Radiation Spectrum of Sequence of Electrons Moving in Spiral in Medium

Integral expressions for spectral-angular and spectral distributions of the radiation power for the sequence of electrons moving in magnetic fields in isotropic transparent medium are investigated using the improved Lorentz`s self-interaction method. Special attention is given to the research of the fine structurre of the spectral distribution of the synchrotron-Cherenkov radiation of one, two, three and four point electrons moving along the spiral in medium. The effects of coherent radiation of harmonics and oscillations in spectrum of the synchrotron-Cherenkov radiation of two, three and four point electrons are established and investigated using the direct numerical method for calculation the function of spectral distributions of the radiation power.


Introduction
Larmor [1][2][3][4] for the first time established that a single charged point particle, which moves with acceleration in vacuum, always radiates electromagnetic waves. The Larmor formula for the power of radiation of point charged particles, which was obtained for the nonrelativistic case, was generalized by the Lienard [5] and Heaviside [6] to the relativistic case.
In 1907, Schott [7][8][9] for the first time strictly within the framework of classical electrodynamics investigated the radiation spectrum of electrons that move in a circle in vacuum. Later G.A. Schott developed classical theory of the radiation of charges moving in the circle and used it for the investigations of a model of atom [9]. His attempt to explain the radiation of the atom on the basis of classical physics was not successful. Due to these reasons, the work of Schott over the course of 40 years has only become an area of academic interest and has practically been forgotten. After 40 years, the Schott formula has been applied to the study of the radiation spectrum of charged particles moving on a macroscopic scale (synchrotron radiation). The main properties of synchrotron radiation of charged particles that move in a magnetic field in vacuum are analyzed in a review of [10] and monographs [11][12][13][14].
The Schott formula [10] being only of academic interest for so long period of time is also related to the fact that Schott only in 1933 [15] established the conditions under which the distributions of charged particles moving with acceleration and performing periodic motion, do not emit electromagnetic waves. Interest in this class of distributions of charged particles and their fields is also due to the possibility of their application to classical, stable models of elementary particles, atoms, and in other cases [16][17][18][19][20].
The radiation spectrum of a sequence of noninteracted charged particles that move along a spiral in vacuum is investigated in [21,22]. Superhigh-power short-wave coherent synchrotron radiation by a sequence of charged particles bunches was studied in [23][24][25].
The classical theory of radiation emitted by charged particles moving with superluminal velocities were traced back to Heaviside [26], Des Condres [27], and Sommerfeld [28][29][30][31]. The classical theory of the Cherenkov phenomenon in a dispersive medium was first formulated by Frank and Tamm in 1937 [32].
The peculiarity in the radiation of charges and multipoles uniformly moving in a medium is analyzed in monographs [33][34][35][36].
The generalized Cherenkov-like effects based on four fundamental interactions have been investigated and classified in [37].
Tsytovich [45], for the first time, examined the case of oscillations in the radiation spectrum of a relativistic charged particle, which moves in a circle, in a constant magnetic field in a medium with dispersion.
The aim of this work is to obtain by the improved Lorentz self-interaction method the basic formulae for the spectral-angular and spectral distributions of the time-average power of the radiation of a sequence (system) moving along an arbitrary given trajectory in a transparent isotropic medium. Using the numerical method of direct numerical calculation of the spectral distribution function of the radiation power of electrons, the fine structure of the radiation spectrum of a sequence of electrons moving along a spiral in a magnetic field in a transparent isotropic medium was studied. Considerable attention is paid to the study of oscillations and coherent radiation near the Cherenkov barrier.

