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+{"text":"\\section{Introduction}\n\nFixing bugs in programs, that is, program repair, is one of the core tasks in software maintenance,\nbut requires effort to\n analyze failed executions, locate the cause of the fault, synthesize a bug fix and validate that the fault has been corrected without introducing new ones \\cite{Mullerburg1983}.\n Automatic program repair \\cite{LeGoues2012, Long2016, Pradel2018, Monperrus:survey} attempts to alleviate most of the manual effort of locating and repairing faults.\nHowever, a major concern in industry is that linters and program repair methods approaches are required to have high precision without risking achieving high enough recall.\nAs an industrial example Google's Tricorder \\cite{Sadowski2015} enforces a false positive rate $< 10\\%$.\n\nOne way to find a ``sweet spot'' of maintaining high precision with adequate recall is to focus on\nrepairing types of simple bugs, such as one-line bugs,\nor  bugs that fall into a small set of templates,\nsuch as mutation operators\n\\cite{LeGoues2012} or other types of predefined templates \\cite{Long2015, Long2016, Pradel2018}.\nHowever, these have been evaluated on either a relatively small numbers of projects, e.g. 69 defects in 8 applications or on synthetic data.\nBecause of this lack of data, it has not previously been possible to estimate the \\emph{recall} of a set of repair templates,\nthat is, the percentage of real-world bugs that can be repaired by one of the templates.\nSimultaneously to the current work, a larger dataset of one-line bugs has been mined \\cite{Sequencer},\nbut even this dataset does not attempt to classify bugs into templates.\n\nAiming to fill this gap, we provide a dataset containing 25,539 single-statement bug-fix changes mined from 100 popular open-source Java Maven projects as well as a larger one containing 153,652 single-statement bug-fix changes mined from 1,000 popular open-source Java projects,\nannotated by whether they match any of a set of 16 bug templates, inspired by state-of-the-art program repair techniques.\nThe chosen templates aim at extracting bugs that compile both before and after repair as such can be quite tedious to manually spot, yet their fixes are so simple that many developers would call them ``stupid'' upon realization.\nWe will refer onwards to these bugs as ``simple stupid bugs'' (SStuBs)\\footnote{The\nacronym is intended to reflect the fact that, for the authors at least, finding such a bug can feel much like stubbing one's toe.} and the corresponding dataset as the Many\\sstubs4J\\ dataset. %\nAutomatic repair of SStuBs\\ is potentially an intermediate step toward more general program repair tools,\nwhile already being useful to developers.\nWe also think that SStuBs\\ might be a good start for the evaluation of machine learning based fault localization and repair methods.\n\n\nAn extra distinctive feature of our dataset is that the smaller version is restricted to projects that can be built automatically\nusing Maven.\nThose that contain a test suite can be built and used to evaluate test based techniques.\nIn an initial analysis, we find that 33.04\\% in the smaller version dataset and 33.47\\% in the larger version of all of the single-statement bugs that we mine match at least one of the SStuB\\ templates resulting in 10,231 and 63,923 SStuB\\ instances respectively.\nThis indicates that a remarkable number of singe-statement bugs can be repaired with a relatively small set of templates.\nIn further analysis we also estimated the frequency in lines of code with which these pattern based and general single-statement bugs appear.\nThis estimation is based on the size of the project's latest version and reveals that in the smaller dataset version SStuBs appear with a frequency of about 1 per 1,600 lines of code and 1 per 2,500 lines of code for the large version.\nWe hope that this dataset can serve as a valuable resource for both future work in program repair and studies in empirical software engineering.\n\n\\section{Methodology}\nWe next describe the methodology we employed to build the dataset.\nOur data generation tools along with documentation and detailed instructions for how to use them\nare available in a public GitHub repository\\footnote{https:\/\/github.com\/mast-group\/mineSStuBs} \nand the dataset is publicly available in Zenodo.\\footnote{DOI: https:\/\/doi.org\/10.5281\/zenodo.3653444}\n\n\\subsection{Selecting Appropriate Java Projects}\nIn order to mine a high quality dataset we opted to selecting high popularity projects.\nFor the small version of the dataset we selected the 100 most popular open source Java Maven \\cite{Miller2010} projects from GitHub up to 1\/4\/2017. \nTo allow evaluation of repair tools that might require building the projects, we selected only Maven ones because\nit is easy to automatically download the required dependencies for every project and build it.\nIn contrast, manual downloading of dependencies would require an immense amount of human effort.\nTo create a ranking for the projects we downloaded the MySQL dump of GHTorrent \\cite{Gousios2013} up to 1\/4\/2017. \nA project's popularity is determined by computing the sum of z-scores of its forks and stars \\cite{Allamanis2015a, Allamanis2016}.\nLastly, we pulled the projects' head commit by 28\/1\/2019 and considered commits until that date.\nThe same approach was used to rank projects for the larger version.\nHowever, the ranking was calculated using a later dump of GHTorrent from 1\/1\/2019.\nA download script along with the list of projects for both variants of the dataset are also available to ensure replicability.\n\n\\subsection{Classifying Commits as Bug-Fixing or not}\nFor every project our tool searches historically through all of its commits to locate bug-fixing ones. \nTo decide if a commit fixes a bug, we checked if its\ncommit message contains at least one of the\nkeywords: `error', `bug', `fix', `issue', `mistake', `incorrect', `fault', `defect', `flaw',\nand `type'. \nThis heuristic was previously used by Ray et al. \\cite{Ray2015} and was shown to achieve 96\\% accuracy on a set of 300 manually verified commits and 97.6\\% on a set of 384 manually verified commits \\cite{Tufano2018}.\nWe sampled 100 random commits containing SStuBs\\ from the small version of the dataset and found it to achieve 94\\% accuracy.\nThe above process produced a total of 115,929 and 883,982 bug-fixing  commits for the small and large dataset variants.\n\n\\subsection{Selecting Single Statement Changes}\nWe have opted to restrict the dataset to small bug fixes that do not require much code modification to fix.\nAdditionally, we are interested in bugs that are not just syntactic errors but cases where the code compiles both before and after the bug was located and repaired.\nAs we are interested in simple bugs that involve only a single statement, we filter out any commits that either add or delete a Java file.\nWe also filter out commits which make a multiple-statement change at any single position in the Java file.\nWe do \\emph{not} filter out commits that make single-line modifications at more than one position in the same file.\nSimilarly to the diff algorithm, we consider a modification as deleting the old lines\/statements and then adding the new ones.\nTo estimate whether a modification spans across multiple statements we calculate the diff for each modified Java file,\nand \nfor each modified chunk, we count how many statements were modified.\nIn the case of blocks each statement in the block's body is counted as a different statement. \nFor \\texttt{if} and \\texttt{while} statements, we count the condition as a separate statement\nfor this purpose.\nThis method allows to us include fixes to single simple statements that span across multiple lines (e.g. due to stylistic reasons)\nas a simple fix, unlike a line-based approach.\nAny commits that modify multiple statements in any single position returned by the diff are dropped while we still maintain commits for which a file's diff contains multiple positions with single statement modifications.\nIn the first case it is not trivial to align the deleted and added statements while it is in the latter.\nFor example, one or more of the deleted statements may have been replaced by multiple of the added ones while simultaneously one or more of the deleted statements may have simply been deleted.\nWe note that our tool ignores any changes to comments, blank lines as well as any formatting changes.\nOur methodology allows cases where the same expression containing a bug appeared multiple times in the file.\nThis filtering produces almost 13,000 and 86,769 commits for the two dataset versions.\nLastly, the employed methodology works in a similar way to the popular SZZ algorithm \\cite{Sliwerski2005} and its extensions \\cite{Kim2006, Williams2008} that have extensively been used to spot fix inducing changes.\n\n\n\\subsection{Creating Abstract Syntax Trees}\n\nEach file in the commit that contains one or more bugs is parsed, yielding an abstract syntax tree (AST) of the file before the repair.\nThen, for each repaired line in the file we extract the AST after applying the repair only on that line and leaving the rest of the lines as is.\nEach extracted pair of ASTs (original and single fix) only differ on the node(s) for the modified line.\nBy performing a simultaneous depth-first traversal on the two ASTs we locate the first node on which the two ASTs differ.\n\n\\subsection{Filtering out Clear Refactorings}\n\nAlthough we filter for bug-fixing changes in Step B,\nthere might still exist changes in the data that do not fix a bug or that do not even produce any behavioural changes.\nThis could happen because the commit-message filter had\na false positive, or because the change is tangled \\cite{herzig2013tangled}, and contains a bug-fixing modification\nalong with unrelated ones to other files.\nTo reduce the number of non-fixing changes in the dataset, we observe that\nthere is a class of refactorings that can produce small changes, namely renamings.\nThese are extracted via the diffs of the modified files.\nOur method spots variable, function, or class renaming as well as any uses of them across other modified files in the commit and excludes them.\n\n\\subsection{SStuB\\ Patterns}\n\nWe next describe the 16 SStuB\\ patterns.\nWe opted to choose patterns that appear often.\nMany of these have been used in pattern-based repair and mutation tools \\cite{LeGoues2012, Long2015, Long2016, Pradel2018}.\nHere we provide a brief description of each pattern. \nDue to page limitations we do not include examples here but in the README of the GitHub repository. %\n\\begin{itemize}\n  \n\\item \\emph{Change Identifier Used}\nChecks whether an identifier appearing in some expression in the statement was replaced with another one.\nIt is easy for developers to by accident utilize a different identifier than the intended one that has the same type.\nCopy pasting code is a potential source of such errors.\nSimilarly named identifiers may further contribute to the occurrence of such errors.\n\n\\item \\emph{Change Numeric Literal}\nChecks whether a numeric literal was replaced with another one.\nIt is easy for developers to mix two numeric values in their program.\n\n\\item \\emph{Change Boolean Literal}\nChecks whether a Boolean literal was replaced. True is replaced with False and vice-versa.\nIn many cases developers use the opposite Boolean value than the intended one.\n\n\\item \\emph{Change Modifier}\nChecks whether a variable, function, or class was declared with the wrong modifiers.\nFor example a developer can forget to declare one of the modifiers.\n\n\\item \\emph{Wrong Function Name}\nChecks if a function with the same parameter list but the wrong name was called. This is a usual pitfall.\n\n\\item \\emph{Same Function More Args}\nChecks whether an overloaded version of the function with more arguments was called.\nFunctions with multiple overload can often confuse developers.\n\n\\item \\emph{Same Function Less Args}\nChecks whether an overloaded version of the function with less arguments was called.\nFor instance, a developer can forget to specify one of the arguments and not realize it if the code still compiles due to function overloading.\n\n\\item \\emph{Same Function Change Caller}\nChecks whether in a function call expression the caller object for it was replaced with another one.\nWhen there are multiple variables with the same type a developer can accidentally perform an operation.\nCopy pasting code or mixing similar variables are common cases of such errors.\n\n\\item \\emph{Same Function Swap Args}\nChecks whether a function was called with two of its arguments swapped.\nWhen multiple function arguments are of the same type, developers %\ncan easily swap two of them without realizing.\nIt was also used in DeepBugs \\cite{Pradel2018}. %\n\n\\item \\emph{Change Binary Operator}\nChecks whether a binary operand was accidentally replaced with another one of the same type.\nFor example, developers very often mix comparison operators in expressions.\nA similar pattern was also used in DeepBugs \\cite{Pradel2018}.\n\n\\item \\emph{Change Unary Operator}\nChecks whether a unary operand was accidentally replaced with another one of the same type\n(e.g., developers often forget the ! operator in a boolean expression).\n\n\\item \\emph{Change Operand}\nChecks whether one of the operands in a binary operation was wrong.\nIt was also used in DeepBugs \\cite{Pradel2018}.\n\n\\item \\emph{More Specific If}\nChecks whether an extra condition (\\&\\& operand) was added  in an \\texttt{if} statement's condition.\n\n\\item \\emph{Less Specific If}\nChecks whether an extra condition which either this or the original one needs to hold ($\\|$ operand) was added in an \\texttt{if} statement's condition.\n\n\\item \\emph{Missing Throws Exception}\nChecks whether the fix added a \\texttt{throws} clause in a function declaration.\n\n\\item \\emph{Delete Throws Exception}\nChecks whether the fix deleted a \\texttt{throws} clause in a function declaration.\n\\end{itemize}\n\n\\subsection{SStuB\\ Pattern Matching}\n\nFinally, each pair of ASTs is automatically checked for fitting any of the SStuB\\ patterns.\nEach pattern is expressed as a mutation operation on the original AST that produces the new one.\nAll instances are added to the single-statement dataset, while only those that match SStuB\\ patterns are saved in the SStuBs\\ one.\n\n\n\\section{Many\\sstubs4J\\ Dataset Statistics}\n\n\n\\begin{table}[tpb]\n\\renewcommand{\\arraystretch}{1.05}\n\\caption{Statistics for each SStuB pattern.}\n\\begin{center}\n\\resizebox{0.935\\columnwidth}{!}{\n\\begin{tabular}{l r r r r}\n\\toprule\n\\textbf{Pattern Name} & \\textbf{\\textit{SStuBs}} & \\textbf{\\textit{Ratio}} & \\textbf{\\textit{SStuBs L}} & \\textbf{\\textit{Ratio L}} \\\\\n\\midrule\nChange Identifier Used \t\t& 3265 \t& 12.78\\%\t& 22668\t& 14.75\\% \\\\\nChange Numeric Literal \t\t& 1137 \t& 4.45\\%\t& 5447\t& 3.55\\% \\\\\nChange Modifier \t\t\t& 1852 \t& 7.25\\%\t& 5011\t& 3.26\\% \\\\\nChange Boolean Literal \t\t& 169 \t& 0.66\\%\t& 1842\t& 1.20\\% \\\\\n\\midrule\nWrong Function Name \t\t& 1486 \t& 5.82\\%\t& 10179\t& 6.62\\% \\\\\nSame Function More Args \t& 758 \t& 2.97\\%\t& 5100\t& 3.32\\% \\\\\nSame Function Less Args \t& 179 \t& 0.70\\%\t& 1588\t& 1.03\\% \\\\\nSame Function Wrong Caller\t& 187 \t& 0.73\\%\t& 1504\t& 0.98\\% \\\\\nSame Function Swap Args \t& 127 \t& 0.50\\%\t& 612\t& 0.39\\% \\\\\n\\midrule\nChange Binary Operator \t\t& 275 \t& 1.08\\%\t& 2241\t& 1.46\\% \\\\\nChange Unary Operator \t\t& 170 \t& 0.67\\%\t& 1016\t& 0.66\\% \\\\\nChange Operand \t\t\t\t& 120 \t& 0.47\\%\t& 807\t& 0.53\\% \\\\\n\\midrule\nLess Specific If \t\t\t& 215 \t& 0.84\\%\t& 2813\t& 1.83\\% \\\\\nMore Specific If\t\t\t& 175 \t& 0.69\\%\t& 2381\t& 1.55\\% \\\\\nMissing Throws Exception \t& 68 \t& 0.27\\%\t& 206\t& 0.13\\% \\\\\nDelete Throws Exception \t& 48 \t& 0.19\\%\t& 508\t& 0.33\\% \\\\\n\\midrule\nTOTAL NO DOUBLE COUNTS\t\t& 8438\t& 33.04\\%\t& 51433\t& 33.47\\% \\\\\nTOTAL \t\t\t\t\t\t& 10231\t& 40.06\\%\t& 63923\t& 41.59\\% \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\label{tab:sstubPercentages}\n\\end{center}\n\\end{table}\n\n\nThe Many\\sstubs4J\\ dataset consists of 10,231 and 63,923 instances of single statement bugs mined from 12,598 and 86,771 bug-fix commits with only single-statement changes respectively for each version.\nConsequently, on average almost 2 single statement bugs and 0.75 SStuBs\\ were mined per valid commit.\nThe data is saved in JSON files and detailed information is available in the GitHub repository.\nEach SStuB\\ instance is also annotated with the SStuB\\ pattern satisfied, the project's name, the Java file's name, the hashes of the fix inducing commit and its parent, the line at which the bug starts, and the AST subtree's location. \nIn some cases a statement might fit more than one patterns.\nIn those cases it is counted as separate instances.\nHowever, in most cases the patterns are distinct.\nThe statistics for each of the 16 SStuB\\ patterns of the Many\\sstubs4J\\ dataset are shown in Table~\\ref{tab:sstubPercentages}.\nPatterns that are similar are grouped together (e.g. patterns that concern functions) and sorted in descending frequency order.\nThe three most common SStuB\\ patterns are \\emph{Change Identifier Used}, \\emph{Wrong Function Name}, and \\emph{Change Numeric Literal}.\n\nWe note that the mined bugs have not been annotated by severity and we expect that to vary.\nSome of the bugs appear in test code.\nAlthough bugs in test code will not reach a final product, they can have significant effect on it as they can potentially mask important bugs in it.\nTest oracle errors can bring confusion that slows down the debugging process while fixing them improves the performance of fault localization algorithms \\cite{Guo2015}.\nSuch bugs might also be quite tedious to locate as it is very rare to test a test suite and even if we follow that logic we would have to endlessly create tests for the tests.\nBy design we do not attempt to restrict the bugs to those that have a failing test case.\nThe goal is to reproduce the situtations that the bugs happen in the wild.\nLastly, as it was recently shown, unit tested code does not appear to be associated with fewer failures while increased coverage is associated with more failures \\cite{chioteli2019does}.\n\n\n\\section{Research Questions} \\label{sec:questions}\n\nAlthough the paper focuses on the dataset, we run a simple analysis to support\nour design decision to focus our new dataset on SStuBs.\nIn order to explore whether the SStuB\\ patterns are useful targets for program repair techniques,\nwe asked two research questions.\n\n\\subsection*{RQ1. Are SStuBs\\ common in open-source code?}%\n\\label{sec:questionPercentage}\nWe measured for each SStuB type the percentage of single statement modifications that are not clear refactorings and fit the pattern.\nThese are visualized in Table~\\ref{tab:sstubPercentages}.\nFor each project $P$ we also estimated the following two densities for the mined SStuBs:\n(a) the number of SStuBs\\ in project $P$ \/ total lines in $P$ at the final snapshot and\n(b) the number of SStuBs\\ in project $P$ \/ total lines added and deleted in $P$ by the final snapshot. \nThus, estimating the frequency per line of code modifications in the project's history. That is counting any line that was added or deleted to the project from the start to its latest version.\nA line modification is counted twice (once as a deletion and once as an addition). \nOnce for deleting the old and once for adding the new line.\nComments and empty lines were excluded from these estimations.\nWe found that in the smaller version of the dataset SStuBs appear with densities of about 2,400 and 30,000 lines of code (LOC) respectively.\n\nWe also estimated the same densities for the larger dataset variant.\nWe found that such bugs appear with a frequency of about 1,600 and 20,000 LOC respectively.\nAs a threat to validity, we acknowledge that the number of LOC in the final snapshot may not be the most informative denominator for a measure of bug density, but developing better ones is a thorny issue left for future work.\n\n\\subsection*{RQ2. Can SStuBs be spotted by existing tools such as static analyzers?} \\label{questionSpot}\n\nWe measure the proportion of bugs in our dataset that can be identified by\nthe popular static analysis tool SpotBugs.\\footnote{https:\/\/spotbugs.github.io\/}\nIf SpotBugs reports any bug for the line containing the SStuB\\ then we consider that SpotBugs successfully detected it.\nWe find that SpotBugs could only locate about $12\\%$ of SStuBs while also reporting more than $200$ million possible bugs when configured to report all warnings, even those with low confidence.\nIn fact, as explained the actual recall is even lower.\nThis is confirmed by a recent study where three static bug detectors including SpotBugs located only $4.5\\%$ of bugs \\cite{Habib2018}.\nThis means that a developer would have to look through hundreds of thousands of warnings produced by SpotBugs to locate a single SStuB.\nThis highlights the necessity for tools that are specifically built to detect SStuBs.\nThe scripts used to run and evaluate SpotBugs are also available in our repository.\n\n\n\n\\section{Related Work}\n\nSeveral previous data sets of real-world bugs have been curated.\nDefects4J \\cite{Just2014} is a popular dataset consisting 395 Java bugs.\nEach bug is fixed in a single commit but the fix may modify multiple source code lines.\nThe ManyBugs dataset \\cite{Goues2015} contains 185 C bugs, a subset of which were used by the GenProg \\cite{LeGoues2012}, Prophet \\cite{Long2016} and SPR \\cite{Long2015} papers.\nBugs.jar \\cite{Saha2018} is comprised of 1,158 Java bugs and their patches.\nThese datasets have the disadvantage of being relatively small.\nMore recently, a few larger-scale data sets of small bugs have been created.\nThe combined datasets are the CodRep dataset \\cite{CodRep} and the Bugs2Fix dataset \\cite{Bugs2Fix} resulting in 40,289 one-line bugs.\nThese datasets are combined into a single dataset of one line bugs in \\cite{Sequencer}.\nOur datasets are of similar size consisting of 25,539 and 153,652 single-statement bugs.\nIn contrast, our dataset focus on estimating the frequency of SStuB\\ templates, motivated by recent\nprogram repair tools and also operates on the statement level, which prevents falsely excluding instances due to formatting or stylistic reasons.\nAlso, the projects from which the small version of our dataset was generated can easily be built using Maven and we provide a list of projects containing tests and which tests fail for each instance (in GitHub repo).\nThus test based methods can be evaluated upon them.\nHowever, unlike Defects4J that aims in comparing test-based patch generation approaches, it aims in techniques that can accurately highlight SStuBs early in development allowing immediate patching since in many cases the fix might be trivial.\nLastly, unlike previous datasets, we take additional steps to filter out refactorings, although we acknowledge that such instances might be rare.\nIn our case however, we were able to filter out almost 5,000 and 35,000 refactored statements for the two dataset versions.\n\n\n\\section{Limitations - Threats to Validity}\n\nAlthough unlikely, it is possible for our SZZ like methodology to extract a pair of aligned statements that are unrelated (i.e. one line was deleted and one was added).\nWe do spot refactorings but there is no guarantee that we have detected 100\\% of them.\nThe heuristic used to spot bug fixing commits could introduce false positives, but this is mitigated by the fact that we focus on single line commits and as already discussed the false positive rate is low.\nOur dataset will not be useful for evaluating whether repair systems are good at fixing larger bugs.\nOur dataset is restricted to Java but could be replicated for other languages by using a parser and creating a module that checks if an AST pair fits any of the SStuB patterns.\nThe precise set of patterns might vary across languages and determining these might be an interesting direction for future work.\n\n\n\n\\section{Conclusions}\n\nWe introduce a new, large-scale dataset of real-world SStuBs, simple one-statement bugs,\nin Java for the evaluation of program repair techniques. The distinguishing feature of our dataset is\nthat where possible, the SStuBs\\ are categorized into one of 16 bug templates, which are inspired\nby those considered in state-of-the-art program repair methods.\nThese types of bugs often result in code that compiles, which means that they are particularly interesting for automated repair. \nWe find that SStuBs\\ occur relatively often --- one per 1,600 LOC in the projects\nwe study --- making them potentially a promising evaluation dataset for repair techniques that could be used to estimate their actual recall.\nThe data could also be used to answer other research questions, such as empirical questions about how and when simple\nbugs are introduced, or about evaluating program repair techniques for small bugs.\nAlso, it can aid in evaluating machine learning systems that learn to localize simple bugs via examples \\cite{Pradel2018} or a language model's entropy \\cite{Karampatsis2020big}.\nLast, coverage information for the maven projects with tests suits in the dataset could be used to estimate how often do tests cover SStuBs.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Accelerator Capabilities} \n\\label{chap:ceo}\n\n\n\\begin{center}\\begin{boldmath}\n\n\\input AccelCap\/authorlist.tex\n\n\n\\end{boldmath}\\end{center}\n\n\\section{Introduction}\n\nAccelerator-based experiments have been the mainstay and driving force of experimental particle physics ever since Rutherford recognized that radioactive sources provided an extremely limited capability to probe the nature of matter at the shortest time and distance scales.  From their earliest beginnings the history of invention in accelerator technology have enabled the history of particle physics discoveries especially on the energy frontier; see Fig.~\\ref{fig:Livingston}.  \n\nYet the figure of merit of energy reach in the constituent center-of-momentum frame is \nfar from the entire story of enabling capabilities. For experiments to be completed in a reasonable amount of time, the rate of interesting experimental events must be sufficiently high.  That is, the luminosity of the accelerator-target combination must be large enough to explore the physics of interest.\n\\begin{equation}\n\\mbox{Interesting\\ Events} = \\mbox{Cross\\ section} \\times \\mbox{Average\\ collision\\ rate} \\times \\mbox{Time} \\ . \n\\eeq{Interesting}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{AccelCap\/Livingston.pdf}\n\\caption{The so-called Livingston plot illustrates how history of discovery on the energy frontier has been enabled by the history of invention (red arrows) in accelerator science and technology.}\n\\label{fig:Livingston}\n\\end{center}\n\\end{figure}\n\n\n\nGenerally high luminosity implies a large charge per bunch, a high collision frequency of beam bunches with the target, and focus of the beam to a small spot. The target can either be fixed or a counter-rotating beam bunch.\nEven energy-reach and accelerator-target luminosity is insufficient for discovery-level experiments. The accelerator should deliver beam particles with appropriate quantum numbers and with a timing structure which maximizes the signal-to-background noise of the experiment. \n\nAssessing the suite of characteristics of present and proposed accelerators in the light of current interests and priorities of particle physicists has been the task of the Accelerator Capabilities Study of the Snowmass process. For proposed or planned capabilities the study has recommended the critical research and development programs needed.\n\nThe study has been a synthesis of individual pre-Snowmass workshops conducted by six working groups culminating with the collective Snowmass meeting of all interested participants. Each working group addressed major challenges foreseen for their respective class of accelerators in their respective pre-Snowmass workshops. The areas of inquiry included but were not limited to the following specific questions:\n\n\\begin{itemize}\n\\item   How high a luminosity is possible for the LHC? \n\\item  \tHow high an energy is possible in the LHC tunnel? \n\\item  \tCould a Higgs factory be built in the LHC tunnel?\n\\item  \tCan the ILC and  CLIC designs be improved using new technologies?  \n\\item  \tCan one design a multi-TeV $\\mu^+\\mu^-$ collider?\n\\item  \tWhat secondary beams are needed for Intensity Frontier experiments?\n\\item  \tWhat proton beams are needed to generate these secondary beams, and can these be made by existing machines?\n\\item  \tWhat accelerator capabilities at heavy flavor factories are required to realize the full range of physics opportunities? \n\\item  \tWhat are new physics opportunities using high-power electron and positron beams?\n\\item  \tWhat is the broad range of test capabilities existing or needed for developing accelerator capabilities?\n\\end{itemize}\n\nThe working groups also considered a set of \"big questions\" regarding accelerator capabilities for the long-term future of high energy physics:\n \\begin{itemize}\n\\item  \tHow can one build a collider at the 10 -- 30 TeV constituent mass scale?\n\\item  \tWhat is the farthest practical energy reach of accelerator-based particle physics?\n\\item  \tHow would one generate ten or more megawatts of proton beam power? \n\\item  \tCan multi-megawatt targets survive and if so, for how long?\n\\item  \tCan plasma-based accelerators achieve energies and luminosities relevant to particle physics?\n\\item  \tCan accelerators be made an order of magnitude cheaper per GeV and\/or per MW?\n\\end{itemize}\n\nThis report is organized as a collection of the executive summaries of each of the six working groups: Energy Frontier proton colliders, Energy Frontier lepton colliders, Intensity Frontier proton sources, Intensity Frontier electron accelerators, electron-ion colliders, and accelerator research and test-beam facilities.\n\n\\section{Energy Frontier proton colliders}\n\nHigh-energy hadron colliders have been the tools for discovery at the highest mass scales of the Energy Frontier for more than a decade. They will remain so, unchallenged for the foreseeable future. The discovery of the Higgs boson at the LHC announced opened a new era for particle physics. Its measured properties are consistent, within the current uncertainties, with those of the Standard Model (SM) Higgs boson. \n\nAfter the discovery of the Higgs boson, understanding what is the real origin of electro-weak symmetry breaking becomes the next key challenge for collider physics. This challenge can be expressed in terms of two questions:  Up to what level of precision does the Higgs boson behave as predicted by the Standard Model? At what energy are the new particles that could offer some insight into the origin of dark matter, the matter-antimatter asymmetry, and neutrino masses? \n\n\\subsection{The LHC and its upgrades}\n\nThe approved LHC program, its future upgrade towards higher luminosities (HL-LHC), and the study of an LHC energy upgrade or of a new proton collider delivering collisions at center-of-mass energies up to 100~TeV, are all essential components of this endeavor.  High luminosity could increase the precision of several key measurements, uncover rare processes, and guide and validate the progress in theoretical modeling, thus reducing systematic uncertainties in the interpretation of the data.  To achieve this end,\n\n{\\it the full exploitation of the LHC is the highest priority of the energy-frontier hadron-collider program. }\n\nThe LHC is expected to restart in Spring 2015 at a center-of-mass energy of 13--14~TeV and reach its  design luminosity of $10^{34}$cm$^{-2}$s$^{-1}$  during 2015.  After 2020, some critical components of the accelerator will reach the radiation damage limit, and others will have lower reliability due to vulnerability to radiation. Furthermore, as the statistical gain in running the accelerator without substantially increased luminosity will become marginal, the LHC will need a substantial increase in luminosity~\\cite{ATLAS}. With a high luminosity upgrade (HL-LHC), the LHC should deliver 3000 fb$^{-1}$  during a decade of operation~\\cite{Zimmermann}. The HL-LHC project is the first priority of Europe, as stated by the Strategy Update for High Energy Physics approved by CERN council in the special session of May 30, 2013 in Brussels.\n\n\nSuperconducting magnets are the single most critical technology for the LHC and for proton colliders beyond the present LHC level of perormance.  Indeed, the present LHC is based on 30 years of development of magnets using NbTi wire. In the LHC NbTi-based magnets are pushed to their limits both in the collider arcs and in the interaction regions designed and built in collaboration with U.S. and Japanese national laboratories. \n\nDelivering the beam brightness required for HL-LHC will be a difficult challenge for the upgraded LHC injector chain; also challenging will be preserving it in the LHC storage ring. In light of this limitation, the preferred route towards increased luminosity is reducing the beta (amplitude) function at the interaction point, \n$\\beta^*$, by means of stronger triplet magnets with larger apertures in the interaction region. The present design of the HL-LHC interaction region requires quadrupoles with an aperture of 150~mm and with a peak field in excess of 12 T --- beyond the capabilities of NbTi conductor. It relies heavily on the success of the advanced Nb$_3$Sn technology developed by an integrated consortium of U.S. national laboratories under the LHC Accelerator Research Program (LARP). In addition to magnets, many other technologies will be involved in HL-LHC, such as crab cavities, advanced collimators, high-temperature superconducting links, and advanced remote handling. LARP plays a vital leadership role in this work to maximize the discovery potential of the extensive LHC infrastructure at CERN.\n\n{\\it A vigorous LARP leading to U.S. participation in the HL-LHC construction project is crucial to the full realization of the potential of LHC to deliver discovery physics to the thousands of U.S. researchers who are engaged at the LHC.}\n\nEnergy and luminosity play complementary roles in exploring the energy frontier with hadron collisions.  Production rates for signals of interest at the collider may be small either because the mass of the produced objects is large or because the coupling strength is small. In the former case, increasing the beam energy is clearly favored. In contrast, higher luminosity may be more effective in probing small couplings at smaller masses. In reaching higher energies, magnet technology will retain a pivotal role.\n\nGiven the progress in magnet technology and the maturity that Nb$_3$Sn is reaching, due largely to the LARP program for HL-LHC, Nb$_3$Sn magnets are expected to reach a limit of 15.5--16~T in operating field (that is with 15--20\\% margin with respect to quench) \nwithin the next decade, thereby opening a path towards a \ncollider with an energy significantly larger than the LHC. This would open the \nway towards very high energy hadron collisions with an exceptional potential for \nprobing the Energy Frontier.\n \n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{AccelCap\/LHCsched.pdf}\n\\caption{LHC baseline plan for the next ten years in terms of energy of the collisions (upper line) \nand of luminosity (lower lines). The first long shutdown (LS) 2013-14 is to allow design parameters \nof beam energy and luminosity. The second LS, 2018, is to increase beam \nintensity and reliability as well as to upgrade the LHC Injectors.}\n\\label{fig:LHCsched}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Options for higher energy hadron colliders}\n\nThe first option is a machine housed in the LHC tunnel. The achievable center-of-mass energy depends on the dipole field strength. An energy of 26 TeV is within the reach of proven Nb$_3$Sn technology, although it requires engineering development. The energy reach would become 33 TeV, if 20 T magnets based on more futuristic high-temperature superconductors (HTS) were practical and affordable, and if they could fit in the limited space available in the LHC tunnel. In a new, longer tunnel, a  100~TeV proton collider is possible. Studies for a very large proton collider able to deliver center-of-mass energies of 100 to 200 TeV have been conducted over the past two decades. The multi-laboratory VLHC study led by Fermilab in 2001~\\cite{VLHC}\nwas noteworthy. The analysis remains valid and argues for a very large (233 km) tunnel that allows the use of NbTi magnet technology.\n\nContinuing an integrated, multi-laboratory program (LARP-like) towards the engineering development of 20~T magnets for HE-LHC or a new collider in a longer tunnel, needs to be an HEP priority. A program focused on engineering readiness, closely coordinated with CERN, would establish the limits of Nb$_3$Sn technology, investigate new conductor materials, and refine our present concepts for how to manage the enormous stresses produced by such high magnetic fields. The experience from RHIC, SSC, and LHC indicates that the dipoles account for about half of the total collider cost. Therefore, investing in magnet technology is critical and should represent a major focus of the research and development process.\n\nWith the renewed interest in the U.S. in a 100~TeV scale collider, the study group recommends participation in the international study for colliders in a large tunnel which CERN is now organizing.  The construction of a new tunnel reduces the demand for high dipole field strength, since there is a tradeoff between the tunnel circumference and required field strength (see Figure 3). The target collision energy is 100 TeV for 20 T dipoles in an 80 km tunnel ($R=13$ km). However, a slightly larger, 100 km circumference tunnel would provide the same collision energy of 100 TeV with field of 16 T, reachable with Nb$_3$Sn technology that relies on a far more mature and less expensive conductor than HTS. A large tunnel would also open up the possibility of a Higgs Factory based upon an $e^+e^-$ storage ring, if the International Linear Collider is not built~\\cite{Bhat}. \n\nThe 100~TeV study should address critical technology and cost feasibility issues and, thus, will provide input for developing a long-term roadmap of high-energy physics. Areas of particular interest to U.S. research groups include beam dynamics, magnets, vacuum systems, and machine protection.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{AccelCap\/Super.pdf}\n\\caption{Role of the superconductor in the energy reach of hadron colliders as a function of field and bending radius, $R$, produced by the dipole magnets.}\n\\label{fig:Super}\n\\end{center}\n\\end{figure}\n\nWhether in the LHC tunnel or in a new, larger tunnel a collider with energy beyond the LHC will have to deal with an additional challenge, the emission by the beam of synchrotron radiation at least 20 times greater per meter than the LHC at its nominal parameters. The beam pipe and beam screen will have to absorb that radiation. Although synchrotron radiation is very beneficial for beam stabilization and will make the higher energy collider the first hadron machine dominated by synchrotron radiation damping, the power dissipated in synchrotron radiation must be removed at cryogenic temperature. In LHC it is removed at 5--10 K. One possible solution for the HE-LHC is based on a beam screen at 40--60 K. This approach has no major drawback in principle, although careful design, engineering and prototyping are needed to validate such a design. Merely relying on a beam screen at 10~K similar to the LHC would be a heavy burden for the cryogenics. Handling synchrotron radiation may be even more complex for a 100 TeV machine.\n\n{\\it Focused engineering development is no substitute for innovative R\\&D. }\n\nIn addition to the focused program of engineering development, the separate programs of long-range research in magnet materials and mechanical structures of magnets must be continued. These programs have provided the intellectual and infrastructure base for the success of LARP.  For example, the long-range future of high-energy physics using p-p collisions may need magnet operating fields beyond 16~T.  In practice, the decision will rest on a tradeoff between tunnel cost --- which scales roughly inversely with dipole field --- and the cost of the dipole magnets that scales roughly linearly with dipole field for long magnets.  The crossover in cost cannot be decided ab initio as the tunnel cost will depend strongly on the geology of the collider site. \nA new generation of magnets using high-temperature superconductors will require new engineering materials with small filament size that are available in multi-kilometer continuous pieces.  Advanced magnets may offer greater temperature margin against quenches due to stray radiation lost from the beam. Higher-field magnets will require innovative stress management techniques, exquisitely sensitive magnet protection schemes, and perhaps novel structural materials.\n\nAlthough synchrotron radiation damping is not dominant in proton colliders, as it is in electron colliders, it is still important to understand beam dynamics and other effects of marginal synchrotron radiation damping. As the energy stored in the beam becomes several gigajoules, control of tenuous beam halos becomes a pressing issue. Likewise machine protection from accidental beam loss and the design of special beam abort dumps become difficult challenges that demand innovation. An effective long-range research and development program must also consider beam physics of the injection chain, effects of noise and ground motion, and design and technology options for the configuration of the interaction regions.  \n\n\n\\section{Energy Frontier lepton colliders}\n\nIn its first three years of running experiments, the Large Hadron Collider (LHC) has begun to explore the energy region up to 1 TeV. The LHC experiments have discovered a Higgs boson which,  within current experimental precision,  is consistent with that of the Standard Model and have measured its mass as about 126~GeV. The discovery of the Higgs boson and the recent release of a technical design report for the International Linear Collider have energized high energy physicists to evaluate in depth the experiments that could be performed with a lepton collider. As described in the report of the Snowmass Energy Frontier Study Group, a future lepton or photon collider would provide a factory for measurements of the properties of the Higgs boson with ultimate precision.  A lepton collider would allow unambiguous searches for new particles that complement searches at the LHC. It would also provide opportunities to probe for and study new physics, both through the production of new particles predicted by models of physics beyond the Standard Model and through the study of indirect effects of new physics on the $W$ and $Z$ bosons, the top quark, and other systems.\n\n\\subsection{International Linear Collider }\n\nIn June 2013 the Global Design Effort (GDE) published a Technical Design Report (TDR)~\\cite{ILCTDR} of the International Linear Collider (ILC), an accelerator that will give these capabilities. The ILC is a superconducting linear $e^+e^-$  collider with a center-of-mass collision energy tunable between 200 and 500 GeV with a luminosity   exceeding 10$^{34}$ cm$^{-2}$s$^{-1}$\\  at 500 GeV, roughly scaling in proportion to the collision energy. The ILC is upgradable in luminosity by a factor of 2 and upgradable  in energy to 1 TeV.  The Japanese high-energy physics community has named the ILC as its first priority. An experienced cadre of U.S. accelerator physicists and engineers is capable and ready to work on this project. \n\n{\\it Our study welcomes the initiative for ILC in Japan. The U.S. accelerator community is capable of contributing as part of a balanced U.S. high-energy physics program.}\n\nThe key characteristics of the ILC accelerator are the relatively long interval between collisions of bunches, narrow beam energy spread, beam position and energy stability, and the ability to polarize both electrons and positrons. The design and technical details of the ILC have been developed over more than 20 years and incorporate extensive US leadership contributions in beam dynamics, damping ring design, electron and positron sources, superconducting RF (SRF) technology,  and beam delivery system design. As described in the TDR, the ILC represents a mature collection of technologies that are ready to proceed to construction.  \nThe 1.2 GeV VUV Free-Electron-Laser \"FLASH\" facility at DESY has provided a critical integrated system test experience. FLASH technology is quite similar to ILC, and ILC operational parameters are within the range of FLASH hardware. Tests at FLASH were done with the ILC bunch number, bunch repetition rate, bunch charge and peak beam current. These tests were successful and encountered no fundamental technology issues with operating a superconducting linac with the ILC design characteristics. The research program has successfully demonstrated the goal of 31.5 MV\/m in installed cryomodules with beam loading, using niobium cavities with no more than two surface-preparation processing cycles. With the specified accelerating gradient, the total length of the 500~GeV ILC is 31~km.  \n\nIndustrial and institutional partners of the GDE advised a strong industrialization program from the outset. A key ingredient was the development, within the GDE, of a project-governance and value-based cost-estimating strategy that was balanced regionally and that took advantage of the intrinsic modularity and relative maturity of the SRF technology. The most costly and time-consuming part of industrialization is the construction and commissioning of heavy infrastructure, notably institutional test facilities. Fabrication of superconducting cavities to the ILC design specifications has been industrialized, with qualified vendors in Europe, North America, and Asia.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{l|l|c|c|c|c|c|c}\n$E_{CM}$ & GeV &   250   &  350 & 500 &  500 up &  250 up &  350 up \\\\ \\hline\nRep. rate  &\tHz  &\t5  &\t5  &\t5  &\t5  &\t10  &\t8   \\\\ \\hline\nBunches\/pulse   &\t &\t1312 &\t1312 &\t1312 &\t2625 &\t2625 &\t2625\\\\ \\hline\nTotal beam power &\tMW\t & 5.3 &\t7.4 &\t10.5 &\t21.0\t &21.0 &\t23.5\\\\ \\hline\nLuminosity  &\t$10^{34}$ cm$^{-2}$s$^{-1}$   &\t0.75\t & 1.0 &\t1.8 &\t3.2 &\t3.0\t &3.2  \\\\ \\hline\n\\end{tabular}\n\\caption{Summary of ILC performance in the range 250 to 500 GeV.   Columns 3--5 give\nthe TDR values. Columns 6--8  indicate scenarios beyond the parameters discussed in the TDR, but still within its technical scope. Units are specified  in the \nsecond column.}\n\\label{tab:ILCparams}\n\\end{center}\n\\end{table}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{l|c|c}\nCenter-of-mass energy   & \\% of TDR cost &\tPower consumption (MW) \\\\ \\hline\n250 GeV \"Higgs factory\" &\t70\\%\t & 120  \\\\  \\hline\n500 GeV\t& 100\\%& \t163 - 204 \\\\ \\hline\n1 TeV upgrade &\t150\\%&\t240\\\\ \\hline \n1.5 TeV upgrade &\t200\\%&\t$> 300$\\\\ \\hline\n\\end{tabular}\n\\caption{Scaling of ILC cost and power consumption scaling as function of center-of-mass energy.}\n\\label{tab:ILCscaling}\n\\end{center}\n\\end{table}\n\n\n\nDuring the research phase, U.S. national laboratories and industry have contributed substantially to cavity performance, cryomodule design, high-level RF performance, demonstrations of beam dynamics, and specific components and systems. For the construction of ILC, it is reasonable to assume the U.S. contributions will continue along these lines. Specifically, construction and testing of completed cryomodules will contribute the greatest economic value to the project. The US community could also provide detailed accelerator design development efforts as an intellectual contribution. It is expected that the US would also contribute to design and construction of one or more linear collider subsystems. Given that the TDR estimates the required labor effort to be 13,000 person years, participation by America's highly experienced accelerator scientists and engineers will be crucial.\n\nExtension of the ILC to 1 TeV is straightforward. It requires lengthened linac tunnels and additional cryomodules, but would use the original ILC sources, damping rings, final focus and interaction regions and beam dumps. No new technological breakthroughs would be required, although research to develop higher-gradient cavities would permit shorter tunnel extensions and thus cost savings.\n\n\\subsection{Alternative $e^+e^-$  colliders}\n\nAlternative approaches to the ILC \u2013 albeit on a much longer time scale \u2013 include approaches with the potential to reach multi-TeV energies. One approach, the Compact Linear Collider (CLIC)~\\cite{CLICCDR}  two-beam accelerator, is based on a high-gradient X-band, warm linac powered by a high current, low energy \n(about 1~GeV) drive beam. CLIC would stretch for 50 km at an ultimate center-of-mass energy of 3 TeV. In contrast, the muon collider --- if proven feasible --- would fit a 3--5~TeV collider onto the present Fermilab site. Moreover, the muon approach has a very large overlap with capabilities needed for Intensity Frontier accelerators. Yet another approach would use wakefields driven either by beams or lasers to achieve accelerating fields of 10 to 100 GeV per meter. Many feasibility and practicality issues need to be solved in order to implement any of these programs.\n\nFor CLIC options at 350 GeV, 1.5 TeV and 3 TeV are currently being studied. The first and second option use a single drive-beam generation complex to feed both linacs, while in option three each linac would be fed by a separate complex. The CLIC design is based on three key technologies, which have been addressed experimentally: high gradient structures, drive beam generation, and power extraction. The normal-conducting accelerating structures in the main linac have a gradient of 100 MV\/m, to limit the length of the machine. The RF frequency of 12 GHz and detailed parameters of the structure have been derived from an overall cost optimization at 3 TeV. Experiments at KEK, SLAC, and CERN have verified the structure design and established its gradient and breakdown rate. The drive beams run parallel to the colliding beams through a sequence of power extraction and transfer structures, where they produce short, high-power RF pulses that are transferred into the accelerating structures. These drive beams are generated in a central complex. The drive-beam generation and power extraction have been demonstrated in a dedicated test facility at CERN.\n\nWakefields in plasma-based accelerators can potentially provide a thousand-fold or greater increase in acceleration gradient over standard technologies.  Two primary approaches are being investigated: beam-driven wakefields (plasma wake field acceleration, or PWFA) and laser driven wakefields (laser plasma acceleration, or LPA). Experiments at SLAC, LBNL, and at European laboratories have shown that plasmas can accelerate and focus high-energy beams at an accelerating gradient in excess of 50 GeV\/m. At present two large R\\&D facilities are spearheading this research in the U.S., FACET at SLAC (PWFA) and BELLA at LBNL (LPA). Smaller efforts are being pursued at university laboratories.\n\nFor any variant of wakefield accelerator to be practical as a linear collider, several feasibility and practicality issues must be resolved in the context of an integrated system test. Most importantly, wakefield accelerators, like standard accelerator modules, must be capable of being staged in a series of phase-locked segments.  Both PWFA and LPA approaches must demonstrate simultaneous positron acceleration and focusing in plasma densities consistent with preserving positron beam quality. Both must demonstrate timing, pointing, and focusing control consistent with the high luminosities required in a lepton collider.  Finally, both must demonstrate that multi-bunch plasma instabilities (such as zero-frequency, convective hose instability) can be overcome with operation at the tens of kHz repetition rate required for high luminosity.  Beyond the feasibility issues are questions of practicality related to overall cost, efficiency, and reliability. These issues may be more demanding for laser-driven concepts as cost and efficiency limitations of drive beam sources are better known.\n\n\nFor the study of the Higgs boson what matters are deliverable luminosity, maturity of design, risk, timescale, and cost. The type of $e^+e^-$collider, linear or circular, is of little importance.  A circular collider might offer a potential alternative to the linear collider. It can benefit from three characteristics of lower-energy circular machines: (1) high luminosity and reliability, (2) the availability of several interaction points, (3) excellent beam energy accuracy. For a given RF power, the luminosity of a storage ring collider rises linearly with its circumference. For a given tunnel size, and assuming the machine operates at the beam-beam tune-shift limit, luminosity rises linearly with the total dissipated synchrotron radiation (SR) power, which is approximately proportional to the total available power. Therefore, the analysis can be scaled to different machine sizes. The energy reach consistent with sufficient luminosity for Higgs studies depends strongly on the machine circumference; a machine of 27~km circumference might reach 240~GeV, while the limit is about \n350~GeV for a machine of 80~km. At a given circumference, the energy reach and luminosity are very strongly coupled; detailed characteristics of the design matter greatly.  \n\nCERN has begun a design study of the TLEP storage ring, a machine with a circumference of 80 to 100 km circumference and dissipated synchrotron radiation power set at 100 MW. The luminosity would be largest at the $Z$ peak, but fall rapidly as center-of-mass increases beyond that point.  The proposed TLEP storage ring would need ~ 1 km of superconducting RF cavities and low $\\beta^*$ insertions, would operate at fixed field and fed by a synchrotron situated in the same tunnel for continuous top-up injection. Multi-bunch operation would be necessary for high luminosity below the top energy, requiring separated beam pipes for electron and positron beams. Beamstrahlung  from the collision process can lead to a short luminosity-lifetime and large instantaneous energy spread in the beam of the order of several percent. Beam losses must be tightly controlled even in a lattice of very large dynamic aperture.  Maintaining vertical to horizontal emittance ratios considerably smaller than light sources needs further study. \n\nThe Chinese high-energy community has also started discussion and  first studies of a circular Higgs factory or a  proton collider in a very large tunnel~\\cite{IHEP}.  Should the ILC not go forward\n over the next decade and should the renewed interest in a very large circumference hadron collider be sustained, the possibility of a circular Higgs factory deserves extensive consideration.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{AccelCap\/Muon.pdf}\n\\caption{ Muon accelerator timeline, including the MAP Feasibility Assessment period. It is anticipated that decision points for moving forward with a Neutrino Factory project supporting Intensity Frontier physics efforts could be reached by the end of this decade, and a decision point for moving forward with a Muon Collider physics effort, supporting a return to the Energy Frontier with a U.S. facility, could be reached by the middle of the next decade.}\n\\label{fig:Muon}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Muon accelerators}\nMuon accelerators~\\cite{Palmer} have the potential to provide world-leading experimental capabilities for physics at center-of-mass energies from the Higgs at 126 GeV up to the multi-TeV scale. A circular muon collider can potentially reach the higher energy range because the larger mass of the muons means they produce far less synchrotron radiation than electrons. For the same reason, beamstrahlung is suppressed leading to a very small spread of the center-of-mass energies in muon collisions. Muons, however, have a lifetime (at rest) of about 2.2 $\\mu$s and therefore decay in flight.  The unstable nature of the muon demands that the sequence of beam creation, manipulation, and acceleration be done rapidly; high-gradient acceleration is essential. Therefore, an Energy Frontier muon collider would necessarily be relatively compact. Even a 5~TeV collider would fit within the Fermilab site.\n\nResearch of critical importance to the performance of a muon collider includes: (1) development of a high-power target station including a high-field capture solenoid (nominally 20~T hybrid normal and superconducting magnet with about 3 GJ stored energy) that is ultimately capable of handling more than 4 MW of power; (2) control of collective effects in the high-intensity beams at the low energy end of the production, capture, and acceleration process; (3) cooling (reduction) of the muon six-dimensional phase space volume by six orders of magnitude to achieve the beam parameters required for the muon collider designs; (4) development of very large-aperture, superconducting dipole magnets with heavy inserts to protect the superconductors from the decay products of the stored muons in the collider. Once items (1) and (2) are resolved positively, a muon accelerator would provide an outstanding high-flux source of well-characterized neutrinos for a range of Intensity Frontier experiments. The potential for muon accelerators to address crucial questions on both the Intensity and Energy Frontiers argues for a robust development program.