation precision is governed by SNR at the receiver. Consequently, we can seperate SD from IA, and dedicate a high processing gain technique for IA and a highly directional antenna for SD.\n\n\n\n\\section{Wide-band DoA Estimation Using Standing-wave Spectrometry}\n\\label{PS}\nIn this section we inaugurate a new idea to estimate DoA of a signal using PDoAs. Here, we propose the source emits a signal with several gigahertz bandwidth in mm-wave, in such a way that the receiver can detect and discriminate between all (line-of-sight (LoS) and none-line-of-sight (NLoS)) paths between the source and the receiver using our proposed PS technique. Now suppose there exists $N_{NL}+1$ paths, 1 LoS path and $N_{NL}$ NLoS paths, between the source and the device. Given the source emits a monotone signal as of (\\ref{sigmod}) with carrier frequency $f$ for the duration $T_p$, received signals at both antennas can be formulated as\n\\begin{equation}\n\\label{interefence}\n     s^{(1)}_{R}(t) = \\!\\underbrace{a_0 e^{j 2\\pi f t}}_{\\text{LoS path}}\\!+\\!\\underbrace{\\sum_{k=1}^{N_{NL}}a_ke^{j 2\\pi f (t-t_k)}}_{\\text{NLoS paths}}+v_1(t)\\;\\;\\; and\\;\\;\\;\n     s^{(2)}_{R}(t)=\\!\\underbrace{a_0 e^{j 2\\pi f (t-\\Delta{t_0})}}_{\\text{LoS path}}\\!+\\!\\underbrace{\\sum_{k=1}^{N_{NL}}a_ke^{j 2\\pi f (t-t_k-\\Delta{t_k}))}}_{\\text{NLoS paths}}+v_2(t)\\,.\n\\end{equation}\nwhere $t_k$ is the delay of signal arrival through NLoS path $k$ to the PIA w.r.t LoS path, and $\\Delta t_k$ and $a_k$ is TDoA of signal between two antennas and amplitude of received signal through path $k$, $k = 0,\\dots,N_{NL}$ (path 0 is the LOS path), respectively. Then, we guide the two received signals into a standing-wave wave-guide (SWWG) via two opposite directions (Figure \\ref{SWR}). Referring to \\cite{jovanov2010standing}, the first and the second paths of signal interact in the SWWG as \n\n\\begin{align}\n     &s^{(1)}_{R}(t) e^{j\\beta(f) x} + s^{(2)}_{R} (t)e^{-j\\beta(f) x}=\\nonumber\\\\\n     &e^{j 2\\pi f t} \\left(\\left(a_0 e^{j\\beta(f) x}+ a_0 e^{-j\\beta(f) x}e^{-j 2\\pi f (\\Delta{t_0})} \\right)+\\left(\\sum_{k=1}^{N_{NL}} a_k e^{-j 2\\pi f t_k} \\left(e^{j\\beta(f) x}+e^{-j\\beta(f) x} e^{-j 2\\pi f (\\Delta{t_k}))} \\right)\\right)\\right)= \\nonumber \\\\\n     &e^{j 2\\pi f t}\\left(2a_0 e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right) \\,, \\nonumber \\\\  \n\\end{align}\n\nwhere $x$ is an arbitrary point along the SWWG, $L$ is the length of the wave-guide and $\\beta(f)=\\frac{2\\pi} {\\lambda_T}=2\\pi \\frac{f}{c_T}$, where $\\beta(f)$, $\\lambda_T$ and $c_T$ are phase constant, wavelength and phase velocity of electro-magnetive wave in the wave-guide, respectively \\cite{steer2019microwave}. As Figure \\ref{SWR} illustrates, using energy detectors along $x-axis$, we have\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6in,height=1.8in]{Figstand.pdf}\n    \\caption{Standing-wave wave-guide. Two waves move in opposite directions interact to form a standing wave, the amplitude of the standing wave is sampled using a group of energy detectors (ED).}\n    \\label{SWR}\n\\end{figure}\n\\begin{align}\n    &E_{sw} (x,f) = \\left| 2a_0 e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}}2a_k e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right|^2 \\nonumber \\\\\n        &= \\: 4 a^{2}_{0} cos^2 (\\beta(f) x + \\pi f \\Delta t_0) +\\sum_{k=1}^{N_{NL}} 4 a^{2}_{k} cos^2 (\\beta(f) x + \\pi f \\Delta t_{k})\\nonumber \\\\ \n    &+\\sum_{k=1}^{N_{NL}} 8  a_0 a_k \\cos{(\\beta(f) x+ \\pi f \\Delta t_0)} \\cos{ (\\beta(f) x + \\pi f \\Delta t_k)} \\cos {(2\\pi f (t_k + \\frac{\\Delta t_k - \\Delta t_0}{2}))} \\nonumber\\\\ \n     &+\\sum_{k=1}^{N_{NL}}\\sum_{l=K+1}^{N_{NL}} 8  a_l a_k \\cos{(\\beta(f) x+ \\pi f \\Delta t_l)} \\cos {(\\beta(f) x + \\pi f \\Delta t_k)} \\cos {(2\\pi f (t_k-t_l + \\frac{\\Delta t_k - \\Delta t_l}{2}))}\\,.\n     \\label{Damaneh}\n\\end{align}\n\nwhere $E_{sw} (x,f)$ is the output of the ED located at $x$. Interestingly, as \\eqref{Damaneh} indicates, we could bypass down-conversion via mixing by using much simpler EDs. Now, suppose that input signal and its DoAs does not change during $T_p$, evidently, sampling rate after EDs can be as low as some $\\frac{1}{T_p}$, if energy detectors provide energy integration of the wave for the whole duration. To simplify (\\ref{Damaneh}), it is clear that $\\Delta t_k \\ll t_l$ and it is very probable that $\\Delta t_k \\ll t_k-t_l$;  $ k = 0,\\dots,N, l = 1,\\dots,N$  \n\\footnote{The experimental results presented in \\cite{rappaport2013millimeter} shows that delays of paths in two urban environment of New York and Austin is an order of several tens of nano seconds, on the other hand, the TDoA of signal between two antennas is a fraction of a nano second if the gap between antennas does not exceed $30 cm$.} Regarding \\eqref{Damaneh}, the first and the second terms have \\emph{cos(.)} components with parameters $\\pi f \\Delta t_0$ and $\\pi f \\Delta t_k$ , while the third and the forth terms have \\emph{cos} components with parameters $\\pi f t_k$ and $\\pi f (t_k-t_l), k \\neq l$, respectively. Given we measure \\eqref{Damaneh} for multiple frequencies and $\\Delta t_k \\ll t_k-t_l$ for all $l,k$, applying Fourier transform over $E_{sw} (x,f)$ across $f$, the third and the forth terms of (\\ref{Damaneh}) can be filtered out using a simple low-pass filter \\footnote{In section \\ref{DoARes} we will show that, this filter can be the same as the matched filter applied for DoA detection.}. The remaining terms after low-pass filtering are denoted by $ \\hat{E}_{sw} (x,f)$\n\n\\begin{align}\n   \\hat{E}&_{sw} (x,f) = \\: 4a_0^2 \\cos^2 (\\beta(f) x + \\pi f \\Delta t_0) +\\sum_{k=1}^{N_{NL}} 4 a^2_k \\cos^2 (\\beta(f) x + \\pi f \\Delta t_k) \\nonumber\\\\\n   =2a_0^2& + 2 a^2 \\cos  (2 \\beta(f) x + 2\\pi f \\Delta t_0)+ \\sum_{k=1}^{N_{NL}}2a^2_k + 2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k) \\,.\n   \\label{phspecforf}\n\\end{align}\n\nThe number NLoS path from the source to the device are very few in mm-wave usually less than 3 path \\cite{heath2016overview}, so $N_{NL} \\le 3$. Here, if we estimate $\\theta_k,a_k$ for $k = 0,\\dots,N$, we can distinguish between all paths from the source to the device and determine signal received power from each path. In section \\ref{DoARes}, \\ref{longSW} and \\ref{FreqCB}, we will discuss two different techniques that that can be used to detect DoAs based on sampling \\eqref{phspecforf} in $f$-domain, and how the angular resolution that can be achived using PS.   \n\nConsider that (\\ref{phspecforf}) is derived by assuming a monotone signal is transmitted by the source. \nNow lets assume, signal is not monotone and has bandwidth $B$, thus signal can be expressed as\n\n\n\n\\begin{equation}\n    s(t) =  \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a(f) e^{j2 \\pi f t} df\\,.\n    \\label{wide-spread}\n\\end{equation}\n\nwhere $a_(f)$ is Fourier transform of $s(t)$. The received signals at the first and the second antennas can be expressed as\n\n\\begin{align}\n\\label{interefence1}\n     &s^{(1)}_{R}(t)= \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_0(f) e^{j 2\\pi f t} df+\\sum_{k=1}^{N_{NL}}\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f)e^{j 2\\pi f (t-t_k)} df+v_1(t)\\nonumber \\\\\n     &s^{(2)}_{R}(t)=\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}}a_0(f) e^{j 2\\pi f (t-\\Delta{t_0})} df+\\sum_{k=1}^{N_{NL}} \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f) e^{j 2\\pi f (t-t_k-\\Delta{t_k}))} df+v_2(t)\\,.\n\\end{align}\n\nAssuming a constant fading over $[f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$, we can express $a_k(f)= \\alpha_k a(f)$, where $\\alpha_k$ denotes the attenuation of path $k$. Consequently, the two signals inside the SWWG can be formulated as \\cite{jovanov2010standing}\n\n\\begin{align}\n\\label{interefence1}\n     &s^{(1)}_{R}(t,x)= \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_0(f) e^{j 2\\pi f t} e^{j \\beta(f) x} df+\\sum_{k=1}^{N_{NL}}\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f)e^{j 2\\pi f (t-t_k)} e^{j \\beta(f) x}df+v_1(t)\\nonumber\\\\\n     &s^{(2)}_{R}(t,x)=\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}}a_0(f) e^{j 2\\pi f (t-\\Delta{t_0})} e^{-j \\beta(f) x} df+\\sum_{k=1}^{N_{NL}} \\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} a_k(f) e^{j 2\\pi f (t-t_k-\\Delta{t_k}))} e^{-j \\beta(f) x} df+v_2(t)\\,.\n\\end{align}\n Finally, the interaction between the two signals ($S_{int}$) in the SWWG can be formulated as \n \\small\n \\begin{align}\n     &S_{int}(t,x)=s^{(1)}_{R}(t,x)+s^{(2)}_{R}(t,x)=\\nonumber\\\\\n     &\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}}  \\left(\\left(a_0(f) e^{j\\beta(f) x}+ a_0(f) e^{-j\\beta(f) x}e^{-j 2\\pi f (\\Delta{t_0})} \\right)+\\left(\\sum_{k=1}^{N_{NL}} a_k(f) e^{-j 2\\pi f t_k} \\left(e^{j\\beta(f) x}+e^{-j\\beta(f) x} e^{-j 2\\pi f (\\Delta{t_k}))} \\right)\\right)\\right) e^{j 2\\pi f t} df= \\nonumber \\\\\n     &\\int_{f_c-\\frac{B}{2}}^{f_c+\\frac{B}{2}} \\left(2a_0(f) e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k(f) e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right)e^{j 2\\pi f t} df \\,.   \n\\end{align}\n\\normalsize\nTherefore, the power spectral density of $S_{int}$ turns out to be \\cite{oppenheim2015signals}\n \\small\n\\begin{align}\n&\\mathscr{{E}}_{sw}(x,f)=\\lim_{T \\to +\\infty}\\mathscr{F} \\left\\{ \\frac{1}{T}\\int_{0}^{T}|S_{int}(t,x)|^2dt\\right\\}\\nonumber\\\\\n&=\\left|2a_0(f) e^{-j\\pi f \\Delta t_0}\\cos{(\\beta(f) x + \\pi f \\Delta t_0)}+\\sum_{k=1}^{N_{NL}} 2 a_k(f) e^{-j 2\\pi f (t_k+\\frac{\\Delta t_k}{2})} \\cos{(\\beta(f) x + \\pi f \\Delta t_k)} \\right|^2 ; \\: f \\in [f_c-\\frac{B}{2},f_c+\\frac{B}{2}]\\,.\n\\label{PSD}\n\\end{align}\n\\normalsize\n\n\n\n\nAs \\eqref{PSD} shows $\\mathscr{{E}}_{sw}$ exactly equals to \\eqref{Damaneh} for $f \\in [f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$. Therefore, similar to the procedure of DoA estimation of a monotone signal, we can estimate all incoming signal DoAs and their power for a non-monotone signal using \\eqref{phspecforf}. Now, lets again consider \\eqref{phspecforf}, we can express $\\hat{E}_{sw} (x,f)$ as summation of two terms\n\\begin{align}\n   &\\hat{E}_{sw} (x,f) = \\sum_{k=0}^{N_{NL}} 2a_k^2  +\\sum_{k=0}^{N_{NL}} 2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k) \\nonumber\\\\\n   &=\\sum_{k=0}^{N_{NL}} 2a_k^2+\\sum_{k=0}^{N_{NL}} 2 a^2_k \\cos (\\frac{4\\pi}{c_T} fx + 2\\pi f \\Delta t_k)  \\,.\n   \\label{phspecforfsimp}\n\\end{align}\nInterestingly, factors $a_0,\\dots,a_{N_{NL}}$ and phases $2\\pi f \\Delta t_0,\\dots,2\\pi f \\Delta t_{{NL}}$, can simply be estimated by applying Fourier transform over $\\hat{E}_{sw} (x,f)$ across $x$.  One useful example of signal as of (\\ref{wide-spread}) is multiple single tones (e.g. 30 monotones) around the center frequency; in section \\ref{FreqCB}, we show that this signal not only provides enough information to estimate all DoAs, but also enables integration to achieve very high SNRs, which results in very high precision DoA estimation.\n\n\\subsection{DoA Detection and Resolution}\n\\label{DoARes}\nAs we proved in the previous section, the interaction between two waves received at each antennas forms a standing-wave and its amplitude can be measured as of (\\ref{phspecforfsimp}) for each frequency $f$, measuring amplitude of the standing-wave, employing a group of EDs. Consider that \\eqref{phspecforfsimp} consists of $2 a^2_k \\cos (2\\beta(f) x + 2\\pi f \\Delta t_k)$ terms. Thus, estimating $\\Delta t_k$ and $\\alpha_k$ for $k = 0,\\dots,N_{NL}$ is equivalent to harmonic decomposition of \\eqref{phspecforfsimp} in $f$-domain. There are several techniques has been introduced for harmonic decomposition, such as Fourier transform, multiple signal classification (MUSIC) \\cite{schmidt1986multiple}, Pisarenco harmonic decomposition \\cite{pisarenko1973retrieval}, to name a few. Here for simplicity we only use a matched filter for DoA estimation. Given far-field assumption we have\n\n\\begin{equation}\n    \\Delta t_k=\\frac{D\\cos{\\theta_k}}{c}\\,.\n\\end{equation}\n\nwhere $\\theta_k$ is DoA of path $k$. Therefore, DoAs can be estimated applying the following matched filter on (\\ref{phspecforf})\n\n\\begin{equation}\n    h(\\theta,f,x)=e^{j2\\pi f \\Delta t_k}e^{j2\\beta(f)x}=e^{j2\\pi f \\frac{Dcos(\\theta)}{c}}e^{j2\\beta(f)x}=e^{j2\\pi f (\\frac{Dcos(\\theta)}{c}+\\frac{4\\pi x}{c_T})}\\,.\n    \\label{matchedfilter}\n\\end{equation}\n(\\ref{matchedfilter}) shows that the matched filter is a single monotone in the $f$-domain. Moreover, as $\\Delta t_k$ increases, the matched filter represents a higher frequency signal in $f$-domain. Therefore, convolving (\\ref{matchedfilter}) with (\\ref{Damaneh}), the third and the forth terms of \\eqref{Damaneh} will be eliminated. To calculate the angular resolution of PS suppose two different paths with two different DoAs $\\theta_1,\\theta'_1$ arrive at PIA and we can completely discriminate between $\\theta_1$ and $\\theta_1'$ using matched filter in (\\ref{matchedfilter}), then we have \n\\begin{align}\n    &\\int_{f-\\frac{B}{2}}^{f+\\frac{B}{2}} e^{j2\\pi f D \\frac{cos(\\theta_1)-cos(\\theta'_1)}{c}} df = 0 \\rightarrow BD\\frac{|cos(\\theta_1)-cos(\\theta'_1)|}{c}=k , k \\in \\mathbb{N} \\rightarrow \\frac{BD}{c} |\\theta_1-\\theta'_1||sin(\\theta_1)|\\approx k \\nonumber \\\\\n    &\\rightarrow |\\theta_1-\\theta'_1| \\approx \\frac{ck}{BD |sin(\\theta)|}\n    \\,.\n    \\label{msdsds}\n\\end{align}\nTherefore the minimum possible angular distance between $\\theta_1$ and $\\theta_1'$ that can be resolved using our proposed technique (referred as DoA estimation resolution) can be approximated as\n\\begin{align}\n    Res(\\theta) \\approx \\frac{c}{BD |sin(\\theta)|}\n    \\,.\n    \\label{Res}\n\\end{align}\nConsequently, DoA estimation resolution is determined merely by $BD$, which means as the gap between two antennas or the signal bandwidth increases the DoA resolution will increase. As we mentioned earlier, in this we mainly use marched filter for DoA detection for simplicity, however, since PDoAs are available in digital domain, future works may consider more complicated signal processing techniques for DoA estimation. Those techniques may result in much better angular resolution than match filtering.\n\n\\subsection{Frequency Resolution}\n\\label{longSW}\n\nConsidering $\\beta=2\\pi \\frac{f}{c_T}$, (\\ref{phspecforfsimp}) clarifies that angle and phase difference of PIMs for any arbitrary frequency inside $[f-\\frac{B}{2},f+\\frac{B}{2}]$ would be easily extracted by applying Fourier transform over $\\mathscr{{E}}_{sw}(x,f)$  across $x$, if we could measure $\\mathscr{{E}}_{sw}(x,f)$ for an infinite length. Unfortunately, in practice we can only measure $\\mathscr{{E}}_{sw}(x,f)$ for a limited length and it enforces a strong limitation on the frequency resolution of the Fourier transform. To calculate of resolution of FFT over $\\mathscr{{E}}_{sw}(x,f)$ across $x$, consider that if we have a signal for length $T$ (in time), the highest FFT resolution possible is $\\frac{1}{T}$ \\cite{oppenheim1999discrete}. Given SWWG length is $L$, referring to (\\ref{phspecforfsimp}), the frequency resolution ($\\delta(f)$) turns out to be\n\\begin{align}\n    Res(f)=\\delta(f) \\rightarrow 2 \\frac{\\delta f}{c_T} L =1 \\rightarrow\n    \\delta(f)=\\frac{c_T}{2L} \\,.\n    \\label{Length}\n\\end{align}\nGiven $c_t\\approx c$, to reach a $1GHz$ frequency resolution  we need a $15cm$ wave-guide and to reach a $100MHz$ resolution we need a $1.5m$ wave-guide. Such a long wave-guide may not be practical specially exploiting PCB or MMIC implementation since it results in a huge attenuation of the signal along the long wave-guide. Thus we may either employing alternative fabrication technologies or the following technique to resolve this issue. \n\\subsection{Frequency Swiping Interferometry (Frequecny Code-book)}\n\\label{FreqCB}\nInstead of spectrometry via a long wave-guide, we can sample PDoAs for a group of frequencies in $[f_c-\\frac{B}{2},f_c+\\frac{B}{2}]$ using a short wave-guide. To this end, we divide the frequency band into $S_f$ frequency steps (also referred as frequency code-book), each step is represented by a monotone (pilot), and measure (\\ref{phspecforf}) for each pilot. We also divide the whole PS duration into $S_f$ time slots and measure PDoA for each pilot at each time slot. Since we measure PDoA for a monotone in each time slot, our approach bypasses the need for a long SWWGs. Consider that, the number of pilots and the distance between them (in $f$-domain) should provide us enough information to detect all DoAs. Referring to (\\ref{phspecforf}), we measure $e^{j 2\\pi f \\Delta t_{in}}$ for each pilot, where $t_{in}$ can potentially changes between $[-\\frac{D}{c},\\frac{D}{c}]$, therefore we should sample the phase difference with at least $\\frac{c}{2D}$ rate (Nyquist rate) in the $f$-domain to capture all information regarding $\\Delta t_{in}$, thence, the code-book should contain at least \n\\begin{equation}\n    \\mathrm{min} \\: S_f= \\frac{B}{\\frac{c}{2D}}=\\frac{2BD}{c}\\,,\n\\end{equation}\npilots (samples in $f$-domain). Consequently, we propose to establish a directional link between two devices, both devices should send pilots, so the other side can estimate DoAs of signal based on measuring PDoAs for all pilots. Using our proposed technique, there is no need for spatial search and all DoAs can be estimated via measuring PDoAs of pilots. Lets $f_0$ denotes the frequency of the first monotone and $\\Delta f_0 = \\frac{c}{2D}$ denotes the distance between pilots in $f-$domain. Thus, the vector of all measured phases for the frequency codebook ($\\boldsymbol{\\Delta \\phi}$) can be expressed as \n\n\\begin{align}\n&\\boldsymbol{\\Delta \\phi}=\n  \\begin{bmatrix}\n      e^{j 2\\pi f_0 \\Delta t_{in}} & e^{j 2\\pi (f_0+\\Delta f_0) \\Delta t_{in}} & \\dots & e^{j 2\\pi (f_0+(S_F-1)\\Delta f_0) \\Delta t_{in}} \n\\end{bmatrix} \n\\nonumber \\\\\n&=e^{j 2\\pi f_0 \\Delta t_{in}}\n\\begin{bmatrix}\n        1 & e^{j 2\\pi (\\Delta f_0) \\Delta t_{in}} & \\dots & e^{j 2\\pi ((S_F-1)\\Delta f_0) \\Delta t_{in}}  \n\\end{bmatrix} \\,,\n\\label{PSULASIM}\n\\end{align}\nwhere $\\Delta t_{in}$ is the TDoA of signal between two antennas. \\eqref{PSULASIM} equals to the vector of phase differences measured by a ULA ($\\boldsymbol{\\Delta \\phi_{u}}$) with $S_F$ elements multiplied by $e^{j 2\\pi f_0 \\Delta t_{in}}$  \n\\begin{align}\n&\\boldsymbol{\\Delta \\phi_{u}}=\n  \\begin{bmatrix}\n      1 & e^{j 2\\pi (\\Delta f_0)\\Delta t_{d}} & \\dots & e^{j 2\\pi ((S_F-1)\\Delta f_0) \\Delta t_{d}} \n\\end{bmatrix} \\,,\n\\label{ULAPSH}\n\\end{align}\nwhere $\\Delta t_{d}$  is the TDoA between of signal between two consecutive elements and $\\Delta f_0$ is the working frequency of ULA. In fact, we reconstruct a ULA that works at frequency $\\Delta f_0$ via PS that works at much higher frequency $f_0$ \\footnote{More interestingly, $f_0$ and $\\Delta f_0$ are independent. $f_{0}$ should be high enough to provide us with enough unused bandwidth required to emulate the ULA. Thus, PS is much more applicable in mm-Wave and Terahertz bands becuase large swaths of spectrum is available.}. As $B$ increases the number of pilots (equivalent to ULA elements) can increase and as $D$ increases $\\Delta f_0$  decreases and again we can increase the number of pilots which results in better angular resolution. In a ULA, usually PDoAs of \\eqref{ULAPSH} are compensated by phase shifters at each elements for different values of possible $\\Delta t_{d}$s to find the best match with $\\boldsymbol{\\Delta \\phi_{u}}$ and detect the DoA (i.e. the spatial search). In our technique, since we measure PDoAs using PS techniques we can find the incoming DoA by digital signal processing. In section \\ref{sim} we will show that output of matched filter of \\eqref{matchedfilter} applied on \\eqref{PSULASIM} is very similar to output of phase shifters applied on \\eqref{ULAPSH} (conventional beamforming). \\footnote{Throughout this work, we only consider a simple matched filter on \\eqref{PSULASIM} to detect DoAs. However, PS provides \\eqref{PSULASIM} in the digital domain, thus, much more complex signal processing techniques can be applied. Future works may consider various frequency sampling and corresponding array signal processing techniques to improve PS performance.}  \n\\subsection{SNR Analysis}\n\\label{SNR}\n\nLong SWWGs is subject to suffering from a huge loss, specially in mmwave. Since SNR is an absolutely critical factor when we deal with millimeter waves, it is more practical not to attenuate the input signal in the receiver by employing long SWWGs. In this section we analyse SNR of the technique that employs a frequency code-book instead of a long SWWG. The block diagram of the receiver using the frequency code-book technique is depicted in Figure \\ref{BD}. As the figure illustrates, input signals pass through 3 stages until DoAs of signal are detected. Each stage may improves SNR. To measure how much the proposed receiver improves SNR we use the processing gain ($G_p$) metric \\cite{rouphael2009rf}. Procesing gain is defined as ratio of the SNR of a processed signal to the SNR of the input signal. $G_p$ of the whole receiver can be expressed as\n\n\\begin{equation}\n    G_p(total)=G_p(stage-1)G_p(stage-2)G_p(stage-3)\\,.    \n\\end{equation}\n\nNow lets calculate the $G_p$ for each stage. We ignore losses caused by hard-wares in our calculation. Consider a very basic formula that governs $G_p$ of any arbitrary process \\cite{dixon1994spread}\n \n\\begin{equation}\nG_p=\\frac{B_{rf}}{B_{info}}=B_{rf}T_{int}\\,,\n\\label{GP}\n\\end{equation}\n\nwhere $B_{rf}$ in input bandwidth, and $B_{info}$ is the information bandwidth and $T_{int}$ is the integration time. This formula states that you can improve SNR of the input signal by integration as long as noise of samples are independent, otherwise integration will amplify the noise the same as signal and SNR won't improve. To make it more clear, suppose that input signal bandwidth is $1 MHz$, and assume that it is sampled by $ 1MHz$ sampling rate. Then we integrate the signal coherently for $1 ms$, in other words, we integrate $1000$ samples of the signal coherently. Consequently, $G_p=1000=\\frac{1Mhz}{1Khz}=1Mhz*1ms$. If we sample the signal with a higher sampling rate, we will have more samples for integration, however, noise of samples are correlated and the integration won't result in higher SNRs.  \nIn view of (\\ref{GP}), lets calculate $G_p$ for the first stage. Given each monotone of the code-book is received for $T_p$, assuming bandwidth of $B_{rf}$ for the BPF, $G_p$ of the first stage can be formulated as, \n\\begin{equation}\n    G_p(stage-1)=B_{rf}T_p\\,.\n\\end{equation}\nConsider that the only information that each ED measures is the amplitude of the standing wave, which is constant during $T_p$, Therefore, the amplitude can be estimated by integrating the input signal for $T_p$.\nTo calculate $G_p$ for the next stage, consider that the wave-guide length is $L$ which is in order a wavelength, as we sample the standing wave through the wave-guide, it is equivalent to sample the standing wave in time with a rate more than $f_c$, since $B_{rf}$ is much less than $f_c$, noise of these samples are not independent and integration at the second stage won't result in  any SNR improvement. \n\nAt the last stage we measure PDoAs for the frequency code-book in different time slots, therefore noise of phase difference measurements at each time slot is independent of all other time slots -even if frequencies of pilots at two different\ntime slots are the same-; therefore, PDoAs can be integrated over all the code-book's pilots and the processing gain of stage-3 can be expressed as     \n \n\\begin{equation}\n   G_p(stage-3)=S_f\\,.\n\\end{equation}%\nFinally, the total $G_p$ (processing gain) of all stages is \n\n\\begin{equation}\n    G_p(total)=B_{rf}T_p S_f\\,.\n    \\label{TPG}\n\\end{equation}\n\n$T_p S_f$ equals total time spent on receiving pilots by the receiver, in other words, using the proposed technique, we can make use of the whole duration of DoA estimation procedure to improve input SNR and consequently, improve DoA estimation precision. As we discussed earlier, directional techniques spend substantial amount of time for spatial search to find the other side of the link, moreover, both sides can not search for each other at the same time which further increases the spatial search duration. Contrarily, employing our proposed technique, both sides are able to search for the other side at the same time and can take advantage of the whole search duration to improve DoA estimation precision.    \n\n\n\n \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=6in,height=3.5in]{Figure_BD.pdf}\n    \\caption{Block diagram of our proposed DoA estimation technique. Input SNR is improved through stages 1 and 3. In stage-1, a monotone signal is received at two antennas and passes through a band-pass-filter (BPF) via each path. After amplification via a low-noise-amplifier (LNA) in each path, both signals enter a wave-guide to form a standing-wave. Amplitude of the standing-wave is measured by a group of energy detector (ED) sensors, which inherently are low-pass filters and therefore, improves the SNR. Then the amplitude is sampled and can be integrated during each monotone time-step ($T_p$). After sampling, signal passes through a phase detector. Finally, PDoAs measured for all frequencies of the code-book are used to estimate DoAs using a matched filter which improves SNR for the second time.}\n    \\label{BD}\n\\end{figure}\n\\subsection{Uplink and Downlink DoA Estimation}\n\\label{scale}\nIn this section we are going to answer the following question: \"How does PS perform in the presence of multiple users? How many devices can find their relative angles simultaneously using PS?\" to answer these questions assume the following scenario: There is a base-station (BS) and $N_d$ devices around it in an environment, all devices require to estimate signal DoAs from the base-station (downlink), and the base-station requires to know DoAs of signals from devices (uplink). In downlink scenario, it is only required that BS sends one common code-book and all devices can find DoA of BS by measureing PDoAs of pilots of the common code-book. However, the uplink scenario is more complicated. If all  devices send the same code-book it is impossible for the BS to distinguish between DoAs. Therefore, devices' code-books have to be orthogonal either in time or frequency. If the BS can split the code-book band ($B$) to $N_{rf}$ sub-bands and uses an exclusive SWR for each sub-band, it can estimate DoA from $N_{rf}$ devices simultaneously (Figure \\ref{Uplink}), since, $N_{rf}$ different frequency code-books can be processed simultaneously at the BS. Considering, the BS can be equipped by antennas with much larger $D$ and more complicated receivers than devices, the BS can estimate DoA from multiple devices simultaneously. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=7in,height=3.5in]{Uplink.pdf}\n    \\caption{In uplink scenario, to be capable of discriminating between DoAs of multiple devices, the BS requires to be equipped with two filter-banks at both lines of it's PIA, and a separate SWR for each frequency sub-band.}\n    \\label{Uplink}\n\\end{figure}\n\n\\subsection{Ultra-fast DoA Estimation}\n\\label{UFast}\nAs we discussed in section \\ref{FreqCB}, we suggest measuring PDoAs for multiple frequencies over multiple time slots to avoid using a large SWWG. In that architecture, we assumed we can only use a single SWR. Therefore, we have to measure PDoA of different pilots at different time slots. Nevertheless, instead of using a single SWR, it is possible to use a cascade of multiple SWRs, discriminating between multiple pilots using a filter bank, and find the PDoA for each monotone exploiting a specific SWR (the architecture is presented in Figure \\ref{Uplink}). Using such an architecture, we can estimate all incoming DoAs in a single time slot, without any negative impact on the processing gain and the DoA estimation precision. Such an ultra-fast DoA estimation has not been previously possible using directional antennas, since those techniques are bound to spatial search. Ultra-fast DoA estimation using PS requires more complex hardwares in comparison to the  technique introduced in section \\ref{FreqCB}, which may make it overpriced or oversized to be implemented on commercial mobile phones. However, it may be very promising for applications such as radar, mm-wave network backhaul, UAV and satellite communications, where more complex and bulky hard-wares can be implemented on devices. \n\n\\section{Simulation Results}\n\\label{sim}\n\nIn this section, the perfromance of the proposed DoA estimation technique for different parameters is studied.\n \\subsection{Simulation Setup and Results}\n \n In the first simulation, we examine a basic scenario where a signal arrive at PIA through only one path, therefore, there is only one DoA to be estimated. We set $f_c=60 GHz$, $B=10GHz$, the steps of the codebook is 40 and pilots are selected equally spaced from $55 GHz$ to $65 GHz$ and $T_p=1 \\mu s$, $c=3 * 10^8 \\frac{m}{s}$, $D = 20 cm$, $L=2.5 mm$ and the number of EDs along the SWWG is set to 30. The received $SNR$ in each antenna is set to $20dB$ and the DoA of the signal is set to $60^o$. Figure \\ref{simfig1}  shows the result of applying the matched filter of (\\ref{matchedfilter}) for differnt $\\theta$. As Figure \\ref{simfig1} illustrates the output shows a distinctive peak at $60^o$. Moreover, Figure \\ref{simfig1} illustrates that PS output pattern is similar to beam-pattern of a ULA with 13 elements. This result may seem contradictory to (\\ref{m\/d}), which indicates that FIM of angle estimation using PIMs equals to a FIM of a ULA with $m$ elements, in which $m$ obeys (\\ref{m\/d}), that results in $m=33$ applying the mentioned parameters. Keep in mind that, (\\ref{CRMPIM}) shows CRLB of angle estimation using PIMs if and only if signal from one source is received at PIA, on the other hand, Figure \\ref{simfig1} shows how PS can discriminate between two or more signals if they are originated from different DoAs. As (\\ref{CRMPIM}) indicates, this bound is only a function of $D$ and SNR, while (\\ref{Res}) shows that DoA estimation resolution is a function of $BD$, which means our technique can discriminate between two incoming DoAs if and only if $B$ is wide enough.                \n \\begin{figure}\n    \\centering\n    \\includegraphics[width=5in,height=3.5in]{Fig11.png}\n    \\caption{The matched filter of (\\ref{matchedfilter}) is applied to phase differences measured for 40 pilots of a frequency code-book that changes between $[55,65] GHz$ and the output is plotted for $\\theta$ between $[0,180]^o$ and is compared with a beam pattern of a ULA with 13 elements \\cite{er1990linear}. $DoA_{in}=60^o$, $f_c=60Ghz$, $SNR_{in}=20dB, \\frac{BD}{c}=6.67$. $D=20cm$ }\n    \\label{simfig1}\n\\end{figure}\n \n In the following simulation we are going to study DoA estimation resolution of the technique. In this simulation, parameters are the same as the first simulation, unless, we assume that the signal received at PIA from two different paths and two different DoAs, we investigate whether the proposed technique can distinguish between these two DoAs or not. Figure \\ref{simfig2} shows the matched filter output for 4 different pairs of DoAs, the gap between 2 DoAs are $20^o,15^o,10^o,5^o$ respectively. As Figure \\ref{simfig2} illustrates, when the gap between two DoAs is $20^o$, two lobs regarding each DoA are completely separated and distinguishable. When the gap resuces to $15^o$, two lobs start merging together, however, two peaks regarding two DoAs are again distinguishable. As the gap further reduces to $10^o$, two lobes merges more and two peaks are hardly distinguishable. And finally when the gap reduces to $5^o$, two lobes completely merge together and two peaks are not distinguishable. With respect to (\\ref{Res}), the DoA resolution with $B=10Ghz$ and $D=20cm$ is approximated to be $17^o$. Since we calculate (\\ref{Res}) assuming matched filters of two DoAs are perpendicular to each other, which means that two lobes are completely separated, thus simulations results are in compliance with (\\ref{Res}). However, it seems that it is a strict metric for DoA resolution, to assume that two DoA are resolvable only if two lobes are completely separated. In practice, we may use $75\\%$ or $50\\%$ of (\\ref{Res}) as a more realistic metric of the resolution. In Figure \\ref{simfig4}, we illustrate matched filter main-lobe width and (\\ref{Res}) versus the parameter $\\frac{BD}{c}$, given $DoA=60^o$. Main-lobe width is defined as the gap between the minimum and the maximum $\\theta$ in which the matched filter output is closer than 3db to its peak. As Figure \\ref{simfig4} expresses, main-lobe width for $\\frac{BD}{c}=6.67$ is $8.5^o$ which is half of the figure calculated by (\\ref{Res}), moreover, this proportion between main-lobe width and (\\ref{Res}) almost holds for every $\\frac{BD}{c}$. Therefore we can use half of (\\ref{Res}) as the DoA estimation resolution if we consider the more practical main-lobe width metric.                         \n \n \\begin{figure}\n    \\centering\n    \\includegraphics[width=7in,height=5in]{fig2.pdf}\n    \\caption{To analyse DoA estimation resoltion, the matched filter outputs are depicted for 4 different pairs of incoming DoAs : (a) $40^o,60^o$, (b) $45^o,60^o$, (c) $50^o,60^o$, (d) $55^o,60^o$. $SNR_{in}=20dB, \\frac{BD}{c}=6.67$. }\n    \\label{simfig2}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in,height=3in]{fig4.png}\n    \\caption{DoA estimation resolution based on the main-lobe width metric and the meric introduced by (\\ref{Res}). $SNR_{in}=20dB$,    $DoA=60^o$ }\n    \\label{simfig4}\n\\end{figure}\nIn the next simulation we analyse the effect of input SNR on DoA estimation error for differnt values of $B$ and $D$. In this simulation, input SNR changes in the interval $[-15,20]dB$. To analyse the error we calculate the root-mean-square error (RMSE) for each input SNR, by repeating the simulation 1000 times and find the average of SE for each SNR. Figure \\ref{simfig3} illustrates RSME of DoA estimation error. As Figure \\ref{simfig3} illustrates angle estimation error depends on $BD$, as $BD$ and SNR increases, error declines.  Similarly, Figure \\ref{CDF} shoes that error CDF of PIAs with equal $BD$ factors are roughly the same. This is consistent with our results on angle resolution. However, it may seems inconsistent with (\\ref{CRMPIM}), which indicates that CRLB of DoA estimation decrease in proportion to $D$ not $BD$, this is because we employ matched filter of (\\ref{matchedfilter}) to find the DoA. To improve the precision, future works may considering using the output of the matched filter only to disambiguate the phase to a valid TDoA and estimate DoA directly based on the TDoA.\n\nIn the next simulation we consider a scenario in which, frequency steps, band-width, antenna gap and integration time is strongly limited. In this scenario, the source can only send 4 pilots at $[59.5,59.83,60.16,60.5]GHz$, $D=1cm$, $T_p=100ns$ and the whole number of available time slots is $M$. The source send those four frequencies in $M$ time slots respectively and repeats sending them until covers the whole $M$ slots. Consequently, the integration time is $MT_p$ -the maximum integration time in this simulation is $16 \\mu s$-. We also assume there is only one incoming DoA at the PIA, since the PIA is not able to discriminate between two DoAs because of limited bandwidth and short antennas' gap. As Figure \\ref{tightcon} shows, the proposed technique is able to estimate DoA with RMSE less than $10^o$ if input SNR is high enough, for $M=160$ input SNR should be above $7dB$ and for $M=20$ input SNR should be above 16dB. Therefore, as input SNR levels decreases we should increase integration time of our technique to provide us with acceptable DoA estimation precision.               \n\nIn the next simulation, parameters are the same, unless there is a NLoS path ($30^o$) besides the LoS ($90^o$) path with a power 15 dB less than LoS path. This simulation is consistant with the experimental results of \\cite{rappaport2013millimeter} on distribution of DoA paths between TX and RX in an urban environment in Brooklyn, New York. In This simulation integration time is set to $40 \\mu s$. As Figure \\ref{tightconSIR} shows existence of the second path does not have a considerable effect on RMSE of the proposed technique. Therefore, it seems that even a very simplified version of the proposed technique (narrow beam-width, short antenna gap) can be used in real world practical mm-wave DoA estimation applications.\n\nIn the next simulation, we investigate DoA estimation precision based on power of NLoS path. Given LoS path arrives at $90^o$ and NLoS path arrives at $30^o$ at the PIA, Figure \\ref{tightconSIR} depicts RMSE of DoA estimation versus power ratio of LoS path to NLoS path. we set the integration time to be $4 \\mu s$, since NLoS path can be considered as a coherent interference, thus SIR won't be improved by integration.  Figure \\ref{tightconSIR} expresses that RMSE drops below $10^o$ when SIR is higher than $8dB$ and $5^o$ when SIR in higher than $12dB$. Referring to \\cite{rappaport2013millimeter}, the  power of the strongest NLoS path expects to be more than 15dB weaker than the LoS path in a dense urban environment, therefore we expect that the proposed technique can estimate DoA of LoS path in an urban environment with error less than $3^{o}$ even when the available bandwidth is very limited (e.g. 1GHz) and the antenna gap is very short (1cm). Such a performance make PS a promising technique for beam initialization requirements of 5G networks, since the required band-width is easily accessible in mm-wave and the PIA size is very small that make it easily implementable on any device. \n\nIn the last simulation we compare the performance of PS technique with  a ULA (beamforming) in terms of DoA estimation precision of a single incoming path. ULA exploits beamforming to steer its beam and compare received power from different angles to find DoA. Figure \\ref{ULAPS} depics RSME of DoA estimation for 3 PIAs with different values of $D$ and $B$ and 3 ULAs with different number of array elements. In this simulation, we suppose that ULA is able to integrate the received signal coherently for $T_p$, we also set $T_p=100ns$ and $M=200$, therefore the total integration time of the PIA is $20 \\mu s$ . The $B_{rf}$ for ULA and PIA is the same and is set to $100MHz$. As the figure illustrates, the performance of the PIA with $D=10cm$ and $B=10GHz$ is approximately equal to ULA with 20 antennas equally spaced with half wavelength gap (array aperture is 5cm) especially for SNR above -9 dB. Moreover, performance of ULA with 4 elements is close to PIA with $BD=10^8$. Consider that for SNRs above -3 dB, the RMSE is less than $5^o$ for an array with 4 elements, while, beam-width of the array is about $30^o$. If such wide beam antenna uses for communication, the angle estimation precision is much more than what is required. As we discussed in section \\ref{DoABW}, angle estimation precision and beamwidth are not coupled and there is no necessity for  antennas of SDMA and IA tasks to be the same.\nMoreover, even when array aperture is small and the number of array elements is few, to obtain a DoA with desirable accuracy a long spatial search is required. For example, to reach an accuracy of $1^o$, any directional antenna with an arbitrary beam-width requires to search at least 180 points to cover a $180^o$ area, in a 2D scenario. On the other hand, to improve PS precision we can simply increase the gap between two antennas and therefore no more complex hardware is required. Furthermore, better precision with ULA requires narrower beams and consequently more time is needed for spatial search to perform the IA task. On the other hand, since no spatial search is required by PS technique, we can obtain an initial guess of DoA very fast, and gradually improve the precision of the estimation by improving SNR through integration.   \n\n\\section{Conclusion}\n\\label{conc}\nIn this paper, we have introduced DoA estimation via SWR. We have shown that how SWR measures phase difference between two antennas for different frequencies named as PDoAs. We have considered two different implementation schemes for PS: 1- using a long wave-guide to measure amplitude of a standing wave, produced by interaction between two waves received at the two antennas 2- measuring the amplitude of the standing wave inside a short wave-guide for different frequencies of a frequency code-book at different time slots. Moreover, for the second scheme, we have explained that we can use a cascade of multiple PS receivers to measure PDoAs at different frequencies concomitantly. We have developed a signal processing method to extract multiple simultaneous DoAs from PDoAs. We have analyzed processing gain of the technique and discussed that we can take advantage of the required time for spatial search essential for directional techniques to improve DoA estimation precision in PS. Finally, we have analyzed that IA and SD tasks of mobile directional communication can be separated and performed via two dedicated antennas; IA can be performed by PS, SD can be by performed by an array. The separation between these two tasks, reduces delay and overhead and increases communication capacity. Our results have shown that, PS can perform similar to an array, while the required receiver is much less complex than the array receiver, and the spatial search required for DoA estimation can be bypassed.  \n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in,height=3in]{FigSNRBD.png}\n    \\caption{RSME of DoA estimation versus input SNR for differnt valus of $B$ and $D$. $DoA=60^o$, $S_f=40$,$B_{rf}T_p=100$.} \n    \\label{simfig3}\n\\end{figure}\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n             \\includegraphics[width=1.2\\textwidth]{CDFminus10dB.png}\n         \\caption{SNR=-10dB}\n         \\label{CDF1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n             \\includegraphics[width=1.2\\textwidth]{CDFminus5dB.png}\n         \\caption{SNR=-5dB}\n         \\label{CDF2}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[width=1.2\\textwidth]{CDF0dB.png}\n         \\caption{SNR=0dB}\n         \\label{CDF3}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[width=1.2\\textwidth]{CDF5dB.png}\n         \\caption{SNR=5dB}\n         \\label{CDF4}\n     \\end{subfigure}\n        \\caption{CDF of angle estimation error for differnt values of $D$, $B$ and input SNR. $DoA=60^o$, $S_f=40$,$B_{rf}T_p=100$.}\n        \\label{CDF}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in,height=3in]{FigSNR.png}\n    \\caption{RSME of DoA estimation. Source only transmits four pilots at $[59.5,59.83,60.16,60.5]GHz$, each in a time slot with duration $T_p$, source repeats emitting these monotones for $M$ time slots. $DoA=60^o$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$.} \n    \\label{tightcon}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \n    \\includegraphics[width=4.5in,height=3in]{FigSIR.png}\n    \\caption{RSME of DoA estimation, signal receives at PIA via two paths, one LoS path ($90^o$) and one NLoS path ($30^o$) in the presence of coherent interference. Source only transmits four pilots at $[59.5,59.83,60.16,60.5]GHz$. $SIR=15dB$, $M=400$, $DoA=60^o$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$.} \n    \\label{tightconSIR}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in,height=3in]{FigvarSIR.png}\n    \\caption{Signal receives at PIA via two paths, one LoS path ($90^o$) and one NLoS path ($30^o$), . $M=40$, $T_p=100ns$, $B_{rf}=100MHz$, $D=1cm$ and source frequency codebook is $[59.5,59.83,60.16,60.5]GHz$.} \n    \\label{tightconvarSIR}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in,height=3in]{BeamformingVsPS.png}\n    \\caption{Comparing PS with ULA with half-wavelength gap between array elements in terms of DoA estimation precision. $M=200$, $T_p=100ns$, $B_{rf}=100MHz$} \n    \\label{ULAPS}\n\\end{figure}\n\n\\newpage\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nLet $G$ be a residually finite group endowed with a word metric given by a finite generating set $X$.\nA subset $S \\subseteq G$ is \\emph{fully detected} by a group $Q$ if there exists a homomorphism $\\varphi: G \\to Q$ such that $\\varphi |_S$ is injective.\nFor a natural number $n$, set $\\G_G^X(n)$ to be the minimal order of a group $Q$ that fully detects the ball of radius $n$ in $G$ (first studied in \\cite{BM11}).\nThe \\emph{full residual finiteness growth of $G$ with respect to $X$}\nis the growth of the function $\\G_{G}^{X}$, that is, its equivalence\nclass under the\nequivalence relation defined by $f\\approx g$ if and only if there is a\nconstant $C$ so that $f(n) \\leq Cg(Cn)$ and $g(n)\\leq Cf(Cn)$ for all\nnatural numbers $n$.\nThe growth of $\\G_G^X$ is independent of choice of generating set $X$ (see Lemma \\ref{lem:containment}).  Therefore\nfull residual finiteness growth is an invariant of a finitely\ngenerated group, and can be denoted simply $\\G_G$.\n\nThis article focuses on finitely generated nilpotent groups. \nWhile it is known that \\emph{word growth} (defined below after Theorem\n\\ref{MainTheorem}) has precisely polynomial growth over this class of\ngroups \\cite{Bass72}, computing other growth functions\nfor this class has proved to be a serious task. \nIndeed, even computing the answer for \\emph{subgroup growth} \\cite{MR1978431} in the two-generated free nilpotent case takes work; see \\cite{MR2342452}.\nThe main difficulty lies in that the structure of $p$-group quotients of a fixed finitely generated nilpotent group can depend heavily on the choice of the prime $p$.\nThat is, it is difficult to draw global behavior (behavior over all finite quotients) from local behavior (behavior over all finite quotients that are $p$-groups).\nMoreover, comparisons between full residual finiteness growth and\n word growth, which is to our knowledge the only nontrivial growth\n function known to have precisely polynomial growth over the class of\n nilpotent groups, do not allow one to immediately draw much information on $\\G_G^X$.\nIn fact, the growth of $\\G_G^X$ is often, but not always, strictly larger than the word growth of $G$ (see Theorem~\\ref{thm:abelian}).\nObtaining \\emph{sharp} control of full residual finiteness growth\nover this class requires new understanding of the structure theory of\nnilpotent groups.\n\nTo present our findings, we begin with some basic examples. In \\S\\ref{sec:examples}, we show that $\\G_G(n) = n^k$ for $G=\\mathbb{Z}^k$ and\n$\\G_G(n) = n^6$ for $G$ equal to the discrete Heisenberg group. The key\nproperty shared by these examples is that the center of $G$ is equal\nto the last term of the lower central series of $G$. In\nfact, we explicitly compute full residual\nfiniteness growths for all groups satisfying a slightly weaker condition.\nTo make this precise we introduce notation: for a nilpotent group $G$\nof class $c$ we denote by $\\gamma_c(G)$ the last nontrivial term of\nits lower central series, by $Z(G)$ its center, and by $\\dim(G)$ its\ndimension. (See \\S\\ref{sec:nilpotent} for more explicit definitions.)\n\n\\begin{introtheorem} \\label{theorem:zl} Let $G$ be a finitely generated\n  nilpotent group of class $c$ with $[Z(G):\\gamma_c(G)] < \\infty$.\n  Then\n  $$\n  \\G_G(n) \\approx n^{c \\dim(G)}.\n  $$\n\\end{introtheorem}\n\nThe conclusion of Theorem \\ref{theorem:zl} does not generally hold when $[Z(G) :\n\\gamma_c(G)] = \\infty$. This is seen by taking $G$ to be the\ndirect product of the discrete Heisenberg group with $\\mathbb{Z}$, which\nsatisfies $\\G_G(n) \\approx n^7$ while $c=2$ and $\\dim(G) = 4$.\nGroups not satisfying the hypothesis of Theorem \\ref{theorem:zl} are\ngenerally more complicated than this example.\nFor instance, in Proposition \\ref{D22example} we provide an example of\na nilpotent group $\\Gamma$ of class $c=3$ with $\\dim(\\Gamma) = 8$ and\n$\\G_\\Gamma(n) \\approx n^{22}$ that does not split as a direct product.\n\nFor general nilpotent group $G$ of class $c$, we introduce methods to find an upper\nbound on the polynomial degree of $\\G_G$.\nDefine a {\\em terraced filtration} of $G$ to be a filtration $1 = H_0 \\leq\nH_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ where each $H_i$ is a maximal normal\nsubgroup of $G$ satisfying $H_i\\cap \\gamma_{i+1}(G) = 1$. \nEvery terraced filtration of $G$ gives an explicit polynomial upper bound\non growth of $\\G_G$.\n\n\\begin{introtheorem} \\label{MainTheorem}\nLet $G$ be a finitely generated nilpotent group of class $c$.\nSuppose $1=H_0\\leq H_1 \\leq \\dotsb H_{c-1}\\leq G$ is a terraced\nfiltration of $G$. Then\n\\[\n\\G_G(n) \\preceq n^{c \\dim(G) - \\sum_{i=1}^{c-1} \\dim(H_i)}.\n\\]\n\\end{introtheorem}\n\\noindent\nThis upper bound generally depends on the choice of terraced\nfiltration. See the comments following the proof of Theorem\n\\ref{MainTheorem} in \\S\\ref{MainProofSection} for an explicit example\ndemonstrating this dependence. It would be interesting to determine\nwhether the lowest upper bound obtained from a terraced filtration\nby Theorem \\ref{MainTheorem} is optimal.\n\nResults on distortion in nilpotent groups from Osin \\cite{MR1872804} and Pittet \\cite{Pittet97} play an important role in all of our proofs.\n\nWe also compare full residual finiteness growth to word growth.\nRecall that the {\\em word growth}, $w_G$, of a finitely generated group $G$ is the\ngrowth of the function $w_G^X(n) = \\left\\lvert B_G^X(n) \\right\\rvert$,\nwhich is independent of $X$.\nGromov \\cite{MR623534} has characterized nilpotent groups in the class\nof finitely generated groups as those for which $w_G$ is polynomial. \nBy applying this theorem, it is shown in \\cite{BM11} (see Theorem 1.3 there) \nthat full residual finiteness growth enjoys the same conclusion.\nIn spite of this similarity, these two growths rarely coincide.\nOur final result characterizes nilpotent groups for which full residual\nfiniteness growth equals word growth.\n\n\\begin{introtheorem} \\label{thm:abelian}\nLet $G$ be a finitely generated nilpotent group.\nThen $\\G_G \\approx w_G$ if and only if $G$ is virtually abelian.\n\\end{introtheorem}\n\\noindent\nIn \\S\\ref{sec:fullresidualfinitenessgrowth} we provide a geometric\ninterpretation of $\\G_G^X$. From this point of view, Theorem\n\\ref{thm:abelian} implies that virtually abelian groups are characterized in the\nclass of finitely generated nilpotent groups solely in terms of the\nasymptotic data of the Cayley graph.\nA non-normal version of full residual finiteness growth,\n  the \\emph{systolic growth}, is studied in \\cite{YC14}.\nThere it is shown that systolic growth matches word growth if and only if the group is \\emph{Carnot}.\n\n\nThis paper is organized as follows: In \\S\\ref{BackgroundSection} we\npresent basic results on nilpotent groups and full residual finiteness\ngrowth, including important lemmas on\nword metric distortion of central subgroups of nilpotent groups\nfollowing from work of Osin \\cite{MR1872804} and Pittet \\cite{Pittet97}. In\n\\S\\ref{sec:abeliangroups} we prove Theorem \\ref{thm:abelian}. In\n\\S\\ref{sec:heisenberg} we compute the full residual\nfiniteness growth of the Heisenberg group and prove\nTheorem \\ref{theorem:zl}. \nIn \\S\\ref{MainProofSection} we give an illustrative example showing that the\nconclusion of Theorem \\ref{theorem:zl} does not hold in general, and\nprove Theorem \\ref{MainTheorem}.\n\nWe finish the introduction with a bit of history.\nThe concept of full residual finiteness growth was first studied by\nBen McReynolds and K.B. in \\cite{BM11}. The full\n  residual finiteness growth of the discrete Heisenberg group is\n  presented in \\cite{YC14}.\nCompare full residual finiteness growth to the concept of \\emph{residual finiteness growth}, which measures\nhow well individual elements are detected by finite quotients,\nappearing in \\cite{B09},\n\\cite{MR2583614}, \\cite{BM13}, \\cite{KM12}, \\cite{R12}, \\cite{BK12}, \\cite{KMS13}.\nAlso compare this with Sarah Black's \\emph{growth function} defined and studied in \\cite{MR1659911}. \nFull residual finiteness growth measures how efficiently the word growth function \ncan be recovered from Black's growth function. See remarks in \\cite{MR1659911} on p.\\ 406 before \\S 2 for further discussion.\n\n\\paragraph*{Acknowledgements}\n\nThe authors are grateful to Benson Farb for suggesting this pursuit.\nThe authors acknowledge useful conversations with Moon Duchin, Michael\nLarsen, Ben McReynolds, Christopher Mooney, and Denis Osin. The authors are\nfurther grateful to Benson Farb and Ben McReynolds for comments on\ndrafts of this paper.\nK.B. gratefully acknowledges support from the AMS-Simons Travel Grant Program.\nThe authors are very grateful to the excellent referee for comments\nand corrections that greatly improved the paper and for suggesting\nProposition \\ref{prop:Treduction}.\n\n\\section{Some background and preliminary results} \\label{BackgroundSection}\n\n\\subsection{Full residual finiteness growth} \\label{sec:fullresidualfinitenessgrowth}\n\nIn this subsection we give a geometric interpretation of full residual\nfiniteness growth for finitely presented groups.\n\nWrite $f \\preceq g$ to mean there exists $C$ such that $f(n) \\leq C g(Cn)$.\nWe write $f \\approx g$ if $f \\preceq g$ and $g \\preceq f$.\nRecall that the \\emph{growth} of a function $f$ is the equivalence class of $f$ with respect to $\\approx$.\n\nWe first prove a lemma that implies that the growth of the function\n$\\G_G^X$ defined in the introduction is independent of generating set\n$X$:\n\n\\begin{lemma} \\label{lem:containment}\nLet $G$ be finitely generated with finitely generated subgroup $H \\leq G$.\nFix finite generating sets $X$ and $Y$ for $G$ and $H$.\nThen $\\G^Y_H \\preceq \\G_G^X$.\n\\end{lemma}\n\n\\begin{proof}\nSince $H \\leq G$, there exists $C > 0$ such that any element in $Y$ can be written in terms of at most $C$ elements in $X$.\nThus, $B_H(n) \\subseteq B_G(Cn)$ for any $n > 1$.\nBecause any homomorphism from $G$ restricts to a homomorphism from\n$H$, this gives\n$$\n\\G_G^X(Cn) \\geq \\G_H^Y(n),\n$$\nas desired.\n\\end{proof}\n\nLemma \\ref{lem:containment} in particular implies that if $X$ and $Y$\nare two finite generating sets of a group $G$, then $\\G_G^X \\approx\n\\G_G^Y$. Let $\\G_G$ denote the equivalence class of $\\G_G^X$ with\nrespect to $\\approx$ for any finite generating set $X$ of $G$.\n\nWe now provide a geometric interpretation of $\\G_G$ in the case that\n$G$ is a finitely presented group.\nLet $G$ be a residually finite group with Cayley graph $\\Gamma$ with\nrespect to a finite generating set $S$.\nEach edge of $\\Gamma$ is labeled by the corresponding generator.\nFor a subset $X \\subseteq \\Gamma$, we set $\\partial X$ to be the collection of edges and vertices of $X$ each of which has closure not contained in the interior of $X$.\nLet $\\{ A_k \\}$ be an increasing sequence of finite connected subsets of $\\Gamma$ with \n$$\nA_{k+1} = \\partial A_{k+1}  \\sqcup A_k.\n$$\nThen the sequence of subsets, $\\{A_k \\}$, is called a \\emph{growing\n  sequence}.  Let $B_{G}^S(n)$ denote the closed ball of radius $n$ in\nthe Cayley graph of $G$ with respect to the word metric induced by\n$S$. We will omit the $S$ from the notation when the generating set is\nunderstood and there is no chance for confusion. The prototypical\nexample of a growing sequence is the sequence that assigns to each\npositive integer $k$ the metric ball $B_{G}^S(k)$ in the Cayley graph\nof $G$ with respect to $S$.\n\nThe \\emph{geometric full residual finiteness growth of $\\Gamma$ with respect to\n  $\\{ A_k \\}$} is the growth of the function, $\\G_{\\Gamma}^{\\{ A_k \\}}: \\mathbb{N} \\to \\mathbb{N}$, given by \n\\begin{eqnarray*}\nn \\mapsto \\min \\{ |Q| : \\text{ $Q$ is a group with $A_n$ isometrically} \\\\\n\\text{embedding in one of its Cayley graphs}\\}.\n\\end{eqnarray*}\n\nOur first lemma demonstrates that the growth of $\\G_G^{\\{A_k \\}}$ does\nnot depend on the growing sequence.\n\n\n\\begin{lemma} \\label{lem:indepofgrowingset}\nLet $\\{ X_k \\}$ and $\\{ Y_k \\}$ be two growing sequences for a finitely generated group $G$.\nThen $\\G_{G}^{\\{X_{k} \\}} \\approx \\G_{G}^{\\{ Y_{k} \\}}$.\n\\end{lemma}\n\n\\begin{proof}\nWe first assume that $\\{ X_k \\}$ and $\\{ Y_k \\}$ are growing sequences from the same Cayley graph realization of $G$.\nThen there exists $K \\in \\mathbb{N}$ such that\n$$\nY_1 \\subseteq X_K \\text{ and } X_1 \\subseteq Y_K.\n$$\nHence, $C_G^{\\{Y_k\\}} (n) \\leq C_G^{\\{X_{k}\\}}(K+i) \\text{ and } C_G^{\\{X_k\\}} (n) \\leq C_G^{\\{Y_{k}\\}}(K+i).$\nThus, we can assume that $\\{ X_k \\}$ and $\\{ Y_k \\}$ are the word metric $k$-balls of $G$ with respect to two different generating sets.\nIt is straightforward to see that there exists $C > 0$ such that $Y_{n} \\subseteq X_{Cn} \\subseteq Y_{C^2n}$ for every natural number $n$.\nHence,\n$$\n\\G_G^{\\{Y_k\\}} (n) \\leq \\G_G^{\\{X_k\\}} (Cn) \\leq \\G_G^{\\{Y_k\\}} (C^2 n),\n$$ as desired.\n\\end{proof}\n\nNext we show that the notions of full residual finiteness growth,\ngiven in the introduction, and geometric full residual\nfiniteness growth, given in this section, agree in the case that the group\n$G$ is finitely presented.\nIt would be interesting to determine if this equivalence holds for all finitely generated groups.\n\n\\begin{lemma} \\label{lem:cayleyvsgirth}\nLet $G$ be a finitely presented group.\nFor any generating set $X$ and growing sequence $\\{ A_k \\}$ we have\n\\[\\G^{\\{ A_k \\}}_G \\approx \\G^{X}_G.\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $X$ be a finite generating set for $G$ and let $R$ be the set of finite relations.\nIt is clear that $\\G^{\\{ A_k \\}}_G \\preceq \\G^{X}_G$. We show the reverse inequality.\nWe can, by Lemma \\ref{lem:indepofgrowingset}, suppose that the growing set $\\{ A_k \\}$ is simply the sequence $ \\{ B_G^X(k) \\}$.\nIt suffices, then, to show that there exists $N \\in \\mathbb{N}$ such that for any $n > N$ and any finite group, $Q$, with $B_G(n)$ isometrically embedding in a Cayley graph realization of $Q$, there exists a homomorphism\n$\\phi: G \\to Q$ with $\\phi |_{B_G(n)}$ being injective.\nSelect $N$ to be the maximal word length of any element in $R$.\nThen since $B_n$ isometrically embeds in a Cayley graph of $Q$, we see that there exists a generating set for $Q$ such that each relator $R$ is satisfied by this generating set.\nThis finishes the proof.\n\\end{proof}\n\nThe next lemma controls some of the full residual finiteness growth of a direct product of groups.\n\n\\begin{lemma} \\label{lem:directproducts}\nLet $G$ and $H$ be finitely generated groups.\nThen \n$$\n\\G_{G \\times H} \\preceq \\G_G \\cdot \\G_H.\n$$\n\\end{lemma}\n\n\\begin{proof}\n  Fix generating sets $X$ and $Y$ for $G$ and $H$. Then $(X \\times \\{\n  1 \\}) \\cup ( \\{1\\} \\times Y )$ is a finite generating set for $G \\times H$.\n  Note that\n  $$\n  B_{G\\times H} (n)  \\subseteq (B_G (n) \\times \\{1\\}) (\\{1\\} \\times B_H(n)).\n  $$\n  Thus, if $Q_1$ is a quotient that fully detects $B_G(n)$ and $Q_2$ a quotient that fully detects $B_H(n)$, then $Q_1 \\times Q_2$ fully detects $B_{G \\times H}(n)$.\n  We see then that $\\G_{G \\times H} \\preceq \\G_G \\G_H$, as desired.\n\\end{proof}\n\\noindent\nCan the conclusion of Lemma \\ref{lem:directproducts} be improved to $\\G_{G\\times H} \\approx \\G_G \\G_H$?\nThis can possibly be false: it is not even true that if $\\varphi : G \\to H$ is a surjective homomorphism, then $\\G_G(n) \\succeq \\G_H(n)$. Consider a free group mapping onto one of Kharlampovich-Sapir's solvable and finitely presented groups of arbitrarily large residual finiteness growth \\cite{KMS13}.\n\nFull residual finiteness growth is well-behaved under taking the\nquotient by a finite normal subgroup:\n\n\\begin{proposition} \\label{prop:Treduction}\n\tLet $G$ be a finitely generated residually finite group.\n\tLet $T$ be a finite normal subgroup of $G$.\n\tThen $\\Phi_G \\approx \\Phi_{G\/T}$.\n\\end{proposition}\n\n\\begin{proof}\n\tFix a generating set $X$ for $G$, and let $Y$ be the image of $X$ under the quotient map $G \\to G\/T$. Let $K$ be the largest length, with respect to $X$, of an element in $T$.\n\tWe first claim $\\Phi_G^X(K+n) \\geq \\Phi_{G\/T}^Y(n)$.\n\tLet $\\phi: G \\to Q$ be a finite quotient of minimal cardinality that fully detects $B_G^X(K+n)$. That is $|Q| = \\Phi_G^X(K+n)$.\n\tDefine $\\psi : G\/T \\to Q\/\\phi(T)$ by $gT \\mapsto \\phi(g) \\phi(T)$.\n\tLet $g \\in B_{G\/T}^Y(n) \\cap \\ker \\psi$. \n\tBy construction, we may lift $g$ to an element $\\tilde g \\in G$ such that $\\tilde g \\in B_{G}^X(n)$ and $\\phi(\\tilde g) \\in \\phi(T)$.\n\tThat is, there exists $t \\in T$, such that $\\phi(g) = \\phi(t)$, which gives\n\t$$\n\t\t\\phi(\\tilde g t^{-1}) = 1.\n\t$$\n\tIf $\\tilde g t^{-1} \\neq 1$, then this contradicts that $\\phi$ fully detects $B_G(K+n)$. Hence, $\\tilde g = t$, and so $\\ker \\psi \\cap B_{G\/T}^Y(K+n)$ is trivial. It follows that $\\psi$ fully detects $B_{G\/T}^Y(n)$, and so $\\Phi_G^X(K+n) \\geq \\Phi_{G\/T}^Y(n)$, as claimed.\n\n\tSince $G$ is residually finite and $T$ is finite, there exists a normal subgroup, $H$, such that $T\\cap H = 1$.\n\tTo finish, we claim that $\\Phi_{G}^X(n) \\leq [G:H]\\Phi_{G\/T}^Y(n)$.\n\tLet $\\psi : G\/T \\to Q$ be a quotient that fully detects $B_{G\/T}^Y(n)$, with $|Q| = \\Phi_{G\/T}^Y(n)$.\n\tLet $\\phi : G \\to Q$ be the natural map $G \\to G\/T \\to Q$.\n\tSet $N = \\ker \\phi \\cap H$.\n\tClearly, $[G: N] \\leq [G: \\ker \\phi] [G:H] = |Q| [G:H]$.\n\tMoreover, if $g \\in B_G^X(n) \\cap N$, then $g \\notin T$.\n\tHence, by the construction of $Y$, we have that $\\phi(g) \\neq 1$.\n\tIt follows that $G\/N$ fully detects $B_G^X(n)$, and so\n\t$\\Phi_{G}^X(n) \\leq [G:H]\\Phi_{G\/T}^Y(n)$, as desired.\n\\end{proof}\n\n\n We finish the section with a lemma that, in some restrictive cases, allows us to pass to finite-index subgroups.\n\n\\begin{lemma} \\label{lem:finiteindex}\nLet $G$ and $H$ be finitely generated nilpotent groups with $H$ normal\nsubgroup in $G$ of finite index.\nIf every normal subgroup of $H$ is normal in $G$, then $\\G_G \\approx \\G_H$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lem:containment}, it suffices to show that $\\G_G \\preceq \\G_H$.\nFix generating sets for $G$ and $H$ so that $B_H(n) \\subseteq B_G(n)$\nfor all $n>0$.\nBecause $H$ is of finite index in $G$ and thus quasi-isometric to $G$, there exists $C > 0$ such that $H \\cap B_G(2n) \\subseteq B_H(Cn)$.\nLet $H\/K$ be a quotient of $H$ that fully detects $B_H(Cn)$.\nBy our assumption, $K$ is normal in $G$ so $G\/K$ is well-defined.\nThen any element in $B_G(2n)$ not in $H$ is mapped nontrivially onto $G\/K$.\nAnd since $H \\cap B_G(2n) \\subseteq B_H(Cn)$, it follows that $B_G(2n)$ is mapped nontrivially onto $G\/K$.\nThus, $B_G(n)$ is fully detected by $G\/K$, and so we are done.\n\\end{proof}\n\n\n \n\\subsection{Nilpotent groups} \\label{sec:nilpotent}\n\nIn this subsection we fix basic notation and present several lemmas that play important roles in our proofs.\nLet $G$ be a group. The \\emph{lower central series} $\\gamma_{k} (G) $ of $G $ is the sequence of subgroups defined by $\\gamma_{1}(G) = G$ and \n$$\\gamma_{k}(G) =[\\gamma_{k-1}(G),G].$$ \nFor any group $H$, let $Z(H)$ denote the center of $H$.\nThe \\emph{upper central series} $\\zeta_k (G) $ of $G $ is\ngiven by $\\zeta_{0}(G)=\\{e\\}$ and the formula \n$$\n\\zeta_{k}(G) \/ \\zeta_{k - 1} (G) =\nZ(G\/\\zeta_{k-1}(G)).\n$$ \nThe group $G$ is said to be \\emph{nilpotent} if $\\gamma_{k}(G) = 1$\nfor some natural number $k$.  Equivalently, $G$ is nilpotent if and\nonly if it is an element of its upper central series. Moreover, $G$ is\nsaid to be \\emph{nilpotent of class $c$} if $\\gamma_c(G) \\neq 1$ and\n$\\gamma_{c+1}(G) = 1$.\n\nIf $G$ is a finitely generated nilpotent group, then the successive quotients of the upper central series of $G$ are abelian groups of finite-rank.\nThus, the upper central series has a refinement\n$$G =  G_{1} \\ge G_{2} \\ge \\ldots G_{n+1} = 1,$$\nsuch that $G_{i} \/ G_{i+1} $ is cyclic for all $i = 1, \\ldots, n$.  \nThe number of infinite cyclic factors in this series does not depend on the series and is called the \\emph{dimension} of $G$, denoted by $\\dim(G)$ \\cite[p. 16, Exercise 8]{MR713786}.\nLet this series be chosen so that $n$ is minimal.\nAn $n$-tuple of elements $(g_{1}, g_{2}, \\ldots , g_{n})\\in G^ n$\nis a \\emph{basis} for $G $ if $g_{i}\\in G_{i}$ and\n$G_i\/G_{i-1}=\\left <g_{i} G_{i - 1} \\right >$ for\neach $i = 1,\\ldots, n $.\nIn the case when $G_{i} \/ G_{i+1} $ is infinite for all $i = 1, \\ldots, n$ we call the $n$-tuple a \\emph{Malcev basis} for $G$.\n\n\nThe set of torsion elements $T$ in a finitely generated nilpotent\ngroup $G$ is a finite normal subgroup, and the quotient $G\/T$ is a\ntorsion-free nilpotent group \\cite[p. 13, Corollary 10]{MR713786}. A\ncorollary of Proposition \\ref{prop:Treduction} is that $G\/T$ has the\nsame full residual finiteness growth as $G$.\n\n\\begin{corollary}\\label{cor:tfreduction}\n  If $G$ is a finitely generated nilpotent group and $T$ is the\n  subgroup of torsion elements then $\\Phi_G \\approx \\Phi_{G\/T}$.\n\\end{corollary}\n\nWe recall a folklore result, used in the proof of the following\nlemmas.\n\n\\begin{lemma} \\label{lem:commutatorproduct}\n  Suppose $G$ is a finitely generated nilpotent group of class $c$. The assignment $(x, y)\n  \\mapsto [x,y]$ defines a homomorphism\n\\[\n\\left( \\zeta_k(G)\/\\zeta_{k-1}(G) \\right) \\times \\left( \\zeta_\\ell(G) \/\n  \\zeta_{\\ell-1}(G) \\right) \\to \\zeta_{k+\\ell-c-1}(G) \/ \\zeta_{k+\\ell-c-2}(G).\n\\]\n\\end{lemma}\n\\begin{proof}\n  This follows immediately from \\cite[Theorem 2.1]{MR2366181}, noting\n  that the upper central series is a central filtration of $G$ when\n  indexed so that the $i^{th}$ term of the filtration is\n  $\\zeta_{c+1-i}(G)$.\n\\end{proof}\n\nIf $G$ is a group generated by a finite set $X$, for $g\\in G$ we use\n$\\| g \\|_X$ to denote the word length of $g$ with respect to $X$. \nLet $G$ be a finitely generated nilpotent group.\nThe following lemma is a consequence of well-known distortion estimates.\n\n\\begin{lemma} \\label{lem:distortion}\nLet $G$ be a nilpotent group of class $c$ generated by a finite set $X$.\nFix a positive integer $i$ and a generating set $X_i$ for $Z(G) \\cap \\gamma_i(G)$.