FEATURES OF INDUCTION ACCELERATION IN AN ELECTROMAGNET WITH STRAIGHT SECTIONS

G. I. DIMOV

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

The one-electron theory of induction acceleration has been developed for axially symmetric magnetic guiding and electric accelerating fields. In doing so, certain requirements on the guiding and accelerating fields were established.

As is known, the requirements on the guiding magnetic field in induction accelerators—betatrons and resonant accelerators, synchrotrons—are the same. The features of electron motion in the guiding field of a synchrotron with straight sections and the requirements on it have been established. Electrons can move stably in such a field along a closed orbit.

Thus, consideration of phenomena directly connected with the guiding field is no longer necessary.

It remains to consider questions connected with the accelerating vortex electric field.

Let there be a stable electron orbit with a nonconstant radius of curvature \(r\):

\[ r = r(x), \]

where \(x\) is the path traversed by the electron.

The amplitude of the magnetic induction must vary along the orbit according to the law:

\[ B_m(x) = \frac{A_m \beta_m}{e c r(x)}, \tag{1} \]

where \(\beta_m\) and \(A_m\) are the relative velocity and total energy of the electron at the amplitude value of the magnetic field,

\(e\) is the charge of the electron,

\(c\) is the speed of light.

The magnetic field varies in time without a change in configuration:

\[ B = B_m(x)\vartheta(t), \tag{2} \]

where \(\vartheta(t)\) is the time dependence.

For an “ideal” electromagnet with four straight sections there will be two radii of curvature: the radius of the quadrants \(r_o\) and the radius of curvature in the straight sections \(r = \infty\).

The increase in the energy of the electron per revolution must correspond to the increase of the guiding magnetic field during the time of one revolution. Quantitatively

this is expressed by the relation between the magnetic induction on the orbit and the magnetic flux enclosed by the orbit.

The magnetic flux enclosed by the orbit is determined by the equation:

\[ \frac{d\Phi_k}{dt}=-\oint E\,dx, \quad *) \tag{3} \]

where \(E\) is the intensity of the vortex electric field on the orbit.

The intensity of the electric field must be proportional to the prescribed rate of change of the electron momentum \(P\):

\[ E=\frac{1}{e}\frac{dP}{dt}. \tag{4} \]

Thus,

\[ d\Phi_k=-\left[\oint E\,dx\right]dt =-\frac{1}{e}\left[\oint \frac{dP}{dt}\,dx\right]dt \]

\[ =-\frac{1}{e}\frac{d}{dt}\left[\oint P\,dx\right]dt =-\frac{1}{e}\,d\left[\oint P\,dx\right]. \quad *) \]

Let at \(t=0\), \(\Phi_k=0\) and \(P=0\). Then

\[ \Phi_k=-\frac{1}{e}\oint P\,dx. \quad *) \tag{5} \]

The momentum must correspond to the magnetic induction on the orbit and vary in accordance with the growth of the latter,

\[ P=-erB=erB_m\vartheta(t). \tag{6} \]

From (1):

\[ rB_m=\mathrm{const}, \]

therefore

\[ \oint P\,dx=-erB_m\vartheta(t)\oint dx=erBL, \quad *) \tag{7} \]

where \(L\) is the length of the orbit.

After substituting (7) into (5), we obtain:

\[ \Phi_k=rBL \]

or

\[ B=\frac{\Phi_k}{rL}. \tag{8} \]

For an “ideal” electromagnet with four straight sections,

\[ L=2\pi r_o+4l, \]

where \(l\) is the length of the straight sections.

Let us denote:

\[ s=\frac{L}{2\pi r_o}. \tag{9} \]

Then the magnetic induction in the quadrants will be expressed as:

\[ B_o=\frac{1}{2s}\frac{\Phi_k}{\pi r_o^2}. \tag{10} \]

\[ \text{*) In formulas under the sign “}\oint\text{” a circular integral should be understood.} \]

For an electromagnet without straight sections, \(s=1\); therefore

\[ B_o=\frac{1}{2}\frac{\Phi_k}{\pi r_o^2}. \]

Thus, the betatron relation connecting the magnetic induction on the orbit and the central magnetic flux changes, with the introduction of straight sections into the electromagnet, by a factor of \(s\) (\(s\) is the orbit-shape coefficient).

