Units

The units of the structure-factor amplitude depend on the incident radiation. For X-ray crystallography they are multiples of the unit of scattering by a single electron (2.82${\displaystyle \times 10^{-15}}$ m); for neutron scattering by atomic nuclei the unit of scattering length of ${\displaystyle 10^{-14}}$ m is commonly used.

The above discussion uses the wave vectors ${\displaystyle |\mathbf {k} |=2\pi /\lambda }$ and ${\displaystyle |\mathbf {q} |=4\pi \sin \theta /\lambda }$. However, crystallography often uses wave vectors ${\displaystyle |\mathbf {s} |=1/\lambda }$ and ${\displaystyle |\mathbf {g} |=2\sin \theta /\lambda }$. Therefore, when comparing equations from different sources, the factor ${\displaystyle 2\pi }$ may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.

Definition of F_hkl

In crystallography, the basis and lattice are treated separately. For a perfect crystal the lattice gives the reciprocal lattice, which determines the positions (angles) of diffracted beams, and the basis gives the structure factor ${\displaystyle F_{hkl}}$ which determines the amplitude and phase of the diffracted beams:

${\displaystyle F_{hk\ell }=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]},}$

(8)

where the sum is over all atoms in the unit cell, ${\displaystyle x_{j},y_{j},z_{j}}$ are the positional coordinates of the ${\displaystyle j}$-th atom, and ${\displaystyle f_{j}}$ is the scattering factor of the ${\displaystyle j}$-th atom.^[4] The coordinates ${\displaystyle x_{j},y_{j},z_{j}}$ have the directions and dimensions of the lattice vectors ${\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }$. That is, (0,0,0) is at the lattice point, the origin of position in the unit cell; (1,0,0) is at the next lattice point along ${\displaystyle \mathbf {a} }$ and (1/2, 1/2, 1/2) is at the body center of the unit cell. ${\displaystyle (hkl)}$ defines a reciprocal lattice point at ${\displaystyle (h\mathbf {a^{*}} ,k\mathbf {b^{*}} ,l\mathbf {c^{*}} )}$ which corresponds to the real-space plane defined by the Miller indices ${\displaystyle (hkl)}$ (see Bragg's law).

${\displaystyle F_{hk\ell }}$ is the vector sum of waves from all atoms within the unit cell. An atom at any lattice point has the reference phase angle zero for all ${\displaystyle hk\ell }$ since then ${\displaystyle (hx_{j}+ky_{j}+\ell z_{j})}$ is always an integer. A wave scattered from an atom at (1/2, 0, 0) will be in phase if ${\displaystyle h}$ is even, out of phase if ${\displaystyle h}$ is odd.

Again an alternative view using convolution can be helpful. Since [crystal structure] = [lattice] ${\displaystyle \ast }$ [basis], ${\displaystyle {\mathcal {F}}}$[crystal structure] = ${\displaystyle {\mathcal {F}}}$[lattice] ${\displaystyle \times {\mathcal {F}}}$[basis]; that is, scattering ${\displaystyle \propto }$ [reciprocal lattice] ${\displaystyle \times }$ [structure factor].

Examples of F_hkl in 3-D

Perfect crystals in one and two dimensions

The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period ${\displaystyle a}$, the reciprocal lattice is an infinite array of points with spacing ${\displaystyle 2\pi /a}$. In two dimensions, there are only five Bravais lattices. The corresponding reciprocal lattices have the same symmetry as the direct lattice. 2-D lattices are excellent for demonstrating simple diffraction geometry on a flat screen, as below. Equations (1)–(7) for structure factor ${\displaystyle S(\mathbf {q} )}$ apply with a scattering vector of limited dimensionality and a crystallographic structure factor can be defined in 2-D as ${\displaystyle F_{hk}=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j})]}}$.

