1. No category

III Crystal Symmetry
3-3 Point group and space group
A.Point group
Symbols of the 32 three dimensional point groups
Rotation axis X
Rotation-Inversion axis X
x
X
X + centre (inversion): Include X for odd
order
X2 or Xm even: only for even rotation
symmetry
X2 + centre; Xm +centre: the same result
(Include Xm for odd order)
Ratation axis with mirror plane normal to it X/m
x
x
mirror
Rotation axis with mirror plane (planes) parallel
to it Xm
x
mirror
x
Rotation axis with diad axis (axes) normal to it
x
X2
x
Rotation-inversion axis with diad axis (axes)
normal to it X2
X
X
Rotation-inversion axis with mirror plane (planes)
parallel to it Xm
X
mirror
X
Rotation axis with mirror plane (planes) normal
to it and mirror plane (planes) parallel to it X/mm
x
mirror
x
mirror
System and
point group
Position in point group symbol
Primary
Secondary
Tertiary
Triclinic
1, 1
Only one symbol which denotes all
directions in the crystal.
Monoclinic
2, m, 2/m
The symbol gives the nature of the
unique diad axis (rotation and/or
inversion).
1st setting: z-axis unique
2nd setting: y-axis unique
Stereographic
representation
1st setting
2nd setting
System and
point group
Orthorhombic
222, mm2,
mmm
Position in point group symbol
Primary
Secondary
Tertiary
Diad (rotation Diad
Diad
and/or
(rotation
(rotation
inversion)
and/or
and/or
along x-axis inversion)
inversion)
along y-axis along z-axis
Tetragonal
Tetrad
Diad
Diad
4, 4, 4/m,
(rotation
(rotation
(rotation
422, 4mm,
and/or
and/or
and/or
42m, 4/mmm inversion)
inversion)
inversion)
along z-axis along x- and along [110]
y-axes
and [110]
axis
Stereographic
representation
System and
point group
Trigonal and
Hexagonal
3, 3, 32, 3m,
3m, 6, 6, 6/m,
622, 6mm,
6m2, 6/mmm
Cubic
23, m3,
432,43m,
m3m
Position in point group symbol
Stereographic
Primary
Secondary
Tertiary
representation
Triad or
Diad
Diad
hexad
(rotation
(rotation
(rotation
and/or
and/or
and/or
inversion)
inversion)
inversion)
along x-, y- normal to x-,
along z-axis and u-axes
y-, u-axes in
the plane
(0001)
Diads or
Triads
Diads
tetrad
(rotation
(rotation
(rotation
and/or
and/or
and/or
inversion)
inversion)
inversion)
along <111> along <110>
along <100> axes
axes
axes
Symmetry Direction
Crystal
System
Primary
Triclinic
None
Monoclinic
[010]
Orthorhombic
Secondary
Tertiary
[100]
[010]
[001]
Tetragonal
[001]
[100]/[010]
[110]
Hexagonal/
Trigonal
[001]
[100]/[010]
[120]/[110]
Cubic
[100]/[010]/
[001]
[111]
[110]
Rotation axis X
Triclinic
1
Rotation-Inversion axis X
1
X + centre
Include X (odd order)
1
For odd order includes X already!
