BETATRON WITH AZIMUTHAL VARIATION OF THE CONTROL MAGNETIC FIELD

A. A. Zvontsov, V. L. Chakhlov, A. A. Filimonov

(Submitted by the Scientific Research Institute of Nuclear Physics, Electronics, and Automation)

1. Features of the betatron acceleration regime in control fields with azimuthal variation

At present, in betatrons for low and medium energies, control magnetic fields are used that are time-varying and azimuthally uniform and that possess, as is known [1], relatively weak focusing properties.

Enhancement of the focusing properties of the control fields is possible only by abandoning the azimuthal uniformity of the control field [1], that is, by forming spatially periodic control fields in the region of the working gap.

Let us consider the features of a betatron control field with spatial (azimuthal) variation, as the most simply formed in accelerators of relatively small dimensions.

We prescribe the control field in the region of the working gap of the betatron in the median plane by one component, taking into account the fundamental harmonic:

\[ H_z(r,\theta)=H_{z0}\left(\frac{<r_0>}{r}\right)^{<n>}\{1+f(r)\sin[\beta(r)-N\theta]\}. \tag{1} \]

where

\(<n>\) is the value of the field index, averaged over the azimuth, on the averaged equilibrium orbit,

\[ <n>=-\frac{r\,dH_{z0}}{H_{z0}dr}, \]

\(f(r)\) is the depth of the field variation;
\(N\) is the number of periodicity elements;
\(\beta(r)\) is the phase of the fundamental harmonic of the field;
\(H_{z0}\) is the field, averaged over the azimuth, at the averaged radius of the equilibrium orbit \(r_0\).

Time-constant control magnetic fields of this type are usually used in isochronous cyclotrons [1]; consequently, the entire developed theory of particle motion proves to be valid also for betatrons with such a control field.

Let us dwell on some features of a betatron with azimuthal variation of the control field.

The equilibrium orbit is found by the method of iteration and, in the first approximation, taking into account only the fundamental harmonic, is described by the following equation:

\[ r_0(\theta)=\langle r_0\rangle\left\{1+\frac{f(\langle r_0\rangle)}{N^2}\sin[\beta(r)-N\theta]\right\}, \tag{2} \]

where

\(\langle r_0\rangle\) is the radius of the equilibrium orbit averaged over azimuth, determined by the expression

\[ \langle r_0\rangle=\frac{mvc}{e\cdot H_{z0}(r)}. \tag{3} \]

\[ \frac{\langle r_0\rangle\cdot f(\langle r_0\rangle)}{N^2}\sin[\beta(r)-N\theta] \]

determines the deviation of the equilibrium orbit from the averaged value.

The maximum deviation of the equilibrium orbit from its averaged value, which is equal to

\[ a=\frac{\langle r_0\rangle\cdot f(\langle r_0\rangle)}{N^2}, \tag{4} \]

amounts, for betatrons of low and medium energies, to only a few percent, so that in practice the equilibrium orbit may be regarded as a circle. Thus, for example, for a betatron model with azimuthal variation of the control field at 6 MeV, manufactured in the sector of portable small-sized betatrons of NIIYaF MGU at TPI, \(a''\) is equal to \(\sim 0.2\) mm at \(N=6\), and \(a\) is equal to \(\sim 0.8\) mm at \(N=3\).

The motion of electrons about the equilibrium orbit is described by the Mathieu–Hill equation:

\[ Y''+[a_y+2g_y\cos 2\xi]Y=0,\qquad Y\to x,z. \tag{5} \]

where \(x, z\) are small deviations from the equilibrium orbit in the \(r\) and \(z\) directions.

\[ 2\xi=\beta(r)-N\theta, \]

\[ a_x=\frac{4}{N^2}\left[1-\langle n\rangle-\frac{f^2\cdot(\langle r_0\rangle\cdot\beta')^2}{2(N^2-1+\langle n\rangle)}\right]; \]

\[ a_z=\frac{4}{N^2}\left[\langle n\rangle+\frac{f^2(\langle r_0\rangle\cdot\beta')^2}{2(N^2-1+\langle n\rangle)}+\frac{f^2\cdot N^2}{2(N^2-1+\langle n\rangle)}\right]. \]

\[ g_x=-\frac{2f}{N^2}\cdot(\langle r_0\rangle\beta');\qquad g_z=\frac{2f}{N^2}\cdot(\langle r_0\rangle\beta'). \tag{6} \]

