Structure factor

In physics, in the area of crystallography, the structure factor of a crystal is a mathematical description of how the crystal scatters incident radiation. The structure factor is a particularly useful tool in the interpretation of interference patterns obtained in X-ray, electron and neutron diffraction experiments.

Scattering from a crystal
A crystal is a periodic arrangement of atoms in a particular pattern. Each of the atoms may scatter incident radiation such as X-rays, electrons and neutrons. Because of the periodic arrangement of the atoms, the interference of waves scattered from different atoms may cause a distinct pattern of constructive and destructive interference to form. This is the diffraction pattern caused by the crystal.

In the kinematical approximation for diffraction, the intensity of a diffracted beam is given by:


 * $$ I_{\Delta\mathbf{k}} = \left | \psi_{\Delta\mathbf{k}} \right |^2 \propto \left | F_{\Delta\mathbf{k}} \right |^2 $$

where $$\psi_{\Delta\mathbf{k}}$$ is the wavefunction of a beam scattered a vector $$\Delta\mathbf{k}$$, and $$F_{\Delta\mathbf{k}}$$ is the so called structure factor which is given by:


 * $$ F_{\Delta\mathbf{k}}=\sum_{j} f_j e^{-i\Delta\mathbf{k} \cdot \mathbf{r}_j}$$

Here, $$\mathbf{r}_j$$ is the position of an atom $$j$$ in the unit cell, and $$f_j$$ is the scattering power of the atom, also called the atomic form factor. The sum is over all atoms in the unit cell. It can be shown that in the ideal case, diffraction only occurs if the scattering vector $$\Delta\mathbf{k}$$ is equal to a reciprocal lattice vector $$\mathbf{K}$$.

The structure factor describes the way in which an incident beam is scattered by the atoms of a crystal unit cell, taking into account the different scattering power of the elements through the term $$f_i$$. Since the atoms are spatially distributed in the unit cell, there will be a difference in phase when considering the scattered amplitude from two atoms. This phase shift is taken into account by the complex exponential term. The atomic form factor, or scattering power, of an element depends on the type of radiation considered. Because electrons interact with matter though different processes than for example X-rays, the atomic form factors for the two cases are not the same.

Structure factors for specific lattice types
To compute structure factors for a specific lattice, compute the sum above over the atoms in the unit cell. Since crystals are often described in terms of their Miller indices, it is useful to examine a specific structure factor in terms of these.

Body-centered cubic (BCC) 

As a convention, the body-centered cubic system is described in terms of a simple cubic lattice with primitive vectors $$a\hat{x}, a\hat{y}, a\hat{z}$$, with a basis consisting of $$\mathbf{r}_0 = \vec{0}$$ and $$\mathbf{r}_1 = (a/2)(\hat{x} + \hat{y} + \hat{z})$$. The corresponding reciprocal lattice is also simple cubic with side $$2\pi/a$$.

In a monoatomic crystal, all the form factors $$f$$ are the same. The intensity of a diffracted beam scattered with a vector $$\mathbf{K}=(2\pi/a)(h\hat{x}^* + k\hat{y}^* + l\hat{z}^*)$$ by a crystal plane with Miller indices $$(hkl)$$ is then given by:


 * $$\begin{matrix}

F_{\mathbf{K}} & = & f \left[ e^{-i\mathbf{K}\cdot\vec{0}} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{y} + \hat{z})} \right] \\ & = & f \left[ 1 + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{y} + \hat{z})} \right] \\ & = & f \left[ 1 + e^{-i\pi(h + k + l)} \right]\\ & = & f \left[ 1 + (-1)^{h + k + l} \right] \\ \end{matrix}$$

We then arrive at the following result for the structure factor for scattering from a plane $$(hkl)$$:

$$ F_{hkl} = \begin{cases} 2f, & h + k + l \ \ \mbox{even}\\ 0, & h + k + l \ \ \mbox{odd} \end{cases} $$

This result tells us that for a reflection to appear in a diffraction experiment involving a body-centered crystal, the sum of the Miller indices of the scattering plane must be even. If the sum of the Miller indices is odd, the intensity of the diffracted beam is reduced to zero due to destructive interference. This zero intensity for a group of diffracted beams is called a systematic absence. Since atomic form factors fall off with increasing diffraction angle corresponding to higher Miller indices, the most intense diffraction peak from a material with a BCC structure is typically the (110). The (110) plane is the most densely packed of BCC crystal structures and is therefore the lowest energy surface for a thin film to expose. Films of BCC materials like iron and tungsten therefore grow in a characteristic (110) orientation.

