Influence of Radial Phase Inhomogeneity on the Configuration of the Magnetic Field in Accelerators

I. G. Leshchenko

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

To ensure proper radial and axial focusing of charged particles in accelerators of the betatron and synchrotron type, it is necessary that the magnetic field in the acceleration zone decrease with radius in inverse proportion to the quantity \(r^n\).

The value of \(n\) is determined, as is known, by the relation

\[ n = -\,\frac{r \cdot dH_z}{H_z \cdot dr}, \tag{1} \]

which is valid at any point of the magnetic field both for the amplitude value and for the instantaneous value of the magnetic-field strength.

Indeed, the instantaneous value of the magnetic-field strength at some point of the interpole space, for its sinusoidal variation and without taking into account the phase shift with respect to the radius of the equilibrium orbit \(r_0\),

\[ H_t = H_m \cdot \sin \omega t = H_{0m} \cdot e^{\displaystyle \int_{r_0}^{r} \frac{-n}{r}\,dr}\,\sin \omega t, \tag{2} \]

where \(H_m = f(H_{0m})\) is obtained from expression (1).

If equation (2) is differentiated with respect to the radius and the resulting equation is divided by equation (2), we find

\[ n = -\,\frac{r}{H_t}\cdot \frac{dH_t}{dr}. \tag{3} \]

This value of the index of decrease of the magnetic-field strength (without taking into account the influence of radial phase inhomogeneity) will hereafter be denoted by \(n_c\).

However, during the process of electron acceleration the quantity \(n\) may vary, and within considerable limits. The reasons for this are the presence of various inhomogeneities of the magnetic field and the saturation of the pole tips in strong magnetic fields. It is quite obvious that, in the presence of magnetic-field inhomogeneities caused by nonuniformly distribut-

losses in the electromagnet due to eddy currents and hysteresis losses, the expression for the field-decay index must be such as to take these phase inhomogeneities into account.

Taking into account the radial phase inhomogeneity, under continuous feeding of the electromagnet with an alternating sinusoidal voltage, the expression for the instantaneous value of the magnetic-field intensity may be represented as:

\[ H_t=H_{0м}\cdot e^{-\int_{r_0}^{r}\frac{n_c}{r}\,dr}\cdot \sin(\omega t-\varphi), \tag{4} \]

where \(\varphi\) is the angle of phase shift of the flux at some radius.

To determine the influence of the radial phase inhomogeneity of the magnetic field on its configuration, i.e., on the value \(n\), at any instant of time, at different points of the electron acceleration zone, from expression (4) we find the instantaneous value of the magnetic-field decay index.

After differentiating expression (4) with respect to the radius,

\[ \frac{dH_t}{dr} = -H_{0м}\cdot e^{-\int_{r_0}^{r}\frac{n_c}{r}\,dr} \cdot \left( \frac{n_c}{r}\cdot \sin(\omega t-\varphi) + \cos(\omega t-\varphi)\frac{d\varphi}{dr} \right) \]

and dividing by equation (4), we obtain:

\[ \frac{dH_t}{H_t\cdot dr} = -n_c\frac{1}{r} - \frac{d\varphi}{\operatorname{tg}(\omega t-\varphi)\,dr}, \]

whence

\[ n_t=n_c+\frac{r}{\operatorname{tg}(\omega t-\varphi)}\cdot\frac{d\varphi}{dr}, \tag{5} \]

or

\[ n_t=n_c+n_{\varphi}, \tag{6} \]

where \(n_{\varphi}\) should be understood as

\[ n_{\varphi}=\frac{r}{\operatorname{tg}(\omega t-\varphi)}\cdot\frac{d\varphi}{dr}. \tag{7} \]

The influence of \(n_{\varphi}\) on the instantaneous value of the magnetic-field decay index \(n_t\) in accelerators can be judged when an experimentally measured dependence \(\varphi=f(r)\) is available. In practice, the value of the phase shift is determined not absolutely, but relative to some point in which there is also a phase shift due to losses in the electromagnet.

The correct choice of the time reference origin (the coordinate origin) is of essential importance.

As is known, in induction accelerators the instant of injection of electrons into the accelerating chamber depends on the voltage on the injector and on the corresponding magnetic-field intensity on the equilibrium orbit \((H_{i0})\), and, when a phase shift is present, the absolute value of the phase shift of the flux on the orbit is also taken into account.

