Classical Electrodynamics · Problem 13.16

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · transition-radiation-magnetic-moment

Problem

A highly relativistic neutral particle of mass mm possessing a magnetic moment μ\mu parallel to its direction of motion emits transition radiation as it crosses at right angles a plane interface from vacuum into a dielectric medium characterized at high frequencies by a plasma frequency ωp\omega_p. The magnetic moment μ\mu is defined in the particle's rest frame. [The particle could be a neutron or, of more potential interest, a neutrino with a small mass.] (a) Show that the intensity of transition radiation is given by (13.79), provided the electric field of the incident particle Ei\mathbf{E}_i is given by (βμ/γze)(\beta\mu/\gamma ze) times the partial derivative in the zz direction of EρE_\rho in (13.80). Note that the electric field actually points azimuthally, but this affects only the polarization of the radiation, not its intensity. (b) Show that in the combined limit of γ≫1\gamma \gg 1 and ω≫ωp\omega \gg \omega_p, the intensity distributions in angle and frequency are given by (13.84) and (13.85), each multiplied by (μω/zeγc)2(\mu\omega/ze\gamma c)^2. (c) By expressing μ\mu in units of the Bohr magneton μB=eℏ/2mec\mu_B = e\hbar/2m_e c and the plasma frequency in atomic units (ℏω0=e2/a0=27.2\hbar\omega_0 = e^2/a_0 = 27.2 eV), show that the ratio of frequency distributions of transition radiation emitted by the magnetic moment to that emitted by an electron with the same speed is dIμ(ν)dIe(ν)=α44(μμB)2(ℏωpℏω0)2⋅ν2\frac{dI_\mu(\nu)}{dI_e(\nu)} = \frac{\alpha^4}{4}\left(\frac{\mu}{\mu_B}\right)^2\left(\frac{\hbar\omega_p}{\hbar\omega_0}\right)^2 \cdot \nu^2 where α=1/137\alpha = 1/137 is the fine structure constant and ν=ω/γωp\nu = \omega/\gamma\omega_p is the dimensionless frequency variable. (d) Calculate the total energy of transition radiation, imposing conservation of energy, that is, ν≤νmax⁡=mc2/ℏωp\nu \leq \nu_{\max} = mc^2/\hbar\omega_p. [This constraint will give only a crude estimate of the energy in the quantum regime where νmax⁡<1\nu_{\max} < 1 because the derivation is classical throughout.] Show that the ratio of total energies for the magnetic moment and an electron of the same speed can be written as IμIe=α420(μμB)2(ℏωpℏω0)2⋅G(νmax⁡)\frac{I_\mu}{I_e} = \frac{\alpha^4}{20}\left(\frac{\mu}{\mu_B}\right)^2\left(\frac{\hbar\omega_p}{\hbar\omega_0}\right)^2 \cdot G(\nu_{\max}) where G≈1G \approx 1 for νmax⁡≫1\nu_{\max} \gg 1 and G≈(10νmax⁡3/π)⋅[ln⁡(1/νmax⁡)−2/3]G \approx (10\nu_{\max}^3/\pi)\cdot[\ln(1/\nu_{\max}) - 2/3] for νmax⁡≪1\nu_{\max} \ll 1. For fixed particle energy and magnetic moment, how does the actual amount of radiated energy vary with the particle's mass for very small mass? Hint: the integrals of Section 2.7 of Gradshteyn and Ryzhik may be of use, although integration by parts is effective.

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