GIANT ACCELERATOR PROJECTS FOR PRODUCING PARTICLES WITH ENERGIES OF \(10^{10}\) eV

Introduction

Obtaining particles of very high energy with the aid of special installations—accelerators—is one of the central problems of modern physics. The greatest development at the present time has been achieved by cyclic resonant accelerators*). In these installations, particles moving in a magnetic field execute identical or nearly identical cycles, receiving energy from a high-frequency electric field as they pass through accelerating gaps. The cyclic frequency \(\dot{\theta}\) of rotation of a particle with total energy \(E\) in a magnetic field \(H\) is equal to:

\[ \dot{\theta}=\frac{ceH}{E}. \tag{1} \]

It follows from (1) that the radius of the orbit is

\[ R=\frac{E\beta}{eH}, \tag{2} \]

where \(\beta=\dfrac{v}{c}\) (\(v\) is the velocity of the particle).

For resonance the frequency of the alternating accelerating electric field \(\omega_\lambda\) must be equal to (or a multiple of) the frequency of rotation of the particle:

\[ \omega_\lambda=q\dot{\theta}, \tag{3} \]

where \(q\) is the order of the resonance (usually \(q=1\)).

Since in the process of acceleration the energy of the particle must on the average increase, in order to satisfy condition (1) either the parameter \(H\), or the parameter \(\omega_\lambda\), or both of these parameters together must change in the corresponding manner.

This is one of the differences between resonant accelerators and the cyclotron, in which the parameters remain unchanged. Therefore the operation of the cyclotron is disturbed as soon as the relativistic effect of the increase in the mass of the accelerated ions begins to manifest itself. For this reason, as is well known, the cyclotron is unsuitable for accelerating electrons. In modern resonant accelerators using the principle of “autophasing,” the relativistic increase of mass does not prevent the action of the accelerator.

Accelerators with a magnetic field \(H\) varying in time and a constant frequency of the accelerating field \(\omega_\lambda\) are called synchrotrons. Synchrotrons are used for accelerating electrons**).

*) For accelerating electrons, alongside resonant accelerators, betatrons also continue to be used. (See UFN 26, 181 (1944); UFN 27, 31 (1945); UFN 30, 119 (1946).) The betatron mode is also used at the initial stage of operation in cyclic resonant electron accelerators (synchrotrons).

**) See, for example, the description of the 80 MeV synchrotron, UFN 37, 501 (1949), as well as the description of the first model of an 8 MeV synchrotron, UFN 31, 584 (1947).

Accelerators with a time-constant magnetic field and a varying (decreasing) frequency of the accelerating electric field have received the name phasotrons. Phasotrons are also used for accelerating heavy particles (protons, deuterons, and α-particles)*.

Finally, if in a resonance accelerator both the frequency of the accelerating electric field and the magnetic field are varied simultaneously, such an accelerator is called a synchrophasotron. In this case the law of variation of \(H\) and \(\omega\) is chosen so that the orbit radius remains constant. The synchrophasotrons being designed are, in most cases, intended for accelerating heavy particles.

Modern resonance accelerators, first proposed by V. I. Veksler\(^1\) in 1944, are based on the property of “autophasing,” or “phase stability.”

In accelerator theory it is customary to call the particle phase \(\varphi\) the phase of the alternating electric voltage

\[ V = V_0 \cos \omega t \]

at the moment when the particles pass through the accelerating gap. For successful operation of the accelerator it is necessary that the particle phase, averaged over many revolutions, be equal to the phase corresponding to an accelerating voltage of sufficient magnitude.

The property of autophasing consists in the fact that, for a certain interval of initial conditions, the particle phase during acceleration performs damped oscillations about a certain stationary phase (the phasing point) \(\varphi_0\), corresponding to the resonant accelerating voltage

\[ V_R = V_0 \cos \varphi_0 . \]

The motion of a particle in resonance with the accelerating voltage is called equilibrium motion, and the corresponding parameters (energy, radius, orbit, etc.) are called equilibrium parameters. With oscillations of the phase (and hence of the energy), oscillations of the orbit radius are also associated (2) about the equilibrium position (the so-called radial-phase oscillations). These oscillations have a frequency considerably smaller than the frequency of revolution of the particle, i.e., in other words, radial-phase oscillations proceed slowly in comparison with the period of revolution of the particles. Superposed on these slow oscillations of the radius are much faster radial and vertical “free” oscillations, well known from the theory of the betatron**. The frequencies of these oscillations \(\omega_r\) and \(\omega_z\) are of the same order as the revolution frequency \(\dot{\theta}\) and are equal, respectively, to:

\[ \omega_r = \dot{\theta}\sqrt{1 - n}, \qquad \omega_z = \dot{\theta}\sqrt{n}, \]

where the quantity

\[ n = - \frac{\partial \ln H_z}{\partial \ln R} \]

—the magnetic-field fall-off index—characterizes the form of the magnetic field. For stability of the motion it is necessary that the quantity \(n\) satisfy the requirement \(0 < n < 1\).

For given initial conditions, phase oscillations take place between the values \(\varphi_1\) and \(\varphi_2\) (see Fig. 1), which are, generally speaking, nonsymmetrically situated with respect to \(\varphi_0\). The limiting values \(\varphi_1\) and \(\varphi_2\), at which an oscillatory regime for the phase is still possible, are respectively equal to

* See the description of the 184-inch Berkeley phasotron, which gives α-particles with an energy of 400 MeV and deuterons with an energy of 200 MeV, UFN 32, 396 (1947).

