View source for Momentum-transfer cross section

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In [[physics]], and especially [[scattering theory]], the '''momentum-transfer cross section''' (sometimes known as the '''momentum-''transport'' cross section'''<ref>{{cite journal|last=Zaghloul|first=Mofreh R.|author2=Bourham, Mohamed A. |author3=Doster, J.Michael |title=Energy-averaged electron–ion momentum transport cross section in the Born approximation and Debye–Hückel potential: Comparison with the cut-off theory | journal=Physics Letters A|date=April 2000|volume=268|issue=4–6|pages=375–381|doi=10.1016/S0375-9601(00)00217-6 | bibcode = 2000PhLA..268..375Z }}</ref>) is an effective [[scattering]] [[cross section (physics)|cross section]] useful for describing the average [[momentum transfer]]red from a particle when it collides with a target.  Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

The momentum-transfer cross section <math>\sigma_{\mathrm{tr}}</math> is defined in terms of an (azimuthally symmetric and momentum independent) [[differential cross section]] <math>\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta)</math> by 
<math display="block">\begin{align}
\sigma_{\mathrm{tr}} &= \int (1 - \cos \theta) \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \, \mathrm{d} \Omega \\
&= \iint   (1 - \cos \theta) \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \sin \theta \, \mathrm{d} \theta \, \mathrm{d} \phi.
\end{align}</math>

The momentum-transfer cross section can be written in terms of the phase shifts from a [[partial wave analysis]] as <ref>{{cite book| last1=Bransden|first1=B.H.|last2=Joachain|first2=C.J.|title=Physics of atoms and molecules|date=2003|publisher=Prentice-Hall|location=Harlow [u.a.]| isbn=978-0582356924|page=584|edition=2.}}</ref>
<math display="block">\sigma_{\mathrm{tr}} = \frac{4\pi}{k^2} \sum_{l=0}^\infty (l+1) \sin^2[\delta_{l+1}(k) - \delta_l(k)].</math>

==Explanation==

The factor of <math>1 - \cos \theta</math> arises as follows.  Let the incoming particle be traveling along the <math>z</math>-axis with vector momentum
<math display="block">\vec{p}_\mathrm{in} = q \hat{z}.</math>

Suppose the particle scatters off the target with polar angle <math>\theta</math> and azimuthal angle <math>\phi</math> plane. Its new momentum is
<math display="block">\vec{p}_\mathrm{out} = q' \cos \theta \hat{z} + q' \sin \theta \cos \phi\hat{x}  + q' \sin \theta \sin \phi\hat{y}.</math>

For collision to much heavier target than striking particle (ex: electron incident on the atom or ion), <math>q'\backsimeq q</math> so
<math display="block"> \vec{p}_\mathrm{out} \simeq q \cos \theta \hat{z} + q \sin \theta \cos \phi\hat{x}  + q \sin \theta \sin \phi\hat{y}</math>

By conservation of momentum, the target has acquired momentum
<math display="block">\Delta \vec{p} = \vec{p}_\mathrm{in} - \vec{p}_\mathrm{out} = q (1 -  \cos \theta) \hat{z} - q \sin \theta \cos \phi\hat{x}  - q \sin \theta \sin \phi\hat{y} .</math>

Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial (<math>x</math> and <math>y</math>) components of the transferred momentum will average to zero.  The average momentum transfer will be just <math>q (1 - \cos \theta) \hat{z}</math>.  If we do the full averaging over all possible scattering events, we get
<math display="block">\begin{align}
\Delta \vec{p}_\mathrm{avg} &= \langle \Delta \vec{p} \rangle_\Omega \\
&= \sigma_\mathrm{tot}^{-1} \int \Delta \vec{p}(\theta,\phi) \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \, \mathrm{d} \Omega \\
&= \sigma_\mathrm{tot}^{-1} \int \left[ q (1 - \cos \theta) \hat{z} - q \sin \theta \cos \phi\hat{x}  - q \sin \theta \sin \phi\hat{y} \right ] \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \, \mathrm{d} \Omega \\
&= q \hat{z} \sigma_\mathrm{tot}^{-1}  \int  (1 -  \cos \theta)  \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \, \mathrm{d} \Omega \\[1ex]
&= q \hat{z} \sigma_\mathrm{tr} / \sigma_\mathrm{tot}
\end{align}</math>
where the total cross section is
<math display="block"> \sigma_\mathrm{tot} = \int \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \mathrm{d} \Omega . </math>

Here, the averaging is done by using [[Expected value|expected value calculation]] (see <math> \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) / \sigma_\mathrm{tot} </math> as a probability density function). Therefore, for a given total cross section, one does not need to compute new integrals for every possible momentum in order to determine the average momentum transferred to a target.   One just needs to compute <math>\sigma_\mathrm{tr}</math>.

==Application==
This concept is used in calculating [[charge radius]] of nuclei such as [[proton]] and deuteron by [[electron scattering]] experiments.

To this purpose a useful quantity called the scattering vector {{mvar|q}} having the dimension of inverse length is defined as a function of energy {{mvar|E}} and scattering angle {{mvar|θ}}:
<math display="block">q = \frac{\frac{2E}{\hbar c} \sin (\theta/2)}{\left[1+ \frac{2E}{Mc^2} \sin^2 (\theta/2)\right]^{1/2}}</math>

==References==
{{reflist}}

[[Category:Momentum]]
[[Category:Scattering theory]]