On the Nonlinear Theory of Phase Oscillations Induced by Quantum Fluctuations of Radiation in Electron Synchrotrons
Abstract Generated abstract
This paper develops a nonlinear treatment of phase oscillations in electron synchrotrons excited by quantum fluctuations of synchrotron radiation, addressing the inadequacy of linear theory for estimating electron losses near the finite phase stability boundaries. Starting from a nonlinear stochastic equation for the electron phase and using an approximate invariant of the corresponding homogeneous motion, it derives a criterion for small beam losses in terms of accelerator parameters, radiation fluctuations, damping, and equilibrium phase. The resulting condition is more restrictive than the earlier linear criterion, by about a factor of 1.7 in the examples considered, and predicts lower limiting energies for weak-focusing synchrotrons. The analysis supports the conclusion that strong focusing becomes necessary for electron synchrotrons at energies of several billion electron-volts.
Full Text
On the Nonlinear Theory of Phase Oscillations Induced by Quantum Fluctuations of Radiation in Electron Synchrotrons
A. N. MATVEEV
(Submitted by Prof. A. A. Vorob’ev, Doctor of Physical and Mathematical Sciences)
As is known [1], the quantum character of radiation causes the excitation of phase oscillations of electrons in synchrotrons. The mechanism of excitation of these oscillations is analogous to the mechanism of excitation of betatron oscillations, which was indicated in [2, 3].
In [1] the mean-square deviation of the phase oscillations of electrons induced by radiation was calculated for a weak-focusing synchrotron. The calculation was carried out in the linear approximation and with neglect of the existence of boundaries for the phase oscillations. This means that this calculation is valid only for small amplitudes of oscillation. A generalization of these results in the same approximation to the case of synchrotrons with strong focusing was carried out in [4, 5]. A method for taking the absorbing boundaries into account was given in [6]. By the method of that work one can calculate not only the mean-square deviation of the oscillations, but also obtain the law of loss of intensity of the electron beam due to phase oscillations induced by quantum fluctuations of the radiation.
The linear theory of phase oscillations, which was considered in [1, 4—6], is valid only for a small value of the deviation of the electron phase from the equilibrium value. Under this assumption, however, it is impossible to answer the basic question that interests us in considering the oscillations, namely: the question of the magnitude of the losses of electrons caused by these oscillations. Indeed, in order for an electron to drop out of the subsequent acceleration process, it is necessary that its phase oscillation go beyond the permissible limits. As is known, the left boundary of the electron phase is the phase angle \(-2\varphi_s\), where \(\varphi_s\) is the equilibrium phase. The right boundary \(\varphi_1\) is determined from the relation:
\[ \sin \varphi_1 + \sin \varphi_s - (\varphi_1 + \varphi_s)\cos \varphi_s = 1. \tag{1} \]
In order to reach the boundaries, the electron phase must deviate by \(2\varphi_s\) to the left or by the amount \(\varphi_1-\varphi_s\) to the right, i.e., it must deviate through rather large angles. For example, at \(\varphi_s = 45^\circ\) these deviations must be respectively equal to \(2\varphi_s \simeq (\pi/2)\) and \((\varphi_1-\varphi_s) \simeq (\pi/4)\), i.e., these deviations are in no sense small. Consequently, for consideration of the principal question in the theory, that of electron losses due to loss of phase, the lin-
linear approximation is insufficient. In this connection there arises the question of the reliability of the predictions of the linear theory, and the need arises to construct a nonlinear theory without the assumption that the deviation of the electron phase from its equilibrium value is small.
As we have already reported, the nonlinear theory leads to considerably more stringent restrictions imposed on the parameters of the synchrotron in comparison with the linear theory. In the present work we shall consider in greater detail the questions of the nonlinear theory of phase oscillations induced by quantum fluctuations of radiation.