Monoclinic
1st setting
X
2
X
2=m
X + centre
Include X (odd order)
2
𝑚
2
mirror
2
mirror
2=m
2
𝑚
Monoclinic
2st setting
X2
12 ≡ 2
Xm
1m ≡ m
X2 or Xm even
X2 + centre, Xm +centre
Include Xm (odd order)
2/m
2/m
1
2
Orthorhombic
Rotation axis X
2/m
Rotation-Inversion axis X
2/m
X + centre
Include X (odd order)
2/m
X2
Xm
X2 or Xm even
222
2mm (2D) = mm2
2/m
X2 + centre, Xm +centre mmm ≡
Include Xm (odd order) 2/m2/m2/m
Already
discussed
Tetragonal
Rotation axis X
4
Rotation-Inversion axis X
4
X + centre
Include X (odd order)
4
𝑚
X2
422
Xm
4mm
X2 or Xm even
42𝑚
X2 + centre, Xm +centre 4/mmm ≡
Include Xm (odd order) 4/m 2/m 2/m
Trigonal
Rotation axis X
3
Rotation-Inversion axis X
2/m
X + centre
Include X (odd order)
3
X2
32
Xm
3m
X2 or Xm even
X2 + centre, Xm +centre
Include Xm (odd order)
2/m
3𝑚 ≡
32 𝑚
Hexagonal
Rotation axis X
6
Rotation-Inversion axis X
6
X + centre
Include X (odd order)
6
𝑚
X2
622
Xm
6mm
X2 or Xm even
6𝑚2
X2 + centre, Xm +centre 6/mmm ≡
Include Xm (odd order) 6/m 2/m 2/m
Cubic
Rotation axis X
Rotation-Inversion axis X
X + centre
Include X (odd order)
23
2/m
m3≡
2/m 3
X2
432
Xm
2/m
X2 or Xm even
43𝑚
X2 + centre, Xm +centre m3m ≡
Include Xm (odd order) 4/m 3 2/m
Examples of point group operation
#1 Point group 222
(1) At a general position [x y z], the symmetry
is 1, Multiplicity = 4
The multiplicity
tells us how many
atoms are generated
by symmetry if we
place a single atom
at that position.
y
x
(2)At a special position [100], the symmetry is
2. Multiplicity = 2
At a special position [010], the symmetry is
2. Multiplicity = 2
At a special position [001], the symmetry is
2. Multiplicity = 2
#2 Point group 4
(1) At a general position [x y z], the symmetry
is 1. Multiplicity = 4
(2) At a special position [001], the symmetry is
4. Multiplicity = 1
#3 Point group 4
(1)At a general position [x y z], the symmetry
is 1. Multiplicity = 4
(2) At a special position [001], the symmetry is
𝑚. Multiplicity = 2
(3) At a special position [000], the
symmetry is 4. Multiplicity = 1
P4
4
2
2
2
1
1
1
1
h
g
f
e
d
c
b
a
1
m
m
m
4
4
4
4
xyz, -x-yz, y-x-z, -yx-z
0 ½ z, ½ 0 -z
½ ½ z, ½ ½ -z
0 0 z, 0 0 -z
½½½
½½0
00½
000
Transformation of vector components
Original vector is P =[𝑝1 , 𝑝2 , 𝑝3 ]= 𝑥, 𝑦, 𝑧
i.e.
P=𝑥x+𝑦y+𝑧z
When symmetry operation transform the
original axes x, y, z to the new axes
x′, y′, z′
New vector after transformation of axes
becomes P ′ = [𝑝′1, 𝑝′2, 𝑝′3 ] = 𝑢, 𝑣, 𝑤
i.e.
P′ = 𝑢 x ′ + 𝑣y′ + 𝑤 z′
The angular relations between the axes may be
specified by drawing up a table of direction
cosines.
x
New
axes
x′
a11
=
y′
a21
=
z′
cosx′x
cosy′x
a31
=
cosz′x
Old axes
y
a12
=
cosx′y
a22
=
cosy′y
a32
=
cosz′y
z
a13
=
cosx′z
a23
=
cosy′z
a33
=
cosz′z
Then
𝑢 = 𝑥 ∗ cosx′x + 𝑦 ∗ cosx′y + 𝑧 ∗ cosx′z
i.e.
𝑝′1 = a11 ∗ 𝑝1 + a12 ∗ 𝑝2 + a13 ∗ 𝑝3
In a dummy notation
𝑝′1 = a1j ∗ 𝑝j
Similarly
𝑝′2 = a2j ∗ 𝑝j
𝑝′3 = a3j ∗ 𝑝j
i.e.
𝑝′i = aij ∗ 𝑝j
Moreover, by repeating the argument for the
reverse transformation and we have
𝑥 = 𝑢 ∗ cosx ′ x + 𝑣 ∗ cosy ′ x + 𝑤 ∗ cosz ′ x
𝑝1 = aj1 ∗ 𝑝′j
Similarly,
𝑝2 = aj2 ∗ 𝑝′j
𝑝3 = aj3 ∗ 𝑝′j
i.e. “old” in terms of “new”
𝑝i = aji ∗ 𝑝′j
For example: #1 Point group 4
The direction cosines for the first operation is
x′= - y
a11 = 0
Old axes
y
a12 =−1
y′= x
z′= z
a21 = 1
a31 = 0
a22 = 0
a32 = 0
x
New
axes
z
a13 = 0
a23 = 0
a33 =1
After symmetry operation, the new position is
[x y z] in new axes.
