RADIATION OF A CHARGED PARTICLE MOVING IN CROSSED ELECTRIC AND MAGNETIC FIELDS

V. G. Bagrov, V. A. Bordovitsyn

The paper theoretically considers the question of the radiation emitted by a charged particle moving in crossed electric and magnetic fields. The radiation spectrum and its polarization are studied. It is shown that the presence of an electric field leads to a shift of the radiation frequencies into the ultraviolet region and to a change in the polarization of the radiation in comparison with synchrotron radiation. This radiation can be observed both in vacuum and in crystals. The results obtained are of undoubted physical interest, and the article is recommended for publication in the collection Radiation Physics of Ionic Crystals.

Polarisation and spectral—angular characteristics of radiation of the charge, moving in constant and uniform ortogonal electric and magnetic fields are considered in the paper.

The case of \(E \leq H\) is investigated. The consideration was performed by the classical theory.

The paper considers the spectral-angular distribution of the radiation of a charge moving in orthogonal constant and uniform electric and magnetic fields \((E < H)\). The linear polarization of the radiation is investigated. The consideration is carried out in the classical theory.

The magnitude of the total radiation intensity for the motion of a charge in constant and uniform orthogonal electric \(\vec E\) and magnetic \(\vec H\) fields for the case \(E < H\) can be found by a simple Lorentz transformation of the known [1] radiation intensity for the case of a purely magnetic field (synchrotron radiation). Such a transformation leads to the following result:

\[ W=\frac{2}{3}\frac{e^4H^2}{m^2c^3}\cdot \frac{1-(\delta^2+\beta_3^2)\cos^2\eta}{\delta^2}. \tag{1} \]

Here the field \(H\) is oriented along the \(Z\) axis, the field \(E\)—along the \(y\) axis, with

\[ E=H\sin\eta\left(-\frac{\pi}{2}\leq\eta\leq\frac{\pi}{2}\right), \]

and the following notation has been introduced:

\[ \delta=\frac{mc^2}{\varepsilon-cp_1\sin\eta},\qquad \beta_3=\frac{p_3}{mc\delta}, \]

where \(\varepsilon\) is the initial kinetic energy of the charge, and \(p\) is its initial momentum. The quantities \(\delta\) and \(\beta_3\) are integrals of motion in our case.

However, the polarization of the radiation, unlike the intensity, does not possess simple covariance properties, and to study it one must solve the radiation problem by direct methods.

The solution of the equations of motion of a charge in our field can, as is known ([2], p. 394), be represented in the following form:

\[ \begin{aligned} x&=-\frac{c p_2}{eH\cos\psi\cos^2\eta}\cos(\omega\xi-\psi) +\frac{c\sin\eta}{\delta\cos^2\eta}\cdot \xi,\\ y&=\frac{c p_2}{eH\cos\psi\cos\eta}\sin(\omega\xi-\psi),\\ z&=\frac{p_3}{m}\cdot \xi,\\ t&=-\frac{p_2\sin\eta}{eH\cos\psi\cos^2\eta}\cos(\omega\xi-\psi) +\frac{\xi}{\delta\cos^2\eta},\\ \operatorname{tg}\psi&=\frac{\varepsilon\sin\eta-cp_1}{c p_2\cos\eta},\qquad \omega=\frac{eH}{mc}\cos\eta . \end{aligned} \tag{2} \]

Formulas (2) describe, in parametric form, the motion of the charge (\(\xi\) is an intermediate parameter).

The vector potential of the radiation field in the wave zone is determined by the known expression ([1], p. 263):

\[ \mathbf A=\frac{e}{cr}\int \mathbf v(\tau)\, \delta\left(\tau-t+\frac{r}{c}-\frac{(\mathbf r\,\mathbf a(\tau))}{rc}\right)\,d\tau, \]

where

\[ \mathbf a=(x,y,z),\qquad \mathbf v=\frac{d\mathbf a}{dt}. \]

The electric field of the radiation, \(\mathbf E=-\frac{1}{c}\frac{\partial \mathbf A}{\partial t}\), determines the radiation intensity

\[ W=\frac{c}{4\pi}\int r^2 E^2\sin\nu\,d\nu\,d\varphi . \tag{3} \]

For the vector \(\mathbf r\) a spherical coordinate system is introduced,

\[ \mathbf r=(r\sin\nu\cos\varphi,\ r\sin\nu\sin\varphi,\ r\cos\nu). \]

