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PHYSICS
A. A. SOKOLOV and I. M. TERNOV
ON A QUASICLASSICAL INTERPRETATION OF QUANTUM EFFECTS IN THE THEORY OF A RADIATING ELECTRON
(Presented by Academician N. N. Bogolyubov, 18 X 1957)
- In the present work we wish to give a quasiclassical interpretation of the quantum effects which, according to the conclusions of quantum theory, should manifest themselves in the motion of an ultrarelativistic electron in a magnetic field \((^{1,2})\).
As is known, the classical equations describing the motion of an ultrarelativistic electron in a synchrotron, with radiation taken into account, have the form
\[ \frac{E\ddot z}{c^2}+\frac{\dot E\dot z}{c^2}-\frac{er\dot\varphi H_r}{c} = -\frac{\dot z W^{\mathrm{cl}}}{c^2}, \tag{1} \]
\[ \frac{E\ddot r}{c^2}+\frac{\dot E\dot r}{c^2}-\frac{Er\dot\varphi^{\,2}}{c^2} +\frac{er\dot\varphi H_z}{c} = -\frac{\dot r W^{\mathrm{cl}}}{c^2}, \tag{2} \]
\[ \dot E=\dot E^{\mathrm{ext}}-W^{\mathrm{cl}}, \tag{3} \]
where \(r, z, \varphi\) are cylindrical coordinates; \(H_z=Br^{-q}\), \(H_r=-qzBr^{-(q+1)}\) are the components of the magnetic-field intensity; \(q\) is the index of decrease of the magnetic field \((0<q<1)\). In the classical case, for the radiation energy we shall have the expression
\[ W^{\mathrm{cl}}=\frac{R^2}{\rho^2}\,\overline{W}^{\mathrm{cl}}, \qquad \overline{W}^{\mathrm{cl}}=\frac{2}{3}\frac{e^2c}{R^2}\left(\frac{E}{m_0c^2}\right)^4, \tag{4} \]
where \(\overline{W}^{\mathrm{cl}}\) is the mean radiation energy, which does not depend on the coordinate \(x=r-R\) of the betatron oscillations, while the radius of the instantaneous equilibrium orbit \(R\) and the radius of curvature \(\rho\) may be found from the equality
\[ E=eBR^{1-q}=eBr^{-q}\rho. \tag{5} \]
Hence, in the linear approximation we find
\[ \rho\simeq R+qx. \tag{6} \]
The expression for the energy received by the electron from the accelerating device (the fundamental harmonic), in the linear approximation of phase oscillations, may be represented in the form
\[ \dot E^{\mathrm{ext}} = \frac{cV_0}{2\pi r}\sin(\varphi-\omega_0 t) = \frac{cV_0}{2\pi r}(\sin\varphi_0+\psi\cos\varphi_0), \tag{7} \]
where \(V_0/e\) is the amplitude of the potential in the accelerating gap; \(\omega_0\) is the angular frequency of the high-frequency field, related to the radius of the synchrotron equilibrium orbit by the relation \(R_0=c/\omega_0\). Here \(\omega_0 t+\varphi_0\) characterizes the uniform rotation of the electron, while \(\psi=\varphi-\omega_0 t-\varphi_0\) contains only the oscillatory part.
Taking the latter relations into account, instead of (1)—(3) we shall have (in the linear approximation):
\[ \ddot z+\gamma \dot z+\omega_z^2 z=-\frac{\overline W^{\mathrm{cl}}}{E}\dot z; \tag{8} \]
\[ \ddot x+\gamma \dot x+\omega_r^2 x=-\frac{\overline W^{\mathrm{cl}}}{E}\dot x-\ddot R-\frac{\dot x\dot R}{R}, \tag{9} \]
\[ \dot rE=(1-q)E\left(\dot R+\frac{x\dot R}{R}\right) =\frac{V_0c}{2\pi}(\sin\varphi_0+\psi\cos\varphi_0)-r\overline W^{\mathrm{cl}}, \tag{10} \]
where \(\omega_z=\sqrt{q}\,c/R,\quad \omega_r=\sqrt{1-q}\,c/R\). Eliminating from the right-hand side of (9) the quantity \(\ddot R+\dot x\dot R/R\), we find, for the description of radial betatron oscillations, the equation
\[ \ddot x+\gamma \dot x+\omega_r^2 x=-\frac{q}{1-q}\frac{\overline W^{\mathrm{cl}}}{E}\dot x, \tag{11} \]
i.e., upon introducing a continuous friction force in the theory of betatron oscillations, along with the “small” damping coefficient \(\gamma=\dot E/E\), one must also take into account the large damping coefficients \((^{3-5})\)
\[ \Gamma_z=\frac{\overline W^{\mathrm{cl}}}{E}\ll\gamma,\qquad \Gamma_r=\frac{q}{1-q}\Gamma_z. \tag{12} \]
Introducing the coordinate \(X=R-R_0\), related to the phase \(\psi\) by means of the relation \(X=-R_0^2\dot\psi/c\), and taking into account that \(\overline W^{\mathrm{cl}}=\left(R/R_0\right)^{2-4q}W_0\) (\(W_0\) corresponds to \(R=R_0\)), we obtain the following equation for determining the phase oscillations:
\[ \ddot\psi+\Gamma\dot\psi+\Omega^2\psi =-\frac{c}{(1-q)R_0^2E_0} \left(\frac{cV_0\sin\varphi_0}{2\pi}-R_0W_0\right)=0, \tag{13} \]
where the point of equilibrium phase must be found from the equation
\[ \frac{cV_0}{2\pi R_0}\sin\varphi_0=W_0; \tag{14} \]
the angular frequency of the slow synchrotron oscillations is equal to
\[ \Omega= \sqrt{\frac{c^2V_0\cos\varphi_0}{2\pi(1-q)R_0^2E_0}} = \sqrt{\frac{cW_0\operatorname{ctg}\varphi_0}{(1-q)R_0E_0}} \ll \omega_r; \tag{15} \]
\[ \Gamma=\frac{3-4q}{1-q}\frac{W_0}{E}\sim \Gamma_r \]
is the “large” damping coefficient of synchrotron oscillations.
