Page 81 - Materials Chemistry, Second Edition

68 2 Solid-State Chemistry

ð10Þ nl = 2d sin y;
where: d ¼ lattice spacing of the crystal planes (e.g.,d 111 would indicate the spacing
between neighboring 111 planes)
l ¼ wavelength of the incoming beam
y ¼ angle of the incident and diffracted beams (i.e., y in ¼ y out )
Illustrated another way, if two waves differ by one whole wavelength, they will

differ in phase by 360 or 2p radians. For instance, the phase difference, f, of the
(hkl) reflection resulting from a wave scattered by an arbitrary lattice atom at
position(x, y, z), and another scattered from an atom at position (0, 0, 0) is:

ð11Þ f ¼ 2pðhu þ kv þ lwÞ;
where the vector (uvw) corresponds to fractional coordinates of (x/a,y/b,z/c).
The two waves may differ in amplitude as well as phase if the two atoms are different.

In particular, for scattering in the 2y ¼ 0 (forward) direction of a wave by an atom
comprised of Z electrons, the waves scattered from all of the electrons in the atom will
be in-phase. [38] Accordingly, the amplitude of the scattered wave is simply Z times the
amplitude of the wave scattered by a single electron. The atomic scattering factor, f,is
used to describe the scattering by an atom in a givendirection (Eq. 12). Asjustdescribed,
f ¼ Z (atomic number) for forward scattering; however, as (sin y)/l increases, f will
decrease due to more destructive interference among the scattered waves.
amplitude of wave scattered by one atom
ð12Þ f ¼
amplitude of wave scattered by one electron
A description of the scattered wavefront resulting from diffraction by a unit cell of
the crystal lattice is significantly more complex. That is, one would need to include
the contribution of waves scattered by all atoms of the unit cell, each with differing
phases and amplitudes in various directions. In order to simplify the trigonometry
associated with adding two waves of varying phases/amplitudes, it is best to
represent individual waves as vectors. [39] Instead of using x and y components for
the vectors in 2-D real space, one may represent the vectors in complex space, with
real and imaginary components. This greatly simplifies the system; that is, the
addition of scattered waves is simply the addition of complex numbers, which
completely removes trigonometry from the determination.
Complex numbers are often expressed as the sum of a real and an imaginary
number of the form a þ ib (Figure 2.42). It should be noted that the vector length
represents the wave amplitude, A; the angle the vector makes with the horizontal
(real) axis represents its phase, f (Eq. 13 – Euler’s equation). The intensity of a
wave is proportional to the square of its amplitude, which may be represented by
Eq. 14 – obtained by multiplying the complex exponential function by its complex
conjugate (replacing i with i).

ð13Þ Ae if ¼ A cos f +Ai sin f
2
if if
ð14Þ Ae if ¼ Ae Ae