ISOTROPY Version 9.0 User`s Manual August 2007
Contents
1. rameter If more than one direction is selected then corresponding irreps must be selected first and the number of directions selected must equal the number of irreps selected The direction of an order parameter can be represented in several ways The most usual way is to simply enter for the parameter the symbol of the direction for example VALUE DIRECTION P1 to selecte the direction P1 VALUE DIRECTION KERNEL selects the order parameter in the most general direction VALUE DIRECTION ONEARM selects order param eters which arise from only one arm of the star of k These are implemented only for non k points of symmetry In addition VALUE DIRECTION ONEARM P1 selects the order parameter P1 among those which arise from only one arm of the star of k VALUE DI RECTION VECTOR A 0 0 selects the order parameter a 0 0 and VALUE DIRECTION VEC TOR O 5A 0 866A selects the order parameter Sa i v3a Note that irrational coefficients and sometimes rational coefficients such as 3 must be given to three decimal places VALUE DOMAIN pl p2 The domain of the isotropy subgroup is selected This command affects the data displayed by DISPLAY ISOTROPY when used with SHOW DOMAIN by DISPLAY DISTORTION when a subgroup has been selected with VALUE DIRECTION and by DISPLAY INVARIANTS when the direction of an order parameter has been selected with VALUE DIRECTION pl is a number indicating the domain The numbering of domains may be obtained by using SH2. without the keywords NEW or OLD then the setting is returned to whichever edition was used the last time the current setting was INTERNATIONAL When ISOTROPY starts the current setting is INTERNATIONAL NEW The SETTING command changes the notation used for irreps unless the setting is changed to INTERNATIONAL SETTING INTERNATIONAL pl This command allows additional options for the setting of space groups in International Tables Some space groups have two choices of origin When ISOTROPY is started origin choice 2 is used This is the choice with the point of inversion at the origin To change the origin choice for a particular space group use SETTING INTERNATIONAL pl ORIGIN p2 where pl is a space group number or symbol and p2 is 1 or 2 The origin choice for all space groups with more than one origin choice can be changed by using ALL for pl Note that the origin choice is changed only for the current setting i e the new or old edition not both For example SETTING INTERNATIONAL 228 ORIGIN 1 changes the setting to origin choice 1 for space group 228 Fd3c SETTING INTERNATIONAL ALL ORIGIN 1 changes the setting of all space groups which have two origin choices to origin choice 1 The monoclinic space groups have two choices for the unique axis When ISOTROPY is started unique axis b is used for the new edition and unique axis c is used for the old edition To change the choice of unique axis for a particular space group use SET
3. FEL pl p2 DIR pl p2 FREQ pl p2 DOM pl DOM P pl p2 DOM SET OR pl DOM SET CL pl DOM SET DIR pl DOM SET UN pl NORM pl p2 p3 POS pl p2 p3 SIZ pl p2 MAX pl SUB MAX SUB SUP LAT PAR pl p6 BAS pl p2 p3 ORI pl FER pl PAR SUB DIR p1 BAS ORI SUB ALT XYZ WY SUBG Summary of ISOTROPY commands Version 9 0 p 3 of 5 show dimension show Landau frequency show Lifshitz frequency show Felix frequency show type of irrep show faintness index for linear coupling with other irreps show frequency and irreps of the Wyckoff position point groups show coordinates of k point show degrees of freedom of k point show the star of k show compatibility relations display data on isotropy subgroups display data on isotropy subgroups of coupled order parameters select parent space group s select isotropy subgroup s select lattice s of subgroup select point group s of subgroup select irreps select k point select parameters a 8 y for the k points select image of irrep select dimension s of irrep select active or non active irreps pl Y N select continuous or discontinuous transitions pl LAN RG NO select Landau frequency frequencies select Lifshitz frequency frequencies select Felix frequency frequencies select directions of order parameter select subduction frequency frequencies select domain select domain pair select order of domain set select class of domain set select
4. Landau and Lifshitz frequencies are zero the Landau and Lifshitz conditions An image is active when at least one active irrep is mapped onto it Note that not all irreps mapped onto active images are active irreps Some of them may fail the Lifshitz condition SHOW BASIS The basis vectors are shown When DISPLAY PARENT is used the primitive basis vectors of the parent space group are shown If LABEL VECTOR CONVENTIONAL is used primitive basis vectors with respect to the conventional unit cell are shown When DISPLAY ISOTROPY is used the basis vectors of the isotropy subgroup are shown When used with LABEL VECTOR PRIMITIVE the primitive basis vectors of the subgroup are given in terms of the primitive basis vectors of the parent space group When used with LABEL VECTOR CONVENTIONAL the conventional basis vectors of the subgroup are given in terms of the conventional basis vectors of the parent space group SHOW CARTESIAN Cartesian coordinates are displayed The cartesian coordinates are defined by the VALUE LATTICE PARAMETER command This is only implemented a few places in ISOTROPY 1 DISPLAY PARENT basis vectors of the primitive lattice SHOW BASIS space group el ements displayed as a rotation matrix followed by a translation SHOW GENERATORS and SHOW ELEMENT and Wyckoff positions SHOW WYCKOFF VECTOR 2 DISPLAY ISOTROPY basis vectors of the primitive lattice SHOW BASIS and space group elements SHOW GENER ATORS and SHOW EL
5. VALUE SHOW and CANCEL commands DISPLAY ISOTROPY Data about isotropy subgroups are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands Isotropy subgroups for irreps associated with k lines and k planes of symmetry and general k vectors are not stored in the data base since they depend on the parameters a 3 y which define the exact location of the k vector The data for these subgroups are read from a file with a name of the form irrxxx_x iso ISOTROPY looks for this file in the user s directory If it is not found the user is prompted and if desired ISOTROPY will then proceed to calculate the requested subgroups and create the file Depending on the order parameter selected by VALUE DIRECTION other files may also be created without notifying the user For example if only the kernel has been selected a file of the form krxxxx_x iso will be created If only subgroups arising from one arm of the star of k has been selected a file of the form arxxxrx_x iso will be created If the direction of the order parameter has been explicitly selected then a file of the form vixrxn_x iso will be created DISPLAY ISOTROPY COUPLED Data about isotropy subgroups of coupled order parameters are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands A parent space group must be selected using VALUE PARENT and two or more irreps must be selected using VALUE IRREP The data for these subgr
6. any other isotropy subgroup for the same irrep VALUE NORMAL pl p2 p3 This command affects the data displayed by DISPLAY ISOTROPY when one of the SHOW TWIN commands is used This command selects the orientation of the plane between a domain pair pl p2 p3 are the Miller indices hkl of the plane All three numbers must be integers Note that if the primitive labeling of vectors is selected by the LABEL VECTORS PRIMITIVE command these Miller indices may not have their usual meaning VALUE ORDER pl p2 The orders of the image are selected This command affects the data displayed by DISPLAY IMAGE The parameters pl and p2 are numbers representing the order of an image number of distinct matrices in the image group If they are both present then orders p1 through p2 are selected If only p1 is present then only order p1 is selected VALUE ORIGIN pl The origin of a subgroup is selected This command affects the data displayed by DISPLAY DIRECTION pl is the position of the origin denoted by three numbers separated by commas The three numbers are coordinates in terms of the basis vectors of the lattice of the parent space group When used with DISPLAY ISOTROPY the alternate origin is selected See SHOW SUBGROUP ALTERNATE VALUE PARENT pl p2 The parent space groups are selected This command affects the data displayed by DISPLAY PARENT DISPLAY INVARIANTS DISPLAY DISTORTION DISPLAY IRREP DISPLAY ISOTROPY DISPLAY KPOINT and DISPLAY SU
7. be used INTERNATIONAL and SCHOENFLIES See the VALUE POINTGROUP command for more explanation about these notations LABEL SPACEGROUP pl The notation for the space group label is changed This affects the label displayed when SHOW PARENT or SHOW SUBGROUP are used The parameter p1 indicates the notation to be used INTERNATIONAL and SCHOENFLIES Also the international notation can be changed to the full symbol or the short symbol using LABEL SPACEGROUP INTERNATIONAL SHORT or LABEL SPACEGROUP INTERNATIONAL FULL respectively The full symbol reflects the choice of cell and unique axis for the monoclinic space groups while the short symbol does not The INTERNATIONAL label also reflects the setting used For example space group 12 is C2 m A2 m B2 m B2 m for the INTERNATIONAL MILLER LOVE BRADLEY CRACKNELL ZAK settings respectively When using the KOVALEV setting only the SCHOENFLIES labeling of space groups is shown LABEL VECTOR pl LABEL VECTOR PRIMITIVE causes the components of vectors and coordinates of points to be displayed in terms of primitive basis vectors LABEL VECTOR CONVENTIONAL causes the components of vectors and coordinates of points to be displayed in terms of conventional basis vectors For example a vector sai 5aj in a face centered cubic lattice would be displayed as 1 2 1 2 0 using conventional basis vectors and as 0 0 1 using primitive basis vectors This vector happens to be chosen for the third primitive basis v