However, recall that real 2-D crystals such as graphene exist in 3-D. The reciprocal lattice of a 2-D hexagonal sheet that exists in 3-D space in the ${\displaystyle xy}$ plane is a hexagonal array of lines parallel to the ${\displaystyle z}$ or ${\displaystyle z^{*}}$ axis that extend to ${\displaystyle \pm \infty }$ and intersect any plane of constant ${\displaystyle z}$ in a hexagonal array of points.

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

The Figure shows the construction of one vector of a 2-D reciprocal lattice and its relation to a scattering experiment.

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

Structure factor of a periodic chain, for different particle numbers ${\displaystyle N}$.

Finite-size effects

For ${\displaystyle S(q)}$ a finite crystal means that the sums in equations 1-7 are now over a finite ${\displaystyle N}$. The effect is most easily demonstrated with a 1-D lattice of points. The sum of the phase factors is a geometric series and the structure factor becomes:

${\displaystyle S(q)={\frac {1}{N}}\left|{\frac {1-\mathrm {e} ^{-iNqa}}{1-\mathrm {e} ^{-iqa}}}\right|^{2}={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}.}$

This function is shown in the Figure for different values of ${\displaystyle N}$. When the scattering from every particle is in phase, which is when the scattering is at a reciprocal lattice point ${\displaystyle q=2k\pi /a}$, the sum of the amplitudes must be ${\displaystyle \propto N}$ and so the maxima in intensity are ${\displaystyle \propto N^{2}}$. Taking the above expression for ${\displaystyle S(q)}$ and estimating the limit ${\displaystyle S(q\to 0)}$ using, for instance, L'Hôpital's rule) shows that ${\displaystyle S(q=2k\pi /a)=N}$ as seen in the Figure. At the midpoint ${\displaystyle S(q=(2k+1)\pi /a)=1/N}$ (by direct evaluation) and the peak width decreases like ${\displaystyle 1/N}$. In the large ${\displaystyle N}$ limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1-D lattice.

In crystallography when ${\displaystyle F_{hkl}}$ is used, ${\displaystyle N}$ is large, and the formal size effect on diffraction is taken as ${\displaystyle \left[{\frac {\sin(Nqa/2)}{(qa/2)}}\right]^{2}}$, which is the same as the expression for ${\displaystyle S(q)}$ above near to the reciprocal lattice points, ${\displaystyle q\approx 2k\pi /a}$. Using convolution, we can describe the finite real crystal structure as [lattice] ${\displaystyle \ast }$ [basis]${\displaystyle \times }$ rectangular function, where the rectangular function has a value 1 inside the crystal and 0 outside it. Then ${\displaystyle {\mathcal {F}}}$[crystal structure] = ${\displaystyle {\mathcal {F}}}$[lattice] ${\displaystyle \times {\mathcal {F}}}$[basis] ${\displaystyle \ast {F}}$[rectangular function]; that is, scattering ${\displaystyle \propto }$ [reciprocal lattice] ${\displaystyle \times }$ [structure factor] ${\displaystyle \ast }$ [ sinc function]. Thus the intensity, which is a delta function of position for the perfect crystal, becomes a ${\textstyle \operatorname {sinc} ^{2}}$ function around every point with a maximum ${\displaystyle \propto N^{2}}$, a width ${\displaystyle \propto 1/N}$, area ${\displaystyle \propto N}$.

Disorder of the first kind

This model for disorder in a crystal starts with the structure factor of a perfect crystal. In one-dimension for simplicity and with N planes, we then start with the expression above for a perfect finite lattice, and then this disorder only changes ${\displaystyle S(q)}$ by a multiplicative factor, to give^[1]

