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 phases which arise from electromagnetic forces between atoms. More generally, the sub ...
and
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, 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 waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
s) obtained in
X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
,
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
and
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
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 ''â ...
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 ...
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 In condensed matter physics, the dynamic structure factor (or dynamical structure factor) is a mathematical function that contains information about inter-particle correlations and their time evolution. It is a generalization of the structure factor ...
.
Derivation of
Consider the
scattering
Scattering is a term used in physics to describe 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 ...
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 a ...
), 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 Ewald sphere is a geometric construction used in electron, neutron, and X-ray crystallography which demonstrates the relationship between:
:* the wavevector of the incident and diffracted x-ray beams,
:* the diffraction angle for a given ref ...
). 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 r ...
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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s and
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
. 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
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 a ...
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 geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
forming a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
. 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 + n_ ...
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
Crystalline materials (mainly metals and alloys, but also stoichiometric salts and other materials) are made up of solid regions of ordered matter (atoms placed in one of a number of ordered formations called Bravais lattices). These regions are kn ...
. 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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
, 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
, 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
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
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 ''â ...
(see
Bragg's law
In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
).
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 federal government that regulates communications by radio, television, wire, satellite, and cable across the United States. The FCC maintains jurisdiction ...
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 with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile met ...
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
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, including α-tin, the sem ...
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. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
(
carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent
In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with o ...
),
tin
Tin is a chemical element with the symbol Sn (from la, stannum) and atomic number 50. Tin is a silvery-coloured 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, t ...
, and most
semiconductors
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
. 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, â„“ are of mixed parity (odd and even values combined) the first (FCC) term is zero, so
* If h, k, â„“ are all even or all odd then the first (FCC) term is 4
** if h+k+â„“ is odd then
** if h+k+â„“ is even and exactly divisible by 4 (
) then
** if h+k+â„“ 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
Caesium chloride or cesium chloride is the inorganic compound with the formula Cs Cl. This colorless salt is an important source of caesium ions in a variety of niche applications. Its crystal structure forms a major structural type where each ...
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 crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on large ...
, 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 + n_ ...
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 an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure. 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.
![square lattice scattering](https://upload.wikimedia.org/wikipedia/commons/6/65/Square_lattice_scattering.png)
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 kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modi ...
,
). 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 physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
.
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
André Guinier (1 August 1911 – 3 July 2000) was a French physicist who did important work in the field of X-ray diffraction and solid-state physics. He worked at the Conservatoire National des Arts et Métiers, then taught at the University o ...
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 o ...
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'HĂ´pital's rule
In calculus, l'HĂ´pital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
) 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(t) = \Pi(t) =
\left\{\begin{array}{r ...
, 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.html" ;"title="sinc.html" ;"title="sinc">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
: