Structure factor

In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material 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.

The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons. Energy-resolved measurements yield the dynamic structure factor.

Derivation

Let us consider a scalar (real) quantity $\phi(\mathbf{r})$ defined in a volume $V$; it may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium. If the scalar function is assumed to be integrable, we can define its Fourier transform $\textstyle \phi(\mathbf{q}) = \int_{V} \phi(\mathbf{r}) \exp (-i \mathbf{q} \mathbf{r}) \, \mathrm{d} \mathbf{r}$. Expressing the field $\phi$ in terms of the spatial frequency $\mathbf{q}$ instead of the point position $\mathbf{r}$ is very useful, for instance, when interpreting scattering experiments. Indeed, in the Born approximation (weak interaction between the field and the medium), the amplitude of the signal corresponding to the scattering vector $\mathbf{q}$ is proportional to $\textstyle \phi(\mathbf{q})$. Very often, only the intensity of the scattered signal $\textstyle I(\mathbf{q})$ is detectable, so that $\textstyle I(\mathbf{q}) \sim \left | \phi(\mathbf{q}) \right |^2$.

If the system under study is composed of a number $N$ of identical constituents (atoms, molecules, colloidal particles, etc.) it is very convenient to explicitly capture the variation in $\phi$ due to the morphology of the individual particles using an auxiliary function $f(\mathbf{r})$, such that:

$\phi(\mathbf{r}) = \sum_{j=1}^{N} f(\mathbf{r} - \mathbf{R}_{j}) = f(\mathbf{r}) \ast \sum_{j=1}^{N} \delta(\mathbf{r} - \mathbf{R}_{j})$,

(1)

with $\textstyle \mathbf{R}_{j}, j = 1, \, \ldots, \, N$ the particle positions. In the second equality, the field is decomposed as the convolution product $\ast$ of the function $f$, describing the "form" of the particles, with a sum of Dirac delta functions depending only on their positions. Using the property that the Fourier transform of a convolution product is simply the product of the Fourier transforms of the two factors, we have $\textstyle \phi(\mathbf{q}) = f(\mathbf{q}) \sum_{j=1}^{N} \exp (-i \mathbf{q} \mathbf{R}_{j})$, such that:

$I(\mathbf{q}) \sim \left | \phi(\mathbf{q}) \right |^2 = \left | f(\mathbf{q}) \right |^2 \times \left ( \sum_{j=1}^{N} \mathrm{e}^{-i \mathbf{q} \mathbf{R}_{j}} \right ) \times \left ( \sum_{k=1}^{N} \mathrm{e}^{i \mathbf{q} \mathbf{R}_{k}} \right )= \left | f(\mathbf{q}) \right |^2 \sum_{jk} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_j - \mathbf{R}_k)}$.

(2)

In general, the particle positions are not fixed and the measurement takes place over a finite exposure time and with a macroscopic sample (much larger than the interparticle distance). The experimentally accessible intensity is thus an averaged one $\textstyle \langle I(\mathbf{q}) \rangle$; we need not specify whether $\langle \cdot \rangle$ denotes a time or ensemble average. We can finally write:

$\langle I(\mathbf{q}) \rangle \sim \langle \left | \phi(\mathbf{q}) \right |^2 \rangle = N \left | f(\mathbf{q}) \right |^2 S(\mathbf{q})$,

(3)

thus defining the structure factor

$S(\mathbf{q}) = \frac{1}{N} \left \langle \sum_{jk} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_j - \mathbf{R}_k)} \right \rangle$.

(4)

Perfect crystals

In a crystal, the constitutive particles are arranged periodically, forming a lattice. In the following, we will consider that all particles are identical (so the above separation in factor and structure factors (3) holds). We also assume that all atoms have an identical environment (i.e. they form a Bravais lattice). The general case of lattice with a basis (see below) is not fundamentally different.

If the lattice is infinite and completely regular, the system is a perfect crystal. In addition, we will neglect all thermal motion, so that there is no need for averaging in (4). As in (2), we can write:

$S(\mathbf{q}) = \frac{1}{N} \left | \sum_{j=1}^{N} \mathrm{e}^{-i \mathbf{q} \mathbf{R}_{j}} \right | ^2$.

The structure factor is simply the squared modulus of the Fourier transform of the lattice, and it is itself a periodic arrangement of points, known as the reciprocal lattice.

One dimension

Structure factor of a periodic chain, for different particle numbers $N$.

The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period $a$, the atom positions $\textstyle R_j = a (j - (N-1)/2)$ (for simplicity, we consider that $N$ is odd). The sum of the phase factors is a simple geometric series and the structure factor becomes:

$S(q) = \frac{1}{N} \left | \frac{1 - \mathrm{e}^{-i N q a}}{1 - \mathrm{e}^{-i q a}} \right | ^2 = \frac{1}{N} \left [ \frac{\sin(N q a/2)}{\sin(q a/2)} \right ] ^2$.

