LETTERS TO THE EDITOR

B R E M S S T R A H L U N G IN NUCLEAR F I S S I O N *

A. I. A l e k s e e v

' The Coulomb interaction between nuclear fragments leads to electromagnetic radiation in the decay of nuclei. ] he calculation of this radiation can be simplified considerably if account is taken of the following. The electromagnetic radiation is most intense while the fragments move in a region with dimensions of the order of the Bohr radius of the mother atom; in this region the moving fragments have not yet been"covered'by the electrons and thus the Coulomb interaction is strongest. The probability that the nuclear fragments will attract electrons of the mother atom is very high [1] since in all cases of nuclear decay the fragment velocity is no greater than the mean velocity of the electrons in the atom. If the fragment velocity is much smaller than the mean velocity of electrons in the outer shell of the mother atom the ionic charge of decay products is essentially zero at the beginning of the trajectory [1], i. e., there is almost complete shielding of the fragments by the elec- trons of the mother atom. In what follows we shalt be interested only in the continuous bremsstrahlung of the fragment and shall neglect the radiation associated with individual photons due to changes of the electronic shells in covering the fragments. A consideration of the mass defects of nuclei shows that in heavy nuclei energies ranging from several million electron volts to 200 Mev are liberated in a single fission event [2]. At these ener- gies the de Broglie wavelength of each fragment is much smaller than the dimensions of the region of intense radiation. Thus, in the decay of a nucleus with charge Z into two fragments with mass numbers A 1 and A z we find

(1)

where Xs is the de Broglie wavelength of thes th fragment (s = 1.2); a is the Bohr radius of the mother atom; m and m e are the masses of the nucleon and electron, respectively; E s is the kinetic energy of the sth fragment after fission; e2/hc = 1/137, In decay of heavy nuclei the energy Esis usually reckoned in the orderof millions of electron volts so that in fission the ratio Xs/a is smaller than 10 "~. As is well known, the fact that the de Brogiie wavelength is small compared with the characteristic dimensions of a charge means that a classical de- scription can be used in analyzing the motion of the particle. Hence, we can use classical considerations to compute the electromagnetic radiation characteristic of the flight of fragments from the point at which they escape through the potential barrier out to infinity. This analysis, however, is not applicable in the small region close to the point at which the fragments pass through the potential barrier (i. e., the point at which the classical momentum of the fragment is zero); in this region the basic requirement for applying a classical description is not satisfied [3]:

d ( a d-7 \ ) - ~ ) << 1, (2)

where p(r) -= ~/2/~(E-- U(r)) is the classical momentum of a particle of mass/J moving with a total energy E in a potential U (r). Estimates show that the regio n of values o f r in which the requirement in (2) is not satisfied is of the order Ar = 10 "9- R (R is the point at which the fragment escapes from the potential barrier, being deter- mined by the equation E - U(r) = 0), so that the fragment radiation in this region is small as compared with the total bremsstrahlung. * The present report is part of a diplomate thesis carried out by the author in 1952 at MIFI under the direction of A. B. Migdal.

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We consider the decay of a nucleus into two fragments with mass numbers A 1 and A= and charges Z 1 and Zg. In moving tn a region with dimensions of order a the fragments Interact tn accordance with Coulomb's law; outside this region the fragments are covered by electrons so that the interaction is described by a more compli- cated relation. However, an estimate Indicates that the bremsstrahlung outside the region of order a is (R/a) s times smaller than the bremsstrahlung Inside this region. We can thus neglect completely the bremsstrahlung outside the region a. It ts sufficient to consider the fragment motion Inside this region. Bocause of the Coulomb interaction the fragments are accelerated. In accordance with the laws of classical electrodynamles a charge which moves wtth accelerated motion must radiate. Under these conditions the time for intense fragment radia- tion is a /v ~ ts the mean relative velocity of the fragment), so that the energy ts radiated mainly as a wave of frequency to which ts smaller than v/a:

(o < v /a . (3)

At these frequencies X ca is large compared with the dimensions of the region of Intense radiation a /k ca = v /e 1 so that most of the fragment radiation is dipole radiation which, tn the present ease, Is characterized by the

following spectral distribution:

4 e, Qz, AxA. z E d ~ ( = ) = ~ - ~ - ~ ~ As Al+A2mc ~l/(=)l~hd=' (4)

where

1 (,o) = , c o ~ ~ + 1 , d~, o

= ( 2(Ax+A2)E"~'/2 2E ~ AxA=ra. .) ZxZ=e a"

(5)

Here d~ (~) is the amount of energy radiated (over the entire trajectory of the fragments) In the form of waves with frequencies lying within the interval dw; E is the kinetic energy of both fragments at Infinity; m ts the mass of the nucleon. In the Integral given In (5) the finite upper limit of Integration, which Is determined by the dimensions of the region of Intense radiation a , is replaced by m. The error which arises in this procedure ts of the order R/a.

At low frequencies ca ~ ca0 the radiated energy is independent of frequency:

�9 , 4 e* (Zx Zz'~= AxA 2 E (6)

At high frequencies ca >> ca0 the energy radiated at a frequency ca is inversely proportional to the square of the frequency:

2___t__ ( e~ (.,) = 3=~z] k.~c ) \A~

Z~ "~

(7)

.The total electromagnetic radiation associated with the fragment bremssurahlung ~ is given by the follow- ing expression:

~=-45 ~.At A=.,) k. Ax+A~,,,) k,,'~-~ ) Z'-'~IZ- ~" (8)

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In writing Eq. (8) we have discarded terms of order (R/a) s and higher.

An estimate shows that the electromagnetic energy which arises from this source ~ may be as high as tens of thousands of electron volts. All the energy $ is radiated by the fragments in moving through a region of order a; outside this region the bremsstrahlung is smaller than $ (R/a) 3

If the charge-to-mass ratios of the fragments are approximately the same, the factor (ZI/A 1 - Z=/A=) 2 tends to suppress the dipole radiation. In these cases, it is necessary to take account of the quadrupole radiation (the magnetic dipole radiation vanishes in any system which consists of only two particles). The energy associated with the quadrupole radiation $ q is given by the following expression:

~q

• \ m e ,9 J ZIZ2 " (9)

It is apparent from F.q. (9) that the energy associated with the quadrupole radiation becomes important when the factor (Z1/A 1 - Z~/A=) 2 is approximately equal to

? n C 2 ~ (lO)

LITERATURE CITED

[1] A. B. Mlgdal, J. Exptl.-Theoret. Phys. (USSR) 9, 1163 (1939).

[2] N. Bohr and J. Wheeler, Phys, Rev. 56,426 (1939).

[3] Landau and Lffshits, Quantum Mechanics (State Tech. Press), p. 158.*

Received December 18, 1957

* In Russian.

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