88 INSTRUMENTAL METHODS
One deduces the space group from the symmetry in the crystal ’ s diffraction
pattern and the systematic absence of specifi c refl ections in that pattern. The
crystal ’ s cell dimensions are derived from the diffraction pattern for the crystal
collected on X - ray fi lm or measured with a diffractometer. An estimation of
Z (the number of molecules per unit cell) can be carried out using a method
calledVM proposed by Matthews.^14 For most protein crystals the ratio of the
unit cell volume and the molecular weight is a value around 2.15 Å^3 /Da. Cal-
culation ofZ by this method must yield a number of molecules per unit cell
that is in agreement with the decided - upon space group.
3.3.3 Theory and Hardware,
The mathematics necessary to understand the diffraction of X rays by a crystal
will not be discussed in any detail here. Chapter 4 of reference 11 contains an
excellent discussion. The arrangement of unit cells in a crystal in a periodic
manner leads to the Laue diffraction conditions shown in equations 3.5 , where
vectorsa , b , and c as well as lattice indices h , k , and l have been defi ned in
Figure 3.5 and S is a vector quantity equal to the difference between the
resultant vectors after diffraction and the incident X - ray beam wave vector
s 0 , so that S = s − s 0.
aS
bS
cS⋅=
⋅=
⋅=
h
k
l(3.5)
The same crystalline arrangement leads to the expression of Bragg ’ s law applied
to X - ray diffraction with incident X - ray beam of wavelength λ as shown in
equation 3.6 and where the terms are defi ned as in Figures 3.5 and 3.6.
(^) λθ= 2 dsin (3.6)
As mentioned above, the formalism of the reciprocal lattice is convenient
for constructing the directions of diffraction by a crystal. In Figure 3.4 the
Ewald sphere was introduced. The radius of the Ewald sphere, also called the
Figure 3.6 Two lattice planes separated by distance d. Incident and refl ected X - ray
beams make the angleθ with the lattice planes. (Adapted with kind permission of
Springer Science and Business Media from Figure 4.17 of reference 11. Copyright 1999,
Springer - Verlag, New York.)
d
θ
θ
θ
θ
θ θ
2 d sin θ