Thick-Target Bremsstrahlung and Target Considerations for Secondary-Particle Production by Electrons

Bremsstrahlung spectra as a function of target thickness up to several radiation lengths, to be used for particle production in high-energy electron machines, are investigated. The shower equations are cast in integral forms which are then solved by iteration. The iterations are performed up to the second-generation photons; and the numerical results show that for most experiments the first-generation photons alone will give sufficiently accurate results. For example, for a target thickness of two radiation lengths and for $\frac{k}{{E}_{0}}=0.5$, where $k$ is the photon energy and ${E}_{0}$ is the incident electron energy, the ratio of the second- to the first-generation photon intensity is 8%. This ratio decreases rapidly as one increases $\frac{k}{{E}_{0}}$ and decreases the target thickness. A very simple formula which approximates the first-generation photon spectra as a function of target thickness is derived. This approximate formula is shown to be accurate enough for estimating the secondary-beam production by electrons. As by-products of our investigation the first- and the second-generation electron and positron spectra were obtained as functions of target thickness. These spectra are useful in estimating the electron and positron background. Some aspects of target considerations for the secondary-beam production are given as an illustration of the use of our formulas.