This function is shown in the Figure below for different values of $N$.

Based on this expression for $S(q)$, one can draw several conclusions: the reciprocal lattice has a spacing $2\pi/a$; the intensity of the maxima increases with the number of particles $S(q = 2 k \pi/a) = N$ (this is apparent from the Figure and can be shown by estimating the limit $S(q \to 0)$ using, for instance, L'Hôpital's rule); the intensity at the midpoint $S(q = (2 k +1) \pi/a) = 1/N$ (by direct evaluation); the peak width also decreases like $1/N$. In the large $N$ limit, the peaks become infinitely sharp Dirac delta functions.

Two dimensions

Diagram of scattering by a square (planar) lattice. The incident and outgoing beam are shown, as well as the relation between their wave vectors $\mathbf{k}_i$, $\mathbf{k}_o$ and the scattering vector $\mathbf{q}$.

In two dimensions, there are only five Bravais lattices. The corresponding reciprocal lattices have the same symmetry as the direct lattice. The Figure shows the construction of one vector of the reciprocal lattice and its relation with a scattering experiment.

A parallel beam, with wave vector $\mathbf{k}_i$ is incident on a square lattice of parameter $a$. The scattered wave is detected at a certain angle, which defines the wave vector of the outgoing beam, $\mathbf{k}_o$ (under the assumption of elastic scattering, $|\mathbf{k}_o| = |\mathbf{k}_i|$). One can equally define the scattering vector $\mathbf{q}=\mathbf{k}_o - \mathbf{k}_i$ and construct the harmonic pattern $\exp (i \mathbf{q}\mathbf{r})$. In the depicted example, the spacing of this pattern coincides to the distance between particle rows: $q = 2\pi /a$, so that contributions to the scattering from all particles are in phase (constructive interference). Thus, the total signal in direction $\mathbf{k}_o$ is strong, and $\mathbf{q}$ belongs to the reciprocal lattice. It is easily shown that this configuration fulfills Bragg's law.

Lattice with a basis

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

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

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 grow. Films of BCC materials like iron and tungsten therefore grow in a characteristic (110) orientation.

Face-centered cubic (FCC)

In the case of a monatomic 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

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

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 gold tend to grow in a (111) orientation with a triangular surface symmetry.

Diamond Crystal Structure

The Diamond cubic crystal structure occurs in 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:

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

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) or 4f(1-i), FF^*=32f²
  • h+k+l is even and exactly divisible by 4 (satisfies h+k+l=4n) then F = 8f
  • h+k+l is even but not exactly divisible by 4(doesn't satisfy h+k+l=4n) then F = 0

Imperfect crystals

Although the perfect lattice is an extremely useful model, real crystals always exhibit imperfections, which can have profound effects on the structure and properties of the material. André Guinier ^[1] proposed a widely employed distinction between imperfections that preserve the long-range order of the crystal (disorder of the first kind) and those that destroy it (disorder of the second kind).

Liquids

In contrast with crystals, liquids have no long-range order (in particular, there is no regular lattice), so the structure factor does not exhibit sharp peaks. They do however show a certain degree of short-range order, depending on their density and on the strength of the interaction between particles. Liquids are isotropic, so that, after the averaging operation in Equation (4), the structure factor only depends on the absolute magnitude of the scattering vector $q = \left |\mathbf{q} \right |$. For further evaluation, it is convenient to separate the diagonal terms $j = k$ in the double sum, whose phase is identically zero, and therefore each contribute a unit constant:

$S(q) = 1 + \frac{1}{N} \left \langle \sum_{j \neq k} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_j - \mathbf{R}_k)} \right \rangle$.

(5)

One can obtain an alternative expression for $S(q)$ in terms of the radial distribution function $g(r)$:^[2]

$S(q) = 1 + \rho \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q}\mathbf{r}} g(r)$.

(6)

Ideal gas

In the limiting case of no interaction, the system is an ideal gas and the structure factor is completely featureless: $S(q) = 1$, because there is no correlation between the positions $\mathbf{R}_j$ and $\mathbf{R}_k$ of different particles (they are independent random variables), so the off-diagonal terms in Equation (5) average to zero: $\langle \exp [-i \mathbf{q} (\mathbf{R}_j - \mathbf{R}_k)]\rangle = \langle \exp (-i \mathbf{q} \mathbf{R}_j) \rangle \langle \exp (i \mathbf{q} \mathbf{R}_k) \rangle = 0$.

High-$q$ limit

Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation (6), since $S(q)-1$ is the Fourier transform of the "regular" function $g(r)$ and thus goes to zero for high values of the argument $q$. This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.