The frequencies of betatron oscillations are approximately determined as follows:

\[ \nu_x^2\simeq 1-\langle n\rangle; \]

\[ \nu_z^2\simeq \langle n\rangle+\frac{1}{2}f^2+f^2\tg^2\eta. \tag{7} \]

where \(\eta\) is the angle between the radius vector and the tangent to the side surface of the shim (ridge).

Expression (7) shows that “gradient” focusing is the principal one, since in installations for low and medium energies with relatively small dimensions it is practically impossible to obtain a significant depth of variation at a sufficiently large angle \(\eta\) (the angle \(\eta\) must be close to \(\pi/2\) for \(\operatorname{tg}^2\eta > 100\)); however, in principle it is possible to use a control field that increases with radius \((<n><0)\).

In betatrons with a time-varying control field, the magnetic flux changing inside the orbit is selected in such a way that the orbit radius remains approximately constant during acceleration [1]; for this it is necessary, during acceleration, to satisfy the so-called betatron relation, equal to \(2:1\) for betatrons with an azimuthally uniform control field.

For betatrons with spatial variation of the control field, this relation differs from \(2:1\):

\[ \frac{\overline{H_z(t)}}{<H_z>(t)} \simeq \frac{2}{1}\cdot \frac{\left[1+\dfrac{k f^2}{4N^2}\right]^2}{1+\dfrac{2f}{N^2}}, \tag{8} \]

where

\[ \overline{H_z(t)} \]
is the field averaged over the area \(S\) enclosed by the orbit;

\[ <H_z>(t) \]
is the field averaged over the perimeter \(\Pi\) of the orbit, \(k=2\div 5\).

Thus, the degree of departure of the betatron relation depends on the depth of field variation \(f\) and on the number of periodicity elements \(N\), but this degree of departure is so small that for low- and medium-energy betatrons this relation may be regarded as practically equal to \(2:1\), since the deviation of the equilibrium orbit from a circle is small, and the values of \(f\) rarely exceed \(0.25 \div 0.35\).

In a betatron with a symmetric control field, the accelerated beam usually has the form of an ellipse, the size of whose semiaxes depends on the ratio \(\nu_x\) to \(\nu_z\), and in the ideal case the shape of the ellipse does not change with azimuth.

In a betatron with azimuthal variation, the behavior of the beam can be studied by means of the envelope method [2]. In the linear approximation, without taking into account the periodic modulation of the phase of betatron oscillations and normalizing the linearly independent solutions of equation (5) so that their Wronskian is equal to unity, one can obtain approximate expressions for the envelopes [3] (neglecting the space charge of the beam):

\[ Q_y=\frac{1}{\sqrt{\nu_y}}\cdot \left\{1-g_y M(\mu_y)\cos[\beta(r)-N\theta]\right\}; \tag{9} \]

where

\[ M(\mu_y)= \frac{1}{(2+\mu)^2-a_y} + \frac{1}{(-2+\mu)^2-a_y}; \]

\[ \mu_y=\frac{2}{N}\nu_y; \]

\(a_y\) and \(g_y\) are the coefficients of the Mathieu equation (5).

In (9), the quantity \(\dfrac{1}{\sqrt{\nu_y}}\) determines the azimuthally averaged beam size, while the beam modulation caused by the periodic structure of the control field is determined by the following quantity:

\[ \frac{g_y \cdot M(\mu_y)}{\sqrt{\nu_y}}\cos[\beta(r)-N\theta]. \tag{10} \]

