Because protein crystals have such a significant amount of the crystal-
lization solution, there is a dark ring due to the solvent. To correct for this
and other contributions, the intensity measurement includes a subtraction
of the average darkness of the region surrounding each spot. For a given
protein crystal, tens to hundreds of thousands of peaks are measured,
yielding a listing of intensities for each hklvalue.
In a diffraction experiment, information concerning the molecules form-
ing the crystals is found by analyzing the intensity of each diffraction point.
The amplitude of each diffraction point, F(hkl), is related to the position
of each atom according to eqn 15.6. This equation can be reversed by
use of a Fourier transform that yields:
(15.7)
In this equation, ρ(r) is theelectron densityat the position r in the crystal,
since in practice the discrete atomic positions are not observed. The inten-
sity of the scattered waves decreases with increasing angle (Figure 15.10),
so measurement of the higher-angle diffraction peaks can be difficult,
especially for proteins crystals that typically diffract X-rays weakly. The
greatest angle for which the diffraction can be measured is identified by
the resolution limit, with the larger angles corresponding to the smaller
resolution limit. To solve the structure of a protein, the resolution limit
should be at least 3 Å. For a lower value of 5 Å, the backbone can be
traced but the positions of the side chains cannot be determined, whereas
a resolution limit of 1 Å allows the determination of the positions of the
protons that are the weakest-diffracting atoms.
ρ()r ( ) π()
V
hkl e
hkl= ihx ky lz
∑(^1) F −++ 2
CHAPTER 15 X-RAY DIFFRACTION AND EXAFS 325
Derivation box 15.1 Phases of complex numbers
The diffraction from a crystal is described in terms of structural factors that are complex
numbers with both an amplitude and phase. A complex number, F, can be considered to
consist of a real component, A, and an imaginary component, iB:F=A+iB wherei= (db15.1)The complex conjugate of a complex number, F*, is defined as:F* =A−iB (db15.2)