\n\n\n\n\\subsection{Photon colliders}\n\nIn a Higgs factory photon collider, two electron beams are accelerated to 80 GeV and converted to about 63 GeV photon beams by colliding with low-energy (3.5 eV photons), high-intensity (5 J per pulse) laser beams via the Inverse Compton Scattering (ICS) process. The two high-energy photon beams then collide and generate Higgs particles through the $s$-channel resonance $\\gamma\\gamma\\to H$. Among various options for a Higgs factory, a photon collider has the distinct advantage that the 80 GeV energy required for the electron beam is lower than for other colliders. Photon colliders have been discussed as options to accompany proposed linear or circular colliders or as a stand-alone facility. \nSince the overall acceleration efficiency of any facility is the product of the individual efficiencies of each of the systems involved in the transfer of power from the wall plug to the beam, each of the systems should be as efficient as possible. Innovative research on efficient RF generation would be extremely beneficial to all lepton collider designs. It should be strongly supported as a key technology to reduce the operating costs of future facilities for both the Energy and Intensity Frontiers.\n\n\\section{Capabilities requiring high-intensity proton beams}\n\nThe beam parameters needed for a variety of measurements of interest in the Intensity Frontier were presented at a community workshop held at Brookhaven National Laboratory in April, 2013~\\cite{Galambos}.\n Experiments included neutrino, kaon, muon, neutron, and proton EDM measurements. From the secondary beam specifications, the workshop identified the proton beam requirements for the different proposed Intensity Frontier measurements. Although the resulting sets of primary proton beam requirements were generally not based on detailed calculations but rather on general guidelines, they can be used as a guide for matching the desired measurements with existing or proposed proton facilities. The study compared 20 existing proton beamlines and 14 planned upgrades with the requirements of 19 secondary beam requests filled out by experiment advocates and drew the following conclusion:\n\n{\\it The next generation of Intensity Frontier experiments will require proton beam intensities and timing structures beyond the capabilities of any existing accelerators.}\n\nThe major proton accelerator facilities surveyed included U.S. capabilities at FNAL, BNL, LANL, and SNS. International representation included CERN, J-PARC, PSI, ISIS, and TRIUMF. New and proposed proton accelerator facilities included the European Spallation Source (ESS), the Chinese Neutron Spallation Source (CSNS), DAE$\\delta$ALUS, and Project X. In addition, existing facilities typically have upgrade proposals --- both near and long term --- which have been included in the study. High proton beam powers of 1 MW or more are desired for most proposed experiments. The beam energy requirement typically exceeds 1 GeV.  The required time structure of the beam varies significantly, depending on the nature of the measurements and the specifics of the measurement conditions; for example, underground detectors have special requirements. \n\nAccelerators with current and anticipated future programs of Intensity Frontier experiments include FNAL, J-PARC, and CERN. The LANL linac is undergoing a refurbishment and in a few years this beam would operate at 100 Hz and provide bunches with a 1 ms duration available for potential users. Cold neutron applications may be able to use this beam structure, as well as others.  Existing accelerators such as J-PARC and SNS might provide MW proton beams, with energy  less than 3 GeV with very low duty factor (less than 10$^{-4}$ to minimize background from cosmic sources) for lower-energy muon and electron neutrino studies.  ISIS (in the UK) has been used for this purpose in the past, and still has a cavern 10 m from their TS-1 target.  The ultimate technical potential of linac technology is suggested by the example of LEDA. The normal-conducting LEDA linac, which produced a 6.7 MeV, 100 mA CW beam, operated at Los Alamos from 1999 to 2001. \n\n\\subsection{Proposed new accelerators}\\label{sec:intenseaccel}\n\nProposed new accelerators for intensity frontier applications include Project X, DAE$\\delta$ALUS, and nuSTORM. Of the three, Project X~\\cite{Holmes}  has the broadest scope and, if required, could be structured as a multi-phase proposal. Project X is a high-intensity proton facility that will support a world-leading Intensity Frontier research program over the next few decades at Fermilab. When compared to other facilities in the planning stages elsewhere in the world, Project X would be unique in its ability to deliver, simultaneously, up to 6 MW of site-wide beam power to multiple experiments, at energies ranging from 233 MeV to 120 GeV, and with flexible beam formats. Project X is designed to support a wide range of experiments based on neutrinos, muons, kaons, nucleons, and nuclei. It will also lay the foundation for the long-term development of a neutrino factory and\/or a muon collider. \n\nProject X capitalizes on the very rapid development of superconducting RF technologies over the last 20 years, and their highly successful application to high-power H\u2212 acceleration at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. As a result of the accelerator research leading to and supporting SNS construction and operation, excellent simulation and modeling tools exist for designing the Project X facility with high confidence that the performance goals can be achieved. The primary supporting technologies required to construct Project X exist today.\n\nThe Project X Collaboration is engaged in a comprehensive development program aimed at mitigating technical and cost risks associated with construction and operations. The major elements of the risk reduction program include an integrated, 25 MeV front-end system test, the Project X Injector Experiment (PXIE). To allow for beam accumulation in the Main Injector, the Project X linac will accelerate H$^-$ ions. Optimizing the superconducting RF cavities will improve the cost effectiveness of the final accelerator design. Achieving the performance required of high-power targets is expected to be challenging. Therefore, high-power target technology will be a key component of research in support of the project.\n\nThe overall scope and goals of the Project X development program are based on being prepared for a 2017 construction start. Essentially all elements described above are required to implement the first stage of Project X.\n\nDAE$\\delta$ALUS~\\cite{Conrad}  is a neutrino research program based on ``decay-at-rest'' sources.  Pions are produced by interaction of 800 MeV protons on a suitable target. This energy is sufficiently above threshold for good pion yield, and low enough that pions will stop in the target before decaying.  The DAE$\\delta$ALUS configuration consists of three nearly identical\n sources of neutrinos. The highest-power accelerator, located 20 km from the\n detector would provide 10 mA of protons at 800 MeV to the neutrino-generating targets. \nA current of 10 mA is approximately a factor of 5 times the highest achieved current at PSI, the world's leading high-power cyclotron today.  Accelerating H$_2^+$  ions rather than\n protons has the potential for reducing space-charge issues at injection. Extraction of \nthe beam at 800 MeV with a stripper foil minimizes the necessity for clean turn separation at the outer radii, only requiring a proton extraction channel with sufficiently large momentum \nacceptance to transport ions stripped from several overlapping turns. The use of H$_2^+$ acceleration represents a novel approach to reducing beam losses at extraction from the\n 800 MeV cyclotron.\n\nTo date, the DAE$\\delta$ALUS feasibility arguments are made by scaling from existing low-energy H$^-$ commercial cyclotrons as well as from the PSI high-power proton cyclotron.  Since the high-power H$_2^+$ concept is quite novel, a systematic research is being conducted to address challenges of meeting the required performance. The most critical elements are (1) ion-source development to achieve very bright, vibrationally cold  H$_2^+$ beams of at least 50 mA CW, (2) injection into cyclotrons with emphasis on bunching efficiency, space requirements and space-charge dynamics, (3) end-to-end simulations to evaluate beam stability and uncontrolled loss, and (4) atomic physics experiments to measure stripping and beam-gas scattering cross sections and possibly techniques for Lorentz dissociation of vibrational states in high-field (about 25~T) magnets in the transport line between the injection and the second-stage cyclotrons. \nAs is the case for Project~X, target systems are challenging and will require sustained research.\n\nThe first phase of DAE$\\delta$ALUS is IsoDAR~\\cite{Bungau}, a compact 600 kW cyclotron proposed to be located 15 m from the KamLAND detector for a definitive search for one or two sterile neutrinos.  The IsoDAR cyclotron would also be a prototypical injector for the superconducting ring cyclotron of DAE$\\delta$ALUS. At present space charge effects at injection are being studied experimentally in a collaboration with industrial partners.\n\nThe nuSTORM facility is designed to deliver beams of electron and muon neutrinos from the decay of a stored muon beam with a central momentum of 3.8 GeV\/c and a momentum acceptance of 10\\%. The facility will be able to (1) conduct highly sensitive searches for sterile neutrinos, (2) provide definitive measurements of neutrino-nucleon scattering cross sections with percent-level precision for the electron and muon neutrinos, and (3) take the crucial first step in developing of muon accelerators as a powerful new technique for particle physics.\n\nnuSTORM is the simplest implementation of the neutrino factory concept.  It utilizes protons from the Main Injector, has a conventional (NuMI-like) target station, pion collection with a NuMI-style horn, a pion transport line, and a 480 m circumference muon decay ring.  The nuSTORM facility can also provide an intense pulses of $10^{10}$  low-energy muons, appropriate for future 6-dimensional muon ionization cooling experiments.  This beam is available during operation of the neutrino physics program and operates in a completely parasitic mode.  Finally the muon decay ring provides the opportunity for accelerator R\\&D on beam instrumentation and potentially on muon acceleration technology.\n\n\\subsection{General accelerator research needs}\n\nIn addition to project-specific needs, the Snowmass workshop identified more general research needed to advance Intensity Frontier capabilities. That research includes but is not limited to the following~\\cite{Shiltsev}:\n\\begin{itemize}\n\\item \tUnderstanding and controlling beam loss: Very precise simulations of beam dynamics involving accurate predictions of beam halos are essential for enabling high-power multi-megawatt-scale machines to run without inducing excessive amounts of radioactivity. To benchmark simulation codes, reliable measurements of very weak beam halos will be required with a large (10$^4$--10$^5$) dynamic range.\n\\item \tSuperconducting RF technology: Most modern high-power proton facilities rely on linacs utilizing superconducting radio frequency (SRF) acceleration. The initial development of SRF technology emphasized maximizing the accelerating gradient to limit the size of the accelerators. Since  many Intensity Frontier applications require the delivery of high-duty-factor beams, the figure of merit is shifting from high-gradient to very high unloaded quality-factor, Q$_0$. Cavity fabrication and processing techniques must minimize cost, while reliably producing cavities with gradients of roughly 20~MV\/m at Q$_0$ in excess of 10$^{10}$ to support future facilities.\n\\item \tHigh-quality, high-current injection systems: Multi-MW linacs and cyclotrons will need high-current, low emittance sources of ion beams, effective beam chopping, and control of strong space-charge effects at low beam energies. While suitable ion sources are close to being available, effective beam chopping is beyond the present state of technology and is under active development. The challenge is to develop choppers to provide full beam deflection on a time scale less than the beam micro-bunch spacing, and\/or to cleanly transport partially deflected beam.\n\\item \tIsochronous ring cyclotrons: Isochronous ring cyclotrons (including fixed-field alternating gradient accelerators) are also good candidates for high continuous (CW) power. Clean extraction of the beam requires well-separated turns. This sets the scale of both the ring diameter and the energy gain per turn. The alternative is to use an ion such as H$_2^+$  that can be extracted with a foil. The large superconducting magnet sectors require some development beyond the present RIKEN magnets before costs can be confirmed. \n\\end{itemize} \n\n\\subsection{High-power targets}\n\nHigh-power target development and its implications for high-power proton accelerators should not be underestimated~\\cite{Hurh}. Three of the present high-power accelerator facilities (SNS, J-PARC-MLF, and FNAL-MINOS) have been limited in beam power during recent operational periods by target concerns, not by the accelerator capability. The cost of these systems is a non-negligible portion of high-power accelerator facilities, and the resources needed to operate and maintain them are also significant (remote maintenance, disposal of radioactive wastes, control of personnel exposure to radiation). Plans for new facilities should include the impacts of handling the proposed high beam powers.\n\nOf the many identified challenges of high-power targets, radiation damage is of special note because it is complex, difficult to evaluate, and time-consuming to investigate. Atom displacements and gas production are the main underlying damage mechanisms, the particulars of which depend on primary beam characteristics, target materials, and specifics of the mechanical design. In the regimes of beam energy and intensity foreseen for Intensity Frontier experiments, one cannot simply scale from experience with nuclear power reactors. Moreover, the formation and growth of defects in target materials in extreme conditions is a mesoscale physics problem that is presently beyond present modeling and computational capabilities.  In addition, techniques of accelerated testing are frequently not validated and\/or appropriate irradiation facilities may not be available.\n\n{\\it Sustained, focused research into target technology in the context of a broad international collaboration of interested laboratories will be essential.}\n\nExperiments on the Intensity Frontier offer a broad range of scientific opportunities accompanied by a similarly broad range of technical challenges that must be overcome to conduct these experiments in acceptable time frames. The technical challenges will require sustained, focused efforts. The time to begin is now.\n\n\\section{High-intensity electron and photon beams}\n\nTo offer new opportunities for physics discoveries heavy flavor factories must accelerate and store beams with characteristics that will be challenging to deliver. Two types of flavor factories are being constructed or proposed worldwide: super $B$ factories and super tau-charm factories. SuperKEKB, a super-high-luminosity $B$ factory, is an upgrade to the KEKB $B$-factory currently under construction in Japan, with commissioning scheduled to commence in January 2015. \nU.S. national labs and universities have made important contributions to the design. Two super tau-charm factories have been proposed: one at Frascati (Tor Vergata) in Italy and one at Novosibirsk in Russia. Both machines are two-ring, symmetric-energy machines, with provisions for longitudinally polarized beams.\n\nFor SuperKEKB to achieve its target luminosity of $8\\times 10^{35}$ cm$^{-2}$s$^{-1}$\\ --- a 40-fold increase over that of KEKB  --- it will require  beam currents approximately twice as high as used at KEKB and vertical bunch sizes at the collision point about 20 times smaller than those achieved at KEKB. Although it would be premature to contemplate increases in the design luminosity of a machine that has not yet started commissioning, there is performance headroom in machine subsystems for increasing luminosity beyond the its baseline \ndesign value~\\cite{Groupfour}. Continuing U.S. collaboration would strengthen the SuperKEKB project, and would support efforts to increase the luminosity to well above that targeted in the existing design. This result would mirror the operational experience with both KEKB and PEP-II.\n\nEnhanced collaboration could also enable polarized beams in the collider. Polarization in SuperKEKB will involve preparing and injecting a beam of polarized electrons in the correct spin orientation parallel to the local precession axis at the injection point. While polarized sources with the required intensity exist (e.g., the SLC polarized gun), they are not low-emittance polarized sources. BNL has developed a candidate gun which is ready for commissioning and testing. The alternative would be to use a polarized DC gun followed by a damping ring to lower the emittance. The physics case needs to be made for both higher luminosity and polarized beams.  \n\nThe technology development for super flavor factories exploits strong synergies with light sources and damping rings. Relevant areas of synergy include high top-up rate to compensate for low lifetimes, beam stability and control, control of electron cloud and fast ion instabilities, coherent synchrotron radiation issues with short bunches, timing jitter and the attendant energy jitter, and beam instrumentation. Many of these areas are important to the design of damping rings for a linear collider and would certainly apply to the design of a circular Higgs factory. Participation of U.S. accelerator personnel in commissioning and machine studies would help to maintain a high level of U.S. expertise with high-current, low-emittance electron storage rings, such as TLEP.\n\nThree plans or proposals address physics opportunities using high-power electron and positron beams. The DarkLight experiment plans to use the 100 MeV, 10 mA electron beam of the Jefferson Lab free electron laser (FEL), impinging on a hydrogen target, to search for gauge bosons associated with Dark Force theories. The gauge bosons are predicted to decay into $e^+e^-$, leading to the final state $e^- + p + e^+ + e^-$. DarkLight will be unique in its ability to detect all four particles in the final state. The leptons will be measured in a large high-rate time-projection chamber (TPC)\n and a silicon detector will measure the protons. A 0.5~T solenoidal magnetic field provides the momentum resolution and focuses the copious M\\o ller scattering background down the beam line, away from the detectors. A first beam test has shown the FEL beam is compatible with the target design and that the hall backgrounds are manageable. The DarkLight experiment has been approved by Jefferson Lab for first running in 2017.\nHigh-intensity electron beams may also be instrumental in fixed target searches for light, weakly-coupled particles including dark matter in the MeV to GeV range. One proposed experiment would require 10$^{20}$ to 10$^{23}$ electrons with energies of $\\geq$1 GeV striking stationary nuclei in a beam dump to produce new weakly interacting states, which pass through shielding material and scatter tens of meters downstream in a relatively small detector, one cubic meter or less.\nA more speculative proposal would use a high-intensity, low-emittance positron beam, impinging on a plasma target, to generate muon\/anti-muon pairs. The beam energy should be just above the $\\mu^+\\mu^-$ production threshold with maximal beam energy asymmetry, i.e., a $\\sim$45 GeV positron beam interacting on an electron target. A low-power experimental test of this muon production rate may be possible at the FACET facility at SLAC.\n\n\\section{Electron-ion colliders}\n\nSeveral designs of future electron-ion colliders have been under consideration in recent years. All of them are based at an existing accelerator facility. The collider configurations include both ring-ring and linac-ring options. Center-of-mass energies of the proposed colliders range from 14 GeV to 2000 GeV.  Most of the collider concepts share enabling technologies. In all designs superconducting RF cavities must be able to operate with high average and high peak beam currents, providing effective damping of high-order modes. The cryomodule design must be able to accommodate containing high beam power. The hadron beam transverse and longitudinal emittances must be small to achieve a high electron-ion collider luminosity. In the designs with medium hadron energy, this requirement calls for the application of powerful cooling techniques.\nThe low \u03b2\u2217 interaction region designs for all proposed colliders face the issues of strong focusing of beams at the collision point and fast separation of beams after the collision. The synchrotron radiation fan produced by electrons in the interaction region magnets must be kept from hitting the beam pipes whether inside or in the vicinity of the detectors or in superconducting magnets.  Requirements imposed by the detector integration must be satisfied and chromatic corrections to the beam optics have to be implemented while maintaining an acceptable dynamic aperture.  Approaches considered include the following:\n\\begin{itemize}\n\\item\tSpecial design of interaction region magnets, including Nb$_3$Sn superconductor technology.\n\\item\tCollimators, absorbers, and masks at appropriate locations to provide protection from synchrotron radiation.\n\\item\tIntegrated dipole field in the detector design.\n\\item\tModified crossing angle geometry (in some designs), including crab-crossings.\n\\end{itemize}\nThe linac-ring designs utilize a high-current polarized electron source, with the average current ranging from 6~mA to 50~mA. A Gatling gun arrangement or large-size cathode gun can likely produce up to 50~mA current. The linac-ring scheme introduces non-standard beam-beam effects, which must be explored to understand the limits on the luminosity and the beam parameters. The effects include the electron beam disruption, the hadron beam kink instability, and the effect of the electron beam parameter fluctuation on the hadron beam.\nOther shared technologies include polarized beam sources and techniques to preserve beam polarization. In ring-ring colliders spin matching and the harmonic correction techniques have to be investigated further with the goal of minimizing the beam depolarization due to synchrotron radiation, especially in the presence of spin rotators and solenoidal detector magnets.\n\n\\section{Accelerator R\\&D and test beam capabilities}\n\nThis study identified a broad range of test capabilities present in existing or needed for proposed Energy and Intensity Frontier accelerators. The first category of test facilities permits testing beam physics or accelerator components essential to manage technical risks in planned projects. A second category integrates accelerator systems to provide proof-of-practicality tests.  The third category provides tests of physics feasibility of concepts and\/or components.  The study identified 35 existing test facilities~\\cite{Groupsix} in the U.S. and overseas both with and without beam testing capability. Although these facilities provide substantial readiness to move forward with the highest priority accelerators for high-energy physics, the long-range future of high energy physics needs a few additional dedicated test capabilities in the near term.\n\n\\subsection{Energy Frontier accelerators}\n\nHadron colliders have dominated the high-energy physics landscape for several decades and are expected to continue in that status over many more years of the LHC operation. Several Energy Frontier hadron colliders are under consideration now, including upgrade of LHC luminosity to $5\\times 10^{34}$  cm$^{-2}$s$^{-1}$, upgrade of the LHC energy to more than 25 TeV, and an 80 to 100 TeV VLHC in a 80--100 km or longer tunnel.\nThe most crucial technology issues for hadron colliders are in the area of superconducting magnets which are the largest drivers of the cost and of the energy and luminosity reach of these colliders. Of greatest challenge are Nb$_3$Sn 15~T arc dipoles and possible HTS superconductors such as YBCO and Bi-2212 capable of supporting dipole fields well above 20 T. For the VLHC, cheaper lower-field magnets and innovative low-cost tunneling technologies are of utmost importance to establish the financial practicality of a collider. Superconducting crab-cavities and very high-power beam dumps need to be developed for beams of such high energy. \n\nMany engineering challenges will be associated with the expected few GJ of beam energy in colliders beyond the present LHC. Increased levels of synchrotron radiation will affect beam dynamics, machine design, and operating costs and complexity. These issues will require research to identify methods of efficient collimation and control, novel vacuum chamber designs, and development of the photon stops to protect the superconducting magnets and other equipment.  \n\nMany requisite technology developments for Energy Frontier hadron colliders \u2013 described previously  \u2013 can be carried out at the existing superconducting magnet and RF test facilities at the DOE national laboratories, as well as at similar facilities overseas, foremost of which is at CERN. Some of these facilities may need upgrades to accommodate the research needs of specific projects (cable tests, transmission line magnet tests, etc.). Many beam dynamics studies concerning collimation and coherent instabilities can be performed at the LHC and its injectors, at the Fermilab Main Injector and Booster, and at RHIC. Most Project X research and development activities and facilities will directly address many issues related to the injector chain for a hadron collider beyond the LHC.  Several promising, novel accelerator schemes \u2013 such as resonance-free, integrable optics, space charge compensation with electron devices, and optical stochastic cooling \u2013 require a dedicated test accelerator for evaluation. The Integrable Optics Test Accelerator (IOTA) at the Advanced Superconducting Test Accelerator (ASTA) facility at Fermilab is a suitable match for the task. \n\nAlthough a primary challenge of any Energy-Frontier $e^+e^-$ linear collider project is value engineering to reduce costs, specific technical challenges remain for design of the positron source and the beam delivery system. These issues are being addressed at facilities including ATF2 at KEK, FLASH at DESY, and ASTA at Fermilab. A CLIC-type system has additional challenges to develop and industrialize the accelerating structures and major components. These questions are being actively addressed at CesrTA, ATF2 and CTF3.\n\nAt energies less than 250 GeV per beam, in tunnels with sufficiently large circumference, circular colliders may potentially provide a high luminosity while drawing on a base of mature technology with rich experience. Primary challenges for rings are the large synchrotron power, beamstrahlung, and large momentum acceptance required. Many of these issues are being addressed in large part at flavor factories, diffraction-limited light sources, and beam test facilities.\n\nPhoton collider designs allow access to the CP property of the Higgs boson, and they can study it at a lower electron beam energy with a high degree of polarization without the need for positrons and ancillary systems. Primary challenges for photon colliders relate to generating sufficient photon flux with high-energy lasers as well as new interaction region design. Appropriate laser technology is being developed within the international consortiums and at LLNL in the U.S.\n\nA muon collider scales well to high energies because the relatively large mass of the muons allows for tight bends and compact accelerator geometry. The primary challenge for muons is creating and cooling them to sufficient phase space density to allow for high enough luminosity. The Muon Accelerator Program is coordinating the R\\&D into the major issues with experiments primarily at FNAL and RAL.