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+{"text":"\\section{Introduction \\label{sec:introduction}}\n\n\nThe bismuth chalcogenides \\ch{Bi2Ch3} (\\ch{Ch} = \\ch{S}, \\ch{Se}, or \\ch{Te}) of the layered tetradymite structure are an interesting class of highly two dimensional narrow bandgap semiconductors.\nStrong spin-orbit coupling inverts the energy ordering of their bands, making them bulk \\glspl{ti} characterized by a single Dirac cone at the Brillouin zone centre and a topologically protected metallic surface state~\\cite{2009-Hsieh-N-460-1101, 2009-Zhang-NP-5-438}.\nThis has augmented longstanding interest in their thermoelectric properties with significant efforts (theoretical and experimental) to understand their electronic properties in detail~\\cite{2013-Cava-JMCC-1-3176, 2017-Heremans-NRM-2-17049}.\nConsisting of weakly interacting \\ch{Ch-Bi-Ch-Bi-Ch} atomic \\glspl{ql} (see \\latin{e.g.}, Figure~1 in Ref.~\\onlinecite{2019-McFadden-PRB-99-125201}), they can also accommodate intercalant species such as \\ch{Li^{+}} in the \\gls{vdw} gap between \\glspl{ql}~\\cite{1988-Paraskevopoulos-MSEB-1-147, 1989-Julien-SSI-36-113, 2010-Bludska-JSSC-183-2813}, similar to the layered \\glspl{tmd}~\\cite{1978-Whittingham-PSSC-12-41, 1987-Friend-AP-36-1}.\nAlthough their bandgaps are \\SI{\\sim 150}{\\milli\\electronvolt}, doping by intrinsic defects, such as \\ch{Ch} vacancies, yields crystals that are far from insulating.\nTo increase the contrast in conductivity between the bulk and the metallic \\gls{tss}, the crystals are often compensated extrinsically.\nFor example, \\ch{Ca} substitution for \\ch{Bi} suppresses the self-doped $n$-type conductivity in \\ch{Bi2Se3}~\\cite{2009-Hor-PRB-79-195208, 2009-Hsieh-N-460-1101}.\nDoping can also be used to modulate their magnetic properties, yielding magnetic \\glspl{ti} where the \\gls{tss} is gapped~\\cite{2010-Chen-S-329-659}, for example, by substitution of \\ch{Bi} with a paramagnetic transition metal~\\cite{2010-Hor-PRB-81-195203, 2019-Tokura-NRP-1-126}.\n\n\nThe intriguing electronic properties of the tetradymite \\glspl{ti} have predominantly been investigated using surface sensitive probes in real and reciprocal space (\\latin{e.g.}, \\gls{sts} and \\gls{arpes}), as well as other bulk methods.\nIn complement to these studies, \\gls{nmr} offers the ability to probe their electronic ground state and low-energy excitations through the hyperfine coupling of the nuclear spin probe to the surrounding electrons.\nSuch a local probe is especially useful when disorder masks sharp reciprocal space features, as is the case in \\ch{Bi2Ch3}.\nThe availability of a useful \\gls{nmr} nucleus, however, is usually determined by elemental composition and natural (or enriched) isotopic abundance, as well as the specific nuclear properties such as the gyromagnetic ratio $\\gamma$ and, for spin $> 1\/2$, the nuclear electric quadrupole moment $Q$.\nWhile the \\ch{Bi2Ch3} family naturally contain several \\gls{nmr} nuclei~\\cite{2013-Nisson-PRB-87-195202, 2014-Koumoulis-AFM-24-1519, 2016-Levin-JPCC-120-25196}, they are either low-abundance or have a large $Q$.\nAs an alternative, here we use an ion-implanted \\gls{nmr} probe at ultratrace concentrations, with detection based on the asymmetric property of radioactive $\\beta$-decay, known as \\gls{bnmr}~\\cite{2015-MacFarlane-SSNMR-68-1}.\n\n\nA key feature of ion-implanted \\gls{bnmr} is the depth resolution\nafforded by control of the incident beam energy~\\cite{2014-Morris-HI-225-173, 2015-MacFarlane-SSNMR-68-1},\nwhich dictates the stopping distribution of the implanted \\gls{nmr} probes.\nAt \\si{\\kilo\\electronvolt} energies,\nthe depth can be varied on the nanometer length-scale and,\nat the lowest accessible energies,\nit may be able to sense the \\gls{tss} in the tetradymite \\glspl{ti},\nwhich is likely confined to depths \\SI{< 1}{\\nano\\meter} below the surface.\nHere one expects Korringa relaxation and Knight shifts from the \\gls{tss} electrons,\nmodified by the phase space restrictions imposed by their chirality.\nThe magnitude of each will depend on both the \\gls{tss} carrier density and\nthe strength of the coupling to the implanted nuclei.\nWhile the motivation to study the \\gls{tss} is strong,\nhere we report the ``bulk'' response of an implanted \\ch{^{8}Li} probe\nin two doped \\ch{Bi2Ch3} \\glspl{ti}.\nThis is an essential step toward detecting the\n\\gls{tss}~\\cite{2014-MacFarlane-PRB-90-214422, 2019-McFadden-PRB-99-125201},\nbut it also demonstrates the sensitivity of the implanted \\ch{^{8}Li} to\nthe carriers in such heavily compensated narrow gap semiconductors.\n\n\nUsing \\gls{bnmr}, we study two single crystals of doped \\ch{Bi2Ch3} --- compensated \\gls{bsc} and magnetic \\gls{btm} --- each with a beam of highly polarized \\ch{^{8}Li^{+}}.\nIn many respects, \\gls{bnmr} is closely related to \\gls{musr}, but the radioactive lifetime is much longer, making the frequency range of dynamics it is sensitive to more comparable to conventional \\gls{nmr}.\nIn addition to purely electronic phenomena, in solids containing mobile species, \\gls{nmr} is also well known for its sensitivity to low frequency diffusive fluctuations~\\cite{1948-Bloembergen-PR-73-679, 1982-Kanert-PR-91-183, 1994-MullerWarmuth-PIR-17-339}, as are often encountered in intercalation compounds.\nAt ion-implantation energies sufficient to probe the bulk of \\gls{bsc} and \\gls{btm}, we find evidence for ionic mobility of \\ch{^{8}Li^{+}} above \\SI{\\sim 200}{\\kelvin}, likely due to \\gls{2d} diffusion in the \\gls{vdw} gap.\nAt low temperature, we find Korringa relaxation and a small temperature dependent negative Knight shift in \\gls{bsc}, allowing a detailed comparison with \\ch{^{8}Li} in the structurally similar \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nIn \\gls{btm}, the effects of the \\ch{Mn} moments predominate, but remarkably the signal can be followed through the magnetic transition.\nAt low temperature, we find a prominent critical peak in the relaxation that is suppressed in a high applied field, and a broad, intense resonance that is strongly shifted.\nThis detailed characterization of the \\ch{^{8}Li} \\gls{nmr} response is an important step towards using depth-resolved \\gls{bnmr} to study the low-energy properties of the chiral \\gls{tss}.\n\n\n\\section{Experiment \\label{sec:experiment}}\n\n\nDoped \\gls{ti} single crystals \\gls{bsc} and \\gls{btm} with nominal stoichiometries \\ch{Bi_{1.99}Ca_{0.01}Se_{3}} and \\ch{Bi_{1.9}Mn_{0.1}Te_{3}} were grown as described in Refs.~\\onlinecite{2009-Hor-PRB-79-195208, 2010-Hor-PRB-81-195203} and magnetically characterized using a Quantum Design \\gls{mpms}.\nIn the \\gls{btm}, a ferromagnetic transition was identified at $T_{C} \\approx \\SI{13}{\\kelvin}$, consistent with similar \\ch{Mn} concentrations~\\cite{2010-Hor-PRB-81-195203, 2013-Watson-NJP-15-103016, 2016-Zimmermann-PRB-94-125205, 2019-Vaknin-PRB-99-220404}.\nIn contrast, the susceptibility of the \\gls{bsc} crystal was too weak to measure accurately,\nbut the data show no evidence for a Curie tail at low-$T$ that could originate from dilute paramagnetic defects.\n\n\n\\gls{bnmr} experiments were performed at TRIUMF's \\gls{isac} facility in Vancouver, Canada.\nDetailed accounts of the technique can be found in Refs.~\\onlinecite{2015-MacFarlane-SSNMR-68-1, 2019-McFadden-PRB-99-125201}.\nA low-energy highly polarized beam of \\ch{^{8}Li^{+}} was implanted into the samples mounted in one of two dedicated spectrometers~\\cite{2014-Morris-HI-225-173, 2015-MacFarlane-SSNMR-68-1}.\nPrior to mounting, the crystals were cleaved in air and affixed to sapphire plates using \\ch{Ag} paint (SPI Supplies, West Chester, PA).\nThe approximate crystal dimensions were \\SI{7.8 x 2.5 x 0.5}{\\milli\\meter} (\\gls{bsc}) and \\SI{5.3 x 4.8 x 0.5}{\\milli\\meter} (\\gls{btm}).\nWith the crystals attached, the plates were then clamped to an aluminum holder threaded into an \\gls{uhv} helium coldfinger cryostat.\nThe incident \\ch{^{8}Li^{+}} ion beam had a typical flux of \\SI{\\sim e6}{ions\\per\\second} over a beam spot \\SI{\\sim 2}{\\milli\\metre} in diameter.\nAt the implantation energies $E$ used here (between \\SIrange{1}{25}{\\kilo\\electronvolt}), \\ch{^{8}Li^{+}} stopping profiles were simulated for \\num{e5} ions using the \\gls{srim} Monte Carlo code (see \\Cref{sec:implantation})~\\cite{srim}.\nFor $E > \\SI{1}{\\kilo\\electronvolt}$, a negligible fraction of the \\ch{^{8}Li^{+}} stop near enough to the\ncrystal surface to sense the \\gls{tss}.\nMost of the data is taken at \\SI{20}{\\kilo\\electronvolt}, where the implantation depth is \\SI{\\sim 100}{\\nano\\meter}, and the results thus reflect the bulk behavior.\n\n\nThe probe nucleus \\ch{^{8}Li} has nuclear spin $I=2$, gyromagnetic ratio $\\gamma \/ 2 \\pi = \\SI{6.3016}{\\mega\\hertz\\per\\tesla}$, nuclear electric quadrupole moment $Q = \\SI[retain-explicit-plus]{+32.6}{\\milli\\barn}$, and radioactive lifetime $\\tau_{\\beta} = \\SI{1.21}{\\second}$.\nThe nuclear spin is polarized in-flight by collinear optical pumping with circularly polarized light~\\cite{2014-Levy-HI-225-165}, yielding a polarization of \\SI{\\sim 70}{\\percent}~\\cite{2014-MacFarlane-JPCS-551-012059}.\nIn each measurement, we alternate the sense of circular polarization\n(left and right) of the pumping light,\nproducing either ``positive'' or ``negative'' helicity in the \\ch{^{8}Li} beam\n(i.e., its nuclear spin polarization is aligned or counter-aligned with the beam).\nData were collected separately for each helicity, which are usually combined,\nbut in some cases, are considered independently to reveal ``helicity-resolved'' properties.\nWhile helicity-resolved spectra are useful to elucidate details of resonance lines\n(see \\Cref{sec:helicities}),\ncombined spectra are helpful to remove detection systematics\n(see \\latin{e.g.},~\\cite{1983-Ackermann-TCP-31-291, 2015-MacFarlane-SSNMR-68-1}).\nThe \\ch{^{8}Li} polarization was monitored after implantation through the anisotropic radioactive $\\beta$-decay, similar to \\gls{musr}.\nSpecifically, the experimental asymmetry $A$ (proportional to the average longitudinal spin-polarization) was measured by combining the rates in two opposed scintillation counters~\\cite{1983-Ackermann-TCP-31-291, 2015-MacFarlane-SSNMR-68-1}.\nThe proportionality constant depends on the experimental geometry and the details of the $\\beta$-decay (here, on the order of \\num{\\sim 0.1}).\n\n\n\\Gls{slr} measurements were performed by monitoring the transient decay of spin-polarization both during and following a pulse of beam lasting several seconds.\nDuring the pulse, the polarization approaches a steady-state value, while after the pulse, it relaxes to essentially zero.\nAt the edge of the pulse, there is a discontinuity in the slope, characteristic of \\gls{bnmr} \\gls{slr} spectra (see \\latin{e.g.}, \\Cref{fig:bsc-slr-spectra}).\nNote that unlike conventional \\gls{nmr}, no \\gls{rf} field is required for the \\gls{slr} measurements.\nAs a result, it is generally more expedient to measure \\gls{slr} than the resonance;\nhowever, there is no spectral resolution of the relaxation, which represents the \\gls{slr} of all the \\ch{^{8}Li}.\nThe temperature dependence of the \\gls{slr} rate was measured at several applied magnetic fields $B_{0}$:\n\\SI{6.55}{\\tesla} parallel to the \\ch{Bi2Ch3} trigonal $c$-axis;\nand at lower fields \\SI{\\leq 20}{\\milli\\tesla} perpendicular to the $c$-axis.\nA typical \\gls{slr} measurement took \\SI{\\sim 20}{\\minute}.\n\n\nResonances were acquired using a \\gls{dc} \\ch{^{8}Li^{+}} beam and a \\gls{cw} transverse \\gls{rf} magnetic field $B_{1}$.\nIn this measurement mode, the \\gls{rf} frequency is stepped slowly through the \\ch{^{8}Li} Larmor frequency\n\\begin{equation*} \\label{eq:larmor}\n   \\omega_{0} = 2 \\pi \\nu_{0} = \\gamma B_{0}\n\\end{equation*}\nand the spin of any on-resonance \\ch{^{8}Li} is rapidly precessed, resulting in a loss in the average time-integrated $\\beta$-decay asymmetry.\nThe resonance amplitudes are determined by several factors:\nthe baseline asymmetry (\\latin{i.e.}, the time integral of the \\gls{slr});\nthe \\gls{rf} amplitude $B_{1}$;\nthe presence of slow, spectral \\ch{^{8}Li} dynamics occurring up to the second timescale (see \\latin{e.g.}, Ref.~\\onlinecite{2017-McFadden-CM-29-10187});\nand, for quadrupole satellite transitions, the relative populations of the magnetic sublevels are somewhat different than conventional pulsed \\gls{nmr}~\\cite{1990-Slichter-PMR, 2015-MacFarlane-SSNMR-68-1, 2019-McFadden-PRB-99-125201}.\nResonances were recorded over a temperature range of \\SIrange{4}{315}{\\kelvin} at both high and low magnetic fields.\nAt high field, the resonance frequency was calibrated against its position in a single crystal \\ch{MgO} at \\SI{300}{\\kelvin}, with the superconducting solenoid persistent.\nA single spectrum typically took \\SI{\\sim 30}{\\minute} to acquire.\n\n\n\\section{Results and analysis \\label{sec:results}}\n\n\n\\subsection{\\ch{Bi2Se3:Ca} \\label{sec:results:bsc}}\n\n\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\gls{bsc}, at both high and low magnetic field, are shown in \\Cref{fig:bsc-slr-spectra}.\nTo aid comparison, $A(t)$ has been normalized by its initial value $A_{0}$ determined from fits described below.\nClearly, the \\gls{slr} is strongly temperature and field dependent.\nAt low field, the \\gls{slr} is very much faster, due to additional relaxation from fluctuations of the host lattice nuclear spins~\\cite{2009-Hossain-PRB-79-144518}.\nThe temperature dependence of the relaxation is non-monotonic, indicating that some of the low frequency fluctuations at $\\omega_{0}$ are frozen out at low temperature.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-slr-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} \\acrlong{slr} spectra in \\ch{Ca} doped \\ch{Bi2Se3} at high and low magnetic field.\n]{ \\label{fig:bsc-slr-spectra}\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\ch{Ca} doped \\ch{Bi2Se3} at high (left) and low (right) magnetic field with \\ch{^{8}Li^{+}} implanted at \\SI{20}{\\kilo\\electronvolt}.\nThe shaded region indicates the duration of the \\ch{^{8}Li^{+}} beam pulse.\nThe relaxation is strongly field dependent, increasing at lower fields, and it increases non-monotonically with increasing temperature.\nThe solid black lines show fits to a stretched exponential described in the text.\nThe initial asymmetry $A_{0}$ from the fits is used to normalize the data which are binned by a factor of \\num{20} for clarity.\n}\n\\end{figure}\n\n\nThe relaxation is non-exponential at \\emph{all} temperatures \\emph{and} fields, so the data were fit with the phenomenological stretched exponential.\nThis approach was also used for \\ch{^{8}Li} in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201} and in conventional \\gls{nmr} of related materials~\\cite{2013-Nisson-PRB-87-195202, 2014-Koumoulis-AFM-24-1519, 2016-Levin-JPCC-120-25196}.\nExplicitly, for a \\ch{^{8}Li^{+}} implanted at time $t^{\\prime}$, the spin polarization at time $t > t^{\\prime}$ follows:\n\\begin{equation} \\label{eq:strexp}\n   R \\left ( t, t^{\\prime} \\right ) = \\exp \\left \\{ - \\left [ \\lambda \\left ( t-t^{\\prime} \\right ) \\right ]^{\\beta} \\right \\},\n\\end{equation}\nwhere $\\lambda \\equiv 1\/T_{1}$ is the \\gls{slr} rate and $0 < \\beta \\leq 1$ is the stretching exponent.\nThis is the simplest model that fits the data well with the minimal number of free parameters, for the entire \\ch{Bi2Ch3} tetradymite family of \\glspl{ti}.\n\n\nUsing \\Cref{eq:strexp} convoluted with the beam pulse, \\gls{slr} spectra in \\gls{bsc}, grouped by magnetic field $B_{0}$ and implantation energy $E$, were fit simultaneously with a shared common initial asymmetry $A_{0}(B_{0}, E)$.\nNote that the statistical uncertainties in the data are strongly time-dependent (see \\latin{e.g.}, \\Cref{fig:bsc-slr-spectra}), which must be accounted for in the analysis.\nUsing custom C++ code incorporating the MINUIT minimization routines~\\cite{1975-James-CPC-10-343} implemented within the ROOT data analysis framework~\\cite{1997-Brun-NIMA-389-81}, we find the global least-squares fit for each dataset.\nThe fit quality is good ($\\tilde{\\chi}_{\\mathrm{global}}^{2} \\approx 1.02$) and a subset of the results are shown in \\Cref{fig:bsc-slr-spectra} as solid black lines.\nThe large values of $A_{0}$ extracted from the fits (\\SI{\\sim 10}{\\percent} for $B_{0} = \\SI{6.55}{\\tesla}$ and \\SI{\\sim 15}{\\percent} for $B_{0} = \\SI{15}{\\milli\\tesla}$) are consistent with the full beam polarization, with no missing fraction.\nThe fit parameters are plotted in \\Cref{fig:bsc-slr-fits}, showing agreement with the qualitative observations above.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-slr-fits.pdf}\n\\caption[\nTemperature and field dependence of the \\ch{^{8}Li} \\acrlong{slr} rate $1\/T_{1}$ and stretching exponent $\\beta$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n]{ \\label{fig:bsc-slr-fits}\nTemperature and field dependence of the \\ch{^{8}Li} \\gls{slr} rate $1\/T_{1}$ and stretching exponent $\\beta$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n$\\beta$ is nearly independent of temperature and field at \\num{\\sim 0.6} (dotted line), except at low field around the large $1\/T_{1}$ peak seen in the bottom panel.\nThe solid and dashed black lines are global fits to \\Cref{eq:rlx}, consisting of a linear $T$-dependence with a non-zero intercept and two \\gls{slr} rate peaks, labelled with index $i$.\nIndependent of the choice of $J_{n}$ used in the analysis, the model captures all the main features of the data.\nThe $T$-linear contribution to $1\/T_{1}$ is shown as the dotted black line.\n}\n\\end{figure}\n\n\nWe now consider a model for the temperature and field dependence of $1\/T_{1}$.\nWe interpret the local maxima in $1\/T_{1}$ in \\Cref{fig:bsc-slr-fits} as \\gls{bpp} peaks~\\cite{1948-Bloembergen-PR-73-679}, caused by a fluctuating field coupled to the \\ch{^{8}Li} nuclear spin with a characteristic rate that sweeps through $\\omega_{0}$ at the peak temperature~\\cite{1948-Bloembergen-PR-73-679, 1979-Richards-TCP-15-141, 1988-Beckmann-PR-171-85}.\nPotential sources of the fluctuations are discussed below.\nThe rate peaks are superposed on a smooth background that is approximately linear, reminiscent of Korringa relaxation in metals~\\cite{1950-Korringa-P-16-601, 1990-Slichter-PMR}.\nThis is surprising, since \\gls{bsc} is a semiconductor, but it is similar to \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nWe discuss this point further in \\Cref{sec:discussion:electronic}.\n\n\nFrom this, we adopt the following model for the total \\gls{slr} rate:\n\\begin{equation} \\label{eq:rlx}\n   1\/T_{1} = a + b T + \\sum_{i} c_{i} \\left ( J_{1,i} + 4J_{2,i} \\right ) .\n\\end{equation}\nIn \\Cref{eq:rlx}, the first two terms account for the $T$-linear contribution with a finite intercept $a$, while the remaining terms describe the $i^{\\mathrm{th}}$ $1\/T_{1}$ peak in terms of a coupling constant $c_{i}$ (proportional to the mean-squared transverse fluctuating field) and the $n$-quantum \\gls{nmr} spectral density functions $J_{n,i}$~\\cite{1988-Beckmann-PR-171-85}.\nIn general, $J_{n,i}$ is frequency dependent and peaked at a temperature where the fluctuation rate matches $\\sim n \\omega_{0}$.\nWhile the precise form of $J_{n,i}$ is not known \\latin{a priori}, the simplest expression, obtained for isotropic \\gls{3d} fluctuations, has a Debye (Lorentzian) form~\\cite{1948-Bloembergen-PR-73-679, 1988-Beckmann-PR-171-85}:\n\\begin{equation} \\label{eq:j3d}\n   J_{n}^{\\mathrm{3D}} = \\frac{\\tau_{c}}{1 + \\left (n \\omega_{0} \\tau_{c} \\right )^{2}} ,\n\\end{equation}\nwhere $\\tau_{c}$ is the (exponential) correlation time of the fluctuations.\nAlternatively, when the fluctuations are \\gls{2d} in character, as might be anticipated for such a layered crystal,\n$J_{n}$ may be described by the empirical expression~\\cite{1979-Richards-TCP-15-141, 1994-Kuchler-SSI-70-434}:\n\\begin{equation} \\label{eq:j2d}\n   J_{n}^{\\mathrm{2D}} = \\tau_{c} \\ln \\left ( 1 + \\left ( n \\omega_{0} \\tau_{c} \\right )^{-2} \\right ).\n\\end{equation}\nFor both \\Cref{eq:j3d,eq:j2d}, we assume that $\\tau_{c}$ is thermally activated, following an Arrhenius dependence:\n\\begin{equation} \\label{eq:arrhenius}\n   \\tau_{c}^{-1} = \\tau_{0}^{-1} \\exp \\left ( - \\frac{ E_{A} }{ k_{B} T } \\right ),\n\\end{equation}\nwhere $E_{A}$ is the activation energy, $\\tau_{0}$ is a prefactor, $k_{B}$ is the Boltzmann constant.\nIf the fluctuations are due to \\ch{^{8}Li^{+}} hopping, $\\tau_{c}^{-1}$ is the site-to-site hop rate.\n\n\nUsing the above expressions, we fit the $1\/T_{1}$ data using a global procedure wherein the kinetic parameters (\\latin{i.e.}, $E_{A, i}$ and $\\tau_{0, i}^{-1}$) are shared at all the different $\\omega_{0}$.\nThis was necessary to fit the data at \\SI{6.55}{\\tesla} where the relaxation is very slow.\nFor comparison, we applied this procedure using both $J_{n}^{\\mathrm{3D}}$ and $J_{n}^{\\mathrm{2D}}$ and the fit results are shown in \\Cref{fig:bsc-slr-fits} as solid ($J_{n}^{\\mathrm{3D}}$) and dashed ($J_{n}^{\\mathrm{2D}}$) lines, clearly capturing the main features of the data.\nThe analysis distinguishes two processes, $i = 1, 2$ in \\Cref{eq:rlx}:\none $(i = 1)$ that onsets at lower temperature with a shallow Arrhenius slope of \\SI{\\sim 0.1}{\\electronvolt}\nthat yields the weaker peaks in $1\/T_1$ at both fields;\nand a higher barrier process $(i = 2)$ with an $E_{A}$ of \\SI{\\sim 0.4}{\\electronvolt} that\nyields the more prominent peak in the low field relaxation,\nwhile the corresponding high field peak must lie above the accessible temperature range.\nThe resulting fit parameters are given in \\Cref{tab:bsc-slr-fits}.\nWe discuss the results in \\Cref{sec:discussion:dynamics}.\n\n\n\\begin{table}\n\\centering\n\\caption[\nArrhenius parameters obtained from the analysis of the temperature dependence of $1\/T_{1}$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n]{ \\label{tab:bsc-slr-fits}\nArrhenius parameters in \\Cref{eq:arrhenius} obtained from the analysis of the temperature dependence of $1\/T_{1}$ in \\ch{Ca} doped \\ch{Bi2Se3} shown in \\Cref{fig:bsc-slr-fits}.\nThe two processes giving rise to the rate peaks are labelled with index $i$.\nGood agreement is found between the $E_{A}$s determined using the spectral density functions $J_{n}$ for \\gls{2d} and \\gls{3d} fluctuations [\\Cref{eq:j2d,eq:j3d}].\n}\n\\begin{tabular}{c S S S S}\n\\toprule\n& \\multicolumn{2}{c}{$i = 1$} & \\multicolumn{2}{c}{$i = 2$} \\\\\n$J_{n}$ & {$\\tau_{0}^{-1}$ (\\SI{e10}{\\per\\second})} & {$E_{A}$ (\\si{\\electronvolt})} & {$\\tau_{0}^{-1}$ (\\SI{e14}{\\per\\second})} & {$E_{A}$ (\\si{\\electronvolt})} \\\\\n\\midrule\n3D & 8.4 \\pm 2.7 & 0.113 \\pm 0.005 & 7 \\pm 5 & 0.395 \\pm 0.015 \\\\\n2D & 9 \\pm 3 & 0.106 \\pm 0.005 & 110 \\pm 90 & 0.430 \\pm 0.016 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nWe now turn to the \\ch{^{8}Li} resonances, with typical spectra shown in \\Cref{fig:bsc-1f-spectra-lf}.\nAs anticipated for a non-cubic crystal, the spectrum is quadrupole split, confirmed unambiguously by the helicity-resolved spectra (see \\Cref{fig:bsc-1f-spectra-helicities} in \\Cref{sec:helicities}).