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| [[File:Cromer-Mann structure factors.svg|thumb|X-ray atomic form factors of oxygen (blue), chlorine (green), Cl<sup>-</sup> (magenta), and K<sup>+</sup> (red); smaller charge distributions have a wider form factor.]]
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| 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 radiation, typically [[X-ray diffraction|X-ray]], [[Electron diffraction|electron]] or [[Neutron diffraction|neutron]]. The common feature of all form factors is that they involve a [[Fourier transform]] of a spatial density distribution of the scattering object from [[real space]] to [[momentum space]] (also known as [[reciprocal space]]). For an object that is spherically symmetric, the spatial density distribution can be expressed as a function of radius, <math>\rho(\mathbf{r})</math>, so that the form factor, <math>f(\mathbf{Q})</math> is defined as :
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| <math>f(\mathbf{Q})=\int \rho(\mathbf{r}) e^{i\mathbf{Q} \cdot \mathbf{r}}\mathrm{d}^3\mathbf{r}</math>,
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| where <math>\rho(\mathbf{r})</math> is the spatial density of the scatterer about the [[center of mass]] of the scatterer (<math>\mathbf{r}=0</math>), and <math>\mathbf{Q}</math> is the [[momentum transfer]]. As a result of the nature of the Fourier transform, the broader the distribution of the scatterer <math>\rho</math> in real space <math>\mathbf{r}</math>, the narrower the distribution of <math>f</math> in <math>\mathbf{Q}</math>; i.e., the faster the decay of the form factor.
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| For crystals, atomic form factors are used to calculate the [[structure factor]] for a given [[Bragg peak]] of a [[crystal]].
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| ==X-ray form factor==
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| [[File:Chlorine scattering factor (real part).svg|thumb|The energy dependence of the [[real part]] of the atomic scattering factor of [[chlorine]].]]
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| X-rays are scattered by the electron cloud of the atom and hence the scattering amplitude of X-rays increases with the [[atomic number]], <math>Z</math>, of the atoms in a sample. As a result X-rays are not very sensitive to light atoms, such as [[hydrogen]] and [[helium]], and there is very little contrast between elements adjacent to each other in the [[periodic table]]. For X-ray scattering, <math>\rho(r)</math> in the above equation is the [[electron]] [[charge density]] 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]].<ref>{{cite book |last= McKie |first= D. |coauthors= C. McKie |title= Essentials of Crystallography |publisher= [[Blackwell Scientific Publications]] |year= 1992 |isbn= 0-632-01574-8}}</ref>
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| In general the X-ray form factor is complex but the imaginary components only become large near an [[absorption edge]]. [[Anomalous X-ray scattering]] 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.
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| ==Electron form factor==
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| The relevant distribution, <math>\rho(r)</math> is the [[Electric potential|potential distribution]] of the atom, and the electron form factor is the Fourier transform of this.<ref>{{cite book |last=Cowley |first=John M. |authorlink=John M. Cowley |title=Diffraction Physics |year=1981 |publisher=[[North-Holland Publishing Company|North-Holland Physics]] Publishing |isbn=0-444-86121-1 |pages=78 }}</ref> The electron form factors are normally calculated from X-ray form factors using the [[Mott-Bethe formula]].<ref>{{cite book |last=De Graef |first=Marc |title=Introduction to Conventional Transmission Electron Microscopy |year=2003 |publisher=[[Cambridge University Press]] |isbn=0-521-62995-0 |pages=113 }}</ref> This formula takes into account both elastic electron-cloud scattering and elastic nuclear scattering.
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| ==Neutron form factor==
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| There are two distinct scattering interactions of [[neutrons]] by [[Atomic nucleus|nuclei]]. Both are used in the investigation structure and dynamics of [[condensed matter]]: they are termed '''nuclear''' (sometimes also termed chemical) and '''magnetic''' scattering.
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| ===Nuclear scattering===
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| Nuclear scattering of the free neutron by the nucleus is mediated by the [[strong nuclear force]]. The [[wavelength]] of thermal (several [[Angstroms]]) 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 [[particle beam|beam]] travel in a [[plane wave]]; for those that undergo nuclear scattering from a nucleus, the nucleus acts as a secondary [[point source]], and [[radiates]] scattered neutrons as a [[spherical wave]]. (Although a quantum phenomenon, this can be visualized in simple classical terms by the [[Huygens–Fresnel principle]].) In this case <math>\rho(r)</math> is the spatial density distribution of the nucleus, which is an infinitesimal point ([[Dirac delta function|delta function]]), with respect to the neutron wavelength. The delta function forms part of the [[Pseudopotential#Fermi_pseudopotential|Fermi pseudopotential]], by which the free neutron and the nuclei interact. The [[Fourier transform#Distributions|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, <math>b</math>, is independent of <math>Q</math>.
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| Since the interaction is nuclear, each isotope has a different scattering amplitude. This Fourier transform is scaled by the [[scattering amplitude|amplitude]] 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 length]], ''b''. Neutron scattering lengths vary erratically between neighbouring elements in the [[periodic table]] and between [[isotopes]] 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.<ref>{{cite book |last= Squires |first= Gordon |title= Introduction to the Theory of Thermal Neutron Scattering |publisher= [[Dover Publications]] |year= 1996 |isbn= 0-486-69447-X | pages=260}}</ref>
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| ===Magnetic scattering===
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| Although neutral, neutrons also have a [[nuclear spin]]. They are a composite [[fermion]] and hence have an associated [[magnetic moment]]. 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 [[atomic orbital|orbitals]] of certain atoms. It is the spatial distribution of these unpaired electrons about the nucleus that is <math>\rho(r)</math> for magnetic scattering.
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| 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.<ref>{{cite book |last= Dobrzynski |first= L. |coauthors= K. Blinowski |title= Neutrons and Solid State Physics|publisher= Ellis Horwood Limited |year= 1994 |isbn= 0-13-617192-3}}</ref> Also, in contrast to nuclear scattering, the magnetic form factor is not isotope dependent, but is dependent on the oxidation state of the atom.
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| ==References==
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| {{reflist}}
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| [[Category:Atomic physics]]
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