On the Nonlinear Theory of Radial-Phase Oscillations
Abstract Generated abstract
This paper develops a nonlinear treatment of radial-phase oscillations in high-energy electron accelerators with synchrotron radiation included. Using phase-plane analysis, separatrix equations, and an adiabatic invariant, it examines damping for finite phase oscillations and then estimates the energy, phase-velocity, and radial spreads produced by the quantum character of radiation. The analysis finds that continuous radiation damping compresses the phase bunch, while quantum fluctuations set a limiting spread that can cause particle loss near the end of acceleration. Compared with the linear theory, the nonlinear theory gives lower limiting energies and stricter requirements on the equilibrium phase, with larger final equilibrium phase, larger orbit radius, higher RF voltage, and lower harmonic number identified as favorable parameter choices.
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ON THE PROBLEM OF THE NONLINEAR THEORY OF RADIAL-PHASE OSCILLATIONS
A. N. Didenko
(Presented by Prof. Dr. of Physico-Mathematical Sciences A. A. Vorob’ev)
1. The influence of the quantum character of radiation on radial-phase oscillations under the assumption that the phase \(\varphi\) differs only slightly from its equilibrium value \(\varphi_s\) (linear theory) has been investigated very thoroughly. It was shown that the existence of oscillations induced by radiation imposes substantial restrictions on the magnitude of the equilibrium phase [1, 2]. However, the linear theory cannot be applied if we are interested in the question of the character of oscillations that are not small, or if the boundaries of the stability region are to be determined. It is therefore of interest to solve the nonlinear equation of phase oscillations with radiation taken into account. This question was first considered by A. N. Matveev.
In the present work, the character of damping of phase oscillations in high-energy electron accelerators is investigated without the assumption that they are small; the question of the influence of the quantum character of radiation on radial-phase oscillations in the nonlinear theory is considered, and a comparison of the results of the linear and nonlinear theories is made by the phase-plane method; several recommendations are made concerning the choice of certain machine parameters.
2. If the quantum character of radiation is not taken into account, i.e., if radiation is considered as a continuous process of energy loss, then the equation of phase oscillations has the following form:
\[ \ddot{\varphi}+\lambda \dot{\varphi}+f^2(\cos\varphi_s-\cos\varphi)=0, \tag{1} \]
where \(\varphi\) is the phase of the accelerating field at the gap at the moment an electron passes through it;
\[ \lambda=\frac{(3-4n)}{(1-n)}\cdot\frac{W_s}{E_s}; \qquad f^2=\frac{1}{(1-n)}\cdot\frac{k}{\lambda^2}\cdot\frac{W_s}{E_s}\cdot\frac{\omega_0}{\cos\varphi_s}. \]
If we introduce the dimensionless quantity \(\zeta=ft\) and assume that during one period of phase oscillations the quantity
\[ \exp\left\{-\frac{1}{2}\frac{\lambda}{f}\zeta\right\} \]
does not change, and neglect small terms, as is done in [3], then for the separatrix equation we obtain the expression:
\[ \varphi'^2=2\left\{\sin\varphi+\sin\varphi_s-(\varphi+\varphi_s)\cos\varphi_s\right\}. \tag{2} \]
From (2) it is seen that, in its external form, the equation of the separatrix does not differ from that which was obtained without taking radiation into account; i.e., radiation does not affect the form of the separatrix to within terms of order \(\frac{\gamma}{f}\), which were neglected in all calculations. Taking the omitted terms into account in the equation for the separatrix leads to the fact that its dimensions decrease somewhat; however, this decrease will be so insignificant (because \(\frac{\gamma}{f} \ll 1\)) that we shall not take it into account further.
