Strong-Focusing Accelerators with a Constant Magnetic Field*

K. R. Symon, D. W. Kerst, L. W. Jones, L. J. Laslett,
K. M. Terwilliger

The principle of strong focusing¹, which ensures a high degree of stability of both radial and vertical betatron oscillations in cyclic accelerators, makes it possible to design many types of accelerators with a constant guiding magnetic field. In such machines there exist stable equilibrium orbits for all particles, from the injection energy up to the maximum. All these orbits can be contained in a narrow ring, as in a synchrotron or betatron; the magnetic field must then vary with radius sufficiently rapidly to ensure the existence of orbits for particles of different energies. If the gradient of the guiding field does not depend on azimuth, then one of the types of betatron oscillations will clearly be unstable. However, by using a magnetic field whose gradient depends on azimuth (strong focusing), stability of both types of betatron oscillations can be ensured, even when the field changes rapidly with radius. Cyclic accelerators of charged particles may be divided into four groups according to the type of guiding magnetic field used in them: a constant field with a constant gradient (the ordinary cyclotron, phasotron, and microtron), a pulsed field with a constant gradient (the weak-focusing synchrotron and betatron), strong focusing with a pulsed field (the strong-focusing synchrotron), and strong focusing with a constant field (the ring phasotron**, betatron, and cyclotron).

In practice, two types of strong-focusing accelerators with a constant field seem the most important. In a machine with radial sectors², strong focusing is provided by the fact that the fields in successive focusing and defocusing sectors vary with radius in the same way, but are opposite in sign (in addition, in certain cases the fields vary in opposite directions). Since the orbit in sectors with the reverse direction of the field tends to move away from the center of the machine, the installation has considerably larger dimensions than an ordinary strong-focusing accelerator for the same energy and with equal magnetic fields. This serious drawback is largely eliminated in the design with spiral sectors³, in which the magnetic field consists of two parts: a field increasing with radius and independent of azimuth, and a field also increasing with radius but periodic in azimuth. The ridges (maxima)

* Phys. Rev. 103, 1837 (1956). One accelerator of this type was proposed in 1953 by A. A. Kolomenskii, V. A. Petukhov, and M. S. Rabinovich, “Some Questions in the Theory of Cyclic Accelerators,” Publishing House of the Academy of Sciences of the USSR, 1955. (Translator’s note.)
** In accordance with the terminology established in our literature, we have replaced the term “strong-focusing synchrotron with a constant guiding field” by “ring phasotron.” (Translator’s note.)

and the valleys (minima) of the periodic field are located on spirals making a small angle with the orbit. The radial distance between the ridges is small in comparison with the radial aperture. Particles crossing the ridges at a small angle experience strong focusing. Since here there is no need for regions with reverse field, the perimeter of this machine may be comparable with the perimeter of an equivalent ordinary strong-focusing accelerator.

The ring phasotron has many important advantages in comparison with the ordinary strong-focusing synchrotron, the chief one being the beam intensity. Since the magnetic field in such machines does not depend on time, the pulse repetition frequency is determined only by the repetition frequency of the cycles of high-frequency modulation (abbreviated h.f.). In an ordinary synchrotron the pulse repetition frequency is limited by the time required to complete the magnetic-field cycle. It is reasonable to suppose that the repetition frequency of the h.f. cycles can be made considerably higher than the reproduction frequency of the magnetic field. Another reason for the high beam intensity is the fact that in strong-focusing accelerators with a constant field a large injection aperture is possible (larger in the design with radial sectors than with spiral ones). Other advantages of the ring phasotron are purely engineering and operational simplifications. A magnet fed by direct current is simpler and cheaper to build and operate than a pulsed one. The magnet can be made solid, since there are no eddy currents, and the residual field and the difficulties associated with saturation are less important than in accelerators with a pulsed field. All field distortions are independent of time. The need for exact matching of the accelerating voltage with the magnetic field disappears, as a result of which we have great freedom in designing the h.f. system. Injection is possible at a lower energy than is assumed for an ordinary synchrotron, owing to the possibility of carrying it out at smaller field values, the ease of the frequency-modulation program, and the large aperture at the injection radius; the complexity of the injection system will thereby be somewhat reduced. The disadvantages of the ring phasotron are the considerable increase in the perimeter for an installation with radial sectors (at least by a factor of 3) and the difficulty of obtaining the required magnetic fields, especially for a machine with spiral sectors.

A betatron with a constant guide field should potentially have a much higher intensity than an ordinary betatron^4. The beam can be injected during a substantial fraction of the cycle if there is a suitable additional accelerating flux, the duration of which can at present be made greater than several tens of microseconds. The only limitation on the beam current at injection is apparently imposed by the influence of space charge, but it can be reduced by using high-voltage injection. In betatrons with a constant guide field there is no problem of coupling the guide field with the accelerating flux; there are also a number of other engineering simplifications, which were mentioned above.

The application of the principle of strong focusing to the cyclotron makes it possible to choose such a dependence of the magnetic field on radius that the period of revolution of the particle remains constant independently of energy even in the relativistic region. In modern cyclotrons (phasotrons) at high energies frequency modulation is necessary in order to compensate for the relativistic increase of mass. A cyclotron with constant frequency should increase the beam output by two orders of magnitude. A cyclotron with radial sectors, in which the field varies periodically with azimuth, was first proposed by Thomas^5. A design with spiral sectors, as applied to the cyclotron, seems more promising.

In the first part of this article we shall examine in detail strong-focusing accelerators with a constant field, both with radial and with spiral sectors. In the second part the theory of orbits in these accelerators is developed. The third part contains a description of a ring phasotron with radial sectors for an energy of 10 Bev and with spiral sectors for an energy of 20 Bev, a betatron with a constant guide field, and a strong-focusing cyclotron.

I. TYPES OF STRONG-FOCUSING ACCELERATORS WITH A CONSTANT FIELD

1. Ring phasotron with radial sectors

A cyclic particle accelerator with radial sectors can be constructed in such a way that the orbits of particles of maximum energy are located at the outer edge of the machine, while the orbits of particles at injection are at the inner edge, and conversely. Here we shall consider installations in which the orbits of particles of maximum energy are located at the outer edge. (We shall analyze the case of the ring phasotron, but most of the conclusions will also be valid for the betatron and the cyclotron.) In a ring phasotron with radial sectors the magnet consists of \(N\) identical elements, each of which is composed of a focusing and a defocusing sector. A sector that focuses in the radial direction, of course, defocuses vertically, and conversely. The azimuthal boundaries of the sectors coincide with radii drawn from the center of the machine (whence the name: construction with radial sectors). The direction of the magnetic field in neighboring sectors is opposite, although the dependence of the field on the radius is the same. The field in the median plane at any azimuth is determined by the function

\[ H \sim \left( \frac{r}{r_0} \right)^k , \tag{1.1} \]

where \(r_0\) is the radius of the equilibrium orbit, measured from the center of the machine, and \(k\) is a constant quantity for the given installation. Figure 1 shows a sample magnetic field of this type. Such a field shape means that the orbits of particles of different energies are similar, i.e., photographically map onto one another. In the ideal case the field along a closed equilibrium orbit is constant in each sector, and the particle trajectories are composed of arcs of circles. In reality such an ideal orbit cannot be obtained, since it is impossible to create sharp field boundaries. However, admitting this idealization, we obtain an especially simple case, when the fields on the orbit of a given energy have one and the same magnitude in sectors with positive and negative field direction. The equilibrium orbit for this case is shown in Fig. 2.

Fig. 1. Transverse section of the magnet for a machine with radial sectors.

It is evident that the deviation of a particle from the equilibrium orbit is determined by the degree of strong focusing. The number of radial \(\nu_x\) and vertical \(\nu_z\) betatron oscillations per revolution depends on the value of \(k\) and on the length of the sectors. \(\nu_x\) and \(\nu_z\) are constant for all energies. Sectors with negative field should preferably be made shorter in order to reduce the radius of the machine. The minimum length of the sectors with negative field is determined by the necessity of preserving the stability of the vertical betatron oscillations. Some vertical

focusing and radial defocusing take place owing to the “scalloping” of the orbit and to the edges of the sectors being crossed not at right angles. If we wish to preserve vertical stability in an installation in which the number of periodicity elements is large and the influence of the “scalloping” of the orbits is small, then the sectors with negative field must be made no shorter than \(2/3\) of the sectors with positive field. This means that the perimeter of the installation, not counting straight intervals, is 5 times greater than that which would be necessary in the absence of sectors with negative field. The ratio (in this case 5) of the radius of the installation to the minimum radius of curvature will be called the enlargement factor of the installation. The steady magnetic field in a ring phasotron can be made considerably larger than the pulsed magnetic field in an ordinary machine. Therefore the dimensions of an installation with radial sectors will in fact be approximately 3 times larger than the dimensions of a strong-focusing accelerator for the same energy with a pulsed field. It is also desirable to make the radial extent of the sectors as small as possible, which requires large field gradients. The permissible gradient is determined by the influence of errors in the assembly of the magnet. With a reasonable choice of gradient the minimum radial aperture is approximately \(2\%\) of the radius of the installation.

Fig. 2. Top view of a magnet with radial sectors.

2. Ring phasotron with spiral sectors

In a ring phasotron with spiral sectors the orbit of particles of maximum energy is located at the outer edge. It is inadvisable to have the orbit of particles of maximum energy inside the machine and to inject the particles from outside, since in that case it is difficult to ensure stability of radial oscillations.

The guiding field in the median plane, if there are no straight intervals, is given by the formula

\[ H=\bar{H}_0\left(\frac{r}{r_0}\right)^k\left\{1+f\cos\left[N\theta-N\tan\xi\ln\left(\frac{r}{r_0}\right)\right]\right\}, \tag{2.1} \]

where \(r\) is the radius drawn from the center of the machine, \(k\) is the magnetic-field index, \(\theta\) is the azimuthal angle with respect to the center of the installation, \(f\) is the inhomogeneity coefficient (relative variation of the field), \(N\) is the number of sectors (the period of the field variation), and \(\xi\) is the spiral inclination angle (the angle between the spiral—the geometrical locus of the values of the maximum field—and the radius).

Figure 3 shows several ridges and valleys forming periodic spirals of the field directed toward the outer side of the machine, and an equilibrium orbit is drawn for this construction. All equilibrium orbits are similar figures, whose linear dimensions are proportional to the radius, and they themselves rotate with the radius owing to the spiral periodicity of the field. Figure 4 shows the dependence of the magnetic field on the radius in the median plane. When the periodic ridges and valleys of the field are crossed at a small angle, the particles first experience a gradient of one sign, then of the other—

as a result of which strong focusing of the betatron oscillations is achieved. Owing to the increase of the field with radius, the negative gradient is smaller than the positive one. To some extent this is compensated by the “scalloping” of the orbits, which causes the particles to travel a longer path in the field with negative gradient than particles that would move along a circle. The degree of betatron focusing depends on the rate of increase of the field with radius, on the nonuniformity coefficient, and on the spiral angle.

Fig. 3. Schematic of a design with spiral sectors.

Fig. 4. Radial dependence of the magnetic field in the median plane.

The minimum radial aperture is limited chiefly by the difficulty of achieving sufficiently strong focusing by a periodic field for a given vertical aperture. If one confines oneself to a sinusoidal variation of the field, then the coefficient \(f = 1/4\), for a given degree of focusing, provides the maximum vertical gap in the case where pole pieces without distributed direct and return windings are used. With such small \(f\), the installation enlargement factor, equal in this case to \((1+f)\), is close to unity. Consequently, the radius of an annular phasotron with spiral sectors is approximately equal to the radius of an ordinary synchrotron for the equivalent energy. With a reasonable choice of parameters, the maximum radial aperture is \(3\%\) of the radius.

3. Other Types of Strong-Focusing Accelerators with a Constant Field

The accelerators considered above have an equilibrium orbit of constant shape, whose scale is proportional to the radius. Many modifications of these accelerators exist. Some of them differ only in having another type of dependence of the field on azimuth.

Changes of this sort do not affect the constancy of the shape of the equilibrium orbits and will have only a very slight effect on other characteristics of the machine. Other modifications keep \(v_x\) and \(v_z\) constant, but violate the similarity property of the equilibrium orbits. The azimuthal boundaries of the focusing and defocusing sectors can be made not coincident with radii, and the fields in the sectors with positive and negative direction can be different functions of radius (a sector with a negative field may even have a field equal to zero). The edges of the sectors can be cut in one and the same direction, thereby approaching a design with spiral sectors. Using special windings, one can pass from a design with spiral sectors at the outer edge of the machine, where a small installation enlargement factor is desirable, to a design with radial sectors, which makes it possible to obtain a large vertical aperture at injection,

on the inner edge. Such an arrangement should have the advantages of both types, but at the cost of a considerable complication of the magnet design.

Another modification is a cyclotron with spiral sectors and constant frequency. In this arrangement the frequency of revolution of the particles can be made independent of the energy even in the relativistic region. However, in this case the orbits will not be similar, and it is impossible to keep \(\nu_x\) and \(\nu_z\) constant.

II. THEORY OF ORBITS

4. Geometry of equilibrium orbits

For constructing a theory of the stability of motion in strong-focusing accelerators with a constant field, it is convenient to characterize them by a system of equilibrium orbits. We shall therefore assume that a system of equilibrium orbits lying in the median plane is specified. If instead the magnetic field is specified, then the equilibrium orbits are determined by integrating the equations of motion.

