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Structure factor

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In <a href="Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> and <a href="Crystallography" title="Crystallography">crystallography</a>, 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 (<a href="Interference_pattern" class="mw-redirect" title="Interference pattern">interference patterns</a>) obtained in <a href="X-ray_diffraction" title="X-ray diffraction">X-ray</a>, <a href="Electron_diffraction" title="Electron diffraction">electron</a> and <a href="Neutron_diffraction" title="Neutron diffraction">neutron</a> <a href="Diffraction" title="Diffraction">diffraction</a> experiments.

Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

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</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy">; 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

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</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hk\ell )}">

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<annotation encoding="application/x-tex">{\displaystyle (hk\ell )}</annotation>

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</math><img src="./c7197952a6d05a7212c55f437a5ca3a575f745aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.329ex; height:2.843ex;" alt="{\displaystyle (hk\ell )}" loading="lazy"> planes of the crystal (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hk\ell )}">

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<annotation encoding="application/x-tex">{\displaystyle (hk\ell )}</annotation>

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</math><img src="./c7197952a6d05a7212c55f437a5ca3a575f745aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.329ex; height:2.843ex;" alt="{\displaystyle (hk\ell )}" loading="lazy"> are the <a href="Miller_index" title="Miller index">Miller indices</a> of the planes) to that produced by a single scattering unit at the vertices of the <a href="Crystal_structure" title="Crystal structure">primitive unit cell</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

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</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> is not a special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

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</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

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</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy"> gives the scattering intensity, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

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</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> gives the amplitude. It is the <a href="Modulus_squared" class="mw-redirect" title="Modulus squared">modulus squared</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F_{hk\ell }|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle |F_{hk\ell }|^{2}}</annotation>

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</math><img src="./f1f119fdb7dd384eceee4cc683cd697ce20e316b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.564ex; height:3.343ex;" alt="{\displaystyle |F_{hk\ell }|^{2}}" loading="lazy"> that gives the scattering intensity. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

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</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> is defined for a perfect crystal, and is used in crystallography, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

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</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy"> is most useful for disordered systems. For partially ordered systems such as <a href="Crystallization_of_polymers" title="Crystallization of polymers">crystalline polymers</a> 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 <a href="Dynamic_structure_factor" title="Dynamic structure factor">dynamic structure factor</a>.

<div class="mw-heading mw-heading2"><h2 id="Derivation_of_S(q)">Derivation of S(q)</h2></div>

Consider the <a href="Scattering" title="Scattering">scattering</a> of a beam of wavelength <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> by an assembly of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> particles or atoms stationary at positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}</annotation>

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</math><img src="./4ffed965a5804edb2dfab7b050b4d87f11a7bc73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.182ex; height:2.843ex;" alt="{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}" loading="lazy">. Assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (<a href="Born_approximation" title="Born approximation">Born approximation</a>), and absorption, refraction and multiple scattering can be neglected (<a href="Kinematic_diffraction" title="Kinematic diffraction">kinematic diffraction</a>). The direction of any scattered wave is defined by its scattering vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} =\mathbf {k_{s}} -\mathbf {k_{o}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =\mathbf {k_{s}} -\mathbf {k_{o}} }</annotation>

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</math><img src="./b9d856f55a8641f19e59b0787908a0fefacf9e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.333ex; height:2.509ex;" alt="{\displaystyle \mathbf {q} =\mathbf {k_{s}} -\mathbf {k_{o}} }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k_{s}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {k_{s}} }</annotation>

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</math><img src="./847deed2b8b174f57f1916eb437e39a045ea2a99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle \mathbf {k_{s}} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k_{o}} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {k_{o}} }</annotation>

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</math><img src="./e4750efe53dac52ea037be87a49c384c6b0ec948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.588ex; height:2.509ex;" alt="{\displaystyle \mathbf {k_{o}} }" loading="lazy"> ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {k_{s}} |=|\mathbf {k_{0}} |=2\pi /\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {k_{s}} |=|\mathbf {k_{0}} |=2\pi /\lambda }</annotation>

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</math><img src="./93a6e82a424a479b35018f18d061ef33f4f65ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.774ex; height:2.843ex;" alt="{\displaystyle |\mathbf {k_{s}} |=|\mathbf {k_{0}} |=2\pi /\lambda }" loading="lazy">) are the scattered and incident beam <a href="Wavevector" class="mw-redirect" title="Wavevector">wavevectors</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is the angle between them. For elastic scattering, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {k} _{s}|=|\mathbf {k_{o}} |}">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {k} _{s}|=|\mathbf {k_{o}} |}</annotation>

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</math><img src="./a69b0e650a697a5a2502445d396f0dcdd433252a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.689ex; height:2.843ex;" alt="{\displaystyle |\mathbf {k} _{s}|=|\mathbf {k_{o}} |}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=|\mathbf {q} |={{\frac {4\pi }{\lambda }}\sin(\theta /2)}}">

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<annotation encoding="application/x-tex">{\displaystyle q=|\mathbf {q} |={{\frac {4\pi }{\lambda }}\sin(\theta /2)}}</annotation>

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</math><img src="./66d2feaf0ef6b7df59bb0313b114ee220ba0db91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.774ex; height:5.343ex;" alt="{\displaystyle q=|\mathbf {q} |={{\frac {4\pi }{\lambda }}\sin(\theta /2)}}" loading="lazy">, limiting the possible range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy"> (see <a href="Ewald_sphere" class="mw-redirect" title="Ewald sphere">Ewald sphere</a>). The amplitude and phase of this scattered wave will be the vector sum of the scattered waves from all the atoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{s}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{s}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}}</annotation>

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</math><img src="./8edce4863d1907d867a14228ae6869e2fe254eb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.626ex; height:7.676ex;" alt="{\displaystyle \Psi _{s}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}}" loading="lazy"> <a href="#cite_note-Warren-1">[1]</a><a href="#cite_note-2">[2]</a>

For an assembly of atoms, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}}">

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</math><img src="./acc195ab3f9d65994b47774eb013601d09217aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.843ex;" alt="{\displaystyle f_{j}}" loading="lazy"> is the <a href="Atomic_form_factor" title="Atomic form factor">atomic form factor</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {q} )=\Psi _{s}(\mathbf {q} )\times \Psi _{s}^{*}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\times \sum _{k=1}^{N}f_{k}\mathrm {e} ^{i\mathbf {q} \cdot \mathbf {R} _{k}}=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}">

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</msub>

<msub>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(\mathbf {q} )=\Psi _{s}(\mathbf {q} )\times \Psi _{s}^{*}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\times \sum _{k=1}^{N}f_{k}\mathrm {e} ^{i\mathbf {q} \cdot \mathbf {R} _{k}}=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}</annotation>

</semantics>

</math><img src="./d519270dc3ecba04e05f02aee71b86c79e369c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.819ex; height:7.676ex;" alt="{\displaystyle I(\mathbf {q} )=\Psi _{s}(\mathbf {q} )\times \Psi _{s}^{*}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\times \sum _{k=1}^{N}f_{k}\mathrm {e} ^{i\mathbf {q} \cdot \mathbf {R} _{k}}=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}" loading="lazy"></td> <td></td> <td class="nowrap">1</td></tr></tbody></table>

The structure factor is defined as this intensity normalized by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\sum _{j=1}^{N}f_{j}^{2}}">

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</munderover>

</mrow>

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</msubsup>

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<annotation encoding="application/x-tex">{\displaystyle 1/\sum _{j=1}^{N}f_{j}^{2}}</annotation>

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</math><img src="./89b8943e244df607ccd3ce1e4642388c6ea39a77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:8.829ex; height:7.676ex;" alt="{\displaystyle 1/\sum _{j=1}^{N}f_{j}^{2}}" loading="lazy"> <a href="#cite_note-3">[3]</a>

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<mfrac>

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</munderover>

<msub>

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</msub>

<msub>

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</msub>

<msup>

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</mrow>

<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{\sum _{j=1}^{N}f_{j}^{2}}}\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}</annotation>

</semantics>

</math><img src="./09b482240670abe0d7244ba405903a01172b3384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.567ex; height:7.843ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{\sum _{j=1}^{N}f_{j}^{2}}}\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}" loading="lazy"></td> <td></td> <td class="nowrap">2</td></tr></tbody></table>

If all the atoms are identical, then Equation (<a href="#math_1">1</a>) becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {q} )=f^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

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</munderover>

</mrow>

</mrow>

</munderover>

<msup>

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</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(\mathbf {q} )=f^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}</annotation>

</semantics>

</math><img src="./8973af2fb979b8dee035d3cf8380dff17303e414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.275ex; height:7.676ex;" alt="{\displaystyle I(\mathbf {q} )=f^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{N}f_{j}^{2}=Nf^{2}}">

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</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{N}f_{j}^{2}=Nf^{2}}</annotation>

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</math><img src="./89b242d21939d62143c8ab0664d52147029aa869.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.654ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{N}f_{j}^{2}=Nf^{2}}" loading="lazy"> so

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<mfrac>

</mfrac>

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</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msup>

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<msub>

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<msub>

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</msub>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}</annotation>

</semantics>

</math><img src="./b076098ace2c52494c7e76885d346a81f5679231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.127ex; height:7.676ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}}" loading="lazy"></td> <td></td> <td class="nowrap">3</td></tr></tbody></table>

Another useful simplification is if the material is isotropic, like a powder or a simple liquid. In that case, the intensity depends on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=|\mathbf {q} |}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle q=|\mathbf {q} |}</annotation>

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</math><img src="./56116ed0e5c6a710eae74c3e26a60767edcae57b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle q=|\mathbf {q} |}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{jk}=|\mathbf {r} _{j}-\mathbf {r} _{k}|}">

<msub>

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</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{jk}=|\mathbf {r} _{j}-\mathbf {r} _{k}|}</annotation>

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</math><img src="./12f51996481204385d9e9f324a2196d51bc1d97b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.25ex; height:3.009ex;" alt="{\displaystyle r_{jk}=|\mathbf {r} _{j}-\mathbf {r} _{k}|}" loading="lazy">. In three dimensions, Equation (<a href="#math_2">2</a>) then simplifies to the Debye scattering equation:<a href="#cite_note-Warren-1">[1]</a>

</mrow>

<mfrac>

<mrow>

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</munderover>

<msub>

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</msub>

<msub>

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</msub>

<mfrac>

<mrow>

<msub>

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<mrow>

<msub>

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</msub>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{\sum _{j=1}^{N}f_{j}^{2}}}\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}{\frac {\sin(qr_{jk})}{qr_{jk}}}}</annotation>

</semantics>

</math><img src="./a6ee63e1c078639186ec94d25b45c566cc89cb4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.418ex; height:7.843ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{\sum _{j=1}^{N}f_{j}^{2}}}\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}f_{k}{\frac {\sin(qr_{jk})}{qr_{jk}}}}" loading="lazy"></td> <td></td> <td class="nowrap">4</td></tr></tbody></table>

An alternative derivation gives good insight, but uses <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> and <a href="Convolution" title="Convolution">convolution</a>. To be general, consider a scalar (real) quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {r} )}</annotation>

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</math><img src="./b691651163c74d532577e18847c7cdd92c7c1b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.297ex; height:2.843ex;" alt="{\displaystyle \phi (\mathbf {r} )}" loading="lazy"> defined in a volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">; 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 <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \psi (\mathbf {q} )=\int _{V}\phi (\mathbf {r} )\exp(-i\mathbf {q} \cdot \mathbf {r} )\,\mathrm {d} \mathbf {r} }">

</mrow>

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \psi (\mathbf {q} )=\int _{V}\phi (\mathbf {r} )\exp(-i\mathbf {q} \cdot \mathbf {r} )\,\mathrm {d} \mathbf {r} }</annotation>

</semantics>

</math><img src="./b46c5994b7cb79c83d0a9396a2f3642169ea9776.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.452ex; height:3.176ex;" alt="{\displaystyle \textstyle \psi (\mathbf {q} )=\int _{V}\phi (\mathbf {r} )\exp(-i\mathbf {q} \cdot \mathbf {r} )\,\mathrm {d} \mathbf {r} }" loading="lazy">. In the <a href="Born_approximation" title="Born approximation">Born approximation</a> the amplitude of the scattered wave corresponding to the scattering vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy"> is proportional to the Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \psi (\mathbf {q} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \psi (\mathbf {q} )}</annotation>

</semantics>

</math><img src="./8a074b586a67adb5a61fddc7da54d10da2645c57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.738ex; height:2.843ex;" alt="{\displaystyle \textstyle \psi (\mathbf {q} )}" loading="lazy">.<a href="#cite_note-Warren-1">[1]</a> When the system under study is composed of a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of identical constituents (atoms, molecules, colloidal particles, etc.) each of which has a distribution of mass or charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {r} )}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {r} )}</annotation>

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</math><img src="./bb64b30ae67dec8ef9cb06c1d3537f00b9a7efed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.19ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {r} )}" loading="lazy"> then the total distribution can be considered the convolution of this function with a set of <a href="Dirac_delta_function" title="Dirac delta function">delta functions</a>.

