Cyclic Electron Accelerator as a Macromodel of an Atom (of the Hydrogen or Positronium Type)

A. A. VOROB'EV

Accelerated electrons come to the Earth in streams from space and are recorded as one variety of cosmic rays. Their action on everything living and nonliving, their action on matter, is of vital interest, especially for cosmonautics. Electron beams treat bodies; for example, they weld pieces of metal, melt it, in large doses kill living organisms, and in small doses can stimulate their vital activity. When electrons are decelerated, their energy is converted into heat, and part of the electron’s energy is emitted in the form of X-rays. The greater the energy of the electrons that have been decelerated, the more penetrating the resulting X-rays are through bodies. The practical application of bremsstrahlung radiation, as well as of accelerated electrons, in science and practice is steadily increasing.

The acceleration of electrons under laboratory conditions is carried out in a vacuum by means of an electric field. The simplest accelerating device is the flight gap between two charged electrodes. Since the operating voltage between the electrodes, which accelerates the electrons, is limited by the finite value of the electric strength of the gap, in one gap an electron can receive only a limited increment of energy. To increase the energy of the accelerated electron considerably, many gaps must be placed in the electron’s path. Such an accelerator becomes very long. For example, in the USA an accelerator has been built whose total length is about four kilometers. Such a structure, and operation with many accelerating gaps arranged along one straight line, are associated with many difficulties. One can construct an installation with a single accelerating gap through which the electron will pass many times. For this it is necessary to turn the electron that has emerged from the accelerating gap, guide it along a circular trajectory, and pass it again through the accelerating gap. And this must be done many times. Such a mechanism is implemented in modern cyclic accelerators of the synchrotron type.

An electron, moving in a circle, will be accelerated only while passing through the gap, which occupies a small part of the circumference, and in modern high-energy accelerators even a very small part of the path. Along the remaining part of the circular path the electron will move with constant velocity.

In nature there is an analogy of such a system in the form of hydrogen-like atoms. The simplest model of the hydrogen atom consists of a proton

and an electron, while the model of a positronium atom consists of a positron and an electron, which rotate about the common center of gravity of these two bodies. Such systems possess many interesting properties, well studied theoretically and experimentally. One of the important properties of the hydrogen atom is the stable motion of the electron in an orbit and the absence of energy losses in this process.

The motion of an electron along stable orbits in the model of a hydrogen-like atom in general, and of the hydrogen and positronium atoms in particular, is a typical problem of motion in a field of central forces. The interaction energy of the electron with the field depends only on the distance of the electron from the center in which the atomic nucleus is located. The problem of the motion of a single electron in the field of a point positive charge is a variety of the Kepler problem. The motion of an electron along a stable orbit in an atom takes place under the action of Coulomb attraction and a central force. In quantum mechanics the state of an electron in a hydrogen atom, according to the Schrödinger equation, is described by the wave function

\[ \psi_{nlm}=C_{nlm}R_{nl}(\rho)Q_{ml}(\nu)e^{im\varphi}, \]

where \(C_{nlm}\) is a normalization factor determined from the condition:

\[ \int_{\infty} |\psi_{nlm}|^{2}\,d\nu=1, \]

\[ R_{nl}=e^{-\frac{1}{2}\rho}\, \frac{d^{2l+1}}{d\rho^{2l+1}} \left\{ e^{\rho}\frac{d^{n+l}}{d\rho^{n+l}} \left(\rho^{n+l}e^{-\rho}\right) \right\}, \]

\[ Q_{ml}(x)=(1-x^{2})^{\frac{|m|}{2}}\cdot \frac{d^{\,l+|m|}}{dx^{\,l+|m|}}(x^{2}-1)^{l}, \]

\[ \rho=\frac{r}{na_{0}},\qquad a_{0}=\frac{h^{2}}{\mu e^{2}},\qquad x=\cos\nu,\qquad h=1.05\cdot10^{-2}\ \text{erg}\cdot\text{sec}. \]

\(\mu\) is the reduced mass of the electron and proton; \(n=1,2,3\ldots;\ l=0,1,2\ldots n-1;\ m=0,\pm1,\pm2,\ldots\pm l.\)

In these stationary states the electron has the energy

\[ E_n=-\frac{\mu e^{4}}{2h^{2}}\cdot\frac{1}{n^{2}}, \]

the square of the angular momentum:

\[ M^{2}=l(l+1)h^{2}, \]

the projection of the angular momentum \(M_z=mh\), and it creates a current with mean density:

\[ \vec{j}=-\frac{ieh}{2\mu} \left(\psi_{nlm}^{*}\nabla\psi_{nlm} -\psi_{nlm}\nabla\psi_{nlm}^{*}\right), \]

where in spherical coordinates \(j_z=j_\nu=0,\ j_\varphi=-\dfrac{ehm}{\mu r\cdot\sin\nu}|\psi_{nlm}|^{2}\).