The geometrical properties of each orbit and the relation between orbits are periodically repeated in azimuth with period \(2\pi/N\). Each orbit is characterized by its equivalent radius \(R\), defined by the relation

\[ S=2\pi R, \tag{4.1} \]

where \(S\) is the length of the orbit. In the general case \(R\) will be somewhat larger than the mean radius \(\langle r\rangle\). Define the azimuthal coordinate \(\Theta\) by the equation

\[ s=\Theta R, \tag{4.2} \]

where \(s\) is the distance measured along the equilibrium orbit from some reference point, say from the azimuthal angle \(\theta_0\). We require that at the reference point the orbit be perpendicular to the radius drawn from the center of the machine, and that the reference points lie on a continuous curve. The parameter \(\Theta\) will be equal to the azimuthal angle \(\theta-\theta_0\) plus a small periodic function with period \(2\pi/N\).

Each orbit will now be determined by the periodic parameter \(\mu(\Theta,R)\), equal to

\[ \mu(\Theta,R)=\frac{R}{\rho(\Theta,R)}. \tag{4.3} \]

Here \(\rho\) is the radius of curvature. The specification of \(\mu(\Theta,R)\), together with the requirement that the center of the orbit lie at the origin of coordinates in the median plane, completely determines the orbit \(R\) for a given reference point. For our purposes it is sufficient to specify the angle \(\zeta(R)\) between the radius drawn from the center of the machine and the coordinate curve \(\Theta=0\) at their point of intersection (Fig. 5). The choice of the parameter \(\mu(\Theta,R)\) is restricted by the condition of periodicity in \(\Theta\) with period \(2\pi/N\) and by the value of the mean

\[ \langle \mu\rangle=\frac{1}{2\pi}\int_0^{2\pi}\mu\,d\Theta =\frac{1}{2\pi}\int_0^S \frac{ds}{\rho}=1. \tag{4.4} \]

In addition, the function \(\mu(\Theta,R)\) is constrained by the requirement that at the point \(\Theta=0\) the orbit \(R\) be perpendicular to the radius drawn from the center of the machine. This requirement leads to a further complication of the constraints imposed on the function \(\mu\). If \(\Theta=0\) is a point of symmetry of the orbit, then it is easy

write the restriction in analytic form

\[ \mu(-\Theta, R)=\mu(\Theta, R). \tag{4.5} \]

If there are no symmetry points, then it is necessary to construct an orbit in order to determine correctly the reference point \(\Theta=0\). Fortunately, errors in the determination of the reference point lead only to small errors (of order \(1/N^2\)) in the equations for betatron oscillations, provided that the angle \(\zeta\) is specified correctly.

We shall also need the parameters \(\eta(\Theta,R)\) and \(\varepsilon(\Theta,R)\), which establish the relation between the perpendicular distance \(dx\) between two neighboring orbits and the increment \(d\Theta\) of the parameter \(\Theta\) along an orthogonal trajectory to the orbits with an increment \(dR\) of the parameter \(R\) (see Fig. 5):

\[ dx=\eta dR, \tag{4.6} \]

\[ d\Theta=\frac{\varepsilon dR}{R}. \tag{4.7} \]

It can be shown\(^6\) that \(\eta,\varepsilon\) satisfy the differential equations

\[ \frac{\partial\varepsilon}{\partial\Theta}=\mu\eta-1, \tag{4.8} \]

\[ \frac{\partial\eta}{\partial\Theta}=-\mu\varepsilon-\int R\frac{\partial\mu}{\partial R}\,d\Theta, \tag{4.9} \]

where three constants of integration are chosen so that \(\varepsilon\) and \(\eta\) are periodic functions of \(\Theta\) (i.e. so that the mean values of the terms on the right-hand side of equations (4.8), (4.9) are zero) and

\[ \left[\frac{\varepsilon}{\eta}\right]_{\Theta=0}=-\operatorname{tg}\zeta. \tag{4.10} \]

Fig. 5. Diagram of equilibrium orbits.

If all equilibrium orbits are geometrically similar, the parameter \(\mu\) depends only on \(\Theta\) and does not depend on \(R\). For simplicity, we shall everywhere confine ourselves to considering machines of this type. If, in addition, \(\zeta\) does not depend on \(R\), then it follows from equations (4.8), (4.9), (4.10) that the parameters \(\eta\) and \(\varepsilon\) also do not depend on \(R\). In this case we shall say that the equilibrium orbits are similar: equilibrium orbits are similar if any system of orbits in the neighborhood of one equilibrium orbit can be obtained by a photographic enlargement or reduction of the system of orbits in the neighborhood of any other equilibrium orbit.

The solutions of equations (4.8) and (4.9) can be obtained by the method of successive approximations. Put

\[ \mu=1+fg(N\Theta), \tag{4.11} \]

where \(g(N\Theta)\) has period \(2\pi\) in \(N\Theta\), has mean value zero, and is normalized so that its mean square is equal to \(1/2\); \(f\) is the coefficient of field inhomogeneity. Since the terms on the right-hand side of equations (4.8) and (4.9) have period \(2\pi/N\) (and mean value zero), they lead to the appearance in \(\eta\) and \(\varepsilon\) of oscillating terms of order \(1/N\). The integral in equation (4.9) is equal to zero if it is assumed that \(\mu\) does not depend on \(R\); in the general case it leads only to very small oscillating terms if \(\mu\) changes insignificantly for a very small relative increase of the radius. The quantity \(\operatorname{tg}\zeta\) is equal to zero in machines with radial sectors and is of order \(N\) in machines

with spiral sectors. We shall therefore take as the zeroth approximation for \(\varepsilon,\eta\)

\[ \eta \simeq 1,\qquad \varepsilon \simeq \operatorname{tg}\zeta, \tag{4.12} \]

which satisfies the conditions imposed on \(\varepsilon\) and \(\eta\).

If \(F(\xi)\) is any periodic function of \(\xi\) with period \(2\pi\), then it is convenient to introduce the notation

\[ \langle F\rangle=\frac{1}{2\pi}\int_0^{2\pi}F(\xi)\,d\xi, \tag{4.13} \]

\[ \{F\}=F(\xi)-\langle F\rangle, \tag{4.14} \]

\[ F'=\frac{dF}{d\xi}, \tag{4.15} \]

\[ F_1=\int \{F\}\,d\xi, \tag{4.16} \]

\[ F_{n+1}=\int F_n\,d\xi, \tag{4.17} \]

where the constants of integration in the last two equations are chosen so that the mean value of \(F_n\) is zero. All functions defined by equations (4.14)—(4.17) have period \(2\pi\) and mean value equal to zero.

Substituting now (4.11) and (4.12) into (4.8) and (4.9) and integrating, we obtain the first approximation

\[ \eta=1-\frac{f\,\operatorname{tg}\zeta}{N}\,g_1(N\Theta), \tag{4.18} \]

\[ \varepsilon=\operatorname{tg}\zeta-\frac{f g_1(0)}{N}\sec^2\zeta+\frac{f}{N}g_1(N\Theta). \tag{4.19} \]

The constants of integration here are chosen in the appropriate manner. [We note that

\[ \langle g_1\cdot g\rangle=\frac{1}{2\pi}\int_0^{2\pi}g_1\,dg_1=0 \tag{4.20} \]

and, if \(g(\xi)\) is an even function, then \(g_1(\xi)\) is odd and, consequently, \(g_1(0)=0\). In any case \(g_1(0)\) is usually small.]

The second approximation can be obtained by substituting \(\eta,\varepsilon\) from (4.18), (4.19) into the right-hand side of equations (4.8), (4.9) and integrating. Each subsequent substitution leads to new terms of order \(1/N^2\) and \(f^2/N^2\) relative to the preceding ones.

5. Betatron Oscillations

For a particle with momentum \(p\), moving along the equilibrium orbit \(R\), according to equation (4.3) we have

\[ pc=eH\rho=\frac{eHR}{\mu}, \tag{5.1} \]

where \(H\) is the magnetic-field strength. Hence we obtain \(H\) as a function

coordinates \(R\) and \(\Theta\)

\[ H(\Theta,R)=\left(\frac{pc}{eR}\right)\mu(\Theta,R). \tag{5.2} \]

Differentiating equation (5.1) with respect to \(x\), where \(x\) is measured in the direction perpendicular to the orbit, we obtain

\[ H\frac{\partial \rho}{\partial x}+\rho\frac{\partial H}{\partial x} =\frac{c}{e}\frac{\partial p}{\partial x}. \tag{5.3} \]

Thus, the field fall-off index is equal to

\[ n=-\left(\frac{\rho}{H}\right)\frac{\partial H}{\partial x} =\frac{\partial \rho}{\partial x}-\rho\frac{\partial \ln p}{\partial x}. \tag{5.4} \]

Using equations (4.3), (4.6), and (4.7), we find

\[ n=-\frac{1}{\eta\mu^{2}} \left[k\mu+\varepsilon\frac{\partial \mu}{\partial \Theta} +R\frac{\partial \mu}{\partial R}\right]. \tag{5.5} \]

Here \(k\) is a parameter characterizing the change in momentum:

\[ k=R\frac{d\ln p}{dR}-1. \tag{5.6} \]

Expressing \(k\) in terms of the mean value of the magnetic field \(\bar H=pc/eR\), we shall have

\[ k=\left(\frac{R}{\bar H}\right)\frac{d\bar H}{dR}. \tag{5.7} \]

Hence it is clear that \(k\) determines the mean index of decrease of the magnetic field. The linearized equations of betatron oscillations about the equilibrium orbit have the form\({}^{7}\)

\[ \frac{d^{2}x}{ds^{2}}-\frac{1-n}{\rho^{2}}x=0, \tag{5.8} \]

\[ \frac{d^{2}z}{ds^{2}}+\frac{n}{\rho^{2}}z=0, \tag{5.9} \]

where \(x\) and \(z\) are deviations from the equilibrium orbit in the radial and axial directions. Hence, using (4.2) and (4.3), we obtain

\[ \frac{d^{2}x}{d\Theta^{2}}+\mu^{2}(1-n)x=0, \tag{5.10} \]

\[ \frac{d^{2}z}{d\Theta^{2}}+\mu^{2}nz=0. \tag{5.11} \]

The character of the betatron oscillations is therefore determined by the functions \(\mu^{2}(\Theta,R)\) and

\[ \mu^{2}n=-\frac{1}{\eta}\left(k\mu+\varepsilon\frac{\partial\mu}{\partial\Theta} +R\frac{\partial\mu}{\partial R}\right). \tag{5.12} \]

Taking into account equations (4.8) and (4.9), one may write (5.12) in the form

\[ \mu^{2}(1-n)=\frac{(k+1)}{\eta}\mu-\frac{1}{\eta}\frac{\partial^{2}\eta}{\partial\Theta^{2}}. \tag{5.13} \]

If the equilibrium orbits are similar, then \(\mu, \eta, \xi\) are functions only of \(\Theta\); according to (5.13), \(\mu^2 n\) will also be a function only of \(\Theta\). Thus, provided \(k\) is constant, the betatron oscillations will be similar. Accelerators with such properties will be called similar. For similar accelerators the following relations will hold:

\[ p=p_0\left(\frac{R}{R_0}\right)^{k+1} \tag{5.14} \]

and

\[ H=\bar H_0\left(\frac{R}{R_0}\right)^k \mu(\Theta). \tag{5.15} \]

6. Approximate solutions of the equations of betatron oscillations

In this section we shall obtain a number of approximate formulas which make it possible to establish the properties of strong-focusing accelerators with a constant field. If the wavelength of the betatron oscillations is large in comparison with the length of a period element (at least equal to the length of four elements), then a smooth approximation is applicable; a detailed analysis of it is given in Appendix A. In this case the equations of “smooth” betatron oscillations take the form

\[ \frac{d^2 X}{d\Theta^2}+\nu_x^2 X=0, \tag{6.1} \]

\[ \frac{d^2 Z}{d\Theta^2}+\nu_z^2 Z=0, \tag{6.2} \]

where, according to equations (5.10), (5.11), and (A.13),

\[ \nu_x^2=\langle \mu^2(1-n)\rangle+\{\mu^2(1-n)\}_1^2, \tag{6.3} \]

\[ \nu_z^2=\langle \mu^2 n\rangle+\{\mu^2 n\}_1^2. \tag{6.4} \]

The solution of equations (6.1) and (6.2) is

\[ X=A\cos \nu_x\Theta+B\sin \nu_x\Theta, \tag{6.5} \]

\[ Z=C\cos \nu_z\Theta+D\sin \nu_z\Theta. \tag{6.6} \]

To these “smooth” solutions there must be added pulsations, which can be determined from equation (A.7). It is clear that \(\nu_x\) and \(\nu_z\) are the numbers of radial and vertical betatron oscillations per revolution. The approximate formulas (6.3) and (6.4) give \(\nu_x\) and \(\nu_z\) with an accuracy of 10% under the condition that \(\nu_x\) and \(\nu_z\) are both less than \(N/4\).

In order that resonant buildup of the betatron oscillations be absent, it is necessary to avoid integral and half-integral values of \(\nu_x\) and \(\nu_z\), as well as integral values of \((\nu_x+\nu_z)^8\). Consequently, \(\nu_x\) and \(\nu_z\) must be the same for all orbits, or approximately so. This is a fundamental limitation for strong-focusing accelerators with a constant field. In accelerators in which the similarity condition is fulfilled, \(\nu_x\) and \(\nu_z\) are inevitably the same for all orbits, which is, of course, an advantage of this type of design.

The relation between the wavelength of the betatron oscillations and the parameters of the installation depends on which terms in equation (5.13) are determining. In a ring phasotron with radial sectors \(\xi=0\); therefore, for a large number of periodicity elements (say \(N>10\)), \(\eta\) is very close to unity, and the second term in equation (5.13) is small, except at the edges

sectors, where it leads to focusing effects. The term \(-(\xi/\eta)(\partial\mu/\partial\Theta)\) in equation (5.12) is responsible for focusing by the edges of the sectors. The mean value of this term is not equal to zero; part of it is contained in the \(\mu\)-term of equation (5.13). Thus, equations (6.7) and (6.8) describe most of the mean focusing edge effects in a machine with radial sectors.