8. by DISPLAY ISOTROPY The parameter pl is a symbol representing the ferroic species FC ferroelectric PF proper ferroelectric IFC improper ferroelectric FS fer roelastic PFS proper ferroelastic IFS improper ferroelastic NF nonferroic OTHER other ferroic VALUE FREQUENCY pl p2 The subduction frequency of the isotropy subgroup is selected This command affects the data displayed by DISPLAY ISOTROPY The parameters pl and p2 are numbers repre senting the subduction frequency If they are both present then subduction frequencies 19 pl through p2 are selected If only pl is present then only subduction frequency pl is selected VALUE GRADIENT pl The number of spatial derivatives in invariant polynomials is selected This command affects the data displayed by DISPLAY INVARIANTS The parameter p1 is a number repre senting the number of derivatives to appear in each invariant polynomial VALUE IMAGE pl The image of the irrep is selected This command affects the data displayed by DISPLAY IMAGE DISPLAY IRREP and DISPLAY ISOTROPY The parameter pl is an image to be selected The image is denoted by a symbol using either the notation of Stokes and Hatch or the notation of Tol dano and Tol dano VALUE IRREP pl p2 The irreps are selected This command affects the data displayed by DISPLAY INVARIANTS DISPLAY DISTORTION DISPLAY IRREP DISPLAY ISOTROPY and DISPLAY KPOINT The parameters pl p2 are irreps to
9. by the VALUE DOMAIN PAIR command Otherwise the pair P P for each domain j will be used In order to display information about twins a plane must be specified by selecting its normal position using the VALUE NORMAL command and a point on the plane using the VALUE POSITION command If pl is INTERSECT then information about the twin intersection group is displayed Each element in this group must obey the following requirements It must be a member of each of the two isotropy subgroups associated with the two domains in the pair This means that this group is a subgroup of the pair intersection group If the element operates on a point in the specified plane the point must stay in the plane This means that the group must be diperiodic If the point operator part of the element operates on a vector perpendicular to the specified plane the direction of the vector must stay the same To display information about twin intersection groups an additional keyword p2 must also be present If p2 is GROUP BASIS ORIGIN GENERATORS or ELEMENTS then the diperiodic space group label the basis vectors of the lattice the origin of the space group the generating elements or the complete list of elements of the twin intersection group are displayed respectively These commands are very similar to the SHOW SUBGROUP SHOW BASIS SHOW ORIGIN SHOW GENERATORS and SHOW ELEMENTS commands used with the DISPLAY ISOTROPY command Information which is disp
10. direction of domain set vector select number of unconnected parts in domain set select Miller indices hkl of plane between twin domains select point on plane between twin domains select size of subgroup s primitive unit cell relative to parent group s select maximal or non maximal subgroups p1 Y N select maximal subgroups of the parent show superspace groups for incommensurate distortions select lattice parameters a b c a B Y select alternate basis vectors select alternate origin select ferroic species pl FC IFS NF OTHER show parent space group symbol show isotropy subgroup symbol show direction of order parameter p1 VEC to show the vector show basis vectors of subgroup show origin of subgroup show alternate basis vectors and origin of subgroup show x y z in subgroup in terms of x y z in parent group show Wyckoff positions in subgroup SH GEN SH EL SH NEW SH SIZ SH IND SH DOM p1 SH DOM SET pl SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH SH ssacgasaasacsasagasaagagaascac v0 DISTINCT PAIR PAIR pl TWIN INT pl TWIN pl TWIN MAX LAT POINT IR pl IM DIM ACT CON LAN LIF FEL FR p1 CART FER DIST PAR pl IR pl KP pl SW pl KVAL pl p2 RANK pl DIR pl p2 DOM pl CELL pl p2 p3 LAT PAR pl p6 WY pl p2 WY IR pl p2 WY XYZ pl p3 POS pl p2 p3 SUB SUP SH MAC pl SH MIC pl p2 S
11. in the table below If only pl is present then only point group p1 is selected The point groups are denoted by a symbol using either the Schoenflies or international notation see table below For example VALUE POINTGROUP OH and VALUE POINTGROUP M 3M both refer to the point group Op or m3m A bar over a number is denoted by a preceding minus sign eg 4 for 4 Point Group Symbols 1C 1 9 lt C4 4 17 C3I 3 25 C ev 6MM DCT ot 10 S4 4 18 D3 32 26 D3H 62M Beco 2 11 C4H 4 19 C3V 3M 27 D6H 6 MMM 4 CS M 12 D4 422 20 D3D 3M 28 T 23 5 C2H 2 M 13 C4V 4MM 21 C6 6 29 TH M3 6 D2 222 14 D2D 42M 22 C3H 6 30 0 432 7 C2V MM2 15 D4H 4 MMM 23 C6H 6 M 31 TD 43M 8 D2H MMM 16 C3 3 24 D6 622 32 0H M 3M VALUE POSITION pl p2 p3 The coordinates of a point are selected This command affects the data displayed by DISPLAY DISTORTION when SHOW LOCAL is used The coordinates are assumed to be in terms of the basis vectors of the lattice using the setting chosen at the time when DISPLAY DISTORTION is used Each coordinate must be given in terms of rational numbers For example 1 2 1 2 1 2 would denote the coordinates at 5 5 4 Also irrational coordi nates can be denoted by x y or z For example x 1 2 x 0 would denote coordinates at x 4 x 0 where x is an arbitrary irrational number The coordinates x y z would denote a general point 23 The command VALUE POSITION also affects the data displayed by DISP
12. of anisotropic gradient terms in the LGW Hamiltonian 10 SHOW FERROIC The ferroic species of the phase transition is shown when DISPLAY ISOTROPY is used See VALUE FERROIC for an explanation of the symbols SHOW FREQUENCY p1 When DISPLAY IRREP is used the point group irreps of the Wyckoff positions which induce the space group irrep are shown along with the subduction frequency If pl is VECTOR only the point group irreps which induce vector irreps of the space group irrep are shown along with the number vector irreps which can be induce This number will be some multiple of the subduction frequency since there may be more than one independent set of vector basis functions for the point group irrep When DISPLAY ISOTROPY is used the irreps which subduce the isotropy subgroup are shown along with the subduction frequency For each irrep there is a direction of an order parameter which remains invariant under the operation of each element in the isotropy subgroup If pl is DIR the direction of that order parameter is shown In parentheses is shown the domain number where you can find that direction if you display the isotropy subgroups for that irrep If pl is GAMMA only the I k 0 irreps are shown If pl is CLASSIFICATION the symbol F for full is shown when the distortions associated with the irrep fully classify the domains of the subgroup Otherwise the symbol P for partial is shown SHOW GENERATORS When use
13. representatives of space group show lattice show point group show basis vectors of lattice show Wyckoff positions p1 VEC POINT ELE CHAR p2 ALL display data on 3 1 dimensional superspace group select superspace group s select basic space group s select lattice s of the basic space group select point group s of the basic space group show superspace group symbol show generating elements of superspace group show coset representatives of superspace group show lattice of superspace group show point group of basic space group D V V V V SH SH SH SH KP PAR pl p2 IR pl KP pl KDEG pl PAR KP KDEG STAR IM IM pl DIM pl p2 ORD pl p2 ACT pl LAN pl p2 TYPE pl IM GEN EL DIM ORD ACT LAN TYPE IR PAR pl p2 IR pl KP pl KDEG pl KVAL pl p2 IM pl EL pl DIM pl p2 ACT pl LAN pl p2 LIF pl p2 FEL pl p2 TYPE pl WY pl p2 WY IR pl COMPAT pl SUB SUP PAR IR pl ACT IM GEN EL CH MAT COMPLEX KER Summary of ISOTROPY commands Version 9 0 p 2 of 5 display data on k points select space group s select irrep select k point select degrees of freedom of k point show space group symbol show coordinates of k point show degrees of freedom of k point show the star of k display data on images of irreps select image select dimension s select order s select active or non active images pl Y N select Landau frequenc
14. selected This command affects the data displayed by DISPLAY DISTORTION pl p2 p3 are the three vectors Each vector is denoted by three numbers separated by commas The three numbers are components of the vector in terms of the basis vectors of the lattice For example 2 0 0 denotes a vector which is two times the first basis vector of the lattice Components may be fractions if the conventional form of the vector has been chosen with LABEL VECTOR CONVENTIONAL Each vector must be a vector of the primitive lattice For example 1 2 1 2 0 would be a vector of a face centered lattice When a super cell has been selected DISPLAY DISTORTION shows all atoms in the super cell VALUE COMPATIBILITY pl The compatibility relations to be shown by DISPLAY IRREP are selected p1 is the label of a k vector See the SHOW COMPATIBILITY command VALUE CONTINUOUS pl Phase transitions which may or may not be allowed to be continuous in Landau theory or in RG theory are selected when DISPLAY ISOTROPY is used VALUE CONTINUOUS RG selects isotropy subgroups to which a phase transition is allowed to be continuous in RG theory VALUE CONTINUOUS LANDAU selects isotropy subgroups to which a phase transition is also allowed to be continuous in Landau theory and VALUE CONTINUOUS NO selects isotropy subgroups to which a phase transition is not allowed to be continuous in either theory In Landau theory a phase transition to a particular subgroup is allowed to be continuous
