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| In [[particle physics]], [[wave|wave mechanics]] and [[optics]], '''momentum transfer''' is the amount of [[momentum]] that one particle gives to another particle.
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| In the simplest example of [[scattering]] of two colliding particles with initial momenta <math>\vec{p}_{i1},\vec{p}_{i2}</math>, resulting in final momenta <math>\vec{p}_{f1},\vec{p}_{f2}</math>, the momentum transfer is given by
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| :<math> \vec q = \vec{p}_{i1} - \vec{p}_{f1} = \vec{p}_{f2} - \vec{p}_{i2} </math>
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| where the last identity expresses [[momentum conservation]]. Momentum transfer is an important quantity because <math>\Delta x = \hbar / |q|</math> is a better measure for the typical distance resolution of the reaction than the momenta themselves.
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| ==Wave mechanics and optics==
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| A wave has a momentum <math> p = \hbar k </math> and is a vectorial quantity. The difference of the momentum of the scattered wave to the incident wave is called ''momentum transfer''. The [[wave number]] k is the [[Absoluteness (mathematical logic)|absolute]] of the [[wave vector]] <math> k = q / \hbar</math> and is related to the [[wavelength]] <math> k = 2\pi / \lambda</math>. Often, momentum transfer is given in wavenumber units in [[reciprocal length]] <math> Q = k_f - k_i </math>
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| ===Diffraction===
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| The momentum transfer plays an important role in the evaluation of [[neutron diffraction|neutron]], [[X-ray diffraction|X-ray]] and [[electron diffraction]] for the investigation of [[condensed matter]]. [[Bragg diffraction]] occurs on the atomic [[crystal lattice]], conserves the wave energy and thus is called [[elastic scattering]], where the [[wave number]]s final and incident particles, <math>k_f</math> and <math>k_i</math>, respectively, are equal and just the direction changes by a [[reciprocal lattice]] vector <math> G = Q = k_f - k_i</math> with the relation to the lattice spacing <math> G = 2\pi / d </math>. As momentum is conserved, the transfer of momentum occurs to [[crystal momentum]].
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| The presentation in <math>Q</math>-space is generic and does not depend on the type of [[radiation]] and wavelength used but only on the sample system, which allows to compare results obtained from many different methods. Some established communities such as [[powder diffraction]] employ the diffraction angle <math> 2\theta </math> as the independent variable, which worked fine in the early years when only a few [[characteristic wavelength]]s such as Cu-K<math>\alpha</math> were available. The relationship to <math>Q</math>-space is
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| <math> Q = \frac {4 \pi \sin \left ( \theta \right )}{\lambda} </math>
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| and basically states that larger <math> 2\theta </math> corresponds to larger <math>Q</math>.
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| ==See also==
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| * [[Mandelstam variables]]
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| *[[Momentum-transfer cross section]]
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| * [[impulse (physics)]]
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| {{DEFAULTSORT:Momentum Transfer}}
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| [[Category:Particle physics]]
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| [[Category:Neutron-related techniques]]
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| [[Category:Synchrotron-related techniques]]
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| [[Category:Diffraction]]
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