We shall call the first term in equation (5.13) the \(\mu\)-term, and the second the \(\eta\)-term. In accelerators with spiral sectors, strong focusing is provided mainly by the \(\eta\)-term. It should be noted that the \(\eta\)-term includes the term \((R/\eta)(\partial\mu/\partial R)\), which appears when the orbits are not similar. It is not difficult to see that in an ordinary strong-focusing synchrotron1 this term is the principal one determining the focusing.

Let us first consider an accelerator with radial sectors, having a large number of periodicity elements, and neglect the \(\eta\)-term. From § 4 it follows that \(\eta=1\) if \(f/N \ll 1\).

We write \(\mu\) in the form (4.11). Then, using (5.13) and putting \(\eta=1\), from equations (6.3) and (6.4) we obtain

\[ \nu_x^2=k+1+\frac{(k+1)^2 f^2}{N^2}\langle g_1^2\rangle, \tag{6.7} \]

\[ \nu_z^2=-k+\frac{f^2}{2}+\frac{(k-1)^2 f^2}{N^2}\langle g_1^2\rangle. \tag{6.8} \]

In equation (6.8) we have neglected a small term containing \(\{g^2\}\). The change in the phase of the betatron oscillations in passing through one element of periodicity is equal to

\[ \sigma=\frac{2\pi\nu}{N}. \tag{6.9} \]

To ensure stability1 \(\sigma\) must be less than \(\pi\), and for the “smooth” approximation to be valid—less than \(\pi/2\). \(k\) and \(f\) can now be expressed in terms of \(\sigma_x\) and \(\sigma_z\), using equations (6.7) and (6.8):

\[ k+1=\frac{N^2}{8\pi^2}\left(\sigma_x^2-\sigma_z^2+b\right), \tag{6.10} \]

\[ f=\frac{4\pi}{\left[2\langle g_1^2\rangle\right]^{1/2}}\, \frac{\left[\sigma_x^2+\sigma_z^2-b\right]^{1/2}}{\left[\sigma_x^2-\sigma_z^2+b\right]}, \tag{6.11} \]

where

\[ b=\frac{4\pi^2}{N^2}\left[1+\frac{f^2}{2}-\frac{4kf^2}{N^2}\langle g_1^2\rangle\right]. \tag{6.12} \]

For sufficiently large \(N\), the quantity \(b\) may be neglected.

With an appropriate choice of \(\sigma_x\) and \(\sigma_z\), the value of \(k\) may be either positive or negative; that is, in a ring phasotron with radial sectors and large \(N\), the orbit corresponding to the maximum energy may lie either at the outer or at the inner edge of the chamber. The term \(b\), which is significant for small \(N\), is positive. Therefore it is more expedient to have an installation with positive \(k\), since in this case, for a given \(N\), larger values of \(k\) and smaller \(f\) can be achieved. The smaller the radial aperture, the larger \(k\) must be, and consequently also \(N\). If we define the increase coefficient of the installation \(C\) as the ratio of the mean radius of curvature of the equilibrium orbit to the minimum radius, then

\[ C=|\mu|_{\max}=\{1+fg(N\Theta)\}_{\max}. \tag{6.13} \]

It is desirable to have \(C\) as small as possible, since for a given maximum magnetic field this leads to the minimum dimensions of the accelerator. From equation (6.11) it follows that, for a given form of the function \(g\), the amplification factor of the installation is the smaller, the larger \(\sigma_x\) is and the smaller \(\sigma_z\) is (and conversely, if \(k\) is negative).

Let us consider “rectangular” oscillations of the field in azimuth, whose root-mean-square value is equal to \(1/2\):

\[ g(\xi)= \begin{cases} \left[\dfrac{1-q}{2q}\right]^{1/2}, & -q\pi<\xi<q\pi, & \text{(I)}\\[6pt] -\left[\dfrac{q}{2(1-q)}\right]^{1/2}, & q\pi<\xi<2\pi-q\pi, & \text{(II)} \end{cases} \tag{6.14} \]

\[ g(\xi+2\pi)=g(\xi). \tag{6.15} \]

This function is shown in Fig. 6. We shall call region 1 the sector with positive field direction, and region 2 the sector with negative field direction. We need the quantity \(\langle g_1^2\rangle\), which is not difficult to calculate:

\[ \langle g_1^2\rangle=\frac{1}{6}\pi^2 q(1-q). \tag{6.16} \]

Fig. 6. Form of the function \(g(\xi)\).

Denote

\[ K=f\left[\langle g_1^2\rangle\right]^{1/2}. \tag{6.17} \]

Then the amplification factor of the installation is equal to the larger of the two quantities

\[ C=1+\frac{\sqrt{3}\,K}{\pi q} \quad \text{or} \quad \frac{\sqrt{3}\,K}{\pi(1-q)}-1. \tag{6.18} \]

The amplification factor of the installation will be minimal if \(q\) is chosen so that the two quantities (6.18) coincide. Then we have

\[ \mu=1+fg(N\Theta)= \begin{cases} C, & -q\pi<N\Theta<q\pi, & \text{(I)}\\ -C, & q\pi<N\Theta<2\pi-q\pi. & \text{(II)} \end{cases} \tag{6.19} \]

The radius of curvature, and consequently also the magnetic field, are constant in magnitude along the equilibrium orbit and are opposite in sign in the two sectors. The ratio of the azimuthal sizes of the sectors is equal to

\[ \Gamma=\frac{q}{(1-q)}=\frac{C+1}{C-1}. \tag{6.20} \]

Using (6.18) and (6.20), we obtain the expression for the minimum amplification factor of the installation:

\[ C=\frac{\Gamma+1}{\Gamma-1} = \left[1+\frac{1}{2}f^2\right]^{1/2}. \tag{6.21} \]

If we take \(\sigma_z=\dfrac{\pi}{6}\), \(\sigma_x=\dfrac{\pi}{2}\), \(b=0\), and use the approximate formulas (6.10) and (6.11), then we shall have

\[ K=\sqrt{45}, \qquad \Gamma=1.31, \qquad C=7.5, \qquad f=10.5, \qquad k=\frac{N^2}{36}. \]

In the next paragraph it will be shown more precisely that the minimum value of \(C\) for large \(N\) is approximately equal to 5.

In an annular phasotron with spiral sectors \(\zeta\) is approximately equal to \(90^\circ\), and the \(\eta\)-term in equation (5.13) is large. Since the inhomogeneity coefficient \(f\) may be taken small, the variable part of the \(\mu\)-term will be small. We shall again assume that \(\mu\) is given by equation (4.11), and \(\eta\) by the approximate expression (4.18). Using the expansion of \(1/\eta\) in a power series, one can easily obtain the relation

\[ \left\langle \left(\frac{\mu}{\eta}\right)\right\rangle = 1+\frac{f^3 \operatorname{tg}^2 \zeta}{N^2}\left\langle g_1^2\right\rangle+\cdots . \tag{6.22} \]

We shall neglect terms of second order of smallness and higher, and also the oscillating part of \(\mu/\eta\). The \(\eta\)-term can be written in the following form:

\[ -\frac{1}{\eta}\frac{\partial^2\eta}{\partial \Theta^2} = \frac{\partial}{\partial\Theta} \left(\frac{1}{\eta}\frac{\partial\eta}{\partial\Theta}\right) + \left(\frac{1}{\eta}\frac{\partial\eta}{\partial\Theta}\right)^2 . \tag{6.23} \]

The first term on the right-hand side of the equation is large and oscillates about a zero mean value; the second term is small, but its mean value is not zero and is positive. Neglecting the oscillating part of the second term, we substitute (6.23) into (5.13) and, using the expression obtained, from equations (6.3), (6.4) find

\[ \nu_x^2=k+1, \tag{6.24} \]

\[ \nu_z^2=-k+\frac{1}{2}f^2+2\left\langle \left(\frac{1}{\eta}\frac{\partial\eta}{\partial\Theta}\right)^2 \right\rangle . \tag{6.25} \]

Let us note that in this approximation the \(\eta\)-term makes no contribution to the radial focusing. Taking \(\eta\) in the form specified by equation (4.18), we obtain

\[ \left\langle \left(\frac{1}{\eta}\frac{\partial\eta}{\partial\Theta}\right)^2 \right\rangle = f^2\operatorname{tg}^2\zeta \left\langle \frac{g^2}{\left(1-fN^{-1}\operatorname{tg}\zeta\,g_1\right)^2} \right\rangle = \]

\[ = f^2\operatorname{tg}^2\zeta \left[ \frac{1}{2} + \frac{2f^2\operatorname{tg}^2\zeta}{N^2} \left\langle \dot g^{\,2} g_1^2\right\rangle +\cdots \right]. \tag{6.26} \]

We shall neglect terms of second and higher order of smallness in the square brackets. Substituting into equations (6.24) and (6.25), we shall have

\[ f^2\operatorname{tg}^2\zeta=\left(\nu_x^2+\nu_z^2-1\right), \tag{6.27} \]

where we have also neglected the term \(f^2/2\). Note that in this approximation the relations (6.24) and (6.27) do not depend on the form of the function \(g(N\Theta)\); only the gain factor of the machine, determined by equation (6.13), depends on the form of \(g(N\Theta)\). Expressing \(\nu_x, \nu_z\) in terms of \(\sigma_x, \sigma_z\), one may write formulas (6.24) and (6.27) in the form

\[ k+1=\frac{N^2\sigma_x^2}{4\pi^2}, \tag{6.28} \]

\[ f^2\operatorname{tg}^2\zeta = \frac{N^2}{4\pi^2}\left(\sigma_x^2+\sigma_z^2\right)-1 . \tag{6.29} \]

The coordinate curve \(\Theta=0\) satisfies, in polar coordinates \(r\) and \(\theta\), the equation

\[ \frac{1}{r}\frac{dr}{d\theta}=\operatorname{ctg}\zeta . \tag{6.30} \]

The distance between ridges (points of maximum magnetic field) in radius, in units of \(r\), is therefore

\[ \lambda=\frac{\Delta r}{r}=\frac{2\pi}{N\operatorname{tg}\zeta}. \tag{6.31} \]

Thus, for fixed \(\sigma_x\), \(\sigma_z\), and \(N\), the ratio \(f/\lambda\) is specified. The greatest possible gap between the poles of the magnet is proportional to \(\lambda\). If the azimuthal field inhomogeneity is produced by shimming the pole tips without using windings, then it can be shown that, for a given \(f/\lambda\), the maximum gap is obtained at \(f \simeq 1/4\) and is approximately equal to \(1/4\lambda r\). Under these conditions the azimuthal field inhomogeneity is very close to sinusoidal,

\[ g(\xi)=\cos \xi, \tag{6.32} \]

and the magnification factor of the installation will be

\[ C=1+f=1.25. \]

As above, let us take \(\sigma_z=\pi/6\), \(\sigma_x=\pi/2\), and set \(f\) equal to \(1/4\); then we obtain \(k+1=N^2/16\), \(\lambda=5.95N^{-2}[1-14.4N^{-2}]^{-1/2}\), \(\operatorname{tg}\zeta=1.05N[1-14.4N^{-2}]^{-1/2}\).

7. Stability of motion in the linear approximation in accelerators with radial sectors

In order to obtain a more accurate relation between the parameters, let us return to the betatron-oscillation equations (5.10) and (5.11). Using formulas (5.12), (4.18), and (4.19) for \(\zeta=0\), we rewrite equations (5.10) and (5.11) for the case of a “rectangular” variation of the field with azimuth (6.19) in the form

\[ \frac{d^2x}{d\Theta^2}\pm kCx=0, \tag{7.1} \]

\[ \frac{d^2z}{d\Theta^2}\mp kCz=0, \tag{7.2} \]

where the upper sign refers to sectors with the positive direction of the field, and the lower sign to those with the negative direction. The term \(\varepsilon\, d\psi/\partial\Theta\) in (5.12) leads, in equations (5.10) and (5.11), to terms determining the focusing at the edges of the sectors. We shall neglect them for the time being. This approximation is valid only when \(N \gg f\), and correspondingly we also neglect unity in comparison with \(n\). When \(N\) is small, edge effects and the higher-order terms in \(\eta\) must be taken into account. The oscillating terms in \(\eta\) reflect the nonequidistance of neighboring equilibrium orbits. For small \(N\), edge effects lead to enhanced vertical focusing and weakened radial focusing, so that it becomes possible to use considerably smaller values of the field inhomogeneity coefficient \(f\) without impairing the vertical stability, if \(k>0\).