Associated with this current is a magnetic field whose projection of the magnetic moment is equal to

\[ M_z^{\mu}=\frac{eh}{2\mu c}\,m. \]

From the theory of the atom represented by these formulas it follows, in particular, that in the normal state of the atom \((\psi_{100})\)

1. \(E_1=-13.6\ \mathrm{eV},\)
2. \(M^2=0,\)
3. \(M_z=0,\)
4. \(\bar{j}=0,\)
5. \(M_z'=0.\)

Consequently, the normal state of the atom is momentless \((M=0)\). It has no classical analogue. The volume density of the probability of finding the electron is written in the form

\[ |\psi_{100}|^2=\frac{1}{\pi a_0^3}\,e^{-\frac{2r}{a_0}}. \]

The distance of the electron from the proton in any direction corresponding to the maximum probability is \(r_1=a_0=0.529\cdot 10^{-8}\ \mathrm{cm};\) the mean distance is \(\bar r_1=\frac{3}{2}a_0.\)

If, in the bound state arising as a result of the interaction of an electron with a proton in the hydrogen atom, Coulomb’s law continues to act in the same form as in macroscopic physics, then at the distance \(r_1\) the Coulomb field of the proton holding the electron will have the strength

\[ \varepsilon_1=\frac{e}{r_1^2}=3.5\cdot 10^{16}\ \mathrm{C.G.S.E.} \]

In a synchrotron the acceleration of the electron takes place in a resonator, whose accelerating gap occupies a small part of the circumference. For the greater part of the circular path the electron moves with constant speed. The stable motion of a particle in a cyclic accelerator and of an electron in a hydrogen-like atom are equally considered with the aid of a potential function of the form

\[ U(r)=r^{-q}, \]

where \(r\) is the radius of the orbit. For a cyclic accelerator the induction of the controlling field varies according to the law: \(B(r)\approx r^{-q}\).

In a hydrogen-like atom and in a cyclic accelerator the particle performs stable motion along an equilibrium orbit. In the case when identical energy relations of the form are satisfied:

\[ U(r)\cdot r^q=\mathrm{const}. \tag{1} \]

Depending on the magnitude of the constant in (1), the radiation will have a quantum character when the quantum numbers are small and the energy levels are discrete, and a classical one when the quantum numbers are large and the energy levels are continuous.

This external analogy, as well as the analogy of the mathematical theory of the motion of an electron in the field of central forces along a closed circle in the case of the simplest model of the hydrogen atom and of a cyclic accelerator, led me as long as a quarter of a century ago to the idea of considering the cyclic accelerator as a large model of the hydrogen atom. In such a consideration it seemed possible to expect that during acceleration phenomena characteristic of the hydrogen atom would be observed; for example, the hydrogen atom emits light in the form of separate lines. Under external actions (by means of magnetic and electric fields, high pressure) one can change the radiation spectrum of hydrogen.

An electron moving in a circle in a homogeneous magnetic field continuously emits electromagnetic waves owing to the presence of centripetal acceleration. The radiation is accompanied by a decrease in the electron’s energy and by its transition to an orbit of smaller radius. In the model of a hydrogen-like atom, an electron moving in a circle in an electrostatic inhomogeneous field does not lose its energy to continuous radiation. Radiation in the hydrogen atom occurs when the electron passes to an orbit of smaller radius and is accompanied by the release of an energy quantum.

Already at the beginning of our century an attempt was made to explain the emission spectra of atoms by means of the classical theory of radiation of an electron moving in a homogeneous magnetic field. This theory may be regarded as an attempt to find an analogy in the description of the radiation of an atom and of an electron in a cyclic accelerator.

Let us note the following. The radiation of an electron moving in a constant magnetic field is accompanied by a decrease in the electron’s energy. In the case of circular motion of an electron in an increasing vortex magnetic field of axial symmetry, the electron’s energy increases at the expense of this field. The electron continuously radiates part of its energy. Consequently, an electron in a cyclic accelerator will radiate the energy received by it from an external source.

The quantum, discrete character of radiation is a feature of the hydrogen atom that has been well studied theoretically and experimentally. In the works of A. A. Sokolov, I. M. Ternov, and D. D. Ivanenko, it was shown by the methods of quantum electrodynamics that an electron moving in a cyclic accelerator, beginning with an energy of 500 MeV, can emit energy quanta. By emitting a photon at some point of the orbit, the electron experiences a recoil impulse that carries it away from the equilibrium orbit. As a result of photon emission the electron undergoes radial and axial oscillations about the equilibrium orbit. A. A. Sokolov’s theory made it possible to find the magnitude of the radiation intensity and the values of the amplitude of the electron’s oscillations. It turned out that these quantities are proportional to Planck’s constant, which is a consequence of the quantum nature of the phenomenon, just as in the hydrogen atom. Therefore a cyclic accelerator at high electron energies may be regarded as a macroscopic model of the hydrogen atom. Thus, the analogy between hydrogen-like atoms and cyclic accelerators proves to be not only external, but also physical in essence.