\nThen there exists $C > 1$ such that for all $g \\in Z(G) \\cap \\gamma_i(G),$\n\\begin{equation} \\label{eq:distortion}\n\\|g \\|_{X} \\leq C \\| g \\|_{X_i}^{1\/i}. \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe first assume $G$ is torsion-free. \nFirst consider the case that $g = x^m$ for some $x\\in X_i$ and $m\\in\n\\mathbb{Z}\\setminus\\{0\\}$. \nAssume without loss of generality that $X_i$ is a basis for the free\nabelian group $Z(G) \\cap \\gamma_i(G)$, so that $\\| g \\|_{X_i} = \\left\\lvert\n  m \\right\\rvert$.\nEmbed $G$ as a cocompact lattice in a simply-connected nilpotent Lie group $N$, which\nidentifies $Z(G) \\cap \\gamma_i(G)$ with a lattice in a\nsimply-connected central subgroup $Z\\leq N$. Fix any left-invariant\nRiemannian metric on $N$, which gives a norm $\\| \\cdot \\|_\\mf{n}$ on\n$\\mf{n}$, the Lie algebra of $N$. Consider the path $\\gamma : [0,\n\\left\\lvert m \\right\\rvert ] \\to Z$ defined so that $\\gamma(\n\\left\\lvert m \\right\\rvert ) = g$ and\n$\\gamma(t) = \\exp( t z )$ for some $z\\in \\mf{n}$. Note that $\\exp(z) =\nx$ if $m>0$ and $\\exp(z) = x^{-1}$ if $m<0$. In particular, $z$ does \\emph{not} depend on $m$.\nBy \\cite[Prop 4.1(1)]{Pittet97}, the length of $\\gamma$ is \n$\\| z \\|_\\mf{n} \\| g\\|_{X_i}$. \nThen applying\n\\cite[Prop 4.1(2)]{Pittet97} to the curve $\\gamma$, there is a\nconstant $C>0$ depending on $z$ so that\n\\begin{equation} \\label{pitteteqn}\nd_N(e, g) \\leq C \\|g\\|_{X_i}^{1\/i}.\n\\end{equation}\nThe quantity $d_N(e,g)$ is uniformly comparable to $\\| g \\|_X$, so\nthis proves the desired inequality for $g$ of the form $x^m$.\n\nNow for any $g\\in Z(G)\\cap \\gamma_i(G)$, write $g = \\prod_{j=1}^k\nx_j^{m_j}$ where $X_i = \\{x_1,\\dotsc, x_k\\}$. Let $C$ be the largest\nconstant appearing in equation \\ref{pitteteqn} as $x$ ranges over\n$x_1,\\dotsc, x_k$. Then there is some $D > 0$ so that\n\\begin{align*}\n  \\| g \\|_X & \\leq \\sum_{j=1}^k \\| x_j^{m_j} \\|_X \\\\\n  & \\leq C \\sum_{j=1}^k \\| x_j^{m_j} \\|_{X_i}^{1\/i} \\\\\n  & \\leq C \\sum_{j=1}^k |m_j|^{1\/i} \\\\\n  & \\leq C k \\left( \\sum_{j=1}^k | m_j | \\right)^{1\/i} \\\\\n  & \\leq CkD \\| g \\|_{X_i}^{1\/i}.\n\\end{align*}\nThe last step follows because $Z(G) \\cap \\gamma_i(G)$ is abelian. \nThe penultimate step follows from the general fact that $(m_1^{1\/i} +\n\\dotsb + m_k^{1\/i})^i \\leq k^i (m_1 + \\dotsb + m_k)$ when $m_j \\geq 1$\nfor all $j$. This completes the proof in the case that $G$ is torsion-free. \n \n\nNow suppose $G$ is an arbitrary finitely generated nilpotent group.\nThere is a torsion-free normal subgroup $H$ of finite index in $G$.  Fix a\ngenerating set $Y$ for $H$.  The map $i : H \\to G$ is a quasi-isometry\nbecause $H$ is finite index in $G$. In fact, because distinct points\nin each of $G$ and $H$ are distance at least 1 and $i$ is injective,\nit is easy to check that $i$ is bi-Lipschitz.  This means that there is\nsome $C\\geq 1$ so that:\n\\begin{enumerate}\n\t\\item For $g,h \\in H$, \n\t$$\n\t\t\\frac{1}{C} \\| g h^{-1} \\|_Y \\leq \\| g h^{-1} \\|_X \\leq C \\| g h^{-1} \\|_Y.\n\t$$\n\t\\item For every element $g \\in G$, there exists $h \\in H$ such that\n\t$$\n\t\t\\| h g^{-1} \\|_X \\leq C.\n\t\t$$\n\\end{enumerate}\n\nFix generating sets $X_i$ for $Z(G) \\cap \\gamma_i(G)$\nand $Y_i$ for $Z(H) \\cap \\gamma_i(H)$. \nWe claim that $Z(H) \\leq Z(G)$.\nIndeed, if not then there exists $h \\in Z(H)$, an integer $r \\geq 1$,\nand elements $x_1, \\ldots, x_r \\in G$ such that $h \\in \\zeta_{r+1}(G)\n\\setminus \\zeta_r(G)$ and\n$$\n[h, x_1, \\ldots, x_r] \\in Z(G) \\setminus \\{1\\}.\n$$\nSince $H$ has finite index in $G$ there exists $n \\in \\mathbb{N}$ such that $x_1^n \\in H$.\nBy Lemma \\ref{lem:commutatorproduct} we have \n\\[\n[h, x_1^n, \\ldots, x_r] = [h, x_1, \\ldots, x_r]^n.\n\\]\nSince $H$ is normal we have $[h, x_1, \\ldots, x_r] \\in H$.\nThis implies $[h, x_1^n, \\ldots, x_r] \\neq 1$ because $H$ is torsion-free.\nTherefore $[h,x_1^n]$ cannot be trivial, which contradicts the fact that $h \\in Z(H)$.\nBy the aforementioned claim, $Z(G) \\cap \\gamma_i(G)$ contains $Z(H) \\cap \\gamma_i(H)$ as a subgroup.\nIn fact, it is not hard to show that $Z(H) \\leq Z(G)$ and $\\gamma_i(H)\n\\leq \\gamma_i(G)$ are, in both cases, subgroups of finite index.\nHence, the inclusion\n$i_2 : Z(H) \\cap \\gamma_i(H) \\to Z(G) \\cap \\gamma_i(G)$ is a\nbi-Lipschitz quasi-isometry with constant $D\\geq 1$.\n\nNow select $C' > 1$ such that inequality \\ref{eq:distortion} holds for all\n$g \\in G$ that are finite order (again, there are only finitely many of them).\nNext, let $g$ be an infinite order element in $G$ with \n$$g \\in Z(G) \\cap \\gamma_i(G).$$\nWe can suppose, without loss of generality, that $X$ contains $X_i$.\nThen since $i_2$ is a $D$-quasi-isometry, there exists $h \\in Z(H) \\cap \\gamma_i(H)$ such that \n$$ \\| h g^{-1} \\|_{X} \\leq  \\| h g^{-1} \\|_{X_i} \\leq D.$$\nSince $H$ is torsion-free, by enlarging $C$ if necessary we have\n$$\n\\| h \\|_Y \\leq C \\| h \\|_{Y_i}^{1\/i}.\n$$\nThus,\n\\begin{equation} \\label{QI1}\n\\| g \\|_X = \\| g h^{-1} h \\|_X \\leq \\| g h^{-1} \\|_X + \\| h \\|_X\n\\leq C + \\| h \\|_X.\n\\end{equation}\nAnd, further,\n\\begin{equation} \\label{QI2}\n\\| h \\|_X \\leq C \\| h \\|_Y \\leq C^2 \\| h \\|_{Y_i}^{1\/i}.\n\\end{equation}\nTo finish,\n\\begin{equation} \\label{QI3}\n\\| h \\|_{Y_i} \\leq D \\| h \\|_{X_i} =  D \\| g g^{-1} h \\|_{X_i} \\leq D(\\| g \\|_{X_i} + D ).\n\\end{equation}\nThe desired inequality follows from equations \\ref{QI1}--\\ref{QI3}, as\nall additive constants can be absorbed into the multiplicative\nconstants.\n\\end{proof}\n\nNext,  we show a technical lemma that will be important in our main proofs:\n\n\\begin{lemma} \\label{lem:technical}\nLet $G$ be a nilpotent group of class $c$ generated by a finite set\n$X$. Fix a number $0<i\\leq c$, and fix generating sets $Y_0$ for $\\zeta_i(G)$ and $Y$ for $Z(G)$.\nThere exists $C_i > 0$ such that for any $g \\in \\zeta_i(G) \\setminus \\zeta_{i-1}(G)$, there exists $x_1, \\ldots, x_{i-1} \\in X$ such that for any $\\gamma \\in \\zeta_{i-1}(G)$,\n$$\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_Y \\leq C_i \\|g \\|_{Y_0}.\n$$\nIn fact, there is some $F_i > 0$ so that\n$$\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X} \\leq F_i \\|g \\|_{Y_0}^{1\/t}, \n$$\nwhere $t$ is the minimal $k$ satisfying $[g \\gamma, x_1, \\ldots, x_{i-1}] \\notin \\gamma_{k+1}(G)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $g \\in \\zeta_i(G) \\setminus \\zeta_{i-1}(G)$ be given.\nSince $G$ is nilpotent, there exists $x_1, \\ldots, x_{i-1} \\in X$ so that\n$$\n[g, x_1, \\ldots, x_{i-1}] \\in Z(G) \\setminus \\{1 \\}.\n$$\nNote that for any $x \\in \\zeta_i(G)$ we have that \n$$\n[x, x_1, \\ldots, x_{i-1}] \\in Z(G).\n$$\nWrite $g = \\prod_{i=1}^n g_i$ where $g_i \\in Y_0$ and $n$ is the  word\nlength of $g$ with respect to $Y_0$.\nApplying Lemma~\\ref{lem:commutatorproduct} repeatedly gives\n\\begin{eqnarray*}\n[g, x_1, \\ldots, x_{i-1}] &=& [g_1 g_2 \\cdots g_n, x_1, \\ldots, x_{i-1}] \\\\\n&=& [g_1, x_1, \\ldots, x_{i-1}][g_2, x_1, \\cdots x_{i-1}] \\cdots [g_n, x_1, \\ldots, x_{i-1}].\n\\end{eqnarray*}\nSet $Y'$ to be $Y$ union the set of all elements of the form $[\\beta, \\alpha_1, \\alpha_2, \\ldots, \\alpha_{i-1}]$ where $\\beta \\in Y_0$ and $\\alpha_i \\in X$.\nNotice that $Y'$ does not depend on $g$. Further, by our above computation, we have\n$$\n\\| [g, x_1, \\ldots, x_{i-1}] \\|_{Y'} \\leq n.\n$$\nBecause $Y'$ is finite, $(Z(G), d_Y)$ is bi-Lipschitz equivalent to\n$(Z(G), d_{Y'})$.\nThis gives $C_i > 0$, depending only on $Y'$, such that\n\\begin{equation} \\label{assertionone}\n0 < \\| [g, x_1, \\ldots, x_{i-1} ] \\|_Y \\leq C_i \\|[g, x_1, \\ldots, x_{i-1}] \\|_{Y'}\n\\leq  C_i n = C_i \\| g \\|_{Y_0}.\n\\end{equation}\nLet $\\gamma \\in \\zeta_{i-1}(G)$ be arbitrary.\nThen as $[g, x_1, \\ldots, x_{i-1}]$ and $[\\gamma, x_1, \\ldots, x_{i-1}]$ are central,\n$$\n[g, x_1, \\ldots, x_{i-1} ] = [g \\gamma, x_1, \\ldots, x_{i-1} ],\n$$\nso the proof of the first assertion is complete. \n\nFix generating sets $X_j$ for $\\gamma_j(G) \\cap Z(G)$ for each $1\\leq\nj \\leq c$. These sets can be chosen independently of $g$ and $i$.\nBy Lemma \\ref{lem:distortion}, for each $j$ we have that there exists $D_j > 1$ such that for all $w \\in \\gamma_j(G) \\cap Z(G)$\n$$\n\\|w \\|_X \\leq D_j \\| w \\|^{1\/j}_{X_j}.\n$$\nSet $D$ to be the maximal such $D_j$. Notice that $D$ only depends on $X$ and $G$.\nSince $\\gamma_t(G) \\cap Z(G)$ is a subset of the abelian group, $Z(G)$, we have that there exists $E > 1$, depending only on $Y$ and the selection of $X_j$, such that\n$$\n\\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X_t} \\leq E \\| [g \\gamma, x_1, \\ldots, x_{i-1}]  \\|_{Y} \\leq E^2  \\| [g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X_t}.\n$$\nCombining these inequalities with Inequality \\ref{assertionone} gives\n\\begin{eqnarray*}\n0 < \\|[g \\gamma, x_1, \\ldots, x_{i-1}] \\|_{X}  &\\leq& D \\| [g \\gamma, x_1, \\ldots, x_{i-1} ] \\|_{X_t}^{1\/t} \\\\\n&\\leq&  E D \\| [g \\gamma, x_1, \\ldots, x_{i-1} ] \\|_{Y}^{1\/t} \\\\\n&\\leq&  E D C_i \\| g \\|_{Y_0}^{1\/t} = F_i \\| g \\|_{Y_0}^{1\/t},\n\\end{eqnarray*}\nfor some constant $F_i$ that depends only on $i$ and our choice of generating sets, as desired.\n\\end{proof}\n\nFor any $g \\in G$ of infinite order, the \\emph{weight $\\nu_G(g)$ of $g$ in the group $G$} is the maximal $k$ such that $\\left< g \\right> \\cap \\gamma_k(G) \\neq \\{ 1 \\}$.\nIf $G$ is a group and $m$ a natural number, let $G^m$ denote the\nnormal subgroup of $G$ generated by all $m^{th}$ powers of elements of\n$G$. When $G$ is nilpotent we have $[G : G^m] < \\infty$ for any $m$\n(see, for instance, \\cite[p.\\ 20, Lemma 4.2]{MR0283083}).\nWe need the following technical result for Lemma \\ref{lem:torsionlengths}.\n\n\\begin{lemma}\n\t\\label{lem:distortion1}\n\tLet $G$ be a nilpotent group generated by a finite set $X$.\nFix a positive integer $i$.\nThen there exists a constant $C > 1$ such that for all $m \\in \\mathbb{N}$ and all $g \\in (Z(G) \\cap \\gamma_i(G))^m$ with $\\nu_G(g) = i$, we have \n$$\nm \\leq C \\| g \\|^i_X.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSelect $Y = \\{ x_1, \\ldots, x_r \\}$ so that the image of $Y$ under\n$$\\pi : (Z(G) \\cap \\gamma_i(G)) \\to (Z(G) \\cap \\gamma_i(G)) \/ \\gamma_{i+1}(G)$$ \ngenerates a free abelian group of rank $r$, where $r$ is the rank of \n$(Z(G) \\cap \\gamma_i(G)) \/ \\gamma_{i+1}(G)$.\nSelect $N$ to be the order of $\\pi(Z(G) \\cap \\gamma_i(G))\/\\pi(\\left<Y\\right>)$.\n\nLet $f$ be the projection $G \\to G\/\\gamma_{i+1}(G)$.\nWe apply \\cite[Theorem 2.2]{MR1872804} to the torsion-free subgroup $\\Pi = f(\\left< x_1, \\ldots,\n  x_r \\right>) \\leq G\/\\gamma_{i+1}(G)$ to get $D > 1$ such that\n$$\n\\sup\\limits_{h \\in \\Pi \\cap B_{f(X)}(n)} \\| h \\|_{f(Y)} \\leq D n^{i}.\n$$\nThus, if $\\| h \\|_{f(X)} = n$, then $\\| h \\|_{f(Y)} \\leq D n^i = D (\\| h \\|_{f(X)})^i$.\nThat is, \n\\begin{equation} \\label{eqn:compareh}\n\\| h \\|_{f(Y)} \\leq D \\| h \\|_{f(X)}^i.\n\\end{equation}\n\nNow suppose $g$ is an element of $(Z(G) \\cap \\gamma_i(G))^m$ with $\\nu_G(g) = i$.\nSince $\\nu_G(g) = i$, we have that $\\pi(g)$ is infinite order.\nThus, we can write $g^N = h \\gamma$ where $h \\in \\left< Y \\right>$ and $\\pi(\\gamma)$ is trivial.\nThus, $\\gamma \\in \\gamma_{i+1}(G)$.\nThe map $f|_{ \\left< Y \\right> }$ is an injection, thus\n$$\n\\| f(h) \\|_{f(Y)} = \\| h \\|_{Y}.\n$$\nFinally, by the fact that $g^N \\equiv h \\mod \\gamma_{i+1}(G)$ and Inequality (\\ref{eqn:compareh}), we have\n\\begin{eqnarray*}\nN \\| g \\|_X &\\geq& \\| g^N \\|_X \\geq \\| f(g^N) \\|_{f(X)} \\\\\n&=& \\| f(h) \\|_{f(X)} \\geq D^{1\/i} \\| f(h) \\|_{f(Y)}^{1\/i}.\n\\end{eqnarray*}\nNotice that for any abelian group, $A$, and $\\ell \\in \\mathbb{N}$ we have\n$$A^{\\ell} = \\left< \\{ x^{\\ell} : x \\in A \\} \\right>.$$\nUsing additive notation, this becomes\n$$\nA^{\\ell} := \\left< \\{ \\ell x : x \\in A \\} \\right>\n= \\ell \\{ x : x \\in A \\} = \\ell A.\n$$\nSo we have\n$$mN A = m ( N A) = m \\{ n x : x \\in A \\}.$$\nTo apply this, note that $A = (Z(G) \\cap \\gamma_i(G))$ is an abelian group, as it is contained in the center.\nFor any element $y \\in N A$, by the definition of $N$, we have\n$f(y)$ is an element of $\\Pi$.\nThus, $f(m y) = m f(y)$ is an element of $\\Pi^m$, and so it follows that\n$$\nf(A^{mN}) \\leq \\Pi^m.\n$$\nIn particular, $g^N \\in (Z(G) \\cap \\gamma_i(G))^{Nm}$, so we have\n$$\nf(g^N) \\in \\Pi^m.\n$$\nSince $m f(Y)$ is a free basis for $\\Pi^m$, $f(Y)$ is a free basis for $\\Pi$, and $f(h) = f(g^N)$, we conclude that\n$$\n\\| f(h) \\|_{f(Y)} = \\| f(g^N) \\|_{f(Y)} \\geq m,\n$$\nso we are done.\n\\end{proof}\n\nWith the previous lemmas in hand, we finish with a proof that gives some control on the word lengths of elements in $G^m$.\n\n\\begin{lemma} \\label{lem:torsionlengths}\nLet $\\tilde G$ be a finitely generated nilpotent group of nilpotence $c$.\nThere exists $f \\in \\mathbb{N}$ such that $G = \\tilde G^f$ is a torsion-free characteristic subgroup of $\\tilde G$ of finite index.\nLet $g \\in G$, $X$, and $t \\in \\mathbb{N}$ be as in Lemma \\ref{lem:technical}.\nThen there exists $C > 1$, $M \\in \\mathbb{N}$, depending only on $G$, such that if $g \\in G^{Mm}$, we have that\n$$\n\\| g \\|_X \\geq C m^{1\/t}.\n$$\n\\end{lemma}\n\n\\begin{proof} \nSet $\\tau(\\tilde G)$ to be the set of all elements of finite order in $\\tilde G$.\nBy \\cite[p.\\ 13, Chapter 1, Corollary 10]{MR713786}, this is a finite characteristic subgroup of $\\tilde G$.\nSince $G$ is residually finite and $\\tau(H)$ is finite, there exists a finite $Q$ that fully detects $\\tau(\\tilde G)$. \nSet $f$ to be the exponent of $Q$ and set $G$ to be the characteristic\nfinite-index subgroup $\\tilde G^{f}$ \\cite[p.\\ 20, Lemma 4.2]{MR0283083},\nThen the map $\\tilde G \\to Q$ factors through $\\tilde G\/G$, and thus\n$\\tau(\\tilde G)$ is fully detected by $\\tilde G\/G$.\nSince $\\tau(\\tilde G)$ contains all the torsion elements in $\\tilde G$, it follows that $G$ is torsion-free.\n\nWe will show by induction on $d$ that for all $n > c$,\n\\begin{equation} \\label{eqn:claim}\n(\\zeta_{d}(G))^{(d)! \\cdots 2! n} \\cap Z(G) \\leq Z(G)^{n}.\n\\end{equation}\nThe base case $\\zeta_1(G) = Z(G)$ is immediate.\nFor the inductive step, set $M = (d)! (d-1)! \\cdots 2!$ and let $H =\n\\zeta_{d}(G) \\leq G$.\nLet $h \\in H^{Mn} \\cap Z(G)$.\nSince $h$ is in $H^{Mn}$ we can write\n$$\nh = g_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn} \\in Z(G),\n$$\nwhere $g_1, \\ldots, g_k$ are elements in $H$.\n\nTo proceed, let $\\tau_n(x_1, x_2, \\ldots, x_k) = \\tau_n(\\overline{x})$ be the $n$th \\emph{Petresco word} \\cite[p. 40]{MR0283083}, which is defined by the recursive formula,\n$$\nx_1^n x_2^n \\cdots x_k^n = \\tau_1(\\overline{x})^n \\tau_2(\\overline{x})^{{n \\choose 2}} \\cdots \\tau_n(\\overline{x})^{n \\choose n-1}.\n$$\nBy the Hall-Petresco Theorem \\cite[p. 41, Theorem 6.3]{MR0283083}, we\nhave that $\\tau_n(H) \\subset \\gamma_{n}(H)$ for all $n \\in \\mathbb{N}$.\nThus, replacing $n$ with $Mn$ and using the Hall-Petresco Theorem, we get:\n$$\ng_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn} = \\tau_1(\\overline{g})^{Mn} \\tau_2(\\overline{g})^{{{Mn} \\choose 2}} \\cdots \\tau_d(\\overline{g})^{{Mn} \\choose d}.\n$$\nBy the Hall-Petresco Theorem, $\\tau_k(\\overline{g}) \\in\n\\zeta_{d-1}(G)$ for all $k>1$, and by definition $h = g_1^{Mn} g_2^{Mn} \\cdots g_k^{Mn}\n\\in \\zeta_{d-1}(G)$. Therefore, because $G\/\\zeta_{d-1}(G)$ is torsion-free, $\\tau_1(\\overline{g})$ is in $\\zeta_{d-1}(G).$\nWe conclude that, for each $1 \\leq k \\leq d$, there exists $z_k \\in \\zeta_{d-1}(G)$ such that\n$$\n\\tau_k(\\overline{g})^{Mn \\choose k} = (z_k)^{\\frac{M}{(d)!}n} \\in \\zeta_{d-1}(G)^{\\frac{M}{(d)!}}.\n$$\nFurther,\n$$\n\\frac{M}{d!} = (d-1)! \\cdots 2!.\n$$\nHence, $h \\in (\\zeta_{d-1}(G))^{(d-1)! (d-2)! \\cdots 2!}$, so by the inductive hypothesis, we must have\n$h \\in Z(G)^{n}$, which completes the proof of equation \\ref{eqn:claim}.\n\nLet $D$ be the product of all finite order elements in $G\/ \\gamma_{n}(G)$ for all $n = 1, \\ldots, c$.\nSelecting $d = c$ in equation \\ref{eqn:claim} we get, for $M = (c)! (c-1)! \\cdots 2! D$, and $n > c$,\n\\begin{equation} \\label{eqn:tough}\nG^{Mn} \\cap Z(G) \\leq Z(G)^{D n}.\n\\end{equation}\n\nNow suppose $g \\in G^{Mm}$.\nBy Lemma \\ref{lem:technical}, there exists $x_1, \\ldots, x_{i-1} \\in X$ such that\n$[g,x_1, \\ldots, x_{i-1}] \\in \\gamma_t(G) \\cap Z(G)$.\nThus, as $G^{Mm}$ is normal, we have, by equation \\ref{eqn:tough},\n$[g,x_1, \\ldots, x_{i-1}] \\in Z(G)^{D m}$.\nHence, by our choice of $D$ and the fact that $Z(G)$ is a free abelian group, we have $\\nu_G([g,x_1, \\ldots, x_{i-1}]) = t$.\nThus, applying Lemma \\ref{lem:distortion1} gives $C_1 > 0$, depending only on $G$, such that\n$$\n\\| [g, x_1, \\ldots, x_{i-1}] \\|_X > C_1 m^{1\/t}.\n$$\nA simple counting argument gives a $C_2 > 0$, depending only on $G$, such that\n$$\n\\| g \\|_X \\geq C_2 \\| [g, x_1, \\ldots, x_{i-1}] \\|_X.\n$$\nThus, we have $C > 0$, depending only on $G$, such that\n$$\n\\| g \\|_X > C m^{1\/t},\n$$\nas desired.\n\\end{proof}\n\n\\section{Some examples and basic results} \\label{sec:examples}\n\n\\subsection{Abelian groups} \\label{sec:abeliangroups}\n\nIn this section we discuss some facts concerning abelian groups and present a proof of Theorem \\ref{thm:abelian}.\nThis begins with the simplest torsion-free group.\nFix $\\{ 1 \\}$ as the generating set $\\mathbb{Z}$.\nThen $B_\\mathbb{Z}(n) = \\{ -n, -n+1, \\ldots , n-1, n \\}$.\nClearly, $B_\\mathbb{Z}(n)$ is fully detected by $\\mathbb{Z}\/(2n+1) \\mathbb{Z}$.\nFurther, any quotient fully detecting $B_\\mathbb{Z}(n)$ has cardinality greater than $2n$.\nSo we get $\\G_\\mathbb{Z}(n) \\approx n$.\nThis result generalizes immediately to all torsion-free finitely\ngenerated abelian groups, and more generally to all finitely generated abelian groups.\n\n\\begin{corollary} \\label{prop:abelian}\nLet $A$ be a finitely generated abelian group.\nThen $\\G_A(n) \\approx n^{\\dim(A)}$.\n\\end{corollary}\n\n\\begin{proof}\nBy Corollary \\ref{cor:tfreduction}, we may assume $A$ is torsion-free.\nThe computation in this case is straightforward.\n\\end{proof}\n\nOne salient consequence of Corollary \\ref{prop:abelian} is that an\nabelian group's full residual finiteness growth $\\G_A$ matches its\nword growth $w_A$. We now prove Theorem~\\ref{thm:abelian}, which shows\nthat this property characterizes abelian groups in the class of\nnilpotent groups.  It also demonstrates that although $\\G_G$ and $w_G$\nshare properties, they are seldom the same.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:abelian}]\nBy Corollary \\ref{cor:tfreduction}, we may assume that $G$ is torsion-free.\nLet's further assume $G$ is not abelian.\nFix a Malcev basis $x_1, \\ldots, x_k$ for $G$.\nFor every $n$, let $Q_n$ be a quotient fully detecting $B_G(n)$.\nLet $c$ be the nilpotent class of $G$.\nFix a tuple $(x_1, \\ldots, x_m)$ consisting of all the basis elements not in $\\zeta_{c-1}(G)$.\nWe claim that there exists $C >0$ such that for any $\\gamma \\in \\zeta_{c-1}(G)$, the image of\n$x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma$\nin $Q_{Cn}$ is nontrivial in $Q_{Cn}$ for any $|k_i| \\leq n^2$ with $\\sum_{i=1}^m |k_i| > 0$.\nIndeed, by the second assertion of Lemma \\ref{lem:technical} there exists $C > 0$ with\n$[x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m}, y_1, y_2, \\ldots, y_{c-1}]$\nbeing nontrivial and having word-length at most $Cn^{2\/{c}} \\leq Cn$ in $G$.\nThus, as nontrivial elements of $B_G(Cn)$ are nontrivial in $Q_{Cn}$,\nthe image of $x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma$ in $Q_{Cn}$\nis nontrivial, so the claim is shown.\n\nConsider the set \n$$\nB^+(n) := \\{ x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m} \\gamma : 1 \\leq k_i \\leq n^2, \\gamma \\in B_G(n) \\cap \\zeta_{c-1}(G) \\}.\n$$\nGiven any $x,y\\in B^+(n)$, the above claim implies that $y^{-1}x$ has\nnontrivial image in $Q_{Cn}$. It follows that $B^+(n)$ is fully detected by $Q_{Cn}$.\nOn the other hand, by comparing with the explicit calculations for\nword growth in \\cite{Bass72} and the appendix of \\cite{MR623534} we see that the set $B^+(n)$ has cardinality at least $n^m w_G(n)$.\nThus, we have\n$$\n\\G_G(n) \\succeq n^m w_G(n),\n$$\nas desired.\n\n\\end{proof}\n\n\\subsection{Some non-abelian groups} \\label{sec:heisenberg}\n\nWe begin this section with the simplest non-abelian example.\nRecall that the \\emph{discrete Heisenberg group} is given by\n$$H_3 = \\left< x, y, z : [x,y] = z , z \\text{ is central } \\right>.$$\n\n\\begin{proposition} \\label{prop:heisenberg}\nWe have $\\G_{H_3}(n) \\approx n^6$.\n\\end{proposition}\n\n\\begin{proof}\nLet $B_{H_3}(n)$ be the ball of radius $n$ in $H_3$ with respect to\nthe generating set $\\{x,y,z \\}$.\nAn exercise in the geometry of $H_3$ show that there is some $D>0$ so\nthat if $x^{\\alpha_1} y^{\\alpha_2} z^{\\alpha_3} \\in B_{H_3}(Dn)$ then\n$\\lvert \\alpha_i \\rvert \\leq\n  n$ for $i=1,2$ and $\\lvert \\alpha_3 \\rvert \\leq n^2$. \nTherefore there is some $C>0$ so that $B_{H_3}(n)$ injects into the quotient\n$H_3 \/ H_3^{Cn^2}$, and so $\\G_{H_3}(n) \\preceq n^6$.\n\nNow note that $B_{H_3}(5n)$ contains $z^i$ for $-n^2 \\leq i \\leq n^2$, as\n$$\n[x^n,y^{j}] z^{k} = z^{nj + k},\n$$\nhas word length at most $5n$ for each $1 \\leq j, k \\leq n$.\nLet $Q_n$ be a quotient detecting $B_{H_3}(5n)$.\nConsider $w = x^a y^b x^c$.\nThen $[w,y] = z^{a}$ and $[w,x] = z^{-b}$. \nIf $w$ is trivial in $Q_n$ then both $[w,y]$ and $[w,x]$ are also trivial.\nIt follows that $w$ has nontrivial image in $Q_n$ for any values $0 <\na,b,c \\leq n^2$.\nThus, $|Q_n| \\geq n^6$, as desired.\n\\end{proof}\n\n\nWe now prove Theorem \\ref{theorem:zl} from the introduction, which\ngeneralizes the conclusion of Proposition \\ref{prop:heisenberg} to a\nlarge class of nilpotent groups.\n\nFor a finite $k$-tuple of elements $X = (x_1, \\ldots, x_k)$ from a group, we will use $B_X^+(n)$ to denote the set\n$$\nB_X^+(n) = \\{ x_1^{\\alpha_1} \\cdots x_k^{\\alpha_k} : 0 \\leq \\alpha_i \\leq n \\} \\subseteq G.\n$$\nNote that this is \\emph{not} generally the same as the semigroup ball\nof radius $n$.\n\n\\begin{proof} [Proof of Theorem \\ref{theorem:zl}]\n  Lemma \\ref{lem:torsionlengths} demonstrates that $B_G(n)$ is fully detected by a quotient of\n  the form $G \/ G^{Mn^c}$ for some $M>0$. We therefore have \n\\[\\G_G(n) \\preceq \\left(\\prod_{i=1}^c n^{\\dim(\\zeta_i(G)\/\\zeta_{i-1}(G))}\\right)^c = n^{\\dim(G) c}.\\]\n\nTo show the reverse inequality, we will show that for any positive integer $n$, there exists set of cardinality approximately $n^{\\dim(G)c}$ that is fully detected by any finite quotient of $G$ that realizes $\\G_G(n)$.  To this end, for each $i$, equip $\\gamma_i(G)$ with a fixed generating set $X_i$.\n Let $Q$ be a quotient of $G$ that realizes $\\G_G(n)$.  By \\cite[Theorem 2.2]{MR1872804}, for any generating\n  set of $\\gamma_c(G)$ there is a constant $C> 0$ such that for every\n  $h, h'$ in $\\gamma_c(G)$, we have\n$$\nd_{\\gamma_c(G)} (h,h') \\leq C [d_G(h,h')]^{c}.\n$$\nThus, the set $B_{\\gamma_c(G)}(n^c \/C)$ must inject into $Q$\nas it is contained in $B_G(n)$.\n\nTo continue, fix a basis $B = (g_1, \\ldots, g_k)$ obtained from the upper central series. \nFor any $i$, let \n\\[\nB_i = \\left\\{ g_{j}\\in B \\mid g_j \\in \\zeta_i(G) \\setminus\n  \\zeta_{i+1}(G), g_j \\text{ nontorsion in } G\/\\zeta_i(G) \\right\\}.\n\\] \nSet $B^t$ to be the tuple consisting of elements from $B_i$ respecting the ordering of the basis.\nThat is, \n$$\nB^t = (g_{a_1}, \\ldots, g_{a_k}),\n$$\nwhere each entry is in some $B_i$ and $a_i < a_{i+1}$.\nWe claim that $B_{B^t}^+(D n^c)$ is fully detected by $Q$ for some $D > 0$.\nTo prove this claim, we will use the fact that if any element in the\nnormal closure of some $g\\in G$ has nontrivial image in $Q$, then $g$\nhas nontrivial image in $Q$.\nLet $x, y \\in B_{B^t}^+(n^c)$ be elements with $x \\neq y$.\nThere is some $i \\leq c$ so that $y^{-1}x \\in \\zeta_i(G) \\setminus \n\\zeta_{i-1}(G)$. \nSet \n\\[\nE = \\max \\{ |B_j| \\} \\cdot \\max \\{ \\| \\gamma \\|_{X_j} : \\gamma \\in\nB_j \\}.\n\\]\nThere is some $\\gamma \\in \\zeta_{i-1}(G)$ so that $\\| y^{-1}x \\gamma\n\\|_{X_i} \\leq E n^c$. \nThis statement follows by reducing the word $y^{-1} x$ to normal form with respect to the basis.\nLet $E_0$ be the largest constant $C_i$ output by Lemma\n\\ref{lem:technical} for $i = 1,\\dotsc, c$.\nBy Lemma \\ref{lem:technical},\n$$\n\\| [y^{-1}x , x_1, \\ldots, x_r ] \\|_{X_c} \\leq E_0 \\| x y^{-1} \\gamma \\|_{X_i} \\leq E_0 E n^c.\n$$\nIt follows then that the set $B_{B^t}^+(n^c\/ (C E_0 E) )$ is fully detected by $Q$.\nSet $D = 1\/(C E_0 E)$.\nBy the definition of a basis we have $|B_{B^t}^+( D n^c)| \\geq (D n)^{c \\dim(G)}$, so we get the desired inequality.\n\\end{proof}\n\n\n\n\n\\section{A general upper bound}\n\\label{MainProofSection}\n\nThe example $H_3 \\times \\mathbb{Z}$, which has full residual finiteness growth $n^7$, demonstrates that the conclusion of Theorem \\ref{theorem:zl} does  not hold for any finitely generated nilpotent group.\nIn this section we prove Theorem \\ref{MainTheorem}, providing a\ntechnique that provides for {\\em any} finitely generated nilpotent group an\nexplicit upper bound of $\\G_G(n)$ of the form $n^d$. We first\nillustrate the technique in an example in Proposition\n\\ref{D22example}, where we show moreover that the upper bound is sharp\nin this example.\n\nLet $U_n$ denote the group of upper triangular unipotent matrices in\n$\\SL_n(\\mathbb{Z})$. \nFor $i\\neq j$, let $e_{i,j}$ denote the elementary matrix differing\nfrom the identity matrix only in that its $ij$-entry is 1.\nWe define the \\emph{coordinates} of the tuple $(x_1, \\ldots, x_k)$ to be the set\n$\\{ x_1, \\ldots, x_k \\}$.\nRecall that a {\\em terraced} filtration of $G$ is a filtration $1 = H_0 \\leq\nH_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ where each $H_i$ is a maximal normal\nsubgroup of $G$ satisfying $H_i\\cap \\gamma_{i+1}(G) = 1$.\n\n\n\\begin{proposition} \\label{D22example}\n  Consider elementary matrices $x = e_{1,4}$ and $y = e_{1,5}$ in\n  $U_5$. Define a normal subgroup $N = \\left< x, y \\right> \\leq\n  U_5$ and set $\\Gamma = U_5 \/ N$. Then $\\G_{\\Gamma}(n) \\approx n^{22}.$\n\\end{proposition}\n\n\\begin{proof}\n  Set $H_3 = \\Gamma$ and $H_2 = \\left< e_{1,2}, e_{1,3} \\right>$, and\n  let $H_0 = H_1 = 1$. \n  Note that $1=H_0\\leq H_1 \\leq H_2 \\leq \\Gamma$\n  forms a terraced filtration of $\\Gamma$.\n  Define two tuples of elements of $\\Gamma$ by $X_3 = (e_{1,3} ,\n  e_{1,2})$ and $X_2 = (e_{2,5}, e_{2,4}, e_{3,5}, e_{2,3}, e_{3,4},\n  e_{4,5})$. For each $i=2,3$, let $Y_i$ be the set of coordinates of\n  $X_i$. Clearly $Y = Y_2 \\cup Y_3$ generates $\\Gamma$. \n\n  To establish the upper bound, let $Q$ be a quotient of $\\Gamma$ detecting $B_\\Gamma(n)$.\n  Each\n  of $H_3^{n^3}$ and $H_2^{n^2}$ is normal in $\\Gamma$, so we can define a normal\n  subgroup $N = H_3^{n^3} H_2^{n^2} \\leq \\Gamma$. \n  A simple induction shows that if $g\\in B_\\Gamma(n)$ then $\\lvert\n  g_{ij} \\rvert \\leq n^{j-i}$. In particular this implies that there\n  is some $C>0$ so that $B_\\Gamma(Cn)$ is fully detected by\n  $G\/N$. Since $\\lvert G\/N \\rvert \\approx n^{22}$, this establishes\n  the desired upper bound on $\\G_\\Gamma(n)$.\n  \n  To establish the lower bound, define the \\emph{depth} of an element\n  $\\gamma \\in \\Gamma$ to be the maximal $i$ with\n  $\\gamma \\notin \\zeta_i(\\Gamma)$.  Order the elements $Y$ in a tuple\n  $(y_1, y_2, \\ldots, y_8)$ of non-increasing depth.  Set $B^+(n)$ to\n  be\n  $$\n  \\left\\{ \\prod_{i=1}^8 y_i^{\\alpha_i}  : 0 \\leq \\alpha_i \\leq n^2 \\text{ if $y_i \\in Y_2$ and } 0 \\leq \\alpha_i \\leq n^3 \\text{ otherwise} \\right\\}.\n  $$\n  We claim that there exists $C >0$ such that any quotient $Q$ in which $B_\\Gamma(Cn)$ embeds restricts to $B^+(n)$ as an injection.\n  This gives the desired lower bound, as $|B^+(n)| \\geq n^{22}$.\n  To see this claim, let $x,y$ be distinct elements in $B^+(n)$.\n  Set $i$ to be the depth of $y^{-1}x$.\n  We break up the rest of the proof of this claim into cases depending on $i$.\n  \n  If $i = 0$, then\n  $y^{-1} x$ is in the center of $\\Gamma$ and we have\n  $$\n  y^{-1} x = e_{1,2}^{a_1} e_{2,5}^{a_2},\n  $$\n  where $|a_1| \\leq n^2$ and $|a_2| \\leq n^3$.\n  Note that $e_{1,2}\\in \\gamma_2(\\Gamma)$ and $e_{2,5}\\in \\gamma_3(\\Gamma)$.\n  Applying Lemma \\ref{lem:distortion} twice, we have that\n  $$\n  \\| y^{-1} x\\|_\\Gamma \\leq \\| e_{1,2}^{a_1} \\|_\\Gamma + \\|\n  e_{2,5}^{a_2} \\|_\\Gamma \\leq C n,\n  $$\n  for some $C > 0$, independent of $n$.\n  Thus $y^{-1} x$ cannot vanish in any quotient that fully detects $B^+(Cn)$.\n  \n  If $i = 1$, then by definition, we may write\n  $$\n  y^{-1} x = e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma,\n  $$\n  where $\\gamma \\in \\zeta_i(\\Gamma)$, $|a_1| \\leq n^2$, $|a_2| \\leq n^3$, and $|a_3| \\leq n^3$.\n  Since this $y^{-1}x$ is not in the center, there exists $z \\in Y$ such that \n  $$\n  [e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z] \\neq 1.\n  $$\n  This element is now in the center. Thus, by Lemma \\ref{lem:commutatorproduct}, we have\n  $$\n  [e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z]\n  = \n  [e_{1,2}, z]^{a_1}\n  [e_{2,4}, z]^{a_2}\n  [e_{3,5}, z]^{a_3}.\n  $$\n  Now by Lemma \\ref{lem:distortion} applied three times, we see that the word length of\n  $[e_{1,2}^{a_1} e_{2,4}^{a_2} e_{3,5}^{a_3} \\gamma, z]$\n  is less than a constant multiple of $n$, where the constant does not depend on $n$.\n  Thus $y^{-1} x$ cannot vanish in any quotient that fully detects $B^+(Cn)$ for some  $C > 0$ independent of $n$.\n  \n  If $i = 2$, then by definition, we may write\n  $$\n  y^{-1} x = e_{2,3}^{a_1} e_{3,4}^{a_2} \\gamma,\n  $$\n  where $\\gamma \\in \\zeta_i(\\Gamma)$, $|a_1|, |a_2| \\leq n^3$.\n  Suppose, without loss of generality, that $a_1 \\neq 0$.\n  Then, using Lemma \\ref{lem:commutatorproduct}, we have that there exists $\\gamma' \\in \\gamma_1(\\Gamma)$ such that\n  $$\n  [y^{-1} x, e_{3,4}] = [e_{2,3}, e_{3,4}]^{a_1} [e_{3,4}, e_{3,4}]^{a_2} \\gamma' = e_{2,4}^{a_1} \\gamma'.\n  $$\n  Now it is clear that there exists $z \\in \\Gamma$ such that\n  $$\n  [[y^{-1} x, e_{3,r}], z] = [e_{2,4}^{a_1} \\gamma', z] \\neq 1.\n  $$\n  We can proceed as in case $i=1$ to achieve the desired conclusion.\n  Indeed, Lemma \\ref{lem:distortion} applies, giving that $y^{-1} x$ is detected if $B(Cn)$ is fully detected for some constant $C > 0$ independent of $n$.\n  That is, we cannot have $y^{-1}x =1$ in $Q$, if $Q$ detects $B_\\Gamma(Cn)$.\n  The claim then follows, ending the proof.\n  \n\\end{proof}\n\n\nWe now prove Theorem \\ref{MainTheorem}.\n\n\\begin{proof}[Proof of Theorem \\ref{MainTheorem}]\nLet $G$ be a finitely generated nilpotent group and suppose $1=H_0\n\\leq H_1 \\leq \\dotsb \\leq H_{c-1}\\leq G$ is a terraced filtration. Set\n$H_c = G$. \n\nChoose a basis $X_1$ of $H_1$. Inductively construct\ntuples $X_2,\\dotsc, X_c$ by setting $X_i$ to be a pull-back of a basis\nfor $H_i \/ H_{i-1}$.\nSet $Y_i$ to be the set of all coordinates of $X_i$ and $Y = \\cup_i Y_i$.\nIt is clear from the construction that $Y$ is generating set for\n$G$. Note also that for any $n \\in \\mathbb{N}$, the subgroup \n\\[\nN(n) = \\prod_{i=1}^c \\left<\n  y^{n^k} : y \\in \\left<  Y_1 \\cup Y_2 \\cup \\dotsb \\cup Y_k\n  \\right> \\right>\n\\]\nis normal in $G$.\n\nWe now claim that there exists a constant $D \\in \\mathbb{N}$ so that for any\n$n \\in \\mathbb{N}$, the ball $B_Y(n)$ is detected by $G\/N(Dn)$.\nTo prove the claim, let $f, M \\in \\mathbb{N}$ be as in Lemma~\\ref{lem:torsionlengths}.\nThen $G^{fM}$ is torsion-free; let $K = G^{f M}$.\nFix a finite generating set $T$ for $K$.\nFor each $i$ and any $n\\in \\mathbb{N}$, Lemma \\ref{lem:torsionlengths} gives\nthat any element $g\\in K^n \\cap H_i$ has word length at least $C_i\nn^{1\/t_i}$ with respect to $T$.\nThus we have that  there exists $D_0 > 0$ such that\n$B_T(D_0 n)$ is fully detected by $K\/N(fMn)$.\nFurther, since $K$ is of finite index in $G$, we have $D_1 > 1$ such that for any $g \\in K$,\n$$\n\\| g \\|_T \\leq D_1 \\| g \\|_Y \\leq D_1^2  \\| g \\|_T.\n$$\nTherefore, as $N(fMn)$ is contained in $K$, any singleton contained in $B_Y(n\/D_1)$ is fully detected by $G\/N(fMn)$ and so $B_Y(n\/(2 D_1))$ is fully detected by $G\/N(fMn)$.\nThis proves the claim, as we can select $D = 2D_1 fM$.\n\nWe will now demonstrate that the order of\n$G\/N(Dn)$ is dictated by a single polynomial of the form $n^b$ for \n\\[\nb = \\sum_{k=1}^c k\\cdot \\dim(H_k \/ H_{k-1}).\n\\]\nSet $G_k = H_k\/ H_{k-1}$. It is apparent from the definition of $N(Dn)$ the index of $N(Dn)$ in $G$ is bounded above by\n$$\n\\prod_{k=1}^{c} | G_k \/ G_k^{D^k n^{k}} |.\n$$\nBy the construction of $D$, the subgroup $G_k^D$ is torsion-free in $G_k$.\nThus, it is clear that $|G_k^D \/ G_k^{D^{k} n^{k}}|$ has order\n$D^{\\dim(G_k) (k-1)}  n^{k \\dim(G_k)}$.\nThis gives an upper bound for the index of $N(Dn)$ in $G$ of the form\n$C_0 n^{\\sum_{k=1}^c k \\dim(G_k)}$, where $C_0 >0$ does not depend on $n$.\n\nOne can check that $b = c\\dim(G) - \\sum_{i=1}^{c-1}\\dim(H_i)$ using\nthe general fact that $\\dim(G\/H) = \\dim(G)- \\dim(H)$ for any finitely\ngenerated nilpotent group $G$ with normal subgroup $H$. This \ncompletes the proof since $G\/N(Dn)$ detects $B_Y(Cn)$.