Let \(N\Theta_0=-q\pi\), \(N\Theta_1=q\pi\), \(N\Theta_2=(2-q)\pi\). Then the solutions of equation (7.1) inside the sectors with positive and negative field direction lead to the following matrix relations between \(x'=dx/d\Theta\) and \(x\) at the points \(\Theta_0\), \(\Theta_1\), \(\Theta_2\):

\[ \begin{pmatrix} x_1\\ x'_1 \end{pmatrix} =M_+ \begin{pmatrix} x_0\\ x'_0 \end{pmatrix}, \qquad \begin{pmatrix} x_2\\ x'_2 \end{pmatrix} =M_- \begin{pmatrix} x_1\\ x'_1 \end{pmatrix}, \tag{7.3} \]

where

\[ \begin{aligned} M_+&= \begin{pmatrix} \cos\psi_+ & (kC)^{-1/2}\sin\psi_+\\ -(kC)^{1/2}\sin\psi_+ & \cos\psi_+ \end{pmatrix},\\ M_-&= \begin{pmatrix} \operatorname{ch}\psi_- & (kC)^{-1/2}\operatorname{sh}\psi_-\\ (kC)^{1/2}\operatorname{sh}\psi_- & \operatorname{ch}\psi_- \end{pmatrix}, \end{aligned} \tag{7.4} \]

\[ \begin{aligned} \psi_+&=\frac{2\pi q}{N}(kC)^{1/2},\\ \psi_-&=\frac{2\pi(1-q)}{N}(kC)^{1/2}. \end{aligned} \tag{7.5} \]

Consequently,

\[ \begin{pmatrix} x_2\\ x'_2 \end{pmatrix} = M \begin{pmatrix} x_0\\ x'_0 \end{pmatrix}, \tag{7.6} \]

where

\[ M=M_-M_+= \begin{pmatrix} \cos\psi_+\,\operatorname{ch}\psi_- - \sin\psi_+\,\operatorname{sh}\psi_-, & (kC)^{-1/2}(\cos\psi_+\,\operatorname{sh}\psi_-+\sin\psi_+\,\operatorname{ch}\psi_-) \\ (kC)^{1/2}(\cos\psi_+\,\operatorname{sh}\psi_- - \sin\psi_+\,\operatorname{ch}\psi_-), & \cos\psi_+\,\operatorname{ch}\psi_-+\sin\psi_+\,\operatorname{sh}\psi_- \end{pmatrix}. \tag{7.7} \]

From this we obtain\(^8\)

\[ \cos\sigma_x=\frac{1}{2}\,Sp(M)=\cos\psi_+\,\operatorname{ch}\psi_-, \tag{7.8} \]

and similarly

\[ \cos\sigma_z=\cos\psi_-\,\operatorname{ch}\psi_+. \tag{7.9} \]

\(\psi_+\) and \(\psi_-\) can be expressed through the ratio \(\Gamma\) of the sector lengths [see (6.20)] and the local field index

\[ n=\frac{k}{C}; \tag{7.10} \]

\[ \begin{aligned} \psi_+&=\left(\frac{2\pi}{N}\right)\left(\frac{\Gamma}{\Gamma-1}\right)n^{1/2},\\ \psi_-&=\left(\frac{2\pi}{N}\right)\left(\frac{1}{\Gamma-1}\right)n^{1/2}. \end{aligned} \tag{7.11} \]

(\(n\) is here regarded as positive.) Formulas (7.5), (7.8), (7.9), and (7.11) were written for \(k>0\). However, they may also be used for \(k<0\); in that case it is convenient to regard \(C\) as negative.

The minimum magnification coefficient of the system is obtained at the maximum value of \(\sigma_x\) and the minimum \(\sigma_z\), or conversely. If we take \(\sigma_x=3/4\pi\), \(\sigma_z=\pi/6\), then from equations (7.8) and (7.9) we obtain

\[ \psi_-=1.32, \]

\[ \psi_+=1.93. \]

From equations (7.11) and (6.21) we shall have

\[ \begin{aligned} \Gamma=\frac{\psi_+}{\psi_-}&=1.46,\\ C&=5.35. \end{aligned} \tag{7.12} \]

The minimum theoretical value of \(C\) is 4.45 for \(\sigma_x=\pi,\ \sigma_z=0\). Therefore \(\sigma_x\) and \(\sigma_z\) must be chosen as close as possible to the boundary of the stability region, but so that the amplitude of the betatron oscillations remains within reasonable limits. For \(\sigma_x=\pi/2,\ \sigma_z=\pi/6\), these more exact formulas give \(\Gamma=1.29,\ C=7.9\). (Compare with the approximate values 1.31 and 7.5 obtained in the preceding paragraph.)

A more general calculation, taking into account straight sections and edge effects, can be carried out in an analogous manner. Suppose that the magnetic field along the equilibrium orbit is constant and of opposite sign in adjacent sectors, and that the field in the straight sections is zero (Fig. 7). Let us denote the relative length of the orbit inside the sectors with positive and negative field direction and in the straight section by \(q_1,\ q_2,\ q_0\), respectively. Thus,

\[ 2q_0+q_1+q_2=1. \tag{7.13} \]

The angles \(\beta_1\) and \(\beta_2\), shown in Fig. 7, are equal to

\[ \beta_1=\frac{2\pi Cq_1}{N}, \qquad \beta_2=\frac{2\pi Cq_2}{N}. \tag{7.14} \]

The number of periodicity elements is

\[ N=\frac{2\pi}{\beta_1-\beta_2}, \tag{7.15} \]

so that the magnification factor of the apparatus is

\[ C=\frac{1}{q_1-q_2}. \tag{7.16} \]

Fig. 7. Equilibrium orbit and notation for a machine with radial sectors and straight sections.

By \(\Phi_1\) and \(\Phi_2\) in Fig. 7 are denoted the angles between the orbit and the normal to the sector edge. It is also convenient to introduce the notation:

\[ \lambda=\frac{2\pi Cq_0}{N}, \tag{7.17} \]

\[ \psi_1=\beta_1(n_1+1)^{1/2}, \qquad \psi_2=\beta_2(n_2-1)^{1/2}, \tag{7.18} \]

\[ \psi_3=\beta_1 n_1^{1/2}, \qquad \psi_4=\beta_2 n_2^{1/2}, \tag{7.19} \]

\(n_1\) and \(n_2\) are the local indices of field falloff at the center of the sectors with positive and negative field direction,

\[ n=\frac{k}{\eta C}, \tag{7.20} \]

where

\[ \eta_1=1-2q_2\left(1-\frac{\sin(\pi Cq_2/N)}{Cq_2\sin(\pi/N)}\right) -2q_0\left(1-\frac{\cos(\pi Cq_2/N)}{(N/\pi)\sin(\pi/N)}\right) \tag{7.21} \]

and

\[ \eta_2=1-2q_1\left(1-\frac{\sin(\pi Cq_1/N)}{Cq_1\sin(\pi/N)}\right) -2q_0\left(1-\frac{\cos(\pi Cq_1/N)}{(N/\pi)\sin(\pi/N)}\right). \tag{7.22} \]

Here unity is not neglected in comparison with \(n\). However, there is neglected

by changing \(\eta\) within the sectors. As a result one obtains:

\[ \begin{aligned} \cos \sigma_x={}&[1+2\delta(\tg\Phi_1+\tg\Phi_2)+2\delta^2\tg\Phi_1\tg\Phi_2]\cos\psi_1\,\ch\psi_2+\\ &+[(n_1+1)^{-1/2}(\tg\Phi_1+\tg\Phi_2+\delta\tg^2\Phi_1+2\delta\tg\Phi_1\tg\Phi_2+\\ &\qquad+\delta^2\tg^2\Phi_1\tg\Phi_2)-(n_1+1)^{1/2}(\delta+\delta^2\tg\Phi_2)]\sin\psi_1\,\ch\psi_2+\\ &+[(n_2-1)^{-1/2}(\tg\Phi_1+\tg\Phi_2+\delta\tg^2\Phi_2+2\delta\tg\Phi_1\tg\Phi_2+\\ &\qquad+\delta^2\tg^2\Phi_2\tg\Phi_1)+(n_2-1)^{1/2}(\delta+\delta^2\tg\Phi_1)]\cos\psi_1\,\sh\psi_2+\\ &+\frac{1}{2}\big[-(n_1+1)^{1/2}(n_2-1)^{1/2}\delta^2-(n_1+1)^{1/2}(n_2-1)^{-1/2}\times\\ &\qquad\times(1+\delta\tg\Phi_2)^2+(n_1+1)^{-1/2}(n_2-1)^{1/2}(1+\delta\tg\Phi_1)^2+\\ &\qquad+(n_1+1)^{-1/2}(n_2-1)^{-1/2}(\tg\Phi_1+\tg\Phi_2+\\ &\qquad+\delta\tg\Phi_1\tg\Phi_2)^2\big]\sin\psi_1\,\sh\psi_2, \end{aligned} \tag{7.23} \]

\[ \begin{aligned} \cos \sigma_z={}&[1-2\delta(\tg\Phi_1+\tg\Phi_2)+2\delta^2\tg\Phi_1\tg\Phi_2]\cos\psi_4\,\ch\psi_3+\\ &+[n_2^{1/2}(-\tg\Phi_1-\tg\Phi_2+\delta\tg^2\Phi_2+2\delta\tg\Phi_1\tg\Phi_2-\\ &\qquad-\delta^2\tg^2\Phi_2\tg\Phi_1)-n_2^{-1/2}(\delta-\delta^2\tg\Phi_1)]\sin\psi_4\,\ch\psi_3+\\ &+[n_1^{-1/2}(-\tg\Phi_1-\tg\Phi_2+\delta\tg^2\Phi_1+2\delta\tg\Phi_1\tg\Phi_2-\\ &\qquad-\delta^2\tg^2\Phi_1\tg\Phi_2)+n_1^{1/2}(\delta-\delta^2\tg\Phi_2)]\cos\psi_4\,\sh\psi_3+\\ &+\frac{1}{2}\big[-n_2^{1/2}n_1^{1/2}\delta^2-n_2^{1/2}n_1^{-1/2}(1-\delta\tg\Phi_1)^2+n_2^{-1/2}n_1^{1/2}\times\\ &\qquad\times(1-\delta\tg\Phi_2)^2+n_2^{-1/2}n_1^{-1/2}(-\tg\Phi_1-\tg\Phi_2+\\ &\qquad+\delta\tg\Phi_1\tg\Phi_2)^2\big]\sin\psi_4\,\sh\psi_3. \end{aligned} \tag{7.24} \]

8. Stability of motion in the linear approximation for an accelerator with spiral sectors

For an accelerator with spiral sectors the increase coefficient of the setting is close to unity, and the attainment of the minimum value \(C\) is no longer the determining condition. The distance between the ridges \(\lambda\), however, is rather small, and, if the gap between the poles of the magnet is maximal, then the change of the field with azimuth in the median plane must be approximately sinusoidal. Therefore we shall assume that the field in the median plane has the form

\[ H=\bar H_0\left(\frac{r}{r_0}\right)^k \left\{1+f\sin\left[N\theta-\left(\frac{1}{w}\right)\ln\left(\frac{r}{r_0}\right)\right]\right\}, \tag{8.1} \]

where

\[ \frac{1}{w}=N\tg\xi=\frac{2\pi}{\lambda}. \tag{8.2} \]

This form of the field has been chosen in order to ensure the condition of similarity.

The linearized equations of betatron oscillations can be obtained by proceeding from the general analysis carried out in the first two sections, although it is more desirable to derive them directly. If one restricts oneself to the linear terms in the differential equations characterizing the deviation of a particle from the coordinate circle of radius

\[ r_1=\frac{cp}{\left[e\bar H_0\left(\dfrac{r_1}{r_0}\right)^k\right]}, \tag{8.3} \]

then we obtain

\[ r''+\left[1+k+\left({f\over w}\right)\cos N\theta\right](r-r_1)\simeq f\cdot r_1\sin N\theta, \tag{8.4} \]

\[ z''-\left[k+\left({f\over w}\right)\cos N\theta\right]z\simeq 0. \tag{8.5} \]

These equations of motion indicate the presence of strong focusing analogous to the usual one, characterized by the Mathieu differential equation; however, in the right-hand side of the equation for the radial motion there is a forcing term, owing to which forced oscillations of the form

\[ r-r_1=-{f\over N^2-(k+1)}\,r_1\sin N\theta \tag{8.6} \]

will be excited.

Because of the forced motion, the frequency of the betatron oscillations may change appreciably, and the nonlinear terms in the equations of motion will be large. It is therefore preferable to write the expansion with respect to the more convenient coordinate curve

\[ x=r-r_1+{f\over N^2-(k+1)}\,r_1\sin N\theta. \tag{8.7} \]

In this way we obtain the following linearized equations:

\[ x''+\left[ k+1-\frac{1}{2}{\dfrac{f^2}{w^2}\over N^2-(k+1)} +{f\over w}\cos N\theta +\frac{1}{2}{\dfrac{f^2}{w^2}\over N^2-(k+1)}\cos 2N\theta \right]x=0, \tag{8.8} \]

\[ z''-\left[ k-\frac{1}{2}{\dfrac{f^2}{w^2}\over N^2-(k+1)} +{f\over w}\cos N\theta +\frac{1}{2}{\dfrac{f^2}{w^2}\over N^2-(k+1)}\cos 2N\theta \right]z=0. \tag{8.9} \]

These equations have the form of the generalized Mathieu equation

\[ {d^2u\over d\tau^2}+(A+B\cdot\cos 2\tau+C\cdot\cos 4\tau)u=0. \tag{8.10} \]

The terms discarded in the coefficients \(A\) and \(C\) of equation (8.10), as is seen from (8.8) and (8.9), are of order \(k^2 w^2\) relative to the principal terms. Therefore, for \(f=1/4\) the permissible error in these coefficients is less than \(2\%\) throughout the entire stability region (Fig. 8). The terms discarded in the coefficients \(B\) are of order \(1/8\,(f/N^2w)^2\) in equation (8.8) and \(1/2\,(f/N^2w)^2\) in equation (8.9), so that the permissible errors throughout the entire stability region will be less than \(2\%\) and \(8\%\), respectively. The coefficient of the third harmonic, which we omitted, is of order \(1/8\,(f/N^2w)^2\) and \(1/2(f/N^2w)^2\) of the coefficient \(B\), respectively. Since the third harmonic makes a contribution to the value of \(\sigma\) proportional to \(1/9\) of the square of the coefficient, its effect may be completely neglected.

Tables of characteristic exponents \((\sigma/\pi)\) of the generalized Mathieu equation (8.10) were computed on an electronic machine by the variational method\(^{9}\). The values of \(A\) are tabulated for \(\sigma\), \(B\), \(C\), covering the main part of the first stability region. These tables also include the case of the ordinary Mathieu equation \((C=0)\). As far as we know, at present there exist unpublished tables of characteristic exponents of the Mathieu equation for the stability region.

