Quantum mechanical scattering of photon and nucleus

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Lua error in package.lua at line 80: module 'strict' not found. In pair production, a photon creates an electron positron pair. In the process of photons scattering in air (e.g. in lightning discharges), the most important interaction is the scattering of photons at the nuclei of atoms or molecules. The full quantum mechanical process of pair production can be described by the quadruply differential cross section given here:[1]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} d^4\sigma &= \frac{Z^2\alpha_\textrm{fine}^3c^2}{(2\pi)^2\hbar}|\mathbf{p}_+||\mathbf{p}_-| \frac{dE_+}{\omega^3}\frac{d\Omega_+ d\Omega_- d\Phi}{|\mathbf{q}|^4}\times \\ &\times\left[- \frac{\mathbf{p}_-^2\sin^2\Theta_-}{(E_--c|\mathbf{p}_-|\cos\Theta_-)^2}\left (4E_+^2-c^2\mathbf{q}^2\right)\right.\\ &-\frac{\mathbf{p}_+^2\sin^2\Theta_+}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)^2}\left (4E_-^2-c^2\mathbf{q}^2\right) \\ &+2\hbar^2\omega^2\frac{\mathbf{p}_+^2\sin^2\Theta_++\mathbf{p}_-^2\sin^2\Theta_-}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)} \\ &+2\left.\frac{|\mathbf{p}_+||\mathbf{p}_-|\sin\Theta_+\sin\Theta_-\cos\Phi}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)}\left(2E_+^2+2E_-^2-c^2\mathbf{q}^2\right)\right]. \\ \end{align}


with

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} d\Omega_+&=\sin\Theta_+\ d\Theta_+,\\ d\Omega_-&=\sin\Theta_-\ d\Theta_-. \end{align}


This expression can be derived by using a quantum mechanical symmetry between pair production and Bremsstrahlung.
Z is the atomic number, \alpha_{fine}\approx 1/137 the fine structure constant, \hbar the reduced Planck's constant and c the speed of light. The kinetic energies  E_{kin,+/-} of the positron and electron relate to their total energies  E_{+,-} and momenta  \mathbf{p}_{+,-} via


E_{+,-}=E_{kin,+/-}+m_e c^2=\sqrt{m_e^2 c^4+\mathbf{p}_{+,-}^2 c^2}.

Conservation of energy yields


\hbar\omega=E_{+}+E_{-}.

The momentum  \mathbf{q} of the virtual photon between incident photon and nucleus is:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} -\mathbf{q}^2&=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right)^2+2|\mathbf{p}_+|\frac{\hbar}{c} \omega\cos\Theta_+ +2|\mathbf{p}_-|\frac{\hbar}{c} \omega\cos\Theta_- \\ &-2|\mathbf{p}_+||\mathbf{p}_-|(\cos\Theta_+\cos\Theta_-+\sin\Theta_+\sin\Theta_-\cos\Phi), \end{align}


where the directions are given via:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \Theta_+&=\sphericalangle(\mathbf{p}_+,\mathbf{k}),\\ \Theta_-&=\sphericalangle(\mathbf{p}_-,\mathbf{k}),\\ \Phi&=\text{Angle between the planes } (\mathbf{p}_+,\mathbf{k}) \text{ and } (\mathbf{p}_-,\mathbf{k}), \end{align}


where  \mathbf{k} is the momentum of the incident photon.

In order to analyse the relation between the photon energy  E_+ and the emission angle  \Theta_+ between photon and positron, Köhn and Ebert integrated [2] the quadruply differential cross section over  \Theta_- and  \Phi . The double differential cross section is:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \frac{d^2\sigma (E_+,\omega,\Theta_+)}{dE_+d\Omega_+} = \sum\limits_{j=1}^{6} I_j \end{align}