I. Time-Averaged Radiation Power of Charged Particles Moving in a Transparent Medium
According to [46,47,54,55] the time-averaged radiation power rad P of charged particles moving in a transparent isotropic medium is determined by the relationship: Here, ( ) t r j , r r is the current density and ( ) t r, r ρ is the charge density. The integration is performed over some volumeτ . According to the hypothesis of Dirac [55,56] Here, α is a positive infinitesimal, which turns into zero after integration, refraction index The instantaneous radiation power ( ) t P rad , which is expressed in terms of the spectral-angular distribution of the radiation power of charged particles, taking into account relations (1), (2), (3), takes the form: The instantaneous radiation power ( ) t P rad , which is expressed in terms of the spectral-angular distribution of the radiation power , can be obtained from (4), (5) using the relation for the Bessel function of the integer index (see p. 416 in [57]): where ( ) is the Bessel function of zero index.. Then we find: The instantaneous radiation power, which is expressed in terms of the spectral distribution of the radiation power, can be obtained from (7), (8) Relation (11) for the spectral distribution of the instantaneous radiation power , is obtained and investigated by Schwinger et al. [58], using the sources theory.

II. Spectral and spectral-angular distributions of the time-average radiation power of sequence of electrons moving along a spiral in a medium
The current density ( ) where ( ) is the Dirac delta-function.
We study the case of the sequence of electrons moving one by one along a spiral in a transparent medium. The law of motion and the velocity of the th l electron in magnetic field are given by the expressions: Here, We will get the time-averaged radiation power of a sequence of electrons N by substituting (12), (13) into (1), (10), (11). Then we find [54,55]: is defined as: Here, 12 t ∆ is the time shift between the first and second electrons.
The coherence factor ( )   (14) and (15) the contribution of separate harmonics to the timeaveraged radiation power can be expressed as: The coherence factor of a single electron is defined

III. Oscillations in the spectrum of synchrotron-Cherenkov radiation of a sequence of electrons moving along a spiral in a medium
The functions of spectral distribution ( ) ω W of the synchrotron-Cherenkov radiation power of one, two, three and four electrons moving along a spiral in a medium are calculated according to (14), (15) 11   cm/s ( Fig. 1-9) has the character of discrete bands. With further increase in harmonics number, the function of spectral distribution of synchrotron-Cherenkov radiation power of electrons takes on a near-periodical character due to the overlapping of the bands of neighboring harmonics and the contribution of other harmonics (Figs. 1-9).
According to relation (21), the expansion of discrete harmonics into bands is due to the Doppler effect.
For small time shifts between the electrons for the system of two, three, and four electrons in the frequency range of , so far as the dimension of this system is smaller in comparison to the radiation wavelength (Figs.1-3). The sequence moving along a spiral radiates as a charged particle with a charge Ne and rest mass 0 Nm , i.e. by a factor 2 N more than a single electron.   The oscillations of the function of spectral distribution of synchrotron-Cherenkov radiation power of one, two, three, and four electrons moving along a spiral in a medium at n c V med / > ⊥ is determined by the contribution of the Bessel functions [59] (Fig. 7-9). The numerical method of direct integration of the function of spectral distribution of radiation power of one, two, three and four electrons moving along a spiral in a medium allowed us to determine the fine structure of the radiation spectrum of these electrons These studies confirm the fact that the synchrotron-Cherenkov radiation of one, two, three, and four electrons is an unified process [53].

Conclusions
1. For a small longitudinal velocity component at low harmonics, the radiation bands of electrons, moving along a spiral in a medium, are discrete.
2. The influence of the Doppler effect determines the boundaries of the bands of individual harmonics in the spectra of synchrotron-Cherenkov radiation of one, two, three and four electrons, moving along a spiral in a medium.
3. For small time shifts between the electrons for the sequence of two, three, and four electrons in the frequency range of so far as the dimension of this system is smaller in comparison to the radiation wavelength. The sequence moving along a spiral radiates as a charged particle with a charge Ne and rest mass 0 Nm , i.e. by a factor 2 N more than a single electron.
4. The oscillations of the function of spectral distribution of synchrotron-Cherenkov radiation power of one, two, three, and four electrons moving along a spiral in a medium at n c V med / > ⊥ is determined by the contribution of the Bessel functions.
5. It is confirmed that the synchrotron-Cherenkov radiation of one, two, three and four electrons is an unified process. The influence of the Doppler effect on the structure of the spectral distribution of the power of the radiation of electrons becomes significant near the Cherenkov barrier.