Face-centered cubic (FCC) 

In the case of a monoatomic FCC crystal, the atoms in the basis are at the origin $$\mathbf{r}_0 = \vec{0}$$ with indices (0,0,0) and at the three face centers $$\mathbf{r}_1 = (a/2)(\hat{x} + \hat{y})$$, $$\mathbf{r}_2 = (a/2)(\hat{y} + \hat{z})$$,  $$\mathbf{r}_3 = (a/2)(\hat{x} + \hat{z})$$ with indices given by (1/2,1/2,0), (0,1/2,1/2), (1/2,0,1/2). An argument similar to the one above gives the expression


 * $$\begin{matrix}

F_{\mathbf{K}} & = & f \left[ e^{-i\mathbf{K}\cdot\vec{0}} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{y})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{y} + \hat{z})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{z})} \right] \\ & = & f \left[ 1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} \right] \\ \end{matrix}$$

with the result

$$ F_{hkl} = \begin{cases} 4f, & h,k,l  \ \ \mbox{all even or all odd}\\ 0, & h,k,l \ \ \mbox{mixed parity} \end{cases} $$

The most intense diffraction peak from a material that crystallizes in the FCC structure is typically the (111). Films of FCC materials like silicon tend to grow in a (111) orientation with a triangular surface symmetry although the surfaces of wafers on which integrated circuits are grown instead have the (100) orientation with a square surface symmetry.

Diamond Crystal Structure

The Diamond cubic crystal structure is possessed by diamond (carbon), most semiconductors and tin. The basis cell contains 8 atoms located at cell positions:

$$\mathbf{r}_0 = \vec{0}$$

$$\mathbf{r}_1 = (a/4)(\hat{x} + \hat{y} + \hat{z})$$

$$\mathbf{r}_2 = (a/4)(2\hat{x} + 2\hat{y})$$

$$\mathbf{r}_3 = (a/4)(3\hat{x} + 3\hat{y} + \hat{z})$$

$$\mathbf{r}_4 = (a/4)(2\hat{x} + 2\hat{z})$$

$$\mathbf{r}_5 = (a/4)(2\hat{y} + 2\hat{z})$$

$$\mathbf{r}_6 = (a/4)(3\hat{x} + \hat{y} + 3\hat{z})$$

$$\mathbf{r}_7 = (a/4)(\hat{x} + 3\hat{y} + 3\hat{z})$$

The Structure factor then takes on a form like this:


 * $$\begin{matrix}

F_{\mathbf{K}} & = & f \left[ \begin{matrix} e^{-i\mathbf{K}\cdot\vec{0}} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{y})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{y} + \hat{z})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{z})} + \\ e^{-i\mathbf{K}\cdot(a/4)(\hat{x} + \hat{y} + \hat{z})} + e^{-i\mathbf{K}\cdot(a/4)(3\hat{x} + \hat{y} + 3\hat{z})} + e^{-i\mathbf{K}\cdot(a/4)(3\hat{x} + 3\hat{y} + \hat{z})} + e^{-i\mathbf{K}\cdot(a/4)(\hat{x} + 3\hat{y} + 3\hat{z})} \end{matrix} \right] \\ & = & f \left[ \begin{matrix} 1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} + \\ (-i)^{h + k + l} + (-i)^{3h + k + 3l} + (-i)^{3h + 3k + l} + (-i)^{h + 3k + 3l} \end{matrix} \right] \\ & = & f \left[ 1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} \right] \cdot \left[ 1 + (-i)^{h + k + l} \right]\\ \end{matrix} $$

with the result


 * for mixed values (odds and even values combined) of h, k, and l, F2 will be 0
 * if the values are unmixed and...
 * h,k,l is odd then F=4f(1+i), FF*=32f2
 * h,k,l is an even(satisfy h+k+l=4n) then F = 8f
 * h,k,l is an even(unsatisfy h+k+l=4n) then F = 0