Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 184　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1970

On the Question of Constructing a Mathematical Model of Particle Capture into Acceleration in a Weak-Focusing Synchrotron

V. V. Tsygankov

(Presented by the scientific seminar of the Nuclear Physics Research Institute)

The solution of the equation of motion of particles in an accelerator can be represented in the form:

\[ x = x_{\mathrm{sv}} + x_{\mathrm{voz}}, \]

where \(x_{\mathrm{sv}}\) denotes free oscillations determined by the type of system and by the initial conditions, and \(x_{\mathrm{voz}}\) denotes the perturbation of the free motion, depending on the magnitude and form of the right-hand side of the inhomogeneous equation. In weak-focusing accelerators \(x_{\mathrm{sv}}\) predominates, since the chamber dimensions are large, whereas in strong-focusing accelerators \(x_{\mathrm{voz}}\) is comparable with \(x_{\mathrm{sv}}\). It is known that automatic-control systems compensate \(x_{\mathrm{voz}}\); therefore the use of systems for automatic compensation of perturbations operating during the accelerator cycle is especially effective in strong-focusing accelerators at ultrahigh energies [2]. An example of the effective application of systems for automatic compensation of perturbations of the radial-phase oscillations of the bunch center in a weak-focusing synchrophasotron at an energy of 10 GeV at JINR, Dubna [3], is explained by the need for recapture into the short-acceleration regime and by the presence of harmonics of the magnetic field whose frequency lies in the region of synchrotron frequencies. Thus, in a weak-focusing accelerator, a system of automatic optimization of the capture condition may be effective, since at the injection level the characteristics of the magnetic field, as well as of the beam of injected particles, do not remain constant from one acceleration cycle to another. The scheme for investigating systems of automatic optimization is as follows: on the basis of experimental or theoretical studies a mathematical model of the object is constructed and, depending on the complexity of the model, synthesis of the system is carried out by methods of optimal-control theory or with the use of computing equipment [4]. Two limiting cases may be indicated: 1 — information about the physical processes is so insignificant that the experimenter is forced, within a limited range of variation of the variables, to represent the dependence between the output quantity and the inputs in the form of a Chebyshev polynomial, whose coefficients are determined from passive-experiment data; 2 — there exists a theoretical model based on exact knowledge of the physical process. In the first case, information about the functional form reduces to zero [5, 6, 7]; in the second case it is known reliably. In practice, intermediate cases are encountered most often. As a rule, the theory is insufficiently developed;

but by carrying out additional investigations the researcher expects, by means of an iterative method, to construct a model in which the principal theoretical aspects of the problem are taken into account. In practice, an excessively cautious approach to the construction of a model predominates. It is believed that until the functional form is known with a high degree of accuracy, the theoretical approach is unjustified. This leads in many cases to the predominance of purely empirical investigations, whereas a semi-theoretical approach, in which the functional form is constructed by means of an iterative process, may prove more effective and productive [5].

The present work is devoted to certain questions in constructing a mathematical model of the process of particle capture into acceleration for synchrotrons, on the basis of a study of the physics of this phenomenon.

Multi-turn injection of particles into a synchrotron is considered. The following assumptions are made: \(n\)—the exponent of the radial decrease of the magnetic field—is constant in azimuth and radius of the accelerator; collective interactions of the particles during injection are not taken into account.

The solution of the equations of betatron oscillations for accelerators of the racetrack type at the azimuth of the inflector \(\sigma\) has the form [1]:

\[ x_k = F_c(\sigma)\cdot \cos(4\mu k+\alpha(\sigma)), \tag{1.1} \]

where

\[ F_c(\sigma)=2|D\cdot \sin\sigma+dD\cdot \cos\sigma|, \]

\[ \alpha(\sigma)=\arg(D\cdot \sin\sigma+dD\cdot \cos\sigma), \tag{1.2} \]

\[ D=\frac{\dfrac{R\gamma}{\chi}\,d^* - x_0}{d^*-d},\qquad d=\frac{c-e^{i\mu}}{s},\qquad d^*=\frac{c-e^{-i\mu}}{s},\qquad c=\cos\frac{\pi\chi}{2}, \]

\[ s=\sin\frac{\pi}{2}\chi,\qquad \mu=\arccos(c-ps),\qquad p=\frac{l\chi}{2R},\qquad \sigma=\frac{S}{R},\qquad \chi=\sqrt{1-n}, \]