\nThis splitting, on the order of a few \\si{\\kilo\\hertz}~\\footnote{Note that the large $A_{0}$ down to low field precludes \\ch{^{8}Li^{+}} sites with very large \\glspl{efg}.},\nis determined by the \\gls{efg} and is a signature of the crystallographic \\ch{^{8}Li} site.\nBesides this, an unsplit component is also apparent, very close to (within \\SI{\\sim 100}{\\hertz}) the centre-of-mass of the four satellites.\nAt low temperature, the ``central'' and split components are nearly equal, but as the temperature is raised, the unsplit line grows to dominate the spectrum, accompanied by a slight narrowing.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-1f-spectra-lf.pdf}\n\\caption[\n\\ch{^{8}Li} resonance spectra in \\acrlong{bsc} at low magnetic field.\n]{ \\label{fig:bsc-1f-spectra-lf}\n\\ch{^{8}Li} resonance spectra in \\gls{bsc} at low magnetic field.\nThe vertical scale is the same for all spectra;\nthey have been normalized to account for changes in intensity due to \\gls{slr}~\\cite{2009-Hossain-PB-404-914}, with their baselines (shown as dashed grey lines) shifted to match the temperature.\nThe spectra consist of a small and nearly temperature-independent quadrupole split pattern, centred about an unsplit Lorentzian line, whose amplitude grows above \\SI{\\sim 150}{\\kelvin}.\nNote the quadrupole pattern of an integer spin nucleus like \\ch{^{8}Li}, has no central satellite (main line).\nThe solid black lines are fits to a sum of this Lorentzian plus and $2I = 4$ quadrupole satellites (see text).\n}\n\\end{figure}\n\n\nThe scale of the quadrupole splitting is determined by the product of the principal component of the \\gls{efg} tensor $eq$ with the nuclear electric quadrupole moment $eQ$.\nWe quantify this with a conventional definition of the quadrupole frequency (for $I=2$)~\\cite{1957-Cohen-SSP-5-321}:\n\\begin{equation*}\n   \\nu_{q} = \\frac{e^{2} q Q}{8 h}.\n\\end{equation*}\nIn high field, a first order perturbation treatment of the quadrupole interaction is sufficient to obtain accurate satellite positions.\nHowever, at low field, where $\\nu_{q} \/ \\nu_{0} \\approx \\SI{6}{\\percent}$, second order terms are required~\\cite{1957-Cohen-SSP-5-321, 1990-Taulelle-NASC-393}.\nBased on the change in satellite splittings by a factor \\num{2} in going from $B_{0} \\parallel c$ to $B_{0} \\perp c$, we assume the asymmetry parameter of the \\gls{efg} $\\eta = \\num{0}$ (\\latin{i.e.}, the \\gls{efg} is axially symmetric).\nThis is reasonable based on likely interstitial sites for \\ch{^{8}Li^{+}}~\\cite{2019-McFadden-PRB-99-125201}.\nPairs of helicity-resolved spectra were fit with $\\nu_{0}$ and $\\nu_{q}$ as shared free parameters, in addition to linewidths and amplitudes.\nAs the difference between the frequency of the unsplit line and the center of the quadrupole split pattern was too small to measure accurately,\nthe fits were additionally constrained to have the same central frequency $\\nu_{0}$.\nThis is identical to the approach used for \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nA subset of the results (after recombining the two helicities) are shown in \\Cref{fig:bsc-1f-spectra-lf} as solid black lines.\n\n\nThe main result is the strong temperature dependence of the resonance amplitude shown in \\Cref{fig:bsc-1f-fits-amp}.\nWhile the satellite amplitudes are nearly temperature independent, the central component increases substantially above \\SI{150}{\\kelvin}, tending to plateau above the $1\/T_{1}$ peak.\nThe other parameters are quite insensitive to temperature.\nTypical linewidths (\\latin{i.e.}, \\acrlong{fwhm}) are \\SI{\\sim 2.2}{\\kilo\\hertz} for the satellites and \\SI{\\sim 3.8}{\\kilo\\hertz} for the central component.\nThe quadrupole frequency $\\nu_{q} \\approx \\SI{5.5}{\\kilo\\hertz}$ varies weakly, increasing slightly as temperature is lowered.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-1f-fits-amp.pdf}\n\\caption[\nThe resonance amplitude as a function of temperature in \\ch{Ca} doped \\ch{Bi2Se3} at \\SI{15}{\\milli\\tesla}.\n]{ \\label{fig:bsc-1f-fits-amp}\nThe resonance amplitude as a function of temperature in \\ch{Ca} doped \\ch{Bi2Se3} at \\SI{15}{\\milli\\tesla}.\nWhile the amplitude of the satellite lines are nearly temperature independent, the central component increases substantially above \\SI{150}{\\kelvin}, plateauing on the high-$T$ side of the $1\/T_{1}$ maximum (grey band).\nThe solid line is a guide, while the dashed line indicates the estimated saturation value for the Lorentizan component.\n}\n\\end{figure}\n\n\nWe also measured resonances at room and base temperature in high field (\\SI{6.55}{\\tesla}) where \\ch{^{8}Li} is sensitive to small magnetic shifts.\nFrom the fits, we use $\\nu_{0}$ to calculate the raw relative frequency shift $\\delta$ in parts per million (\\si{\\ppm}) using:\n\\begin{equation} \\label{eq:shift}\n   \\delta = 10^{6} \\left ( \\frac{\\nu_{0} - \\nu_{\\ch{MgO}} }{ \\nu_{\\ch{MgO}} } \\right ) ,\n\\end{equation}\nwhere $\\nu_{\\ch{MgO}}$ is the reference frequency position in \\ch{MgO} at \\SI{300}{\\kelvin}.\nThe shifts are small:\n\\SI[retain-explicit-plus]{+12 \\pm 2}{\\ppm} at room temperature and \\SI{-17 \\pm 3}{\\ppm} at \\SI{5}{\\kelvin}, the latter considerably smaller in magnitude than in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nBecause \\ch{^{8}Li} \\gls{nmr} shifts are generally so small, it is essential to account for the demagnetization field of the sample itself.\nFrom $\\delta$, the corrected shift $K$ is obtained by the \\gls{cgs} expression~\\cite{2008-Xu-JMR-191-47}:\n\\begin{equation}\n   K = \\delta + 4 \\pi \\left ( N - \\frac{1}{3} \\right ) \\chi_{v}\n\\end{equation}\nwhere $N$ is the dimensionless demagnetization factor that depends only on the shape of the sample and $\\chi_{v}$ is the dimensionless (volume) susceptibility.\nFor a thin film, $N$ is \\num{1}~\\cite{2008-Xu-JMR-191-47}, but for the thin platelet crystals used here, we estimate $N$ is on the order of \\num{\\sim 0.8}, treating them as oblate ellipsoids~\\cite{1945-Osborn-PR-67-351}.\nFor the susceptibility, we take the average of literature values reported for pure \\ch{Bi2Se3}~\\cite{1958-Matyas-CJP-8-309, 2003-Kulbachinskii-PB-329-1251, 2015-Pafinlov-JPCM-27-456002, 2016-Chong-JAC-686-245}, giving $\\chi_{v}^{\\mathrm{CGS}} \\approx \\SI{-2.4e-6}{\\emu\\per\\centi\\meter\\cubed}$.\nNote that we have excluded several reports~\\cite{2008-Janicek-PB-403-3553, 2012-Young-PRB-86-075137, 2014-Zhao-NM-13-580} whose results disagree by an order of magnitude from those predicted by Pascal's constants~\\cite{2008-Bain-JCE-85-532}.\nApplying the correction for \\gls{bsc} yields $K$s of \\SI[retain-explicit-plus]{-2 \\pm 2}{\\ppm} and \\SI{-31 \\pm 3}{\\ppm} at room and base temperature, respectively.\nWe discuss this below in \\Cref{sec:discussion:electronic}.\n\n\n\\subsection{\\ch{Bi2Te3:Mn} \\label{sec:results:btm}}\n\n\nTypical \\ch{^{8}Li} \\gls{slr} spectra at high and low field in the magnetically doped \\gls{btm} are shown in \\Cref{fig:btm-slr-spectra}.\nIn contrast to nonmagnetic \\gls{bsc}, the relaxation at high field is fast, typical of paramagnets with unpaired electron spins~\\cite{2015-MacFarlane-PRB-92-064409, 2016-Cortie-PRL-116-106103,2011-Song-PRB-84-054414}.\nThe fast high field rate produces a much less pronounced field dependence.\nAt low field, the \\gls{slr} rate is peaked at low temperature.\nThe relaxation is also non-exponential and fits well using \\Cref{eq:strexp}, with a stretching exponent systematically smaller than in the nonmagnetic \\gls{bsc} or \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nWe analyzed the data with the same global approach, obtaining good quality fits ($\\tilde{\\chi}_{\\mathrm{global}}^{2} \\approx 1.01$) demonstrated by the solid black lines in \\Cref{fig:btm-slr-spectra}.\nThe shared values of $A_{0}$ from the fits are large (\\SI{\\sim 10}{\\percent} for $B_{0} = \\SI{6.55}{\\tesla}$ and \\SI{\\sim 15}{\\percent} for $B_{0} = \\SI{20}{\\milli\\tesla}$), consistent with the full beam polarization, implying that there is remarkably no magnetic wipeout from very fast relaxation~\\cite{2015-MacFarlane-PRB-92-064409}, even at low field.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-slr-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} \\acrlong{slr} spectra in \\ch{Mn} doped \\ch{Bi2Te3} at high and low magnetic field.\n]{ \\label{fig:btm-slr-spectra}\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\ch{Mn} doped \\ch{Bi2Te3} at high (left) and low (right) magnetic field for \\ch{^{8}Li^{+}} implantation energies of \\SI{20}{\\kilo\\electronvolt} and \\SI{8}{\\kilo\\electronvolt}, respectively.\nThe shaded region denotes the duration of the \\ch{^{8}Li^{+}} beam pulse.\nThe \\gls{slr} is substantial and orders of magnitude faster than in \\gls{bsc} (see \\Cref{fig:bsc-slr-spectra}) at high field.\nThe field dependence to the \\gls{slr} is much weaker than in the nonmagnetic tetradymites.\nThe solid black lines are fits to a stretched exponential convoluted with the \\ch{^{8}Li^{+}} beam pulse as described in the text.\nThe initial asymmetry $A_{0}$ from the fit is used to normalize the spectra.\nThe high and low field spectra have been binned for by factors of \\num{20} and \\num{5}, respectively.\n}\n\\end{figure}\n\n\nThe fit parameters are shown in \\Cref{fig:btm-slr-fits}.\nAt all temperatures, especially at high field, the \\gls{slr} rate $1\/T_{1}$ is orders of magnitude larger than in the nonmagnetic analogs.\nNo clear $1\/T_{1}$ \\gls{bpp} peaks can be identified between \\SIrange[range-units=single,range-phrase=--]{100}{300}{\\kelvin};\nhowever, in the low field data, a critical divergence is evident at the magnetometric transition at about \\SI{13}{\\kelvin}.\nIn high field, this feature is largely washed out, with a remnant peak near \\SI{50}{\\kelvin}.\nAbove \\SI{200}{\\kelvin}, the \\gls{slr} rate increases very rapidly and is well-described by $1\/T_{1} \\propto \\exp [ - E_{A} \/ ( k_{B} T )]$,\nwith $E_{A} \\approx \\SI{0.2}{\\electronvolt}$ at both fields.\nWe discuss this below in \\Cref{sec:discussion:dynamics}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-slr-fits.pdf}\n\\caption[\nTemperature dependence of the \\ch{^{8}Li} \\acrlong{slr} rate $1\/T_{1}$ in \\ch{Mn} doped \\ch{Bi2Te3} at high and low field.\n]{ \\label{fig:btm-slr-fits}\nTemperature dependence of the \\ch{^{8}Li} \\gls{slr} rate $1\/T_{1}$ in \\ch{Mn} doped \\ch{Bi2Te3} at high and low field.\nAt low field, $1\/T_{1}$ shows a critical peak at the ferromagnetic transition at $T_{C} \\approx \\SI{13}{\\kelvin}$, as the \\ch{Mn} spin fluctuations freeze out.\nAbove \\SI{200}{\\kelvin}, the \\gls{slr} rate increases exponentially in manner nearly independent of applied field.\nThe solid grey lines are drawn to guide the eye.\n}\n\\end{figure}\n\n\nIn contrast to \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, the resonance in \\gls{btm} consists of a single broad Lorentzian with none of the resolved fine structure (see \\Cref{fig:btm-1f-spectra}).\nSurprisingly, the very broad line has significant intensity, dwarfing the quadrupole pattern in \\gls{bsc} in both width and amplitude.\nIn addition, there is a large negative shift at \\SI{10}{\\kelvin}.\nAt room temperature, the line is somewhat narrower, and the shift is reduced in magnitude.\nQuantitative results from Lorentzian fits are summarized in \\Cref{tab:btm-1f-fits}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-1f-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} resonances in \\ch{Mn} doped \\ch{Bi2Te3} and \\ch{Ca} doped \\ch{Bi2Se3} at high magnetic field.\n]{ \\label{fig:btm-1f-spectra}\nTypical \\ch{^{8}Li} resonances in \\ch{Mn} doped \\ch{Bi2Te3} and \\ch{Ca} doped \\ch{Bi2Se3} at high magnetic field.\nThe vertical scale is the same for both spectra;\nthey have been normalized to account for changes in intensity and baseline~\\cite{2009-Hossain-PB-404-914}.\nThe lineshape in the magnetic \\gls{btm} is well-described by a broad Lorentzian (solid black line) with no quadrupolar splitting.\nA large negative shift is also apparent for \\gls{btm} with respect to the reference frequency in \\ch{MgO} (vertical dashed line).\nThe dotted vertical line indicates the expected resonance position due to demagnetization, revealing a large positive hyperfine field (\\SI{\\sim 38}{\\gauss}) at \\SI{5}{\\kelvin} in the magnetic state.\n}\n\\end{figure}\n\n\n\\begin{table}\n\\centering\n\\caption[\nResults from the analysis of the \\ch{^{8}Li} resonance in \\acrlong{btm} at high and low temperature with $B_{0} = \\SI{6.55}{\\tesla} \\parallel (001)$.\n]{ \\label{tab:btm-1f-fits}\nResults from the analysis of the \\ch{^{8}Li} resonance in \\gls{btm} at high and low temperature with $B_{0} = \\SI{6.55}{\\tesla} \\parallel (001)$.\nThe (bulk) magnetization $M$ measured with $\\SI{1.0}{\\tesla} \\parallel (001)$ is included for comparison.\n}\n\\begin{tabular}{S S S S[retain-explicit-plus] S}\n\\toprule\n{$T$ (\\si{\\kelvin})} & {$\\tilde{A}$ (\\si{\\percent})} & {\\acrshort{fwhm} (\\si{\\kilo\\hertz})} & {$\\delta$ (\\si{\\ppm})} & {$M$ (\\si{\\emu\\per\\centi\\meter\\cubed})} \\\\\n\\midrule\n294 & 33 \\pm 5     & 16.2 \\pm 1.2 &  +10 \\pm 9 &  0.076 \\\\\n 10 & 14.4 \\pm 0.8 & 41.7 \\pm 1.6 & -206 \\pm 12 & 7.698 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Discussion \\label{sec:discussion}}\n\n\nThe \\ch{^{8}Li} \\gls{nmr} properties of nonmagnetic \\gls{bsc} are quite similar to previous measurements in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nThe resonance spectra show a similar splitting ($\\nu_q$ is about \\SI{25}{\\percent} smaller in \\gls{bsc}), indicating a similar site for \\ch{^{8}Li}.\nThe resemblance of the spectra extends to the detailed temperature dependence, including the growth of the unsplit line approaching room temperature.\nSurprisingly, the \\gls{bsc} spectra are better resolved than \\gls{bts}, implying a higher degree of order, despite the \\ch{Ca} doping.\nThis is also evident in the \\gls{slr}, with a stretching exponent $\\beta$ closer to unity in \\gls{bsc} than in \\gls{bts}.\nThis likely reflects additional disorder in \\gls{bts} from \\ch{Bi\/Te} anti-site defects~\\cite{2012-Scanlon-AM-24-2154} that are much more prevalent than for \\ch{Bi\/Se}, due to the difference in radii and electronegativity.\nThe sharp quadrupolar pattern indicates a well-defined crystallographic \\ch{^{8}Li^{+}} site, and the corresponding small \\gls{efg} suggests it is in the \\gls{vdw} gap.\n\\Gls{dft} calculations of the \\gls{efg} may enable a precise site assignment.\nThe high field \\gls{slr} is also similar to \\gls{bts}:\nit is slow and near the lower limit measurable due to the finite \\ch{^{8}Li} lifetime $\\tau_\\beta$ and comparable to the \\gls{vdw} metal \\ch{NbSe2}~\\cite{2006-Wang-PB-374-239}, where the carrier concentration is much higher, but significantly slower than the \\gls{ti} alloy \\ch{Bi_{1-x}Sb_{x}}~\\cite{2014-MacFarlane-PRB-90-214422}.\nThe low field enhancement of $1\/T_1$ is also similar, so it cannot be essentially related to the dilute \\ch{^{125}Te} moments that are absent in \\gls{bsc}, but probably determined primarily by the \\SI{100}{\\percent} abundant \\ch{^{209}Bi}.\nFrom such a detailed similarity, it is clear that a quantitative comparison with \\gls{bts} and other \\gls{vdw} materials will be useful.\n\n\nWith these similarities in mind, the rest of the discussion is organized as follows: in \\Cref{sec:discussion:dynamics},\nwe consider evidence of mobility of the \\ch{^{8}Li^{+}} ion;\nin \\Cref{sec:discussion:electronic}, electronic effects at low temperature in \\gls{bsc};\nand the magnetic properties of \\gls{btm} in \\Cref{sec:discussion:magnetic}.\n\n\n\\subsection{Dynamics of the \\ch{Li^{+}} ion \\label{sec:discussion:dynamics}}\n\nIn \\gls{bts}, we considered if the evolution of the spectrum with temperature (similar to \\Cref{fig:bsc-1f-spectra-lf}) was the result of a site change transition from a meta-stable quadrupolar \\ch{^{8}Li^{+}} site to a lower energy site with very small \\gls{efg} at higher temperature~\\cite{2019-McFadden-PRB-99-125201}, similar to elemental \\ch{Nb}~\\cite{2009-Parolin-PRB-80-174109}.\nWe now consider an alternative explanation; namely, dynamic averaging of the quadrupolar interaction due to \\ch{^{8}Li^{+}} motion.\nExamples of this are found in conventional \\ch{^{7}Li} \\gls{nmr}, where, unlike \\ch{^{8}Li}, the $I = 3\/2$ quadrupole spectrum has a main line (the $m = \\pm 1\/2$ satellite) overlapping the averaged resonance~\\cite{2012-Galven-CM-24-3335, 2012-Indris-JPCC-116-14243}.\nDynamic averaging is suggested by the onset near, but below, the \\gls{slr} rate peak (see \\Cref{fig:bsc-1f-fits-amp}).\nHowever, for hopping between equivalent interstitial sites (probably the quasi-octahedral Wyckoff $3b$ site in the \\gls{vdw} gap --- see Figure~12 in Ref.~\\onlinecite{2019-McFadden-PRB-99-125201}),\none does not expect that the \\gls{efg} will average to a value near zero (required to explain the unsplit line).\nA point charge estimate reveals the quasi-tetrahedral (Wyckoff $6c$) site, thought to be the saddle point in the potential for \\ch{Li^{+}} between adjacent $3b$ sites~\\cite{2016-Gosalvez-PRB-93-075429}, has an \\gls{efg} of opposite sign to the $3b$ site.\nIf $6c$ is instead a shallow minimum, the \\ch{^{8}Li^{+}} residence time there may be long enough that the \\gls{efg} averages to near zero.\nIn the fast motion limit at higher temperatures, one would then expect the quadrupole splitting to re-emerge when the residence time in the $6c$ ``transition'' site becomes much shorter~\\cite{2012-Indris-JPCC-116-14243}.\n\n\nWe now examine the two kinetic processes causing the $1\/T_{1}$ peaks in \\gls{bsc}.\nIt is surprising to find two distinct thermally activated processes sweeping through the \\gls{nmr} frequency, especially since only a single process was found in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nFirst, we consider the weaker feature, the low temperature $(i=1)$ peaks.\nIn layered materials, small intercalates can undergo highly localized motion at relatively low temperatures below the onset of free diffusion~\\cite{1982-Kanert-PR-91-183, 1994-MullerWarmuth-PIR-17-339}.\nSuch local motion may be the source of the \\gls{slr} rate peak, but it is quite ineffective at narrowing the resonance, consistent with the absence of any lineshape changes in the vicinity of the $i=1$ peaks.\nCaged local motion is usually characterized by a small activation barrier, comparable to the \\SI{\\sim 0.1}{\\electronvolt} observed here.\nSimilar phenomena have been observed at low temperature, for example, in neutron activated \\ch{^{8}Li} \\gls{bnmr} of \\ch{Li} intercalated graphite, \\ch{LiC12}~\\cite{1989-Schirmer-SM-34-589, 1995-Schirmer-ZNA-50-643}.\nIt is not clear why such motion would be absent in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, which has a larger \\gls{vdw} gap than \\gls{bsc} (\\SI{2.698}{\\angstrom} vs.\\ \\SI{2.568}{\\angstrom}).\nAlternatively, this feature in the relaxation may have an electronic origin,\nperhaps related to the emergent low-$T$ magnetism in \\ch{MoTe2} observed\nby \\gls{musr}~\\cite{2018-Guguchia-SA-4-eaat3672} and \\ch{^{8}Li} \\gls{bnmr}~\\cite{Krieger-tbp}.\n\n\nIn contrast, the \\gls{slr} rate peak above \\SI{200}{\\kelvin} $(i = 2)$ is almost certainly due to \\gls{efg} fluctuations caused by stochastic \\ch{^{8}Li^{+}} motion.\nFrom the data, we cannot conclude that this is long-range diffusion, but the room temperature \\ch{Li^{+}} intercalability of \\ch{Bi2Se3}~\\cite{1988-Paraskevopoulos-MSEB-1-147, 1989-Julien-SSI-36-113, 2010-Bludska-JSSC-183-2813} suggests it is.\nIts barrier, on the order of \\SI{\\sim 0.4}{\\electronvolt}, is comparable to other \\gls{vdw} gap layered ion conductors, but it is about twice as high as in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, possibly a result of the \\ch{Se} (rather than \\ch{Te}) bounded \\gls{vdw} gap, which provides less space between neighbouring \\glspl{ql}.\n\n\nWe now consider the Arrhenius law prefactors $\\tau_{0}^{-1}$, that, for ionic diffusion, are typically in the range \\SIrange[range-phrase=--,range-units=single]{e12}{e14}{\\per\\second}.\nFor the low-$T$ process ($i = 1$), independent of the form of $J_{n}$ (see \\Cref{tab:bsc-slr-fits}), $\\tau_{0}^{-1} \\approx \\SI{9e10}{\\per\\second}$ is unusually low.\nIn contrast, for the high-$T$ ($i = 2$) process, it is much larger and depends strongly on $J_{n}$.\nFor \\gls{3d} diffusion, it is in the expected range, while the \\gls{2d} model yields an extremely large value, \\SI{\\sim e16}{\\per\\second},\nin the realm of prefactor anomalies~\\cite{1983-Villa-SSI-9-1421} and opposite to the small value expected for\nlow dimensional diffusion~\\cite{1978-Richards-SSC-25-1019}.\nSimilar behaviour was observed recently in \\ch{^{7}Li} \\gls{nmr} of \\ch{LiC6}~\\cite{2013-Langer-PRB-88-094304},\nwhere surprisingly, $J_{n}^{\\mathrm{2D}}$ was concluded to be less appropriate than $J_{n}^{\\mathrm{3D}}$,\nsuggesting that \\ch{Li} motion in the \\gls{vdw} gap is not as ideally \\gls{2d} as might be expected.\nIn \\gls{bsc}, the anomaly may be related to local dynamics that onset at lower $T$, imparting some \\gls{3d} character to the motion.\n\n\nGiven the evidence for long-range \\ch{Li^{+}} motion in \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, the absence of a relaxation peak in \\gls{btm} may seem unexpected.\nBoth \\ch{Ca^{2+}} and \\ch{Mn^{2+}} dopants (substitutional for \\ch{Bi^{3+}}) have an effective $-1$ charge yielding an attractive trapping potential for the positive interstitial \\ch{^{8}Li^{+}}, but the \\ch{Mn} concentration is an order of magnitude larger.\nThe high trap density in \\gls{btm} will suppress \\ch{Li^{+}} mobility.\nThe exponential increase in $1\/T_{1}$ above \\SI{200}{\\kelvin} may be the onset of a diffusive \\gls{bpp} peak, but, in this case, one does not expect it to be so similar between the two very different magnetic fields.\nThis may reflect a trade-off between the increase in $\\omega_{0}$ that shifts the peak to higher temperature, slowing the relaxation on its low-$T$ flank, and the increased polarization of the \\ch{Mn} moments by the field that amplifies local magnetic inhomogeneities.\nA motional origin for this increase is consistent with the apparent $E_{A} \\sim \\SI{0.2}{\\electronvolt}$, similar to \\ch{^{8}Li^{+}} in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, which also has a \\ch{Te} bounded \\gls{vdw} gap of similar size to \\gls{btm} (\\SI{2.620}{\\angstrom}).\nHowever, it may have a different explanation, see below in \\Cref{sec:discussion:magnetic}.\n\n\n\\subsection{Electronic effects at low temperature \\label{sec:discussion:electronic}}\n\n\nBismuth chalcogenide (\\ch{Bi2Ch3}) crystals exhibit substantial bulk conductivity, despite a narrow gap in the \\gls{3d} band structure, making it difficult to distinguish effects of the metallic \\gls{tss}.\nThis is due to native defects (\\latin{e.g.}, \\ch{Ch} vacancies) that are difficult or impossible to avoid~\\cite{2013-Cava-JMCC-1-3176}.\nExtrinsic dopants, such as substitutional \\ch{Ca\/Bi}, can be used to compensate the spontaneous $n$-type doping.\nBrahlek \\latin{et al.}\\ have argued~\\cite{2015-Brahlek-SSC-215-54} that, even for the most insulating compensated samples, the carrier densities far exceed the Mott criterion, making them heavily doped semiconductors in the metallic regime.