ON THE INFLUENCE OF RADIATION ON RADIAL-PHASE OSCILLATIONS IN HIGH-ENERGY ELECTRON CYCLIC ACCELERATORS

A. N. DIDENKO

(Presented by Prof. A. A. Vorob’ev, Doctor of Physical and Mathematical Sciences)

1. The question of the motion of an electron with allowance for the quantum character of radiation is of great theoretical and practical importance and is therefore widely discussed in the literature. From the theoretical point of view it is of interest as an example of the quantum-mechanical excitation of macroscopic oscillations; from the practical point of view this question is important because it makes it possible to design accelerator installations with allowance for new effects that should already manifest themselves at energies attainable with present-day technology, and to estimate the upper limit of the energies attainable in ordinary cyclic accelerators.

The influence of the quantum character of radiation on the trajectory of an electron was first predicted and investigated in the works of A. A. Sokolov and I. M. Ternov [1, 2]. They showed that, when an electron moves in an axially symmetric constant magnetic field, radius fluctuations associated with the quantum character of radiation lead to the excitation of radial and azimuthal betatron oscillations that grow rapidly with time. When an electron moves in a synchrotron, these same quantum fluctuations of radiation must also influence the so-called radial-phase oscillations. In the works of M. Sands [3], A. A. Sokolov, I. M. Ternov, and G. M. Strakhovskii [4], A. N. Matveev [5], and A. A. Kolomenskii [6], formulas were obtained that determine the magnitude of the mean-square phase deviation for ordinary accelerators and accelerators with strong focusing, and concrete calculations were carried out. It turned out that the phase oscillations induced by radiation are fairly significant. The existence of these oscillations leads to the result that the equilibrium phase $\varphi$ cannot be chosen smaller than a certain minimum value if we wish to avoid additional electron losses caused by phase oscillations. Thus, carrying out in [5] calculations of the mean-square phase deviation for a machine with quite definite parameters $(E = 1 \text{ Bev},\ R = 3.5 \text{ m})$ and assuming that this quantity reaches its maximum value at the end of the acceleration cycle, the author showed that, for accelerators with weak focusing, the minimum value of the phase should not be less than $0.2\pi$.

In the present work the character of radiation-induced radial-phase oscillations in accelerators with weak focusing is investigated.

over the entire acceleration cycle, depending on various operating regimes, and the question of the maximum energies attainable in such machines is discussed.

2. The equation of radial-phase oscillations with allowance for radiation is well known and can be obtained by various methods [3, 4, 5, 6]. It is simplest to obtain it from the energy-balance equation:

\[ \frac{d}{dt}(E-E_s)=\frac{\omega_s}{2\pi}\left\{ eV_0(\cos\varphi-\cos\varphi_s)-(I-I_s)\right\}, \tag{1} \]

where \(E\) is the particle energy, \(V_0\) is the amplitude of the high-frequency voltage at the gap, \(\varphi\) is the phase at which the electron passes through the accelerating gap,

\[ I=\frac{4\pi}{3}\frac{e^2}{R}\left(\frac{E}{mc^2}\right)^4 \]

is the energy radiated by the electron in one revolution. The subscript \(s\) denotes the values of these same quantities for the equilibrium particle. In the general case \(V_0\) and \(\varphi_s\) are functions of time. Denoting by \(L\) the length of all straight sections in the synchrotron, and by \(\psi=\varphi-\varphi_s\) the deviation of the phase from its equilibrium value, and assuming that acceleration takes place at the \(k\)-th harmonic of the high-frequency field, we obtain the relation

\[ \frac{\Delta E}{E}=(1-n)\frac{\lambda^2}{k}\frac{\psi}{\omega_0}, \tag{2} \]

where

\[ \lambda=1+\frac{L}{2\pi R_s}; \qquad \omega_0=\omega\lambda=\frac{c}{R}. \]

Substituting (2) into (1), we obtain the equation for phase oscillations:

\[ \ddot{\psi}+\gamma\dot{\psi}+\Omega^2\psi=0, \tag{3} \]

where

\[ \gamma=\left(\frac{\dot E}{E}+\frac{(3-4n)}{(1-n)}\frac{W}{E}\right); \qquad \Omega^2=\frac{k\omega_0}{\lambda}\frac{\operatorname{tg}\varphi_s}{(1-n)}\frac{(\dot E+W)}{E}, \]

\(\dot E\) is the energy acquired by the electron per unit time; \(W\) is the radiation power.