The method of obtaining the nonlinear stochastic equation describing phase oscillations induced by radiation is completely analogous to the method of obtaining the linear stochastic equation (2) of work [6]. This nonlinear stochastic equation has the following form:
\[ \ddot{\psi}+\gamma \dot{\psi}+f^{2}\left[\cos \varphi_{s}-\cos \left(\varphi_{s}+\psi\right)\right] = -\frac{k\omega\alpha}{\lambda E_{s}} \left[W_{s}-\sum_{i}\varepsilon_{i}\,\delta(t-t_{i})\right], \tag{2} \]
where the following notation has been used: \(\psi=\varphi-\varphi_s\), where \(\varphi\) is the phase of passage of the electron through the high-frequency field, and \(\varphi_s\) is the equilibrium value of this phase; \(\varepsilon_i\) is the energy of the quantum emitted at the time characterized by the index of this quantity; \(k\) is the harmonic number of the high-frequency field on which acceleration takes place; \(R_0\) is the radius of curvature of the curvilinear sections of the synchrotron; \(\lambda=1+z/2\pi R_0\), where \(z\) is the total length of all rectilinear sections of the synchrotron in one revolution;
\[ \omega=\frac{c}{R_{0}\lambda}, \qquad f^{2}=\frac{k\omega^{2}\alpha}{2\pi\lambda}\,\frac{e v_{0}}{E_{s}}, \qquad \gamma=(4-z)\,\frac{2\omega r_{0}}{3R_{0}}\left(\frac{E_{s}}{mc^{2}}\right)^{3}; \]
\[ r_{0}=\frac{e^{2}}{mc^{2}}, \qquad \frac{\delta\langle R\rangle}{\langle R\rangle}=\alpha\,\frac{\delta E}{E}, \qquad W_{s}=\frac{2ce^{2}}{3R_{0}^{2}}\left(\frac{E_{s}}{mc^{2}}\right)^{4}; \]
\(e v_{0}\) is the amplitude of the high-frequency field.
Under the assumption that \(\psi\) is small, equation (2) can be linearized, and then the linear equation (2) of work [6] is obtained.
The general method of solving the problem posed consists, as is known, in the following. One must find the solution of the homogeneous equation (2) in the form of a Fourier expansion, the coefficients and frequencies of which depend on a certain parameter. Then the right-hand side of the equation must be treated as a small perturbation, the influence of this perturbation on the small parameter on which the solutions of the homogeneous equation depend must be found, and then the influence of this perturbation on the solutions found must be determined. Taking into account the random character of the small perturbation makes it possible to determine the statistical properties of the desired solutions.
However, the practical implementation of the indicated method of solution is hardly possible, in view of the very large amount of computational work even within the framework of machine techniques. Therefore we shall apply another method.
First of all, in equation (2) we pass to a new independent variable \(\xi\) and a new function \(z\) by the following formulas:
\[ \xi=\int f\,dt,\qquad \psi=uz,\qquad u=\exp\left(-\frac{1}{2}\int_{0}^{\xi}q\,d\xi\right),\qquad q=\frac{f'}{f^{2}}+\frac{\gamma}{f}, \tag{3} \]
where primes denote derivatives with respect to \(\xi\). Then equation (2) takes the form:
\[ z''+\frac{1}{u}\,[\cos \varphi_s-\cos(\varphi_s+uz)] -\left(\frac{1}{4}q''+\frac{1}{2}q'\right)Z = \frac{k\omega\alpha}{\lambda E_s u f} \left[W_{1s}-\sum_i \varepsilon_i \delta(\xi-\xi_i)\right], \tag{4} \]
where
\[ W_{1s}=W_s/f. \]
In this equation the term
\[ -\left(\frac{1}{4}q''+\frac{1}{4}q'\right)z \tag{5} \]
can always be neglected. Let us denote by \(\Delta_c E_s\) the magnitude of the change in the equilibrium energy over the period of one synchrotron oscillation. We obviously have:
\[ \frac{f'}{f}\sim \frac{\Delta_c E_s}{E_s}\ll 1, \qquad \frac{\alpha}{f}\sim \left[ \frac{r_0}{R_0} \left(\frac{E}{mc^2}\right)^3 \right]^{1/2}\ll 1 . \tag{6} \]
Consequently, \(|q|\ll 1\). Similarly, \(|q'|\ll 1\). Therefore the term (5) can always be neglected. The homogeneous equation corresponding to equation (4) can now be written in the following form:
\[ z''+\frac{1}{u}\,[\cos\varphi_s-\cos(\varphi_s+uz)]=0. \tag{7} \]
Next note that \(u'=-\frac{1}{2}qu\ll u\). Therefore, multiplying equation (7) by \(z'\) and integrating the result obtained, taking account only of the inequality just indicated, we obtain the constant of motion in the following form:
\[ Q=\frac{z'^2}{2} +\frac{1}{u^2}\,[uz\cos\varphi_s-\sin(\varphi_s+uz)]. \tag{8} \]
It follows from (8) that bound states will exist only for the following values of the constant of motion:
\[ -\frac{1}{u^2}\sin\varphi_s = Q_{\min}\le Q_{\max}>Q = \frac{1}{u^2}\,(-2z_s\cos\varphi_s+\sin\varphi_s). \tag{9} \]
If the value of \(Q\) for any reason goes outside the indicated limits, the particle will go off to infinity, i.e. it drops out of the subsequent acceleration process and will be lost.