Induction acceleration in an axially symmetric guiding magnetic field \(B(x)=\mathrm{const}\) assumes a uniform distribution of the accelerating electric field along the orbit:

\[ \frac{\partial}{\partial x}E=0. \]

In this case, over the course of one revolution, at every point of the orbit the increase in the electron energy corresponds exactly to the increase in the magnetic field.

It is necessary to determine the distribution of the electric field \(E(x)\), satisfying the indicated requirement, in the case of a nonuniform distribution of the magnetic induction \(B(x)\) along the orbit

\[ E=\frac{1}{e}\frac{dP}{dt} =\frac{1}{e}\frac{d}{dt}(erB) =B\frac{dr}{dt}+r\frac{dB}{dt} \]

\[ =B\frac{dr}{dx}\frac{dx}{dt} +r\left(\frac{\partial B}{\partial x}\frac{dx}{dt} +\frac{\partial B}{\partial t}\right). \tag{11} \]

If

\[ \frac{\partial B}{\partial t}=0, \]

then \(E=0\) and

\[ B\frac{dr}{dx}\frac{dx}{dt} +r\frac{\partial B}{\partial x}\frac{dx}{dt}=0. \tag{12} \]

For \(\dfrac{\partial B}{\partial t}\ne 0\), the last equation remains valid by virtue of (2).

Therefore

\[ E=r\frac{\partial B}{\partial t}. \tag{13} \]

Since

\[ \frac{\partial B}{\partial t}=B_m(x)\frac{d}{dt}\vartheta(t), \]

then

\[ E=r(x)B_m(x)\frac{d}{dt}\vartheta(t). \tag{14} \]

From (1)

\[ r(x)B_m(x)=\frac{\beta_m A_m}{ec}. \]

Therefore

\[ E=\frac{A_m\beta_m}{ec}\frac{d}{dt}\vartheta(t). \tag{15} \]

Thus,

$$ \frac{\partial E}{\partial x}=0. $$

For any distribution of the magnetic induction along the orbit, uniform motion with acceleration requires constancy of the tangential component of the electric field along the orbit.

In the electromagnet designs used in practice, the condition

$$ \frac{\partial E}{\partial x}=0 $$

is, as a rule, not fulfilled.

Even in “round” electromagnets, where the guiding magnetic field is axially symmetric—for example, in a Kerst-type W-shaped betatron electromagnet—the accelerating vortex electric field is not uniformly distributed along the orbit.

In an electromagnet with straight sections, the electric field is distributed along the orbit in an obviously nonuniform manner.

The calculation of the vortex electric field can be carried out by the method used for calculating the magnetic field. This is evident from Maxwell’s equations:

$$ \operatorname{rot} E=-\frac{\partial B}{\partial t}, $$

$$ \operatorname{rot} H=j. \tag{16} $$

The rate of change of the magnetic induction in the electromagnet coincides in direction with the magnetic induction and is proportional to it. If the magnetic field of currents distributed in the same way as the magnetic induction in the electromagnet is found, then the problem of determining the vortex electric field along the orbit will thereby be solved. In Fig. 1 we give the curve \(E(x)\) for an “ideal” electromagnet with straight sections, obtained by the indicated method. We see that on the straight sections the vortex electric field drops sharply.

Fig. 1

And in the case \(\partial E/\partial x \ne 0\), when the betatron relation is satisfied, there is correspondence between the increase of energy per revolution and the increase of the guiding magnetic field during a revolution. On individual sections of the orbit, however, the electron receives too little or too much energy from the vortex electric field (an analogous phenomenon occurs in a synchrotron; over a small segment of its path the electron receives the energy that it must receive over the entire revolution). The magnetic field grows uniformly as the electron moves; the increase in its energy is uniform only on average over a revolution, while during the revolution the rate of energy increase fluctuates.

This phenomenon should cause additional radial oscillations of the electrons. The amplitude of these oscillations is probably small.

Let us calculate the oscillations caused by the nonuniformity of the vortex electric field along the orbit for an electromagnet with four straight sections.

Let the electric field be distributed along the orbit as shown in Fig. 2. Such a distribution is less favorable in comparison with that observed in practice (Fig. 1).

Fig. 2

During the passage of an electron through the rectilinear section, the magnetic field in the quadrants increases by the amount

\[ \delta B=\frac{s-1}{4s}\,\Delta B, \tag{17} \]

where \(\Delta B\) is the increase of the magnetic field per revolution.