\n\\end{proof}\n\nWe conclude with an example that shows that the upper bound to $\\G_G$\ngiven by Theorem \\ref{MainTheorem} generally may depend on choice of\nterraced filtration. Consider the group\n$\\tilde G = U_3 \\times U_4 \\times U_5$, which is nilpotent of class\n$c=4$.  There is an isomorphism\n$Z(\\tilde G) \\cong Z(U_3) \\times Z(U_4) \\times Z(U_5)$. Under\nidentifications $Z(U_3)\\cong Z(U_4) \\cong Z(U_5) \\cong \\mathbb{Z}$, define an\ninfinite cyclic subgroup\n\\[\nZ = \\{ (x,y,z)\\in Z(U_3)\\times Z(U_4)\\times Z(U_5) \\mid x=y=z \\} \\leq\nZ(\\tilde G).\n\\]\nLet $G = \\tilde G \/ Z$ and let $\\pi: \\tilde G \\to G$ be the quotient\nmap. Then $\\pi$ restricts to an isomorphism $Z(U_3)\\times Z(U_4) \\cong\nZ(G)$. Under this identification, the last term of the lower central\nseries of $G$ is\n\\[\n\\gamma_4(G) = \\{ (x,y)\\in Z(U_3)\\times Z(U_4) \\mid x=y \\}.\n\\]\nSince $\\gamma_3(G)$ contains the image of $Z(U_4)$, we see that\n$Z(G) \\leq \\gamma_3(G)$. Since $H\\cap\nZ(G)$ is nontrivial for any nontrivial normal subgroup $H\\leq G$, it follows that\n$H_2$ is trivial for any terraced filtration of $G$.\n\nNow define $H_0 = H_1 = H_2 = 1$ and $H_3 = \\pi(U_3)$, and\n$H_0' = H_1' = H_2' = 1$ and $H_3' = \\pi(U_4)$. It is easy to see that\nboth $\\pi(U_3)$ and $\\pi(U_4)$ are maximal normal subgroups of $G$\nwhose intersection with $\\gamma_4(G)$ is trivial. It follows from the\nabove comments that\n\\[\nH_0\\leq H_1 \\leq H_2 \\leq H_3 \\leq G \\quad \\text{ and } \\quad H_0'\n\\leq H_1' \\leq H_2' \\leq H_3' \\leq G\n\\]\nare terraced filtrations of $G$. However these filtrations give\ndifferent upper bounds for $\\G_G$ because $\\dim( \\pi(U_3) ) = 3$ while\n$\\dim( \\pi(U_4) ) = 6$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe  gauge-gravity correspondence\n \\cite{Maldacena:1997re, Gubser:1998bc, Witten:1998qj}\nhas got a nonrelativistic version  where\n strongly coupled quantum theories at critical points\n can be studied  \\cite{Son:2008ye, Balasubramanian:2008dm, Herzog:2008wg, Hartnoll:2008vx, Hartnoll:2008kx, Maldacena:2008wh, Denef:2009tp, Kachru:2008yh, Singh:2009tq, Singh:2010rt, Balasubramanian:2010uk, Singh:2010zs, Singh:2012un, Narayan:2012hk, Singh:2013iba, Narayan:2012ks, Singh:2017wei, Singh:2018ibp, Mishra:2018tzj, Taylor:2015glc}.\nSome of these quantum systems  involve  strongly coupled fermions \n at finite density or it may simply be a gas of ultra-cold atoms \n\\cite{Son:2008ye, Balasubramanian:2008dm}. \nIn the studies involving \n`nonrelativistic' Schr\\\"odinger spacetimes the 4-dimensional\n  spacetime geometry generally requires supporting\n Higgs like field such as  massive vector field \n \\cite{Herzog:2008wg, Denef:2009tp, Son:2008ye} or a tensor field. \nThe  spacetimes possessing a Lifshitz symmetry\n provide  similar holographic dual description of  nonrelativistic \nquantum theories living on their  boundaries \\cite{Kachru:2008yh}, see \\cite{Taylor:2015glc} for a review.  \n\nIn this work we shall mainly study entanglement entropy of the \nexcitations in asymptotically \n$Lif_4^{(a=2)}\\times S^1\\times S^5$ background. The latter is \na Lifshitz vacua in massive type IIA (mIIA) theory \\cite{Singh:2017wei, Singh:2018ibp} with dynamical exponent of time being $a=2$. The massive type IIA theory \\cite{ROMANS1986374} is a ten-dimensional maximal supergravity where the antisymmetric \ntensor field is explicitly massive. The theory also includes a positive cosmological constant related to mass parameter. Due to this structure the mIIA theory provides a unique setup to study Lifshitz   solutions. Particularly the  $Lif_4^{(2)}\\times S^1\\times S^5$\nsolution is a background generated by the bound state of $(F1,D2,D8)$ branes \\cite{Singh:2017wei}\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{sol2a9n}\n&&ds^2= L^2\\left(- {dt^2\\over  z^4} +{dx_1^2+dx_2^2\\over z^2}+{dz^2\\over\nz^2}  +{dy^2\\over  q^2} + d\\Omega_5^2 \\right) ,\\nonumber\\\\\n&&e^\\phi=g_0 ,\n ~~~~~C_{(3)}= -{1 \\over g_0}\n{L^3\\over  z^4}dt\\wedge dx_1\\wedge dx_2, \\nonumber\\\\ &&\nB_{(2)}=  { L^2\\over q  z^2}dt\\wedge dy  \n\\eea \nThe metric and the form fields have explicit invariance under constant scalings (dilatation); \n$z\\to \\lambda z,~t\\to \\lambda^2 t,~\nx_i\\to \\lambda x_i, ~y\\to y$. The dynamical exponent of time is $2$ here.\nThe background describes a strongly coupled nonrelativistic \nquantum theory at the UV critical point. \n\\footnote{Analogous T-dual  solution do also exist in type IIB  theory with \nconstant axion flux  switched \non \\cite{Balasubramanian:2010uk}}  \n\nIt is worthwhile to study excitations of the\n$Lif_4^{(2)}\\times S^1\\times S^5$\n vacua as it immediately provides us a prototype  \n$Lif_4^{(2)}$ background in four dimensions which is holographic dual to 3-dimensional Lifshitz theory on its boundary. The  excitations would tell us how this Lifshitz theory behaves\nnear its critical point. Particularly we shall study a class of string like excitations which themselves form solutions of massive IIA sugra \nand explicitly involve $B$-field \\cite{Singh:2018ibp}. \nThese also induce running of  dilaton as well. It is observed that the resulting RG flow in the deep IR can be described simply by ordinary type IIA  theory. The reason for this is due to the fact that the contributions of massive stringy modes decouple from the low energy  dynamics of the theory in the IR, far away from UV critical point \\cite{Singh:2018ibp}. \n\nIn this report we aim to study holographic entanglement entropy (HEE) \\cite{Ryu:2006bv, *Ryu:2006ef, *Hubeny:2007xt} of the excited Lifshitz subsystems which are either a disc or a strip in a perturbative framework. A critical observation is that for small sized systems the entanglement entropy density remains constant at first order. That is, \nthe first order contributions to the entropy density remain independent of the\nsize ($\\ell$) of the subsystem. This is a peculiarity and quite unlike  relativistic CFTs where usually the entropy density (of excitations) is linearly proportional to the typical size of the subsystem \\cite{Bhattacharya:2012mi}. \nWe discover that the resolution lies in the nature\nof the chemical potential ($\\mu_E$) for the Lifshitz  system. We gather  evidence that \nsuggests that  energy density (of excitations) falls off with the size of system as $\\propto 1\/\\ell^2$. Furthermore the $1\/\\ell^{2}$ dependence is exactly same as the entanglement temperature behaviour in the Lifshitz theory. \nNotwithstanding these peculiarities,  \n the entropy of excitations consistently \nfollows the first law of entanglement\nthermodynamics \\cite{Bhattacharya:2012mi,Allahbakhshi:2013rda} up to first order.\n\\par In addition, we also carry out a calculation of entanglement entropy at second order for both disc and strip subsystems. Contributions arising at this order bestow an explicit $\\ell$ dependence upon the entropy. We argue how the first law can still be obeyed by modifying our chemical potential $(\\mu_E)$ and entanglement temperature $(T_E)$. A similar argument was put forward in \\cite{Mishra:2015cpa} for asymptotically AdS spacetime.\n\nThe unusual symmetry of Lifshitz spacetime makes it a good background to study novel features of entanglement in a non-relativistic quantum theory at zero temperature \\cite{Son:2008ye, Balasubramanian:2008dm, Kachru:2008yh}. It is well known that for such systems, e.g. a particle in a one-dimensional box the momentum of the particle scales with the length as $ p \\propto \\frac{1}{\\ell}$ and the energy $\\mathcal{E} \\propto \\frac{1}{\\ell^2}$; our calculations of entanglement entropy also support this explicit size dependence of energy, as shown in equation \\eqref{enr34}. We hope our work will help shed some light on holographic treatment of non-relativistic quantum systems at strong coupling that are often interesting in e.g. condensed matter theory.\n\n\\par The rest of the paper is organized as follows: in section \\ref{sec2} we review salient features of $Lif^{(2)}_4\\times S^1\\times S^5$ vacua with IR excitations in mIIa theory. The holographic entanglement entropy for a disc subsystem is calculated in section \\ref{sec3}. In section \\ref{sec4} we carry out similar analysis for strip subsystem at first and second orders, section \\ref{sec5} contains the conclusion.\n\n   \n\\section{$Lif^{(2)}_4\\times S^1\\times S^5$ vacua and excitations}\\label{sec2}\n \n The massive type IIA supergravity theory is the only known maximal \nsupergravity in ten dimensions which allows  massive string $B_{\\mu\\nu}$  \nfield and a mass dependent cosmological constant \\cite{ROMANS1986374}.\nThe cosmological constant \n generates a  nontrivial  potential term  for the dilaton \nfield. The mIIA theory  does not admit  flat  Minkowski \n solutions. \nNonetheless the theory gives rise to  well known  \nFreund-Rubin type vacua $AdS_4\\times S^6$   \\cite{ROMANS1986374},  \nthe  supersymmetric domain-walls \nor D8-branes \\cite{Polchinski:1995mt, Bergshoeff:1996ui,Witten:2000mf, Hull:1998vy, Haack:2001iz}, \n  $(D6,D8)$, $(D4,D6,D8)$ \nbound states \\cite{Singh:2001gt, Singh:2002eu} and Galilean-AdS geometries \\cite{Singh:2009tq, Singh:2010rt}. \n In all of these massive tensor field\nplays a key role.\nUnder the `massive' T-duality \\cite{Bergshoeff:1996ui} the D8-branes  \n can be mapped  over to the axionic D7-branes of type IIB string theory\nand vice-versa.  \nThe  $B$-field also plays  important role in obtaining  \nnon-relativistic Lifshitz solutions \\cite{Singh:2017wei, Singh:2018ibp}. \nThe latter solutions are of no surprise in mIIA theory,\nas an  observed  feature in four-dimensional AdS gravity theories\nhas been that in order to obtain  non-relativistic \nsolutions one needs to include \nmassive (Proca)  gauge fields in the gravity theory \\cite{Son:2008ye}. \nOther  different situations where massless vector fields \ncan give rise to  non-relativistic vacua, \n  involve  boosted black D$p$-branes\ncompactified along lightcone direction \\cite{Singh:2010zs, Singh:2012un}. These latter class of solutions are also called \nhyperscaling (or conformally) Lifshitz  vacua \\cite{Narayan:2012hk}.   \n\nParticularly the $a=2$ Lifshitz vacua with IR excitations in mIIA theory\ncan be written as \\cite{Singh:2018ibp} \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{sol2a9}\n&&ds^2= L^2\\left(- {dt^2\\over  z^4h} +{dx_1^2+dx_2^2\\over z^2}+{dz^2\\over\nz^2}  +{dy^2\\over  q^2h} + d\\Omega_5^2 \\right) ,\\nonumber\\\\\n&&e^\\phi=g_0 h^{-1\/2},\n ~~~~~C_{(3)}= -{1 \\over g_0}\n{L^3\\over  z^4}dt\\wedge dx_1\\wedge dx_2, \\nonumber\\\\ &&\nB_{(2)}=  { L^2\\over q  z^2}h^{-1}dt\\wedge dy  \\ ,\n\\eea \nwhere the harmonic function $h(z)= 1+{z^2\\over z_{I}^2}. $\nThe parameter $z_I$ is related to the charge of the NS-NS \nstrings. \nThe excitations  involve $g_{tt}$ and $g_{yy}$\n metric components, and leaving the  $x_1,x_2$ \nplane (worldvolume directions of D2-branes) \nunaffected.\\footnote{\nHere  $L={2\\over g_0  m l_s}$, \nand $m$ being the mass parameter in the mIIA action. (We would set $l_s=1$ and\n $g_0=1$.) \n The  constant $q$ is a free (length)  parameter \nand $g_0$ is weak string coupling.\nNote $L$ is dimensionless parameter, it\ndetermines overall radius of curvature of the spacetime.\n Therefore Romans' theory with \n  $m \\ll{ 2\\over g_0 l_s}$ would be preferred here \nso that  $L\\gg 1$ in  the solutions \\eqn{sol2a9},  else\n  these classical vacua cannot be trusted. Also, from  the D8\nbrane\/domain-wall correspondence  in \\cite{Bergshoeff:1996ui}, one typically\nexpects $m \\approx {g_0 N_{D8} \\over l_s}$, a value which is definitely\nwell within ${ 2\\over g_0 l_s}$ for a finite number of $D8$ branes, $N_{D8}$, \nin these backgrounds. }\n The  excitations do also  induce a running of dilaton field. \nThe $B_{ty}$ component of the string field  \n is also coupled to the  excitations. Since\n$h\\sim 1$ as  $z\\to 0$, \nthese excitations form normalizable modes ($z_I$ would correspond to \nadding relevant operators in the boundary Lifshitz theory). \nThe solution \\eqn{sol2a9} asymptotically flows to  weakly coupled  \n  regime  in the UV (note that the string coupling,\\ $g_0<1$). \nWhile, in the\ndeep IR region, with $z\\gg z_I$ where $h\\approx {z^2\\over z_I^2}$,\nthe   vacua is  driven to another  \nweakly coupled Lifshitz regime.  For $z\\gg z_I$,  \nthe  IR geometry transforms  to dilatonic\n  $Lif_4^{(3)}\\times S^1\\times S^5$ solution.\nThis solution enables us to study  the effect \nof the excitations in \n $a=2$  Lifshitz theory. Note the $z_I$ dependent excitations \nat zero temperature\nare mainly in the form of charge excitations, along with nontrivial \nentanglement chemical potential, as we would see next. \n \n\n\\section{ Entanglement  of a disc subsystem}\\label{sec3}\n \nFor asymptotically AdS space-time dual to a CFT, the entanglement entropy can be calculated by the Ryu-Takayanagi formula \\cite{Ryu:2006bv, *Ryu:2006ef}. We assume the same is true for an asymptotically Lifshitz space-time, dual to a non-relativistic field theory with Lifshitz scaling symmetry. We consider a round disc of radius $\\ell$ at the center of the $x_1,x_2$ plane with its boundary identified with the corresponding boundary of  $2d$  Ryu-Takayanagi surface lying inside the Lifshitz bulk geometry \\eqn{sol2a9}. We shall assume $y$ is a compactified direction \n\\begin{equation}\n\ty\\sim y +2\\pi r_y\\;.\n\\end{equation}\nIn  radial coordinates $(r=\\sqrt{x_1^2+x_2^2})$ \nthe Ryu-Takayanagi area functional \\cite{Ryu:2006bv, Ryu:2006ef} \nfor static bulk surface is given by \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{areaint1}\n{\\cal A}_\\gamma = 2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz \n {r\\sqrt{1+r'^2}\\over z^2} h^{1\\over2}\\,,\n\\eea\nwhere, $r'={dr\\over dz},~h(z)=\\left(1+ {z^2\\over z_I^2 }\\right)$ and \n$\\epsilon \\ll \\ell$ is UV cut-off  \nof the Lifshitz theory. We need to extremize the area integral by solving the Euler-Lagrange equation for $r(z)$\n\\begin{multline}\n\t2zrr''h(z) - 4rr'^3h(z) - 4rr'h(z) - 2zr'^2h(z) -2zh(z) - zrr'^3h'(z) - zrr'h'(z) = 0\\,,\n\\end{multline}\nIt is impossible to analytically calculate the full area integral \\eqref{areaint1}. To facilitate our job, therefore, we restrict ourselves to small subsystems, with  $\\ell\\ll z_I$. In this domain, we can make a perturbative expansion and obtain solutions order by order in the dimensionless ratio ${\\ell\\over z_I}$; such that $r(z)=r_{(0)}+ r_{(1)}+ \\cdots$, and correspondingly we would write $${\\cal A}_\\gamma ={\\cal A}_0 +{\\cal A}_1 +\\cdots\\;,$$ for small  $\\ell$. Our immediate interest is in calculating terms up to leading order and \nfirst order only in the ${\\ell\\over z_I} $ expansion. \n\\par The equation at zeroth order is\n\\begin{equation}\n\tzr_{(0)}r_{(0)}^{\\prime \\prime} - 2r_{(0)}r_{(0)}^{\\prime 3} - 2r_{(0)}r_{(0)}^{\\prime} - zr_{(0)}^{\\prime 2} - z = 0\\,,\n\\end{equation}\nfor which  $r_{(0)}=\\sqrt{\\ell^2-z^2}$ defines the extremal surface (half circle) \\cite{Ryu:2006ef, Blanco:2013joa} with the boundary conditions $r_{(0)}(0)=\\ell$, and $r_{(0)}(z_\\ast)=0$, where $z=z_*$ is the point of return that lies at $z_\\ast= \\ell$. One then finds that the area\n\n\\begin{align}\n{\\cal A}_0 &= 2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz \\frac{r_{(0)}\\sqrt{1 + r_{(0)}^{\\prime 2}}}{z^2},\\nonumber \\\\ \n&= 2\\pi L^2 \\left(\\frac{\\ell}{\\epsilon} - 1 \\right).\n\\end{align}\n\n${\\cal A}_0$ being a ground state contribution it obviously remains independent of the parameter $z_I$ of the bulk geometry. This only means that there is no effect of excitations on the leading term. As explained in \\cite{Blanco:2013joa}, the first order contribution can be evaluated using only the tree level embedding function and is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{area1}\n{\\cal A}_1 &=& \n2\\pi L^2 \\int^{z_\\ast}_{\\epsilon} dz r_{(0)} \n{\\sqrt{1+r_{(0)}'^2}\\over 2\\,z_I^2},\\nonumber\\\\\n&=& \\pi L^2  \\left({\\ell^2\\over z_I^2}\\right).\n\\eea\nFrom here the complete expression of entanglement entropy of \na  disc shaped subsystem up to first\norder becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{gb12}\nS_E^{Disc}[\\ell,z_I]&\\equiv& {{\\cal A}_\\gamma \\over 4 G_{4}},\\nonumber\\\\\n&=& S_{E}^{(0)}+\n {\\pi L^2 \\over  4\\,G_{4}}  \\left( \n   {\\ell^2\\over z_I^2} \\right),\n\\eea\nwhere the Newton's constant in $4$D and $5$D are related to the 10-dimensional Newton's constant by $\\frac{1}{G_{4}} = \\frac{L\\,2\\pi r_y}{G_5}$ and $\\frac{1}{G_5}\\equiv {L^5 Vol(S^5)\\over G_{10}}$. We shall be using $G_4$ and $G_5$ back and forth in our calculation.\\\\\nThe ground state entropy contribution is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nS_{E}^{(0)}= {\\pi L^2 \\over  2G_{4}}  \\left( \n{\\ell\\over \\epsilon} -1\\right)\\ .\n\\eea\nThe equation \\eqref{gb12} is a  meaningful expression for entanglement entropy only if we maintain   $\\ell\\ll z_I$. The first order term explicitly depends on $z_I$, so  small fluctuations of the bulk quantities, like $\\delta z_I$, would result in corresponding change in entropy. For a fixed size $\\ell$, one could express these variations of the entropy density as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{ent34}\n{\\delta s_E^{Disc}}\n={\\delta S_E^{Disc}\\over \n\\pi \\ell^2}\n=  {L^2 \\over 4 G_4} \n   \\delta \\left( {1\\over  z_I^2} \\right),\n\\eea\nwhere $\\pi \\ell^2$ is the disc area.\nEquation \\eqref{ent34} provides a complete expression  up to  first order. \nAt second order the entropy\nwill receive  new $z_I$ dependent contributions.\n\nNext, we note that\nthe right hand side of equation \\eqref{ent34} is actually independent of the disc size $\\ell$!\nOn first hand observation this appears very surprising because, according to the first law of \nentanglement thermodynamics \\cite{Bhattacharya:2012mi}, we expected that the entropy density of excitations would  have had  $\\ell^2$ dependence, namely in the form of inverse temperature (usually entanglement\ntemperature goes as $T_E^{-1}\\propto \\ell^a$; and the dynamical exponent of time in our Lifshitz background  is $a=2$). Especially this aspect of the  first law has been found to remain true  in a variety of relativistic CFTs, where entanglement temperature is given by $T_E \\propto {1 \\over \\pi  \\ell}$; see for example \\cite{Bhattacharya:2012mi, Allahbakhshi:2013rda, Mishra:2015cpa, Mishra:2016yor, Mishra:2018tzj, Ghosh:2017ygi, Bhattacharya:2019zkb}. What, then, is so different for the Lifshitz system described by equation \\eqn{ent34}? To understand this phenomenon we first need to get an estimate of the energy associated with the excitations in our system.\n\n\\subsection {Energy, winding charge and chemical potential}\n \nWe now turn to find the energy of excitations of the `massive strings' due to which we have a configuration in equation \\eqn{sol2a9},\nwhere we can express $B_{ty}\\simeq B_{ty}^{massive}\n+B_{ty}^{excitation}$.\nNote that we are treating $y$ as a compact direction.\nThe Scherk-Schwarz compactification \\cite{Scherk:1978ta, *Scherk:1979zr, Lavrinenko:1996mp} of the Lifshitz background \\eqn{sol2a9} on a circle \nalong $y$ gives rise to the following 1-form potential\n\\be\nA_{(1)}={L^2\\over q z^2}\\left(1+ {z^2\\over z_I^2}\\right)^{-1} dt.\n\\ee\nIt represents a gauge field in the lower dimensional supergravity whose only non-zero component is $A_t$. It can be determined from here that due to  string excitations the net change in the $U(1)$ charge (due to winding strings) is\n\\be\\label{rho12}\n \\bigtriangleup \\rho \n= {N\\over V_2}= {\\bigtriangleup Q\\over 2\\pi r_y V_2}=\n{2 L\\over  G_5 z_I^2},\\ee\nwhere $V_2$ is the area element of $x_1,x_2$ plane, see a calculation in the appendix. The entanglement chemical potential, with the prescription in \\cite{Mishra:2015cpa}, \ncan be obtained by measuring gauge  field \nat the turning point, \n namely\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{chempotdef}\n\\mu_E \\equiv A_t|_{z=z_\\ast} = {L^2r_y \\over q z_\\ast^2}+ \\cdots \\,,\n\\eea  \nwhere ellipses denote  sub-leading terms which are not required at first order. This is a logical guess inspired by black hole thermodynamics, where the value of the one form at the black hole horizon is known to give the chemical potential conjugate to the U(1) charge. Even for backgrounds with non-relativistic conformal symmetry as considered in \\cite{Maldacena:2008wh}, the Kaluza-Klein gauge field measured at the horizon produces the correct thermal chemical potential. There's no horizon in our bulk space-time; instead, we use the critical point $z_*$ associated with the entanglement wedge.\n\nAt leading order we have $z_\\ast \\simeq \\ell$, hence essentially this thermodynamic variable gets uniquely fixed by the \nLifshitz ground state \\eqn{sol2a9n}. \nSo for small  $\\ell ~ (> 0)$ the chemical potential remains quite important, and we obtain\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \n\\mu_E\\cdot \\bigtriangleup \\rho \\simeq \n{L^2 \\over \\pi G_4}{1\\over z_I^2\\ell^2}. \n\\eea\n There are no other excitations except the winding strings, the energy density due to\n the excitations can be estimated to be\n\\begin{equation} \\label{enr34}\n\\bigtriangleup {\\cal E}={\\cal E}-{\\cal E}_0 \\simeq \n\\frac{1}{2}\\mu_E \\bigtriangleup \\rho = \\frac{L^3 r_y}{q G_5} \\frac{1}{z_I^2 \\ell^2} = {L^2 \\over 2\\pi G_4}\\frac{1}{z_I^2 \\ell^2}\\,,\n\\end{equation}\n where ${\\cal E}_0$ is the (normalized) energy of the\nground state of our Lifshitz theory\\footnote{We do notice an explicit dependence of energy density on the system size; which is unlike relativistic CFT but is a familiar feature in non-relativistic theories, the particle in a box being an immediate example.}. This is the only meaningful deduction we can make from here, particularly in absence of a direct method to evaluate full stress-energy tensor of the Lifshitz theory.\\footnote{ There is an early work \\cite{Ross:2009ar} but it does not include dilatonic scalar field excitations like in our background. In contrast in asymptotically AdS spacetimes one knows how to obtain stress-energy tensor by doing Fefferman-Graham expansion near  AdS boundary \\cite{Balasubramanian:1999re,*Kraus:1999di,*Bianchi:2001kw}. Perhaps something similar could also be done in the Lifshitz case involving dilaton field.} Assuming that the entanglement temperature of the 3-dimensional\n$a=2$ Lifshitz system  faithfully behaves as \\cite{Bhattacharya:2012mi}\n \\be\\label{spheretemp1}\nT_E = {4\\over \\pi \\ell^2}\\,,\\ee\nwe determine that the ratio \n$${\\mu_E \\over T_E} = {\\pi L^2 r_y\\over 4  q}\\,,$$ \n is indeed independent of  $\\ell$. \nEssentially this  ratio seems to get uniquely fixed by the Lifshitz ground state \\eqn{sol2a9n} at the leading order. Note that the excitations seem to have no effect on it. The analysis also implies that the energy density and the entanglement temperature both fall off with the system size $\\ell$ at the same rate, and the ratio\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n{\\bigtriangleup {\\cal E} \\over T_E}= \n{ \\pi L^3  r_y \\over 4 q G_5 z_I^2} \n\\equiv {1\\over 2} {k_E N\\over  V_2}, \n\\eea\nstays fixed for small discs. However this ratio does depend on the excitations \nnamely through $z_I$. \nIn the second equality we have preferred to view dimensionless quantity\n $k_E={\\pi  L^2 r_y\\over 8 q}$ as being \nanalogous to the Boltzmann constant in usual \nthermodynamics. (For example, we could have expressed total energy of disc as $\\bigtriangleup { E}= {1\\over 2} N k_E T_E $ with out affecting anything.) {\\it Hence it can be concluded that the entanglement\nentropy  per unit disc area is fixed \nfor small discs of radii  $\\ell\\ll z_I$}. It is \n also confirmed that the entropy of excitations \\eqref{ent34} \nfollows the first law relation \\cite{Bhattacharya:2012mi, Allahbakhshi:2013rda, Wong:2013gua, Pang:2013lpa, Mishra:2015cpa, Mishra:2016yor, Ghosh:2016fop, Ghosh:2017ygi, Bhattacharya:2019zkb}\n\\be\n\\delta s_E=  {1 \\over T_E} (\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho),\n\\ee\nunder infinitesimal changes in the bulk quantity, $\\delta z_I$.\n\nWe summarize  our main observations at first order;\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nT_E\\propto {1\\over \\ell^2}, ~~~~\n\\bigtriangleup s_E = \\text{Fixed}, ~~~~\n\\mu_E\\propto r_y T_E, ~~~~\n\\bigtriangleup {\\cal E}\\propto N T_E, ~~~~\n\\Delta \\rho= \\text{Fixed}, ~~~~\n\\eea\nat a given entanglement temperature.\n\n\\subsection{Entanglement entropy of a disc at second order}\nLet us now consider corrections to holographic entanglement entropy at next higher order. It is somewhat easier to calculate when one chooses $z(r)$ parameterization, so let us rewrite the integral as\n\\begin{equation}\n\t\\mathcal{A}_{\\gamma} = 2\\pi L^2\\int_{0}^{1}dr \\frac{r\\sqrt{1+z'^2}}{z^2}h^{\\frac{1}{2}}\\,,\n\\end{equation}\nwhere we rescaled $r$ and $z$ to the dimensionless variables $\\frac{r}{\\ell}$ and $\\frac{z}{\\ell}$. It suffices to obtain the embedding up to first order to get the entanglement at second order \\cite{Blanco:2013joa, Bhattacharya:2019zkb}. So, we expand $z(r)$ as $z(r) = z_{(0)} + z_{(1)} + \\cdots$, where $z_{(0)} = \\sqrt{1 - r^2}$ and $z_{(1)}$ satisfies the equation\n\\begin{equation}\\label{eqnsec}\n\tz_{(1)}'' + \\frac{1-2r^2}{r(1-r^2)}z_{(1)}'-\\frac{2}{(1-r^2)^2}z_{(1)} = \\frac{1}{\\sqrt{1-r^2}}\\,,\n\\end{equation}\nwith the boundary conditions: $z_{(1)}'(0) = 0$ and $z_{(1)}(\\ell) = 0$. One can check that a consistent solution to equation \\eqref{eqnsec} is\n\\begin{equation}\\label{embdsec}\n\tz_{(1)} = - \\frac{1-r^2-2\\sqrt{1-r^2}+2\\ln \\left(1+\\sqrt{1-r^2}\\right)}{2\\sqrt{1-r^2}}.\n\\end{equation}\nTherefore, the area integral now acquires a new contribution $\\mathcal{A}_\\gamma = \\mathcal{A}_0 + \\mathcal{A}_{1} + \\mathcal{A}_2$ where\n\\begin{equation}\\label{area2}\n\t\\mathcal{A}_2 = 2\\pi L^2\\frac{\\ell^4}{z_I^4}\\left(\\frac{5}{8} - \\ln 2\\right),\n\\end{equation}\nwhich is negative as expected. The area difference from pure $AdS$ at both orders is plotted in figure \\ref{spherearea} .\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.75]{spherearea-eps-converted-to.pdf}\n\t\\caption{Area difference from $AdS$ ground state for spherical subsystem, the second order correction is negative. Plot drawn by choosing $z_I^2 = 2$ and $L = r_y = q = 1$.}\n\t\\label{spherearea}\n\\end{figure}\nTotal entropy of the disc at this order will be\n\\begin{equation}\\label{discent2}\n\tS_E^{(2)} = S_E^{(0)} + \\frac{\\pi L^2}{4 G_4}\\frac{\\ell^2}{z_I^2}\\left(1 + \\frac{\\ell^2}{z_I^2}\\left(\\frac{5}{4} - 2\\ln 2\\right)\\right).\n\\end{equation}\nSo that the variation of entropy density, at second order, becomes:\n\\begin{equation}\\label{entden2}\n\t\\delta s_E^{(2)} = \\frac{L^2}{4 G_4}\\left(1 + \\frac{\\ell^2}{z_I^2}\\left(\\frac{5}{2} - 4\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right).\n\\end{equation}\nAs previous, we wish to express \\eqref{entden2} as a `first law' like relationship. We find that one way to achieve this is to absorb all second order corrections to a modified temperature and chemical potential, this method was first used in \\cite{Mishra:2015cpa} although they worked with differences rather than variation as we do. To this end, we first note that the turning point $z_*$ should be corrected at $\\mathcal{O}\\left(\\frac{\\ell^2}{z_I^2}\\right)$ as\n\\begin{equation*}\n\tz_* \\equiv z(0) = \\ell + \\frac{\\ell^3}{z_I^2}\\left(\\frac{1}{2} - \\ln 2\\right).\n\\end{equation*}\nThe chemical potential, defined in equation \\eqref{chempotdef}, can be expressed including $\\mathcal{O}(\\frac{\\ell^2}{z_I^2})$ corrections as\n\\begin{align}\n\t\\mu_E^{(1)} \\simeq& \\frac{L^2r_y}{q\\ell^2}\\left(1+\\frac{\\ell^2}{z_I^2}\\left(\\frac{1}{2} - \\ln 2\\right)\\right)^{-2}\\left(1+\\frac{\\ell^2}{z_I^2}\\right)^{-1}\\,, \\nonumber \\\\\n\t=& \\frac{L^2r_y}{q\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right).\n\\end{align}\nSo we get\n\\begin{align*}\n\t\\mu_E^{(1)}\\delta\\Delta \\rho ~=&~ \\frac{2 L^3r_y}{qG_5\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right),\\\\\n\t=&~ \\frac{L^2}{\\pi G_4\\ell^2}\\left(1 - \\frac{\\ell^2}{z_I^2}\\left(2 - 2\\ln 2\\right)\\right)\\delta\\left(z_I^{-2}\\right),\n\\end{align*} \nwhile the energy remains the same as defined in \\eqref{enr34}. From equation \\eqref{entden2}, a bit of paperwork then leads to the following result\n\\begin{equation}\\label{spherelaw2}\n\t\\delta s_E^{(2)} = \\frac{1}{T_E^{(2)}}\\left(\\delta\\Delta\\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta\\Delta\\rho \\right),\n\\end{equation}\nwhere $T_E^{(2)}$ denotes the `entanglement temperature' at second order, which is given by\n\\begin{align}\\label{spheretemp2}\n\tT_E^{(2)} &= \\frac{\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho}{\\delta \\Delta s_E^{(2)}},\\nonumber \\\\\n\t&=\\frac{\\frac{L^2}{\\pi G_4\\ell^2}\\left[1 - \\frac{\\ell^2}{z_I^2}\\left(1 - \\ln 2\\right) \\right]}{\\frac{L^2}{4 G_4}\\left[1 - \\frac{\\ell^2}{z_I^2}\\left(4\\ln 2 - \\frac{5}{2}\\right) \\right]},\\nonumber \\\\\n\t&\\simeq T_E^{(1)}\\left[1 + \\frac{\\ell^2}{z_I^2}\\left(5\\ln 2 - \\frac{7}{2}\\right) \\right],\n\\end{align}\nwhere $T_E^{(1)}$ stands for the first order temperature, defined in eqn. \\eqref{spheretemp1}. The term in parentheses is a negative number, so second order correction to `entanglement temperature' results in its sharper fall. See figure \\ref{fig1} for an illustration of this behaviour.\n\n\\begin{figure}[h!]\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{spheremu-eps-converted-to.pdf}\n\t\t\\caption{$\\mu_E$ vs. $l$}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{spheretemp-eps-converted-to.pdf}\n\t\t\\caption{$T_E$ vs. $l$}\n\t\\end{subfigure}\n\t\\caption{The unbroken and dashed curves display the behaviour of the uncorrected and corrected quantities, respectively; both the entanglement temperature and chemical potential decrease due to higher order corrections. The plots were drawn by setting $z_I^2 = 2$ and $L = r_y = q = 1$.}\n\t\\label{fig1}\n\\end{figure}\n\n\\par Some comments are in order to justify equation \\eqref{spherelaw2}, we have seen that for small enough subsystem size $(\\ell \\ll z_I)$, the change in entanglement entropy at first order in our perturbative calculation follows a relationship akin to the first law of thermodynamics. If one considers this relationship an actual `law' for entanglement entropy, one must find a consistent way to describe new contributions at higher orders. Equation \\eqref{spheretemp2} proposes that at second order, the chemical potential as well as the entanglement temperature should be corrected to keep the law intact. In fact, we expect this procedure to work at all higher orders. It could be thought that a more accurate measure of these quantities are obtained as one climbs the perturbation ladder.\n\\section{ Entanglement entropy of narrow  strip}\\label{sec4}\n \nWe now consider a strip like subsystem with  coordinate\nwidth $-\\ell\/2\\le x_1\\le \\ell\/2$, and the  range of $x_2\\in[0,l_2]$, such that\n$l_2\\gg \\ell$. The straight line  boundary of the two-dimensional strip\nis identified with the boundary  of the  RT surface in the bulk \nat constant time. \nThe area functional of this static  surface is \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{str10}\n{\\cal A}_\\gamma=2 L^2 l_2  \\int^{z_\\ast}_{\\epsilon} dz \n{\\sqrt{1+x_1'^2}\\over z^2} h^{1\\over2}\\,. \n\\eea\nFor small width  $\\ell\\ll z_I$, we \nmake a perturbative expansion of the integrand.\nThe extremal surface satisfies the following equation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \nx_1'={z^2\\over z_\\ast^2} {1\\over \\sqrt{{h\\over h_\\ast}-{z^4\\over z_\\ast^4}}}\\,,\n\\eea\nwhere $h_\\ast\\equiv h(z_\\ast)$.\nWe have  specific boundary conditions such that\nnear  the  spacetime boundary \n$x_{1}|_{z=0}=\\ell\/2$ and  \nthe turning point is given by $x_{1}|_{z\\sim z_\\ast}=0$. \nThis leads to the first integral of the following type\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{str11} \n\\ell=2\\int^{z_\\ast}_0 dz{z^2\\over z_\\ast^2} {1\\over \\sqrt{{h\\over h_\\ast}-{z^4\\over z_\\ast^4}}}\\,,\n\\eea\nwhich gives rise to a perturbative expansion in ${z_\\ast \\over z_I}$ \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{str12}\n\\ell=z_\\ast\n\\left(b_0 +{z_\\ast^2 \\over2 z_I^2} I_1 + \\cdots\\right)\\,,\n\\eea\nwhere coefficients are expressible as Beta-functions $b_0=\n{1\\over 4} B\\left(\\frac{3}{4}, \\frac{1}{2}\\right)$ and $I_1= \n{1\\over 4}(B(\\frac{3}{4},-\\frac{1}{2}) -B(\\frac{5}{4},-\\frac{1}{2}))$.\nThe equation \\eqn{str12} can be inverted and expressed as a perturbative expansion of the turning point\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} z_\\ast=z_\\ast^{(0)} \\left(1 -{z_\\ast^{(0)2} \\over z_I^2} {I_1\\over2 b_0} + \\cdots \\right),\n\\eea\nwhere $z_\\ast^{(0)}\\equiv {\\ell \\over 2b_0}$ is the turning point in the \nabsence of excitations.\n\nThe leading area of strip can be evaluated using the tree level values\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n{\\cal A}_0 &&= \n2 L^2 l_2 \\int^{z_\\ast^{(0)}}_{\\epsilon} dz \n{\\sqrt{1+x_{1(0)}'^2}\\over z^2}\\,,\\nonumber\\\\\n&&={2 L^2 l_2\\over z_\\ast^{(0)}} \\int^{1}_{\\epsilon \\over z_\\ast^{(0)}} \nd\\zeta {1\\over\\zeta^2 \\sqrt{1-\\zeta^4}}\\,,\n\\nonumber\\\\ &&\n=2 L^2 l_2 \\left({1\\over\\epsilon}-{2(b_0)^2\\over \\ell}\\right).