Contradictions arose during the development of the theory. V. V. Vladimirskii and others at first believed that in cyclic accelerators quantum effects should be neglected in the region where the total energy of the electron is considerably greater than the energy of the emitted photon. Therefore quantum corrections would play a practically noticeable role only in the region of electron energy \(10^{13}\) eV, which exceeds by several orders of magnitude the energy limit indicated by A. A. Sokolov’s theory (about 600 MeV).

What is essential is that only the discrete, quantum character of radiation is inherent in the hydrogen atom. An electron accelerated in a cyclic accelerator radiates continuously in the region of all energies, while photons only in the region of high energies. The classical continuous radiation of an electron in an accelerator causes a decrease in its energy, orbit radius, and amplitude of betatron oscillations. Quantum radiation of an electron in an orbit in an accelerator causes a decrease in its energy and, consequently, also of the orbit radius and damping of oscillations, while the occurrence of recoil in the emission of quanta is accompanied by an increase in the amplitude of oscillations. Since the phenomena under consideration

are being investigated as applied to a synchrotron, one must also take into account the change in energy due to phase oscillations. The resulting effect must be composed of the damping of betatron oscillations due to the influence of classical radiation and the increase of the amplitude of betatron and phase oscillations due to the emission of quanta.

The quantum character of the radiation of electrons in a cyclic accelerator was discovered and investigated in the theoretical works of A. A. Sokolov and I. M. Ternov. In these same works calculations were made of the acceleration conditions under which, owing to the emission of quanta, electrons would deviate strongly from the equilibrium orbit, strike the walls of the accelerating chamber and, consequently, fall out of the beam. Physical limits were determined up to which acceleration in such installations is practically possible. This was a very important result, concerning the physical limit of operation of synchrotrons. In the works of other scientists the conclusions of A. A. Sokolov provoked serious objections.

When electrons are accelerated in a homogeneous magnetic field, automatic stabilization of the motion is absent. Electron acceleration is then generally unstable. The emission of a quantum by electrons disrupts the stability and, in the absence of transverse stability of the system, leads to a buildup of oscillations and to the loss of electrons from the beam.

A. A. Kolomenskii and A. N. Lebedev considered the acceleration of electrons in a focusing inhomogeneous magnetic field, which automatically provides transverse stability of the motion of electrons within certain limits of deviation of the orbit from the equilibrium one. According to their data, in focusing fields there occurs damping of the oscillations of electrons that arise upon the emission of a quantum of radiation. The discussion made it possible to clarify many questions and, in particular, indicated a new, higher limit of the energies of electron acceleration in synchrotrons.

From the ideas of classical theory it was to be expected that, with an increase in the energy of the electrons, the electron beam, as a result of the superposition of the magnetic fields of parallel currents, should contract. On the other hand, the interaction of relativistic electrons in the beam leads to transverse instability in the motion of the electrons and to broadening of the beam. According to the macroatom model and A. A. Sokolov’s quantum theory, the beam in the region of high energies should expand upon emitting light quanta, just as in the hydrogen atom, be displaced from the stable orbit and thus broaden. It was important to verify the formulas of the theory of electron acceleration and to establish whether, in the high-energy region, the electron beam contracts, as follows from the classical theory of the interaction of magnetic fields of parallel currents, or expands, as follows from the theory of the macroatom.

During the construction of the Tomsk synchrotron for an energy of 1500 MeV, it was proposed to investigate the above-mentioned contradictions in the theory of electron acceleration; to test the macroatom hypothesis; and to determine the physical limit of the attainable energies of electrons in cyclic synchrotron accelerators with weak focusing of the “Sirius” type.

In practice, the change in the oscillation amplitude should affect the size of the spot diameter. With an increase in the total energy of an electron moving on the equilibrium orbit, classical continuous radiation, as well as quantum radiation, is accompanied by a decrease in its energy, a decrease in the amplitude of oscillations, and a decrease in the orbital radius. The recoil experienced by the electron causes an increase in the amplitude of oscillations with the growth of the electron’s total energy. The resultant amplitude of the radial oscillations of the electron, as its total energy grows, will be composed of a component that decreases with the grow-

energy, and a component whose magnitude increases with an increase in the total energy of the electron. The superposition of two such dependences in a certain energy region may give a constant value. The amplitude of the radial oscillations of the electrons in some energy interval may turn out to be a constant quantity, independent of the total energy of the electron. The energy region in which “quantum” fluctuations of the radiation begin may be shifted by adjusting the parameters of the accelerator. In this way the energy region in which the independence of the amplitude of the radial oscillations of the electrons from the energy will be observed may also be shifted.