15. to the origins of the space groups When used with LABEL VECTOR PRIMITIVE the coordinates are given in terms of primitive basis vectors When used with LABEL VECTOR CONVENTIONAL the coordinates are given in terms of conventional basis vectors In the case of centered Bravais lattices the conventional basis vectors are not primitive VALUE ACTIVE pl When DISPLAY IMAGE is used VALUE ACTIVE YES selects active images and VALUE ACTIVE NO selects images which are not active When DISPLAY IRREP or DISPLAY ISOTROPY is 16 used VALUE ACTIVE YES selects active irreps and VALUE ACTIVE NO selects irreps which are not active An irrep is active when both its Landau and Lifshitz frequencies are zero the Landau and Lifshitz conditions An image is active when at least one active irrep is mapped onto it Note that not all irreps mapped onto active images are active irreps Some of them may fail the Lifshitz condition VALUE BASIS pl p2 p3 The basis vectors of a subgroup are selected This command affects the data displayed by DISPLAY DIRECTION pl p2 p3 are the three vectors Each vector is denoted by three numbers separated by commas The three numbers are components of the vector in terms of the basis vectors of the lattice of the parent space group See VALUE CELL below for examples When used with DISPLAY ISOTROPY the alternate basis vectors are selected See SHOW SUBGROUP ALTERNATE VALUE CELL pl p2 p3 The basis vectors of a super cell are
16. 007 24 VALUE TYPE pl The irrep type 1 2 3 is selected This command affects the data displayed by DISPLAY IRREP and DISPLAY IMAGE A type 1 irrep is real A type 2 irrep is complex but its characters are real A type 3 irrep is complex and its characters are also complex VALUE WYCKOFF pl p2 Wyckoff positions are selected This command affects the data displayed by DISPLAY PARENT DISPLAY IRREP and DISPLAY DISTORTION The parameters pl p2 are the single letter symbols of the positions as given in International Tables VALUE WYCKOFF IRREP pl p2 Irreps of the point group of a Wyckoff position are selected This command affects the data displayed by DISPLAY PARENT DISPLAY IRREP and DISPLAY DISTORTION The pa rameters pl p2 are the irrep symbols using the convention of Bradley and Cracknell VALUE WYCKOFF XYZ pl p2 p3 The parameters x y z of the Wyckoff position are selected pl p2 p3 are the values of X Y z respectively 25 Q sc PA NOUODAQAAAQAAaA D SE SE SE SE SE SE ssacgaaaa Sc 0JPrrreeer SH D V V V V SH SH SH SH SH pl GE pl Veit V ALL SH SH ALL V SH SET T pl T I pl OR p2 T I pl AX p2 T I pl CELL p2 T MAG T NOMAG SP pl EL pl VEC pl IM pl PO pl LAT pl PAR PAR pl p2 LAT PAR pl p6 LAT pl p2 POINT pl p2 WY pl p2 WY IR pl p2 WY XYZ pl p3 PAR CART GEN EL LAT POINT BAS WY pl p2 SUP SUP pl p2 P
17. 3 VALUE KVALUE O 1 1 4 selects values for two k vectors the first one at a k point of symmetry and the second one at a k line of symmetry Each value must be given as a ratio of two integers as shown in the examples VALUE LANDAU pl p2 The Landau frequency of the image is selected This command affects the data displayed by DISPLAY IMAGE DISPLAY IRREP and DISPLAY ISOTROPY The parameters pl and p2 are numbers representing the Landau frequency of an image the number of independent third degree invariants If they are both present then Landau frequencies p1 through p2 are selected If only pl is present then only Landau frequency p1 is selected VALUE LATTICE pl p2 When DISPLAY PARENT is used the Bravais lattices of the space group are selected When DISPLAY ISOTROPY is used the Bravais lattices of the isotropy subgroup are selected When DISPLAY SUPERSPACEGROUP is used the Bravais lattices of the basic space group of the 3 1 dimensional superspace group are selected The parameters pl and p2 are lattices If they are both present then lattices pl through p2 are selected using the order shown in the table below If only p1 is present then only lattice p1 is selected The lattices are denoted by a symbol using either the Schoenflies notation or the Pearson notation see table below For example VALUE LATTICE TI and VALUE LATTICE Q V both select the body centered tetragonal lattice Lattice Symbols Lattice Symbols 1 triclin
18. AR pl p2 LAT pl p2 POINT pl p2 SUP GEN EL LAT POINT Summary of ISOTROPY commands Version 9 0 p 1 of 5 quit width of screen number of lines displayed at a time cancel a V command cancel all previously used V commands cancel a SH command cancel all previously used SH commands display all V commands in effect display all SH commands in effect help can be inserted in place of any keyword display the current space group setting change the setting pl I I NEW I OLD MI K B Z origin choice p1 space group or ALL and p2 1 2 axis choice pl space group or ALL and p2 B C HEX RH cell choice p1 space group or ALL and p2 1 2 3 magnetic space groups return to Federov space groups change the labeling of space groups pl SCH I I SHORT I FULL change the labeling of space group elements p1 I M K B Z X1 XYZT change the form of the vectors pl CON PRIM use image notation of Tol dano and Tol dano p1 TOL NOTOL change the labeling of point groups p1 SCH I change the labeling of lattices p1 SCH P display data on parent space group select space group s select lattice parameters a b c a B Y select lattice s select point group s select wyckoff positions select point group irrep of wyckoff position select parameters x y z of wyckoff position show space group symbol show Cartesian coordinates show generating elements of space group show coset
19. EMENT 3 DISPLAY DISTORTION Wyckoff positions and displacement vectors SHOW MICROSCOPIC In the case of Wyckoff positions the values of the param eters x y z must be selected by VALUE WYCKOFF XYZ if needed Vectors must be given in terms of conventional basis vectors not primitive The cartesian coordinate system is chosen so that the x axis lies along the a axis and the y axis lies in the ab plane SHOW CHARACTER The irrep character for an element of the parent space group is shown when DISPLAY IRREP is used The SHOW ELEMENT command is also automatically executed The element is selected with the VALUE ELEMENT command The character is the trace of the matrix onto which the irrep maps the element of the space group SHOW COMPATIBILITY The compatibility relations for an irrep is shown when DISPLAY IRREP is used The irrep of the little group of k is decomposed into irreps of the little group of k where k is some k vector with more degrees of freedom than k and contains k as a subspace of its domain For example if k is a point of symmetry then k may be a line of symmetry which contains that point k may be selected by the VALUE COMPATIBILITY command If not selected then relations for every possible k with one degree of freedom less than that of k are shown SHOW COMPLEX The complex form of an irrep is shown when DISPLAY IRREP is used SHOW CONTINUOUS The phase transitions allowed to be continuous in Landau theory
20. H LOC p1 p2 SH PAR SH IR pl Summary of ISOTROPY commands Version 9 0 p 4 of 5 show elements of parent group which generate subgroup show elements of parent group which are elements of subgroup show new fractionals in subgroup show relative size of subgroup s primitive unit cell to parent group s show index of subgroup in parent group show domains p1 GEN to show generators show domain sets p1 GR to show set group INT to show intersection group BAS to show basis vectors OR to show origin GEN to show generators ELE to show elements ALL to show equivalent sets or directions EQ to show equivalence operators DIR to show domain set vectors show which domains are distinct show which pairs of domains are equivalent show pair intersection group pl I pair group pl GR and its basis vectors pl B its origin pl 0 its generating elements pl GE and its elements pl E show twin intersection group its label pl GR its basis vectors pl B its origin pl 0 its generating elements pl GE and its elements pl E show twin group its label p1 GR its basis vectors pl B its origin pl 0 its generating elements pl GE and its elements pl E show switching elements in twin group pl SIDE NORMAL BOTH show whether or not the subgroup is maximal show lattices of parent space group and of subgroup show point groups of parent space group and of subgroup show irr
21. ISOTROPY Version 9 0 User s Manual August 2007 Harold T Stokes Dorian M Hatch and Branton J Campbell Department of Physics and Astronomy Brigham Young University ISOTROPY version 9 0 August 2007 ISOTROPY is a program which displays information about space groups irreducible repre sentations isotropy subgroups and phase transitions The concepts and alogrithms used in ISOTROPY were developed by Harold T Stokes Dorian M Hatch and Branton J Camp bell The actual computer source code was written by Harold T Stokes Questions about using ISOTROPY may be directed to Harold T Stokes via e mail at stokesh byu edu This software may be distributed without restriction but if it is used in research that results in publications the use of this program should be acknowledged with reference to H T Stokes D M Hatch and B J Campbell To run ISOTROPY type iso After a pause the program will respond with a message and then a prompt character indicating that it is ready for a command The user then enters commands which causes ISOTROPY to display the desired data All entered commands as well as the displayed data are copied into a file iso log in the current default directory This file can be later examined or printed out ISOTROPY has access to six different space group settings found in the following references 1 International Tables for Crystallography Vol A edited by T Hahn Reidel Boston 1983 I