\n\\eea\nwhile the first order contribution is evaluated as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{foa}\n{\\cal A}_1 &&= \n2 L^2 l_2 \\int^{z_\\ast}_{0} dz  \n{\\sqrt{1+x_{1(0)}'^2}\\over 2 z_I^2}\\,,\\nonumber\\\\\n&& = L^2 l_2\\left({a_1 z_\\ast^{(0)} \\over z_I^2 }\\right).\n\\eea\nwhere the coefficient $a_1={1\\over 4} B\\left(\\frac{1}{4},\\frac{1}{2}\\right)$.\nThe entanglement entropy of small strip  up to first order is then given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\nS_E^{strip} = \\frac{\\mathcal{A}_0 + \\mathcal{A}_1}{4G_5} = {L^2 l_2 \\over 2 G_{4}}  \n\\left({1\\over \\epsilon} -{2b_0^2\\over \\ell}\n+{a_1\\over 4b_0}~\\frac{\\ell}{z_I^2} \\right).\n\\eea\n\nNow any small  change in the bulk parameter \n($\\delta z_I$) will necessarily effect the entanglement entropy at first order.For a fixed width $\\ell$, we find the change in entropy per unit area of the strip as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{jh4}\n\\delta s_E^{strip} \\equiv {\\delta S_E^{strip}\\over l_2 \\ell}\n=  {L^2 \\over 8 G_{4}} {a_1\\over b_0}\n\\delta \\left( {z_I^{-2}} \\right),\n\\eea\nwhich is complete expression up to first order. Once again we find that the right hand side is independent of $\\ell$, as it was also in the case of a disc. Following from the  disc case in the previous section, the effective chemical potential for strip subregion is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\n\\mu_E=\n{L^2 r_y\\over q z_\\ast^2}\\simeq \n{4 b_0^2 L^2 r_y\\over q \\ell^2}\\,.\n\\eea\nFrom here and \\eqref{rho12}, let us define for the strip\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array}\\label{enr35} \n\\bigtriangleup {\\cal E} \\equiv \\frac{1}{2}\n\\mu_E . \\bigtriangleup \\rho = {4 L^3r_y \\over  G_5\\,q}\\frac{b_0^2}{z_I^2 \\ell^2 } = \\frac{2 L^2}{\\pi G_4}\\frac{b_0^2}{z_I^2\\ell^2}.\n\\eea\nThis is like the disc result in \\eqn{enr34}, $i.e.$ \n$\\bigtriangleup {\\cal E}\\propto T_E$.\nUsing  \\eqn{enr35} we conclude  that the entanglement\nentropy density \\eqn{jh4} of the strip subsystems \nalso conforms to the first law relation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\def\\ba{\\begin{array} \\label{flfo}\n\\delta s_E={1\\over T_E} \\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho\\right). \\eea\nwhere for the strip, entanglement temperature is\ndefined as $T_E={8 b_0^3 \\over a_1}{4\\over \\pi \\ell^2}$ \nin  3-dimensional  Lifshitz theory.\n\n\\subsection{Strip entropy at second order}\n\nIt is instructive to find out the change in entanglement \nentropy at higher orders in $\\frac{\\ell^2}{z_I^2}$ and \ninterpret its thermodynamic property, here we include the \nresults at $\\mathcal{O}\\left(\\frac{\\ell^4}{z_I^4}\\right)$. \\\\ \n\\par The turning point $z_*$, as discussed before in \\eqn{str11} and \\eqn{str12}, could be related to the strip-width $\\ell$ as\n\t\\begin{equation}\\label{turnpt2}\n\t\tz_* = \n\\frac{z_*^{(0)}}{1 + \\frac{z_*^{(0) 2}}{2z_I^2} \\frac{I_1}{b_0} - \n\\frac{z_*^{(0) 4}}{8z_I^4} \\left(\\frac{I_2}{b_0} + \\frac{4I_1^2}{b_0^2}\\right)}\\,,\n\t\\end{equation}\n\twhere the new co-efficient $I_2$ can be expressed as: \n$I_2 = \\frac{1}{8}\\big(2B(\\frac{3}{4}, -\\frac{3}{2}) - \n3B(\\frac{5}{4}, -\\frac{3}{2})\\big)$. With the help of \n\\eqref{turnpt2}, the area integral \\eqn{str10} now reads\n \t\t$\\mathcal{A}_{\\gamma} = \\mathcal{A}_0 + \\mathcal{A}_1 + \\mathcal{A}_2$,\n where $\\mathcal{A}_0$ and \n$\\mathcal{A}_1$ are as obtained before.\n  The second order contribution is\n\t\\begin{equation}\n\\mathcal{A}_2=\n- \\frac{2 L^2 l_2}{z_*^{(0)}} \\frac{z_*^{(0)4}}{8z_I^4}\\left(\\frac{4a_0I_1^2}{b_0^2} + \\frac{2I_1J_1}{b_0}\\right).\n\t\\end{equation}\n\tThe new coefficients introduced in above expression are listed below\n\t\\begin{align*}\n\t\ta_0 &= -\\frac{1}{4}B\\left(\\frac{3}{4}, \\frac{1}{2}\\right) = -b_0\\,, \\\\\n\t\tJ_1 &=\\frac{1}{4}\\left(B\\left(\\frac{3}{4}, -\\frac{1}{2}\\right) \n+ 3B\\left(\\frac{1}{4}, -\\frac{1}{2}\\right) \\right).\n\t\\end{align*}\n\nAfter some simplification the contribution to the area of the RT \nsurface at second order turns out to be\n\t\\begin{equation}\n\t\t\\mathcal{A}_2 = -\\frac{L^2 l_2\\ell}{64} \\frac{\\ell^2}{z_I^4}\\frac{1}{b_0^2}\\left(\\frac{a_1^2}{b_0^2} - 1 \\right).\n\t\\end{equation}\n\tThe coefficient $a_1$ has already been defined in eq. \\eqref{foa}. Hence, the total entanglement entropy density, at second order in perturbation theory, becomes\n\t\\begin{equation}\\label{kl23}\n\t\ts_E^{(2)} = s_E^{(0)} + \\frac{L^2}{8 G_4}\\frac{1}{z_I^2}\n\\frac{a_1}{b_0}\\left(1 - \\frac{\\ell^2}{z_I^2} \\frac{1}{32b_0^2}\\left(\\frac{a_1^2}{b_0^2} - 1\\right)\\right).\n\t\\end{equation}\n\tThe area difference including second order correction has been shown in figure \\ref{striparea}.\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.75]{striparea-eps-converted-to.pdf}\n\t\t\\caption{The area difference at first and second order of perturbation analysis for strip subsystem, plots drawn by choosing $z_I^2 = 2$ and $L=r_y=q=l_2=1$.}\n\t\t\\label{striparea}\n\t\\end{figure}\n\tTo write down the `first law' we need to rewrite the expression for $s_E^{(2)}$ in terms of variation in $\\mathcal{E}$ and $\\mu_E \\Delta \\rho$; recall that the chemical potential was defined as the value of the gauge potential at the turning point. Here, it is sufficient to compute $\\mu_E$ up to first order\n\t\\begin{equation*}\n\t\t\\mu_E^{(1)} \\simeq \\frac{L^2}{z_*^2}\\left(1 - \\frac{z_*^2}{z_I^2} \\right) = \\frac{L^2r_y}{qz_*^{(0)2}}\n\\left(1 + \\frac{z_*^{(0)2}}{z_I^2}(\\frac{I_1}{b_0^2} - 1) \\right).\n\t\\end{equation*}\n\tSo that,\n\t\\begin{align*}\n\t\t\\mu_E^{(1)}\\delta \\Delta \\rho ~=&~ \\frac{L^3r_y}{qG_5}\\frac{8b_0^2}{\\ell^2}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{8b_0^2}\\left(\\frac{a_1}{b_0} - 3\\right) \\right]\\delta\\left(z_I^{-2}\\right),\\\\\n\t\t=&~ \\frac{L^2}{2\\pi G_4}\\frac{8b_0^2}{\\ell^2}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{8b_0^2}\\left(\\frac{a_1}{b_0} - 3\\right) \\right]\\delta\\left(z_I^{-2}\\right).\n\t\\end{align*}\nA little effort, then, allows us to write\n\t\\begin{equation}\\label{law2}\n\t\t\\delta s_E^{(2)} = \\frac{1}{T_E^{(2)}}\\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho\\right).\n\t\\end{equation}\n\tHere, $T_E^{(2)}$ stands for the modified entanglement temperature at second order.\n\t\\begin{align}\\label{temp2}\n\t\tT_E^{(2)} &= \\frac{\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E^{(1)}\\delta \\Delta \\rho}{\\delta \\Delta s_E^{(2)}},\\nonumber \\\\\n\t\t\t\t  &= \\frac{4}{\\pi \\ell^2}\\frac{8b_0^3}{a_1}\n\t\t\t\t\t\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{16b_0^2} \\left(\\left(\\frac{a_1}{b_0} - 3 \\right) + \\left(\\frac{a_1^2}{b_0^2} - 1\\right) \\right) \\right] \\nonumber \\\\\n\t\t\t\t  &= T_E^{(1)}\\left[1 + \\frac{\\ell^2}{z_I^2}\\frac{1}{16b_0^2}\\left(\\left(\\frac{a_1}{b_0} - 1\\right)\\left(\\frac{a_1}{b_0} + 2 \\right) - 2 \\right) \\right]\n\t\\end{align}\nWhere by $T_E^{(1)}$, we refer to the temperature at first order defined in equation \\eqref{flfo}, the numerical value of $\\frac{a_1}{b_0} \\approx 2.188$, so the correction at this order results in an increase of $T_E$, albeit by a tiny amount. The uncorrected and corrected temperatures are plotted in figure \\ref{fig2}.\n\n\\begin{figure}\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{Lif4mu-eps-converted-to.pdf}\n\t\t\\caption{$\\mu_E$ vs. $\\ell$}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}{0.475\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=4.75 cm, width=\\textwidth]{Lif4temp-eps-converted-to.pdf}\n\t\t\\caption{$T_E$ vs. $\\ell$}\n\t\\end{subfigure}\n\t\\caption{The unbroken and dashed curves display the behaviour of the uncorrected and corrected quantities, respectively; the entanglement temperature is found to increase due to higher order corrections while the chemical potential decreases.The plots were drawn by setting $z_I = 2$ and $L = r_y = q = G_5 = 1$.}\n\t\\label{fig2}\n\\end{figure}\n\n\\subsection{Numerical results for strip subsystem}\nWe end this section with a comparison of our perturbative results with some numerical analysis. For the numerical computation we chose $z_I = 4$ and used \\eqref{str11} to obtain corresponding lengths $\\ell$ of the sub-region for different choices of the turning point $z_{*}$. We also obtain the area difference $\\Delta \\cal{A}$ from \\eqref{str10} for the same $z_{*}$ values and plot the two sets against each other. The output is summarized in figure \\ref{stripnumeric}. From the graph we conclude that a second order perturbation series analysis is trustworthy for small strip-width.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.5]{StripNumeric.pdf}\n\t\\caption{Numerical plot of area difference from $AdS$ ground state for strip subsystem and comparison with second order perturbation series analysis. The pre-factor in \\eqref{str10} was ignored in the plot.}\n\t\\label{stripnumeric}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec5}\n\nThe Lifshitz background  \n$Lif_4^{(2)}\\times {S}^1\\times S^5$ of the massive type IIA theory\nallows exact  excitations which couple to massless modes of string in the IR. We calculated the entanglement entropy of the theory\nat the boundary of these spacetimes, both for strip as well as disc shaped systems. At leading order, we found that the entropy density of the excitations remains fixed and does not grow with $\\ell$, the subsystem size, so long as $\\ell\\ll z_I$. We find that this behaviour is consistent with the fact that energy density of the excitations itself behaves as  $\\bigtriangleup{\\cal E} \\propto 1\/\\ell^2$, which is in agreement with $\\bigtriangleup{\\cal E}\n\\simeq \\frac{1}{2}\\mu_E \\bigtriangleup\\rho $. Note that the entanglement temperature itself goes as $T_E\\propto{1\\over \\ell^2}$.\n\nBut this entanglement behaviour is quite different in comparison to the relativistic CFTs, where the entropy density of excitations grows linearly with the subsystem size, while the energy density of excitations remains fixed. Nevertheless we have found that the first law of entanglement thermodynamics\n\\begin{equation}\n\t\\delta s_E = \\frac{1}{T_E}\\left(\\delta \\Delta \\mathcal{E} + \\frac{1}{2}\\mu_E\\delta \\Delta \\rho\\right),\n\\end{equation} \nholds good if we accept the hypothesis that the energy of a subsystem in the Lifshitz background \\eqref{sol2a9} is given by\n$$\\bigtriangleup{ E}\\simeq \\mu_E N \n\\simeq {1\\over 2} N k_E T_E\\,. $$\nOur results appear to indicate an \nequipartition nature of the entanglement thermodynamics for non-relativistic Lifshitz\nsystem. But this is perhaps true only for \nthe high entanglement temperature regime (i.e. small $\\ell\\ll z_I$).\n\\par Further, we studied what happens to the first law of entanglement if we assume it to remain valid beyond the leading order. There is lack of consensus on this aspect, despite there being enough evidence for it to be a natural feature at first order. We discussed how the first law could be extended up to second order by making use of appropriately modified chemical potential and entanglement temperature. We think this is necessary because otherwise, we need to look for a new quantity at each higher order to account for the corrections; while the entanglement entropy, like its thermal counterpart should depend only on the energy and conserved charges of the theory. Such redefinition should work at all orders, thereby allowing the `first law of entanglement thermodynamics' to be obeyed quite generally, irrespective of the degree of perturbation theory.\n\\par It would be interesting to obtain the HEE numerically for ball subsystems and compare with our perturbative results. This, however, involves solving boundary value problem and proves to be non-trivial. Another interesting problem is to consider shape dependence of holographic entanglement entropy in similar spirit to \\cite{Fonda:2015nma, Cavini:2019wyb}. We hope to return to these problems in future.\n\\vskip 0.5cm\n\\begin{acknowledgements} \nHS is thankful to the organisers of the \nAdS\/CFT@20 workshop at ICTS Bangaluru \nand the STRINGS-2019 at Brussels for the exciting meetings and warm hospitalities. SM would like to thank Aranya Bhattacharya for useful discussions and help with Mathematica.\n\\end{acknowledgements} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPrimitive meteorites contain presolar dust grains, grains that formed \nin stellar outflows and supernova (SN) ejecta and survived not only a \nlong history as interstellar (IS) grains in the interstellar medium \n(ISM), but also the formation of the solar system and conditions in the \nmeteorites' parent bodies (Anders \\& Zinner 1993; Ott 1993). The stellar \norigin of these grains is indicated by their isotopic compositions, which \nare completely different from those of material in the solar system and \nreflect the compositions of their stellar sources. Their study in the \nlaboratory provides information on stellar nucleosynthesis, Galactic \nchemical evolution (hereafter GCE), physical and chemical properties \nof stellar atmospheres and ejecta, and conditions in the early solar \nsystem (Bernatowicz \\& Zinner 1997; Zinner 1998).\n\nThe following types of presolar grains have been identified to date: \ndiamond  (Lewis et al. 1987), silicon carbide (SiC, carborundum) \n(Bernatowicz et al. 1987), graphite (Amari et al. 1990), aluminum \noxide (Al$_2$O$_3$, corundum), spinel (MgAl$_2$O$_3$) (Hutcheon et al. \n1994; Nittler et al. 1994) and silicon nitride (Si$_3$N$_4$) \n(Nittler et al. 1995). In addition, graphite and SiC contain tiny \nsubgrains of Ti carbide \n(graphite contains also Zr and Mo carbide) that were identified by \ntransmission electron microscopy (Bernatowicz et al. 1991, 1996; \nBernatowicz, Amari, \\& Lewis 1992). The carbonaceous phases diamond, \nSiC and graphite were discovered because they carry isotopically \nanomalous noble gases (Tang \\& Anders 1988a; Lewis, Amari, \\& Anders \n1990, 1994; Huss \\& Lewis 1994a,b; Amari, Lewis, \\& Anders 1995a); \nthey can be extracted from meteorites in almost pure form by chemical \nand physical processing (Tang et al. 1988; Amari, Lewis, \\& Anders 1994). \nThis results in relatively large amounts (micrograms) of samples that \ncan be analyzed in great detail. In contrast, presolar corundum and \nsilicon nitride have been discovered by ion microprobe isotopic \nmeasurements of individual grains from chemically resistant residues \nand only a limited number of grains ($\\sim$ 100 for corundum) have been \nmeasured to date.\n\nSilicon carbide, graphite, corundum, and Si$_3$N$_4$ grains are large \nenough (up to several $\\mu$m in diameter for corundum and Si$_3$N$_4$, \nup to $>$ 10 $\\mu$m for SiC and graphite) to be analyzed individually \nfor their elemental and isotopic compositions. The ion microprobe makes \nit possible to measure isotopic ratios of major and some minor elements \nin single grains down to $\\sim$ 0.5 $\\mu$m in diameter (e.g., Hoppe et al.\n1994, 1995; Huss, Hutcheon, \\& Wasserburg 1997; Travaglio et al. 1999). \nIon microprobe isotopic measurements have been made for C, N, O, Mg, Si, \nK, Ca, and Ti on many (for the major elements on thousands of) grains  \n(see, e.g., Zinner 1998). Single grain isotopic measurements, albeit on \na limited number of grains, have also been made for Zr, Mo, and Sr by \nresonance ionization mass spectrometry (Nicolussi et al. 1997, 1998a,b,c; \nDavis et al. 1999; Pellin et al. 1999); and for He and Ne by laser \nextraction and noble gas mass spectrometry (Nichols et al. 1991, 1994;  \nKehm et al. 1996). Diamonds, instead, are too small ($\\sim$ 2 nm) for \nsingle grain analysis and isotopic measurements have been made on ``bulk\nsamples'', collections of many grains. Bulk analyses have also been made \nin SiC and graphite of the noble gases (Lewis et al. 1994; Amari et al. \n1995a) and of trace elements such as Sr, Ba, Nd, Sm, and Dy in SiC (see, \ne.g., Anders \\& Zinner 1993; Hoppe \\& Ott 1997; Zinner 1998).\n\nBased on their isotopic compositions, several stellar sources have been \nidentified for presolar grains. Most corundum grains are believed to have \noriginated in low-mass red giant (RG) and asymptotic giant branch (AGB) \nstars. This identification rests on the O isotopic ratios and inferred \n$^{26}$Al\/$^{27}$Al ratios (Huss et al. 1994; Nittler et al. 1997; Nittler\n1997; Choi et al. 1998; Nittler \\& Alexander 1999a). Low-density graphite \ngrains, a subtype of SiC grains termed X grains, and Si$_3$N$_4$ grains have \nisotopic signatures indicative of a SN origin (Nittler et al. 1995; \nTravaglio et al. 1999). A SN origin has also been proposed for a few \ncorundum grains (Nittler et al. 1998; Choi et al. 1998). In addition, \na few rare SiC grains with large $^{30}$Si excesses and low \n$^{12}$C\/$^{13}$C and $^{14}$N\/$^{15}$N ratios are possibly of a nova origin\n(Gao \\& Nittler 1997).\n\nMost SiC grains, in particular the ``mainstream'' component, which \naccounts for $\\sim$ 93\\% of all SiC (Hoppe et al. 1994; Hoppe \\& Ott 1997), \nare believed to come from carbon stars, thermally pulsing (TP) AGB stars\nduring late stages of their evolution. \nThe best evidence for such an origin are the $s$-process (slow neutron \ncapture process) isotopic patterns displayed by the heavy elements Kr, \nSr, Zr, Mo, Xe, Ba, Nd, and Sm  (Lewis et al. 1994; Hoppe \\& Ott 1997; \nGallino, Raiteri, \\& Busso 1993; Gallino, Busso, \\& Lugaro 1997;  \nNicolussi et al. 1997, 1998a; Pellin et al. 1999) and the presence of  \nNe-E(H), almost pure $^{22}$Ne (Lewis et al. 1990, 1994; Gallino et al. \n1990). The C and N isotopic compositions as well as $^{26}$Al\/$^{27}$Al\nratios of individual mainstream SiC grains are by and large consistent \nwith a carbon star origin  (Hoppe et al. 1994; Huss et al. 1997; Hoppe \\& \nOtt 1997).\n\nIn contrast to C, N, Ne, Al and the heavy elements, the variations in the \nSi (and Ti) isotopic ratios measured in single mainstream grains cannot be \nexplained in terms of nucleosynthesis in AGB stars (Gallino et al. 1990, \n1994; Brown \\& Clayton 1992a). They have been interpreted to indicate that \nmany stellar sources (Clayton et al. 1991; Alexander 1993), whose \ninitial compositions vary because of GCE, contributed SiC grains to the \nsolar system (Gallino et al. 1994; Timmes \\& Clayton 1996; Clayton \\&  \nTimmes 1997a). However, a fundamental problem with this interpretation \nis the fact that the metallicity implied by the Si isotopic compositions \nof the mainstream grains is higher than that of the sun. This would mean \nthat the grains are younger than the solar system. A solution to this \npuzzle has been proposed by Clayton (1997) who considered the possibility \nthat the sun and the AGB stars that were the sources of the mainstream \ngrains did not originate in the same Galactic region but changed their \npositions because of Galactic diffusion. Alexander \\& Nittler (1999), \non the other hand, used the Si and Ti isotopic compositions of the \nmainstream grains themselves (Figs. 3 and 4) to infer the metallicity of the \nISM at the time of solar system formation. From this exercise they \nconcluded that the sun has an atypical Si isotopic composition.\n\nIn this paper we will revisit the Si isotopic compositions of the \nmainstream SiC grains. In \\S 2 we will first describe the isotopic \nproperties of these grains in greater detail, discuss their AGB origin, \nand review previous attempts to understand their Si isotopic ratios. \nAfter presenting new calculations for the nucleosynthesis of Si (and Ti) \nin AGB stars (\\S 3), we will present a new approach to the problems \nof the distribution of Si isotopic ratios (\\S 4): instead of assuming an\naverage monotonic relationship of metallicity with time in the Galaxy, \nwe investigate how \nheterogeneities in the Si isotopic ratios could result from fluctuations \nin the contributions from various types of nucleosynthetic sources to \nthe low-mass stars that, in their AGB phase, produced the SiC grains. \nPreliminary accounts can be found in Lugaro et al. (1999a,b).\n\nGalactic chemical evolution models of the Galactic disk that, \nalthough in different ways, \ndeal with compositional inhomogeneities in the ISM have previously \nbeen presented by Malinie et al. (1993), Wilmes \\& K\\\"oppen (1995), \nCopi (1997) and van den Hoek \\& de Jong (1997). We \nwill show that our approach is consistent with the \nresults obtained by inhomogeneous GCE models, which, however, did \nnot address the evolution of isotopic compositions.\n\n\\section{Meteoritic SiC, the mainstream component and the Si isotope\npuzzle}\n\nAmong all presolar grains, SiC has been most widely studied because it \nis relatively abundant (6 ppm in the Murchison and in similar primitive \nmeteorites) and is present in various classes of meteorites (Huss \\& \nLewis 1995). Ion microprobe isotopic analyses of single grains have \nrevealed several distinct classes. This is shown in Figs. 1 and 2, which \ndisplay the C, N, and Si isotopic ratios. For historical reasons the Si \nisotopic ratios are expressed as $\\delta$-values, deviations in permil\n($^o\\!\/\\!_{oo}$) from the solar isotopic ratios of ($^{29}$Si\/$^{28}$Si)$_{\\odot}$ =\n0.0506331 and ($^{30}$Si\/$^{28}$Si)$_{\\odot}$ = 0.0334744 (Zinner, Tang, \\&\nAnders 1989):\n\n$\\delta$$^{29}$Si\/$^{28}$Si = [($^{29}$Si\/$^{28}$Si)$_{meas}$\/\n($^{29}$Si\/$^{28}$Si)$_{\\odot} -$ 1] $\\times$ 1000,\n\n$\\delta$$^{30}$Si\/$^{28}$Si = [($^{30}$Si\/$^{28}$Si)$_{meas}$\/\n($^{30}$Si\/$^{28}$Si)$_{\\odot} -$ 1] $\\times$ 1000. \n\nAccording to their C, N and Si isotopic compositions five different \ngroups of grains can be distinguished and these groups are indicated \nin the figures. Also indicated in the figures are the abundances of the \ndifferent groups. \nBy far the most common grains are the mainstream grains. It should \nbe noted that the frequency distributions of grains in the plots of \nFigs. 1 and 2 do not correspond to their abundances in meteorites, \nbut that rare grain types, located by automatic imaging in the ion \nprobe (Nittler et al. 1995; Amari et al. 1996) are over-represented. \nAn additional grain with extreme $^{29}$Si and $^{30}$Si excesses \nhas been found (Amari, Zinner, \\& Lewis 1999); its composition\n($\\delta$$^{29}$Si\/$^{28}$Si = 2678 $^o\\!\/\\!_{oo}$, $\\delta$$^{30}$Si\/$^{28}$Si =\n3287 $^o\\!\/\\!_{oo}$) lies outside the boundaries of the plot in Fig. 2. \n\nThe possible stellar sources of the different groups of SiC grains \nhave been discussed elsewhere (e.g., Zinner 1998). Here we wish to \nconcentrate on grains of the mainstream \ncomponent (Hoppe et al. 1994; Hoppe \\& \nOtt 1997). Their $^{12}$C\/$^{13}$C ratios lie between 15 and 100 and their \n$^{14}$N\/$^{15}$N ratios between the solar ratio of 272 and 10,000 (Fig. 1). \nTheir Si isotopic ratios plot along a line of slope 1.31 in a \n$\\delta$$^{29}$Si\/$^{28}$Si vs. $\\delta$$^{30}$Si\/$^{28}$Si 3-isotope \nplot (Fig. 3). Most grains show large $^{26}$Mg excesses attributed to the \npresence of $^{26}$Al ($T_{1\/2}$ = 7 $\\times 10^5$ yr), now extinct, at\nthe\ntime of their formation. Inferred $^{26}$Al\/$^{27}$Al ratios range up to \n10$^{-2}$ (Hoppe et al. 1994; Huss et al. 1997). Much more limited isotopic \ndata exist for Ti. In Fig. 4 the measurements by Hoppe et al. \n(1994) and Alexander \\& Nittler (1999) are plotted as\n$\\delta$$^{i}$Ti\/$^{48}$Ti values against the $\\delta$$^{29}$Si\/$^{28}$Si \nvalues of these grains. The correlation between the Ti ratios, especially\n$\\delta$$^{46}$Ti\/$^{48}$Ti, and \nthe Si isotopic ratios has already been noticed by Hoppe et al. \n(1994).\n\nThere are many pieces of evidence that indicate that mainstream \nSiC grains come from carbon stars. Carbon stars are TP-AGB stars \nwhose spectra are dominated by lines of C compounds such as C$_2$, \nCH, and CN, indicating that C $>$ O in their envelopes (Secchi 1868). \nThey become C-rich because of the recurrent third\ndredge up (TDU) episodes mixing with the envelope newly synthesized   \n$^{12}$C~ from the He shell where it is produced by the triple-$\\alpha$ \nreaction  \n(Iben \\& Renzini 1983). For high-temperature carbonaceous phases, \nsuch as SiC, to condense from a cooling gas the condition C $>$ O has \nto be satisfied (Larimer \\& Bartholomay 1979; Sharp \\& Wasserburg \n1995; Lodders \\& Fegley 1997a). Carbon stars experience substantial \nmass loss by stellar winds and have extended atmospheres with \ntemperatures of 1500 - 2000 K, at which SiC is expected to condense. \nIn fact, carbon stars are observed to have circumstellar dust \nshells that show the 11.3 $\\mu$m emission feature of SiC (Cohen 1984; \nLittle-Marenin 1986; Martin \\& Rogers 1987; Speck, Barlow, \\& \nSkinner 1997). Recently, Clayton, Liu, \\& Dalgarno (1999) proposed\nthat in a SN environment with high levels of ionizing  $\\gamma$-rays carbon \ndust can condense from a gas of O $>$ C. However, even if this should \nbe possible, it would not apply to the expanding atmospheres of \ncarbon stars. In contrast to such atmospheres, the solar system \nis characterized by O $>$ C and phases such as SiC are not believed \nto be able to form under these conditions. This apparently is the \nreason why all SiC grains found in primitive meteorites are of presolar\norigin according to their isotopic compositions. This is in marked \ncontrast to presolar corundum grains, which make up only a small fraction\n($\\sim$1\\%) of all meteoritic corundum grains.\n\nIsotopic compositions of presolar grains are the most diagnostic \nindicators of their stellar sources. As already mentioned in \n\\S 1, the $s$-process patterns of the heavy elements exhibited \nby mainstream SiC grains constitute the most convincing argument \nfor their origin in carbon stars, which are the major source of the  \n$s$-process elements in the Galaxy. The envelopes of these stars show \nlarge enhancements of typical $s$-elements such as Sr, Y, Zr, Ba, La, \nCe, and Nd (Smith \\& Lambert 1990). In SiC grains $s$-process patterns \nare seen in the elements Kr, Xe, Sr, Ba, Nd, Sm, and Dy (Lewis et al. \n1990, 1994; Ott \\& Begemann 1990a,b; Prombo et al. 1993; Richter, Ott, \n\\& Begemann 1993, 1994; Zinner, Amari, \\& Lewis 1991; Podosek et al. \n1999), which have been measured in bulk samples. Although these \nsamples were collections of all  SiC grain types, there is little \ndoubt that the isotopic results must have been dominated by \ncontributions of mainstream grains.\n\nIn addition, isotopic analyses of Sr, Zr, and Mo by resonance \nionization mass spectrometry (RIMS) have been made on a limited \nnumber of single SiC grains. Although most of these measurements \nwere made on SiC grains for which no C and Si isotopic data had \nbeen obtained (Nicolussi et al. 1997, 1998a,b), for statistical \nreasons essentially all of these grains must have been mainstream \ngrains. They display characteristic $s$-process patterns with large \ndepletions in the $p$-only isotopes $^{84}$Sr, $^{92}$Mo, and \n$^{94}$Mo, and in the $r$-only isotope $^{100}$Mo. Large depletions \nare also seen in $^{96}$Zr, indicating that neutron densities in the \nstellar sources of these grains must have been low, compatible with \n$^{13}$C~ being the major neutron source in AGB stars (Gallino et al. \n1998a). The depletions in $^{96}$Zr are consistent with isotopic \nabundance data\nobtained by spectroscopic observations of the ZrO bandheads in AGB star \nenvelopes (Lambert et al. 1995). Recent RIMS analysis of Mo in SiC grains \nthat had been identified as mainstream grains on the basis of their \nC, N, and Si isotopic ratios confirmed that mainstream grains \nindeed carry $s$-process signatures in this element (Pellin et al. \n1999). Gallino et al. (1997) could successfully reproduce the measured \n$s$-process compositions of the heavy elements in mainstream SiC grains\nwith models of low-mass \nAGB stars of close-to-solar metallicity, in which neutrons are primarily \nproduced by the $^{13}$C~ source during the radiative interpulse period. In\naddition to the isotopic patterns of the heavy elements, the large \noverabundances of refractory $s$-process elements such as Zr, Y, Ba, \nand Nd in single SiC grains (Amari et al. 1995b) are further evidence \nfor an AGB-star origin (Lodders \\& Fegley 1995, 1997a,b, 1998).\n\nEvidence for a carbon star origin is also obtained from the light \nelements. It should be noted that as far as neutron-capture \nnucleosynthesis is concerned, because of their large abundance \nthe light elements (elements lighter than Fe) in AGB stars are \nconsidered to be neutron poisons for the synthesis of the \nheavy elements. However, because of their relatively small cross \nsections, they are only marginally affected by neutron capture. \nElements up to Mg are also affected by charged particle reactions \nwith H and He. The most important light-element signature of SiC \nis the Ne isotopic composition, which is dominated by $^{22}$Ne  \n(Lewis et al. 1990, 1994). In fact, it has been the presence of \nthis Ne component, Ne-E(H), that, together with the so-called \nXe-S component  (Srinivasan \\& Anders 1978; Clayton \\& Ward 1978) \nled to the isolation of presolar SiC (Tang \\& Anders 1988a). \nGallino et al. (1990, 1994) showed that Ne-E(H) matches the \npredicted isotopic composition of Ne in the He shell of AGB stars. \nAlmost all initial CNO nuclei are first converted to $^{14}$N during \nshell H burning and then to $^{22}$Ne via the chain $^{14}$N($\\alpha$,$\\gamma$)$^{18}$F($\\beta^+\\nu$)$^{18}$O~($\\alpha$,$\\gamma$)$^{22}$Ne~ in the \nHe shell during thermal pulses. \nAnother piece of evidence is obtained from the distribution of the  \n$^{12}$C\/$^{13}$C ratios in mainstream grains that is very similar to that \nmeasured astronomically in carbon stars (Dominy \\& Wallerstein \n1987; Smith \\& Lambert 1990; see also Fig. 14 in Anders \\& Zinner \n1993).\n\nThe ranges of $^{12}$C\/$^{13}$C and $^{14}$N\/$^{15}$N ratios \nmeasured in mainstream grains roughly agree with the ranges predicted \nby theoretical \nmodels of AGB stars. Proton captures occurring in the deep envelope \nduring the main sequence phase followed by first (and second) \ndredge-up as well as shell He burning and the TDU during the \nTP-AGB phase affect the C and N isotopes in the envelope.\n$^{12}$C\/$^{13}$C ratios\npredicted by canonical stellar evolution models range from $\\sim$ 20 \nat first dredge-up in the RG phase to $\\sim$ 300 \nin the late TP-AGB phases (Iben 1977a; Bazan 1991; Gallino et al. \n1994). Predicted $^{14}$N\/$^{15}$N ratios are 600 - 1,600 (Becker \\& \nIben 1979; El Eid 1994), falling short of the range observed in the grains. \nHowever, the assumption of deep mixing (``cool bottom processing'' or \nCBP) of envelope material to deep hot regions in $M \\lesssim$ 2.5 \n$M_{\\odot}$~ stars during their RG and AGB phases (Charbonnel 1995; Wasserburg,  \nBoothroyd, \\& Sackmann 1995; see also Langer et al. 1999a for \nrotationally induced mixing) results in partial H burning, with\nhigher $^{14}$N\/$^{15}$N and lower $^{12}$C\/$^{13}$C ratios in the envelope\nthan in canonical models (see also \nHuss et al. 1997). \nAs a matter of fact, CBP mechanisms have been introduced to explain the\nobserved \n$^{12}$C\/$^{13}$C ratios in RG stars of low mass \n(Gilroy 1989; Gilroy \\& Brown 1991;  \nPilachowski et al. 1997), which are lower than those \npredicted by canonical models.\n\nIn contrast to the heavy elements, and the light elements C, N,  \nNe and Al, the Si isotopic ratios of mainstream SiC grains cannot \nbe explained by nuclear processes taking place in a single star. \nIn Fig. 3 we plotted the Si isotopic data measured in SiC grains \nfrom three different size fractions isolated from the Murchison \ncarbonaceous (CM2) meteorite (Hoppe et al. 1994, 1996a) and data \nfrom the Orgueil (CI) meteorite (Huss et al. 1997). The three \nMurchison size fractions are KJE (0.5 - 0.8 $\\mu$m in diameter), \nKJG (1.5 - 3 $\\mu$m), and KJH (3 - 5 $\\mu$m) (Amari et al. 1994). \nOf the smallest Murchison grain size fraction KJE we plotted only \ndata points with errors smaller than 15 $^o\\!\/\\!_{oo}$. The distributions of \nSi isotopic ratios measured in SiC grains from other meteorites \nare very similar to that shown in Fig. 3 (Alexander 1993; Huss, \nFahey, \\& Wasserburg 1995;  Gao et al. 1995). Also plotted in \nFig. 3 is the correlation line obtained from a fit to the grain \ndata. This line, which does not go through the solar composition \nbut passes slightly to the right of it, has a slope of 1.31 and \nan intercept of the ordinate at $\\delta$$^{29}$Si\/$^{28}$Si$_{int}$ \n= $-$ 15.9 $^o\\!\/\\!_{oo}$, a little different from the parameters determined from \nthe KJG and KJH dataset only (Hoppe et al. 1994).\n\nThere have been various attempts to explain the Si isotopic \ndistribution of the mainstream SiC grains. Zinner et al.  (1989) \nalready realized that the scatter in the Si isotopic ratios \nindicate several stellar sources. Stone et al. (1991) first \nnoticed the correlation line of Si isotopic ratios in SiC grains \nfrom Orgueil and proposed an origin in a single AGB star \nwith mixing of two components but they did not address the question of \nhow the end components could be generated by nucleosynthetic processes \nin a single star. The only nuclear reactions in AGB stars that are \nbelieved to substantially affect the Si isotopes are neutron captures \nin the He shell. However, it has been determined early on (Gallino et \nal. 1990; Obradovic et al. 1991; Brown \\& Clayton 1992b), and will be \nseen in more detail in the \\S 3, that neutron captures shift the \nSi isotopic ratios along a line with a slope that varies, depending \non mass and metallicity, from 0.35 to 0.75 in a \n$\\delta$$^{29}$Si\/$^{28}$Si vs. $\\delta$$^{30}$Si\/$^{28}$Si \n3-isotope plot. Furthermore, in low-mass AGB stars of \nclose-to-solar metallicity predicted shifts of envelope material \nare only on the order of 20 $^o\\!\/\\!_{oo}$, an order of magnitude less than \nthe range seen in mainstream grains. Nuclear processes in a single \nAGB star therefore cannot produce the mainstream distribution and \nthis led to the conclusion that several stars with varying initial \nSi isotopic compositions must have contributed SiC grains to the \nsolar system  (Clayton et al. 1991; Alexander 1993). The situation \nis similar for Ti. It should be emphasized that there is a \nfundamental qualitative difference between the isotopic compositions \nof C, N, Ne, and the heavy elements, and those of Si and Ti. While \nthe former are dominated by RG and AGB nucleosynthesis, the effect \nof stellar nucleosynthesis on the Si and Ti ratios is relatively \nsmall and cannot explain the compositions observed in grains; the \npresence of an extra component has to be invoked.\n\nBrown \\& Clayton (1992b, 1993) proposed a single-star model by \nconsidering Mg burning at elevated temperatures in the He-burning \nshell. In this model ($\\alpha$,n) reactions on Mg in a 5.5 $M_{\\odot}$~ \nAGB star produce a neutron-rich Si isotopic composition at the far \nend of the mainstream correlation line. Mixing with the original, \nclose to solar, composition in the envelope combined with variable \nmass loss from this star could lead to the observed distribution. \nHowever, to accomplish this, the temperature in the He shell \nhas to be raised by 10\\% above that produced by the standard AGB \nmodels (Iben 1977b). This leads to serious problems with other \nprocesses and with the general question of energy generation and \nstellar structure. Moreover, the Ti isotopic variations and \nespecially the correlation with the Si isotopic ratios (Fig. 4) \ncannot be explained in this way because the temperatures required \nfor Mg burning in the He shell do not affect at all the Ti \nisotopes, nor would the $s$-process isotopic signatures be compatible \nwith such a situation.\n\nThis leaves variations in the initial Si isotopic compositions of \nthe AGB stars that contributed SiC grains to the solar system as \nthe most likely explanation for the Si isotope distribution of \nmainstream grains. Variations of the initial Si compositions in \nturn are expected as the result of GCE. Low-mass stars, \nwhich became AGB stars at the end of their evolution and \ncontributed grains to the protosolar nebula, are likely to \nhave been born at different times before solar system formation \nand thus reflect the isotopic composition of the Galaxy in \ndifferent earlier epochs. An explanation of the Si isotopic \ncompositions in mainstream grains thus requires an understanding \nof the evolution of the Si isotopic ratios throughout Galactic \nhistory.\n\nGallino et al. (1994) approached this problem by assuming that \n$^{29}$Si and $^{30}$Si are mostly primary isotopes that are \nproduced by SNe of Type II, together with a major fraction of \n$^{28}$Si, and that substantial contributions to Galactic Si \nin the form of almost pure $^{28}$Si from SNe of Type Ia late \nin Galactic history determine the Si isotopic evolution reflected \nby the grains. \nThese authors also advanced a tentative interpretation of the \nTi isotopes. They noticed the correlation between the Ti and \nSi isotopic compositions (Fig. 4) and concluded that, as for \nSi, neutron-capture nucleosynthesis cannot explain the Ti \nisotopes nor the correlation with Si and they would have to be \ninterpreted within the framework of the chemical evolution of \nthe Galaxy.\n\nTimmes \\& Clayton (1996) and Clayton \\& \nTimmes (1997a,b) constructed a detailed model of the Galactic \nhistory of the Si isotopes that is based on the GCE model of Timmes, Woosley \n\\& Weaver (1995). The SN production yields were obtained from \nthe Type II SN models of Woosley \\& Weaver (1995, henceforth WW95)  \nand from the popular W7 Type Ia SN model by Thielemann, Nomoto, \\& \nYokoi (1986). According to the WW95 models, $^{29}$Si and \n$^{30}$Si in Galactic disk stars are predominantly \nsecondary isotopes, i.e. their production in massive Type II SNe \nincreases with increasing metallicity, since it \nrequires the prior presence of primary isotopes such as $^{12}$C,\n$^{14}$N and $^{16}$O.\nAs a consequence, early SNIIe produced mostly pure $^{28}$Si, \nwhereas later SNIIe added more and more $^{29}$Si and $^{30}$Si \nto the ISM. This resulted in a continuous increase of the \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios throughout \nGalactic history and the distribution of the grains' Si isotopic ratios \napparently reflects this change. According to the Timmes \\& Clayton \nmodel, the spread in the isotopic compositions of the mainstream grains \ncorresponds to variations in the birth dates of their parent stars. \nSpecifically, the birth dates of the parents stars range over $\\sim$ \n5 Gyr (see Fig. 6 in Timmes \\& Clayton 1996).\n\nHowever, this model suffers from a fundamental problem. Because \nmost mainstream grains have $^{29}$Si\/$^{28}$Si and \n$^{30}$Si\/$^{28}$Si ratios that \nare larger than those of the solar system (Fig. 3), they are \ninferred to be younger than the sun. This absurd corollary of \nthe model led Clayton (1997) to consider the possibility that \nthe mainstream grains originated from stars that were born at \ndifferent Galactic radii than the sun. Indeed, according to \nWielen, Fuchs, \\& Dettbarn (1996), scattering \nby massive molecular clouds may lead to the diffusion of those stars \nfrom central metal-rich regions of the Galaxy (for which higher \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios are predicted \nthan those in the present solar neighborhood) to the region where \nthey, once they became AGB stars, shed their SiC grains into the \nprotosolar cloud. \nAlexander \\& Nittler (1999) took a different approach to resolve the \nparadox that the grains are apparently younger than the sun. They \nfitted the Si and Ti isotopic compositions of the mainstream grains \nto contributions from nucleosynthesis in the parent AGB stars and \nthe stars' original isotopic compositions. However, in contrast to \nthe Timmes \\& Clayton (1996) model, they tried to determine the \nevolution of the Si and Ti isotopes from the grain data themselves. \nFrom this fit they concluded that most mainstream grains \ndo not come from stars with higher than solar metallicities but that \nthe sun has an atypical Si isotopic composition.\n\nCommon to the GCE models listed above is that they predict a \nmonotonic relationship between the Si isotopic composition and \nGalactic time for the ISM at a given Galactic radius. In other \nwords, a given age of the Galaxy or, if different Galactic radii \nare considered, a given function of age and Galactic radius \ncorresponds to a given isotopic composition (this functional \nrelationship is implied in the Clayton \\& Timmes (1997a) model \nbut has never been explicitly worked out).\n\nHowever, one has to consider that the Si isotopic composition \nof the Galaxy, and in particular that of different regions where \nindividual low-mass stars are born, evolves as the result of \ncontributions from discrete stellar sources, mostly SNe. It \nis unlikely that these discrete contributions are instantly \nmixed with preexisting material so that the isotopic compositions \nof large Galactic regions are completely homogenized. The work \nof Edvardsson et al. (1993) and others has shown that stars \nfrom any given Galactic epoch and Galactic radius display a \nconsiderable spread in metallicity and, more generally, in elemental abundances. \nNumerous attempts have been made to explain these observations: \nstellar orbital diffusion, chemical condensation processes \nand\/or thermal diffusion in stellar atmospheres, incomplete mixing of \nstellar ejecta, sequential stellar enrichment and local infall of \nmetal-deficient gas (see van den Hoek \\& de Jong 1997 for review and \ndiscussion of these different attempts).\n\nThe study of meteorites provided ample evidence that the protosolar \nnebula was isotopically not homogenized. In addition to the survival \nof pristine stardust, isotopic anomalies are found in material that \napparently was processed in the solar nebula (see, e.g., Clayton, \nHinton, \\& Davis 1988; Lee 1988; Wasserburg 1987). Examples include \n$^{16}$O excesses (up to 5\\%) and large deficits and excesses of the \nn-rich isotopes of the Fe-peak elements such as $^{48}$Ca and $^{50}$Ti \nin refractory inclusions. These anomalies indicate the survival of \nisotopic signatures from different nucleosynthetic reservoirs. Another \nindication of local isotopic heterogeneity comes from the value of the \n$^{17}$O\/$^{18}$O ratio, which is 5.26 in the solar system but \n3.65 $\\pm$ 0.15 in diverse molecular clouds (Penzias 1981; Wannier \n1989; Henkel \\& Mauersberg 1993, see also discussion concerning \nthe $^{14}$N\/$^{15}$N ratio by Chin et al. 1999).\n\nWe therefore want to explore to what extent the Si isotopic \nspread of mainstream SiC grains can be explained by local \nheterogeneities in the regions from which the low-mass parent \nstars of the grains originally formed. Before doing this we \nwill examine in more detail the nucleosynthesis of the Si \nand Ti isotopes in AGB stars. However, because of the largely \nuncertain astrophysical origin of all Ti isotopes  (see  \nTimmes et al. 1995; Woosley 1996; Woosley et al. 1997), the \nsituation concerning the Ti isotopes in presolar SiC \ngrains and their interpretation in terms of SN contributions \nand GCE is a complicated issue in itself. It will be treated \nin a separate paper.\n\n\\section{Nucleosynthesis of the Si and Ti isotopes in AGB stars}\n\nDuring all the evolutionary phases of low-mass stars ($M <$ 10 $M_{\\odot}$) the\nmaximum temperature in the inner regions never reaches high enough values\nto allow the burning of any element heavier than He. \nConsequently, only the production of $^{12}$C~ and of $^{16}$O from\ninitial H or He nuclei is possible and, in particular, there are no \ncharged-particle interactions that involve the nucleosynthesis of \nSi and Ti.\nThe initial isotopic compositions of these elements can be nevertheless  \nmodified by slow neutron capture (the $s$ process), which occurs\nin the tiny region between the H shell and the He shell (hereafter He\nintershell) during the AGB phase. \n\nAccording to the AGB models of low-mass ($M =$ 1.5 $-$ 3 $M_{\\odot}$) stars with \nmetallicities in the range from half solar to solar obtained with the FRANEC\ncode and discussed in detail \nby Straniero et al. (1997) and Gallino et al. (1998a), neutrons are \nreleased in the He intershell by two different sources, $^{13}$C~ and \n$^{22}$Ne. The maximum temperature achieved during the \nrecurrent thermal instabilities (or thermal pulses: TP) of the He \nshell is not high enough to consume $^{22}$Ne to an appreciable extent, so \nthat the $^{13}$C~ source has to play the major role. However, \nthe number of $^{13}$C~ nuclei left behind by the H-burning shell is too \nsmall to account for the $s$-element enhancements observed in carbon \nstars. A special mechanism has to be invoked in order to build up a \nsufficient amount of $^{13}$C~ in the He intershell. In AGB stars of mass \n$M \\ga 1.5$ $M_{\\odot}$, after a limited number \nof thermal pulses, soon after the quenching of a given instability, \nthe convective envelope penetrates into the top layers of\nthe He intershell and mixes with the envelope  \nmaterial enriched in $^{12}$C~ and $s$-process elements. This TDU \nphenomenon leaves a \nsharp H\/He discontinuity, where some kind of hydrodynamical mixing, \npossibly driven by rotation, occurs \n(Herwig et al. 1997; Singh, Roxburgh, \\& Chan \n1998; Langer et al. 1999a,b). In these conditions, a small \namount of protons penetrates from the \nenvelope into the He intershell (see Gallino et al. 1998a for \ndiscussion). At H reignition, these protons are captured by the \nabundant $^{12}$C~ present in the intershell as \na consequence of partial He burning that occurred during the previous \nthermal instability. Consequently, a so-called $^{13}$C {\\it pocket} \nis formed in a small region at the top of the \nHe intershell. Before the onset of the next pulse, the progressive \ncompression and heating of these layers cause all $^{13}$C~ to burn\nradiatively in the interpulse period \nvia the $^{13}$C($\\alpha$,n)$^{16}$O~ reaction, at a temperature of around 8 keV. \nThe neutron exposure, or time integrated neutron flux \n$\\delta \\tau = \\int {\\it N}_{\\rm n}$ $v_{th}~ dt$, experienced\nduring the interpulse period may reach quite a \nhigh value, $\\delta \\tau_1 (8  {\\rm keV})$ of up to 0.4 mbarn$^{-1}$, \ndepending on the initial amount of $^{13}$C~. The maximum neutron density in this \nradiative phase remains low: $N_{{\\rm n}, max}$ $\\sim$ 10$^7$ n\/cm$^{3}$.\n\nThe material that experienced neutron captures in the $^{13}$C~ pocket \nis engulfed and diluted ($\\sim$1\/20) \nby the next growing convective thermal pulse, which extends over almost \nthe whole He intershell, and is mixed with material already $s$-processed \nduring the previous pulses, together with ashes of the H-burning \nshell. Among them are Si and Ti in their initial abundances. For advanced \npulses, the overlapping factor between subsequent pulses becomes $r$ \n$\\approx$ 0.4 \nand the mass of the convective pulse is slightly smaller than 10$^{-2}$ $M_{\\odot}$~ \n(Gallino et al. 1998a). A small neutron burst, at around 23 keV, is\nreleased in convective thermal pulses, during the latest phases of the\nAGB evolution, when the bottom temperature in the He shell is sufficiently\nhigh to marginally activate the $^{22}$Ne($\\alpha$,n)$^{25}$Mg~ reaction. The $^{22}$Ne is provided by\nthe \n$^{14}$N($\\alpha$,$\\gamma$)$^{18}$F($\\beta^+\\nu$)$^{18}$O~($\\alpha$,$\\gamma$)$^{22}$Ne~ chain starting from $^{14}$N present in the ashes of H burning.\nSome extra $^{14}$N derives from primary $^{12}$C that is dredged up into the\nenvelope and partly converted to $^{14}$N by H-shell burning. The neutron\nexposure provided by the $^{22}$Ne neutron source during the thermal \npulse is low, reaching at most\n$\\delta \\tau_2 (23 {\\rm keV}) =$ 0.03 mbarn$^{-1}$.\nHowever, the peak neutron density can reach 10$^{10}$ n\/cm$^{3}$.\n\nExposure to the two neutron fluxes is repeated through the \npulses with TDU. \nThe $^{13}$C-pocket features are kept constant pulse after pulse, \nwhile the small neutron exposure from the $^{22}$Ne neutron source \nduring thermal pulses increases \nwith pulse number, reflecting the slight increase of the strength of the \nthermal instability with core mass.\n\nWe have performed new calculations for the nucleosynthesis\ndue to neutron capture in AGB stars. \nThese calculations are based on the\nstellar models and the nuclear network described in detail by Gallino\net al. (1998a). We want to follow here in particular the modifications of \nthe Si and Ti isotope abundances arising by neutron capture \nin the intershell region during the neutron fluences in the \n$^{13}$C~ pocket and in the thermal pulses. Then we will follow the isotopic \ncompositions of Si and Ti in the envelope as they are modified \nduring the whole TP-AGB phase by the mixing of He intershell material \ndue to TDU episodes. The envelope itself is progressively eroded by stellar\nwinds and by the growth of the H-burning shell.\n\nIn Table 1 the Maxwellian averaged neutron capture cross sections (in the\nform of $\\sigma_{code}$=$\\left< \\sigma v \\right>\/v_{th}(30 {\\rm\nkeV})$, expressed in mbarn) are listed for selected isotopes. \nValues are reported for the two typical temperatures \n(8 keV and 23 keV) at which neutrons are released by the $^{13}$C~ and by the\n$^{22}$Ne~ neutron source, and for the standard temperature of 30 keV at \nwhich these cross sections are currently given in the literature. The\nquoted values have been \ntaken from the compilation by Beer, Voss, \\& Winters (1992), with three \nexceptions: $^{28}$Si, for which a renormalization to the 30 keV \nrecommended value of Bao \\& K\\\"appeler (1987) has been considered \n(Beer 1992, private communication), $^{150}$Sm, from Wisshak et al. (1993), \nand the $^{33}$S(n,$\\alpha$)$^{30}$Si cross section from Schatz et al. \n(1995). The use of $\\sigma_{code}$ demonstrates how the cross section\nat any given energy departs from the usual $1\/v$ rule. For a perfect \n$1\/v$ dependence, $\\sigma_{code}$ at different $kT$ values should remain\nconstant. In reality, with the exception of $^{49}$Ti, strong \ndepartures from this rule are shown by all Si \nand Ti isotopes. The $^{30}$Si abundance \nresulting from a given neutron exposure is strongly dependent on the\nreaction rate of $^{32}$S(n,$\\gamma$)$^{33}$S, where $^{33}$S is subsequently quickly transformed to $^{30}$Si via $^{33}$S(n,$\\alpha$)$^{30}$Si.\n\nThe neutron capture cross sections of two typical \nheavy $s$-only nuclei, $^{100}$Ru and $^{150}$Sm, have also been included\nin Table 1  \nin order to show that Si and Ti (this is also true for all \nother elements lighter that Fe) are not as much affected by neutron \ncapture as the heavier elements. The neutron capture cross sections of \nlight elements are much smaller (by as much as three \norders of magnitude) than \nthose of typical heavy isotopes. However, because of their large \ninitial abundances, isotopes lighter than $^{56}$Fe act as important \nneutron poisons for the build-up of the heavy elements. \nIn the last column, the relative uncertainties of the 30 keV cross sections are \nreported. The cross sections of all Si and Ti isotopes still \nsuffer from large uncertainties, around 10\\%, \nwhereas for many heavy isotopes recent experiments have achieved a \nprecision of the order of 1\\% (see K\\\"appeler 1999). \n\nTable 2 shows the production factors with respect to solar \nof the Si and Ti isotopes, as well as that of two \n$s$-only nuclei $^{100}$Ru and $^{150}$Sm, \nin the He intershell at different phases of the 15$^{\\rm th}$ thermal \npulse for an AGB star of 1.5 $M_{\\odot}$~ and solar metallicity, with the \nstandard choice of the $^{13}$C~ pocket (case ST of Gallino et al 1998a). \nColumns 2 and 3 show the effect of the $^{13}$C~ neutron source on the Si \nand Ti isotopic abundances inside the $^{13}$C~ pocket. The values of\ncolumn 4 were calculated at the time when the $s$-enriched pocket\nhas been engulfed by the growing convective pulse and diluted\nwith both $s$-processed material from\nthe previous pulse and material from the H-burning ashes, containing in\nparticular Si and Ti of initial composition. \nThe difference between the values of columns 4 and 5 \nexpresses the effect of the $^{22}$Ne neutron source activated during the\n15$^{\\rm th}$ pulse. Note how $^{100}$Ru and $^{150}$Sm production factors are up \nto three orders of magnitude larger than those of Si and Ti. \nFrom the results given in Table 2, it is easily recognized that \nneutron captures only marginally modify the initial Si and Ti \nisotopic compositions. $^{28}$Si tends to be slightly consumed (by \n15 $^o\\!\/\\!_{oo}$, and by 20 $^o\\!\/\\!_{oo}$, respectively) after both neutron exposures. \nActually, during \nthe high neutron exposure from the $^{13}$C~ source $^{29}$Si is more \nefficiently consumed (by a factor 1.6) than produced, because of the \nvery low cross section of $^{28}$Si. Also $^{30}$Si is consumed (by \na factor 2.2) during this phase. In contrast, the abundances of both \nneutron-rich Si isotopes grow during the thermal pulse, by factors\nof about 1.3 and 1.4, respectively, relative \nto their initial values in the convective \npulse. These features are mainly due to the fact that the neutron \ncapture cross sections strongly depart from the  1\/$v$ trend. Note \nthat $\\sigma_{code}(^{28}$Si) is almost an order of magnitude greater at\n23 keV than at 8 keV, which explains why in the TP phase this isotope is \ndestroyed to a larger extent than in the $^{13}$C~ pocket. \nAs a consequence, we observe the \ngrowth of $^{29}$Si during the TP. \n\nAmong the Ti isotopes, $^{50}$Ti is a neutron magic nucleus ($N$ $=$\n28) and its neutron capture cross section is very small compared to \nthose of the other Ti isotopes. It shows a strong departure from the \n1\/$v$ trend (see Table 1), being a factor of 4 greater at 23 keV than \nat 8 keV. As shown in Table 2, $^{50}$Ti accumulates during the \n$^{13}$C~ neutron exposure because of its very low neutron capture cross \nsection, which makes this isotope a bottleneck of the abundance flow. \n$^{49}$Ti is produced in both phases, by a larger factor during the \npulse, while $^{48}$Ti is consumed. Note that during the high neutron \nexposure by the $^{13}$C~ neutron source $^{48}$Ti is only marginally \nmodified, despite its relatively large cross section. This results \nfrom abundance flow starting at $^{40}$Ca, an isotope of large initial \nabundance and of cross section $\\sigma_{code}$(8 keV) $=$ 6.11 mbarn, \nwhich is consequently consumed (by a factor $\\approx$ 6) in this phase. \nThe $^{46}$Ti, $^{47}$Ti and $^{48}$Ti isotopes, similarly to the \nSi isotopes, suffer almost negligible variations. \n\nBecause of their relatively small cross sections, a behavior similar to \nthat of Si and Ti is shown by other light elements below Fe, among \nthem S and Ca. It should be emphasized that the final \nSi isotope composition mostly depends on the small neutron exposure by \nthe $^{22}$Ne neutron source in the convective pulse rather than on \nthe very large neutron exposure by the $^{13}$C~ neutron source taking \nplace in the tiny radiative $^{13}$C~ pocket.  \n\nTable 2, column 6 shows the production factors in the envelope \nimmediately after the TDU that follows the\nquenching of the 15$^{\\rm th}$ thermal pulse. At this stage, the star has\nbecome a C star, with C\/O $=$ 1.3, and the isotopic composition of the\nenvelope results from the mixing of the $s$-processed and $^{12}$C-enriched \nmaterial cumulatively carried into the envelope by previous TDU episodes.\n\nPredictions for the Si and Ti isotopic compositions in the envelope of AGB\nstars of solar metallicity and initial mass of 1.5 and 3 $M_{\\odot}$~ during\nrepeated TDUs in the TP phase are shown in \nFig. 5. Fig. 6 reports predictions for the resulting Ti vs. Si correlation:\nas in Fig. 4, Ti ratios are plotted as function of the\n$^{29}$Si\/$^{28}$Si ratio. They are all reported in the form of \n$\\delta$-values for three different choices of the amount of $^{13}$C~ in the \nHe intershell. The standard case (ST) of Gallino et al. (1998a) \ncorresponds to an average mass fraction of $^{13}$C of 6 $\\times$ \n10$^{-3}$ distributed over a tiny layer of a few 10$^{-4}$ $M_{\\odot}$~ at the top \nof the He intershell, case d3 corresponds to the amount of case ST \ndivided by 3 and case u2 is an upper limit corresponding to the amount \nof case ST multiplied by 2. As already mentioned in Busso et al. (1999a) \n(see also Busso, Gallino, \\& Wasserburg 1999b), a spread in the $^{13}$C \namount in stars of different metallicities\nis required by spectroscopic observations of  \n$s$-enhanced stars, and conceivably depends on the initial stellar mass or \nother physical characteristics (such as stellar rotation).\nThe measurements of Zr, Mo, and Sr isotopic ratios in individual SiC \ngrains have confirmed this spread in the $^{13}$C~ amount: all single \ngrain compositions can be matched by low-mass AGB models of about solar \nmetallicity if we consider different amounts of $^{13}$C~ (Gallino et al. \n1998b; Nicolussi et al. 1998b). Open symbols are for \nenvelopes with C\/O$>$1, the condition for SiC condensation. \nNote that case ST for solar metallicity best reproduces the $s$-process\nisotopic distribution of bulk SiC grains, which is slightly different from\nthe solar main component (for a general discussion see Gallino et al. 1997; \nBusso et al. 1999b).\n\nNot surprisingly the $^{29}$Si\/$^{28}$Si, $^{30}$Si\/$^{28}$Si, and \n$^{47}$Ti\/$^{48}$Ti ratios are only a few percent (up to 25 $^o\\!\/\\!_{oo}$, \n40 $^o\\!\/\\!_{oo}$, and \n14 $^o\\!\/\\!_{oo}$, respectively) higher than the corresponding solar ratios. The \n$^{46}$Ti\/$^{48}$Ti and $^{49}$Ti\/$^{48}$Ti ratios are \nup to 70 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ higher than the solar ratios. \nIn agreement with the results shown in Table 2, the\nonly ratio that is affected to a significant extent (up to 500 $^o\\!\/\\!_{oo}$~\nhigher than solar)\nis the $^{50}$Ti\/$^{48}$Ti ratio. Note that the \n$\\delta^{50}$Ti$\/^{48}$Ti values range from $+$100 $^o\\!\/\\!_{oo}$~ to $+$500 \n$^o\\!\/\\!_{oo}$, depending on the $^{13}$C~ amount. $^{50}$Ti is a\nmagic nucleus whose abundance is very sensitive to \nthe high neutron exposure in the $^{13}$C~ pocket. \nThe fact that the $^{50}$Ti\/$^{48}$Ti ratio is significantly changed during \nthe AGB phase is, in a way, consistent with the $^{50}$Ti\/$^{48}$Ti ratios \nmeasured in SiC grains. Model predictions do not reproduce the spread of the \nmeasured Si and Ti compositions, nor could they ever explain the \nnegative $\\delta$-values measured in some grains; \nthe calculated $^{50}$Ti\/$^{48}$Ti ratio, \nthough, reaches $\\delta$-values that are higher than those of \nall the other Si and Ti $\\delta$-values \nboth in AGB model predictions and in single \nSiC grain measurements (up to 300 $^o\\!\/\\!_{oo}$, \nsee Fig. 4). \n\nWe also investigated two other TP-AGB models: a case of $M=5$ $M_{\\odot}$~ of\nsolar metallicity (Fig. 7) and a case of $M=3$ $M_{\\odot}$~ of 1\/3 solar\nmetallicity (Fig. 8). An interesting feature is common to both models:\nthe maximum temperature at the bottom of the He convective shell is\nsomewhat higher than in the models described above. As a consequence, \nthe production factors for $^{29}$Si and\n$^{30}$Si, whose production is most sensitive to the $^{22}$Ne~ neutron \nsource (see Table 2), at the end of the 15$^{\\rm th}$ pulse reach 3.2 and \n5.9, respectively, for the $M=5$ $M_{\\odot}$~ star of solar metallicity, and 3.1 \nand 6.5, \nrespectively, for the $M$ $=$ 3 $M_{\\odot}$~ star of $Z$ $=$ $Z_{\\odot}$\/3. \n\nThis results in an increase of up to 80 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ in  \n$\\delta^{29}$Si$\/^{28}$Si and $\\delta^{30}$Si$\/^{28}$Si, \nrespectively, in the envelope of the $M=5$ $M_{\\odot}$~ model (Fig. 7), \nand of up to 100 $^o\\!\/\\!_{oo}$~ and 200 $^o\\!\/\\!_{oo}$~ in the \n1\/3 $Z_{\\odot}$ model (Fig. 8). The largest $^{29}$Si and $^{30}$Si excesses\nmeasured in SiC are reproduced, however with a slope of about 0.5\nfor the mixing line, whereas the slope of the mainstream correlation line in \nthe Si 3-isotope plot is 1.31 (Fig. 4). \nNote the extremely high values \n(up to 2000 $^o\\!\/\\!_{oo}$) reached by $\\delta^{50}$Ti$\/^{48}$Ti in the last \ncase (Fig. 8). They\nresult from the fact that, in our AGB model, the neutron exposure in the\n$^{13}$C~ pocket is very sensitive to metallicity: it grows with\ndecreasing metallicity (see Gallino et al. 1999).\n\nAs for all the cases above, the initial isotopic composition of the star \nhas been assumed to be solar, including  the $Z = Z_{\\odot}\/3$ case. \nIn principle, some enhancement for \nisotopes produced by $\\alpha$ captures (such as $^{16}$O, $^{20}$Ne, \n$^{24}$Mg, $^{28}$Si, $^{40}$Ca and $^{48}$Ti) as well as complex \nsecondary-like trends of many other nuclei should be taken into account \nin the initial composition of low-metallicity stars. This is a tricky point, \nfor these variations have to be deduced from GCE models together with \nspectroscopic observations and are, in many cases, not well defined. \nAn exercise of this kind, in connection with a possible explanation for \nthe Si isotopic composition of SiC of type Z, can be found in \nHoppe et al. (1997). For the AGB model of\n$Z$ $=$ $Z_{\\odot}$\/3, we made some tests by assuming a small enhancement of the \ninitial $^{28}$Si and $^{32}$S, as well as of other $\\alpha$-rich \nisotopes according to the spectroscopic evidence by\nEdvardsson et al. (1993), and  small depletions in the initial\nabundance of  the secondary-like isotopes $^{29,30}$Si. It turned out that\nthe resulting Si isotope composition in the He intershell as a consequence \nof neutron captures was quite insensitive to the above variations, being\ndominated by the most abundant $^{28}$Si. \n\nIt has to be remarked here that several features of the predicted Si and\nTi ratios (e.g., the slope in the Si 3-isotope plot) depend on the\nneutron capture cross sections which, for the Si as well as the\nTi isotopes, are still quite uncertain, as shown in\nthe last column of Table 1. New measurements are highly desirable for \nobtaining the best possible AGB model predictions. \n\n\\section{The stellar sources of Si and Galactic heterogeneity}\n\n\\subsection{Supernova sources}\n\nAs Timmes \\& Clayton (1996) have pointed out, SNe of Type II are \nthe dominant sources of Si in the Galaxy, especially in its early \nstages. At later Galactic times, SNe of Type Ia also contribute $^{28}$Si. \nThe SNII models of WW95 show that $^{28}$Si is a primary isotope \nwhereas $^{29}$Si and $^{30}$Si are predominantly secondary isotopes (see also \nTimmes \\& Clayton 1996 for details). This means that $^{28}$Si \ncan be synthesized in early Type II SNe from a pure H and He \ncomposition, whereas the production of $^{29}$Si and $^{30}$Si \nrequires the prior presence of primary isotopes such as $^{12}$C, \n$^{14}$N and $^{16}$O. As a \nconsequence, the $^{29,30}$Si\/$^{28}$Si ratios of the ejecta of \nSNIIe increase with the metallicity of the stars. While $^{28}$Si \nis the product of explosive O burning, both $^{29}$Si and $^{30}$Si \nare synthesized in a narrow region by explosive Ne burning. \nActually, $^{29}$Si production is restricted to the outer region \nof the Ne burning shell.\n\nFig. 9 shows the $^{29,30}$Si\/$^{28}$Si ratios (plotted as \n$\\delta$-values) of the averages of the yields of SNII models of \ndifferent metallicities. The cases of metallicity \n$Z$  = 0.1 $Z_{\\odot}$ and $Z$ = $Z_{\\odot}$ are taken from \nWW95, the cases $Z$ = 0.5 $Z_{\\odot}$ and \n$Z$ = 2 $Z_{\\odot}$ are from more recent, unpublished \ncalculations by Weaver \\& Woosley. To obtain the averages we took the \ninitial mass function for massive stars into account by weighing the \ncontributions from SNIIe of different masses according to $M^{-2.35}$ \nper unit mass interval, the Salpeter initial mass function. Fig. 9\nshows that in the WW95 models there exists a fairly good linear \nrelationship between the $^{29,30}$Si\/$^{28}$Si ratios of the \naverage SN yields and the metallicity, demonstrating the secondary \nnature of the heavy Si isotopes. \nThe GCE of the Si isotopes is thus believed to have progressed from small \n$^{29,30}$Si\/$^{28}$Si ratios at early Galactic times to larger and \nlarger ratios as the metallicity of the whole Galaxy increased and \nSNIIe of increasing metallicity contributed their Si to the ISM \n(Timmes \\& Clayton 1996).\n\nThis process is expected to have resulted in the Si isotopic ratios at the \ntime and place of solar formation. However, closer inspection of Fig. \n9 and especially Fig. 10, where the $\\delta$-values of the\n$^{29,30}$Si\/$^{28}$Si ratios of the averages of SNII yields are \nplotted in a Si 3-isotope plot, reveals that the SNII models by Weaver \n\\& Woosley do not exactly produce the solar Si isotopic composition. \nIt is evident that $^{29}$Si in the presently available \nmodels is under-produced and the isotopic evolution expected from SNII \ncontributions misses the solar isotopic composition (Fig. 10). This \nis a long-recognized problem: Type II SN models under-produce $^{29}$Si \nrelative to $^{30}$Si as compared to the solar isotopic ratio (Timmes \net al. 1995; Timmes \\& Clayton 1996; Thielemann, Nomoto, \\& Hashimoto \n1996; Nomoto et al. 1997). \nThis fact is also demonstrated by a comparison of model predictions and \nthe Si isotopic ratios of type X SiC, Si$_3$N$_4$, and low-density \ngraphite grains, all of which are believed to originate from Type II SNe  \n(Nittler et al. 1995; Travaglio et al. 1999). The Si isotopic ratios \nof these grains have systematically higher $^{29}$Si\/$^{30}$Si ratios \nthan those predicted by SN models (Zinner et al. 1998;  Travaglio et al. \n1999). In order to achieve the solar ratios, Timmes \\& Clayton  (1996) \nproposed multiplying the $^{29}$Si yields of SNII models by a\nfactor \nof $\\sim$ 1.5. We will do likewise in this paper and multiply the \n$^{29}$Si yields by the same factor to obtain the best fit to the \nsolar isotopic ratios or to the grain data.