In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
and
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, 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 pattern
In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude (constructive int ...
s) obtained in
X-ray
An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
,
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
and
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
experiments.
Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written
; it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written
or
and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the
planes of the crystal (
are the
Miller indices
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...
of the planes) to that produced by a single scattering unit at the vertices of the
primitive unit cell.
is not a special case of
;
gives the scattering intensity, but
gives the amplitude. It is the
modulus squared
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square o ...
that gives the scattering intensity.
is defined for a perfect crystal, and is used in crystallography, while
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
Consider the
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
of a beam of wavelength
by an assembly of
particles or atoms stationary at positions
. Assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (
Born approximation
Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named ...
), and absorption, refraction and multiple scattering can be neglected (
kinematic diffraction). The direction of any scattered wave is defined by its scattering vector
.
, where
and
(
) are the scattered and incident beam
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
s, and
is the angle between them. For elastic scattering,
and
, limiting the possible range of
(see
Ewald sphere). The amplitude and phase of this scattered wave will be the vector sum of the scattered waves from all the atoms
For an assembly of atoms,
is the
atomic form factor
In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident ...
of the
-th atom. The scattered intensity is obtained by multiplying this function by its complex conjugate
The structure factor is defined as this intensity normalized by
If all the atoms are identical, then Equation () becomes
and
so
Another useful simplification is if the material is isotropic, like a powder or a simple liquid. In that case, the intensity depends on
and
. In three dimensions, Equation () then simplifies to the Debye scattering equation:
[
An alternative derivation gives good insight, but uses ]Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
s and convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
. To be general, consider a scalar (real) quantity defined in a volume ; 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 mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
as . In the Born approximation
Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named ...
the amplitude of the scattered wave corresponding to the scattering vector is proportional to the Fourier transform .[ When the system under study is composed of a number of identical constituents (atoms, molecules, colloidal particles, etc.) each of which has a distribution of mass or charge then the total distribution can be considered the convolution of this function with a set of delta functions.
with the particle positions as before. 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 , so that:
This is clearly the same as Equation () with all particles identical, except that here is shown explicitly as a function of .
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 ; we need not specify whether denotes a time or ]ensemble average
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
. To take this into account we can rewrite Equation () as:
Perfect crystals
In a crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
, the constitutive particles are arranged periodically, with translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.
Analogously, an operato ...
forming a lattice. The crystal structure can be described as a Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 ...
with a group of atoms, called the basis, placed at every lattice point; that is, rystal structure= attice asis If the lattice is infinite and completely regular, the system is a perfect crystal. For such a system, only a set of specific values for can give scattering, and the scattering amplitude for all other values is zero. This set of values forms a lattice, called the reciprocal lattice
Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
, which is the Fourier transform of the real-space crystal lattice.
In principle the scattering factor 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
: and .
The structure factor is then simply the squared modulus of the Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the lattice, and shows the directions in which scattering can have non-zero intensity. At these values of the wave from every lattice point is in phase. The value of the structure factor is the same for all these reciprocal lattice points, and the intensity varies only due to changes in with .
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 m); for neutron scattering by atomic nuclei the unit of scattering length of m is commonly used.
The above discussion uses the wave vectors and . However, crystallography often uses wave vectors and . Therefore, when comparing equations from different sources, the factor may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.
Definition of
In crystallography, the basis and lattice are treated separately. For a perfect crystal the lattice gives the reciprocal lattice
Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
, which determines the positions (angles) of diffracted beams, and the basis gives the structure factor which determines the amplitude and phase of the diffracted beams:
where the sum is over all atoms in the unit cell, are the positional coordinates of the -th atom, and is the scattering factor of the -th atom. The coordinates have the directions and dimensions of the lattice vectors . 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 and (1/2, 1/2, 1/2) is at the body center of the unit cell. defines a reciprocal lattice
Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
point at which corresponds to the real-space plane defined by the Miller indices
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...
(see Bragg's law
In many areas of science, Bragg's law — also known as Wulff–Bragg's condition or Laue–Bragg interference — is a special case of Laue diffraction that gives the angles for coherent scattering of waves from a large crystal lattice. It descr ...
).
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 since then is always an integer. A wave scattered from an atom at (1/2, 0, 0) will be in phase if is even, out of phase if is odd.
