Synchrotron as a "macroatom"
Abstract Generated abstract
The paper develops the analogy, attributed to A. A. Vorob'ev, between electron motion in atoms and in cyclic accelerators, treating the synchrotron as a possible "macroatom." It argues from the author's self-oscillatory deterministic quantum mechanics that classical trajectories with superposed quantum oscillations can be meaningfully applied to accelerator electrons, unlike in standard probabilistic quantum mechanics. Focusing on betatron oscillations in the final stages of synchrotron acceleration, the paper derives oscillator energy levels for radial motion and estimates, for a representative synchrotron, an oscillatory quantum energy of about 10^-7 eV, corresponding to radiation near 10^8 s^-1 and a wavelength of about 300 cm. The analysis suggests that each accelerator should have a characteristic limiting oscillatory quantum energy and that analogous quantum effects may be considered.
Full Text
Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Volume 170 1969
THE SYNCHROTRON AS A “MACROATOM”
B. N. Rodimov
As early as 1948, Prof. A. A. Vorob'ev formulated the idea that between the motion of electrons in the hydrogen atom and the motion of electrons in cyclic accelerators there exists a certain similarity, which may be not merely formal. Figuratively speaking, an accelerator in certain respects may be regarded as a “macroatom.” Prof. Vorob'ev suggested to the author of these lines that he take up this problem, but at that time the author treated it skeptically. Later, proceeding from this idea, Prof. A. A. Sokolov and his collaborators carried out a large cycle of work on the study of the radiation of electrons in electron synchrotrons at high energies (of the order of billions of electron-volts) and on the identification of the effect of quantum phenomena on the motion of electrons in a synchrotron.
The similarity between such a “macroatom” and an ordinary “microatom” has a deeper physical basis than could have been expected from ordinary quantum-mechanical notions. In considering the motion of electrons in an accelerator, one uses the classical Maxwell theory and assumes that the electrons in the accelerator move along quite definite trajectories. Of the trajectories of an electron in an atom, ordinary quantum mechanics refuses to speak on principle. The Bohr orbits of the electron in the hydrogen atom are mentioned only in the historical sense or for “popularization.”
However, if one recalls that near the nucleus of the hydrogen atom, at the radius of the first Bohr orbit, the intensity of the Coulomb electric field has a magnitude of the order of a billion volts per centimeter, then it may be supposed that the definite spatial distribution of the probability of the electron’s presence, which is predicted by ordinary quantum mechanics, has a deep physical basis that is not investigated by ordinary quantum mechanics.
It can be shown that from Einstein’s theory of relativity there follows the necessity for the existence of still another quantum mechanics, which, in contrast to ordinary quantum mechanics, having a statistical probabilistic character, is a deterministic theory [1]. From this quantum mechanics it follows that the Bohr elliptical orbits, taken with a certain refinement, exist as the classical basis for real, more complex trajectories obtained as a result of the superposition, upon the classical motion, of a special quantum oscillatory motion. These orbits can be calculated by this mechanics with the same degree of reliability with which the probability distribution of the electron’s presence is calculated by means of the Schrodinger equation.
The “Chinese wall” between micro- and macro-phenomena, which arises if one takes into account the existence of only one
quantum mechanics, in fact, does not exist. Hence it follows that the study of the concrete motion of electrons in atoms and molecules can also contribute much to the understanding of many features of particle motion in such manifestly macroscopic objects as charged-particle accelerators.
Let us dwell briefly on some specific questions concerning the motion of electrons in synchrotrons.
From the standpoint of the second, self-oscillatory quantum mechanics, we may expect a number of definite quantum effects connected with the fact that, superimposed on any classical motion of particles, there is also a special quantum motion. The apparatus of ordinary quantum mechanics also makes it possible to investigate these effects (see the works of A. A. Sokolov, Ternov, and others), but there this turns out to be fundamentally inconsistent, since in ordinary quantum mechanics one cannot speak of electron trajectories.
