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UDC 535
PHYSICS
A. A. SOKOLOV, I. M. TERNOV
ON THE THEORY OF INDUCED TRANSITIONS IN THE RADIATION OF A LUMINOUS ELECTRON
(Presented by Academician N. N. Bogolyubov on 26 VI 1965)
Let us consider the radiation of an electron moving in a constant and homogeneous magnetic field, occurring under the influence of an incident external electromagnetic wave. As is known (${}^{1}$), the probability of induced transitions of an electron from the state $E_n$ to the state $E_{n'}$ is given by the expression:
$$ W_{nn'}= \frac{2\pi N(\vec{\varkappa})e^2c}{\hbar L^3\varkappa} \left\{(\vec{\alpha}^{+}\vec{\alpha})-(\vec{\alpha}^{+}\vec{\varkappa}^{0})(\vec{\alpha}\vec{\varkappa}^{0})\right\} \left|\int_0^t e^{-ict(\varkappa_{nn'}-\varkappa)}\,dt\right|^2, \tag{1} $$
where $\omega_{nn'}=c\varkappa_{nn'}=|E_n-E_{n'}|/\hbar$ is the resonance transition frequency; $N(\vec{\varkappa})$ is the number of incident quanta with momentum $\hbar\vec{\varkappa}=\hbar\varkappa\vec{\varkappa}^{0}$; $\alpha_\mu$ is the matrix element of the Dirac matrix, related to the wave functions of the electron in the magnetic field by the relation
$$ \alpha_\mu=\int \psi_{n'}^{+} e^{-i\vec{\varkappa}\vec{x}} \alpha_\mu \psi_n\,d^3x . \tag{2} $$
In the case of damping, caused in particular by the finite time (on the average equal to $\tau/2$) of the electron’s stay at the initial level, the right-hand side of equality (1) can be generalized by introducing the decaying time factor $\exp(-t/\tau)$. Then the integral
$$ I=\left|\int_0^t e^{-ict[\varkappa_{nn'}-\varkappa]}e^{-t/\tau}\,dt\right|^2 = \frac{1+e^{-t/\tau}-2e^{-t/\tau}\cos(\varkappa_{nn'}-\varkappa)t} {c^2(\varkappa_{nn'}-\varkappa)^2+1/\tau^2} \tag{3} $$
for a time interval $t\gg\tau$ goes over into the expression
$$ I=\tau g(\omega)=\tau\frac{\tau}{1+\tau^2(\omega_{nn'}-\omega)^2}, \tag{4} $$
in which $\omega=c\varkappa$ is the frequency of the incident quanta. Therefore, for large time intervals we may introduce the mean probability per unit time of quantum transitions
$$ w_{nn'}=2\frac{W_{nn'}}{\tau} = \frac{2\pi N(\vec{\varkappa})e^2c}{\hbar\varkappa L^3} \left\{\vec{\alpha}^{+}\vec{\alpha}-(\vec{\alpha}^{+}\vec{\varkappa}^{0})(\vec{\alpha}\vec{\varkappa}^{0})\right\}g(\omega). \tag{5} $$
The wave functions of an electron moving in a constant and homogeneous magnetic field and satisfying the Dirac equation were given in our earlier works ((${}^{2}$), see also (${}^{1}$)). The energy spectrum of the electron is then given by the expression:
$$ E_n=c\hbar K_n=c\hbar\sqrt{k_0^2+4n\gamma+k_z^2}, \tag{6} $$
in which $k_0=m_0c/\hbar$, $n$ is the principal quantum number $(n=0,1,2\ldots)$; $\gamma=e_0H/2c\hbar$, $k_z$ is the momentum of the electron along the field.
Determining the frequency of the forced resonant transitions, we find
\[ \omega_{n,n\mp\nu}=\nu\Omega\,\frac{m_0c^2}{E}\left\{1\pm\frac{\beta^2\nu}{4n}\right\} =\nu\Omega\,\frac{m_0c^2}{E}\left\{1\pm\nu\,\frac{\hbar\Omega}{2m_0c^2}\left(\frac{m_0c^2}{E}\right)^2\right\}. \tag{7} \]
Here \(\nu\) is the harmonic number; \(\Omega=e_0H/m_0c\) is the cyclotron frequency. In this formula and in all subsequent calculations we shall retain, in the expansions, terms of order \(\nu/n\) no higher than the first, which, as is evident from (7), is equivalent to an expansion in Planck’s constant \(\hbar\).