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</munderover>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {r} )=\sum _{j=1}^{N}f(\mathbf {r} -\mathbf {R} _{j})=f(\mathbf {r} )\ast \sum _{j=1}^{N}\delta (\mathbf {r} -\mathbf {R} _{j}),}</annotation>

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</math><img src="./80e0fc77f120aee7d6a41b6ba0e3c13446febad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.666ex; height:7.676ex;" alt="{\displaystyle \phi (\mathbf {r} )=\sum _{j=1}^{N}f(\mathbf {r} -\mathbf {R} _{j})=f(\mathbf {r} )\ast \sum _{j=1}^{N}\delta (\mathbf {r} -\mathbf {R} _{j}),}" loading="lazy"></td> <td></td> <td class="nowrap">5</td></tr></tbody></table>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}</annotation>

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</math><img src="./4ffed965a5804edb2dfab7b050b4d87f11a7bc73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.182ex; height:2.843ex;" alt="{\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \psi (\mathbf {q} )=f(\mathbf {q} )\times \sum _{j=1}^{N}\exp(-i\mathbf {q} \cdot \mathbf {R} _{j})}">

</mrow>

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</munderover>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \psi (\mathbf {q} )=f(\mathbf {q} )\times \sum _{j=1}^{N}\exp(-i\mathbf {q} \cdot \mathbf {R} _{j})}</annotation>

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</math><img src="./34fb9ef4970dd09026e0c468a10bf114daf55bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.012ex; height:3.843ex;" alt="{\displaystyle \textstyle \psi (\mathbf {q} )=f(\mathbf {q} )\times \sum _{j=1}^{N}\exp(-i\mathbf {q} \cdot \mathbf {R} _{j})}" loading="lazy">, so that:

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<msup>

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<mrow>

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<msup>

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<msub>

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</munderover>

<msup>

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<msup>

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<mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle I(\mathbf {q} )=\left|f(\mathbf {q} )\right|^{2}\times \left(\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right)\times \left(\sum _{k=1}^{N}\mathrm {e} ^{i\mathbf {q} \cdot \mathbf {R} _{k}}\right)=\left|f(\mathbf {q} )\right|^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}.}</annotation>

</semantics>

</math><img src="./778e2fbcc756991e82202848ae74059d54243628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:77.142ex; height:7.676ex;" alt="{\displaystyle I(\mathbf {q} )=\left|f(\mathbf {q} )\right|^{2}\times \left(\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right)\times \left(\sum _{k=1}^{N}\mathrm {e} ^{i\mathbf {q} \cdot \mathbf {R} _{k}}\right)=\left|f(\mathbf {q} )\right|^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}.}" loading="lazy"></td> <td></td> <td class="nowrap">6</td></tr></tbody></table>

This is clearly the same as Equation (<a href="#math_1">1</a>) with all particles identical, except that here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is shown explicitly as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy">.

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \langle I(\mathbf {q} )\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \langle I(\mathbf {q} )\rangle }</annotation>

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</math><img src="./3ea7f906a3178f2ebb0538b27da8a69f97145438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle \textstyle \langle I(\mathbf {q} )\rangle }" loading="lazy">; we need not specify whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot \rangle }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot \rangle }</annotation>

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</math><img src="./47e28d0cdb6d5e1c9af01324e09276f06e4d44c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \langle \cdot \rangle }" loading="lazy"> denotes a time or <a href="Ensemble_average" class="mw-redirect" title="Ensemble average">ensemble average</a>. To take this into account we can rewrite Equation (<a href="#math_3">3</a>) as:

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left\langle \sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle .}</annotation>

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</math><img src="./6c125cb9ab43b59385153a29eea31cee30756b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.908ex; height:7.676ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left\langle \sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle .}" loading="lazy"></td> <td></td> <td class="nowrap">7</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Perfect_crystals">Perfect crystals</h2></div>

In a <a href="Crystal" title="Crystal">crystal</a>, the constitutive particles are arranged periodically, with <a href="Translational_symmetry" title="Translational symmetry">translational symmetry</a> forming a <a href="Crystal_lattice" class="mw-redirect" title="Crystal lattice">lattice</a>. The crystal structure can be described as a <a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a> with a group of atoms, called the basis, placed at every lattice point; that is, [crystal structure] = [lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

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<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

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</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"> [basis]. If the lattice is infinite and completely regular, the system is a <a href="Perfect_crystal" title="Perfect crystal">perfect crystal</a>. For such a system, only a set of specific values for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy"> can give scattering, and the scattering amplitude for all other values is zero. This set of values forms a lattice, called the <a href="Reciprocal_lattice" title="Reciprocal lattice">reciprocal lattice</a>, which is the Fourier transform of the real-space crystal lattice.

In principle the scattering factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

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</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy"> 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 (<a href="#math_1">1</a>) can be written as

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<annotation encoding="application/x-tex">{\displaystyle I(\mathbf {q} )=f^{2}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}}</annotation>

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</math><img src="./a32e086d9e1b1fe19a790df67364ccd9b032d8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.66ex; height:8.343ex;" alt="{\displaystyle I(\mathbf {q} )=f^{2}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}}</annotation>

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</math><img src="./aefb96aaaccecc577b313e6e2eebcb366b70bf78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.512ex; height:8.343ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}}" loading="lazy">.</dd></dl>

The structure factor is then simply the squared modulus of the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the lattice, and shows the directions in which scattering can have non-zero intensity. At these values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

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</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Units">Units</h3></div>

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<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times 10^{-15}}">

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<annotation encoding="application/x-tex">{\displaystyle \times 10^{-15}}</annotation>

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</math><img src="./837f026844d86fdca8f4acee384259e863d34db5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.288ex; height:2.676ex;" alt="{\displaystyle \times 10^{-15}}" loading="lazy"> m); for neutron scattering by atomic nuclei the unit of scattering length of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-14}}">

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<annotation encoding="application/x-tex">{\displaystyle 10^{-14}}</annotation>

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</math><img src="./92a08bbb40a2400004de406f706eb549de0d1e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.48ex; height:2.676ex;" alt="{\displaystyle 10^{-14}}" loading="lazy"> m is commonly used.

The above discussion uses the wave vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {k} |=2\pi /\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {k} |=2\pi /\lambda }</annotation>

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</math><img src="./fe26b5317fcbe0b9b442212bb76ff41202a42b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.815ex; height:2.843ex;" alt="{\displaystyle |\mathbf {k} |=2\pi /\lambda }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {q} |=4\pi \sin \theta /\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {q} |=4\pi \sin \theta /\lambda }</annotation>

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</math><img src="./8e5d1c7331eff1db11d6ffef3bd63e440ff2031d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.54ex; height:2.843ex;" alt="{\displaystyle |\mathbf {q} |=4\pi \sin \theta /\lambda }" loading="lazy">. However, crystallography often uses wave vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {s} |=1/\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {s} |=1/\lambda }</annotation>

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</math><img src="./d144c15c7af575e569591c6351b434c55ec23209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.128ex; height:2.843ex;" alt="{\displaystyle |\mathbf {s} |=1/\lambda }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {g} |=2\sin \theta /\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {g} |=2\sin \theta /\lambda }</annotation>

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</math><img src="./8f9d0e53e1414b10eebb2d166258e20837134543.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.129ex; height:2.843ex;" alt="{\displaystyle |\mathbf {g} |=2\sin \theta /\lambda }" loading="lazy">. Therefore, when comparing equations from different sources, the factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.

<div class="mw-heading mw-heading3"><h3 id="Definition_of_Fhkl">Definition of Fhkl</h3></div>

In crystallography, the basis and lattice are treated separately. For a perfect crystal the lattice gives the <a href="Reciprocal_lattice" title="Reciprocal lattice">reciprocal lattice</a>, which determines the positions (angles) of diffracted beams, and the basis gives the structure factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hkl}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hkl}}</annotation>

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</math><img src="./40a256c8688cd3af0a00fb8e6dbabf5554b3bd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.02ex; height:2.509ex;" alt="{\displaystyle F_{hkl}}" loading="lazy"> which determines the amplitude and phase of the diffracted beams:

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]},}</annotation>

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</math><img src="./16ee72ccd9ba16412dd9d2a2c62a6041d740f2fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.52ex; height:7.676ex;" alt="{\displaystyle F_{hk\ell }=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]},}" loading="lazy"></td> <td></td> <td class="nowrap">8</td></tr></tbody></table>

where the sum is over all atoms in the unit cell, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},y_{j},z_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j},y_{j},z_{j}}</annotation>

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</math><img src="./1f5c76ffcd87bb9ca49bd3f372cf6b5168bd78fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.347ex; height:2.343ex;" alt="{\displaystyle x_{j},y_{j},z_{j}}" loading="lazy"> are the positional coordinates of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">-th atom, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{j}}</annotation>

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</math><img src="./acc195ab3f9d65994b47774eb013601d09217aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.843ex;" alt="{\displaystyle f_{j}}" loading="lazy"> is the scattering factor of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">-th atom.<a href="#cite_note-4">[4]</a> The coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},y_{j},z_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j},y_{j},z_{j}}</annotation>

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</math><img src="./1f5c76ffcd87bb9ca49bd3f372cf6b5168bd78fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.347ex; height:2.343ex;" alt="{\displaystyle x_{j},y_{j},z_{j}}" loading="lazy"> have the directions and dimensions of the lattice vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }">

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</math><img src="./98735198279ea2237902abe353cfc8156f2eea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.041ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }" loading="lazy">. 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation>

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</math><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"> and (1/2, 1/2, 1/2) is at the body center of the unit cell. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hkl)}">

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<annotation encoding="application/x-tex">{\displaystyle (hkl)}</annotation>

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</math><img src="./926597b1a24ff22533b8eebfd578abdddf0b8a2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle (hkl)}" loading="lazy"> defines a <a href="Reciprocal_lattice" title="Reciprocal lattice">reciprocal lattice</a> point at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h\mathbf {a^{*}} ,k\mathbf {b^{*}} ,l\mathbf {c^{*}} )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (h\mathbf {a^{*}} ,k\mathbf {b^{*}} ,l\mathbf {c^{*}} )}</annotation>

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</math><img src="./07c2dd6d71aa8cb751011eac45268462e3df3996.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.626ex; height:2.843ex;" alt="{\displaystyle (h\mathbf {a^{*}} ,k\mathbf {b^{*}} ,l\mathbf {c^{*}} )}" loading="lazy"> which corresponds to the real-space plane defined by the <a href="Miller_index" title="Miller index">Miller indices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hkl)}">

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<annotation encoding="application/x-tex">{\displaystyle (hkl)}</annotation>

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</math><img src="./926597b1a24ff22533b8eebfd578abdddf0b8a2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle (hkl)}" loading="lazy"> (see <a href="Bragg's_law" title="Bragg's law">Bragg's law</a>).

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

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</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle hk\ell }">

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<annotation encoding="application/x-tex">{\displaystyle hk\ell }</annotation>

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</math><img src="./d326d35816e810e335242d7a63aa075ff8940339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.52ex; height:2.176ex;" alt="{\displaystyle hk\ell }" loading="lazy"> since then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hx_{j}+ky_{j}+\ell z_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle (hx_{j}+ky_{j}+\ell z_{j})}</annotation>

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</math><img src="./a35ffc75a0716de5781aa0db45f0c5d9dcc327cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.289ex; height:3.009ex;" alt="{\displaystyle (hx_{j}+ky_{j}+\ell z_{j})}" loading="lazy"> is always an integer. A wave scattered from an atom at (1/2, 0, 0) will be in phase if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> is even, out of phase if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> is odd.

Again an alternative view using convolution can be helpful. Since [crystal structure] = [lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"> [basis], <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">[crystal structure] = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">[lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./ed04111e72d462a6d7c79b52c118d5741b24489f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.735ex; height:2.176ex;" alt="{\displaystyle \times {\mathcal {F}}}" loading="lazy">[basis]; that is, scattering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto }</annotation>

</semantics>

</math><img src="./0e3a55007ba2f092d6cafe6d33598e0608b81150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.676ex;" alt="{\displaystyle \propto }" loading="lazy"> [reciprocal lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>

</semantics>

</math><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"> [structure factor].

