miscellaneous
settings
labeling
space groups
k points
images
irreps
isotropy subgroups
distortions
invariant polynomials
bushes
order parameters
Q: quit
SC p1: width of screen
PAGE p1: number of lines displayed at a time
C V ...: cancel a V command
C V ALL: cancel all previously used V commands
C SH ...: cancel a SH command
C SH ALL: cancel all previously used SH commands
D V: display all V commands in effect
D SH: display all SH commands in effect
?: help (can be inserted in place of any keyword)
D SET: display the current space-group setting
SET p1: change the setting (p1 = I, I NEW, I OLD, MI, K, B, Z)
SET I p1 OR p2: origin choice (p1 = space group or ALL and p2 = 1, 2)
SET I p1 AX p2: axis choice (p1 = space group or ALL and p2 = B, C, HEX, RH)
SET I p1 CELL p2: cell choice (p1 = space group or ALL and p2 = 1, 2, 3)
SET MAG: magnetic space groups
SET NOMAG: return to Federov space groups
L SP p1: change the labeling of space groups (p1 = SCH, I, I SHORT, I FULL)
L EL p1: change the labeling of space group elements (p1 = I, M, K, B, Z)
L VEC p1: change the form of the vectors (p1 = CON, PRIM)
L IM p1: use image notation of Toledano and Toledano (p1 = TOL, NOTOL)
L POINT p1: change the labeling of point groups (p1 = SCH, I)
L LAT p1: change the labeling of lattices (p1 = SCH, P)
D PAR: display data on parent space group
V PAR p1 [p2]: select space group(s)
V LAT PAR p1...p6: select lattice parameters a, b, c, alpha, beta, gamma
V LAT p1 [p2]: select lattice(s)
V POINT p1 [p2]: select point group(s)
V WY p1 [p2...]: select wyckoff positions
V WY IR p1 [p2...]: select point-group irrep of wyckoff position
V WY XYZ p1 p2 p3: select parameters x,y,z of wyckoff position
SH PAR: show space group symbol
SH CART: show Cartesian coordinates
SH GEN: show generating elements of space group
SH EL: show coset representatives of space group
SH LAT: show lattice
SH POINT: show point group
SH BAS: show basis vectors of lattice
SH WY [p1 p2]: show Wyckoff positions (p1 = VEC, POINT, ELE, CHAR, p2 = ALL)
D KP: display data on k points
V PAR p1 [p2]: select space group(s)
V IR p1: select irrep
V KP p1: select k point
V KDEG p1: select degrees of freedom of k point
SH PAR: show space group symbol
SH KP: show coordinates of k point
SH KDEG: show degrees of freedom of k point
SH STAR: show the star of k
D IM: display data on images of irreps
V IM p1: select image
V DIM p1 [p2]: select dimension(s)
V ORD p1 [p2]: select order(s)
V ACT p1: select active or non-active images (p1 = Y, N)
V LAN p1 [p2]: select Landau frequency/frequencies
V TYPE p1: select type of irrep
SH IM: show image symbol
SH GEN: show generating matrices
SH EL: show all matrices
SH DIM: show dimension
SH ORD: show order
SH ACT: show whether image is active or not
SH LAN: show Landau frequency
SH TYPE: show type of irrep
D IR: display data on irreps
V PAR p1 [p2]: select space group(s)
V IR p1: select irrep
V KP p1: select k point
V KDEG p1: select degrees of freedom of k point
V KVAL p1 [p2...]: select parameters alpha, beta, gamma for the k point
V IM p1: select image of irrep
V EL p1: select space-group element (to display the character or matrix)
V DIM p1 [p2]: select dimension(s)
V ACT p1: select active or non-active irreps (p1 = Y, N)
V LAN p1 [p2]: select Landau frequency/frequencies
V LIF p1 [p2]: select Lifshitz frequency/frequencies
V FEL p1 [p2]: select Felix frequency/frequencies
V TYPE p1: select type of irrep
V WY p1 [p2...]: select wyckoff positions
V WY IR p1: select point-group irrep of wyckoff position
V COMPAT p1: select k point for compatibility relation
SH PAR: show space group
SH IR [p1]: show irrep (p1=other settings: M, K, B, Z, P)
SH ACT: show whether the irrep is active or not