We can express it in old axes by
𝑝i = aji ∗ 𝑝′j = 𝑝′j ∗ aji
0 −1 0
𝑝3 ] = [𝑥 𝑦 𝑧] 1 0 0
0 0 1
= [𝑦 𝑥 𝑧 ]
i.e.
[𝑝1
𝑝2
or
𝑝1
y
0 1 0 x
𝑝2 = −1 0 0 y = x
𝑝3
z
0 0 1 z
14 plane lattices + 32 point groups
 230 Space groups
Triclinic
Bravais
Lattices
P
Monoclinic
P, C
2, m, 2/m
Orthorhombic
P, C, F, I
222, mm2, 2/m 2/m 2/m
Trigonal
P, R
Hexagonal
P
Tetragonal
P, I
3, 3, 32, 3m, 32/m
6, 6, 6/m, 622, 6mm, 6m2,
6/m 2/m 2/m
4, 4, 4/m, 422, 4mm, 42m,
4/m 2/m 2/m
Isometric
P, F, I
Crystal Class
Point Groups
1, 1
23, 2/m3, 432, 43m, 4/m32/m
B. Space group
Table for all space groups
Look at the notes!
Good web site to read about space group
http://www.uwgb.edu/dutchs/SYMMETRY/3d
SpaceGrps/3dspgrp.htm
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
Symmetry elements in space group
(1)Point group
(2)Translation symmetry + point group
Translational symmetry operations
The first character:
P: primitive
A, B, C: A, B, C-base centered
F: Face centered
I: Body centered
R: Romohedral
Glide plane also exists for 3D space group with more possibility
Symmetry planes normal to the plane of projection
Symmetry plane
Graphical symbol Translation
Symbol
Reflection plane
None
m
Glide plane
1/2 along line
1/2 normal to
plane
1/2 along line &
1/2 normal to
plane
1/2 along line &
1/2 normal to
plane
1/4 along line &
1/4 normal to
plane
a, b, or c
Glide plane
Double glide
plane
Diagonal glide
plane
Diamond glide
plane
a, b, or c
e
n
d
Symmetry planes normal to the plane of projection
Projection plane
Symmetry planes parallel to plane of projection
Symmetry plane
Graphical symbol Translation
Symbol
Reflection plane
None
m
Glide plane
1/2 along arrow
a, b, or c
Double glide
plane
1/2 along either
arrow
e
Diagonal glide
plane
1/2 along the
arrow
n
Diamond glide
plane
1/8 or 3/8 along
the arrows
d
3/8
1/8
The presence of a d-glide plane automatically implies a
centered lattice!
Glide planes
---- translation plus reflection across the glide
plane
* axial glide plane (glide plane along axis)
---- translation by half lattice repeat plus
reflection
---- three types of axial glide plane
i. a glide, b glide, c glide (a, b, c)
1
2
1
2
along line in plane ≡ along line parallel to
projection plane
e.g. b glide
,
b
--- graphic symbol for the axial glide plane
along y axis
c.f. mirror (m)
graphic symbol for mirror
1
2
If the axial glide plane is normal to projection
plane, the graphic symbol change to
ẑ c
ŷ
b
x̂
c glide
a
glide plane⊥𝑧 axis
If b glide plane is ⊥𝑧 axis
ẑ
glide plane symbol
ŷ
x̂
b
,
underneath the glide plane
𝑐
2
c glide: along z axis
or
𝑎+𝑏+𝑐
2
along [111] on rhombohedral axis
ii. Diagonal glide (n)
𝑎+𝑏 𝑏+𝑐 𝑎+𝑐
,
,
2
2
2
or
𝑎+𝑏+𝑐
2
(tetragonal, cubic system)
If glide plane is perpendicular to the drawing
plane (xy plane), the graphic symbol is
If glide plane is parallel to the drawing plane,
the graphic symbol is
iii. Diamond glide (d)
𝑎+𝑏 𝑎+𝑏+𝑐
,
4
4
(tetragonal, cubic system)
Symbols of symmetry axes
Symmetry
Element
Identity
2-fold ⊥ page