<div class="mw-heading mw-heading3"><h3 id="Examples_of_Fhkl_in_3-D">Examples of Fhkl in 3-D</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Body-centered_cubic_(BCC)">Body-centered cubic (BCC)</h4></div>

For the body-centered cubic Bravais lattice (cI), we use the points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0)}</annotation>

</semantics>

</math><img src="./28c6fd55d5621fd95ca93549660fbb355fd9bd22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,0)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}</annotation>

</semantics>

</math><img src="./a053970be40a2729265ad03f207299b888458eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.852ex; height:3.509ex;" alt="{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}" loading="lazy"> which leads us to

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=\sum _{j}f_{j}e^{-2\pi i(hx_{j}+ky_{j}+\ell z_{j})}=f\left[1+\left(e^{-i\pi }\right)^{h+k+\ell }\right]=f\left[1+(-1)^{h+k+\ell }\right]}</annotation>

</semantics>

</math><img src="./04e041de2449b496dd9c30c6713e3a989ca5a7c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.247ex; height:6.343ex;" alt="{\displaystyle F_{hk\ell }=\sum _{j}f_{j}e^{-2\pi i(hx_{j}+ky_{j}+\ell z_{j})}=f\left[1+\left(e^{-i\pi }\right)^{h+k+\ell }\right]=f\left[1+(-1)^{h+k+\ell }\right]}" loading="lazy"></dd></dl>

and hence

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }={\begin{cases}2f,&h+k+\ell ={\text{even}}\\0,&h+k+\ell ={\text{odd}}\end{cases}}}</annotation>

</semantics>

</math><img src="./2affd4832dca490fb4842cc8149e9480f2e2ed82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.103ex; height:6.176ex;" alt="{\displaystyle F_{hk\ell }={\begin{cases}2f,&h+k+\ell ={\text{even}}\\0,&h+k+\ell ={\text{odd}}\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Face-centered_cubic_(FCC)">Face-centered cubic (FCC)</h4></div>

The <a href="Face-centred_cubic" class="mw-redirect" title="Face-centred cubic">FCC</a> lattice is a Bravais lattice, and its Fourier transform is a body-centered cubic lattice. However to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }}</annotation>

</semantics>

</math><img src="./51f775ff9aa7b2d6a537af8627265f8efa039961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.216ex; height:2.509ex;" alt="{\displaystyle F_{hk\ell }}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},y_{j},z_{j}=(0,0,0)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j},y_{j},z_{j}=(0,0,0)}</annotation>

</semantics>

</math><img src="./26703a0f3bca98e3cf070a8bb52ceca6129f3daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.81ex; height:3.009ex;" alt="{\displaystyle x_{j},y_{j},z_{j}=(0,0,0)}" loading="lazy"> and at the three adjacent face centers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j},y_{j},z_{j}=\left({\frac {1}{2}},{\frac {1}{2}},0\right)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j},y_{j},z_{j}=\left({\frac {1}{2}},{\frac {1}{2}},0\right)}</annotation>

</semantics>

</math><img src="./a9468782eb1695886f24c42573066c45712b713d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.094ex; height:6.176ex;" alt="{\displaystyle x_{j},y_{j},z_{j}=\left({\frac {1}{2}},{\frac {1}{2}},0\right)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0,{\frac {1}{2}},{\frac {1}{2}}\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

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<mfrac>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(0,{\frac {1}{2}},{\frac {1}{2}}\right)}</annotation>

</semantics>

</math><img src="./e16e1d0b58d1fad68e3a763d66ef393c491dbaf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.649ex; height:6.176ex;" alt="{\displaystyle \left(0,{\frac {1}{2}},{\frac {1}{2}}\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1}{2}},0,{\frac {1}{2}}\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{2}},0,{\frac {1}{2}}\right)}</annotation>

</semantics>

</math><img src="./11c14daf05800675d60e640d107f49b84679b643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.649ex; height:6.176ex;" alt="{\displaystyle \left({\frac {1}{2}},0,{\frac {1}{2}}\right)}" loading="lazy">. Equation (<a href="#math_8">8</a>) becomes

<msub>

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</msub>

</mrow>

</mrow>

</munderover>

<msup>

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<msub>

</mrow>

</msub>

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</msub>

<msub>

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</msup>

<mrow>

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<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

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<mrow>

<mrow>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=f\sum _{j=1}^{4}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=f\left[1+\mathrm {e} ^{[-i\pi (h+k)]}+\mathrm {e} ^{[-i\pi (k+\ell )]}+\mathrm {e} ^{[-i\pi (h+\ell )]}\right]=f\left[1+(-1)^{h+k}+(-1)^{k+\ell }+(-1)^{h+\ell }\right]}</annotation>

</semantics>

</math><img src="./04c11968ee184e993a75bdbc1eb6ad5c6091b351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:113.923ex; height:7.676ex;" alt="{\displaystyle F_{hk\ell }=f\sum _{j=1}^{4}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=f\left[1+\mathrm {e} ^{[-i\pi (h+k)]}+\mathrm {e} ^{[-i\pi (k+\ell )]}+\mathrm {e} ^{[-i\pi (h+\ell )]}\right]=f\left[1+(-1)^{h+k}+(-1)^{k+\ell }+(-1)^{h+\ell }\right]}" loading="lazy"></dd></dl>

with the result

<msub>

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</msub>

<mrow>

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<mtd>

</mstyle>

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<mtd>

<mtext>mixed parity</mtext>

</mstyle>

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</mtable>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }={\begin{cases}4f,&h,k,\ell \ \ {\mbox{all even or all odd}}\\0,&h,k,\ell \ \ {\mbox{mixed parity}}\end{cases}}}</annotation>

</semantics>

</math><img src="./5d4cc451f8098a7d95af55393d9b5a7b498f480b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.61ex; height:6.176ex;" alt="{\displaystyle F_{hk\ell }={\begin{cases}4f,&h,k,\ell \ \ {\mbox{all even or all odd}}\\0,&h,k,\ell \ \ {\mbox{mixed parity}}\end{cases}}}" loading="lazy"></dd></dl>

The most intense diffraction peak from a material that crystallizes in the FCC structure is typically the (111). Films of FCC materials like <a href="Gold" title="Gold">gold</a> tend to grow in a (111) orientation with a triangular surface symmetry. A zero diffracted intensity for a group of diffracted beams (here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h,k,\ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h,k,\ell }</annotation>

</semantics>

</math><img src="./3057e78e129593dc65dbd3ab02f5c21c7bb1e16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\displaystyle h,k,\ell }" loading="lazy"> of mixed parity) is called a systematic absence.

<div class="mw-heading mw-heading4"><h4 id="Diamond_crystal_structure">Diamond crystal structure</h4></div>

The <a href="Diamond_cubic" title="Diamond cubic">diamond cubic</a> crystal structure occurs for example <a href="Diamond" title="Diamond">diamond</a> (<a href="Carbon" title="Carbon">carbon</a>), <a href="Tin" title="Tin">tin</a>, and most <a href="Semiconductors" class="mw-redirect" title="Semiconductors">semiconductors</a>. 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

<mtr>

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<mfrac>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{j},y_{j},z_{j}=&(0,\ 0,\ 0)&\left({\frac {1}{2}},\ {\frac {1}{2}},\ 0\right)\ &\left(0,\ {\frac {1}{2}},\ {\frac {1}{2}}\right)&\left({\frac {1}{2}},\ 0,\ {\frac {1}{2}}\right)\\&\left({\frac {1}{4}},\ {\frac {1}{4}},\ {\frac {1}{4}}\right)&\left({\frac {3}{4}},\ {\frac {3}{4}},\ {\frac {1}{4}}\right)\ &\left({\frac {1}{4}},\ {\frac {3}{4}},\ {\frac {3}{4}}\right)&\left({\frac {3}{4}},\ {\frac {1}{4}},\ {\frac {3}{4}}\right)\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./531e9433353c2d14060601808224e7cc834ce61c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:72.394ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}x_{j},y_{j},z_{j}=&(0,\ 0,\ 0)&\left({\frac {1}{2}},\ {\frac {1}{2}},\ 0\right)\ &\left(0,\ {\frac {1}{2}},\ {\frac {1}{2}}\right)&\left({\frac {1}{2}},\ 0,\ {\frac {1}{2}}\right)\\&\left({\frac {1}{4}},\ {\frac {1}{4}},\ {\frac {1}{4}}\right)&\left({\frac {3}{4}},\ {\frac {3}{4}},\ {\frac {1}{4}}\right)\ &\left({\frac {1}{4}},\ {\frac {3}{4}},\ {\frac {3}{4}}\right)&\left({\frac {3}{4}},\ {\frac {1}{4}},\ {\frac {3}{4}}\right)\\\end{aligned}}}" loading="lazy"></dd></dl>

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 (<a href="#math_8">8</a>) becomes:

<msub>

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</msub>

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</munderover>

<msup>

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<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }({\rm {{basis})=f\sum _{j=1}^{2}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=f\left[1+\mathrm {e} ^{[-i\pi /2(h+k+\ell )]}\right]=f\left[1+(-i)^{h+k+\ell }\right]}}}</annotation>

</semantics>

</math><img src="./fec74575f85a623a6d38001ff0d15abc5f0ab1cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.388ex; height:7.676ex;" alt="{\displaystyle F_{hk\ell }({\rm {{basis})=f\sum _{j=1}^{2}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=f\left[1+\mathrm {e} ^{[-i\pi /2(h+k+\ell )]}\right]=f\left[1+(-i)^{h+k+\ell }\right]}}}" loading="lazy"></dd></dl>

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)

<msub>

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</msub>

<mrow>

<mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=f\left[1+(-1)^{h+k}+(-1)^{k+\ell }+(-1)^{h+\ell }\right]\times \left[1+(-i)^{h+k+\ell }\right]}</annotation>

</semantics>

</math><img src="./b671c5a02c9f1655e5bd9e2e93552c4b32ca9c9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.934ex; height:3.343ex;" alt="{\displaystyle F_{hk\ell }=f\left[1+(-1)^{h+k}+(-1)^{k+\ell }+(-1)^{h+\ell }\right]\times \left[1+(-i)^{h+k+\ell }\right]}" loading="lazy"></dd></dl>

with the result

<ul><li>If h, k, ℓ are of mixed parity (odd and even values combined) the first (FCC) term is zero, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F_{hk\ell }|^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle |F_{hk\ell }|^{2}=0}</annotation>

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</math><img src="./2f9bee3891e86a3dbc554b3e0d5b812f7385d4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.825ex; height:3.343ex;" alt="{\displaystyle |F_{hk\ell }|^{2}=0}" loading="lazy"></li>

<li>If h, k, ℓ are all even or all odd then the first (FCC) term is 4

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=4f(1\pm i),|F_{hk\ell }|^{2}=32f^{2}}</annotation>

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</math><img src="./2cd8b14b45607d238b24ffe8ee7189fb1b92cb32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.766ex; height:3.343ex;" alt="{\displaystyle F_{hk\ell }=4f(1\pm i),|F_{hk\ell }|^{2}=32f^{2}}" loading="lazy"></li>

<li>if h+k+ℓ is even and exactly divisible by 4 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h+k+\ell =4n}">

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<annotation encoding="application/x-tex">{\displaystyle h+k+\ell =4n}</annotation>

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</math><img src="./649c95f0b8217f2bfb8e31c526a132661171cfe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.856ex; height:2.343ex;" alt="{\displaystyle h+k+\ell =4n}" loading="lazy">) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk\ell }=4f\times 2,|F_{hk\ell }|^{2}=64f^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=4f\times 2,|F_{hk\ell }|^{2}=64f^{2}}</annotation>

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</math><img src="./6177ddd3ae792ef6e3aaaac6a794010e9bcb0ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.154ex; height:3.343ex;" alt="{\displaystyle F_{hk\ell }=4f\times 2,|F_{hk\ell }|^{2}=64f^{2}}" loading="lazy"></li>

<li>if h+k+ℓ is even but not exactly divisible by 4 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h+k+\ell \neq 4n}">

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<annotation encoding="application/x-tex">{\displaystyle h+k+\ell \neq 4n}</annotation>

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</math><img src="./f49f9354c74add11ac6d867afb25affb36b3ea47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.856ex; height:2.676ex;" alt="{\displaystyle h+k+\ell \neq 4n}" loading="lazy">) the second term is zero and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F_{hk\ell }|^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle |F_{hk\ell }|^{2}=0}</annotation>

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</math><img src="./2f9bee3891e86a3dbc554b3e0d5b812f7385d4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.825ex; height:3.343ex;" alt="{\displaystyle |F_{hk\ell }|^{2}=0}" loading="lazy"></li></ul></li></ul>

These points are encapsulated by the following equations:

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }={\begin{cases}8f,&h+k+\ell =4N\\4(1\pm i)f,&h+k+\ell =2N+1\\0,&h+k+\ell =4N+2\\\end{cases}}}</annotation>

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</math><img src="./d5a9db179a9f161d5347df9fa43d6c9fa589a305.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:41.685ex; height:8.509ex;" alt="{\displaystyle F_{hk\ell }={\begin{cases}8f,&h+k+\ell =4N\\4(1\pm i)f,&h+k+\ell =2N+1\\0,&h+k+\ell =4N+2\\\end{cases}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Rightarrow |F_{hk\ell }|^{2}={\begin{cases}64f^{2},&h+k+\ell =4N\\32f^{2},&h+k+\ell =2N+1\\0,&h+k+\ell =4N+2\\\end{cases}}}</annotation>

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</math><img src="./adfaad1832f6cca25010e44b9d9b3d664dd02a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:42.646ex; height:8.509ex;" alt="{\displaystyle \Rightarrow |F_{hk\ell }|^{2}={\begin{cases}64f^{2},&h+k+\ell =4N\\32f^{2},&h+k+\ell =2N+1\\0,&h+k+\ell =4N+2\\\end{cases}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is an integer.