SH IM: show image of irrep
SH GEN: show space-group elements that map onto the image generators
SH EL: show space-group element selected
SH CH: show character of selected space-group element
SH MAT: show matrix of selected space-group element
SH COMPLEX: show complex form of irrep matrix and character
SH KER: show generators of kernel
SH DIM: show dimension
SH LAN: show Landau frequency
SH LIF: show Lifshitz frequency
SH FEL: show Felix frequency
SH TYPE: show type of irrep
SH FAINT: show faintness index for linear coupling with other irreps
SH FR: show frequency and irreps of the Wyckoff-position point groups
SH KP: show coordinates of k point
SH KDEG: show degrees of freedom of k point
SH STAR: show the star of k
SH COMPAT: show compatibility relations
D IS: display data on isotropy subgroups
D IS COUP: display data on isotropy subgroups for coupled order parameters
V PAR p1 [p2]: select parent space group(s)
V SUB p1 [p2]: select isotropy subgroup(s)
V LAT p1 [p2]: select lattice(s) of subgroup
V POINT p1 [p2]: select point group(s) of subgroup
V IR p1 [p2...]: select irreps
V IR MAG p1 [p2...]: indicate which irreps
are magnetic
V KP p1: select k point
V KVAL p1 [p2...]: select parameters alpha, beta, gamma for the k points
V IM p1: select image of irrep
V DIM p1 [p2]: select dimension(s) of irrep
V ACT p1: select active or non-active irreps (p1 = Y, N)
V CON p1: select continuous or discontinuous transitions (p1 = LAN, RG, NO)
V LAN p1 [p2]: select Landau frequency/frequencies
V LIF p1 [p2]: select Lifshitz frequency/frequencies
V FEL p1 [p2]: select Felix frequency/frequencies
V DIR p1 [p2...]: select directions of order parameter
V FREQ p1 [p2]: select subduction frequency/frequencies
V DOM p1: select domain
V DOM P p1 p2: select domain pair
V DOM SET OR p1: select order of domain set
V DOM SET CL p1: select class of domain set
V DOM SET DIR p1: select direction of domain set vector
V DOM SET UN p1: select number of unconnected parts in domain set
V NORM p1 p2 p3: select Miller indices (hkl) of plane between twin domains
V POS p1 p2 p3: select point on plane between twin domains
V SIZ p1 [p2]: select size of subgroup's primitive unit cell relative to parent group's
V MAX p1: select maximal or non-maximal subgroups (p1 = Y, N)
V SUB MAX : select maximal subgroups the of parent
V LAT PAR p1...p6: select lattice parameters a, b, c, alpha, beta, gamma
V BAS p1 p2 p3: select alternate basis vectors
V ORI p1: select alternate origin
V FER p1: select ferroic species (p1 = FC, PFC, IFC, FS, PFS,
IFS, NF, OTHER,
SH PAR: show parent space group symbol
SH SUB: show isotropy subgroup symbol
SH DIR [p1]: show direction of order parameter (p1=VEC to show the vector)
SH BAS: show basis vectors of subgroup
SH ORI: show origin of subgroup
SH SUB ALT: show alternate basis vectors and origin of subgroup
SH XYZ: show x,y,z in subgroup in terms of x,y,z in parent group
SH WY SUBG: show Wyckoff positions in subgroup
SH GEN: show elements of parent group which generate subgroup
SH EL: show elements of parent group which are elements of subgroup
SH NEW: show new fractionals in subgroup
SH SIZ: show relative size of subgroup's primitive unit cell to parent group's
SH IND: show index of subgroup in parent group
SH DOM [p1]: show domains (p1=GEN to show generators)
SH DISTINCT: show which domains are distinct
SH PAIR: show which pairs of domains are equivalent
SH PAIR p1: show pair intersection group (p1=I), pair group (p1=GR) and its basis vectors (p1=B),
its origin (p1=O), its generating elements (p1=GE), and its elements (p1=E).
SH TWIN INT p1: show twin intersection group: its label (p1=GR), its basis vectors (p1=B), its origin (p1=O),
its generating elements (p1=GE), and its elements (p1=E).
