Soft-x-ray Cherenkov radiation generated by a charged particle moving near a finite-size screen

This paper demonstrates the application of a polarization current approach to estimate the yield of soft x-ray Cherenkov radiation when a charged particle travels in the vicinity of a finite-size screen. To account for resonant anomalous dispersion of the complex permittivity, we apply the formalism for the atomic scattering factor. We derived the simple analytical expression for the radiation field, which allows us to determine all properties for a wide energy range of emitted radiation of a charged particle. The influence of screen sizes on Cherenkov radiation characteristics is determined by macroscopic and microscopic approaches.

Figure 1

A schematic view of the radiation geometry for a charged particle moving in the vicinity of a finite-size screen.

Figure 2

Fresnel equations as a function of observation angle $\theta$. The solid green and dashed red curves correspond, respectively, to Fresnel equations for electric and magnetic components of an incident wave impinging on an aluminum-vacuum boundary. The x-ray energy near the $L$-edge absorption of aluminum is 72.6 eV. The dot-dashed black and the dotted blue curves correspond, respectively, to Fresnel equations for electric and magnetic components of an incident wave impinging on a silicon-vacuum boundary, respectively. The x-ray energy near the $L$-edge absorption of silicon is 99.8 eV.

Figure 3

Spectral-angular distribution of PR in $\theta$, emitted by an electron of energy $\gamma =50$ from various target material. The photon energies: x-ray near absorption $L$ edge of silicon is 99.8 eV (red dashed line), x-ray near absorption $L$ edge of aluminum is 72.6 eV (blue line). Parameters are $a=0.1$ mm, $d=10$ mm, $b=1\mu \mathrm{m}$, and $\varphi =0$. Notice that distributions have scale factors for better presentation: ${10}^6$ for silicon and ${10}^3$ for aluminum.

Figure 4

The PR intensity emitted by an electron of energy $\gamma =50$ at maximum of spectra-angular distribution in $\theta$ as a function of longitudinal size $a$ of aluminum foil. The red solid and the blue dashed curves correspond to the forward DR ($\theta =0$) and ChR (${\theta }_c=15.{31}^{\circ }$), respectively. Parameters are $d=10$ mm, $b=1\mu \mathrm{m}, \varphi =0$, the energy of x ray near the absorption $L$ edge of aluminum is 72.6 eV.

Figure 5

The PR intensity emitted by an electron of energy $\gamma =50$ at maximum of a spectral-angular distribution in $\theta$ as a function of transverse size $d$ of aluminum foil. The red solid and the blue dashed curves correspond to the forward DR ($\theta =0$) and ChR (${\theta }_c=15.{31}^{\circ }$), respectively. Parameters are $a=0.1$ mm, $b=1\mu \mathrm{m}, \varphi =0$, and the x-ray energy near absorption of the $L$ edge of aluminum is 72.6 eV.

Figure 6

The PR intensity emitted at the maximum of spectral-angular distribution in $\theta$ as a function of an electron energy $\gamma$. The red solid and the blue dashed curves correspond to the forward DR ($\theta =0$) and ChR (${\theta }_c=15.{31}^{\circ }$), respectively. Parameters are $a=0.1$ mm, $d=10$ mm, $b=1\mu \mathrm{m}, \varphi =0$, and the energy of x ray near the absorption $L$ edge of aluminum is 72.6 eV. Note that the $Y$ axis is logarithmic.

Figure 7

The real and imaginary components of the aluminum scattering factor for x rays as a function of photon energy. The blue diamonds and red circles correspond, respectively, to the real and imaginary parts of the aluminum scattering factor. Similarly, the dashed and solid curves correspond, respectively, to the approximation functions for the real and imaginary parts of scattering factor.

Figure 8

Spectral-angular distribution of PR in $\theta$ emitted by an electron of energy $\gamma =48$ from aluminum foil for the different cases of averaged photon energy. The black solid, the red dotted, and blue dashed curves correspond, respectively, to $\mathrm{\Delta }\hslash \omega =69–74$ eV, $\mathrm{\Delta }\hslash \omega =69–71$ eV, and $\mathrm{\Delta }\hslash \omega =71.6–73.6$ eV, the averaging photon energy. Parameters are $a=5\mu \mathrm{m}, d=10$ mm, $b=1\mu \mathrm{m}$, and $\varphi =0$.

Figure 9

Spectral-angular distribution of PR in $\theta$ emitted by an electron of energy $\gamma =50$ from aluminum foil. Parameters are $a=0.8\mu \mathrm{m}, d=8$ nm, $b=1\mu \mathrm{m}, \varphi =0$, and the x-ray energy is 80 eV.