X-RAY CRYSTALLOGRAPHY 85
symmetry axis in an X - ray beam path and observing the X - ray pattern as the
crystal oscillates over a small angle will produce refl ection spots arranged in
concentric circles. The concentric circle pattern of diffraction spots can be
thought of as the intersection of a series of parallel planes with a sphere as
shown in Figure 3.4. The planes represent the three - dimensional lattice that is
not actually the crystal lattice but instead its reciprocal lattice. The unit dis-
tances in the lattice are reciprocally related to unit distances in the crystal,
hence the name reciprocal lattice. The direction in which the X - ray beams are
diffracted depends on two factors: the unit cell distances in the crystal and the
X - ray wavelength. To be detected, the diffraction spots must be on, or pass
through, the surface of the sphere represented by the circle in Figure 3.4. The
radius of the sphere, called the sphere of refl ection or the “ Ewald sphere, ” is
reciprocal to the X - ray radiation wavelength — that is, 1/ λ. When the crystal is
rotated, the reciprocal lattice rotates with it and different points within the
lattice are brought to diffraction. The diffracted beams are called “ refl ections ”
because each of them can be regarded as a refl ection of the primary X - ray
beam against planes in the crystal.
A short discussion of crystalline habits and their important diffraction
characteristics follows here; reference 11 contains much greater detail. Mole-
cules of organic, organometallic, or inorganic materials, when precipitating
from solution, attempt to reach their lowest free energy state. Frequently, this
is accomplished by packing the molecules in a regular way in a crystalline
habit. Regular packing of molecules in a crystal will defi ne a unit cell through
generation of three repeating vectorsa (the x axis), b , (the y axis), and c (the
z axis), with angles α , β , and γ between them. Planes can be constructed
Figure 3.4 X - ray beam passing through the “ Ewald Sphere ” and diffracted by planes
in a single crystal produces refl ection spots. (Adapted with kind permission of Springer
Science and Business Media from Figure 1.13 of reference 11. Copyright 1999, Springer -
Verlag, New York.)
X-ray beamdetectorEwald sphere