SH TWIN p1: show twin group: its label (p1=GR), its basis vectors (p1=B), its origin (p1=O),
its generating elements (p1=GE), and its elements (p1=E).
SH TWIN SW p1: show switching elements in twin group (p1=SIDE, NORMAL, BOTH).
SH DOM SET [p1]: 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)
SH MAX: show whether or not the subgroup is maximal
SH LAT: show lattices of parent space group and of subgroup
SH POINT: show point groups of parent space group and of subgroup
SH IR [p1]: show irrep (p1=other settings: M, K, B, Z, P)
SH IM: show image of irrep
SH DIM: show dimension
SH ACT: show whether the irrep is active or not
SH CON: show whether or not the phase transition is continuous
SH LAN: show Landau frequency
SH LIF: show Lifshitz frequency
SH FEL: show Felix frequency
SH FR [p1]: show frequencies and irreps that subduce the subgroup (p1=DIR, GAM, CL)
SH CART: show Cartesian coordinates
SH FER: show ferroic species
D DIST: display distortions
V PAR p1: select parent space group
V IR p1: select irrep
V KP p1: select k point
V KVAL p1 [p2...]: select parameters alpha, beta, gamma for the k point
V RANK p1: select rank and symmetries of macroscopic tensor
V DIR p1 [p2...]: select directions of order parameter
V DOM p1: select domain
V CELL p1 p2 p3: select super cell
V LAT PAR p1...p6: select lattice parameters a, b, c, alpha, beta, gamma
V WY p1 [p2...]: select wyckoff positions
V WY IR p1 [p2...]: select point-group irrep of wyckoff position
V WY XYZ p1 p2 p3: select parameters x,y,z of wyckoff position
V POS p1 p2 p3: select coordinates of point
SH MAC [p1]: show macroscopic distortion (p1=PSEUDO)
SH MIC [p1] [p2]: show microscopic distortion (p1=SC, VEC, p2=PSEUDO)
SH LOC [p1] [p2]: show local distortion (p1=SC, VEC, p2=PSEUDO)
SH PAR: show parent space group symbol
SH IR [p1]: show irrep (p1=other settings: M, K, B, Z, P)
SH DIR: show direction of order parameter
SH DOM: show domain
SH WY [p1]: show Wyckoff positions (p1=IR)
SH POS IR: show point group irrep of point about Wyckoff position
SH UNIT: show distortions in unit cell of parent
SH CART: show Cartesian coordinates
D INV: display invariant polynomials
V PAR p1: select parent space group
V IR p1 [p2...]: select irreps
V KVAL p1 [p2...]: select parameters alpha, beta, gamma for the k point
V DIR p1 [p2...]: select directions of order parameter
V DOM p1 [p2...]: select domains
V DEG p1 [p2]: select degree(s) of polynomials
V GRAD p1: select number of spatial derivatives
SH PAR: show parent space group symbol
SH IR [p1]: show irrep (p1=other settings: M, K, B, Z, P)
SH DIM: show dimension
D BUSH: display data on bush of modes
V PAR p1: select parent space group
V IR p1: select irrep
V KVAL p1 [p2...]: select parameters alpha, beta, gamma for the k point
V DIR p1: select direction of order parameter
V WY p1 [p2...]: select Wyckoff positions
V DEG p1 [p2]: select degree(s) of invariant polynomials
SH MODES: show atomic displacements
SH INV: show invariant polynomials in free energy
D DIR: display directions of order parameters
V PAR p1: select parent space group
V SUBG p1: select subgroup
V BASIS p1 p2 p3: select basis vectors of subgroup lattice
V ORIGIN p1: select origin of subgroup
SH KPOINT: show k vectors
SH SUB: show subgroup symbol
SH SIZ: show relative size of subgroup's primitive unit cell to parent group's
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.
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).
2. International Tables for X-Ray Crystallography, Vol. I, edited by N. F. M. Henry and K. Lonsdale (Kynoch Press, Birmingham, 1965).
3. 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).
4. O. V. Kovalev, Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Corepresentations (Gordon and Breach, Amsterdam, 1993).
5. C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, Oxford, 1972).
6. 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 all k 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.) and the default irrep labeling is Miller-Love.
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 six words.
In the following material, each command 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 p1 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 automatically executed.