Since there will be no corresponding increase in the electron energy in the rectilinear section, the electron energy will deviate from the required value by

\[ A\beta=ecrB;\qquad \delta(A\beta)=\frac{s-1}{4s}\,\Delta(A\beta), \tag{18} \]

where \(\Delta(A\beta)\) is the deviation of the energy, multiplied by the relative velocity, over one revolution.

The deviation of the electron energy from the required value will cause a corresponding deviation of the electron radius in the quadrant. The relation between the deviations of energy and radius is found from the following relations:

\[ n=-\frac{\partial B}{B}\frac{r}{\partial r},\qquad \frac{\partial B}{B}=-n\frac{\partial r}{r},\qquad A\beta=ecrB, \]

\[ \delta(A\beta)=ec(r\,\delta B+B\,\delta r) =ecrB\left(\frac{\delta B}{B}+\frac{\delta r}{r}\right), \]

\[ \delta A(\beta)=A\beta(1-n)\frac{\delta r}{r}. \tag{19} \]

Thus

\[ \frac{\delta r}{r} =\frac{s-1}{4s}\,\frac{1}{1-n}\,\frac{\Delta(A\beta)}{A\beta} =\frac{s-1}{4s}\,\frac{1}{1-n}\,\frac{\Delta B}{B}. \tag{20} \]

The equivalent angular frequency of free oscillations in an electromagnet with four rectilinear sections is determined by [1]

\[ \omega'_{\beta}=\frac{1+s}{2s}\,\omega_{\beta}. \tag{21} \]

where \(\omega_\beta\) is the angular frequency of free oscillations in the quadrants. For radial oscillations,

\[ \omega_{\beta r}=\sqrt{1-n}\,\frac{\beta c}{r_0}. \tag{22} \]

The electron revolution frequency in the presence of straight sections is

\[ \omega'=\frac{1}{s}\,\frac{\beta c}{r_0}. \tag{23} \]

From (21), (22), (23), and (9) we obtain

\[ k'=\frac{\omega'_{\beta r}}{\omega'}=\frac{1+s}{2}\sqrt{1-n}. \tag{24} \]

The equation of the electron’s radial oscillations may be written in the form

\[ x=r-r_0=\delta r\sum_{i=0}^{\nu}\sin\left(\omega'_{\beta r}t-\alpha_i\right), \tag{25} \]

where \(\alpha_i\) is the phase of the perturbation of the oscillations,
\(\nu\) is the number of perturbations from the beginning of the oscillations.

The perturbations of the oscillations are repeated after each quarter of the revolution period \(T'\); therefore:

\[ \alpha_0=0;\qquad \alpha_1=2\pi\frac{1}{4}\frac{T'}{T'_{\beta r}} =2\pi\frac{1}{4}\frac{\omega'_{\beta r}}{\omega'} =\frac{1}{4}\,2\pi k', \]

\[ \alpha_2=\frac{2}{4}\,2\pi k';\qquad \alpha_3=\frac{3}{4}\,2\pi k';\qquad \alpha_4=\frac{4}{4}\,2\pi k', \]

\[ \alpha_i=i\,\frac{\pi}{2}\,k'. \tag{26} \]

Thus,

\[ x=\delta r\sum_{i=0}^{\nu}\sin\left(\omega'_{\beta r}t-i\,\frac{\pi}{2}k'\right). \]

Using the well-known equality

\[ \sum_{i=0}^{\nu}\sin(y+iz) = \sin\left(y+\frac{\nu}{2}z\right)\, \sin\left(\frac{\nu+1}{2}z\right)\, \operatorname{cosec}\frac{z}{2}, \tag{27} \]

we obtain the equation of the oscillations in the following form:

\[ x= \frac{\delta r}{\sin\left(k'\frac{\pi}{2}\right)} \cdot \sin\left(\omega'_{\beta r}t-\frac{\nu\pi}{4}\right) \cdot \sin\left(\frac{\nu+1}{4}\pi k'\right). \tag{28} \]

Hence the amplitude of the radial oscillations is

\[ x_m = -\frac{\delta r}{\sin\left(k'\frac{\pi}{4}\right)} . \tag{29} \]

Calculations in specific cases show that the radial oscillations due to the nonuniformity of the vortex electric field are small. Induction acceleration in an electromagnet with straight sections is of interest as preliminary acceleration in a synchrotron.

REFERENCES

1. Blachman and Courant, Rev. Sci. Instr., 20, 596—601, 1949.