\nIn this case, we expect metallic \\gls{nmr} characteristics~\\cite{1978-Holcomb-SUSSP-19-251}, namely a magnetic Knight shift $K$, proportional to the carrier spin susceptibility $\\chi_{s}$~\\footnote{%\nThe \\gls{nmr} shift $K$ is comprised of several terms including both the spin and orbital response of all the surrounding electrons. In metals, where the local carrier density is substantial,\nthe Fermi contact coupling with the electron spins often dominates, as we have assumed here.%\n}.\nIn the simplest (isotropic) case,\n\\begin{equation} \\label{eq:knight-shift}\n   K = A \\chi_{s},\n\\end{equation}\nwhere $A$ is the hyperfine coupling constant, which is accompanied by a \\gls{slr} rate following the Korringa law~\\cite{1950-Korringa-P-16-601, 1990-Slichter-PMR},\n\\begin{equation} \\label{eq:korringa-rate}\n   \\frac{1}{T_{1}} = 4 \\pi \\hbar A^{2} \\gamma_{n}^{2} \\left ( \\frac{\\chi_{s}}{g^{*}\\mu_{B}} \\right )^{2} k_{B} T.\n\\end{equation}\nHere, $\\gamma_{n}$ is the nuclear gyromagnetic ratio, $g^{*}$ is the carrier $g$-factor, and $\\mu_{B}$ is the Bohr magneton.\nCombining \\Cref{eq:knight-shift,eq:korringa-rate}, we obtain the Korringa product, which is independent of the value of $A$,\n\\begin{equation} \\label{eq:korringa-product}\n   T_{1}TK^{2} = \\frac{\\hbar (g^{*} \\mu_B)^{2}}{4 \\pi k_{B} \\gamma_{n}} = S(g^{*}).\n\\end{equation}\nFor \\ch{^{8}Li},\n\\begin{equation*}\n   S(g^{*}) \\approx 1.20 \\times 10^{-5} \\left ( \\frac{ g^{*} }{g_0} \\right )^{2}~\\si{\\second\\kelvin},\n\\end{equation*}\nwhere, unlike in metals, we have allowed for an effective $g$-factor that may be far from its free electron value\n$g_0 \\approx 2$~\\cite{1972-Look-PRB-5-3406}.\nIndeed, recent \\gls{epr} measurements in \\ch{Bi2Se3} find $g^{*} \\approx 30$~\\cite{2016-Wolos-PRB-93-155114}.\n\n\nAccording to Ref.~\\onlinecite{2015-Brahlek-SSC-215-54}, \\gls{bts} and \\gls{bsc} lie on opposite sides of the Ioffe-Regel limit, where the carrier mean free path is equal to its Fermi wavelength, with \\gls{bsc} having a higher carrier density and mobility.\nA comparative Korringa analysis could test this assertion and, to this end, using \\Cref{eq:korringa-product} we define the dimensionless Korringa ratio as\n\\begin{equation} \\label{eq:korringa-constant}\n   \\mathscr{K} = \\frac{T_{1}TK^2}{S(g^{*})}.\n\\end{equation}\nBelow the Ioffe-Regel limit, the autocorrelation function of the local hyperfine field at the nucleus, due to the carriers (that determines $T_{1}$) becomes limited by the diffusive transport correlation time.\nThis has been shown to enhance the Korringa rate (\\latin{i.e.}, shortening $T_{1}$)~\\cite{1971-Warren-PRB-3-3708, 1983-Gotze-ZPB-54-49}.\nFrom this, one expects $\\mathscr{K}$ would be smaller in \\gls{bts} than in \\gls{bsc}.\n\n\nThere are, however, significant difficulties in determining the experimental $\\mathscr{K}$.\nFirst, the Korringa slope depends on magnetic field.\nAt low fields, this is due to coupling with the host nuclear spins, a phenomenon that is quenched in high fields where the \\ch{^{8}Li} \\gls{nmr} has no spectral overlap with the \\gls{nmr} of host nuclei.\nFor example, in simple metals, we find the expected field-independent Korringa slope at high fields in the Tesla range~\\cite{2015-MacFarlane-SSNMR-68-1}.\nIn contrast, in \\gls{bts}, the slope decreases substantially with increasing field, even at high fields~\\cite{2019-McFadden-PRB-99-125201}.\nWe suggested this could be the result of magnetic carrier freeze-out.\nWhile we do not have comparably extensive data in \\gls{bsc}, we can compare the slope at the same field, \\SI{6.55}{\\tesla} (see \\Cref{tab:korringa}).\nHere, in both materials, the relaxation is extremely slow, exhibiting no curvature in the \\gls{slr} during the \\ch{^{8}Li} lifetime\n(\\Cref{fig:bsc-slr-spectra}), so the uncertainties in $1\/T_{1}T$ are likely underestimates.\nThe larger slope is, however, consistent with a higher carrier density $n$ in \\gls{bsc}.\nThe Korringa slopes should scale~\\cite{1972-Look-PRB-5-3406} as $n^{2\/3}$.\nUsing $n \\sim \\SI{1e19}{\\per\\centi\\meter\\cubed}$ in \\gls{bsc}~\\cite{2009-Hor-PRB-79-195208} and \\SI{\\sim 2e17}{\\per\\centi\\meter\\cubed} in \\gls{bts}~\\cite{2011-Jia-PRB-84-235206}, the slopes should differ by a factor of \\num{\\sim 14}, while experimentally the ratio is \\num{\\sim 5}.\n\n\nThe next difficulty is accurately quantifying the shift $K$, which is quite small with a relatively large demagnetization correction.\nExperimentally, the zero of shift, defined by the calibration in \\ch{MgO}, differs from the true zero (where $\\chi_{s} = 0$) by the difference in chemical (orbital) shifts between \\ch{MgO} and the chalcogenide.\nHowever, because \\ch{Li} chemical shifts are universally very small, this should not be a large difference, perhaps a few \\si{\\ppm}.\nThe negative low temperature shift is also somewhat surprising.\nThe hyperfine coupling $A$ for \\ch{Li} is usually determined by a slight hybridization of the vacant $2s$ orbital with the host conduction band.\nAs the $s$ orbital has density at the nucleus, the resulting coupling is usually positive, with the $d$ band metals \\ch{Pd} and \\ch{Pt} being exceptional~\\cite{2015-MacFarlane-SSNMR-68-1}.\nFor a positive $A$, the sign of $K$ is determined by the sign of $g^{*}$, which has not yet been\nconclusively measured in either \\gls{bsc} or \\gls{bts}.\nA more serious concern is that $K$ depends on temperature (in contrast to simple metals) and, at least in \\gls{bts}, also on applied field~\\cite{2019-McFadden-PRB-99-125201}.\nTo avoid ambiguity from the field dependence, we similarly restrict comparison to the same field, \\SI{6.55}{\\tesla}.\nA similarly temperature dependent shift (for the \\ch{^{207}Pb} \\gls{nmr}) was found in the narrow band semiconductor \\ch{PbTe}~\\cite{1973-Hewes-PRB-7-5195}, where it was explained by the temperature dependence of the Fermi level $E_{F}$~\\cite{1970-Senturia-PRB-1-4045}.\nAt low-$T$ in the heavily $p$-type \\ch{PbTe}, $E_{F}$ occurs in the valence (or a nearby impurity) band, but with increasing temperature, moves upward into the gap, causing a reduction in $|K|$.\nWith this in mind, we assume the low temperature shift is the most relevant for a Korringa comparison.\nWithout a measured $g^{*}$ in \\gls{bts}, we simply assume it is the same as \\gls{bsc}, and use the $g^{*}_{\\parallel}$ from \\gls{epr}~\\cite{2016-Wolos-PRB-93-155114} to calculate the values of $\\mathscr{K}$ in \\Cref{tab:korringa}.\n\n\n\\begin{table}\n\\centering\n\\caption[\nKorringa analysis of \\acrlong{bsc} and \\acrlong{bts} at \\SI{6.55}{\\tesla} and low temperature.\n]{ \\label{tab:korringa}\nKorringa analysis of \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201} at \\SI{6.55}{\\tesla} and low temperature.\nTo calculate $\\mathscr{K}$, we take $S(g^{*})$ in \\Cref{eq:korringa-constant} to be \\SI{2.69e-3}{\\second\\kelvin}, using the $g_{\\parallel}^{*}$ from \\gls{epr}~\\cite{2016-Wolos-PRB-93-155114}.\n}\n\\begin{tabular}{l S S S}\n\\toprule\n    & {$1\/(T_{1}T)$ (\\SI{e-6}{\\per\\second\\per\\kelvin})} & {$K$ (\\si{\\ppm})} & {$\\mathscr{K}$} \\\\\n\\midrule\n\\acrlong{bsc} & 9.5  \\pm 0.8  &  -31 \\pm 3 &   0.038 \\pm 0.008 \\\\\n\\acrlong{bts} & 1.79 \\pm 0.07 & -115 \\pm 3 &   2.78  \\pm 0.18  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nThe values of $\\mathscr{K}$ are just opposite to the expectation of faster relaxation for diffusive \\gls{bts} compared to metallic\n\\gls{bsc}~\\cite{2015-Brahlek-SSC-215-54}.\nElectronic correlations can, however, significantly alter the Korringa ratio to an extent that depends on disorder~\\cite{1994-Shastry-PRL-72-1933}.\nThere should be no significant correlations in the broad bulk bands of the chalcogenides, but in narrow impurity bands, this is certainly a possibility.\nWe note that $\\mathscr{K}$ is also less than \\num{1} in \\ch{PbTe}~\\cite{1973-Alexander-JN-1-251}, similar to \\gls{bsc}.\nAt this stage, without more data and a better understanding of the considerations mentioned above, it is premature to draw further conclusions.\n\n\n\\subsection{Magnetism in \\acrlong{btm} \\label{sec:discussion:magnetic}}\n\n\nIn the \\ch{Mn} doped \\ch{Bi2Te3} at low field, the relaxation from magnetic \\ch{Mn^{2+}} becomes faster as the spin fluctuations slow down on cooling towards $T_{C}$.\nIn particular, the low-$T$ increase in $1\/T_{1}$ occurs near where correlations among the \\ch{Mn} spins become evident in \\gls{epr}~\\cite{2016-Zimmermann-PRB-94-125205, 2017-Talanov-AMR-48-143}.\nIn remarkable contrast to ferromagnetic \\ch{EuO}~\\cite{2015-MacFarlane-PRB-92-064409}, the signal is not wiped out in the vicinity of $T_{C}$, but $1\/T_{1}$ does become very fast.\nThis is likely a consequence of a relatively small hyperfine coupling consistent with a \\ch{Li} site in the \\gls{vdw} gap.\n\n\nHigh applied field slows the \\ch{Mn} spins more continuously starting from a higher temperature, suppressing the critical peak and reducing $1\/T_{1}$ significantly, a well-known phenomenon in \\gls{nmr} and \\gls{musr} at magnetic transitions (see \\latin{e.g.}, Ref.~\\onlinecite{2001-Heffner-PRB-63-094408}).\nThis also explains the small critical peak in the \\ch{^{8}Li} \\gls{slr} in the dilute magnetic semiconductor, \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}.\nAs in \\ch{GaAs}, \\ch{Mn} in \\ch{Bi2Te3} is both a magnetic and electronic dopant.\nAt this concentration, \\gls{btm} is $p$-type with a metallic carrier density of \\SI{\\sim 7e19}{\\per\\centi\\meter\\cubed}~\\cite{2010-Hor-PRB-81-195203, 2016-Zimmermann-PRB-94-125205}.\nHowever, the difference in scale of $1\/T_{1}$ at high field between \\Cref{fig:btm-slr-fits,fig:bsc-slr-fits} shows that the \\ch{Mn} spins completely dominate the carrier relaxation.\n\n\nIt is also remarkable that the resonance is so clear in the magnetic state, in contrast to \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}.\nThe difference is not the linewidth, but rather the enhanced amplitude.\nThis may be due to slow \\ch{^{8}Li} spectral dynamics occurring on the timescale of $\\tau_{\\beta}$ that effectively enhance the amplitude, \nfor example, slow fluctuations of the ordered \\ch{Mn} moments, not far below $T_{C}$.\nSimilar behavior was found in rutile \\ch{TiO2} at low temperature, where it was attributed to field fluctuations due to a nearby electron polaron~\\cite{2017-McFadden-CM-29-10187}.\nEnhancement of the \\gls{rf} field at nuclei in a ferromagnet~\\cite{1959-Gossard-PRL-3-164} may also play a role.\n\n\nAbove \\SI{200}{\\kelvin}, the activated increase in $1\/T_{1}$ may indicate the onset of diffusive \\ch{^{8}Li^{+}} motion, similar to the nonmagnetic analogs, with the additional effect that the local magnetic field from the \\ch{Mn} spins is also modulated by \\ch{^{8}Li} hopping (not just the \\gls{efg}), similar to the ordered magnetic \\gls{vdw} layered \\ch{CrSe2}~\\cite{2020-Ticknor-RSCA-10-8190}.\nHowever, it may instead mark the onset of thermal excitation of electrons across the bandgap that is narrowed by \\ch{Mn} doping\\cite{2010-Hor-PRB-81-195203}.\nThermally excited carriers may also explain the increase in $1\/T_1$ at comparable temperatures in \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}, a very different medium for \\ch{Li^+} diffusion.\nThermally increased carrier density would strengthen interaction between the \\ch{Mn} moments, extend their effects via \\gls{rkky} polarization, and increase $1\/T_{1}$.\nMeasurements at higher temperatures may be able to discriminate these possibilities, but it is likely that both processes will contribute.\n\n\nThe stretched \\gls{slr} and field-dependent $1\/T_{1}$\nare characteristic of the \\gls{nmr} response in a disordered glassy magnetic state, consistent with\nthe random Mn\/Bi site disorder. The magnetic properties of \\gls{btm}~\\cite{2010-Hor-PRB-81-195203, 2013-Watson-NJP-15-103016, 2016-Zimmermann-PRB-94-125205, 2019-Vaknin-PRB-99-220404} are similar to the dilute (ferro)magnetic semiconductors that include a uniform magnetization of the carriers. In this context, our data provides a \\emph{local} characterization of the inhomogeneous magnetic state of \\gls{btm} that will be useful in developing a detailed microscopic understanding of its magnetism.\nHaving established the effects of \\ch{Mn} magnetism in the bulk, it would be interesting to use lower implantation energies to study how they may be altered in the surface region by coupling to the \\gls{tss}.\n\n\n\\section{Conclusion \\label{sec:conclusion}}\n\n\nUsing implanted \\ch{^{8}Li} \\gls{bnmr}, we have studied the electronic and magnetic properties of the doped \\glspl{ti} \\gls{bsc} and \\gls{btm}.\nFrom \\gls{slr} measurements, we find evidence at temperatures above \\SI{\\sim 200}{\\kelvin} for site-to-site hopping of isolated \\ch{Li^{+}} with an Arrhenius activation energy of \\SI{\\sim 0.4}{\\electronvolt} in \\gls{bsc}.\nAt lower temperature the electronic properties dominate, giving rise to Korringa-like relaxation and negative Knight shifts, similar to isostructural \\gls{bts}.\nA quantitative comparison reveals Korringa ratios opposite to expectations across the Ioffe-Regel limit.\nIn \\gls{btm}, the magnetism from dilute \\ch{Mn} moments dominates all other spin interactions, but the \\gls{bnmr} signal remains measurable through the magnetic transition at $T_{C}$, where a critical peak in the \\gls{slr} rate is observed.\nThe \\SI{\\sim 0.2}{\\electronvolt} activation energy from the high temperature increase in the \\gls{slr} may be related to \\ch{Li} mobility or to thermal carrier excitations.\n\n\nWith these new results, a more complete picture of the implanted \\ch{^{8}Li} \\gls{nmr} probe of the tetradymite \\ch{Bi} chalcogenides (and other \\gls{vdw} chalcogenides) is beginning to emerge.\nAt high temperatures, isolated \\ch{^{8}Li^{+}} has a tendency to mobilize, providing unique access to the kinetic parameters governing \\ch{Li^{+}} diffusion in the ultra-dilute limit.\nAt low temperature, \\ch{^{8}Li} is sensitive to the local metallic and magnetic properties of the host.\nWith the bulk \\gls{nmr} response now established in \\ch{Bi2Ch3} \\glspl{ti}, the prospect of directly probing the chiral \\gls{tss} with the depth resolution provided by \\gls{bnmr} remains promising.\n\n\n\\begin{acknowledgments}\nWe thank:\nR.\\ Abasalti, D.\\ J.\\ Arseneau, S.\\ Daviel, B.\\ Hitti, K.\\ Olchanski, and D.\\ Vyas for their excellent technical support;\nM.\\ H.\\ Dehn, T.\\ J.\\ Parolin, O.\\ Ofer, Z.\\ Salman, Q.\\ Song, and D.\\ Wang for assistance with early measurements;\nas well as D.\\ E.\\ MacLaughlin, S.\\ D.\\ Senturia, and A.\\ Wolos for useful discussions.\nThis work was supported by NSERC Discovery grants to R.F.K.\\ and W.A.M.\nAdditionally, R.M.L.M.\\ and A.C.\\ acknowledge support from their NSERC CREATE IsoSiM Fellowships.\nThe crystal growth at Princeton University was supported by the ARO-sponsored MURI on topological insulators, grant number W911NF1210461.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Parameters of the electron number density\nprofiles.}\\label{table_neCC} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{lcccccc}\n    \\hline\n   \n    Cluster &  $n_{e0}$ & $\\beta$ & $\\theta_{c1}$ & $\\theta_{c2}$ & $f$ \\\\\n      & (10$^{-2}$cm$^{-3}$) &  & (arcsec) & (arcsec) &  \\\\\n    \\hline\n    \\hline\n    A1413 & $3.89\\pm0.54$ & $0.535\\pm0.016$ & $6.7\\pm1.4$ & $40.1\\pm4.1$ & $0.760\\pm0.020$ \\\\\n    A1689  & $4.15\\pm0.31$ & $0.871\\pm0.040$ & $21.6\\pm1.0$ & $104.5\\pm5.3$ & $0.870\\pm0.010$ \\\\\n    A1835  & $11.3\\pm0.4$ & $0.802\\pm0.015$ & $9.3\\pm0.2$ & $63.8\\pm1.6$ & $0.940\\pm0.001$ \\\\\n    A2204  & $20.4\\pm1.1$ & $0.716\\pm0.028$ & $7.5\\pm0.3$ & $67.6\\pm1.9$ & $0.959\\pm0.004$ \\\\\n    A2261  & $4.07\\pm0.59$ & $0.631\\pm0.024$ & $10.2\\pm1.8$ & $39.1\\pm5.9$ & $0.760\\pm0.050$ \\\\\n    MS1358.4+6245  & $9.63\\pm0.79$ & $0.676\\pm0.017$ & $3.3\\pm0.2$ & $37.0\\pm1.8$ & $0.934\\pm0.003$ \\\\\n    RXJ1347.5-1145  & $28.5\\pm1.4$ & $0.632\\pm0.009$ & $4.0\\pm0.2$ & $23.3\\pm1.6$ & $0.942\\pm0.004$ \\\\\n    ZW3146  & $16.9\\pm0.3$ & $0.669\\pm0.005$ & $4.4\\pm0.1$ & $25.8\\pm0.6$ & $0.882\\pm0.004$ \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nClusters of galaxies are the largest gravitationally-bound objects\narising thus far from the process of hierarchical structure\nformation (\\cite{Voit05}). As the most recent and most massive\nobjects of the Universe, clusters are excellent probes for studying\nits formation and evolution. The observed state of gas within a\ncluster is determined by a combination of shock heating during\naccretion, radiative cooling, and thermal feedback produced by the\ncooling itself, so the density and temperature of the ICM represent\nthe full thermal history of clusters' formation. To better\nunderstand the physics of ICM, it is necessary to have sufficient\nknowledge of the gas density and temperature distributions. Though\nclusters are the ideal target objects for X-ray observations of the\nhot ICM, millimeter and sub-millimeter measurements provide\nindependent and complementary tools for studying the same ICM by\nexploiting the Sunyaev Zel'dovich (SZ) effect (\\cite{Sunyaev72}).\n\nThe SZ effect is the Comptonization of the cosmic microwave\nbackground (CMB) photons, coming from the last scattering surface,\nby the hot electrons population of the ICM. The photon energy\nvariation, which is caused by the scattering process, can be\nexpressed as CMB temperature variations\n\n\\begin{equation}\\label{eq_SZ}\n\\Delta T_{SZ}=yT_{CMB}f(x)(1+\\delta_n(x,\\theta_e))+\\Delta T_{kin}\n\\end{equation}\nwhere\n\n\\begin{equation}\\label{eq_y}\ny=\\int\\theta_ed\\tau_e=\\int\\left(\\frac{k_BT_e}{m_ec^2}\\right)\\sigma_Tn_edl\\propto\\int\nP_edl\n\\end{equation}\nrepresents the comptonization parameter, $x=(h\\nu)\/(k_BT_{CMB})$ the\ndimensionless frequency, $h$ and $k_B$ are respectively the Planck\nand Boltzmann constants, $T_{CMB}$, $m_e$ and $\\sigma_T$, the CMB\ntemperature at $z=0$, the electron mass at rest and the Thomson\ncross section, $\\theta_e$ represents the dimensionless thermal\nenergy of the ICM, $\\tau_e$ is the electron optical depth. The\nparameters $n_e$, $T_e$, and $P_e$ are the electron number density,\ntemperature, and pressure of the ICM,\n$\\delta_n(x,\\theta_e)=f_n(x)\\theta_e^n\/f(x)$ is the relativistic\ncorrection term that accounts for the thermal energy of the\nelectrons involved in the scattering processes, where\n$f(x)=x[(e^x+1)\/(e^x-1)-4]$ is a dimensionless quantity that\ndescribes the spectral signature of the effect, and the subscript\n$n$ indicates the maximum order of the relativistic correction\n($n=4$ in this work, \\cite{Nozawa95}). The last term of Eq.\n\\ref{eq_SZ} is the kinematic component of the SZ effect, which\ncontains the contribution from the bulk motion of the electron\npopulation with respect to the last scattering surface reference\nframe. For the purpose of this paper, this term is omitted, assuming\nthat it is disentangled from the thermal component by\nmulti-frequency observations, together with the signal from the\nprimary CMB emission.\n\nThe SZ effect is redshift independent and, for this reason, it is\npossible to detect distant clusters without any existing X-ray or\noptical observations. This is the case of the ongoing ground-based\nexperiments such as SPT (\\cite{Ruhl04}), ACT (\\cite{Kosowsky03}),\nand the all sky survey like Planck (\\cite{Planck11a}) or the\nupgraded MITO (\\cite{DePetris07}) and OLIMPO (\\cite{Masi08}) with\nnew spectroscopic capabilities and the proposed 30-m diameter C-CAT\n(\\cite{Sebring06}). However, some assumptions on cluster physics\nstill have to be made in order to directly extract cluster\nobservables.\n\nEstimates of cluster's total mass can be derived by SZ observations\nwhen X-ray or lensing measurements are available or by empirically\ncalibrated scaling relations linking the SZ flux to the total mass\n(e.g. \\cite{Vikhlinin09, Arnaud10, Planck11b, Comis11}). Total mass\ncan also be determined by SZ observations alone when applying\nthermal energy constraints (\\cite{Mroczkowski11}).\n\nTo accurately reproduce the gas inside the cluster, an ICM universal\nmodel is mandatory (e.g. \\cite{Nagai07, Arnaud10}). In this paper we\nconfirm that the simple isothermal \\textit{beta}-model is clearly an\ninappropriate cluster representation for total mass recovery by SZ\nobservations, particularly in the presence of relaxed cool core (CC)\nclusters. These objects show a well studied peaked density profile\nwith a temperature decrement in the core region (\\cite{Jones84}). In\nthe local universe this class of clusters is observationally a\nsignificant percentage of the total cluster population\n(\\cite{Eckert11}). Even if the X-ray estimated CC fraction is biased\nby selection effects in flux-limited samples, recently a 35\\% of\nclusters have picked up in the SZ high signal-to-noise ratio Planck\nearly cluster data-set (\\cite{Planck11c}) are CC clusters. Large\nscatter in mass estimates of CC clusters has been highlighted\npreviously using numerical simulations by Hallman et al. (2006) and\nHallman et al. (2007).\n\nWe investigate the bias on the mass in a limited sample of eight\nnearby ($0.1<z<0.5$) and high-mass ($M>10^{14}$ $M_{\\odot}$) CC\nclusters observed by Chandra. The SZ maps of these clusters, which\nare expressed in thermodynamic temperature units and convolved with\nseveral instrumental beams, are dealt with by applying the\nisothermal \\textit{beta}-model. The total mass is derived in three\ndifferent ways: by assuming hydrostatic equilibrium and a fixed gas\nfraction and by applying a self similar scaling relation. To focus\nonly on the consequences of the employed ICM model, in our analysis\nwe neglect all the contaminants present in the sky by assuming in\nthis way the best situation to recover cluster total mass.\n\nIn Sect. \\ref{sec_ne_Te}, we discuss the electron number density\nradial profile and follow self-similar studies to characterize a\nuniversal electron temperature radial profile of a limited sample of\neight CC clusters observed by Chandra. In Sect. \\ref{sec_MCMC} we\ngenerate maps of the SZ effect in thermodynamic temperature. In\nSect. \\ref{sec_total_mass} we evaluate cluster total mass under\ndifferent sets of assumptions. The bias on the recovered mass is\ndescribed in Sect. \\ref{sec_MB}, which discusses the main\ncontributions. Conclusions are summarized in Sect.\n\\ref{sec_conclusions}.\n\n\\section{Electron number density and temperature\nprofiles}\\label{sec_ne_Te}\n\n\\begin{table*}\n\\begin{minipage}[hbpt]{\\textwidth}\n\\caption{CC galaxy clusters properties used in the analysis to\ngenerate a universal $T_e$ profile for this class of clusters.}\n\\label{table_te} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{l c c c c c c}\n    \\hline\n   \n    Cluster & z & $D_A$ & $r_{500}$\\footnote{\\cite{Morandi07}} & $\\theta_{500}$ & $T_X$\\footnote{Temperature scale  calculated by a weighted mean of Bonamente et al. (2006) data.}& $T_{e0}$\\footnote{Electron temperature of the cluster obtained by fitting the isothermal \\textit{beta}-model to X-ray data (\\cite{Bonamente06}).