From (2), in accordance with the existing relation between the rate of change of the phase and the change in the radius of the equilibrium orbit, one can determine the admissible spread in radius for particles that have not yet fallen out of the acceleration regime:
\[ \frac{\Delta R}{R} = \frac{\dot{k}}{k}\,\frac{f}{\omega_0} \left| 2\left\{\sin\varphi+\sin\varphi_s-(\varphi+\varphi_s)\cos\varphi_s\right\} \right| \tag{3} \]
- If we now take into account that the quantities \(\gamma\), \(f\), and \(\exp\left\{-\frac{1}{2}\frac{\gamma}{f}\,\zeta\right\}\) vary slowly with time, then by the known theorem on the adiabatic invariant one can obtain an expression characterizing the damping of phase oscillations. In our case it is written as follows:
\[ \int_{-\varphi_0}^{\varphi} \sqrt{\sin\varphi+\sin\varphi_s-(\varphi+\varphi_s)\cos\varphi_s}\,d\varphi = \frac{\mathrm{const}}{fE}\,e^{-\gamma t}. \tag{4} \]
The representation of the separatrices on the phase plane (Fig. 1) leads to the conclusion that they very strongly resemble parts of a lemniscate. Therefore
Fig. 1
the equation of the separatrix can, with small errors, be approximated by the expression
\[ r^2=(\varphi_1+\varphi_s)^2\cos 2\vartheta_s, \tag{5} \]
where \(\vartheta_s\) is the angle formed by the tangent to the separatrix at the point \(\varphi=-\varphi_s\)
with the \(\varphi\)-axis. We note that \(\theta_s\) changes with a change in \(\varphi_s\), although this change is insignificant.
Taking (5) into account, we obtain:
\[ (\varphi_1+\varphi_s)^2 \sin 2\theta_s=\frac{\mathrm{const}}{E}\,e^{-\gamma t} \tag{6} \]
or
\[ (\varphi_1+\varphi_s)=\mathrm{const}'\sqrt[4]{\frac{1}{E V_0(t)\sin^2 2\theta_s}}\,e^{-\frac{1}{2}\gamma t}. \tag{7} \]
It follows from (7) that, for a quite definite value of the equilibrium phase, in the expression characterizing the damping of oscillations, along with the old, well-studied factor \(\sqrt[4]{\frac{1}{E V_0(t)}}\), which takes into account the weak damping of phase oscillations, there appears the factor \(\exp\{-\frac{1}{2}\gamma t\}\), which leads to compression of the beam of accelerated particles into bunches of very small extent in a time of order \(\frac{1}{\gamma}\). Therefore, in what follows we shall assume that, for high-energy electron accelerators, the dimensions of the beam are determined only by external factors and by the quantum nature of the radiation, and not by its initial extent. We note that for the time being we regard radiation as a continuous process of energy loss.
- The mean-square spread in energies due to the quantum character of the radiation can be obtained if the quantities characterizing the change of energy in some act of radiation, multiplied by the corresponding probabilities of radiation and exponential factors, are summed over all acts of radiation and over time:
\[ \left(\frac{\overline{\Delta E}}{E}\right)^2 = \frac{1}{2}\int_0^\infty e^{-\gamma t}\,dt \int_0^\infty \left(\frac{\varepsilon_\nu}{E}\right)^2 W_\nu\,d\nu, \tag{8} \]
where \(\varepsilon_\nu\) is the energy of the emitted quantum corresponding to the \(\nu\)-th harmonic. The factor \(1/2\) takes into account the fact that the phase of emission is a random quantity.
Substituting the values of the quantities entering this formula and carrying out the integration, we obtain:
\[ \left(\frac{\overline{\Delta E}}{E}\right)^2 = \frac{55\sqrt{3}}{96}\, \frac{(1-n)}{(3-4n)}\, \frac{\hbar\omega_0}{mc^2} \left(\frac{E}{mc^2}\right)^2 . \tag{9} \]
From the well-known relations connecting the energy spread with the spread in radius and in the rate of change of phase, it follows that the mean-square energy spread determined by (9) corresponds to definite mean-square spreads in the rate of change of phase
\[ \overline{\dot{\varphi}^{\,2}_{\mathrm{q}}} = \frac{55\sqrt{3}}{64}\, \frac{k}{\lambda}\, \frac{1}{(3-4n)}\, \frac{hc}{e^2} \left(\frac{mc^2}{E}\right)\cos\varphi_s \tag{10} \]
and in radius
\[ \left(\frac{\overline{\Delta R}}{R}\right)^2_{\mathrm{q}} = \frac{\lambda^4}{k^2}\, \frac{f^2}{\omega_0^2}\, \overline{\varphi^{\,2}_{\mathrm{q}}}. \tag{11} \]
Comparing (11) with (3), we see that the particles do not fall out of the acceleration regime if
\[ \varphi'^2 \gg \varphi_{kb\,\max}^{\prime\,2}. \]
In practice it is assumed that the beam will not be lost if the permissible spread in radius exceeds by a factor of 2–3 the total spread caused by various reasons. Therefore the inequality may also be written as
\[ \varphi'^2=\alpha^2\varphi_{kb\,\max}^{\prime\,2}, \tag{12} \]
where \(\alpha\) is the safety factor.