In Fig. 8 the stability region of betatron oscillations in a ring phasotron with spiral sectors for \(k \gg 1\) is shown, calculated on the basis of the formulas written above and the tabulated solutions of equation (8.10). If \(k \gg 1\), the coefficients \(A\), \(B\), \(C\) depend only on \(k/N^2\) and \(f/N^2 w\). Therefore the lines of constant \(\sigma_x\) and \(\sigma_z\) are represented by us in the coordinates \(k/N^2\) and \(f/N^2 w\). If we put \(\sigma_z=\pi/6\), \(\sigma_x=\pi/2\) and \(f=1/4\), then we obtain \(k=0.057\,N^2\), \(f/N^2 w=0.25\) and \(\lambda=6.3\,N^{-2}\). (Compare with the approximate values \(k=0.062\,N^2\), \(f/N^2 w=0.265\) and \(\lambda=5.95\,N^{-2}\), obtained at the end of § 6.)

9. Nonlinear effects

The preceding analysis of betatron oscillations was based on expanding the equations of motion in powers of the displacement from the equilibrium orbit, retaining only the linear terms. Small amplitudes of betatron oscillations \(x\) and \(z\) then satisfied linear differential equations with coefficients periodic in \(\Theta\).

In an ideal accelerator the only periodicity is associated with \(N\) identical elements, and the period of the coefficients is \(2\pi/N\). In a real machine various perturbations are unavoidable, so that the coefficients will be strictly periodic in \(\Theta\) with period \(2\pi\) and approximately periodic with period \(2\pi/N\). With the periodicity \(2\pi/N\) there is connected the requirement that \(\sigma_x\) and \(\sigma_z\) should not be integral or half-integral multiples of \(2\pi\). In reality \(\sigma\) must be less than \(\pi\), since otherwise the tolerances for the construction and assembly of the magnet become very severe. To avoid resonances with perturbations, \(\nu_x\) and \(\nu_z\) must not be integers or half-integers. Moreover, if perturbations can lead to coupling of the \(x\) and \(z\) motions, then \(\nu_x+\nu_z\) must not be an integer. These requirements are associated with the periodicity \(2\pi\).

Fig. 8. Dependence of \(\sigma_x\) and \(\sigma_z\) inside the stability region on the parameters of spiral sectors for \(k \gg 1\).

The study of the influence of nonlinear terms in the equations of motion has not yet advanced as far as the study of the linearized equations. Approximate analytical methods for solving nonlinear equations were developed by Moser\(^{10}\), Sturrock\(^{8}\), and Hagedorn\(^{11}\). Their results reduce to the following: if the coefficients have period \(2\pi\) in \(\Theta\) and \(\nu_x\), \(\nu_z\) are the numbers of betatron oscillations per period \(2\pi\), then resonances occur when

\[ n_x\nu_x+n_z\nu_z=\text{an integer}, \tag{9.1} \]

\[ n_x,n_z=0,1,2,\ldots \]

Let

\[ n_x+n_z=q. \tag{9.2} \]

Then, if \(q=1\) or \(q=2\), the motion is unstable even in the linear approximation.

approximation (this rule was established in the preceding section). If \(q=3\), then, generally speaking, the influence of the quadratic terms in the differential equations is such that the motion becomes unstable even at very small amplitudes. If \(q=4\), then the motion may be stable or unstable depending on the form of the cubic (and linear) terms. If \(q>4\), then, generally speaking, the motion is stable for sufficiently small amplitudes of the betatron oscillations. In any case, if \(q>4\) and the equations of motion are nonlinear, there exists a limiting amplitude of the betatron oscillations, beginning with which the oscillations become unstable, at least in the sense that the particles leave the chamber.

Numerical calculations carried out on an electronic computer, and experiments carried out by the Brookhaven group\(^{12}\) on an electronic analogue of a strong-focusing accelerator, apparently confirm these conclusions.

If this criterion is applied to periodicities by elements with period \(2\pi/N\), then \(\nu_x\) and \(\nu_z\) in (9.1) must be replaced by \(\sigma_x/2\pi\), \(\sigma_z/2\pi\) (the numbers of betatron oscillations per periodicity element). In this case we arrive, for example, at the conclusion that values of \(\sigma_x\) or \(\sigma_z\) close to \(2\pi/3\) are dangerous, as are values for which \(\sigma_x+2\sigma_z\) or \(\sigma_z+2\sigma_x\) are close to \(2\pi\). We shall call these resonances, associated with the periodicity of the magnet construction, “sector” resonances.

Numerical calculations have shown that the limiting amplitude of betatron oscillations in accelerators with spiral sectors becomes very small when \(\sigma\) approaches \(2\pi/3\). Applying this criterion to periodicity \(2\pi\), we find the values \(\nu_x\) and \(\nu_z\) (Fig. 9) excluded by this rule. In Fig. 9, \(\nu_x\) is represented by horizontal lines and \(\nu_z\) by vertical lines. The lines denoted \(q=1, 2, 3, 4\) represent the values of \(\nu\) excluded by this rule. The lines \(q=1\) depict integral resonances, the lines \(q=2\)—half-integral resonances (verticals and horizontals) and coupling resonances (diagonals), and the lines \(q=3,4\)—third- and fourth-order fractional resonances. It is still unclear how serious the third- and fourth-order fractional resonances are, since they arise only because of nonlinearities. Experiments on the electronic analogue at Brookhaven\(^{12}\) show that, in order to detect these resonances, it is necessary specially to introduce nonlinear perturbations. This, of course, is not true for the resonances \(\sigma=2\pi/3\) considered in the preceding section, which are resonances associated with the periodicity of the magnet construction. At present, it is apparently more prudent to avoid, as far as possible, all the forbidden lines shown in Fig. 9.

Fig. 9. Linear and nonlinear resonances in a strong-focusing accelerator. \(M_x\) and \(M_z\) are integers.

It has been shown that the nonlinear terms in accelerators with radial sectors are small; in order of magnitude they are no larger than in ordinary strong-focusing accelerators. However, the nonlinear terms in accelerators with spiral sectors are much larger, and they play a more important role in determining the character of the betatron oscillations. Numerical calculations show—

show that, although the influence of nonlinear effects in a ring phasotron with spiral sectors is considerable, the limiting amplitudes are nevertheless sufficiently large for betatron oscillations to be stable, provided that \(\sigma\) is not too close to \(2\pi/3\) (say, \(\sigma_x < 0.6\pi\)).

10. Phase Stability

The momentum \(p(R)\) is determined by integrating equation (5.6)

\[ p = p_0 \exp \left[ \int_{R_0}^{R} \frac{k+1}{R}\, dR \right]. \tag{10.1} \]

If \(k\) does not depend on \(R\), this equality goes over into (5.14). Thus, the momentum and energy are determined as functions of the parameter \(R\). Since \(R\), generally speaking, is close to the mean radius of the orbit, the radial aperture required for given initial and final momenta can be determined from equation (10.1). It is clear that, for a given momentum interval, the radial aperture decreases as \(k\) increases. If \(k \gg 1\), then the radial aperture is much smaller than \(R\), and for constant \(k\) we approximately obtain

\[ \frac{R_1 - R_0}{R_0} \simeq \left( \frac{1}{k+1} \right) \ln \left( \frac{p_1 - p_0}{p_0} \right). \tag{10.2} \]

The angular velocity of a particle in the orbit \(R\) is equal to

\[ \omega = \frac{d\Theta}{dt} = \frac{\beta c}{R} = \frac{pc^2}{ER}, \tag{10.3} \]

where \(E\) is the total energy, including the rest energy. Squaring (10.3) and differentiating, we shall have

\[ \frac{E}{\omega}\frac{d\omega}{dE} = \frac{1}{\left( \dfrac{E^2}{E_0^2} \right) - 1} - \frac{1}{\left( \dfrac{R}{E} \right)\left( \dfrac{dE}{dR} \right)} . \tag{10.4} \]

Let us now differentiate the equation

\[ E^2 = p^2 c^2 + E_0^2 . \tag{10.5} \]

Then, using equation (5.6), we obtain

\[ \frac{E}{\omega}\frac{d\omega}{dE} = \frac{(k+1)E_0^2 - E^2}{(E^2 - E_0^2)(k+1)} . \tag{10.6} \]

If \(k=\mathrm{const}\), this equation can be integrated. Then

\[ \frac{\omega}{\omega_1} = \frac{E_1}{E} \left( \frac{E^2 - E_0^2}{E_1^2 - E_0^2} \right)^{\frac{k}{2(k+1)}} , \tag{10.7} \]

where \(\omega_1\) is the angular frequency of revolution of a particle with energy \(E_1\). The graph of \(\omega/\omega_1\) is shown in Fig. 10, where \(\omega_t\) is the angular frequency corresponding to the critical energy; \(k\) has been taken equal to 99. If the critical energy is defined as

\[ E_t = (k+1)^{1/2} E_0, \tag{10.8} \]

then for \(E<E_t\) the value \(d\omega/dE\) is positive, whereas for \(E>E_t\) the value \(d\omega/dE\) is negative. If the particles are accelerated by a high-frequency electric field applied to one or more gaps, then the theories of phase stability in strong-focusing accelerators with a constant field are analogous to the theory for an ordinary cyclotron or synchrotron[^13]. When \(d\omega/dE\) is positive, the particle can execute stable oscillations about an equilibrium phase located on the rising part of the h.f. wave. When \(d\omega/dE\) is negative, the equilibrium phase lies on the falling part of the h.f. wave. At \(E=E_t\) there is no phase stability. In order to accelerate particles to energies greater than the critical energy, it is necessary at the appropriate moment to transfer the equilibrium phase from the rising to the falling part of the wave.

Fig. 10. Dependence of the revolution frequency on energy.

In a cyclotron the revolution frequency \(\varphi/2\) must be the same for all energies, and equation (10.6) then gives the relation between \(k\) and \(E\) in the form

\[ k+1=\frac{E^2}{E_0^2}. \tag{10.9} \]

In a cyclotron \(k\) must increase with energy, and therefore the betatron oscillations are dissimilar, even if the equilibrium orbits are similar.

III. APPLICATION OF THE THEORY TO ACCELERATOR DESIGN

11. Ring phasotron

As an example of the application of the developed theory to the design of high-energy accelerators, possible parameters of a ring phasotron with radial and spiral sectors are given below. In choosing the parameters of a ring phasotron, one is guided in many respects by the same considerations as for a strong-focusing accelerator with a pulsed field1: resonances, assembly tolerances, scattering on gas. Injection and acceleration will apparently differ rather substantially from injection and acceleration in an ordinary synchrotron of comparable energy.

Whereas for a proton strong-focusing synchrotron of energy 25 Bev it is proposed to use, for injection, a linear accelerator of energy 50 Mev, for a ring phasotron one may use a Van de Graaff electrostatic generator of energy 5 Mev. As an injector, an electrostatic generator has a number of advantages over a linear accelerator: a large pulsed current, simplicity, lower cost, energy stability, and smaller beam dimensions. Although the most probable injection is during a single revolution, carried out by means of a pulsed inflector which passes a current of several milliamperes in a pulse, it is also possible to obtain large currents in the injection beam over many revolutions.

The possibility of injection at low energies was evident already at the very inception of the principle of strong focusing in a constant field. A large interval of pulses in an accelerator with a constant field requires a large area of the pole tips, operating at a very low flux density.

However, the unprofitability of using iron at low beam-current densities was demonstrated. It was therefore proposed to use a sequence of accelerators with a high current density in the iron and with regenerative extraction of the beam, employed for injecting particles from one accelerator into the next one at higher energy. Such an extraction system was used on betatrons and has recently been implemented on a cyclotron; the time of orbit rearrangement makes it possible to use this system for injection, at the same time ensuring both the motion of the injected beam away from the magnetic perturbation and the damping of the oscillations excited in it. However, this scheme will require very careful adjustment. The feasibility of such a system was proved by extensive theoretical work by Teng2 and others at Argonne National Laboratory. Teng emphasizes that the use of high-energy injection makes it possible to avoid to a considerable extent the problem of frequency modulation and of controlling the shape of the small magnetic fields that arises at low injection energy. The problem of frequency modulation, in particular in constant-field accelerators, has many interesting possible solutions that are absent in pulsed-field accelerators.

An arbitrary dependence of frequency on time makes it possible to use, in a ring phasotron, a mechanical system for frequency modulation in a resonator with a high quality factor \(Q\). At large \(Q\), possible in an unloaded cavity, the required energy gain per turn can be imparted to the particles by a single resonator consuming a reasonable power. The modulation may be carried out by means of a moving diaphragm that tunes the volume of the resonator. Model experiments show that such a system makes it possible to change the frequency by a factor of 3. If the injection energy is \(5\ \mathrm{MeV}\), then, in order to reach relativistic velocities, a frequency change by a factor of 10 is required. Therefore one can use one resonator, operating as a self-excited generator, to accelerate particles to an energy of about \(50\ \mathrm{MeV}\). Then the voltage on this resonator is turned off and a second resonator is turned on, continuing to accelerate the particles further. The switching can be controlled by comparing the frequencies between the resonators. The relative phases of the resonators can be controlled by a weak coupling between them. (On the electron synchrotron of the University of Michigan, whose high-frequency system consists of two resonators, it was shown that a transition can be made from one resonator to the other without noticeable particle losses.) If desired, a third resonator can be added and the second transition made in the region of relativistic energies, where the particle revolution frequency changes little (see Fig. 10). The third resonator must be designed so as to provide a very large voltage over a small frequency range. Fine frequency tuning can be carried out with the aid of reactance tubes feeding the resonator.

This high-frequency system apparently makes it possible to accelerate protons to an energy of \(20\ \mathrm{GeV}\) with a repetition rate of several pulses per second. Although the system described above, proposed on the basis of experiments, is already being built at present, in the future other high-frequency systems may prove more advantageous, for example the following:

1. A large number of low-voltage ferrite-loaded resonators with small \(Q\), in which the required law of frequency variation is achieved by changing, according to a definite law, the magnetizing current in the ferrites. This system is planned for the CERN proton synchrotron*) and for Brookhaven’s strong-focusing synchrotron with a pulsed field.