2-fold in page
2 sub 1 ⊥ page
2 sub 1 in page
3-fold
3 sub 1
3 sub 2
4-fold
4 sub 1
4 sub 2
4 sub 3
6-fold
6 sub 1
6 sub 2
6 sub 3
Graphical Symbol
Translation
Symbol
None
None
None
None
1/2
1/2
None
1/3
2/3
None
1/4
1/2
3/4
None
1/6
1/3
1/2
1
2
2
21
21
3
31
32
4
41
42
43
6
61
62
63
Symmetry
Element
6 sub 4
6 sub 5
Inversion
3 bar
4 bar
6 bar
2-fold and
inversion
2 sub 1 and
inversion
4-fold and
inversion
4 sub 2 and
inversion
6-fold and
inversion
6 sub 3 and
inversion
Graphical Symbol
Translation
Symbol
2/3
5/6
None
None
None
None
64
65
1
3
4
6 = 3/m
None
2/m
None
21/m
None
4/m
None
42/m
None
6/m
None
63/m
i. All possible screw operations
screw axis --- translation τ plus rotation
screw Rn along c axis
= counterclockwise rotation 360/R o +
translation n/R c
2
21
4
41
3
31
42
43
32
6
61
62
63
64
65
62

T
Symmorphic space group is defined as a space
group that may be specified entirely by
symmetry operation acting at a common point
(the operations need not involve τ) as well as
the unit cell translation. (73 space groups)
Nonsymmorphic space group is defined as a
space group involving at least a translation τ.
•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), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc)
•Hexagonal – The primary symmetry symbol will always be a 6, (-6),
61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm)
•Trigonal – The primary symmetry symbol will always be a 3, (-3) 31
or 32 (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, Cmc21, 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, P21/n)
•Triclinic – The lattice descriptor will be followed by either a 1 or a (1).
Examples
Space group P1
P1
No. 1
P1
1 Triclinic
C11
Origin on 1
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
1
a
1
x, y, z
No
conditions
http://img.chem.ucl.ac.uk/sgp/large/001az1.htm
Space group P1
P1
No. 2
Ci1
P1
1 Triclinic
Origin on 1
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
2
i
1
x, y, z；x, y, z
General:
No
conditions
1
h
1
1
g
1
1
f
1
1
e
1
1
d
1
1
c
1
b
1
a
1
1
1
1
2
1
, ,
1
2 2
1 1
0, ,
1
2
1
2
1
2
2 2
1
, 0,
2
1
, ,0
2
, 0, 0
1
0, , 0
2
0, 0,
1
2
0, 0, 0
Special:
No
conditions
http://img.chem.ucl.ac.uk/sgp/large/002az1.htm
Space group P112
P112
C21
No. 3
Ist setting
P112
2 Monoclinic
Origin on 2; unique axis c
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
2
e
1
d
1
1
x, y, z; x, y, z
General:
hkl
hk0
00l
No conditions
2
1
Special:
c
2
2
1
, ,z
, 0, z
No conditions
1
b
2
1
a
2
2
1
2
1
0, , z
2
0, 0, z
http://img.chem.ucl.ac.uk/sgp/large/003az1.htm
Space group P121
P121
C21
No. 3
P121
2
Monoclinic
nd
Origin on 2; unique axis b
2
setting
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
2
e
1
x, y, z; x, y, z
General:
hkl
h0l
0k0
No
conditions
1
d
2
1
1
c
2
2
1
1
b
2
1
a
2
2
, y,
1
2
, y, 0
0, y,
1
2
0, y, 0
Special:
No
conditions
http://img.chem.ucl.ac.uk/sgp/large/003ay1.htm
Space group P1121
P21
C22
No. 4
Ist setting
P1121
2 Monoclinic
Origin on 21; unique axis c
