Crystallography and Diffraction. Theory and Modern Methods of Analysis
Workshop 1
Symmetry and Space Group Determination Answers
Unit cells and crystal systems for a range of inorganic solids
Formula
a (Å)
b (Å)
c (Å)
 ()
 ()
 ()
Ti5O9
Al
Zn
V2O5
NaSb
Cu2O
TiO2
CaCO3
Fe3C
P4S10
Pb2O3
-U
WO3
MgO
Al2O3
FeCl3
AgNO2
H2O
CaCl2
CaC2
5.57
4.05
2.66
11.5
6.80
4.27
4.59
6.36
4.52
9.07
7.05
10.59
7.27
4.44
5.13
6.76
3.53
4.52
6.24
3.87
7.12
4.05
2.66
3.56
6.34
4.27
4.59
6.36
5.09
9.18
5.62
10.59
7.50
4.44
5.13
6.76
5.17
4.52
6.43
3.87
8.86
4.05
4.95
4.37
12.48
4.27
2.95
6.36
6.74
9.19
3.86
5.63
3.82
4.44
5.13
6.76
6.17
7.37
4.20
6.37
97.5
90
90
90
90
90
90
46.1
90
92.4
90
90
90
90
55.3
53.2
90
90
90
90
112.3
90
90
90
117.7
90
90
46.1
90
101.2
80.1
90
89.9
90
55.3
53.2
90
90
90
90
108.5
90
120
90
90
90
90
46.1
90
110.5
90
90
90
90
55.3
53.2
90
120
90
90
2.
Crystal
system
triclinic
cubic
hexagonal
orthorhombic
monoclinic
cubic
tetragonal
trigonal
orthorhombic
triclinic
monoclinic
tetragonal
monoclinic
cubic
trigonal
trigonal
orthorhombic
hexagonal
orthorhombic
tetragonal
(i)
P21/m P = Primitive cell. 21 = 2-screw axis parallel to b. / = centre of
symmetry. m = mirror plane perpendicular to b.
(ii)
Equivalent positions:
First apply the translational symmetry (21 axis) to coordinates x, y, z
(1) x,y,z 
(2) –x, y + 1/2 , -z
Then apply the centre of symmetry
(3) -x, -y, -z (4) x, -y + 1/2, z
(iii)
systematic absences for 21 axis, i.e. 0 k 0 absent when k = 2n + 1 (i.e.
odd)
3. WO2Cl2
(i)
(ii)
Space group C2/c
Systematic absences: C
c
hkl absent when h + k odd
h0l absent when l odd
Equivalent positions
First apply the translational symmetry (c-glide)
(1) x, y, z
(2) x, -y, z + ½
Then apply the centre of symmetry to these
(3) -x, -y, -z
(4) –x, y, -z + ½
Finally we need to generate the positions associated with the extra point of
origin at ½ , ½ , 0
(1) x, y, z
(2) x, -y, z + ½
(3) -x, -y, -z
(4) –x, y, -z + ½
(5) x + ½ , y + ½ , z (6) x + ½ , -y + ½ , z + ½
(7) -x + ½ , -y + ½ , -z (8) –x + ½ , y + ½ , -z + ½
(4) Braggs law n = 2dsin
Use Bragg's law to calculate d-spacings in Å.
h k
1 1
2 0
2 2
3 1
2 2
l
1
0
0
1
2
2()
36.91
42.88
62.27
74.61
78.61
d(Å)
2.435
2.109
1.491
1.272
1.217
a-axis from 2 0 0 reflection d200 = a/2, therefore a = 2.109 x 2 = 4.218 Å
get b axis from 2 2 0 reflection using the equation
1
h2 k 2 l 2



d 2 a 2 b2 c2
Therefore:
1
22
22


0
(1491
. ) 2 (4.218) 2 b 2
solving for b
b = 4.216Å
Similarly use 1 1 1 reflection to get c-axis.
1
h2 k 2 l 2



d 2 a 2 b2 c2
Therefore:
1
1
1
1


 2
2
2
2
(2.435)
(4.218)
(4.216)
c
solving for c
c = 4.218Å
i.e cubic within the probable error.
5.
(i) YBa2CuO5 orthorhombic
First look for general absences (affecting all hkl)
None so must be primitive P
Then look for translation symmetry on each axis
a axis
h00 absent when h odd => 21 parallel to a
0kl absent when k odd b glide perpendicular to a
b axis
0k0 absent when k odd => 21 parallel to b
h0l absent when h + l odd => n glide perpendicular to b
c axis
00l absent when l odd => 21 parallel to c
hk0 no absences
Possible space groups P21/b, 21/n, 21 or P21/b, 21/n, 21/m
i.e. in short form Pbn21 or Pbnm (actually Pbnm)
(ii) BICOVOX tetragonal
Use determinative tables here
General absences
hkl absent when h+k+l odd i.e. I centred
h00, 0k0 and 00l, absent when h, k, l odd (implicit from I condition)
hhl absent when l odd (implicit from I condition)
Possible space groups I4, I-4, I4/m, I422, I4mm, I-42m, I4/mmm
(actually I4/mmm)
(iii) ZnO hexagonal
Use determinative tables here
Look for reflections of the type hkil where (i = h+k)
h-h0l absent completely
hh-2hl absent when l odd
000l absent when l odd
Possible space groups
P63mc, P-62c, P63/mmc (actually P63/mmc)
(iv) Al2O3 Trigonal
Use determinative tables
Note i = -(h + k)
hkil condition –h+k+l = 3n
000l condition l = 6n
R3c or R-3c (actually R3c)