Accounting for Radiation Braking in Calculating the Displacement of Betatron-Accelerated Electrons onto the Target

V. M. Razin

(Presented by the scientific seminar of the Physics and Technology Faculty)

In large betatrons, as a result of the electromagnetic radiation of electrons moving with enormous acceleration along an orbit at high energies, of the order of 70 MeV and above, the radius of the electron orbit decreases. In calculating the displacement of accelerated electrons onto the target, this effect must be taken into account. In solving this question we shall follow the method indicated by Artsimovich and Pomeranchuk [1].

The change in the orbit radius is calculated in [1] for the case when it is small in comparison with \(r_0\). As the initial equations of motion, [1] uses the following equations:

\[ \frac{dp}{dt}=eE-\Delta W, \tag{1} \]

\[ \frac{pc}{e}=Hr, \tag{2} \]

where
\[ \begin{aligned} p &\text{— electron momentum,}\\ r &\text{— electric-field strength,}\\ E &\text{— radius of the trajectory,}\\ \Delta W &\text{— force of radiation braking, numerically equal to the energy lost to radiation per }1\text{ cm of path.} \end{aligned} \]

\(\Delta W\) is determined by the expression

\[ \Delta W=\frac{2}{3}\rho_0^2\left[\vec{\beta}\,\vec{H}\right]^2 \left(\frac{W}{m_0c}\right)^2, \tag{3} \]

where
\[ \begin{aligned} W &\text{— electron energy, including its rest energy,}\\ m_0 &\text{— rest mass of the electron,}\\ c &\text{— speed of light,}\\ \rho_0=\frac{e^2}{m_0c^2} &\text{— classical electron radius,}\\ H &\text{— magnetic-field strength,}\\ e &\text{— electron charge.} \end{aligned} \]

in an electron accelerator

\[ H=H_m(r)\psi(t)=H_m(r)\sin \omega t=H_m(r)\sin\varphi, \tag{4} \]

where \(H_m(r)\) is the maximum value of the magnetic-field intensity at the given point.

The solution of the equations of motion (1) and (2) is given in [1] in the form:

\[ \frac{x}{r_0}=-3.8\cdot10^{-17}\, \frac{H_m^2(r_0)W_m}{1-n}\, \frac{1}{\psi(t)} \int_0^t \psi^4(t)\,dt. \tag{5} \]

Here \(x=r-r_0\),
\(W_m\) is the maximum value of the electron energy in electron-volts,
\(H\sim r^{-n}\) near the stable orbit.

Taking into account (4) and the fact that \(\omega=2\pi f\), let us pass to the new variable \(\varphi=\omega t\)

\[ \frac{x}{r_0}=-6\cdot10^{-16}\, \frac{H_m^2(r_0)W_m}{f(1-n)}\, \frac{1}{\sin\varphi} \int_0^\varphi \sin^4\varphi\,d\varphi, \tag{6} \]

where

\[ \frac{1}{\sin\varphi}\int_0^\varphi \sin^4\varphi\,d\varphi = \frac{1}{\sin\varphi} \left( \frac{3}{8}\varphi-\frac{1}{4}\sin 2\varphi+\frac{1}{32}\sin 4\varphi \right). \tag{7} \]

This change in the orbit radius must be taken into account in calculating the displacement, by substituting into the calculation formulas, as the required displacement, not the distance from the equilibrium calculated orbit to the target, but the distance from the actual position of the orbit at the moment of ejection to the target.

In high-energy betatrons, the reduction of the orbit radius as a result of radiation losses due to emission becomes so large that it becomes necessary, at the end of acceleration, to create an additional accelerating vortex electric field by increasing the magnetic flux within the orbit circle. Such compensation of radiation losses prevents the orbit radius from decreasing to the point where the accelerated electrons strike the inner wall of the chamber. The problem of compensating radiation losses is, to some extent, analogous to the problem of displacing electrons onto the target. Indeed, in the first case the problem is to prevent a reduction of the orbit radius by means of an additional vortex electric field; in the second, to increase the orbit radius until the electrons collide with the target, and in the particular case this increase in radius is also obtained as a result of the action of an additional vortex electric field.

The same method can be applied to solving these problems, namely: an additional increase of the magnetic field within the orbit circle. This suggests the possibility of combining the processes of compensating losses due to radiation braking and expanding the orbit for the purpose of ejecting the accelerated electrons onto the target.