** See, for example, Ya. P. Terletskii, Journ. Phys. USSR 9, 159 (1945), and also UFN 27, 31 (1945).

—\(\varphi_0\) and \(\varphi_3\) (the latter value is determined from a certain transcendental equation and also depends on \(\varphi_0\)).

Autophasing is accomplished with a sufficiently slow (“adiabatic”) change in the accelerator parameters (the magnetic-field strength \(H\), the frequency of the accelerating electric field \(\omega_\lambda\), etc.).

Fig. 1. Variable accelerating electric voltage and the behavior of the phase of a particle in a resonant accelerator.

As theory shows,^2 the behavior of the phase of particles in resonant accelerators is described by an equation coinciding with the equation of a physical pendulum with an external torque and adiabatically varying coefficients

\[ \frac{d}{dt}(a\dot{\varphi}) + b\cos\varphi = c, \tag{4} \]

where \(a\), \(b\), \(c\) are slowly varying functions of time. The behavior of the phase is conveniently investigated on the phase plane, in the coordinates \((\varphi,\dot{\varphi})\). An approximate picture on the phase plane for equation (4) is shown in Fig. 2 (the notation is the same as in Fig. 1). Closed oval orbits surrounding the point \(\varphi_0\) correspond to a periodic variation of the phase about \(\varphi_0\). Such is the behavior of the phase for particles that are captured into the accelerating regime and, on the average, are resonantly accelerated.

The boundary curve (separatrix) separates closed periodic orbits from unclosed phase trajectories. The latter correspond to particles that are not captured into the accelerating regime, successively pass into the various phases (including also the corresponding deceleration), and cannot attain large energy values. As is seen, the “phase region of capture” lies within \((-\varphi_0,\varphi_3)\).

The presentation of the theory of particle motion in a resonant accelerator is beyond the scope of the present review, which is specifically devoted to the design of synchrotron devices intended for obtaining protons with an energy of \(10\,000\) MeV.

Fig. 2. Picture on the phase plane for the phase of a particle in a resonant accelerator.

However, in view of the subsequent exposition, a few more remarks must be made. According to (1), (2), and (3), the radius of the equilibrium orbit in the synchrotron grows proportionally to the particle velocity (and ceases to grow only in the relativistic region, for \(\beta \simeq 1\)):

\[ R(t)=\frac{cq}{\omega_\lambda}\,\beta(t). \tag{5} \]

In the phasotron the equilibrium radius increases considerably faster because of the decrease in the frequency \(\omega_\lambda(t)\) (and continues to increase also in the relativistic region):

\[ R = cq \cdot \frac{\beta(t)}{\omega_\lambda(t)} . \tag{5a} \]

In order to keep the equilibrium radius constant during acceleration \((R = R_0 = \mathrm{const}.)\), as happens in the synchrophasotron, it is necessary to increase the frequency \(\omega_\lambda(t)\) with time, in proportion to the velocity \(\beta(t)\):

\[ \omega_\lambda = \frac{cq}{R_0}\,\beta(t) = \frac{cq}{R_0}\sqrt{1-\left(\frac{E_0}{E(t)}\right)^2}, \tag{6} \]

where \(E_0\) is the rest energy of the particles. In other words, the frequency of the electric field \(\omega_\lambda\) must change in the same way as the frequency of a particle moving in the betatron along an equilibrium orbit with the same radius and the same magnetic field as in the synchrotron.

On the other hand, according to (1), for resonance it is necessary that the magnetic field increase in proportion to \(\omega_\lambda(t)E(t)\). At the same time, from the point of view of the constancy of the equilibrium radius, no restrictions are imposed on the change in the magnitude of \(H\).

Consequently, with a simultaneous change of the magnetic field, one can ensure that the radius of the equilibrium orbit remains constant, while resonance acceleration is carried out. For an arbitrary (provided it is sufficiently slow) increase in the intensity of the magnetic field, the frequency of the accelerating electric field \(\omega_\lambda\) must vary with time according to a perfectly definite law, in accordance with the change of \(H(t)\). This law can be obtained at once by combining (1), (3), and (6):

\[ \omega_\lambda(t) = \frac{c}{R_0\left[1+\left(\frac{E_0}{eR_0H(t)}\right)^2\right]^{1/2}} . \tag{7} \]

The total change in frequency required in the process of acceleration is determined by the initial \(E_i\) and final \(E_f\) energies of the particle. A large change of the frequency \(\omega_\lambda\) in the process of acceleration presents considerable technical difficulties. For a given value of the final energy, the ratio of the final \(\omega_{\lambda f}\) and initial \(\omega_{\lambda i}\) frequencies depends only on the initial energy and is equal to

\[ k = \frac{\omega_{\lambda f}}{\omega_{\lambda i}} = \frac{\sqrt{1-\left(\frac{E_0}{E_f}\right)^2}} {\sqrt{1-\left(\frac{E_0}{E_i}\right)^2}} . \tag{8} \]

Thus, the higher the injection energy of the particles \(E_i\) (the injection energy), the less it is necessary to modulate the frequency \(\omega_\lambda\) in the synchrophasotron. As is evident from (8), in the relativistic case \((E_f \gg E_0)\) the quantity \(k\) practically no longer depends on the final energy \(E_f\) and is entirely determined by the value of the initial energy.