Taking into account only “gradient” focusing and with \(\beta(r)=0\), one can obtain the following expressions:

\[ \begin{aligned} M(\mu_x) &\simeq \frac{1}{2\left[1-\dfrac{4(1-\langle n\rangle)}{N^2}\right]}; \\[6pt] M(\mu_z) &\simeq \frac{1}{2\left(1-\dfrac{4\langle n\rangle}{N^2}\right)} . \end{aligned} \tag{11} \]

\[ Q_x=\frac{1}{\sqrt[4]{1-\langle n\rangle}} \left[ 1+\frac{f^2}{N^2}\cdot \frac{1}{1-\dfrac{4(1-\langle n\rangle)}{N^2}} \cdot \cos N\theta \right]; \]

\[ Q_z=\frac{1}{\sqrt[4]{\langle n\rangle}} \left[ 1-\frac{f^2}{N^2}\cdot \frac{1}{\left(1-\dfrac{4\langle n\rangle}{N^2}\right)} \cos N\theta \right]. \tag{12} \]

Then the ratio of the maximum value of the envelope to its mean value will determine the degree of beam “modulation”:

\[ \Gamma=\frac{Q_{\max}(\theta)}{\overline{Q}} . \tag{13} \]

Here \(\Gamma\) is the form factor of the envelope.

Consequently, from (12) and (13) we obtain

\[ \Gamma_x=1+\frac{f}{N^2}\cdot \frac{1}{\left[1-\dfrac{4(1-\langle n\rangle)}{N^2}\right]} \simeq 1+\frac{f}{N^2}; \tag{14} \]

\[ \Gamma_z=1+\frac{f}{N^2}\cdot \frac{1}{\left(1-\dfrac{4\langle n\rangle}{N^2}\right)} \simeq 1+\frac{f}{N^2}. \]

Thus, the beam modulation amounts to only a few percent, and the total number of accelerated electrons in the equilibrium beam can be determined as follows:

\[ P=2\pi^2\langle r_0\rangle (r_i-\langle r_0\rangle)^2 \frac{\nu_x}{\nu_z}\cdot \rho_{\mathrm{eq}} . \tag{15} \]

where

\(r_i\) is the radius at which the injector is installed;

\(\rho_{\mathrm{eq}}\) is the equilibrium charge density, determined by the expression

\[ \rho_{\mathrm{eq}}= \frac{m_0 \cdot \omega_0^2(1-\langle n\rangle)} {e\cdot 2\pi(1-\beta^2)^{3/2}}, \tag{16} \]

where

\(\omega_0\) is the angular frequency of revolution;

\(\beta\) is the relativistic factor.

Knowing the envelopes of the betatron oscillations and using the relation between the deflecting and the coordinates of the particle’s departure from the injector, one can determine the variation of the optimal injection angle as a function of the azimuthal position of the injector [1, 2].

Since

\[ \psi_{io}(\theta)= \frac{r_i-\langle r_0\rangle}{\langle r_0\rangle} \cdot \frac{\partial Q_x(\theta)/\partial \theta}{Q_x(\theta)}; \tag{17} \]

and \(Q(\theta)\) is determined by equations (9) and (12), we obtain

\[ \psi_{io}(\theta)\simeq \frac{r_i-\langle r_0\rangle}{\langle r_0\rangle} \cdot \frac{f}{N}\sin[\beta(r)-N\theta]. \tag{18} \]

Calculations show that, for betatrons with azimuthal variation of the control field, the change in the optimal injection angle is small, and this change lies within the angular divergence of the injected electron beam, which practically ensures capture of the electrons into acceleration when the injector is installed at any azimuth. The foregoing considerations also indicate the geometrical meaning of the optimal injection angle: the direction of optimal admission of particles into the accelerator chamber coincides with the direction of the tangent to the envelope of the radial betatron oscillations (if the injector is located in the median plane).

In analyzing the dynamics of electron motion it was assumed that the control field is constant in time, although it is known that, in an azimuthally uniform control field, phase nonuniformity has a significant effect on the dynamics. The phase nonuniformity depends in many respects on the design of the betatron electromagnet, and in designing betatrons with an azimuthally uniform control field one seeks to reduce the phase nonuniformity as much as possible.

In a betatron with spatial variation of the control field, azimuthal-phase nonuniformity can be represented in the form of a periodic component with a period equal to the periodicity of the fundamental harmonic of the field, which is present even in the ideal case, upon which there will be superimposed the variation of the phase nonuniformity caused by the design of the electromagnet and by the technology of its manufacture.

