Structure factor (crystallography)

The structure factor, F, is a function that expresses the sum of amplitudes of scattered X-rays from all atoms in a crystal.

We have, in the previous section, discussed the scattering factor of a single atom, which is the sum of amplitudes of waves scattered by the electrons in the atom. We shall now describe the sum of amplitudes of scattered X-rays from all atoms in a unit cell.

Consider a one-dimensional lattice along the a-axis consisting of two types of atoms (see diagram above). If the Laue equation along the a-axis is satisfied for R₁ and R₃ as well as for R₂and R₄, we have:

$\delta _{11\: atoms}=\delta _{22\: atoms}=\frac{a}{h}(cos\alpha -cos\alpha _0)=n\lambda$

However, if the Laue equation along the -axis is not satisfied for R₁ and R₂ (or for R₂and R₃) we have:

$\delta _{12\: atoms}=x(cos\alpha -cos\alpha _0)$

Combining the two equations above and letting n = 1,

$\delta _{12\: atoms}=\frac{\lambda hx}{a}\; \; \; \; \; \; \; (34)$

Substitute eq28 in eq34,

$\phi =\frac{2\pi hx}{a}\; \; \; \; \; \; \; (35)$

Recall from the previous section that the amplitude of the scattered X-ray received by the detector from an atom is equivalent to the scattering factor $f=4\pi \int_{0}^{\infty }\rho \frac{sinkr}{kr}r^2dr$ . If we have two atoms, the resultant amplitude F at the detector is

$F=f_1+f_2e^{i\phi }\; \; \; \; \; \; \; (36)$

The second term in eq36 includes the factor e^iΦ as the scattered rays from the second atom may have a phase difference from that from the first atom. If Φ = 0, F = f₁+ f₂ .

Question

Show that the intensity detected from eq36 is consistent with the intensity associated to the resultant amplitude of two general complex waves having a phase difference of Φ.

Answer

Let the waves be y₁ = Ae^i(kx-ωt)and y₂ = Be^i(kx-ωt+Φ) . The resultant amplitude is y = Ae^i(kx-ωt)+ y₂ = Be^i(kx-ωt+Φ). As we know, the intensity of the resultant wave is proportional to the square of magnitude of the amplitude of the wave:

$I\propto\left | y \right |^2=y^*y=\left [ Ae^{-i(kx-\omega t)}+Be^{-i(kx-\omega t+\phi )} \right ]\left [ Ae^{i(kx-\omega t)}+Be^{i(kx-\omega t+\phi )} \right ]$

$I\propto A^2+ABe^{i\phi }+ABe^{-i\phi}+B^2$

Since $e^{\pm i\phi}=cos\phi \pm isin\phi$

$I\propto A^2+B^2+2ABcos\phi$

Similarly, for eq36

$I \propto (f_1+f_2e^{-i\phi})(f_1+f_2e^{i\phi})$

$I\propto {f_{1}}^{2}+{f_{2}}^{2}+2 {f_{1}}{f_{2}}cos\phi$

Substitute eq35 in eq36,

$F=f_1+f_2e^{i\frac{2\pi hx}{a}}\; \; \; \; \; \; \; (37)$

In three dimensions, eq37 becomes

$F_{hkl}=f_1+f_2e^{i\frac{2\pi hx_2}{a}}+...+f_je^{i\frac{2\pi hx_j}{a}}+$

$f_1e^{i\frac{2\pi hy_1}{a}}+f_2e^{i\frac{2\pi hy_2}{a}}+ ...+f_je^{i\frac{2\pi hy_j}{a}}+$

$f_1e^{i\frac{2\pi hz_1}{a}}+f_2e^{i\frac{2\pi hz_2}{a}}+ ...+f_je^{i\frac{2\pi hz_j}{a}}$

So,

$F_{hkl}=\sum_{j}f_je^{i2\pi (\frac{hx_j}{a}+\frac{ky_j}{b}+\frac{lz_j}{c})}\; \; \; \; \; \; \; (38)$

The first term in eq38 is $f_1e^{i\frac{2\pi hx_1}{a}}$ in one-dimension. Since there is no phase difference between the resultant X-ray of the first atom with itself, $f_1e^{i\frac{2\pi hx_1}{a}}=f_1$ and eq38 is consistent with eq37, i.e. the phases of scattered X-rays of atoms are relative to that of the first atom at the coordinate of x₁ = 0. Eq38 can be rewritten in terms of fractional coordinates:

$F_{hkl}=\sum_{j}f_je^{i2\pi ({hx_j}'+{ky_j}'+{lz_j}')}\; \; \; \; \; \; \; (39)$

where ${x_j}'=\frac{x_j}{a}$ (i.e. x_jin units of a), ${y_j}'=\frac{y_j}{b}$ , ${z_j}'=\frac{z_j}{c}$ .