${\displaystyle S(q)={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}\exp \left(-q^{2}\langle \delta x^{2}\rangle \right)}$

where the disorder is measured by the mean-square displacement of the positions ${\displaystyle x_{j}}$ from their positions in a perfect one-dimensional lattice: ${\displaystyle a(j-(N-1)/2)}$, i.e., ${\displaystyle x_{j}=a(j-(N-1)/2)+\delta x}$, where ${\displaystyle \delta x}$ is a small (much less than ${\displaystyle a}$) random displacement. For disorder of the first kind, each random displacement ${\displaystyle \delta x}$ is independent of the others, and with respect to a perfect lattice. Thus the displacements ${\displaystyle \delta x}$ do not destroy the translational order of the crystal. This has the consequence that for infinite crystals (${\displaystyle N\to \infty }$) the structure factor still has delta-function Bragg peaks – the peak width still goes to zero as ${\displaystyle N\to \infty }$, with this kind of disorder. However, it does reduce the amplitude of the peaks, and due to the factor of ${\displaystyle q^{2}}$ in the exponential factor, it reduces peaks at large ${\displaystyle q}$ much more than peaks at small ${\displaystyle q}$.

The structure is simply reduced by a ${\displaystyle q}$ and disorder dependent term because all disorder of the first-kind does is smear out the scattering planes, effectively reducing the form factor.

In three dimensions the effect is the same, the structure is again reduced by a multiplicative factor, and this factor is often called the Debye–Waller factor. Note that the Debye–Waller factor is often ascribed to thermal motion, i.e., the ${\displaystyle \delta x}$ are due to thermal motion, but any random displacements about a perfect lattice, not just thermal ones, will contribute to the Debye–Waller factor.

Disorder of the second kind

However, fluctuations that cause the correlations between pairs of atoms to decrease as their separation increases, causes the Bragg peaks in the structure factor of a crystal to broaden. To see how this works, we consider a one-dimensional toy model: a stack of plates with mean spacing ${\displaystyle a}$. The derivation follows that in chapter 9 of Guinier's textbook.^[6] This model has been pioneered by and applied to a number of materials by Hosemann and collaborators^[7] over a number of years. Guinier and they termed this disorder of the second kind, and Hosemann in particular referred to this imperfect crystalline ordering as paracrystalline ordering. Disorder of the first kind is the source of the Debye–Waller factor.

To derive the model we start with the definition (in one dimension) of the

${\displaystyle S(q)={\frac {1}{N}}\sum _{j,k=1}^{N}\mathrm {e} ^{-iq(x_{j}-x_{k})}}$

To start with we will consider, for simplicity an infinite crystal, i.e., ${\displaystyle N\to \infty }$. We will consider a finite crystal with disorder of the second-type below.

For our infinite crystal, we want to consider pairs of lattice sites. For large each plane of an infinite crystal, there are two neighbours ${\displaystyle m}$ planes away, so the above double sum becomes a single sum over pairs of neighbours either side of an atom, at positions ${\displaystyle -m}$ and ${\displaystyle m}$ lattice spacings away, times ${\displaystyle N}$. So, then

${\displaystyle S(q)=1+2\sum _{m=1}^{\infty }\int _{-\infty }^{\infty }{\rm {d}}(\Delta x)p_{m}(\Delta x)\cos \left(q\Delta x\right)}$

where ${\displaystyle p_{m}(\Delta x)}$ is the probability density function for the separation ${\displaystyle \Delta x}$ of a pair of planes, ${\displaystyle m}$ lattice spacings apart. For the separation of neighbouring planes we assume for simplicity that the fluctuations around the mean neighbour spacing of a are Gaussian, i.e., that

${\displaystyle p_{1}(\Delta x)={\frac {1}{\left(2\pi \sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-a\right)^{2}/(2\sigma _{2}^{2})\right]}$

and we also assume that the fluctuations between a plane and its neighbour, and between this neighbour and the next plane, are independent. Then ${\displaystyle p_{2}(\Delta x)}$ is just the convolution of two ${\displaystyle p_{1}(\Delta x)}$s, etc. As the convolution of two Gaussians is just another Gaussian, we have that

${\displaystyle p_{m}(\Delta x)={\frac {1}{\left(2\pi m\sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-ma\right)^{2}/(2m\sigma _{2}^{2})\right]}$

The sum in ${\displaystyle S(q)}$ is then just a sum of Fourier transforms of Gaussians, and so