Again an alternative view using convolution can be helpful. Since rystal structure= attice asis rystal structure= attice asis that is, scattering eciprocal lattice tructure factor
Examples of in 3-D
Body-centered cubic (BCC)
For the body-centered cubic Bravais lattice (''cI''), we use the points and which leads us to
:
and hence
:
Face-centered cubic (FCC)
The FCC
The Federal Communications Commission (FCC) is an independent agency of the United States government that regulates communications by radio, television, wire, internet, wi-fi, satellite, and cable across the United States. The FCC maintains ju ...
lattice is a Bravais lattice, and its Fourier transform is a body-centered cubic lattice. However to obtain without this shortcut, consider an FCC crystal with one atom at each lattice point as a primitive or simple cubic with a basis of 4 atoms, at the origin and at the three adjacent face centers, , and . 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
Gold is a chemical element; it has chemical symbol Au (from Latin ) and atomic number 79. In its pure form, it is a brightness, bright, slightly orange-yellow, dense, soft, malleable, and ductile metal. Chemically, gold is a transition metal ...
tend to grow in a (111) orientation with a triangular surface symmetry. A zero diffracted intensity for a group of diffracted beams (here, of mixed parity) is called a systematic absence.
Diamond crystal structure
The diamond cubic
In crystallography, the diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, in ...
crystal structure occurs for example diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...
(carbon
Carbon () is a chemical element; it has chemical symbol, symbol C and atomic number 6. It is nonmetallic and tetravalence, tetravalent—meaning that its atoms are able to form up to four covalent bonds due to its valence shell exhibiting 4 ...
), tin
Tin is a chemical element; it has symbol Sn () and atomic number 50. A silvery-colored metal, tin is soft enough to be cut with little force, and a bar of tin can be bent by hand with little effort. When bent, a bar of tin makes a sound, the ...
, and most semiconductors
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping levels ...
. 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 of mixed parity (odd and even values combined) the first (FCC) term is zero, so
* If h, k, ℓ are all even or all odd then the first (FCC) term is 4
** if h+k+ℓ is odd then
** if h+k+ℓ is even and exactly divisible by 4 () then
** if h+k+ℓ is even but not exactly divisible by 4 () the second term is zero and
These points are encapsulated by the following equations:
:
:
where is an integer.
Zincblende crystal structure
The zincblende structure is similar to the diamond structure except that it is a compound of two distinct interpenetrating fcc lattices, rather than all the same element. Denoting the two elements in the compound by and , 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
:
We then arrive at the following result for the structure factor for scattering from a plane :
:
and for scattered intensity,
Hexagonal close-packed (HCP)
In an HCP crystal such as graphite
Graphite () is a Crystallinity, crystalline allotrope (form) of the element carbon. It consists of many stacked Layered materials, layers of graphene, typically in excess of hundreds of layers. Graphite occurs naturally and is the most stable ...
, the two coordinates include the origin and the next plane up the ''c'' axis located at ''c''/2, and hence , which gives us
:
From this it is convenient to define dummy variable , 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
The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period , the reciprocal lattice is an infinite array of points with spacing . In two dimensions, there are only five Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 ...
s. 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 apply with a scattering vector of limited dimensionality and a crystallographic structure factor can be defined in 2-D as .
However, recall that real 2-D crystals such as graphene
Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...
exist in 3-D. The reciprocal lattice of a 2-D hexagonal sheet that exists in 3-D space in the plane is a hexagonal array of lines parallel to the or axis that extend to and intersect any plane of constant 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.
A parallel beam, with wave vector is incident on a square lattice of parameter . The scattered wave is detected at a certain angle, which defines the wave vector of the outgoing beam, (under the assumption of elastic scattering
Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the internal states of the Elementary particle, particles involved stay the same. In the non-relativistic case, where ...
, ). One can equally define the scattering vector and construct the harmonic pattern . In the depicted example, the spacing of this pattern coincides to the distance between particle rows: , so that contributions to the scattering from all particles are in phase (constructive interference). Thus, the total signal in direction is strong, and belongs to the reciprocal lattice. It is easily shown that this configuration fulfills Bragg's law
In many areas of science, Bragg's law — also known as Wulff–Bragg's condition or Laue–Bragg interference — is a special case of Laue diffraction that gives the angles for coherent scattering of waves from a large crystal lattice. It descr ...
.
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
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.
In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one ...
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.
The generally applicable structure factor can be used to include the effect of any imperfection. In crystallography, these effects are treated as separate from the structure factor , 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
For a finite crystal means that the sums in equations 1-7 are now over a finite . 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:
:
This function is shown in the Figure for different values of .
When the scattering from every particle is in phase, which is when the scattering is at a reciprocal lattice point , the sum of the amplitudes must be and so the maxima in intensity are . Taking the above expression for and estimating the limit using, for instance, L'Hôpital's rule
L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application (or repeated application) of the rule often converts an indeterminate form ...
) shows that as seen in the Figure. At the midpoint (by direct evaluation) and the peak width decreases like . In the large limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1-D lattice.
In crystallography when is used, is large, and the formal size effect on diffraction is taken as , which is the same as the expression for above near to the reciprocal lattice points, . Using convolution, we can describe the finite real crystal structure as attice asismath>\times rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname\left(\frac\right) = \Pi\left(\frac\ri ...