Using formulas (2) and carrying out the calculations in the same way as they are performed in the theory of synchrotron radiation (see [1], pp. 264–266), after simple but cumbersome computations one can obtain the following expressions for the components of the radiation field in the spherical coordinate system:

\[ \begin{aligned} E_\varphi &=-\frac{2e\Omega\cos\eta} {rcA\left(1-\beta_3\cos^2\eta\cos\nu-\sin\eta\sin\nu\cos\varphi\right)} \times\\ &\quad\times \sum_{n=1}^{\infty} n\Bigg\{ \sqrt{1-(\delta^2+\beta_3^2)\cos^2\eta}\, (\sin\nu-\sin\eta\cos\varphi) I_n'(nx)\cos n(\Omega t-s_0) \\ &\qquad -\frac{\sin\eta\cos\eta\cos\nu\sin\varphi}{A} \left[\cos\nu-\beta_3(1-\sin\eta\sin\nu\cos\varphi)\right]\times \\ &\qquad\qquad\times I_n(nx)\sin n(\Omega t-s_0) \Bigg\}, \\[1ex] E_\nu &=-\frac{2e\Omega\cos\eta} {rcA\left(1-\beta_3\cos^2\eta\cos\nu-\sin\eta\sin\nu\cos\varphi\right)} \times \end{aligned} \]

\[ \times \sum_{n=1}^{\infty} n \left\{ \sqrt{1-(\delta^2+\beta_3^2)\cos^2\eta}\,\sin\eta\cos\vartheta\sin\varphi\, I_n'(nx)\cos n(\Omega t-s_0) \right. \]
\[ \left. -\frac{\cos\eta}{A}(\sin\vartheta-\sin\eta\cos\varphi) [\cos\vartheta-\beta_3(1-\sin\eta\sin\vartheta\cos\varphi)] \right. \]
\[ \left. \times I_n(nx)\sin n(\Omega t-s_0) \right\}. \]

Here the following notation has been introduced:

\[ A=\sqrt{(1-\sin\eta\sin\vartheta\cos\varphi)^2-\cos^2\eta\cos^2\vartheta}, \]

\[ x=\frac{A\sqrt{1-(\delta^2+\beta_3^2)\cos^2\eta}} {1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi}, \]

\[ \Omega=\frac{eH}{mc}\cdot \frac{\delta\cos^3\eta} {1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi}, \]

\(s_0\) is a certain phase constant in time, the specific value of which is immaterial for our problem. Substituting these expressions into formula (3) and averaging over time, we obtain the spectral-angular distribution of the radiation intensity of the “\(\sigma\)” (\(dW_2\)) and “\(\pi\)” (\(dW_3\)) components of the linear polarization of the radiation

\[ dW_2=\frac{e^4H^2}{2\pi m^2c^3}\cdot \frac{\delta^2\cos^8\eta\cdot n^2} {A^2(1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi)^3} \times \]
\[ \times\left\{ [1-(\delta^2+\beta_3^2)\cos^2\eta] (\sin\vartheta-\sin\eta\cos\varphi)^2 I_n'^2(nx) +\right. \]
\[ \left. +\frac{\sin^2\eta\cos^2\eta\cos^2\vartheta\sin^2\varphi}{A^2} [\cos\vartheta-\beta_3(1-\sin\eta\sin\vartheta\cos\varphi)]^2 I_n^2(nx) \right\}, \tag{4} \]

\[ dW_3=\frac{e^4H^2}{2\pi m^2c^3}\cdot \frac{\delta^2\cos^8\eta\cdot n^2} {A^2(1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi)^3} \times \]
\[ \times\left\{ [1-(\delta^2+\beta_3^2)\cos^2\eta]\sin^2\eta\cos^2\vartheta\sin^2\varphi\, I_n'^2(nx) +\right. \]
\[ \left. +\frac{\cos^2\eta}{A^2}(\sin\vartheta-\sin\eta\cos\varphi)^2 [\cos\vartheta-\beta_3(1-\sin\eta\sin\vartheta\cos\varphi)]^2 I_n^2(nx) \right\}. \]

The total spectral-angular distribution has the form:

\[ dW=\frac{e^4H^2}{2\pi m^2c^3}\cdot \frac{\delta^2\cos^8\eta\cdot n^2} {(1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi)^3} \times \tag{5} \]
\[ \times\left\{ [1-(\delta^2+\beta_3^2)\cos^2\eta] I_n'^2(nx) +\right. \]
\[ \left. +\frac{\cos^2\eta}{A^2} [\cos\vartheta-\beta_3(1-\sin\eta\sin\vartheta\cos\varphi)]^2 I_n^2(nx) \right\}. \]

Let us note that the averaging over time must be carried out in the coordinate system where the motion is periodic, i.e., where there is a purely magnetic field. This is equivalent to the appearance in expressions (4) and (5) of the additional factor
\(1-\beta_3\cos^2\eta\cos\vartheta-\sin\eta\sin\vartheta\cos\varphi\).