<div class="mw-heading mw-heading4"><h4 id="Zincblende_crystal_structure">Zincblende crystal structure</h4></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, the resulting structure factor is

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }={\begin{cases}4(f_{A}+f_{B}),&h+k+\ell =4N\\4(f_{A}\pm if_{B}),&h+k+\ell =2N+1\\4(f_{A}-f_{B}),&h+k+\ell =4N+2\\\end{cases}}}</annotation>

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</math><img src="./7efdc7836acf66be745f7aeaa074dfab67197ad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:44.467ex; height:8.843ex;" alt="{\displaystyle F_{hk\ell }={\begin{cases}4(f_{A}+f_{B}),&h+k+\ell =4N\\4(f_{A}\pm if_{B}),&h+k+\ell =2N+1\\4(f_{A}-f_{B}),&h+k+\ell =4N+2\\\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Cesium_chloride">Cesium chloride</h4></div>

<a href="Cesium_chloride" class="mw-redirect" title="Cesium chloride">Cesium chloride</a> 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 (<a href="#math_8">8</a>) becomes

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=\sum _{j=1}^{2}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=\left[f_{Cs}+f_{Cl}\mathrm {e} ^{[-i\pi (h+k+\ell )]}\right]=\left[f_{Cs}+f_{Cl}(-1)^{h+k+\ell }\right]}</annotation>

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</math><img src="./2c30153f90296fb37f56f345a9d5d3cfb1555d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:80.24ex; height:7.676ex;" alt="{\displaystyle F_{hk\ell }=\sum _{j=1}^{2}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=\left[f_{Cs}+f_{Cl}\mathrm {e} ^{[-i\pi (h+k+\ell )]}\right]=\left[f_{Cs}+f_{Cl}(-1)^{h+k+\ell }\right]}" loading="lazy"></dd></dl>

We then arrive at the following result for the structure factor for scattering from a plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (hk\ell )}">

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<annotation encoding="application/x-tex">{\displaystyle (hk\ell )}</annotation>

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</math><img src="./c7197952a6d05a7212c55f437a5ca3a575f745aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.329ex; height:2.843ex;" alt="{\displaystyle (hk\ell )}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }={\begin{cases}(f_{Cs}+f_{Cl}),&h+k+\ell &{\text{even}}\\(f_{Cs}-f_{Cl}),&h+k+\ell &{\text{odd}}\end{cases}}}</annotation>

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</math><img src="./67a463dc22c449c5ca1831f36d79c8679a77982d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.038ex; height:6.176ex;" alt="{\displaystyle F_{hk\ell }={\begin{cases}(f_{Cs}+f_{Cl}),&h+k+\ell &{\text{even}}\\(f_{Cs}-f_{Cl}),&h+k+\ell &{\text{odd}}\end{cases}}}" loading="lazy"></dd></dl>

and for scattered intensity,

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<annotation encoding="application/x-tex">{\displaystyle |F_{hk\ell }|^{2}={\begin{cases}(f_{Cs}+f_{Cl})^{2},&h+k+\ell &{\text{even}}\\(f_{Cs}-f_{Cl})^{2},&h+k+\ell &{\text{odd}}\end{cases}}}</annotation>

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</math><img src="./6cb0309abfc59825b15f8dd6322adceabdfc9cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.44ex; height:6.176ex;" alt="{\displaystyle |F_{hk\ell }|^{2}={\begin{cases}(f_{Cs}+f_{Cl})^{2},&h+k+\ell &{\text{even}}\\(f_{Cs}-f_{Cl})^{2},&h+k+\ell &{\text{odd}}\end{cases}}}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Hexagonal_close-packed_(HCP)">Hexagonal close-packed (HCP)</h4></div>

In an HCP crystal such as <a href="Graphite" title="Graphite">graphite</a>, the two coordinates include the origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0,0,0\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(0,0,0\right)}</annotation>

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</math><img src="./60d2869010c5314182a13b5878679c61786f3028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle \left(0,0,0\right)}" loading="lazy"> and the next plane up the c axis located at c/2, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1/3,2/3,1/2\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(1/3,2/3,1/2\right)}</annotation>

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</math><img src="./aa1ea7c2aec044e00ddeb7c7a9ac97b10949713d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.339ex; height:2.843ex;" alt="{\displaystyle \left(1/3,2/3,1/2\right)}" loading="lazy">, which gives us

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<annotation encoding="application/x-tex">{\displaystyle F_{hk\ell }=f\left[1+e^{2\pi i\left({\tfrac {h}{3}}+{\tfrac {2k}{3}}+{\tfrac {\ell }{2}}\right)}\right]}</annotation>

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</math><img src="./a4ef61f01b9a8073319184e64f373da4e483abf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.272ex; height:7.676ex;" alt="{\displaystyle F_{hk\ell }=f\left[1+e^{2\pi i\left({\tfrac {h}{3}}+{\tfrac {2k}{3}}+{\tfrac {\ell }{2}}\right)}\right]}" loading="lazy"></dd></dl>

From this it is convenient to define dummy variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\equiv h/3+2k/3+\ell /2}">

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<annotation encoding="application/x-tex">{\displaystyle X\equiv h/3+2k/3+\ell /2}</annotation>

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</math><img src="./2c3960fad3140fd7541657641fac611f4a318d2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.416ex; height:2.843ex;" alt="{\displaystyle X\equiv h/3+2k/3+\ell /2}" loading="lazy">, and from there consider the modulus squared so hence

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<annotation encoding="application/x-tex">{\displaystyle |F|^{2}=f^{2}\left(1+e^{2\pi iX}\right)\left(1+e^{-2\pi iX}\right)=f^{2}\left(2+e^{2\pi iX}+e^{-2\pi iX}\right)=f^{2}\left(2+2\cos[2\pi X]\right)=f^{2}\left(4\cos ^{2}\left[\pi X\right]\right)}</annotation>

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</math><img src="./d213a802f67c2f42886cdb86f5c54ac22aa461ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:100.592ex; height:3.509ex;" alt="{\displaystyle |F|^{2}=f^{2}\left(1+e^{2\pi iX}\right)\left(1+e^{-2\pi iX}\right)=f^{2}\left(2+e^{2\pi iX}+e^{-2\pi iX}\right)=f^{2}\left(2+2\cos[2\pi X]\right)=f^{2}\left(4\cos ^{2}\left[\pi X\right]\right)}" loading="lazy"></dd></dl>

This leads us to the following conditions for the structure factor:

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<annotation encoding="application/x-tex">{\displaystyle |F_{hk\ell }|^{2}={\begin{cases}0,&h+2k=3N{\text{ and }}\ell {\text{ is odd,}}\\4f^{2},&h+2k=3N{\text{ and }}\ell {\text{ is even,}}\\3f^{2},&h+2k=3N\pm 1{\text{ and }}\ell {\text{ is odd,}}\\f^{2},&h+2k=3N\pm 1{\text{ and }}\ell {\text{ is even}}\\\end{cases}}}</annotation>

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</math><img src="./f78ac3fc3dbf948efa462f65850c990c80907dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:49.054ex; height:11.509ex;" alt="{\displaystyle |F_{hk\ell }|^{2}={\begin{cases}0,&h+2k=3N{\text{ and }}\ell {\text{ is odd,}}\\4f^{2},&h+2k=3N{\text{ and }}\ell {\text{ is even,}}\\3f^{2},&h+2k=3N\pm 1{\text{ and }}\ell {\text{ is odd,}}\\f^{2},&h+2k=3N\pm 1{\text{ and }}\ell {\text{ is even}}\\\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Perfect_crystals_in_one_and_two_dimensions">Perfect crystals in one and two dimensions</h3></div>

The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, the reciprocal lattice is an infinite array of points with spacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi /a}</annotation>

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</math><img src="./053c019d2fcf1a284e405447e2df67be7eb99444.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.887ex; height:2.843ex;" alt="{\displaystyle 2\pi /a}" loading="lazy">. In two dimensions, there are only five <a href="Bravais_lattice" title="Bravais lattice">Bravais lattices</a>. 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

</semantics>

</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy"> apply with a scattering vector of limited dimensionality and a crystallographic structure factor can be defined in 2-D as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hk}=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j})]}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hk}=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j})]}}</annotation>

</semantics>

</math><img src="./5ffdca0e7ed5ef03fd54745d58831056abbde955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.744ex; height:7.676ex;" alt="{\displaystyle F_{hk}=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j})]}}" loading="lazy">.

However, recall that real 2-D crystals such as <a href="Graphene" title="Graphene">graphene</a> exist in 3-D. The reciprocal lattice of a 2-D hexagonal sheet that exists in 3-D space in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

</semantics>

</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> plane is a hexagonal array of lines parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{*}}</annotation>

</semantics>

</math><img src="./a5b376dccffe5ae946dcdb7e98bf41beae28dc9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle z^{*}}" loading="lazy"> axis that extend to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</math><img src="./c586ae37f8efec026b8a4ea3f6a5253576c2c4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle \pm \infty }" loading="lazy"> and intersect any plane of constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{i}}</annotation>

</semantics>

</math><img src="./bdd674709e68fa8231280c6d1f5a07161d84aa9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.509ex;" alt="{\displaystyle \mathbf {k} _{i}}" loading="lazy"> is incident on a square lattice of parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. The scattered wave is detected at a certain angle, which defines the wave vector of the outgoing beam, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} _{o}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{o}}</annotation>

</semantics>

</math><img src="./9640fad5fb8280397e08652a123efa0f5f6f76c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.441ex; height:2.509ex;" alt="{\displaystyle \mathbf {k} _{o}}" loading="lazy"> (under the assumption of <a href="Elastic_scattering" title="Elastic scattering">elastic scattering</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {k} _{o}|=|\mathbf {k} _{i}|}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mathbf {k} _{o}|=|\mathbf {k} _{i}|}</annotation>

</semantics>

</math><img src="./cdf7d0d0df5b244f3fea839aac79b4260df8386e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.337ex; height:2.843ex;" alt="{\displaystyle |\mathbf {k} _{o}|=|\mathbf {k} _{i}|}" loading="lazy">). One can equally define the scattering vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} =\mathbf {k} _{o}-\mathbf {k} _{i}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =\mathbf {k} _{o}-\mathbf {k} _{i}}</annotation>

</semantics>

</math><img src="./c27471337b2fcb31719f8addb1cb78ef2e04b335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.006ex; height:2.509ex;" alt="{\displaystyle \mathbf {q} =\mathbf {k} _{o}-\mathbf {k} _{i}}" loading="lazy"> and construct the harmonic pattern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(i\mathbf {q} \mathbf {r} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(i\mathbf {q} \mathbf {r} )}</annotation>

</semantics>

</math><img src="./6b28194f185a51a79857f0b8c73d8e6a5b53ccbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:2.843ex;" alt="{\displaystyle \exp(i\mathbf {q} \mathbf {r} )}" loading="lazy">. In the depicted example, the spacing of this pattern coincides to the distance between particle rows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=2\pi /a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2\pi /a}</annotation>

</semantics>

</math><img src="./d890d7ab69f4c90ecf55fcc4b0ec77bfae8ae669.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.055ex; height:2.843ex;" alt="{\displaystyle q=2\pi /a}" loading="lazy">, so that contributions to the scattering from all particles are in phase (constructive interference). Thus, the total signal in direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} _{o}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} _{o}}</annotation>

</semantics>

</math><img src="./9640fad5fb8280397e08652a123efa0f5f6f76c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.441ex; height:2.509ex;" alt="{\displaystyle \mathbf {k} _{o}}" loading="lazy"> is strong, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

</semantics>

</math><img src="./7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" loading="lazy"> belongs to the reciprocal lattice. It is easily shown that this configuration fulfills <a href="Bragg's_law" title="Bragg's law">Bragg's law</a>.

<div class="mw-heading mw-heading2"><h2 id="Imperfect_crystals">Imperfect crystals</h2></div>

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. <a href="Andr%C3%A9_Guinier" title="André Guinier">André Guinier</a><a href="#cite_note-5">[5]</a> proposed a widely employed distinction between imperfections that preserve the <a href="Long-range_order" class="mw-redirect" title="Long-range order">long-range order</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {q} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )}</annotation>

</semantics>

</math><img src="./31ea4c5c7094b826de0f89baec66eff13b4815b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {q} )}" loading="lazy"> can be used to include the effect of any imperfection. In crystallography, these effects are treated as separate from the structure factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hkl}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hkl}}</annotation>

</semantics>

</math><img src="./40a256c8688cd3af0a00fb8e6dbabf5554b3bd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.02ex; height:2.509ex;" alt="{\displaystyle F_{hkl}}" loading="lazy">, 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.

