BioPHYSICAL chemistry

CHAPTER 15 X-RAY DIFFRACTION AND EXAFS 327

according to its atomic scattering factor, fi(hkl), which has both an amplitude fi(hkl) and a
phase φι(hkl):

(db15.6)

The amplitude fi(hkl) depends upon the type of atom (Figure 15.10) and the phase φι(hkl)
depends upon the position of the atom in the unit cell. Notice that φι(hkl) is the phase for
the scattering of the ith atom and is distinct from α(hkl), which is the phase of the hklreflection
(Figure 15.12).
To derive an expression for the phase φι(hkl), consider the simple example of a diatomic
molecule that forms an orthorhombic crystal (Figure 15.13). Atom 1 is positioned at the
origin and atom 2 is positioned in the xyplane at (x,y,0). Each atom will scatter the X-rays,
with the waves scattered from all atom 1 being in phase and the waves from all atom 2
being in phase, but the waves from the two sets of atoms are not necessarily in phase. The
phase of the hklreflection will shift by 2πhif atom 2 is positioned a distance of aaway from
atom 1. Since atom 2 is shifted in the xdirection by a distance xrelative to atom 1, the
phase difference for this shift is given by this factor reduced by x/a:

(db15.7)

Considering all three dimensions yields a total phase difference of:

(db15.8)

The expression for the structure factor can now be simplified by introducing the diffraction
9 ector, S, and theposition 9 ector, r:

φπ()hkl

hx

a

ky

b

lz

c

=++

⎛

⎝

⎜⎜

⎞

⎠

2 ⎟⎟

φπxhkl h

x

a

()=

⎛

⎝

⎜⎜

⎞

⎠

2 ⎟⎟

Ff() ()hkl F hkl eihkl() i()hkl f
i

N

===i

=

∑

α

1

(()hkl e()

i

N

ihkl

=

∑

1

φ

Shifted

Waves

from

red

atoms

are in

phase

Waves

from

blue

atoms

are in

phase

Figure 15.13The scattering from the
lattice of each atom type is in phase but
the waves from the two lattices are not
in phase.