2. The use of drift tubes (or one or several sectors serving as them) operating at a high harmonic of the revolution frequency. In this case, tuning over a broad frequency range is apparently difficult.

) European Council for Nuclear Research under UNESCO. (Translator’s note.*)

1. Several h.f. schemes have also been proposed in which several groups of particles of different energies can be accelerated simultaneously in the chamber. If some of them prove practicable, it will become possible to increase the duty factor*) substantially and, consequently, the intensity.

In accelerators with strong focusing, when the critical energy \(E_t\), determined by equation (10.8), is reached, phase stability disappears. In a ring phasotron with radial sectors and negative \(k\) (the orbits of maximum energy are located inside the machine) there is no critical energy. Accelerators with spiral sectors and negative \(k\) are apparently impossible. Figure 11 shows the dependence of frequency on energy for a strong-focusing accelerator with a pulsed field and for an accelerator with a constant field at \(k>0\) and \(k<0\).

Fig. 11. Dependence of h.f. on energy in strong-focusing accelerators with pulsed and constant magnetic fields.

12. A ring phasotron with radial sectors for an energy of 10 Bev

To illustrate the properties of a high-energy ring phasotron, let us consider a design with radial sectors. Of all the designs proposed at present, it has been studied most thoroughly, although the design with spiral sectors is apparently more economical. From equations (7.23) and (7.24) one can find \(\sigma_x\) and \(\sigma_z\) for given \(N\), \(n\), \(\beta_1\), \(\beta_2\), and \(\delta\). Table I gives typical parameter values for an accelerator with 64 periodicity elements.

Table I

Approximate parameters of an accelerator with radial sectors

\(N = 64\)	\(\beta_1 = 15.00^\circ\)	\(\sigma_x = 122.1^\circ\)
\(n_1 = n_2 = 36\)	\(\beta_2 = 9.37^\circ\)	\(\sigma_z = 22.0^\circ\)
\(C = 5.35\)	\(\delta = 0.05^\circ\)	\(\nu_x = 21.7\)
\(k = 192.5\)	\(\Phi_1 = \Phi_2 = 5.74^\circ\)	\(\nu_z = 3.91\)

For this example we have chosen a maximum proton energy of 10 Bev and a maximum magnetic field of 20,000 gauss. The restrictions on the degree of radial and vertical focusing are determined by the tolerances which the machine parameters (for example, \(n\)) must satisfy in order to avoid resonances. Since \(\nu_x\), for constant \(\sigma_x\), is approximately proportional to the square root of \(n\), weakening the focusing relaxes these tolerances. In the case where expressions (7.8) and (7.9) are valid, the tolerance on \(n\) for \(\Delta \nu = 1/2\) is determined by the expression

\[ \frac{dn}{n} = \frac{2\pi \sin \sigma} {N\left(\psi_1 \sin \psi_1 \operatorname{ch}\psi_2 - \psi_2 \cos \psi_1 \operatorname{sh}\psi_2\right)} . \tag{12.1} \]

) By duty factor is meant the ratio of the duration of the pulse of accelerated particles to the acceleration time. Translator’s note.*

For the design we have chosen, the tolerance on \(n\) is approximately \(1\%\). For a constant field the value of \(n\) can be maintained with a higher degree of accuracy than in the case of a pulsed field, since all distortions are independent of time.

Incorrect installation of sectors in strong-focusing accelerators leads to large distortions of the equilibrium orbits[^15]. It can be shown[^15] that, in a ring phasotron with radial sectors, the distortion of the equilibrium orbit for a given root-mean-square error in the installation of the sectors will be greater than in an ordinary strong-focusing accelerator with the same number of periodicity elements and comparable values of \(\nu_x,\ \nu_z\), approximately in the ratio of the perimeters. Here a simplifying assumption was made that in each periodicity element the sectors are set ideally, while the errors in the installation of the elements themselves are random and independent. For this accelerator a root-mean-square error in the installation of 128 sectors equal to \(0.02\ \mathrm{mm}\) should lead to a maximum distortion of the equilibrium orbit of \(\pm 2.0\ \mathrm{cm}\).

The influence of space charge and scattering in gas was considered by Blachman and Courant[^16] and others[^17]. In the present case the beam injected from the Van de Graaff electrostatic generator will, after scattering, occupy an aperture of \(\pm 10\ \mathrm{cm}\). Adiabatic damping of the betatron oscillations with an increase of the impulse by a factor of 100 will reduce these oscillations to \(\pm 1.0\ \mathrm{cm}\). With a reasonable choice of the energy gain per revolution (\(75\ \mathrm{keV}\)) one can obtain \(3\cdot 10^{11}\) protons in the pulse.

The values of the physical quantities corresponding to the parameters of Table I are given in Table II.

Fig. 12. Cross section of the magnet and coils for an accelerator with radial sectors.

Table II

Physical parameters of an accelerator with radial sectors.
The subscript “0” refers to the maximum energy, the subscript “\(i\)” to the injection energy.

\(E_0 = 10\ \mathrm{Bev}\)	\(E_i = 5\ \mathrm{Mev}\)	kinetic energy of protons
\(r_0 = 97.3\ \mathrm{m}\)	\(r_i = 95\ \mathrm{m}\)	radius of the ring phasotron
\(B_0 = 20\,000\ \mathrm{gs}\)	\(B_i = 200\ \mathrm{gs}\)	guide magnetic field
\(\rho_0 = 18.2\ \mathrm{m}\)	\(\rho_i = 17.8\ \mathrm{m}\)	radius of curvature
\(Z_0 = 3.0\ \mathrm{cm}\)	\(Z_i = 15.0\ \mathrm{cm}\)	vertical semi-aperture
\(|r_0-r_i| = 2.3\ \mathrm{m}\)	—	radial aperture
\(E_t = 12\ \mathrm{Bev}\)	—	critical energy
\(Z_i = 2.5\ \mathrm{cm}\)	—	vertical half-height of the injected beam
\(\delta_i = \pm 0.001\ \mathrm{rad}\)	—	angular divergence of the injected beam
\(p = 5\cdot 10^{-6}\ \mathrm{mm\ Hg}\)	—	pressure in the vacuum chamber

Figure 12 shows a transverse section of a possible magnet design. Most of the field changes can be accomplished with the aid of reverse coils.

turns located on the pole tips. Table III gives the characteristics of the sectors for an accelerator whose parameters are given in Tables I and II.

When using the system described above, the repetition frequency of the pulses is determined only by the magnitude of the applied r.f. voltage and by the permissible rate of mechanical modulation of the frequency.

Table III

Characteristics of the sectors of a 10-Bev accelerator

Total weight of iron . . . . . . . . . .	9650 t
Total weight of copper . . . . . . . . .	670 t
Required current . . . . . . . . . . . .	112 000 ampere-turns
Power for magnet excitation . . .	5.5 Mw

The use of this r.f. system will apparently make it possible to obtain 1–3 pulses per second, with \(3 \cdot 10^{11}\) protons per pulse.

13. A Ring Phasotron with Spiral Sectors for an Energy of 20 Bev

Let us consider a ring phasotron in which the magnetic field is specified by formula (8.1). The motion of particles in such a field was considered in the second part. Equations (6.24), (6.27), and (6.31) show that, in the “smooth” approximation,

\[ \nu_x^2 = 1 + k, \tag{13.1} \]

\[ \nu_z^2 = -k + \left(\frac{f}{wN}\right)^2 + \frac{1}{2} f^2, \tag{13.2} \]

where \(w=\lambda/2\pi\) and \(\lambda\) is the relative radial distance between two adjacent ridges.

The parameters of a ring phasotron with spiral sectors for an energy of 20 Bev will be obtained from the “smooth” approximation and the condition \(\sigma=2\pi\nu/N<\pi\) (the stability condition for solutions of Hill’s equation). Then the change in these parameters will be shown on the basis of the exact solution of the linearized differential equations with the aid of an electronic computer.

Of the many types of injectors, we may choose a 50-Mev linear accelerator, a cyclotron, or, for much lower energy, an electrostatic Van de Graaff generator. Suppose that in the accelerator all equilibrium orbits are present, corresponding to a particle-energy interval from 5 Mev at the inner edge of the working region up to 20 Bev at the outer edge. We may choose \(k=82.5\), \(r_0=5000\) cm, where \(r_0\) is the mean orbit radius at maximum energy, and the mean field strength on the orbit is 14 000 gauss. This gives, for the mean radius at an energy of 5 Mev, the value \(r_i=4688\) cm. The required radial aperture is approximately \(d=r_0-r_i=312\) cm. The ratio of the mean field on the orbit corresponding to the maximum energy to the mean field on the injection orbit is \(\overline{H}_0/\overline{H}_i=203\).

Since \(k=82.5\), according to the “smooth” approximation \(\nu_x=9.15\). To remain within the stability region of the solutions of the linearized differential equations with variable coefficients, we must have \(2\nu<N\). We choose the number of sectors \(N\), or ridges crossed in one revolution, equal to 31. This gives \(\sigma_x=0.6\pi\). We can then choose \(\sigma_z=0.268\pi\), so that \(\nu_z=4.15\). Such a choice of \(\nu_x\) and \(\nu_z\) makes it possible to avoid za

for the forbidden lines in Fig. 9. In this case the probable operating point lies in one of the two large squares in the plane \(\nu_x,\nu_z\). The characteristics of the ridges can now be found from the second equation of the “smooth” approximation (13.2), which, for the chosen values of \(N\) and \(k\), gives \(f/w=218\).

Thus, if one takes \(f=1/4\), then \(\lambda=0.00506\), and the radial distance between neighboring ridges at the outer edge is \(25.3\ \mathrm{cm}\). This result is approximate.

For the case in which the magnetic field in the median plane is represented by formula (8.1), Fig. 8 shows the stability region calculated on the basis of exact solutions of the linearized equations. According to this diagram, at \(\sigma_x=0.615\pi\) and \(\sigma_z=0.25\pi\) one has \(f/wN^2=0.303\) and \(k/N^2=0.075\). If the number of periodicity elements is chosen as \(N=33\), then \(\nu_x=10.15,\ \nu_z=4.15\). Both quantities now lie in the middle of a large square bounded by the integer, half-integer, and third-integer fractional resonances (in order to be at the center of the admissible maximum square, the operating point \(\nu_x,\nu_z\) must be 0.15 above or below the integer-resonance lines for both dimensions). If again one takes \(f=1/4\), then \(w=1/1320\), and the radial distance between ridges is \(\lambda r_0=23.8\ \mathrm{cm}\).

Let us consider the question of attaining the specified value of \(f\). The shape of the equipotential surfaces corresponding to the field inhomogeneity coefficient \(f=1/4\) at \(k=150\) is shown in Fig. 13. Figure 13 gives several equipotential curves for different values of the magnetic potential.

Fig. 13. Shape of equipotentials in a magnet with spiral sectors for \(k=150\) and \(f=0.25\). The abscissa and ordinate are in the same units.

These curves were obtained by numerical calculations. They indicate the existence of deep slots in the surface of the pole tips when the ridges are at a distance of approximately \(0.13\lambda\) from the median plane. Apparently, slots appear in the surface when the vertical gap between ridges exceeds \(1/4\) of the radial distance between them. The appearance of these slots means that, for a large vertical gap, poles of opposite polarity are required in order to ensure the required value of the field inhomogeneity coefficient \(f\). In order not to cut into the poles regions with a reverse field, the vertical gap between the ridges must be made less than \(1/4\) of the radial distance between them. The same result was obtained analytically.

Figure 14 shows the shape of the equipotential surfaces for the case \(f=1/4\). The dependence of the vertical gap on \(f\) is presented in Fig. 15, where \(G\) is the maximum vertical gap between the vertices of the ridges in the case,

when there are no additional straight windings. If constancy of \(\nu_x\) and \(\nu_z\) is required, then the quantity \(f/w\) must also be constant. Thus one obtains the dependence \(G\cdot f/w\) on \(f\), shown in Fig. 15. We see that, under the condition of constant focusing, i.e. for \(f/w=\mathrm{const}\), the value of \(f\)

Fig. 14. Magnetic potential \(V=Z/W+f\sin(X/W)\sin[H(Z/W)]\) for \(k=0\) and \(f=0.25\). The poles corresponding to \(V=\pm 1.1\) give the maximum vertical gap without slots in the pole surface.

corresponding to the maximum vertical gap between the ridges is equal to \(1/4\), and the maximum vertical gap in units of \(\lambda r\) is \(G=0.275\). It also follows from the graphs that, in the absence of slots in the pole tips, for \(f\) in the interval from 0.14 to 0.36 the vertical gap is less than the maximum by only 10%. This analytical result agrees, as already mentioned, with the result obtained by numerical calculations.

Fig. 15. Maximum gap \(G\), multiplied by \((f/\lambda)\), as a function of \(f\). The use of the critical slot depth in the poles. The variation of the field in the median plane is sinusoidal.

For the example under consideration, \(\lambda r_0=2\pi w r_0=23.8\ \mathrm{cm}\). This means that if \(G=0.275\,\lambda r_0\) is chosen, then at the injection radius \(G=6.15\ \mathrm{cm}\), and at maximum energy \(6.6\ \mathrm{cm}\).