}\\\\\n    name &  & (Gpc) & (kpc) & (arcsec) & (keV)& (keV)\\\\\n    \\hline\n    \\hline\n    $A1413$ & 0.142 & 0.52 & $1195\\pm232$ & $321\\pm62$ & $6.6\\pm0.6$& $7.3\\pm0.2$\\\\\n    $A1689$ & 0.183 & 0.63 & $1402\\pm260$ & $377\\pm70$ & $8.7\\pm0.9$& $10.0\\pm0.3$\\\\\n    $A1835$ & 0.252 & 0.81 & $1439\\pm414$ & $387\\pm111$ & $10.5\\pm1.0$& $8.4\\pm0.2$\\\\\n    $A2204$ & 0.152 & 0.55 & $1796\\pm320$ & $483\\pm86$ & $11.3\\pm1.8$& $6.5\\pm0.2$\\\\\n    $A2261$ & 0.224 & 0.74 & $1201\\pm168$ & $323\\pm45$ & $7.0\\pm1.0$& $7.2\\pm0.4$\\\\\n    $MS1358.4+6245$ & 0.327 & 0.97 & $1633\\pm885$ & $439\\pm238$ & $8.4\\pm1.1$& $8.3\\pm0.6$\\\\\n    $RXJ1347.5-1145$ & 0.451 & 1.19 & $1734\\pm170$ & $466\\pm46$ & $14.8\\pm1.5$& $13.5\\pm0.5$\\\\\n    $ZW3146$ & 0.291 & 0.90 & $1804\\pm344$ & $485\\pm92$ & $8.7\\pm0.4$& $6.6\\pm0.1$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nA general parametric model for the cluster atmosphere must be\ndefined to forecast the shape of cluster SZ signals in matched\nfilter techniques for detecting clusters in blind surveys. ACT has\ndetected new clusters assuming a two-dimensional Gaussian profile as\nfilter (\\cite{Sehgal10}), while SPT has detected a projected\nspherical \\textit{beta} profile (\\cite{Vanderlinde10}) and Planck a\nuniversal pressure profile (\\cite{Melin11}).\n\nThe approach for determining the total mass cluster can be\ndifferent. High-quality X-ray data allows an accurate modeling of\ncluster morphology, but in the case of low angular resolution and\/or\nlow signal-to-noise ratio a simple isothermal \\textit{beta}-model is\nstill applied (e.g. \\cite{Marriage10, Sayers11}).\n\nIn this work we analyze this model (\\cite{Cavaliere78}), which is\nbased on the very general assumption that the electron temperature\n$T_e$ is constant along the whole considered cluster radial\nextension and that the electron number density follows a spherical\ndistribution as\n\n\\begin{equation}\\label{eq_beta}\nn_{e,ISO}(r)=n_{e0}{\\left(1+\\frac{r^2}{r_c^2}\\right)}^{-\\frac{3}{2}\\beta},\n\\end{equation}\nwhere $n_{e0}$ is the central electron number density, $r_c$ the\ncore radius, and $\\beta$ the power law index. The subscript ISO\nindicates, hereafter, the isothermal \\textit{beta}-model. The proved\ninadequacy of this model is compensated for by the advantage of\nextracting a simple analytic expression for the $y$ parameter along\nthe off-axis angular separation, $\\theta$,\n\n\\begin{equation}\\label{eq_ybeta}\ny_{ISO}(\\theta)=y_0{\\left(1+\\frac{\\theta^2}{\\theta_c^2}\\right)}^{\\frac{1}{2}-\\frac{3}{2}\\beta}\n\\end{equation}\nwhere\n\n\\begin{equation}\\label{eq_y0}\ny_0=n_{e0}\\frac{k_BT_e}{m_ec^2}\\sigma_T\nr_c\\sqrt{\\pi}\\frac{\\Gamma\\left(\\frac{3}{2}\\beta-\\frac{1}{2}\\right)}{\\Gamma\\left(\\frac{3}{2}\\beta\\right)},\n\\end{equation}\nand $\\theta_c=r_c\/D_A$, with $D_A$ the angular diameter distance,\nwhich has been calculated for each cluster by using\n\n\\begin{equation}\nD_A=\\frac{c}{H_0(1+z)}\\int^z_0\\frac{dz'}{E(z')},\n\\end{equation}\nwhere $H_0$ is the Hubble constant and\n$E(z)={\\left[\\Omega_{M}(1+z)^3+\\Omega_{\\Lambda}\\right]}^{1\/2}$. We\nadopt a $\\Lambda CDM$ cosmology with $H_0=70$ km\/s\/Mpc,\n$\\Omega_M=0.3$, and $\\Omega_{\\Lambda}=0.7$.\n\nIt is easy to find clusters that are not relaxed or that display\nstructures that are very difficult to describe with this model.\nTherefore, it is realistic to assume that many newly discovered\nclusters in blind SZ surveys exhibit such significant deviations as\nwell. To explore a particular ICM gas morphology, we focus on CC\nclusters. We started analyzing a small sample of eight objects\nextracted from the Chandra dataset investigated in Bonamente et al.\n(2006).\n\nThe study of a central region, commonly known as the core region,\nhas been challenged by high-resolution numerical simulations\n(\\cite{Navarro95, Borgani04, Kay04, Nagai07, Henning09}). These\nworks lead to an agreement on whether there is a cooling core in the\nvery central denser gas region ($r<0.1r_{500}$) of some clusters, as\nwell as a slower decline in the temperature at large radii\n($r>0.2r_{500}$). As usual we refer to $r_{500}$ as the radius of\nthe cluster that defines a volume with mean density $500$ times the\ncritical density $\\rho_{crit}$ at cluster redshift. The choice of\n$r_{500}$ is motivated by simulation results from Evrard et al.\n(1996) showing the gas within this radius relaxed and in hydrostatic\nequilibrium.\n\nMoreover, many observational studies in X-rays have shown that the\nX-ray surface brightness, hence the underlying density, cannot be\nrepresented correctly by a \\textit{beta} profile. A second component\nshould be added or a peaked central part introduced in order to\nproperly fit the observation. The observed deprojected density\nprofiles are peaked for CC systems and flatter for morphologically\ndisturbed clusters.\n\nThe $n_e$ cluster profile can be described by a double\n\\textit{beta}-model profile\n\n\\begin{eqnarray}\\label{eq_2beta}\nn_{e,CC}(r)&=&n_{e0}\\left[f{\\left(1+\\frac{r^2}{r_{c1}^2}\\right)}^{-\\frac{3}{2}\\beta}+(1-f)\n{\\left(1+\\frac{r^2}{r_{c2}^2}\\right)}^{-\\frac{3}{2}\\beta}\\right],\\nonumber\\\\\n\\end{eqnarray}\nwhere the parameters' data have been taken from Bonamente et al.\n(2006) and adapted to this work to have a symmetric standard\ndeviation (D'Agostini, 2003).\n\nThis distribution is a generalization of a double\n\\textit{beta}-model profile of the electron number density,\ndeveloped by La Roque et al. (2005), but instead using the same\n$\\beta$ parameter for both the central region and the outskirts, as\nin Bonamente et al. (2006). The $r_{c1}=\\theta_{c1}\/D_A$ and\n$r_{c2}=\\theta_{c2}\/D_A$ are the core radii of the inner and outer\ndistributions, and $f$ is a parameter defined between 0 and 1 that\nrepresents how the core region dominates the outer region. These\nparameters, together with $n_{e0}$, are taken from Bonamente et al.\n(2006) and summarized in Table \\ref{table_neCC} for the selected\nclusters.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Contefg1.eps}\\\\\n  \\caption{\\itshape Radial electron temperature profiles of a Chandra-selected sample of CC galaxy clusters (\\cite{Bonamente06}) and the best\n  fit (solid line) of the electron temperature profile with the 1$\\sigma$ error\n  (dotted lines) proposed in this work. Temperatures and radii\n  are expressed in terms of $T_X$ and $r_{500}$, respectively, which\n  are used as scale quantities throughout this work.}\\label{fig_fitTe}\n\\end{figure}\n\nWe describe the temperature profile as\n\n\\begin{eqnarray}\\label{eq_Te_general}\n\\frac{T_{e,CC}(r)}{T_X}=\\left\\{\n\\begin{array}{rl}\nA_1{\\left(\\frac{r}{r_{500}}\\right)}^{m_1} & \\mbox{  for } \\frac{r}{r_{500}}<0.1 \\\\\nA_2{\\left(\\frac{r}{r_{500}}\\right)}^{m_2} & \\mbox{  for }\n\\frac{r}{r_{500}}>0.2\n\\end{array}\n\\right.\n\\end{eqnarray}\nwhere $A_{1,2}$ are determined by fixing the position at which the\ntwo power laws, described by the $m_1$ and $m_2$ parameters,\nintersect each other, as explained in Appendix \\ref{sec_appendix}.\n\nThe search for a universal temperature profile in the cluster halo\nregion has been a target of several works based on observations\n(e.g. \\cite{Markevitch98, DeGrandi02, Zhang04, Vikhlinin05,\nVikhlinin06, Sanderson06, Zhang07, Pratt07, Zhang08}) and\nhydrodynamical simulations (e.g., \\cite{Loken02, Borgani04, Kay04,\nPiffaretti08}).\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Parameters of the single \\textit{beta}-model, estimated\nusing the MCMC procedure described in the text, as best fit of the\nCC cluster maps.}\\label{table_neISO} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llccccc}\n    \\hline\n   \n     Cluster & FWHM fov & $\\Delta T_{SZ0}$ & $\\beta$ & $\\theta_c$  \\\\\n     & (arcmin) & ($\\mu$K) &   & (arcsec)  \\\\\n    \\hline\n    \\hline\n   $Cl_{A1413}$ & 1& $-273\\pm9$ & $0.731\\pm0.006$ &  $66.1\\pm3.0$   \\\\\n    & 4.5& $-224\\pm14$ & $0.743\\pm0.009$ &  $81.7\\pm7.3$   \\\\\n    & 7& $-195\\pm11$ & $0.758\\pm0.011$ &  $96.4\\pm8.0$   \\\\\n    \\hline\n    $Cl_{A1689}$& 1& $-554\\pm8$ & $0.979\\pm0.006$ &  $80.2\\pm1.5$   \\\\\n    & 4.5& $-430\\pm12$ & $0.994\\pm0.004$ &  $93.7\\pm2.2$   \\\\\n    & 7& $-592\\pm84$ & $0.909\\pm0.018$ &  $61.4\\pm8.3$   \\\\\n    \\hline\n    $Cl_{A1835}$& 1& $-709\\pm10$ & $0.951\\pm0.006$ &  $56.8\\pm1.1$   \\\\\n    & 4.5& $-468\\pm29$ & $0.980\\pm0.013$ &  $76.0\\pm5.0$   \\\\\n    & 7& $-818\\pm88$ & $0.870\\pm0.010$ &  $36.0\\pm3.6$   \\\\\n    \\hline\n    $Cl_{A2204}$& 1& $-727\\pm8$ & $0.848\\pm0.003$ &  $68.4\\pm0.9$   \\\\\n    & 4.5& $-551\\pm18$ & $0.865\\pm0.007$ &  $86.0\\pm3.6$   \\\\\n    & 7& $-514\\pm39$ & $0.859\\pm0.017$ &  $86.9\\pm9.0$   \\\\\n    \\hline\n    $Cl_{A2261}$ & 1& $-399\\pm10$ & $0.830\\pm0.006$ &  $52.3\\pm1.7$   \\\\\n    & 4.5& $-273\\pm17$ & $0.849\\pm0.008$ &  $71.6\\pm4.6$   \\\\\n    & 7& $-323\\pm32$ & $0.816\\pm0.013$ &  $55.5\\pm6.4$   \\\\\n    \\hline\n    $Cl_{MS1358.4+6245}$& 1& $-304\\pm12$ & $0.857\\pm0.011$ &  $49.6\\pm2.7$   \\\\\n    & 4.5& $-245\\pm28$ & $0.856\\pm0.013$ &  $56.0\\pm6.1$   \\\\\n    & 7& $-1246\\pm116$ & $0.753\\pm0.009$ &  $8.7\\pm1.2$   \\\\\n    \\hline\n    $Cl_{RXJ1347.5-1145}$& 1& $-1683\\pm15$ & $0.835\\pm0.002$ &  $37.7\\pm0.5$   \\\\\n    & 4.5& $-893\\pm30$ & $0.860\\pm0.005$ &  $62.7\\pm2.5$   \\\\\n    & 7& $-739\\pm41$ & $0.873\\pm0.011$ &  $73.1\\pm5.1$   \\\\\n    \\hline\n    $Cl_{ZW3146}$& 1& $-645\\pm13$ & $0.839\\pm0.005$ &  $42.4\\pm1.1$   \\\\\n    & 4.5& $-404\\pm38$ & $0.857\\pm0.013$ &  $61.3\\pm6.7$   \\\\\n    & 7 & $-585\\pm77$ & $0.801\\pm0.011$ &  $36.5\\pm5.2$   \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe profile, proposed in this paper specifically for CC clusters,\nfollows both the central drop and the outer decline of the gas\ntemperature. The function is formalized in the\n$\\log\\left(r\/r_{500}\\right)-\\log\\left(T_e(r)\/T_X\\right)$ plane on\nwhich the power laws of Eq. \\ref{eq_Te_general} become linear\nfunctions (see Appendix \\ref{sec_appendix} for a complete\ntreatment). To fix the parameters $A_1$ and $A_2$ of the radial\nelectron temperature profile and to confirm the power laws indices\n$m_1$ and $m_2$, we fit a co-adding of Chandra electron temperature\nnormalized to $T_X$ data, of the CC clusters selection with the\nproposed function. $T_X$ represents the average temperature value in\nthe range $(0.1\\div1.0)r_{500}$, and it is used to scale each\ncluster, in order to fit the universal temperature function to the\nmeasured profiles. In Table \\ref{table_te} we report the cluster\nredshift, $z$, and the angular diameter distance, $D_A$. The scale\nradius $r_{500}$ and temperature $T_X$ are also collected. The\nchosen radial range, which is used to calculate the scale\ntemperature $T_X$, corresponds to a cut in the central region\n($r<0.1r_{500}$). Obviously this value cannot be compared easily\nwith results of other works because it strictly depends on the data\nradial extension. In fact, an important source of bias is the\ntemperature definition (\\cite{Vikhlinin06}). Here, we use the\nspectroscopic temperature $T_X$. In Figure \\ref{fig_fitTe} the\ntemperature data of our cluster sample with the best fit are\nplotted. By following Appendix \\ref{sec_appendix}, the resulting\n$T_e$ profile parameters are $A_1=2.41\\pm0.14$, $A_2=0.55\\pm0.10$,\n$m_1=0.38\\pm0.02$ and $m_2=-0.29\\pm0.11$, which univocally define\nthe $T_e(r)$ function. We note that, even if the power law that\ndescribes the outskirts of the cluster temperature distribution\nsuffers larger uncertainties, $m_1$ and $m_2$ are both compatible,\nwithin one standard deviation, with estimates available in the\nliterature. For example, Zhang et al. (2008) find $m_1=0.38\\pm0.04$,\nfor $r<0.2r_{500}$, in agreement with Sanderson et al. (2006), who\nproposes $m_1=0.4$, for $r<0.1r_{500}$. For radii larger then\n$0.2r_{500}$, Zhang et al. (2008) fitted a selected sample of data\nfrom XMM-Newton finding structurally similar behavior to ours with\n$m_2=-0.28\\pm0.19$.\n\n\\section{Pipeline of cluster simulation}\\label{sec_MCMC}\n\nThe analysis reported in this paper can be summarized in the\nfollowing steps:\n\\begin{itemize}\n    \\item construction of an SZ signal distortion profile $\\Delta T_{SZ}\n    (\\theta)$, using the ICM information coming from existing X-ray\n    observations;\n    \\item convolution of the cluster SZ map with several instrumental beam profiles;\n    \\item extraction of the ICM parameters as in the assumptions of the isothermal \\textit{beta}-model;\n    \\item estimation of the cluster total mass $M_{tot}$;\n    \\item calculation of the bias on cluster total mass due to\n    the incorrect description of the ICM.\n\\end{itemize}\n\nIn the first step, we generate angular profiles of the SZ signal\ndistortion $\\Delta T_{SZ} (\\theta)$, assuming an observing frequency\nof 150 GHz. The electron number density $n_e$ and temperature $T_e$\nprofiles are constructed using the equations presented in Sect.\n\\ref{sec_ne_Te}, which we assume to be a good representation of a\ncool core ICM. The angular profile of the comptonization parameter\nis then evaluated by projecting the electron pressure profile on the\nplane orthogonal to the cluster line of sight. The SZ signal is\nobtained using Eq. \\ref{eq_SZ}.\n\nThe $\\Delta T_{SZ} (\\theta)$ profiles are then convolved by\nconsidering three different instrumental beam profiles modeled as a\nGaussian, corresponding to large single dishes (SPT or ACT) with\n$FWHM=1$ arcmin, medium size telescopes (MITO or OLIMPO) with\n$FWHM=4.5$ arcmin, and small apertures (Planck) with $FWHM=7$\narcmin.\n\nThe errors associated to the convolved $\\Delta T_{SZ}(\\theta)$\nprofiles are treated as only due to instrumental noise. An\noptimistic choice of the sensitivity, for all the observatories, is\n6 $\\mu$K\/beam, corresponding to the Planck channel at 143 GHz\n(\\cite{Planck11c}) assuming the necessary integration time on source\nfor the other experiments. Contaminants are not included in the\nstudy since we wish to assess our ability to extract the mass of the\nclusters under ideal conditions.\n\nTo simulate the missing knowledge of X-ray observational results, we\nignore cluster morphology and assume the most general model for it:\nan isothermal \\textit{beta}-model, that expressed in temperature is\n\n\\begin{equation}\\label{eq_DeltaT_beta}\n\\Delta T_{SZ}=\\Delta\nT_{SZ0}{\\left(1+\\frac{\\theta^2}{\\theta_c^2}\\right)}^{\\frac{1}{2}-\\frac{3}{2}\\beta}.\n\\end{equation}\n\nWe apply the Metropolis Hastings (M-H) algorithm Monte Carlo Markov\nChain (MCMC) procedure to fit this equation (after a convolution\nwith the corresponding instrumental beam) on the simulated profiles\nto extract the parameters $\\Delta T_{SZ0}$, $\\beta$ and $\\theta_c$.\nFor each cluster, at a fixed field of view (\\textit{fov}), we\nanalyze the accepted set of parameters derived by the MCMC\nprocedure. The degeneracy among the extracted \\emph{beta}-model\nparameters affects their uncertainties.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\columnwidth]{Contefg2.eps}\\\\\n  \\caption{\\itshape Radial profiles of the electron temperature, $T_e$ (top),\n  number density, $n_e$ (middle) and  pressure, $P_e$ (bottom) for\n  the cluster ZW3146. The red dashed curve describes the CC template\n  while the black solid curve represents the ISO model (the shadowed regions define\n  1$\\sigma$ uncertainties), with parameters extracted by MCMC analysis\n  considering $\\Delta T_{SZ}(\\theta)$ profiles convolved with a beam\n  of 7 arcmin (FWHM).}\\label{fig_ne_Te_pe_profiles1}\n\\end{figure}\n\nWe obtain the \\textit{beta}-model parameter set that is most\nconsistent with the $\\Delta T_{SZ}(\\theta)$ profile, given the\nassumed instrumental noise and beam sizes. All the parameters\nresulting from the MCMC analysis are collected in Table\n\\ref{table_neISO} for each cluster and for each \\textit{fov}. Figure\n\\ref{fig_ne_Te_pe_profiles1} shows the electron temperature (top),\nnumber density (middle), and pressure (bottom) profiles for only the\ncluster ZW3146, as an example, of both the original CC ICM and the\nrecovered ISO model. The errors associated to the curves account for\nthe 1$\\sigma$ uncertainties on the parameters.\n\n\\section{Cluster total mass estimation}\\label{sec_total_mass}\n\nWe want to stress the consequences of the assumptions on the ICM\nphysics when we miss X-ray information. A quantity, such as the\ntotal mass $M_{tot}$, can be biased by a different physical state of\nthe ICM (i.e. mergers or cooling flow mechanisms). In particular we\nestimate the mass for both the ICM discussed templates ($CC$ and\n$ISO$), by using the following different approaches:\n\n\\begin{itemize}\n    \\item hydrostatic equilibrium assumption for the cluster\n    gas (hydrostatic equilibrium, HE);\n    \\item gas fraction independence of cluster physical state (fixed gas fraction, FGF);\n    \\item $M_{tot}-Y$ scaling relation (scaling law,\n    SL), as derived in the standard self-similar collapse scenario.\n\\end{itemize}\n\nThe masses are calculated, in particular, within a fixed integration\nradius $r_{int}$ (aperture radius), which we arbitrarily choose\nequal to the $r_{500}$ values as reported in Morandi et al. (2007,\nsee Table \\ref{table_te}). This choice is motivated by the need to\nfix an aperture radius within which to estimate integrated\nquantities. We point out that $r_{int}$ does not always correspond\nto the same overdensity, due to the different assumed ICM templates.\nIt is clear that, hereafter, results associated to the clusters\nsimulated in this work cannot be considered as describing the true\nICM physics of the observed objects. We choose, however, to maintain\nthe link with the ``native'' cluster in the name ($NAME \\rightarrow\nCl_{NAME}$).\n\n\\subsection{Hydrostatic equilibrium}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Total cluster masses calculated considering the HE and FGF\napproaches.} \\label{table_mass HE_FGF} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llcc}\n    \\hline\n   \n    Cluster & ICM template & $M_{tot,HE}$ & $M_{tot,FGF}$\\footnote{derived by $M_{gas}$ assuming\n$f_{gas}=0.1$}  \\\\\n     & (FWHM fov)  & (10$^{14}$ $M_{\\odot}$)& (10$^{14}$ $M_{\\odot}$)  \\\\\n    \\hline\n    \\hline\n    A1413 & CC   &$3.85\\pm0.77$& $8.73\\pm2.03$\\\\\n    $Cl_{A1413}$& ISO (1')   &$6.59\\pm0.60$&$6.21\\pm0.35$\\\\\n    & ISO (4.5')   &$6.67\\pm0.59$&$6.02\\pm0.59$\\\\\n    & ISO (7')   &$6.66\\pm0.58$&$5.93\\pm0.89$\\\\\n    \\hline\n    A1689 & CC  & $8.96\\pm1.97$&$12.72\\pm2.33$\\\\\n    $Cl_{A1689}$& ISO (1')  &$13.56\\pm1.41$& $9.88\\pm3.12$\\\\\n    & ISO (4.5')   &$13.61\\pm1.33$&$9.14\\pm4.08$\\\\\n    & ISO (7')   &$12.66\\pm1.33$&$8.86\\pm2.01$\\\\\n    \\hline\n    A1835 & CC  & $10.23\\pm2.30$&$12.72\\pm1.18$\\\\\n    $Cl_{A1835}$& ISO (1') &$16.21\\pm1.61$& $10.08\\pm0.33$\\\\\n    & ISO (4.5')   &$16.32\\pm1.55$&$9.13\\pm0.94$\\\\\n    & ISO (7')   &$15.08\\pm1.40$&$7.99\\pm1.44$\\\\\n    \\hline\n    A2204 & CC  & $12.91\\pm3.30$&$14.84\\pm2.96$\\\\\n    $Cl_{A2204}$& ISO (1') &$19.88\\pm3.27$&$10.13\\pm0.24$\\\\\n    & ISO (4.5')   &$20.33\\pm3.05$&$10.87\\pm0.67$\\\\\n    & ISO (7')   &$19.89\\pm2.86$&$10.32\\pm1.54$\\\\\n    \\hline\n    A2261 & CC  & $4.79\\pm1.11$&$10.55\\pm3.48$\\\\\n    $Cl_{A2261}$& ISO (1')  &$7.88\\pm1.09$&$7.80\\pm3.50$\\\\\n     & ISO (4.5')   &$7.84\\pm1.18$&$7.24\\pm1.22$\\\\\n    & ISO (7')   &$7.81\\pm1.08$&$6.94\\pm1.09$\\\\\n   \\hline\n    MS1358.4+6245 & CC  & $8.09\\pm1.75$&$10.23\\pm1.65$\\\\\n    $Cl_{MS1358.4+6245}$& ISO (1')   &$13.27\\pm1.77$&$7.74\\pm0.63$\\\\\n    & ISO (4.5')   &$13.17\\pm1.88$&$7.27\\pm1.22$\\\\\n    & ISO (7')   &$11.90\\pm1.51$&$6.12\\pm1.14$\\\\\n    \\hline\n    RXJ1347.5-1145 & CC  & $14.66\\pm3.02$&$32.34\\pm4.05$\\\\\n    $Cl_{RXJ1347.5-1145}$& ISO (1')  &$24.45\\pm2.49$& $25.04\\pm0.48$\\\\\n    & ISO (4.5')   &$24.47\\pm2.54$&$22.08\\pm1.30$\\\\\n    & ISO (7')   &$24.66\\pm2.49$&$21.07\\pm1.84$\\\\\n    \\hline\n    ZW3146 & CC  & $9.06\\pm1.64$&$17.88\\pm1.05$\\\\\n    $Cl_{ZW3146}$& ISO (1')  &$15.05\\pm0.60$& $13.77\\pm0.58$\\\\\n    & ISO (4.5')   &$15.14\\pm0.67$&$12.72\\pm1.97$\\\\\n    & ISO (7')   &$14.44\\pm0.64$&$12.03\\pm2.61$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe first approach, HE, assumes a spherical symmetry for the\ncluster, so that the hydrostatic equilibrium equation can be written\n(\\cite{Sarazin88}) as\n\n\\begin{equation}\\label{eq_hyd_eq}\n\\frac{dP_{gas}(r)}{dr}=-\\rho_{gas}(r)G\\frac{M_{tot}(<r)}{r^2}\n\\end{equation}\nwhere $M_{tot}(<r)$ is the total cluster mass within radius $r$ and\nunder the ideal gas assumption, $P_{gas}=(\\rho_{gas}k_BT_{gas})\/(\\mu\nm_p)$, with $\\rho_{gas}=\\mu_e m_p n_e$, $\\mu$ and $\\mu_e$ are the\ntotal and electron mean molecular weights (i.e. the mean particle\nmass per electron in units of the proton mass $m_p$), $G$ is the\ngravitational constant and $T_{gas}=T_i=T_e$, the ion and electron\ntemperatures respectively, because the system is assumed to be in\nthermal equilibrium. To adapt this equation to the SZ physics, we\nsubstitute the gas pressure, which accounts for both electrons and\nions, with the simple electron pressure $P_e$, which causes the SZ\neffect, so we have\n\n\\begin{equation}\\label{eq_hyd_eq}\n\\frac{dP_e(r)}{dr}=-\\mu m_p G\\frac{M_{tot}(<r)}{r^2}n_e(r).\n\\end{equation}\nThe total mass can be derived as\n\n\\begin{equation}\\label{eq_hyd_eq_Mtot}\nM_{tot}(<r)=-\\frac{kT_e(r)}{G\\mu\nm_p}r\\left[\\frac{\\partial\\ln(n_e(r))}{\\partial\\ln(r)}+\\frac{\\partial\\ln(kT_e(r))}{\\partial\\ln(r)}\\right],\n\\end{equation}\nwhich reduces, in the simple case of the isothermal\n\\textit{beta}-model, to\n\n\\begin{equation}\\label{eq_hyd_eq_Mtot_iso}\nM_{tot}(<r)=\\frac{3\\beta kT_e}{G\\mu m_p}\\frac{r^3}{r^2_c+r^2}.\n\\end{equation}\nThe previous equations yield $M_{tot,HE,CC}$ and $M_{tot,HE,ISO}$,\nrespectively. If we also calculate the cluster gas mass by\n\n\\begin{equation}\\label{eq_Mgas}\nM_{gas}=\\mu_em_p\\int n_edV,\n\\end{equation}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{Contefg3.eps}\\\\\n  \\caption{\\itshape Radial profiles of the cumulative total (top) and\n  gas (middle) cluster masses and the gas fraction (bottom) for the cluster $Cl_{ZW3146}$. The\n  plots refer to the two ICM templates (CC and ISO) discussed\n  in the text, under the hydrostatic hypothesis. The ISO model parameters have been extracted by MCMC analysis\n  considering $\\Delta T_{SZ}(\\theta)$ profiles convolved with a beam\n  of 7 arcmin (FWHM).}\\label{fig_Mtot_Mgas_log}\n\\end{figure}\nit is easy to estimate the gas fractions, $f_{gas,HE,CC}$, and\n$f_{gas,HE,ISO}$. The plots in Fig. \\ref{fig_Mtot_Mgas_log} refer,\nstill as an example, to the cluster $Cl_{ZW3146}$ and show the\nradial profiles of the cumulative cluster total and gas masses for\nboth the ICM templates and the gas fraction. The $1\\sigma$ lines are\nderived by considering the uncertainties of the ICM parameters.\n\n\\subsection{Fixed gas fraction}\n\nThe second approach, FGF, assumes that the gas fraction, estimated\nat $r_{int}$, is independent of the cluster dynamical state, as\ndeduced by simulations (e.g. \\cite{Rasia06, Lau09}) and as recently\nderived by XMM-Newton and Subaru observations (\\cite{Zhang10}).\nUnder this hypothesis, we can derive the cluster total masses\ndirectly by gas masses as $M_{tot,FGF}=M_{gas}\/f_{gas}$.\n\nBecause the integrated quantities are calculated by referring to a\nfixed radius $r_{int}$, defined in the previous section, it is\nevident that, for each cluster and for each considered ICM model, we\nare dealing with different overdensities. We calculate these\noverdensities by\n\n\\begin{equation}\\label{eq_delta}\n\\Delta_{int}=\\frac{2G}{H_0^2E(z)^2r_{int}^3}M_{tot}(r_{int}).