The intensity of a diffraction signal is proportional to the square of the magnitude of the three-dimensional structure factor, i.e. $I \propto \left | F_{hkl} \right |^2$ and therefore systematic absences appear when $F_{hkl}=0$ .

For example, the body centred cubic unit cell (where a = b = c) has fractional coordinates shown in the diagram below.

As the unit cell is composed of the same atoms, the structure factor for each atom is the same. If an atom is shared by m neighbouring unit cells, we multiply the scattering factor of the atom with 1/m. Therefore, eq39 becomes:

$F_{hkl}=\frac{1}{8}f[e^{i2\pi (0+0+0)}+e^{i2\pi (0+k+0)}+e^{i2\pi (h+k+0)}+e^{i2\pi (h+0+0)}+$

$e^{i2\pi (0+0+l)}+e^{i2\pi (0+k+l)}+e^{i2\pi (h+k+l)}+e^{i2\pi (h+0+l)}]+fe^{i2\pi (\frac{h}{2}+\frac{k}{2}+\frac{l}{2})}$

Since e^i2π = cos2π + isin2π = 1, e^iπ = cosπ + isinπ = -1 and e^ab = (e^a)^b

$F_{hkl}=\frac{1}{8}f[1+1^k+1^{h+k}+1^h+1^l+1^{k+l}+1^{h+k+l}+1^{h+l}]+f(-1^{h+k+l})$

Since 1^a = 1

$F_{hkl}=f[1+(-1^{h+k+l})]$

If h + k + l is even, F_hkl = 2f. If h + k + l is odd, F_hkl = 0. Hence, systematic absences occur when h + k + l is odd. A primitive cubic unit cell is different from a BCC in that it does not have an atom at $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ . Its structure factor is F_hkl = f, which means the diffraction intensities of a cubic P cell has no restriction other than the scattering factor of the atom, which is dependent on the Bragg’s angle θ and hence Bragg’s law of $sin\theta =\frac{\lambda \sqrt{h^2+k^2+l^2}}{2a}$ .

For a face-centred cubic unit cell, the coordinates of the atoms are the same as a cubic P unit cell plus six other atoms on the six faces: $(\frac{1}{2},\frac{1}{2},0)$ , $(\frac{1}{2},0,\frac{1}{2})$ , $(0,\frac{1}{2},\frac{1}{2})$ , $(\frac{1}{2},\frac{1}{2},1)$ , $(\frac{1}{2},1,\frac{1}{2})$ , $(1,\frac{1}{2},\frac{1}{2})$ . These atoms are each shared by two neighbouring cells. The structure factor of a face-centred cubic unit cell is:

$F_{hkl}=f+\frac{1}{2}f[e^{i2\pi (\frac{h}{2}+\frac{k}{2}+0)}+e^{i2\pi (\frac{h}{2}+0+\frac{l}{2})}+e^{i2\pi (0+\frac{k}{2}+\frac{l}{2})}+$

$e^{i2\pi (\frac{h}{2}+\frac{k}{2}+l)}+e^{i2\pi (\frac{h}{2}+k+\frac{l}{2})}+e^{i2\pi (h+\frac{k}{2}+\frac{l}{2})}]$

$F_{hkl}=f+\frac{1}{2}f[(-1^{h+k})+(-1^{h+l})+(-1^{k+l})+(-1^{h+k+2l})+$

$(-1^{h+2k+l})+(-1^{2h+k+l})]$

$F_{hkl}=f+\frac{1}{2}f[(-1^{h+k})+(-1^{h+l})+(-1^{k+l})+(-1^{h+k})(-1^{2l})+$

$(-1^{h+l})(-1^{2k})+(-1^{k+l})(-1^{2h})]$

$F_{hkl}=f+f[(-1^{h+k})+(-1^{h+l})+(-1^{k+l})]$

If h, k, l are all even or all odd, F_hkl = 4f. If one index is odd and two are even, or vice versa, F_hkl = 0. Therefore, a face-centred cubic unit cell does not have systematic absences when h, k, l are all even or all odd.

Structure factor (crystallography)

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