From the known expressions

\[ H_{i0}=\frac{3.46\sqrt{U_i}}{r_0} \quad \text{and} \quad H_{i0}=H_{0м}\cdot \sin\omega t_i \tag{8} \]

for an injection phase not exceeding four electrical degrees. When \(\sin \omega t_i = \omega t_i\), the value of \(t_i\) is determined as

\[ t_i=\frac{3.46\sqrt{U_i}}{\omega\cdot r_0\cdot H_{0m}}. \tag{9} \]

Substituting in the denominator

\[ \omega=2\pi f \quad\text{and}\quad r_0H_{0m}=\frac{A}{300\sin\alpha}, \]

we obtain:

\[ t_i\simeq \frac{150\sqrt{U_i}}{f\cdot A}, \tag{10} \]

where
\(U_i\) is the voltage at the injector, in volts,
\(r_0\) is the radius of the equilibrium orbit, in cm,
\(f\) is the frequency of variation of the magnetic field, in Hz,
\(A\) is the maximum energy of the accelerated particles (electrons), in MeV,
\(\alpha\) is the phase of particle extraction from the equilibrium orbit \((65^\circ—85^\circ)\).

In the presence of a phase shift at the equilibrium orbit, the magnetic-field strength required at the moment of injection, corresponding to the voltage at the injector, can be expressed as

\[ H_{i0}=H_{0m}\cdot \sin(\omega t-\varphi_0), \tag{11} \]

where \(\omega t-\varphi_0\) is the injection phase, which must satisfy conditions (8) and (10), i.e.

\[ \omega t_i=\omega t-\varphi_0. \]

Thus, expression (7) for \(n_\varphi\) at the radius of the equilibrium orbit can be represented as:

\[ n_\varphi=-\frac{r_0}{\operatorname{tg}\omega t_i}\cdot \frac{d\varphi_0}{dr_0}, \tag{12} \]

i.e., at the radius of the equilibrium orbit and at the azimuth where the injector is located, the relative phase shift should be taken as equal to zero.

Thus, the temporal origin of the coordinates corresponds to the point lying on the curve \(\varphi=f(r)\) at the radius of the equilibrium orbit.

In deriving equation (5), the general case of phase shift was considered without taking into account the signs in each particular case.

Analogously to the magnetic-field falloff index \(n_c\) (without allowance for phase inhomogeneity), the plus sign before \(n_\varphi\) is taken in the region where \(\varphi\) increases in the direction of phase lag with increasing radius, and the minus sign where the phase shift decreases.

The sign before \(\varphi\) in the denominator of expression (7) corresponds to the sign of the shift: for a lagging field, “\(-\)”; for a leading field, “\(+\)”.

For ordinary voltages on the electron gun \((30—60\ \text{kV})\), at \(f=50\ \text{Hz}\), for accelerators with energies from units to hundreds of MeV and higher, the injection phase varies within the range from tens to units of microseconds, which corresponds to less than \(1—2\) electrical degrees. The phase shift is usually still smaller than the injection phase \((\omega t_i)\); therefore expression (7), as for small angles, with an error of less than \(0.1\%\), can be replaced by another:

\[ n_\varphi=\frac{r}{\omega t-\varphi}\cdot \frac{d\varphi^{1)}}{dr}. \tag{13} \]

\(^{1)}\) A similar expression can be obtained from equation (4), taking into account that \(\sin(\omega t-\varphi)=\omega t-\varphi\) for angles less than four electrical degrees, with an error of less than \(0.1\%\).

As is known from the condition for electron oscillations at the beginning of acceleration in the betatron regime, owing to the azimuthal phase inhomogeneity of the magnetic field, \(\varphi\) is allowed to be less than \(0.1\,\omega t_i\) [3]; therefore, for the subsequent analysis, equation (14) may be written in the form

\[ n_t = n_c + \frac{r}{\omega t_i}\cdot \frac{d\varphi}{dr}. \tag{14} \]

For the expression

\[ n_{\varphi i} = \frac{r}{\omega t_i}\cdot \frac{d\varphi}{dr} \]

it is easy to see that, for the same phase shifts \(d\varphi/dr\) and voltages at the injector, \(n_{\varphi i}\) increases in proportion to the square of the radius of the equilibrium orbit,

\[ n_{\varphi i} = \frac{r_0}{\dfrac{3.46\sqrt{U_i}}{r_0 H_0}}\cdot \frac{d\varphi}{dr} = \frac{H_0 r_0^2}{3.46\sqrt{U_i}}\cdot \frac{d\varphi}{dr} \simeq r_0^2, \tag{15} \]

where \(n_{\varphi i}\) is the value of \(n_{\varphi}\) at the instant of injection.