\n\nIt should be mentioned that there still exist major problems associated \nwith the synthesis of the Si isotopes in massive stars. \nAs Arnett \\& Bazan (1997) pointed out, heterogeneous mixing \nbetween different layers during the late evolutionary stages \nmight have a major effect on the nucleosynthesis of \ncertain elements. Bazan \\& Arnett (1998) used a two-dimensional \nhydrodynamic code to investigate convective O-shell burning in a 20  \n$M_{\\odot}$~ star. They concluded that the results of these calculations differ \nin many ways from those of one-dimensional models and that corresponding \nchanges in the nucleosynthesis of Si during this stage are to be \nexpected. It remains to be seen whether full nucleosynthetic calculations \nin two- or three-dimensional models can shed light on the problem of \nrelative yields of the Si isotopes. Another problem is the relative \ncontribution of Type Ia and Type II SNe to the GCE of the heavy elements, \nin particular Fe. Whereas in the Timmes et al. (1995) GCE model Type II \nSNe were assumed to contribute 2\/3 of the Fe in the solar system, Woosley \net al. (1997) favored a more important role of Type Ia SNe, letting them \ncontribute as much as half of the solar Fe. This would indicate somewhat \nhigher contributions by Type Ia SNe to the Galactic $^{28}$Si relative\nto SNIIe. In addition to possible uncertainties in the nuclear physics \nand in the treatment of the various convective zones affecting the \nproduction of the three Si isotopes, problems are related to the \neffect of mass loss from the most massive stars and to Galactic \nenrichment by close binary massive stars (Woosley, Langer, \\& Weaver \n1993, 1995), to the effect of rotation (Heger, Langer, \\& Woosley 1999), \nand to the still uncertain development of the explosion (WW95, \nThielemann et al. 1996).\n\nWhile in Fig. 9 only averages of SNII models of different metallicities \nhave been plotted, it has to be realized that SN models of different masses \nyield very different Si isotopic ratios. In Fig. 10, in addition to\naverages, we also plotted the isotopic ratios of individual SNII models \nof different masses for the $Z$ = 0.1 $Z_{\\odot}$ and the \n$Z$ = $Z_{\\odot}$ case. The yields and Si isotopic ratios for the\n$Z_{\\odot}$ case are also given in Table 3. As can be seen, the isotopic \nratios of different mass SNIIe span a wide range. There are variations \nnot only in the $^{29,30}$Si\/$^{28}$Si ratios but also in the\n$^{29}$Si\/$^{30}$Si ratio. The last column in Table 3 shows the latter \nratio (already readjusted by augmenting the $^{29}$Si yield) for SNIIe of different\nmass. Note that the ratio is smaller than unity for most SNIIe of lower mass \nbut larger than unity for the two most massive models. In Fig. 11 we plotted \nagain the average Si isotopic ratios for the $Z_{\\odot}$ case together with\nthe averages for the SN models with masses $M$ $\\leq$ 25 $M_{\\odot}$~ and 30 $M_{\\odot}$~ \n$\\leq$ $M$ $\\leq$ 40 $M_{\\odot}$. This time the theoretical $^{29}$Si yield was\nincreased \nby the same factor 1.5 for Type II SNe of all masses in such a way that the \nweighted average ratios plot on the slope-one line or, in other words, \nthat the average $^{29}$Si\/$^{30}$Si ratio is solar. As can be seen, the\nlow-mass average falls slightly below the slope-one line through the origin \n(pure $^{28}$Si) and the solar isotopic composition and the high-mass \naverage falls above this line.\n\nWhile the evolution of the Si isotopes of the Galaxy as a whole and of \nthe average of material in an annulus of a given Galactic radius \nundoubtedly followed the slope-one line, we expect certain variations \nin the Si isotopic ratios even at a given time and a given Galactic radius \nin relatively small regions from which low-mass stars formed. The reason \nis that the addition of contributions from individual SNe, which are \nresponsible for the Si isotopic ratios of a certain region, is a stochastic \nprocess and we do not expect that material from these contributions is \ninstantly homogenized with preexisting material. Fluctuations result \nfrom the fact that individual SN sources have yields with very different \nSi isotopic compositions as is clearly shown in Figs. 10 and 11. In \naddition to the isotopic ratios of Type II SNe, in Figs. 10 and 11 \nas well as in Table 3 we show also the ratios of the W7 SNIa model \n(Thielemann et al. 1986; updated by Nomoto et al. 1997) and the SNIa model \noriginating from sub-Chandrasekhar (hereafter sub-Ch) white dwarfs accreting \nHe from \na binary companion (Woosley \\& Weaver 1994). These SN types are believed \nto be the major sources of Si in the Galaxy around the time of solar \nsystem formation (Timmes \\& Clayton 1996; Woosley et al. 1997). \n\nLet us now consider the effect on the Si isotopic ratios of the \nadmixture of material from one of these SN sources to material with a \ngiven isotopic composition (Fig. 11). Three-isotope plots such as those \nshown in Figs. 10 and 11 have the property that the isotopic composition \nof a mixture between two components lies on a straight line connecting \nthe isotopic ratios of the two components. For the sake of demonstration \nwe arbitrarily selected as starting composition the Si isotopic \ncomposition of the sun. Admixture of different SN sources (we chose  \nSNIa W7, SNIa sub-Ch, and, again for the sake of demonstration, the low- and high-mass averages of the \n$Z_{\\odot}$ Type II SN models of WW95) will shift the starting \ncomposition in the directions of the arrows in the figure (see also Figs. \n8 of Timmes \\& Clayton 1996 for mixtures between average ISM and \nejecta from individual SNe. These authors also mentioned the possibility \nof reproducing a larger-than-unity slope for the Si isotopic ratios \nof the mainstream SiC grains but did not systematically develop the \nlocal heterogeneity picture as it is done in this work). Thus \nadmixture of material from a SNIa sub-Ch will shift the \ncomposition toward the origin (pure $^{28}$Si). We note that the average of\nthe high-mass SNII sources (with adjusted $^{29}$Si yield) lies above the\nslope-one line from the origin through the solar composition. This means \nthat admixture of material from these sources will shift, on average, \nthe original solar composition along a line with a slope larger than one. \nLikewise, because the average of the low-mass sources lies below the \nslope-one line, admixture from these sources will again, on average, \nresult in a shift along a line with a slope larger than one. The same \nis true, even though to a lesser extent, if material from the Type Ia \nW7 SN model (containing essentially pure $^{28}$Si) is added to the solar \ncomposition.\n\n\\subsection{Monte Carlo calculations}\n\nIf contributions from a limited number of such sources are \nconsidered, the resulting Si isotopic compositions will fluctuate from \none mix to the next because of the statistical nature of these \ncontributions. We have developed a simplified Monte Carlo (MC) model in \nwhich we add material from a limited number of discrete SN sources \nin a statistical way to material with an arbitrary (but reasonable) \nstarting isotopic composition in order to see whether the Si isotopic \ndistribution of the mainstream SiC grains can be explained as the \nresult of statistical fluctuations or, in other words, local \nheterogeneities in the regions where low-mass stars - as AGB stars \nthe sources of mainstream grain - were born. A detailed description of \nour MC model can be found in the Appendix.\n\nWe have performed different calculations with different assumed starting \ncompositions. As first test we took the average Si isotopic ratios of the \nmainstream SiC grains corrected for AGB contributions \n($\\delta$$^{29}$Si\/$^{28}$Si$_{mean}$ = 30.4 $^o\\!\/\\!_{oo}$~ and\n$\\delta$$^{30}$Si\/$^{28}$Si$_{mean}$ = 27 $^o\\!\/\\!_{oo}$~ - see Appendix) as starting \ncomposition. In other words, we considered the mean composition of the \nmainstream grains' parent stars as a possible ``standard'' composition \nof the ISM from which these stars were born. We then randomly added $N$~ \nSN contributions and randomly chose the sign of each contribution (i.e., \nthe sign of the parameter $a$, the constant factor by \nwhich the total mass ejected by each SN is multiplied) in order \nto simulate material that could \nhave seen more or less from each kind of SN source relative to the \nchosen standard ISM composition. In this way we generated 200 different \nmixtures, whose isotopic ratios are plotted Fig. 12a. The plot shows the \ncase for $N$~ = 100. \nFor $a$, the fraction taken from each SN \nsource, we obtained $a$ = 1.5 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$. Note that\nbecause \nwe computed abundances for each isotope $i$ in the form of mass fractions \n$X_i$ (see Appendix), the contributing terms $a \\times M_{ejected}$ do \nnot have a \ndimension and the parameter $a$ has the dimension of the inverse of a \nmass. As explained in the Appendix, \nbecause of the limited number (200) of cases, a certain range of the parameter \n$N$~ is expected to yield a good fit. Other similarly good matches\nare obtained for values of $N$~ ranging between $\\sim$ 50 and $\\sim$ 200, \nand $a$ accordingly from $\\sim$ 1.9 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ \nand $\\sim$ 0.95 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$. In Fig. 12a we also took \nthe modification of the Si isotopes by \nnucleosynthesis in the AGB parent stars into account, adding the average \nisotopic shift given above to the results of the Monte Carlo calculation. \nAs can be seen from the figure, the 200 different mixtures generated with \nthese parameters by MC in a random fashion match the distribution of the \ngrains surprisingly well. We note that the slope of the correlation line \nof the MC points is larger than unity but somewhat smaller than the slope \nof 1.31 of the mainstream correlation line. The slope of the MC \ncompositions reflects the distribution of the SN sources, mostly the \nSNII sources of solar metallicity. As has been already pointed out above \nin the discussion of Fig. 11, these sources are aligned with an average \nslope that is greater than one.\n\nIf we choose starting compositions different from the average of the mainstream \nSiC grains, it turns out that for a wide \nrange of starting Si isotopic ratios, as long as they are constrained to \nbe compositions expected for the Galactic evolution of the Si isotopes \n(i.e., compositions that in Si 3-isotope plots such as those in Figs. 10 \nand 11 lie on the slope-one line between the origin representing pure \n$^{28}$Si and the solar isotopic composition), values for the parameters \n$N$~ and $a$ can be found that let us achieve a good match with the Si \nisotope distribution of the mainstream SiC grains, albeit with different \nchoices of the parameters $N$~ and $a$ for each case. \nWe investigated three more cases for which we chose $a$ always to be \npositive. The results are shown in Figs. 12b, 12c, and 12d. In the \nfirst of these cases the starting isotopic composition is solar, in \nthe second case Si is depleted in the heavy isotopes by 100 $^o\\!\/\\!_{oo}$~ \n($\\delta$$^{29}$Si\/$^{28}$Si$_{init}$ = $-$ 100 $^o\\!\/\\!_{oo}$~ and\n$\\delta$$^{30}$Si\/$^{28}$Si$_{init}$ = $-$ 100 $^o\\!\/\\!_{oo}$) and \nin the third by 200 $^o\\!\/\\!_{oo}$. The best-fit parameters are $N$~ = 70 and \n$a$ = 1.7 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ for the first case, \n$N$~ = 420 and $a$ = 1.1 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ for the second case,\nand \n$N$~ = 600 and $a$ = 1.5 $\\times 10^{-5}$ $M_{\\odot}$$^{-1}$ \nfor the third case. Also in these cases, we obtain good fits for a range \nof $N$~ and $a$ values.\n\nIt is clear that the addition of SN material will change the concentration \nof other elements as well. As the statistical nature of these additions \nresults in a range of Si isotopic ratios, we expect it to result in a \ncorresponding range of elemental ratios as well and these variations can \nbe compared with astronomical observations in stars. In Fig. 13a we\nplotted the scatter in elemental ratios obtained by the MC calculation \nfor the case with $\\delta$$^{29}$Si\/$^{28}$Si$_{init}$ =\n$\\delta$$^{30}$Si\/$^{28}$Si$_{init}$ = 0. In different Galactic \nregions elemental ratios relative to H are expected to be affected by newly \ninfalling gas and not only by the contributions from stellar nucleosynthesis. \nFor this reason we plotted ratios relative to Fe and normalized to the\nsolar system abundances (i.e., [Elem\/Fe] = \nlog[(Elem\/Fe)\/(Elem\/Fe)$_{\\odot}$]). The spreads in the theoretical \nelemental ratios are quite modest, especially if compared with ratios \nobserved in stars. Edvardsson et al. (1993) measured elemental abundances \nin a large number of stars from the Galaxy and concluded that stars \nfrom a given epoch (i.e. of a given age) and from a given Galactic \nradius show a considerable spread in metallicity. That this spread is \nnot simply the result of variations in the amount of newly infalling \nmaterial is shown by the fact that also elemental ratios between elements, \nin particular relative to Fe, show considerable variations (Fig. 13b). \nEdvardsson et al. (1993) pointed out that, despite observational errors \n(including a typical uncertainty of 1 - 2 Gyr in the age of dwarf stars), \nthese variations are intrinsic (see also Timmes et al. 1995). \nA comparison of Figs. 13a and 13b shows that the spread in elemental \nratios obtained from a model of statistical fluctuations in the \ncontributions from various SN sources is smaller than the spread \nobserved in stars. Since we do not know exactly how much of the spread \nin the Edvardsson et al. (1993) data is due to experimental errors, we \njust want to emphasize that the MC spread is not larger than that in \nstars. The models by Copi (1997) and van den Hoek \\& de Jong (1997), which \nmake use of stochastic approaches in the study of GCE, are able to \naccount for these elemental spreads. The mass of a well-mixed \nregion (a sort of mixing scale) that in the Copi (1997) model \nyields a good fit to the spread of the abundances \nof $\\alpha$-elements (such as Si) in stars \nis $M \\sim 10^5$ $M_{\\odot}$. If we interpret our constant \nparameter $a$ as the inverse of the total mass of the region in which the \nSN ejecta are expected to be well mixed, we find for this region a mass of $M \n= 1\/a \\sim 10^5$ $M_{\\odot}$, a number remarkably similar to that found by Copi (1997).\n\nWe conclude that local heterogeneities in Galactic regions that \ncan explain the variations in Si isotopic ratios observed in the \nmainstream SiC grains imply variations in elemental ratios that are \ncompatible with those observed in stars. In principle, \nsuch heterogeneities could be the cause of the mainstream isotopic \nvariations.\n\n\\subsection{Discussion}\n\nIt has to be emphasized that our model for explaining the Si isotopic\nvariations in mainstream SiC grains does not pretend to fully\nsimulate the isotopic compositions of the SiC mainstream grains. \nIt uses theoretical yields of the Si isotopes ejected from SNe that \nneeded adjustment to explain the composition of the solar system (see\nsection 4.1). In addition, the model is overly simplistic and at \nthis point should only be understood as a demonstration that local \nheterogeneities due to the statistical nature of SN contributions \ncan in principle successfully reproduce these variations. In reality \nthe situation is expected to be much more complicated:\n\n\\begin{itemize}\n\n\\item {1) There will be a statistical spread in the individual SN \ncontributions;}\n\\item {2) There will be a spread in the initial composition;} \n\\item {3) There will be contributions from SNe with a range of \nmetallicities;} \n\\item {4) There will be a range in ages of the AGB stars because \nof differences in their mass.}\n\n\\end{itemize}\n\nWe will discuss these points in turn. \n\n\\begin{itemize}\n\n\\item {1) We assumed that all the SN sources that add material to a given \nGalactic region contribute the same amount as expressed by a single \nvalue of the parameter $a$. In reality different SNe will contribute \ndifferent amounts and in some extreme cases one SN will completely \ndominate the local mix. In our MC calculations we have also assumed \ndifferent statistical distributions for the parameter $a$ and could \nachieve essentially the same final results as those shown in Figs. 12.}\n\n\\item {2) We have shown that different initial Si isotopic compositions can \nproduce distributions close to that of the mainstream grains if the \nparameters for the admixture of SN material (number of SN sources,  \n$N$, and fraction of Si ejected by a SN, $a$) are chosen appropriately. \nIn reality we have to expect a whole range of initial compositions \nreflecting different times and different degrees of homogenization \nof matter in the Galaxy. We expect that material at a given Galactic \nradius is homogenized on a time scale of less than 10$^8$ yr, the period \nof Galactic rotation. Any complete homogenization will destroy the \nlocal heterogeneities in which we are interested. The real local \nisotopic compositions will represent some balance between heterogeneity \nand processes of homogenization. The overall result will be the GCE of \nthe elements and the isotopes, the overall trend being modified by \nlocal fluctuations.}\n\n\\item {3) In our model we have considered only Type II SNe of solar \nmetallicity. In reality there will be a range of metallicities. \nThis is for two reasons. First, we expect to encounter some range \nin age for AGB stars as will be discussed in the next section. \nSecond, if local regions are highly contaminated with previous \nSN contributions, new SNe from such regions will have higher-than-average \nmetallicities. According to the WW95 SNII models the addition of  \nSNIIe ejecta of a given metallicity to an ISM parcel of the same \nmetallicity will result in higher $^{29}$Si\/$^{28}$Si and \n$^{30}$Si\/$^{28}$Si ratios than those of the starting material. This \nreflects the fact that $^{29}$Si and $^{30}$Si are secondary \nisotopes. The enrichment of the heavy Si isotopes in Type II SN \nejecta over the average ISM material has first been pointed out by \nClayton (1988) on the basis of an idealized GCE model. The enhancement of $^{29}$Si and $^{30}$Si in the SNII \nejecta over the starting composition is clearly seen in Fig. 11 for \nSNIIe of solar metallicity, where the average value plots to the upper right \nof the solar Si isotopic composition (this includes an assumed \nenhanced production of $^{29}$Si). However, a minimum metallicity is \nrequired for the contributing SNIIe in order to achieve the average Si isotopic \nratios of the protosolar nebula or those of the mainstream grains. We \nconclude from Fig. 9 that a metallicity of $Z$ $>$ 0.75 $Z_{\\odot}$ \nis required for the average Si isotopic composition of SNII \nejecta to be heavier than 150 $^o\\!\/\\!_{oo}$, the maximum of the \nmainstream grains. This, \nhowever, is a lower limit since in reality low-mass \nstars do not form from pure SN ejecta.}\n\n\\item {4) It has to be clear that the Si isotopic compositions of the \nmainstream grains reflect those of their parent stars at the time of their \nbirth and it is these compositions that we want to explain. However, it is \nalso clear that, depending on their mass, different stars were born at \ndifferent Galactic times, even if they all produced SiC grains at \nthe same time during their AGB phase (and even this last assumption is \nnot strictly valid because different SiC grains could have different IS \nlife times between their formation and the birth of the solar system). \nSo far we have made the implicit assumption that grains came only from \nstars in the 1.5 - 3 $M_{\\odot}$~ range when we computed the inferred Si isotopic \nratios of the mainstream grains without any AGB contributions (i.e., the \ninitial isotopic ratios of the parent stars). In the following subsection\nwe want to explore \nthis question in more detail.}\n\n\\end{itemize}\n\nThe processes leading to heterogeneity in the Si isotopic ratios \nin the ISM are much more complex than the simple mixing assumed in \nour MC model. However, and this is the most important conclusion of \nour tests, it is practically certain that the ISM at a given Galactic \ntime and at a given Galactic radius is not characterized by a unique \nSi isotopic composition but by a range of compositions. This \ndistribution of Si ratios will shift toward heavier isotopic \nratios during the evolution of the Galaxy. Note that presently we do not know \nthe exact distribution of the Si isotopes at a given Galactic time nor the \nrelationship between Galactic time and the mean of the Si isotopic ratios \nof the distributions. For the time being we assume that the \nlatter is the same \nas that of the Timmes \\& Clayton (1996) model, but this point will be discussed \nin more detail in \\S 4.5.\n\n\\subsection{The mass of the SiC parent stars}\n\nIn \\S 3 and Figs. 5 - 8 we showed that the shift in Si isotopic \ncompositions due to neutron capture in the He shell of AGB stars depends \non stellar mass and metallicity. As we demonstrated in that section, \nthe changes in the Si isotopic composition due to the $s$-process \nin AGB stars depend almost entirely on the small \nneutron exposure from the $^{22}$Ne source, with the $^{13}$C pocket \nhaving no influence, independent \nof the magnitude of its strength. The effects on stellar mass and \nmetallicity we are discussing in this section actually are the results \nof complete stellar evolutionary calculations of the AGB phases \nusing the FRANEC code. \nAt solar metallicity, from a 5 $M_{\\odot}$~ complete AGB evolutionary model we \nfind a somewhat higher maximum temperature at the bottom of the He thermal \npulses than in lower mass stars (1.5 $M_{\\odot}$~ to 3 $M_{\\odot}$). Because the $\\alpha$-capture reaction rate is proportional to \n$T^{21}$, this higher temperature \nincreases the efficiency of the neutron burst from the $^{22}$Ne \nsource, correspondingly changing the predicted final Si ratios \nas illustrated in the figures.\n\nThe same tendency is found in stellar evolutionary calculations \nwith the FRANEC code for AGB stars of different mass and\na metallicity of 1\/3 $Z_{\\odot}$, as shown in Table 4. The maximum\ntemperature at the bottom of the thermal pulse increases \nslightly from pulse to pulse, starting from about \n2.65 $\\times 10^8$ K for the pulse \nwhen TDU occurs for the first time. The temperature during  the \npulse rises in a very rapid burst; subsequently the bottom \ntemperature decreases more \nor less exponentially from its maximum, with a total duration time \n(at $T >$ 2.5 $\\times 10^8$ K) of a few years.\n\nWhereas for 1.5 $M_{\\odot}$~ and 3 $M_{\\odot}$~ stars of \nsolar metallicity the maximum shifts in $\\delta$$^{30}$Si\/$^{28}$Si are \nonly 26 $^o\\!\/\\!_{oo}$~ and 37 $^o\\!\/\\!_{oo}$, respectively, the maximum shift for a 5 $M_{\\odot}$~ \nstar is 180 $^o\\!\/\\!_{oo}$. In Figs. 14a-d we plotted the results of the MC\ncalculations \nfor the $-$100 $^o\\!\/\\!_{oo}$~ case if we add the shifts expected for AGB stars of \nmasses 1.5 $M_{\\odot}$, 3 $M_{\\odot}$, and 5 $M_{\\odot}$~ with solar metallicity and of 3 $M_{\\odot}$~ \nwith $Z$ = 0.006.  The ranges of shifts were added in a random, \nstatistical fashion in our MC test. As can be seen, only the 1.5 $M_{\\odot}$~ and \n3 $M_{\\odot}$~ stars of solar metallicity give results in reasonable agreement \nwith the grain data, while 5 $M_{\\odot}$~ stars as well as stars with $Z$ = 0.006 \nshift the Si isotopic compositions far to the right of the grain data \nand the solar composition. Especially for the low-metallicity case of \nFig. 14d the predicted shifts have a much wider spread. \nIt is a remarkable result of our heterogeneity model \nthat, without any AGB contributions, the solar composition is one of \nthe possible compositions and at the same time the mainstream data \ncan be reproduced if the AGB shifts are small, as for the 1.5 $M_{\\odot}$~ and \n3 $M_{\\odot}$~ star models of close-to-solar metallicity. This is not true anymore if the AGB shifts are as \nlarge as those for 5 $M_{\\odot}$~ stars.\n\nFrom the above discussion, it is reasonable, even if not proven, to assume \nthat mostly low-mass stars (with $M$ $\\leq$ 3 $M_{\\odot}$) contributed SiC to the solar system. Actually, \nthere are many pieces of evidence that indicate that this is indeed the \ncase:\n\n\\begin{itemize}\n\n\\item {1) Feast (1989) performed a study of the kinematics of peculiar red \ngiants including S, SC, and C stars. On the basis of 427 C stars he \nestimated their mean mass to be 1.6 $M_{\\odot}$. Although this estimate \nneeds to be improved, if SiC grains came from average \nC stars, they came from low-mass stars.}\n\n\\item {2) Another argument for low masses of carbon stars is based on a \ncomparison of the observed luminosities of AGB stars in the Magellanic \nclouds with predicted luminosities. Theory predicts intermediate-mass \nstars of 5 - 8 $M_{\\odot}$~ to have $M_v$ of less than - 6.5 but typical  \nluminosities of S and C stars are much lower, indicating low-mass stars \n(Mould \\& Reid 1987;  Frogel, Mould, \\& Blanco 1990; Van Loon et al. \n1998).}\n\n\\item {3) Another argument is based on theoretical predictions about the \noccurrence of hot bottom burning (HBB) in intermediate-mass (5 - 8 $M_{\\odot}$) \nstars. HBB takes place when the bottom layers of the convective envelope \nare hot enough for some proton capture nucleosynthesis to occur. In this \ncase, most $^{12}$C~ dredged up from the He \nshell during the TP-AGB phase is converted to $^{14}$N, \npreventing the star from becoming a carbon \nstar. There are several theoretical studies that indicate that HBB occurs in\nstars of $\\gtrsim$ 5 $M_{\\odot}$~ of solar metallicity and in stars with \n$\\gtrsim$ 4 $M_{\\odot}$~ of lower metallicity (Boothroyd, Sackmann, \\& Wasserburg \n1995; Forestini \\& Charbonnel 1997; Lattanzio et al. 1997). For solar \nmetallicity stars, the FRANEC code finds HBB in a 7 $M_{\\odot}$~ but not a 5 $M_{\\odot}$~  \nstar. The situation is complicated by the finding that if HBB stops while \nthermal pulses and TDUs continue in a star with mass loss, the star can \nbecome C-rich (Frost et al. 1998; Lattanzio \\& Forestini 1999). However, this happens only in \nlow-metallicity stars. Furthermore, if superwinds during the advanced \nAGB phase erode the envelope quickly, it is possible that TP \ncease before the star becomes C-rich. Thus, by and large \nit is not very likely that there are a substantial number \nof C-rich intermediate-mass stars that could have contributed SiC \nto the solar system.}\n\n\\item {4) We also obtain constraints on the mass and the metallicity of the \nparent stars from the isotopic compositions measured in presolar SiC \ngrains when these compositions are compared with model calculations:\n\n\\begin{itemize}\n\n\\item {i) It has already been pointed out that the heavy elements patterns \nmeasured in presolar SiC are well reproduced by models of neutron-capture \nnucleosynthesis in AGB stars (Gallino et al. 1997) . However, this \nagreement exists only for low-mass AGB stars of close-to-solar \nmetallicity and not for intermediate-mass stars, or of AGB stars of low\nmetallicity. A particularly \ndiagnostic isotopic ratio is the $^{96}$Zr\/$^{94}$Zr ratio. The large  \n$^{96}$Zr depletions measured in mainstream SiC grains (Nicolussi et al. \n1997; Pellin et al. 1999) are well reproduced only with models of AGB \nstars of 1.5 - 3 $M_{\\odot}$~ and about solar metallicity (Gallino et al. 1998b), but \nhigher-mass stars and stars with low metallicity are predicted to \nproduce huge $^{96}$Zr excesses.} \n\n\\item {ii) Gallino et al. (1990) pointed out that the He and Ne isotopic \ndata of presolar SiC grains are best explained in terms of \nnucleosynthesis in low-mass AGB stars of close-to-solar metallicity \n(see their Fig. 1). Another important observation is that SiC grains \ndo not show large $^{25}$Mg excesses (within relatively large errors). \nThis again indicates low-mass stars in which $^{22}$Ne does not burn. \nIndeed, the FRANEC code yields $^{25}$Mg excesses of up to $\\sim$ 200 \n$^o\\!\/\\!_{oo}$~ in the envelope of 1.5 and 3 $M_{\\odot}$~ AGB stars of solar metallicity, \nwhereas predicted \nexcesses are an order of magnitude larger in the 5 $M_{\\odot}$~ model of $Z$ = \n0.02 and in the 3 $M_{\\odot}$~ model of $Z$ = 0.006.} \n\n\\item {iii) The situation is similar with regard to the \n$^{12}$C\/$^{13}$C ratios, where the observed range is best \nreproduced by low-mass AGB models of close-to-solar metallicity (Gallino et \nal. 1990;  Bazan 1991). New results from the FRANEC code confirm \nthese earlier conclusions: the best agreement is obtained for 1.5 $M_{\\odot}$~ \n($^{12}$C\/$^{13}$C = 40 - 60) and 3 $M_{\\odot}$~ ($^{12}$C\/$^{13}$C = 90 - 100) AGB \nstars of solar metallicity, the 5 $M_{\\odot}$~ model of solar metallicity\nand the 3 $M_{\\odot}$~ model of low metallicity ($Z$ = \n0.006) yield much higher ratios ($^{12}$C\/$^{13}$C = 90 - 120 and \n100 - 700, respectively). In low-mass star models with $M \\lesssim$ 2.5 $M_{\\odot}$,\nthe presence of \ncold bottom processing (CBP) (Charbonnel 1995; Wasserburg et al. 1995;\nBoothroyd \\& Sackmann 1999) \nlowers the initial $^{12}$C\/$^{13}$C ratio at the beginning of the TP-AGB \nphase.} \n\n\\item {iv) Cold bottom processing is also important for the \n$^{14}$N\/$^{15}$N ratio. The high ratios observed in many individual \nmainstream SiC grains (Fig. 1) can only be explained by CBP (Huss et al. \n1997) operating in low-mass stars.}\n\n\\end{itemize}}\n\n\\item {5) A lower limit on the masses of carbon stars can be \nobtained from models with TDU. Existing models predict TDU only for \nstars with $M \\gtrsim$ 1.5 $M_{\\odot}$~ (Lattanzio 1989; \nStraniero et al. 1997; Gallino \net al. 1998a;  Busso et al. 1999b). In the FRANEC \ncode the limit depends on the value of Reimer's parameter $\\eta$ used. \nFor a star of $M =$ 1.5 $M_{\\odot}$~ of solar metallicity the limit is \n$\\eta$= 0.3 for TDU to occur and for producing C\/O $>$ 1 in the \nenvelope during the advanced stages of the \nAGB phase. The fact that there is a \nminimum mass below which TDU does not occur is of great \nimportance, since because of it SiC grains cannot originate \nfrom long-lived stars of low mass and low metallicity. \nIt is worth noting that increasing the metallicity above \nsolar works against an AGB star to become C-rich. In our \ncalculations the star remains O-rich at $Z = 2 \\times$ $Z_{\\odot}$ \nfor $M = $ 1.5 $M_{\\odot}$, and already at $Z = 1.25 \\times$ \n$Z_{\\odot}$ for $M = $ 3 $M_{\\odot}$.}\n\n\\end{itemize}\n\nFrom all these considerations it appears that most presolar SiC grains \ncome from AGB stars of 1.5 - 3 $M_{\\odot}$~ and close-to-solar metallicity. Of \ncourse, it is not said that the mainstream SiC grains have to come from \ntypical carbon stars. It is possible that these very large grains \npreferentially originated from stars with very dense winds and thus from\nstars having masses at the upper end of the above range. However, stars \nwith masses much larger than 3 $M_{\\odot}$~ can definitely be excluded.\n\n\\subsection{Star lifetimes, grain lifetimes and Galactic chemical \nevolution}\n\nLet us now return to the question of lifetimes of the possible source \nstars for SiC grains. Whereas the calculated lifetime of a 5 $M_{\\odot}$~ star \nof solar metallicity is 1.1 $\\times 10^8$ yr (Schaller et al. 1992), \nthose of the 3 $M_{\\odot}$~ and 1.5 $M_{\\odot}$~ stars are 4.4 $\\times 10^8$ and \n2.9 $\\times 10^9$ yr, respectively. Especially the latter lifetime \nwould result in a non-negligible difference in the Si isotopic ratios \ndue to the overall temporal evolution of the Si isotopes in the Galaxy. \nAccording to the model of Timmes \\& Clayton (1996) a time difference of \n2.9 $\\times 10^9$ yr corresponds to a difference of 125 $^o\\!\/\\!_{oo}$~ in the \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios, which is almost the \nwhole range covered by the mainstream grains. This means that a star that\nwas born 2.9 $\\times 10^9$ yr before the sun should have \n$^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios that are, on average, \n125 $^o\\!\/\\!_{oo}$~ smaller than the solar ratios. It also means that if stars of \ndifferent mass and therefore different lifetimes contributed SiC grains \nto the solar system, these grains are expected to have different Si isotopic \ncompositions. There are several factors \nthat play a role here. One is the initial mass function for stars. There \nare more stars of lower mass and we have already pointed out that the \nestimated mean mass of carbon stars is $\\sim$1.6 $M_{\\odot}$~ (Feast 1989). On the other hand, \nthe fact that SiC grains are relatively large \nsuggests that more massive AGB stars with very dense winds, meaning \nhigh mass loss at low speed, were selected as the SiC grains' parent \nstars.