with

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} I_1&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+}} \\ &\times \ln\left(\frac{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1+\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2}{-(\Delta^{(p)}_2) ^2-4p_+^2p_-^2\sin^2\Theta_+ -\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1-\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2 }\right) \\ &\times\left[-1-\frac{c\Delta^{(p)}_2}{p_-(E_+-cp_+\cos\Theta_+)}+\frac{p_+^2c^2\sin^2\Theta_+} {(E_+-cp_+\cos\Theta_+)^2}-\frac{2\hbar^2\omega^2p_-\Delta^{(p)}_2}{c(E_+-cp_+\cos \Theta_+)((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)}\right], \\ I_2&=\frac{2\pi Ac}{p_-(E_+-cp_+\cos\Theta_+)}\ln\left( \frac{E_-+p_-c}{E_--p_-c}\right), \\ I_3&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+ }} \\ &\times\ln\Bigg(\Big((E_-+p_-c)(4p_+^2p_-^2\sin^2\Theta_+(E_--p_-c)+(\Delta^{(p)}_1+\Delta^{(p)}_2) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\ &-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)\Big((E_--p_-c) (4p_+^2p_-^2\sin^2\Theta_+(-E_--p_-c) \\ &+(\Delta^{(p)}_1-\Delta^{(p)}_2) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)^{-1}\Bigg) \\ &\times\left[\frac{c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{p_-(E_+-cp_+\cos\Theta_+)}\right.\\ &+\Big[((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)(E_-^3+E_-p_-c)+p_-c(2 ((\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)E_-p_-c \\ &+\Delta^{(p)}_1\Delta^{(p)}_2(3E_-^2+p_-^2c^2))\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1} \\ &+\Big[-8p_+^2p_-^2m^2c^4\sin^2\Theta_+(E_+^2+E_-^2)-2\hbar^2\omega^2p_+^2\sin^2\Theta_+p_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\ &+2\hbar^2\omega^2p_- m^2c^3(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)\Big] \Big[(E_+-cp_+\cos\Theta_+)((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)\Big]^{-1} \\ &+\left.\frac{4E_+^2p_-^2(2(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2-4m^2c^4p_+^2p_-^2\sin^2\Theta_+)(\Delta^{(p)}_1E_-+\Delta^{(p)}_2p_-c)}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}\right], \\ I_4&=\frac{4\pi Ap_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}+\frac{16\pi E_+^2p_-^2 A(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}, \\ I_5&=\frac{4\pi A}{(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)} \\ &\times\left[\frac{\hbar^2\omega^2p_-^2}{E_+cp_+\cos\Theta_+} \Big[E_-[2(\Delta^{(p)}_2)^2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+8p_+^2p_-^2\sin^2\Theta_+((\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2)] \right.\\ &+p_-c[2\Delta^{(p)}_1\Delta^{(p)}_2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+16\Delta^{(p)}_1\Delta^{(p)}_2p_+^2p_-^2\sin^2\Theta_+]\Big]\Big[(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\ &+ \frac{2\hbar^2\omega^2 p_{+}^2 \sin^2\Theta_+(2\Delta^{(p)}_1\Delta^{(p)}_2 p_-c+2(\Delta^{(p)}_2)^2E_-+8p_+^2p_-^2\sin^2\Theta_+ E_-)}{E_+-cp_+\cos\Theta_+}\\ &-\Big[2E_+^2p_-^2\{2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2 +8p_+^2p_-^2\sin^2\Theta_+[((\Delta^{(p)}_1)^2+(\Delta^{(p)}_2)^2)(E_-^2+p_-^2c^2)\\ &+4\Delta^{(p)}_1\Delta^{(p)}_2E_-p_-c]\}\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\ &-\left.\frac{8p_+^2p_-^2\sin^2\Theta_+(E_+^2+E_-^2)(\Delta^{(p)}_2p_-c +\Delta^{(p)}_1 E_-)}{E_+-cp_+\cos\Theta_+}\right], \\ I_6&=-\frac{16\pi E_-^2p_+^2\sin^2\Theta_+ A}{(E_+-cp_+\cos\Theta_+)^2 (-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)} \end{align}


and

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} A&=\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}\frac{|\mathbf{p}_+||\mathbf{p}_-|}{\omega^3},\\ \Delta^{(p)}_1&:=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right) + 2\frac{\hbar}{c}\omega|\mathbf{p}_+|\cos\Theta_+,\\ \Delta^{(p)}_2&:=2\frac{\hbar}{c}\omega|\mathbf{p}_i|-2|\mathbf{p}_+||\mathbf{p}_-| \cos\Theta_+ + 2. \end{align}


This cross section can be applied in Monte Carlo simulations. An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon.

References

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  1. Bethe, H.A., Heitler, W., 1934. On the stopping of fast particles and on the creation of positive electrons. Proc. Phys. Soc. Lond. 146, 83–112
  2. Koehn, C., Ebert, U., Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams, Atmos. Res. (2014), vol. 135-136, pp. 432-465