Bevatrons

In recent years there has been intensive construction of modern accelerators of various types, some of which are already operating successfully. Alongside the construction of accelerators designed to produce particles with energies of the order of hundreds of MeV, projects have appeared for the creation of much more powerful installations, in which particles are to be accelerated to energies of the order of several thousand million (billions, in American terminology) electron-volts. Accelerators designed to produce particles in this energy interval have received in the American literature the name bevatrons (from \(10^9 eV = 1\) BeV = 1 billion electron-volts). In 1947 the literature described a project for a proton accelerator at the University of Birmingham³ for \(1.3 \cdot 10^9 eV\). Here projects will be described for two bevatrons at 10 BeV, the development of which is being carried out in the USA at the Brookhaven National Laboratory under Livingston’s direction and at the University of Berkeley under Brobeck’s direction.

The energy value of 10 BeV, as Livingston⁴ indicates, was chosen as exceeding the threshold for the formation of pairs of nucleons—5.6 BeV⁵. This value lies in the range of energies possessed by a certain fraction of primary cosmic rays. In the case of electrons, losses due to radiation*), in the opinion of many authors, set a limit for the maximum attainable energy of electrons in cyclic accelerators of the order of 1–2 BeV. For protons the radiation losses, which are inversely proportional to the fourth power of the rest energy of the particle, are negligibly small for energies of tens of BeV. Therefore, in the case of heavy particles, the maximum attainable energy, if one speaks of the particle-acceleration methods now employed, is limited only by technical possibilities and economic considerations.

Of the four principally possible types of accelerators: the phasotron, betatron, synchrophasotron, and linear accelerator, the synchrophasotron has the greatest advantage at energies of this order. The phasotron requires a solid magnet of enormous radius, with a magnetic field whose magnitude and form must satisfy fairly stringent requirements. The betatron requires a large amount of iron to create the powerful central flux necessary to satisfy the 2:1 condition. A linear accelerator for protons to such energies would have to have a colossal length, not to mention the fact that difficulties would have to be overcome in focusing the particle beam and synchronizing the system along its entire length⁶. The synchrophasotron with its ring magnet is therefore apparently the most practically suitable for obtaining protons with an energy of 10 BeV.

Next, it is necessary to choose the type of magnet—with an iron core or without one. The advantages of an air-core magnet are the absence of a limitation on the magnitude of the limiting field strength due to iron saturation and the saving of a large quantity of iron. However, such a type of magnet leads to impermissibly large values of reactive power and creates very great difficulties in the sense of mechanical design and strength, as well as in obtaining the required current distribution to create a magnetic field of suitable configuration. It is therefore no accident that both projects mentioned contemplate the use of an accelerator of the synchrophasotron type having a magnet with an iron core. At the same time, of course, the known advantages of the synchrophasotron do not mean that in the design and construction of such colossal installations it will not be necessary to overcome a whole series of very serious difficulties connected with cyclotron problems—

) On the radiation of electrons in accelerators see UFN 33, 277 (1947); UFN 34*, 398 (1948).

synchronous, and betatron techniques. Here one has to deal with scales unprecedented up to now (suffice it to say that an energy of 10 BeV exceeds the maximum energy so far obtained in accelerators by 25 times).

The amount of data published on questions of the design and construction of bevatron[s] is still small. For the Brookhaven Laboratory project there are only a few brief notes concerning individual questions of construction^4,7,8,9. The project of the University of California at Berkeley, led by Professor Brobeck^10 and appearing after a series of preliminary communications, is described in more detail. However, in this latter case as well, the data given are tentative, and most of the problems have only been posed, not solved. We shall consider these two projects separately.

The Brookhaven Laboratory Bevatron Project

The annular magnet of the accelerator will probably contain rectilinear sections, i.e. it will be of the “racetrack” type*) (see Fig. 3). The magnet is made up of plates each 1.25 cm thick. The maximum magnetic field on

Fig. 3. Schematic drawing of the bevatron—a projected 10 BeV proton accelerator of the synchrophasotron type. In the right corner is shown the Van de Graaff generator used for the high-voltage injection of protons. The beam of high-energy particles emerges on the left near the human figure, shown to indicate the size of the installation.

the orbit is 15,000 H. Correspondingly, the radius of the equilibrium orbit is \(R_0 = 24.4\) m. During operation of the accelerator, an energy of \(5 \cdot 10^7\) joules is stored in its magnetic circuit. The transverse section of the magnet has a C-shaped form with

*) The name “racetrack,” meaning in translation “hippodrome,” was given to this type of magnet because of its oval shape. Rectilinear sections, in which the particle moves in the absence of a magnetic field, are introduced for ease of installation and servicing of the apparatus, for accommodating the bulky accelerating system, for ensuring the injection and extraction of the particle beam from the accelerator, etc.

with an air gap on the outer side. The air gap has a transverse cross section of \(120 \times 30\) cm. The properties of such a magnet are being investigated on a model at \(1/4\) of full size, in the form of a \(6^\circ\) arc. The injection energy is \(W_i = 4\) MeV. The frequency of the accelerating electric field varies from \(0.18\) Mc/s to \(2\) Mc/s, i.e. by more than a factor of 11. The use of resonators, like those used in electron accelerators\(^{11}\), therefore encounters great difficulties in the present case. The resonator would have to be of very large dimensions; tuning it over the indicated wide frequency interval is extremely difficult, and when operating with a detuned resonator currents of the order of thousands of amperes are required. It is therefore proposed to use an accelerating system of the “transformer” type, i.e. to use the induction method of acceleration, but not over the whole turn, as in a betatron, but in a narrow accelerating gap. One of the rectilinear sections of the chamber is surrounded by a ring of laminated or powdered ferromagnetic material, the weight of which reaches several tons. The winding of this ring core is the primary, and the beam itself serves as the secondary winding. The latter is accelerated by the vortex electric field produced in the chamber inside the ring-shaped magnet. In such a system it proves considerably more convenient to vary the frequency of the electric field over a large interval. The energy increment per turn should be \(\Delta W = 5.5\) keV. The acceleration time is about \(1\) sec. Methods of synchronizing the accelerating electric field and the controlling magnetic field are to be tested on an electron model \(1\) m in diameter.

