Radial-phase motion of electrons in a synchrotron
Abstract Generated abstract
This paper develops an analytic description of radial phase motion of electrons in a synchrotron using energy conservation and the phase of an electron relative to the traveling wave, rather than the usual separation into equilibrium, slow, and fast motions. Starting from coupled equations for energy, phase, and radius, the authors derive an approximate fourth order polynomial form for the radial deviation near the equilibrium orbit and express the resulting motion through elliptic integrals and Weierstrass functions. The analysis gives conditions for stable oscillatory motion in terms of the discriminant of the associated cubic, provides the period of radial oscillations, and obtains a corresponding expression for the electron phase as a function of azimuth. The resulting formulas are presented as useful for studying electron bunching in a synchrotron.
Full Text
RADIAL-PHASE MOTION OF ELECTRONS IN A SYNCHROTRON
A. N. DIDENKO, A. S. CHUMAKOV
To study the motion of electrons in a synchrotron, one usually writes the equations of motion in a time-dependent axially symmetric field and then considers separately three types of motion [1, 2]:
- Motion of an electron along the equilibrium orbit.
- Slow radial-phase oscillations about the equilibrium orbit. In the process of phase oscillations the electrons move along almost closed instantaneous orbits.
- Fast free oscillations about the instantaneous orbit.
However, radial-phase motion can be described on the basis of more general considerations of the law of conservation of energy and the concept of the phase of an electron relative to the traveling wave.
In this case the equations of motion have the form:
\[ \frac{dE}{d\Theta}=\frac{eV_0}{2\pi}\cos \zeta , \tag{1} \]
\[ \frac{d\zeta}{dt}=\left(\frac{\omega r}{\beta c}-1\right), \tag{2} \]
\[ \frac{dE}{dr}=(1-n)\frac{E}{r}. \tag{3} \]
The quantities entering into (1–3) are related to one another by the following relations:
\[ \zeta=\omega t-\Theta;\qquad \Theta=\frac{\beta c}{r}. \tag{4} \]
In these expressions the following notation is used:
\(E\)—electron energy,
\(\Theta\)—azimuth of the electron’s position,
\(\zeta\)—phase of the electron relative to the traveling wave,
\(V_0\)—amplitude of the voltage on the resonator,
\(r\)—instantaneous radius of the electron,
\(\beta c\)—velocity of the electron,
\(n\)—index of decrease of the magnetic field.
Integrating (3) and jointly the equations (1) and (2), we obtain
\[ E = E_{b.x}\left(\frac{r}{r_{b.x}}\right)^{1-n}, \tag{5} \]
\[ \frac{eV_0}{2\pi}\sin z = E_{b.x}\left(\frac{r}{r_{b.x}}\right)^{1-n} \left[ \frac{\omega r}{\beta c}\frac{(1-n)}{(2-n)}-1 \right]+C, \tag{6} \]
where
\[ C= -\frac{eV_0}{2\pi}\sin z_{b.x} - E_{b.x} \left[ \frac{\omega r_{b.x}}{\beta c}\frac{(1-n)}{(2-n)}-1 \right]. \]
With the aid of (5) and (6), equation (1) can be written in the form
\[ d\Theta = \frac{dr}{ \left\{ \left(\frac{eV_0}{2\pi}\right)^2 r^{2n} \frac{r_{b.x}^{2(1-n)}}{(1-n)^2 E_{b.x}^{\,2}} - \left[ \frac{\omega r^2}{(1-n)(2-n)\beta c} - \frac{r}{(1-n)} + \frac{C r^n r_{b.x}^{1-n}}{(1-n)E_{b.x}} \right]^2 \right\}^{1/2} }. \tag{7} \]
Equation (7) is not integrable in the general case. However, it can be integrated with a high degree of accuracy if one puts \(r=r_0+\rho\), and expands \(r^n\) and \(r^{2n}\) in a binomial series in the small deviation from the equilibrium radius \(\rho\).
