NaCl crystallizes into a lattice, which has alternate
Na
and
Cl
atoms along all the edges of the cube as
well as through the perpendicular lines running along the middle of the faces. If the atoms were identical
then it would be a simple cubic with half the lattice constant.
The basis atoms are at (referred to the cubic unit cell):
Na :
d
1
=
(0
,
0
,
0)
d
2
=
parenleftbigg
1
2
,
1
2
,
0
parenrightbigg
d
3
=
parenleftbigg
1
2
,
0
,
1
2
parenrightbigg
d
4
=
parenleftbigg
0
,
1
2
,
1
2
parenrightbigg
Cl :
d
5
=
parenleftbigg
1
2
,
1
2
,
1
2
parenrightbigg
d
6
=
parenleftbigg
0
,
0
,
1
2
parenrightbigg
d
6
=
parenleftbigg
0
,
1
2
,
0
parenrightbigg
d
6
=
parenleftbigg
1
2
,
0
,
0
parenrightbigg
2.2.4
Diamond lattice
Carbon, Silicon, Germanium and compounds like GaAs, InSb, ZnS, SiC crytallize in a lattice that is fcc
with a two atom basis.
The cubic unit cell has 8 atoms in it, each of the four fcc lattice points contributing two atoms.
The
second atom is displaced along the diagonal of the cube by
1
4
th
of the cubes diagonal. So we have for the
basis in terms of the cubic lattice vectors
a
i

2.2. A FEW COMMON LATTICE TYPES
21
Figure 2.9: The
NaCl
lattice. The cubic lattice constant
a
= 5
.
6
˚
A and for
KCl
,
KBr
,
LiH
,
AgBr
,
PbS
all have similar lattices.
d
1
=
(0
,
0
,
0)
d
2
=
parenleftbigg
1
2
,
1
2
,
0
parenrightbigg
d
3
=
parenleftbigg
0
,
1
2
,
1
2
parenrightbigg
d
4
=
parenleftbigg
1
2
,
0
,
1
2
parenrightbigg
d
5
=
parenleftbigg
1
4
,
1
4
,
1
4
parenrightbigg
d
6
=
parenleftbigg
3
4
,
3
4
,
1
4
parenrightbigg
d
7
=
parenleftbigg
1
4
,
3
4
,
3
4
parenrightbigg
d
8
=
parenleftbigg
3
4
,
1
4
,
3
4
parenrightbigg
(2.17)

22
CHAPTER 2. DIFFRACTION AND BASICS OF CRYSTAL STRUCTURE
Figure 2.10: The diamond lattice. On the left column, we have used the cubic unit cell, on the right we
use the FCC primitive vectors. The cubic lattice constant
a
= 3
.
57
˚
A and for Si
a
= 5
.
43
˚
A

2.2. A FEW COMMON LATTICE TYPES23PROBLEM : Whichhklreflections would be cut off by this basis, following eqns. 2.18 and 2.19? Considerthe general reciprocal lattice vector to be of the formG=hb1+kb2+lb3Can you see the advantage of making the basis as small as possible?

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