According to some reports\(^{12}\), a bevatron for accelerating protons to an energy of \(3\) BeV is also being designed at the Brookhaven Laboratory. Only the following data have been published about this project:

\[ R_0 = 915\ \text{cm}, \qquad W_i = 3\ \text{MeV}, \qquad \Delta W = 1.15\ \text{KeV}, \qquad A = 8\ \text{cm}, \]

where \(A\) is the half-height of the vacuum chamber.

Project for a Bevatron at the University of Berkeley

The problems considered in this project, to a considerable extent, concern not only this particular installation, but are common to all large installations designed to obtain particles with energies in the hundreds and thousands of MeV.

One of the chief difficulties in designing a bevatron (of the synchrophasotron type) is the need to modulate the frequency of the accelerating electric field many times in accordance with the change in the proton velocity [see (7)]. Therefore, according to (8), it is desirable to begin the synchrophasotron regime at as large a value of the initial velocity as possible. In estimating the value of the energy at which the synchrophasotron regime should be begun, the author of the project proceeds from the fact that at present frequency modulation by a factor of two has been achieved in the phasotron and that modulation by a factor of three should soon be achieved\(^{13}\). It is therefore proposed first to accelerate the protons in the betatron regime to a velocity equal to one third of the speed of light, and then, having brought them to the final energy, to continue the acceleration in the synchrophasotron regime. In this case, as will be seen below, this circumstance (the presence of an initial betatron regime) will not require an additional increase in the quantity of iron, thanks to the selected form of the transverse section of the ring magnet (see Fig. 5).

When injecting particles at the beginning of the betatron regime, it is also necessary, if possible, to use high-voltage injection in order to reduce losses associated with scattering of particles and beam divergence due to Coulomb repulsion, and also to avoid the influence of the azimuthal asymmetry of the magnetic field.

By azimuthal asymmetry of the magnetic field is meant a deviation of the configuration of the controlling magnetic field \(H_z\) from axial symmetry, i.e. the presence of a dependence of \(H_z\) on the azimuth \(\theta\). If \(H_z(\theta)\) is represented in the form of a Fourier series:

\[ H=H(R)\left[1+\sum_{l=1}^{\infty} h_l \cos(l\theta+\alpha_l)\right], \tag{9} \]

then, according to Bohm and Foldy\(^2\), the correction \(\delta R\) to the radius of the instantaneous orbit \(R_i\), introduced by the azimuthal asymmetry \(\delta H\), is equal to:

\[ \delta R=R_i\sum_{l=1}^{\infty}\frac{h_l}{l^2+n-1}\cos(l\theta+\alpha_l), \tag{10} \]

where \(n\) is the exponent of decrease of the magnetic field:

\[ n=-\frac{\partial \ln H_z}{\partial \ln R}. \tag{11} \]

As is seen from (10), the circular orbit is distorted, and the lower harmonics (small values of \(l\)) have the greatest importance. Therefore, approximately one may take that

\[ \frac{\delta R}{R_i}\simeq \frac{1}{n}\frac{\delta H}{H}. \tag{12} \]

The azimuthal asymmetry in the present case is dangerous because of the comparatively small value of \(\frac{\Delta R}{R_0}\)—the ratio of the width of the working region to the radius of the equilibrium orbit—which, according to (12), imposes rather stringent requirements on the magnitude of the azimuthal variation of the magnetic field \(\frac{\delta H}{H}\). Injection is proposed to be carried out with the aid of a Van de Graaff generator at an energy of 4 MeV, which corresponds to a sufficiently large initial value of the magnetic-field strength on the orbit, 120 gauss. The successive stages of the acceleration process are presented in Table I.

Table I

In order to bring the proton beam from the Van de Graaff generator tangentially to the orbit, it is proposed to use an electric deflector with a radius of the deflecting plates equal to \(4.5\) m. In this case the electric field between the plates must be equal to \(18\ \frac{\mathrm{kV}}{\mathrm{cm}}\). The effective width of the “injector,” which presents a danger of particle losses due to collisions, is determined in this case by the thickness of the inner deflecting plate. At the end of the acceleration cycle the proton beam must be displaced onto the outer or inner target, either by disrupting the resonance relation between the frequency of the accelerating field and the magnitude of the magnetic-field strength, or by applying a pulsed deflecting electric field. The resulting beam from the target is then to be used for various investigations.

The general layout of the accelerator is shown in Fig. 4. As is evident, this is a “racetrack”-type accelerator, composed of four arc sections,

connected by four small rectilinear intervals. The radius of the equilibrium orbit \(R_0\) is determined by the final energy \(E_m\) and the maximum attainable value of the magnetic-field intensity \(H_m\), according to the formula \(E \simeq 300\,HR\) (in electron-volts), and is \(24.4\) m. The length of each of the rectilinear sections is \(6.1\) m. According to the theory of the “racetrack,”\({}^{14}\) with the chosen value of the magnetic-field fall-off index \(n = 0.7\), such rectilinear intervals disturb the stable motion of the particles very little. Injection is to take place in two rectilinear intervals.

Fig. 4. Diagram of the bevatron being designed at the University of California: \(A\)—chamber; \(B\)—magnet; \(G\)—motor-generators; \(D\)—Van de Graaff generator; \(N\)—vacuum pumps; \(M\)—target; \(R\)—distributing device, transformers and rectifier; \(U\)—accelerating section.

