Physics

B. K. KERIMOV

ENERGY DISTRIBUTION OF BREMSSTRAHLUNG FROM A LONGITUDINALLY POLARIZED ELECTRON

(Presented by Academician N. N. Bogolyubov on 2 July 1960)

1. The theoretical ((^{1-6,\ 11})) and experimental ((^{7-9})) study of the polarization properties of particles in bremsstrahlung is of great interest in connection with the discovery of longitudinally polarized electrons in processes due to weak interactions ((^{10})). One of the experimental methods for determining the degree of longitudinal polarization of electrons is the study of the circular polarization of the bremsstrahlung of a polarized electron in the Coulomb field of a nucleus. The results of investigations show that only electrons polarized along (or opposite to) the direction of motion can produce bremsstrahlung photons possessing circular polarization; moreover, for a high-energy electron ((E \gg m_0 c^2)) the degree of circular polarization of the radiation at the upper end of the spectrum reaches a maximum value of (\sim 100\%).

In ((^2)) an expression was given for the differential cross section of bremsstrahlung from a longitudinally polarized electron only for the case of zero photon emission angle, and in this same special case the degree of polarization of the radiation was calculated as a function of the photon energy.

In ((^1)) we calculated the differential cross section of bremsstrahlung from a longitudinally polarized relativistic electron, applicable for an arbitrary photon emission angle in the Born approximation; from the general formula found for the angular dependence of the cross section (see ((^1)), formula (17)), in the special case one obtains an expression for the degree of circular polarization for zero angle of emission of the bremsstrahlung photon. However, the graph we obtained for the degree of circular polarization of the radiation (see ((^1)), Fig. 2) does not agree with the graph in ((^2)). In a subsequent paper ((^{11})) McVoy gives a new, corrected graph of the dependence of the degree of circular polarization of bremsstrahlung on the photon energy, replacing the erroneous one given in ((^2)). The corrected result of McVoy ((^{11})), as well as subsequent papers ((^{4-6})) on the study of bremsstrahlung from polarized electrons, confirmed our result ((^1)) for the dependence of the degree of circular polarization of bremsstrahlung from a longitudinally polarized electron on the energy and angle of photon emission (see ((^1)), formula (20)). It should be noted that the erroneous graph of bremsstrahlung polarization from paper ((^2)) is also reproduced in the book ((^{12})) (see Fig. 26). In paper ((^3)) it was shown that near the upper limit of the spectrum the circular polarization of bremsstrahlung from extremely relativistic electrons completely polarized in the direction of motion reaches almost (25\%); this result of paper ((^3)) on the magnitude of the circular polarization contradicts both experimental data ((^{7-9})) and the results of subsequent theoretical investigations of bremsstrahlung polarization ((^{1,\ 4-6,\ 11})).

In ((^{1-6})) the differential cross section of bremsstrahlung of a polarized electron was calculated, integrated only over the directions of the electron momentum after radiation. However, for comparison with experiment—

therefore the integral cross section for circularly polarized bremsstrahlung is of the greatest interest.

In the present work we have obtained the energy distribution of the bremsstrahlung process of a longitudinally polarized electron, which, in addition to the known Bethe–Heitler cross section, contains a correction arising from taking into account the polarization states of the initial electron and the emitted photon (the term (\sim ls) in (2), (5)).

1. In the Born approximation, for the angular distribution of bremsstrahlung from a longitudinally polarized relativistic electron we have the expression (see (1), formulas (17)—(19))

[
d\sigma_{ls}(\theta)\,d\Omega
=
\frac{1}{2}\,d\sigma_{\mathrm{B-H}}(\theta)\,d\Omega
+
ls\,d\sigma_{1}(\theta)\,d\Omega .
\tag{1}
]

Here (d\sigma_{\mathrm{B-H}}(\theta)) is the known expression for the angular distribution of bremsstrahlung without taking into account the polarization states of the electron and photon; see ((^{13})) and (1), formula (18)); (d\sigma_{1}(\theta)) characterizes the angular distribution of circularly polarized bremsstrahlung produced by a longitudinally polarized electron and is determined by formula (19) of work ((^{1})); (d\Omega=\sin\theta\,d\theta\,d\varphi) is the solid angle of emission of the photon. The quantity (s=\pm 1) is the eigenvalue of the projection operator (\boldsymbol{\sigma}\mathbf{p}/|\mathbf{p}|), which characterizes the longitudinal polarization of the incident electron ((^{14,1})). For (s=1) the electron spin is directed along the momentum (right-polarized electron), and for (s=-1), opposite to the momentum (left-polarized electron). The quantity (l=\pm 1) determines the polarization of the emitted photon; (l=1) corresponds to a photon polarized in the right-handed circular sense, and (l=-1) in the left-handed circular sense.