22. LAY ISOTROPY when one of the SHOW TWIN commands are used In this case a point on the plane between the pair of domains is selected VALUE RANK pl The rank of a macroscopic tensor is selected This command affects the data displayed by DISPLAY DISTORTION when SHOW MACROSCOPIC is used pl shows the indices of the tensor in numerical order For example pl 1234 indicates a tensor of rank 4 Indices to be symmetrized are enclosed by square brackets For example pl 12 indicates a totally symmetrized tensor of rank 2 Indices to be antisymmetrized are enclosed by curly brackets For example pl 1 23 indicates a tensor of rank 3 which is antisymmetric with respect to the 2nd and 3rd indices The value of the rank cannot exceed 6 VALUE SIZE pl p2 The relative sizes of the primitive unit cells of the parent space group and the isotropy subgroup are selected This command affects the data displayed by DISPLAY ISOTROPY The parameters pl and p2 are numbers representing relative size If they are both present then sizes p1 through p2 are selected If only pl is present then only size p1 is selected VALUE SUBGROUP pl p2 The isotropy subgroup space groups are selected This command affects the data displayed by DISPLAY ISOTROPY The parameters pl and p2 are space groups If they are both present then space groups pl through p2 are selected If only pl is present then only space group p1 is selected The space groups can be denoted by a n
23. OW DOMAINS with DISPLAY ISOTROPY Normally DISPLAY ISOTROPY used with SHOW DOMAIN causes all domains to be displayed VALUE DOMAIN causes only one domain to be displayed Normally DISPLAY DISTORTION uses the direction of the order parameter in the first domain VALUE DOMAIN along with VALUE DIRECTION uses the direction of the order parameter in selected domain When used with DISPLAY INVARIANTS the number of domains selected must be equal to the number of irreps selected by the VALUE IRREP command VALUE DOMAIN PAIR pl p2 A pair of domains of the isotropy subgroup is selected This command affects the data displayed by DISPLAY ISOTROPY when used with SHOW PAIRS or SHOW TWIN commands 18 VALUE DOMAIN SETS CLASS pl The class of the domain set is selected This command affects the data displayed by DISPLAY ISOTROPY pl is a number which is displayed in the class column when SHOW DOMAIN SETS is used VALUE DOMAIN SETS DIRECTION pl The direction of the domain set vector s is selected This command affects the data displayed by DISPLAY ISOTROPY pl is a number which is displayed in the dir column when SHOW DOMAIN SETS is used with SHOW DOMAIN SETS DIRECTION VALUE DOMAIN SETS ORDER pl The order of the domain set is selected This command affects the data displayed by DISPLAY ISOTROPY pl is a number representing the number of domains present in the set VALUE DOMAIN SETS UNCONNECTED pl The number of unconnected parts of the dom
24. PERSPACEGROUP The parameters pl and p2 are space groups If they are both present then space groups pl through p2 are selected If only pl is present then only space group pl is selected The space groups can be de noted by a number or by a symbol using either the Schoenflies or international notation 22 either short or full symbol For example VALUE PARENT 124 refers to space group 124 D2 or P4 mec or P4 m2 c2 c This command could have also been entered as VALUE PARENT D4H 2 or VALUE PARENT P4 MCC or VALUE PARENT P4 M2 C2 C Note that since parameters are delimited by space characters they should not contain any space characters themselves In the international notation bars over numbers are denoted by a preceding minus sign eg P 3C1 for P3cl and subscripts are denoted by a preceding underline character eg P4_2 MCM for P42 mcm When used with DISPLAY SUPERSPACEGROUP the VALUE PARENT command selects the basic space groups of the 3 1 dimensional superspace groups VALUE POINTGROUP pl p2 When DISPLAY PARENT is used the point groups of the space group are selected When DISPLAY ISOTROPY is used the point groups of the isotropy subgroup are selected When DISPLAY SUPERSPACEGROUP is used the point groups of the basic space group of the 3 1 dimensional superspace group are selected The parameters pl and p2 are space groups If they are both present then point groups pl through p2 are selected using the order shown
25. REP DISPLAY ISOTROPY and DISPLAY DISTORTION This command also selects the k vector displayed by DISPLAY KPOINT The parameter pl is the k point to be selected This command cancels the effect of any VALUE IRREP command previously used The k point is denoted by a symbol using either the notation of Miller and Love Kovalev Bradley and Cracknell or Zak For example Y K8 Z and Z all refer to the same k point using each of the notations respectively in space group 12 A2 m Note that Z and A are equivalent k points in the Zak notation for this space group When there are equivalent k points any of them may be used for pl GM is used for I As with irrep 20 notation the k point notation used for p1 must agree with the current space group setting See the VALUE IRREP command for a further discussion of this point VALUE KVALUE pl p2 The values of the parameters a 3 y defining the k vectors are selected The parameters pl p2 each represent a set of parameters for a k vector The number of sets selected must equal the number of irreps selected These parameters need only be selected if one or more of the irreps are associated with k vectors which are not at k points of symmetry For example a k vector on a k line of symmetry is defined by a single parameter a which gives the position of the vector on the line In that case VALUE KVALUE 1 1 4 would select one value a L VALUE KVALUE 2 1 4 3 8 selects two values and 8
26. SPLAY PARENT is used the Bravais lattice of the space group is shown When DISPLAY SUPERSPACEGROUP is used the Bravais lattice of the 3 1 dimensional super space group is shown When DISPLAY ISOTROPY is used the Bravais lattices of both the parent space group and isotropy subgroup are shown SHOW LIFSHITZ The Lifshitz frequency of the irrep is shown when DISPLAY IRREP or DISPLAY ISOTROPY is used The Lifshitz frequency is the number of times that the vector representation is contained in the antisymmetrized cube of the irrep SHOW LOCAL pl p2 The local microscopic distortions at a point about a Wyckoff position are displayed when DISPLAY DISTORTION is used The coordinates of the point must be selected with the command VALUE POSITION If pl is not present then the distortions are shown as linear combinations of basis functions of the irrep of the point group associated with the point The command SHOW LOCAL SCALAR displays distortions of scalar functions such as occupa tion probabilities The command SHOW LOCAL VECTOR displays vector distortions such as 12 atomic displacements The command SHOW LOCAL VECTOR PSEUDO displays pseudo vector distortions such as molecular rotations or magnetic moments SHOW MACROSCOPIC PSEUDO The macroscopic distortions tensor components are displayed when DISPLAY DISTORTION is used The type of tensor must be specified by the VALUE RANK command The parameter PSEUDO indicates a pseudo tensor such
27. TING INTERNATIONAL pl AXIS p2 where pl is a space group number or symbol and p2 is B or C The choice of unique axis for all monoclinic space groups can be changed by using ALL for pl Note that the axis choice is changed only for the current setting i e the new or old edition not both For example SETTING INTERNATIONAL 5 AXIS C changes the setting to unique axis c for space group 5 A2 SETTING INTERNATIONAL ALL AXIS C changes the setting of all monoclinic space groups to unique axis c The base centered monoclinic space groups in the new edition have three cell choices When ISOTROPY is started cell choice 1 is used To change the cell choice for a particular space group use SETTING INTERNATIONAL pl CELL p2 where pl is a space group number or symbol and p2 is 1 2 or 3 The cell choice for all base centered monoclinic space groups can be changed by using ALL for p1 Note that the cell choice is changed only if the current setting is the new edition For example SETTING INTERNATIONAL 5 CELL 3 changes the setting to cell choice 3 for space group 5 A2 SETTING INTERNATIONAL ALL CELL 3 changes the setting of all base centered monoclinic space groups to cell choice 3 The trigonal space groups have two choices for axes hexagonal and rhombohedral When ISOTROPY is started the hexagonal axes are used To change the choice of axes for a par ticular space group use SETTING INTERNATIONAL pl AXIS p2 where pl is a space group number or symbo
28. UE DOMAIN SETS DIRECTION SHOW DOMAIN SETS BASIS The basis vectors of the domain set group are shown when DISPLAY ISOTROPY is used When used with SHOW DOMAIN SETS INTERSECT the basis vectors of the intersection group are shown SHOW DOMAIN SETS DIRECTION The directions of the domain set vectors are shown when DISPLAY ISOTROPY is used See the tutorial on domains SHOW DOMAIN SETS ELEMENTS The elements of the domain set group are shown when DISPLAY ISOTROPY is used When used with SHOW DOMAIN SETS INTERSECT the elements of the intersection group are shown SHOW DOMAIN SETS EQUIVALENT Operators which take us to equivalent domain sets or equivalent directions are shown when DISPLAY ISOTROPY is used This command is used with SHOW DOMAIN SETS ALL SHOW DOMAIN SETS GENERATORS The generating elements of the domain set group are shown when DISPLAY ISOTROPY is used When used with SHOW DOMAIN SETS INTERSECT the generating elements of the intersection group are shown SHOW DOMAIN SETS GROUP The space group symmetry of the domain set is shown when DISPLAY ISOTROPY is used SHOW DOMAIN SETS INTERSECT The space group symmetry of the intersection of the symmetries of every domain in the set is shown when DISPLAY ISOTROPY is used SHOW DOMAIN SETS ORIGIN The origin of the domain set group is shown when DISPLAY ISOTROPY is used When used with SHOW DOMAIN SETS INTERSECT the origin of the intersection group is shown SHOW ELEMENTS When u