Let us consider the picture of phenomena connected with betatron oscillations of electrons in a synchrotron at the final stages of acceleration.
The focusing magnetic field of the synchrotron may be expressed by the formula [2]:
\[ V_{\mathrm{rel}}=\frac{m_0c^2}{e}\left[1+\frac{2eV_{\mathrm{mo}}}{m_0c^2}\right]^{\frac12}. \tag{1} \]
An electron moving in a circular orbit and oscillating in radius in this potential well will constitute an oscillating rotator. The radial focusing force acting on the electron is written as
\[ \frac{d}{dt}[mr]=-e\frac{\partial V_{\mathrm{rel}}}{\partial r}. \tag{2} \]
For a small rate of change of the mass,
\[ m\frac{d^2r}{dt^2}=\sqrt{1-\frac{v^2}{c^2}}\cdot\frac{\partial V_{\mathrm{mo}}}{\partial r}. \tag{3} \]
Since, for small deviations from the equilibrium orbit,
\[ V_{\mathrm{mo}}=\frac{eH^2R^2}{2m_0c^2}\left\{1+(1-n)\frac{x^2}{R^2}+h\frac{z^2}{R^2}\right\}, \tag{4} \]
then
\[ \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\frac{d^2r}{dt^2} + \frac{e^2H^2R^2}{2m_0c^2}\sqrt{1-\frac{v^2}{c^2}}\cdot\frac{2x}{R^2}. \tag{5} \]
or
\[ m\ddot{x}=-kx;\qquad (x=r-R), \]
where
\[ k=\frac{e^2H^2}{2m_0c^2}\sqrt{1-\frac{v^2}{c^2}}. \tag{6} \]
Taking the expression for the energy of the oscillatory levels of the oscillator in the form
\[ E_{\mathrm{osc}}=\frac{h}{2\pi}\sqrt{\frac{k}{m}}\left(n+\frac12\right), \tag{7} \]
or
\[ E_{\mathrm{osc}}=\frac{h}{2\pi}\sqrt{\frac{K}{m}}\,n, \tag{8} \]
we obtain, for the following data:
\[ R=423\ \mathrm{cm},\quad H=10^{4}\ \mathrm{Oe},\quad E_{\mathrm{kin}}=2\cdot10^{9}\ \mathrm{eV};\quad m=3\cdot10^{-24}\ \mathrm{g}. \]
\[ K=10^{-8}. \]
Hence the energy of the oscillatory quanta is:
\[ \Delta E_{\mathrm{osc}}=10^{-19}\ \mathrm{erg}=10^{-7}\ \mathrm{eV}=h\nu \tag{9} \]
and the corresponding frequency and wavelength of the radiation are:
\[ \nu=10^{8}\ \mathrm{s}^{-1},\quad \lambda=3\cdot10^{2}\ \mathrm{cm}. \]
Since
\[ k=\frac{e^{2}H^{2}}{m_{0}c^{2}}\sqrt{1-\frac{v^{2}}{c^{2}}},\quad H=\frac{mvc}{ek}, \tag{10} \]
then
\[ \frac{k}{m}=\frac{v^{2}}{R^{2}}\simeq\frac{c^{2}}{R^{2}}=\mathrm{const}. \tag{11} \]
Thus
\[ \Delta E_{\mathrm{osc}}=\sqrt{\frac{k}{m}}=\frac{c}{R}\,\mathrm{const}. \tag{12} \]
That is, for each accelerator there must exist its own limiting energy of the oscillatory quanta.
In an analogous way, other quantum effects may also be investigated, up to, as Prof. A. A. Vorob’ev believes, the Stark and Zeeman effects.
References
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B. N. Rodimov. Self-Oscillatory Quantum Mechanics, Tomsk University Press, 1967.
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B. N. Rodimov and P. A. Cherdantsev. Articles in Izv. TPI, vol. 87, 1957.