59-553 Structure Factors 78

Until now, we have only typically considered reflections

arising from planes in a hypothetical lattice containing
one atom in the asymmetric unit. In practice we will
generally deal with structures of molecules containing
several atoms. In a crystal, each of the (independent)
sets of atoms will make lattices so we need a way to
determine the relationship between these lattices and
the X-rays that we can observe – this information is
provided by the Structure Factors.
 59-553 Structure Factors 79
Structure factors, F(hkl), are the useful result of the diffraction data we
collect. The F(hkl) values are complex numbers that express both the
amplitude and phase of the reflection off of the (hkl) family of planes –
structure factors are the quantities that we need to obtain for each of the
reciprocal lattice points. The set of structure factors that we obtain for a
crystal are the data that we use to model the unit cell contents.

The actual data that we use looks something like this:

2 0 2 7.66 0.70 In a real data set, there are usually

2 0 2 7.25 0.70 thousands of structure factors. Note
-2 0 -2 6.35 0.69 that these are in the format:
-2 0 -2 6.93 0.70
1 1 2 1000.00 89.30 h k l |Fo2| (Fo2)
1 1 2 985.38 88.65
1 1 2 928.03 88.64 Where the magnitude of the structure
-1 -1 -2 871.24 88.63
factor and the error in its measurement
0 2 2 7.88 0.72
0 2 2 6.63 0.72
are recorded for each of the reciprocal
0 -2 -2 7.91 0.73 lattice points (hkl).
 59-553 Atomic Scattering Factors 80
To understand how structure factors are constructed, one must first examine
the atomic scattering factors. Atomic scattering factors, f, provide a measure
of the scattering power of all of the electrons in an atom of a given type.
Let the electron density at a distance r from the center of an atom be ρ(r).
Consider the wave scattered at the position r in a direction s relative to the
incident beam of radiation in the direction so. The scattered intensity
depends on the phase difference, , which is 2 /  times the path length
difference, is given by:
 = (2 / )[r · (s - so)] = 2 /  r · S
where S = (s - so). The vector S is a vector in reciprocal space. The wave
scattered by the volume element dv at r will have an amplitude with a
maximum of ρ(r) dv. Combining this with the phase ( exp (i) ) then the
amplitude of the wave at r must be ρ(r) exp(2i r · S) dv. The total scattering
power of the atom is given by summing over all volume elements dv of the
atom giving:
f(S) = ∫ ρ(r) exp(2i r · S) dv

ei = cos() – isin()

 59-553 Atomic Scattering Factors 81

Displacement Factor
The expression for the scattering factor function represents the scattering by an atom
at 0 K. Changes in temperature affect the thermal motion of atoms, and this in turn
affects the scattered intensities. In 1913 Peter Debye originally proposed and later
Ivar Waller modified a relation describing the effect of the thermal motion of atoms on
intensity. The Debye-Waller equation assumes the form:

f = f ° exp[-B(sin2θ/2)]

where f is the corrected scattering factor for a given atom type; f ° is the scattering
factor for a given atom calculated at zero Kelvin; B = 82 u2 and u2 is the mean
square displacement of the atoms. This factor only reduces the intensity of the peaks
and does not change the sharpness or shape of the peaks. This displacement factor
was used originally to correct calculated intensities for thermal motion of the atoms.
However, this factor also takes into account a variety of other factors such as static
disorder, absorption, how tightly an atom is bound in the structure, wrong scaling of
measurements, and incorrect atomic scattering functions. When the displacement
parameter for a given atom is expressed as a single term B, it is said to represent an
isotropic model of motion. Atoms that do not vibrate the same amount in all directions
may be represented with an ellipsoidal anisotropic model rather than the spherical
isotropic model. Ellipsoid models require six displacement variables for each atom –
we will look at this when we discuss refinement.
 59-553 82
Physical interpretation of s·r
To get a more intuitive feel for the meaning of the structure factor equation, which
will be developed below, it is useful to consider the physical interpretation of s·r.
Remember that a dot product can be interpreted as the projection of one vector on
the other (the component of one vector that is parallel to the other vector),
multiplied by the length of the other vector.

s·r = |s| |r| cos

In this figure, |r| cos is the component of the position vector r in the direction of s, which is
perpendicular to the Bragg planes. Since the length of the diffraction vector, |s|, is equal to 1/d,
s·r is equal to the fraction of the distance from one Bragg plane to the next that the
position vector r has travelled from the origin. (Of course, s·r can be any real number, so it
can be greater than one.) We define a wave diffracted from the Bragg plane passing through
the origin to have a phase of zero. Waves diffracted from the next Bragg plane have a phase
of 2 (which is equivalent to a phase of zero) and, in general, diffraction from any point r will
have a phase of 2s·r.
 59-553 83
The derivation of the phase (with respect to the origin) found
in the handout, is simply a proof of the s·r relationship earlier
in the notes. In this derivation, xyz are not in fractional
coordinates.