The solution of this equation

\[ \psi=\psi_0 e^{-\frac12\int_0^t \gamma\,dt'}\cos(\Omega t+\varphi_0) \tag{4} \]

characterizes the change of phase with time caused by the nonquantum character of the radiation. The quantum character of the radiation may be taken into account either by adding the corresponding fluctuation terms to the right-hand side and solving the resulting inhomogeneous equation [7], or by summing over all frequencies the quantities characterizing the phase change caused by an individual act of radiation, multiplied by the probability of radiation, followed by integration of the resulting expression over time. In both cases, for the mean square deviation we obtain the expression:

\[ \begin{aligned} \overline{\psi^2(t)} &=\frac{55}{48\sqrt{3}}\frac{k}{\lambda^3} \frac{e^2\frac{h}{2\pi}c}{(1-n)R^4m} \int_0^t e^{-\int_{t'}^{t}\gamma\,dt''} \left(\frac{E}{mc^2}\right)^6 \frac{\operatorname{ctg}\varphi_s(t')}{(\dot E+W)}\,dt' \\ &=\frac{55}{48\sqrt{3}}\frac{k}{\lambda^3} \frac{e^2\frac{h}{2\pi}c}{(1-n)R^4m} \frac{mc^2}{E(t)} e^{-\int_0^t \frac{(3-4n)}{(1-n)}\frac{W}{E}\,dt'} \int_0^t e^{\int_0^{t'}\frac{(3-4n)}{(1-n)}\frac{W}{E}\,dt''} \\ &\qquad {}\times \left(\frac{E}{mc^2}\right)^7 \frac{\operatorname{ctg}\varphi_s(t')}{(\dot E+W)} \,dt'. \end{aligned} \tag{5} \]

If in (5) one neglects \(\dot E\) in comparison with \(W\) and substitutes a specific value for \(W\), then one obtains a formula for \(\overline{\psi^2}(t)\) that coincides with the formula obtained in [5].

Formula (5) makes it possible to investigate how the mean square deviation of the phase changes with time.

Let us consider the case, important for practice, in which the energy of the particles increases according to a linear law, i.e. \(E = E_0 \dfrac{t}{T} = E_0 \xi\), where \(E_0\) is the energy of the particles at the end of acceleration, and \(T\) is the acceleration time of the particle. Then:

\[ \psi^2(\xi)= \frac{55}{48\sqrt{3}}\, \frac{k}{\gamma^{2}}\, \frac{e^2 \dfrac{h}{2\pi} cT}{(1-n)R^4m}\, \frac{mc^2}{E(\xi)}\, e^{\alpha \xi^4} \int_0^\xi e^{-\alpha \xi'^4}\, \frac{\operatorname{ctg}\varphi_s(\xi')}{(E+W)} \,d\xi', \tag{6} \]

where

\[ \alpha = \frac{2}{3}\, \frac{(3-4n)}{(1-n)}\, \frac{r_0 cT}{\lambda R^2} \left(\frac{E_0}{mc^2}\right)^3; \qquad r_0=\frac{e^2}{mc^2}. \]

1. If \(\dot E \ll W\) and \(\operatorname{ctg}\varphi_s=\text{const}\), then

\[ \psi^2(\xi)= \frac{55\sqrt{3}}{64}\, \frac{k}{\lambda}\, \frac{\dfrac{h}{2\pi}c}{e^2}\, \frac{\operatorname{ctg}\varphi_s}{(3-4n)}\, \frac{mc^2}{E(\xi)} \left(1-e^{-\alpha \xi^4}\right). \tag{7} \]

Extending the limits of integration to infinity, from (7) we obtain Sands’s formula. Let us investigate the character of this function. Differentiating with respect to \(\xi\) and equating the resulting expression to zero, we obtain the equation

\[ 1+\alpha \xi^4-e^{\alpha \xi^4}=0, \tag{8} \]

from which we find the extremal value of \(\xi\). It is not difficult to verify that this will be a maximum of the function. Solving equation (8), we obtain:

\[ \xi=\sqrt[4]{\frac{28}{3\alpha}}. \tag{9} \]

For a machine with the parameters given in Sands’s article [3]
\((E=1.5\ \text{Bev};\ R=375\ \text{cm};\ n=0.6;\ \lambda=1.25)\), we obtain: \(\alpha=242.88\), \(\xi_{\max}=0.447\); \(E_{\max}=670\ \text{Mev}\). It follows from this that if the value of the equilibrium phase remained constant throughout the entire acceleration cycle, then the mean square deviation would be greatest not at the end of acceleration, but at \(\xi=0.447\).

Therefore, the minimum value of the equilibrium phase should have been determined not from the value of the mean square deviation at the end of acceleration, as was done by the authors of the above-mentioned papers [3, 5], but from its value at \(\xi=0.447\). True, the value of the mean square deviation at the maximum differs little from its value at the end of acceleration for a machine with the parameters indicated in [3]:

\[ \psi^2_{\max}=2.05\,\psi^2_{\text{end}}; \qquad \sqrt{\psi^2_{\max}}=1.43\sqrt{\psi^2_{\text{end}}}, \]

but since by this time the phase oscillations not associated with the quantum nature of the radiation will not yet have had time to damp, the absence of losses could be ensured only when the minimum value of the phase exceeds \(0.2\pi\) by a factor of 2–3. Let us note that the existence of a maximum of the function \(\psi^2(\xi)\) was first pointed out by A. A. Kolomenskii [7].

