First make sure you understand Equation 6
.
the planes corresponding to the smallest reciprocal lattice vector yield the smallest angle
for the ring. Thus we know that the angle at which the
j
th
diﬀraction ring occurs is
determined by the ratio
sin(
φ
j
/
2)
sin(
φ
1
/
2)
=
d
j
d
1
where
d
j
is the magnitude of the the
j
th
reciprocal lattice vector arranged according to
increasing magnitudes.
Since we need the lengths of the reciprocal lattice vectors we do that ﬁrst. This is
easy since the reciprocal lattice of the fcc lattice is bcc and vice versa. So we just need
to ﬁnd the distances in these lattices in real space.
bcc lattice:
The lattice sites arranged in order from the origin have coordinates (0
,
0
,
0),
(1
/
2
,
1
/
2
,
1
/
2), (1
,
0
,
0), (1
,
1
,
0), (3
/
2
,
1
/
2
,
1
/
2),(1
,
1
,
1), (2
,
0
,
0) and their symmetry coun-
terparts. The distances from the origin are
√
3
2
,
1
,
√
2
,
√
11
2
,
√
3
,
2
which yields the ratios