\n\nThe main challenges for plasma accelerators include demonstrating a concept for positron acceleration and demonstrating staged acceleration with the required beam brightness and efficiency, and those are being addressed through multi-institution collaborations conducting experiments at dedicated and proposed facilities BELLA\/BELLA-II at LBNL and FACET\/FACET-II at SLAC. \n\nA major challenge in applying plasma wakefield accelerator concepts to high-energy colliders is staging successive drive beams to accelerate a sufficiently intense witness beam to the desired energy. Staging requires the generation of high-intensity, multiple-drive electron\/laser beams capable of high-repetition-rate operation. Tailoring the longitudinal characteristics of a drive beam may effectively increase the energy transfer efficiency from the drive beam to the witness beam, thus reducing the number of required stages. Issues of wakefield acceleration in dielectric structures include identifying materials with appropriate breakdown voltage characteristics, identifying optimal structure geometries, and demonstrating staging with the required efficiency. Relevant research is being conducted at the AWA at ANL, ATF at BNL and FACET at SLAC. The NLCTA facility at SLAC hosts a dedicated beamline studying the primary issues for laser-driven microstructures including damage limits, coupling geometry, staging, and maintaining phase tolerances.\n\nSeveral facilities in the U.S. are exploring wakefield accelerators. AWA at Argonne combines multiple 75 MeV drive beams with a 15 MeV witness beam. BELLA at LBL uses a 1 PW laser to conduct experiments that synchronize two laser beamlines to accelerate a single, sub-GeV electron beam. FACET at SLAC uses a 30~GeV beam from the SLAC linac to investigate positron acceleration, high-transformer ratio configurations and the ``afterburner'' concept. Multiple drive beam generation and pulse shaping is one of the major goals for the proposed FACET II upgrade.\n\n\\subsection{Intensity Frontier accelerators}\n\nThe short- and medium-term needs of the U.S. Intensity Frontier program will employ Fermilab proton accelerators --- power-upgraded Booster and Main Injector augmented\n by the ``muon campus'' rings ($g-2$, mu2e, nuSTORM, etc.). Longer-term aspirations require construction of the proposed Project X, described in Section~\\ref{sec:intenseaccel}.\n Project X could support a wide range of experiments based on neutrinos, muons, kaons, \nnucleons, and nuclei and also can form a basis of the front-end of a future Neutrino\n Factory. The proposed DAE$\\delta$ALUS experiment calls for construction of several \nvery high-power, 1 to 5~MW, \n800~MeV proton cyclotrons.    \nThe primary design constraints in high-power hadron accelerators for the Intensity Frontier are minimizing beam loss ($<$ 1 watt\/m, or $<$ 0.01\\% total beam loss), and flexible beam delivery in experiment-specific time formats ranging from quasi-CW for rare particle decays to single few-ns-long bunches. Additional challenges facing designers of Intensity Frontier accelerators include: (1) generating high-current, high-quality beams, (2) efficiently accelerating high beam currents to high energies, (3) controlling emittance and minimizing beam halo, (4) manipulating beams on a bunch-by-bunch basis for parallel experiments, (5) maintaining low loss during beam extraction, and (6) creating target systems for extreme power densities and extreme radiation environments.\n\nProject X R\\&D activities at the PXIE facility and beam studies with the SRF cryomodules at Fermilab's ASTA facility will directly address many issues listed above. Many beam experiments on efficient collimation, coherent instabilities, and novel extraction methods can be performed at Fermilab's Main Injector and Booster, at the ORNL's SNS, or at the LHC and its injectors. Studies of promising novel accelerator techniques --- like halo suppression in the integrable optics and space charge compensation with electron devices ---\n can be performed at the IOTA at the ASTA facility. A dedicated facility is needed for development and tests of high-power targets, though some target-related issues can be explored at existing accelerators. Many tests and technology demonstrations to support the DAE$\\delta$ALUS experiment design work can be performed at the existing facilities, such as INFN-Catania, RIKEN, ORNL, or in the private sector at medical cyclotron companies.\n\n\\section{Summary remarks}\n\nThe approved LHC program, its future upgrade towards higher luminosities (HL-LHC), and the study of an LHC energy upgrade or of a new proton collider delivering collisions at center-of-mass energies up to 100 TeV, are all essential components of exploring the Energy Frontier with proton colliders. Therefore the priority recommendation of our study is full exploitation of LHC.  Doing so requires a strong LHC Accelerator Research Program sponsored by the OHEP that transitions to a US-LHC high luminosity construction project. During the project period we recommend continuing a focused, integrated, laboratory program that emphasizes the engineering readiness of technologies suitable for a 26 TeV upgrade of the LHC or a machine of higher energy in a larger tunnel. The most critical technology development toward higher-energy hadron colliders is the next generation high-field Nb$_3$Sn magnets (limited to about 15 T) and adequate beam control technology to assure machine protection.\n\nOur study welcomes the initiative for ILC in Japan. The U.S. accelerator community is capable of contributing  as part of a balanced U.S. high-energy physics program. As described in its Technical Design Report (TDR), the ILC is technically ready to proceed to construction. The TDR incorporates leadership U.S. contributions to machine physics and technology in superconducting RF (SRF), high-power targets for positron production, beam delivery, damping ring design, and beam dynamics such as electron cloud effects. \nPlasma-based accelerators (beam or laser driven) providing GV\/m accelerating fields would open the way to linear colliders in the muli-TeV range with a relatively small footprint if the preservation of beam quality during multi-stage acceleration can be demonstrated. A circular muon collider, if feasible, could provide discovery-level experimental capabilities at energies from the Higgs $s$-channel threshold at 126 GeV up to the multi-TeV scale. Such a collider could probe narrow physics features with extreme precision, because the larger mass of the muon suppresses synchrotron radiation during the collision process that would spread the energy of the beams. \n\nRequired accelerator capabilities for Intensity Frontier  experiments are more diverse than for the energy frontier. The common characteristics required are (1) average beam power $>$ 1 MW delivered \nwith (2) a flexible, experiment-dependent time structure. The overarching conclusion of the study is that the next generation of intensity frontier experiments requires beam intensities and timing structures beyond the capabilities of any existing accelerator. Fermilab's proposed, multi-stage Project X would yield a world-leading facility that would simultaneously deliver a flexible ``on-demand'' beam structure that could serve multiple experiments over an energy range 0.25 -- 120 GeV. Our study heard exciting possibilities for\n DAE$\\delta$ALUS, a capability of narrower experimental scope than Project X. For both proposed \ncapabilities high-power targets are a difficult challenge that limits facility performance. One cannot \nsimply scale the performance with high-energy beams from experience with nuclear reactors. \nComputed radiation effects in inhomogeneous materials, subject to time-varying irradiation, \nneed validation with controlled, instrumented in-beam tests. \n\nThe relevant technologies of super flavor-factories exploit strong synergy with light sources and damping rings for lepton colliders. Doubtless the operation of such machines such as SuperKEKB, the super-high-luminosity $B$ factory,  will inform the final design of damping rings for a energy frontier lepton collider. \nAll electron-ion colliders studied in recent years would be based at an existing accelerator facility. The collider configurations include both ring-ring and linac-ring options; center-of-momentum energies \nrange from 14~GeV to 2000~GeV.  Most of the collider concepts share enabling technologies. \nSuperconducting RF cavities must be able to operate with high average and high peak beam \ncurrents, providing effective damping of high-order modes. \n\nAlthough existing accelerator test-bed facilities provide substantial readiness to advance the \nhighest priority accelerators for high-energy physics, the long range future of high energy physics \nneeds a few additional dedicated test capabilities in the near term, most notably, for high power \ntargetry, muon accelerators, and high-intensity accelerator systems.  The group did not examine \nthe process in basic accelerator physics and modeling that must accompany any long-range research \nprogram. Such research bears fruit beyond the confines of specific projects and is essential to the\n long-term health of the field.\n\nThe Snowmass Accelerator Capabilities Study offers the following summary findings and recommendations:\n\\begin{itemize}\n\\item \tTo maximize the exploration of the Energy Frontier, the full exploitation of the LHC is the highest priority of the hadron-collider program. \n\\item\tAs described in its Technical Design Report (TDR), the ILC is technically ready to proceed to construction.\n\\item \tA vigorous, integrated U.S. research program toward demonstrating feasibility of a muon collider is highly desirable. The current funding level is inadequate to assure timely progress.\n\\item \tThe overarching conclusion is that the next generation of Intensity Frontier experiments requires beam intensities and timing structures beyond the capabilities of any existing accelerator.\n\\item \tFermilab's proposed, multi-stage Project X would yield a world-leading facility that could serve multiple experiments over an energy range 0.25 -- 120 GeV.\n\\item \tSustained, focused research into high-power target technology in the context of a broad international collaboration of interested laboratories will be essential to frontier accelerator facilities.\n\\item \tThe relevant technologies of super flavor-factories exploit strong synergy with light sources and damping rings for lepton colliders.\n\\item \tAll electron-ion colliders studied in recent years would be based at an existing accelerator facility with center-of-momentum energies range from 14 GeV to 2000 GeV.\n\\item \tInnovations in acceleration and beam transport techniques such as plasma and dielectric wakefield acceleration have a significant potential to reduce the size of future accelerator facilities. \n\\item \tLong-term research including fundamental accelerator and beam physics theory and simulation will expand the technical options for any future accelerator-based facility. Focused engineering development is no substitute for innovative R\\&D. \n\\end{itemize}\n\n{\\bf Acknowledgements: }\nThis report has been partially supported by the Office of High Energy Physics and the Office of Nuclear Physics of the U.S. Department of Energy and the National Science Foundation. The conveners thank all those who have participated in the Snowmass 2013 process, both at workshops and by submitting white papers.  \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe strain energy of a piece of homogeneous anisotropic elastic material depends on the orientation of the material relative to the directions of principal stress, although the orientation dependence vanishes trivially for isotropic solids.  This property is therefore an inherently anisotropic feature of elasticity, and it raises the question of how to find  the material orientation (if any) which minimizes the strain energy for a given state of stress or  strain. New results are presented in this paper on the determination of optimal orientations for both general and specific types of anisotropy. \n\nThe general  problem of determining optimal orientations in anisotropic elasticity has been the subject of several studies in the last two decades, beginning with the work of Seregin and Troitskii \\cite{Seregin81} in the context of  orthotropic solids. They determined the important {\\it coaxiality condition}: a minimum or maximum of strain energy requires that the stress and strain share common principal axes.  The coaxiality condition was subsequently and independently obtained by others: first, by  Rovati and Taliercio \\cite{Rovati91} who considered 3D  elastic materials with orthotropic and cubic symmetries (although their derivation is not restricted to these symmetries but is applicable to general anisotropy), and later by    \nCowin \\cite{Cowin94}.  Cowin  derived the coaxiality condition  independent of material symmetry considerations.  He showed that the commutativity of the stress and strain is a consequence of the stationarity condition of the strain energy with respect to rotations of the moduli.  Vianello \\cite{Vianello96a} provided a more formal derivation of coaxiality in linear elasticity.  He used the  tangent space of the rotation group to show that there are at least two orientations of the moduli that simultaneously make the energy stationary and stress and strain coaxial, a result later generalized to  hyperelasticity \\cite{Vianello96b} (it was subsequently shown that at least three such orientations exist for both linear elasticity \\cite{Sgarra97a} and for hyperelasticity \\cite{Sgarra97b}).  There is a slight distinction between the problems considered by Cowin and by Vianello that is important to note for our purposes \\cite{Cowin97}.  Thus, Cowin \\cite{Cowin94} considered  stress states with fixed principal directions but arbitrary amplitudes, whereas Vianello \\cite{Vianello96a} assumed a specific state of stress.  The former constraint  defines a smaller set of possible elastic moduli for which coaxility can be attained, because it requires that optimal condition be simultaneously satisfied by a family of coaxial stresses.  Not surprisingly, Cowin found that only materials with orthotropic symmetry meet these conditions.   In this paper the stress state is taken as given, in the same spirit as \\cite{Vianello96a} and also of Rovati and Taliercio \\cite{Rovati03}.   While the emphasis here is on three dimensional elasticity, the optimality problem has also been \n addressed within  the context of  two dimensional elasticity \\cite{Pedersen89,Pedersen90,Gea04}.\nCowin and Yang  \\cite{Cowin00} considered a related but more general question of optimality with respect to Kelvin modes, rather than simply the freedom to orient a given material.  \n  A more extensive review of the literature can be found in the recent paper of Rovati and Taliercio \\cite{Rovati03}.  \n\nIt is interesting to note that the coaxiality condition has been derived in a variety of  different ways: for particular symmetries (e.g. orthotropic) \\cite{Seregin81}, using Lagrange multipliers \\cite{Cowin94}, from general analytic considerations \\cite{Vianello96a}, and even using the 6-dimensional eigenvector properties of the elasticity tensor \\cite{Banichuk96}. The derivation of the coaxiality condition  presented here differs from all these previous methods.    Our starting point is a representation of the rotation matrix due to Mehrabadi, Cowin and Jaric \\cite{mcj}.  This formulation also enables derivation of conditions for minima or maxima, in a simpler  and more general form than that obtained by Cowin \\cite{Cowin94}.  Section \\ref{sec2} begins with the problem definition and notation.  The stationarity and min\/max conditions are discussed in Section \\ref{sec3}.  Specific conditions for particular material symmetries are derived in Section \\ref{sec4}, and we conclude in Section \\ref{sec5} by defining the {\\it strain deviation angle}, a concept which could have application in practical circumstances in anisotropic elasticity. \n\n\n\\section{Problem definition and notation}\\label{sec2}\n\n\\subsection{Optimal orientation of anisotropic solids}\n\nConsider a fixed coordinate system $\\{ \\bf{e}_1,\\, \\bf{e}_2,\\, \\bf{e}_3 \\}$ coincident with the principal axes of stress.  The stress tensor is therefore  \n\\beq{eq1}\n\\bosy{\\sigma} = \\sigma_{\\rm I}  \\,\\bf{e}_1 \\otimes  \\bf{e}_1 + \\sigma_{\\rm II} \\, \\bf{e}_2 \\otimes  \\bf{e}_2 + \\sigma_{\\rm III} \\, \\bf{e}_3 \\otimes  \\bf{e}_3, \n\\end{equation}\nwhere $\\sigma_{\\rm I}$, $\\sigma_{\\rm II}$ and $\\sigma_{\\rm III}$ are the principal stresses, in no particular order.  Alternative expressions for the stress include the $3\\times 3$ matrix representation, \n\\beqa{s1}\n \\bosy{\\sigma}    = \n\\left[ \\begin{array}{ccc}\n\\sigma_{\\rm I} & 0 & 0 \\\\\n0 & \\sigma_{\\rm II} &  0 \\\\\n0 & 0 &  \\sigma_{\\rm III}  \n\\end{array} \\right] , \n\\end{eqnarray} \nand indicial\\footnote{Lower case Latin suffices take on the values 1, 2, and 3, and the summation convention on repeated indices is assumed unless noted otherwise.} notation, \n\\beq{t2}\n\\sigma_{ij}=  \\sigma_{\\rm I}\\,  \\delta_{i1} \\delta_{j1} +  \\sigma_{\\rm II}\\,  \\delta_{i2} \\delta_{j2}+  \\sigma_{\\rm III}\\,  \\delta_{i3} \\delta_{j3}. \\quad\n\\end{equation}\nOur goal is to find the orientation or orientations which minimize the energy function for fixed stress\n\\beq{s4}\n{\\cal E}  \\equiv  \\sigma_{ij} \\sigma_{kl} \\, s_{ijkl}\\, . \n\\end{equation} \nHere, $s_{ijkl}$ are the components of the fourth order compliance tensor relative to $\\{ \\bf{e}_1,\\, \\bf{e}_2,\\, \\bf{e}_3 \\}$.  Think of the material as being free to be oriented in  such a way that the energy ${\\cal E}$ depends upon the orientation of the {\\it material}  with respect to the fixed principal axes of the stress. \nThe material moduli for stiffness and compliance are ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}^{(0)}$ and $\\cS^{(0)}$ when aligned with the fixed axes. It is not necessary  to  specify at this stage whether or not the moduli possess any symmetry with respect to these axes.  The main point is that  the material is free to orient in arbitrary directions with {\\it oriented} moduli  ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}$ and $\\cS$ while the stress  orientation remains  fixed. \n\n\\subsection{Notation and tensor rotation}\n Hooke's law  relating stress $\\sigma_{ij}$ and strain $\\varepsilon_{ij}$ is \n\\beq{00}\n\\sigma_{ij} = c_{ijkl}\\varepsilon_{kl}, \n\\qquad\n\\varepsilon_{ij} = s_{ijkl}\\sigma_{kl}. \n\\end{equation}\nHere $c_{ijkl}$ denote the components of the stiffness tensor, inverse to the compliance: \n$c_{ijkl}s_{klpq} = I_{ijpq}$, where $I_{ijpq} = (\\delta_{ip} \\delta_{jq} +\\delta_{iq}\\delta_{jp})\/2$ is the fourth order identity tensor.   \nThe rotated elasticity components could be expressed in terms of the unrotated components $c^{(0)}_{ijkl}$ and $s^{(0)}_{ijkl}$, using Euler angles, for instance.  The concise Voigt notation is used to represent the elements of the elasticity tensor in the fixed basis.     Thus, the compliance is  \n$\\cS = \\left[ S_{IJ} \\right],\\  I,J=1,2,\\ldots 6$, with $S_{12}= s_{1122}$, $S_{16}= s_{1112}$,  $S_{44}= s_{2323}$, etc.,   \n\\beqa{a1a}\n\\cS  = \\left[ \n\\ba{cccccc}\nS_{11} & S_{12} & S_{13} & \n S_{14} &  S_{15} & S_{16} \n\\\\ & & & & & \\\\\n & S_{22} & S_{23} & \n S_{24} &  S_{25} &  S_{26} \n\\\\ & & & & & \\\\\n & & S_{33} & \n  S_{34} &  S_{35} &  S_{36} \n\\\\ & & & & & \\\\\n &   &   \n & S_{44} & S_{45} & S_{46}\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{55} & S_{56}\n\\\\ & & & & & \\\\\n&&&& & S_{66}\n\\end{array} \\right] .  \n\\end{eqnarray}\n\nAn alternative representation for the elasticity tensor, closely related to \\rf{a1a}, is the $6\\times 6$ matrix   $\\hbS$ with elements $ \\left[ \\hS_{IJ}\\right]$  defined as \n\\beqa{defs}\n\\hbS = {\\bf T} \\, \\cS\\, {\\bf T},\n\\quad \\mbox{where  }\n{\\bf T} \\equiv \n\\left[ \n\\ba{cc}\n~ {\\bf I}~ & 0  \\\\ & \\\\\n0 & \\sqrt{2} {\\bf I}\n\\end{array} \\right] .\n\\end{eqnarray}\nExplicitly,   \n\\beqa{a1}\n\\hbS  \\equiv \\left[ \n\\ba{cccccc}\nS_{11} & S_{12} & S_{13} & \n 2^{\\frac12} S_{14} & 2^{\\frac12} S_{15} & 2^{\\frac12} S_{16} \n\\\\ & & & & & \\\\\nS_{12} & S_{22} & S_{23} & \n 2^{\\frac12} S_{24} & 2^{\\frac12} S_{25} & 2^{\\frac12} S_{26} \n\\\\ & & & & & \\\\\nS_{13} & S_{23} & S_{33} & \n 2^{\\frac12} S_{34} & 2^{\\frac12} S_{35} & 2^{\\frac12} S_{36} \n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{14} & 2^{\\frac12} S_{24} & 2^{\\frac12} S_{34} \n & 2S_{44} & 2S_{45} & 2S_{46}\n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{15} & 2^{\\frac12} S_{25} & 2^{\\frac12} S_{35} \n & 2S_{45} & 2S_{55} & 2S_{56}\n\\\\ & & & & & \\\\\n 2^{\\frac12} S_{16} & 2^{\\frac12} S_{26} & 2^{\\frac12} S_{36} \n & 2S_{46} & 2S_{56} & 2S_{66}\n\\end{array} \\right] , \n\\end{eqnarray}\nThis representation is useful in taking advantage of the fact that  fourth order elasticity tensors in 3 dimensions are \nequivalent to second order symmetric tensor of 6 dimensions \\cite{c3}.\nSimilar equations follow for ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}$ and  $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} = {\\bf T} \\, {\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}\\, {\\bf T}$.  Define \n\\beqa{eqs2}\n\\widehat{\\bosy{\\sigma}}  \\equiv \\left[ \n\\ba{c}\n\\sigma_{11} \\\\  \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sqrt{2} \\sigma_{23} \\\\  \\sqrt{2}\\sigma_{31} \\\\ \\sqrt{2}\\sigma_{12}\n \\end{array} \\right] , \n \\qquad\n \\widehat{\\bosy{\\varepsilon}}  \\equiv \\left[ \n\\ba{c}\n\\varepsilon_{11} \\\\  \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ \\sqrt{2}\\varepsilon_{23} \\\\  \\sqrt{2}\\varepsilon_{31} \\\\ \\sqrt{2}\\varepsilon_{12}\n \\end{array} \\right] , \n\\end{eqnarray}\nthen the stress strain relations \\rf{00} become \n\\beqa{s3}\n\\widehat{\\bosy{\\sigma}}   = \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\widehat{\\bosy{\\varepsilon}} , \n \\qquad\n \\widehat{\\bosy{\\varepsilon}}  = \\hbS \\widehat{\\bosy{\\sigma}}. \n\\end{eqnarray}\nNote that $\\hbS$ and $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}$ are the matrix inverse of each other; \n$\\hbS \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} = \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\hbS = \\hbI$ where $\\hbI=$diag$(1,1,1,1,1,1)$. \n\nThe rotation about ${\\bf n}$, $|{\\bf n} | = 1$  by angle $\\phi$ is defined as ${\\bf Q} ({\\bf n} , \\phi )\\in O(3)$, such that vectors (including the basis vectors) transform as ${\\bf v} \\rightarrow {\\bf v}' = {\\bf Q}{\\bf v}$. By considering small rotations, it may be readily seen that  ${\\bf Q} ({\\bf n} , \\phi )$ can be expressed in terms a skew symmetric tensor ${\\bf P}$ that is linear in $\\bf n$. Thus,  \n\\beq{qsat1}\n\\frac{d {\\bf Q}}{d\\phi} ({\\bf n} , \\phi )  = {\\bf P}\\, {\\bf Q} ({\\bf n} , \\phi ), \\quad \\mbox{where}\\quad P_{ij}({\\bf n})= e_{ijk}n_k\\, ,  \n\\end{equation}\nand hence \n\\beq{qeq}\n{\\bf Q} = e^{\\phi {\\bf P}},  \n\\end{equation}\nNote that    ${\\bf Q}$ possesses alternate well-known expressions\n\\beqa{a1a1} {\\bf Q}  ({\\bf n} , \\phi ) &=& {\\bf I} + \\sin\\phi\\, {\\bf P}+ (1 -  \\cos\\phi) {\\bf P}^2 \\nonumber \\\\\n&=& {\\bf n}\\otimes{\\bf n} + \\cos\\phi \\, ({\\bf I} -{\\bf n}\\otimes{\\bf n} )+\n\\sin\\phi\\, {\\bf P}\\, . \n\\end{eqnarray}\nIn particular for our needs here, the small angle expansion is \n\\beq{qsat2}\n{\\bf Q} ({\\bf n} , \\phi )  = {\\bf I} +  \\phi {\\bf P} + \\mbox{O}(\\phi^2)\\, .  \n\\end{equation}\n\nUnder the change of basis associated with ${\\bf Q} ({\\bf n} , \\phi )$, second order tensors (including stress and strain) transform as $\\bosy{\\sigma} \\rightarrow \\bosy{\\sigma} '$, where\n\\beq{qsat3}\n\\sigma_{ij} ' = Q_{ir}Q_{js} \\, \\sigma_{rs} \n\\quad \\Leftrightarrow \\quad \\sigma_{ij} ' = {\\cal Q}_{ijrs}\\, \\sigma_{rs}. \n\\end{equation}\nThe fourth order ``rotation\" tensor follows from \\rf{qsat3} as \n\\beq{qsat4}\n{\\cal Q}_{ijrs} = \\frac12 \\, \\big( Q_{ir}Q_{js} + Q_{is}Q_{jr}\\big),  \n\\end{equation}\nand eqs. \\rf{qsat1} and \\rf{qsat4} imply\n\\beq{qsat5}\n\\frac{d {\\cal Q}_{ijrs} }{d\\phi} ({\\bf n} , \\phi )  = {\\cal P}_{ijpq}{\\cal Q}_{pqrs}, \n\\quad \\mbox{with}\\quad {\\cal Q}_{ijrs}  ({\\bf n} , 0 ) = I_{ijpq}, \n\\end{equation}\nwhere\n\\beq{qsat6}\n{\\cal P}_{ijpq} = \n\\frac12 \\, \\big( \\delta_{ip}P_{jq}  +\\delta_{iq}P_{jp} +\\delta_{jp}P_{iq} + \\delta_{jq}P_{ip}\n\\big)\\, . \n\\end{equation}\nThe formal solution of \\rf{qsat5}, with meaning that should be clear, is \n\\beq{qsat7}\n{\\bf {\\cal Q}} = e^{\\phi {\\bf {\\cal P}}},   \n\\end{equation}\nand the small angle approximation is\n\\beq{qsat8}\n{\\cal Q}_{ijpq} = I_{ijpq}+\\phi  {\\cal P}_{ijpq} + \\mbox{O}(\\phi^2)\\, . \n\\end{equation}\n\nMehrabadi et al. \\cite{mcj}  derived an elegant expression for ${\\bf {\\cal Q}}$ analogous to the representation  for ${\\bf Q}({\\bf n} , \\phi ) $. The key is the characteristic equation of ${\\bf {\\cal P}}$, (${\\bf {\\cal P}}^5 +5{\\bf {\\cal P}}^3+4{\\bf {\\cal P}}=0$ where ${\\cal P}^2_{ijkl} = {\\cal P}_{ijpq}{\\cal P}_{pqkl}$ etc.) which permits the exponential expression \\rf{qsat7} to be simplified.  The result is most simply presented in terms of the $6\\times 6$ rotation matrix $\\hbQ $  introduced by Mehrabadi et al. \\cite{mcj}, and defined in the same manner as before.  Thus, \n$\\hbQ  = {\\bf T} {\\cal Q} {\\bf T} $  and $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}  = {\\bf T} {\\cal P} {\\bf T} $  where ${\\bf T}$ is defined in \\rf{defs} and ${\\cal Q}$ and ${\\cal P}$ are the $6\\times 6$ Voigt matrices.   Explicitly,  $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}$ is a skew symmetric six dimensional tensor  linear in  $\\bf n$, \n\\beqa{a1b}\n\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} ({\\bf n}) =  \\left[ \n\\ba{cccccc}\n0 & 0 & 0 & \n        0 & \\sqrt{2} n_2 & -\\sqrt{2} n_3  \n\\\\ & & & & & \\\\\n0 & 0 & 0 & \n        -\\sqrt{2} n_1 &0 &  \\sqrt{2} n_3\n\\\\ & & & & & \\\\\n0 & 0 & 0 & \n        \\sqrt{2} n_1 &  -\\sqrt{2} n_2 &0\n\\\\ & & & & & \\\\\n0 & \\sqrt{2} n_1 & -\\sqrt{2} n_1   & 0 & n_3 & -n_2\n\\\\ & & & & & \\\\\n-\\sqrt{2} n_2 &0 &  \\sqrt{2} n_2  & -n_3 & 0 & n_1\n\\\\ & & & & & \\\\\n\\sqrt{2} n_3 &  -\\sqrt{2} n_3 &0 &n_2 & -n_1 & 0 \n\\end{array} \\right] . \n\\end{eqnarray}\n $\\hbQ  ({\\bf n} , \\phi )$ is  an orthogonal second order tensor of six \ndimensions, satisfying $\\hbQ\\hbQ^T = \\hbQ^T\\hbQ = \\hbI$.  \nEquation \\rf{qsat7} becomes\n\\beq{they}    \n \\hbQ ({\\bf n} , \\phi ) =  e^{\\phi  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n})} ,\n \\end{equation}\n and  has the explicit expansion  \\cite{mcj}\n\\beq{mc2}\n\\hbQ ({\\bf n} , \\phi )= \\hbI + \n\\sin\\phi\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}\n+ (1 -  \\cos\\phi)\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \n+\\frac13 \\sin\\phi (1 -  \\cos\\phi)\\,( \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^3)\n+\\frac16  (1 -  \\cos\\phi)^2\\,( \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^4)\\, . \\quad\n\\end{equation}\nFinally, we note that  fourth order tensors ${\\mathbb C}} \t\t\t\t\\newcommand{\\cS}{{\\mathbb S}  $ transforms as $\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} \\rightarrow \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S} ' = \\hbQ\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}\\hbQ^T$.\n\n\\subsection{Orientation function revisited}\n \n\nDenote the matrix of rotation from the fixed to the ``current''  axes as $\\hbQ $.  Thus, \n\\beq{s3b} \n\\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}   = \\hbQ \\widehat{\\bf C}}  \t\\newcommand{\\hbS}{\\widehat{\\bf S}^{(0)} \\hbQ^T , \n\\qquad\n\\hbS   = \\hbQ \\hbS^{(0)} \\hbQ^T . \n\\end{equation}\nHence the objective function of   \\rf{s4} for the stress-based energy minimization becomes \n\\beq{s5}\n{\\cal E} = \\widehat{\\bosy{\\sigma}}^T\\,  \\hbS\\,  \\widehat{\\bosy{\\sigma}}\\, . \n\\end{equation}\nThis is the starting point in the next section to derive conditions necessary for a minimum.  It is important to emphasize the initial assumption that the stress is aligned with the fixed axes, \\rf{s1}, or in terms of $\\widehat{\\bosy{\\sigma}}$,     \n\\beqa{s1a}\n\\quad \n\\widehat{\\bosy{\\sigma}}  \\equiv \\left[ \n\\ba{c}\n\\sigma_{\\rm I} \\\\  \\sigma_{\\rm II} \\\\ \\sigma_{\\rm III} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] .\n\\end{eqnarray} \nThis ensures that the energy varies as the material axes are rotated (if the stress were also rotated then the energy would be, trivially,  unchanged). \n\n\\section{Stationarity and min\/max conditions}\\label{sec3}\n\n\\subsection{Angular derivatives of the strain energy}\n\nConsider the energy  $\\cal E$ of \\rf{s5} as a function of the rotation $\\hbQ$.  A  stationary value is obtained if $\\cal E$ is unchanged  with respect to additional small rotations of $\\hbS$.  This requires calculating the first derivative with respect to rotation angle for arbitrary rotation. The second derivative is needed  to distinguish the stationary point as a minimum or maximum, or otherwise. \n\nThe strain energy obtained by arbitrary rotation  of the material about the axis $\\bf n$ is \n\\beq{s6}\n{\\cal E} ( {\\bf n}, \\phi) = \\widehat{\\bosy{\\sigma}}^T\\, \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) \\,  \\widehat{\\bosy{\\sigma}}\\, .   \n\\end{equation}\nThe first derivative can be expressed, \n\\beqa{s8q}\n \\frac{\\partial \\, }{\\partial \\phi }{\\cal E} ( {\\bf n}, \\phi) &= &\n\\widehat{\\bosy{\\sigma}}^T\\, \\big[ \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi )  + \\hbQ({\\bf n} , \\phi ) \\hbS^T  \\hbQ^T ({\\bf n} , \\phi )\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T ({\\bf n}) \\big] \\,  \\widehat{\\bosy{\\sigma}}\n\\nonumber \\\\\n&= & 2\\widehat{\\bosy{\\sigma}}^T\\,  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi )  \\,  \\widehat{\\bosy{\\sigma}}\n\\, .  \n\\end{eqnarray}\nThis follows from eqs. \\rf{qsat1}, \\rf{they} and \\rf{s6}, using the fact that $\\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}$ and $\\hbQ$ commute.  Similarly, the second derivative follows as \n\\beq{s8qq}\n \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} ( {\\bf n}, \\phi) \n= 2\\widehat{\\bosy{\\sigma}}^T\\, \\big[  \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbQ({\\bf n} , \\phi ) \\hbS \\hbQ^T ({\\bf n} , \\phi ) \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T ({\\bf n})\n\\big]  \\,  \\widehat{\\bosy{\\sigma}} \n\\, .  \\quad\n\\end{equation}\n\n\n\\subsection{Condition for stationary strain energy}\n \nAssume,  with no loss in generality, that the  stationary orientation is at $\\phi = 0$. \nIf  $\\hbS$ is at a stationary point, then the energy should be unchanged regardless of the axis $\\bf n$, or \n\\beq{s7}\n{\\cal E} \\mbox{ stationary} \\quad \\Leftrightarrow \\quad \n  \\left. \\frac{\\partial \\, }{\\partial \\phi } {\\cal E} ( {\\bf n}, \\phi)\\right|_{\\phi = 0} = 0, \\quad \\forall \\, |{\\bf n} |=1.\n\\end{equation}\nThis becomes, using \\rf{s8q} evaluated at  $\\phi = 0$,  \n\\beq{s9}\n  \\left. \\frac{\\partial \\, }{\\partial \\phi }{\\cal E} ( {\\bf n}, \\phi)\\right|_{\\phi = 0} = \n2\\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, .  \n\\end{equation}\nWe now take advantage of the fact that the stress is aligned with the fixed axes.  Thus, eqs.   \\rf{a1b} and \\rf{s1a} give \n\\beq{r1}\n\\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n})\n = \n \\bigg( 0,\\, 0,\\, 0,\\, \\sqrt{2} (\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 ,\\,  \\sqrt{2} (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 ,\\, \n \\sqrt{2} (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3  \\bigg) . \n \\end{equation}\n Hence, \n\\beqa{r2}\n  \\widehat{\\bosy{\\sigma}}^T\\,   \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}({\\bf n}) \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, \n   =   \n2  \n\\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] \n \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\  \\\\ \n \\sigma_{\\rm II} \\\\  \\\\ \n \\sigma_{\\rm III} \\end{array} \\right] .\n   \\end{eqnarray}\nThis must vanish for arbitrary direction ${\\bf n}$, hence the energy $\\cal E$  is stationary if \n\\beqa{r3}\n   \\left[ \\begin{array}{ccc}\n\\sigma_{\\rm III} - \\sigma_{\\rm II} & 0 & 0 \\\\\n && \\\\\n 0 & \\sigma_{\\rm I} - \\sigma_{\\rm III} & 0 \\\\\n && \\\\\n 0 & 0 & \\sigma_{\\rm II} - \\sigma_{\\rm I} \\end{array} \\right]\n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] \n \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\  \\\\ \n \\sigma_{\\rm II} \\\\  \\\\ \n \\sigma_{\\rm III} \\end{array} \\right]  = \n \\left[ \\begin{array}{c}\n0 \\\\  \\\\ \n0 \\\\  \\\\ \n0 \\end{array} \\right] .\n   \\end{eqnarray}\n\nLet us assume for simplicity that the state of stress is triaxial, so that $\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}$ are distinct.  The left matrix in \\rf{r3} can be removed, implying a linear condition in the stress:\n\\beq{4r}\n{\\bf E} \\, \\big(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}\\big)^T = \\big(0,\\, 0,\\, 0 \\big)^T, \n\\end{equation}\nwhere ${\\bf E}$ involves moduli (compliances) only\n\\beqa{r4}\n   {\\bf E}    = \n \\left[ \\begin{array}{ccc}\n S_{14} &  S_{24} &  S_{34} \\\\\n && \\\\ \n S_{15} &  S_{25} &  S_{35} \\\\\n && \\\\\n S_{16} &  S_{26} &  S_{36} \n \\end{array} \\right] .\n   \\end{eqnarray}\nThus, the energy function ${\\cal E}$ is stationary if \n $\\big(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\, \\sigma_{\\rm III}\\big)$ is a right null vector of the $3\\times 3$ matrix $\\bf E$. \nBased on \\rf{s3} and \\rf{s1}, the condition \\rf{4r}  is equivalent to the requirement that the off-diagonal elements of the strain vanish: \n\\beq{e12}\n\\varepsilon_{23}=\\varepsilon_{31}=\\varepsilon_{12}=0\\, ,  \n\\end{equation}\nor, \n\\beqa{r6}\n{\\cal E} \\mbox{ stationary} \\quad \\Leftrightarrow \\quad \n\\widehat{\\bosy{\\varepsilon}}  = \\left[ \n\\ba{c}\n\\varepsilon_{1} \\\\  \\varepsilon_{2} \\\\ \\varepsilon_{3} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] = \\left[ \n\\ba{c}\n\\varepsilon_{\\rm I} \\\\  \\varepsilon_{\\rm II} \\\\ \\varepsilon_{\\rm III} \\\\ 0 \\\\ 0 \\\\ 0\n \\end{array} \\right] .\n\\end{eqnarray} \nwhere $\\varepsilon_{\\rm I}, \\varepsilon_{\\rm II}\\varepsilon_{\\rm III}$ are the  principal strains.  \nWe have therefore derived the simple but important general result: \n\n\\medskip \n\\noindent\n{\\bf Result 1}: \\textcolor{blue}{The energy  \n$ \\textcolor{blue} {\\cal E}$  is stationary iff the stress and strain are coaxial. }\n\\medskip \n\nEquation \\rf{e12} states that the 3-vector $ (\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\,\\sigma_{\\rm III})$ is a right null-vector of ${\\bf E}$. \nThis  requires as a {\\it necessary but not sufficient condition} that \n\\beq{v1}\n\\mbox{det} \\, {\\bf E} = 0.  \n\\end{equation}\nConsequences of this condition were explored in detail by Rovati and Taliercio \\cite{Rovati03} for particular material symmetries: cubic, transversely isotropic and tetragonal.  A different approach is taken in Section \\ref{sec4} below, where the strain energy will be minimized directly.  \n\nWhile Result 1 is not new but has been derived by several authors \n\\cite{Seregin81,Rovati91,Cowin94,Vianello96a,Banichuk96,Rovati03}, \nthe present derivation is novel and explicit.  In particular, it allows us to go further and obtain the condition necessary for a minimum or maximum.  This is  explored next. \n\n\\subsection{Condition for an energy  minimum}\n\nThe second derivative of $\\cal E$ at the stationary point follows from \\rf{s8qq} evaluated at $\\phi = 0$, \n\\beq{cm1}\n  \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E}(  {\\bf n}, \\phi)\\right|_{\\phi = 0} = \n2\\widehat{\\bosy{\\sigma}}^T\\, \\big[ \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \\hbS  + \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} \\hbS \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T  \\big] \\,  \\widehat{\\bosy{\\sigma}}\\, .  \n\\end{equation}\nEach term on the right hand side will be examined in turn.  Using \\rf{r1}, it follows that \n\\beqa{cm4}\n  \\widehat{\\bosy{\\sigma}}^T\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^2 \\hbS  \\,  \\widehat{\\bosy{\\sigma}}\\, = 2\\, \n\n   \\left[ \\begin{array}{c}\n \\sigma_{\\rm I} \\\\ \\\\  \\sigma_{\\rm II} \\\\ \\\\ \\sigma_{\\rm III}\n \\end{array} \\right]^T\\, \n \\left[ \n\\ba{cccccc}\n S_{11} &  S_{12} &  S_{13} & \n    S_{14} &    S_{15} &    S_{16} \n\\\\ & & & & & \\\\\n S_{12} &  S_{22} &  S_{23} & \n    S_{24} &    S_{25} &    S_{26} \n\\\\ & & & & & \\\\\n S_{13} &  S_{23} &  S_{33} & \n    S_{34} &    S_{35} &    S_{36} \n\\end{array} \\right]\\, \n\\left[ \n\\ba{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm I})n_2^2 + (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3^2\\\\\n\\\\\n(\\sigma_{\\rm I} - \\sigma_{\\rm II})n_3^2 + (\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1^2\\\\\n\\\\\n(\\sigma_{\\rm II} - \\sigma_{\\rm III})n_1^2 + (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2^2\\\\\n\\\\\n(\\sigma_{\\rm II} +\\sigma_{\\rm III} - 2\\sigma_{\\rm I})n_2n_3 \\\\\n\\\\\n(\\sigma_{\\rm III} +\\sigma_{\\rm I} - 2\\sigma_{\\rm II})n_3n_1 \\\\\n\\\\\n(\\sigma_{\\rm I} +\\sigma_{\\rm II} - 2\\sigma_{\\rm III})n_1n_2 \\\\\n\\end{array} \\right],\\quad \n\\end{eqnarray}\nand\n\\beqa{cm2}\n  \\widehat{\\bosy{\\sigma}}^T\\, \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q} \\hbS \\widehat{\\bf P}}  \t\\newcommand{\\hbQ}{\\widehat{\\bf Q}^T  \\,  \\widehat{\\bosy{\\sigma}}\\, =  \n   4\\,\n  \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n    \\left[ \\begin{array}{ccc}\n S_{44} &  S_{45} &  S_{46} \\\\\n && \\\\ \n S_{45} &  S_{55} &  S_{56} \\\\\n && \\\\\n S_{46} &  S_{56} &  S_{66} \n \\end{array} \\right]\\, \n \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]\\, . \n\\end{eqnarray}\nThus, \n\\beq{cm6}\n   \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} (  {\\bf n}, \\phi)\\right|_{\\phi = 0} =   4\\, {\\bf n}^T {\\bf F} {\\bf n} , \n\\end{equation}\nwhere the symmetric $3\\times 3$ matrix ${\\bf F}$ has elements  \n\\beqa{G}\nF_{11} &=& (\\sigma_{\\rm III} - \\sigma_{\\rm II})\\, \\big[ 2 S_{44} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) +\n\t(S_{12}-S_{13})\\sigma_{\\rm I} + (S_{22}-S_{23})\\sigma_{\\rm II} + (S_{32}-S_{33})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nF_{22} &=& (\\sigma_{\\rm I} - \\sigma_{\\rm III})\\, \\big[ 2 S_{55} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\n\t(S_{13}-S_{11})\\sigma_{\\rm I} + (S_{23}-S_{21})\\sigma_{\\rm II} + (S_{33}-S_{31})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nF_{33} &=& (\\sigma_{\\rm II} - \\sigma_{\\rm I})\\, \\big[ 2 S_{66} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) +\n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II} + (S_{31}-S_{32})\\sigma_{\\rm III} \\big], \n\\nonumber \t \\\\\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I}) + \\frac12 \\,\n(\\sigma_{\\rm II} +\\sigma_{\\rm III}- 2\\sigma_{\\rm I})\\, \\big(\n\tS_{14}\\sigma_{\\rm I} +S_{24}\\sigma_{\\rm II} +S_{34}\\sigma_{\\rm III}\\big) ,\n\\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  + \\frac12 \\,\n(\\sigma_{\\rm III} +\\sigma_{\\rm I}- 2\\sigma_{\\rm II})\\, \\big(\n\tS_{15}\\sigma_{\\rm I} +S_{25}\\sigma_{\\rm II} +S_{35}\\sigma_{\\rm III}\\big),\n\\nonumber \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\\frac12 \\,\n(\\sigma_{\\rm I} +\\sigma_{\\rm II}- 2\\sigma_{\\rm III})\\, \\big(\n\tS_{16}\\sigma_{\\rm I} +S_{26}\\sigma_{\\rm II} +S_{36}\\sigma_{\\rm III}\\big)\\, . \\nonumber\n\\end{eqnarray}\n \nThe second derivative \\rf{cm6} must be positive for all directions ${\\bf n}$ at an orientation where $\\cal E$ is a local minimum.  Noting that \n\\beqa{G4}\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I}) + \\frac12 \\,\n(\\sigma_{\\rm II} +\\sigma_{\\rm III}- 2\\sigma_{\\rm I})\\, \\varepsilon_{23} ,\n\\nonumber \\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  +\\frac12 \\,\n(\\sigma_{\\rm III} +\\sigma_{\\rm I}- 2\\sigma_{\\rm II})\\, \\varepsilon_{31},\n \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) +\\frac12 \\,\n(\\sigma_{\\rm I} +\\sigma_{\\rm II}- 2\\sigma_{\\rm III})\\, \\varepsilon_{12} , \\nonumber\n\\end{eqnarray}\nit follows that at a stationary point the off-diagonal elements of $\\bf F$ become, \n\\beqa{G5}\nF_{23} &=& 2 S_{56} (\\sigma_{\\rm I} - \\sigma_{\\rm III}) (\\sigma_{\\rm II} - \\sigma_{\\rm I})  ,\n\\nonumber \\\\\nF_{31} &=& 2 S_{46} (\\sigma_{\\rm II} - \\sigma_{\\rm I}) (\\sigma_{\\rm III} - \\sigma_{\\rm II})  ,\n \\\\\nF_{12} &=& 2 S_{45} (\\sigma_{\\rm III} - \\sigma_{\\rm II}) (\\sigma_{\\rm I} - \\sigma_{\\rm III}) . \\nonumber\n\\end{eqnarray}\n Equivalently, by pre- and post-multiplication of $\\frac 12 \\bf F$ by the diagonal matrix\n diag$[(\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1},\\, (\\sigma_{\\rm I} - \\sigma_{\\rm III})^{-1},\\, (\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}]$, it follows that $\\bf G$ must be  positive definite, where \n \\beqa{G6}\nG_{11} &=&  S_{44}+ \\frac12 (\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1}\\, \\big[  \n\t(S_{12}-S_{13})\\sigma_{\\rm I} + (S_{22}-S_{23})\\sigma_{\\rm II} + (S_{32}-S_{33})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nG_{22} &=&  S_{55} + \\frac12 (\\sigma_{\\rm I} - \\sigma_{\\rm III})^{-1}\\, \\big[   \n\t(S_{13}-S_{11})\\sigma_{\\rm I} + (S_{23}-S_{21})\\sigma_{\\rm II} + (S_{33}-S_{31})\\sigma_{\\rm III} \\big], \n\t\\nonumber \\\\\nG_{33} &=&  S_{66}+ \\frac12 (\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[   \n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II} + (S_{31}-S_{32})\\sigma_{\\rm III} \\big], \n\\nonumber \t \\\\\nG_{23} &=&  S_{56}  \\,  ,\n\\\\\nG_{31} &=&  S_{46} \\,  ,\n\\nonumber \\\\\nG_{12} &=&  S_{45} \\, . \\nonumber\n\\end{eqnarray}\nNote that $G_{11} =  S_{44}- \\frac12 (\\sigma_{\\rm III} - \\sigma_{\\rm II})^{-1}\\, (\\varepsilon_3 - \\varepsilon_2)$, etc. or, using \\rf{r6},  \n\\beqa{G7}\n {\\bf G} = \\left[ \\begin{array}{ccc}\n   S_{44}- \\frac12 \\big( \\frac{\\varepsilon_{\\rm III} - \\varepsilon_{\\rm II} }{\\sigma_{\\rm III} - \\sigma_{\\rm II}}\\big)  &   S_{45} &   S_{46} \\\\\n && \\\\ \n   S_{45} &  S_{55} -\\frac12 \\big( \\frac{\\varepsilon_{\\rm I} - \\varepsilon_{\\rm III} }{\\sigma_{\\rm I} - \\sigma_{\\rm III}}\\big) &   S_{56}  \\\\\n && \\\\ \n   S_{46}&    S_{56} &   S_{66} - \\frac12 \\big(\\frac{\\varepsilon_{\\rm II} - \\varepsilon_{\\rm I} }{\\sigma_{\\rm II} - \\sigma_{\\rm I}} \\big)\n  \\end{array} \\right]\\, . \n  \\end{eqnarray}\nIn summary, \n\\beqa{cmsum}\n \\left. \\frac{\\partial^2 \\, }{\\partial \\phi^2 }{\\cal E} (  {\\bf n}, \\phi)\\right|_{\\phi = 0}\n =  \n   8\\,\n  \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]^T\\, \n    {\\bf G}\\, \n \\left[ \\begin{array}{c}\n(\\sigma_{\\rm III} - \\sigma_{\\rm II})n_1 \\\\ \\\\  (\\sigma_{\\rm I} - \\sigma_{\\rm III})n_2 \\\\ \\\\ (\\sigma_{\\rm II} - \\sigma_{\\rm I})n_3\n \\end{array} \\right]\\, . \n\\end{eqnarray}\nThis must hold for arbitrary $\\bf n$, $|{\\bf n}|=1$, and  again assuming that the principal stresses are distinct, it follows that  $\\bf G$ must be positive definite.  Combined with Result 1 for the existence of a stationary point, this gives  \n\n\\medskip \n\\noindent\n{\\bf Result 2}:  \\textcolor{blue}{The energy\n${\\cal E}$  is a local minimum if the stress and strain are coaxial and the symmetric matrix ${\\bf G}$ of \\rf{G7} is positive definite.  }  \n \n This can be rewritten (with obvious notation)\n\\beqa{G7ex}\n  \\left[ \\begin{array}{ccc}\n  S_{44} &  S_{45} & S_{46} \\\\\n && \\\\ \nS_{45} &  S_{55}  &  S_{56}  \\\\\n && \\\\ \n  S_{46}&    S_{56} &   S_{66} \n  \\end{array} \\right]\\, > \\, \n  \\left[ \\begin{array}{ccc}\n   \\frac12 \\big( \\frac{\\varepsilon_{\\rm III} - \\varepsilon_{\\rm II} }{\\sigma_{\\rm III} - \\sigma_{\\rm II}}\\big)  &   0 &   0 \\\\\n && \\\\ \n   0 &  \\frac12 \\big( \\frac{\\varepsilon_{\\rm I} - \\varepsilon_{\\rm III} }{\\sigma_{\\rm I} - \\sigma_{\\rm III}}\\big) &   0  \\\\\n && \\\\ \n   0&    0 &   \\frac12 \\big(\\frac{\\varepsilon_{\\rm II} - \\varepsilon_{\\rm I} }{\\sigma_{\\rm II} - \\sigma_{\\rm I}} \\big)\n  \\end{array} \\right]  \\, . \n  \\end{eqnarray}\n The left matrix is positive definite because of the positive definite properties of the moduli.  The minimum condition therefore requires that this latter matrix be greater than\\footnote{The matrix $\\bf U$ is greater than the matrix $\\bf V$ if \n${\\bf r}^T {\\bf U} {\\bf r} > {\\bf r}^T {\\bf V} {\\bf r}$ for all nonzero  ${\\bf r} \\in R^3$.}  the right hand diagonal matrix defined by the principal stresses and strains.  The requirement that the full matrix  is positive definite can be relaxed if the stationarity is restricted in  orientation axis $\\bf n$.  Thus, only the single scalar quantity ${\\bf n}^T{\\bf G}{\\bf n}$ needs to be considered in the important special case of rotation about a single axis.  This situation is examined in detail in the Appendix. \n\n\n\n\n\\section{Optimal orientation of particular material symmetries}\\label{sec4}\n\n\n\\subsection{Partition of the energy}\nBefore considering specific material symmetries, several general results are presented which help focus attention on the anisotropic part of the strain energy.  Separate  contributions to the energy function ${\\cal E}$ of \\rf{s4} from  isotropic and anisotropic parts of the elastic moduli may be \ndistinguished as follows, \n\\beqa{fa}\n{\\cal E}&=& {\\cal E}^{\\rm (is)} + {\\cal E}^{\\rm (an)} \n\\nonumber \\\\\n &=&  \\sigma_{ij} \\sigma_{kl} \\, s^{\\rm (is)}_{ijkl} +\\sigma_{ij} \\sigma_{kl} \\, s^{\\rm (an)}_{ijkl}.  \n\\end{eqnarray}\nThe moduli are partitioned into isotropic and anisotropic parts \n\\beq{wr}\ns_{ijkl} =  s^{\\rm (is)}_{ijkl} + s^{\\rm (an)}_{ijkl},\n\\qquad\nc_{ijkl} =  c^{\\rm (is)}_{ijkl} + c^{\\rm (an)}_{ijkl},  \n\\end{equation}\nwith the  isotropic moduli defined by \n\\beq{wr2}\ns^{\\rm (is)}_{ijkl} = \\frac{1}{3\\kappa_s} \\, J_{ijkl} + \\frac{1}{2\\mu_s} \\, K_{ijkl},\n\\qquad\nc^{\\rm (is)}_{ijkl} = 3\\kappa_c \\, J_{ijkl} + 2\\mu_c \\, K_{ijkl}.  \n\\end{equation}\nHere,   \n\\beq{wr3}\n J_{ijkl}  = \\frac{1}{3}\\, \\delta_{ij} \\delta_{kl}, \n\\qquad\nK_{ijkl}= I_{ijkl}- J_{ijkl} \\, , \n\\end{equation}\nand $I_{ijkl}$ are the elements of the fourth order identity. \nThe effective ``bulk\" and ``shear\" moduli $\\kappa_s,\\, \\mu_s$ and $\\kappa_c,\\, \\mu_c$  are obtained as \n\\beqa{km}\n && \\frac{1}{\\kappa_s} =  3 s_{ijkl}J_{ijkl} = \nS_{11}+S_{22}+S_{33}+2S_{12}+2S_{13}+2S_{23}, \n\\\\\n&& \\frac{15}{4 \\mu_s} =  \\frac{3}{2}s_{ijkl}K_{ijkl} = S_{11}+S_{22}+S_{33} - S_{12}-S_{23}-S_{31} +3S_{44}+3S_{55}+3S_{66} ,\n\\qquad \n\\\\\n&& 9 \\kappa_c =  3 c_{ijkl}J_{ijkl} = \nC_{11}+C_{22}+C_{33}+2C_{12}+2C_{13}+2C_{23}, \n\\\\\n&& 15 \\mu_c =  \\frac{3}{2}c_{ijkl}K_{ijkl} = C_{11}+C_{22}+C_{33} - C_{12}-C_{23}-C_{31} +3C_{44}+3C_{55}+3C_{66} .\n\\qquad \n\\end{eqnarray}\nNote that in general $\\kappa_c \\ne \\kappa_s$ and $\\mu_c\\ne \\mu_s$.  The anisotropic parts of the moduli in \\rf{wr} are defined as the remainder after subtracting the isotropic parts, \n$s^{\\rm (an)}_{ijkl} = s_{ijkl} - s^{\\rm (is)}_{ijkl}$, etc. \n\nThe energy associated with the isotropic part of the moduli becomes \n\\beq{fis}\n{\\cal E}^{\\rm (is)} = \\frac{1}{\\kappa_s}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_s} \\sigma_{ij}'\\sigma_{ij}',\n\\end{equation}\nwhere $\\overline{\\sigma}$ and ${\\bosy{\\sigma}}'$ are the hydrostatic and deviatoric stress, respectively, \n\\beq{fis2}\n\\overline{\\sigma}=  \\frac13 \\sigma_{kk}, \\qquad \\sigma_{ij}' = \\sigma_{ij} - \\overline{\\sigma}\\delta_{ij}.\n\\end{equation}\nThese may be written explicitly in terms of the principal stresses, from \\rf{s1}, as  \n\\beqa{sdev}\n\\overline{\\sigma} &=&  \\frac13\\, \\big( \\sigma_{\\rm I}+\\sigma_{\\rm II}+\\sigma_{\\rm III}\\big), \\\\\n \\bosy{\\sigma}'    &=& \\frac13\\, \n\\left[ \\begin{array}{ccc}\n2 \\sigma_{\\rm I}- \\sigma_{\\rm II}- \\sigma_{\\rm III} & 0 & 0 \\\\ && \\\\\n0 & 2 \\sigma_{\\rm II}- \\sigma_{\\rm III}- \\sigma_{\\rm I} &  0 \\\\ && \\\\\n0 & 0 &  2 \\sigma_{\\rm III}- \\sigma_{\\rm I}- \\sigma_{\\rm II}  \n\\end{array} \\right] \n\\equiv \\left[ \\begin{array}{ccc}\n\\sigma_{\\rm I} ' & 0 & 0 \\\\ && \\\\\n0 &  \\sigma_{\\rm II}' &  0 \\\\ && \\\\\n0 & 0 &   \\sigma_{\\rm III}'  \n\\end{array} \\right] .\\qquad \n\\end{eqnarray} \nThe energy associated with  the anisotropic part of the moduli is\n\\beq{fan}\n{\\cal E}^{\\rm (an)} = \\overline{\\sigma}^2 s^{\\rm (an)}_{jjkk} + 2\\overline{\\sigma} \\sigma_{ij}' \\, s^{\\rm (an)}_{ijkk} + \n\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\nBy definition, the scalar quantity $s^{\\rm (an)}_{jjkk}$ is zero, and accordingly, the anisotropic energy simplifies to \n\\beq{fq}\n{\\cal E}^{\\rm (an)} =2\\overline{\\sigma} \\sigma_{ij}' \\, s^{\\rm (an)}_{ijkk} + \n\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\n\nIt may be shown that the $3\\times 3$ matrices $\\bf E$ and $\\bf G$ of \\rf{r4} and \\rf{G7} vanish  for isotropic materials.  In general, they depend upon the anisotropic part of the material moduli. \n\n\n\n\\subsection{Materials with cubic symmetry}  \n\nIn the fixed coordinate system of the principal stress axes, the  elastic compliance for  a material with cubic symmetry is \n\\beqa{cub1}\n\\cS^{(0)}  = \\left[ \n\\ba{cccccc}\nS_{11}^{(0)} & S_{12}^{(0)} & S_{12}^{(0)} & \n 0&0&0 \n\\\\ & & & & & \\\\\n & S_{11}^{(0)} & S_{12}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n & & S_{11}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n &   &   \n & S_{44}^{(0)} & 0 & 0\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{44}^{(0)}&  0 \n\\\\ & & & & & \\\\\n&&&&& S_{44}^{(0)}\n\\end{array} \\right] . \n\\end{eqnarray}\nThere are three independent  moduli, $\\kappa$, $\\mu_1$ and $\\mu_2$, where\n\\beq{cub2}\n\\frac{1}{3\\kappa} =   S_{11}^{(0)}+2  S_{12}^{(0)},\\quad \n\\frac{1}{2\\mu_1} = 2S_{44}^{(0)},\\quad\n\\frac{1}{2\\mu_2} =   S_{11}^{(0)}-  S_{12}^{(0)}\\, . \n\\end{equation} \nThe associated fourth order tensors  can be expressed succinctly using the irreducible tensor  notation of Walpole \\cite{c1}, as\n\\beq{wr5}\nc^{(0)}_{ijkl} = 3\\kappa \\, J_{ijkl} + 2\\mu_1 \\, L^{(0)}_{ijkl}+ 2\\mu_2 \\, M^{(0)}_{ijkl}, \n\\qquad\ns^{(0)}_{ijkl} = \\frac{1}{3\\kappa} \\, J_{ijkl} + \\frac{1}{2\\mu_1} \\, L^{(0)}_{ijkl}+ \\frac{1}{2\\mu_2} \\, M^{(0)}_{ijkl}. \n\\end{equation}\nHere $L^{(0)}_{ijkl}= I_{ijkl}- D^{(0)}_{ijkl}$,  $M^{(0)}_{ijkl}= D^{(0)}_{ijkl}- J_{ijkl}$, and \n\\beq{Ddef}\nD^{(0)}_{ijkl} = \\delta_{i1} \\delta_{j1}\\delta_{k1} \\delta_{l1}\n+\\delta_{i2} \\delta_{j2}\\delta_{k2} \\delta_{l2} +\\delta_{i3} \\delta_{j3}\\delta_{k3} \\delta_{l3}\\, . \n\\end{equation}\nThis format makes it relatively straightforward to determine the effective isotropic moduli, \n\\beq{wr6}\n\\kappa_c = \\kappa_s = \\kappa, \\qquad 5\\mu_c = 3\\mu_1+2\\mu_2, \\qquad\n\\frac{5}{\\mu_s} = \\frac{3}{\\mu_1} + \\frac{2}{\\mu_2}\\, . \n \\end{equation}\nThus, \n\\beq{w8}\ns^{(an,0)}_{ijkl} = \\frac{1}{10}\\bigg(\\frac{1}{\\mu_1}- \\frac{1}{\\mu_2}\\bigg)\\, \\big(\n5J_{ijkl} + 2K_{ijkl} - 5D^{(0)}_{ijkl} \\big). \n\\end{equation}\n\nThe anisotropic  part of the energy, \\rf{fq}, depends only upon the deviatoric part of the stress,  \n\\beq{fq2}\n{\\cal E}^{\\rm (an)} =\\sigma_{ij}' \\sigma_{kl}' \\, s^{\\rm (an)}_{ijkl} .\n\\end{equation}\nThe reason is that the second order tensor $s^{(an,0)}_{ijkk}$ is identically zero for cubic symmetry, and hence  remains zero in the rotated material axes: $s^{\\rm (an)}_{ijkk} = 0$.  The first term in \\rf{fq} therefore vanishes, leaving the simpler expression \\rf{fq2}.  The isotropic tensors $J_{ijkl}$ and $K_{ijkl}$ are  unchanged under rotation, and consequently, from \\rf{fis}, \\rf{w8} and \\rf{fq2}, \n\\beq{fq3}\n{\\cal E} = \\frac{1}{\\kappa}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_1} \\sigma_{ij}'\\sigma_{ij}' + {\\cal E}^{\\rm (ex)},\n\\qquad\n{\\cal E}^{\\rm (ex)} = \\frac{1}{2}\\bigg(\\frac{1}{\\mu_2}- \\frac{1}{\\mu_1}\\bigg)\\,\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} ,  \n\\end{equation}\nwhere $D_{ijkl}$ is the rotated version of $D^{(0)}_{ijkl}$.  In order to avoid ambiguity, let $\\sigma_{(rot)kl}$ denote the stress in the {\\it rotated} coordinate system, then it follows that\n\\beq{fq4}\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} = \\sigma_{(rot)11}^{'2} + \\sigma_{(rot)22}^{'2} +\\sigma_{(rot)33}^{'2}  \\ . \n\\end{equation}\nThe scalar second invariant of the deviatoric stress is \n\\beqa{in}\n\\sigma_{ij}'\\sigma_{ij}' &=& {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2\n\\nonumber \\\\\n&=& \\sigma_{(rot)11}^{'2} + \\sigma_{(rot)22}^{'2} +\\sigma_{(rot)33}^{'2}\n+ 2\\sigma_{(rot)23}^{'2} + 2\\sigma_{(rot)31}^{'2} +2\\sigma_{(rot)12}^{'2} .  \n\\end{eqnarray}\nTherefore, the function ${\\cal E}^{\\rm (ex)}$ of \\rf{fq3} is stationary when either the right member of \\rf{fq4} or \n\\beq{fq6}\n\\Gamma = \\sigma_{(rot)23}^{'2} + \\sigma_{(rot)31}^{'2} +\\sigma_{(rot)12}^{'2},\n\\end{equation}\nare stationary.  Furthermore, \n\\beq{fq7}\n{\\cal E}^{\\rm (ex)} ={\\cal E}^{\\rm (ex,0)}  +  \\bigg(\\frac{1}{\\mu_1}- \\frac{1}{\\mu_2}\\bigg)\\,\n\\big[ \\sigma_{(rot)23}^{'2} + \\sigma_{(rot)31}^{'2} +\\sigma_{(rot)12}^{'2}\\big] \\, , \n\\end{equation}\nwhere ${\\cal E}^{\\rm (ex,0)}$ is the unrotated or fixed value, which follows from \\rf{sdev} as\n\\beq{q9}\n{\\cal E}^{\\rm (ex,0)} = \\frac{1}{2}\\bigg(\\frac{1}{\\mu_2}- \\frac{1}{\\mu_1}\\bigg)\\,  \n\\big( {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2 \\big)\\, . \n\\end{equation}\n Hence, \n\\beq{wq3}\n \\mu_1 > \\mu_2 \\quad \\Rightarrow \\quad {\\cal E}^{\\rm (ex)}\\le  {\\cal E}^{\\rm (ex,0)}, \n \\end{equation}\nwith equality when the material and stress axes are aligned.  Thus, \\textcolor{blue}{a local minimum that is not aligned with the stress axes occurs iff $\\mu_1 > \\mu_2$ and occurs when $\\Gamma$ of \\rf{fq6} achieves a local maximum}.   It will be shown below that the maximum value is $\\frac12 \\sigma_{ij}'\\sigma_{ij}'$, or equivalently, that ${\\cal E}^{\\rm (ex)}$ is zero at the stationary point. \n\n\nAs  the material axes are rotated to transform $s^{(an,0)}_{ijkl} \\rightarrow s^{\\rm (an)}_{ijkl}$, the only part that contributes to the anisotropic strain energy  is $D_{ijkl}^{(0)} \\rightarrow D_{ijkl}$.  \nConditions for obtaining the stationary value of strain energy are next derived by focusing on the dependence upon $D_{ijkl}$.  The $6\\times 6$   matrix associated with the unrotated tensor $D^{(0)}_{ijkl}$ is\n\\beqa{D1}\n\\widehat{\\bf D}^{(0)}    = \n\\left[ \\begin{array}{ccc}\nI_{3\\times 3} & &0_{3\\times 3}  \\\\ && \\\\ \n0_{3\\times 3} && 0_{3\\times 3} \n\\end{array} \\right] .\n\\end{eqnarray} \nIt is convenient to split $ \\hbQ $ as follows into $3\\times 3$ matrices, \n\\beqa{D2}\n \\hbQ   = \n\\left[ \\begin{array}{ccc}\n\\hbQ_1 & &\\hbQ_2  \\\\ && \\\\ \n\\hbQ_3 && \\hbQ_4 \n\\end{array} \\right] ,   \n\\end{eqnarray} \nso that the rotated tensor    \n$\\widehat{{\\bf D}}  = \\hbQ \\widehat{{\\bf D}}^{(0)} \\hbQ^T$,  follows from \\rf{D1} and \\rf{D2} as \n\\beqa{D2a}\n\\widehat{{\\bf D}}  = \n\\left[ \\begin{array}{ccc}\n\\hbQ_1 \\hbQ_1^T & &\\hbQ_1 \\hbQ_3^T  \\\\ && \\\\ \n\\hbQ_3 \\hbQ_1^T && \\hbQ_3 \\hbQ_3^T \n\\end{array} \\right] .  \n\\end{eqnarray} \nThe term associated with the rotated energy  follows from \\rf{s1a} and \\rf{D2a} as  \n\\beq{dq2}\n\\sigma_{ij}' \\sigma_{kl}' \\, D_{ijkl} =   (\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') \\, \\hbQ_1\\hbQ_1^T \\, (\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}')^T\\, .   \n\\end{equation}\n  Thus,  any  stress that is a null vector of $\\hbQ_1^T$\nalso  yields the minimum or maximum value for ${\\cal E}^{\\rm (ex)}$ of \\rf{fq3}, i.e. zero.  \nThis suggests $\\hbQ_1$ as the focus of attention, and implies    \n the important result: \\textcolor{blue}{Every stress state which is a null vector of $\\hbQ_1^T$ corresponds to a global minimum (maximum) of ${\\cal E}$ if $\\mu_1 >\\mu_2$ ($\\mu_2 >\\mu_1$)}.  We therefore search for null vectors of $\\hbQ_1^T$. \n\nBefore deriving two alternative methods to find null vectors of $\\hbQ_1^T$ in the next two subsections, note that the quantity \\rf{fq4} vanishes at a stationary orientation, and hence $\\sigma_{(rot)11} = \\sigma_{(rot)22}  = \\sigma_{(rot)22}$.  Thus, the  stress in each of the three rotated axial directions is equal, a result previously obtained by Rovati and Taliercio \\cite{Rovati91,Rovati03}.   Furthermore, at the stationary point it may be easily shown that the following identities hold:\n\\beq{idh}\n{\\cal D}\\bosy{\\sigma} = \\overline{\\sigma}\\, {\\bf I} , \\quad\n{\\cal L}\\bosy{\\sigma} = \\bosy{\\sigma}', \\quad \n{\\cal M}\\bosy{\\sigma} = 0\\, ,\n\\end{equation}\nwhere ${\\cal D}, {\\cal L}, {\\cal M}$ are the (rotated) tensors with components $D_{ijkl}, L_{ijkl}, M_{ijkl}$, respectively.  Hence, the strain at optimal orientation is simply\n\\beq{idh2}\n\\bosy{\\varepsilon} = \\frac{1 }{3\\kappa} \\, \\overline{\\sigma} \\, {\\bf I}   \\quad\n+ \\frac{ 1 }{2\\mu_1}\\, \\bosy{\\sigma}'\\, \\qquad (\\mbox{optimal orientation only}).\n\\end{equation}\nThis is clearly coaxial with the stress, which follows from the commutation property of coaxial second order tensors \n($\\bosy{\\sigma} \\bosy{\\varepsilon} - \\bosy{\\varepsilon}  \\bosy{\\sigma}=0$ in this case). \n\nIt is also worth remarking that we do not seek null vectors of  the matrix  $\\bf E$, although this approach is feasible and has been used to  advantage by Rovati and Taliercio \\cite{Rovati03}. Some comments on $\\bf E$ are in order though.  The $3\\times 3$ matrix follows from  eqs. \\rf{D1} through \\rf{D2a} as \n\\beq{D3}\n{\\bf E} = \\hbQ_3\\hbQ_1^T\\, ,  \n\\end{equation}\nand the condition \\rf{v1}  is satisfied if either  $\\mbox{det} \\, \\hbQ_1$ or $\\mbox{det} \\, \\hbQ_3$ vanish.  These can be made more explicit in terms of the elements of the rotation matrix.  \nLet \n\\beqa{qq}\n{\\bf Q} = \n\\left[ \\begin{array}{ccc}\nq_{11} & q_{12} & q_{13} \\\\\nq_{21} & q_{22} & q_{23} \\\\\nq_{31} & q_{32} & q_{33} \n\\end{array} \\right] ,  \n\\end{eqnarray}\nthen using the   the explicit representation of the $6\\times 6$ rotation matrix from Auld  \\cite{AuldI} or otherwise, the condition  \\rf{v1} implies \n\\beqa{D5}\n\\mbox{det} \\, \\hbQ_1 = \\left| \\begin{array}{cccccc}\nq_{11}^2 & q_{12}^2 & q_{13}^2    \\\\\n&& \\\\\nq_{21}^2 & q_{22}^2 & q_{23}^2    \\\\\n&& \\\\\nq_{31}^2 & q_{32}^2 & q_{33}^2  \n\\end{array} \\right| = 0,\n\\quad \n\\mbox{or}\\quad    \\mbox{det} \\, \\hbQ_3 = (2)^{3\/2}\\, \n\\left| \\begin{array}{cccccc}\n q_{21}q_{31} & q_{22}q_{32}& q_{23}q_{33} \\\\\n&& \\\\\n q_{31}q_{11} & q_{32}q_{12}& q_{33}q_{13} \\\\\n&& \\\\\n q_{11}q_{21} & q_{12}q_{22}& q_{13}q_{23}\n\\end{array} \\right| = 0.