Let us now return to equation (4). From this equation it is seen that the mechanism exciting phase oscillations consists in jump-like changes in the velocity of motion of the particle phase, caused by acts of photon emission. From this equation it is seen that the magnitude of the jump-like change \(\Delta z'\), caused by the emission of a photon of energy \(\varepsilon\), is equal to
\[ \Delta z'=-\frac{k\omega\alpha}{\lambda E_s u f}\,\varepsilon . \tag{10} \]
On the other hand, from relation (8) it is seen that the mean change of the constant of motion, caused by random jumps \(\Delta z'\), is given by the equality
\[ \langle \Delta Q\rangle=\langle z'\Delta z'\rangle+\frac{1}{2}\langle(\Delta z')^2\rangle =\frac{1}{2}\langle(\Delta z')^2\rangle, \tag{11} \]
which, taking (10) into account, may be rewritten as follows:
\[ \langle \Delta Q\rangle =-\frac{1}{2}\,\frac{k^2\omega^2 a^2}{\lambda^2 E_S^{\,2}u^2 f^2}\, \langle \varepsilon^2\rangle . \tag{12} \]
It follows from this that if at \(\xi=0\) the electrons were at rest in the lowest energy state at the bottom of the potential well determined by relation (9), then the mean value of the quantity \(Q\) at other values of \(\xi\) is given by the equality:
\[ \langle Q\rangle =-\frac{1}{u^2}\sin\varphi_S +\frac{k^2\omega^2 a^2}{2\lambda^2} \int_0^\xi \frac{\langle \varepsilon^2\rangle}{E_S^{\,2}u^2 f^2}\,d\xi . \tag{13} \]
It is obvious that, in order to ensure a small amount of losses due to phase oscillations induced by radiation, it is necessary to require that \(\langle Q\rangle\ll Q_{\max}\). On the basis of (13), this condition of small electron losses may be written in the following form:
\[ \frac{k^2\omega^2 a^2 u^2}{4\lambda^2} \int_0^\xi \frac{\langle \varepsilon^2\rangle}{E_S^{\,2}f^2u^2}\,d\xi \ll (\sin\varphi_S-\varphi_S\cos\varphi_S). \tag{14} \]
The stronger inequality (14) is, the less significant are the electron losses due to phase oscillations induced by radiation. It is clear that for \(\langle Q\rangle=Q_{\max}\) all electrons will be practically lost.
Carrying out the calculations indicated in (14), this condition may be written in the following form:
\[ -\frac{55\pi}{48\sqrt{3}}\, \frac{ka\omega}{\lambda} \left(\frac{r_e}{R_0}\right)^2 \frac{hc}{e^2} \sqrt{\frac{E_S}{ev_0}}\, \int_0^t \exp\!\left(-\int_{\tau}^{t}\gamma\,d\tau\right) \left(\frac{E_S}{mc^2}\right)^5 \sqrt{\frac{E_S}{ev_0}}\,d\tau \ll (\sin\varphi_S-\varphi_S\cos\varphi_S). \tag{15} \]
Taking into account that \(1/\gamma\) is much greater than the period of oscillations of the system under consideration and much smaller than the acceleration period, one can prove the validity of the relation:
\[ \int_0^t \exp\!\left(-\int_{\tau}^{t}\gamma\,d\tau\right) \left(\frac{E_S}{mc^2}\right)^5 \sqrt{\frac{E_S}{ev_0}}\,d\tau \ll \frac{3R_0}{2(4-\alpha)r_0\omega}\, \sqrt{\frac{E_S}{ev_0}} \left(\frac{E_S}{mc^2}\right)^2, \tag{16} \]
where equality is attained only in the following cases:
a) if the left-hand side increases monotonically, then equality is attained only asymptotically at infinity;
b) if the left-hand side has a maximum, then equality is attained at this maximum. Let us note that in real accelerators case (a) is realized.
