TheInfoList

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, the atomic form factor, or atomic scattering factor, is a measure of the
scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. The latter is described by the wavefunction : \psi(\mathbf) = ...
of a wave by an isolated atom. The atomic form factor depends on the type of
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 Light or visible light is electromagnetic radiation within the portion of the electromagnetic ...

, which in turn depends on the nature of the incident radiation, typically
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 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency ...
,
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

or
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 behav ...
. The common feature of all form factors is that they involve a
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
of a spatial density distribution of the scattering object from to
momentum space In physics and geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties ...
(also known as
reciprocal space In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fun ...

). For an object with spatial density distribution, $\rho\left(\mathbf\right)$, the form factor, $f\left(\mathbf\right)$, is defined as $f\left(\mathbf\right)=\int \rho\left(\mathbf\right) e^\mathrm^3\mathbf$, where $\rho\left(\mathbf\right)$ is the spatial density of the scatterer about its
center of mass In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
($\mathbf=0$), and $\mathbf$ is the
momentum transfer In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the n ...
. As a result of the nature of the Fourier transform, the broader the distribution of the scatterer $\rho$ in real space $\mathbf$, the narrower the distribution of $f$ in $\mathbf$; i.e., the faster the decay of the form factor. For crystals, atomic form factors are used to calculate the
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 o ...
for a given
Bragg peak to the left, unlike the x-ray beam below. The Bragg peak is a pronounced peak on the ''Bragg curve'' which plots the energy loss of ionizing radiation during its travel through matter. For protons, Alpha particle, α-rays, and other ion rays, t ...
of a
crystal A crystal or crystalline solid is a solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformatio ...

.

# X-ray form factors

X-rays are scattered by the electron cloud of the atom and hence the
scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. The latter is described by the wavefunction : \psi(\mathbf) = ...
of X-rays increases with the
atomic number The atomic number or proton number (symbol ''Z'') of a chemical element In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. ...
, $Z$, of the atoms in a sample. As a result, X-rays are not very sensitive to light atoms, such as
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the che ...

and
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

, and there is very little contrast between elements adjacent to each other in the
periodic table The periodic table, also known as the periodic table of (the) chemical elements, is a tabular display of the chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistry is ...

. For X-ray scattering, $\rho\left(r\right)$ in the above equation is the
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

about the nucleus, and the form factor the Fourier transform of this quantity. The assumption of a spherical distribution is usually good enough for
X-ray crystallography X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract into many specific directions. By measurin ...

. In general the X-ray form factor is complex but the imaginary components only become large near an
absorption edgeAn absorption edge, absorption discontinuity or absorption limit is a sharp discontinuity in the Absorption spectroscopy, absorption spectrum of a substance. These discontinuities occur at Wavelength, wavelengths where the energy of an absorbed photo ...
.
Anomalous X-ray scatteringAnomalous X-ray scattering (AXRS or XRAS) is a non-destructive determination technique within X-ray diffraction X-ray crystallography (XRC) is the experimental science determining the atomic and molecular structure of a crystal A crystal or ...
makes use of the variation of the form factor close to an absorption edge to vary the scattering power of specific atoms in the sample by changing the energy of the incident x-rays hence enabling the extraction of more detailed structural information. Atomic form factor patterns are often represented as a function of the magnitude of the ''scattering vector'' $Q = 4\pi \sin \left(\theta \right) / \lambda$. Here $2\theta$ is the angle between the incident x-ray beam and the detector measuring the scattered intensity, and $\lambda$ is the wavelength of the X-rays. One interpretation of the scattering vector is that it is the ''resolution'' or ''yardstick'' with which the sample is observed. In the range of scattering vectors between $0 < Q < 25$ Å−1, the atomic form factor is well approximated by a sum of Gaussians of the form :$f\left(Q\right) = \sum_^ a_i \exp\left\left(-b_i \left\left(\frac\right\right)^2\right\right) + c$ where the values of ai, bi, and c are tabulated here.

# Electron form factor

The relevant distribution, $\rho\left(r\right)$ is the of the atom, and the electron form factor is the Fourier transform of this. The electron form factors are normally calculated from X-ray form factors using the Mott–Bethe formula. This formula takes into account both elastic electron-cloud scattering and elastic nuclear scattering.