The periodic component of the phase nonuniformity is not dangerous, since its influence is inversely proportional to the square of the number of elements of the field periodicity and reduces to an insignificant temporal change in the depth of variation of the field and, consequently, also of the field gradients at different azimuths.

A much greater danger is posed by the component of the phase nonuniformity caused by the design of the electromagnet and by the technology of its manufacture and assembly; however, the tolerances for this component are no stricter than the tolerances for the azimuthal-phase nonuniformity of an azimuthally uniform control field.

2. Determination of the Technical-and-Economic Ratios of Betatron Electromagnets with Azimuthal Variation of the Control Field

In developing various versions of electromagnets that provide the formation of a control field with azimuthal variation, it is necessary to know the technical-and-economic indices of such electromagnets, which show what factors, and to what extent, affect the economy of a betatron installation [4, 5].

Those technical-and-economic ratios that are used in comparing different versions of betatrons with an azimuthally uniform control field [5] must be supplemented, since in betatrons with a periodic control field the structure of the field used and the methods of forming it impose their own special features on the economy of the installation.

Let us consider possible ways of comparing betatron electromagnets with different structures of the control field.

One of the important indices characterizing a betatron electromagnet is the reactive energy stored in its interpole space, the magnitude of which depends on the total magnetic flux circulating in the electromagnet and on the magnetomotive force necessary to obtain the specified intensity of the control field in the working gap [4].

The total flux circulating in the electromagnet, independently of the structure of the control field, can be represented in the form of three of its components:

1. The magnetic flux required for accelerating the electrons to the specified energy.
2. The magnetic flux required for controlling the trajectory of electron motion.
3. The leakage magnetic flux.

The magnetic flux required for accelerating the electrons to the specified energy (its magnitude should be as large as possible) is enclosed inside the orbit (usually a “wave-shaped” one for accelerators with azimuthal variation of the control field [1]); and since the betatron ratio in practice differs only very little from \(2:1\), its magnitude is determined simply:

\[ \Phi_{\langle r_0\rangle}=2\pi \langle r_0^2\rangle B_{oz}, \tag{19} \]

where \(B_{oz}\) is the control-field induction, averaged over the azimuth, at the averaged radius of the equilibrium orbit \(\langle r_0\rangle\).

But since

\[ B_{oz}\simeq \frac{W_k}{300\langle r_0\rangle}, \tag{20} \]

then

\[ \Phi_{r_0}\simeq \frac{2\pi \langle r_0\rangle W_k}{300}, \tag{21} \]

where \(W_k\) is the final kinetic energy of the accelerated electrons.

Thus, the magnitude of the magnetic flux required for accelerating electrons to a specified energy is determined by the average values of the radius of the equilibrium orbit and of the induction, and depends linearly on the final energy of the accelerated electrons. Since, for equal radii of the equilibrium orbits of the compared variants, the magnitude of this flux, at a given energy of the accelerated electrons, is a stable quantity, it is most expedient to compare all the above-mentioned components of the total magnetic flux with the magnitude of the magnetic flux required for acceleration purposes; and the linear dependence of \(\Phi_{\langle r_0\rangle}\) on \(W_k\) makes it possible to carry out such a comparison at different energies of the accelerated electrons.

The magnetic flux required for controlling the trajectory of motion of the accelerated electron beam may be represented as consisting of two parts, one of which is a component of the flux necessary for acceleration, while the second part circulates in the annular region of the pole between the equilibrium orbit and the outer radius of the pole \((r_n)\); and since this part does not affect the attainable energy of the accelerated electrons, it should, if possible, be smaller, provided that the specified focusing forces are obtained in the specified region. Consequently, the magnitude of this part of the flux depends on the type of control field and on the width of the annular region between \(r_n\) and \(r_0\).

The leakage magnetic flux \((\Phi_{\mathrm{leak}})\) is distributed beyond the boundary of the pole (beyond \(r_n\)); and since this flux is an additional load for the ferromagnetic material of the return magnetic circuit, its magnitude, which has a noticeable effect on the dimensions and weight of the return magnetic circuit, should be as small as possible. To a considerable extent, the magnitude of this flux depends on the type of control field.