Low-$q$ limit

In the low-$q$ limit, as the system is probed over large length scales, the structure factor contains thermodynamic information, being related to the isothermal compressibility $\chi _T$ of the liquid by the compressibility equation:

$\lim _{q \rightarrow 0} S(q) = \rho k_\text{B}T \chi _T= k_\text{B}T\left(\frac{\partial \rho}{\partial p}\right)$.

Hard-sphere liquids

Structure factor of a hard-sphere fluid, calculated using the Percus-Yevick approximation, for volume fractions $\Phi$ from 1% to 40%.

In the hard sphere model, the particles are described as impenetrable spheres with radius $R$; thus, their center-to-center distance $r \geq 2R$ and they experience no interaction beyond this distance. Their interaction potential can be written as:

$V(r) = \left \lbrace \begin{array}{l l} \infty \, &\text{for} \,\, r < 2 R\\ 0\, &\text{for} \,\, r \geq 2 R\\ \end{array} \right .$

This model has an analytical solution^[3] in the Percus–Yevick approximation. Although highly simplified, it provides a good description for systems ranging from liquid metals^[4] to colloidal suspensions.^[5] In an illustration, the structure factor for a hard-sphere fluid is shown in the Figure, for volume fractions $\Phi$ from 1% to 40%.

Polymers

In polymer systems, the general definition (4) holds; the elementary constituents are now the monomers making up the chains. However, the structure factor being a measure of the correlation between particle positions, one can reasonably expect that this correlation will be different for monomers belonging to the same chain or to different chains.

Let us assume that the volume $V$ contains $N_c$ identical molecules, each composed of $N_p$ monomers, such that $N_c N_p = N$ ($N_p$ is also known as the degree of polymerization). We can rewrite (4) as:

$S(\mathbf{q}) = \frac{1}{N_c N_p} \left \langle \sum_{\alpha \beta = 1}^{N_c} \sum_{jk = 1}^{N_p} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_{\alpha j} - \mathbf{R}_{\beta k})} \right \rangle = \frac{1}{N_c N_p} \left \langle \sum_{\alpha = 1}^{N_c} \sum_{jk = 1}^{N_p} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_{\alpha j} - \mathbf{R}_{\alpha k})} \right \rangle + \frac{1}{N_c N_p} \left \langle \sum_{\alpha \neq \beta = 1}^{N_c} \sum_{jk = 1}^{N_p} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_{\alpha j} - \mathbf{R}_{\beta k})} \right \rangle$,

(7)

where indices $\alpha , \beta$ label the different molecules and $j, k$ the different monomers along each molecule. On the right-hand side we separated intramolecular ($\alpha = \beta$) and intermolecular ($\alpha \neq \beta$) terms. Using the equivalence of the chains, (7) can be simplified:^[6]

$S(\mathbf{q}) = \underbrace{\frac{1}{N_p} \left \langle \sum_{jk = 1}^{N_p} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_{j} - \mathbf{R}_{k})} \right \rangle}_{S_1(q)} + \frac{N_c - 1}{N_p} \left \langle \sum_{jk = 1}^{N_p} \mathrm{e}^{-i \mathbf{q} (\mathbf{R}_{1 j} - \mathbf{R}_{2 k})} \right \rangle$,

(8)

where $S_1 (q)$ is the single-chain structure factor.

See also

• R-factor (crystallography)
• Patterson function

Notes

1. See Guinier, chapters 6-9
2. See Chandler, section 7.5.
3. Wertheim, M. (1963). "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres". Physical Review Letters 10 (8): 321. Bibcode:1963PhRvL..10..321W. doi:10.1103/PhysRevLett.10.321.
4. Ashcroft, N.; Lekner, J. (1966). "Structure and Resistivity of Liquid Metals". Physical Review 145: 83. Bibcode:1966PhRv..145...83A. doi:10.1103/PhysRev.145.83.
5. Pusey, P. N.; Van Megen, W. (1986). "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres". Nature 320 (6060): 340. doi:10.1038/320340a0.
6. See Teraoka, Section 2.4.4.

References

1. Als-Nielsen, N. and McMorrow, D. (2011). Elements of Modern X-ray Physics (2nd edition). John Wiley & Sons.
2. Guinier, A. (1963). X-ray Diffraction. In Crystals, Imperfect Crystals, and Amorphous Bodies. W. H. Freeman and Co.
3. Chandler, D. (1987). Introduction to Modern Statistical Mechanics. Oxford University Press.
4. Hansen, J. P. and McDonald, I. R. (2005). Theory of Simple Liquids (3rd edition). Academic Press.
5. Teraoka, I. (2002). Polymer Solutions: An Introduction to Physical Properties. John Wiley & Sons.

External links

• Structure Factor Tutorial located at the University of York.