\\quad\n\\end{eqnarray}\nUsing the fact that the column vectors of a transformation matrix form an orthonormal triad, it follows that \n\\beq{D7a}\n \\hbQ_1^T\\, (1,\\, 1,\\, 1)^T =  (1,\\, 1,\\, 1)^T, \\quad  \\hbQ_3\\,(1,\\, 1,\\, 1)^T =  (0,\\, 0,\\, 0)^T.   \n\\end{equation}\nThat is, $(\\sigma_{\\rm I},\\, \\sigma_{\\rm II},\\,\\sigma_{\\rm III}) = \\lambda\\, (1,\\, 1,\\, 1)$ is a   null vector of $\\bf E$ for any $\\lambda$.  Hence, $\\bf E$ is not of full rank, implying that det$\\, {\\bf E}$ is always zero.  However, this   is not of interest as the null space corresponds to hydrostatic stress, for which the energy function is independent of material orientation.  The implications of \\rf{D5} are not considered further, and we return to the simpler task of finding null vectors of $\\hbQ_1^T$ alone. \n\n\nTwo  methods for achieving the minimum energy ${\\cal E}^{\\rm (ex)} = 0$ are  described, both using explicit forms of the rotation.  The first involves a single rotation about an arbitrary axis, and the second is in terms of standard Euler angles.   \n\n\\subsubsection{Minimum energy state with a single rotation}\n\nThe range of transformations which correspond to energy minima can be obtained using Euler's Theorem \\cite{Baruh} which states that any transformation matrix ${\\bf Q}$ can be represented in the form \\rf{qeq} for some axis ${\\bf n}$, $|{\\bf n}|=1$,  and angle $\\phi$.  Thus, \n\\beqa{qeq1}\n{\\bf Q}({\\bf n},\\, \\phi) = \n\\left[ \\begin{array}{ccc}\n \t\t1-2s^2(n_2^2+n_3^2) & 2s(sn_1n_2-cn_3) & 2s(sn_1n_3+cn_2)\n  \\\\ && \\\\\n \t 2s(sn_1n_2+cn_3)\t& ~ 1-2s^2(n_3^2+n_1^2) ~&  2s(sn_2n_3 -cn_1)\n \\\\ && \\\\\n \t 2s(sn_1n_3 -cn_2)   &  2s(sn_2n_3+cn_1) & 1-2s^2(n_1^2+n_2^2)  \n\\end{array} \\right] ,\\quad\n\\end{eqnarray}\nwhere $c=\\cos \\frac{\\phi}{2}$, $s=\\sin \\frac{\\phi}{2}$.\nand the elements of the associated $\\hbQ_1$ are determined by squaring each element in \\rf{qeq1}. It may be shown that \n\\beq{qeq2}\n\\mbox{det} \\, \\hbQ_1\\ ({\\bf n},\\, \\phi) = \\cos 2\\phi + 2(2+ 3 \\cos\\phi) (1-\\cos\\phi)^2 \\,  (\nn_1^2n_2^2 + n_2^2n_3^2 + n_3^2n_1^2 )+ 6(1-\\cos\\phi)^3 \\,  \nn_1^2n_2^2 n_3^2 \\, .  \\quad   \n\\end{equation}\nNote that $n_1^2n_2^2 + n_2^2n_3^2 + n_3^2n_1^2 \\le 1\/3$ and $n_1^2n_2^2 n_3^2 \\le 1\/27$ with equality when $n_1^2= n_2^2 =n_3^2 = 1\/3$.  \n\nFor a given $n_3^2$ and angle $\\phi$, \n\\beq{n1n2}\nn_1^2, n_2^2 =  \\frac{1}{2} (1-n_3^2) \\pm \\big[ \\frac{1}{4} (1-n_3^2)^2- g\\big]^{1\/2}, \n\\end{equation}\nwhere \n\\beq{geq}\ng(n_3^2, \\phi) = \n\\frac{-\\cos 2\\phi - 2(2+ 3 \\cos\\phi) (1-\\cos\\phi)^2 \\,  n_3^2(1-n_3^2 )}{ \n2 (1-\\cos\\phi)^2 [ (2+ 3 n_3^2 +3\\cos\\phi(1 - n_3^2)]}\\, .    \n\\end{equation}\n\n\nThe null vector of $\\hbQ_1$ is such that \n\\beqa{nv1}\n\\sigma_{\\rm I} ' Q_{11}^2 + \\sigma_{\\rm II} ' Q_{12}^2 + \\sigma_{\\rm III} ' Q_{13}^2 &=&0, \n\\nonumber \\\\\n\\sigma_{\\rm I} ' Q_{21}^2 + \\sigma_{\\rm II} ' Q_{22}^2 + \\sigma_{\\rm III} ' Q_{23}^2 &=&0,\n\\\\\n\\sigma_{\\rm I} ' Q_{31}^2 + \\sigma_{\\rm II} ' Q_{32}^2 + \\sigma_{\\rm III} ' Q_{33}^2 &=&0.\n\\nonumber\n\\end{eqnarray}\nUsing the fact that this is a deviatoric stress, we replace  $ \\sigma_{\\rm III} ' = -  \\sigma_{\\rm I} '- \\sigma_{\\rm II} '$ in the final equation of \\rf{nv1}, to get\n\\beq{nv2}\n\\sigma_{\\rm I} ' \\bigg(Q_{31}^2 - Q_{33}^2\\bigg) + \\sigma_{\\rm II} ' \\bigg(Q_{32}^2 - Q_{33}^2\\bigg) = 0.\n\\end{equation}\nHence, \n\\beq{nv3}\n\\sigma_{\\rm I} '  = a_0 \\bigg(Q_{32}^2 - Q_{33}^2\\bigg) ,\\quad  \\sigma_{\\rm II} ' = a_0\\bigg(Q_{33}^2 - Q_{31}^2\\bigg) ,\n\\end{equation}\nfor arbitrary  $a_0 \\ne 0$. Once again using the fact that $ \\sigma_{\\rm III} ' = -  \\sigma_{\\rm I} '- \\sigma_{\\rm II} '$ gives\n\\beq{nv3a}\n\\sigma_{\\rm III} '  = a_0 \\bigg(Q_{31}^2 - Q_{32}^2\\bigg) . \n\\end{equation}\nIn the same way, using the other equations in \\rf{nv1}, three alternative expressions for the null vector are found:\n\\beqa{3alt}\n\\big(\\sigma_{\\rm I} ',\\, \\sigma_{\\rm II} ',\\,  \\sigma_{\\rm III} '\\big)  \n&=& a_1 \\bigg(Q_{32}^2 - Q_{32}^2,\\,  Q_{33}^2 - Q_{31}^2,\\, Q_{31}^2 - Q_{32}^2 \\bigg) \\\\\n&=& a_2 \\bigg(Q_{12}^2 - Q_{12}^2,\\,  Q_{13}^2 - Q_{11}^2,\\, Q_{11}^2 - Q_{12}^2 \\bigg) \\\\\n&=& a_3 \\bigg(Q_{22}^2 - Q_{22}^2,\\,  Q_{23}^2 - Q_{21}^2,\\, Q_{21}^2 - Q_{22}^2 \\bigg) ,  \n\\end{eqnarray}\nfor some constants $a_1$, $a_2$, $a_3$.  Thus, from the first expression, with $a_1=1$, \n\\beqa{s3a}\n\\sigma_{\\rm I} '  &=&  \\big[   (1+n_1^2)(n_2^2-n_3^2)(1-\\cos \\phi) +n_3^2-n_2^2 - 4 n_1n_2n_3\\sin\\phi\\big](1-\\cos \\phi) , \\\\\n\\sigma_{\\rm II} '  &=&   \\big\\{   [(1-n_1^2)(n_1^2-n_3^2) -2n_2^2](1-\\cos \\phi) +n_2^2-n_1^2 +1 +2 n_1n_2n_3\\sin\\phi\\big\\}(1-\\cos \\phi) - 1, \\\\\n\\sigma_{\\rm III} '  &=&   \\big\\{   [(1-n_1^2)(n_2^2-n_1^2) +2n_3^2](1-\\cos \\phi) +n_1^2-n_3^2 -1 +2 n_1n_2n_3\\sin\\phi \\big\\}(1-\\cos \\phi) + 1, \\qquad \\label{s3c}\n\\end{eqnarray}\nThese equations provide us with a two parameter set of stress states, described by $0< n_3^2< 1 $ and $\\phi$.  The two are independent insofar as  $n_1^2$ and $n_2^2$ of \\rf{n1n2} lie in $(0,1)$.  This in turn requires that $g$ of \\rf{geq} satisfies \n\\beq{n1n2q}\n0 \\le g(n_3^2, \\phi) \\le  \\frac{1}{4} (1-n_3^2)^2, \n\\end{equation}\nwhich defines the range of $0< n_3^2< 1 $ and $\\phi$. \n\n\\subsubsection{Minimum energy using Euler angles}\n\nThe standard three Euler angles  $(\\theta_1 ,\\, \\theta_2 ,\\, \\theta_3)$ are used to transform from \n$\t\\{ {\\bf e}_1, {\\bf e}_2, {\\bf e}_3\\} \n\\rightarrow \n\t\\{ {\\bf e}_1', {\\bf e}_2', {\\bf e}_3' = {\\bf e}_3\\}\n\\rightarrow \n\t\\{ {\\bf e}_1''={\\bf e}_1', {\\bf e}_2'', {\\bf e}_3'' \\}\n\\rightarrow \n\t\\{ {\\bf e}_1''', {\\bf e}_2''', {\\bf e}_3'''={\\bf e}_3'' \\}\n$.  That is,  first rotate about the ${\\bf e}_3$ axis by $\\theta_1$, then about the ${\\bf e}_1'$ axis by $\\theta_2$, and finally about the ${\\bf e}_3''$ axis by $\\theta_3$.  The transformation matrix is \n\\beqa{eul}\n&&{\\bf Q} (\\theta_1,\\theta_2,\\theta_3) \\nonumber \\\\ && \\nonumber \\\\ && =  \n\\left[ \\begin{array}{ccc}\n \t\t\\cos \\theta_1 \\cos \\theta_3 - \\sin \\theta_1 \\cos \\theta_2 \\sin \\theta_3 \n & \t\\sin \\theta_1\\cos \\theta_3 + \\cos \\theta_1 \\cos \\theta_2  \\sin \\theta_3\n & \t\\sin \\theta_2 \\sin \\theta_3\n  \\\\ && \\\\\n \t\t-\\cos \\theta_1\\sin \\theta_3 - \\sin \\theta_1 \\cos \\theta_2 \\cos \\theta_3 \n & \t-\\sin \\theta_1\\sin \\theta_3 + \\cos \\theta_1 \\cos \\theta_2 \\cos \\theta_3 \n & \t\\sin \\theta_2 \\cos \\theta_3\n \\\\ && \\\\\n \t\\sin \\theta_1 \\sin \\theta_2 \n & \t-\\cos \\theta_1 \\sin \\theta_2  & \\cos \\theta_2\n\\end{array} \\right] , \\qquad\n\\end{eqnarray}\nand it follows from this and $\\rf{D5}_1$  that \n\\beq{detex}\n\\mbox{det} \\, \\hbQ_1\\ (\\theta_1,\\theta_2,\\theta_3) = \\cos 2\\theta_1\\, \\cos 2\\theta_2\\, \\cos 2\\theta_3\n-\\frac14\\, \\sin 2\\theta_1\\,  \\sin 2\\theta_3\\, \\cos \\theta_2\\, (3\\cos 2 \\theta_2 + 1)\\, .    \\quad\n\\end{equation}\nThe condition that this vanish  is equivalent to eq. (90) of Rovati and Taliercio \\cite{Rovati03}, although their result is obtained in a different manner.\n\n\nConsider, for instance,  $\\theta_3=0$, for which \n\\beq{D8}\n\\mbox{det} \\, \\hbQ_1\\ (\\theta_1,\\theta_2,0) = \\cos 2\\theta_1\\, \\cos 2\\theta_2,   \n\\end{equation}\nand hence there are null spaces associated with $\\hbQ_1(\\theta_1, \\pi\/4,0)$ and $\\hbQ_1(\\pi\/4,\\theta_2,0)$.  The  null spaces are lines in the stress space, \nwhich follow from the simplified form of $\\hbQ_1^T$, \n\\beqa{td}\n\\hbQ_1^T (\\theta_1,\\theta_2,0) = \n\\left[ \\begin{array}{ccc}\n \\cos^2 \\theta_1 ~&~  \\sin^2 \\theta_1 \\cos^2 \\theta_2  ~&~  \\sin^2 \\theta_1  \\sin^2 \\theta_2\n  \\\\ && \\\\\n \\sin^2 \\theta_1     & \\cos^2 \\theta_1\\cos^2 \\theta_2  \n & \\cos^2 \\theta_1 \\sin^2 \\theta_2\n \\\\ && \\\\\n0 &  \\sin^2 \\theta_2  & \\cos^2 \\theta_2\n\\end{array} \\right] . \\quad\n\\end{eqnarray}\nThe possible states of deviatoric stress are:  \n$(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\lambda \\, (0, \\, -1,\\, 1)$ if $\\theta_2 = \\pi\/4,\\, \\theta_3 = 0$, and \n$(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\lambda \\, (\\cos 2\\theta_2, \\, -\\cos^2\\theta_2,\\, \n\\sin^2\\theta_2)$, if $\\theta_1 = \\pi\/4,\\, \\theta_3 = 0$. The first family of stresses correspond to a 2D elasticity problem (see Appendix):  $\\sigma_{\\rm I}'=0,\\, \\sigma_{\\rm II}'+ \\sigma_{\\rm III}'=0$, and it is also a null vector of $\\hbQ_1 (0,  \\pi\/4,\\theta_3)$.  The second is also a null vector of $\\hbQ_1 (0, \\theta_2 , \\pi\/4)$.  Similarly, $\\lambda \\, ( 1,\\, -1,\\, 0)$ is a null vector of $\\hbQ_1( \\pi\/4,\\pi\/2, \\theta_3)$ and \n$\\lambda \\, ( -\\cos^2\\theta_1,\\, \\sin^2\\theta_1 , \\cos 2\\theta_1 )$ is a null vector of $\\hbQ_1( \\theta_1, \\pi\/2,\\pi\/4)$. \n\nConversely, an orientation which provides a minimum in energy can be found for a given state of stress.  Assume with no loss in generality that \n$\\sigma_{\\rm II}' < 0 < \\sigma_{\\rm III}'$.  Define the angle  $\\theta_2$ by \n\\beq{lp}\n\\tan^2 \\theta_2 = -\\frac{\\sigma_{\\rm III}'}{\\sigma_{\\rm II}'},\n\\end{equation}\nthen the deviatoric stress may be expressed\n\\beq{exp}\n(\\sigma_{\\rm I}',\\, \\sigma_{\\rm II}',\\,\\sigma_{\\rm III}') = \\big( \\sigma_{\\rm III}' - \\sigma_{\\rm II}'\\big)\\, (\\cos 2\\theta_2, \\, -\\cos^2\\theta_2,\\, \n\\sin^2\\theta_2). \n\\end{equation}\nIt follows from the above example  that this deviatoric stress is a null vector of $\\hbQ_1(\\pi\/4,\\theta_2,0)$. \n\n\nIt is instructive to  examine further the example of eq. \\rf{exp}.  The rotated material axes,  denoted  $\\{ {\\bf e}_1', {\\bf e}_2',{\\bf e}_3'\\}$, are given by the columns of ${\\bf  Q}(\\pi\/4,\\theta_2,0)$: \n\\beqa{inst1}\n{\\bf e}_1' = \\frac{1}{\\sqrt{2}}\n\\left[ \\begin{array}{r}\n 1 \\\\ \\\\\n -c  \\\\ \\\\\n s   \n\\end{array} \\right], \n\\quad\n{\\bf e}_2' = \\frac{1}{\\sqrt{2}}\n\\left[ \\begin{array}{r}\n 1  \\\\ \\\\\n c  \\\\ \\\\\n -s   \n\\end{array} \\right], \n\\quad\n{\\bf e}_3' = \n\\left[ \\begin{array}{c}\n0 \\\\ \\\\\n s \\\\ \\\\\n c  \n\\end{array} \\right], \n\\quad\n\\end{eqnarray}\nwhere $s=\\sin \\theta_2$, $c=\\cos \\theta_2$, or from \\rf{lp},\n\\beq{inst2}\ns = \\sqrt{ \\frac{\\sigma_{\\rm III}'}{\\sigma_{\\rm III}' - \\sigma_{\\rm II}'}}, \n\\quad\nc = \\sqrt{ \\frac{-\\sigma_{\\rm II}'}{\\sigma_{\\rm III}' - \\sigma_{\\rm II}'}}. \n\\end{equation}\nThe rotated tensor $D_{ijkl}$ is\n\\beq{inst3}\n{\\cal D} = {\\bf e}_1'\\otimes{\\bf e}_1'\\otimes{\\bf e}_1'\\otimes{\\bf e}_1'\n+ {\\bf e}_2'\\otimes{\\bf e}_2'\\otimes{\\bf e}_2'\\otimes{\\bf e}_2'\n+ {\\bf e}_3'\\otimes{\\bf e}_3'\\otimes{\\bf e}_3'\\otimes{\\bf e}_3'\\, , \n\\end{equation}\nand hence\n\\beq{inst4}\n{\\cal D} \\bosy{\\sigma}' = \\sum \\limits_{k=1}^3 {\\bf e}_k'\\otimes{\\bf e}_k'\\, \n ({\\bf e}_k'\\cdot \\bosy{\\sigma}' {\\bf e}_k' )\\, , \n\\end{equation}\nIt may be seen by direct calculation that the three scalars ${\\bf e}_k'\\cdot \\bosy{\\sigma}' {\\bf e}_k'$ (no sum) are identically zero by virtue of \\rf{inst1} and \\rf{inst2}.  This demonstrates  explicitly that \n\\beq{inst5}\n{\\cal D} \\bosy{\\sigma}' = 0 , \n\\end{equation}\nat the optimal orientation.  The identities \\rf{idh} follow accordingly. \n\n\\subsubsection{Summary for cubic symmetry}\n\n The extreme values of the energy for cubic materials are ${\\cal E}_1$ and ${\\cal E}_2$, where \n\\beq{ex12}\n{\\cal E}_j = \\frac{1}{\\kappa}\\overline{\\sigma}^2 + \\frac{1}{2\\mu_j}\n\\big( {\\sigma_{\\rm I}'}^2 + {\\sigma_{\\rm II}'}^2 + {\\sigma_{\\rm III}'}^2 \\big), \\qquad j=1,\\, 2. \n\\end{equation}  \nThe \nfixed axes are always one of the stationary orientations, since $\\bf E$ of \\rf{r4} vanishes.  The stationary value for the unrotated axes is ${\\cal E}_2$, which is the global minimum (maximum) if $\\mu_2 > \\mu_1$ ($\\mu_2 < \\mu_1$). The stationary value ${\\cal E}_1$ occurs at some rotated axes, \nthe existence of which is not in doubt for a material of cubic symmetry (or any material symmetry for that matter). The important point to note is that it is possible to explicitly determine  such orientations. Thus, we have shown  \nby direct construction the material orientation that achieves the stationary energy value ${\\cal E}_1$ for any state of stress.  This is a global minimum (maximum) if $\\mu_2 < \\mu_1$ ($\\mu_2 > \\mu_1$).   \n\nIt is interesting that the expressions for the extreme values in  \\rf{ex12} have the form of the energy for an isotropic solid, but with different shear moduli.  This is evident by writing ${\\cal E}_1$ and ${\\cal E}_2$ in terms of the invariants of the stress tensor: \n\\beq{ex122}\n{\\cal E}_j = \\frac{1}{9\\kappa}\\big({\\rm tr}\\, \\bosy{\\sigma}\\big)^2\n+ \\frac{1}{2\\mu_j}\\big[ {\\rm tr}\\, \\bosy{\\sigma}^2  - \\frac13 \\big({\\rm tr}\\, \\bosy{\\sigma}\\big)^2 \\big], \\qquad j=1,\\, 2. \n\\end{equation}\n\n\\subsubsection{Example materials}\nNoting that $\\mu_1 = c_{44}$ and $\\mu_2 = (c_{11}-c_{12})\/2$  allows us to determine the sign of $(\\mu_2 - \\mu_1)$. \nTable A.1 of Musgrave \\cite{Musgrave} provides data for $c^* = 2(\\mu_2 - \\mu_1)$ for a multitude of materials.  These show $c^*$ to be negative for most elemental and engineering materials with cubic symmetry and different lattice structures: aluminum, nickel, copper, silver, gold (all f.c.c structure), iron (b.c.c), brass (f.c.c. and b.c.c), diamond, silicon, germanium (all diamond structure), and GaSb, InSb (both zinc-blende).  Hence, for all of these cubic materials there exist optimal orientations of the axes that achieve the lowest energy state possible.  Materials with positive $c^*$ include crystalline compounds of potassium, sodium and silver with rock-salt structure: KF, KCL, KBr, KI, NaF, NaCl, NaBr, NaI, AgCl, AgBr; plus  caesium compounds with structure related to b.c.c.  For these, the orientation associated with \\rf{exp}, for instance, gives maximum strain energy.  The minimum energy is achieved by no rotation. \n\n\\subsection{Transverse isotropy} \nMaterials with hexagonal symmetry, or equivalently, transverse isotropy (TI), are characterized by five moduli.  In the coordinate system of the principal axes, the elements of the compliance are  \n\\beqa{TI1}\n\\cS^{(0)}  = \\left[ \n\\ba{cccccc}\nS_{11}^{(0)} & S_{12}^{(0)} & S_{13}^{(0)} & \n 0&0&0 \n\\\\ & & & & & \\\\\n & S_{11}^{(0)} & S_{13}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n & & S_{33}^{(0)} & \n 0&0&0\n\\\\ & & & & & \\\\\n &   &   \n & S_{44}^{(0)} & 0 & 0\n\\\\ & & & & & \\\\\n   S &Y  &M & & S_{44}^{(0)}&  0 \n\\\\ & & & & & \\\\\n&&&&& S_{66}^{(0)}\n\\end{array} \\right] ,\n\\end{eqnarray}\nwith $S_{66}^{(0)} = \\frac12 ( S_{11}^{(0)} - S_{12}^{(0)} )$.  The TI material is characterized by an axis of symmetry, defined by the unit vector ${\\bf n}$, which is here chosen as the ${\\bf e}_3$ axis.  In general, the strain energy depends only upon the orientation of ${\\bf n}$ with respect to the stress axes, and  the problem is  formulated as one of selecting ${\\bf n}$ to minimize $\\cal E$. \n\nFirst note that two of the five moduli can be ascribed to the isotropic part of the elasticity; or conversely, an isotropic part   may be subtracted from the compliance  tensor $s_{ijkl}$ according to \\rf{wr}, \\rf{wr2} and \\rf{km}, where \n \\beq{km3}\n \\frac{1}{\\kappa_s} =  \n2S_{11}^{(0)}+S_{33}^{(0)}+2S_{12}^{(0)}+4S_{13}^{(0)}, \\qquad\n \\frac{15}{4 \\mu_s} =   2S_{11}^{(0)}+S_{33}^{(0)} - S_{12}^{(0)} - 2S_{13}^{(0)}+6S_{44}^{(0)}+3S_{66}^{(0)} ,\n \\end{equation}\n  leaving a tensor $s_{ijkl}^{\\rm (an)}$ with three constants.  The anisotropic compliance depends upon the orientation of the axis of symmetry as follows  \n\\beqa{t1}\ns_{ijkl}^{\\rm (an)} &=& a\\, n_in_jn_kn_l + b\\, (\\delta_{ij}n_kn_l+ \\delta_{kl}n_in_j) \n+ \\frac{c}{2}\\, (\\delta_{ik}n_jn_l+ \\delta_{il}n_jn_k + \\delta_{jk}n_in_l+ \\delta_{jl}n_in_k) \n\\nonumber \\\\ && - \\frac13 (a+6b+2c) J_{ijkl} - \\frac{2}{15}(a+5c) K_{ijkl} .\\quad\n\\end{eqnarray}\nThe tensors $J_{ijkl}$ and $K_{ijkl}$ are defined in \\rf{wr3}, and \n the elastic constants $a$, $b$ and $c$ follow from \\rf{TI1}-\\rf{t1} as \n\\beq{abc}\na=S_{11}^{(0)} + S_{33}^{(0)} - 2S_{13}^{(0)} - 4S_{44}^{(0)}, \\quad \nb=S_{13}^{(0)} - S_{12}^{(0)}, \\quad \nc=2S_{44}^{(0)} - 2S_{66}^{(0)}. \\quad \n\\end{equation}\n  The stress is, as usual, aligned with the fixed axes, so that the total strain energy follows from \\rf{fa}, \\rf{t1} and \\rf{t2}, as \n\\beq{t3a}\n{\\cal E} =  \\big[ \\frac{1}{\\kappa_s} - \\frac13 (a+6b+2c)\\big] \n\\overline{\\sigma}^2 + \\big[  \\frac{1}{2\\mu_s} - \\frac{2}{15}(a+5c)\\big]\\sigma_{ij}'\\sigma_{ij}' + \n{\\cal E}^{\\rm (ex)} , \n\\end{equation}\nwhere  $\\overline{\\sigma}$  is defined in \\rf{fis2} and the extra energy term is   \n\\beq{t4}\n{\\cal E}^{\\rm (ex)} =a\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm I})2\\sigma_{\\rm I}n_1^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm II})2\\sigma_{\\rm II}n_2^2\n+ (3b\\overline{\\sigma}+ c\\sigma_{\\rm III})2\\sigma_{\\rm III}n_3^2\\, . \n\\end{equation}\nThe latter shows that the anisotropic part of the energy ${\\cal E}^{\\rm (ex)}$ depends upon the TI axis orientation through the three parameters \n$n_1^2$, $n_2^2$ and $n_3^2$, which satisfy $n_1^2+n_2^2+n_3^2=1$. Since each of  $n_1^2$, $n_2^2$ and $n_3^2$ must be non-negative, the permissible set is the equilateral  triangular area $A$ of the  plane $n_1^2+n_2^2+n_3^2-1=0$   bounded by the lines $L_1:\\, n_2^2+n_3^2=1$, $L_2:\\,n_3^2+n_1^2=1$ and $L_3:\\,n_1^2+n_2^2=1$. \n\n\nConsider first the possibility that the optimal orientation lies  \non one of the lines $L_i$, $I=1,2,3$.  Thus, along $L_3$, a simple calculation using $n_3^2 = 0$ shows that \n\\beq{t8}\n{\\cal E}^{\\rm (ex)}  = a (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\big( n_1^2 - N_1\\big)^2 + 2\\sigma_{\\rm II} \\big( \n3b\\overline{\\sigma} + c \\sigma_{\\rm II}\\big)  - a (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, N_1^2\\, \\quad \\mbox{on }L_3, \n\\end{equation}\nwhere\n\\beq{t7}\nN_1 \\equiv \\frac{a \\sigma_{\\rm II} +c (\\sigma_{\\rm I} + \\sigma_{\\rm II}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm II} - \\sigma_{\\rm I})}\\, . \n\\end{equation}\nThus, $n_1^2 = N_1$, is a {\\em possible} optimal orientation.  It must first be checked whether or not $N_1$  lies in $(0,1)$. If this is so, and if $a>0$, then an energy minimum occurs at the point $n_1^2=N_1$, $n_2^2=1-N_1$ on $L_3$.  \nSimilarly, a minimum occurs on $L_1$ at $n_2^2=N_2$, $n_3^2=1-N_2$ if $N_2 \\in (0,1)$ and $a>0$, and on $L_2$ at $n_3^2=N_3$, $n_1^2=1-N_3$ if $N_3 \\in (0,1)$ and $a>0$, where\n\\beqa{t9}\nN_2 &=& \\frac{a \\sigma_{\\rm III} +c(\\sigma_{\\rm II} + \\sigma_{\\rm III}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm III} - \\sigma_{\\rm II})},\n\\\\\\\nN_3 &=& \\frac{a \\sigma_{\\rm I} +c(\\sigma_{\\rm III} + \\sigma_{\\rm I}) +3b\\overline{\\sigma} }{a(\\sigma_{\\rm I} - \\sigma_{\\rm III})} \\, . \n\\end{eqnarray} \n\nNow consider the possibility of the minimum occurring in the interior of $A$.   Substitute $n_3^2=1 -n_1^2-n_2^2$ into \\rf{t4} and setting the partial derivatives with respect to $n_1^2$ and $n_2^2$ to zero, yields a pair of simultaneous conditions\n\\beqa{t5}\n\\big( \\sigma_{\\rm I} - \\sigma_{\\rm III}\\big)\\, \\big[ \na\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)\n+ 3b\\overline{\\sigma}+ c(\\sigma_{\\rm I}+ \\sigma_{\\rm III}) \\big] &=&0,\n\\\\\n\\big( \\sigma_{\\rm II} - \\sigma_{\\rm III}\\big)\\, \\big[ \na\\, \\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big)\n+ 3b\\overline{\\sigma}+ c(\\sigma_{\\rm II}+\\sigma_{\\rm III})\\big] &=&0 \\, . \n\\end{eqnarray}\nThese cannot be satisfied in general if the three principle stresses are distinct.   We therefore conclude that \\textcolor{blue}{\\em the optimal $\\bf n$ will lie inside $A$ iff the stress is biaxial}.  \nFor instance, if $\\sigma_{\\rm II} $ and $ \\sigma_{\\rm III}$ are equal, then eq. \\rf{t5} combined with $n_3^2=1 -n_1^2-n_2^2$ imply that ${\\cal E}^{\\rm (ex)}$ of \\rf{t4} can be expressed as a function of   $n_1^2$ alone, and the expression is identical in form to that given in \\rf{t8}. Thus, the existence of a minimum inside $A$ requires biaxiality $(\\sigma_{\\rm II}=\\sigma_{\\rm III})$ and that $0 < N_1 < 1$.  The associated optimal direction is not unique, but is defined by the cone $n_1^2 = N_1$, $n_2^2+n_3^2 = 1-N_1$ (note that $N_3 = 1-N_1$ when $\\sigma_{\\rm II}=\\sigma_{\\rm III}$). \nAgain, \\rf{t8} indicates that the optimal orientation corresponds to a minimum (maximum) in energy if $a>0$ $(a<0)$.   Thus, the sign of the elastic compliance  $a$ is crucial in determining whether the stationary point  is a minimum or a maximum.  \n\nThese conclusions may also be confirmed by the coaxiality condition for the stress and strain. Thus, for arbitrary orientation\n\\beqa{od}\n\\varepsilon_{23} &=& n_2n_3\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm II} + \\sigma_{\\rm III} \\big)\\big]\\, , \n\\nonumber \\\\\n\\varepsilon_{31} &=& n_3n_1\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm III} + \\sigma_{\\rm I} \\big)\\big]\\, , \n\\\\\n\\varepsilon_{12} &=& n_1n_2\\, \\big[ \na\\big( \\sigma_{\\rm I}n_1^2+ \\sigma_{\\rm II}n_2^2+ \\sigma_{\\rm III}n_3^2\\big) \n+ 3b\\overline{\\sigma}+ c\\big( \\sigma_{\\rm I} + \\sigma_{\\rm II} \\big)\\big]\\, . \n\\nonumber \n\\end{eqnarray}\nThe requirement that these simultaneously  vanish is identical to the  above conditions for the existence of the minimum inside $A$ or along its perimeter. \n\n\nIn summary, $a>0$ is a necessary condition that an energy minimum occurs at points inside $A$ or along the lines $L_j$, $j=1,2,3$.  A minimum is achieved iff one or more of $N_1$, $N_2$ or $N_3$ lies in $(0,1)$.  The minimum occurs on the associated bounding line $L_j$ or on a cone of directions for biaxial states of stress.   Otherwise, the global energy minimum corresponds to one of the  vertices  of $A$, i.e. at $n_1^2 = 1$ or $n_2^2 = 1$ or $n_3^2 = 1$.  In this default case the TI axis of symmetry is aligned with one of the stress axes.  These findings are in agreement with those of Rovati and Taliercio \\cite{Rovati03}, who stated the condition as follows: At least one of the principal axes of stress must lie in a plane of transverse isotropy, or alternatively, the TI axis must lie in a plane defined a pair of principal axes of stress. \n\n\\subsection{Tetragonal symmetry} \n\nThe moduli have the same general form as in \\rf{TI1}, except that there is no relation between $S_{66}^{(0)}$, $S_{11}^{(0)}$ and $S_{12}^{(0)}$.  In this sense,  tetragonal symmetry is the same as TI but with  one additional elastic constant.  The isotropic moduli are given by eq. \\rf{km3}, and the anisotropic part of the compliance is \n\\beqa{t1tet}\ns_{ijkl}^{\\rm (an)} &=& a'\\, n_in_jn_kn_l + b'\\, (\\delta_{ij}n_kn_l+ \\delta_{kl}n_in_j) \n+ \\frac{c'}{2}\\, (\\delta_{ik}n_jn_l+ \\delta_{il}n_jn_k + \\delta_{jk}n_in_l+ \\delta_{jl}n_in_k) \n\\nonumber \\\\ && \n+ d\\, (p_ip_j- q_iq_j)(p_kp_l - q_kq_l) \n- \\frac13 (a'+6b'+2c') J_{ijkl} - \\frac{2}{15}(a'+5c' + 3d) K_{ijkl} .\\quad \\qquad\n\\end{eqnarray}\nThe additional fourth order tensor as compared to TI is  $ (p_ip_j- q_iq_j)(p_kp_l - q_kq_l)$, \nwhere $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$ form an orthonormal triad.  \nThe elastic constants $a'$, $b'$, $c'$  and $d$ are\n\\beqa{abcd}\n&&a'=\\frac12 S_{11}^{(0)} +\\frac12 S_{12}^{(0)}+ S_{33}^{(0)} + S_{66}^{(0)}  - 2S_{13}^{(0)} - 4S_{44}^{(0)}, \n\\nonumber \\\\ \n&& b'=S_{13}^{(0)} - S_{12}^{(0)} - \\frac12 \\big( S_{11}^{(0)}- S_{12}^{(0)}\\big)  + S_{66}^{(0)}, \\quad \nc'=2S_{44}^{(0)} - 2S_{66}^{(0)},  \\\\\n&&\nd=\\frac12 \\big( S_{11}^{(0)}- S_{12}^{(0)}\\big)  - S_{66}^{(0)} . \\quad \\nonumber\n\\end{eqnarray}\nCompared to the TI constants $a$, $b$, $c$ of \\rf{abc}, \n\\beq{abcom}\na'=a-d,\\quad b'=b-d,\\quad c'=c. \n\\end{equation}\n\n\nThe strain energy of the tetragonal material can be split into a component similar in form to that for a TI material,  and an additional term proportional to the constant $d$.  The minimization of the TI part of the energy is as before (with $a',b',c'$ instead of $a,b,c$), and depends  upon the orientation of $\\bf n$ but not $\\bf p$ and $\\bf q$.   The additional energy term depends  on the deviatoric part of the stress and on these directions, \n\\beq{tet2}\n{\\cal E}^{\\rm tet} = d \\big[ (p_ip_j-q_iq_j)\\sigma_{ij}'\\big]^2 - \\frac25\\, d \\sigma_{ij}'\\sigma_{ij}', \n\\end{equation}\nor, in terms of the principal stresses,  \n\\beq{tet3}\n{\\cal E}^{\\rm tet} = d\\, \\big( \\sigma_{\\rm I} '\\Delta_1'  + \\sigma_{\\rm II} '\\Delta_2' + \\sigma_3 '\\Delta_{\\rm III}' \\big)^2 - \\frac25\\, d \\sigma_{ij}'\\sigma_{ij}' \\, , \n\\end{equation}\nwhere\n\\beq{di}\n \\Delta_i'  = p_i^2- q_i^2 \\, \\quad \\mbox{(no sum)} .\\quad \n\\end{equation}\nThe final term in \\rf{tet3} is independent of $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$, and it is only  necessary to  consider the quantity\n\\beq{tq1}\n{\\cal E} '  = d\\, f^2,\\quad \\mbox{where } \nf = \\sigma_{\\rm I} '\\Delta_1'  + \\sigma_{\\rm II} '\\Delta_2' + \\sigma_3 '\\Delta_{\\rm III}'\\, . \n\\end{equation}\nThe orientation dependence is captured by the quantity $f$.  \n\nIt is now demonstrated that for any given $\\bf n$, there is at least one set  of $ {\\bf p}, {\\bf q}$ orthogonal to $\\bf n$  which make ${\\cal E} '$ vanish.  Let $ {\\bf p}^{(0)}, {\\bf q}^{(0)}$ be an arbitrary pair of unit vectors such that $\\{ {\\bf n}, {\\bf p}^{(0)}, {\\bf q}^{(0)}\\}$ form an orthonormal triad, then every possible   set $\\{ {\\bf n}, {\\bf p}, {\\bf q}\\}$ is defined by the pair $ {\\bf p}, {\\bf q}$ obtained by rotation about $\\bf n$ by angle $\\phi$:\n\\beq{tet5}\n{\\bf p}(\\phi) = \\cos \\phi \\, {\\bf p}^{(0)} - \\sin \\phi \\, {\\bf q}^{(0)}, \n\\qquad \n{\\bf q}(\\phi) = \\sin \\phi \\, {\\bf p}^{(0)} + \\cos \\phi \\, {\\bf q}^{(0)}. \n\\end{equation}\nIt may then be readily verified that \n \\beq{tet6}\n\\Delta_i'(\\phi) = \\Delta_i'(0)\\, \\cos 2\\phi - \\Delta_i'(\\frac{\\pi}{4})\\, \\sin 2\\phi , \\quad i=1,2,3. \n\\end{equation}\nEquation \\rf{tet6} implies that \n\\beq{tet7}\n{\\cal E} ' (\\phi) = d\\, \\big[ f(0)\\, \\cos 2\\phi - f(\\frac{\\pi}{4})\\, \\sin 2\\phi\\big]^2 \\, , \n\\end{equation}\nand hence \n\\beq{tet8}\n{\\cal E} ' (\\phi^*) = 0 , \\quad \\mbox{where } \\tan 2\\phi^* = f(0)\/f(\\frac{\\pi}{4}) \\, , \n\\end{equation}\nThus, if $d>0$, the situation for tetragonal symmetry is a simple addition to the TI situation: First find $\\bf n$ which minimizes  the TI part of the energy.  Then, select the pair  $  {\\bf p}, {\\bf q}$ such that they satisfy \\rf{tet8}.  The minimum energy is then exactly that achieved by the TI part of the moduli (although it depends upon $a',b',c'$ rather than $a,b$ and $c$). \n\nIf $d<0$ then the situation is more complicated, and the sequential minimization of first the  TI energy and then  the additional energy ${\\cal E} '$ does not work, although these do define stationary points for the strain energy.  The $d-$term must be taken into account when optimizing with respect to $\\bf n$, and a more complicated minimization problem is involved. \n\nTetragonal symmetry represents a demarcation between the simpler higher material symmetries for which explicit results can be obtained, and the  lower material symmetries which require numerical resolution, in general.  Exceptions may occur, however, it is useful and instructive to distinguish the cubic, TI and tetragonal symmetries from those of, for example, monoclinic symmetry with 13 independent moduli to consider. \n\n\\section{Strain deviation angle}\\label{sec5}\n\n\\subsection{Definition of the strain deviation angle}\n\nA necessary condition for an energy minimum is that the stress and strain are coaxial.  This is always the case in isotropic media,  whereas it will be the exception rather than the rule under conditions of  general anisotropy and arbitrary stress.  According to Euler's theorem \\cite{Baruh} the transformation from one set of principal axes to the other can be reduced to an axis of rotation $\\bf n$, $|{\\bf n}|=1$, and an angle of rotation $\\phi$. The stress axes  have been assumed to coincide with the fixed axes ${\\bf e}_j$, $j=1,2,3$.   Let  ${\\bf e}_j'$ be the orthonormal axes of the strain tensor, then it follows that the rotation matrix is simply the matrix composed of the three unit vectors as columns, \n\\beq{qr}\n{\\bf Q} = \\big[ {\\bf e}_1 '\\, {\\bf e}_2 '\\, {\\bf e}_3' \\big] \\, . \n\\end{equation}\nLet $\\bf Q$ be represented by \\rf{qeq}, then it follows from the latter that \n\\beq{qr2}\ne_{ijk}Q_{jk} = 2 \\sin \\phi\\, n_i.