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
2
a
1
x, y, z; x, y,
1
2
+z
General:
hkl: No
conditions
hk0: No
conditions
00l: l=2n
http://img.chem.ucl.ac.uk/sgp/large/004az1.htm
Explanation:
Condition limiting possible reflections
#1 Consider the diffraction condition from plane (h k 0)
Two atoms at x, y, z; x, y,
1
2
+z
The diffraction amplitude F can be expressed as
fi ∗ e−2π i [h k l]∗[x y z ]
F=
i
fi ∗ e−2π i [h k 0]∗[x y z ]
=
i
= fi ∗ e−2π i [h k 0]∗[x y z ] + fi ∗ e−2π i [h k 0]∗[ x
= fi ∗ e−2π i(hx +ky ) + fi ∗ e−2π i −hx −ky
= fi ∗ e−2π i(hx +ky ) + e2π i(hx +ky )
y 1/2 +z ]
= fi ∗ 2 cos 2πi hx + ky
= 2fi
Therefore, no conditions can limit the (h, k, 0) diffraction
#2 For the planes (00l)
Two atoms at x, y, z; x, y, 12+z
The diffraction amplitude F can be expressed as
fi ∗ e−2π i [h k l]∗[x i y i z i ]
F=
i
fi ∗ e−2π i [0 0 l]∗[x i y i z i ]
=
i
= fi ∗ e−2π i [0 0 l ]∗[x y z ] + fi ∗ e−2π i [0 0 l ]∗[ x
−2π ilz
−2π i
= fi ∗ e
+ fi ∗ e
= fi ∗ e−2π ilz ∗ 1 + e−πil
= fi ∗ 1 + e−πil
y 1/2 +z ]
l
+lz
2
If l=2n, then F=2fi
If l=2n+1, then F=0
Therefore, the condition l=2n limit the (0, 0 ,l) diffraction.
Space group P1211
P21
C22
No. 4
P1211
2
Monoclinic
nd
Origin on 21; unique axis b
2
setting
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
2
a
1
1
x, y, z; x, +y, z
2
General:
hkl: No
conditions
h0l: No
conditions
0k0: k=2n
Space group B112
B2
No. 5
B112
2 Monoclinic
C23
Ist setting
Origin on 2; unique axis c
Number
Wyckoff
Point
Coordinates of equivalent
Condition
of
notation
symmetry
positions
limiting
positions
possible
reflections
4
c
1
x, y, z; x, y, z
General:
hkl: h+l=2n
+(
1
1 1
0
2 2
hk0: h=2n
00l: l=2n
2
b
2
0, , z
Special:
2
a
2
0, 0, z
as above only
2
(1/2+𝑥,𝑦,1/2+z)
(𝑥,𝑦,z)
(x,y,z)
y
x
(1/2+x,y,1/2+z)
What can we do with the space group information
contained in the International Tables?
1. Generating a Crystal Structure from its
Crystallographic Description
2. Determining a Crystal Structure from
Symmetry & Composition
Example: Generating a Crystal Structure
http://chemistry.osu.edu/~woodward/ch754/sym_itc.
htm
Description of crystal structure of Sr2AlTaO6
Space Group = Fm3m; a = 7.80 Å
Atomic Positions
Atom
Sr
Al
Ta
O
x
0.25
0.0
0.5
0.25
y
0.25
0.0
0.5
0.0
z
0.25
0.0
0.5
0.0
From the space group tables
http://www.cryst.ehu.es/cgibin/cryst/programs/nph-wp-list?gnum=225
32
f
3m
24
e
4mm
24
d
mmm
8
4
4
c
b
a
43m
m3m
m3m
xxx, -x-xx, -xx-x, x-x-x,
xx-x, -x-x-x, x-xx, -xxx
x00, -x00, 0x0, 0-x0,00x,
00-x
0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼,
¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0
¼¼¼,¼¼¾
½½½
000
Sr 8c; Al 4a; Ta 4b; O 24e
40 atoms in the unit cell
stoichiometry Sr8Al4Ta4O24  Sr2AlTaO6
F: face centered
 (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)
(000) (½½0) (½0½) (0½½)
Sr
8c: ¼ ¼ ¼  (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾)
¼ ¼ ¾  (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼)
Al
¾ + ½ = 5/4 =¼