The use of two injectors—two Van de Graaff generators—should increase the intensity of the beam of accelerated protons and ensure reliability of operation. In the third rectilinear interval an accelerating device is placed. Finally, the fourth, last, is intended to facilitate the release of the products obtained in the collision of the accelerated proton beam with the target. In addition, vacuum pumps are connected to each rectilinear interval (see Fig. 4). The weight of the iron required for the magnet is 13,000 tons.

The transverse section of the magnet is shown in Fig. 5. The most important here are the dimensions of the transverse section (height and width) of the air gap. These dimensions determine to a considerable degree the cost of the entire installation as a whole, since the working volume in which the required magnetic field must be created depends on them.

The choice of the gap height is determined by the following factors: a) the angular divergence of the injected beam, b) displacement of the plane of symmetry of the magnetic field, c) the thickness of the chamber walls and the magnitude of the permissible gap between the walls and the magnet, d) scattering of ions on the residual gas.

The injected beam can be focused to such an extent that the maximum amplitude of the vertical oscillations will not exceed 7.5 cm. For the walls of the chamber and the clearance it is sufficient to allow 5 cm. The vertical displacement of the median plane can be corrected by means of auxiliary coils in the straight sections. An essential factor affecting the choice of the gap height is, apparently, scattering by the residual gas. At a pressure of \(10^{-5}\) mm Hg, a gap height available to the motion of the protons equal to 15 cm, and an energy increment per revolution \(\Delta W = 6300\) eV, about 10% of the injected particles will reach the maximum energy (if other factors besides scattering are not taken into account). The height of the air gap, from the above considerations, is chosen equal to 20 cm.

In the general case, the fraction represented by the number of ions that have reached the maximum energy (if only scattering is taken into account), out of the number of injected ions, is approximately proportional to the quantity

\[ \exp \left[-\frac{K R_0 N Z^2}{A^2 \Delta W W_i}\right], \tag{13} \]

where \(W_i\) is the kinetic energy of injection;

\(Z\) and \(N\) are the atomic number and the number of atoms per unit volume of the residual gas,

\(\Delta W\) is the increment of the ion energy per revolution,

\(R_0\) is the radius,

\(K\) is a constant,

\(2A\) is the vertical dimension of the vacuum chamber, which is assumed to be much smaller than the horizontal dimension*). Owing to the exponential factor, a twofold change in pressure may increase the particle losses by a factor of five, which indicates the sensitivity of the magnitude of the output current to the value of the pressure in the chamber. It should be noted, however, that the energy losses due to incomplete vacuum are very small: thus, at a pressure of \(10^{-5}\) mm Hg, the energy losses of a proton with an energy of 4 MeV amount to 25 eV per revolution, and at 10 BeV the losses fall to a value of less than 1 eV per revolution.

Fig. 5. Cross section of the ring magnet of a betatron: \(A\) — main winding; \(Б\) — vacuum chamber; \(В\) — forvacuum region; \(Г\) — auxiliary (betatron) winding.

The choice of the radial dimension (the width of the working region) presents a more difficult problem than the choice of the vertical dimension, and is determined by a large number of factors. These factors are: a) the angular divergence of the injected beam, b) radial oscillations and compression of the orbit, determining the “transparency” of the injector (i.e., the possibility of passing the injector during the acceleration process), c) displacement of the orbit caused by violation of the 2:1 condition in betatron acceleration, d) radial-phase oscillations during the synchrophasotron regime and, in particular, during the transition regime (in the transition from betatron acceleration), e) azimuthal asymmetry of the magnetic field, f) inaccuracy in satisfying the resonance relation (7) between the frequency of the accelerating field and the magnetic-field intensity during the synchrophasotron regime, g) spatial inhomogeneities of the magnet, h) scattering by the residual gas in the vacuum chamber.

* On the influence of particle scattering by the residual gas in the synchrophasotron chamber, see the detailed work of Blachman and Courant[^12].

FROM CURRENT LITERATURE

The change in the orbit radius associated with the relative error \(a\) in the betatron condition 2:1 is equal to\(^5\)

\[ \frac{a}{1-n}\left(1-\frac{H_i}{H}\right)R_0, \]

where \(H_i\) is the magnetic field at injection, and \(n=0.7\) is the index of decrease of the magnetic field. For \(a=0.1\%\) this change in radius during betatron acceleration may reach \(6.25\ \text{cm}\).

The change in radius associated with the relative inaccuracy \(f\) in satisfying the synchrophasotron condition (7) is equal to

\[ -\frac{f}{n+\beta^2(1-n)}R_0. \]

For \(f=0.1\%\) this change may reach a maximum of \(3.5\ \text{cm}\). The change in radius due to radial-phase oscillations is equal to\(^2\)

\[ \left[ \frac{\operatorname{tg}\varphi_0}{4\pi n(1-n)} \frac{\Delta W}{qW} \right]^{1/2} R_0\Delta\varphi, \]

where \(W\) is the kinetic energy of the ion, \(\Delta\varphi\) is the amplitude of the phase oscillations, \(\varphi_0\) is the value of the equilibrium phase (the phasing point), and \(q\) is the number of periods of the accelerating field during one revolution of the particle (the resonance multiplicity). This formula is characterized by the presence of the factor \(q^{-1/2}\). The appearance of this factor is connected with the fact that in the given accelerator the frequency of the alternating accelerating electric field exceeds the particle revolution frequency by a factor of \(q\) (\(q=6\)). At the beginning of synchrophasotron acceleration \(\Delta\varphi\) may reach 1 radian. With \(\Delta W=6300\ \text{eV}\) and \(q=6\), this gives, for the maximum deviation of the radius from \(R_0\), the value \(7.5\ \text{cm}\). Azimuthal asymmetry, if it reaches any appreciable magnitude at the chosen injection energy, may be corrected by shimming or by the corresponding construction of the current lead in separate quadrants of the “racetrack.” Since the width of the gap is several times greater than its height, losses due to scattering are in this case (with vessels having vertical walls) small.