CANCEL SHOW IRREP [p1]
Cancels the effect of SHOW IRREP. The parameter p1 is the name
of the one the settings (MILLER-LOVE, KOVALEV,
BRADLEY-CRACKNELL, or ZAK). If p1 is present, then
only the notation specified by p1 is removed from the display.
If p1 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. When 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 displayed.
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 derivatives. 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
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 alpha, beta, gamma which define the exact location
of the k vector.
The data for these subgroups are read from a special file.
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.
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 subgroups are read from a special file.
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. 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 VALUE
All VALUE commands currently in effect are shown.
LABEL ELEMENT p1
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 p1
indicates the notation to be used: INTERNATIONAL, MILLER-LOVE,
KOVALEV, BRADLEY-CRACKNELL and ZAK for the notations
of the international tables, 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.
LABEL IMAGE p1
LABEL IMAGE TOLEDANO adds the notation of Toledano and Toledano
for images whenever image symbols are displayed.
LABEL IMAGE NOTOLEDANO removes this notation.
LABEL LATTICE p1
The notation for Bravais lattices is changed. This affects the label
displayed when SHOW LATTICE is used.
The parameter p1 indicates the notation to be used: SCHOENFLIES
and PEARSON. See the VALUE LATTICE command for more
explanation about these notations.
LABEL POINTGROUP p1
The notation for point groups is changed. This affects the label
displayed when SHOW POINTGROUP is used.
The parameter p1 indicates the notation to be used: INTERNATIONAL
and SCHOENFLIES. See the VALUE POINTGROUP command for more
explanation about these notations.
LABEL SPACEGROUP p1
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 p1
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 (1/2)i+(1/2)j 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 vector for that lattice.)
LABEL VECTOR PRIMITIVE cannot be used with the
INTERNATIONAL setting.
PAGE p1
This command sets the number of lines that can be displayed 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 p1 is
NOBREAK, all lines are displayed at one time.
QUIT
The program ISOTROPY exits.
SCREEN p1
The width of the display is changed to p1 characters wide. The default
width 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 p1
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 setting can be displayed using the
DISPLAY SETTING command.
Additional options for the settings in International
Tables are discussed under the command, SETTING INTERNATIONAL p1.
If the SETTING INTERNATIONAL command is used (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 p1
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 p1 ORIGIN p2, where p1 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 p1. (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 SETTING INTERNATIONAL p1 AXIS p2, where p1 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 p1. (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 p1 CELL p2, where p1 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 particular space group, use SETTING INTERNATIONAL p1 AXIS p2, where p1 is a space-group number or symbol and p2 is HEXAGONAL or RHOMBOHEDRAL. The choice of axes for all trigonal space groups can be changed by using ALL for p1. (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 R-3c. 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 associated 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', RI32.
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 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 elements, 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 GENERATORS and SHOW ELEMENT).
(3) DISPLAY DISTORTION: Wyckoff positions and displacement vectors
(SHOW MICROSCOPIC).
In the case of Wyckoff positions, the values of the parameters 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 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 INVARIANTS 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 [p1]
The symbol for the direction of the order parameter is shown when
DISPLAY ISOTROPY or
DISPLAY DISTORTION is used.
If p1 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 X3+ and P5 of space group D4h17. We would find that one of the subgroups is D2d4 with order parameter direction P1(1)P4(3). This means that this subgroup is an intersection of the first domain of D4h16 [irrep X3+, direction P1=(a,0)] and the third domain of D2d11 [irrep P5, direction P4=(a,0,- a,0)]. The direction of the order parameter for D2d4 is denoted (a,0,b,0,-b,0), the first two components associated with irrep X3+ and the remaining four components associated with irrep P5. 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 VALUE 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 used 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.
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
q1n1n2n3
where q1 is the order parameter associated with R2+ and
n1,n2,n3 are components of the order parameter n
associated with
R4+. The coupling is linear in q1, and the degree of the part of
the polynomial containing components of n 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 of anisotropic gradient terms in the LGW
Hamiltonian.
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 p1 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 p1 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 p1 is GAMMA, only the gamma (k=0) irreps are shown. If p1 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 used 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.
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 of the
current space-group setting. The parameter p1 allows the irrep
symbol to also be shown in other notations. p1 can be MILLER-LOVE,
KOVALEV, BRADLEY-CRACKNELL, or ZAK, referring to the
irrep notations of Miller and Love, Kovalev, Bradley and Cracknell, and
Zak, respectively. p1 can also be POINTGROUP, in which case the
conventional labels for point-group irreps are shown for irreps at the
gamma point (k=0).
Even when p1 is present, this command causes the
irrep to be shown in the notation of the current setting in addition
to the setting specified by p1. 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 DISPLAY PARENT is used, the Bravais lattice of the 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 [p1] [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 p1 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 occupation probabilities.
The command SHOW LOCAL VECTOR displays vector
distortions, such as 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 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 [p1] [p2]
The microscopic distortions are displayed when DISPLAY DISTORTION is
used. If p1 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 probabilities.
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
which 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.
SHOW PAIRS [p1]
This command shows information about pairs of domains of isotropy subgroups
when DISPLAY ISOTROPY is used.
We denote a pair of domains by (Pi,Pj), where
Pi 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 (P1,Pj) is considered
for each domain j.
When the pair has not been selected, equivalence classes of domain pairs are displayed. Any class of pairs has an element of the form (P1,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 p1 is INTERSECT, the intersection of the isotropy groups belonging to the two domains in the domain pair is 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 (Pi,Pj), this element takes Pi to Pj and it also takes Pj to Pi. 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 p1 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 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 ISOTROPY is used.
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 consistent 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 TWIN p1 [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 by the
VALUE DOMAIN PAIR command. Otherwise, the pair (P1,Pj) 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 p1 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 displayed about the twin group is controlled by p1. 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, 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 p1 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 shown 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 [p1,p2]
When DISPLAY PARENT is used,
the symbol for the Wyckoff position is shown.
If p1 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 p1 is ELEMENTS, the space-group elements which belong to the
point group of the Wyckoff position are shown.
If p1 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 x,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 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 p1
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 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 p1 p2 p3
The basis vectors of a subgroup are selected. This command affects the data
displayed by DISPLAY DIRECTION. p1,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 p1 p2 p3
The basis vectors of a super cell are selected. This command affects the data
displayed by DISPLAY DISTORTION. p1,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 p1
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 p1
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 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 p1 [p2]
The degrees of invariant polynomials are selected.
This command affects the data displayed by DISPLAY INVARIANTS.
The parameters, p1 and p2, are numbers representing the degree of the
polynomial. If they are both present,
then degrees p1 through p2 are selected. If only p1 is
present, then only degree p1 is selected. If this command has not been used
yet, or if the CANCEL VALUE DEGREE command has been used, ISOTROPY
displays the invariant polynomials of degrees 1 through 4 by default.
VALUE DIMENSION p1 [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, p1 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 p1 [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
parameter. 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 select the direction P1). VALUE DIRECTION KERNEL selects the order parameter in the most general direction. VALUE DIRECTION ONEARM selects order parameters 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 DIRECTION VECTOR,A,0,0 selects the order parameter (a,0,0), and VALUE DIRECTION VECTOR,0.5A,0.866A selects the order parameter (1/2a,1/2sqrt(3)a). Note that irrational coefficients (and sometimes rational coefficients such as 1/3) must be given to three decimal places.
VALUE DOMAIN p1 [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.
p1 is a number indicating the domain. (The numbering of domains may
be obtained by using SHOW 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 p1 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.
VALUE DOMAIN SETS CLASS p1
The class of the domain set is selected.
This command affects the data displayed by DISPLAY ISOTROPY.
p1 is a number which is displayed in the "class" column when
SHOW DOMAIN SETS is used.
VALUE DOMAIN SETS DIRECTION p1
The direction of the domain set vector is selected.
This command affects the data displayed by DISPLAY ISOTROPY.
p1 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 p1
The order of the domain set is selected.
This command affects the data displayed by DISPLAY ISOTROPY.
p1 is a number representing the number of domains present in the set.
VALUE DOMAIN SETS UNCONNECTED p1
The number of unconnected parts of the domain set is selected.
This command affects the data displayed by DISPLAY ISOTROPY.
p1 is a number.
VALUE ELEMENT p1
The element of the parent space group is selected.