To carry out this combination, we propose the following method. At the end of acceleration, when the reduction of the orbit becomes appreciable, one should introduce, in comparison with the usual displacement pulses, a longer time pulse of accelerating magnetic flux, of sinusoidal half-wave form, so that instead of a reduction of the radius of the equilibrium orbit, a gradual expansion of the orbit begins. This expansion must be calculated in such a way that, at the desired moment in time, the accelerated electrons reach the target.

Let us proceed to the mathematical analysis of the proposed method. We write the following equations of motion:

\[ \frac{dp}{dt}=e(E+E_k)-\Delta W, \tag{8} \]

\[ \frac{pc}{e}=Hr. \tag{9} \]

Our equations differ from equations (1) and (2) only by the addition of \(E_k\)—the field strength of the compensating vortex electric field; therefore it seems possible to make use, in the main, of the method of analysis mentioned above [1].

Eliminating \(p\) from (8) and (9), we obtain the trajectory equation

\[ \frac{e}{c}\,\frac{d}{dt}(Hr)=e(E+E_k)-\Delta W. \tag{10} \]

For \(\Delta W=0\) and \(E_k=0\), the orbit radius is independent of time and is equal to \(r_0\). Denoting the compensating magnetic flux by \(\Phi\), we shall assume that

\[ \Phi=\pi r_1^{2}B, \tag{11} \]

where \(B\) is the mean value of the induction in a circle of radius \(r_1<r_0\).

Let us prescribe the character of the change in the induction of the compensating magnetic flux as follows:

\[ B=B_m\sin(\omega_0t-\alpha). \tag{12} \]

Here the angular frequency \(\omega_0\) is determined by the duration of the pulse of the compensating magnetic flux, and the phase \(\alpha\) by the moment at which this pulse begins.

The quantity \(E_k\) is determined by the expression

\[ E_k=\frac{1}{2\pi rc}\,\frac{d\Phi}{dt}. \tag{13} \]

Taking (11) into account, we may write:

\[ E_k=\frac{r_1^{2}}{2rc}\,\frac{dB}{dt}. \tag{14} \]

The total electric-field strength as a function of the radius will be determined by the equation

\[ E(r)=E+E_k=\frac{1}{rc}\,\frac{d}{dt}\int_{0}^{r}Hr\,dr+\frac{r_1^{2}}{2rc}\,\frac{dB}{dt}. \tag{15} \]

On the basis of (4), the expression for \(E\) may be represented in the form:

\[ E=\frac{1}{rc}\,\psi'(t)\int_{0}^{r}H_m(r)r\,dr. \tag{16} \]

For small changes of \(r\), all quantities entering into equation (10) may be expanded in powers of \(x=r-r_0\) and restricted to the first terms-

expansion. Let us note, first of all, that in the expressions for \(E\) the term containing \(x\) to the first power vanishes, since \(\dfrac{d}{dr}E(r_o)=0\).

To prove this, differentiate (16):

\[ \frac{d}{dr}E(r)=\frac{1}{c}\,\psi'(t) \left[ H_m(r)-\frac{1}{r^2}\int_0^r H_m(r)\,dr \right]_{r=r_o}. \tag{17} \]

The second term in the brackets is evidently equal to \(\dfrac{1}{2}\overline{H_m(r_o)}\). The bar denotes the mean value of the field strength in a circle of radius \(r_o\). According to the betatron condition \(H(r_o)=\dfrac{\overline H}{2}\), the expression for \(\dfrac{d}{dr}E(r)\) becomes zero at \(r=r_o\).

Bearing this in mind and expanding both sides of equality (10) in powers of \(x\), we have

\[ \frac{e}{c}\frac{d}{dt} \left[ H(r_o)r_o+xH(r_o)+r_ox\left(\frac{d}{dr}H(r)\right)_{r=r_o} \right] = eE(r_o)+ \]

\[ +\frac{er_1^2}{2r_oc}\left(1-\frac{x}{r_o}\right)\frac{dB}{dt}-\Delta W. \tag{18} \]

Here \(E_k\) is represented in the form

\[ E_k=\frac{r_1^2}{2c}\frac{dB}{dt}\frac{1}{r} \simeq \frac{r_1^2}{2r_oc}\frac{dB}{dt}\left(1-\frac{x}{r_o}\right). \tag{19} \]