\n\\end{equation}\n\nThe results obtained for the HE and FGF approaches are collected in\nTable \\ref{table_mass HE_FGF}. In order to derive all the quantities\ncorresponding to the ISO template from the parameters obtained with\nthe MCMC procedure, we need to convert the $\\Delta T_{SZ0}$ values\nto central electron number densities. This is possible using\nequation\n\n\\begin{equation}\\label{eq_n_e0}\nn_{e0}=\\frac{\\Delta T_{SZ0}}{T_{CMB}f(x)}\\frac{m_ec^2}{\\sigma_T\n\\theta_cD_A\\sqrt{\\pi}}\\frac{\\Gamma(\\frac{3}{2}\\beta)}{\\Gamma(\\frac{3}{2}\\beta-\\frac{1}{2})}\\frac{1}{T_{e0}},\n\\end{equation}\nwhich is derived by expressing Eq. \\ref{eq_y0} in terms of $\\Delta\nT_{SZ}$. Of course this expression introduces a degeneracy between\ntemperature and density, possibly producing the same $\\Delta\nT_{SZ0}$ value. For this reason the masses presented in this work\nhave been obtained by fixing the electron temperature to the $T_X$\nvalues (Table \\ref{table_te}) and assuming $f_{gas}=0.1$.\n\n\\subsection{$M_{tot}-Y$ scaling relation}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Total mass as derived by applying the $M_{tot}-Y$ scaling\nlaw as in Eq. \\ref{eq_scaling_M-Ycyl}.} \\label{table_SL} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llcccc}\n    \\hline\n   \n    Cluster &ICM template& $D_A^2Y$ &$f_{gas}$ & $\\Delta_{int}$& $M_{tot,SL}$ \\\\\n    name & (FWHM fov)& (kpc$^2$) & & & (10$^{14}$ $M_{\\odot}$) \\\\\n    \\hline\n    \\hline\n    A1413 & CC& $100\\pm1$ & $0.23\\pm0.05$ & $345\\pm69$& $1.94\\pm0.23$ \\\\\n    $Cl_{A1413}$& ISO (1')& $95\\pm5$ & $0.10\\pm0.01$ & $591\\pm53$& $2.81\\pm0.18$ \\\\\n     & ISO (4.5')& $92\\pm12$ & $0.09\\pm0.01$ & $598\\pm53$& $2.86\\pm0.30$ \\\\\n     & ISO (7')& $90\\pm9$ & $0.09\\pm0.01$ & $597\\pm52$& $2.81\\pm0.26$ \\\\\n    \\hline\n    A1689 & CC& $177\\pm2$ & $0.15\\pm0.03$ & $477\\pm105$& $3.34\\pm0.38$ \\\\\n     $Cl_{A1689}$&ISO (1')& $152\\pm6$ & $0.07\\pm0.01$ & $722\\pm75$& $4.15\\pm0.23$ \\\\\n     &ISO (4.5')& $142\\pm6$ & $0.07\\pm0.01$ & $724\\pm71$& $4.17\\pm0.23$ \\\\\n     &ISO (7')& $139\\pm32$ & $0.07\\pm0.02$ & $674\\pm71$& $4.16\\pm0.82$ \\\\\n    \\hline\n    A1835 & CC& $263\\pm2$ & $0.13\\pm0.03$ & $468\\pm105$& $4.48\\pm0.36$ \\\\\n     $Cl_{A1835}$&ISO (1')& $186\\pm7$ & $0.06\\pm0.01$ & $741\\pm74$& $5.05\\pm0.25$ \\\\\n      &ISO (4.5')& $171\\pm17$ & $0.06\\pm0.01$ & $746\\pm71$& $5.12\\pm0.51$ \\\\\n      &ISO (7')& $152\\pm28$ & $0.05\\pm0.01$ & $689\\pm64$& $5.00\\pm0.85$ \\\\\n    \\hline\n     A2204 & CC& $271\\pm2$ & $0.12\\pm0.04$ & $337\\pm86$& $5.28\\pm0.72$ \\\\\n     $Cl_{A2204}$&ISO (1')& $241\\pm6$ & $0.06\\pm0.01$ & $520\\pm86$& $6.78\\pm0.47$ \\\\\n     &ISO (4.5')& $231\\pm15$ & $0.05\\pm0.01$ & $531\\pm80$& $6.89\\pm0.59$ \\\\\n      &ISO (7')& $222\\pm31$ & $0.05\\pm0.01$ & $520\\pm75$& $6.88\\pm0.99$ \\\\\n    \\hline\n    A2261 & CC& $134\\pm2$ & $0.22\\pm0.06$ & $388\\pm90$& $2.31\\pm0.38$ \\\\\n     $Cl_{A2261}$&ISO (1')& $110\\pm5$ & $0.10\\pm0.02$ & $639\\pm89$& $2.87\\pm0.19$ \\\\\n    &ISO (4.5')& $104\\pm9$ & $0.09\\pm0.02$ & $635\\pm95$& $2.93\\pm0.29$ \\\\\n     &ISO (7')& $99\\pm16$ & $0.09\\pm0.02$ & $633\\pm88$& $2.93\\pm0.48$ \\\\\n    \\hline\n    MS1358.4+6245& CC& $166\\pm4$ & $0.13\\pm0.03$ & $233\\pm50$& $3.82\\pm0.35$ \\\\\n   $Cl_{MS1358.4+6245}$&ISO (1')& $127\\pm11$ & $0.06\\pm0.01$ & $381\\pm51$& $4.67\\pm0.45$ \\\\\n   &ISO (4.5')& $113\\pm21$ & $0.06\\pm0.01$ & $379\\pm54$& $4.56\\pm0.76$ \\\\\n   &ISO (7')& $96\\pm18$ & $0.05\\pm0.01$ & $342\\pm43$& $4.36\\pm0.76$ \\\\\n     \\hline\n   RXJ1347.5-1145 & CC& $1165\\pm4$ & $0.23\\pm0.03$ & $305\\pm63$& $8.09\\pm0.52$ \\\\\n      $Cl_{RXJ1347.5-1145}$&ISO (1')& $719\\pm14$ & $0.10\\pm0.01$ & $509\\pm52$& $8.68\\pm0.40$ \\\\\n      &ISO (4.5')& $654\\pm41$ & $0.09\\pm0.01$ & $509\\pm52$& $8.86\\pm0.60$ \\\\\n      &ISO (7')& $649\\pm66$ & $0.09\\pm0.01$ & $513\\pm52$& $9.11\\pm0.80$ \\\\\n     \\hline\n   ZW3146 & CC& $312\\pm4$ & $0.20\\pm0.04$ & $201\\pm37$& $4.42\\pm0.31$ \\\\\n     $Cl_{ZW3146}$&ISO (1')& $224\\pm11$ & $0.09\\pm0.01$ & $334\\pm13$& $5.21\\pm0.23$ \\\\\n     &ISO (4.5')& $208\\pm38$ & $0.08\\pm0.01$ & $336\\pm15$& $5.31\\pm0.81$ \\\\\n     &ISO (7')& $205\\pm43$& $0.08\\pm0.02$ & $321\\pm14$ & $5.41\\pm0.11$ \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe total cluster mass can be inferred by applying a self-similar\nrelation that links it to the integrated comptonization parameter.\nThe $M_{tot}-Y$ scaling law is usually calibrated for a fixed\noverdensity value, while in this work we are dealing with masses\ncalculated at different overdensities. For this reason in the\nscaling law, we make the dependence on overdensity explicit. Simple\nconsiderations, based on the assumption that cluster evolution is\ncompletely determined by gravitational processes, lead to an easy\nscaling relation that connects the total mass of a cluster of\ngalaxies, $M_{tot}$, to its temperature, $T_e$, considering an\nisothermal structure for the ICM. Following Kravtsov et al. (2006)\nand Bryan \\& Norman (1998) and keeping the overdensity dependence\nexplicit, we have\n\n\\begin{equation}\\label{eq_scaling_M-T}\nk_BT_e=\\mu m_p{\\left[\\frac{27}{16}\\Delta_{int}\nG^2H_0^2E(z)^2\\right]}^{1\/3}M_{tot}^{2\/3}.\n\\end{equation}\n\nTo connect a spherically integrated quantity (e.g. cluster total\nmass) to one derived by a cylindrical integration (e.g. integrated\nSZ flux), we consider the spherical analogous to the SZ flux,\n$Y_{S}$. This quantity is directly proportional to the cluster total\nmass as\n\n\\begin{equation}\\label{eq_scaling_M-Y-T}\nY_{S}=\\frac{k_B\\sigma_T}{m_ec^2}\\int\nn_eT_edV=\\frac{k_B\\sigma_T}{m_ec^2}T_{mw}\n\\frac{f_{gas}M_{tot}}{\\mu_e m_p}\n\\end{equation}\nwhere $M_{tot}$ is the cluster total mass inside a sphere with\nradius equal to $r_{int}$, and $T_{mw}$ is the gas mass-weighted\nmean temperature.\n\nBy combining the previous equations, we get\n\n\\begin{equation}\\label{eq_scaling_M-Ysph}\nM_{tot}=0.248{\\left[f_{gas}^3E(z)^2\\Delta_{int}\\right]}^{-1\/5}Y_{S}^{3\/5}\\mbox{$10^{14}$\n$M_{\\odot}$}.\n\\end{equation}\nThis means that, by estimating $Y_{S}$, we can easily infer the\ncorresponding  total cluster mass. Furthermore, in order to derive\nthe integrated SZ flux, we need to solve the equation\n\n\\begin{equation}\nY=2\\pi\\int^{\\theta_{int}}_0 y(\\theta)\\theta d\\theta,\n\\end{equation}\nwhere $\\theta_{int}=r_{int}\/D_A$ and $y(\\theta)$ is extracted by the\nsimulated SZ temperature decrement profiles (in the CC case) or\ncalculated directly from Eq. \\ref{eq_ybeta}, by considering the\nparameters as derived from the MCMC analysis in Sect. \\ref{sec_MCMC}\n(in the ISO case). The two defined integrated Comptonization\nparameters ($Y_S$ and $Y$) are connected by the dimensionless\nquantity $C=D_A^2Y\/Y_{S}$.\n\nWe averaged the $C$ factors, assuming different cylindric depths\nequal to 2, 5, and 10 $r_{int}$, obtaining $C_2=1.52\\pm0.07$,\n$C_5=1.85\\pm0.14$, and $C_{10}=2.00\\pm0.19$, respectively,\nconverging to the value in Bonamente et al. (2008).\n\nThe scaling law can be rewritten as\n\n\\begin{equation}\\label{eq_scaling_M-Ycyl}\nM_{tot}=0.248{\\left[f_{gas}^3E(z)^2\nC_{\\infty}^3\\Delta_{int}\\right]}^{-1\/5}(D_A^2Y)^{3\/5}\\mbox{$10^{14}$\n$M_{\\odot}$},\n\\end{equation}\nwhich can directly give an estimation of cluster total mass\n$M_{tot,SL}$. All the quantities included in the scaling law are\nlisted in Table \\ref{table_SL}\n\n\\section{Bias on the total mass}\\label{sec_MB}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Biases on the cluster total mass estimate for three\ndifferent approaches as derived by three different fovs.}\n\\label{table_bias} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{l l c c c}\n    \\hline\n   \n    Cluster& FWHM fov & $MB_{HE}$ & $MB_{FGF}$ & $MB_{SL}$ \\\\\n    name& (arcmin) & & &\\\\\n    \\hline\n    \\hline\n    $Cl_{A1413}$& 1 & $0.75\\pm0.31$ & $-0.29\\pm0.19$ & $0.58\\pm0.32$\\\\\n    & 4.5 & $0.79\\pm0.35$ & $-0.30\\pm0.15$ & $0.59\\pm0.34$\\\\\n    & 7 & $0.81\\pm0.31$ & $-0.33\\pm0.17$ & $0.61\\pm0.31$\\\\\n    \\hline\n    $Cl_{A1689}$& 1 & $0.56\\pm0.31$ & $-0.24\\pm0.14$ & $0.43\\pm0.24$\\\\\n    & 4.5 & $0.59\\pm0.33$ & $-0.30\\pm0.15$ & $0.45\\pm0.28$\\\\\n    & 7 & $0.50\\pm0.30$ & $-0.35\\pm0.20$ & $0.55\\pm0.45$\\\\\n    \\hline\n    $Cl_{A1835}$& 1 & $0.63\\pm0.31$ & $-0.26\\pm0.07$ & $0.44\\pm0.16$\\\\\n    & 4.5 & $0.64\\pm0.28$ & $-0.32\\pm0.09$ & $0.47\\pm0.21$\\\\\n    & 7 & $0.51\\pm0.32$ & $-0.41\\pm0.11$ & $0.45\\pm0.29$\\\\\n    \\hline\n    $Cl_{A2204}$& 1 & $0.59\\pm0.32$ & $-0.25\\pm0.15$ & $0.45\\pm0.27$\\\\\n    & 4.5 & $0.62\\pm0.30$ & $-0.29\\pm0.15$ & $0.50\\pm0.27$\\\\\n    & 7 & $0.63\\pm0.30$ & $-0.31\\pm0.18$ & $0.47\\pm0.33$\\\\\n    \\hline\n    $Cl_{A2261}$&1  & $0.75\\pm0.30$ & $-0.23\\pm0.25$ & $0.52\\pm0.36$\\\\\n    & 4.5 & $0.74\\pm0.33$ & $-0.28\\pm0.29$ & $0.53\\pm0.40$\\\\\n    & 7 & $0.69\\pm0.32$ & $-0.30\\pm0.25$ & $0.54\\pm0.40$\\\\\n    \\hline\n    $Cl_{MS1358.4+6245}$& 1 & $0.71\\pm0.31$ & $-0.28\\pm0.12$ & $0.50\\pm0.21$\\\\\n    & 4.5 & $0.69\\pm0.34$ & $-0.32\\pm0.17$ & $0.55\\pm0.34$\\\\\n    & 7 & $0.52\\pm0.29$ & $-0.43\\pm0.14$ & $0.47\\pm0.32$\\\\\n    \\hline\n    $Cl_{RXJ1347.5-1145}$& 1 & $0.79\\pm0.32$ & $-0.26\\pm0.08$ & $0.51\\pm0.17$\\\\\n    & 4.5 & $0.77\\pm0.30$ & $-0.35\\pm0.08$ & $0.56\\pm0.19$\\\\\n    & 7 & $0.82\\pm0.38$ & $-0.38\\pm0.10$ & $0.61\\pm0.26$\\\\\n    \\hline\n    $Cl_{ZW3146}$& 1 & $0.66\\pm0.32$ & $-0.28\\pm0.06$ & $0.47\\pm0.16$\\\\\n    & 4.5 & $0.67\\pm0.32$ & $-0.33\\pm0.12$ & $0.53\\pm0.27$\\\\\n    & 7 & $0.62\\pm0.33$ & $-0.40\\pm0.14$ & $0.57\\pm0.31$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe results clearly show that a general interpretation of cluster\nphysics, as assumed in a pure self-similar scaling law like\n$M_{tot}-Y$, can infer a wrong estimate of cluster total mass. To\nemphasize  and to quantify this point, we define a mass bias as\n\n\\begin{equation}\\label{eq_MB}\nMB=\\frac{M_{tot,ISO}-M_{tot,CC}}{M_{tot,CC}}.\n\\end{equation}\nThis quantifies the difference between the SZ derived mass, as\nresults from the isothermal \\textit{beta}-model assumption (ISO) and\nthe X-ray derived mass (CC), which we deduce from the HE, FGF, and\nSL approaches. We notice no significant bias dependence on the beam\nsize used to convolve the SZ signals, under the assumption of the\npresence of instrumental noise alone. Therefore, all plots refers to\nresults obtained with a \\textit{fov} of 7 arcmin FWHM, considering\n$T_X$ values as in Table \\ref{table_te}. We calculate the mass bias\nfor all the analyzed approaches.\n\nSince the $\\Delta T_{SZ0}$ parameter does not give us unique\ninformation on the physics of the ICM, we have to study different\npairs of the parameters $n_{e0}$ and $T_x$. Thus, we select values\nof $T_x$ that describe a reliable range of electron temperatures\n(from 5 keV to 15 keV) and calculate the corresponding $n_{e0}$. The\nmass biases are plotted in Figure \\ref{fig_bias}a, where the three\ncases (HE, FGF, and SL) are shown all together in the plots\ncorresponding to clusters $Cl_{A1689}$, $Cl_{A2204}$, and\n$Cl_{RXJ1347.5-1145}$, representing a wide span in the electron\ntemperature values. Table \\ref{table_bias} lists the mass biases, as\nestimated at the X-ray derived electron temperature ($T_X$ in Table\n\\ref{table_te}).\n\nA check of the goodness of the procedure was done by simulating the\nobservation of a cluster having an isothermal \\textit{beta}-model\n$\\Delta T_{SZ}(\\theta)$ profile instead of a CC cluster. For this\nvalidation procedure, named \\textit{Test}, we used the same\nanalytical expression of the SZ signal to extract the parameters.\nThe assumed electron temperature is the one obtained by Bonamente et\nal. (2006) and reported in the last column in Table \\ref{table_te}.\nIn Figure \\ref{fig_bias}b we represent the cluster total mass bias\nwith these assumptions. While the bias is always zero in the HE\napproach, for the FGF and SL cases, as expected, we notice a mass\nbias dependence on $T_e$. Due to the degeneracy between the electron\ntemperature and number density, the mass bias varies with $n_e$. In\nall cases, it is worth noting that these biases nullify for electron\ntemperatures equal to $T_{e0}$ values, thus proving that the method\nis not affected by systematics.\n\nThe degeneracy between the ICM parameters, resulting from yielding\nthe same SZ signal, produces different trends on FGF and SL biases\nwith electron temperature. For increasing values of electron\ntemperature it underestimates the $M_{gas}$. Considering the FGF\napproach, this implies an underestimation of the total mass, too.\nFor the SL method this would instead produce an increasing trend in\nthe mass bias with temperature because an $M_{gas}$ underestimation\ncorresponds to a decreasing gas fraction.\n\nThe net result is a mass bias that is always different from zero for\nthe HE approach, as well as for the FGF and SL ones, unless it is\nfor an electron temperature suitable to correctly modeling the CC\ncluster as an isothermal one but generally different from the true\nICM temperature (see the crossovers in mass biases in Figure\n\\ref{fig_bias}).\n\nThe mass bias reflects a combination of different contributions\n(\\cite{Piffaretti08}). In particular for this work, the bias can be\nreduced to an \\textit{integration bias} and a \\textit{modeling\nbias}, which are estimated and discussed separately. The\n\\textit{integration bias} refers to the mass bias due to the\ndifferent integration ranges along the line of sight. An infinite\nintegration range is considered for the analytic $\\Delta T_{SZ}$\nexpression, while a finite range is assumed in the numerical\nprojection method, which is limited by the missing knowledge of the\nelectron pressure profile outside the region defined by the data.\nThe \\textit{integration bias} that can be evaluated by still\napplying the whole procedure to an ISO template (Figure\n\\ref{fig_bias}c). The \\textit{modeling bias} reflects the dependence\nof the bias on the assumed model for the ICM. In order to highlight\nit and to cancel the \\textit{integration bias}, we apply the\nprocedure to the CC cluster, recovered as an isothermal one. In this\ncase both the cluster signals (CC mock data and ISO recovered data)\nare obtained by projecting the three-dimensional electron pressure\nprofile. For disentangling the \\textit{integration bias}, the ISO\nprofile is also integrated over a limited range along the line of\nsight. The HE bias ranges between $50\\%$ and $80\\%$, always implying\na mass overestimation that is independent of the electron\ntemperature.\n\n\\begin{figure*}\n\\centering\n  \\begin{flushleft}\\hspace{2.5cm} \\Large{(a)} \\hspace{4.6cm} \\Large{(b)} \\hspace{3.0cm} \\Large{(c)}  \\hspace{3.0cm} \\Large{(d)}\\\\\\end{flushleft}\n  \\includegraphics[width=\\textwidth]{Contefg4.eps}\n  \\caption{\\itshape Total mass biases estimated\n  with different approaches, HE (black), FGF (red), SL (blue), for three representative\n  clusters of our sample (from top to bottom: $Cl_{A1689}$,\n  $Cl_{A2204}$, and\n  $Cl_{RXJ1347.5-1145}$). The ISO parameters used for the bias\n  estimation are associated to SZ signals convolved with a beam\n  of 7 arcmin (FWHM). In panel (a) the total mass bias is shown, while in\n  panel (b) the reliability of bias estimation has been tested. The two\n  bias contributions are disentangled: integration (c) and modeling (d)\n  biases. Assuming no prior for $T_e$, the mass biases are plotted\n  versus a reliable range of electron temperatures (5-15 keV).}\\label{fig_bias}\n\\end{figure*}\n\n\\section{Conclusions}\\label{sec_conclusions}\n\nWe studied the bias that affects the estimate of cluster total mass\nby SZ observations when an isothermal \\textit{beta}-model is assumed\nto describe the ICM physical properties, specifically in the case of\nCC clusters when X-ray and\/or lensing information is missing.\n\nIt is well known that, rather than the central Comptonization\nparameter $y_0$, affected by the choice of cluster profile modeling,\nan integrated quantity, like the parameter $Y$, appears to be a more\nrobust mass proxy. Nevertheless, we have shown that CC clusters can\ngenerate observed $y$ maps in which the ICM morphology could still\nbe substantially hidden, even for the current most sensitive\nexperiments operating from the largest available mm\/submm\ntelescopes.\n\nWhile a general assumption of cluster morphology is efficient at\ndetecting them in blind SZ maps, the possible mismatch with the\nactual cluster profile results in a mass bias. In fact simple ICM\nmodels, like the isothermal \\textit{beta}-model, applied to SZ\nobservations can wrongly estimate cluster total mass in the presence\nof peculiar ICM dynamics as in CC clusters, which are studied in the\ncurrent analysis, and mergers.\n\nWe analyzed the mass bias as derived in a limited sample of eight CC\nclusters observed by Chandra, both nearby ($0.1<z<0.5$) and with\nhigh mass ($M>10^{14}$ $M_{\\odot}$). Under the assumption of an\nisothermal \\textit{beta}-model, the cluster total mass was derived\napplying three different approaches: the hydrostatic equilibrium\nequation, a fixed gas fraction, and a self-similar $M_{tot}-Y$\nrelation.\n\nAssuming we had no information from X-ray observations, we reported\nthe bias on the derived total mass as dependent on electron gas\ntemperature. Only in the case of hydrostatic equilibrium does this\nbias appear almost constant for the considered clusters in the range\nof 50-80 \\%. Incidentally, we notice that an electron temperature\nvalue exists for which the FGF and SL mass biases vanish. This could\nbe the only case in which a simple isothermal \\textit{beta}-model\naccurately reproduces the mass of CC clusters.\n\nThe large biases on total cluster mass recovery in CC clusters\nrepresent another reason to definitely discard the isothermal\n\\textit{beta}-model for this purpose and to firmly support more\nsophisticated models, with universal pressure profiles (e.g.\n\\cite{Arnaud10}). This is already employed for modeling cluster\natmospheres in almost all the present blind-survey data reduction\n(SPT and Planck), and it is planned in the next future for ACT\nobservations.\n\n\\begin{acknowledgements}\nPart of this work has been supported by funding from Ateneo\n2009-C26A09FTJ7. We thank S. Borgani and the anonymous referee for\ntheir useful comments and suggestions.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe {\\it mapping class group} of a compact connected, possibly with boundary, surface  \n$S$, $\\mod_S$, is defined as \nthe group of isotopy\nclasses of homeomorphisms $S\\rightarrow S$. The algebraic investigation\nof that group goes as far back as Dehn and Nielsen, who first showed that the mapping class\ngroup of an orientable surface is embedded into the group of outer automorphisms of the\nfundamental group of $S$, ${\\rm Out}(\\pi_1(S))$. This embedding was later extended to a wider \nclass of 2-orbifolds. Maclachlan and Harvey \\cite{mh} proved that $\\mod_S\\le {\\rm Out}(\\pi_1(S))$ \nfor a surface $S$ obtained from a compact surface of genus $k$ by deleting $s$ points,\n$t$ discs and $r$ marked points. Recently, Fujiwara \\cite{fujiwara} extended the above \nembedding to the case of hyperbolic 2-orbifolds (orientable or non-orientable) with finite volume. \n\nGrossman \\cite{Gr} was the first to show that if $S_k$ is a compact\norientable surface of genus $k$, then $\\mod_{S_k}$\nis residually finite. Recently, Allenby, Kim and Tang \\cite{akt} extended the result\nof Grossman for non-orientable closed surfaces. Ivanov \\cite{ivanov} gave a geometric proof\nof the result of Grossman which seems that can easily be extended to the non-orientable \ncase. The proofs of both results in \\cite{Gr} and \\cite{akt} are using\ncombinatorial group theory arguments. One of the key ingredients of both results\nis the fact that the group of conjugating automorphisms of the fundamental group of the surface\ncoincides with the group of its inner automorphisms. If $G$ is a group, then an automorphism \n$f$ of $G$ is called a {\\it conjugating automorphism} if  $f(g)$ is a conjugate of $g$ for \nevery $g\\in G$. The conjugating automorphisms of a group $G$ form a  subgroup of the group \nof  automorphisms of $G$, ${\\rm Aut}(G)$, which\nwe denote by ${\\rm Conj}(G)$. Moreover, ${\\rm Conj}(G)$ is normal in ${\\rm Aut}(G)$ and \ncontains the subgroup of inner automorphisms of $G$, ${\\rm Inn}(G)$. \n\n\nIn the present note we investigate the residual finiteness of ${\\rm Out}(G)$ for hyperbolic\ngroups $G$. Our study is, as well, based on the investigation of the group of conjugating \nautomorphisms of $G$.\nIn fact, using the powerful geometric ideas developed by Paulin \\cite{Pau} and \nBestvina \\cite{bestvina} in the context of ultralimits of metric spaces, \nwe show that in a hyperbolic group $G$, the quotient group ${\\rm Conj}(G)\/{\\rm Inn}(G)$ is\nalways finite. The main theorem shows that every virtually torsion-free subgroup of the\nouter automorphism group of a conjugacy separable, hyperbolic group is residually\nfinite.\n\nAs a result, we obtain the residual finiteness of the outer automorphism group \nof finitely generated Fuchsian groups and of free-by-finite groups. Consequently the \nmapping class groups of hyperbolic 2-orbifolds with finite volume are \nresidually finite. Hence, we retrieve the results of Grossman and of\nAllenby, Kim and Tang as special cases of our corollaries.\n\nIn an addendum at the end of the paper we show how to generalize the main\nresults for relatively hyperbolic groups. \n\n\n\\section{Main Results}\n\nThe next lemma is what really proved in ~\\cite[Theorem 1]{Gr}. We\nreproduce here the proof for the reader's convenience.\n\n\\begin{lem}\\label{lem:Gr} Let $G$ be a finitely generated, conjugacy separable\ngroup. Then the quotient group ${\\rm Aut}(G)\/{\\rm Conj}(G)$ is residually\nfinite.\n\\end{lem}\n\n\\begin{proof}\nSince $G$ is finitely generated, we can find a family\n$(K_{i})_{i\\in I}$ of characteristics subgroups of finite index in\n$G$ such that for each normal subgroup $N$ of finite index in $G$\nthere is a subgroup $K_{i}$ in the above family, contained in $N$. \nEvery automorphism $f$ of $G$ induces an\nautomorphism $f_{i}$ of $G\/K_{i}$ which acts as a permutation on the\nconjugacy classes of $G\/K_{i}$. Therefore, for each $i\\in I$ we have a\nhomomorphism $\\pi_{i}:{\\rm Aut}(G)\\rightarrow S_{i}$, where $S_{i}$ denotes the\npermutation group on the set of conjugacy classes of the finite group\n$G\/K_{i}$. Since each conjugating automorphism preserves conjugacy\nclasses, the group ${\\rm Conj}(G)$ is contained in the intersection\n$\\bigcap _{i\\in I}$ Ker$(\\pi_{i})$ of the kernels of the $\\pi_{i}$'s.\nNow let $f\\in \\bigcap _{i\\in I}$ Ker$(\\pi_{i})$. Then for every $g\\in G$, \nthe elements $f(g)$ and $g$ are conjugate in each quotient\n$G\/K_{i}$. In particular, the elements $f(g)$ and $g$ are\nconjugate in each finite quotient of $G$ and thus they are\nconjugate in $G$, by the conjugacy separability of $G$. This means\nthat $f$ is a conjugating automorphism of $G$ and hence\n${\\rm Conj}(G)=\\bigcap_{i\\in I}$ Ker$(\\pi_{i})$.\n\\end{proof}\n\nThe following material on ultralimits of metric spaces can be\nfound in detail in the paper of Kapovich and Leeb \\cite{kl}.