<div class="mw-heading mw-heading3"><h3 id="Finite-size_effects">Finite-size effects</h3></div>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> a finite crystal means that the sums in equations 1-7 are now over a finite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. 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:

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)={\frac {1}{N}}\left|{\frac {1-\mathrm {e} ^{-iNqa}}{1-\mathrm {e} ^{-iqa}}}\right|^{2}={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}.}</annotation>

</semantics>

</math><img src="./a16c0940db45302937cbdcc5096e09394d3b3d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.102ex; height:6.843ex;" alt="{\displaystyle S(q)={\frac {1}{N}}\left|{\frac {1-\mathrm {e} ^{-iNqa}}{1-\mathrm {e} ^{-iqa}}}\right|^{2}={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}.}" loading="lazy"></dd></dl>

This function is shown in the Figure for different values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">.

When the scattering from every particle is in phase, which is when the scattering is at a reciprocal lattice point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=2k\pi /a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2k\pi /a}</annotation>

</semantics>

</math><img src="./5408ccb604bca1d4bb70a627a781d1ff52bff9c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle q=2k\pi /a}" loading="lazy">, the sum of the amplitudes must be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto N}</annotation>

</semantics>

</math><img src="./5516d046d551b419d572f1ed9a4a31d7e4a3d412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.517ex; height:2.176ex;" alt="{\displaystyle \propto N}" loading="lazy"> and so the maxima in intensity are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto N^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto N^{2}}</annotation>

</semantics>

</math><img src="./2380240c551044a9eb76473427702b8f65e6c748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.631ex; height:2.676ex;" alt="{\displaystyle \propto N^{2}}" loading="lazy">. Taking the above expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> and estimating the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q\to 0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q\to 0)}</annotation>

</semantics>

</math><img src="./65fc60c0e2b1199cb2ef2fd8b84d2a91acfc0f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.155ex; height:2.843ex;" alt="{\displaystyle S(q\to 0)}" loading="lazy"> using, for instance, <a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's rule</a>) shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q=2k\pi /a)=N}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q=2k\pi /a)=N}</annotation>

</semantics>

</math><img src="./e0ce7211122d58c396d9cd43b06926cadf625502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.737ex; height:2.843ex;" alt="{\displaystyle S(q=2k\pi /a)=N}" loading="lazy"> as seen in the Figure. At the midpoint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q=(2k+1)\pi /a)=1/N}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q=(2k+1)\pi /a)=1/N}</annotation>

</semantics>

</math><img src="./57166cda153d5338927fae14f9c350920fab3c41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.874ex; height:2.843ex;" alt="{\displaystyle S(q=(2k+1)\pi /a)=1/N}" loading="lazy"> (by direct evaluation) and the peak width decreases like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/N}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/N}</annotation>

</semantics>

</math><img src="./aa5c2544725c51dfe75eea07ee1f487feb8664c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.389ex; height:2.843ex;" alt="{\displaystyle 1/N}" loading="lazy">. In the large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1-D lattice.

In crystallography when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hkl}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{hkl}}</annotation>

</semantics>

</math><img src="./40a256c8688cd3af0a00fb8e6dbabf5554b3bd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.02ex; height:2.509ex;" alt="{\displaystyle F_{hkl}}" loading="lazy"> is used, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is large, and the formal size effect on diffraction is taken as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\sin(Nqa/2)}{(qa/2)}}\right]^{2}}">

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\sin(Nqa/2)}{(qa/2)}}\right]^{2}}</annotation>

</semantics>

</math><img src="./38d2910fac064c5c2c6c4010de4e47aaa70b21ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.698ex; height:6.843ex;" alt="{\displaystyle \left[{\frac {\sin(Nqa/2)}{(qa/2)}}\right]^{2}}" loading="lazy">, which is the same as the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> above near to the reciprocal lattice points, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\approx 2k\pi /a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\approx 2k\pi /a}</annotation>

</semantics>

</math><img src="./ea2bf5caff529a6642cfb4ccbebbbcfd6faef0b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle q\approx 2k\pi /a}" loading="lazy">. Using convolution, we can describe the finite real crystal structure as [lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"> [basis]<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>

</semantics>

</math><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"> <a href="Rectangular_function" title="Rectangular function">rectangular function</a>, where the rectangular function has a value 1 inside the crystal and 0 outside it. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">[crystal structure] = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">[lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./ed04111e72d462a6d7c79b52c118d5741b24489f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.735ex; height:2.176ex;" alt="{\displaystyle \times {\mathcal {F}}}" loading="lazy">[basis] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast {F}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast {F}}</annotation>

</semantics>

</math><img src="./e312fc42c54ed54131e08c2f10ffd63612d51174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.903ex; height:2.176ex;" alt="{\displaystyle \ast {F}}" loading="lazy">[rectangular function]; that is, scattering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto }</annotation>

</semantics>

</math><img src="./0e3a55007ba2f092d6cafe6d33598e0608b81150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.676ex;" alt="{\displaystyle \propto }" loading="lazy"> [reciprocal lattice] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>

</semantics>

</math><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"> [structure factor] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"> [ <a href="Sinc" class="mw-redirect" title="Sinc">sinc</a> function]. Thus the intensity, which is a delta function of position for the perfect crystal, becomes a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {sinc} ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {sinc} ^{2}}</annotation>

</semantics>

</math><img src="./5aa3f27751c23e15872dcbb08c917f012706236f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.942ex; height:2.676ex;" alt="{\textstyle \operatorname {sinc} ^{2}}" loading="lazy"> function around every point with a maximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto N^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto N^{2}}</annotation>

</semantics>

</math><img src="./2380240c551044a9eb76473427702b8f65e6c748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.631ex; height:2.676ex;" alt="{\displaystyle \propto N^{2}}" loading="lazy">, a width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto 1/N}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto 1/N}</annotation>

</semantics>

</math><img src="./ff7bb668311fb13067786b9e63a4d816a2ced5af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.842ex; height:2.843ex;" alt="{\displaystyle \propto 1/N}" loading="lazy">, area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \propto N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \propto N}</annotation>

</semantics>

</math><img src="./5516d046d551b419d572f1ed9a4a31d7e4a3d412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.517ex; height:2.176ex;" alt="{\displaystyle \propto N}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Disorder_of_the_first_kind">Disorder of the first kind</h3></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> by a multiplicative factor, to give<a href="#cite_note-Warren-1">[1]</a>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}\exp \left(-q^{2}\langle \delta x^{2}\rangle \right)}</annotation>

</semantics>

</math><img src="./06fe65c55c7a9ead569dfeb164efdec6e5c0e150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.327ex; height:6.843ex;" alt="{\displaystyle S(q)={\frac {1}{N}}\left[{\frac {\sin(Nqa/2)}{\sin(qa/2)}}\right]^{2}\exp \left(-q^{2}\langle \delta x^{2}\rangle \right)}" loading="lazy"></dd></dl>

where the disorder is measured by the mean-square displacement of the positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

</semantics>

</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"> from their positions in a perfect one-dimensional lattice: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(j-(N-1)/2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(j-(N-1)/2)}</annotation>

</semantics>

</math><img src="./bb8f3370c5e8e8e77605c7f8eb388186dae9391d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.038ex; height:2.843ex;" alt="{\displaystyle a(j-(N-1)/2)}" loading="lazy">, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}=a(j-(N-1)/2)+\delta x}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}=a(j-(N-1)/2)+\delta x}</annotation>

</semantics>

</math><img src="./631717fc35b29e628f1037a0c5fb9891c5fe292a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.595ex; height:3.009ex;" alt="{\displaystyle x_{j}=a(j-(N-1)/2)+\delta x}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation>

</semantics>

</math><img src="./d22318bef6d7358b79bd993321d65d7c1d3db9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}" loading="lazy"> is a small (much less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">) random displacement. For disorder of the first kind, each random displacement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation>

</semantics>

</math><img src="./d22318bef6d7358b79bd993321d65d7c1d3db9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}" loading="lazy"> is independent of the others, and with respect to a perfect lattice. Thus the displacements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation>

</semantics>

</math><img src="./d22318bef6d7358b79bd993321d65d7c1d3db9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}" loading="lazy"> do not destroy the translational order of the crystal. This has the consequence that for infinite crystals (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\to \infty }</annotation>

</semantics>

</math><img src="./e23159ea0d291e21c5709a6dd7486bed7f18febe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\to \infty }" loading="lazy">) the structure factor still has delta-function Bragg peaks – the peak width still goes to zero as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\to \infty }</annotation>

</semantics>

</math><img src="./e23159ea0d291e21c5709a6dd7486bed7f18febe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\to \infty }" loading="lazy">, with this kind of disorder. However, it does reduce the amplitude of the peaks, and due to the factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q^{2}}</annotation>

</semantics>

</math><img src="./024d4dbdf3feb09055609f33baa8a7ae23aef1d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:3.009ex;" alt="{\displaystyle q^{2}}" loading="lazy"> in the exponential factor, it reduces peaks at large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> much more than peaks at small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">.

The structure is simply reduced by a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> 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 <a href="Debye%E2%80%93Waller_factor" title="Debye–Waller factor">Debye–Waller factor</a>. Note that the Debye–Waller factor is often ascribed to thermal motion, i.e., the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta x}</annotation>

</semantics>

</math><img src="./d22318bef6d7358b79bd993321d65d7c1d3db9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta x}" loading="lazy"> are due to thermal motion, but any random displacements about a perfect lattice, not just thermal ones, will contribute to the Debye–Waller factor.

<div class="mw-heading mw-heading3"><h3 id="Disorder_of_the_second_kind">Disorder of the second kind</h3></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. The derivation follows that in chapter 9 of Guinier's textbook.<a href="#cite_note-:1-6">[6]</a> This model has been pioneered by and applied to a number of materials by Hosemann and collaborators<a href="#cite_note-7">[7]</a> 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 <a href="Paracrystalline" class="mw-redirect" title="Paracrystalline">paracrystalline</a> ordering. Disorder of the first kind is the source of the <a href="Debye%E2%80%93Waller_factor" title="Debye–Waller factor">Debye–Waller factor</a>.

To derive the model we start with the definition (in one dimension) of the

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle S(q)={\frac {1}{N}}\sum _{j,k=1}^{N}\mathrm {e} ^{-iq(x_{j}-x_{k})}}</annotation>

</semantics>

</math><img src="./b1a05ea2c52f7e980e67e4d829565bd9e031070f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.121ex; height:7.676ex;" alt="{\displaystyle S(q)={\frac {1}{N}}\sum _{j,k=1}^{N}\mathrm {e} ^{-iq(x_{j}-x_{k})}}" loading="lazy"></dd></dl>

To start with we will consider, for simplicity an infinite crystal, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\to \infty }</annotation>

</semantics>

</math><img src="./e23159ea0d291e21c5709a6dd7486bed7f18febe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\to \infty }" loading="lazy">. 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> planes away, so the above double sum becomes a single sum over pairs of neighbours either side of an atom, at positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -m}</annotation>

</semantics>

</math><img src="./75f684a4a70b26b6b5018ee63ee0f47af6f34e0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.849ex; height:2.176ex;" alt="{\displaystyle -m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> lattice spacings away, times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. So, then

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1+2\sum _{m=1}^{\infty }\int _{-\infty }^{\infty }{\rm {d}}(\Delta x)p_{m}(\Delta x)\cos \left(q\Delta x\right)}</annotation>

</semantics>

</math><img src="./051d75dec68774285cbf57e5e9386bda5fb72eeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.722ex; height:6.843ex;" alt="{\displaystyle S(q)=1+2\sum _{m=1}^{\infty }\int _{-\infty }^{\infty }{\rm {d}}(\Delta x)p_{m}(\Delta x)\cos \left(q\Delta x\right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}(\Delta x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{m}(\Delta x)}</annotation>

</semantics>

</math><img src="./1792aaa8be36268c436683af3f7d882fd4544715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.009ex; height:2.843ex;" alt="{\displaystyle p_{m}(\Delta x)}" loading="lazy"> is the probability density function for the separation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation>

</semantics>

</math><img src="./f3890eb866b6258d7a304fc34c70ee3fb3a81a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" loading="lazy"> of a pair of planes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> 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

<msub>

</mrow>

</msub>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1}(\Delta x)={\frac {1}{\left(2\pi \sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-a\right)^{2}/(2\sigma _{2}^{2})\right]}</annotation>

</semantics>

</math><img src="./95f487305139e7ede9041c508aa8464a41365bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-left: -0.089ex; width:45.69ex; height:7.009ex;" alt="{\displaystyle p_{1}(\Delta x)={\frac {1}{\left(2\pi \sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-a\right)^{2}/(2\sigma _{2}^{2})\right]}" loading="lazy"></dd></dl>

and we also assume that the fluctuations between a plane and its neighbour, and between this neighbour and the next plane, are independent. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}(\Delta x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{2}(\Delta x)}</annotation>