Since

\[ \frac{e}{c}\frac{d}{dt}\left[H(r_o)r_o\right] = \frac{e}{2\pi r_oc}\frac{d}{dt}\left[2\pi r_o^2H(r_o)\right] = \frac{e}{2\pi r_oc}\frac{d\Phi_1}{dt} = eE(r_o), \tag{20} \]

then

\[ \frac{d}{dt} \left[ xH(r_o)+r_ox\left(\frac{d}{dr}H(r)\right)_{r=r_o} \right] = -\frac{r_1^2}{2r_o}\left(1-\frac{x}{r_o}\right)\frac{dB}{dt} -\frac{c}{e}\Delta W. \tag{21} \]

Integrating, we have

\[ xH(r_o)(1-n) = -\frac{r_1^2}{2r_o}B -\frac{c}{e}\int_0^t \Delta W\,dt. \tag{22} \]

In integrating we omit the term \(\dfrac{x}{r_o}\ll 1\) and take into account that, near the equilibrium orbit,

\[ H\sim r^{-n}; \qquad \frac{dH}{dr}\sim -nr^{-n-1}, \]

i.e.,

\[ H(r_o)+r_o\left[\frac{d}{dr}H(r)\right]_{r=r_o}=H(r_o)(1-n). \]

The integral appearing on the right-hand side of equation (22) was evaluated in [1].

Passing to the variable \(\varphi=\omega t\) and substituting the numerical values of the constants, we write the integral in the following form:

\[ \frac{c}{e}\int_{0}^{t}\Delta W\,dt = 6\cdot 10^{-16}\, \frac{H_m^{3}(r_0)r_0 W_m}{f} \int_{0}^{\varphi}\psi^{3}(\varphi)\,d\varphi . \tag{23} \]

The energy \(W_m\) here is expressed in electron-volts.

Now expression (22), taking into account (4) and (12), can, after simple algebraic transformations, be represented in the form

\[ \frac{x}{r_0} = \frac{r_1^{2}}{2r_0^{2}}\, \frac{B_m \sin(\omega_0 t-\alpha)} {H_m(r_0)(1-n)\sin\varphi} - \]

\[ - 6\cdot 10^{-16}\, \frac{H_m^{2}(r_0)W_m}{f(1-n)} \frac{1}{\sin\varphi} \int_{0}^{\varphi}\sin^{4}\varphi\,d\varphi . \tag{24} \]

Introducing the notation

\[ K_1= \frac{r_1^{2}B_m} {2r_0^{2}H_m(r_0)(1-n)}, \tag{25} \]

\[ k_1(t)= \frac{\sin(\omega_0 t-\alpha)} {\sin\omega t}, \tag{26} \]

\[ K_2= 6\cdot 10^{-16}\, \frac{H_m^{2}(r_0)W_m} {f(1-n)}, \tag{27} \]

\[ k_2(t)= \frac{1}{\sin\omega t} \int_{0}^{\omega t}\sin^{4}\omega t\,d(\omega t), \tag{28} \]

we obtain:

\[ \frac{x}{r_0}=K_1k_1(t)-K_2k_2(t). \tag{29} \]

Equation (29) characterizes the change in the radius of the orbits in the presence of compensation for radiation losses by the method proposed above, and at the same time takes into account the influence of radiation braking on the process of change of the equilibrium-orbit radius. We note that the function \(f(t)=K_2k_2(t)\) coincides exactly with expression (6), differing from the latter only in sign.

The character of the change in the betatron orbit radius as a function of the phase of the accelerating field, when compensation of radiation losses and displacement onto the target are combined, is presented for several particular cases in Fig. 1.

The dashed lines in Fig. 1 indicate, in relative units, the position of the walls of the vacuum chamber, and the dotted line indicates the relative position of the target. As is seen from the graphical constructions of Fig. 1, by choosing \(B_m\), \(\omega_0\), and \(\alpha\) in the corresponding manner, one can obtain collision of the accelerated electrons with the target in the desired phase of the accelerating field and at the same time prevent electrons from striking the chamber wall as a result of contraction of the electron orbit due to the presence of radiation braking.

Fig. 1

The work was carried out under the supervision of Professor Doctor [[unclear: surname]] A. A.

REFERENCES

1. Artsimovich L. and Pomeranchuk I. Radiation of fast electrons in a magnetic field, ZhETF, 16, 379, 1946.