\(S\) is the distance from the edge of the sector to the inflector in the direction of particle motion; \(R\) is the radius of the equilibrium orbit; \(l\) is the length of the straight section; \(\gamma\) is the angle between the direction of departure of the particle and the tangent to the equilibrium orbit at the place where the inflector is installed. Taking (1.2) into account, equation (1.1) can be represented in the form:

\[ x_k=x_0\cdot B(k)+\frac{\gamma R}{\chi}A(k), \tag{1.3} \]

where

\[ A(k)=a\cdot \cos 4\mu k+\frac{s(b+pa)+2pbc}{\sin\mu}\cdot \sin 4\mu k, \]

\[ B(k)=b\cdot \cos 4\mu k-\frac{s(a+pb)}{\sin\mu}\cdot \sin 4\mu k, \tag{1.4} \]

\[ a=\sin\sigma,\qquad b=\cos\sigma, \]

\(x_0\) is the distance from the point of departure of the particle to the instantaneous orbit, \(x_k\) is the value of \(x\) on the \(k\)-th revolution \((k=1,2,3,\ldots)\). In expression (1.3), \(A(k)\) and \(B(k)\) are analogous to the effective “sine” and “cosine” [8], but do not coincide with them, since in (1.3) \(x_0\) and \(\dfrac{\gamma R}{\chi}\) are separated. For \(p=0\), \(A(k)=\sin 4\mu k\), \(B(k)=\cos 4\mu k\), and \(\sigma\) should then be considered equal to zero. For \(p\ne 0\) and \(\sigma\ne 0\) we obtain an expression for \(x_k\) coinciding with \(x_k\) [9].

We shall henceforth consider collisions of particles only with the inflector; taking account of collisions with the vertical walls will affect only the absolute value of the captured charge. The condition for particles to bypass the inflector has the form:

\[ (g+u)\cdot B(\kappa)+\frac{R}{\kappa}\cdot A(\kappa)\leq \Delta R_{\mathrm{ob}}\cdot \kappa+u, \tag{1.5} \]

where \(u\) is the distance from the inner plate of the inflector to the instantaneous orbit, \(g\) is the distance from the inner plate to the point of departure of the particle from the inflector (the thickness of the plate is neglected), \(\Delta R_{\mathrm{ob}}=\left|\frac{dR}{dt}\right|\cdot T\) is the contraction of the instantaneous orbit per revolution, \(\left|\frac{dR}{dt}\right|\) is the rate of contraction of the instantaneous orbit, and \(T\) is the revolution period. From (1.5) it follows:

\[ 1)\qquad g_1(\kappa)\leq \frac{1-B(\kappa)}{B(\kappa)}\cdot u -\frac{A(\kappa)}{B(\kappa)}\cdot \frac{R}{\kappa}\cdot \gamma +\frac{\kappa}{B(\kappa)}\cdot \Delta R_{\mathrm{ob}}, \tag{1.6} \]

if \(B(\kappa)>0\).

\[ 2)\qquad g_2(\kappa)\geq \frac{1-B(\kappa)}{B(\kappa)}\cdot u -\frac{A(\kappa)}{B(\kappa)}\cdot \frac{R}{\kappa}\cdot \gamma +\frac{\kappa}{B(\kappa)}\cdot \Delta R_{\mathrm{ob}}, \tag{1.7} \]

if \(B(\kappa)<0\).

In (1.6) and (1.7), for fixed values of \(\Delta R_{\mathrm{ob}}, \gamma, u\), among \(\kappa=1,2,3,\ldots\), let us find such a value \(\kappa=\kappa'\) for which \(g_1(\kappa')\) is minimal, and introduce the notation \(g_1(\kappa')=g_{\min}(u)\), and such a value \(\kappa=\kappa''\) for which \(g_2(\kappa'')\) is maximal. Then the region of the inflector from which, on leaving it, particles will not collide with the inflector is equal to \((g_2(\kappa''),\,g_1(\kappa'))\), with allowance for the inequality

\[ 0\leq g_2(\kappa'')\leq g_1(\kappa')\leq g_{\max}, \tag{1.8} \]

where \(g_{\max}\) is the width of the inflector.

There are no analytical methods for investigating the function \(g(\kappa)\) of the discrete argument \(\kappa\); therefore the operation of finding \(\kappa'\) and \(\kappa''\) has to be carried out numerically.