# Neutron form factor

There are two distinct scattering interactions of
neutrons The neutron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass ...

by
nuclei ''Nucleus'' (plural nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...
. Both are used in the investigation structure and dynamics of
condensed matter 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 which arise from electromagnetic forces between atoms. More ge ...
: they are termed nuclear (sometimes also termed chemical) and magnetic scattering.

## Nuclear scattering

Nuclear scattering of the free neutron by the nucleus is mediated by the
strong nuclear force In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
. The
wavelength In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

of thermal (several
ångström The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...

s) and cold neutrons (up to tens of Angstroms) typically used for such investigations is 4-5 orders of magnitude larger than the dimension of the nucleus ( femtometres). The free neutrons in a
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized group ...
travel in a
plane wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

; for those that undergo nuclear scattering from a nucleus, the nucleus acts as a secondary
point source A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources can us ...
, and radiates scattered neutrons as a
spherical wave The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. I ...

. (Although a quantum phenomenon, this can be visualized in simple classical terms by the
Huygens–Fresnel principle The Huygens–Fresnel principle (named after Netherlands, Dutch physicist Christiaan Huygens and France, French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the Far-field diffraction p ...
.) In this case $\rho\left(r\right)$ is the spatial density distribution of the nucleus, which is an infinitesimal point (
delta function File:Dirac function approximation.gif, frame, The Dirac delta function as the limit as a \rightarrow 0 (in the sense of distribution (mathematics), distributions) of the sequence of zero-centered normal distributions \delta_a(x) = \frac \mathrm ...
), with respect to the neutron wavelength. The delta function forms part of the Fermi pseudopotential, by which the free neutron and the nuclei interact. The Fourier transform of a delta function is unity; therefore, it is commonly said that neutrons "do not have a form factor;" i.e., the scattered amplitude, $b$, is independent of $Q$. Since the interaction is nuclear, each isotope has a different scattering amplitude. This Fourier transform is scaled by the
amplitude The amplitude of a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Period, a descriptor for ...
of the spherical wave, which has dimensions of length. Hence, the amplitude of scattering that characterizes the interaction of a neutron with a given isotope is termed the
scattering lengthThe scattering length in quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundat ...
, ''b''. Neutron scattering lengths vary erratically between neighbouring elements in the
periodic table The periodic table, also known as the periodic table of (the) chemical elements, is a tabular display of the chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistry is ...

and between
isotopes Isotopes are two or more types of atoms that have the same atomic number 300px, The Rutherford–Bohr model of the hydrogen atom () or a hydrogen-like ion (). In this model it is an essential feature that the photon energy (or frequency) of ...
of the same element. They may only be determined experimentally, since the theory of nuclear forces is not adequate to calculate or predict ''b'' from other properties of the nucleus.

## Magnetic scattering

Although neutral, neutrons also have a
nuclear spin In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a nuclear model, model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The f ...
. They are a composite
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
and hence have an associated
magnetic moment The magnetic moment is the magnetic strength and orientation of a or other object that produces a . Examples of objects that have magnetic moments include: loops of (such as s), permanent magnets, s (such as s), various s, and many astronomical ...

. In neutron scattering from condensed matter, magnetic scattering refers to the interaction of this moment with the magnetic moments arising from unpaired electrons in the outer orbitals of certain atoms. It is the spatial distribution of these unpaired electrons about the nucleus that is $\rho\left(r\right)$ for magnetic scattering. Since these orbitals are typically of a comparable size to the wavelength of the free neutrons, the resulting form factor resembles that of the X-ray form factor. However, this neutron-magnetic scattering is only from the outer electrons, rather than being heavily weighted by the core electrons, which is the case for X-ray scattering. Hence, in strong contrast to the case for nuclear scattering, the scattering object for magnetic scattering is far from a point source; it is still more diffuse than the effective size of the source for X-ray scattering, and the resulting Fourier transform (the magnetic form factor) decays more rapidly than the X-ray form factor. Also, in contrast to nuclear scattering, the magnetic form factor is not isotope dependent, but is dependent on the oxidation state of the atom.

# References

{{reflist Atomic physics