This command affects the data displayed by DISPLAY IRREP.
p1 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 0 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 character. In the other notations, the point operation comes first,
followed by the fractional, each part separated by a space character.
The notation used for p1 must agree with the point-operation notation
selected for elements. The
LABEL ELEMENT command changes the notation selected.
VALUE FELIX p1 [p2]
The Felix frequency of the image
is selected. This command affects the data displayed
by DISPLAY IRREP and DISPLAY ISOTROPY.
The parameters, p1 and p2, are numbers representing 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 p1
The ferroic species of the phase transition
is selected. This command affects the data displayed
by DISPLAY ISOTROPY.
The parameter p1 is a symbol representing the ferroic species:
FC (ferroelectric), PF (proper ferroelectric),
IFC (improper ferroelectric), FS (ferroelastic),
PFS (proper ferroelastic), IFS (improper ferroelastic),
NF (nonferroic), OTHER (other ferroic).
VALUE FREQUENCY p1 [p2]
The subduction frequency of the isotropy subgroup
is selected. This command affects the data displayed
by DISPLAY ISOTROPY.
The parameters, p1 and p2, are numbers representing the subduction
frequency.
If they are both present,
then subduction frequencies p1 through p2 are selected. If only p1 is
present, then only subduction frequency
p1 is selected.
VALUE GRADIENT p1
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 representing the number of derivatives to
appear in each invariant polynomial.
VALUE IMAGE p1
The image of the irrep is selected. This command affects the data displayed
by DISPLAY IMAGE, DISPLAY IRREP, and DISPLAY ISOTROPY.
The parameter p1 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 Toledano and Toledano.
VALUE IRREP p1 [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 p1,p2,... are irreps to be selected.
Only the first irrep listed (p1) is used by DISPLAY DISTORTIONS,
DISPLAY IRREP, and
DISPLAY ISOTROPY. This command cancels 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, Y1+, 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
p1. GM is used for gamma. 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 p1 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 IRREP MAGNETIC p1, [p2...]
Irreps are designated to be either magnetic YES or
nonmagnetic NO. This command affects the data displayed by
DISPLAY ISOTROPY COUPLED when SETTING MAGNETIC is used.
VALUE KDEGREE p1
The degrees of freedom of the k vector are selected.
This command affects the data displayed
by DISPLAY IRREP and DISPLAY KPOINT.
VALUE KPOINT p1
Irreps which arise from a given k point are selected.
This command affects the data displayed
by DISPLAY IRREP, DISPLAY ISOTROPY and DISPLAY DISTORTION.
This command also selects the k vector displayed by DISPLAY KPOINT.
The parameter p1 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
p1. GM is used for gamma.
As with irrep 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 p1 [p2 ...]
The values of the parameters alpha,beta,gamma defining the k vectors
are selected. The parameters p1,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 alpha which gives
the position of the vector on the line. In that case,
VALUE KVALUE 1,1/4 would select one value, alpha=1/4. VALUE KVALUE 2,1/4,3/8 selects two values, alpha=1/4 and beta=3/8.
VALUE KVALUE 0 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 p1 [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, p1 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 p1 is
present, then only Landau frequency
p1 is selected.
VALUE LATTICE p1 [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.
The parameters, p1 and p2 are lattices. If they are both present, then
lattices p1 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. triclinic | 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 | O | OP | 11. Hexagonal | H | HP |
| 5. base-centered orthorhombic | O-B | OC | 12. primitive cubic | C | CP |
| 6. body-centered orthorhombic | O-V | OI | 13. face-centered cubic | C-F | CF |
| 7. face-centered orthorhombic | O-F | OF | 14. body-centered cubic | C-V | CI |
VALUE LATTICE PARAMETER p1 p2 p3 p4 p5 p6
The lattice parameters are selected. The parameters p1 through p6
are the values of a,b,c,alpha,beta,gamma, respectively. a,b,c
are the lengths of the 3 basis vectors a,b,c of the conventional
lattice, alpha is the angle between b and c, beta is the angle
between a and c, and gamma is the angle between a and b.
VALUE LIFSHITZ p1 [p2]
The Lifshitz frequency of the irrep
is selected. This command affects the data displayed
by DISPLAY IRREP and DISPLAY ISOTROPY.