, where the rectangular function has a value 1 inside the crystal and 0 outside it. Then rystal structure= attice asis ectangular function that is, scattering eciprocal lattice tructure factor sinc function">sinc.html" ;"title="sinc">sinc function Thus the intensity, which is a delta function of position for the perfect crystal, becomes a function around every point with a maximum , a width , area .
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 by a multiplicative factor, to give[
:
where the disorder is measured by the mean-square displacement of the positions from their positions in a perfect one-dimensional lattice: , i.e., , where is a small (much less than ) random displacement. For disorder of the first kind, each random displacement is independent of the others, and with respect to a perfect lattice. Thus the displacements do not destroy the translational order of the crystal. This has the consequence that for infinite crystals () the structure factor still has delta-function Bragg peaks – the peak width still goes to zero as , with this kind of disorder. However, it does reduce the amplitude of the peaks, and due to the factor of in the exponential factor, it reduces peaks at large much more than peaks at small .
The structure is simply reduced by a 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 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 . 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
In materials science, paracrystalline materials are defined as having short- and medium-range ordering in their lattice (similar to the liquid crystal phases) but lacking crystal-like long-range ordering at least in one direction.
Origin and d ...
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
:
To start with we will consider, for simplicity an infinite crystal, i.e., . 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 planes away, so the above double sum becomes a single sum over pairs of neighbours either side of an atom, at positions and lattice spacings away, times . So, then
:
where is the probability density function for the separation of a pair of planes, 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
:
Finite crystals with disorder of the second kind
For a one-dimensional crystal of size N
:S(q)=1+2\sum_^N\left(1-\frac\right)r^m\cos\left(mqa\right)
where the factor in parentheses comes from the fact the sum is over nearest-neighbour pairs (m=1), next nearest-neighbours (m=2), ... and for a crystal of N planes, there are N-1 pairs of nearest neighbours, N-2 pairs of next-nearest neighbours, etc.
Liquids
In contrast with crystals, liquids have no long-range order
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.
In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one ...
(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
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.
In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one ...
, depending on their density and on the strength of the interaction between particles. Liquids are isotropic, so that, after the averaging operation in Equation (), the structure factor only depends on the absolute magnitude of the scattering vector q = \left , \mathbf \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:
One can obtain an alternative expression for S(q) in terms of the radial distribution function
In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.
If ...
g(r):
Ideal gas
In the limiting case of no interaction, the system is an ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
and the structure factor is completely featureless: S(q) = 1, because there is no correlation between the positions \mathbf_j and \mathbf_k of different particles (they are independent random variables
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of ...
), so the off-diagonal terms in Equation () average to zero: \langle \exp i \mathbf (\mathbf_j - \mathbf_k)rangle = \langle \exp (-i \mathbf \mathbf_j) \rangle \langle \exp (i \mathbf \mathbf_k) \rangle = 0.
High- limit
Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation (), since S(q)-1 is the Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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- 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
In statistical mechanics and thermodynamics the compressibility equation refers to an equation which relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid. It reads:kT\left(\frac\right)=1+\rho \int_V \ ...
:
: \lim _ S(q) = \rho \, k_\mathrmT\, \chi _T = k_\mathrmT \left(\frac\right).
Hard-sphere liquids
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) = \begin
\infty &\text r < 2 R, \\
0 &\text r \geq 2 R.
\end
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. 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
A polymer () is a chemical substance, substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeat unit, repeating subunits derived from one or more species of monomers. Due to their br ...
systems, the general definition () holds; the elementary constituents are now the monomer
A monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or two- or three-dimensional network in a process called polymerization.
Classification
Chemis ...
s 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
The degree of polymerization, or DP, is the number of structural unit, monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymeriza ...
). We can rewrite () as:
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, () can be simplified:[See Teraoka, Section 2.4.4.]
where S_1 (q) is the single-chain structure factor.
See also
*R-factor (crystallography)
In crystallography, the R-factor (sometimes called residual factor or reliability factor or the R-value or RWork) is a measure of the disagreement between the crystallographic model and the experimental X-ray diffraction data - lower the R value lo ...
*Patterson function The Patterson function is used to solve the phase problem in X-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam ...
*Ornstein–Zernike equation In statistical mechanics the Ornstein–Zernike (OZ) equation is an integral equation introduced by Leonard Ornstein and Frits Zernike that relates different correlation functions with each other. Together with a closure relation, it is used to ...
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
The University of York (abbreviated as or ''York'' for Post-nominal letters, post-nominals) is a public Collegiate university, collegiate research university in York, England. Established in 1963, the university has expanded to more than thir ...
.
Definition of F_
by IUCr
Learning Crystallography, from the CSIC
{{Authority control
Crystallography