Without the use of additional windings distributed over the pole surfaces, it would be necessary to change the vertical gap by a factor of 203 in order for the magnetic field to change by the same factor. By placing additional windings between the ridges, it is possible to preserve the full aperture at all radii. Thus, by a corresponding selection of the windings and of the currents in them, \(G(r)\) can always be made approximately equal to 0.275 of the radial distance between the ridges, which is practically constant. However, it is not at all necessary to have the vertical gap constant for all radii, since the amplitudes of betatron oscillations decay with time as \(p^{-1/2}\). If the momentum increases by approximately a factor of 203, then the aperture required for betatron oscillations decreases by approximately \(\sqrt{200}\sim 14\) times. Consequently, it is desirable to have the aperture at the injection radius about 10 times

larger than at the maximum energy, and it is advantageous for the beam to fill all of it during injection. In reality, the vertical gap must be greater than that required when damping is taken into account, in order to avoid particle losses due to distortion of the equilibrium orbit as a result of unavoidable errors in assembling the magnet. If the maximum possible vertical gap is maintained, i.e., if \(G\) is kept equal to \(0.275kr\), the aperture will in fact even increase somewhat during acceleration owing to the small increase in \(r\). Therefore, in order to increase the aperture at the injection radius, it is necessary to introduce reverse poles at this location. This can be accomplished by means of additional straight turns placed between the ridges, as indicated in Fig. 16. In this case the poles can be moved apart and the vertical aperture thereby increased. Thus, apparently, it is possible to double the vertical gap at the injection radius.

Fig. 16. Construction of a magnet with spiral sectors.

The configuration of the ridges and turns providing the proper field shape is shown in Fig. 16. The iron contours coincide with equipotential surfaces. The figure also shows the arrangement of the turns between the ridges. The current in these turns determines the position of certain equipotential surfaces, making it possible to set the pole tips so that the vertical gap between the ridges is constant. Since the magnetic field decreases between all neighboring ridges by the same amount, the number of reverse turns in a groove decreases by the same number relative to the neighboring one. Thus, a larger number of ampere-turns is concentrated in the groove on the ridge of maximum field. Fig. 16 also shows how the use of direct and reverse currents makes it possible to increase the vertical gap at the injection radius. Excitation of such a magnet requires a power of approximately \(1.8\) MW.

With such a method of creating the required field configuration, it is necessary to pass the windings through the ridges, since the ridges are arranged along a spiral. Straight-line gaps between sectors make it possible to return the turns back to the same radius. Since the field changes from ridge to ridge by approximately \(35\%\), the vertical gap between the ridge tops must also change by \(35\%\) from one end of a sector to the other. The variation of the vertical gap along the ridges will be less significant if the sectors, having a length of about \(9.76\) m, are divided by straight-line gaps into, say, 3 parts each approximately \(3\) m long. Then the vertical gap along the ridges will vary by approximately \(12\%\), and the turns between the ridges can return to the same radius every \(\sim 3.4\) m (Fig. 17).

The introduction of straight-line gaps, in which the field is approximately zero, however, complicates the problem. If the boundaries of the straight-line gaps coincide with radii, then the machine and the orbits are not similar. Consequently, \(\delta\) changes during acceleration. This problem, studied by the MURA technical group\(^*\), is one of the most important. It was shown that the arrangement of the straight-line gaps discussed above leads to a minimal change in \(\delta\) for a reasonable gap length.

* Midwestern accelerator group, USA. Translator’s note.

There is another method for obtaining a magnetic field of a given configuration, which simplifies certain problems. It was studied by the MURA technical group on magnetic models. The magnet configuration is shown in Fig. 18. The average dependence of the field on the radius \((r)^k\) is achieved by means of return turns located on the poles of the magnet, in a manner analogous to that used in an accelerator with radial sectors. The azimuthal dependence of the field is provided by introducing iron shims of a definite configuration. Since the shims are arranged along spirals, they must be separated by several nonmagnetic spacers (for example, copper inserts), so that the magnetic flux does not propagate along the shims. Such ridges and the required fields were obtained on a model by Peterson and Elph in the MURA technical group.

Fig. 17. Diagram of the arrangement of windings in a magnet with spiral sectors.

Experiments showed that in a ring phasotron with spiral sectors it is possible to relax the requirements on the vertical gap. Peterson and Elph were able to increase \(f\) considerably above \(1/4\) without reducing the vertical gap and without using reverse poles. This was achieved

Fig. 18. Construction of a magnet with movable shims.

by small deviations from a simple sinusoidal form of variation of the field with azimuth. A value \(f \sim 0.38\) was obtained without large harmonic distortions of the field in the median plane. Further study of this possibility is required in order to show how strongly the strongly focusing term in \(\nu_z\) increases for attainable field shapes. Any increase in focusing in the \(z\)-direction will make it possible to increase the vertical aperture.

Before choosing the magnitude of the vertical gap, one more question must be answered. As was indicated in § 9, there exists a limiting amplitude of stable betatron oscillations, after reaching which the particles begin to oscillate about a second equilibrium orbit, located inside or outside the chamber. If the second orbit is inside the chamber, then

particles will be lost only in the case when the amplitude of the oscillations is greater than the shortest distance from the orbit to the chamber wall. For radial oscillations the value of the limiting amplitude may lie in the interval from 0.1 to 0.3 of the radial distance between ridges, and for vertical oscillations it is considerably smaller. In the example given, the value of the limiting amplitude is small, since \(\sigma_x \sim 2\pi/3\). It therefore makes sense to reduce the influence of certain nonlinearities, since this will make it possible to increase the vertical aperture beyond 0.275 of the radial distance between ridges. Numerical calculations have shown that for some types of nonlinearities it is not worthwhile to increase the vertical aperture, since the limiting amplitude does not exceed the dimensions of the chamber. At present, in connection with the design of an accelerator with spiral sectors and a large useful vertical aperture, the sources of nonlinear effects are being investigated. Generally speaking, if the angle \(\zeta\) is made smaller, so that the oscillations change little over the length of a sector, the limiting amplitude increases.

Fig. 19. View of the magnetic gap with rectangular shims.

The most promising method of reducing \(\zeta\), and consequently the nonlinearities, is the use of such magnets as provide large values of \(f\). We shall indicate the two most important magnet designs.

Rectangular iron shims, the vertical gap between which is \(1/4\) of the gap between the poles without them (Fig. 19). Taking account of the leakage flux, for the case \(A=2\) and \(B=9\) one may write \(f=\sqrt{2\langle\Delta H^2\rangle}/\bar H=0.71\), where \(A\) and \(B\) are given in units of half the gap. If, as in the preceding example, \(f/w=330\), then \(w=0.00215\). Such a design makes it possible to obtain a good vertical aperture at the injection radius:

\[ G=\left[\frac{4\pi w}{(A+B)}\right]\cdot 4688\ \text{cm}=11.1\ \text{cm}. \]

The magnification factor of the installation is less than

\[ (A+B)/(A+\tfrac14 B)=2.3, \]

i.e. quite acceptable for the injection radius. If similarity of the equilibrium orbits is not required, then the proportions of the ridges can be changed, and the magnification factor of the installation can be reduced at the radius corresponding to the maximum energy. For example, it is possible to make a continuous transition to \(A=9,\ B=8\), and \(G=7\ \text{cm}\) for the same value of \(f/w\), taking account of leakage flux. The magnification factor of the installation, taking account of edge effects, will then be 1.38.

Fig. 20. View of the field along the circumference in a magnet with separated spiral sectors at a radius of \(10\,000\ \text{cm}\), with a gap of \(30\ \text{cm}\).

The second design, which possesses many advantages, is a magnet with separated spiral sectors. By winding each ridge separately with straight coils and placing the coils on the surface of the poles, as shown in Fig. 12 for a magnet with radial sectors, the ridges can be separated so that the field between them decreases almost to zero. This makes it possible greatly to increase \(f\). For the field shown in Fig. 20, the magnification factor of the installation is 2, the field nonuniformity coefficient \(f=1.28\), and the verti-

The vertical aperture may be about 30 cm. The angle between the edges of the sectors and the orbit is still sufficiently large to provide a large limiting amplitude of betatron oscillations (an amplitude of up to 90 cm is possible). The shape of the sectors is shown in Fig. 21.

To reduce the power consumed, the vertical aperture in the region of maximum field can be made much smaller than 30 cm. However, it is highly desirable to retain the aperture at the injection radius.

Fig. 21. View of a magnet with separated spiral sectors.
Each ridge has its own windings.

Although such a system leads to a large perimeter for the installation, it nevertheless has many advantages in comparison with the systems already described. Such a magnet is simpler to construct, and the vacuum chamber is easier to build. Access to the targets is also facilitated. In those places where large straight sections are required, the sectors can be moved apart without destroying similarity. The limiting amplitude of the betatron oscillations is sufficiently large, and therefore the working aperture at the injection radius is substantially increased.

14. Betatron with a Constant Guiding Field

A constant guiding field makes it possible to greatly increase the injection time when using betatron acceleration.^4 Particles can be injected during the entire time while the central magnetic flux is increasing. In the process of acceleration the particles spiral outward, entering the region of maximum field. After the change in the flux penetrating the particle orbit becomes equal to the value \(\Delta \Phi\) corresponding to the increase of the momentum to its final value, the particle reaches the target (or the ejector radius). Particles continue to arrive at the target as long as the flux continues to increase by the amount \(\Delta \Phi\). If the value of \(\Delta \Phi\) is less than the maximum central flux \(\Phi_0\), then the time of useful injection and ejection may amount to 25% of the time of variation of the central flux from \(-\Phi_0/2\) to \(+\Phi_0/2\). In the case where the core is supplied with alternating current, the duty factor \(D\) (the relative time of useful injection) is expressed by the formula (cf. Fig. 22)

\[ D=\frac{1}{2\pi}\arccos\left[\frac{2\Delta\Phi}{\Phi_0}-1\right]. \tag{14.1} \]

In order that particles not be lost because of collision with the injector, a certain minimum rate of increase of the magnetic flux at the moment of injection is required; in fact, this leads to a decrease in the duty factor.

Since the equilibrium orbit is not circular and its radius changes during acceleration, the relation between \(\Delta \Phi\) and the increment of momentum

differs from that in an ordinary betatron. The voltage per revolution, in the Gaussian system, is equal to

\[ V=\frac{1}{c}\frac{d\Phi}{dt}, \tag{14.2} \]

where \(\Phi\) is the flux in the betatron core.

The energy gain per unit time is determined by the formula

\[ \frac{dE}{dt}=\left(\frac{e\omega}{2\pi c}\right)\frac{d\Phi}{dt}. \tag{14.3} \]

Here \(\omega/2\pi\) is the frequency of revolution of the particle [equation (10.3)]. Hence we obtain

\[ R\cdot dp=\frac{dE}{\omega}=\left(\frac{e}{2\pi c}\right)d\Phi . \tag{14.4} \]

Consequently, the required change in the accelerating flux will be determined by the relation

\[ \Phi_2-\Phi_1=\frac{2\pi c\overline{R}}{e}(p_2-p_1), \tag{14.5} \]

where

\[ \overline{R}=\frac{1}{p_2-p_1}\int_{p_1}^{p_2}R\,dp. \tag{14.6} \]

If \(k=\mathrm{const}\), then from equation (5.14) it follows that

\[ \overline{R}=R_2\cdot\left(\frac{k+1}{k+2}\right) \frac{1-(p_1/p_2)^{\frac{k+2}{k+1}}}{1-(p_1/p_2)}. \]

For \(p_1\ll p_2\) this equation simplifies to

\[ \overline{R}=\left(\frac{k+1}{k+2}\right)R_2. \tag{14.7} \]

Fig. 22. Dependence of the betatron flux on time.
Figure labels: Output; Injection; Time; \(\Phi/2\); \(-\Phi/2\); flux in core; \(\Delta\Phi\); \(\Delta\Phi'\).

The use of a constant guide field in the energy range from 20 to 300 MeV makes it possible to increase the duty factor by more than \(10^4\) times in comparison with existing betatrons and synchrotrons. The increase in beam current will apparently be smaller because of the influence of space charge during injection.

In betatrons with a pulsed guide field, a large amount of energy is dissipated in the magnet gap, and therefore it is necessary to use equipment capable of producing large currents and voltages. In betatrons with a constant guide field, only the accelerating core operates in pulsed mode. It may be made in the form of a closed iron ring. In this case the dissipation is considerably reduced and, consequently, the required power is reduced. The equipment for such an installation is simpler.

For accelerating electrons up to several hundred MeV, a betatron with either radial or spiral sectors may be used. The construction requirements remain the same as for the ring phasotron. Since the change of the central flux for a specified interval of momentum variation is proportional to the period of revolution of the particle, for a betatron the problem of reducing the perimeter of an installation with radial sectors becomes doubly

important. Therefore, for betatrons it is apparently more reasonable to choose \(N\) in the interval from 10 to 30.

The accelerated electron beam in a strong-focusing betatron with a constant field is approximately monoenergetic, and the pulse duration corresponds to the duty factor discussed above. Modern betatrons and synchrotrons give an extremely monoenergetic beam and a long pulse duration. Thus, a betatron with a constant guiding field, without practically worsening the beam characteristics, makes it possible, in addition, to obtain average beam currents of the order of several milliamperes. Therefore it is a very advantageous installation for accelerating electrons to energies from several tens of MeV to several hundreds of MeV.

15. Strong-focusing cyclotron

In order to make semirelativistic particles revolve in a cyclotron with constant frequency and in approximately circular orbits, it is necessary to create a field increasing with radius. The instability of motion in the axial direction that then arises can be eliminated by using the principle of strong focusing. There are many configurations of the magnetic field that make it possible to realize such a cyclotron. A cyclotron of this type was first proposed by Thomas \(^{5}\). The Thomas cyclotron is, in essence, a strong-focusing accelerator with a constant field varying sinusoidally with azimuth. It consists of three or more radial sectors. Thomas showed that in such a machine there exist stable orbits for energies below a limiting value determined by the number of sectors. Extensive experimental work on the Thomas cyclotron was carried out at the University of California. It culminated in the successful construction and testing of two experimental models accelerating electrons to a speed equal to half the speed of light \(^{18}\). We shall briefly discuss here the principal features of the strong-focusing cyclotron, placing the main emphasis on the construction with spiral sectors.