Thus, on the basis of all the foregoing, in addition to the coefficient \(\sigma_n\), equal to \([4, 6]\)

\[ \sigma_n=\frac{\Phi_n}{\Phi_{r_n}}, \tag{22} \]

it is necessary to introduce the following coefficients:

\[ \kappa_{r_n-r_0} = \frac{\Phi_{r_n}}{\Phi_{\langle r_0\rangle}} = \frac{\Phi_{r_0}+\Phi_{r_n-r_0}}{\Phi_{\langle r_0\rangle}} = 1+\frac{\Phi_{r_n-\langle r_0\rangle}}{\Phi_{\langle r_0\rangle}}, \tag{23} \]

\[ \kappa_n = \frac{\Phi_n}{\Phi_{\langle r_0\rangle}} = \frac{\Phi_{r_0}+\Phi_{r_n-r_0}+\Phi_{\mathrm{leak}}}{\Phi_{r_0}} = \kappa_{r_n-r_0} + \frac{\Phi_{\mathrm{leak}}}{\Phi_{\langle r_0\rangle}} = \sigma_n\cdot \kappa_{r_n-\langle r_0\rangle}, \tag{24} \]

where

\[ \Phi_n \]
is the total magnetic flux;

\[ \Phi_{r_n} \]
is the magnetic flux in the circle of the outer radius of the poles, under the condition that the specified focusing forces are obtained in the specified region;

\[ \Phi_{r_n-\langle r_0\rangle} \]
is the magnetic flux in the annular region between \(\langle r_0\rangle\) and \(r_n\), under the condition that the specified distribution of the azimuthally averaged decay index and the variation depth of the control field in this region is obtained;

\(k_{\mathrm{p}}\) is a coefficient taking into account the ratio of the total flux to the flux required for accelerating the electrons. This coefficient characterizes the degree of utilization of the magnetic flux. Usually \(k_{\mathrm{p}}\) is considerably greater than 1. The less \(k_{\mathrm{p}}\) differs from 1, the more “economical” the installation is;

\(k_{r_{\mathrm{n}}-\langle r_0\rangle}\) is a coefficient taking into account the ratio of the magnetic flux in the circle of the outer radius of the poles to the flux required for accelerating the electrons,

\(k_{r_{\mathrm{n}}-\langle r_0\rangle}\) indirectly characterizes the magnitude of the magnetic flux that is required for controlling the trajectory of electron motion.

Let us note that the theoretical determination of the above-mentioned coefficients is associated with considerable difficulties, owing to the complex structure of the field. It is much simpler to determine these coefficients experimentally; for this purpose it is necessary to place turns at the corresponding radii and, from the magnitude of the emf induced in these turns, determine the values of the coefficients.

Knowledge of these coefficients makes it possible to compare easily the magnetic fluxes (their “share” in relative units) circulating in the corresponding regions, independently of the type of controlling field and the geometrical parameters of the electromagnets. It must be taken into account that the stored reactive energy is also determined by the magnitude of the magnetomotive force \(F_{\mathrm{no}}\), which depends on the structure of the controlling field and on the vertical aperture of the interpolar gap.

Therefore the ratio

\[ \frac{F_{\mathrm{n.o.}1}}{F_{\mathrm{m.o.}2}} = \frac{\delta_{01}}{\delta_{02}}\cdot \frac{H_{0z1}}{H_{0z2}} = \kappa_F, \tag{25} \]

where

\(\delta_{01}\) and \(\delta_{02}\) are the values of the interpolar gap at the radii of the equilibrium orbits;

\(H_{0z1}\) and \(H_{0z2}\) are the mean field strengths of the controlling field at these radii, determines the degree of increase in ampere-turns for one electromagnet variant in comparison with another.

For a betatron with spatial variation of the controlling field one may write:

\[ 0.8\,H_{z\max}(\langle r_0\rangle)\cdot \bar{\delta}_0 = 0.8\,H_{0z}\cdot \bar{\delta}_0, \tag{26} \]

where \(H_{z\max}(\langle r_0\rangle)\) is the maximum value of the strength of the controlling field at \(\langle r_0\rangle\);

\(\bar{\delta}_0\) is the design length of the magnetic line of force at \(\langle r_0\rangle\), averaged over azimuth.