An important and complex question is the problem of the “transparency” of the injector during acceleration, i.e., the problem of particles passing through the injector during acceleration. As Widerøe and Touschek have shown, the “transparency” of the injector depends on the rate of contraction of the ion orbit during injection. The amount by which the orbit radius decreases in one revolution is equal to

\[ \frac{\Delta W}{2W_i}X. \]

Even for \(X=1\ \text{m}\), the amount of contraction is only \(0.075\ \text{cm}\). With so small a contraction, the fraction of captured ions is estimated at \(1\%\). It is possible, however, to increase the capture efficiency if the value of the ratio \(\overline{H}/H\) is changed in the corresponding manner, where \(\overline{H}\) is the mean value of the magnetic field inside the orbit, and \(H\) is the value of the magnetic field on the orbit. For example, if the value of \(\overline{H}\) is kept constant during the injection period, then the orbit radius will decrease by \(2.5\ \text{cm}\) in one revolution.

On the basis of all the factors enumerated, for the radial size of the working region a value of \(120\ \text{cm}\) is chosen, which amounts to about \(5\%\) of the equilibrium radius. We note that, in order to ensure the required value \(n=0.7\), the height of the gap over a length of \(120\ \text{cm}\) must change by only \(0.7\ \text{cm}\).

At larger values of the magnetic-field intensity, the width of the working region may be chosen smaller; this is connected with the fact that the ampli-

from which the free oscillations decay as \(H^{-1/2}\), while the amplitude of the phase oscillations decreases in the course of acceleration in proportion to \(W^{-1/2} \div W^{-3/4}\), where \(W\) is the kinetic energy (the meanings are given for the nonrelativistic and relativistic cases, respectively).

In accordance with this, for large values of the magnetic-field intensity it is proposed to reduce the volume in which the magnetic field is created, by saturating special pointed edges of the pole pieces (see Fig. 5). In this way a considerable reduction in the magnitude of the required magnetic energy can be achieved. The stored magnetic energy can be calculated from the formula

\[ M=\frac{10}{4\pi}\,10^{-8}\frac{H_m l\Phi}{\eta}, \]

where \(H_m\) is the maximum field in the gap, \(l\) is the length of the gap, \(\Phi\) is the magnetic flux through the windings, and \(\eta\) is the fraction of the number of ampere-turns that create the field in the gap, out of the total number of ampere-turns. Taking the field intensity in the iron to be 17,000 gauss, and in the gap 15,000 gauss, and \(\eta=0.8\), we obtain for \(M\) a value of 48 megajoules. The most important parameter in the problem of feeding a high-power magnet is the interval of time during which the magnetic field increases from zero to its maximum value. As this interval of time increases, the cost of the feeding devices decreases. However, too great an increase in the acceleration time is limited by the following factors: a) loss of energy in the residual gas, b) scattering in the residual gas, c) divergence of the ion beam due to space charge, d) the need to pass the injector. These factors were discussed above, with the exception of the effect of space charge. The limiting value of the current in the beam associated with the space charge turns out to be about \(10^{-8}\) ampere, i.e. much higher than may be expected from other considerations. Therefore the influence of space charge is insignificant.

The acceleration time, or, what is the same thing, the rise time of the magnetic field, is chosen equal to 1 sec.

Another important question in magnet feeding is the method of creating a huge store of energy. Generally speaking, this energy can be stored in capacitors, storage batteries, or rotating flywheels. The first two methods prove disadvantageous in the present case because of the bulkiness and complexity of the equipment. It is proposed to use three-phase 60-cycle alternating-current generators connected to the magnet through ignitrons, which serve as rectifier-inverters. In order to deliver \(48\cdot 10^6\) joules of energy to the magnetic circuit in 1 sec., generators of 60,000 kilowatts are required. To reduce the reaction force on the ground it is desirable to use two machines of 30,000 kilowatts each, the rotations of which should occur in opposite directions.

With the aid of a rectifier-inverter, the 60-cycle alternating current is converted into a unidirectional current that rises to its maximum value in 1 sec. and falls to zero in the same time. During the rise of the current the energy enters the magnetic circuit from the generators. After the maximum value of the current has been reached, the process begins to proceed in the opposite direction and the energy begins to be returned back to the generators, which now operate as electric motors, rotating the heavy flywheels. Suitable for rectifier-inverter action are pentode ignitrons GL 506, operating at a maximum current value of 900 amperes with a reverse-voltage value of 20,000 volts. With the value of the voltage in the magnetic circuit equal to 20,000 volts, the amplitude value of the current must reach 4,800 amperes, for which 36 ignitrons are required in a three-phase two-half-period rectifier. With the described method of feeding the magnetic circuit, the current in it will rise and fall linearly with time.

From Current Literature

At a rotor speed in the generator equal to 1800 rpm, in order to maintain the speed within 10% of this value, a flywheel 4.5 m in diameter and a dynamo also 4.5 m long and weighing 30 tons are required.