The proof of this inequality (16) and of the assertions made concerning it is evident from the following identity:
\[ \frac{d}{dx}\int_{0}^{x}\exp\left(-\int_{z}^{x}\gamma\,d\tau\right)f(z)\,dz = f(x)-\gamma\int_{0}^{x}\exp\left(-\int_{z}^{x}\gamma\,d\tau\right)f(z)\,dz . \tag{17} \]
Taking (16) into account, condition (15) can be written in the following form, convenient for applications:
\[ \frac{55\pi}{32\sqrt{3}}\, \frac{k\alpha}{\lambda(4-\alpha)}\, \frac{r_{0}}{R_{0}}\, \frac{\hbar c}{e^{2}}\, \frac{mc^{2}}{ev_{0}}\, \left(\frac{E_{s}}{mc^{2}}\right)^{3} \ll (\sin\varphi_{s}-\varphi_{s}\cos\varphi_{s}) . \tag{18} \]
This is the basic criterion of the nonlinear theory for the smallness of electron losses due to phase oscillations induced by quantum fluctuations of radiation.
Let us compare the criteria of the linear and nonlinear theories. Denote by \(F(E)\) the quantity
\[ F(E)= \frac{55\pi}{16\sqrt{3}}\, \frac{k\alpha}{\lambda(4-\alpha)}\, \frac{r_{0}}{R_{0}}\, \frac{\hbar c}{e^{2}}\, \frac{mc^{2}}{ev_{0}}\, \left(\frac{E_{s}}{mc^{2}}\right)^{3}. \tag{19} \]
Then the criterion of the nonlinear theory may be written in the form
\[ F(E)\ll f_{1}(\varphi_{s});\qquad f_{1}(\varphi_{s})=2(\sin\varphi_{s}-\varphi_{s}\cos\varphi_{s}). \tag{20} \]
The criterion of the linear theory, on the basis of formula (8) of work [5], is written as follows:
\[ F(E)<\frac{l^{2}}{8}\sin\varphi_{s}=f_{2}(\varphi_{s}), \tag{21} \]
where \(l\) denotes the region of admissible phase oscillations. The values of the functions \(f_{1}\) and \(f_{2}\) for a number of values of \(\varphi_{s}\) are equal to
| \(\varphi_{s}\) | \(15^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) |
|---|---|---|---|---|
| \(f_{1}(\varphi_{s})\) | 0.011 | 0.09 | 0.3 | 0.7 |
| \(f_{2}(\varphi_{s})\) | 0.018 | 0.15 | 0.5 | 1.2 |
Thus, approximately, the relation
\[ \frac{f_{2}}{f_{1}}\simeq 1.7 \tag{22} \]
is satisfied.
This means that criterion (20) of the nonlinear theory is stronger than criterion (21) of the linear theory.
As an example, let us consider a weak-focusing synchrotron without straight sections with the following parameter values:
\[ n=0.6,\quad R_{0}=400\ \text{cm},\quad ev_{0}=100\ \text{kev},\quad k=4. \]
Then, expressing \(E\) in billions of electron-volts, we have:
\[ F(E)=0.16\,E^{3}. \]
Let us take, for definiteness, the equilibrium phase \(\varphi_s = 45^\circ\). Then it is evident from the formulas that, according to the criterion of the linear theory, the left-hand side of inequality (21) will be approximately two times smaller than the right-hand side at an energy of \(1.3\ \mathrm{BeV}\). By the criterion of the linear theory (20), it is likewise seen that the left-hand side of this inequality will be two times smaller already at an energy equal to approximately \(1\ \mathrm{BeV}\). Therefore, if on the basis of the linear theory we conclude that, under the conditions considered, the synchrotron ceases to operate somewhere in the range of energies equal to \(1.3\ \mathrm{BeV}\), then the nonlinear theory tells us that the synchrotron actually ceases to operate at energies in the range of \(1\ \mathrm{BeV}\).
Thus we see that the nonlinear theory substantially corrects the results of the linear theory. As for the importance of its predictions, it is obvious and requires no explanation.
Thus, the nonlinear theory of phase oscillations induced by quantum fluctuations of radiation confirms the conclusion of our works [5, 6, 7] concerning the inevitability of a transition to strong focusing at electron energies of several billion electron-volts.
References
- M. Sands, Phys. Rev. 97, 470 (1955).
- A. A. Sokolov and I. M. Ternov. DAN SSSR, 92, 537 (1953); ZhETF, 25, 698 (1953).
- A. A. Sokolov and I. M. Ternov. DAN SSSR, 97, 823 (1954); ZhETF, 28, 432 (1955).
- A. A. Kolomenskii. ZhETF, 30, 207 (1956).
- A. N. Matveev. DAN SSSR, 108, 432 (1956).
- A. N. Matveev. DAN SSSR, 109, 495 (1956).
- A. N. Matveev. DAN SSSR, 107, 671 (1956).