4a: 0 0 0  (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)
(000) (½½0) (½0½) (0½½)
Ta
4b: ½ ½ ½  (½½½) (00½) (0½0) (½00)
O
(000) (½½0) (½0½) (0½½)
x00
24e: ¼ 0 0  (¼00) (¾½0) (¾0½) (¼½½)
-x00 ¾ 0 0  (¾00) (¼½0) (¼0½) (¾½½)
0x0 0 ¼ 0  (0¼0) (½¾0) (½¼½) (½¾½)
0-x0 0 ¾ 0  (0¾0) (½¼0) (½¾½) (0¼½)
00x 0 0 ¼  (00¼) (½½¼) (½0¾) (0½¾)
00-x 0 0 ¾  (00¾) (½½¾) (½0¼) (0½0¼)
Bond distances:
Al ion is octahedrally coordinated by six O
Al-O distance
d = 7.80 Å 
0.25 − 0 2 + 0 − 0 2 + 0 − 0 2 = 1.95 Å
Ta ion is octahedrally coordinated by six O
Ta-O distance
d = 7.80 Å 
0.25 − 0.5 2 + 0.5 − 0.5 2 + 0.5 − 0.5
= 1.95 Å
Sr ion is surrounded by 12 O
Sr-O distance: d = 2.76 Å
2
Determining a Crystal Structure from
Symmetry & Composition
Example:
Consider the following information:
Stoichiometry = SrTiO3
Space Group = Pm3m
a = 3.90 Å
Density = 5.1 g/cm3
First step:
calculate the number of formula units per unit
cell :
Formula Weight SrTiO3 = 87.62 + 47.87 + 3
(16.00) = 183.49 g/mol (M)
Unit Cell Volume = (3.9010-8 cm)3 = 5.93 
10-23 cm3 (V)
(5.1 g/cm3)(5.93  10-23 cm3) : weight in a
unit cell
(183.49 g/mole) / (6.022 1023/mol) : weight
of one molecule of SrTiO3
 (5.1 g/cm3)(5.93  10-23 cm3)/
(183.49 g/mole/6.022 1023/mol) = 0.99
 number of molecules per unit cell : 1 SrTiO3.
From the space group tables (only part of it)
6
e
4mm
3
3
1
1
d
c
b
a
4/mmm
4/mmm
m3m
m3m
x00, -x00, 0x0,
0-x0,00x, 00-x
½ 0 0, 0 ½ 0, 0 0 ½
0½½,½0½,½½0
½½½
000
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wplist?gnum=221
Sr: 1a or 1b; Ti: 1a or 1b
 Sr 1a Ti 1b or vice verse
O: 3c or 3d
Evaluation of 3c or 3d:
Calculate the Ti-O bond distances:
d (O @ 3c) = 2.76 Å (0 ½ ½)
D (O @ 3d) = 1.95 Å (½ 0 0, Better)
Atom
Sr
Ti
O
x
0.5
0
0.5
y
0.5
0
0
z
0.5
0
0
Another example from the note
The usage of space group for crystal structure
identification
Space group P 4/m 3 2/m
Reference to note chapter 3-2 page 26
From the space group tables (only part of it)
6
e
4mm
3
3
1
1
d
c
b
a
4/mmm
4/mmm
m3m
m3m
x00, -x00, 0x0,
0-x0,00x, 00-x
½ 0 0, 0 ½ 0, 0 0 ½
0½½,½0½,½½0
½½½
000
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wplist?gnum=221
#1 Simple cubic
Number of
Wyckoff
Point
positions
notation
symmetry
1
A
m3m
Coordinates of equivalent positions
0, 0, 0
#2 CsCl structure
atoms
Number of
Wyckoff
Point
Coordinates of equivalent
positions
notation
symmetry
positions
Cl
1
a
m3m
0, 0, 0
Cs
1
b
m3m
1
2
1
1
2
2
, ,
CsCl Vital Statistics
Formula
Crystal System
Lattice Type
Space Group
Cell Parameters
Atomic Positions
Density
CsCl
Cubic
Primitive
Pm3m, No. 221
a = 4.123 Å, Z=1
Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5
(can interchange if desired)
3.99
#3 BaTiO3 structure
atoms
Number of
Wyckoff
Point
Coordinates of equivalent
positions
notation
symmetry
positions
Ba
1
a
m3m
0, 0, 0
Ti
1
b
m3m
1
O
3
c
4/mmm
0, , ;
2
1
2
1
2
183 K
rhombohedral
(R3m)
Temperature
278 K
Orthorhombic
(Amm2)
1
, ,
1
2 2
1 1
2
2
1
, 0, ;
1
2
, ,0
2
393 K
Tetragonal
(P4mm).
Cubic
(Pm3m)