Classical Electrodynamics · Problem 15.7

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-nuclear-fission

Problem

Consider the radiation emitted in nuclear fission by the sudden creation of two fragments of charge and mass

(Z_1 e, A_1 m)

and

(Z_2 e, A_2 m)

recoiling in opposite directions with total c.m.s. kinetic energy

E

. Treat the nuclei as point charges and their motion after the very short initial period of acceleration is nonrelativistic, but keep terms up to second order in

1/c

, as in Problem 15.1. For simplicity, assume that the fragments move with constant speeds in opposite directions away from the origin for

t > 0

. The relative speed is

c\beta

. (a) Using the appropriate generalization of Problem 15.1a, show that the intensity of radiation per unit solid angle and per unit photon energy in the c.m. system is

\frac{d^2I}{d(\hbar\omega)\, d\Omega} = \frac{\alpha\beta^2\sin^2\theta}{4\pi^2}|p + q\beta\cos\theta|^2

where

\theta

is the angle between the line of recoil and the direction of observation, and

\alpha = \frac{e^2}{\hbar c}; \qquad p = \frac{Z_1 A_2 - Z_2 A_1}{A_1+A_2}; \qquad q = \frac{Z_1 A_2^2 + Z_2 A_1^2}{(A_1+A_2)^2}

Show that the radiated energy per unit photon energy is

\frac{dI}{d(\hbar\omega)} = \frac{2\alpha\beta^2}{3\pi}\left(p^2 + \frac{\beta^2 q^2}{5}\right)

where the first term is the electric dipole and the second the quadrupole radiation. (b) As an example of the asymmetric fission of

{}^{235}

U by thermal neutrons, take

Z_1 = 36

A_1 = 95

(krypton),

Z_2 = 56

A_2 = 138

(barium) (three neutrons are emitted during fission), with

E = 170

MeV and

mc^2 = 931.5

MeV. What are the values of

p^2

and

q^2

? Determine the total amount of energy (in MeV) radiated by this ``inner bremsstrahlung'' process, with the substitution

\beta^2\to\beta^2(1-\hbar\omega/E)

, as a crude way to incorporate conservation of energy. What are the relative amounts of energy radiated in the dipole and quadrupole modes? In actual fission, roughly 7 MeV of electromagnetic energy is radiated within

10^{-8}

s. How does your estimate compare? If it is much smaller or larger, attempt to explain.

Related theory

Nuclear Fission

Browse the problem library