2. As is known, in the radiation of a luminous electron it is necessary to estimate the influence of the discreteness factor of the radiation, since, according to quantum theory, during one revolution a comparatively small number of quanta is emitted,
\(N\sim \alpha\,E/(m_0c^2)\) \((^2)\), especially when \(E<E_{1/5}\).
To describe the discrete character of the radiation, we introduce the fluctuation force
\[ \mathbf F^{\mathrm{fluct}} =-\frac{\mathbf v}{c^2}\sum_i \left\{ \frac{R}{\rho}\,\varepsilon\,\delta(t-t_i) -\frac{R^2}{\rho^2}\overline W^{\mathrm{cl}} \right\}. \tag{16} \]
The introduction of the fluctuation force is formally equivalent to the fact that in equations (8), (9), (10) one should replace the quantities according to the scheme (\(\mathbf v\)— no-
number of the emitted harmonic):
\[ \overline{W}^{\mathrm{cl}} \to \sum_i \overline{\varepsilon}\delta(t-t_i) = \sum_i \hbar \frac{c}{R}\,\gamma\delta(t-t_i), \]
\[ W^{\mathrm{cl}} \to \frac{R}{\rho}\sum_i \overline{\varepsilon}\delta(t-t_i) = \sum_i \left(1-q\frac{x}{R}\right)\hbar\frac{c}{R}\,\gamma\delta(t-t_i). \tag{17} \]
Then these equations take the form
\[ \ddot z+\gamma\dot z+\omega_z^2 z = -\sum_i \frac{\dot z}{E}\,\overline{\varepsilon}\delta(t-t_i) = -\sum_i \frac{c\cos\theta}{E}\,\overline{\varepsilon}\delta(t-t_i); \tag{18} \]
\[ \ddot x+\gamma\dot x+\omega_r^2 x = -\sum_i \frac{\dot x}{E}\,\overline{\varepsilon}\delta(t-t_i) -\ddot R-\frac{\dot x\dot R}{R}, \tag{19} \]
\[ r\dot E=(1-q)E\left(\dot R+\frac{x\dot R}{R}\right) = \frac{cV_0}{2\pi}(\sin\varphi_0+\psi\cos\varphi_0) -\sum_i R\overline{\varepsilon}c\left[1+(1-q)\frac{x}{R}\right]\delta(t-t_i). \tag{20} \]
Equation (18) characterizes axial oscillations with allowance for quantum fluctuations.