${\displaystyle S(q)=1+2\sum _{m=1}^{\infty }r^{m}\cos \left(mqa\right)}$

for ${\displaystyle r=\exp[-q^{2}\sigma _{2}^{2}/2]}$. The sum is just the real part of the sum ${\displaystyle \sum _{m=1}^{\infty }[r\exp(iqa)]^{m}}$ and so the structure factor of the infinite but disordered crystal is

${\displaystyle S(q)={\frac {1-r^{2}}{1+r^{2}-2r\cos(qa)}}}$

This has peaks at maxima ${\displaystyle q_{p}=2n\pi /a}$, where ${\displaystyle \cos(q_{P}a)=1}$. These peaks have heights

${\displaystyle S(q_{P})={\frac {1+r}{1-r}}\approx {\frac {4}{q_{P}^{2}\sigma _{2}^{2}}}={\frac {a^{2}}{n^{2}\pi ^{2}\sigma _{2}^{2}}}}$

i.e., the height of successive peaks drop off as the order of the peak (and so ${\displaystyle q}$) squared. Unlike finite-size effects that broaden peaks but do not decrease their height, disorder lowers peak heights. Note that here we assuming that the disorder is relatively weak, so that we still have relatively well defined peaks. This is the limit ${\displaystyle q\sigma _{2}\ll 1}$, where ${\displaystyle r\simeq 1-q^{2}\sigma _{2}^{2}/2}$. In this limit, near a peak we can approximate ${\displaystyle \cos(qa)\simeq 1-(\Delta q)^{2}a^{2}/2}$, with${\displaystyle \Delta q=q-q_{P}}$ and obtain

${\displaystyle S(q)\approx {\frac {S(q_{P})}{1+{\frac {r}{(1-r)^{2}}}{\frac {\Delta q^{2}a^{2}}{2}}}}\approx {\frac {S(q_{P})}{1+{\frac {\Delta q^{2}}{[q_{P}^{2}\sigma _{2}^{2}/a]^{2}/2}}}}}$

which is a Lorentzian or Cauchy function, of FWHM ${\displaystyle q_{P}^{2}\sigma _{2}^{2}/a=4\pi ^{2}n^{2}(\sigma _{2}/a)^{2}/a}$, i.e., the FWHM increases as the square of the order of peak, and so as the square of the wave vector ${\displaystyle q}$ at the peak.

Finally, the product of the peak height and the FWHM is constant and equals ${\displaystyle 4/a}$, in the ${\displaystyle q\sigma _{2}\ll 1}$ limit. For the first few peaks where ${\displaystyle n}$ is not large, this is just the ${\displaystyle \sigma _{2}/a\ll 1}$ limit.

Ideal gas

In the limiting case of no interaction, the system is an ideal gas and the structure factor is completely featureless: ${\displaystyle S(q)=1}$, because there is no correlation between the positions ${\displaystyle \mathbf {R} _{j}}$ and ${\displaystyle \mathbf {R} _{k}}$ of different particles (they are independent random variables), so the off-diagonal terms in Equation (9) average to zero: ${\displaystyle \langle \exp[-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})]\rangle =\langle \exp(-i\mathbf {q} \mathbf {R} _{j})\rangle \langle \exp(i\mathbf {q} \mathbf {R} _{k})\rangle =0}$.

High-q limit

Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation (10), since ${\displaystyle S(q)-1}$ is the Fourier transform of the "regular" function ${\displaystyle g(r)}$ and thus goes to zero for high values of the argument ${\displaystyle q}$. This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.

Low-q limit

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

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

Hard-sphere liquids

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

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

${\displaystyle V(r)={\begin{cases}\infty &{\text{for }}r<2R,\\0&{\text{for }}r\geq 2R.\end{cases}}}$

This model has an analytical solution^[9] in the Percus–Yevick approximation. Although highly simplified, it provides a good description for systems ranging from liquid metals^[10] to colloidal suspensions.^[11] In an illustration, the structure factor for a hard-sphere fluid is shown in the Figure, for volume fractions ${\displaystyle \Phi }$ from 1% to 40%.