29. a ca U0 Summary of ISOTROPY commands Version 9 0 p 5 of 5 show direction of order parameter show domain show Wyckoff positions p1 IR show point group irrep of point about Wyckoff position show distortions in unit cell of parent show Cartesian coordinates display invariant polynomials select parent space group select irreps select parameters a 3 for the k point select directions of order parameter select domains select degree s of polynomials select number of spatial derivatives show parent space group symbol show irrep pl other settings M K B Z P show dimension display data on bush of modes select parent space group select irrep select parameters a 3 for the k point select direction of order parameter select Wyckoff positions select degree s of invariant polynomials show atomic displacements show invariant polynomials in free energy display directions of order parameters select parent space group select subgroup select basis vectors of subgroup lattice select origin of subgroup show k vectors show subgroup symbol show relative size of subgroup s primitive unit cell to parent group s
30. ain set is selected This command affects the data displayed by DISPLAY ISOTROPY pl is a number VALUE ELEMENT pl The element of the parent space group is selected This command affects the data dis played by DISPLAY IRREP pl is a space group element The element is denoted by a symbol using the notation of International Tables Miller and Love Kovalev Bradley and Cracknell or Zak For example X 1 2 Y Z 2 0 1 2 0 H2 0 1 2 0 C2X 0 1 2 0 and UX O 1 2 0 all refer to the same element using each of the notations respectively In the notation of the International Tables the x y z parts are separated by a space char acter In the other notations the point operation comes first followed by the fractional each part separated by a space character The notation used for pl must agree with the point operation notation selected for elements The LABEL ELEMENT command changes the notation selected VALUE FELIX pl p2 The Felix frequency of the image is selected This command affects the data displayed by DISPLAY IRREP and DISPLAY ISOTROPY The parameters pl and p2 are numbers repre senting the Felix frequency of an irrep the number of anisotropic gradient terms in the LGW Hamiltonian If they are both present then Felix frequencies p1 through p2 are selected If only p1 is present then only Felix frequency p1 is selected VALUE FERROIC pl The ferroic species of the phase transition is selected This command affects the data displayed
31. and the default irrep labeling is Miller Love The user can set up his own defaults by creating a file iso ini containing commands If this file exists in the current default directory ISOTROPY will execute these commands when it starts before giving the initial prompt ISOTROPY Commands A command consists of a series of keywords and parameters Different keywords and parameters are separated by a space character in the command All keywords may be entered using either upper or lower case letters All keywords may be abbreviated to the first one or more characters depending on the ambiguity of the meaning of different commands that start with the same letter s ISOTROPY tells the user when a keyword is ambiguous If a question mark is entered in place of one of the keywords all valid keywords that could be entered at the position of the question mark will be displayed For example if a simple is entered in place of a command ISOTROPY will indicate that the valid keywords are CANCEL DISPLAY LABEL PAGE QUIT SCREEN SETTING SHOW and VALUE This means that the first keyword in a command must be one of these words As another example if LABEL is entered as a command ISOTROPY will indicate that the valid keywords are ELEMENT IMAGE LATTICE POINTGROUP SPACEGROUP and VECTOR This means that if the first keyword in a command is LABEL then it must be followed by one of those three words In the following material each comma
32. as a magnetic moment For hexagonal crystals the tensors are given with respect to an orthogonal coordinate system where the y axis is chosen to lie along the hexagonal b axis SHOW MATRIX The irrep matrix for an element of the parent space group is shown when DISPLAY IRREP is used The command SHOW ELEMENT is also automatically executed The element is selected with the VALUE ELEMENT command SHOW MAXIMAL Maximal isotropy subgroups are indicated when DISPLAY ISOTROPY is used An isotropy subgroup is maximal when it is not a subgroup of any other isotropy subgroup for the same irrep SHOW MICROSCOPIC pl p2 The microscopic distortions are displayed when DISPLAY DISTORTION is used If pl is not present then the distortions are shown as linear combinations of basis functions of the irrep of the point group associated with the Wyckoff position The command SHOW MICROSCOPIC SCALAR displays distortions of scalar functions such as occupation probabili ties The command SHOW MICROSCOPIC VECTOR displays vector distortions such as atomic displacements The command SHOW MICROSCOPIC VECTOR PSEUDO displays pseudo vector distortions such as molecular rotations or magnetic moments SHOW MODES The atomic displacements in each vibration mode of a bush are displayed when DISPLAY BUSH is used SHOW NEWFRACTIONALS The new fractionals in the unit cell of the isotropy subgroup are shown when DISPLAY ISOTROPY is used These new fractionals are vectors wh
33. be selected Only the first irrep listed p1 is used by DISPLAY DISTORTIONS DISPLAY IRREP and DISPLAY ISOTROPY This command can cels the effect of any VALUE KPOINT command previously used The irrep is denoted by a symbol using either the notation of Miller and Love Kovalev Bradley and Cracknell or Zak For example Yi K8T1 Z1 and Z1 all refer to the same irrep using each of the notations respectively in space group 12 A2 m Note that Z1 and A1 are the same irrep in the Zak notation for this space group When there are more than one symbol for the same irrep any of them may be used for pl GM is used for I When a physically irreducible representation is constructed from two complex conjugate irreps the notation indicates this for example Z1Z2 or K22T1T2 in space group 30 Pnc2 mc2 The irrep notation used for pl must agree with the current space group setting The SETTING command changes the current setting When SETTING INTERNATIONAL is used the current irrep notation does not change When ISOTROPY is first started the irrep notation is Miller and Love See the SETTING command for an explanation of how the command affects the irrep selected by VALUE IRREP VALUE KDEGREE pl The degrees of freedom of the k vector are selected This command affects the data displayed by DISPLAY IRREP and DISPLAY KPOINT VALUE KPOINT pl Irreps which arise from a given k point are selected This command affects the data dis played by DISPLAY IR
34. ce group of the 3 1 dimensional superspace group is displayed When DISPLAY ISOTROPY is used the point groups of both the parent space group and isotropy subgroup are shown SHOW POSITION IRREP The irrep of the point group associated with a point about a Wyckoff position is shown when DISPLAY DISTORTION is used with SHOW LOCAL SHOW SIZE The relative sizes of the primitive unit cells of the parent space group and the isotropy subgroup is shown when DISPLAY ISOTROPYis used 14 SHOW STAR When DISPLAY KPOINT or DISPLAY IRREP is used the star of k is displayed SHOW SUBGROUP The space group symbol of the isotropy subgroup is shown when DISPLAY ISOTROPY is used SHOW SUBGROUP ALTERNATE An alternate basis vector and origin are shown when DISPLAY ISOTROPY is used If the basis vectors selected by VALUE BASIS and the origin selected by VALUE ORIGIN are con sistent with the subgroup symmetry being displayed the selected values will be shown instead of those in the data base This provides a way for you to try some nicer choices of basis vectors and origins and find out whether these choices still describe the same subgroup symmetry SHOW SUPERSPACEGROUP The superspace group is displayed when DISPLAY SUPERSPACEGROUP is used SHOW TWIN pl p2 This command displays information about twin intersection groups and twin groups when DISPLAY ISOTROPY is used This information is shown for a particular domain pair if one has been selected
35. d with DISPLAY PARENT the generators of the parent space group are shown Generators of the lattice are actually not shown explicitly When used with DISPLAY IMAGE the generating matrices of the image are shown When used with DISPLAY IRREP elements of the parent space group which are mapped onto the generating matrices of the irrep s image are shown When used with DISPLAY ISOTROPY the elements of the parent space group which generate the isotropy subgroup are shown When used with DISPLAY SUPERSPACEGROUP the generators of the 3 1 dimensional su perspace group are shown Generators of the lattice are actually not shown explicitly SHOW IMAGE The symbol of the image of the irrep is shown when DISPLAY IMAGE DISPLAY IRREP or DISPLAY ISOTROPY is used SHOW INDEX The index of the isotropy subgroup in the parent space group is shown when DISPLAY ISOTROPY is used The index is the size of the parent space group relative to the subgroup SHOW INVARIANTS The invariant polynomials in the free energy expansion of a bush of modes are displayed when DISPLAY BUSH is used SHOW IRREP p1 The symbol of the irrep is shown when DISPLAY INVARIANTS DISPLAY DISTORTION DISPLAY IRREP or DISPLAY ISOTROPY is used The symbol shown uses the irrep notation 11 of the current space group setting The parameter p1 allows the irrep symbol to also be shown in other notations pl can be MILLER LOVE KOVALEV BRADLEY CRACKNELL or ZAK referring to the
