Page 42 - Instant notes

A6
DIFFRACTION BY SOLIDS

Key Notes
Diffraction takes place when a wave interacts with a lattice
whose dimensions are of the same order of magnitude as that of
the wavelength of the wave. At these dimensions, the lattice
scatters the radiation, so as to either enhance the amplitude of the
radiation through constructive interference, or to reduce it
through destructive interference. The pattern of constructive and
destructive interference yields information about molecular and
crystal structure. The most commonly used radiation is X-rays,
which are most strongly scattered by heavy elements. High
velocity electrons behave as waves, and are also scattered by the
electron clouds. Neutrons slowed to thermal velocities also
behave as waves, but are scattered by atomic nuclei.
In crystallographic studies, the different lattice planes which are
present in a crystal are viewed as planes from which the incident
radiation can be reflected. Constructive interference of the
reflected radiation occurs if the Bragg condition is met: nλ=2d
sinθ. For most studies, the wavelength of the radiation is fixed,
and the angle θ is varied, allowing the distance between the
planes, d, to be calculated from the angle at which reflections are
observed.
For a crystalline solid, the distance between the lattice planes is
easily obtained from the Miller indices, and the unit cell
dimensions. The relationship between these parameters can be
used to modify the Bragg condition. In the simple case of a
primitive cubic unit cell, the allowed values for θ as a function of
2 1/2
2
2
h, k, and l are given by: sinθ= (h +k +l ) λ/2a. Some whole
numbers (7, 15, 23, for example) cannot be formed from the sum
of three squared numbers, and the reflections corresponding to
2
2
2
these values of (h +k +l ) are missing from the series. In other
unit cells, missing lines occur as a result of the symmetry of the
unit cell. Simple geometric arguments show that for a body
centered cubic unit cell, h+k+l must be even, and that for a face
centered cubic unit cell, h, k and l must be all even or all odd for
reflections to be allowed. The forbidden lines are known as
systematic absences.
In the powder diffraction method, the crystalline sample is
ground into a powder, so that it contains crystals which are
oriented at every possible angle to the incident beam. In this way,
the Bragg condition for every lattice plane is simultaneously