Since the magnet requires 30,000 ampere-turns, and the amplitude value of the current is 4800 amperes, the number of turns of the main winding must be 62. Taking the current density in the conductor as 160 amperes/cm², we obtain a conductor cross-sectional area equal to 1.7 cm² for the effective value of the current

\[ \frac{1}{\sqrt{3}}\,4800 = 2760 \text{ amperes.} \]

The weight of copper in the windings turns out to be 325 tons, and the active losses \((I^2R)\) under continuous operation reach 1360 kilowatts. The required motor power may be estimated from the active losses in the winding, the losses in the iron, and the losses in the generator. For the losses in the iron the requirement is imposed that they not exceed 1% of the energy stored per cycle, or 480,000 joules every two seconds, i.e., 240 kilowatts. The losses in the generators must amount to about 3% of the reactive power, i.e., 1800 kilowatts. Consequently, the motor power must be equal to 3400 kilowatts, or 5000 horsepower. The motors must be asynchronous, with lagging rotation of the rotor (large relative slip), so as to give an approximately constant value of the power when the speed of rotation varies within 10%.

The power requirements can be reduced by operating with a lower pulse repetition rate. For example, if instead of thirty pulses per minute one takes three pulses, then the required power of the motors and generators can be reduced by a factor of \(\sqrt{10}\). However, this saving apparently will not justify itself, since the cost of the power-supply equipment constitutes only a small fraction of the total cost of the installation; moreover, in order to benefit from operation at a low pulse repetition rate, the generators must be designed for large instantaneous overload values of capacitance.

The change in the central magnetic flux necessary for realization of the betatron regime is achieved by means of an additional winding around the inner or outer vertical yoke (see Fig. 5). Control of the central flux is most conveniently performed by maintaining the appropriate ratio between the voltage values on the main and auxiliary (betatron) windings. This can be achieved by feeding the auxiliary winding from a rectifier connected to the main generator through a transformer with adjustable voltage (variac).

The cross-sectional area of the inner yoke is 4.5% of the area enclosed by the orbit. Therefore, while the field at the orbit changes from 120 to 450 gauss (this corresponds to acceleration in the betatron regime—see Table I), the mean intensity of the magnetic field (magnetic flux density) in the inner vertical yoke, according to the betatron condition 2:1, must change by the amount

\[ \frac{2(450-120)}{0.045}=14\,000 \text{ gauss.} \]

The magnetic flux density at injection is taken as equal to \(-7000\) gauss in a direction opposite to the direction of the magnetic flux in the air gap. During the betatron regime the flux in the inner yoke changes its direction to the opposite one and again reaches a density of 7000 gauss, so that the change in flux density turns out to be equal to the required value of 14,000 gauss (see Fig. 6). This value corresponds to the change in magnetic flux density in pole tips—

nickel, equal to 500 gauss, and the change in the field intensity in the gap, equal to 330 gauss.

To create a magnetic-flux density in the inner core of 7000 gauss, 2400 ampere-turns are required in the auxiliary winding, or 600 amperes with 4 turns. The duration of the betatron regime is

\[ \frac{450-120}{15\,000}=0.022\ \text{sec}, \]

which corresponds to a frequency of 25 cycles. At such a frequency the losses due to Foucault currents in the transformer iron (plates of thickness 0.15 cm) reach 0.45 W/kg, or 4800 kW in continuous operation. During one half-cycle these losses are equal to

\[ \frac{4800}{50}=96\ \text{kilojoules}. \]

With a voltage on the auxiliary winding of 20,000 volts, to replenish these losses a current amplitude of 480 amperes is required. Thus, the total current through the auxiliary winding must have an amplitude of \(600+480=1100\) amperes.

At the beginning of the synchrophasotron regime the flux through the inner core again begins to decrease and by the end of the acceleration reaches a maximum in the opposite direction, at a magnetic-flux density of \(-15000\) gauss (see Fig. 6). For this, a voltage of the order of 500 volts must be applied to the auxiliary winding in the direction opposite to that applied in the betatron regime, i.e., creating a field that is somewhat retarding. However, owing to the slowness of the decrease in magnetic-flux density, this effect is weak and can easily be compensated by increasing the amplitude of the accelerating synchrophasotron voltage by several percent.

Fig. 6. Graph of the change in magnetic-field intensity in the air gap and of magnetic-flux density in the inner vertical core of the betatron magnet: \(A\)—betatron regime 0.02 sec; \(Б\)—synchrophasotron regime; \(В\)—magnetic-field intensity in the air gap; \(Г\)—magnetic-flux density in the inner core.

Further, in the next half-cycle (idle with respect to acceleration), a voltage of the order of 200 volts must be applied to the auxiliary winding in order to bring the magnetic-flux density in the core to the value \(-7000\) gauss, which is the initial value for the next cycle (see Fig. 6). It is obvious that a decrease in the magnetic-flux density in the inner core from \(+7000\) to \(-15000\) gauss (during the synchrophasotron regime) will produce an inverse betatron action, i.e., create a somewhat retarding field. However, this effect is weak owing to the slowness of the decrease in magnetic-flux density and can easily be compensated by increasing the amplitude of the accelerating synchrophasotron voltage by several percent.

Fig. 7. Diagram of the accelerating system of the betatron, placed in one of the straight sections.

The accelerating system is placed in one of the straight sections, as shown in Fig. 7, and consists of a resonator 3.66 m long, extended along the section. Modulation of the frequency of the electric field in the resonator is produced by changing the capacitance with the aid of rotating capacitors, also shown in Fig. 7. With the ratio of the maximum and minimum capacitance in the rotating capacitors equal to

16, the frequency may vary from 3 to 12 Mc/s. Since in the process of acceleration the period of revolution of the ions changes from 1.8 μsec to 0.6 μsec, then, when operating on the sixth harmonic (\(q=6\)), the frequency interval used will be from 3.3 to 10 Mc/s.