Restricting ourselves to the fourth power of the deviation \(\rho\), equation (7), after integration, can be represented in the form of an elliptic integral
\[ \int_0^{\Theta} d\Theta = \int_{r_{b.x}}^{\rho} \frac{d\rho}{ \sqrt{a\rho^4+b\rho^3+c\rho^2+d\rho+e} }. \tag{8} \]
In this integral the coefficients \(a,b,c,d,e\) are determined by the initial conditions of the electron \(r_{b.x}, E_{b.x}, z_{b.x}\) and by the accelerator parameters \(n,\omega,r_0,V_0\), and have the form:
\[ \begin{aligned} a={}& \left[ \left(\frac{eV_0}{2\pi}\right)^2-C^2 \right] \frac{ 2n(2n-1)(2n-2)(2n-3) }{ 4!(1-n)^2 E_{b.x}^{\,2} } r_0^{2n-4} r_{b.x}^{2(1-n)} \\ &- \left\{ \frac{2\omega C}{(1-n)(2-n)\beta c} \frac{r_{b.x}^{1-n}}{E_{b.x}} \right\} \frac{(2-n)(1+n)n(n-1)}{4!} r_0^{\,n-2} \\ &+ \left[ \frac{2C r_{b.x}^{1-n}}{(1-n)^2 E_{b.x}} \right] \frac{(1+n)n(n-1)(n-2)}{4!} r_0^{\,n-3} - \left[ \frac{\omega^2}{(2-n)^2(\beta c)^2} \right]; \end{aligned} \]
\[ \begin{aligned} b={}& \left[ \left(\frac{eV_0}{2\pi}\right)^2-C^2 \right] \frac{ 2n(2n-1)(2n-2) }{ 3!(1-n)^2 E_{b.x}^{\,2} } r_0^{2n-3} r_{b.x}^{2(1-n)} \\ &- \left[ \frac{2\omega C}{(1-n)(2-n)\beta c} \frac{r_{b.x}^{1-n}}{E_{b.x}} \right] \frac{(2-n)(1-n)n}{3!} r_0^{\,n-1} \;+\; \end{aligned} \]
\[ \begin{aligned} &\cdots \left[ \frac{2C}{(1-n)^2}\frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right] \frac{(1+n)n(n-1)}{3!}\rho_0^{\,n-2} - \left[ \frac{4\omega^2}{(2-n)^2(\beta c)^2} \right]\rho_0 + \left[ \frac{2\omega}{(1-n)(2-n)\beta c} \right]; \\[6pt] c={}& \left[ \left(\frac{eV_0}{2\pi}\right)^2-C^2 \right] \frac{2n(2n-1)}{2!(1-n)^2E_{b.x}^{\,2}} \rho_0^{\,2n-2}\rho_{b.x}^{\,2(1-n)} \\ &- \left[ \frac{2\omega C}{(1-n)(2-n)\beta c} \frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right] \frac{(2-n)(1+n)}{2!}\rho_0^{\,n} \\ &+ \left[ \frac{2C}{(1-n)^2}\frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right] \frac{(1+n)n}{2!}\rho_0^{\,n-1} - \left[ \frac{6\omega^2}{(2-n)^2(\beta c)^2} \right]\rho_0^2 \\ &+ \left[ \frac{6\omega}{(1-n)(2-n)\beta c} \right]\rho_0 - \frac{1}{(1-n)^2}; \\[6pt] d={}& \left[ \left(\frac{eV_0}{2\pi}\right)^2-C^2 \right] \frac{2n\rho_0^{\,2n-1}}{(1-n)^2E_{b.x}^{\,2}} \rho_{b.x}^{\,2(1-n)} \\ &+ \left[ -\frac{2\omega C}{(1-n)(2-n)\beta c} \frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right](2+n)\rho_0^{\,1+n} + \left[ \frac{2C}{(1-n)^2}\frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right](1+n)\rho_0^{\,n} \\ &- \left[ \frac{4\omega^2}{(2-n)^2(\beta c)^2} \right]\rho_0^2 + \left[ \frac{6\omega}{(1-n)(2-n)\beta c} \right]\rho_0^2 - \frac{2\rho_0}{(1-n)^2}; \\[6pt] e={}& \left[ \left(\frac{eV_0}{2\pi}\right)^2-C^2 \right] \frac{\rho_0^{\,2}\rho_{b.x}^{\,2(1-n)}}{(1-n)^2E_{b.x}^{\,2}} - \left[ \frac{2\omega C}{(1-n)(2-n)\beta c} \frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right]\rho_0^{\,2+n} \\ &+ \left[ \frac{2C}{(1-n)^2}\frac{\rho_{b.x}^{\,1-n}}{E_{b.x}} \right]\rho_0^{\,1+n} + \left[ \frac{\omega^2}{(2-n)^2(\beta c)^2} \right]\rho_0^4 \\ &+ \left[ \frac{2\omega}{(1-n)(2-n)\beta c} \right]\rho_0^3 - \frac{\rho_0^2}{(1-n)^2}. \end{aligned} \]
Integral (8) may be rewritten as
\[ \Theta+\Theta_{b.x}=\int_0^\rho \frac{d\rho}{\sqrt{R(\rho)}}, \tag{9} \]
where
\[ \Theta_{b.x}=\int_0^{\rho_{b.x}}\frac{d\rho}{\sqrt{R(\rho)}};\qquad R(\rho)=a\rho^4+b\rho^3+c\rho^2+d\rho+e. \tag{10} \]
Physically, \(\Theta_{b.x}\) denotes the angle through which an electron with initial deviation \(\rho_{b.x}\) must pass in order to reach the equilibrium radius.
