BioPHYSICAL chemistry

detailed analysis since the contribution from each atom of

the molecule must be considered in the analysis. Each type

of atom present in a molecule will scatter the X-rays dif-

ferently (Figure 15.10). Since the X-rays are scattered by

the electrons of the atoms, the primary effect is that the

scattering is larger for atoms with more electrons, so that

at an angle of 0°, the amplitude of the scattering, fj, increases

as the atomic number increases. For every atom, there is

a similar drop off in the amplitude as the angle between

the initial beam and the measuring point increases.

A molecule is considered to be composed of atoms, each

of which has a certain scattering amplitude, fj, at a point

in space denoted by xj, yj, and zj. These molecules pack

inside the unit cell of a Bravais lattice. For any two neigh-

boring atoms, A and B, whether the scattered waves add

constructively or destructively depends upon the relative

phases that differ according to the pathlength difference of

the two scattered waves. If the phase of each scattered wave

is denoted by φj, then the sum of the two waves Fis:

F=fAeiφA+fBeiφB (15.5)

To determine the scattering from a

molecule, the scattering from each atom

is added together as the product of the

amplitude and phase factor according to:

(15.6)

The quantity F(hkl) is called the struc-

ture factor, with the indices hkl being

used to specify the diffraction peaks. In a

typical measurement, the protein crystal

is oscillated through a small angle and the

diffraction peaks form a series of circles

(Figure 15.11). The position of each peak

is determined by the space group. By

measuring the diffraction for a series of

different angles it is possible to identify

the space group by comparing the posi-

tions of the peaks at the different angles.

After the space group is found, each

peak is identified with a hklvalue and

the intensity of each spot is measured.

F()hkl f ejihkl()

j

=∑ φj

324 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

O

C

H

Fe^2 

30

10

20

0

0 0.2 0.4 0.6 0.8 1.0 1.2

f

(sin θ)/λ

Figure 15.10Dependence of the
scattering amplitude, f, on the
scattering angle, θ, for different
atoms.

Figure 15.11Oscillation photograph of the diffraction
from crystals of the small metal-binding protein. The
dark ring is due to solvent present in the crystal.