The electron density
The problem we now address is how to interpret the data from a diffractometer in
terms of the detailed structure of a crystal. To do so, we must go beyond Bragg's
law. If a unit cell contains several atoms with scattering factorsand coordinates
(xja, yjb, zjc), then the overall amplitude of a wavediffracted by the {hkl} planes is
given by
The sum is over all the atoms in the unit cell. The quantity Fhkl is called the
structure factor.
Consider the crystal shown schematically in Fig.1. The reflection corresponds to
two waves from adjacent A planes, the phase difference of the waves being
2π. If there is a B atom at a fraction x of the distance between the two A planes,
then it gives rise to a wave with a phase difference 2 π x relative to an A
reflection. To see this conclusion, note that, if x = 0, there is no phase difference;
if x = 1/2 the phase difference is π; if x = 1, the B atom lies where the lower A
atom is and the phase difference is 2 π. Now consider a (200) reflection. There is
now a 2 x 2 π difference between the waves from the two A layers, and if B were
to lie at x = 0.5 it would give rise to a wave that differed in phase by 2 π from the
wave from the upper A layer.
Thus, for a general fractional position x, the phase difference for a (200)
reflection is 2 x (2 π x). For a general (h00) reflection, the phase difference is
therefore h x (2 π x).
For three dimensions, this result generalizes to the equation above..
The A and B reflections interfere destructively when the phase difference is π,
and the total intensity is zero if the atoms have the same scattering power. For
example, if the unit cells are cubic I with a B atom at x = y = z = 1/2, then the A,B
phase difference is (h + k + l ) π. Therefore, all reflections for odd values of h + k
+ l vanish because the waves are displaced in phase by π. Hence the diffraction
pattern for a cubic I lattice can be constructed from that for the cubic P lattice (a
cubic lattice without points at the centre of its unit cells) by striking out all
reflections with odd values of h + k + l. Recognition of these systematic absences
in a powder spectrum immediately indicates a cubic I lattice (Fig. 2).
Fig.1 Diffraction from a crystal containing two kinds of atoms. (a) For a (100) reflection from the A planes, there is a phase difference of 2 π between waves reflected by neighbouring planes. (b) For a (200) reflection, the phase difference is 4 π. The reflection from a B plane at a fractional distance x a from an A plane has a phase that is x times these phase differences.
Fig. 2. The powder diffraction patterns and the systematic absences of three versions of a cubic cell. Comparison of the observed pattern with patterns like these enables the unit cell to be identified. The locations of the lines give the cell dimensions.
If the amplitude of the waves scattered from A is fA at the detector, that of the
waves scattered from B is f BeiΦhkl , with Φhkl the phase difference. The
total amplitude at the detector is therefore
Fhkl = fA + fBei Φhkl
Because the intensity is proportional to the square modulus of the amplitude of the
wave, the intensity, Ihkl the detector is
Ihkl α F*hkl Fhkl = ( fA + fBe- i Φhkl) (fA + fBei Φhkl)
This expression expands to
Ihkl α f2A + f2B + fA fB (e- iΦhkl + eiΦhkl) = f2A + f2B + 2fA fBcosΦhkl
The cosine term either adds to or subtracts from f2A + f2B
depending on the value of Φhkl which in turn depends on h, k, and l and x, y, and z. Hence, there is a variation in the intensities of the lines with different hkl.
Calculate the structure factor for NaCl
Self-test Which reflections cannot be observed for a cubic I lattice?
[for h + k+ l odd, Fhkl= 0].
The intensity of the (hkI) reflection is proportional to IFhkll2 , so in principle we can
determine the structure factors experimentally by taking the square root of the
corresponding intensities . Then, once we know all the structure factors Fhkl , we can calculate the electron density distribution, Ρ(r), in the unit cell by
using the expression
where V is the volume of the unit cell. This equation is called a Fourier synthesis of the electron density.
Calculating an electron density by Fourier synthesis
Answer
The results are plotted in Fig. 3 (blue line). The positions of three
atoms can be discerned very readily. The more terms there are included, the
more accurate the density plot. Terms corresponding to high values of h (short
wavelength cosine terms in the sum) account for the finer details of the electron
density; low values of h account for the broad features.
Fig. 3 The plot of the electron density calculated in The last example (blue) and the Self-test below (purple).
Self-test Use mathematical software to experiment with different structure
factors (including changing signs as well as amplitudes). For example, use the same
values of Fh as above, but with positive signs for h ≥ 6.