</semantics>

</math><img src="./89fa9b9b9870e08eeecf5cb0ea89c6e6b71a4cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.388ex; height:2.843ex;" alt="{\displaystyle p_{2}(\Delta x)}" loading="lazy"> is just the convolution of two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}(\Delta x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1}(\Delta x)}</annotation>

</semantics>

</math><img src="./e16136f5196de7f542b92d2479a614f108cea1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.388ex; height:2.843ex;" alt="{\displaystyle p_{1}(\Delta x)}" loading="lazy">s, etc. As the convolution of two Gaussians is just another Gaussian, we have that

<msub>

</mrow>

</msub>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{m}(\Delta x)={\frac {1}{\left(2\pi m\sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-ma\right)^{2}/(2m\sigma _{2}^{2})\right]}</annotation>

</semantics>

</math><img src="./3ee45e99d1a13e4d17ebd8a6f2639b586d6b7934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-left: -0.089ex; width:52.432ex; height:7.009ex;" alt="{\displaystyle p_{m}(\Delta x)={\frac {1}{\left(2\pi m\sigma _{2}^{2}\right)^{1/2}}}\exp \left[-\left(\Delta x-ma\right)^{2}/(2m\sigma _{2}^{2})\right]}" loading="lazy"></dd></dl>

The sum in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> is then just a sum of Fourier transforms of Gaussians, and so

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1+2\sum _{m=1}^{\infty }r^{m}\cos \left(mqa\right)}</annotation>

</semantics>

</math><img src="./32a3fbc34080b5c029eedf4539ba11398e540077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.33ex; height:6.843ex;" alt="{\displaystyle S(q)=1+2\sum _{m=1}^{\infty }r^{m}\cos \left(mqa\right)}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\exp[-q^{2}\sigma _{2}^{2}/2]}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=\exp[-q^{2}\sigma _{2}^{2}/2]}</annotation>

</semantics>

</math><img src="./e5ffdc407cac909aa989eff76edb7b62a4030e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.645ex; height:3.343ex;" alt="{\displaystyle r=\exp[-q^{2}\sigma _{2}^{2}/2]}" loading="lazy">. The sum is just the real part of the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{m=1}^{\infty }[r\exp(iqa)]^{m}}">

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{\infty }[r\exp(iqa)]^{m}}</annotation>

</semantics>

</math><img src="./82729dc14555ad7f0af25342a751afbf3c43b598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.411ex; height:6.843ex;" alt="{\displaystyle \sum _{m=1}^{\infty }[r\exp(iqa)]^{m}}" loading="lazy"> and so the structure factor of the infinite but disordered crystal is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)={\frac {1-r^{2}}{1+r^{2}-2r\cos(qa)}}}</annotation>

</semantics>

</math><img src="./2fcf3e7d435e9a597a2f872ad0df72bd4352bbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.077ex; height:6.509ex;" alt="{\displaystyle S(q)={\frac {1-r^{2}}{1+r^{2}-2r\cos(qa)}}}" loading="lazy"></dd></dl>

This has peaks at maxima <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{p}=2n\pi /a}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q_{p}=2n\pi /a}</annotation>

</semantics>

</math><img src="./655e7cc8276978715af0ab58cfa0a567036177db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.476ex; height:3.009ex;" alt="{\displaystyle q_{p}=2n\pi /a}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(q_{P}a)=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(q_{P}a)=1}</annotation>

</semantics>

</math><img src="./b6da37d1aa81c312f65ddaa3d8b847665f0d08c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.915ex; height:2.843ex;" alt="{\displaystyle \cos(q_{P}a)=1}" loading="lazy">. These peaks have heights

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q_{P})={\frac {1+r}{1-r}}\approx {\frac {4}{q_{P}^{2}\sigma _{2}^{2}}}={\frac {a^{2}}{n^{2}\pi ^{2}\sigma _{2}^{2}}}}</annotation>

</semantics>

</math><img src="./d204ca2d879bf217a437be10d101594cfd0e1b73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.778ex; height:6.843ex;" alt="{\displaystyle S(q_{P})={\frac {1+r}{1-r}}\approx {\frac {4}{q_{P}^{2}\sigma _{2}^{2}}}={\frac {a^{2}}{n^{2}\pi ^{2}\sigma _{2}^{2}}}}" loading="lazy"></dd></dl>

i.e., the height of successive peaks drop off as the order of the peak (and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">) squared. Unlike finite-size effects that broaden peaks but do not decrease their height, disorder lowers peak heights. Note that here we assuming that the disorder is relatively weak, so that we still have relatively well defined peaks. This is the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\sigma _{2}\ll 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\sigma _{2}\ll 1}</annotation>

</semantics>

</math><img src="./f709f4ebeff82105f81915c56d49856c72443ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.228ex; height:2.509ex;" alt="{\displaystyle q\sigma _{2}\ll 1}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\simeq 1-q^{2}\sigma _{2}^{2}/2}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\simeq 1-q^{2}\sigma _{2}^{2}/2}</annotation>

</semantics>

</math><img src="./2d816214a34adb25aa7a9fa7b3300583c0c55d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.993ex; height:3.343ex;" alt="{\displaystyle r\simeq 1-q^{2}\sigma _{2}^{2}/2}" loading="lazy">. In this limit, near a peak we can approximate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(qa)\simeq 1-(\Delta q)^{2}a^{2}/2}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(qa)\simeq 1-(\Delta q)^{2}a^{2}/2}</annotation>

</semantics>

</math><img src="./f9341a22dd11abc6050e3be48a9a2aa01fb24f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.799ex; height:3.176ex;" alt="{\displaystyle \cos(qa)\simeq 1-(\Delta q)^{2}a^{2}/2}" loading="lazy">, with<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta q=q-q_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta q=q-q_{P}}</annotation>

</semantics>

</math><img src="./5221cb6f5917be3071d7db0e98aa6dcccfa2fca4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.517ex; height:2.509ex;" alt="{\displaystyle \Delta q=q-q_{P}}" loading="lazy"> and obtain

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)\approx {\frac {S(q_{P})}{1+{\frac {r}{(1-r)^{2}}}{\frac {\Delta q^{2}a^{2}}{2}}}}\approx {\frac {S(q_{P})}{1+{\frac {\Delta q^{2}}{[q_{P}^{2}\sigma _{2}^{2}/a]^{2}/2}}}}}</annotation>

</semantics>

</math><img src="./d0d0f0ab7105dcad599811fcf8f4a71ce3e49282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:41.133ex; height:9.009ex;" alt="{\displaystyle S(q)\approx {\frac {S(q_{P})}{1+{\frac {r}{(1-r)^{2}}}{\frac {\Delta q^{2}a^{2}}{2}}}}\approx {\frac {S(q_{P})}{1+{\frac {\Delta q^{2}}{[q_{P}^{2}\sigma _{2}^{2}/a]^{2}/2}}}}}" loading="lazy"></dd></dl>

which is a <a href="Cauchy_distribution" title="Cauchy distribution">Lorentzian or Cauchy function</a>, of FWHM <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{P}^{2}\sigma _{2}^{2}/a=4\pi ^{2}n^{2}(\sigma _{2}/a)^{2}/a}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q_{P}^{2}\sigma _{2}^{2}/a=4\pi ^{2}n^{2}(\sigma _{2}/a)^{2}/a}</annotation>

</semantics>

</math><img src="./f208f91c800bde1167ce825db1e7c1b4471ed48f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.408ex; height:3.343ex;" alt="{\displaystyle q_{P}^{2}\sigma _{2}^{2}/a=4\pi ^{2}n^{2}(\sigma _{2}/a)^{2}/a}" loading="lazy">, i.e., the FWHM increases as the square of the order of peak, and so as the square of the wave vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> at the peak.

Finally, the product of the peak height and the FWHM is constant and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4/a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4/a}</annotation>

</semantics>

</math><img src="./154a87345f59afd234d08e240e9274d077b9c547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle 4/a}" loading="lazy">, in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\sigma _{2}\ll 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\sigma _{2}\ll 1}</annotation>

</semantics>

</math><img src="./f709f4ebeff82105f81915c56d49856c72443ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.228ex; height:2.509ex;" alt="{\displaystyle q\sigma _{2}\ll 1}" loading="lazy"> limit. For the first few peaks where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is not large, this is just the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}/a\ll 1}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{2}/a\ll 1}</annotation>

</semantics>

</math><img src="./2e76c2d27644c2d83daa8be714d40ae2a2a34deb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.55ex; height:2.843ex;" alt="{\displaystyle \sigma _{2}/a\ll 1}" loading="lazy"> limit.

<div class="mw-heading mw-heading4"><h4 id="Finite_crystals_with_disorder_of_the_second_kind">Finite crystals with disorder of the second kind</h4></div>

For a one-dimensional crystal of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1+2\sum _{m=1}^{N}\left(1-{\frac {m}{N}}\right)r^{m}\cos \left(mqa\right)}</annotation>

</semantics>

</math><img src="./2744a8540a6b2c93f40afb6dfadead302dee62d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.395ex; height:7.343ex;" alt="{\displaystyle S(q)=1+2\sum _{m=1}^{N}\left(1-{\frac {m}{N}}\right)r^{m}\cos \left(mqa\right)}" loading="lazy"></dd></dl>

where the factor in parentheses comes from the fact the sum is over nearest-neighbour pairs (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=1}</annotation>

</semantics>

</math><img src="./b6100c5ebd48c6fd848709f2be624465203eb173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}" loading="lazy">), next nearest-neighbours (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=2}</annotation>

</semantics>

</math><img src="./b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}" loading="lazy">), ... and for a crystal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> planes, there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N-1}</annotation>

</semantics>

</math><img src="./86aeb216b214f70df1341f34ce273cd3582ce2aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-1}" loading="lazy"> pairs of nearest neighbours, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N-2}</annotation>

</semantics>

</math><img src="./9cdfb930783ca4ca0062df54535c6e35d555dd0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-2}" loading="lazy"> pairs of next-nearest neighbours, etc.

<div class="mw-heading mw-heading2"><h2 id="Liquids">Liquids</h2></div>

In contrast with crystals, liquids have no <a href="Long-range_order" class="mw-redirect" title="Long-range order">long-range order</a> (in particular, there is no regular lattice), so the structure factor does not exhibit sharp peaks. They do however show a certain degree of <a href="Short-range_order" class="mw-redirect" title="Short-range order">short-range order</a>, depending on their density and on the strength of the interaction between particles. Liquids are isotropic, so that, after the averaging operation in Equation (<a href="#math_4">4</a>), the structure factor only depends on the absolute magnitude of the scattering vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\left|\mathbf {q} \right|}">

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=\left|\mathbf {q} \right|}</annotation>

</semantics>

</math><img src="./b96b9afde3c31c2225218ab327e2a2a3d8ff0f98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle q=\left|\mathbf {q} \right|}" loading="lazy">. For further evaluation, it is convenient to separate the diagonal terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j=k}</annotation>

</semantics>

</math><img src="./b954ae5b92d2350b5af4f8d1b0e91d26a01bbe70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.294ex; height:2.509ex;" alt="{\displaystyle j=k}" loading="lazy"> in the double sum, whose phase is identically zero, and therefore each contribute a unit constant:

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

<msup>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1+{\frac {1}{N}}\left\langle \sum _{j\neq k}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle }</annotation>

</semantics>

</math><img src="./01850db12c42ce7672d2dccb5113e301c232ad33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.331ex; height:7.843ex;" alt="{\displaystyle S(q)=1+{\frac {1}{N}}\left\langle \sum _{j\neq k}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle }" loading="lazy">.</td> <td></td> <td class="nowrap">9</td></tr></tbody></table>

One can obtain an alternative expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

</semantics>

</math><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"> in terms of the <a href="Radial_distribution_function" title="Radial distribution function">radial distribution function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(r)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(r)}</annotation>

</semantics>

</math><img src="./e09e9ec780782afb0b2ae8c172811dba1e4eb63c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.974ex; height:2.843ex;" alt="{\displaystyle g(r)}" loading="lazy">:<a href="#cite_note-8">[8]</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1+\rho \int _{V}\mathrm {d} \mathbf {r} \,\mathrm {e} ^{-i\mathbf {q} \mathbf {r} }g(r)}</annotation>

</semantics>

</math><img src="./76115c6c7324f2355206f59f91ec1d0853662a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.891ex; height:5.676ex;" alt="{\displaystyle S(q)=1+\rho \int _{V}\mathrm {d} \mathbf {r} \,\mathrm {e} ^{-i\mathbf {q} \mathbf {r} }g(r)}" loading="lazy">.</td> <td></td> <td class="nowrap">10</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Ideal_gas">Ideal gas</h3></div>