36. displayed We call this the pair intersection group This group is always one of the other isotropy subgroups belonging to the same irrep Therefore we display simply the direction of that isotropy subgroup The domain of that subgroup is also shown in parentheses if it is not the first domain If p1 is SWITCH an element of the space group which switches the two domains in the pair is displayed For the domain pair P P this element takes P to P and it also takes P to P When the two domains are identical we consider the switching element to not exist If p1 is GROUP the space group label of the pair group is displayed This group consists of all elements in the pair intersection group plus all elements that switch the two domains in the pair If pl is BASIS ORIGIN GENERATORS or ELEMENTS then the basis vectors of the lattice the origin of the space group the generating elements or the complete list of elements of the pair group are displayed respectively These commands are very similar to the SHOW BASIS SHOW ORIGIN SHOW GENERATORS and SHOW ELEMENTS commands used with the DISPLAY ISOTROPY command SHOW PARENT The parent space group is displayed when DISPLAY PARENT DISPLAY INVARIANTS DISPLAY DISTORTION DISPLAY IRREP DISPLAY ISOTROPY or DISPLAY KPOINT is used SHOW POINTGROUP When DISPLAY PARENT is used the point group of the space group is displayed When DISPLAY SUPERSPACEGROUP is used the point group of the basic spa
37. ector for that lattice LABEL VECTOR PRIMITIVE cannot be used with the INTERNATIONAL setting PAGE pl This command sets the number of lines that can be displayed on the terminal screen at one time When the output to a particular DISPLAY command requires more lines than p1 only p1 lines are displayed at a time When ISOTROPY starts the default number of lines is 22 If pl is NOBREAK the output does not stop until the next prompt is issued QUIT The program ISOTROPY exits SCREEN pl The width of the crt screen is changed to pl characters wide The default width of the screen is 80 characters If the data to be displayed require more than p1 characters in a line ISOTROPY first tries to arrange the last column of data to form more than one line on the screen If this cannot be done the line is truncated and an asterisk appears at the right edge of the screen SETTING pl The current space group setting is changed The parameter p1 is the name of the setting INTERNATIONAL NEW INTERNATIONAL OLD MILLER LOVE KOVALEV BRADLEY CRACKNELL and ZAK refer to the setting of International Tables new ed International Tables old ed Miller and Love Kovalev Bradley and Cracknell and Zak respectively The current set ting can be displayed using the DISPLAY SETTING command Additional options for the set tings in International Tables are discussed under the command SETTING INTERNATIONAL pl If the SETTING INTERNATIONAL command is used
38. ep pl other settings M K B Z P show image of irrep show dimension show whether the irrep is active or not show whether or not the phase transition is continuous show Landau frequency show Lifshitz frequency show Felix frequency show frequencies and irreps that subduce the subgroup p1 DIR GAM CL show Cartesian coordinates show ferroic species display distortions select parent space group select irrep select k point select parameters a 3 for the k point select rank and symmetries of macroscopic tensor select directions of order parameter select domain select super cell select lattice parameters a b c a B Y select wyckoff positions select point group irrep of wyckoff position select parameters x y z of wyckoff position select coordinates of point show incommensurate distortions show macroscopic distortion pl1 PSEUDO show microscopic distortion pl1 SC VEC p2 PSEUDDO show local distortion p1 SC VEC p2 PSEUDO show parent space group symbol show irrep pl other settings M K B Z P SH DIR SH DOM SH WY pl SH POS IR SH UNIT SH CART INV PAR pl IR pl p2 KVAL pl p2 DIR pl p2 DOM pl p2 DEG pl p2 GRAD pl SH PAR SH IR pl SH DIM BUSH PAR pl IR pl KVAL pl p2 DIR pl WY pl p2 DEG pl p2 SH MODES SH INV D DIR V PAR pl V SUBG pl V BASIS pl p2 p3 V ORIGIN pl SH KPOINT SH SUB SH SIZ ssaacscaaca csc U0 ssaaca
39. he subgroups is D5 with order parameter direction P1 1 P4 3 This means that this subgroup is an intersection of the first domain of Dj irrep Xy direction P a 0 and the third domain of D5 irrep Ps direction Py a 0 0 The direction of the order parameter for De is denoted a 0 b 0 b 0 the first two components associated with irrep X n and the remaining four components associated with irrep Ps The irreps associated with each part of the order parameter may be seen by using SHOW IRREP SHOW DISTINCT The domains for distinct subgroups are shown when DISPLAY ISOTROPY is used SHOW DOMAINS GENERATORS The possible domains arising from the phase transition are shown when DISPLAY ISOTROPY is used The number of possible domains is equal to the index of the isotropy subgroup in the parent space group When the parameter GENERATORS is present the element of the parent space group which generates the domain is also shown When DISPLAY DISTORTIONS is used SHOW DOMAINS will cause the domain selected by VALUE DOMAIN to be shown SHOW DOMAIN SETS The possible multidomain structures domain sets are shown when DISPLAY ISOTROPY is used See the tutorial on domains SHOW DOMAIN SETS ALL Equivalent domain sets are shown when DISPLAY ISOTROPY is used and when a class has been selected with VALUE DOMAIN SETS CLASS When used with SHOW DOMAIN SETS DIRECTION equivalent directions are shown when a direction has been selected with VAL
40. hown when DISPLAY IMAGE or DISPLAY IRREP is used A type 1 irrep is real A type 2 irrep is complex but its characters are real A type 3 irrep is complex and its characters are also complex SHOW UNITCELL Only distortions in the unit cell of the parent are shown when DISPLAY DISTORTIONS is used with SHOW MICROSCOPIC SHOW WYCKOFF pl p2 When DISPLAY PARENT is used the symbol for the Wyckoff position is shown If pl is VECTOR the coordinates of the Wyckoff position are shown In addition if p2 is ALL the coordinates of all of the points associated with the Wyckoff position are shown If p1 is POINTGROUP the point group of the Wyckoff position is shown If pl is ELEMENTS the space group elements which belong to the point group of the Wyckoff position are shown If pl is CHARACTER the characters of the point group irreps are shown for each space group element which belongs to the point group of the Wyckoff position When DISPLAY DISTORTION is used the symbol for the Wyckoff position is shown If p1 is IRREP the irrep of the point group of the Wyckoff position is shown SHOW WYCKOFF SUBGROUP When DISPLAY ISOTROPY is used the Wyckoff positions of the atoms in the subgroup are shown SHOW XYZ The z y z coordinates of a point in the unit cell of the isotropy subgroup are given in terms of the x y z coordinates of a point in the unit cell of the parent space group when DISPLAY ISOTROPY is used The coordinates are given with respect
41. ic T AP 8 primitive tetragonal Q TP 2 primitive monoclinic M MP 9 body centered tetragonal Q V TI 3 base centered monoclinic M B MC 10 Trigonal RH HR 4 primitive orthorhombic 0 OP 11 Hexagonal H HP 5 base centered orthorhombic 0 B OC 12 primitive cubic C CP 6 body centered orthorhombic 0 V OI 13 face centered cubic C F CF 7 face centered orthorhombic 0 F OF 14 body centered cubic C V CI 21 VALUE LATTICE PARAMETER pl p2 p3 p4 p5 p The lattice parameters are selected The parameters pl through p6 are the values of a b c a B Y respectively a b c are the lengths of the 3 basis vectors a b c of the conven tional lattice a is the angle between b and c is the angle between a and c and y is the angle between a and b VALUE LIFSHITZ pl p2 The Lifshitz frequency of the irrep is selected This command affects the data displayed by DISPLAY IRREP and DISPLAY ISOTROPY The parameters pl and p2 are numbers repre senting the Lifshitz frequency of an irrep the number of times that the vector representa tion is contained in the antisymmetrized cube of the irrep If they are both present then Lifshitz frequencies pl through p2 are selected If only pl is present then only Lifshitz frequency pl is selected VALUE MAXIMAL pl When DISPLAY ISOTROPY is used VALUE MAXIMAL YES selects maximal isotropy subgroups and VALUE MAXIMAL NO selects subgroups which are not maximal An isotropy subgroup is maximal when it is not a subgroup of
