As an example, let us consider octahedral tilting in the perovskite structure. These structures were originally classified by A. M. Glazer [Acta Cryst. (1972), B28, 3384] and more recently by C. J. Howard and H. T. Stokes [Acta Cryst. (1998), B54, 782].
We will begin with the structure of the parent phase already entered. Start at stokes.byu.edu/isodistort.html and click on the link labeled "Start with a cubic perovskite parent structure". You should see a new page with the title:
From the information at the top of the page, we see that the parent structure has cubic space-group symmetry #221 Pm-3m with lattice parameter a=4.2 Angstroms. Sr atoms are at the Wyckoff 1b positions, Ti atoms are at the Wyckoff 1a positions, and O atoms are at the Wyckoff 3d positions. This is the cubic perovskite structure of SrTiO3. We also see information about nearest-neighbor distances between every pair of unique atomic positions:
Nearest neighbor distances: Sr-Sr= 4.20000, Sr-Ti= 3.63731, Sr-O= 2.96985, Ti-Ti= 4.20000, Ti-O= 2.10000, O-O= 2.96985
This information can be useful for checking whether the atomic positions were entered correctly.
We will now enter the value for the k point in the first Brillouin zone. (This is "Method 2: General method".) In-phase octahedral tilts (+) produce structures at the M point, and out-of-phase tilts (-) produce structures at the R point. We will explore out-of-phase tilts and select the R point, as shown below. We leave the values for a,b,g blank since R is a special k point.
Specify k point: a= b= g= # of incommensurate modulations= (help)
As an alternate method ("Method 1: Search over all special k points") to choosing the k point, you may simply generate a list of possible symmetries of the distorted structure (called the isotropy subgroup of the parent structure). You may restrict the symmetries in the list by selecting one or more crystal systems, the space-group symmetry, or the conventional lattice. This is also a very useful tool if you don't know which k point to select (or which IR to select on the next page) but you do know something about the symmetry of the distorted structure you want to obtain.
Click on the "OK" button following the k point you entered. You should see a new page with the title:
In the drop-down list labeled "IR", we see the irreducible representations of the parent space-group at the R point. Actually, only the IRs which induce atomic displacements or atomic ordering in the parent structure are listed. IRs R2+, R1-, R3-, and R5- are missing from the list because they do not induce any atomic displacements or atomic ordering in the perovskite structure. We happen to know that octahedral tilts are induced by the IR R4+. If we didn't know this, we could try each IR in the list, one by one, and from the graphics display (which we will soon see), we could identify visually which distortions are octahedral tilts. Enter the IR R4+ as shown below.
Next choose an IR: (help)
Click on the "OK" button. You should see a new page with the title:
The order parameter is a vector in representation space. Since the IR R4+ is a three-dimensional IR, the order parameter here is a three-dimensional vector. In the drop-down list are shown various choices for the direction of the order parameter: (a,0,0), (a,a,0), (a,a,a), etc. Associated with each order parameter direction is an isotropy subgroup. These are the set of space-group operators that leave the order parameter direction invariant. The most general direction (a,b,c) results in distortions with triclinic symmetry, space group #2 P-1. We will begin by selecting the simplest direction, (a,0,0), as shown below. This will produce distortions with tetrahedral symmetry, space group #140 I4/mcm.
Finish selecting the distortion mode by choosing an order parameter direction
(help)
Click on the "OK" button. You should see a new page with the title:
Click on the "OK" button. You should see a new page in another new window with the title:
Click the "OK" button to save an ascii data file (an "isoviz" file) containing the information required to visualize the symmetry-mode parameters of the distortion. Then open the isoviz file with the standalone ISOVIZ program. See the ISOVIZ page of the ISOTROPY Suite for instructions on how to install and run ISOVIZ. In ISOVIZ, drag the mouse across the drawing, causing it to rotate, until it appears something like the drawing below.
The heavy lines in the drawing indicate the outlines of the unit
cell of the distorted structure (I4/mcm) and the unit
cell of the original parent structure (Pm-3m). To the
right of the drawing are a set of slide bars.