The parameters, p1 and p2, are numbers representing the Lifshitz frequency
of an irrep (the number of times that the vector representation is contained
in the antisymmetrized cube of the irrep).
If they are both present,
then Lifshitz frequencies p1 through p2 are selected. If only p1 is
present, then only Lifshitz frequency
p1 is selected.
VALUE MAXIMAL p1
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 any other
isotropy subgroup for the same irrep.
VALUE NORMAL p1 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.
p1 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 p1 [p2]
The orders of the image
are selected. This command affects the data displayed
by DISPLAY IMAGE.
The parameters, p1 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 p1
The origin of a subgroup is selected. This command affects the data
displayed by DISPLAY DIRECTION. p1 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 p1 [p2]
The parent space groups are selected. This command affects the
data displayed by DISPLAY PARENT, DISPLAY INVARIANTS,
DISPLAY DISTORTION,
DISPLAY IRREP, DISPLAY ISOTROPY,
and DISPLAY KPOINT.
The parameters, p1 and p2, are space groups.
If they are both present, then space groups p1 through p2
are selected. If only p1 is present, then only space group p1
is selected. The space groups can be denoted by a number or by a symbol,
using either the Schoenflies or international notation (either short or full
symbol). For example,
VALUE PARENT 124 refers to space group #124 (D24h or
P4/mcc 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)
and subscripts are denoted by a preceding underline character (eg.,
P4_2/MCM for P42/mcm).
VALUE POINTGROUP p1 [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.
The parameters, p1 and p2 are space groups. If they are both present,
then point groups p1 through p2 are selected (using the order shown in
the table below). If only p1 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 Oh or m-3m. A bar over a number is denoted by a preceding
minus sign (eg., -4 for "4 bar").
| 1. C1 | 1 | 9. C4 | 4 | 17. C3I | -3 | 25. C6V | 6MM |
| 2. CI | -1 | 10. S4 | -4 | 18. D3 | 32 | 26. D3H | -62M |
| 3. C2 | 2 | 11. C4H | 4/M | 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 | M-3 |
| 6. D2 | 222 | 14. D2D | -42M | 22. C3H | -6 | 30. O | 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. OH | M-3M |
VALUE POSITION p1 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
(1/2,1/2,1/2).
Also, irrational coordinates can be denoted by x, y, or z.
For example, x 1/2-x 0 would denote coordinates at
(x,1/2-x,0), where x is an arbitrary irrational number.
The coordinates x y z would denote a general point.
The command VALUE POSITION also affects the data displayed by DISPLAY 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 p1
The rank of a macroscopic
tensor is selected. This command affects the data displayed
by DISPLAY DISTORTION when SHOW MACROSCOPIC is used.
p1 shows the indices of the tensor in numerical order. For example,
p1=1234 indicates a tensor of rank 4. Indices to be symmetrized are
enclosed by square brackets. For example, p1=[12] indicates
a totally-symmetrized tensor of rank 2. Indices to be antisymmetrized are
enclosed by curly brackets. For example,
p1=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 p1 [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, p1 and p2, are numbers representing relative size.
If they are both present,
then sizes p1 through p2 are selected. If only p1 is
present, then only size p1 is selected.
VALUE SUBGROUP p1 [p2]
The isotropy subgroup space groups are selected. This command affects the
data displayed by
DISPLAY ISOTROPY.
The parameters, p1 and p2, are space groups.
If they are both present, then space groups p1 through p2
are selected. If only p1 is present, then only space group p1
is selected. The space groups can be denoted by a number 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 displayed by DISPLAY ISOTROPY.
VALUE TYPE p1
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 p1 [p2 ...]
Wyckoff positions are selected. This command affects the data displayed by
DISPLAY PARENT, DISPLAY IRREP, and DISPLAY DISTORTION.
The parameters p1,p2,... are the single-letter
symbols of the positions, as given in International Tables.
VALUE WYCKOFF IRREP p1 [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 parameters p1,p2,... are the irrep symbols, using the convention of
Bradley and Cracknell.
VALUE WYCKOFF XYZ p1 p2 p3
The parameters x,y,z of the Wyckoff position are selected. p1,p2,p3
are the values of x,y,z, respectively.