In order to obtain the final equations for radial oscillations, we must substitute into the right-hand side of (19), instead of \(\ddot R+\frac{x\dot R}{R}\), the value from (20), while discarding terms proportional to \(\psi\). Then we find:
\[ \ddot x+\gamma\dot x+\omega_r^2 x = \frac{R}{E(1-q)}\frac{d}{dt}\sum_i \overline{\varepsilon}\delta(t-t_i). \tag{21} \]
Let us note that the energy of the fluctuational radiation is inversely proportional to \(\rho\) (6), and not to \(\rho^2\), as in the classical case. However, it should be taken into account that the transition probability is also inversely proportional to \(\rho\):
\[ w(\gamma,t_i)=\frac{R}{\rho}\,\overline{w}(\gamma,t_i), \tag{22} \]
where
\[ \overline{w}(\gamma,t_i) = \frac{e^2}{\pi\sqrt{3}\,c\hbar}\frac{c}{R} \left(\frac{m_0c^2}{E}\right)^2 \int_{\frac{2}{3}\gamma\left(m_0c^2/E\right)^3}^{\infty} K_{5/3}(\xi)\,d\xi . \tag{23} \]
Therefore, in passing to continuous radiation we again obtain the classical formula
\[ \lim_{\Delta t_i\to 0}\sum_i \overline{\varepsilon}\delta(t-t_i) = \int_0^\infty d\gamma\int_0^\infty \frac{R}{\rho}\,\overline{\varepsilon}\,\frac{R}{\rho}\, \overline{w}(\gamma,t_i)\delta(t-t_i)\,dt_i = \left(\frac{R}{\rho}\right)^2\overline{W}^{\mathrm{cl}}. \tag{24} \]
In exactly the same way, the “large” damping coefficients can be obtained by the limiting transition from quantum fluctuational forces to continuous radiation:
\[ \lim_{\Delta t_i\to 0} \frac{R}{E(1-q)}\frac{d}{dt}\sum_i \overline{\varepsilon}\delta(t-t_i) = \]
\[ = \frac{R}{E(1-q)}\frac{d}{dt} \int_0^\infty d\gamma\int_0^\infty \overline{\varepsilon}\left(1-\frac{qx}{R}\right) \overline{w}\delta(t-t_i)\,dt_i = -\Gamma_r\dot x \tag{25} \]
and so on.
For the square of the amplitude of radial oscillations, taking into account the fluctuation force, from (21) we find the value
\[ a^{2}=a_{0}^{2}\exp\left[-\int_{0}^{t}\gamma(t')\,dt'\right]+ \]
\[ +\frac{55}{24\sqrt{3}(1-q)^{2}}\frac{e^{2}}{m_{0}c}\frac{\hbar}{m_{0}cR_{0}} \int_{0}^{t}\exp\left[-\int_{t'}^{t}\gamma(t'')\,dt''\right] \left(\frac{E(t')}{m_{0}c^{2}}\right)^{5}dt' . \tag{26} \]
To obtain the formula given in (2,6), one should take into account that
\[ \exp\left[-\int_{t'}^{t}\gamma(t'')\,dt''\right]=\frac{E(t')}{E(t)}. \]
For axial oscillations the corresponding expression will be approximately \((E/m_{0}c^{2})^{2}\) times smaller. In the same papers (3–5), it is proposed, along with the introduction of the fluctuation force acting on betatron oscillations, also to retain the “large” classical value for the damping coefficients \((\Gamma_z,\Gamma_r)\).
Let us next examine synchrotron oscillations in the presence of quantum fluctuations. Setting in formula (20) \(x=0\) and \(X=-\frac{R_{0}^{2}}{c}\dot{\psi}\), we find
\[ R\dot{E}=-\frac{E_{0}R_{0}^{2}(1-q)}{c}\ddot{\psi} =\frac{E_{0}R_{0}^{2}(1-q)\Omega^{2}}{c}\psi +\frac{V_{0}c\sin\varphi_{0}}{2\pi} -\sum_i \hbar\nu c\,\delta(t-t_i). \tag{27} \]
Taking into account that the period of synchrotron oscillations \(T=2\pi/\Omega \gg 2\pi/\omega_r\) is much greater than the mean time interval \(\Delta t_i=t_{i+1}-t_i\) between two successive acts of radiation, we can average expression (27) over a time interval \(\Delta t\) satisfying the condition \(\Delta t_i\ll\Delta t\ll T\). Then for the phase oscillations we obtain the classical equation, with the fluctuation quantum force turning into a friction force with a “large” damping coefficient \(\Gamma\)*. This is apparently connected with the circumstance that phase oscillations are caused by a change in the total energy, on which quantum effects should have an influence only at very large values \(E\sim E_{l_2}\) (1), whereas the action of the fluctuation force on radial betatron oscillations should manifest itself at comparatively low energies \(E\sim E_{l_s}\) (2).
Moscow State University
named after M. V. Lomonosov
Received
14 X 1957
CITED LITERATURE
- A. A. Sokolov, N. P. Klepikov, I. M. Ternov, ZhETF, 24, 249 (1953).
- A. A. Sokolov, I. M. Ternov, ZhETF, 25, 698 (1953); DAN, 97, 823 (1954).
- A. A. Kolomensky, A. N. Lebedev, CERN, Symposium, 1, Geneva, 1956, p. 447.
- M. S. Livingston, CERN, Symposium, 1, Geneva, 1956, p. 439.
- I. Henry, Phys. Rev., 106, 1057 (1957).
- A. N. Matveev, DAN, 107, 671 (1956).
- M. Sands, Phys. Rev., 97, 470 (1955).
- A. N. Matveev, Nuovo Cim., 5, 1782 (1957).
* In papers (3, 7, 8), devoted to phase oscillations, both the fluctuation force and the “large” damping coefficient are taken into account.