Page 86 - Materials Chemistry, Second Edition

73
2.3. The Crystalline State
34 .
where: h ¼ Planck’s constant (6.626 10 J s); p ¼ momentum
Accordingly, if we represent a wave as a vector as described previously, the
wavevector, k, will have a magnitude of:

p jj 2p
ð23Þ k jj ¼ ¼ ;
l
h
where: h ¼ the reduced Planck’s constant (h/2p)
As one can see from Eq. 23, the magnitude of the wavevector, k, will have units
proportional to [1/distance] – analogous to reciprocal lattice vectors. This is no

accident; in fact, solid-state physicists refer to reciprocal space as momentum space
(or k-space), where long wavevectors correspond to large momenta and energy, but
small wavelengths. This definition is paramount to understanding the propagation of
electrons through solids, as will be described later.
In order to determine which lattice planes give rise to Bragg diffraction, a
geometrical construct known as an Ewald sphere is used. This is simply an applica-
tion of the law of conservation of momentum, in which an incident wave, k,
impinges on the crystal. The Ewald sphere (or circle in two-dimensional) shows
which reciprocal lattice points, (each denoting a set of planes) which satisfy
Bragg’s Law for diffraction of the incident beam. A specific diffraction pattern is
recorded for any k vector and lattice orientation – usually projected onto a two-
dimensional film or CCD camera. One may construct an Ewald sphere as follows
(Figure 2.44):
(i) Draw the reciprocal lattice from the real lattice points/spacings.
(ii) Draw a vector, k, which represents the incident beam. The end of this vector
should touch one reciprocal lattice point, which is labeled as the origin (0, 0, 0).
(iii) Draw a sphere (circle in 2-D) of radius |k| ¼ 2p/l, centered at the start of the
incident beam vector.
(iv) Draw a scattered reciprocal lattice vector, k’, from the center of the Ewald
sphere to any point where the sphere and reciprocal lattice points intersect.
(v) Diffraction will result for any reciprocal lattice point that crosses the Ewald
sphere such that k’ ¼ k þ g, where g is the scattering vector. Stated more
succinctly, Bragg’s Law will be satisfied when g is equivalent to a reciprocal
lattice vector. Since the energy is conserved for elastic scattering of a photon,
the magnitudes of k’ and k will be equivalent. Hence, using the Pythagorean
theorem, we can re-write the diffraction condition as:
2 2 2
ð24Þ ðkþgÞ ¼ k jj ) 2k gþg ¼ 0
Since g ¼ ha* þ kb* þ lc*, with a magnitude of 2p/d hkl for the (hkl) reflection,
Eq. 24 can be re-written in terms that will yield Bragg’s Law in more familiar terms
(c.f. Eq. 1):

2p 2p 4p 2siny 1
2
ð25Þ 2 sin y ¼ 2 ) ¼
l d hkl d hkl l d hkl