\n\\end{equation}\nThis provides a formula to determine both the angle $\\phi$ and the axis of rotation $\\bf n$.    \n\nThe strain deviation angle $\\phi$ is defined as the angle of rotation between the stress and strain axes. This angle is identically zero in isotropic materials for all stress states.  In anisotropic materials it depends on both the material constants and the state of stress.   However,  the above analysis tells us that $\\phi = 0$ is a necessary condition for energy minimization. Therefore, the magnitude of $\\phi$ provides, through a single parameter, the degree to which the given state of stress and material orientation are optimal.  It does so without requiring any calculation of the energy locally or globally.   It requires only that the principal strain axes are determined, and from those $\\phi$  can be immediately computed. \n\nFor a given material, stress and strain, the  strain deviation angle can be obtained from \\rf{qr2}.  A more explicit method is to use the  general identity for integer $m$: \n\\beq{ged}\n  \\cos m\\phi =  \\frac12\\, {\\rm tr} \\big({\\bf Q}^m\\big) - \\frac12. \n\\end{equation}\nThis follows from, for example, eqs. \\rf{qeq} and \\rf{a1a1}, which imply\n\\beq{ged2} {\\bf Q}^m  ({\\bf n} , \\phi ) = {\\bf n}\\otimes{\\bf n} + \\cos m\\phi \\, ({\\bf I} -{\\bf n}\\otimes{\\bf n} )+\n\\sin m\\phi\\, {\\bf P}\\, . \n\\end{equation}\nFor instance, $m=1$ gives the  strain deviation angle explicitly in terms of the first invariant of $\\bf Q$: \n\\beq{qr3}\n  \\phi = \\cos^{-1}\\big[ \\frac12 \\big({\\rm tr}{\\bf Q}  -1\\big)\\big].\n\\end{equation}\n\n\n\\subsection{Weak anisotropy}\nLet $\\varepsilon_{j}^{(0)}$, $j=1,2,3$ be the principal strains for the isotropic medium, i.e. the principal axes of \n$s_{ijkl}^{\\rm (is)}\\sigma_{kl}$, where $\\sigma_{ij}$ is given by \\rf{s1} and\/or \\rf{t2}.  In order to determine the strain deviation angle we first need to find the principal axes of strain.  \nIt is useful to express the strain as \n\\beq{usf}\n\\varepsilon_{ij} = s_{ijkl}^{\\rm (is)}\\sigma_{kl}+\\gamma_{ij},\n\\end{equation}\n where \n\\beqa{w0}\n\\gamma_{ij} &=& s_{ijkl}^{\\rm (an)}\\sigma_{kl}\n\\nonumber \\\\ &=&\ns_{ij11}^{\\rm (an)}\\sigma_{\\rm I} + s_{ij22}^{\\rm (an)}\\sigma_{\\rm II}+s_{ij33}^{\\rm (an)}\\sigma_{\\rm III}   \n\\end{eqnarray}\nIt is assumed that $\\bosy{\\gamma}$ is small, so that standard perturbation analysis may be used to find the first correction to the directions of principal strain, \n$\\{ {\\bf e}_1', {\\bf e}_2',{\\bf e}_3'\\}$ , which to leading order are coincident with the stress directions: \n\\beq{wa1}\n{\\bf e}_i' = {\\bf e}_i + \\sum\\limits_{j\\ne i}\\, \\big( \\varepsilon_{i}^{(0)} - \\varepsilon_{j}^{(0)}\\big)^{-1}\\, \n\\big( {\\bf e}_i\\cdot \\bosy{\\gamma}{\\bf e}_j\\big) \\, {\\bf e}_j, \\, \\quad \\mbox{no sum on }i. \n\\end{equation}\nLet $E$ and $\\nu$ be the isotropic Young's modulus and Poisson's ratio characterizing $s_{ijkl}^{\\rm (is)}$, then \\rf{wa1} implies that, to leading order, \n\\beq{wa2}\nQ_{ij} = - Q_{ji} \\approx \\frac{E}{1+\\nu}\\, \\frac{\\gamma_{ij}}{ (\\sigma_j - \\sigma_i)},\\quad i\\ne j.\n\\end{equation}\nHence, the strain deviation angle for weak anisotropy is \n\\beq{wa3}\n\\phi \\approx \\sin \\phi \\approx \n \\frac{E}{1+\\nu}\\, \\bigg[ \\frac{\\gamma_{12}^2}{ (\\sigma_{\\rm I} - \\sigma_{\\rm II})^2}\n + \\frac{\\gamma_{23}^2}{ (\\sigma_{\\rm II} - \\sigma_{\\rm III})^2} + \\frac{\\gamma_{31}^2}{ (\\sigma_{\\rm III} - \\sigma_{\\rm I})^2}\\bigg]^{1\/2}\\, . \n\\end{equation}\nIt is useful to write the stress dependence explicitly by eliminating $\\bosy{\\gamma}$, \n\\beqa{wa31}\n\\phi &\\approx &\n \\frac{E}{1+\\nu}\\, \\bigg[  \n \\bigg( \\frac{\\sigma_{\\rm I}s_{14}^{\\rm (an)}+\\sigma_{\\rm II}s_{24}^{\\rm (an)}+\\sigma_{\\rm III}s_{34}^{\\rm (an)}  }{ \\sigma_{\\rm II} - \\sigma_{\\rm III}}\\bigg)^2\n +\\bigg( \\frac{\\sigma_{\\rm I}s_{15}^{\\rm (an)}+\\sigma_{\\rm II}s_{25}^{\\rm (an)}+\\sigma_{\\rm III}s_{35}^{\\rm (an)}  }{ \\sigma_{\\rm III} - \\sigma_{\\rm I}}\\bigg)^2\n \\nonumber \\\\  && \\qquad \\qquad\n + \\bigg(\\frac{\\sigma_{\\rm I}s_{16}^{\\rm (an)}+\\sigma_{\\rm II}s_{26}^{\\rm (an)}+\\sigma_{\\rm III}s_{36}^{\\rm (an)}  }{ \\sigma_{\\rm I} - \\sigma_{\\rm II}}\\bigg)^2\n \\bigg]^{1\/2}\\, . \n\\end{eqnarray}\nThis shows that the strain deviation angle depends upon the same $9$ moduli that appear in the matrix $\\bf E$ of eq. \\rf{r4}. \n\nThe above formula breaks down for biaxial stress.  In this case the choice of fixed axes is arbitrary since any orthogonal pair in the the plane spanned by the equal principal stresses are valid.  However, the choice can be made {\\it a posteriori} such that the term that would otherwise be singular is zero.  For instance, if $\\sigma_{\\rm I} = \\sigma_{\\rm II}$, then the axes ${\\bf e}_1$ and ${\\bf e}_2$ can be selected such that $\\gamma_{12}= 0$.  \n\n\\section{Conclusions}\n\nThe 6-dimensional notation of Mehrabadi et al.  \\cite{mcj} is well suited to the problem of finding optimal orientations of anisotropic solids.  It leads quite naturally to the main results of the paper, which we recapitulate:\n\n\\medskip \n\\noindent\n{\\bf Result 1}: {The energy  \n$ \\textcolor{blue} {\\cal E}$  is stationary iff the stress and strain are coaxial. }\n\n\\noindent\n{\\bf Result 1a}: {A necessary (but not sufficient) condition for this to hold is that det ${\\bf E}=0$, where $\\bf E$ is defined in \\rf{r4}}\n\n\\noindent\n{\\bf Result 2}:  {The energy\n${\\cal E}$  is a local minimum if the stress and strain are coaxial and the symmetric matrix ${\\bf G}$ of \\rf{G7} is positive definite.  } \n\nResult 2 provides for the first time an explicit set of conditions that must be satisfied if  the stationary condition is a minimum or a maximum.  \n\nSpecific  results are given for materials of cubic, transversely isotropic  and tetragonal symmetries.  In each case the existence of a minimum or maximum depends on the sign of a single elastic constant.  For cubic symmetry we have several new findings.  For instance,  eqs. \\rf{s3a} - \\rf{s3c} provide a two parameter set of stress states which minimize or maximize the strain energy if a material of cubic symmetry is rotated  about an arbitrary axis $\\bf n$ by angle $\\phi$ (subject to the constraint \\rf{n1n2q}).  Alternatively, eqs. \\rf{lp} and \\rf{exp} provide a means to find the optimal orientation for a given state of stress.   In particular, the rotation of the material axes depends only upon the deviatoric stress.  This demonstrates that the stationary (minimum or maximum) value of energy can always be achieved for cubic materials.  Furthermore, it shows that \nthe optimal orientation of a solid with cubic material symmetry is not normally aligned with the symmetry directions.  \n\nThe remainder of the new results concern the optimal orientation of  TI and tetragonal materials,   and  are in general agreement with results of  Rovati and Taliercio \\cite{Rovati03} obtained by a different procedure.  However, the results obtained here are more direct and provide considerable insight into the nature of the optimal states for these material symmetries.  In particular,  the problem for tetragonal symmetry is very similar to that for TI, with an additional energy term that can be simply minimized or maximized (depending on the sign of the constant $d$ of \\rf{abcd}.   \n\nFinally, we have defined and  introduced the strain deviation angle.  The strain deviation angle is inherently anisotropic, and directly related to the problem of energy minimization since the angle   defines the degree to which a state of stress or strain is not optimal.  Future work will explore other consequences of this new concept.\n\n\n\\section*{Appendix: Two dimensional elasticity}\n\n\nOptimal orientation for  two dimensional elastic anisotropy  is an important special case of the general problem.  It was recently considered by  Gea and Luo \\cite{Gea04}, and is reconsidered here  in the context of the present theory.  We will see that some of the features Gea and Luo  obtained transfer to the 3D problem: in particular, the dependence of the minimization upon a single elastic constant. \n\nThe two dimensional strain energy function is \n\\beq{q1}\n{\\cal E}(\\theta ) = S_{11} \\sigma_{\\rm I}^2 + S_{22} \\sigma_{\\rm II}^2 + 2S_{12} \\sigma_{\\rm I} \\sigma_{\\rm II}\\, . \n\\end{equation}\nwhere $S_{11}$, $S_{22}$ and $S_{12}$ depend upon the angle $\\theta$ by which the material is rotated relative to the fixed $\\bf{e}_3$ axis.  Consider an orthotropic material with compliance elements \n$S^{(0)}_{11},\\,  S^{(0)}_{22},\\,  S^{(0)}_{12},\\,  S^{(0)}_{66}$ in the unrotated (fixed) axes.  \nUsing the standard relations \\cite{Christensen} for the transformation of the moduli, it may be shown that \n\\beq{q2}\n{\\cal E}(\\theta ) = \\frac{1 }{4} \\, d_0\\, ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\big( \n\\cos 2\\theta -  \\Lambda \\big)^2\\,  + b_0 ,\n\\end{equation}\nwhere $\\Lambda$ is a combination of stress and moduli, \n\\beq{x6}\n \\Lambda  = \\bigg( \\frac{ \\sigma_{\\rm II} + \\sigma_{\\rm I} }{ \\sigma_{\\rm II} - \\sigma_{\\rm I} } \\bigg) \\,\n \\frac{ c_0 }{ d_0 }\\, , \n \\end{equation}\n $c_0$ and $d_0$ are   moduli, \n\\beq{x1a}\nc_0 = S^{(0)}_{11} - S^{(0)}_{22}, \\qquad \nd_0 = S^{(0)}_{11} + S^{(0)}_{22} - 2S^{(0)}_{12} - 4S^{(0)}_{66} , \n\\end{equation}\nand $b_0$ is a constant,\n\\beq{b0}\nb_0 = ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, S^{(0)}_{66} + \\frac14 ( \\sigma_{\\rm II} + \\sigma_{\\rm I})^2\\, \\big(\nS^{(0)}_{11} + S^{(0)}_{22}+2 S^{(0)}_{12} -\\frac{c_0^2}{d_0}\\big) . \n\\end{equation}\n\nIt can be easily seen that the energy ${\\cal E}$ of \\rf{q2} is  stationary with respect to $\\theta$ when \n\\beq{st}\n\\cos 2\\theta = 1, \\mbox{ and  } \\cos 2\\theta = -1\n\\quad \\Leftrightarrow \\quad \\theta =0, \\mbox{ and  } \\theta = \\pi\/2, \n\\end{equation}\nrespectively.  Which of these yields the  smaller value for ${\\cal E}$ depends upon the sign of $d_0\\Lambda$, or equivalently, the sign of $ (\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0$.  Specifically, the minimum is at $\\cos 2\\theta = $sgn$[(\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0]$. A third stationary value is possible if $-1< \\Lambda<1 $, and occurs at \n\\beq{ste}\n\\cos 2\\theta = \\Lambda, \n\\quad \\Leftrightarrow \\quad \\theta  =\\pm \\theta^*, \n\\end{equation}\nwhere $\\theta^* = \\frac12\\, \\cos^{-1}\\Lambda$.  If this stationary point occurs, it follows from explicit evaluation that it corresponds to a global minimum or maximum of the energy.  \nThus, \n\\beqa{q3}\n\\left. \\ba{c}\n{\\cal E}(0)\\\\ \\\\\n{\\cal E}(\\frac{\\pi}{2})\\\\ \\\\\n{\\cal E}(\\theta^*)\n\\end{array} \\right\\} = b_0 + d_0\\, ( \\sigma_{\\rm II} - \\sigma_{\\rm I})^2\\, \\times \\, \n\\left\\{ \\ba{l}\n \\sin^4  \\theta^* \\, , \\\\ \\\\\n \\cos^4  \\theta^* \\, , \\\\ \\\\\n0\\, . \n\\end{array}\\right. \n\\end{eqnarray}\nIt is clear that $\\theta = \\pm \\theta^*$ is a repeated global minimum (maximum) if  $d_0 >0\\, (d_0 <0)$. This is the fundamental result of  Gea and Luo \\cite{Gea04} (although their conclusion is slightly different since they do not start with the stress in the principal axes frame): \n\\textcolor{blue}{ If  $-1< \\Lambda<1 $ and $d_0 >0$ then $ \\theta  =\\pm \\theta^*$ is a repeated global minimum of ${\\cal E}(\\theta)$.  Otherwise, the minimum occurs when $\\cos 2\\theta = $sgn$[(\\sigma_{\\rm II}^2 - \\sigma_{\\rm I}^2)c_0]$.\n}\n\nThe results of Gea and Luo are now reconsidered within the context of the general theory applied to 2D.   Based on the general theory for 3D, the 2D condition for a stationary orientation is \n\\beq{21}\n\\varepsilon_{12} = 0\\, , \n\\end{equation}\nor, in terms of the stress, assuming for simplicity that $\\sigma_{\\rm III} = 0$: \n\\beq{22}\nS_{16}\\sigma_{\\rm I} + S_{26}\\sigma_{\\rm II} = 0\\, . \n\\end{equation}\nThe latter is consistent with the general formulation, i.e. eq. \\rf{r3}, under the assumption that $\\sigma_{\\rm I}$ and $\\sigma_{\\rm II}$ are distinct (if they are not distinct, then the stress-based energy function is constant for all material orientations). \nThe additional condition that the stationary orientation is a local minimum follows from \n\\beq{cm8}\n\\left. \\frac{d^2 {\\cal E}(\\theta)}{d\\theta^2 } \\right|_{\\theta = 0} =  4 F_{33} , \n\\end{equation}\nas \n\\beq{h3}\nF_{33} = 2 (\\sigma_{\\rm II} - \\sigma_{\\rm I})^2 \\, G_{33} > 0 . \n\\end{equation}\nThe exact form of $G_{33}$  follows from  \\rf{G6}   with $\\sigma_{\\rm III} = 0$ as  \n\\beq{h4}\nG_{33} = S_{66}  + \\frac12 \n(\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[ \n\t(S_{11}-S_{12})\\sigma_{\\rm I} + (S_{21}-S_{22})\\sigma_{\\rm II}  \\big]\\, . \n\\end{equation}\nRearrangement gives \n\\beq{h5}\nG_{33} = \\frac14 \\,\n(\\sigma_{\\rm II} - \\sigma_{\\rm I})^{-1}\\, \\big[ (\\sigma_{\\rm I} + \\sigma_{\\rm II}) c + \n(\\sigma_{\\rm I} - \\sigma_{\\rm II}) d \\big]\\, , \n\\end{equation}\nwhere $c$ and $d$ are \n\\beq{x1}\nc = S_{11} - S_{22}, \\qquad \nd = S_{11} + S_{22} - 2S_{12} - 4S_{66} . \n\\end{equation}\n\nThe specific case of an orthotropic material is considered next.  \nIt may be shown by use of standard relations \\cite{Christensen} that the combinations of moduli in \\rf{x1} transform according to \n\\beq{x2}\nc (\\theta) = c_0 \\, \\cos 2\\theta , \\qquad \nd (\\theta) = d_0 \\, \\cos 4\\theta ,  \n\\end{equation}\nwhere \n$c_0 = c(0)$ and $d_0 = d(0)$ are the same moduli defined in \\rf{x1a}. \nAlso, \n\\beqa{x3}\nS_{16} &=&-\\frac{1}{4} \\big( c_0 +  d_0 \\cos 2\\theta \\big) \\, \\sin 2\\theta ,\n\\\\ \nS_{26} &=&-\\frac{1}{4} \\big( c_0 -  d_0 \\cos 2\\theta \\big) \\, \\sin 2\\theta . \n\\end{eqnarray}\nHence, \n\\beq{x4}\n\\varepsilon_{12} = -\\frac{1}{4}  \\sin 2\\theta \\, \n\\big[ (\\sigma_{\\rm I} + \\sigma_{\\rm II}) c_0 + (\\sigma_{\\rm I} - \\sigma_{\\rm II}) d_0 \\cos 2\\theta \\big]\\, . \n\\end{equation}\nThe strain $\\varepsilon_{12}$ vanishes if \n\\beq{x5}\n \\sin 2\\theta = 0,\\quad \\mbox{or}\\quad \n \\cos 2\\theta = \\Lambda .\n \\end{equation}\nThus, the stationary points are $\\theta = 0,\\, \\pi\/2$ and $\\pm \\theta^*$ where $\\cos 2\\theta^* = \\Lambda $, in agreement with the results of Gea and Luo \\cite{Gea04}. \n\nUsing the same notation, \\rf{h5} becomes\n\\beq{qwa}\nG_{33} (\\theta) =    \\frac14 \\,d_0\\,  \\big( \\Lambda\n\\cos 2\\theta -  \\cos 4\\theta \\big)\\, .\n\\end{equation}\nIn particular, if $- 1<\\Lambda <1$, then \n\\beq{qw}\nG_{33}(\\theta^*)  =   \\frac14 \\, d_0\\,  \\sin^2 2\\theta^*\\, .\n\\end{equation}\nThis implies that $\\theta= \\pm \\theta^*$ is a local minimum of ${\\cal E}$ iff $d_0$ is positive. The general analysis for 3D optimal orientation does not provide an explicit statement about global minima.   In order to show that it is a global maximum one must compare the value of the energy  ${\\cal E}$ at $\\theta= \\pm \\theta^*$ with its value at the other local minimum, as done in \\rf{q3}.  \n\n\\medskip\n\\noindent\n{\\bf Acknowledgment}\\,   I would like to acknowledge Hae Chang Gea for useful suggestions.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nCavity opto-mechanics describes a paradigmatic system for quantum\nmetrology: a massive object with mechanical degrees of freedom is\ncoupled to and measured by a bosonic field. Interest in this generic\nsystem is motivated by several considerations.  For one, the system\nallows one to explore and address basic questions about quantum\nlimits to measurement. In this context, quantum limits to quadrature\nspecific and non-specific measurements, both for those performed\ndirectly on the mechanical object and also those performed through\nthe mediation of an amplifier \\cite{brag95qmbook}. Second, as a\ndetectors of weak forces, cavity opto-mechanical systems in the\nquantum regime may yield improvements in applications ranging from\nthe nanoscale (e.g.\\ for atomic or magnetic force microscopies) to\nthe macroscale (e.g.\\ in ground- or space-based gravity wave\nobservatories).  Finally, such systems, constructed with ever-larger\nmechanical objects, may allow one to test the validity of quantum\nmechanics for massive macroscopic objects.  Striking developments in\nthis field were presented at ICAP 2008 by Harris and Kippenberg.\n\nOur contribution to this developing field is the realization that a\ncavity opto-mechanical system can be constructed using a large gas\nof ultracold atoms as the mechanical object.  Having developed an\napparatus that allows quantum gases to be trapped within the optical\nmode of a high-finesse Fabry-Perot optical resonator, we are now\nable to investigate basic properties of opto-mechanical systems.\nSeveral of these investigations are described below.  The\natoms-based mechanical oscillator may be considered small by some,\nwith a mass ($\\simeq 10^{-17}$ g) lying geometrically halfway\nbetween the single-atom limit explored at the quantum regime in ion\nand atom traps ($10^{-22}$ g) \\cite[for example]{meek96,bouc99}, and\nthe small ($\\simeq 10^{-12}$ g) nanofabricated systems now\napproaching quantum limits \\cite{laha04approach,rega08}.\nNevertheless, our system offers the advantages of immediate access\nto the quantum mechanical regime, of the \\emph{ab initio}\ntheoretical basis derived directly from quantum optics and atomic\nphysics, and of the tunability and amenability to broad new probing\nmethods that are standard in ultracold atomic physics.  Our\nmotivation for probing cavity opto-mechanics with our setup is not\njust to poach the outstanding milestones of this field (e.g.\\\nreaching the motional ground state or observing measurement\nbackaction and quantum fluctuations of radiation pressure with a\nmacroscopic object \\cite{murc08backaction}).  Rather, we hope to\ncontribute to the development of macroscopic quantum devices by\nclarifying experimental requirements and the role of and limits to\ntechnical noise, developing optimal approaches to signal analysis\nand system control, exploring the operation and uses of multi-mode\nquantum devices, and defining different physical regimes for such\nsystems. Also, our opto-mechanical system may have direct\napplication as part of an atom-based precision (perhaps\ninterferometric) sensor.\n\n\\section{Collective modes of an intracavity atomic ensemble}\n\nThe theoretical reasoning for considering a trapped atomic gas\nwithin a high-finesse optical resonator as a macroscopic cavity\nopto-mechanical system is laid out in recent work\n\\cite{murc08backaction}. Recapping that discussion, we consider the\ndispersive coupling of an ensemble of $N$ identical two-level atoms\nto a single standing-wave mode of a Fabry-Perot cavity, obtaining\nthe spectrum of ``bright'' eigenstates of the atoms-cavity system\naccording to the following Hamiltonian:\n\\begin{equation}\n\\mathcal{H} = \\hbar \\omega_c \\hat{n} + \\sum_{i} \\frac{\\hbar g^2(z_i)}{\\Delta_{ca}}\n+ \\mathcal{H}_{a} + \\mathcal{H}_{in\/out}\n\\end{equation}\nHere $\\hat{n}$ is the cavity photon number operator, $\\Delta_{ca} = \\omega_c\n- \\omega_a$ is the difference between the empty-cavity and atomic\nresonance frequencies, and $g(z_i) = g_0 \\sin(k_p z)$ is the\nspatially dependent atom-cavity coupling frequency with $z_i$ being\nthe position of atom $i$ and $k_p$ being the wavevector at the\ncavity resonance.  The term $\\mathcal{H}_a$ describes the energetics of\natomic motion while $\\mathcal{H}_{in\/out}$ describes the electromagnetic\nmodes outside the cavity.  Note that this expression already treats\nthe atom-cavity coupling to second order in $g$. Repeating this\nanalysis starting from the first-order term does not change our\nconclusions substantially.\n\nNow, let us assume that all the atoms are trapped in harmonic\npotentials with ``mechanical'' trap frequency $\\omega_z$ and neglect\nmotion along directions other than the cavity axis.  Further, we\ntreat the atomic motion only to first order in atomic displacements,\n$\\delta z_i$, from their equilibrium positions, $\\bar{z}_i$;  i.e.\\\nwe assume atoms to be confined in the Lamb-Dicke regime with $k_p\n\\delta z_i \\ll 1$. We now obtain the canonical cavity\nopto-mechanical Hamitonian \\cite{kipp08sciencereview} as\n\\begin{equation}\n\\mathcal{H} = \\hbar \\omega_c^\\prime \\hat{n} + \\hbar \\omega_z \\hat{a}^\\dagger \\hat{a}\n- F \\hat{Z} \\hat{n} + \\mathcal{H}^\\prime_{a} + \\mathcal{H}_{in\/out} \\label{eq:hami}\n\\end{equation}\n\nWe make several steps to arrive at this expression. First, we allow\nthe cavity resonance frequency to be modified as $\\omega_c^\\prime =\n\\omega_c + \\sum_i g^2(\\bar{z}_i) \/ \\Delta_{ca}$, accounting for the cavity\nresonance shift due to the atoms at their equilibrium positions.\nSecond, we introduce the collective position variable $\\hat{Z} =\nN_{\\mbox{\\scriptsize{eff}}}^{-1} \\sum_i \\sin(2 k_p \\bar{z}_i) \\delta z_i$ that, along with\na weighted sum $\\hat{P} = \\sum_i \\sin(2 k_p \\bar{z}_i) p_i$ of the\natomic momenta $p_i$, describes the one collective motion within the\natomic ensemble that is coupled to the cavity-optical field. The\noperators $\\hat{a}$ and $\\hat{a}^\\dagger$ are defined conventionally for this\nmode. In our treatment, absent the presence of light within the\noptical cavity, this mode is harmonic, oscillating at the mechanical\nfrequency $\\omega_z$, and endowed with a mass $M$ equal to that of\n$N_{\\mbox{\\scriptsize{eff}}} = \\sum_i \\sin^2(2 k_p \\bar{z}_i)$ atoms. Third, we summarize\nthe opto-mechanical coupling by the per-photon force $F = N_{\\mbox{\\scriptsize{eff}}}\n\\hbar k g_0^2 \/ \\Delta_{ca}$ that acts on the collective mechanical mode.\nFinally, we lump all the remaining atomic degrees of freedom, and\nalso the neglected higher order atom-cavity couplings, into the term\n$\\mathcal{H}^\\prime_a$.\n\nWith this expression in hand, we may turn immediately to the\nliterature on cavity opto-mechanical systems to identify the\nphenomenology expected for our atoms-cavity system.  Several such\nphenomena are best described by referring to the opto-mechanical\nforce on the collective atomic mode, given as\n\\begin{equation}\n\\hat{\\mathcal{F}}_{\\mbox{\\scriptsize{opto}}} = - M \\omega_z^2 \\hat{Z} + F \\hat{n}. \\label{eq:optoforce}\n\\end{equation}\n\nWe consider the following effects:\n\\begin{itemize}\n\\item{} If we allow the state of the cavity to follow the atomic motion adiabatically ($\\omega_z \\ll \\kappa$),\nneglect quantum-optical fluctuations of the cavity field, and assume\nthe collective atomic displacement remains small, the linear\nvariation of $\\langle \\hat{n} \\rangle$ with $\\hat{Z}$ modifies the\nvibration frequency of the collective atomic motion.  Here, $\\kappa$\nis the cavity half-linewidth.  This modification, known as the\n``optical spring,'' has been observed in various opto-mechanical\nsystems and has been used to trap macroscopic objects optically\n\\cite{shea04spring,corb06spring,corb07gram}. We have made\npreliminary observations of the optical spring effect in our system\nas well.\n\n\\item{} For larger atomic displacements, the opto-mechanical force\nmay become notably anharmonic, and even, under suitable conditions,\nbistable \\cite{dors83bistability}.  Our observations of the\nresulting opto-mechanical bistability \\cite{gupt07nonlinear} are\ndiscussed in Sec.\\ \\ref{sec:bistab}.\n\n\\item{} When the cavity field no longer follows the atomic motion\nadiabatically, the opto-mechanical potential is no longer\nconservative.  The dramatic effects of such non-adiabaticity are the\ncavity-induced damping or coherent amplification of the mechanical\nmotion \\cite{brag67}.  Such effects of dynamical backaction have\nbeen detected in several micro-mechanical systems\n\\cite{kipp05anal,arci06,giga06cooling} and also for single\n\\cite{maun04cooling} or multiple atoms \\cite{chan03} trapped within\na cavity.\n\n\\item{} Finally, we consider also the effects of quantum-optical\nfluctuations of the intracavity photon number and, thereby, of the\noptical forces on the atomic ensemble.  It can be shown that these\nforce fluctuations represent the backaction of quantum measurements\nof the collective atomic position \\cite{murc08backaction}, as\ndescribed in Sec.\\ \\ref{sec:backaction}.\n\\end{itemize}\n\n\\section{Collective atomic modes in various regimes}\n\nThe theoretical treatment described above is suitable in the\nLamb-Dicke regime of atomic confinement and under the condition that\nthe linear opto-mechanical coupling term ($F \\hat{Z} \\hat{n}$) is\ndominant (i.e.\\ that the intracavity atomic gas is not tuned to\npositions of exclusively quadratic sensitivity).  These conditions\nare met in our experiments at Berkeley, where an ultracold gas of\nabout $10^5$ atoms of $^{87}$Rb is transported into the mode volume\nof a high-finesse Fabry-Perot optical resonator. The resonator\nlength is tuned so that the resonator supports one TEM$_{00}$ mode\nwith wavelength $\\lambda_T = 850$ nm (trapping light) and another\nwithin a given detuning $\\Delta_{ca}$ (in the range of 100's of GHz) of the\nD2 atomic resonance line (probe light). Laser light with wavelength\n$\\lambda_T$ is sent through the cavity to generate a 1D optical\nlattice potential in which the cold atomic gas is trapped (Fig.\\\n\\ref{fig:scheme}). The gas is strewn across over $> 100$ contiguous\nsites in this 1D optical lattice. Within each well, atoms are\nbrought by evaporative cooling to a temperature $T \\sim 700$ nK. At\nthis temperature, the atoms lie predominantly in the ground state of\nmotion along the cavity axis, with $\\hbar \\omega_z \/ k_B \\simeq 2 \\,\n\\mu\\mbox{K} \\gg T$, and the Lamb-Dicke condition is satisfied with\nrespect to the wavevector of probe light ($k_p \\simeq 2 \\pi \/ (780\n\\, \\mbox{nm})$) used to interrogate the atomic motion.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file=scheme.eps,width=4.5in}\n\\end{center}\n\\caption{Scheme for opto-mechanics with ultracold atoms in the\nLamb-Dicke confinement regime.  A high finesse cavity supports two\nlongitudinal modes -- one with wavelength of about 780 nm that is\nnear the D2 resonance of $^{87}$Rb atoms trapped within the\nresonator, and another with wavelength of about 850 nm.  Light at\nthe 850 nm resonance produces a one-dimensional optical lattice,\nwith trap minima indicated in orange, in which atoms are confined\nwithin the lowest vibrational band.  These atoms induce frequency\nshifts on the 780 nm cavity resonance.  The strength of this shift,\nand of its dependence on the atomic position, varies between the\ndifferent sites of the trapping optical lattice, as shown.\nNevertheless, in the Lamb-Dicke confinement regime, the complex\natoms-cavity interactions reduce to a simple opto-mechanical\nHamiltonian wherein a single collective mode of harmonic motion,\ncharacterized by position and momentum operators $\\hat{Z}$ and\n$\\hat{P}$, respectively, is measured, actuated, and subjected to\nbackaction by the cavity probe.} \\label{fig:scheme}\n\\end{figure}\n\nThe opto-mechanics picture of atomic motion in cavity QED has also\nbeen considered recently by the Esslinger group in Z\\\"{u}rich\n\\cite{bren08optonote}. There, a continuous Bose-Einstein condensate\nof $^{87}$Rb is trapped in a large-volume optical trap within the\ncavity volume. Yet, in spite of the stark differences in the\nexternal confinement and the motional response of the condensed gas,\na similar opto-mechanical Hamiltonian emerges.  In our prior\ndescription of the Lamb-Dicke regime, optical forces due to cavity\nprobe light are found to excite and, conversely, to make the cavity\nsensitive to a specific collective motion in the gas.  In the case\nof a continuous condensate, the cavity optical forces excite atoms\ninto a specific superposition of the $\\pm 2 \\hbar k_p$ momentum\nmodes. Interference between these momentum-excited atoms and the\nunderlying condensate creates a spatially (according to $k_p$) and\ntemporally (according to the excitation energy) periodic density\ngrating that is sensed via the cavity resonance frequency.  Thus,\nidentifying operators $\\hat{a}$ and $\\hat{a}^\\dagger$ with this\nmomentum-space excitation and the operator $\\hat{Z}$ with the density\nmodulation, we arrive again at the Hamiltonian of Eq.\\\n\\ref{eq:hami}.\n\nWe can attempt to bridge these two opto-mechanical treatments by\ntracking the response of an extended atomic gas to spatially\nperiodic optical forces (due to probe light at wavelength 780 nm) as\nwe gradually turn up the additional optical lattice potential (due\nto trapping light at wavelength 850 nm).  In the absence of the\nlattice potential, a zero-temperature Bose gas forms a uniform\nBose-Einstein condensate.  The excitations of this system are\ncharacterized by their momentum and possess an energy determined by\nthe Bogoliubov excitation spectrum; in Fig.\\ \\ref{fig:bands}(a), we\npresent this spectrum as a free-particle dispersion relation,\nneglecting the effects of weak interatomic interactions. The\nspatially periodic optical force of the cavity probe excites a\nsuperposition of momentum excitations as described above.