The magnitude of the required increase of energy per revolution can be found from the expression

\[ \Delta W = e 2\pi R_0^2 \frac{dH}{dt} 10^{-8}, \tag{14} \]

where \(\frac{dH}{dt} = 15\,000\) gauss per sec. This gives the value \(\Delta W = 6300\) eV. The equilibrium phase \(\varphi_0\) is chosen approximately equal to \(60^\circ\), or, in other words, the amplitude of the accelerating voltage is \(12\,000\) volts. This is close to the known operating conditions of the existing phasotron, whence the required power of the generator may be estimated as being of the order of 100 kilowatts. The relation (7) between \(\omega_\lambda\) and \(H\), which must be satisfied with an accuracy up to 0.1%, can be automatically maintained in the following way. On the rotating condenser a “comb” is fastened, which drives a rheostat or some other device, permitting one to obtain a voltage proportional to the required magnetic field. This voltage is compensated by a voltage taken from a coil rotating in a magnetic field. The difference of the voltages is supplied to a control device in order to produce the corresponding small change in the frequency of the resonator. It is proposed to give the plates of the rotating condenser such a form that the change of frequency should take place according to a law as close as possible to (7), so that the required frequency correction will in fact be small.

The vacuum chamber is assembled from porcelain sections, each 30 cm wide, having a cross section of \(120 \times 20\) cm and a wall thickness of 1.875 cm (\(3/4\) inch).

The joints between the separate sections are sealed with rubber gaskets, which are covered with thin metallic plates. The vacuum pumps are arranged in pairs on each rectilinear section (see Fig. 4). Two rings of plastic, with a diameter of the order of the diameter of the orbit, fastened with pointed edges to pole tips, close the space about the vacuum chamber (these rings in cross section are shown in Fig. 5). The volume bounded by these rings and the pole tips is pumped out in order to reduce the pressure on the vacuum chamber. This volume may contain hydrogen or helium at low pressure. Thus, the gas penetrating into the chamber will have a small atomic number, as a result of which the losses due to scattering will be reduced.

The expected current of accelerated ions at the exit of the accelerator may be estimated approximately in the following way. The time interval \(\Delta t\), during which the instantaneous orbits at injection are within the limits of the working region, is determined by the formula 17

\[ \Delta t = (1-n)\frac{H_i}{\dot H_i}\frac{\Delta R}{R_0}, \tag{15} \]

where \(H_i\) is the value of the magnetic field in the injection period, \(\Delta R\) is the width of the working region. In the present case

\[ H_i = 120 \text{ gauss}, \quad \frac{\Delta R}{R_0} = 5\%, \quad \dot H_i = 15\,000 \text{ gauss/sec}, \quad n = 0.7. \]

Hence the maximum capture time is \(\Delta t = 120\) μsec. Assuming that each of the two Van de Graaff generators gives a current of 50 microamperes and

Taking into account that capture occurs every two seconds, and that for various reasons only 2% of the maximum possible value is captured during injection, we obtain an average value of the current captured as a result of injection equal to \(1.2 \cdot 10^{-10}\) amperes. If, furthermore, one takes into account that, owing to scattering, 10% of this amount reaches the end of the acceleration, then the average value of the current at the output is of the order of \(10^{-11}\) amperes. This average current is sufficient for experimental purposes, especially if one takes into account that it consists of separate pulses with much larger instantaneous values of the current.

The project also gives some data on the costs connected with the construction of the bevatron, which may be of some interest. The cost of the magnet is estimated at 10 million dollars, the cost of the devices feeding the magnet with power at 1 million dollars. The total cost of the installation is estimated at 15–20 million dollars.

In conclusion we present a table containing the values of the principal parameters of the projected installation—a synchrophasotron for obtaining protons with an energy of \(10^{10}\) eV.

Table II

K. Andreev

CITED LITERATURE

1. V. I. Veksler, DAN SSSR 43, 316 (1944); DAN SSSR 44, 393, (1944); Journ. Phys. USSR 9, 153 (1945).
2. D. Bohm and L. Foldy, Phys. Rev. 70, 249 (1946).
3. M. L. Oliphant, J. S. Gooden and G. S. Hibe, Proc. Phys. Soc. 59, 666 (1947).
4. M. S. Livingston, Phys. Rev. 73, 1258 (1948).
5. H. Feshbach and L. Schiff, Phys. Rev. 72, 254 (1947).

1. J. C. Slater, UFN 37 (1949), 38 (1949).
2. J. P. Blewett, Phys. Rev. 73, 1258 (1948).
3. G. K. Green, Phys. Rev. 73, 1258 (1948).
4. W. H. Moore and J. P. Blewett, Phys. Rev. 73, 1258 (1948).
5. W. M. Brobeck, Rev. Sci. Instr. 19, 545 (1948).
6. F. R. Elder, A. M. Gurewitsch, R. V. Langmuir and H. C. Pollock, Journ. Appl. Phys. 18, 810 (1947).
7. N. M. Blachman and E. D. Courant, Phys. Rev. 74, 140 (1948).
8. K. R. McKenzie, Phys. Rev. 73, 540 (1948); 74, 104 (1948).
9. D. M. Dennison and T. H. Berlin, Phys. Rev. 70, 764 (1946).
10. E. Amaldi and B. Ferretti, Rev. Sci. Instr. 17, 389 (1946).
11. R. Wideroe, Journ. Appl. Phys. 18, 783 (1947).
12. D. W. Kerst and R. Serber, Phys. Rev. 60, 53 (1942).