\n\nLet us consider two extreme cases. First we consider the case that all \nor most of the \nmainstream SiC grains came from AGB stars of approximately the same \nmass and therefore also the same time of formation. In this \ncase time differences do not play a role and \nthe distribution of the Si isotopic ratios of the mainstream grains \ncan in principle be explained as having an origin in \nlocal isotopic heterogeneities \ndue to the statistical nature of SN contributions to the ISM. This is \nschematically shown in Fig. 15a where the whole range of Si isotopic \ncompositions exhibited by the SiC mainstream grains is interpreted \nas the spread in Si isotopes that existed at the time of formation \nof the grains' parent stars. These parent stars have to have \napproximately the same mass but at this point it is not said whether \nit is low (1.5 $M_{\\odot}$) or high (3$M_{\\odot}$).\n\nThe second case to be considered is one in which AGB stars of \na {\\it considerable} mass range contributed the \nmainstream grains to the solar system. In this case the \nrange in Si isotopic shifts\ndue to the formation time difference between stars of 1.5 $M_{\\odot}$~ and of 3 $M_{\\odot}$~ \nis of the same order of magnitude \nas the spread of the mainstream grains. \nVariations in the Si \nisotopic ratios of individual grains are expected to arise from \nthe age differences \nof their parent stars (which in turn vary because of GCE) \nand local \nheterogeneities of the Si isotopes play a complementary role. \nThis situation is schematically depicted in \nFig. 15b, where the \nspread of the mainstream grains' Si isotopic ratios is interpreted \nas a superposition of local heterogeneity distributions representative \nof different Galactic times.\n\nAnother factor that plays a role here is the lifetime of the SiC grains \nin the ISM. This lifetime has to be added to the lifetime of the AGB \nparent stars in terms of time differences between the birth of these \nstars and the formation of the solar system and the implication of \nthese time differences for the Si isotopic ratios. Unfortunately, at \npresent we do not have any good direct measure of grain lifetimes. \nAttempts have been made to determine IS grain lifetimes from the \nmeasurement of  cosmogenic $^{21}$Ne produced in the grains from the  \nspallation of Si by Galactic cosmic rays (Tang \\& Anders 1988b; \nLewis et al. 1994). Estimates obtained in this way range up to \n1.3 $\\times 10^8$ yr. However, besides poor knowledge of the flux of \nGalactic cosmic rays, there are many other uncertainties associated \nwith this approach. Single grain measurements showed that only $\\sim$ \n5\\% of all SiC grains are rich in $^{22}$Ne (Nichols et al. 1991, 1992, \n1993). If one assumes that outgassing is the reason that the other \ngrains lack measurable amounts of $^{22}$Ne and that the same process \nremoved cosmogenic $^{21}$Ne from these grains, one arrives at much higher \nestimates for IS grain lifetimes. However, it is unclear that outgassing \nis indeed the cause for the large variations of $^{22}$Ne among single SiC \ngrains. Another problem is the determination of spallation recoil \nloss from the grains. From experimental measurements of spallation \nrecoil Ott \\& Begemann (1997, 1999) concluded that a determination of presolar \nexposure ages from cosmogenic $^{21}$Ne is not feasible. These authors \npropose the use of spallation Xe as more promising but before this is \ndone we do not have any reliable IS lifetimes for presolar grains. An \nalternative way is to use model ages derived from theoretical \ndestruction rates of IS grains by SN shocks and collisions (see, e.g., \nWhittet 1992; Jones et al. 1997). Estimates range up to $\\sim 10^9$ yr \nbut there are also large uncertainties in this approach.\n\n\\subsection{SiC grains and the Si isotopic composition of the sun}\n\nSo far we have discussed differences in the formation time of AGB \nstars of different masses that possibly contributed SiC grains to \nthe solar system. We have not discussed yet the relationship \nbetween the formation time of the grains' parent stars relative to \nthat of the solar system and implications for their relative Si \nisotopic compositions. In the Timmes \\& Clayton (1996) model the \nfact that most mainstream grains have isotopically heavier compositions \nthan the solar system (implying that they are younger) \nbut must have formed before the sun presents a fundamental problem. \nOur heterogeneity model alleviates this fundamental problem, because \nin principle it can explain the spread in the Si isotopic compositions \nof the mainstream grains as inhomogeneities of the Si isotopes in the \nISM at a given time (see Fig. 15a). However, the grains' parent stars \nmust have formed before the solar system and we must \ndiscuss the effect of this time difference on their Si isotopic \ncompositions.\n\nFor the sake of discussion we again consider two extreme cases. First we consider the case that most \nmainstream SiC grains came from AGB stars of 3 $M_{\\odot}$. The \nevolution time of such stars is 4.4 $\\times 10^8$ yr. According to \nTimmes et al. (1995) and Timmes \\& Clayton (1996), such a time \ndifference corresponds to a shift of \n19 $^o\\!\/\\!_{oo}$~ of the Si isotopic ratios. If we assume that the spread in \nSi isotopic ratios at the time of the birth of the parent stars \ncoincides with that of the mainstream grains, the Si isotopic distribution \nat the time of solar system formation 4.4 $\\times 10^8$ yr later is \nisotopically heavier by 19 $^o\\!\/\\!_{oo}$~ (Fig. 16a). This shift is \nrelatively small compared to the range of the mainstream grains. The Si \nisotopic composition of the sun, while falling at the outer edge of \nthis distribution, lies still within the range of compositions \nexpected to be present at the time of solar system formation. \nThe fact that the sun has a \ncomposition that differs from those of most of the grains, is not a \nfundamental problem in this case. It simply means that the sun, \nas many other SiC \nparent stars, has an unusual composition but one that is not incompatible with expectations.\n\nLet us next consider the other extreme, that all grains come from stars of \n1.5 $M_{\\odot}$. The evolution time of these stars is 2.9 $\\times 10^9$ yr. \nAccording to Timmes \\& Clayton (1996) this time difference corresponds to \na shift of the Si isotopic ratios by 125 $^o\\!\/\\!_{oo}$~. This means that the distributions \nof the Si isotopes at the time of star formation and at the time of solar \nsystem formation 2.9 $\\times 10^9$ yr later are shifted by this amount \nrelative to one another. This is shown in Fig. 16b where, again, we assume \nthat the Si isotopic distribution of the stars coincides with that of \nthe mainstream grains. This time, the inferred distribution \n2.9 $\\times 10^9$ yr later is shifted so much that it seems quite\nimpossible \nthat the solar system composition observed today can be explained as \nbeing part of this distribution. In other words, in the extreme case in which only stars of \n$M = $ 1.5 $M_{\\odot}$~ contributed SiC grains, we are faced with the \nsame fundamental problem as the Timmes \\& Clayton (1996) model, namely that \nthe actual solar system composition is much too light compared to the \ndistribution predicted for the time of solar system formation if the mainstream grains came from old stars. \n\nClayton (1997) has addressed this problem and \nhas proposed a solution in terms of a systematic difference in the \nGalactic radius at which the parent stars of the mainstream grains \non the one hand and the sun on the other hand formed. The parent stars \nare assumed to have formed at smaller Galactic radii where the \nmetallicity and $^{29}$Si\/$^{28}$Si and $^{30}$Si\/$^{28}$Si ratios are \nbelieved to be higher. His model involves the diffusion of stars from \nsmaller to larger Galactic radii due to scattering on IS clouds, \nfollowing the stellar \norbital diffusion model by Wielen, Fuchs, \\& Dettbarn (1996). Such a \nmodel had been proposed in order to explain the spread in elemental \nabundances observed in stars. However, a more detailed quantitative \ntreatment by Nittler \\& Alexander (1999b) shows that with reasonable \nassumptions a diffusion model cannot account for the isotopically heavy \nSi compositions of the grains relative to the sun. Furthermore, van \nden Hoek \\& de Jong (1997) pointed out that stellar orbital diffusion \ncannot sufficiently explain the elemental abundance variations. \nWhile we do not want to discard the orbital diffusion model, we \nhope that the heterogeneity explanation will give a more definitive answer \nto this important question. This would require a model that, \nby computing the Galactic evolution of the Si isotopes \nby taking into account incomplete mixing of \ndifferent stellar yields, overcomes the problems discussed in \n\\S 4.3 and all those connected \nto the overly simplistic nature of our approach.\n\nIt should be noted that in our estimates of the Si isotopic shifts \nassociated with time differences we have used the Si isotopic evolutions \nvs. time relationship given by Timmes et al. (1995) and \nTimmes \\& Clayton (1996). This relationship crucially depends on the \nrelative proportion in which Type Ia and Type II SNe contribute to the \nenrichment of the ISM in Si isotopes. \nTimmes et al. (1995) attributed a dominant role to Type II \nSNe by assuming that at the time of solar system formation they \ncontributed 2\/3 of the Fe. Woosley et al. (1997), on the other hand \nestimated that this fraction would be 1\/2. This would mean that \nthe $^{28}$Si contribution from Type Ia SNe is higher and therefore \nthe Si isotopes evolve more slowly toward heavier compositions. \nThis in turn would mean that a Si isotopic shift corresponding to \na given time difference (Fig. 16) is smaller than what we assumed. \n\nIn conclusion, there are still large uncertainties as to the masses \nof the parent AGB of the grains, the ISM life times of the grains, and the \ntime dependence of the evolution of the Si isotopes in the Galaxy. \nAll of these uncertainties have to be clarified before we can hope to \nsolve the problem of the difference of the Si isotopic compositions of \nthe mainstream SiC grains and that of the solar system.\n\n\\section{Conclusions}\n\nMainstream SiC grains are the major group of presolar SiC grains found in \nmeteorites. Although there \nis overwhelming evidence that mainstream grains have an origin in the \nexpanding atmospheres of AGB stars, their Si isotopic ratios show a \ndistribution (Fig. 3) that cannot be explained by nucleosynthesis in \nAGB stars. The theoretically predicted Si isotopic shifts in the \nenvelope of AGB stars are either much smaller than the range observed \nin the mainstream grains (for AGB models of $M$ = 1.5 and 3 $M_{\\odot}$~ and \nsolar metallicity) or (for AGB models of $M$ = 5 $M_{\\odot}$~ and solar \nmetallicity and $M$ = 3 $M_{\\odot}$~ and $Z$  = 0.006) show a slope of $\\sim$ 0.5 \ncorrelation between the $\\delta$$^{29}$Si\/$^{28}$Si and \n$\\delta$$^{30}$Si\/$^{28}$Si values instead of the slope 1.31 correlation \nline exhibited by the grains.\n\nThe distribution of the Si isotopic ratios of the mainstream grains \nhas previously been interpreted to be the result of GCE of the Si \nisotopes. In this interpretation the grains' parent stars are expected \nto have a range of different Si isotopic ratios if they were born at \ndifferent times. In this paper we proposed an alternative explanation \nfor the Si isotope distribution by invoking isotopic heterogeneities \ndue to the statistical nature of the contributions of a limited number \nof SN sources to the IS material from which the grains' parent stars \nformed. The Si isotopic ratios of the ejecta of possible SN sources, \nclassical Type Ia SNe, Type Ia SNe from sub-Ch white dwarfs, \nand Type II SNe of different masses, span a wide range. We developed a \nsimple Monte Carlo model in which contributions from these SN sources \nwere admixed in a random way to material with a given Si isotopic \ncomposition. As long as this composition lies on the theoretically \nexpected GCE line going through the solar Si isotopic composition, \nwe could show that, with the right choice of parameters, the \ndistribution of the Si isotopic ratios in the mainstream grains can \nbe successfully reproduced for a wide range of starting compositions. \nThe parameters to be adjusted are the total number of SN sources \nselected and the fraction of the material ejected from each SN that \nis admixed to the starting material. In addition, an adjustment of the SN \nyield of $^{29}$Si by a factor of 1.5 is necessary to achieve the \nSi isotopic ratios of the solar system. \nAstronomical observations of variations of \nelemental ratios in stars are compatible with the predictions \nfrom our MC model.\n\nThese results demonstrate that, in principle, the mainstream distribution \ncan be explained as the result of local fluctuation in the ISM due to \nthe admixture of material from a limited number of SN sources to the \npreexisting IS matter. If the AGB stars that contributed SiC grains to \nthe protosolar nebula were born within a short period of time (by having a \nnarrow range of masses and the grains having short IS lifetimes), such \nfluctuations must have been the dominant cause of the mainstream \ndistribution. If, however, the AGB parent stars had a large range of \nmasses and therefore a large range of lifetimes and\/or the grains themselves \nexperienced a large range of residence times in the ISM, the parent \nstars must have been born at different Galactic eras and their initial \nSi isotopic ratios must show considerable variations because of the \nvarying average composition of the ISM due to the GCE of the Si isotopes. \nIn this case we still expect that local fluctuations will be \nsuperimposed on these average compositions. To simulate these complex \nprocesses it will be necessary to apply to the Si isotopes a\nGalactic evolution model that is able to take both components properly \ninto account. \nHowever, a successful model would require knowledge of the mass \ndistribution of AGB stars that contributed SiC grains (at least in \nthe size range of the single grains whose data are plotted in Fig. 3) \nand the distribution of the IS lifetimes of these grains. Both pieces \nof information are presently unknown and we can only hope that further \nprogress in the study of the grains and their origin will get us \ncloser to an answer.\n\nWe are grateful to Peter Hoppe for providing isotopic data on the \nMurchison KJE size fraction, to Gary Huss for providing his Orgueil \ndata, and to Stan Woosley for providing unpublished results of the Si \nyields from his and Weaver's $Z$ = 0.5 $Z_{\\odot}$ and \n$Z$ = 2 $Z_{\\odot}$ supernova models. \nWe are deeply indebted to Oscar Straniero, Maurizio Busso, Alessandro \nChieffi and Marco Limongi for all \ntheir scientific input and thank Don Clayton for ideas and discussions. \nThe detailed and thoughtful review by Don Clayton substantially \ncontributed to the final version of this paper. \nML gratefully acknowledges the invaluable help of John Lattanzio. \nEZ deeply appreciates the hospitality extended to him by Roberto Gallino\nduring a visit to the Dipartimento di Fisica Generale of the University \nof Torino and the support for this visit provided by the Gruppo Nazionale di\nAstronomia del CNR. SA acknowledges the support for a visit to the same \nDepartment provided by the University of Torino.\nThis work was supported by an Overseas Postgraduate Research Scheme \naward (ML), NASA grant NAG5-8336 (SA and EZ) and by MURST Cofin98 \nProgetto Evoluzione Stellare (RG).\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent\n\nThe isospin-symmetry violation in atomic nuclei is predominantly\ndue to the Coulomb interaction that exerts long-range polarizations\non neutron and proton states. To consistently take into account\nthis polarization, one needs to\nemploy huge configuration  spaces. For that reason, an accurate description of isospin impurities in atomic nuclei, which\nis  strongly motivated by the recent high-precision measurements of the $0^+\n\\rightarrow 0^+$ Fermi superallowed $\\beta$-decay rates, is difficult to\nbe obtained in  shell-model\napproaches, and  specific approximate methods are required.\\cite{[Orm95a],[Tow08]}\n\n\nThe long-range polarization effects can be included within the\nself-consistent mean-field (MF) or DFT\napproaches, which are practically the only microscopic frameworks\navailable for heavy, open-shell\nnuclei with many valence particles. These approaches, however, apart from\nthe  {\\it physical\\\/} contribution to the isospin mixing, mostly caused by\nthe Coulomb field and, to a much lesser extent, by\nisospin-non-invariant components of the nucleon-nucleon force,  also introduce the spurious isospin mixing  due to the {\\it spontaneous\\\/} isospin-symmetry breaking.\\cite{[Eng70],[Cau80],[Cau82]}\n\n\nHereby, we present results on the isospin mixing and\nisospin symmetry-breaking corrections to the superallowed Fermi $\\beta$-decay\nobtained by using the newly developed isospin- and angular-momentum-projected DFT\napproach without pairing.\\cite{[Sat09],[Sat09a],[Sat10],[Sat10a]}\nThe model employs symmetry-restoration techniques to remove\nthe spurious isospin components and restore angular momentum symmetry, and\ntakes advantage of the natural ability of  MF to describe self-consistently\nthe subtle balance between the Coulomb force making  proton and\nneutron wave functions different and the isoscalar part of the strong\ninteraction producing the opposite effect.\n\nThe paper is organized as follows. In Sec.~\\ref{theo}, we describe the main theoretical\nbuilding blocks of the isospin- and angular-momentum-projected DFT.\nSection~\\ref{isomix} presents some preliminary applications of the formalism to the isospin symmetry-breaking corrections  to the Fermi superallowed\n$\\beta$-decay matrix elements, whereas Sec.~\\ref{symm} discusses applications of the\nisospin-projected DFT to nuclear symmetry energy. The summary is contained in  Sec.~\\ref{summary}.\n\n\n\\section{The projected DFT framework}\n\\label{theo}\n\n\n\n\n\nThe building block of the isospin-projected DFT  is the  Slater determinant,\n$|\\Phi\\rangle$, representing the self-consistent Skyrme-HF solution provided\nby the HF solver HFODD.\\cite{[Dob09d]} Self-consistency ensures that\nthe balance between the long-range Coulomb force and short-range strong\ninteraction, represented in our model by the Skyrme energy density functional (EDF), are properly taken\ninto account. The unphysical isospin mixing is taken care of by the\nrediagonalization of the entire Hamiltonian in the good isospin basis, $|T,T_z\\rangle$,\nas described in Refs.\\cite{[Sat10],[Sat10a]}\nThis yields the eigenstates:\n\\begin{equation}\\label{mix2}\n|n,T_z\\rangle\n= \\sum_{T\\geq |T_z|}a^n_{T,T_z}|T,T_z\\rangle\n\\end{equation}\nnumbered by an index $n$.  The so-called isospin-mixing\ncoefficients (or, equivalently, isospin impurities)\nare defined for the $n-$th eigenstate as\n\\begin{equation}\n\\alpha_C^n = 1 - |a^n_{T,T_z}|_{\\text{max}}^2 ,\n\\end{equation}\nwhere $|a^n_{T,T_z}|_{\\text{max}}^2$ stands for the dominant amplitude in the wave function\n$|n,T_z\\rangle$.\n\nWithin the isospin- and angular-momentum-projected DFT, we\nuse the normalized basis of states $|I,M,K; T,T_z\\rangle$\nhaving both good angular momentum and good\nisospin.\\cite{[RS80]}\nHere, $M$ and $K$ denote the angular-mo\\-men\\-tum components\nalong the laboratory and intrinsic $z$-axes, respectively. The  $K$ quantum\nnumber is not conserved. In order to avoid problems with overcompleteness of\nthe basis, the $K$-mixing is performed by rediagonalizing\nthe Hamiltonian in the so-called {\\it collective space}, spanned for each $I$\nand $T$ by the {\\it natural states\\\/}, $|IM;TT_z\\rangle^{(i)}$, as described\nin Refs.\\cite{[Dob09d],[Zdu07a]} Such a rediagonalization yields the\neigenstates:\n\\begin{equation}   \\label{KTmix}\n|n; IM; T_z\\rangle =\n\\sum_{i,T\\geq |T_z|}\n   a^{(n)}_{iIT} | IM; TT_z\\rangle^{(i)} ,\n\\end{equation}\nwhich are labeled by the index $n$ and by the conserved quantum numbers $I$, $M$, and\n$T_z=(N-Z)\/2$ [compare Eq.~(\\ref{mix2})].\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig1.eps}\n\\caption[T]{\\label{fig1}\nThe absolute values of the norm kernels, $|{\\cal N}(\\beta_T; \\alpha, \\beta, \\gamma )|\n= |\\langle \\Phi | \\hat{R}(\\beta_T ) \\hat{R}(\\alpha, \\beta, \\gamma )\n|\\Phi\\rangle|$, for a state in $^{14}$N calculated with the SLy4 EDF, plotted versus the rotation angle in the isospace $\\beta_T$.\nThe solid curve, exhibiting the single singularity at $\\beta_T = \\pi $, corresponds to\nthe pure isospin-projected DFT theory, which is regular for all\nSkyrme-type functionals.\\protect\\cite{[Sat10]}\nThe dotted lines correspond to two fixed sets of the Euler\nangles in space, with $\\alpha =\\gamma \\approx 0.314$, and\n$\\beta \\approx 0.229$ (left curve)\nand  $\\beta \\approx 1.414$ (right curve). The poles that appear\ninside the integration region, $0<\\beta_T<\\pi$, give rise to singularities in\n the projected DFT approach.}\n\\end{center}\\end{figure}\n\n\nThe isospin projection does not produce singularities in energy kernels; hence, it can be safely used with all commonly used EDFs.\\cite{[Sat10]} Coupling the isospin and angular-momentum\nprojections, however, leads to singularities in both the norm (see\nFig.~\\ref{fig1}) and energy kernels. This fact narrows the\napplicability of the model to Hamiltonian-driven EDFs which,\nfor  Skyrme-type functionals, leaves only one option: the  SV\nparametrization.\\cite{[Bei75]} The alternative would be to use an\nappropriate regularization scheme, which is currently under\ndevelopment.\\cite{[Lac09],[BD10]}\n\n\n\n\\section{Isospin-mixing and isospin-breaking corrections to superallowed\n$\\beta$-decay}\n\\label{isomix}\n\n\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig2.eps}\n\\caption[T]{\\label{fig2}\nIsospin impurities in the ground states of $^{40}$Ca (upper panel) and\n$^{100}$Sn (lower panel), plotted as functions of the excitation energy of\nthe doorway state for a set of commonly used Skyrme EDFs.\\cite{[Ben03]} Results of the linear fits and the\ncorresponding regression\ncoefficients, $R$, are also shown.}\n\\end{center}\\end{figure}\n\n\nEvaluation of  $\\alpha_C$ is\na prerequisite to calculate isospin corrections to reaction  and decay rates.\nAs is well known,\\cite{[Aue83]}  isospin impurities are\nthe largest in $N=Z$ nuclei, increase along the $N=Z$ line with increasing\nproton number, and are strongly quenched with increasing $|T_z|=|N-Z|\/2$.\nSuch characteristics were also early estimated based on the perturbation\ntheory\\cite{[Sli65]} or hydrodynamical model.\\cite{[Boh67]} Quantitatively,\nafter getting rid of the spurious mixing, which lowers the true $\\alpha_C$ by as\nmuch as 30\\%,\\cite{[Sat09a]}\nthe isospin impurity increases from a fraction of a percent in very light\n$N=Z$ nuclei to  $\\sim$0.9\\%  in $^{40}$Ca, and $\\sim$6.0\\% in $^{100}$Sn,\nas shown in Fig.~\\ref{fig2}. In the particular case of $^{80}$Zr, the\ncalculated impurity of 4.4\\% agrees well with the empirical value  deduced from\nthe giant dipole resonance $\\gamma$-decay studies.\\cite{[Cam10a]} This  makes us believe that our model  is indeed capable of\ncapturing essential physics associated with the isospin mixing. Unfortunately,\ncurrent experimental errors are too large to discriminate between different\nparametrizations of the Skyrme functional. The variations between EDFs in Fig.~\\ref{fig2} result in $\\sim$10\\% uncertainty in calculated\nvalues of $\\alpha_C$.\n\n\nThe magnitude of theoretical $\\alpha_C$ is quite well correlated\nwith the excitation energy, $E_{T=1}$, of the $T=1$ doorway state,\nsee Fig.~\\ref{fig2}. However, in order to make a precise determination of $E_{T=1}$, spectroscopic quality EDFs are needed, and this  is not yet the case.\\cite{[Kor08]} This explains why the values of $\\alpha_C$\ndo not  correlate well with basic EDF characteristics, including the isovector and isoscalar effective mass,\nsymmetry energy, binding energy per particle, and\nincompressibility (see discussion in Ref.\\cite{[Sat10a]}).\n\n\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.54\\textwidth,clip]{kazi10_fig3.eps}\n\\caption[T]{\\label{fig3}\nValues of $|V_{ud}|$ deduced from the superallowed $\\beta$-decay\n(full circles) for three different sets of the $\\delta_C$ corrections calculated in:\nRef.\\protect\\cite{[Tow08]} (a);  Ref.\\protect\\cite{[Lia09]} with NL3 and\nDD-ME2 Lagrangians (b); and in the present work (c).\nTriangles mark values of $|V_{ud}|$ obtained from the\npion-decay\\protect\\cite{[Poc04]} and neutron-decay\\protect\\cite{[Ams08]} studies,\nrespectively. The open circle shows the value deduced from the\n$\\beta$-transitions in $T=1\/2$ mirror nuclei.\\protect\\cite{[Nav09a]}\n}\n\\end{center}\\end{figure}\n\n\nIncreasing demand on precise values of  isospin impurities has been\nstimulated by the recent high-precision measurements of superallowed\n$\\beta$-decay rates.\\cite{[Har05c],[Tow08]}\nA reliable determination of the corresponding\nisospin-breaking correction, $\\delta_C$,\nrequires the isospin- and angular-momentum-projected DFT.\\cite{[Sat10a]}.\nThis correction is obtained by calculating\nthe $0^+  \\rightarrow 0^+$ Fermi\nmatrix element of the isospin raising\/lowering operator $\\hat T_{\\pm}$ between\nthe ground state (g.s.) of the even-even nucleus  $| I=0, T\\approx 1, T_z = \\pm 1 \\rangle$\nand its isospin-analogue partner in the $N=Z$ odd-odd nucleus, $|I=0, T\\approx\n1, T_z = 0 \\rangle$:\n\\begin{equation}\\label{fermime}\n|\\langle I=0, T\\approx 1,\nT_z = \\pm 1 | \\hat T_{\\pm} | I=0, T\\approx 1, T_z = 0 \\rangle |^2 \\equiv 2 (\n1-\\delta_C ).\n\\end{equation}\n\n\nTo determine the  $|I=0, T\\approx 1, T_z = 0 \\rangle$ state\nin the odd-odd $N=Z$ nucleus, we\nfirst compute the so-called\nantialigned g.s.\\ configuration, $|\\bar \\nu \\otimes \\pi \\rangle$ (or  $| \\nu\n\\otimes \\bar \\pi \\rangle$), by placing the odd neutron and the odd proton in\nthe lowest available time-reversed (or signature-reversed)\nHF orbits.\nThen, to correct for the fact that the antialigned\nconfigurations manifestly break the isospin symmetry,\\cite{[Sat10]} that is,\n$|\\bar \\nu \\otimes \\pi \\rangle \\approx \\frac{1}{\\sqrt 2} (|T=0 \\rangle + |T=1\n\\rangle )$, we apply the isospin and angular-momentum projections to create\nthe  basis $|I,M,K,T,T_z=0 \\rangle$, in which the total Hamiltonian is rediagonalized (see  Sec.~\\ref{theo}).\nA similar scheme is used to compute the\n$| I=0, T\\approx 1, T_z = \\pm 1 \\rangle$ states in the even-even nuclei.\n\n\nOur studies indicate\\cite{[Sat10a]} that to obtain a fair estimate\nof $\\delta_C$ for $A<40$ and $A>40$ nuclei, one needs to use large\nharmonic oscillator bases consisting of at least $N=10$ and 12\nfull shells, respectively. Even then, the results\nare subject to systematic errors due to the basis cut-off, which can be\nestimated to be $\\sim$10\\%.\nDespite the fact that  not all $N=12$ calculations in heavy ($A> 40$) nuclei\nhave yet been completed, and that owing to the\n shape-coexistence effects, there are still some\nambiguities concerning the global minima, our  preliminary results point\nto  encouraging conclusions. Namely,  the mean value of the structure-independent\nstatistical-rate function $\\bar{{\\cal F}}t$,\\cite{[Har05c]} obtained for 12 out of\n13 transitions known empirically with high precision (excluding the\n$^{38}$K$\\rightarrow$$^{38}$Ar case), equals $\\bar{{\\cal F}}t = 3069.4(10)$,\nwhich gives the value of the CKM matrix element equal to $|V_{ud}| =  0.97463(24) $.\nThese values match well those obtained by Towner and Hardy in their\nrecent compilation\\cite{[Tow08]} (see Fig.~\\ref{fig3}).\nBecause of a poor spectroscopic quality of the SV parameterization, the confidence\nlevel\\cite{[Tow10]} of our results is poor. Nevertheless, it should be\nstressed that our method is quantum-mechanically consistent (see\ndiscussion in Refs.\\cite{[Mil08],[Mil09]}) and contains no adjustable free parameters.\n\n\n\n\\section{Symmetry energy}\\label{symm}\n\n\n\\begin{figure}\\begin{center}\n\\includegraphics[angle=0,width=0.46\\textwidth,clip]{kazi10_fig4.eps}\n\\caption[T]{\\label{fig4}\nTop: schematic illustration of the isospin-symmetry-breaking\nmechanism in MF of odd-odd $N=Z$  nuclei. Bottom:\n $E_{\\text{sym}}^{\\text{(int)}}$ in odd-odd $N=Z$ nuclei calculated with\nSLy4, SV, SLy4$_L$, and SkM$^*_L$ EDFs. See text for details.}\n\\end{center}\\end{figure}\n\n\n\nThe spontaneous violation of isospin symmetry in all but isoscalar MF configurations of\n$N=Z$ nuclei offers a way to study the nuclear symmetry\nenergy.  The idea, which is schematically\nsketched in the upper portion of Fig.~\\ref{fig4}, invokes  the mixed-symmetry\nantialigned $|\\bar\\nu \\otimes \\pi\\rangle$ (or $|\\nu \\otimes \\bar\\pi\\rangle$)\nconfiguration in an odd-odd $N=Z$ nucleus. By applying the isospin projection to the\nHF state $|\\bar\\nu \\otimes \\pi\\rangle$, one decomposes  it into the isoscalar $T=0$ and isovector\n$T=1$ parts. As argued below, the magnitude of the splitting, $E_{\\text{sym}}^{\\text{(int)}}$,\ndepends on the isovector channel of a given EDF, i.e., its symmetry energy.\n\n\nFor the Skyrme-type EDFs, the  symmetry energy\nin the nuclear matter limit can be decomposed as:\\cite{[Sat06w2]}\n\\begin{equation}\\label{nmsym}\na_{\\text{sym}} = \\frac{1}{8}\\varepsilon_{FG} \\left( \\frac{m}{m_0^\\star}\n\\right) +\n\\left[ \\left( \\frac{3\\pi^2}{2}\\right)^{2\/3} C_1^\\tau \\rho^{5\/3}\n+ C_1^\\rho \\rho \\right]\n\\equiv  a_{\\text{sym}}^{\\text{(kin)}} + a_{\\text{sym}}^{\\text{(int)}}.\n\\end{equation}\nThe first term in Eq.~(\\ref{nmsym}) is associated with the isoscalar part of the nucleon-nucleon interaction\nand primarily depends on the mean single-particle level spacing at the Fermi energy. This term\nis scaled by the inverse isoscalar effective mass. The second\n(interaction) term, is related to the isovector part of the Skyrme-EDF:\n$\\delta {\\cal H}_{t=1} = C_1^\\rho \\rho_1^2 + C_1^\\tau \\rho_1 \\tau_1$\n(for definitions, see Ref.\\cite{[Ben03]} and references quoted therein).\n\n\nThe value of $E_{\\text{sym}}^{\\text{(int)}}$ appears to be mainly sensitive to the\ninteraction term, which is illustrated in Fig.~\\ref{fig4}.\nIndeed, despite the fact that\nSLy4 and SV EDFs have similar values of  $a_{\\text{sym}}$ (equal\nto 32\\,MeV and 32.8\\,MeV, respectively), the\ncorresponding energy splittings $E_{\\text{sym}}^{\\text{(int)}}$ differ substantially.\nThe reduced values of $| E_{\\text{sym}}^{\\text{(int)}} |$ in  SV\nare due to its small value of $a_{\\text{sym}}^{\\text{(int)}} = 1.4 $\\,MeV,\\footnote{\nThis small value shows how unphysical are the consequences of\nthe saturation mechanism built into  SV through the strong\nmomentum dependence and results in an unphysically low\n isoscalar effective mass  $m^*\/m\\approx 0.38$. Although  SV  has\na relatively reasonable global strength of the symmetry energy $a_{\\text{sym}}$, its physical origin is incorrect.}\n which is\nan order of magnitude smaller than the corresponding SLy4 value:\n $a_{\\text{sym}}^{\\text{(int)}} = 14.4$\\,MeV.\n\n\nAn interesting aspect of  our analysis of\n$E_{\\text{sym}}^{\\text{(int)}}$ relates to its dependence on the\ntime-odd terms, which  are poorly constrained for Skyrme EDFs.\nTo quantify this dependence, we have performed\ncalculations by using the SLy4$_L$ and SkM$^*_L$ functionals, which\nhave the spin coupling constants adjusted to the Landau parameters.\\cite{[Ben02],[Zdu05]}\nThese EDFs have different  values of $a_{\\text{sym}}$ but the same\n$a_{\\text{sym}}^{\\text{(int)}} = 14.4$\\,MeV. The similarity of the calculated energy splittings shown in Fig.~\\ref{fig4} confirms that this quantity\nprimarily depends on the isovector terms of the functional. Moreover, its\nsignificant dependence on the time-odd terms opens up new options for adjusting\nthe corresponding coupling constants to experimental data. This will certainly require the simultaneous restoration of isospin and\nangular-momentum symmetries, as presented in this study.\n\n\n\n\n\\section{Summary}\\label{summary}\n\n\nIn summary, the isospin- and angular-momentum-projected DFT\ncalculations have been performed to estimate  the isospin-breaking\ncorrections to $0^+ \\rightarrow 0^+$ Fermi superallowed $\\beta$-decays.\nPreliminary results for the average value of the nucleus-independent $\\bar{\\cal\nF}t = 3069.4(10)$ and the amplitude $|V_{ud}| = 0.97463(24)  $ were found\nto be  consistent with the recent estimates by  Towner and Hardy,\\cite{[Tow08]}\nnotwithstanding a  low spectroscopic quality of the Skyrme EDF SV used.\n\nApplicability of the isospin-projected DFT  to analyze\nthe nuclear symmetry energy has also been discussed. It has been demonstrated\nthat the isospin projection offers a  rather unique opportunity to study the\ninteraction part of the symmetry energy\nin the odd-odd $N=Z$ nuclei and that this quantity is influenced\nby time-odd fields of the energy density functional.\n\n\n\nThis work was supported in part by the Polish Ministry of Science\nunder Contract Nos.~N~N202~328234 and N~N202~239037, Academy of Finland and\nUniversity of Jyv\\\"askyl\\\"a within the FIDIPRO programme, and by the Office of\nNuclear Physics,  U.S. Department of Energy under Contract Nos.\nDE-FG02-96ER40963 (University of Tennessee) and\nDE-FC02-09ER41583  (UNEDF SciDAC Collaboration).\nWe acknowledge the CSC - IT Center for Science Ltd, Finland for the\nallocation of computational resources.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}