quantum mechanical scattering of photon and nucleus

&\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}_+

\end{align}

with

: $$

\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. {{br}}$Z$ is the atomic number, $\backslash alpha\_\{fine\}\backslash approx\; 1/137$ the fine structure constant, $\backslash hbar$ the reduced Planck 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 $\backslash mathbf\{p\}\_\{+,-\}$ 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 $\backslash mathbf\{q\}$ of the virtual photon between incident photon and nucleus is:

: $$

\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}_+

\end{align}

where the directions are given via:

: $$

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

In order to analyse the relation between the photon energy $E\_+$ and the emission angle $\backslash Theta\_+$ between photon and positron, Köhn and Ebert integrated 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 the quadruply differential cross section over $\backslash Theta\_-$ and $\backslash Phi$. The double differential cross section is:

: $$

\begin{align}

\frac{d^2\sigma (E_+,\omega,\Theta_+)}{dE_+d\Omega_+} =

\sum\limits_{j=1}^{6} I_j

\end{align}

with

: $$

\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

: $$

\begin{align}

A&=\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}\frac{|\mathbf{p}_+