Transition radiation from a single boundary

 The simplest situation creating transition radiation is with only one boundary as indicated in fig. 3.1. In both media the solution to the inhomogeneous Maxwell equations including the particle gives rise to a continuous energy loss (dE/dx) as will be described in detail in section 3.4.2. However, to fulfil the boundary conditions on the surface between the two media it is necessary to add solutions to the homogenous Maxwell equations: this homogenous solution is the transition radiation.

  

Figure 3.1: Transition radiation created at a single boundary between 2 media. The angle $\theta$ is highly exaggerated.

For the electric field at the boundary, in the limit of ultra-relativistic particles, the electric field is apart from some numeric constants

 

$$\vec{e} \omega \vec{\theta} {\frac{\vec{\theta}}{\gamma^{-2}+\theta^2+\xi_1^2}} {\frac{\vec{\theta}}{\gamma^{-2}+\theta^2+\xi_2^2}}$$ (48)

with a similar formula for the magnetic field $\vec{h}$($\omega$,$\vec{\theta}$) . $\vec{\theta}$ is the difference between the unit vectors in the direction of the particle and the radiation, $\gamma$ is the relativistic gamma factor of the particle and $\omega$ the frequency of the radiation. For the two media $\xi_{i}^{}$ is defined as

 

$$\xi_{i}^{} {\frac{\omega_{{P}_i}}{\omega}}$$ (49)

where $\omega_{{P}_i}^{}$ is the plasma frequency of a material i with electron density n_{e i} considered as an electron gas,

 

$$\omega_{{P}_i}^{2} {\frac{4\pi\alpha n_{{e}_i}}{m_{e}}} \simeq {\frac{Z \rho}{A}}$$ (50)

with Z , A the atomic number and weight, $\alpha$ the fine structure constant and $\rho$ the density. The definition of $\theta$ can be seen in fig. 3.1.

  

Figure: The angular distribution ${\frac{{d}^2W}{{d}\omega{d}\theta}}$ of transition radiation from a single boundary between polypropylene and air for a 4 GeV electron ( $\gamma$ = 8000 ).

The energy radiated per solid angle per unit frequency takes the form
 

$${\frac{{d}^2W}{{d}\omega{d}\Omega}} \left\vert \vec{e}(\omega,\vec{\theta}) \times \vec{h}(\omega,\vec{\theta}) \right\vert$$ =

$${\frac{\alpha}{\pi^2}} \left\vert \frac{\theta}{\gamma^{-2}+\theta^2+\xi_1^2} - \frac{\theta}{\gamma^{-2}+\theta^2+\xi_2^2} \right\vert^$$ = (51)

The angular distribution is illustrated in fig. 3.2.

Performing the angular integration in (3.4) the total energy radiated per unit frequency is
  $$\begin{align} \frac{{d}W}{{d}\omega} &= \frac{\alpha}{\pi} \left( \frac{\xi_1^... ...ation over frequency} W &= \frac{2 \alpha \gamma \omega_{{P}_1}}{3},\end{align}$$
under the assumption $\omega_{{P}_2}^{}$ $\ll$ $\omega_{{P}_1}^{}$ .

In this ideal situation the transition radiation is proportional to $\gamma$ but the proportionality cannot be preserved in a practical detector.

Taking into account a low energy cutoff on the detected photons, which will exist in all detectors, the number of photons emitted from a single boundary is

 

$$\gamma \int_{\omega_{{min}}}^{\infty} {\textstyle\frac{1}{\omega}} {\frac{{d}W}{{d}\omega}} \omega \simeq \alpha$$ (52)

with $\omega_{{min}}^{}$ = 0.15$\gamma$$\omega_{{P}}^{}$ as an example. The main problem for detectors based on transition radiation is (3.7), the number of emitted photons from a single surface is low, and hence many surfaces are needed. However, many closely packed surfaces also offer the possibility to use interference to create a threshold detector.