Sadiq IPAC 2013

Title: Linac Based Positron Production

Positron production using S-band High Repetition Rate Linac (HRRL) was performed at Idaho State University's Idaho Accelerator Center (IAC). Positrons were produced by impinging electrons to a tungsten foil. Bremsstrahlung photons generated in the tungsten foil pair produces electron and positrons. In this paper, we describe the production, transportation and detection of positrons when electron beam energy is 15 MeV.

convert to latex

The Cross section formula is given in Formula 3Cs, pg 928 of reference H.W. Koch & J.W Motz, Rev. Mod. Phys., vol 31 (1959) pg 920 as

Note

Bethe & Heitler first calculated this radiation in 1934 which is why you will sometimes hear Bremsstrahlung radiation refererd to as Bethe-Heitler.

[math]d \sigma = 4 Z^2r_e^2 \alpha \frac{d \nu}{\nu} \left \{ \left (1 + \left( \frac{E}{E_0} \right )^2 \right ) \left [ \frac{\phi_1(\gamma)}{4} - \frac{1}{3} \ln Z -f(Z)\right ] - \frac{2E}{3E_0} \left [ \frac{\phi_2(\gamma)}{4} - \frac{1}{3} \ln Z -f(Z)\right ] \right \} [/math]

where

[math]E_0[/math] = initial total energy of the electron

[math]E[/math] = final total energy of the electron

[math]\nu = \frac{E_0-E}{h}[/math] = energy of the emitted photon

[math]Z[/math] = Atomic number = number of protons in target material

[math]\gamma = \frac{100 m_ec^2 h \nu}{E_0 E Z^{1/3}}[/math] = charge screening parameter

Coulomb correction to using the Born approximation (approximation assumes the incident particle is a plane wave interacting with a static E-field the correction accounts for changes iin the plane wave due to the presence of the field) Charge screening and the coulomb correction are different effects that have been shown to be additive/independent. File:Haug 2008.pdf

[math]f(Z) = (Z \alpha)^2 \sum_1^{\infty} \frac{1}{ n [ n^2 + (Z \alpha)^2]}[/math]

[math]\sim (Z \alpha)^2 \left \{ \frac{1}{1+(Z \alpha)^2} +0.20206 - 0.0369(Z \alpha)^2 + 0.0083 (Z \alpha)^4 - 0.002 (Z \alpha)^6\right \}[/math]

[math]\alpha = \frac{1}{137}[/math]

[math]\phi_1[/math] and [math]\phi_2[/math] = screening functions that depend on Z

if [math]Z \ge 5[/math]

[math]\phi_1(\gamma) = 20.863 - 2 \ln[1+(0.55 \gamma)^2] - 4[1-0.6e^{-0.98} - 0.4e^{-3 \gamma/2}][/math]

[math]\phi_2(\gamma) = \phi_1(\gamma) - \frac{2}{3}(1+6.5 \gamma + 6 \gamma^2)[/math]

For Z<5 see Tsai, Rev.Mod. Phys., vol 46 (1974) pg 815

if [math]3 \ge Z \lt 5[/math] use Equation 3.46 and 3.47

if [math] Z \lt 2[/math] use Equation 3.25 and 3.26