4. If no assumptions are made regarding the smallness of \(\dot E\) and the constancy of the equilibrium phase, then for accelerators with energy increasing according to a linear law,

\[ \psi^2(\xi)= b e^{-\frac{\alpha\nu}{4}} \int_{0}^{\chi} e^{\frac{\alpha\chi'}{4}} \frac{\chi'\gamma_1(\chi')}{\sqrt{1-\mu^2\chi'^{\,2}}}\,d\chi' - b\nu e^{-\frac{\alpha\nu}{4}} \int_{0}^{\chi} e^{\frac{\alpha\chi'}{4}} \frac{\gamma_1(\chi')}{\sqrt{1-\mu^2\chi'^{\,2}}}\,d\chi' , \tag{10} \]

where

\[ b=\frac{55}{192\sqrt{3}}\, \frac{k}{\lambda^3}\, \frac{e^2\frac{h}{2\pi}cT}{(1-n)R^4m} \left(\frac{E_0}{mc^2}\right)^7 \frac{mc^2}{E(\xi)}\,e^{-\frac{\alpha\xi}{4}}; \]

\[ \alpha=\frac{2}{3}\left(\frac{3-4n}{1-n}\right) \frac{r_0cT}{\lambda R^2} \left(\frac{E_0}{mc^2}\right)^3; \qquad \delta=\frac{2}{3}\frac{e^2c}{\lambda R^2} \left(\frac{E_0}{mc^2}\right)^4; \]

\[ \nu=\frac{E}{\delta}; \qquad \mu=\frac{2\pi\lambda\dot E}{\omega_0 e V_0(\xi)} = \frac{2\pi\lambda\dot E}{\omega_0 e V_0(0)(1+\beta\xi)} = \frac{\mu_0}{(1+\beta\xi)}; \]

\[ \gamma_1=\frac{2\pi\lambda}{\omega_0 e V_0(\xi)}; \qquad \chi=\eta+\nu; \qquad \eta=\xi^4. \]

In the general case the integrals entering into (10) cannot be evaluated. However, using the fact that the quantity \(\mu^2\chi'^{\,2}\) is considerably less than unity, and taking the function \(V_0(\xi)\) to be known at each instant of time, and restricting ourselves in the expansion of

\[ \frac{1}{\sqrt{1-\mu^2\chi'^{\,2}}} \]

to the first two terms (i.e., assuming that

\[ \operatorname{ctg}\varphi_s=\cos\varphi_s+\frac{1}{2}\cos^3\varphi_s, \]

which is valid for \(90^\circ>\varphi_s>60^\circ\) and gives an error of order \(10\%\), with an underestimate at \(\varphi_s=45^\circ\)), we obtain:

\[ \psi^2(\xi)= \frac{a}{\xi(1+\beta\xi)} \left\{ \eta\left(1+A\frac{\mu^2}{2}\right) - \frac{4}{\alpha}\left(1-e^{-\frac{\alpha\eta}{4}}\right) \left(1+B\frac{\mu^2}{2}\right) \right\}, \tag{11} \]

where

\[ a=\frac{55\sqrt{3}}{96}\, \frac{k}{\lambda}\, \frac{2\pi}{(3-4n)}\, \frac{hc}{eV_0(0)R} \left(\frac{E_0}{mc^2}\right)^3; \]

\[ A=(\nu+\eta)^2-\frac{16\nu}{\alpha} +\frac{96}{\alpha^2} -\frac{12\eta}{\alpha}; \]

\[ B=\nu^2-\frac{16\nu}{\alpha}+\frac{96}{\alpha^2}. \]

Substituting the specific values of the quantities entering into formula (11) \((\alpha=242.88;\ \mu_0^2=144.72;\ \nu=0.00625;\ a=4.444;\ \beta=20;\ k=4)\) and carrying out the numerical calculation, we obtain the values for the mean square and the square root of the mean square deviations, \(\psi^2(\xi)\) and \(\sqrt{\psi^2(\xi)}\) (Figs. 1 and 2).

From Fig. 2 it is seen that \(\sqrt{\psi^2(\xi)}\), beginning at the time \(\xi=0.2\), increases according to a linear law, reaching the value \(\sqrt{\psi^2(\xi)}=0.493\) at the end of acceleration.