Substituting into (12) the specific expression for \(\varphi_{\varphi-\varphi_s}^{\prime\,2}\) and \(\varphi_{kb\,\max}^{\prime\,2}\), we obtain an expression for the energy at which a strong decrease begins in the number of accelerated particles, owing to their falling out of the acceleration regime:
\[ E_{\mathrm{pred}}=mc^2 \sqrt[3]{ \frac{\dfrac{2(\sin\varphi_s-\varphi_s\cos\varphi_s)}{\alpha^2}} {\dfrac{55\sqrt{3}\pi}{48}\,\dfrac{k}{\lambda}\,\dfrac{1}{(3-4n)}\,\dfrac{hc}{e^2}\,\dfrac{e^2}{R}\,\dfrac{1}{eV_0(t)}} }. \tag{13} \]
The function \(F(\varphi_s)=\sin\varphi_s-\varphi_s\cos\varphi_s\) increases very rapidly with increasing \(\varphi_s\). Therefore the most important factor affecting the increase of \(E_{\mathrm{pred}}\) is the choice of a large value of the equilibrium phase at the end of acceleration. In addition, it follows from (13) that an increase in the radius of the equilibrium orbit, an increase in the amplitude of the high-frequency voltage, and a decrease in the harmonic number are desirable.
- If similar calculations are carried out according to the linear theory, then for \(\varphi'^2\) and \(E_{\mathrm{pred}}\) we obtain the following expressions:
\[ \varphi'^2=2\sin\varphi_s \left|\,2\varphi_s^2-\frac{(\varphi-\varphi_s)^2}{2}\,\right|, \tag{14} \]
\[ E_{\mathrm{pred}}= \sqrt[3]{ \frac{2\varphi_s^2\sin\varphi_s}{\alpha^2}\cdot \frac{48}{55\sqrt{3}\pi}\cdot \frac{\lambda}{k}\cdot \frac{R}{hc}\,eV_0(t) }, \tag{15} \]
i.e.,
\[ \frac{E_{\mathrm{pred,\,lin}}}{E_{\mathrm{pred,\,nonlin}}} = \sqrt[3]{ \frac{\varphi_s^2\sin\varphi_s} {\sin\varphi_s-\varphi_s\cos\varphi_s} }. \tag{16} \]
From (16) it is evident that the linear theory gives excessively high values for \(E_{\mathrm{pred}}\).
Let us note that the values of \(E_{\mathrm{pred}}\) calculated according to the linear and nonlinear theories do not coincide even for small values of \(\varphi_s\). This is a consequence of the fact that, using such a device as cutting off the curve \(\varphi'=\varphi'(\varphi)\) for the linear equation at the point \(\varphi=-\varphi_s\), we obtain an expression for determining the stability region which does not coincide with the corresponding expression for the nonlinear equation for any values of the equilibrium phase.
All this can be represented in the phase plane. Figure 1 shows separatrices for various values of \(\varphi_s\), calculated according to the nonlinear theory, and the separatrix for \(\varphi_s=60^\circ\), calculated according to the linear theory (curve 1). The straight line \(\varphi'=\mathrm{const}\) characterizes the noise level caused by the quantum nature of the radiation at the end of acceleration for a machine with the parameters given in the paper by Sands [1].
From Fig. 1 and from formulas (13) and (15) it follows that for \(\alpha = 2\text{–}3\) the loss of particles will begin at energies of the order of \(1\,Bev\), and that the nonlinear theory imposes more stringent requirements on the choice of the equilibrium phase at the end of acceleration.
Analysis of the formulas given above makes it possible to choose the parameters in such a way that the particle losses at the end of acceleration are minimal, although it apparently will not be possible to reduce them to zero.
REFERENCES
- M. Sands, Phys. Rev. 97, 470 (1955).
- Matveev A. N., DAN SSSR, vol. 108, issue 3, 1956.