Solid State Physics
265
where u′, v′ and w′ are integers. For example, the point E in the two dimensional
lattice in Fig. 8.2 is represented by 2a + b. The direction of the vector is then
represented by a set of numbers [u, v, w] which is obtained by multiplying u′, v′
and w′ by the lowest common denominator. If some of the components are
negative, this is indicated by a bar over the number, e.g., if u is negative, the
direction is given by [| |, , ]
u v w .
A crystal plane is characterized by the intercepts u′, a, v′ b and w′ c along
the three axes. The reciprocals 1/u′, 1/v′ and 1/w′ are reduced to the simplest
integers h, k, l by multiplying the reciprocals by their lowest common
denominator. The plane is then denoted by the set of numbers (h, k, l), called
the Miller indices. If some of the intercepts are along the negative axes, this is
indicated by a bar over the corresponding index, e.g., if u′ is negative, the Miller
indices are (| |, , )
h k l . It is clear from this that parallel planes (with the
corresponding intercepts having the same sign) are represented by the same
Miller indices. When an intercept is at infinity, the corresponding Miller index
is zero. For example, (1, 0, 0) represents a plane parallel to the yz plane, with a
positive intercept along the x-axis, while a plane with intercepts – 2a, 3b and
parallel to the z-axis, is denoted by the Miller indices ( 3 , 2, 0).
Diffraction by a Lattice
Crystal structures are determined experimentally by analysing the diffraction
of x-rays by a crystal. Since x-rays have a wavelength of about 1 Å, the atoms in
a crystal serve as a grating and produce diffraction maxima. The measurement
of the positions and intensities of these maxima gives information about the
crystal structure. The conditions for the maxima were obtained in terms of the
Bragg condition in Eq. (2.38) by considering the superposition of scattering
from different planes. This is an over-simplified picture. Here a more rigorous
and complete derivation of the maxima is presented by looking at the
superposition of scattered waves from different atoms.
Consider a radiation described by B exp [i (k 0 .r – ωt)], | k 0 | = 2π/λ, incident
on a lattice with atoms at points
r n = ua + vb + wc, u = 0, ..., n 1 ; v = 0, ..., n 2 ;
w = 0, ..., n 3
(8.12)
The scattered wave will have the same wavelength as the incident beam,
but will propagate in some other direction. Since the scattered beam has the
same phase as the incident beam at the point of scattering, it is given by
A =
0
exp [ . ] exp [ .(
)],
′
−
∑
n
n
n
B
i
i
k r
k r r
| k | = | k 0 | =
2π
λ
265
where u′, v′ and w′ are integers. For example, the point E in the two dimensional
lattice in Fig. 8.2 is represented by 2a + b. The direction of the vector is then
represented by a set of numbers [u, v, w] which is obtained by multiplying u′, v′
and w′ by the lowest common denominator. If some of the components are
negative, this is indicated by a bar over the number, e.g., if u is negative, the
direction is given by [| |, , ]
u v w .
A crystal plane is characterized by the intercepts u′, a, v′ b and w′ c along
the three axes. The reciprocals 1/u′, 1/v′ and 1/w′ are reduced to the simplest
integers h, k, l by multiplying the reciprocals by their lowest common
denominator. The plane is then denoted by the set of numbers (h, k, l), called
the Miller indices. If some of the intercepts are along the negative axes, this is
indicated by a bar over the corresponding index, e.g., if u′ is negative, the Miller
indices are (| |, , )
h k l . It is clear from this that parallel planes (with the
corresponding intercepts having the same sign) are represented by the same
Miller indices. When an intercept is at infinity, the corresponding Miller index
is zero. For example, (1, 0, 0) represents a plane parallel to the yz plane, with a
positive intercept along the x-axis, while a plane with intercepts – 2a, 3b and
parallel to the z-axis, is denoted by the Miller indices ( 3 , 2, 0).
Diffraction by a Lattice
Crystal structures are determined experimentally by analysing the diffraction
of x-rays by a crystal. Since x-rays have a wavelength of about 1 Å, the atoms in
a crystal serve as a grating and produce diffraction maxima. The measurement
of the positions and intensities of these maxima gives information about the
crystal structure. The conditions for the maxima were obtained in terms of the
Bragg condition in Eq. (2.38) by considering the superposition of scattering
from different planes. This is an over-simplified picture. Here a more rigorous
and complete derivation of the maxima is presented by looking at the
superposition of scattered waves from different atoms.
Consider a radiation described by B exp [i (k 0 .r – ωt)], | k 0 | = 2π/λ, incident
on a lattice with atoms at points
r n = ua + vb + wc, u = 0, ..., n 1 ; v = 0, ..., n 2 ;
w = 0, ..., n 3
(8.12)
The scattered wave will have the same wavelength as the incident beam,
but will propagate in some other direction. Since the scattered beam has the
same phase as the incident beam at the point of scattering, it is given by
A =
0
exp [ . ] exp [ .(
)],
′
−
∑
n
n
n
B
i
i
k r
k r r
| k | = | k 0 | =
2π
λ