In the limiting case of no interaction, the system is an <a href="Ideal_gas" title="Ideal gas">ideal gas</a> and the structure factor is completely featureless: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)=1}</annotation>

</semantics>

</math><img src="./fac4023b33b1b115e87623378883cdcce8d450e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.639ex; height:2.843ex;" alt="{\displaystyle S(q)=1}" loading="lazy">, because there is no correlation between the positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{j}}</annotation>

</semantics>

</math><img src="./c9df4175b0af5c5ccaeb66a6ebf4997169d79173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.913ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} _{j}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{k}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{k}}</annotation>

</semantics>

</math><img src="./8065bf1aeb85ec21fb2bd66b1687befaa076479a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.092ex; height:2.509ex;" alt="{\displaystyle \mathbf {R} _{k}}" loading="lazy"> of different particles (they are <a href="Independent_random_variables" class="mw-redirect" title="Independent random variables">independent random variables</a>), so the off-diagonal terms in Equation (<a href="#math_9">9</a>) average to zero: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \exp[-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})]\rangle =\langle \exp(-i\mathbf {q} \mathbf {R} _{j})\rangle \langle \exp(i\mathbf {q} \mathbf {R} _{k})\rangle =0}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \exp[-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})]\rangle =\langle \exp(-i\mathbf {q} \mathbf {R} _{j})\rangle \langle \exp(i\mathbf {q} \mathbf {R} _{k})\rangle =0}</annotation>

</semantics>

</math><img src="./03f0c0d741438cd1d54b86e1ab5d43498f96aca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.287ex; height:3.009ex;" alt="{\displaystyle \langle \exp[-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})]\rangle =\langle \exp(-i\mathbf {q} \mathbf {R} _{j})\rangle \langle \exp(i\mathbf {q} \mathbf {R} _{k})\rangle =0}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="High-q_limit">High-q limit</h3></div>

Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation (<a href="#math_10">10</a>), since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(q)-1}</annotation>

</semantics>

</math><img src="./8ba89ec6fb38d6613ae606a949570f1f6a38680c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.381ex; height:2.843ex;" alt="{\displaystyle S(q)-1}" loading="lazy"> is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the "regular" function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(r)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(r)}</annotation>

</semantics>

</math><img src="./e09e9ec780782afb0b2ae8c172811dba1e4eb63c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.974ex; height:2.843ex;" alt="{\displaystyle g(r)}" loading="lazy"> and thus goes to zero for high values of the argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.

<div class="mw-heading mw-heading3"><h3 id="Low-q_limit">Low-q limit</h3></div>

In the low-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> limit, as the system is probed over large length scales, the structure factor contains thermodynamic information, being related to the <a href="Isothermal_compressibility" class="mw-redirect" title="Isothermal compressibility">isothermal compressibility</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{T}}</annotation>

</semantics>

</math><img src="./9e015142e40031e07b12cb961cde2ed201e15e84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.844ex; height:2.009ex;" alt="{\displaystyle \chi _{T}}" loading="lazy"> of the liquid by the <a href="Compressibility_equation" title="Compressibility equation">compressibility equation</a>:

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{q\rightarrow 0}S(q)=\rho \,k_{\mathrm {B} }T\,\chi _{T}=k_{\mathrm {B} }T\left({\frac {\partial \rho }{\partial p}}\right)}</annotation>

</semantics>

</math><img src="./9e5c19ac869d48cca4d8aed3c0e93bfb4e4d679b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.663ex; height:6.176ex;" alt="{\displaystyle \lim _{q\rightarrow 0}S(q)=\rho \,k_{\mathrm {B} }T\,\chi _{T}=k_{\mathrm {B} }T\left({\frac {\partial \rho }{\partial p}}\right)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Hard-sphere_liquids">Hard-sphere liquids</h3></div>

In the <a href="Hard_sphere" class="mw-redirect" title="Hard sphere">hard sphere</a> model, the particles are described as impenetrable spheres with radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">; thus, their center-to-center distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq 2R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\geq 2R}</annotation>

</semantics>

</math><img src="./85c90f43178d9fca079bb35097aaece598953a29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.074ex; height:2.343ex;" alt="{\displaystyle r\geq 2R}" loading="lazy"> and they experience no interaction beyond this distance. Their interaction potential can be written as:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(r)={\begin{cases}\infty &{\text{for }}r<2R,\\0&{\text{for }}r\geq 2R.\end{cases}}}</annotation>

</semantics>

</math><img src="./7da7ca007ca318c4bdb3257955e8a35f3e4913db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.972ex; height:6.176ex;" alt="{\displaystyle V(r)={\begin{cases}\infty &{\text{for }}r<2R,\\0&{\text{for }}r\geq 2R.\end{cases}}}" loading="lazy"></dd></dl>

This model has an analytical solution<a href="#cite_note-9">[9]</a> in the <a href="Percus%E2%80%93Yevick_approximation" title="Percus–Yevick approximation">Percus–Yevick approximation</a>. Although highly simplified, it provides a good description for systems ranging from liquid metals<a href="#cite_note-10">[10]</a> to colloidal suspensions.<a href="#cite_note-11">[11]</a> In an illustration, the structure factor for a hard-sphere fluid is shown in the Figure, for volume fractions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> from 1% to 40%.

<div class="mw-heading mw-heading2"><h2 id="Polymers">Polymers</h2></div>

In <a href="Polymer" title="Polymer">polymer</a> systems, the general definition (<a href="#math_4">4</a>) holds; the elementary constituents are now the <a href="Monomer" title="Monomer">monomers</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{c}}</annotation>

</semantics>

</math><img src="./0fbd8c6f4282c5a1b26587f751f68cd74d9dbc5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.81ex; height:2.509ex;" alt="{\displaystyle N_{c}}" loading="lazy"> identical molecules, each composed of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{p}}</annotation>

</semantics>

</math><img src="./2fb238cb4527a90799df004d5b6879317369a4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.925ex; height:2.843ex;" alt="{\displaystyle N_{p}}" loading="lazy"> monomers, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{c}N_{p}=N}">

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<annotation encoding="application/x-tex">{\displaystyle N_{c}N_{p}=N}</annotation>

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</math><img src="./537d3262392ab8d23daccc78ee2b1c98e8c35778.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.898ex; height:2.843ex;" alt="{\displaystyle N_{c}N_{p}=N}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle N_{p}}</annotation>

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</math><img src="./2fb238cb4527a90799df004d5b6879317369a4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.925ex; height:2.843ex;" alt="{\displaystyle N_{p}}" loading="lazy"> is also known as the <a href="Degree_of_polymerization" title="Degree of polymerization">degree of polymerization</a>). We can rewrite (<a href="#math_4">4</a>) as:

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )={\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha \beta =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\beta k})}\right\rangle ={\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\alpha k})}\right\rangle +{\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha \neq \beta =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\beta k})}\right\rangle }</annotation>

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</math><img src="./efea14a9d154a687f328c3e4476ed26e64c46073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:110.1ex; height:8.009ex;" alt="{\displaystyle S(\mathbf {q} )={\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha \beta =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\beta k})}\right\rangle ={\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\alpha k})}\right\rangle +{\frac {1}{N_{c}N_{p}}}\left\langle \sum _{\alpha \neq \beta =1}^{N_{c}}\sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{\alpha j}-\mathbf {R} _{\beta k})}\right\rangle }" loading="lazy">,</td> <td></td> <td class="nowrap">11</td></tr></tbody></table>

where indices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation>

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</math><img src="./e4b46b57cfa0011b643037751809904d915c1b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }" loading="lazy"> label the different molecules and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j,k}">

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<annotation encoding="application/x-tex">{\displaystyle j,k}</annotation>

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</math><img src="./d23e18a251a10a993e66d41e8dbcaf858ba4fa5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle j,k}" loading="lazy"> the different monomers along each molecule. On the right-hand side we separated intramolecular (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\beta }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =\beta }</annotation>

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</math><img src="./ef6894a6c2f414b03c984a1c7f0639063b0020ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.918ex; height:2.509ex;" alt="{\displaystyle \alpha =\beta }" loading="lazy">) and intermolecular (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \neq \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \neq \beta }</annotation>

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</math><img src="./d0fa83832e27c24569e10eb40f1296faadab48ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.918ex; height:2.676ex;" alt="{\displaystyle \alpha \neq \beta }" loading="lazy">) terms. Using the equivalence of the chains, (<a href="#math_11">11</a>) can be simplified:<a href="#cite_note-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbf {q} )=\underbrace {{\frac {1}{N_{p}}}\left\langle \sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle } _{S_{1}(q)}+{\frac {N_{c}-1}{N_{p}}}\left\langle \sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{1j}-\mathbf {R} _{2k})}\right\rangle }</annotation>

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</math><img src="./d8fd50119cb2739fabff04ee5df5b52930103f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:61.871ex; height:11.843ex;" alt="{\displaystyle S(\mathbf {q} )=\underbrace {{\frac {1}{N_{p}}}\left\langle \sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{j}-\mathbf {R} _{k})}\right\rangle } _{S_{1}(q)}+{\frac {N_{c}-1}{N_{p}}}\left\langle \sum _{jk=1}^{N_{p}}\mathrm {e} ^{-i\mathbf {q} (\mathbf {R} _{1j}-\mathbf {R} _{2k})}\right\rangle }" loading="lazy">,</td> <td></td> <td class="nowrap">12</td></tr></tbody></table>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}(q)}">

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<annotation encoding="application/x-tex">{\displaystyle S_{1}(q)}</annotation>

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</math><img src="./48abb43de51506583737722b91160633192724c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.358ex; height:2.843ex;" alt="{\displaystyle S_{1}(q)}" loading="lazy"> is the single-chain structure factor.

<ul><li><a href="R-factor_(crystallography)" title="R-factor (crystallography)">R-factor (crystallography)</a></li>

<li><a href="Patterson_function" title="Patterson function">Patterson function</a></li>

<li><a href="Ornstein%E2%80%93Zernike_equation" title="Ornstein–Zernike equation">Ornstein–Zernike equation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFWarren1969" class="citation book cs1">Warren, B. E. (1969). <a rel="nofollow" class="external text" href="https://archive.org/details/xraydiffraction00warr">X-ray Diffraction</a>. Addison Wesley.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFCowley1992" class="citation book cs1">Cowley, J. M. (1992). Electron Diffraction Techniques Vol 1. Oxford Science. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780198555582</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFEgamiBillinge2012" class="citation book cs1">Egami, T.; Billinge, S. J. L. (2012). Underneath the Bragg Peaks: Structural Analysis of Complex Material (2nd ed.). Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080971339</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="4" class="citation web cs1"><a rel="nofollow" class="external text" href="http://reference.iucr.org/dictionary/Structure_factor">"Structure Factor"</a>. Online Dictionary of CRYSTALLOGRAPHY. IUCr. Retrieved 15 September 2016.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> See Guinier, chapters 6-9

</li>

<li id="cite_note-:1-6"><a href="#cite_ref-:1_6-0">^</a> <cite id="CITEREFGuinier1963" class="citation book cs1">Guinier, A (1963). X-Ray Diffraction. San Francisco and London: WH Freeman.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFLindenmeyerHosemann1963" class="citation journal cs1">Lindenmeyer, PH; Hosemann, R (1963). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160817105147/http://scitation.aip.org/content/aip/journal/jap/34/1/10.1063/1.1729086">"Application of the Theory of Paracrystals to the Crystal Structure Analysis of Polyacrylonitrile"</a>. Journal of Applied Physics. 34 (1): 42. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963JAP....34...42L">1963JAP....34...42L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1729086">10.1063/1.1729086</a>. Archived from <a rel="nofollow" class="external text" href="http://scitation.aip.org/content/aip/journal/jap/34/1/10.1063/1.1729086">the original</a> on 2016-08-17.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> See Chandler, section 7.5.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFWertheim1963" class="citation journal cs1">Wertheim, M. (1963). "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres". Physical Review Letters. 10 (8): 321–323. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PhRvL..10..321W">1963PhRvL..10..321W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.10.321">10.1103/PhysRevLett.10.321</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFAshcroftLekner1966" class="citation journal cs1">Ashcroft, N.; Lekner, J. (1966). "Structure and Resistivity of Liquid Metals". Physical Review. 145 (1): 83–90. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966PhRv..145...83A">1966PhRv..145...83A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.145.83">10.1103/PhysRev.145.83</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFPuseyVan_Megen1986" class="citation journal cs1">Pusey, P. N.; Van Megen, W. (1986). "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres". Nature. 320 (6060): 340. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986Natur.320..340P">1986Natur.320..340P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F320340a0">10.1038/320340a0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4366474">4366474</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> See Teraoka, Section 2.4.4.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ol><li>Als-Nielsen, N. and McMorrow, D. (2011). Elements of Modern X-ray Physics (2nd edition). John Wiley & Sons.</li>

