Wikipedia / Structure factor
Wikipedia / 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 critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments.
are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell gives the amplitude. It is the modulus squared is most useful for disordered systems. For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed.
The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons. Energy-resolved measurements yield the dynamic structure factor.
Derivation of S(q)
Consider the scattering. Assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (Born approximation), and absorption, refraction and multiple scattering can be neglected (kinematic diffraction) are the scattered and incident beam wavevectors (see Ewald sphere
is the atomic form factor-th atom. The scattered intensity is obtained by multiplying this function by its complex conjugate
If all the atoms are identical, then Equation ( so
. In three dimensions, Equation () then simplifies to the Debye scattering equation:
An alternative derivation gives good insight, but uses Fourier transforms and convolution; this may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium. If the scalar function is integrable, we can write its Fourier transform. In the Born approximation. then the total distribution can be considered the convolution of this function with a set of delta functions.
, so that:
This is clearly the same as Equation (.
denotes a time or ensemble average. To take this into account we can rewrite Equation () as:
Perfect crystals
In a crystal, the constitutive particles are arranged periodically, with translational symmetry forming a lattice. The crystal structure can be described as a Bravais lattice [basis]. If the lattice is infinite and completely regular, the system is a perfect crystal can give scattering, and the scattering amplitude for all other values is zero. This set of values forms a lattice, called the reciprocal lattice, which is the Fourier transform of the real-space crystal lattice.
can be used to determine the scattering from a perfect crystal; in the simple case when the basis is a single atom at the origin (and again neglecting all thermal motion, so that there is no need for averaging) all the atoms have identical environments. Equation () can be written as
.
The structure factor is then simply the squared modulus of the Fourier transform.
Units
m is commonly used.
may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.
Definition of Fhkl
In crystallography, the basis and lattice are treated separately. For a perfect crystal the lattice gives the reciprocal lattice which determines the amplitude and phase of the diffracted beams:
-th atom. defines a reciprocal lattice which corresponds to the real-space plane defined by the Miller indices (see Bragg's law).
is odd.
[structure factor].
Examples of Fhkl in 3-D
Body-centered cubic (BCC)
For the body-centered cubic Bravais lattice (cI which leads us to
and hence
Face-centered cubic (FCC)
The FCC. Equation () becomes
with the result
The most intense diffraction peak from a material that crystallizes in the FCC structure is typically the (111). Films of FCC materials like gold of mixed parity) is called a systematic absence.
Diamond crystal structure
The diamond cubic crystal structure occurs for example in diamond (carbon), tin, and most semiconductors. There are 8 atoms in the cubic unit cell. We can consider the structure as a simple cubic with a basis of 8 atoms, at positions
But comparing this to the FCC above, we see that it is simpler to describe the structure as FCC with a basis of two atoms at (0, 0, 0) and (1/4, 1/4, 1/4). For this basis, Equation () becomes:
And then the structure factor for the diamond cubic structure is the product of this and the structure factor for FCC above, (only including the atomic form factor once)
with the result
If h, k, ℓ are all even or all odd then the first (FCC) term is 4
These points are encapsulated by the following equations:
is an integer.
Zincblende crystal structure
, the resulting structure factor is
Cesium chloride
Cesium chloride is a simple cubic crystal lattice with a basis of Cs at (0,0,0) and Cl at (1/2, 1/2, 1/2) (or the other way around, it makes no difference). Equation () becomes
:
Hexagonal close-packed (HCP)
In an HCP crystal such as graphite and the next plane up the c axis located at c, which gives us
, and from there consider the modulus squared so hence
This leads us to the following conditions for the structure factor:
Perfect crystals in one and two dimensions
. In two dimensions, there are only five Bravais lattices.
However, recall that real 2-D crystals such as graphene in a hexagonal array of points.
The Figure shows the construction of one vector of a 2-D reciprocal lattice and its relation to a scattering experiment.
(under the assumption of elastic scattering belongs to the reciprocal lattice. It is easily shown that this configuration fulfills Bragg's law.
Imperfect crystals
Technically a perfect crystal must be infinite, so a finite size is an imperfection. Real crystals always exhibit imperfections of their order besides their finite size, and these imperfections can have profound effects on the properties of the material. André Guinier proposed a widely employed distinction between imperfections that preserve the long-range order of the crystal that he called disorder of the first kind and those that destroy it called disorder of the second kind. An example of the first is thermal vibration; an example of the second is some density of dislocations.
, so separate factors for size or thermal effects are introduced into the expressions for scattered intensity, leaving the perfect crystal structure factor unchanged. Therefore, a detailed description of these factors in crystallographic structure modeling and structure determination by diffraction is not appropriate in this article.
Finite-size effects
. 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:
using, for instance, L'Hôpital's rule limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1-D lattice.
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 by a multiplicative factor, to give
.
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 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
. The derivation follows that in chapter 9 of Guinier's textbook. This model has been pioneered by and applied to a number of materials by Hosemann and collaborators 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
. We will consider a finite crystal with disorder of the second-type below.
. So, then
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
s, etc. As the convolution of two Gaussians is just another Gaussian, we have that
is then just a sum of Fourier transforms of Gaussians, and so
and so the structure factor of the infinite but disordered crystal is
. These peaks have heights
and obtain
which is a Lorentzian or Cauchy function at the peak.
limit.
Finite crystals with disorder of the second kind
pairs of next-nearest neighbours, etc.
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 ( in the double sum, whose phase is identically zero, and therefore each contribute a unit constant:
in terms of the radial distribution function:
Ideal gas
In the limiting case of no interaction, the system is an ideal gas of different particles (they are independent random variables), so the off-diagonal terms in Equation (.
High-q limit
Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation ( is the Fourier transform. This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.
Low-q limit
limit, as the system is probed over large length scales, the structure factor contains thermodynamic information, being related to the isothermal compressibility of the liquid by the compressibility equation:
.
Hard-sphere liquids
In the hard sphere and they experience no interaction beyond this distance. Their interaction potential can be written as:
This model has an analytical solution in the Percus–Yevick approximation. Although highly simplified, it provides a good description for systems ranging from liquid metals to colloidal suspensions. from 1% to 40%.
Polymers
In polymer systems, the general definition () 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.
is also known as the degree of polymerization). We can rewrite () as: the different monomers along each molecule. On the right-hand side we separated intramolecular) and intermolecular) terms. Using the equivalence of the chains, () can be simplified: is the single-chain structure factor.
See also
Notes
References
Als-Nielsen, N. and McMorrow, D. (2011). Elements of Modern X-ray Physics (2nd edition). John Wiley & Sons.
Guinier, A. (1963). X-ray Diffraction. In Crystals, Imperfect Crystals, and Amorphous Bodies. W. H. Freeman and Co.
Chandler, D. (1987). Introduction to Modern Statistical Mechanics. Oxford University Press.
Hansen, J. P. and McDonald, I. R. (2005). Theory of Simple Liquids (3rd edition). Academic Press.
Teraoka, I. (2002). Polymer Solutions: An Introduction to Physical Properties. John Wiley & Sons.
External links
Structure Factor Tutorial located at the University of York.