Complete treatment of single-photon emission in planar channeling

Approximate solutions of the Dirac equation are found for ultrarelativistic particles moving in a periodic potential, which depends only on one coordinate, transverse to the largest component of the momentum of the incoming particle. As an example, we employ these solutions to calculate the radiation emission of positrons and electrons trapped in the planar potential found between the (110) planes in silicon. This allows us to compare with the semiclassical method of Baier, Katkov, and Strakhovenko, which includes the effect of spin and photon recoil but neglects the quantization of the transverse motion. For high-energy electrons, the high-energy part of the angularly integrated photon energy spectrum calculated with the found wave functions differs from the corresponding one calculated with the semiclassical method. However, for lower particle energies, it is found that the angularly integrated emission energy spectra obtained via the semiclassical method is in fairly good agreement with the full quantum calculation except that the positions of the harmonic peaks in photon energy and the photon emission angles are shifted.

Figure 1

The Feynman diagrams corresponding to the process under study. The double fermion lines correspond to the electron solutions of the Dirac equation in the background field of the inter planar crystal potential, which in the perturbative picture corresponds to including all orders of interactions with this background field.

Figure 2

The potential $\phi (y)$ along with the positron energy bands in the first Brillouin zone for the energy $ϵ=100\mathrm{MeV}$. This energy was chosen to have a reasonable number of levels so that the plot is not cluttered.

Figure 3

A plot of the expectation value of $p_y$ as function of the quantum numbers $k_B$ and $n$ for a 100 MeV positron, corresponding to Fig. 1.

Figure 4

The total power, relative to the average, of a 250 GeV electron/positron incident at the position $y_0$ with zero angle.

Figure 5

The differential power (probability per unit time multiplied by the photon energy) for the case of a 20 GeV electron in the state $n=20$ between the (110) planes of silicon. The label “quantum” is the calculation carried out by employing the wave functions found in this paper, “Baier SC” is the semiclassical method of Baier et al. for a particle with the same transverse mechanical energy, and “classical” is using the classical formula for radiation emission.

Figure 6

The differential power (probability per unit time multiplied by the photon energy) for the case of a 10 GeV positron in the state $n=150$ between the (110) planes of silicon. The label “quantum” is the calculation carried out by employing the wave functions found in this paper, and “Baier SC” is the semiclassical method of Baier et al. for a particle with the same transverse mechanical energy.

Figure 7

The differential power (probability per unit time multiplied by the photon energy) for the case of a 250 GeV electron in the state $n=45$ between the (110) planes of silicon. The label “quantum” is the calculation carried out by employing the wave functions found in this paper, “Baier SC” is the semiclassical method of Baier et al. for a particle with the same transverse mechanical energy, “classical” is using the classical formula for radiation emission, and LCFA is the local constant field approximation.

Figure 8

The differential power (probability per unit time multiplied by the photon energy) for the case of a 1 TeV electron in the state $n=120$ between the (110) planes of silicon. The labels have the same meaning as in Fig. 7.

Figure 9

The differential power (probability per unit time multiplied by the photon energy) for the case of a 1 TeV positron in the state $n=1512$ between the (110) planes of silicon. The labels have the same meaning as in Fig. 7.