<li><a href="Andr%C3%A9_Guinier" title="André Guinier">Guinier, A.</a> (1963). X-ray Diffraction. In Crystals, Imperfect Crystals, and Amorphous Bodies. W. H. Freeman and Co.</li>

<li><a href="David_Chandler_(chemist)" title="David Chandler (chemist)">Chandler, D.</a> (1987). Introduction to Modern Statistical Mechanics. Oxford University Press.</li>

<li><a href="Jean-Pierre_Hansen" title="Jean-Pierre Hansen">Hansen, J. P.</a> and McDonald, I. R. (2005). Theory of Simple Liquids (3rd edition). Academic Press.</li>

<li>Teraoka, I. (2002). Polymer Solutions: An Introduction to Physical Properties. John Wiley & Sons.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.ysbl.york.ac.uk/~cowtan/sfapplet/sfintro.html">Structure Factor Tutorial</a> located at the <a href="University_of_York" title="University of York">University of York</a>.</li>

<li><a rel="nofollow" class="external text" href="http://reference.iucr.org/dictionary/Structure_factor">Definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{hkl}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{hkl}}</annotation>

</semantics>

</math><img src="./40a256c8688cd3af0a00fb8e6dbabf5554b3bd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.02ex; height:2.509ex;" alt="{\displaystyle F_{hkl}}" loading="lazy"></a> by <a href="International_Union_of_Crystallography" title="International Union of Crystallography">IUCr</a></li>

<li><a rel="nofollow" class="external text" href="https://www.xtal.iqf.csic.es/Cristalografia/index-en.html">Learning Crystallography, from the CSIC</a></li></ul>

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</style><div id="Crystallography625" style="font-size:114%;margin:0 4em"><a href="Crystallography" title="Crystallography">Crystallography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Timeline_of_crystallography" title="Timeline of crystallography">Timeline of crystallography</a>

<ul><li>Crystallographers</li></ul></li>

<li><a href="Metallurgy" title="Metallurgy">Metallurgy</a></li>

<li>Biocrystallography</li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Crystal_structure" title="Crystal structure">Structure</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li>

<li><a href="Miller_index" title="Miller index">Miller index</a></li>

<li><a href="Crystallographic_point_group" title="Crystallographic point group">Point group</a></li>

<li><a href="Reciprocal_lattice" title="Reciprocal lattice">Reciprocal lattice</a></li>

<li><a href="Crystallographic_restriction_theorem" title="Crystallographic restriction theorem">Restriction theorem</a></li></ul></li>

<li><a href="Periodic_table_(crystal_structure)" title="Periodic table (crystal structure)">Periodic table</a></li>

<li><a href="Crystal_structure_prediction" title="Crystal structure prediction">Structure prediction</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Systems26" scope="row" class="navbox-group" style="width:1%"><a href="Crystal_system" title="Crystal system">Systems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cubic_crystal_system" title="Cubic crystal system">Cubic</a></li>

<li><a href="Hexagonal_crystal_family" title="Hexagonal crystal family">Hexagonal</a></li>

<li><a href="Monoclinic_crystal_system" title="Monoclinic crystal system">Monoclinic</a></li>

<li><a href="Orthorhombic_crystal_system" title="Orthorhombic crystal system">Orthorhombic</a></li>

<li><a href="Tetragonal_crystal_system" title="Tetragonal crystal system">Tetragonal</a></li>

<li><a href="Triclinic_crystal_system" title="Triclinic crystal system">Triclinic</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Crystal_growth" title="Crystal growth">Growth</a>

<ul><li><a href="Crystallite" title="Crystallite">Crystallite</a></li>

<li><a href="Equiaxed_crystal" title="Equiaxed crystal">Equiaxed</a></li></ul></li>

<li><a href="Crystal_twinning" title="Crystal twinning">Twinning</a>

<ul><li><a href="Fiveling" title="Fiveling">Fiveling</a></li></ul></li>

<li><a href="Aperiodic_crystal" title="Aperiodic crystal">Aperiodic crystal</a>

<ul><li><a href="Quasicrystal" title="Quasicrystal">Quasicrystal</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Phase_transition" title="Phase transition">Phase transition</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Phase_diagram" title="Phase diagram">Phase diagram</a>

<ul><li><a href="Eutectic_system" title="Eutectic system">Eutectic</a></li>

<li><a href="Miscibility_gap" title="Miscibility gap">Miscibility gap</a></li>

<li><a href="Crystal_polymorphism" title="Crystal polymorphism">Polymorphism</a></li>

<li><a href="Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li></ul></li>

<li><a href="Phase_transformation_crystallography" title="Phase transformation crystallography">Phase transformation crystallography</a></li>

<li><a href="Precipitation_hardening" title="Precipitation hardening">Precipitation</a></li>

<li><a href="Segregation_(materials_science)" title="Segregation (materials science)">Segregation</a></li>

<li><a href="Spinodal_decomposition" title="Spinodal decomposition">Spinodal decomposition</a></li>

<li><a href="Supersaturation" title="Supersaturation">Supersaturation</a></li>

<li><a href="Ostwald_ripening" title="Ostwald ripening">Ostwald ripening</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Crystallographic_defect" title="Crystallographic defect">Defects</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Grain_boundary" title="Grain boundary">Grain boundary</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Disclination" title="Disclination">Disclination</a></li>

<li><a href="Grain_growth" title="Grain growth">Growth</a></li>

<li><a href="Abnormal_grain_growth" title="Abnormal grain growth">Abnormal growth</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Perfect_crystal" title="Perfect crystal">Perfect crystal</a></li>

<li><a href="Stacking_fault" title="Stacking fault">Stacking fault</a></li>

<li><a href="Dislocation" title="Dislocation">Dislocation</a>

<ul><li><a href="Burgers_vector" title="Burgers vector">Burgers vector</a></li>

<li><a href="Partial_dislocation" title="Partial dislocation">Partial dislocation</a></li>

<li><a href="Cross_slip" title="Cross slip">Cross slip</a></li>

<li><a href="Frank%E2%80%93Read_source" title="Frank–Read source">Frank–Read source</a></li>

<li><a href="Cottrell_atmosphere" title="Cottrell atmosphere">Cottrell atmosphere</a></li>

<li><a href="Peierls_stress" title="Peierls stress">Peierls stress</a></li>

<li><a href="Lomer%E2%80%93Cottrell_junction" title="Lomer–Cottrell junction">Lomer–Cottrell junction</a></li></ul></li>

<ul><li><a href="Slip_bands_in_metals" title="Slip bands in metals">Slip bands</a></li></ul></li>

<li><a href="Interstitial_defect" title="Interstitial defect">Interstitials</a>

<ul><li><a href="Bjerrum_defect" title="Bjerrum defect">Bjerrum defect</a></li>

<li><a href="Frenkel_defect" title="Frenkel defect">Frenkel defect</a></li>

<li><a href="Wigner_effect" title="Wigner effect">Wigner effect</a></li></ul></li>

<li><a href="Vacancy_defect" title="Vacancy defect">Vacancy</a>

<ul><li><a href="Schottky_defect" title="Schottky defect">Schottky defect</a></li>

<li><a href="F-center" title="F-center">F-center</a></li></ul></li>

<li><a href="Stone%E2%80%93Wales_defect" title="Stone–Wales defect">Stone–Wales defect</a></li>

<li><a href="Crystallographic_defects_in_diamond" title="Crystallographic defects in diamond">Defects in diamond</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Laws</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Bragg's_law" title="Bragg's law">Bragg's law</a></li>

<li><a href="Friedel's_law" title="Friedel's law">Friedel's law</a></li>

<li><a href="Law_of_constancy_of_interfacial_angles" title="Law of constancy of interfacial angles">Steno's law (constancy of interfacial angles)</a></li>

<li><a href="Law_of_rational_indices" title="Law of rational indices">Law of rational indices</a></li>

<li><a href="Law_of_symmetry_(crystallography)" title="Law of symmetry (crystallography)">Law of symmetry</a></li>

</div></td></tr></tbody></table><div>

<ul><li><a href="Bragg_plane" title="Bragg plane">Bragg plane</a></li>

<li><a href="Ewald's_sphere" title="Ewald's sphere">Ewald's sphere</a></li>

<li><a href="Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Hermann–Mauguin notation</a></li>

<li><a href="Thermal_ellipsoid" title="Thermal ellipsoid">Thermal ellipsoid</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Characterisation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Electron_crystallography" title="Electron crystallography">Electron</a>

<ul><li><a href="Electron_diffraction" title="Electron diffraction">Diffraction</a></li>

<li><a href="Electron_scattering" title="Electron scattering">Scattering</a></li></ul></li>

<li><a href="Neutron_crystallography" class="mw-redirect" title="Neutron crystallography">Neutron</a>

<ul><li><a href="Neutron_diffraction" title="Neutron diffraction">Diffraction</a></li>

<li><a href="Neutron_scattering" title="Neutron scattering">Scattering</a></li></ul></li>

<li><a href="Nuclear_magnetic_resonance_crystallography" title="Nuclear magnetic resonance crystallography">Nuclear magnetic resonance</a></li>

<ul><li><a href="X-ray_diffraction" title="X-ray diffraction">Diffraction</a></li>

<li><a href="X-ray_scattering" class="mw-redirect" title="X-ray scattering">Scattering</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Direct_methods_(crystallography)" title="Direct methods (crystallography)">Direct methods</a></li>

<li><a href="Isomorphous_replacement" title="Isomorphous replacement">Isomorphous replacement</a></li>

<li><a href="Molecular_replacement" title="Molecular replacement">Molecular replacement</a></li>

<li><a href="Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li>

<li><a href="Patterson_map" class="mw-redirect" title="Patterson map">Patterson map</a></li>

<li><a href="Phase_retrieval" title="Phase retrieval">Phase retrieval</a>

<ul><li><a href="Gerchberg%E2%80%93Saxton_algorithm" title="Gerchberg–Saxton algorithm">Gerchberg–Saxton</a></li></ul></li>

<li><a href="Single_particle_analysis" title="Single particle analysis">Single particle analysis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Software</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="CrystalExplorer" title="CrystalExplorer">CrystalExplorer</a></li>

<li><a href="OctaDist" title="OctaDist">OctaDist</a></li>

<li><a href="ShelXle" title="ShelXle">SHELX</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;"><a href="Crystallographic_database" title="Crystallographic database">Databases</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bilbao_Crystallographic_Server" title="Bilbao Crystallographic Server">Bilbao Crystallographic Server</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Journals</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Crystal_Growth_%26_Design" title="Crystal Growth & Design">Crystal Growth & Design</a></li>

<li><a href="Crystallography_Reviews" title="Crystallography Reviews">Crystallography Reviews</a></li>

<li><a href="Journal_of_Chemical_Crystallography" title="Journal of Chemical Crystallography">Journal of Chemical Crystallography</a></li>

<li><a href="Journal_of_Crystal_Growth" title="Journal of Crystal Growth">Journal of Crystal Growth</a></li>

<li><a href="Kristallografija" title="Kristallografija">Kristallografija</a></li>

<li><a href="Zeitschrift_f%C3%BCr_Kristallographie_%E2%80%93_Crystalline_Materials" title="Zeitschrift für Kristallographie – Crystalline Materials">Zeitschrift für Kristallographie – Crystalline Materials</a></li>

<li><a href="Zeitschrift_f%C3%BCr_Kristallographie_%E2%80%93_New_Crystal_Structures" title="Zeitschrift für Kristallographie – New Crystal Structures">Zeitschrift für Kristallographie – New Crystal Structures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Awards</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carl_Hermann_Medal" title="Carl Hermann Medal">Carl Hermann Medal</a></li>

<li><a href="Ewald_Prize" title="Ewald Prize">Ewald Prize</a></li>

<li><a href="Gregori_Aminoff_Prize" title="Gregori Aminoff Prize">Gregori Aminoff Prize</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">History</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemical_crystallography_before_X-rays" title="Chemical crystallography before X-rays">Chemical crystallography before X-rays</a></li>

<li><a href="Physical_crystallography_before_X-rays" title="Physical crystallography before X-rays">Physical crystallography before X-rays</a></li>

<li><a href="Timeline_of_crystallography" title="Timeline of crystallography">Timeline of crystallography</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Organisation</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="International_Organization_for_Biological_Crystallization" title="International Organization for Biological Crystallization">IOBCr</a></li>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Associations12" scope="row" class="navbox-group" style="width:1%">Associations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="European_Crystallographic_Association" title="European Crystallographic Association">Europe</a>

<ul><li><a href="French_Crystallographic_Association" title="French Crystallographic Association">France</a></li>

<li><a href="German_Crystallographic_Society" title="German Crystallographic Society">Germany</a></li>

<li><a href="Crystallographic_Society_of_Japan" title="Crystallographic Society of Japan">Japan</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="https://commons.wikimedia.org/wiki/Category:Crystallography" class="extiw external" title="commons:Category:Crystallography">Commons</a></li></ul>

</div></td></tr></tbody></table></div>

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