In the Sirius synchrotron, A. V. Kozhevnikov measured the width of the beam of accelerated electrons by optical methods. In agreement with the theory of A. A. Sokolov, up to an energy of 600 MeV a decrease in the transverse dimensions of the electron beam was observed. In the electron-energy interval from 600 to 700 MeV the transverse dimensions of the beam remained approximately constant. A further increase in the total energy of the electrons, owing to the predominance of the role of the quantum character of the radiation, was accompanied by a broadening of the electron beam. The experimental results over a wide energy range correspond to theoretical expectations. This makes it possible to regard a cyclic accelerator as a quantum system and as a macroscopic model of a hydrogen-like atom. In this system one can simultaneously model both Bohr’s postulates concerning the existence of a discrete set of orbits (in the region of low energies) and the propositions of later quantum mechanics concerning the smearing of Bohr orbits (in the region of high energies).

The synchrotron is the product of human creation. Consequently, phenomena in the hydrogen atom can be modeled according to a prescribed program with the aid of a synchrotron. For example, axial and radial oscillations of electrons in a synchrotron or storage ring will probably affect the radiation spectrum of electrons in cyclic accelerators, modeling the action of an external electric and magnetic field on the radiation spectrum of a hydrogen-like atom.

It is of interest to investigate in greater detail the analogy at lower particle energies in cyclic accelerators, when the quantum character of the radiation does not manifest itself and the loss of energy by the particles occurs continuously. It is therefore interesting to determine what phenomenon in the hydrogen atom will be analogous to this phenomenon: either the processes of transition from one energy level to another at very large principal quantum numbers, when the energy of the measured quanta will be small, or whether, in cyclic accelerators, such conditions are possible under which electrons will move without radiating, as in their motion along Bohr orbits. If the latter is correct, then the creation of a radiationless cyclic accelerator is possible.

In the two systems under consideration—the atom and the macroatom—one should speak not only of analogies when scale relations are observed, but also of differences when the scales are not observed. In the atom, motion takes place in three-dimensional space in the Coulomb field of central forces. For such an atomic system the postulate is valid that an electron, moving in an orbit, does not radiate, despite the presence of centripetal acceleration of very large magnitude.

According to the arguments presented, at high energies one may expect the emission of gravitons by masses moving according to the Kepler–Newton laws during the transition of a particle from one orbit to another. In the region of low energies, masses moving in a cyclic accelerator must emit gravitational waves continuously.

In cyclic accelerators at high energies, the radiation spectrum of the electrons will be quasi-continuous, and two neighboring energy states of the electron will differ little from one another. Since the electron energy is large, this small value of the electron energy proves to be sufficiently large and will correspond to the energy of quanta in the visible or ultraviolet regions of the spectrum.

It is known that the wave and corpuscular nature of electromagnetic radiation have long been opposed to one another.

In the case of a cyclic accelerator there is the interesting feature that an electron accelerated to high energy radiates energy continuously, while at certain moments of its motion it also emits individual quanta.

In cyclic acceleration, as the energy of the electrons increases, against the background of their continuous radiation there appears discrete, quantum radiation. The accumulation of a parameter—the electron energy—leads to the emergence of a new quality of the system in the form of discrete radiation.

Thus, this system possesses both classical and quantum properties. Accordingly, one may speak of the unity of the corpuscular and wave properties of the radiation of this material system.

In the present case there is an example in which the discrete and the continuous are properties of the system that do not exclude one another. In the case under consideration, intermittent and continuous radiation can appear together.

Continuous radiation is observed in the region of lower energies. It is possible to propose that in the region of high parameters of the system, as in the case of the atom \((\varepsilon \approx 10^{16}\ \text{c. d. s. e.})\), the accelerated electron will not radiate continuously. The radiating system will produce only discrete radiation. The electron in the accelerator will be free from continuous loss of energy. In such a system, the electrons must be distributed among states satisfying the Pauli principle. At small principal quantum numbers and with a large number of accelerated electrons, this may lead to the beam of accelerated electrons having a finite radial size, due to the finiteness of the states with different \(n\) in which the electrons are found.

The author expresses his gratitude to Doctor of Physical and Mathematical Sciences A. N. Didenko, Professor A. A. Sokolov, and Professor I. M. Ternov for their participation in the discussion of the questions considered in the article and for the comments they made.