\n\nA $\\textsl{filter}$ on the set of natural numbers $\\mathbb{N}$ is\na non-empty set $\\omega$ of subsets of $\\mathbb{N}$ with the\nfollowing properties:\n\n\\begin{enumerate}\n\\item The empty set is not contained in $\\omega$.\n\\item $\\omega$ is closed under finite intersections.\n\\item If $A\\in \\omega$ and $A\\subseteq B$, then $B\\in \\omega$.\n\\end{enumerate}\n\nA filter $\\omega$ is called $\\textsl{ultrafilter}$ if it is\nmaximal. An ultrafilter is called \\textsl{non-principal} if it\ncontains the complements of the finite subsets of $\\mathbb{N}$.\nLet $\\omega$ be an ultrafilter on $\\mathbb{N}$ and\n$a:\\mathbb{N}\\rightarrow [0,\\infty)$ a sequence of non-negative real\nnumbers. Then there is a unique point $l$, which we denote by\n$\\textrm{lim}_{\\omega}a_{i}$, in the one-point compactification\n$[0,\\infty]$ of $[0,\\infty)$ such that for each neighborhood $U$\nof $l$, the inverse image $a^{-1}(U)$ of $U$ under $a$ is\ncontained in $\\omega$.\n\nLet $(X_{i},d_{i},x_{i}^{0})_{i\\in \\mathbb{N}}$ be a sequence of\nbased metric spaces and let $\\omega$ be an ultrafilter on\n$\\mathbb{N}$. On the subspace $Y$ of $\\prod_{i\\in\n\\mathbb{N}}X_{i}$ consisting of all sequences $(x_{i})$ for which\n$\\textrm{lim}_{\\omega}d_{i}(x_{i},x_{i}^{0})<\\infty$, we define a\npseudo-metric $d_{\\omega}$ by\n$d_{\\omega}\\big((x_{i}),(y_{i})\\big)=\\textrm{lim}_{\\omega}\\big(d_{i}(x_{i},y_{i})\\big)$.\nThe \\textsl{based ultralimit}\n$(X_{\\omega},d_{\\omega},x_{\\omega})$, where\n$x_{\\omega}=(x_{i}^{0})_{i\\in \\mathbb{N}}$, is the associated\nmetric space.\n\n\\bigskip\n\nLet $X$ be a geodesic metric space and $\\delta$ a non-negative real number.\nThe space $X$ is called {\\it  $\\delta$-hyperbolic\\\/} if for every triangle $\\Delta\\subset X$\nwith geodesic sides, each side is contained in the $\\delta$-neighbourhood of the\nunion of the two other sides.\n\nLet $G$ be a finitely generated group and let $X=X(G,S)$ be the Cayley graph of\n$G$ with respect to a finite generating set $S$ for $G$, closed under inverses.\nThe group $G$ is called {\\it $\\delta$-hyperbolic\\\/} if $X$ is a $\\delta$-hyperbolic\nspace with respect to the word metric. \n\n\nIn the next lemma we use various results concerning actions of groups on $\\mathbb R$-trees.\nFor details, we refer the reader \nto the paper of Morgan and Shalen \\cite{ms}.\n\nThe key point of Lemma \\ref{lem:finite} is the following statement.\nAssume that an action of a hyperbolic group $G$ on a real tree $Y$ is obtained \nas a limit of a sequence of actions of $G$ on its Cayley graph $X$, where each \naction in the sequence is the natural action of $G$ on $X$ twisted by an \nautomorphism of $G$. If the conjugacy class of an element $g$ in $G$ is periodic \nunder these automorphisms, then $g$ is elliptic when acting on the limit tree $Y$. \n\n\nAs it was pointed out by the referee, the above statement seems\nto be well known to the experts (see \\cite{bestvina2}). Nonetheless, the\nauthors failed to track down a reference for its proof.  So, a proof of it,\nis included in the lemma for the reader's convenience and completeness. \n\n\\begin{lem}\\label{lem:finite} Let $G$ be a hyperbolic group. Then the group \n${\\rm Inn}(G)$ of inner automorphisms of $G$ is of\nfinite index in ${\\rm Conj}(G)$.\n\\end{lem}\n\n\\begin{proof} Suppose to the contrary that ${\\rm Inn}(G)$ is of infinite\nindex in ${\\rm Conj}(G)$. Fix an infinite sequence\n$f_{1},f_{2},\\dots,f_{n},\\dots$ of conjugating automorphisms of\n$G$ representing pairwise distinct cosets of ${\\rm Inn}(G)$ in ${\\rm Conj}(G)$. We\napply the method of Bestvina and Paulin, using ultrafilters, to\nconstruct an $\\mathbb{R}$-tree on which $G$ acts by isometries.\n\nLet $X=X(G,S)$ be the Cayley graph of $G$ with respect to a finite\ngenerating set $S$ closed under inverses. Then $X$ with the associated\nword metric $d$ is a $\\delta$-hyperbolic metric space for some $\\delta\\ge 0$. \nEach $f_{i}$ gives an\naction $\\rho_{i}$ by isometries of $G$ on $X$ by\n$\\rho_i(g,x)=f_{i}(g)x$. The outer automorphism group of a\nvirtually cyclic group is finite. Thus, we may assume that $G$ is\nnon-elementary. In that case the action of $G$ on the boundary\n$\\partial X$ of $X$ is non-trivial and Lemma 2.1 in ~\\cite{Pau}\napplies to any action $\\rho_{i}$, yielding a sequence $x_{i}^{0}$\nof elements of $X$ such that\n\\begin{equation} \\label{eq:min} \\underset{g\\in\nS}{\\textrm{max}}\\,d\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)\\leq\n\\underset{g\\in S}{\\textrm{max}}\\,d\\big(x,f_{i}(g)x\\big)\n\\end{equation}\nfor all $x$ in $X$. Let $\\lambda_{i}=\\underset{g\\in\nS}{\\textrm{max}}\\,d\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)$. Then\n\\begin{equation} \\label{eq:trineq}\nd\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)\\leq \\lambda_{i} \\|g\\|\n\\end{equation}\nfor all $g$ in $G$, where $\\|g\\|$ denotes the word-length of $g$\nwith respect to $S$. If the sequence $(\\lambda_{i})$ contains a\nbounded subsequence, then the argument in ~\\cite[Case 1, p.\n338]{Pau} shows that there are indices $i$ and $j$ with $i\\neq j$\nsuch that the automorphisms $f_{i}$ and $f_{j}$ differ by an inner automorphism\nof $G$ and consequently they give rise to the same coset of ${\\rm Inn}(G)$, \ncontradicting the choice of the $f_{i}$. It\nfollows that\n$\\underset{i\\rightarrow\\infty}{\\textrm{lim}}\\lambda_{i}=\\infty$.\nWe consider the sequence $(X_{i},d_{i},x_{i}^{0})$ of based metric\nspaces, where $X_{i}=X$ and $d_{i}=\\frac{d}{\\lambda_{i}}$. Note\nthat the space $(X_{i},d_{i},x_{i}^{0})$ is\n$\\frac{\\delta}{\\lambda_{i}}$-hyperbolic for all $i$.\nFor any non-principal\nultrafilter $\\omega$ on $\\mathbb{N}$, let\n$(X_{\\omega},d_{\\omega},x_{\\omega})$ be the corresponding based\nultralimit. The fact that the distance $d_{\\omega}(x,y)$ of two\npoints $x=(x_{i})$ and $y=(y_{i})$ of $X_{\\omega}$ is approximated\nby the distances $d_{i}(x_{i},y_{i})$ for infinitely many $i$,\nimplies that $X_{\\omega}$ is a $0$-hyperbolic space (i.e., an\n$\\mathbb{R}$-tree), since $\\underset{i\\rightarrow\n\\infty}{\\textrm{lim}}\\frac{\\delta}{\\lambda_{i}}=0$. The action of\n$G$ on $X_{\\omega}$ is given by\n$g\\cdot(x_{i})=\\big(f_{i}(g)x_{i}\\big)$. Inequality\n~(\\ref{eq:trineq}) ensures that the action is well-defined. We will\nshow that the action of $G$ on $X_{\\omega}$ is trivial (i.e. there\nis a global fixed point).\n\nLet $g$ be an element of $G$ which acts as a hyperbolic isometry\non $X_{\\omega}$, and let $\\t_{\\omega}(g)$ denote its translation\nlength. Fix $x=(x_{i})\\in X_{\\omega}$ such that $x$ lies on the axis of $g$. Then\n$$\\t_{\\omega}(g)=d_{\\omega}(gx,x)=\\left(\\textrm{lim}_{\\omega}\nd_i(f_i(g)x_i,x_i)=\\right)\n\\displaystyle\\frac{d_{\\omega}(g^{n}x,x)}{n}$$ for all positive integers $n$.\nIn particular,\n$d_{\\omega}(gx,x)=\\displaystyle\\frac{d_{\\omega}(g^{2}x,x)}{2}$ and thus\n\\begin{equation}\\label{eq:2}\n\\textrm{lim}_{\\omega}\\Big(2d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)-d_{i}\n\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\Big)\n=0.\n\\end{equation}\nFor each $i$, we fix an element $y_{i}$ of $X$ on which the\ndisplacement function of $f_{i}(g)$ attains its minimum\n$\\t(f_{i}(g))$, i.e.\n$\\t(f_{i}(g))=d\\big(f_{i}(g)y_{i},y_{i}\\big)=\\textrm{inf\\,}\\{d\\big(f_{i}(g)y,y\\big)|\\,y\\in\nX\\}$.\\footnote{The reader should not confuse this with the algebraic translation length of \nthe elements of a group $G$.} \nSince $X$ is a $\\delta$-hyperbolic space, there is a non-negative\nconstant $K(\\delta)$ depending only on $\\delta$ such that\n\\begin{equation}\\label{ineq:a}\nd\\big(f_{i}(g)x_{i},x_{i}\\big)\\geq\n2d(x_{i},y_{i})+\\t(f_{i}(g))-K(\\delta).\n\\end{equation}\nThen,\n\\begin{equation} \\label{ineq:b}\n\\begin{array}{ccl} A & = &\n2d\\big(f_{i}(g)x_{i},x_{i}\\big)-d\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\\\\n & \\geq &\n 4d(x_{i},y_{i})+2\\t(f_{i}(g))-2K(\\delta)-d\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\\\\n & \\geq & 4d(x_{i},y_{i})+2\\t(f_{i}(g))-2K(\\delta)-2d(x_{i},y_{i})-2\\t(f_{i}(g))\\\\\n & = & 2d(x_{i},y_{i})-2K(\\delta)\\,,\n \\end{array}\\end{equation}\nwhere the second inequality follows from the triangle one. Working\nin a similar way, we see that\n\\begin{equation}\n2d(x_{i},y_{i})-K(\\delta)\\leq\nd\\big(f_{i}(g)x_{i},x_{i}\\big)-\\t(f_{i}(g))\\leq 2d(x_{i},y_{i})\n\\end{equation}\nand hence we can say that \n\\begin{equation}\\label{ineq:d}\n\\big|d\\big(f_{i}(g)x_{i},x_{i}\\big)-\\t(f_{i}(g))\\big|\\leq\n2d(x_{i},y_{i})+K(\\delta).\n\\end{equation}\nConsequently,\n\\[\\begin{array}{ccl}\n\\Big|\\t_{\\omega}(g)-\\frac{\\t(f_{i}(g))}{\\lambda_{i}}\\Big| & \\leq &\n\\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+\n\\Big|d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)-\n\\frac{\\t(f_{i}(g))}{\\lambda_{i}}\\Big| \\\\\n & \\leq &\n \\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+2\\displaystyle\\frac{d(x_{i},y_{i})}\n{\\lambda_{i}}+\\frac{K(\\delta)}{\\lambda_{i}}\\\\\n & \\leq &\n \\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+\\displaystyle\\frac{|A|}{\\lambda_{i}}+\n 3\\frac{K(\\delta)}{\\lambda_{i}}\\,,\\\\\n\\end{array}\\]\nwhere the second inequality follows from ~(\\ref{ineq:d}) and the third\none from ~(\\ref{ineq:b}). The $\\omega$-limit of each term in\nthe right-hand side of the above inequality is $0$ (for the second term\nsee ~(\\ref{eq:2})). Therefore\n\\[\\t_{\\omega}(g)=\\textrm{lim}_{\\omega}\\frac{\\t(f_{i}(g))}{\\lambda_{i}}=0\\,,\\]\nsince $\\t(f_{i}(g))=\\t(g)$ for all $i$ ($f_{i}$ being a conjugating\nautomorphism). Hence, each element $g$ of the finitely generated\ngroup $G$ acts as an elliptic isometry on $X_{\\omega}$. This means\nthat the action of $G$ on $X_{\\omega}$ has a global fixed point, say $z=(z_i)$. \nIt follows that for every $\\varepsilon>0$ and every\nfinite subset $F$ of $G$, there is an $\\Omega\\in \\omega$ such that\n\\[d_{i}\\big(z_{i},f_{i}(g)z_{i}\\big)=\\frac{d(z_{i},f_{i}(g)z_{i})}{\\lambda_{i}}<\n\\varepsilon,\\] for all $i\\in \\Omega$ and $g\\in F$. This\ncontradicts to the minimality of $\\lambda_{i}$ (see ~(\\ref{eq:min})).\n\\end{proof}\n\nWe should mention here that this is the best possible result in that generality.\nIndeed, as shown by Burnside \\cite{burnside} and subsequently by several other\nauthors (see also Sah \\cite{sah}), there are finite groups that posses non-trivial \nouter conjugating automorphisms\n(known also as outer, class preserving automorphisms). Therefore one can \neasily construct free-by-finite groups with outer conjugating automorphisms by\nconsidering the direct product of the above mentioned finite groups by free groups.  \n\nWe are now able to show our main theorem.\n\n\\begin{thm}\\label{theorem}\nLet $G$ be a conjugacy separable, hyperbolic group. Then each\nvirtually torsion-free subgroup of the outer automorphism group\n${\\rm Out}(G)$ of $G$ is residually finite.\n\\end{thm}\n\\begin{proof} It suffices to show that each torsion-free subgroup $H$\nof ${\\rm Out}(G)$ is residually finite. We consider the following short\nexact sequence\n\\[1 \\rightarrow {\\rm Conj}(G)\\big\/{\\rm Inn}(G)\\stackrel{i}{\\rightarrow} {\\rm Aut}(G)\\big\/{\\rm Inn}(G)\\stackrel{\\pi}{\\rightarrow} \n{\\rm Aut}(G)\\big\/{\\rm Conj}(G) \\rightarrow 1\\,.\\]\n By Lemma ~\\ref{lem:finite} the first term is finite. This implies that the \nrestriction of $\\pi$ on $H$\nis a monomorphism. It follows that $H$ is residually finite being\nisomorphic to a subgroup of ${\\rm Aut}(G)\\big\/{\\rm Conj}(G)$, which is\nresidually finite by Lemma ~\\ref{lem:Gr}.\n\\end{proof}\n\n\nIn view of the above theorem, given a group $G$ it is natural to\nseek conditions under which the outer automorphism group ${\\rm Out}(G)$\nof $G$ is virtually torsion-free. Recently, Guirardel and Levitt ~\\cite{gl} have shown that the\nouter automorphism group of a hyperbolic group $G$ is virtually\ntorsion-free, provided that $G$ is virtually torsion-free. Thus, by\nTheorem ~\\ref{theorem}, the outer automorphism group of a\nconjugacy separable hyperbolic group is residually finite, since \na residually finite hyperbolic group is virtually torsion-free. \n\nThe next lemma is more or less\nknown (see ~\\cite{gl,Mc}).\n\n\n\\begin{lem}\\label{lem:vir} Let $G$ be a finitely generated group containing a\nnormal subgroup $N$ of finite index whose center is trivial. If\n${\\rm Out}(N)$ is virtually torsion-free, then so is ${\\rm Out}(G)$.\n\\end{lem}\n\n\\begin{proof} Let ${\\rm Aut}_N(G)$ denote the subgroup of ${\\rm Aut}(G)$\nconsisting of those automorphisms of $G$ which fix $N$ and induce\nthe identity on $G\/N$. The restriction map $\\phi: {\\rm Aut}_N(G) \\rightarrow {\\rm Aut}(N)$ is an\ninjection. Indeed, let $g\\in G$ and $f\\in {\\rm Aut}_N(G)$. Then\n$f(g)=gh_{g}$ for some $h_{g}\\in N$. Suppose now that $f$ is in\nthe kernel of the restriction map. Then for each $h$ in $N$ we\nhave $ghg^{-1}=f(ghg^{-1})=gh_{g}hh_{g}^{-1}g^{-1}$. This implies\nthat $h_{g}$ is in the center of $N$, which is trivial, and\ntherefore $f$ is the identity. From the injectivity of $\\phi$ we\nget $\\phi\\big({\\rm Inn}_N(G)\\big)={\\rm Inn}(N)$, where ${\\rm Inn}_N(G)$ denotes\nthe (normal) subgroup of ${\\rm Inn}(G)$ consisting of all inner\nautomorphisms of $G$ induced by elements of $N$. We conclude that\nthe quotient group ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ embeds into ${\\rm Out}(N)$. In\nparticular, ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is virtually torsion-free.\n\nNow the restriction of the natural projection $\\pi:{\\rm Aut}(G) \\rightarrow\n{\\rm Out}(G)$ to ${\\rm Aut}_N(G)$ induces a map $\\tilde{\\pi}:{\\rm Aut}_N(G)\/{\\rm Inn}_N(G) \n\\rightarrow {\\rm Out}(G)$. The kernel of $\\tilde{\\pi}$ (which is equal to\n${\\rm Inn}(G)\/{\\rm Inn}_N(G)$) is finite, since $N$ is of finite index in\n$G$, while its image has finite index in ${\\rm Out}(G)$, since ${\\rm Aut}_N(G)$ \nis of finite index in ${\\rm Aut}(G)$, by ~\\cite[Lemma 1]{Mc}. It\nfollows that each finite-index, torsion-free subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ \nembeds as a subgroup of finite index in ${\\rm Out}(G)$,\nwhich proves the lemma.\n\\end{proof}\n\n\\begin{cor}\\label{cor:free-by-finite}\nThe outer automorphism group of a finitely generated,\nfree-by-finite group is residually finite.\n\\end{cor}\n\\begin{proof} It is well-known that every finitely generated,\nfree-by-finite group $G$ is hyperbolic. Furthermore, $G$ is\nconjugacy separable by ~\\cite{Dy}. Therefore Theorem\n~\\ref{theorem} applies to $G$. On the other hand, it is also known\nthat the outer automorphism group of a free group is virtually\ntorsion-free. If $G$ is virtually infinite cyclic, then ${\\rm Out}(G)$\nis finite. In the case where $G$ is not virtually cyclic, the\ncenter of a free subgroup of finite index in $G$ is trivial, and\nthe result follows from Lemma ~\\ref{lem:vir}.\n\\end{proof}\n\nRecall that a Fuchsian group is a discrete subgroup of the group of \nisometries of the hyperbolic plane $\\mathbb{H}^{2}$.\n\n\\begin{cor} The outer automorphism group of a finitely generated,\nFuchsian group is residually finite.\n\\end{cor}\n\n\\begin{proof} Every finitely generated Fuchsian group $G$ contains  \na normal torsion-free subgroup of finite \nindex, say $N$, which is either a free group or the fundamental group of an\norientable surface group of genus $g\\ge 2$. So $N$ is hyperbolic since it is either free or \nquasi-isometric to $\\H^2$. Moreover $N$ is conjugacy separable \\cite[Theorem 3.3]{stebe} and \nits outer automorphism group ${\\rm Out}(N)$ is virtually torsion-free and so $N$ \nsatisfies the hypotheses of\nTheorem \\ref{theorem}. Hence, ${\\rm Out}(N)$ is residually finite.\n\nIf $N$ is cyclic then $G$ is virtually cyclic and so ${\\rm Out}(G)$ is finite. In all other\ncases, $N$ is a normal subgroup of finite index in $G$ with trivial centre (since it is\na non-cyclic torsion-free hyperbolic group) and so from the proof of \nLemma \\ref{lem:vir} we have that ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is a subgroup of ${\\rm Out}(N)$. \nHence, every subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is residually finite. But, again from the proof of \nLemma \\ref{lem:vir}, there is a torsion-free subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ \nthat  embeds as a finite index subgroup \nin ${\\rm Out}(G)$. Therefore, ${\\rm Out}(G)$ is residually finite.\n\\end{proof}\n\n\nThe corollary below generalizes the results of Grossman \\cite{Gr} and of \nAllenby, Kim and Tang \\cite{akt}. Notice that for the exceptional cases of the Torus and\nthe Klein bottle it is easily verified that the result still holds.\n\\begin{cor}\nThe mapping class group $\\mod_S$ of a hyperbolic 2-orbifold $S$ with finite volume\nis residually finite.\n\\end{cor}\n\n\\begin{proof}\nThe fundamental group of a hyperbolic 2-orbifold with finite volume is a\nfinitely generated Fuchsian group. Hence, the proof is an immediate consequence \nof the results of Fujiwara \\cite{fujiwara}, the\nabove corollaries and the fact that subgroups of residually finite groups\nare residually finite. \n\\end{proof}\n\n\n\\subsubsection*{Addendum. Added, October 8, 2005.} Only recently the authors found out that their \nmain result can be\ngeneralized to relatively hyperbolic groups by using \\cite[Theorem\n1.1]{bs}.\n\nRelatively hyperbolic groups were introduced by\nGromov in \\cite{gromov}, in order to generalize notions such as\nthe fundamental group of a complete, non-compact, finite volume \nhyperbolic manifold and to give a hyperbolic version of small \ncancellation theory over free groups by adopting the geometric language \nof manifolds with cusps. \n\nThis notion has been \ndeveloped by several authors and, in particular, various \ncharacterizations of relatively hyperbolic groups\nhave been given (see \\cite{bow,osin} and \\cite{ds} \nand references therein).\n\nWe recall here one of Bowditch's equivalent definitions. A finitely\ngenerated group $G$ is {\\it hyperbolic relative to a\nfamily of finitely generated subgroups\\\/} $\\mathcal{G}$ if $G$ admits a \nproper, discontinuous and isometric action on a proper, hyperbolic \npath metric space $X$ such that $G$ acts on the ideal boundary of \n$X$ as a geometrically\nfinite convergence group and the elements of $\\mathcal G$\nare the  maximal parabolic subgroups of $G$.\n\n We should mention here that Farb\n\\cite{farb} introduced a weaker notion of relative hyperbolicity \nfor groups using constructions on the Cayley graph of the groups.\n\n\n\n\n\nExcept of the fundamental groups of hyperbolic manifolds\nof finite volume, another interesting family of relatively \nhyperbolic groups are the fundamental groups of graphs of finitely generated\ngroups with finite edge groups which are hyperbolic relative to the \nfamily of vertex groups (since their action on the Bass-Serre tree \nsatisfies definition 2 in \\cite{bow}).\n\nThe key lemma of the paper, Lemma 2.2, generalizes to\nrelatively hyperbolic groups.\n\n\\setcounter{section}{2} \\setcounter{thm}{1}\n\n\\begin{lem}\\hskip -.2cm{$\\mathbf '$}\nLet $G$ be a relatively hyperbolic group. Then the group\nInn$(G)$ of inner automorphisms of $G$ is of finite\nindex in Conj$(G)$.\n\\end{lem}\n\n\\noindent {\\it Sketch of proof.} In this case the Cayley graph\nof $G$ is replaced by the $\\d$-hyperbolic metric space $X$ on\nwhich $G$ acts by isometries.\n\nLet $\\l_i=\\inf\\limits_{x\\in X}\\max\\limits_{s\\in S} d(x,f_i(s)x)$\nwhere $S$ is a fixed finite generating set $G$ and let $x_i^0\\in X$\nsuch that $\\max\\limits_{s\\in S}d(x_i^0,f_i(s)x_i^0)\\le \\l_i+\\frac 1i$.\n\nBy the proof of Theorem 1.1 in \\cite{bs}, the sequence $\\l_i$\nconverges to infinity. Hence for every non-primitive ultrafilter\n$\\omega$ on $\\mathbb N$ the based ultralimit $(X_{\\omega},d_{\\omega},x_{w}^0)$ of the\nsequence of based metric spaces $(X,\\frac{d}{\\l_i},x_i^0)$ is an\n$\\mathbb R$-tree. Moreover, there is an induced non-trivial isometric\n$G$-action on $(X_{\\omega},d_{\\omega},x_{w}^0)$, given by $g\\cdot\nx_i=f_i(g)x_i$.\n\nFollowing step-by-step the proof of Lemma 2.2 we arrive again at the\ncontradiction that the action has a global fixed point.\n\nThe reader should be careful in the following.\n\\begin{enumerate}\n\\item The elements $y_i$ of $X$ are chosen such that\n$$\\t(f_i(g))\\le d(f_i(g)y_i,y_i) \\le \\t(f_i(g))+\\frac 1i.$$\n\\item The inequalities (5), (6) and (7) become\n$$ A\\ge 2d(x_i,y_i)-2K(\\d) -\\frac 2i \\eqno (5') $$\n$$ 2d(x_i,y_i)-K(\\d) \\le d(f_i(g)x_i,x_i)-\\t(f_i(g))\\le 2d(x_i,y_i)+\\frac 1i \\eqno (6')$$\n$$ \\textrm{and} \\quad |d(f_i(g)x_i,x_i)-\\t(f_i(g))| \\le 2d(x_i,y_i)+K(\\d)+\\frac 1i, \\eqno (7')$$\n\\noindent respectively.\n\n\\item Finally, the last inequality becomes\n$$\\left|\\t_{\\omega}(g) - \\frac{\\t(f_i(g))}{\\l_i}\\right|\\le\n\\left| \\t_{\\omega}(g)- d_i(f_i(g)x_i,x_i)\\right|+\\frac{|A|}{\\l_i}+3\\frac{K(\\d)}{\\l_i}+\\frac{3}{i\\l_i}.$$\n\\end{enumerate}\n\\hfill$\\Box$\n\nConsequently, Theorem 2.3 generalizes as follows.\n\n\\setcounter{thm}{2}\n\\begin{thm}\\hskip -.2cm{$\\mathbf '$}\nLet $G$ be a conjugacy separable, relatively hyperbolic group.\nThen each virtually torsion-free subgroup of the outer\nautomorphism group {\\rm Out}$(G)$ of $G$ is residually finite.\n\\end{thm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe Thomas-Ehrman Level Displacement formalism (TELD)~\\cite{bib:ehr51,bib:tho52} is an established technique for calculating the level displacement between mirror pairs. It \nis found to be particularly useful in situations where a reaction proceeds via a proton resonant state in a proton-rich nucleus.\nLargely, this usefulness derives from the fact that such states are above the particle decay threshold, usually resulting in proton partial\nwidths too narrow to be measured experimentally. Thus, by appealling to the charge symmetry of the nuclear force, one may make use of\nrelatively abundant spectroscopic data of analogue states in the mirror nucleus to determine the properties of the astrophysically interesting states. \nExamples in the literature are the $^{21}$Ne-$^{21}$Na~\\cite{bib:mar57}, $^{20}$F-$^{20}$Na~\\cite{bib:lan86}, \n$^{18}$O-$^{18}$Ne~\\cite{bib:wie88}, $^{22}$Ne-$^{22}$Mg~\\cite{bib:rui03} and $^{46}$Ti-$^{46}$Cr~\\cite{bib:hor02,bib:he07} mirror nuclear pairs.\nHowever, a survey of the literature finds inconsistency in the definition of critical parameters, leading to errors in the calculations. \nIn the present work a complete and consistent TELD formalism is presented and made available for wider use. \n\n\\section{The Wave Function}\nHere we reproduce and expand upon the original work of Thomas~\\cite{bib:tho52}, using, for consistency, exactly the same terminology. The channel radius\nof two interacting bodies is defined as $a_c=1.44\\times(A_{1c}^{1\/3}+A_{2c}^{1\/3})$ fm, with $A_{1c}$ and $A_{2c}$ being the mass numbers of the bodies\nof the pair; the reduced mass is $M_c= A_{1c} A_{2c}\/(A_{1c}+A_{2c})$; the energy of relative motion is $\\epsilon_c$, which may be positive or negative.\nThe subscript $c$ is used to describe all of the features of the channel, unless it is necessary to distinguish the positive-energy ($\\epsilon_{c+}>0$)\nfrom the negative-energy ($\\epsilon_{c-}<0$) channels in which case the symbols $c+$ and $c-$ are used, respectively.