42. ich are lattice vectors in the parent space group but are not lattice vectors in the isotropy subgroup SHOW ORDER The order of the image is shown when DISPLAY IMAGE is used SHOW ORIGIN The origin of the isotropy subgroup with respect to the parent space group is shown when DISPLAY ISOTROPY is used When used with LABEL VECTOR PRIMITIVE the coordinates of the origin are given in terms of the primitive basis vectors of the parent space group When used with LABEL VECTOR CONVENTIONAL the coordinates of the origin are given in terms of the conventional basis vectors of the parent space group 13 SHOW PAIRS pl This command shows information about pairs of domains of isotropy subgroups when DISPLAY ISOTROPY is used We denote a pair of domains by P P where P is the ith domain of an isotropy subgroup The pair to be considered can be selected by the VALUE DOMAIN PAIRS command If the pair has not been selected then the pair Pi Pj is considered for each domain 7 When the pair has not been selected equivalence classes of domain pairs are displayed Any class of pairs has an element of the form P Pj Each pair is considered for every domain j The number in the pair column numbers the class of pairs to which the pair belongs For example domain 3 pair 2 means the pair 1 3 is in the second equivalence class of pairs If pl is INTERSECT the intersection of the isotropy groups belonging to the two domains in the domain pair is
43. if the irrep is active and the order parameter is a possible minimum of the free energy expanded to fourth degree RG theory imposes the additional constraint that the coefficients of the free energy expansion lie within the attractor basin of a stable fixed point VALUE DEGREE pl p2 The degrees of invariant polynomials are selected This command affects the data displayed by DISPLAY INVARIANTS The parameters pl and p2 are numbers representing the degree of the polynomial If they are both present then degrees pl through p2 are selected If only pl is present then only degree p1 is selected If this command has not been used yet 17 or if the CANCEL VALUE DEGREE command has been used ISOTROPY displays the invariant polynomials of degrees 1 through 4 by default VALUE DIMENSION pl p2 The dimensions of matrices in the irrep s image are selected This command affects the data displayed by DISPLAY IMAGE DISPLAY IRREP and DISPLAY ISOTROPY The parameters pl and p2 are numbers representing the dimension of matrices in an image If they are both present then dimensions p1 through p2 are selected If only p1 is present then only dimension p1 is selected VALUE DIRECTION pl p2 The direction of the order parameter is selected This command affects the data displayed by DISPLAY ISOTROPY DISPLAY DISTORTION DISPLAY INVARIANTS DISPLAY BUSH and DISPLAY DIRECTION The parameters p1 p2 each represent the direction of an order pa
44. irrep notations of Miller and Love Kovalev Bradley and Cracknell and Zak respectively pl can also be POINTGROUP in which case the conventional labels for point group irreps are shown for irreps at the I point k 0 Even when pl is present this command causes the irrep to be shown in the notation of the current setting in ad dition to the setting specified by pl When DISPLAY INVARIANTS or DISPLAY ISOTROPY COUPLED is used the symbols of all irreps selected are shown in sequence on the same line SHOW KDEGREE The degrees of freedom of the k vector are shown For example a k point of symmetry has 0 degrees of freedom and a k line of symmetry has 1 degree of freedom SHOW KERNEL The generating elements of the kernel of the irrep are shown when DISPLAY IRREP is used The kernel is the set of all elements in the parent space group which map onto the unit matrix in the image SHOW KPOINT The coordinates of the k vector are shown when DISPLAY KPOINT DISPLAY IRREP or DISPLAY DIRECTION is used The coordinates are given in terms of the reciprocal lattice vectors derived from the basis vectors of the direct lattice The form depends on the space group setting as well as the form of the vectors used primitive or conventional SHOW LANDAU The Landau frequency of the image is shown when DISPLAY IMAGE DISPLAY IRREP or DISPLAY ISOTROPY is used The Landau frequency is the number of independent third degree invariants SHOW LATTICE When DI
45. l and p2 is HEXAGONAL or RHOMBOHEDRAL The choice of axes for all trigo nal space groups can be changed by using ALL for pl Note that the axis choice is changed only for the current setting i e the new or old edition not both For example SETTING INTERNATIONAL 167 AXIS RHOMBOHEDRAL changes the setting to rhombohedral axes for space group 167 R3c SETTING INTERNATIONAL ALL AXIS RHOMBOHEDRAL changes the setting of all trigonal space groups to rhombohedral axes SETTING MAGNETIC Data about the isotropy subgroups of the grey magnetic space groups are displayed when DISPLAY ISOTROPY is used All coordinates are expressed in terms of the primitive basis vectors in the setting of Miller and Love Magnetic space groups are displayed in the following format 1 the number of the associ ated Fedorov space group in parentheses 2 the Belov number and 3 the symbol given in Miller and Love Magnetic space groups are selected using the number or symbol of the corresponding Fedorov space group For example VALUE PARENT 155 or VALUE PARENT R32 selects the grey magnetic space group R321 The command VALUE SUBGROUP 155 selects any of the four space groups associated with R32 i e R32 R321 R32 R732 SETTING NOMAGNETIC The command SETTING MAGNETIC is cancelled SHOW ACTIVE Active images are indicated when DISPLAY IMAGE is used Active irreps are indicated when DISPLAY IRREP or DISPLAY ISOTROPY is used An irrep is active when both its
46. layed about the twin group is is controlled by pl Each element in the twin group must obey the following requirements It must either keep both domains in the pair invariant or switch them This means that this group is a subgroup of the pair group If the elements operates on a point in the specified plane the point must stay in the plane This means that the group must be diperiodic If the point operator part of the element operates on a vector perpendicular to the specified plane the direction of the vector either stays the same or reverses its direction If p1 is GROUP BASIS ORIGIN 15 GENERATORS or ELEMENTS then the diperiodic space group label the basis vectors of the lattice the origin of the space group the generating elements or the complete list of elements of the twin group are displayed respectively These commands are very similar to the SHOW SUBGROUP SHOW BASIS SHOW ORIGIN SHOW GENERATORS and SHOW ELEMENTS commands used with the DISPLAY ISOTROPY command If pl is SWITCH then a switching element is displayed If p2 is SIDE then the element switches the two domains but does not change the direction of the vector normal to the plane If p2 is NORMAL then the element reverses the direction of the vector normal to the plane but does not switch the two domains If p2 is BOTH then the element switches both the two domains and the direction of the vector normal to the plane SHOW TYPE The irrep type 1 2 3 are s
47. n used with SHOW INVARIANTS the invariant polynomials in the free energy expansion of the bush of modes are displayed DISPLAY DIRECTION The irreps and directions of order parameters are displayed given the parent space group subgroup basis vectors of the subgroup lattice and origin of the subgroup DISPLAY DISTORTION Symmetry allowed distortions are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands DISPLAY IMAGE Data about images of the irreps are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands DISPLAY INVARIANTS Invariant polynomials in the representation space of one or more coupled irreps are dis played The displayed data are controlled by VALUE SHOW and CANCEL commands A parent space group must be selected using VALUE PARENT and one or more irreps must be selected using VALUE IRREP Each invariant polynomial and its degree are displayed without needing to use any SHOW commands VALUE GRADIENT may be used to display invariant polynomials containing spatial deriva tives The value selected by this command determines the number of derivatives each polynomial will contain Note that polynomials which vanish in a volume integral are also displayed The user must inspect these himself and discard them by hand The degree of the polynomial must be selected using VALUE DEGREE DISPLAY IRREP Data about irreps are displayed The displayed data are controlled by
48. nd is denoted by keywords and parameters p1 p2 which may be a number or another keyword A parameter in brackets eg p2 is optional and does not need to be present for a command to be valid CANCEL pl p2 p3 The keyword CANCEL can be put in front of any SHOW or VALUE command below to cancel the effect of that command These cancel commands will not be explicitly listed here except for a few commands which require some additional explanation CANCEL SHOW ALL Cancels the effect of all SHOW commands previously used CANCEL SHOW ELEMENT This command also causes CANCEL SHOW CHARACTER and CANCEL SHOW MATRIX to be au tomatically executed CANCEL SHOW IRREP pl Cancels the effect of SHOW IRREP The parameter pl is the name of the one the settings MILLER LOVE KOVALEV BRADLEY CRACKNELL or ZAK If pl is present then only the notation specified by pl is removed from the display If pl is not present then all irrep symbols are removed from the display Note that the irrep symbol in the current setting can only be removed from the display by removing all irrep symbols with CANCEL SHOW IRREP CANCEL VALUE ALL Cancels the effect of all VALUE commands previously used DISPLAY BUSH Data about a bush of vibrational modes are displayed The parent space group irrep direction of the order parameter and Wyckoff positions of the atoms must be selected When used with SHOW MODES the atomic displacements in each mode is displayed Whe