Integrating expression (1) over (d\Omega), with the aid of formulas (18), (19) from (1) we obtain the following expression for the integral bremsstrahlung cross section at fixed longitudinal polarizations of the particles participating in the process:

[
d\sigma_{ls}(E,E')
=
\frac{1}{2}\,d\sigma_{\mathrm{B-H}}(E,E')
+
ls\,d\sigma_{1}(E,E').
\tag{2}
]

Here

[
\begin{aligned}
d\sigma_{1}(E,E')={}&
\bar{\varphi}\,\frac{p'\,d\varepsilon_{\gamma}}{p\varepsilon_{\gamma}}
\Bigg{
-\frac{\varepsilon_{\gamma}(3EE' + c^{2}p'^{2})}{3c^{3}pp'^{2}}
\
&\quad
+\frac{\varepsilon_{\gamma}\mu^{2}}{cp}
\left[
-\frac{6\mu^{2}}{c^{4}p^{4}}
-\frac{4p'^{2}+p^{2}}{c^{2}p^{2}p'^{2}}
+\varepsilon_{0}\frac{E(E^{2}+2\mu^{2})}{c^{5}p^{5}}
+\varepsilon\frac{E+E'}{2c^{3}p'^{3}}
\right]
\
&\quad
+L\left[
\frac{\varepsilon_{\gamma}(EE'+3c^{2}p^{2})}{3c^{3}p^{2}p'}
+\frac{\varepsilon_{\gamma}\mu^{2}}{c^{3}p^{2}p'}
\left(
\frac{3\mu^{2}(EE'-\mu^{2})}{c^{4}p^{4}}
+\frac{EE'-3\mu^{2}}{c^{2}p^{2}}
\right.\right.
\
&\qquad\left.\left.
+\frac{p^{2}+p'^{2}}{2p'^{2}}
-\varepsilon\frac{E'p^{2}+Ep'^{2}}{4cp'^{3}}
+\varepsilon_{0}\frac{2\mu^{2}E(3\varepsilon_{\gamma}E-c^{2}p^{2})-c^{4}p^{4}(E+E')}{4c^{5}p^{5}}
\right)
\right]
\Bigg},
\tag{3}
\end{aligned}
]

where

[
\varepsilon_{0}=\ln\frac{E+cp}{E-cp};
\qquad
\varepsilon=\ln\frac{E'+cp'}{E'-cp'};
\qquad
\mu=m_{0}c^{2};
]

[
L=\ln\frac{EE'+c^{2}pp'-\mu^{2}}{EE'-c^{2}pp'-\mu^{2}};
\qquad
\bar{\varphi}=\frac{Z^{2}r_{0}^{2}}{137};
\qquad
r_{0}=\frac{e^{2}}{m_{0}c^{2}},
]

[
E=c\sqrt{p^{2}+m_{0}^{2}c^{2}},\ p
\quad\text{and}\quad
E'=c\sqrt{p'^{2}+m_{0}^{2}c^{2}},\ p'
]

are, respectively, the total energy and momentum of the electron before and after bremsstrahlung; (\varepsilon_{\gamma}=E-E') is the energy of the emitted (\gamma)-quantum. Formula (2) describes the energy dependence of the effective bremsstrahlung cross section of a longitudinally polarized electron with allowance for the correlation of the polarizations of the emitted (\gamma)-quantum and the initial electron (the term (\sim ls\,d\sigma_{1})). Upon averaging over the initial polarization states of the electron ((s=\pm 1))

and summing over the polarization states of the (\gamma)-quantum ((l=\pm 1)), formula (2) goes over into the well-known Bethe—Heitler formula (d\sigma_{\mathrm{B}-\mathrm{H}}(E,E')) for bremsstrahlung (formula (25.16) of the book ({}^{13})).

From (2), for the energy dependence of the degree of circular polarization of the bremsstrahlung from a longitudinally polarized electron, we obtain the formula

[
P_c(E,E')=
\frac{{d\sigma_{ls}}{l=1}-{d\sigma}{l=-1}}
{{d\sigma}{l=1}+{d\sigma}{l=-1}}
=
s\,\frac{2\,d\sigma_1(E,E')}{d\sigma.}-\mathrm{H}}(E,E')
\tag{4}
]

In the case of extremely relativistic energies (E,E'\gg m_0c^2), formulas (2) and (4) are greatly simplified and become the following:

[
d\sigma_{ls}(E,E')=
\bar{\varphi}\,
\frac{E'\,d\varepsilon_\gamma}{E\varepsilon_\gamma}
\left(2\ln\frac{2EE'}{\varepsilon_\gamma m_0c^2}-1\right)
\left{
\left(\frac{E^2+E'^2}{EE'}-\frac{2}{3}\right)
+
ls\,\frac{\varepsilon_\gamma(3E+E')}{3EE'}
\right},
\tag{5}
]

[
P_c(E,E')=
s\,\frac{\varepsilon_\gamma(3E+E')}{3(E^2+E'^2)-2EE'}
=
s\,\frac{\varepsilon_\gamma(4-\varepsilon_\gamma/E)}
{E\left(4-4\varepsilon_\gamma/E+3\varepsilon_\gamma^{\,2}/E^2\right)}.
\tag{6}
]

Here (E=cp) and (E'=cp') are the kinetic energies of the ultrarelativistic electron in the initial and final states, respectively.

Formula (5) is a generalization of the well-known Bethe—Heitler formula for the bremsstrahlung of an extremely relativistic electron (see ({}^{13}), formula (25.21)), with allowance for the longitudinal polarization of the incident electron and the circular polarization of the emitted (\gamma)-quantum (the term (\sim ls)). It follows from (6) that at the upper end of the spectrum, when (\varepsilon_\gamma=E), the circular polarization of the bremsstrahlung is maximal and is complete: (P_c=s=\pm1).

In conclusion, I express my gratitude to Prof. A. A. Sokolov for discussion of the work.

Moscow State University
named after M. V. Lomonosov

Received
28 VI 1960

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