For practice, what is important is not the value \(\overline{\psi^{2}}(\xi)\) itself, but its amplitude value \(\overline{\psi_0^{\,2}}\), which determines the conditions under which the electrons do not fall out of the acceleration regime. Taking \(\overline{\psi_0^{\,2}}=2\overline{\psi^{2}}(\xi)\), and \(\varphi_{smin}=\dfrac{\sqrt{\psi^{2}}}{\sqrt{2}}\), we obtain an expression

Fig. 1

Fig. 2

for the minimum value of the phase: \(\varphi_{smin}=19.9^\circ\). If the multiplicity \(k=6\) is chosen, then under the same conditions \(\varphi_{smin}=24^\circ\).

The oscillations of the mean radius, characterized by the mean-square deviation, are associated with the phase oscillations described by equation [5]. From formulas (2) and (4) it follows that

\[ \left(\frac{\Delta r}{r}\right)^2 = \frac{2}{3}\,\frac{\lambda}{k}\,\frac{r_0}{R}\, \frac{\operatorname{tg}\varphi_s(\xi)}{(1-n)} \left(\frac{E_0}{mc^2}\right)^3 \xi^3 \left(1+\frac{\nu}{\xi^4}\right) \overline{\psi^2}(\xi). \tag{12} \]

Carrying out a direct calculation, we find that at the end of acceleration
\(\sqrt{\overline{(\Delta r)^2}}\simeq 0.7\ \text{cm}\). Hence it is seen that the oscillations of the mean radius are small and do not impose significant restrictions on the parameters of the machine.

1. From what has been considered above it is clear that the form of the function \(\overline{\psi^2}(\xi)\) depends essentially on whether the equilibrium phase \(\varphi_s\) is constant or varies with time. This is evident from simple considerations:

\[ \overline{\psi^2}(\xi)\sim \frac{1}{E(\xi)}\,\operatorname{ctg}\varphi_s\,e^{-\xi^2/4}. \]

This function has a maximum if \(\varphi_s=\text{const}\), and has none if, with increasing \(\xi\), \(\operatorname{ctg}\varphi_s\) grows faster than \(\xi\) (i.e. \(\operatorname{ctg}\varphi_s\sim \xi^n;\ n>1\)). Since in practice \(\varphi_s\) is usually not a constant quantity, all calculations carried out for constant \(\operatorname{ctg}\varphi_s\) are hardly of great significance. Moreover, even if such a case could be realized, it would prove extremely undesirable, since in that case it is necessary to carry out the calculation for the time \(\xi=0.477\) and to take into account that the usual phase oscillations by this time have not yet had time to damp out; they must therefore be taken into account together with the phase oscillations induced by the quantum character of the radiation.

Let us also note that if one performs calculations for a machine with the same parameters, but in which the particles reach maximum energies in a time approximately an order of magnitude shorter than that given above \((T=0.03)\), the calculations lead to the following: quantum fluctuations begin to exert an influence at higher energies \((\xi=0.4)\), and then \(\sqrt{\overline{\psi^2}(\xi)}\) also grows according to a linear law, reaching at the end of acceleration a value very little different from that given above.

Finally, one can determine at what exit energies \(E_0\) from the accelerator the phase oscillations caused by the quantum character of the radiation will be so large that, in order to avoid particle losses, it will be necessary to choose the equilibrium phase near the value \(\pi/2\) (i.e., when the particle receives nothing from the high-frequency system and the synchrotron would have to cease accelerating the particles). The calculations lead to the following expression for the energy:

\[ E_0 = mc^2 \sqrt{ \frac{24\pi}{55\sqrt{3}}\, \frac{e^2}{\dfrac{h}{2\pi}c}\, (3-4n)\, \frac{\lambda}{k}\, \frac{V_0(1)}{H r_0} }, \tag{13} \]

where \(V_0(1)\) is the amplitude of the high-frequency voltage at the end of acceleration; \(H\) is the magnetic-field strength; \(r_0=e^2/mc^2\) is the electron radius. For reasonable values of the quantities entering into (13), we obtain \(E_0\simeq 10\) BeV, i.e., obtaining energies above 10 BeV in ordinary accelerators with weak focusing is impossible because of the rapid growth of phase oscillations caused by the quantum character of the radiation.

References

1. Sokolov A. A., Ternov I. M. Dokl. Akad. Nauk SSSR, 92, 537 (1953); ZhETF, 25, 698, 1953.
2. Sokolov A. A., Ternov I. M. Dokl. Akad. Nauk SSSR, 97, 823 (1954); ZhETF, 28, 432, 1955.
3. M. Sands, Phys. Rev., 97, 470, 1955.
4. Sokolov A. A., Ternov I. M., Strakhovskii G. M. ZhETF, 31, 439, 1956.
5. Matveev A. N. Dokl. Akad. Nauk SSSR, 108, No. 3, 1956.
6. Kolomenskii A. A. ZhETF, 30, 207, 1956.
7. Kolomenskii A. A. Report at the Geneva Conference, 1956.