Classical Electrodynamics · Problem 15.9

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-screened-coulomb

Problem

A particle of charge zeze, mass mm, and nonrelativistic velocity vv is deflected in a screened Coulomb field, V(r)=Zze2e−αr/rV(r) = Zze^2 e^{-\alpha r}/r, and consequently emits radiation. Discuss the radiation with the approximation that the particle moves in an almost straight-line trajectory past the force center. (a) Show that, if the impact parameter is bb, the energy radiated per unit frequency interval is dIdω(ω,b)=83πZ2e2c(z2e2mc2)2(cv)2α2K12(αb)\frac{dI}{d\omega}(\omega, b) = \frac{8}{3\pi}\frac{Z^2 e^2}{c}\left(\frac{z^2 e^2}{mc^2}\right)^2\left(\frac{c}{v}\right)^2 \alpha^2 K_1^2(\alpha b) for ω≪v/b\omega \ll v/b, and negligible for ω≫v/b\omega \gg v/b. (b) Show that the radiation cross section is dχ(ω)dω≈−163Z2e2c(z2e2mc2)2(cv)2{x22[K02(x)−K12(x)+2K0(x)K1(x)x]}x1x2\frac{d\chi(\omega)}{d\omega} \approx -\frac{16}{3}\frac{Z^2 e^2}{c}\left(\frac{z^2 e^2}{mc^2}\right)^2\left(\frac{c}{v}\right)^2\left\{\frac{x^2}{2}\left[K_0^2(x) - K_1^2(x) + \frac{2K_0(x)K_1(x)}{x}\right]\right\}_{x_1}^{x_2} where x1=αbmin⁡x_1 = \alpha b_{\min}, x2=αbmax⁡x_2 = \alpha b_{\max}. (c) With bmin⁡=ℏ/mvb_{\min} = \hbar/mv, bmax⁡=v/ωb_{\max} = v/\omega, and α−1=1.4a0Z−1/3\alpha^{-1} = 1.4a_0 Z^{-1/3}, determine the radiation cross section in the two limits, x2≪1x_2 \ll 1 and x2≫1x_2 \gg 1. Compare your results with the ``screening'' and ``no screening'' limits of the text.

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