[转载]Wyckoff Positions
2014-12-02 11:00阅读：
wyckoff position 各种晶格群的表格
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list
下面是从其他网站转的，读过后，对wyckoff position
可以有大致的了解了。
Space Groups & The International Tables for
Crystallography
From examination of a space group in “The International Tables for
Crystallography” Vol. A, you should be able to ascertain the
following information:
·
Herman-Mauguin (HM) Symbol (Long, Short)
·
Point
Group (HM, Schoenflies)
·
Locate
and identify symmetry elements
·
Understand Wyckoff site multiplicity and symmetry
·
Distinguish general and special positions
·
Extinction
conditions
·
Identify
possible subgroups and supergroups
Understanding the Herman-Mauguin Space Group Symbol
Space groups are typically identified by their short Herman-Mauguin
symbol (i.e. Pnma, I4/mmm, etc.).
The symmetry elements
contained in the short symbol are the minimum number needed to
generate all of the remaining symmetry elements.
This
symbolism is very efficient, condensed form of noting all of the
symmetry present in a given space group.
We won’t go into
all of the details of the space group symbol, but I will expect you
to be able to determine the Crystal system, Bravais Lattice and
Point group from the short H-M symbol. You should also be able to
determine the presence and orientation of certain symmetry elements
from the short H-M symbol and vice versa.
The HM space group symbol can be derived from the symmetry elements
present using the following logic.
The first letter identifies the centering of the lattice, I will
hereafter refer to this as the lattice descriptor :
·
P ®
Primitive
·
I
® Body centered
·
F ® Face
centered
·
C ®
C-centered
·
B ®
B-centered
·
A ®
A-centered
The next three symbols denote symmetry elements present in certain
directions, those directions are as follows:
Crystal System
|
Symmetry Direction
|
Primary
|
Secondary
|
Tertiary
|
Triclinic
|
None
|
|
|
Monoclinic
|
[010]
|
|
|
Orthorhombic
|
[100]
|
[010]
|
[001]
|
Tetragonal
|
[001]
|
[100]/[010]
|
[110]
|
Hexagonal/
Trigonal
|
[001]
|
[100]/[010]
|
[120]/[1`1 0]
|
Cubic
|
[100]/[010]/
[001]
|
[111]
|
[110]
|
[100] – Axis parallel or plane perpendicular to the x-axis.
[010] – Axis parallel or plane perpendicular to the y-axis.
[001] – Axis parallel or plane perpendicular to the z-axis.
[110] – Axis parallel or plane perpendicular to the line running at
45° to the x and y axes.
[1`1 0] – Axis parallel or plane perpendicular to the long face
diagonal of the ab face of a hexagonal cell.
[111] – Axis parallel or plane perpendicular to the body
diagonal.
For a better understanding see specific examples from class notes.
However, with no knowledge of the symmetry diagram we can
identify the crystal system from the space group symbol.
·
Cubic – The secondary symmetry symbol will always be either
3 or –3 (i.e. Ia3, Pm3m, Fd3m)
·
Tetragonal – The primary symmetry symbol will always be
either 4, (-4), 4
1, 4
2 or 4
3 (i.e.
P4
12
12, I4/m, P4/mcc)
·
Hexagonal – The primary symmetry symbol will always be a 6,
(-6), 6
1, 6
2, 6
3, 6
4 or
6
5 (i.e. P6mm, P6
3/mcm)
·
Trigonal – The primary symmetry symbol will always be a 3,
(-3) 3
1 or 3
2 (i.e P31m, R3, R3c, P312)
·
Orthorhombic – All three symbols following the lattice
descriptor will be either mirror planes, glide planes, 2-fold
rotation or screw axes (i.e. Pnma, Cmc2
1, Pnc2)
·
Monoclinic – The lattice descriptor will be followed by
either a single mirror plane, glide plane, 2-fold rotation or screw
axis or an axis/plane symbol (i.e. Cc, P2, P2
1/n)
·
Triclinic – The lattice descriptor will be followed by
either a 1 or a (-1).
The point group can be determined from the short H-M symbol by
converting glide planes to mirror planes and screw axes to rotation
axes. For example:
Space Group = Pnma ® Point Group = mmm
Space Group = I`4c2 ® Point Group =`4m2
Space Group = P4
2/n ® Point Group = 4/m
Wyckoff Sites
One of the most useful pieces of information contained in the
International Tables are the Wyckoff positions. The Wyckoff
positions tell us where the atoms in a crystal can be found.
To understand how they work consider the monoclinic space
group Pm. This space group has only two symmetry elements,
both mirror planes perpendicular to the b-axis. One at y =
0 and one at y = ½ (halfway up the unit cell in the b direction).
Now let’s place an atom in the unit cell at an arbitrary position,
x,y,z. If we now carry out the symmetry operation
associated with this space group a second atom will be generated by
the mirror plane at x,-y,z. However, if we were to place
the atom on one of the mirror planes (its y coordinate would have
to be either 0 or ½) then the reflection operation would not create
a second atom.
All of the information in the proceeding paragraph is contained
in Wyckoff positions section of the International Tables.
Pm has three Wyckoff sites as shown in the table below:
Multiplicity
|
Wyckoff Letter
|
Site Symmetry
|
Coordinates
|
2
|
c
|
1
|
(1) x,y,z (2)
x,-y,z
|
1
|
b
|
m
|
x,½,z
|
1
|
a
|
m
|
x,0,z
|
The multiplicity tells us how many atoms are generated by symmetry
if we place a single atom at that position.
In this case
for every atom we insert at an arbitrary position (x,y,z) in the
unit cell a second atom will be generated by the mirror plane at
x,-y,z.
This corresponds to the uppermost Wyckoff position
2c.
The letter is simply a label and has no physical
meaning.
They are assigned alphabetically from the bottom
up.
The symmetry tells us what symmetry elements the atom
resides upon.
The uppermost Wyckoff position, corresponding
to an atom at an arbitrary position never resides upon any symmetry
elements.
This Wyckoff position is called the general
position.
The coordinates column tells us the coordinates
of all of the symmetry related atoms (two in this case).
All of the remaining Wyckoff positions are called special
positions.
They correspond to atoms which lie upon one of
more symmetry elements, because of this they always have a smaller
multiplicity than the general position.
Furthermore, one or
more of their fractional coordinates must be fixed.
In this
case the y value must be either 0 or ½ or the atom would no longer
lie on the mirror plane.
Generating a Crystal Structure from its Crystallographic
Description
Using the space group information contained in the
International Tables we can do many things. One powerful
use is to generate an entire crystal structure from a brief
description.
Consider the following description of the crystal structure of
Sr2AlTaO6.
Space Group = Fm`3m
a = 7.80 Å
Atomic Positions
Atom X
Y Z
Sr 0.25
0.25