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file=bands.eps,width=4.5in}\n\\end{center}\n\\caption{Influence of band structure on the opto-mechanical response\nof an ultracold atomic gas confined within a high-finesse\nFabry-Perot optical resonator.  We consider the relevant macroscopic\nexcitation produced by cavity probe light at wavevector $k_p = 2 \\pi\n\/ (780 \\, \\mbox{nm})$ within a Bose gas confined within a\none-dimensional optical lattice formed by light at wavevector $k_t =\n2 \\pi \/ (850 \\, \\mbox{nm})$ and with variable depth.  The gas is\ncooled to zero temperature, non-interacting, and extended evenly\nacross many lattice sites.  Energies are scaled by the recoil energy\n$E_r = \\hbar^2 k_t^2 \/ 2 m$ and wavevectors by $k_t$.  (a) With a\nweak lattice applied ($2 E_r$), the band structure for atomic\nexcitations (black lines) is slightly perturbed from the\nfree-particle excitations in the absence of a lattice (gray).  The\ncavity probe excites atoms primarily to states with quasi-momenta\n$\\pm 2 k_p$ within the second excited band, corresponding closely to\nmomentum eigenstates in the lattice-free regime.  (b) In a deep\nlattice ($15 E_r$), energy bands show little dispersion and are\nspaced by energies scaling as the square root of the lattice depth.\n(c) Lines show the energies of the three lowest energy states at\nquasi-momentum $2 k_p$ as a function of the lattice depth.  The\nrelative probability for excitation by cavity probe light to each of\nthese states, taken as the square of the appropriate matrix element,\nis shown by the width of the shaded regions around each line.  At\nzero lattice depth, cavity probe light excites the second excited\nband exclusively.  At large lattice depth, the excitation\nprobability to the first excited band grows while excitation to\nhigher bands is suppressed. At intermediate lattice depths, several\nexcited states are populated, indicating the onset of complex\nmulti-mode behaviour.} \\label{fig:bands}\n\\end{figure}\n\n\nAdding the lattice potential changes both the state of the\nBose-Einstein condensate, which now occupies the lowest Bloch state,\nand also the state of excitations, which are now characterized by\ntheir quasi-momentum and by the band index.  There are now many\nexcitations of the fluid that may be excited at the quasi-momentum\nselected by the spatially periodic cavity probe.  In the case that\nthe lattice is very shallow, shown in Fig.\\ \\ref{fig:bands}(a), the\ncavity probe will still populate only one excited state nearly\nexclusively.  Given the relation between the wavelengths of the\ntrapping (850 nm) and cavity-probe light (780 nm), this excited\nstate lies in the second excited band.  As the lattice is deepened,\nhowever, matrix elements connecting to quasi-momentum states on\nother bands will grow (shown in Fig.\\ \\ref{fig:bands}(c)).  Now our\nsimple opto-mechanical picture is made substantially more complex,\nwith multiple mechanical modes oscillating with differing mechanical\nfrequencies all influencing the optical properties of the cavity.\n\nContinuing to deepen the optical lattice this complexity will be\nalleviated when we reach the Lamb-Dicke regime, i.e.\\ as the\nLamb-Dicke parameter $k_p \\delta z$ becomes ever smaller, the\nprobabilities of excitation from the ground state via the cavity\nprobe hone in on the first excited band.  We calculate such\nprobabilities as $p_i \\propto \\left| <2 k_p; i| \\cos(2 k_p z) | g>\n\\right|^2$ where the ket is the $2 k_p$ quasi-momentum Bloch state\nin the $i$th band, and the bra is the ground state in the lattice\nconsidered. Here, we interpret excitations to higher bands as being\ncontrolled by terms of higher order in the Lamb-Dicke parameter,\ne.g.\\ excitations to the second excited band result from couplings\nthat are quadratic in the atomic positions.\n\nThus, we confirm that a simple opto-mechanics picture emerges for\nthe collective atomic motion within a cavity both in the shallow-\nand deep-lattice limits.  We note, however, that these limits differ\nin two important ways.  First, we see that the mechanical\noscillation frequency for the collective atomic motion is\nconstrained to lie near the bulk Bragg excitation frequency in the\nshallow-lattice limit, whereas it may be tuned to arbitrarily high\nfrequencies (scaling as the square root of the lattice depth) in the\ndeep-lattice limit.  The ready tunability of the mechanical\nfrequency in the latter limit may allow for explorations of quantum\nopto-mechanical systems in various regimes, e.g.\\ in the the\nresolved side-band regime where ground-state cavity cooling and also\nquantum-limited motional amplification are possible\n\\cite{marq07sideband,wils07groundstate}. Second, we see that the\nmechanical excitation frequency has a significant quasi-momentum\n(Doppler) dependence in the shallow-lattice limit. This dependence\nmakes it advantageous to use low-temperature Bose-Einstein\ncondensates for experiments of opto-mechanics, as indeed achieved in\nthe Z\\\"{u}rich experiments, so as to minimize the Doppler width of\nthe Bragg excitation frequency. In contrast, the excitation\nbandwidth is dramatically reduced (exponentially with the lattice\ndepth) in the deep-lattice limit. Thus, one can conduct\nopto-mechanics experiments with long-lived mechanical resonances in\nthe deep-lattice limit without bothering to condense the atomic gas.\nNevertheless, we note that variations in the mechanical frequency\ndue to the presence of significant radial motion (not considered in\nthis one-dimensional treatment) do indeed limit the mechanical\nquality factor in the Berkeley experiments.\n\n\\section{Effects of the conservative optomechanical potential: optomechanical bistability}\n\\label{sec:bistab}\n\nThe observation of cavity nonlinearity and bistability arising from\ncollective atomic motion is described in recent work\n\\cite{gupt07nonlinear}.  Briefly, we find that the optical force due\nto cavity probe light will displace the equilibrium collective\natomic position $\\langle \\hat{Z} \\rangle$, leading to a\nprobe-intensity-dependent shift of the cavity resonance frequency.\nBy recording the cavity transmission as cavity probe light was swept\nacross the cavity resonance, we observed asymmetric and shifted\ncavity resonance lines, and also hallmarks of optical bistability.\n\nRefractive optical bistability is well studied in a variety of\nexperimental systems \\cite{boyd03}.  One unique aspect of our\nexperiment is the observation of both branches of optical\nbistability at average cavity photon numbers as low as $0.02$.  The\nroot of such strong optical nonlinearities is the presence within\nthe cavity of a medium that (1) responds significantly to the\npresence of infrequent cavity photons (owing to the strong\ncollective cooperativity) and (2) recalls the presence of such\nphotons for long coherence times (here, the coherence is stored\nwithin the long-lived collective motion of the gas).  It is\ninteresting to consider utilizing such long-lived motional\ncoherence, rather than the shorter-lived internal state coherence\ntypically considered, for the various applications of cavity QED and\nnonlinear optics in quantum information science, e.g.\\ photon\nstorage and generation, single-photon detection, quantum logic\ngates, etc.\n\nSuch motion-induced cavity bistability can also be understood in the\ncontext of the opto-mechanical forces described by Eq.\\\n\\ref{eq:optoforce}.  Neglecting the non-adiabatic following of the\ncavity field to the collective motion (essentially taking $\\kappa \/\n\\omega_z \\rightarrow \\infty$ so that dynamical backaction effects\nare neglected) and also the quantum fluctuations of the cavity\nfield, we may regard atomic motion in an optically driven cavity to\nbe governed by an opto-mechanical potential of the form\n\\begin{equation}\nU(Z) = \\frac{1}{2} M \\omega_z^2 Z^2 + n_{\\mbox{\\scriptsize{max}}} \\hbar \\kappa \\,\n\\arctan\\left( \\frac{\\Delta_{pc} - F Z \/ \\hbar}{\\kappa}\\right)\n\\label{eq:ucoll}\n\\end{equation}\nHere $\\Delta_{pc}$ is the detuning of the constant frequency probe from the\nmodified cavity resonance frequency $\\omega_c^\\prime$, and $n_{\\mbox{\\scriptsize{max}}}$\nis the average number of cavity photons when the cavity is driven on\nresonance.\n\nThe form of this potential is sketched in Fig.\\\n\\ref{fig:bistablepotential} for different operating conditions of\nthe atoms-cavity system.  Cavity bistability \\cite{gupt07nonlinear}\nis now understood as reflecting an effective potential for the\ncollective atomic variable $Z$ that has two potential minima.\nRemarkably, these potential minima may be separated by just\nnanometer-scale displacements in $Z$.  Even though the inherent\nquantum position uncertainty of each individual atom (10's of nm) is\nmuch larger than this separation, the reduced uncertainty in the\ncollective variable $Z$ allows for these small displacements to\nyield robust and distinct experimental signatures in the cavity\ntransmission.\n\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file = bistablepotential.eps, height =2in}\n\\end{center}\n\\caption{\\small For different stable regimes of cavity operation,\nthe cavity-relevant collective mode of the intracavity atomic\nensemble is trapped in a particular minimum of the effective\npotential $U(Z)$ (bottom).  In the regime of bistability, the two\nstable cavity states reflect the presence of two potential minima.}\n\\label{fig:bistablepotential}\n\\end{figure}\n\n\n\\section{Quantum fluctuations of the optomechanical potential: measurement backaction}\n\\label{sec:backaction}\n\nAside from the conservative forces described above, the intracavity\natomic medium is subject also to dipole force fluctuations arising\nfrom the quantum nature of the intracavity optical field. Indeed,\nshould these force fluctuations be especially large, the picture of\ncavity optical non-linearity and bistability described in the\nprevious section, in which we implicitly assume that the collective\natomic motion may follow adiabatically into a local minimum of an\nopto-mechanical potential, must be dramatically modified.  To assess\nthe strength of such force fluctuations, let us consider the impact\non the atomic ensemble of a single photon traversing the optical\ncavity.  During its residence time of $\\sim 1 \/ 2 \\kappa$, such a\nphoton would cause a dipole force that imparts an impulse of $\\Delta\nP = f\/(2 \\kappa)$ on the atomic medium, following which the\ncollective mode is displaced by a distance $\\Delta Z = \\Delta P \/ (M\n\\omega_z)$; in turn, this displacement will shift the cavity\nresonance frequency by $F \\Delta Z \/ \\hbar$.  Comparing this\nsingle-photon-induced, transient frequency shift with the cavity\nhalf-linewidth leads us to define a dimensionless ``granularity\nparameter'' as\n\\begin{equation}\n\\epsilon = \\sqrt{ \\frac{F \\Delta Z}{\\hbar \\kappa}} = \\frac{F\nZ_{\\mbox{\\scriptsize{ho}}}}{\\hbar \\kappa},\n\\end{equation}\nwhere $Z_{\\mbox{\\scriptsize{ho}}} = \\sqrt{\\hbar \/ 2 M \\omega_z}$ is the harmonic\noscillator length for the atomic collective mode.  The condition\n$\\epsilon > 1$ marks the granular (or strong) opto-mechanical\ncoupling regime in which the disturbance of the collective atomic\nmode by single photons is discernible both in direct quantum-limited\nmeasurements of the collective atomic motion and also in subsequent\nsingle-photon measurements of the cavity resonance frequency.  In\nour experiments, the granularity parameter is readily tuned by\nadjusting frequency difference between the cavity and atomic\nresonance $\\Delta_{ca}$.  Under conditions of our recent work, the\ngranular regime is reached at  $|\\Delta_{ca}|\/(2\\pi) < 27$ GHz.\n\n\nIn recent work, we have focused on effects of fluctuations of the\ndipole force in the non-granular regime, attained at atom-cavity\ndetunings in the 100 GHz range.  As described in our work\n\\cite{murc08backaction}, and also derived in earlier treatments\n\\cite{hora97,vule00,murr06,marq07sideband}, these fluctuations will\ncause the motional energy of the collective atomic mode to vary\naccording to the following relation:\n\\begin{equation}\n\\frac{d}{d t} \\langle a^\\dagger a \\rangle = \\kappa^{2} \\epsilon^2\n\\left[S_{nn}^{(-)} + \\left(S_{nn}^{(-)} - S_{nn}^{(+)}\\right) \\langle\na^\\dagger a \\rangle \\right] \\label{eq:adaga}\n\\end{equation}\nHere, the relevant dipole force fluctuations are derived from the\nspectral density of intracavity photon number fluctuations at the\nmechanical frequency $\\omega_z$, calculated for a\ncoherent-state-driven cavity as $S_{nn}^{(\\pm)} = 2 \\langle n \\rangle\n\\kappa (\\kappa^{2} + (\\Delta_{pc} \\pm \\omega_z)^{2})^{-1}$.  Eq.\\\n\\ref{eq:adaga}, which can be derived readily from a rate-equation\napproach \\cite{marq07sideband}, reveals two manners in which the\nmechanical oscillator responds to a cavity optical probe: momentum\ndiffusion, which raises the mechanical oscillator energy at a\nconstant rate, and the dynamic backaction effects of cavity-based\ncooling or amplification of the mechanical motion, described by an\nexponential gain or damping.\n\nThe mechanical momentum diffusion in an opto-mechanical system plays\nthe essential metrological role of providing the backaction\nnecessary in a quantum measurement, as discussed, for example, by\nCaves in the context of optical interferometry \\cite{cave81}.  For\n$\\omega_z \\ll \\kappa$, we see that, at constant circulating power in\nthe cavity, this diffusion is strongest for probe light at the\ncavity resonance and weaker away from resonance.  This dependence on\nthe intensity and detuning of the cavity probe light precisely\nmatches the rate of information carried by a cavity optical probe on\nthe state of the mechanical oscillator.  To elucidate this point, we\nrecall that, under constant drive by a monochromatic input field,\nthe intracavity electric field oscillates at the input field\nfrequency with complex amplitude $E_{cav} = \\eta \/ (\\kappa - i\n\\Delta_{pc})$.   A displacement by $\\Delta Z$ of the mechanical oscillator\nvaries the probe-cavity detuning by $F \\Delta Z \/ \\hbar$. In\nresponse, the electric field in the cavity varies as\n\\begin{equation}\nE_{cav} \\simeq E_0 \\left( 1 + \\frac{i}{\\kappa - i \\Delta_{pc}} \\frac{F\n\\Delta Z}{\\hbar} \\right) = E_0 + E_{sig},\n\\end{equation}\nwhere $E_0$ is the cavity field with the cantilever at its\nequilibrium position and we expand to first order in $\\Delta Z$. The\nsensitivity of the cavity field to the cantilever displacement, at\nconstant intracavity intensity (constant $E_0$), is determined by\nthe magnitude of $|E_{sig}\/E_0|^2 \\propto 1 \/ (1 + \\Delta_{pc}^2 \/\n\\kappa^2)$; this functional dependence matches that of the momentum\ndiffusion term, supporting its representing  measurement backaction.\n\nTo measure this backaction heating, we take advantage of several\nfeatures of our experiment.  First, by dint of the low temperature\nof our atomic ensemble, we ensure that the effects of dynamical\nbackaction (cooling and amplification) are negligible.  Second, the\nlow quality-factor of our mechanical oscillator ensures that the\nmomentum diffusion of the collective atomic motion leads to an\noverall heating of the atomic ensemble, allowing us to measure this\ndiffusion bolometrically.  Third, the large single-atom\ncooperativity in our cavity QED system implies that this backaction\nheating of the entire atomic ensemble dominates the single-atom\nheating due to atomic spontaneous emission.  And, fourth, owing to\nthe finite, measured depth of our intracavity optical trap,\nbackaction heating can be measured via the light-induced loss rate\nof atoms from the trap.  The measured light-induced heating rate was\nfound to be in good agreement with our predictions, providing the\nfirst quantification of measurement backaction on a macroscopic\nobject at a level consistent with quantum metrology limits.\n\n\\section{Future developments: cavity QED\/atom chips}\n\nWhile continuing explorations of quantum opto-mechanics in our\nexisting apparatus, we are also developing an experimental platform\nthat integrates the capabilities of single- and many-atom cavity QED\nonto microfabricated atom chips.  Similar platforms have been\ndeveloped recently by other groups \\cite{tepe06res,colo07}. Aside\nfrom enabling myriad applications in quantum atom optics and atom\ninterferometry, we anticipate the cavity QED\/atom chip to provide\nnew capabilities in cold-atoms-based opto-mechanics. For instance,\nthe tight confinement provided by microfabricated magnetic traps\nwill allow atomic ensembles to be confined into single sites of the\nintracavity optical lattice potential, providing a means of tuning\nthe opto-mechanical coupling between terms linear or quadratic in\n$\\hat{Z}$.  As emphasized by Harris and colleagues\n\\cite{thom08membrane}, a purely quadratic coupling may allow for\nquantum non-demolition measurements of the energy of the macroscopic\nmechanical oscillator.\n\n\\begin{figure}\n\\begin{center}\n\\psfig{file = atom_chip_figure.eps, height = 2 in}\n\\end{center}\n\\caption{An integrated cavity QED\/atom chip. (a) A top view of the\nmicrofabricated silicon chip shows etched trenches, later\nelectroplated with copper and used to tailor the magnetic field\nabove the chip surface.   The left portion of the image shows wire\npatterns used for producing the spherical-quadrupole field of a\nmagneto-optical trap and also the Ioffe-Pritchard fields for\nproducing stable magnetic traps.  Serpentine wires spanning the\nentire chip form a magnetic conveyor system to translate atoms to\nthe optical cavities that are located in the right half of the\nimage.  (b) A detailed view shows the serpentine wires and also a\ntwo-wire waveguide surrounding a central, rectangular hole that\npierces the atom chip.  (c) Fabry-Perot cavities are formed by\nmirrors straddling the atom chip. Between the mirrors, the chip is\nthinned to below 100 $\\mu$m -- outlines of the thinned areas are\nseen also in (a).  The cavity mode light passes unhindered through\nthe chip via the microfabricated holes shown in (b).}\n\\label{fig:atomchip}\n\\end{figure}\n\n\n\n\\bibliographystyle{ws-procs9x6}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} The CFHT Adaptive Optics Bonnette (AOB)\nprovides excellent correction using much fainter guide stars than\nprevious systems so that observations of many extragalactic targets are\nnow possible, using either a nearby reference star or the object\nitself. A wide variety of extragalactic objects have now been observed\nwith AOB and some of these are used as examples to discuss some of the\nproblems encountered and to demonstrate the potential of adaptive\noptics for studies of the high redshift universe.\n\n\\section{The Adaptive Optics Bonnette} \nThe CFHT AOB is based on a curvature wavefront system (Roddier et al. 1991)\nwith 19 subapertures.  A complete description of the system is given by\nArsenault et al. (1994), and a comprehensive discussion of its\nperformance by Rigaut et al. (1997).  If the target is to be observed in\nthe infrared, a dichroic is used to reflect the visible light to\nthe wavefront sensor (WFS) while transmitting the IR light to the\nscience detector, but if the target is to be observed in the visible, a\nbeamsplitter must be used to send a percentage of the light to the\nWFS.  The delivered image quality depends on the brightness of the\nreference star, its distance from the target, the seeing at the time of\nobservation and the wavelength of observation. To gain an appreciation\nof how these variables affect the images and what can be expected, the\nreader is encouraged to visit the ``performance meter'' at\nhttp:\/\/www.cfht.hawaii.edu\/manuals\/aob\/psf.html  To achieve FWHM\n$\\sim$0\\farcs12 imaging in the near-IR (diffraction-limited) and  at\n$I$ under median seeing conditions, the reference star must be brighter\nthan R $\\sim$14 and within $\\sim$30\\arcsec\\ of the target. However, the\nsystem will provide good correction on stars as faint as R = 17 under\ngood seeing conditions, and the reference star can be located anywhere\nwithin the 90\\arcsec\\ diameter field.  AOB offers considerable\nadvantages in terms of operational efficiency, since it is literally a\npush-button operation  that eliminates all the overhead of focussing\nand guiding associated with conventional observing.\n\nThe observations reported here were all made with the University of\nMontreal infrared camera MONICA (Nadeau et al. 1994) which was modified\nto give a pixel scale of 0\\farcs034 and hence a field of\n8\\farcs8$\\times$8\\farcs8.  This very small field is a significant\nhandicap for many observations, as is the fact that the detector\nsuffers from persistence problems, i.e., a bright source leaves a\nresidual image that slowly decays with time.  Since the site seeing is\nvariable, observations of relatively bright stars usually {\\it have} to\nbe carried out in order to monitor the PSF (point-spread function). The\nsequence of observations of very faint sources thus has to be carefully\nplanned to minimize such problems while adequately monitoring the PSF\nto attain the scientific goals.  V\\'eran et al. (1997) have recently\ndemonstrated that statistics of the wavefront sensor signals can be\nused to derive an excellent model of the average PSF. This is obviously\nideal in that  the synthetic PSF is simultaneous with the the target\nobservation and it obviates the requirement of directly observing the\nbright star.\n\n \n\\section{Examples of Self-referencing Targets} \nObjects such as the nuclei of nearby galaxies, seyferts and AGN can be\nusually guided on directly, i.e., the nucleus is often sufficiently\npoint-like to enable the WFS to function properly as long as there is\nsufficient flux.  For example guiding on the nucleus of M31 was\nstraightforward, resulting in diffraction-limited images (FWHM =\n0\\farcs12) of the stars in the nuclear bulge region, despite the fact\nthat it is double with a fainter component at a distance of\n$\\sim$0\\farcs5. The near-IR image quality is comparable to that of\nWFPC on $HST$, enabling accurate magnitudes and colors to be\ndetermined for a significant number of stars in the bulge (Davidge et\nal.  1997).  During the commissioning period, other nearby galaxies,\nAGN and Seyfert 1 type galaxies with point-like nuclei and magnitudes\nin the range m = 11--14 (e.g., bright ones like NGC 4151 to fainter\nones like NGC 6814) were observed with AOB, guiding on the bright\nnuclei themselves. However, guiding was  not successful on more diffuse\nobjects such as the V = 13 nucleus of the Seyfert 2 galaxy Mrk 266, or\nthe large bright elliptical galaxy which hosts 3C296.\n\nThe small MONICA  field  complicates observations of such targets since\nneither the sky nor PSF stars are normally included in the frames.  An\naccurate estimate of the PSF is essential for these sources in order to\nadequately subtract the effects of the bright point source from the\nnuclear region, so a nearby reference star of similar brightness must\nbe monitored throughout the sequence of observations.\n\n\\section{Targets with guide stars}\nNearby guide stars must be used for targets which are not sufficiently\nconcentrated or bright enough for the WFS to perform adequately. In\nthis case, not only are observations of the guide star itself required\nto monitor the PSF variations, but measurements of starfields (e.g.,\nglobular cluster fields) are also required to estimate the degradation\nof the PSF due to anisoplanatic effects.  Steinbring (1997) has developed\nsemi-empirical software that models this degradation over the field as\na function of the seeing and produces an ``off-axis PSF\" that can be\nused to help analyse the target data. For some projects this is not\nimportant, but for the observations of quasar host galaxies, for\nexample, it is vital.  Studies of the latter are notoriously difficult\ndue both to the faintness of the galaxy itself plus the superposition\nof the bright nuclear source. Some quasars have already been\nobserved with AOB, both in the visible region (at $I$) with a CCD detector and\nin the near infrared (mostly at $H$).  Analysis of the z = 1.1 quasar,\n1055.3+019, which is well-resolved in both $I$ and $H$, has recently\nbeen published by Hutchings et al. (1998) who suggest that the host\ngalaxy has been undergoing a close encounter or merger event.\n\nThe components of multiply-imaged gravitational lenses formed by\ngalaxies in the line of sight to distant quasars typically have\nsubarcsecond separations and consequently their study and monitoring\ncould significantly benefit from the improved resolution offered by\nadaptive optics.  At present, about $\\sim$35 such lenses are known\n(Kochanek 1997) but virtually none of them have\nnearby guide stars that are sufficiently bright to give good\ncorrection. One exception is SBS 1520+530, a\ndoubly-imaged BAL quasar with a separation of 1\\farcs6 which happens to\nbe only 13\\arcsec\\ from a m$\\sim$12 star. $H$ band images with FWHM =\n0\\farcs15 taken with AOB reveal the lensing galaxy 0\\farcs40 from the\nfainter component, offset 0\\farcs12 from the line joining components A\nand B (Crampton, Schecter and Beuzit 1997).\n\n\\section{Registering invisible targets}\n\nTargets which are very faint, particularly those that are diffuse,\noften present additional problems since the MONICA field is so small\nthat there aren't any objects or features that can be used to register\nthe images. In general, near-IR images have to be dithered to remove the\neffects of bad pixels and to improve the flat-field, offsets are often\nrequired to enlarge the area surveyed, and differential flexure between\nthe WFS and the detector may produce additional shifts. Consequently,\nregistering and superimposing the images is not trivial. The AOB WFS\ncoordinates are recorded in the FITS headers and observations of star\nfields have been used to calibrate these positions in terms of pixel\nlocation on the detector and\/or equatorial coordinates.  Repeated\nobservations of the same point source (e.g., a quasar) over several\nhours show that the WFS positions can be used to register frames with a\ndispersion of 1.4 pixels or 0\\farcs05. Since the delivered images in\nthe near-IR usually have FWHM $\\sim$0\\farcs12, this is clearly\ninadequate, so future systems should be designed with a low-order WFS\nincorporated close to the detector  to minimize flexure and allow for\nsub-pixel ``drizzling\" and registration.  Ideally, the coordinates of\nthis WFS should allow  diffraction-limited images to be registered to\nbetter than $\\sim$10\\% of their FWHM; for 8m telescopes this will be of\nthe order of a few mas.\n\nAlthough the 0\\farcs034 pixel scale is necessary to properly sample the\n$\\sim$0\\farcs1 resolution delivered by AOB, many extragalactic projects\nwould benefit from a larger field. KIR, a new 1K$\\times$1K camera with\nthree times higher sensitivity and a field of 36\\arcsec\\ has just been\ncommissioned, so registration will be easier. Even so, astrometry to\nthe precision that is possible with 0\\farcs1 images will be\nchallenging:  relative positions can be measured to an accuracy of a\nfew mas, but tying these to an absolute reference frame is difficult.\nSince many extragalactic targets are diffuse, a larger pixel scale\nwould not be a disadvantage for most investigations, and the larger\nfield would significantly improve the probability of including stars,\nproviding even better registration and astrometry.\n\n\nThe jet of 3C273 is an example of an object which is barely visible,\neven on background-subtracted 300s exposures at $H$ and $K$.  The jet\nis located between 11 -- 21\\arcsec\\ from 3C273 which, of magnitude R =\n13, was used as the guide star for AOB.  Given the small field of the\ndetector, the jet had to be observed at two different locations along\nthe jet and then all the frames were registered using WFS positions.\nUnfortunately the site seeing was poor on average and very variable\n(from 0\\farcs5 to 1\\farcs2) during the observations so, combined\nwith registration uncertainties, the resulting (preliminary)\nimage has a resolution of only $\\sim$0\\farcs3. Observations\nwith the new KIR camera should be significantly better.\n\n\n\n\\section{Summary}\nOur experience with AOB demonstrates that adaptive optic systems can be\nmade robust, reliable, efficient and user-friendly and will routinely\ndeliver 0\\farcs1 resolution images from $I$ to $K$ with substantial\ngains in the visible. AOB observations of many extragalactic targets\nare now feasible and competitive with $HST$.  Images of most of the\nobjects discussed in this article can be found on the web at\nhttp:\/\/www.hia.nrc.ca\/science\/instrumentation\/optical\/pueo2\/pueo2.html .\nSpectroscopic observations using instruments designed to exploit the\nAOB image quality are just beginning, although to obtain reasonable\nspectral resolution simultaneously with  high spatial resolution will\nrequire adaptive optic systems on 8--10m telescopes for the majority\nof extragalactic targets.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}