For accelerators with small radii of the equilibrium orbit, when the injection of electrons into the accelerating chamber occurs at \(t=t_i\), which is usually equal (from equation 10) to several tens of microseconds, and when \(d\varphi \ll \omega t_i\), the value of \(n_{\varphi}\) is very close to zero.

In this case \(n_{\varphi}\) need not be taken into account in tuning and starting accelerators. In practice these are betatrons and synchrotrons up to 25 MeV \((n_{\varphi}<0.1\,n_c)\).

When the question of the configuration of the magnetic field concerns accelerators with large radii, the quantity \(n_{\varphi}\) may prove to be very significant, far exceeding the static (required) value of the magnetic-field fall-off index \(n_c\).

For lagging values of the phase shift \((-\varphi)\), when the difference \(\omega t-\varphi\) tends to zero, \(n_{\varphi}\) tends to infinity and, consequently, the required pattern of magnetic-field distribution in the acceleration zone is completely disturbed. The indicated change of the field concerns to a greater extent the region of radii near the outer rim of the poles, where the phase shift is considerably greater than at \(r_0\).

Fig. 1

We experimentally determined the influence of radial phase inhomogeneity on the configuration of the magnetic field for a betatron with radius \(r_0=50\) cm [9].

The dependence \(n_c=f(r)\) (Fig. 1) was obtained by measuring the magnetic-field fall-off index with an \(n\)-meter, which in practice almost does not take \(n_{\varphi}\) into account [8].

To determine \(n_{\varphi}\), the radial phase inhomogeneity at the same azimuth was measured with a microphasometer (Fig. 2).

From the obtained curve \(\varphi=f(r)\), similarly to the determination of \(n_c\) by a graphical method, the value of \(n_\varphi\) was calculated at radii from 47 to 53 cm for various times from the start of acceleration, using the expression

\[ n_\varphi=-\frac{r}{(t-\varphi)}\cdot\frac{\Delta\varphi}{\Delta r}, \tag{16} \]

obtained from (13) on passing to finite increments \(d\varphi\) and \(dr\).

Since the phase shift is measured with the microphasometer in microseconds, the phase \(\omega t\) in equation (13) is also taken in microseconds.

Fig. 2

The results of determining \(n_t=f(r,t)\) have been confirmed experimentally and are presented in Fig. 3. Such a distribution of the magnetic field in the gap of the interpole space at the beginning of acceleration leads to a considerable reduction of the working region between the poles.

The theoretical and experimental data make it possible to assert that radial phase inhomogeneity has a very serious influence on the change in the configuration of the magnetic field at the initial moment of particle acceleration. The construction and operation of accelerators for high energies (hundreds and thousands of MeV), in the presence of even a slight phase inhomogeneity along the radius, will always require correction of the topography of the magnetic field at the initial moment of acceleration.

Fig. 3

To weaken the influence of radial phase inhomogeneity on the configuration of the magnetic field, it is necessary:

1) to use an increased voltage on the electron gun, since \(t_i\) is proportional to \(\sqrt{U_i}\);

2) to use a reduced frequency of the electromagnet supply, since \(t_i\) is inversely proportional to the frequency \(f\) (equation 10);

3) to reduce the magnitude of the phase inhomogeneity by the proper choice of the design and manufacturing technology of the electromagnet.

The influence of radial phase inhomogeneity on the process of particle acceleration has been briefly reported in the foreign literature.

There is a report [4] on the harmful effect of eddy currents in a flat model of the magnet of a 3500 MeV proton synchrotron on the magnetic field of the gap. Eddy currents in the 12.7 mm plates of which the model is assembled affect the field configuration during the first 30 msec, whereas proton acceleration must continue for 1 sec; moreover, as the radius increases, \(n\) increases.

A curve \(n_t = f(t)\) is given. Correction of the field was carried out by concentric coils placed at different radii in the working zone of the interpolar space.