In the particular case \(\eta=0,\ \beta_3=0\), we obtain results known from the theory of synchrotron radiation (see [3]).

Summation over the spectrum and integration over the angles in formula (5), naturally, leads to expression (1).

For the individual components of polarization the following formulas are obtained:

\[ W_2 = W\left[\frac{6+q^2}{8}-Q\right], \]

\[ W_3 = W\left[\frac{2-q^2}{8}+Q\right], \qquad q=\sqrt{\frac{1-(\delta^2+\beta_3^2)\cos^2\eta}{1-\beta_3^2\cos^2\eta}} . \tag{6} \]

The quantity \(Q\) in the general case has a complicated expression, which in the particular case \(\beta_3=0\) is somewhat simplified and has the form:

\[ Q= \frac{\delta^2\cos^2\eta}{16(1-\delta^2\cos^2\eta)} \left\{ 2(2+\delta^2\cos^2\eta) - \right. \]

\[ \left. -\cos\eta\, \frac{ 4+\delta^2\left[2+(11+5\delta^2)\sin^2\eta+\delta^2(10+3\delta^2)\sin^4\eta+3\delta^4\sin^6\eta\right] }{ (1+\delta^2\sin^2\eta)^{5/2} } \right\}. \tag{17} \]

Let us consider in more detail the most important case of relativistic motion of the charge. We shall call relativistic such a charge whose velocity is close to the speed of light in that coordinate system in which it moves along a circle. In the case \(\beta_3=0\), relativistic motion is determined by the criterion \(\delta\ll 1\).

Under this condition the quantity \(Q\) takes the form:

\[ Q=\frac{1}{2}\,\delta^2\cos^2\eta\,\sin^2\frac{\eta}{2}. \]

The maximum value \(Q_{\max}=\dfrac{\delta^2}{27}\) is small. Thus, the polarization of the radiation of a relativistic charge in our case differs only slightly from the polarization of synchrotron radiation. For \(\eta=\dfrac{\pi}{2}\) (motion in orthogonal fields equal in magnitude), it follows from formula (7) that \(Q=0\), and the polarization coincides exactly with the polarization of synchrotron radiation.

Finally, let us consider the question of the spectral density of the radiation. Integrating by means of known integrals (see [1], p. 273) in formula (5), we obtain the spectral density in the form

\[ dW= \frac{e^4H^2\delta^2\cos^4\eta}{m^2c^3q}\, n\left\{ 2q^2 I'_{2n}(2nq) - (1-q^2)\int_0^{2nq} I_{2n}(x)\,dx \right\}. \]

This formula passes into Schott’s formula for \(\eta\simeq 0\). In the relativistic case, using the approximation of Bessel functions by Macdonald functions (see [1], p. 278), we obtain

\[ dW= \frac{3\sqrt{3}}{4\pi}\cdot \frac{e^4H^2}{m^2c^3}\cdot \frac{1}{\delta^2}\, y\,dy \left\{ 2K_{2/3}(y)-\int_y^\infty K_{1/3}(x)\,dx \right\}. \]

Here it is denoted that

\[ y=\frac{\omega}{\omega_c}, \qquad \omega=n\,\frac{eH}{mc}\,\delta\cos^3\eta, \qquad \omega_c=\frac{3}{2}\,\frac{eH}{mc}\,\frac{1}{\delta^2}. \]

In form, the spectral distribution obtained is that for synchrotron radiation; however, the frequencies undergo a shift \(\omega \sim \cos^3 \eta\), i.e., the radiation maximum is displaced toward shorter wavelengths.

Thus, the polarization and spectral properties of the radiation considered, while resembling the properties of synchrotron radiation, also possess specific distinctions that are of physical interest.

REFERENCES

1. D. D. Ivanenko, A. A. Sokolov. Classical Field Theory. GITTL, 1951.
2. V. V. Batygin, I. N. Toptygin. Collection of Problems in Electrodynamics. Fizmatgiz, 1962.
3. A. A. Sokolov, A. I. Matveev, I. M. Ternov. DAN SSSR, 102, 65, 1959.