Calculation of radiation yield

The number of photons per MeV per incident electron per $g/cm^2$ of radiator (Z,A) is given by [*]:

$\frac{d^2n}{d\kappa dt} = \frac{3.495 \times 10^{-4}}{A\kappa}[Z^2\Phi_n(Z,E_0,k)+Z\Phi_e(Z,E_0,k)](MeV^{-1}g^{-1}cm^2)$,

where $\kappa$ - photon kinetic energy in MeV;

$E_0$ - incident electron total energy (in units of the electron rest mass);

$k$ - incident photon energy (in units of the electron rest mass);

Calculation of $\Phi_e(Z,E_0,k)$

$\Phi_e(Z,E_0,k) = C_B\{2[1-\frac{2E}{3E_0}+(\frac{E}{E})^2][L-\sqrt{\eta}]+\sqrt{\eta}[1-\frac{L^2}{2\rho}-\frac{1}{\rho^2}(\frac{1}{2}L-[\frac{\rho(\rho+2)(E_0+1)}{E_0-1}]^{\frac{1}{2}})^2]\}$;

$E = E_0 - k$;

$\rho = E_0 -k(1+E_0-\sqrt{E^2 - 1})$;

$\eta = \rho/(\rho+2)$;

$L = 2 ln(\frac{(E_0-1)^{\frac{1}{2}}+[\eta(E_0+1)^{\frac{1}{2}}]}{(E_0-1)^{\frac{1}{2}}-[\eta(E_0+1)^{\frac{1}{2}}]})$;

$C_B = \frac{\frac{1}{4}\psi(\varepsilon)-1-lnZ^{\frac{2}{3}}}{3.798-ln\varepsilon-lnZ^{\frac{2}{3}}}$;

$\varepsilon = 100k/E_0EZ^{2/3}$;

Case A: For $\varepsilon \geq 0.88$ the screening effect is negligible, $\psi(\varepsilon)=19.19-4ln\varepsilon$ (free electron form) and in this case $C_B = 1$.

Case B: For $\varepsilon \lt 0.88$ we have $\psi(\varepsilon) = 19.70 + 4.117(0.88-\varepsilon)-3.806(0.88-\varepsilon)^2 + 31.84(0.88-\varepsilon)^3-58.63(0.88-\varepsilon)^4+40.77(0.88-\varepsilon)^5$

Calculation of $\Phi_n(Z,E_0,k)$

$\gamma(=100k/E_0EZ^{1/3}) \leq 15$ , $k\lt k_x$:

$\Phi_n(Z,E_0,k) = 4([1+ (\frac{E}{E_0})^2][\frac{1}{4}\phi_1(\gamma)-\frac{1}{3}lnZ - f(Z)]-\frac{2E}{3E_0}[\frac{1}{4}\phi_2(\gamma)-\frac{1}{3}lnZ-f(Z)])$

$\gamma \gt 15$, $k\lt k_x$:

$\Phi_n(Z,E_0,k) = \frac{p}{p_0}(\frac{4}{3}-2EE_0(\frac{p^2+p^2_0}{p^2p^2_0})+\frac{\omega_0E}{p^3_0}+\frac{\omega E_0}{p^3}-\frac{\omega\omega_0}{pp_0}+l{\frac{k}{2pp_0}})$

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Reference: [*] J.L. Matthews, R.O. Owens, Accurate Formulae For the Calculation of High Energy Electron Bremsstrahlung Spectra, NIM III (1973) I57-I68.