BIOINORGANIC CHEMISTRY A Short Course Second Edition

X-RAY CRYSTALLOGRAPHY 89

sphere of refl ection, is reciprocal to the wavelength of X - ray radiation — that
is, 1/ λ. The reciprocal lattice rotates exactly as the crystal. The direction of the
beam diffracted from the crystal is parallel toMP in Figure 3.7 and corre-
sponds to the orientation of the reciprocal lattice. The reciprocal space vector
S ( h k l ) = OP ( h k l ) is perpendicular to the refl ecting plane h k l , as defi ned
for the vectorS. This leads to the fulfi llment of Bragg ’ s law as | S ( h k l )| =
2 (sin )/ λ = 1/ d.
Molecules and atoms within a crystal are not in static positions but vibrate
around an equilibrium position. Atoms around the periphery of a molecule
will vibrate to a greater extent, whereas central atoms will have relatively fi xed
positions. The resultant weakening of X - ray beam intensity, especially at high
scattering angles, is expressed as the temperature factor. In the simplest case,
components of the atom ’ s vibration are all in the same direction, called the
isotropic case. The component perpendicular to the refl ecting plane and thus
alongS is equal for each ( h k l ), and the temperature correction factor for
isotropic atomic scattering is given by equation 3.7 :

TB

BB

d

()exp

sin

exp

sin

iso=−⎡ exp

⎣⎢

⎤

⎦⎥

=−⎡ ( )

⎣⎢

⎤

⎦⎥

=−( )

2

2

2

4

2

4

θ 1

λ

θ

λ

22

⎡

⎣⎢

⎤

⎦⎥

(3.7)

Assuming isotropic and harmonic vibration the thermal parameter B becomes
the quantity shown in equation 3.8 , where^2 is the mean square displacement
of the atomic vibration.

Bu=× 8 π^22 (3.8)

For anisotropic vibration the temperature factor is more complex because^2
now depends on the direction ofS. The anisotropic temperature factor is often

Figure 3.7 The Ewald sphere used to construct the direction of the scattered beam.
The sphere has radius 1/ λ. The origin of the reciprocal lattice is O. The incident X - ray
beam is labeleds 0 and the scattered beam is labeled s. (Adapted with kind permission
of Springer Science and Business Media from Figure 4.19 of reference 11. Copyright
1999, Springer - Verlag, New York.)

s (^0) o
s
1
λ
reciprocal
lattice
2θ
M
P
O
S
o - - - - O =
1
λ
ū
ū