The colors of the slide bars correspond to colors of the atoms in the
drawing: red for Sr, green for Ti, and blue for O. Only the O has
slide bars since the IR R4+ only induces
atomic displacements and ordering
for that atom and not for the others. Move the
blue slide bar labeled R4+[O:d]Eu(a)
back and forth. This controls the amplitude of the
atomic displacements. You can clearly see that the O atoms are
rotating about the Ti atoms in an out-of-phase pattern: the rotation
is counterclockwise in the righthand plane, clockwise in the middle
plane, and counterclockwise again in the lefthand plane. This is the
Glazer
a0a0c- tilt
system.
The two black slide bars at the bottom affect the strain. GM1+strain(a)
is simply a volume strain. Its slide bar zooms the image
in and out. GM3+strain(a) is a tetrahedral strain. Its slide
bar increases the lattice parameters a and b while
decreasing c and vice versa.
Delete this window and return to the previous window with the title,
"distortion". You can make a CIF file of this
I4/mcm structure by selecting "CIF file" instead
of "Save interactive distortion" (which was the default selection). Also, you may
enter an amplitude for the atomic displacements in the box labeled
"[O:d]Eu(a)" (the default value is zero). This number has the same
meaning as the number next to the slide bar.
Next we will view a different tilt system. Delete this window and
return to the previous window with the title,
In the drop-down list, select the order parameter in the (a,a,0) direction,
as shown below.
Choose an order parameter direction
(help)
This order parameter produces distortions with orthorhombic symmetry,
space group #74, Imma. Click on the "OK" button. You should
see a new page with the title:
Here we see three boxes for amplitudes of atomic displacements: two
for the O atoms and one for the Sr atoms. We also see three boxes for
strain amplitudes. The boxes are separated by the
headings, "Primary:" and "Secondary:".
The primary order parameter always belongs to the IR you selected, in
this case, R4+. Secondary order parameters belong to other IRs, in this
case, R5+, GM1+, GM3+, and GM5+.
Distortions produced by secondary order parameters
exhibit symmetries which are equal or greater that those produced by
the primary order parameter. In this case, the symmetry produced by
R5+ is Imma, the same as that produced by R4+.
Click on the "OK" button. You should see a new page in another new
window with the title:
After saving the ISOVIZ file and opening it in ISOVIZ,
You now see seven slide bars: one for Ti displacements, two for O
displacements, one for O ordering, and three for strains. The slide
bar labeled R4+ is the primary order parameter, and the other slide
bars are the secondary order parameters. You should be able to easily
convince yourself that the R4+ slide bar for the O atoms causes a
rigid rotation of the O atoms about the Ti atoms. Furthermore, you
should be able to see that the rotation axis is along one of the cubic
<110> axes. This is the Glazer
a0b-b- tilt
system.
The most common distorted perovskite structure found in nature is the
Glazer
a+b-b- tilt
system, space group Pnma. This is a superposition of an
out-of-phase tilt system
(a0b-b-) with an
in-phase tilt system
(a+b0b0).
We obtain the distortions leading to this structure by
coupling two IRs together, R4+ for
a0b-b- and M3+
for a+b0b0. No
single IR will take the structure to
a+b-b-.
Delete the present window and the previous window, and then return to
previous pages until you see the page with the title:
Under method 2, enter 2 for the number of superposed IRs, as shown
below.
Specify number of IRs if superposed IRs are needed:
(help)
Click on the "OK" button. You should
see a new page with the same title as before:
This time, there are places to enter two different k points.
Enter the R and M points, as shown below.
k vector 1:
a=
b=
g=
(help)
Click on the "OK" button. You should
see a new page with the title:
Enter the two IRs, R4+ and M3+, as shown below.
IR 1:
(help)
Click on the "OK" button. You should
see a new page with the title:
Select the Pmna entry in the drop-down list, as shown below.
From this point on, you can generate the distortions and view a graphical
rendition just as before.
ISODISTORT: order parameter direction
ISODISTORT: distortion
ISODISTORT: save interactive distortion
Superposed IRs
ISODISTORT: search
ISODISTORT: search
k vector 2:
a=
b=
g=ISODISTORT: irreducible representation
IR 2:
ISODISTORT: order parameter direction