49. nternational Tables for X Ray Crystallography Vol I edited by N F M Henry and K Lonsdale Kynoch Press Birmingham 1965 S C Miller and W F Love Tables of Irreducible Representations of Space Groups and Co Representations of Magnetic Space Groups Pruett Boulder 1967 This is essentially the same as A P Cracknell B L Davies S C Miller and W F Love Kronecker Product Tables Vol 1 Plenum New York 1979 O V Kovalev Representations of the Crystallographic Space Groups Irreducible Repre sentations Induced Representations and Corepresentations Gordon and Breach Amster dam 1993 C J Bradley and A P Cracknell The Mathematical Theory of Symmetry in Solids Clarendon Oxford 1972 The Irreducible Representations of Space Groups edited by J Zak Benjamin New York 1969 These references will be referred to as International Tables new ed International Tables old ed Miller Love Kovalev Bradley Cracknell and Zak respectively ISOTROPY has access to the irrep labeling of Miller Love Kovalev Bradley Cracknell and Zak for all physically irreducible representations arising from k points of symmetry Also ISOTROPY has access to the irrep labeling of Miller Love for all physically irreducible representations arising from allk points including k lines and k planes of symmetry and general k vectors When ISOTROPY starts the default space group setting is International Tables new ed
50. or in RG theory are indicated when DISPLAY ISOTROPY is used In Landau theory a phase transition to a particular subgroup is allowed to be continuous if the irrep is active and the order parameter is a possible minimum of the free energy expanded to fourth degree RG theory imposes the additional constraint that the coefficients of the free energy expansion lie within the attractor basin of a stable fixed point SHOW DIMENSION The dimension of the image of the irrep is shown when DISPLAY IMAGE DISPLAY INVARIANTS DISPLAY IRREP or DISPLAY ISOTROPY is used In the cases of DISPLAY IN VARIANTS and DISPLAY ISOTROPY COUPLED the dimension of the reducible representation is shown the sum of the dimensions of each of the irreps selected SHOW DIRECTION pl The symbol for the direction of the order parameter is shown when DISPLAY ISOTROPY or DISPLAY DISTORTION is used If pl is VECTOR then the vector form of the order parameter is also shown When DISPLAY DISTORTION is used the direction is shown only when one has been selected by VALUE DIRECTION Also the parameter p1 is ignored When DISPLAY ISOTROPY COUPLED is used the symbol for the order parameter direction of each irrep is shown followed by a number in parentheses indicating which domain of the uncoupled isotropy subgroup is involved For example suppose we display the isotropy subgroups for the coupled order parameters of irreps X and P of space group D17 We would find that one of t
51. oups are read from a file with a name of the form Sxxxx_LX iso ISOTROPY looks for this file in the user s directory If it is not found the user is prompted and if desired ISOTROPY will then proceed to calculate the re quested subgroups and create the file Note that not commands implemented for DISPLAY ISOTROPY are implemented for DISPLAY ISOTROPY COUPLED DISPLAY KPOINT Data about k vectors in the first Brillouin zone are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands DISPLAY PARENT Data about space groups are displayed The word PARENT refers to parent space group The displayed data are controlled by VALUE SHOW and CANCEL commands DISPLAY SETTING The current space group setting being used is displayed DISPLAY SHOW All SHOW commands currently in effect are shown DISPLAY SUPERSPACEGROUP Data about 3 1 dimensional superspace groups are displayed The displayed data are controlled by VALUE SHOW and CANCEL commands DISPLAY VALUE All VALUE commands currently in effect are shown LABEL ELEMENT pl The notation for the space group elements is changed This affects the label displayed whenever elements of a space group are displayed It also affects the way in which elements are to be entered with the VALUE ELEMENT command The parameter pl indicates the notation to be used INTERNATIONAL MILLER LOVE KOVALEV BRADLEY CRACKNELL and ZAK for the notations of the international table
52. s Miller and Love Kovalev Bradley and Cracknell and Zak respectively See the VALUE ELEMENT command for an explanation of these notations Note that the SETTING command does not change the notation of space group elements For example it is possible to show elements of a space group in the setting of Zak using Bradley and Cracknell s notation for the space group elements When ISOTROPY is started the default notation for elements is BRADLEY CRACKNELL The parameters X1X2X3X4 and XYZT affect the notation used for elements of the 8 1 dimensional superspace groups The parameter X1X2X3X4 default causes them to be displayed in 71 2 3 4 coordinates using the international notation only The param eter XYZT causes them to be displayed in x y z t coordinates using any of the available notations BRADLEY CRACKNELL by default LABEL IMAGE pl LABEL IMAGE TOLEDANO adds the notation of Tol dano and Tol dano for images whenever image symbols are displayed LABEL IMAGE NOTOLEDANO removes this notation LABEL LATTICE pl The notation for Bravais lattices is changed This affects the label displayed when SHOW LATTICE is used The parameter pl indicates the notation to be used SCHOENFLIES and PEARSON See the VALUE LATTICE command for more explanation about these notations LABEL POINTGROUP pl The notation for point groups is changed This affects the label displayed when SHOW POINTGROUP is used The parameter p1 indicates the notation to
53. sed with DISPLAY PARENT the coset representatives of the parent space group with respect to its translation subgroup are shown When used with DISPLAY IMAGE all of the matrices of the image are shown When used with DISPLAY IRREP the element of the parent space group selected by VALUE ELEMENT is shown When used with DISPLAY ISOTROPY the coset representatives of the subgroup with respect to its translation subgroup are shown When used with DISPLAY SUPERSPACEGROUP the coset representatives of the 3 1 dimensional superspace group with respect to its translation subgroup are shown SHOW FAINTNESS The nonzero faintness indices for all of the other irreps of the parent space group are shown when DISPLAY IRREP is used Invariant polynomials exist which linearly couple these irreps to the selected irrep The minimum degree of the part of the polynomial associated with the selected irrep is the faintness index For example if the selected irrep is R4 we find a faintness index 3 for irrep R2 The form of the invariant polynomial in this case is qi71723 where q is the order parameter associated with R2 and 1 72 173 are components of the order parameter 7 associated with R4 The coupling is linear in q and the degree of the part of the polynomial containing components of 77 is 3 the faintness index SHOW FELIX The Felix frequency of the image is shown when DISPLAY IRREP or DISPLAY ISOTROPY is used The Felix frequency is the number
54. umber or by a symbol See VALUE PARENT for an explanation of the symbols used VALUE SUBGROUP MAXIMAL The maximal subgroups of the parent are selected This command affects the data dis played by DISPLAY ISOTROPY VALUE SUBGROUP SUPERSPACEGROUP For non k points of symmetry we consider isotropy subgroups which have the symme try of 3 1 dimensional superspace groups and allow incommensurate distortions This command affects the data displayed by DISPLAY IRREP DISPLAY ISOTROPY and DISPLAY DISTORTION VALUE SUPERSPACEGROUP pl p2 The 3 1 dimensional superspace groups are selected This command affects the data dis played by DISPLAY SUPERSPACEGROUP The parameters pl and p2 are superspace groups If they are both present then superspace groups pl through p2 are selected If only pl is present then only superspace group p1 is selected The superspace groups can be denoted by a number or by a symbol using the notation in International Tables Vol C For exam ple VALUE SUPERSPACEGROUP 49 8 refers to superspace group 49 8 Pmaa 0 7 This command could have also been entered as VALUE SUPERSPACEGROUP PMAA 01 2G Note that since parameters are delimited by space characters they should not contain any space characters themselves In the international notation bars over numbers are denoted by a preceding minus sign eg P 3C1 00G for P3c1 00y and subscripts are denoted by a preceding underline character eg P4_2 MCM 00G for P42 mcm
55. y frequencies select type of irrep show image symbol show generating matrices show all matrices show dimension show order show whether image is active or not show Landau frequency show type of irrep display data on irreps select space group s select irrep select k point select degrees of freedom of k point select parameters a 3 for the k point select image of irrep select space group element to display the character or matrix select dimension s select active or non active irreps pl Y N select Landau frequency frequencies select Lifshitz frequency frequencies select Felix frequency frequencies select type of irrep select wyckoff positions select point group irrep of wyckoff position select k point for compatibility relation show truncated irrep matrices used for incommensurate distortions show space group show irrep pl other settings M K B Z P show whether the irrep is active or not show image of irrep show space group elements that map onto the image generators show space group element selected show character of selected space group element show matrix of selected space group element show complex form of irrep matrix and character show generators of kernel DIM LAN LIF FEL TYPE FAINT FR KP KDEG STAR COMPAT Is IS COUP PAR pl p2 SUB pl p2 LAT pl p2 POINT pl p2 IR pl p2 KP pl KVAL pl p2 IM pl DIM pl p2 ACT pl CON pl LAN pl p2 LIF pl p2
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