\n\nFor external wave functions, a radial factor (Equ. 1 of ref~\\cite{bib:tho52}) may be written that satisfies the wave equation\n\\begin{equation}\n\\overline{F}_c^{\\prime\\prime} + (2M_c\/\\hbar^2)(\\epsilon_c - \\mho_c)\\overline{F}_c = 0,\n\\label{eq:1}\n\\end{equation}\nwhere a prime signifies differentiation with respect to $r$ (in the following descriptions, all the derivatives are with respect to $r$ unless stated\notherwise). The interaction potential may be written\n\\begin{equation}\n\\mho_c = Z_{1c}Z_{2c}e^2r_c^{-1} + (\\hbar^2\/2M_c)\\ell(\\ell+1)r_c^{-2},\n\\label{eq:2}\n\\end{equation}\nwhere the nuclear potential term disappears in the external region. In the notation of Yost, Wheeler, and Breit~\\cite{bib:yos36}, the positive-energy\nsolution, which is regular at the origin, is designated by ${F(kr)}$ and has the asymptotic form for large $r$,\n\\begin{equation}\nF_{c+} \\thicksim \\sin(x-\\frac{1}{2}\\ell \\pi -\\eta \\mathrm{ln}2x+ \\sigma).\n\\label{eq:3}\n\\end{equation}\nLikewise, there is a solution which is linearly independent of $F$ and irregular at the origin which is conveniently taken with the asymptotic form\nfor large $r$,\n\\begin{equation}\nG_{c+} \\thicksim \\cos(x-\\frac{1}{2}\\ell \\pi -\\eta \\mathrm{ln}2x+ \\sigma).\n\\label{eq:4}\n\\end{equation}\nThe quantities entering Equ.~\\ref{eq:3} and~\\ref{eq:4} are\n\\begin{eqnarray}\nx_{c\\pm} & = & kr \\nonumber \\\\\nk_{c\\pm} & = & p\/\\hbar =(2M_c|\\epsilon|\/\\hbar^2)^{1\/2} \\nonumber,\n\\end{eqnarray}\nwith Sommerfeld parameter\n\\begin{eqnarray}\n\\eta_{c\\pm}=M_cZ_{1c}Z_{2c}e^2\/\\hbar^2k =Z_{1c}Z_{2c}e^2\/\\hbar v \\nonumber,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\sigma_{c+}=\\mathrm{arg}\\Gamma(1+\\ell+i\\eta) \\nonumber.\n\\end{eqnarray}\nIt is worth noting that $x$ is replaced with $\\rho$ in some formulations.\n\nThe general solution of this equation, $\\overline{F}(r)$, is a linear combination of $F$ and $G$. The Wronskian relation for these two particular\nsolutions, which directly follows from Equ.~\\ref{eq:1}-~\\ref{eq:4} is\n\\begin{equation}\nF^\\prime G-G^\\prime F=k_{c+}.\n\\label{eq:5}\n\\end{equation}\nExtensive tables~\\cite{bib:blo50} and several computer codes~\\cite{bib:bar74,bib:bar82,bib:sea02} have been developed for evaluating $F$ and $G$ and\ntheir derivatives when $\\eta>0$.\n\nFor the $c-$ channels, only the solution to Equ.~\\ref{eq:1}, vanishing at large distances from the origin, can occur; it is the\nWhittaker function~\\cite{bib:whi35,bib:mag48},\n\\begin{widetext}\n\\begin{eqnarray}\nW_{-\\eta,\\ell +\\frac{1}{2}}(2x_{c-})=\n\\frac{e^{-x-\\eta \\mathrm{ln}2x}}{\\Gamma (1+\\ell+\\eta)}\\int_0^\\infty t^{\\ell+\\eta}e^{-t}\\left (1+\\frac{t}{2x} \\right) ^{\\ell-\\eta}dt.\n\\label{eq:8}\n\\end{eqnarray}\n\\end{widetext}\nWhittaker function and its derivative may be accurately calculated using the {\\tt whittaker\\_w}~\\cite{bib:nob04} computer code.\nHowever, it is useful to note that if there is no Coulomb interaction in a $c-$ channel, one has from Equ.~\\ref{eq:8} for $s$, $p$, $d$, and $f$\norbitals the simpler relations\n\\begin{eqnarray}\n& & W_{0,\\frac{1}{2}}(2x)=e^{-x} \\label{eq:9} \\\\\n& & W_{0,\\frac{3}{2}}(2x)=(1+x^{-1})e^{-x} \\label{eq:10} \\\\\n& & W_{0,\\frac{5}{2}}(2x)=(1+3x^{-1}+3x^{-2})e^{-x} \\label{eq:11} \\\\\n& & W_{0,\\frac{7}{2}}(2x)=(1+6x^{-1}+15x^{-2}+15x^{-3})e^{-x} \\label{eq:12}\n\\end{eqnarray}\nwhich can be used for checking the results from a more complicated code.\n\nIn discussing conditions at the nuclear surface, one needs to evaluate the real and imaginary parts of the logarithmic derivatives,\n$g_c=E^\\prime\/E$, and these are \\cite{bib:tho52},\n\\begin{eqnarray}\n& & g_{c+}^{Re}=(FF^\\prime+GG^\\prime)(F^2+G^2)^{-1} \\label{eq:13} \\\\ \n& & g_{c+}^{Im}=k(F^2+G^2)^{-1} \\label{eq:14} \\\\\n& & g_{c-}^{Re}=W^\\prime W^{-1} \\label{eq:15} \\\\\n& & g_{c-}^{Im}=0 \\label{eq:16} \n\\end{eqnarray}\nwhere $g_c=g^{Re}+\\mathrm{i}g^{Im}$ and $r_c=a_c$. Although the simple WKB approximation \\cite{bib:tho52,bib:mar57} can perform well in calculating the\nlogarithmic derivatives of the Coulomb and Whittaker functions in specified regions, modern computer codes perform essentially exact calculations and are preferred.\nFor example, the difference between the WKB approximation and the exact evaluation of $g_{c-}$, performed using the code {\\tt whittaker\\_w}~\\cite{bib:nob04},\nin the region 0.1$<x<$5.0, 0.1$<\\eta<$10.0 (roughly corresponding to $Z_{1c}Z_{2c}\\leq$20 and bound nucleon\nenergy 0.1$<|E_b|<$10.0 MeV), can be as large as 3\\% ($\\ell$=1),\n1\\% ($\\ell$=2) and 0.6\\% ($\\ell$=3), respectively, and becomes\nsmaller as $\\eta$ increases. In the case that $\\ell$=0, the differences\ncan be as much as 10\\% when $x<$0.1 and $\\eta<$0.9,\nor $x<$0.2 and $\\eta<$0.5, or $x<$0.5 and $\\eta<$0.3. \nIn addition, it should be noted that a frequently used subroutine, {\\tt COULFG}~\\cite{bib:bar82}, was unable to reproduce the results\n($F$, $G$ and their derivatives) of the subroutine {\\tt RCWFN} near values of $\\eta=1.70x+5.13$ unless the\n{\\tt ACCUR} variable was set to be less than $10^{-17}$ (of course the smaller the safer, {\\em e.g.}, $10^{-30}$\nif possible). \n\n\\section{Calculation of level displacements}\n\nUnder the assumption of charge symmetry of nuclear forces, the $nn$ and $pp$ nuclear interactions are identical. The difference in the excitation energy\nof levels in mirror nuclei is therefore due to differences in the Coulomb, electromagnetic spin-orbit, and mass energies. As suggested\nby~\\cite{bib:tho52}, one can evaluate this by considering $H_p-H_n \\equiv V$. Furthermore, in the one-level approximation, irrespective of the\nboundary conditions, in the internal region the proton and neutron wavefunctions are the same to within a multiplicative constant.\nUnder these assumptions, one obtains (Equ. 30a in ref.~\\cite{bib:tho52})\n\\begin{equation}\nE_{n}-E_{p}=-\\langle V \\rangle_\\tau + \\Delta_{\\lambda n} - \\Delta_{\\lambda p},\n\\label{eq:18}\n\\end{equation}\nwhere there is a boundary condition that satisfies $\\overline{b}_{nc}=\\overline{b}_{pc}$ (defined in Equ. 24c of~\\cite{bib:tho52}).\nHere, $\\langle V \\rangle_\\tau$ is the mean value of $V$ in the internal region, and $E_n$ and $E_p$ are the Eigenvalues satisfying\n$H_{n(p)}\\Psi_{n(p)}=E_{n(p)}\\Psi_{n(p)}$ of the nucleus with the odd neutron or proton, respectively.\nThe difference of the level displacements of the neutron and proton states (see Equ. 30b of~\\cite{bib:hor02}),\n\\begin{equation}\n\\Delta_{\\lambda}=\\Delta_{\\lambda n} - \\Delta_{\\lambda p} = -\\sum_{c\\pm}\\gamma_c^2(g_{nc}^{Re}-g_{pc}^{Re}),\n\\label{eq:19}\n\\end{equation}\nis referred to as the boundary condition level displacement.\nThe energy difference of corresponding levels of mirror nuclei, following from Equ.~\\ref{eq:18} and ~\\ref{eq:19}, can be written as\n\\begin{equation}\n(E_{n}^{\\ast}-E_{n}^{g.s.})-(E_{p}^{\\ast}-E_{p}^{g.s.})=\\Delta_{\\lambda}^{\\ast}-\\Delta_{\\lambda}^{g.s.},\n\\label{eq:20}\n\\end{equation}\nwhere it is assumed that the quantity $\\langle V \\rangle_\\tau$ is the same in the excited state as in the ground state. The scripts $\\ast$ and {\\em g.s.}\ndenote the corresponding quantities are being evaluated with respect to the excited state and the ground state, respectively. Assuming that the level\ndisplacements of the two ground states are the same simplifies this relation further,\n\n\\begin{equation}\nE^\\ast(n)-E^\\ast(p)=\\Delta_{\\lambda}^{\\ast},\n\\label{eq:21}\n\\end{equation}\nwhere $E^\\ast(n)$ and $E^\\ast(p)$ are the corresponding excitation energies in the mirror nuclei.\nTherefore, the observed energy difference between mirror nuclear states is due to different level displacements,\n$\\Delta_{\\lambda n}$ and $\\Delta_{\\lambda p}$, in the two nuclei.\n\n\\begin{table*}\n\\caption{\\label{table1} Various definitions of $\\Gamma_{\\lambda c}$,$\\gamma_{\\lambda c}^2$, $\\theta_{\\lambda c}^2$ and\n$\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60} of a level $\\lambda$ in the literature.}\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n \n   & $\\Gamma_{\\lambda c}$ & $\\gamma_{\\lambda c}^2$ & $\\theta_{\\lambda c}^2$ & $\\theta_{sp}^2$ \\\\\n  \\hline\n  Thomas~\\cite{bib:tho52}          & 2$P_c(\\gamma_{\\lambda c}^2\/a_c)$ & $\\frac{\\hbar^2}{2M_c a_c}\\theta_{\\lambda c}^2$    & $a_c\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$                    &  \\\\\n  Lane and Thomas~\\cite{bib:lan58,bib:lan60} & 2$P_c\\gamma_{\\lambda c}^2$       & $\\frac{\\hbar^2}{M_c a_c^2}\\theta_{\\lambda c}^2$   & C$^2$S$\\theta_{sp}^2$ & $\\frac{a_c}{2}\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$ \\\\\n  French~\\cite{bib:fre60}          & 2$P_c\\gamma_{\\lambda c}^2$       & $\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{\\lambda c}^2$ & C$^2$S$\\theta_{sp}^2$ & $\\frac{a_c}{3}\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$ \\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\nWe find that the definitions used by various authors of partial width $\\Gamma_{\\lambda c}$, reduced width $\\gamma_{\\lambda c}^2$, dimensionless reduced\nwidth $\\theta_{\\lambda c}^2$ and dimensionless single-particle reduced width $\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60}, of a level $\\lambda$, are not\nconsistent, as shown in Table~\\ref{table1}. Following the previous work, the definitions of French~\\cite{bib:fre60} are adopted in the present work,\nthough it is possible to achieve consistency using the alternative definitions~\\cite{bib:tho52,bib:lan58}.\nOne should pay attention that the definition of $\\gamma_{\\lambda c}^2$~\\cite{bib:fre60} is different from that of~\\cite{bib:tho52} by a factor of $3\/a_c$.\nThus, for the positive-energy channel,\n\\begin{equation}\n-\\gamma_{c+}^2g_{c+}^{Re}=-\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{c+}^2P_c(FF_{x}^\\prime+GG_{x}^\\prime)\n\\label{eq:22}\n\\end{equation}\nwith the Coulomb penetrability $P_c=x\/(F^2+G^2)$; and for the negative-energy channel,\n\\begin{equation}\n-\\gamma_{c-}^2g_{c-}^{Re}=-\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{c-}^2 x \\frac{W_{x}^\\prime}{W}.\n\\label{eq:23}\n\\end{equation}\nWhere $F_x^\\prime$,$G_x^\\prime$ and $W_x^\\prime$ represent differentiation with respect to $x$. Equ.~\\ref{eq:22} \\&~\\ref{eq:23} were defined as\n$\\Delta_b$ and $\\Delta_r$ in previous literature by assuming $\\overline{b}_{nc}=\\overline{b}_{pc}=0$ (see Fig.~\\ref{prcfig1}).\n\nThe reasonable assumption that the reduced widths, $\\gamma_{c}^2$, are the same for the mirror levels leads to an\nassumption of $\\theta_{c}^2=\\theta_{p}^2=\\theta_{n}^2$. Thus, the excitation-energy displacements of mirror nuclei can be expressed as\n\\begin{equation}\n\\Delta_{\\lambda}^{\\ast}=\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_c^2 \\left\\{\\left[P_c(FF_{x}^\\prime +GG_{x}^\\prime)\\right]_{\\mid E=E_r}- \\left(x\\frac{W_{x}^\\prime}{W}\\right)_{\\mid E=E_b} \\right\\},\n\\label{eq:24}\n\\end{equation}\nwhere $E_r$ [=$E^\\ast(p)-S_p^p$] and $E_b$ [=$|E^\\ast(n)-S_n^n|$] are the energies relative to the respective nucleon thresholds ($S_p^p$ and $S_n^n$;\nthe superscripts $p$ and $n$ refer to the odd-proton (or proton-rich) and odd-neutron (or neutron-rich) nuclei, respectively; the subscripts $p$ and $n$\ndenote the corresponding nucleon separation energies). A pictorial representation of the physical meanings of the parameters described here are illustrated  \nin Fig.~\\ref{prcfig1} which shows the case of the 6.424~MeV state in $^{46}$Ti and its analogue state in $^{46}$Cr~\\cite{bib:hjj07}. It is clear from the \nfigure that the level displacement may equally well be written as,\n\\begin{equation}\nE^\\ast(n)-E^\\ast(p)=[E^\\ast(n)-S_p^p]-[E^\\ast(p)-S_p^p]=E_b^\\prime-E_r,\n\\label{eq:25}\n\\end{equation}\nwhere the quantity $E_b^\\prime$ is different from that of $E_b$ defined above. It appears that confusion \nover these definitions has, in part, been the source of errors in the past. For example, in several previous \nworks~\\cite{bib:lan86,bib:wie88,bib:hor02,bib:rui03} the level displacement has been written as \n$E^\\ast(n)-E^\\ast(p)=E_b-E_r$ in contrast to the correct expression of $E^\\ast(n)-E^\\ast(p)=E^\\prime_b-E_r$ \nwhich follows from Marion {\\em et al.}~\\cite{bib:mar57}. However, the pressent work validates that the correct\nexpression was used in the calculations~\\cite{bib:gor06}.\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[width=7cm]{mirror.eps}\n\\end{center}\n\\caption{\\label{prcfig1} Definitions and calculation results of relevant quantities in case of the $^{46}$Ti-$^{46}$Cr mirror pair. The calculated values\n(in units of MeV) are with respect to the analogue states at 6.424 MeV (in $^{46}$Ti) and 6.246 MeV (in $^{46}$Cr)~\\cite{bib:hjj07}.}\n\\end{figure}\n\nThe final ingredient which is needed to allow calculation of the level shift is the dimensionless reduced width, $\\theta_c^2$. \nThis can be calculated according to the relation $\\theta_c^2$=C$^2$S$\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60}, \nwhere the dimensionless single-particle reduced width $\\theta_{sp}^2$ has already been\ndetermined~\\cite{bib:ili97,bib:bar98}, and the factor C$^2$S is calculated via a shell-model code such as {\\tt OXBASH}~\\cite{bib:bro92}. \n\n\\section{Online resources}\nThe algorithms developed here have been made available online~\\cite{bib:heweb}.\nThe Visual Basic tool requires the {\\tt TELD\\_VB} and {\\tt TELD} executables be in the same directory. Opening the {\\tt TELD} \napplication provides a window in which one defines the parameters of the mirror states to be considered \n($A$, $Z$, $S_n$, $S_p$ and $E_x(n)$). One also could change values for the channel radius, the orbital angular momentum of the\nsingle particle and spectroscopic factor for the state. \nActivation of the ``calc\" button then performs the TELD formalism and returns the resulting excitation of the \nproton-rich analogue state and the proton decay width for this state. This last parameter is calculated using the relation\n\\begin{equation}\n\\Gamma_p=\\frac{3\\hbar^2P_{c}\\theta_p^2}{M_c a_c^2}.\n\\label{eq:29}\n\\end{equation}\nIn addition, a Fortran source code, which allows users to make changes whenever required, is also encloded in the TELD.rar package.\nFigure~\\ref{VBfig} shows a screenshot from the {\\tt TELD} program with all relevant parameters for the case of the \n6.424~MeV state in $^{46}$Ti and its analogue state in $^{46}$Cr~\\cite{bib:hjj07}. \n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[width=7cm]{vbfig.eps}\n\\end{center}\n\\caption{\\label{VBfig} Screenshot from the {\\tt TELD} program for the case of the 6.424~MeV state in $^{46}$Ti and its analogue state \nin $^{46}$Cr~\\cite{bib:hjj07}.} \n\\end{figure}\n\n\n\\section{Summary}\nA complete and consistent Thomas-Ehrman Level Displacement formalism has been presented and made available on the World Wide Web. With this, \nif one has knowledge (or makes a reasonable assumption) of the quantity $\\theta_c^2$ and spectroscopic factor $S$, one may estimate the\nlocation of the mirror to a known excited state. Alternatively, experimental measurement of the mirror level displacement\nprovides a route to determining $\\theta_c^2$ (or $S$ factor).\n\n\\begin{center}\n\\textbf{Acknowledgments}\n\\end{center}\n\nThe authors acknowledge support from the Engineering and Physical Sciences Research Council, UK.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent proposals to perform quantum computation in nuclear spin systems\nusing nuclear magnetic resonance (NMR)\ntechniques~\\cite{Gershenfeld97,Cory97x,Cory97b,Chuang97e}, coupled logic\noperations are performed using spin-spin couplings that occur naturally in\nmolecular systems.\nWhile this is straightforward for small systems with a few spins,\ngeneralization to complex molecular structures has been challenging.  \nThe complication is caused by the many spin-spin couplings which occur along\nwith the desired one.\nThis fundamental task to turn off spurious evolution is so difficult that,\ncoercing a complex system to {\\em do nothing}~\\cite{Jozsa98} -- ceasing all\nevolution -- can be just as difficult as making it do something\ncomputationally useful.\n\nThe task of turning off all couplings is known in the art of NMR as {\\em\ndecoupling}; doing this for all but a select subset of couplings is known as\n{\\em selective recoupling}.  \nA common method to perform these tasks is to interrupt the free evolution by\ncarefully chosen pulses.  These pulses are single spin-1\/2 (qubit) operations\nthat transform the hamiltonian in the time between pulses in such a manner\nthat unwanted evolutions in consecutive time intervals cancel out each other.\n\nPulse sequences which perform selective recoupling are generally difficult to\nfind for a large system.  Each pulse simultaneously affects many coupling\nterms in the hamiltonian.  To turn off all but one of the coupling terms,\nthese pulses have to satisfy many simultaneous requirements.\nIngenious sequences have been found for usual NMR\napplications~\\cite{Ernst94,Slichter,Mansfield} but they do not address the\nproblems relevant to quantum computation.  In usual NMR applications, the\nstructure of the spin systems is not known a-priori.  Therefore, pulse\nsequences are designed to address all the spins together rather than\nindividual spins.\nQuantum computation brings new requirements, and initial\nefforts~\\cite{Linden98} have been made to develop pulse sequences to satisfy\nthese needs; however, to-date, schemes have necessitated resources (such as\ntotal number of pulses applied) exponential in the number of spins being\ncontrolled.\n\nIn this paper, we present an {\\em efficient} scheme to perform selective\nrecoupling.\nIn contrast to the situation with traditional NMR, this scheme addresses the\nproblem relevant to NMR quantum computation, in which the molecular structure\nis assumed to be well-known, and spins are individually addressible.\nThe method is related to a class of well-known matrices called {\\em Hadamard\nmatrices}.  \nWe derive from any $n \\times n$ Hadamard matrix a pulse scheme that decouples\n$n$ spins using $n$ time intervals and ${\\cal O}(n^2)$ pulses.\nThis decoupling scheme can easily be modified  \n(i) to remove Zeeman evolution and \n(ii) to implement selective recoupling.  \nThis completes the construction for $n$ spins whenever $n \\times n$ Hadamard\nmatrices exist. \nWhen $n \\times n$ Hadamard matrices do not exist, a scheme for $n$ spins can\nstill be constructed using larger existing Hadamard matrices.  \nIn doing so, an extra amount of effort is required, but this can be bounded\nusing existence properties of Hadamard matrices.\nAltogether, the scheme requires $cn$ time intervals and less than \n$cn^2$ pulses, where $c \\approx 1$ with upper bound $c \\leq 2$.\nOur method applies whenever the spins couple pairwise and have very\ndifferent Zeeman frequencies, such as in heteronuclear spin systems.\n\nThe paper is structured as follows.  \nIn Section~\\ref{sec:statement}, we review relevant concepts in NMR quantum\ncomputing and re-state the problem precisely. \nIn Section~\\ref{sec:scheme}, we first motivate the construction of the\ndecoupling scheme with examples, and then describe the general construction\nrelated to Hadamard matrices.  Important properties of Hadamard matrices \nare summarized.  Modifications of the decoupling scheme to perform selective\nrecoupling are described.\nWe conclude with some general remarks and discussions of various properties \nand limitations of the scheme.  \n\n\\section{NMR quantum computing and the statement of the problem} \n\\label{sec:statement}\n\nIn this section, we describe the NMR system and describe how a\nuniversal set of (non-fault tolerant)\noperations~\\cite{Div95,Barenco95,Deutsch95}, namely, the single qubit\noperations and the controlled-NOT gate, can be realized using basic NMR\nprimitives.\n\nWe shall consider a physical system which consists of a solution of identical\nmolecules.  Each molecule has $n$ non-magnetically equivalent nuclear spins\nwhich serve as qubits.\nA static magnetic field is applied externally along the $+\\hat{z}$ direction.\nThis magnetic field splits the energy levels of the spin states aligned with\nand against it.  This is described in the hamiltonian by the Zeeman terms,\nwhich, in the energy eigenbasis, are given by\n\\begin{equation}\n\t{\\cal H}_{\\rm Z} \n\t= -\\frac{1}{2} \\sum_i \\hbar \\omega_i \\sigma_z^{(i)} \n\\,,\n\\label{eq:ham:zeeman}\n\\end{equation}\nwhere $i$ is the spin index, $\\omega_i\/2\\pi$ is the {\\em Zeeman\nfrequency} for the $i$-th spin, and $\\sigma_z^{(i)}$ is the Pauli\nmatrix operating on the $i$-th spin.  \nThe convention $\\hbar = 1$ is used for the rest of the paper.\nThe spins have very different Zeeman frequencies, a situation loosely termed \nas ``heteronuclear'' in this paper.\n\nNuclear spins can interact via the dipolar coupling~\\cite{Slichter,Abragam}.\nThis is given by the hamiltonian\n\\begin{equation}\n\t{\\cal H}_{\\rm d} = \\sum_{i<j} g_{ij}^{\\rm d}  \\left[\\rule{0pt}{2.4ex}  \n\t\t\\vec{\\sigma}^{(i)} \\cdot \\vec{\\sigma}^{(j)} \n\t\t- 3 (\\hat{r}_{ij} \\cdot \\vec{\\sigma}^{(i)})  \n\t\t\\otimes (\\hat{r}_{ij} \\cdot \\vec{\\sigma}^{(j)})  \\right] \n\\label{eq:dipolarfull}\n\\,,\n\\end{equation}\nwhere $\\hat{r}_{ij}$ denotes the unit displacement vector from the $i$-th to\nthe $j$-th spin, and $g_{ij}^{\\rm d}$ denotes the coupling constant between\nthem.\nSpin-spin coupling can also be mediated by coupling to electrons.  This\nindirect coupling has a tensor part and a scalar part.  The tensor part is\nusually of the same form as ${\\cal H}_{\\rm d}$.  The scalar part is given by\nthe hamiltonian \n\\begin{equation} \n\t{\\cal H}_{\\rm s} = \\sum_{i<j} g_{ij}^{\\rm s} \n\t\t\t\\vec{\\sigma}^{(i)} \\cdot \\vec{\\sigma}^{(j)} \n\\label{eq:scalar}\n\\,.  \n\\end{equation} \nIf the molecules tumble fast and isotropically, dipolar coupling and\nindirect tensor coupling will be averaged away; otherwise, the physics \ncan be more complicated.  \nHowever, in the presence of a strong external magnetic field, only the {\\em\nsecular} part (terms that commute with ${\\cal H}_{\\rm Z}$) is\nimportant~\\cite{Slichter,Abragam}.\nFor a heteronuclear system, the resulting coupling becomes \n\\begin{equation} \n\t{\\cal H}_{\\rm c} = \\sum_{i<j} g_{ij}^{\\rm c} \\sigma_z^{(i)} \\otimes\n\t\\sigma_z^{(j)}\n\\label{eq:dipolar}\n\\,, \n\\end{equation} \nindependent of the original form of coupling.  \n\nSingle qubit operations are performed by applying {\\em pulsed} radio frequency\n(RF) magnetic fields along some directions $\\hat{\\eta}$ perpendicular to the\nstatic field.\nTo address the $i$-th spin, the frequency of the RF field is chosen to\napproximate $\\omega_i\/2\\pi$.\nWhen the $\\omega_i$'s are very different, very short pulses can be used, so\nthat during the pulses, all other evolutions are negligible except for the\nrotation operator $e^{-i \\frac{\\theta}{2} \\vec{\\sigma}^{(i)} \\cdot\n\\hat{\\eta}}$ where $\\theta$ is proportional to the pulse duration and the\npower.  The Lie group of all single qubit operations can be generated by\nrotations about $\\hat{x}$ and $\\hat{y}$.\nOur scheme uses only rotations of $\\theta = \\pi$ along $\\hat{x}$, which\nimplement $\\sigma_x$ (up to an irrelevant overall phase) on the spins being\naddressed.  We denote this operation by $X$, superscripted by the spin index\nwhenever appropriate.