Since OR/d = 2, d = OR/2, OQ/OR = 2(hx/a + ky/b)

Thus, in 3-D, the phase of the scattering of an atom at xyz is:
(note if xyz is in fractional coordinates, this is just the expression
2(hx + ky + lz) that is listed in the equations I have shown you.
 59-553 Atomic Scattering Factors 84

The scattering factors are scaled with respect to the scattering that would be
expected from a single electron and the intensity of the scattering is
dependent on the scattering angle. At a scattering angle () of 0 (i.e. straight
through the atom in the same direction as the incident beam), the scattering
factor of the atom is equal to the number of electrons on the atom.

The atomic scattering factors are calculated

values and the scattering factors for each of the
elements are contained in most of the structure
solution software packages. For most
diffraction experiments, the electron density
around an atom is assumed to be spherical –
this can be a poor assumption for easily
polarized atoms.

Note that the scattering factor is independent of the wavelength of the X-

radiation used and depends only on the scattering angle and the electron
density of the atom (note that Ca+2 and Cl- have different behaviour at higher
values of  because the electron density is different!)
59-553 Atomic Scattering Factors 85

The decrease in the intensity of the scattering

with increasing  can be understood in a
manner similar to that which we examined
earlier in the context of systematic absences.
If distance between the electron density close
to the nucleus (O) and the density at P is
small compared the repeat distance (d) then
the scattered waves will have roughly the
same phase. As the repeat distance
becomes smaller, the waves will be more out-
of-phase so destructive interference will
reduce the overall intensity of the beam.
 59-553 Structure Factors 86

Consider that an atom j is located at rj from the origin in a unit cell of a

crystal. This shift in origin from the center of the individual atom means that
the distance r in the equation for the scattering by an atom becomes r + rj.
Thus the scattering by atom j becomes:
fj = ∫ ρ(r) exp[2i (r + rj) · S] dv
= fj exp(2i rj · S)
where fj = ∫ ρ(r) exp(2i r · S) dv
This is just another way of saying that an atom will have the same scattering
power no matter where it is located in the unit cell – the “exp(2i rj · S)”
factor merely accounts for the phase change that occurs upon translation of
the atom away from the origin of the cell.

Again, S is the scattering vector and is the bisector of so, a unit vector in the
incident beam direction, and s, a unit vector in the diffracted beam direction.
The angle between so and s = 2θ, is called the scattering angle.
|S| = 2 sin θ /  = 1/dhkl.
 59-553 Structure Factors 87

Similar expressions may be derived for all of the other atoms in the unit cell.
The total scattering power of all of the atoms is given by the sum of the
individual scattering amplitudes.
F(S) = ∑ fj
= ∑ fj exp(2i rj · S)

Bragg's Law requires that the phase difference between the waves
scattered by successive unit cells must be equal to an integral multiple of
2/. Since the scattered wave may be considered as coming from 1/n
multiples of the cell edge vectors a, b, and c, then
(2 / ) (a · S) = 2 h / 
(2 / ) (b · S) = 2 k / 
(2 / ) (c · S) = 2 l / 
This is another way of expressing the Laue equations.
 59-553 Structure Factors 88

The coordinates of atoms, in fractional coordinates, for the jth atom are
labelled xj, yj, and zj. The vector from the origin, rj, may be written as:
rj = axj + byj + czj
Thus the product rj · S may be written as:
rj · S = xj a · S + yj b · S + zj c · S
= h xj + k yj + l zj
Finally, the total scattering power for all of the atoms in the unit cell may be
written as:

F(hkl) = ∑ fj exp[2 i (h xj + k yj + l zj)]

The relation above is known as the structure factor expression. This relation may be
recast in terms of its amplitude, |F(hkl)|, and its phase angle, (hkl) or in terms of its
real, A, and imaginary, B, components in the following expressions.
F(hkl) = |F(hkl)| exp [i (hkl)]
F(hkl) = A + iB
 59-553 Structure Factors and Electron Density 89
Electron Density
If the structure factor expression is rewritten as a continuous summation
over the volume of the unit cell then the expression becomes:

F(S) = ∫ ρ(r) exp (2 i rj · S) dv

By multiplying both sides by exp (-2 i rj · S) and integrating over the volume
of diffraction space, dvr, we get an expression for the electron density of the
unit cell.

ρ(r) = ∫ F(S) exp (-2 i rj · S) dvr

Since F(S) is nonzero only at the lattice points, The integral may be written
as discrete sums over the three indices h, k, and l:

ρ(xyz) = 1/V ∑ ∑ ∑ F(hkl) exp [-2i (h x + k y + l z)]

or
ρ(xyz) = 1/V ∑ ∑ ∑ |F(hkl)| exp -2i [(h x + k y + l z – (hkl))]

where the three summations run over all values of h, k, and l.