But since

\[ H_{0z} = \frac{H_{z\max}(\langle r_0\rangle)} {1+f(\langle r_0\rangle)}, \tag{27} \]

then

\[ \bar{\delta}_0 = \delta_0\left[1+f(\langle r_0\rangle)\right], \tag{28} \]

where \(f(\langle r_0\rangle)\) is the value of the depth of variation of the control field on the equilibrium orbit.

The depth of variation is readily determined from the known parameters of the poles (the width and number of ridges, \(\langle r_0\rangle\), \(r_{\mathrm{н}}\), the variation of \(\delta\) along the radial coordinate, and the radius of the central core) [7].

It follows from (10) that, in the case of equal interpolar gaps, in a betatron with azimuthal variation the equivalent (calculated) air gap is \([1+f(\langle r_0\rangle)]\) times larger than the air gap of a betatron with an azimuthally uniform control field, which leads to a corresponding increase in \(k_F\).

Let us note that in a betatron with spatial variation of the control field, the vertically focusing force acting on that part of the accelerated electron beam which is at a considerable distance from the median plane is greater than the same force acting in a betatron with an azimuthally uniform control field, because the depth of field variation increases along the vertical coordinate. Thus, the interpolar gap at \(\langle r_0\rangle\) in a betatron with spatial variation of the control field may be chosen smaller than in a betatron with an azimuthally uniform control field, and, as a result, \(k_P\) may be close to unity.

Knowing the ratios between the total magnetic fluxes and magnetomotive forces of the variants being compared, one can also compare them in terms of the amount of reactive energy stored:

\[ \frac{Q_1}{Q_2}=\frac{\Phi_{\mathrm{п}1}}{\Phi_{\mathrm{п}2}}\cdot k_P; \tag{29} \]

As an example, Table 1 compares several variants of betatron electromagnets with azimuthal variation of the control field with the electromagnet of the serially produced PMB-6 betatron. The compared variants had equal radii of the equilibrium orbits, equal radii of the central cores, and equal outer radii of the poles.

It follows from the table that the use of control fields with azimuthal variation makes it possible to reduce stray fields and reduce the ne—

Table 1

not only the total magnetic flux, but also the flux in the annular region between the equilibrium orbit and the outer radius of the pole, while simultaneously reducing the stored reactive energy. The table also shows that a considerable increase in the inter-pole gap (from 41 to 44.5 mm) leads to a substantial increase in the leakage fluxes and, as a consequence, to an increase in the total flux and in the stored reactive energy.

Similar criteria may be used to evaluate the techno-economic characteristics of betatrons with control fields of another structure; however, when fields of opposite direction are used in individual sections of the orbit, their absolute values must be taken into account.

Conclusion

The successful commissioning for radiation of a betatron with a control field similar to that described above gives grounds to hope that the use of control fields with azimuthal variation will open up new possibilities for further improvement of electron induction accelerators.

References

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2. A. M. Baldin, V. V. Mikhailov, M. S. Rabinovich. The envelope method for investigating free oscillations in accelerators. Journal of Experimental and Theoretical Physics, issue 6, 12, 1956, p. 993.

3. V. P. Dmitrievskii, T. M. Prilipko, V. S. Rybalko. Influence of local inhomogeneities of the magnetic field on the motion of particles in accelerators with spatial variation. JINR Preprint, Dubna, 1967, No. R-9-3434-1.

4. L. M. Anan’ev, A. A. Vorob’ev, V. I. Gorbunov. Electron Induction Accelerator—Betatron. Gosatomizdat, 1961.

5. V. L. Chakhlov. Candidate’s dissertation. Tomsk, TPI, 1964.

6. M. F. Filippov. Approximate determination of the scattering coefficient of the magnetic flux of the poles of a betatron electromagnet. Izvestiya TPI, vol. 87, Tomsk, TSU Publishing House, 1957.

7. A. A. Zvontsov, V. L. Chakhlov. Determination of the variation depth of the magnetic field of a betatron. Proceedings of the Jubilee Scientific and Technical Conference of the Faculty of Automatic Systems. Tomsk, 1970.