In the 300 MeV synchrotron of the Massachusetts Institute of Technology [5], in order to eliminate the harmful effect of eddy currents, windings were placed above and below the accelerating chamber, through the nine turns of which a specified current was passed.

In the cited literature [4, 5] and in the article [6], devoted to the magnetic problems of a 1000 MeV cosmotron, no equations or method are given for determining the index of decrease of the magnetic field with allowance for the presence of radial phase inhomogeneity of the magnetic field.

The equations derived by us make it possible without great difficulty to determine the dynamics of the influence of phase inhomogeneities on the configuration of the magnetic field throughout the entire region of the interpolar space, i.e.

\[ n_t = f(r, z, t). \]

The expressions (5, 16, etc.) for determining the dynamics of the configuration of the magnetic field are valid within certain limits:

a) for continuous excitation of the electromagnet from \(t = 0\) to \(t = t_n\). \(t_n\) corresponds to the moment at which the index of decrease of the magnetic field begins to change appreciably because of saturation of the pole corrective teeth;

b) for pulsed excitation of the electromagnet from \(t = t_{v.x}\) to \(t = t_n\), where \(t_{v.x}\) is the time of establishment of eddy currents in the steel (the time of establishment of the phase shift \(\varphi\)).

For \(t < t_{v.x}\) the phase shift \(\varphi\) is a quantity depending not only on the radius but also on time. In practice this will refer to the region of transient magnetic processes, in which \(n_t\) will be expressed by a very complicated function. In addition, together with the change in the configuration of the magnetic field in the acceleration zone, there occurs a violation of the 2:1 condition for the betatron acceleration regime [7]. Consequently, from this point of view as well, it is undesirable that, in an accelerator operating on pulses, the moment of injection of the accelerated particles be less than \(t_{v.x}\).

The magnitude of the eddy-current establishment time can be determined from the expressions given in the literature [1, 2, 7]. Thus, for sheet steel of grade E41—0.35 with 4.6% silicon content, \(t_{v.x} = 6 \cdot 8\ \mu\text{sec}\). Thus, the time of establishment of eddy currents is the initial boundary for considering the dynamics of the process of change of \(n_t\) in partial cycles. This same time determines the necessary minimum voltage on the electron gun; for example, in a 100 MeV betatron the voltage on the gun must be not less than

\[ U_i = \left(\frac{t_{v.x} \cdot f \cdot A}{150}\right)^2 = \left(\frac{7 \cdot 50 \cdot 100}{150}\right)^2 \approx 55\ \text{kV}. \]

In conclusion, we note that with continuous excitation of the electromagnet the radial phase inhomogeneity, especially at the rims of the magnetic poles, depends to a considerable extent on the magnetizing windings. On

On partial cycles, the influence of the magnetizing windings on the radial phase inhomogeneity is reduced because the time for eddy currents to be established in copper is, as a rule, much longer than the time for eddy currents to be established in steel. Therefore, an analytical calculation of the dynamics of the magnetic-field configuration must be carried out for the known radial phase inhomogeneity of the magnetic field for the given operating mode of the installations. With known dynamics of the radial phase inhomogeneity on partial cycles, the limits of applicability of expressions (5), (14), etc., are \(t = 0\) and \(t = t_{н}\).

REFERENCES

1. Nikhoki L. S., Pulse Engineering, Sovetskoe radio, 1949.

2. Arkad'ev V. K., Electromagnetic Processes in Metals, Part II, ONTI USSR, 1936.

3. Goward F. K. Effect of azimuthal inhomogeneities in the Magnetic Field of a Betatron or Sinchrotron, Pros. Phys. Soc., 61, 345, 284—293, 1948.

4. Livingston M. S., Blewett J. P., Green G. K. and Haworth L. I. Desing study for three-Bev Proton Accelerator, Rev. Sci. Instr., 21, 1, 7—22, 1950.

5. The 300-Mev Sinchrotron at the Massachusetts Institute of Technology, Eng., 191, 4967, 440—442, 1951.

6. Blewett M. H., Eddy-Current Phenomena in the Cosmotron, Rev. Sci. Instr., 24, 9, 856—860, 1953.

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8. Leshchenko I. G. Methods for measuring the magnetic-field fall-off index in accelerator installations, Izvestiya TPI, vol. 87, 1957.

9. Leshchenko I. G. Calculation, construction, and experimental study of the electromagnet of a 100-MeV betatron, Dissertation, TPI, 1955.