3.6 Reciprocal Lattice
57
The vectors b i fulfill the conditions
a i · b j = 2π δ i j .
(3.17)
Thus, it is clear that (3.14) is fulfilled. For an arbitrary reciprocal lattice vector G = k 1 b 1 + k 2 b 2 + k 3 b 3
and a vector R = n 1 a 1 + n 2 a 2 + n 3 a 3 in direct space we find
G · R = 2π (n 1 k 1 + n 2 k 2 + n 3 k 3 ) .
(3.18)
The number in brackets is an integer. Additionally, we note that the reciprocal lattice of the reciprocal
lattice is again the direct lattice. The reciprocal lattice of the fcc is bcc and vice versa. The reciprocal
lattice of hcp is hcp (rotated by 30
◦ with respect to the direct lattice).
For later, we note two important theorems. A (sufficiently well behaved) function f (r) that is
periodic with the lattice, i.e. f (r) = f (r + R) can be expanded into a Fourier series with the reciprocal
lattice vectors according to
f (r) =
a G exp (i G · r) ,
(3.19)
where a G denotes the Fourier component of the reciprocal lattice vector G, a G =
V f (r) exp(−i G · r)
d
3 r. If f (r) is lattice periodic, the integral given in (3.20) is zero unless G is a reciprocal lattice vector.
V
f (r) exp (−i G · r) d
3 r =
a G
0, G /
∈ G
.
(3.20)
3.6.2 Miller Indices
The Miller indices [246] form a triplet of integer numbers to denote directions and lattice planes in
the crystal. A vector R in direct space is denoted with its components h, k, and l relative to the lattice
vectors a i , i.e. h = a 1 · R and so forth. In order to arrive at the set of smallest possible integer numbers,
the values possibly must be divided by a suitable fraction. Directions are denoted in square brackets
[hkl], and go into the direction of ha 1 + ka 2 + la 3 . A set of crystallographically equivalent directions
is denoted with hkl. For negative indices, [−1 00] can also be written with a bar, [ ¯
100].
A lattice plane is the set of all lattice points in a plane spanned by two independent lattice vectors R 1
and R 2 . Lattice planes are denoted as (hkl) with parentheses; a set of equivalent planes is denoted with
curly brackets as {hkl}. The lattice points on that plane form a two-dimensional Bravais lattice. The
entire lattice can be generated by shifting the lattice plane along its normal n = (R 1 × R 2 )/|R 1 × R 2 |.
The plane belongs to the reciprocal lattice vector G n = 2π n/d, d being the distance between planes.
This correspondence between reciprocal lattice vectors and sets of planes allows the orientation of
planes to be described in a simple manner. The shortest reciprocal lattice vector perpendicular to the
plane is used. The coordinates with respect to the primitive translation vectors of the reciprocal space b i
form a triplet of integer numbers and are called Miller indices of the plane, i.e. G n = h b 1 + k b 2 +l b 3 .
The plane described by G n ·r = A fulfills the condition for a suitable value of A. The plane intersects
the axes a i at the points x 1 a 1 , x 2 a 2 and x 3 a 3 . Thus we find G n x i a i = A for all i. From (3.18) follows
G n · a 1 = 2π h, G n · a 2 = 2π k and G n · a 3 = 2π l, where h, k and l are integers. The Miller indices
(hkl) are thus proportional to the reciprocal values 1/x i of the axis intersections of the plane with the
lattice vectors of the direct lattice. An example is shown in Fig. 3.34.
57
The vectors b i fulfill the conditions
a i · b j = 2π δ i j .
(3.17)
Thus, it is clear that (3.14) is fulfilled. For an arbitrary reciprocal lattice vector G = k 1 b 1 + k 2 b 2 + k 3 b 3
and a vector R = n 1 a 1 + n 2 a 2 + n 3 a 3 in direct space we find
G · R = 2π (n 1 k 1 + n 2 k 2 + n 3 k 3 ) .
(3.18)
The number in brackets is an integer. Additionally, we note that the reciprocal lattice of the reciprocal
lattice is again the direct lattice. The reciprocal lattice of the fcc is bcc and vice versa. The reciprocal
lattice of hcp is hcp (rotated by 30
◦ with respect to the direct lattice).
For later, we note two important theorems. A (sufficiently well behaved) function f (r) that is
periodic with the lattice, i.e. f (r) = f (r + R) can be expanded into a Fourier series with the reciprocal
lattice vectors according to
f (r) =
a G exp (i G · r) ,
(3.19)
where a G denotes the Fourier component of the reciprocal lattice vector G, a G =
V f (r) exp(−i G · r)
d
3 r. If f (r) is lattice periodic, the integral given in (3.20) is zero unless G is a reciprocal lattice vector.
V
f (r) exp (−i G · r) d
3 r =
a G
0, G /
∈ G
.
(3.20)
3.6.2 Miller Indices
The Miller indices [246] form a triplet of integer numbers to denote directions and lattice planes in
the crystal. A vector R in direct space is denoted with its components h, k, and l relative to the lattice
vectors a i , i.e. h = a 1 · R and so forth. In order to arrive at the set of smallest possible integer numbers,
the values possibly must be divided by a suitable fraction. Directions are denoted in square brackets
[hkl], and go into the direction of ha 1 + ka 2 + la 3 . A set of crystallographically equivalent directions
is denoted with hkl. For negative indices, [−1 00] can also be written with a bar, [ ¯
100].
A lattice plane is the set of all lattice points in a plane spanned by two independent lattice vectors R 1
and R 2 . Lattice planes are denoted as (hkl) with parentheses; a set of equivalent planes is denoted with
curly brackets as {hkl}. The lattice points on that plane form a two-dimensional Bravais lattice. The
entire lattice can be generated by shifting the lattice plane along its normal n = (R 1 × R 2 )/|R 1 × R 2 |.
The plane belongs to the reciprocal lattice vector G n = 2π n/d, d being the distance between planes.
This correspondence between reciprocal lattice vectors and sets of planes allows the orientation of
planes to be described in a simple manner. The shortest reciprocal lattice vector perpendicular to the
plane is used. The coordinates with respect to the primitive translation vectors of the reciprocal space b i
form a triplet of integer numbers and are called Miller indices of the plane, i.e. G n = h b 1 + k b 2 +l b 3 .
The plane described by G n ·r = A fulfills the condition for a suitable value of A. The plane intersects
the axes a i at the points x 1 a 1 , x 2 a 2 and x 3 a 3 . Thus we find G n x i a i = A for all i. From (3.18) follows
G n · a 1 = 2π h, G n · a 2 = 2π k and G n · a 3 = 2π l, where h, k and l are integers. The Miller indices
(hkl) are thus proportional to the reciprocal values 1/x i of the axis intersections of the plane with the
lattice vectors of the direct lattice. An example is shown in Fig. 3.34.