Let us now suppose that the incident electromagnetic wave is linearly polarized, with the electric-field vector lying in the plane of the electron’s orbit of rotation and directed along the radius toward its center (the so-called \(\sigma\)-component \(((^3),\) see also \((^1), \S 28)\). For simplicity we shall also assume that the angle of incidence of the external electromagnetic wave in the spherical coordinate system is \(\theta\sim\pi/2\), i.e., the momentum vector also lies in the plane of the electron’s orbit of rotation. Then for the transition probability (5) we obtain (see \((^2)\))
\[ w_{nn'}=\frac{2\pi Ne^2c}{\hbar\chi L^3}\,\bar{\alpha}_1^+\bar{\alpha}_1\,g(\omega), \tag{8} \]
where
\[ \bar{\alpha}_1^+\bar{\alpha}_1 = \frac{KK'-k_0^2}{4KK'}\left(I_{n,n'-1}^2+I_{n-1,n'}^2\right) - 2\,\frac{4\gamma\sqrt{nn'}}{4KK'}\,I_{n,n'-1}I_{n-1,n'}. \tag{9} \]
In this expression the Laguerre function, for \(n>n'\),*
\[ I_{nn'}(y)=\frac{1}{\sqrt{n!\,n'!}}\,e^{-y/2}y^{(n-n')/2}Q_{n'}^{\,n-n'}(y) \tag{10} \]
depends on the argument \(y=\chi^2/4\gamma\), where \(\chi\) is the frequency of the incident photon. Expanding the coefficients in formula (9) with respect to \(\nu/n\), we obtain
\[ \bar{\alpha}_1^+\bar{\alpha}_1 = \frac{\gamma}{K_n^2} \left\{n\mp\frac{\nu}{2}(1-\beta^2)\right\} \left\{ \begin{array}{l} R_{n,n-\nu}^2,\\ R_{n,n+\nu}^2 \end{array} \right. \tag{11} \]
where
\[ R_{n,n'}=I_{n,n-1}(y)-I_{n-1,n'}(y). \tag{12} \]
Let us now write the expression for the energy absorbed per unit time by the electron in resonant transitions (the absorption power)
\[ P_n=\hbar\omega_{n,n+\nu}w_{n,n+\nu} -\hbar\omega_{n,n-\nu}w_{n,n-\nu} = \frac{4\pi N(\vec{\chi})\hbar\omega}{L^3}\,\nu\,\frac{\Omega^2}{\omega^2}\, \frac{e^2}{2m_0} \left(\frac{m_0c^2}{E}\right)^3\tau\Phi, \tag{13} \]
where
\[ \Phi= \frac{\left[n+\frac{1}{2}\nu(1-\frac{3}{2}\beta^2)\right]R_{n,n+\nu}^2} {1+\tau^2(\omega_{n,n+\nu}-\omega)^2} - \frac{\left[n-\frac{1}{2}\nu(1-\frac{3}{2}\beta^2)\right]R_{n,n-\nu}^2} {1+\tau^2(\omega_{n,n-\nu}-\omega)^2}. \tag{14} \]
We transform the denominator in the following way:
\[ \tau(\omega_{n,n-\nu}-\omega) = \tau(\omega_{n,n+\nu}-\omega) + \tau(\omega_{n,n-\nu}-\omega_{n,n+\nu}) = x+\frac{\nu}{2n}Q\beta^2, \tag{15} \]
where \(Q=\omega\tau\) and the quantity \(\omega\simeq\nu\Omega m_0c^2/E\) is close to the resonant frequency.
Next we consider the expression for \(R_{n,n\pm\nu}^2\). As was shown earlier (see, for example, \((^2)\)), the Laguerre functions \(I_{nn'}\) admit a good approxi-
* In the case \(n'>n\), we must interchange the indices \(n\) and \(n'\) and multiply the entire expression by \((-1)^{n-n'}\).
...mation through Bessel functions
\[ I_{nn'}(y) \simeq J_\nu(2\sqrt{yn})-\frac{(\nu-1)\sqrt{4ny}}{4n}\,J_\nu'(2\sqrt{yn}), \tag{16} \]
where \(\nu=n-n'\).
Retaining terms of order \(\nu/n\) no higher than the first, we find, with the aid of this approximation,
\[ R_{n,n\pm\nu}^{2} = 4J_\nu'^2(\nu\beta) \left\{ 1-\frac{1\mp\nu}{n}\sqrt{yn}\, \frac{J_\nu''(\nu\beta)}{J_\nu'(\nu\beta)} \right\}, \tag{17} \]
where the second derivative \(J_\nu''\) is not difficult to eliminate with the aid of Bessel’s equation.
Substituting this entire expansion into formula (14), we find that
\[ \Phi = \frac{4\nu J_\nu'^2(\nu\beta)}{1+x^2} \left\{ \frac{\nu(1-\beta^2)}{\beta}\, \frac{J_\nu(\nu\beta)}{J_\nu'(\nu\beta)} -\frac{3}{2}\beta^2 +\frac{\beta^2 x Q}{1+x^2} \right\}. \tag{18} \]
Finally, the absorbed energy (13) can be expressed in terms of the electric-field strength of the incident wave. Indeed, proceeding from the expression for the energy density \(\mathcal{E}^2/4\pi=\hbar\omega N/L^3\), we obtain
\[ P_n = \frac{e^2\mathcal{E}^2\tau}{2m_0}\, \frac{m_0c^2}{\nu E}\,\Phi . \tag{19} \]
The formula obtained is applicable for any harmonic number \(\nu\) and has no restrictions associated with the velocity of the electron.
From (18), in the particular case of a nonrelativistic electron \(\beta\ll 1\), for dipole transitions, when in (18) one may put
\[ J_1(\nu\beta)=\nu\beta/2,\qquad J_1'(\nu\beta)=1/2,\qquad E\sim m_0c^2, \]
we find the result obtained in (4):
\[ P_n = \frac{e^2\mathcal{E}^2\tau}{2m_0}\, \frac{1}{1+x^2} \left[ 1+\beta^2\frac{Qx}{1+x^2} \right]. \tag{20} \]
Moscow State University
named after M. V. Lomonosov
Received
25 VI 1965
REFERENCES
- A. A. Sokolov, Introduction to Quantum Electrodynamics, Moscow, 1958.
- A. A. Sokolov, I. M. Ternov, ZhETF, 25, 698 (1953).
- A. A. Sokolov, I. M. Ternov, ZhETF, 31, 473 (1956).
- J. Schneider. Phys. Rev. Letters, No. 2, 504 (1959).