\n\nCoupled operations such as controlled-phase-shift or controlled-NOT acting on\nthe $i$-th and the $j$-th spins can be performed given the primitive, \n\\begin{equation} \n      \tZ\\!\\!Z_{ij} = \n\te^{-i \\frac{\\pi}{4} \\sigma_z^{(i)} \\otimes \\sigma_z^{(j)}} \n\\label{eq:zzij}\n\\,.\n\\end{equation} \nFor instance, a controlled-NOT from the $i$-th to the $j$-th spin\ncan be implemented by\n\\begin{equation}\n\te^{-i \\frac{\\pi}{4} \\sigma_y^{(i)}} \n\te^{i \\frac{\\pi}{4} \\sigma_x^{(i)}} \n\te^{i \\frac{\\pi}{4} \\sigma_y^{(i)}} \n\te^{-i \\frac{\\pi}{4} \\sigma_x^{(j)}} \n\te^{i \\frac{\\pi}{4} \\sigma_y^{(j)}}\n\tZ\\!\\!Z_{ij} \n\te^{-i \\frac{\\pi}{4} \\sigma_y^{(j)}}  \n\\,.\n\\end{equation}\n\nThe ultimate goal is to be able to efficiently realize arbitrary quantum\noperations on an $n$ spin system with arbitrary couplings.  In this paper, we\nconsider a more limited objective, which can now be stated precisely, using\nthe definitions of Eq.(\\ref{eq:ham:zeeman}), Eq.(\\ref{eq:dipolar}), and\nEq.(\\ref{eq:zzij}):\n\\begin{quote} \nGiven a heteronuclear system of $n$ spins with free evolution $e^{-i ({\\cal\nH}_{\\rm Z} + {\\cal H}_{\\rm c}) t}$, controlled using typical RF pulses, how\ncan $Z\\!\\!Z_{ij}$ be implemented efficiently?\n\\end{quote}\n\n\\section{Construction of the scheme} \n\\label{sec:scheme} \n\nWe will first construct a decoupling scheme to remove the entire coupling term\n${\\cal H}_{\\rm c}$ in the total evolution.  The scheme is derived from\nHadamard matrices, which will be reviewed.\nMethods to remove ${\\cal H}_{\\rm Z}$ and to implement selective recoupling\nwill be described afterwards.\n\n\\subsection{Construction of the decoupling scheme} \n\\label{sec:examples} \n\nTo construct the decoupling scheme, we only consider ${\\cal H}_{\\rm c}$ in\nthe evolution.  Effects of ${\\cal H}_{\\rm Z}$ can be included later since all\nmatrix exponents commute.\n\nTo motivate the general construction, we analyze the simplest example of\ndecoupling two spins.  The evolution operator for an arbitrary duration $t$ is\ngiven by $\\tau = e^{-i g_{12}^{\\rm c} t \\sigma_z^{(1)} \\otimes\n\\sigma_z^{(2)}}$.  The sequence of events, $\\tau X^{(2)} \\tau X^{(2)}$, known\nas {\\em refocusing} in NMR, will first couple and then decouple the spins.\nThis can be described mathematically as:\n\\begin{eqnarray}\n\t&   & \\tau (X^{(2)} \\tau  X^{(2)}) \n\\label{eq:refocus}\n\\\\\n\t& = & e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t    (\\sigma_x^{(2)} \n\t      e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t     \\sigma_x^{(2)})  \n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t\te^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_x^{(2)} \n\t\t\\sigma_z^{(2)} \\sigma_x^{(2)}}\n\\label{eq:third}\n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\te^{-i \\theta \\sigma_z^{(1)} \\otimes (-\\sigma_z^{(2)})}\n\\label{eq:fourth}\n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t\te^{i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\\\\\n\t& = &  I \n\\end{eqnarray}\nwhere $\\theta = g_{12}^{\\rm c} t$.  Eq.(\\ref{eq:third}) is obtained using\nTaylor series expansion of the matrix exponents and using $\\sigma_x^2 = I$,\nand Eq.(\\ref{eq:fourth}) is obtained using anticommutivity of $\\sigma_x$ and\n$\\sigma_z$.\n\nThe essential features of this decoupling procedure are:\n\\\\ \\noindent {\\bf 1.} The pair of $X^{(2)}$ negates the sign of\n$\\sigma_z^{(2)}$ in the evolution between the pulses.\n\\\\ \\noindent {\\bf 2.} The $X$ pulses make the signs of the $\\sigma_z$\nmatrices of the two spins disagree for exactly half of the time.\n\\\\ \\noindent {\\bf 3.} Since the coupling is bilinear in the $\\sigma_z$\nmatrices, it is unchanged (negated) when the signs of the $\\sigma_z$\nmatrices agree (disagree).  The $X$ pulses therefore negate the coupling for\nexactly half of the time.\n\\\\ \\noindent {\\bf 4.} Since the matrix exponents commute, negating the\ncoupling for exactly half of the time suffices for the evolution to be \ncancelled out. \n\nThe most crucial point leading to decoupling is that, the signs of the\n$\\sigma_z$ matrices of the coupled spins, controlled by pairs of $X$ pulses,\ndisagree for half of the time.\n\nIn general, we consider pulse sequences which concatenate equal-time intervals\nand use $X$ pulses to control the signs of the $\\sigma_z$ of each spin.\nThe essential information on the signs can be represented by \na ``sign matrix'' defined as follows.\nThe ``sign matrix'' of a pulse scheme for $n$-spins with $m$ time intervals is\nthe $n \\times m$ matrix with the $(i,a)$ entry being the {\\em sign} of\n$\\sigma_z^{(i)}$ in the $a$-th time interval.\nThese sign matrices have one-to-one correspondence with our restricted class\nof pulse sequences.\nWe denote any sign matrix for $n$ spins by $S_n$. \nFor example, the sequence in Eq.(\\ref{eq:refocus}) can be represented by \nthe sign matrix\n\\begin{equation}\n\tS_2 = \\left[ \\begin{array}{cc}\n\t{+}&{+}\n\\\\\t{+}&{-} \n\t\\end{array} \\right] \n\\label{eq:s2}\n\\,. \n\\end{equation} \nThe general construction of decoupling scheme is now reduced to finding \nsign matrices such that every two rows disagree in half of the entries.  \n\nAs a second example, we construct a decoupling scheme for four spins.  We\nfirst find a correct sign matrix following the previous observations, and\nthen derive the corresponding pulse sequence.  For example, a possible \nsign matrix is given by \n\\begin{equation} \n\tS_4 = \\left[ \\begin{array}{cccc}\n\t{+}&{+}&{+}&{+}\n\\\\\t{+}&{+}&{-}&{-}\n\\\\\t{+}&{-}&{-}&{+}\n\\\\\t{+}&{-}&{+}&{-}\n\t\\end{array} \\right] \n\\label{eq:s4}\n\\,, \n\\end{equation} \nin which any two rows disagree in exactly two entries.  \n$S_4$ can be converted to a pulse scheme by converting each column to a time\ninterval before and after which $X$ pulses are applied to spins (rows) given\nby $-$'s.  No pulses are applied to spins (rows) with $+$'s.  The resulting\nsequence,\n\\begin{eqnarray}\n\t\\tau \n\t(X^{(3)} X^{(4)} \\tau X^{(3)} X^{(4)})  \n\t(X^{(2)} X^{(3)} \\tau X^{(2)} X^{(3)})  \\times \n\\nonumber\n\\\\\n\t\\hfill (X^{(2)} X^{(4)} \\tau X^{(2)} X^{(4)})  \n\\label{eq:dec4}\n\\,, \n\\end{eqnarray}\nis the identity by construction and this can also be verified directly.  \nNote that ${\\cal H}_{\\rm c}$ in $\\tau = e^{-i {\\cal H}_{\\rm c} t}$ now denotes\nthe sum of six possible coupling terms for four spins.\nNote also Eq.(\\ref{eq:dec4}) is written in such a way that it corresponds\nvisually to the sign matrix, though the evolutions are actually in reverse\ntime order relative to $S_4$.  However, such ordering is irrelevant for\ncommuting evolutions.  Since $X^{(i)} X^{(i)} = I$, Eq.(\\ref{eq:dec4}) can be\nsimplified to\n\\begin{equation}\n\t\\tau \n\t(X^{(3)} X^{(4)} \\tau X^{(4)})  \n\t(X^{(2)} \\tau X^{(3)})  \n\t(X^{(4)} \\tau X^{(2)} X^{(4)})  \n\\label{eq:dec4short}\n\\,.\n\\end{equation}\nThis simplified pulse sequence can also be obtained directly from\nEq.(\\ref{eq:s4}) by converting columns to time intervals and inserting\n$X^{(i)}$ between intervals whenever the $i$-th row changes sign or \nwhenever a $-$ sign reaches either end of the row.  Pulse sequences for\nEq.(\\ref{eq:dec4}) and Eq.(\\ref{eq:dec4short}) are shown in\nFig.~\\ref{fig:pulseseq}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=s24.eps,width=3.3in}}\n\\vspace*{2ex}\n\\caption{(a) Pulse sequence corresponding to Eq.(\\ref{eq:dec4}).  From $S_4$,\neach ``$-$'' sign in the $i$-th row and $a$-th column translates to two $X$\npulses at $\\omega_i$ before and after the $a$-th time interval.\n(b) Pulse sequence obtained from simplifying (a).  This corresponds to\nEq.(\\ref{eq:dec4short}), and can be constructed directly from $S_4$ by\ntranslating each change of sign in the $i$-th row to an $X$ pulse at\n$\\omega_i$.  A ``$-$'' sign at the end of the row also gives rise to an $X$\npulse at end of the last time interval.}\n\\label{fig:pulseseq}\n\\end{center}\n\\end{figure}\n\nThe above scheme can be generalized to decouple $n$ spins with $m$ time\nintervals as follows:\n\\begin{quote} \nConstruct the $n \\times m$ sign matrix $S_n$, with entries $+$ or $-$, such\nthat {\\em any} two rows disagree in exacly half of the entries.  For\neach $-$ sign \nin the $i$-th row and the $a$-th column, apply $X^{(i)}$ before and after the\n$a$-th time interval.\n\\end{quote}\nBecause of the pulses, the sign of the $\\sigma_z$ matrix for each spin in\neach time interval is as given by the sign matrix.  The $\\sigma_z$ matrices\nof any two spins therefore have opposite signs for half of the time, so\nthat their coupling is negated for exactly half of the time, and the \nevolution is always cancelled.\n\nFor $n$ spins, $n \\times m$ sign matrices which correspond to decoupling\nschemes do not necessarily exist for arbitrary $m$, but they always exist for\nlarge and special values of $m$.  A possible structure is:\n\\begin{eqnarray}\n\\nonumber\n\tS_n = \\left[ \\begin{array}{cccccccccccc}\n\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}\n\\\\\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}&{-}&{\\cdots}&{+}\n\\\\\t{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }\n\\\\\t{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}&{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}\n\\\\\t{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ } \n\\\\\t{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}\n\t\\end{array} \\right] \n\\,, \n\\end{eqnarray} \nin which intervals are bifurcated when rows (spins) are added.  Such\nbifurcation takes place whenever it is impossible to add an extra row that is\northogonal to all the existing ones (``depletion'').  If such depletion occurs\nfrequently, the sign matrix will have exponential number of columns, and\ndecoupling will take exponential number of steps as $n$ increases.\nThe challenge is to find correct sign matrices with subexponential number of\ncolumns. \nThis difficult problem turns out to have solutions given by the Hadamard \nmatrices, which will be decribed next.  \n\n\\subsection{Hadamard matrices}\n\\label{sec:Hadamard}\n\nHadamard matrices have applications in many areas such as the construction of\ndesigns, error correcting codes and Hadamard\ntransformations~\\cite{CRC96,Sloanehp,vanLint92,MacWilliams77}.\n\nA Hadamard matrix of order $n$, denoted by $H(n)$, is an $n \\times n$ matrix\nwith entries $\\pm 1$, such that\n\\begin{equation}\n\tH(n)H(n)^{T} = n I\n\\label{eq:ortho}\n\\,.\n\\end{equation}\nThe rows are pairwise orthogonal, therefore any two rows agree in exactly half\nof the entries.  Likewise columns are pariwise orthogonal.  We abbreviate\n``$\\pm 1$'' as ``$\\pm$''.  $S_2$ and $S_4$ in Eqs.(\\ref{eq:s2}) and\n(\\ref{eq:s4}) are simple examples of $H(2)$ and $H(4)$.  An example of $H(12)$\nis given by\n\\begin{eqnarray}\n\\nonumber\n\tH(12) = \\left[ \\begin{array}{cccccccccccc}\n\t{+}&{+}&{+}&{+}&{+}&{+}& {-}&{+}&{+}&{+}&{+}&{+}\n\\\\\t{+}&{+}&{+}&{-}&{-}&{+}& {+}&{-}&{+}&{-}&{-}&{+}\n\\\\\t{+}&{+}&{+}&{+}&{-}&{-}& {+}&{+}&{-}&{+}&{-}&{-}\n\\\\\t{+}&{-}&{+}&{+}&{+}&{-}& {+}&{-}&{+}&{-}&{+}&{-}\n\\\\\t{+}&{-}&{-}&{+}&{+}&{+}& {+}&{-}&{-}&{+}&{-}&{+} \n\\\\\t{+}&{+}&{-}&{-}&{+}&{+}& {+}&{+}&{-}&{-}&{+}&{-}\n\\\\\t{-}&{+}&{+}&{+}&{+}&{+}& {-}&{-}&{-}&{-}&{-}&{-}\n\\\\\t{+}&{-}&{+}&{-}&{-}&{+}& {-}&{-}&{-}&{+}&{+}&{-}\n\\\\\t{+}&{+}&{-}&{+}&{-}&{-}& {-}&{-}&{-}&{-}&{+}&{+}\n\\\\\t{+}&{-}&{+}&{-}&{+}&{-}& {-}&{+}&{-}&{-}&{-}&{+}\n\\\\\t{+}&{-}&{-}&{+}&{-}&{+}& {-}&{+}&{+}&{-}&{-}&{-} \n\\\\\t{+}&{+}&{-}&{-}&{+}&{-}& {-}&{-}&{+}&{+}&{-}&{-}\n\t\\end{array} \\right] \n\\,.\n\\end{eqnarray} \n\nThe following is a list of useful facts about Hadamard matrices (details and \nproofs omitted):\n\n\\begin{enumerate} \n\\item {\\em Equivalence}~~Permutations or negations of rows or columns of\nHadamard matrices leave the orthogonality condition invariant.  Two Hadamard\nmatrices are equivalent if one can be transformed to the other by a series of\nsuch operations.  Each Hadamard matrix is equivalent to a {\\em normalized}\none, which has only $+$'s in the first row and column.  For instance, $H(12)$\nshown previously can be {\\em normalized} by negating the 7-${\\rm th}$ row\nand column.\n\n\\item {\\em Necessary conditions}~~$H(n)$ exists only for $n = 1$, $n = 2$ or\n$n \\equiv 0 \\bmod 4$.  This is obvious if the matrix is normalized, and the\ncolumns are permuted so that the first three rows become: \n\\begin{eqnarray}\n\\nonumber\n\t\\left[ \\begin{array}{cccccccccccc}\n\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}& {+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}\n\\\\\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}& {-}&{\\cdots}&{-}&{-}&{\\cdots}&{-}\n\\\\\t{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}& {+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}\n\\\\\t{\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}&   {\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}\n\t&{\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}\n\t\\end{array} \\right] \n\\,.\n\\end{eqnarray} \n\n\\item {\\em Hadamard's conjecture}~\\cite{Hadamard93}~~$H(n)$ exists for every\n$n \\equiv 0 \\bmod 4$.  This famous conjecture is verified for all $n < 428$.  \n\n\\item {\\em Sylvester's construction}~\\cite{Sylvester67}~~If $H(n)$ and $H(m)$ \nexist, then $H(nm)$ can be constructed as $H(n) \\otimes H(m)$. \nIn particular, $H(2^r)$ can be constructed as $H(2)^{\\otimes r}$, which is\nproportional to the matrix representation of the Hadamard transformation\nfor $r$ qubits.\n\n\\item {\\em Paley's construction}~\\cite{Paley33}~~Let $q$ be an odd prime\npower.  If $q \\equiv 3 \\bmod 4$, then $H(q+1)$ exists; if $q \\equiv 1 \\bmod 4$,\nthen $H(2(q+1))$ exists.  \n\n\\item {\\em Numerical facts}~\\cite{CRC96}~~\nFor an arbitrary integer $n$, let $\\underline{n}$ and $\\overline{n}$ be the\nlargest and smallest integers that satisfy $\\underline{n} < n \\leq\n\\overline{n}$ with {\\em known} $H(\\overline{n})$ and $H(\\underline{n})$.  We\ndefine the ``gap'', $\\delta_n$, to be $\\overline{n} - \\underline{n}$ (see\nFig.~\\ref{fig:gap}).\nFor $n\\leq 1000$, $H(n)$ is known for every possible order except for 6 cases,\nand the maximum gap is 8.  For $n \\leq 10000$, $H(n)$ is unknown for 192\npossible orders and the maximum gap is 32.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=gap.eps,width=1.4in}}\n\\vspace*{2ex}\n\\caption{The gap $\\delta_n$ between two existing orders of Hadamard matrices.}\n\\label{fig:gap}\n\\end{center}\n\\end{figure}\n\n\\end{enumerate}\n\nThe importance of the full connection to Hadamard matrices will become \nclear after we construct the scheme for an arbitrary number of spins \nin the next section.  \n\n\\subsection{General Construction of Scheme} \n\nIn this section, a decoupling scheme for an arbitrary number of spins is\nconstructed.  Variations of the scheme to remove Zeeman evolution and to\nimplement selective recoupling are also constructed.  The requirements of the\nscheme are discussed.\n\nRecall that $\\overline{n}$ is the minimum integer no smaller than $n$ with\nknown $H(\\overline{n})$, and $m$ is the number of columns in a sign matrix.\nFor any variations of the scheme for $n$ spins, $m_n$ is defined to be the\nminimum number of columns in any valid sign matrix and it represents the\nnumber of time intervals needed for the intended operation.\n\nTo construct a decoupling scheme for $n$ spins when $H(n)$ does not\nnecessarily exist, any $H(\\overline{n})$ with $\\overline{n}-n$ rows omitted\ncan be used as the sign matrix since subsets of rows of $H(\\overline{n})$ are\nstill pairwise orthogonal.\nIn other words, any $n \\times \\overline{n}$ submatrix of $H(\\overline{n})$ is\na valid sign matrix for decoupling $n$ spins, and $m_n = \\overline{n}$.\n\nTo remove both ${\\cal H}_{\\rm Z}$ and ${\\cal H}_{\\rm c}$, we use the fact that\nthe Zeeman term for the $i$-th spin is linear in $\\sigma_z^{(i)}$, and\nnegating $\\sigma_z^{(i)}$ for half of the time results in no net Zeeman\nevolution for the $i$-th spin.  Therefore, Zeeman evolution for all spins can\nbe removed if the sign matrix has identically zero row sum.\nSuch a sign matrix can be constructed by starting with a {\\em normalized}\n$H(\\overline{n})$ and excluding the first row of $H(\\overline{n})$ in the sign\nmatrix.  Since a normalized $H(\\overline{n})$ has only $+$'s in the first row,\nall other rows have zero row sums by orthogonality.\nSuch construction is possible unless $n = \\overline{n}$, in which case \nconstruction should start with $H(\\overline{n+1})$.  Therefore, $m_n =\n\\overline{n}$ if $n < \\overline{n}$ and $m_n = \\overline{n+1}$ otherwise.\n\nTo implement selective recoupling between the $i$-th and the $j$-th spins, the\nsign matrix should have equal $i$-th and $j$-th rows but any other two rows\nshould be orthogonal.\nThe coupling term $g_{ij}^{\\rm c} \\sigma_z^{(i)} \\otimes \\sigma_{\\rm z}^{(j)}$\nnever changes sign and that coupling is implemented, while all other couplings\nare removed.\nThe sign matrix can be obtained from a normalized $H(\\overline{n})$ by\nreplacing the $1$-st row with the $j$-th row, and replacing the $j$-th row\nwith the $i$-th row.  This scheme also removes ${\\cal H}_{\\rm Z}$ and $m_n =\n\\overline{n}$.\nTo implement $Z\\!\\!Z_{ij}$, the duration of each interval $t$ is chosen\nto satisfy $g_{ij}^{\\rm c} \\overline{n} t = \\pi\/4$.  Note that the total time\nused to implement $Z\\!\\!Z_{ij}$ is the shortest possible, since the coupling\nis always ``on''.\n\nThe scheme requires $m_n$ time intervals.  Consequently, it requires at\nmost $n m_n$ pulses, since $XX=I$ and the $X$ pulses are only used in pairs.  \nThe remaining question is: how does $m_n$ depend on $n$?\nFor simplicity, we consider $\\overline{n}$ in place of $m_n$. \nIf Hadamard matrices exist and can be constructed for all orders,\n$\\overline{n} = n$.\nHowever, some Hadamard matrices are missing, either because no construction\nmethods are known or they simply cannot exist.\nDue to missing Hadamard matrices, $\\overline{n} = cn$ where $c \\geq 1$.  \nWe argue in the following that the scheme is still very efficient.  \nFirst of all, we prove $c < 2$.  For each $n$, there exists $r$ such that \n$2^{r-1} \\!  \\leq n < 2^r$.  Since $H(2^r)$ exists by Sylvester's\nconstruction, $cn = \\overline{n} \\leq 2^r \\!\\! < 2n$.\nWe now show that $c$ is close to the ideal value 1 in most cases, due to the\nexistence of Hadamard matrices of orders other than powers of 2.\nThis is why the full connection to Hadamard matrices is important.  \nIn Fig.~\\ref{fig:c}, $c$ as a function of $n$ is plotted for $n \\leq 10000$.  \nWithin this technologically relevant range of $n$, $c$ deviates significantly\nfrom 1 only for a few exceptional values of $n$ when $n$ is small.\nOne arrives at the same conclusion by considering the gap $\\delta_n =\n\\overline{n} - \\underline{n} > \\overline{n} - n$, which bounds the extra\nnumber of intervals needed in the scheme caused by missing Hadamard matrices.\nFor $n \\leq 10000$, $\\delta_n\/n \\ll 1$ except for the few exceptional values\nof $n$ as a numerical fact.\nFor completeness, we present arguments for $c \\approx 1$ for {\\em arbitrarily\nlarge} $n$ in Appendix~\\ref{sec:largen}.  \nThis is based on Paley's construction and the prime number theorem. \nFinally, if Hadamard's conjecture is proven, $\\delta_n \\leq 3$ $\\forall n$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=cvsnsmall.eps,width=1.7in}\\psfig{file=cvsn.eps,width=1.74in}}\n\\vspace*{2ex}\n\\caption{Plots of $c$ vs $n$, where $cn = \\overline{n} = m_n$ is the minimun\nnumber of time intervals required to perform decoupling or selective\nrecoupling for an $n$-spin system.  $c$ for $n \\leq 100$ and $101 \\geq n \\leq\n10000$ are plotted separately.}\n\\label{fig:c}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion} \n\nWe reduce the problem of decoupling and selective recoupling in heteronuclear\nspin systems to finding sign matrices which is further reduced to finding\nHadamard matrices.\nWhile the most difficult task of constructing Hadamard matrices is not\ndiscussed in this paper, solutions already exist in the literature. \nEven more important is that the connection to Hadamard matrices results in\nvery efficient schemes.\n\nSome properties of the scheme are as follows.  \nFirst of all, the scheme is optimal in the following sense.  \nThe rows of Hadamard matrices and their negations form the codewords of first\norder Reed-Muller codes, which are {\\em perfect\ncodes}~\\cite{vanLint92,MacWilliams77}.\nIt follows that, for each Hadamard matrix, it is impossible to add an extra\nrow which is orthogonal to all the existing ones.\nTherefore, for a given $n$, $m_n = \\overline{n}$ is in fact the minimum number\nof time intervals necessary for decoupling or recoupling, if one restricts \nto the class of pulse sequences considered.\nSecond, the scheme applies for arbitrary duration of the time intervals.  \nThis is a consequence of the commutivity of all the terms in the hamiltonian,\nwhich in turn comes from the large separations of the Zeeman frequencies\ncompared to the coupling constants.  \nSpin systems can be chosen to satisfy this condition.\nFinally, disjoint pairs of spins can couple in parallel. \n\nWe outline possible simplifications of the scheme for systems with restricted\nrange of coupling.\nFor example, a linear spin system with $n$ spins but only $k$-nearest\nneighbor coupling can be decoupled by a scheme for $k$ spins only.  \nThe $i$-th row of the $n \\times \\overline{k}$ sign matrix can be chosen to be\nthe $r$-th row of $H(\\overline{k})$, where $i \\equiv r \\bmod k$.\nSelective recoupling can be implemented using a decoupling scheme for $k+1$\nspins.  The sign matrix is constructed as in decoupling using\n$H(\\overline{k+1})$ but the rows for the spins to be coupled are chosen to be\nthe $k+1$-th row different from all existing rows~\\cite{kplus1}.\nThis method involving periodic boundary conditions generalizes to other\nspatial structures.  The size of the scheme depends on $k$ and the exact \nspatial structure but not on $n$.\n\nThe scheme has several limitations.  \nFirst of all, it only applies to systems in which spins can be individually\naddressed by short pulses and coupling has the simplified form given by\nEq.(\\ref{eq:dipolar}).\nThese conditions are essential to the simplicity of the scheme.  They can all\nbe satisfied if the Zeeman frequencies have large separations.\nSecond, generalizations to include couplings of higher order than bilinear\nremain to be developed.\nFurthermore, in practice, RF pulses are inexact and have finite\ndurations, leading to imperfect transformations and residual errors.\n\nThe present discussion is only one example of a more general issue, that the\nnaturally occuring hamiltonian in a system does not directly give rise to\nconvenient quantum logic gates or other computations such as simulation of\nquantum systems~\\cite{Terhal98}.\nEfficient conversion of the given system hamiltonian to a useful form is\nnecessary and is an important challenge for future research.\n\n\\section{Acknowledgments}\n\nThis work was supported by DARPA under contract DAAG55-97-1-0341 and Nippon\nTelegraph and Telephone Corporation (NTT).  D.L. acknowledges support of an\nIBM Co-operative Fellowship.  We thank Hoi-Fung Chau, Kai-Man Tsang, Hoi-Kwong\nLo, Alex Pines, Xinlan Zhou and Lieven Vandersypen for helpful comments.\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}