In § 10 the relation (10.9) between the total energy \(E\) and the mean field index \(k\) for a cyclotron was obtained. In § 7 an approximate relation was also obtained connecting \(k\) with the frequency of betatron oscillations. For accelerators with spiral sectors the simple formula (6.24) is valid

\[ \nu_x = (1+k)^{1/2}. \tag{15.1} \]

According to formula (10.9), \(\nu_x\) is directly related to the energy

\[ \nu_x \approx \frac{E}{E_0}. \tag{15.2} \]

The motion of particles in such an accelerator, as in an ordinary cyclotron, begins at the center of the installation at \(E \sim E_0\) and \(\nu_x \sim 1\). As the energy increases, as follows from equation (15.2), the particles must successively pass through integral or half-integral resonances at energy values equal to an integral or half-integral multiple of \(E_0\). If the first integral resonance is regarded as determining the energy of the particles in the cyclotron, then the maximum kinetic energy will be approximately equal to the rest energy (according to more accurate calculations, even somewhat less \(^{19}\)). With a sufficiently large voltage on the dees and small field distortions, it will apparently be possible to carry the particles through the resonances sufficiently rapidly and thereby avoid a noticeable increase in the oscillation amplitude. In any case, to ensure stability, \(\nu_x\) must be less than \(\frac{1}{2}N\), and the attainable energy cannot exceed

\(\frac{1}{2}NE_0\). The third fractional resonance (see § 9) at \(\sigma_x = 2\pi/3\) \((\nu_x = N/3)\) may sharply reduce the value of the attainable energy \(E\) for a given number of sectors \(N\).

In an accelerator with radial sectors, when the number of periodicity elements is small, \(N\) \((N < 8)\), strong focusing is provided mainly by the \(\eta\)-term in equation (5.13). Consequently, relations (15.1) and (15.2) remain approximately valid, and the preceding qualitative discussion is correct. This applies in particular to the Thomas cyclotron.

As follows from equations (6.24) and (6.25), in a cyclotron for which the \(\eta\)-term in equation (5.13) is dominant, the focusing depends on \(k\) and on the quantity

\[ F = 2\left\langle \left(\frac{1}{\eta}\cdot \frac{\partial \eta}{\partial \Theta}\right)^2 \right\rangle + \frac{1}{2}f^2 . \tag{15.3} \]

The focusing parameter \(F\), according to equations (6.24) and (6.25), is equal to

\[ F = \nu_x^2 + \nu_z^2 - 1 . \tag{15.4} \]

In § 6 it was noted that in accelerators with spiral sectors the optimum value of the field inhomogeneity coefficient \(f\) is approximately \(1/4\). With the aid of equation (6.26), the focusing parameter \(F\) for this value of \(f\) can be written in the form

\[ F \simeq \frac{1}{16}\left(\operatorname{tg}^2 \xi + \frac{1}{2}\right). \tag{15.5} \]

In Fig. 23, circles of constant \(F\) are plotted in the coordinates \(\nu_x, \nu_z\). The vertical lines correspond to constant \(k\) and, consequently, constant energy \(E\). Lines corresponding to integer and half-integer resonances \((\nu_x, \nu_z\)—integer or half-integer) and coupling resonances \((\nu_x + \nu_z\)—integer) are also shown. In the course of acceleration the operating point \(\nu_x, \nu_z\) will move along the curve connecting the line \(k = 0\) with the line \(k = (E/E_0)^2 - 1\). The shape of this curve will depend on the manner in which \(F\) changes with radius. In a real ma-

Fig. 23. Working region for a cyclotron with spiral sectors. \(F\) is the focusing parameter.

width at the center \(F\) is approximately equal to zero, so that the curve will begin near the point \(\nu_x=1,\ \nu_z=0\). In accelerating particles one may expect difficulties when the operating point crosses some resonance line; integral resonances or resonances with vertical oscillations are especially dangerous, since the vertical aperture is not large. It follows from the graph in Fig. 23 that the particles will necessarily pass through a half-integer resonance with radial oscillations near \(E=E_0+\frac{1}{2}E_0\) and a coupling resonance before reaching the energy \(E=2E_0\).

Let us dwell in more detail on the main characteristics of a strong-focusing cyclotron. The frequency of revolution of particles in the cyclotron is equal to

\[ \frac{\omega}{2\pi}=\frac{\beta c}{2\pi R}=\frac{c}{2\pi\lambda}, \tag{15.6} \]

where \(2\pi\lambda\) is the rf wavelength (it is assumed that acceleration proceeds on the first harmonic). Hence we obtain

\[ \frac{E}{E_0}=\frac{\lambda}{(\lambda^2-R^2)^{1/2}} . \tag{15.7} \]

The momentum \(p(R)\) is equal to

\[ p=\frac{mcR}{(\lambda^2-R^2)^{1/2}}, \tag{15.8} \]

and, consequently, the average magnetic field is determined by the formula

\[ \overline{H}=\frac{pc}{eR}=\frac{mc^3/e}{(\lambda^2-R^2)^{1/2}} . \tag{15.9} \]

Then from equation (10.9) for \(k\) we shall have

\[ k=\frac{R^2}{\lambda^2-R^2}. \tag{15.10} \]

Relations (15.6)—(15.10) are exact. In order to determine the shape of the spiral ridges, it is necessary to solve the equations for betatron oscillations. An approximate picture can be constructed on the basis of equations (15.1), (15.4), and (15.5). Together with equation (15.10), these formulas give

\[ \operatorname{tg}^2 \xi=\frac{16R^2}{\lambda^2-R^2}+16\nu_z^2-\frac{1}{2}. \tag{15.11} \]

Suppose that the operating point in Fig. 23 moves along the horizontal line \(\nu_z=1/\sqrt{32}\). Then

\[ \operatorname{tg}\xi \simeq \frac{4R}{(\lambda^2-R^2)^{1/2}} . \tag{15.12} \]

If the scalloping of the equilibrium orbit is neglected, then \(R\) may be replaced by the radius \(r\). Substituting then (15.12) into (6.30), we obtain the equation of the spiral ridge in polar coordinates

\[ \theta_0=4\arcsin\left(\frac{r}{\lambda}\right). \tag{15.13} \]

If it is assumed that the azimuthal variation of the field is sinusoidal, then the function \(\mu\) has the form

\[ \mu=1+\frac{1}{4}\cos[N(\theta-\theta_0)]. \tag{15.14} \]

and the magnetic field is given by the formula

\[ H=\overline{H}_\psi= \frac{mc^2/e}{(\lambda^2-R^2)^{1/2}} \left\{1+\frac{1}{4}\cos\left[N\theta-4N\arcsin\left(\frac{r}{\lambda}\right)\right]\right\}. \tag{15.15} \]

The number of sectors \(N\) in this approximation was considered arbitrary. If the energy of the accelerated particles is \(E=2E_0\) (kinetic energy for protons \(\sim 1\) Bev), then \(\nu_x \simeq 2\) at the outer radius. In order that the stability of betatron oscillations not be disturbed, \(N\) in this case must be no less than 4, and in order to avoid the third subresonance at \(\sigma_x=2\pi/3\), it is apparently necessary to choose \(N=6\).

Fig. 24. Arrangement of hills in a cyclotron with 6 spiral sectors.

Fig. 25. Dependence of the total energy and magnetic field on the radius in a cyclotron with constant frequency (\(E_0\)—rest energy and \(2\pi\lambda\)—wavelength of the oscillations).

Figure 25 presents \(E\) and \(\overline{H}\) as functions of \(R\) for such a cyclotron. Figure 24 shows the arrangement of hills, given by equation (15.13), for a cyclotron with six spiral sectors at the maximum energy \(E=2E_0\).

APPENDIX A

Smooth approximation

Let the equation of motion in one of the dimensions have the form

\[ \frac{d^2x}{d\theta^2}=f(x,\theta), \tag{A.1} \]

where \(f(x,\theta)\) is a periodic function with period \(2\pi/N\) in \(\theta\). We shall assume that \(N\gg \nu\), i.e., the wavelength of the betatron oscillations is large compared with the length of an element of periodicity. It is therefore reasonable to seek an approximate solution in the form

\[ x=X+\xi(X,\theta), \tag{A.2} \]

where \(X(\theta)\) are “smooth” oscillations satisfying the equation

\[ \frac{d^2X}{d\theta^2}=F(X), \tag{A.3} \]

which does not depend on the periodic structure of the magnet; \(\xi(X,\theta)\) are “ripples,” periodic in \(\theta\) with period \(2\pi/N\). The mean value of the ripples for fixed \(X\) is equal to zero. We shall also assume that the ripples \(\xi\) and the derivatives \(dX/d\theta\), \(d^2X/d\theta^2\) are small. The meaning of the smallness of these quantities will be clarified shortly.

Substituting (A.2) into (A.1), we obtain

\[ X''+\xi_{\theta\theta}+2\xi_{X\theta}X' + \xi_{XX}X'^2+\xi_X X''=f(X+\xi,\theta), \tag{A.4} \]

where primes denote derivatives with respect to \(\theta\). We now average with respect to \(\theta\), regarding \(X, X', X''\) as fixed. Taking into account that \(\langle \xi\rangle=0\), we obtain the equation corresponding to (A.3):

\[ \frac{d^2X}{d\theta^2}=\langle f(X+\xi,\theta)\rangle. \tag{A.5} \]

Next, using definition (4.14), we may write

\[ \xi_{\theta\theta}=\{f(X+\xi,\theta)\}-2\xi_{X\theta}X' -\xi_{XX}X'^2-\xi_X X''. \tag{A.6} \]

It follows that the last two terms are of order \((\sigma/2\pi)^2\) relative to the first and are therefore negligibly small when \(N\gg \nu\). The second term is of order \(\sigma/\pi\) relative to the first, but it can be shown that, in the first approximation, it does not affect the solution of equation (A.5). Therefore we neglect the last three terms in equation (A.6) and replace \(\{f(X+\xi,\theta)\}\) by \(\{f(X,\theta)\}\), i.e., we assume that \(\{\xi f_X\}\ll \{f\}\). We can now integrate equation (A.6) and obtain the first approximation for the pulsations

\[ \xi=f_2(X,\theta). \tag{A.7} \]

Here the notation (4.16), (4.17) has been used. Substituting the expression for the pulsations (A.7) into equation (A.5) and restricting ourselves to first order in \(\xi\), we find the smooth approximation

\[ \frac{d^2X}{d\theta^2}=\langle f\rangle+\langle f_2\cdot f_X\rangle. \tag{A.8} \]

(Essentially the same result was obtained by Sigurgeirsson\(^{20}\).) We obtain an approximate solution of equation (A.1) by adding to the solution of equation (A.8) the pulsations determined by expression (A.7). The second term on the right-hand side of equation (A.8) may be integrated by parts and the equation written in the form

\[ \frac{d^2X}{d\theta^2}=\langle f\rangle-\langle f_1\cdot f_{X_1}\rangle. \tag{A.9} \]

If the right-hand side in (A.1) depends linearly on \(x\),

\[ f(x,\theta)=g(\theta)x, \tag{A.10} \]

then equation (A.9) is the linear equation

\[ \frac{d^2X}{d\theta^2}=\bigl[\langle g\rangle-\langle g_1^2\rangle\bigr]X. \tag{A.11} \]

The approximate solution of this equation is expressed in terms of Floquet functions

\[ x=e^{\pm i\nu\theta}\bigl(1+g_2(\theta)\bigr), \tag{A.12} \]

where

\[ \nu^2=\langle g_1^2\rangle-\langle g\rangle. \tag{A.13} \]

This result can be generalized to the case of two dimensions

\[ \left. \begin{aligned} \frac{d^2x}{d\theta^2} &= f(x,z,\theta),\\ \frac{d^2z}{d\theta^2} &= g(x,z,\theta). \end{aligned} \right\} \tag{A.14} \]

We seek solutions in the form

\[ \left. \begin{aligned} x &= X+\xi,\\ z &= Z+\zeta. \end{aligned} \right\} \tag{A.15} \]

Whence we obtain the approximate equations

\[ \left. \begin{aligned} \xi &= f_2(X,Z,\theta),\\ \zeta &= g_2(X,Z,\theta), \end{aligned} \right\} \tag{A.16} \]

where \(X\) and \(Z\) satisfy the equations

\[ \left. \begin{aligned} \frac{d^2X}{d\theta^2} &= \langle f\rangle+\langle f_2\cdot f_X\rangle+\langle g_2\cdot f_Z\rangle,\\ \frac{d^2Z}{d\theta^2} &= \langle g\rangle+\langle f_2 g_X\rangle+\langle g_2\cdot g_Z\rangle. \end{aligned} \right\} \tag{A.17} \]

The averaging over \(\theta\) here has been carried out under the assumption that \(X, Z\) are constant.

As was shown, equations (A.13) give the values \(\nu\), or \(\sigma = -\dfrac{2\pi\nu}{N}\), with an accuracy of \(10\%\) of \([\langle g_1^2\rangle]^{1/2}\), provided that \([\langle g_1^2\rangle]^{1/2} \ll \dfrac{N}{4}\). We have also considered certain nonlinearities and have shown that equations (A.8) and (A.17) are in good agreement with more exact calculations, except for the neighborhood of the boundary of the stability region. The boundaries themselves of the stability region, where the wavelength of the betatron oscillations is equal to \(\infty\), are also determined sufficiently accurately from equations (A.8) and (A.17). The resonant lines corresponding to “sector” resonances (which are of greater interest), where the wavelength of the betatron oscillations becomes equal to the length of a small number of sectors, are not predicted at all by the “smooth” equations.

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