Inversion of the elliptic integral (9) gives an expression for \(\rho\) as a function of \(\theta\) for an electron with specified initial conditions [3]:
\[ \rho=-\frac{b}{a}+\frac{1}{2\sqrt{a}}\, \frac{ \zeta'\!\left(\theta+\theta_{\mathrm{in}}-\frac{\gamma}{2}\right) +\zeta'\!\left(\theta-\theta_{\mathrm{in}}-\frac{\gamma}{2}\right) }{ \zeta\!\left(\theta-\theta_{\mathrm{in}}+\frac{\gamma}{2}\right) -\zeta\!\left(\theta+\theta_{\mathrm{in}}-\frac{\gamma}{2}\right) }. \tag{11} \]
The Weierstrass function \(\zeta\) and its derivative \(\zeta'\), entering into (11), have invariants \(g_2\) and \(g_3\), which are determined as follows:
\[ g_2=ae-4bd+3c^2, \]
\[ g_3= \begin{vmatrix} a & b & c\\ b & c & d\\ c & d & e \end{vmatrix}. \tag{12} \]
The parameters \(\gamma\) and \(\theta_{\mathrm{in}}\) are found from the solution of the equations [4]:
\[ \gamma(0)=\frac{b^2-ac}{a}, \]
\[ \zeta(\theta_{\mathrm{in}})= -\frac{ \zeta'\!\left(\frac{\gamma}{2}\right) }{ \zeta''\!\left(\frac{\gamma}{2}\right) } \, \frac{1}{ \rho_{\mathrm{in}}+\frac{b}{a} -\frac{1}{\sqrt{a}}\, \frac{ \zeta''\!\left(\frac{\gamma}{2}\right) }{ \zeta'\!\left(\frac{\gamma}{2}\right) } } +\zeta\!\left(\frac{\gamma}{2}\right). \tag{13} \]
Before proceeding to the analysis of expression (11), let us note that, in the theory of elliptic functions, instead of the fourth-degree polynomial \(R(\rho)\), there often appears a third-degree polynomial in canonical form
\[ F(z)=4z^3-g_2z-g_3. \]
To pass from \(R(\rho)\) to the canonical form it is necessary to subject the variable \(\rho\) to the fractional-linear transformation
\[ \rho=\alpha+ \frac{R'(\alpha)} { 4\left\{z-\frac{1}{24}R''(\alpha)\right\} }, \]
where
\[ \alpha=-\frac{b}{a}+\frac{1}{\sqrt{a}}\, \frac{ \zeta''\!\left(\frac{\gamma}{2}\right) }{ \zeta'\!\left(\frac{\gamma}{2}\right) } \]
is one of the roots of the polynomial \(R(\rho)\).
The polynomial \(F(z)\) has roots, in decreasing order, \(e_1, e_2, e_3\), and its discriminant is equal to
\[ \Delta=g_2^3-27g_3^2. \]
Returning to (11), we see, first, that \(\rho\) is a periodic function of \(\Theta\), and, second, that stable oscillatory motion of the electrons is possible only for \(\Delta>0\). The period of the radius oscillations is
\[ 2\omega_1=\frac{2K}{\sqrt{e_1-e_3}}, \tag{14} \]
where \(K\) is the complete elliptic integral of the first kind, evaluated with modulus
\[ k=\sqrt{\frac{e_2-e_3}{e_1-e_3}}. \]
Substituting (11) into (2), the latter equation can be integrated with the aid of the addition theorem for the function \(\zeta\) [3]:
\[ \varphi-\varphi_{\mathrm{in}} = \left[\frac{\omega r_0}{\xi c}-1\right]\Theta - \frac{\omega}{\xi c}\cdot\frac{b}{a}\,\Theta + \frac{\omega}{\xi c\sqrt{a}} \ln \frac{ \sigma\left(\Theta+\Theta_{\mathrm{in}}-\dfrac{v}{2}\right) }{ \sigma\left(\Theta+\Theta_{\mathrm{in}}+\dfrac{v}{2}\right) } + \tag{15} \]
\[ + \frac{\omega}{\beta c}\, \frac{\Theta+\Theta_{\mathrm{in}}}{\sqrt{a}}\, \zeta(v). \]
In expression (15), \(\sigma\) and \(\zeta\) are the Weierstrass sigma- and zeta-functions.
The expression obtained in (15) makes it possible to determine the phase of any electron relative to the traveling wave at any azimuth. Like the radius, the electron phase is a periodic function with period
\[ 2\omega_1=\frac{2K}{\sqrt{e_1-e_3}}. \]
Expression (15) can be used in considering the question of electron bunching in a synchrotron.
References
- Frank N. H. Phys. Rev., 70, 177, 1946.
- M. S. Rabinovich. Proceedings of FIAN. Vol. X, 1958.
- N. I. Akhiezer. Elements of the Theory of Elliptic Functions. Gostekhizdat, 1948.
- S. V. Kovalevskaya. Scientific Work. Publishing House of the Academy of Sciences of the USSR, 1948.