Let us consider injection of a parallel beam of particles (\(\gamma=0\)); then case (1.7) drops out, since for \(B(\kappa)<0\) inequality (1.5) is always satisfied, and from (1.6) we obtain:

\[ g_{\min}(u)=g_1(\kappa')= \frac{1-B(\kappa')}{B(\kappa')}\cdot u +\frac{\kappa'}{B(\kappa')}\cdot \Delta R_{\mathrm{ob}}. \tag{1.9} \]

Suppose that the instantaneous orbit was at a distance \(u\) from the inner plate of the inflector when a charge \(I(t)\,dt\) was injected into the accelerator chamber, where \(I(t)\) is the inflector current. The charge captured into the orbit \(u\) is equal to:

\[ dQ=q(u)\,du=\int_{0}^{g_{\min}(u)} q(g)\,dg, \tag{1.10} \]

where \(g_{\min}(u)\) is determined by expression (1.9), and \(q(g)\) is the distribution function of the injected charge over the width of the inflector. In the case of a uniform distribution of the injected charge with respect to \(g\): \(q(g)=\frac{I(t)\,dt}{g_{\max}}\). Instead of the variable \(t\), introduce \(u\): \(dt=\)

\[ =\frac{du}{\left|\dfrac{dR}{dt}\right|}, \]

then from (1.10) it follows that the charge distribution function

Table 1

\[ \Delta R_{\mathrm{ob}}\ [\mathrm{cm}] \]

Table 2

\[ \Delta R_{\mathrm{ob}}[\mathrm{cm}] \]

with respect to the coordinate \(u\) or to the amplitudes of betatron oscillations (since \(\gamma=0\)) is equal to:

\[ q(u)=\frac{I(u)\cdot g_{\min}(u)}{\left|\dfrac{dR}{dt}\right|\cdot g_{\max}}, \]

where \(I(u)\) is the injector current as a function of the duration of the injection pulse. It follows from (1.2) that for \(I(u)=I=\mathrm{const}\) the distribution function of the particles with respect to the coordinate \(u\) after the first stage of injection, which

Fig. 1. Dependence of the charge captured at the first stage of injection for a parallel beam of particles on \(\Delta R_{\mathrm{ob}}\) and \(n\) for the TPI 1.5 GeV synchrotron.

lasts until the rf field is switched on, coincides, up to a constant, with \(g_{\min}(u)\). The charge captured at the 1st stage of injection is equal to:

\[ Q_1=\int_{0}^{u_{\max}}\int_{0}^{g_{\min}(u)} q(g)\,dg\,du = \frac{I}{\left|\dfrac{dR}{dt}\right|\cdot g_{\max}} \times \]

\[ \times \left(\frac{1-B(\kappa')}{B(\kappa')}\cdot \frac{u_{\max}^{2}}{2} +\frac{\Delta R_{\mathrm{ob}}\cdot \kappa'}{B(\kappa')}\cdot u_{\max}\right), \tag{1.12} \]

where \(u_{\max}\) is the half-width of the working region of the magnetic field in radius.

Using this procedure, a calculation was made of the values of \(\kappa^1\) (Tables 1, 2) and \(Q_1\) (Figs. 1, 2) as functions of \(\Delta R_{\mathrm{ob}}\) and \(n\) for the case of injection of a parallel particle beam into the 1.5 GeV synchrotron of the Tomsk Polytechnic Institute and into the 10 GeV synchrophasotron of JINR, Dubna. From Figs. 1, 2 and Tables 1, 2 it is seen that, in the region of small values of \(\Delta R_{\mathrm{ob}}\), the charge captured at the first stage of injection depends substantially on the index

Fig. 2. Dependence of the charge captured at the first stage of injection for a parallel beam on \(\Delta R_{\mathrm{ob}}\) and \(n\) for the 10 GeV synchrophasotron of JINR, Dubna

of the magnetic-field falloff, and this dependence is more strongly expressed in the case when the inflector is located in the middle of a sector. Apparently, fluctuations of the captured charge by a factor of \(1.5 \div 2\) at the 680 MeV synchrotron of the Physics Institute of the USSR Academy of Sciences can also be explained by particle collisions with the inflector due to instability of \(n\).

REFERENCES

1. M. S. Rabinovich. Fundamentals of the Theory of the Synchrophasotron. Proceedings of the Physics Institute, USSR Academy of Sciences, vol. X, 1958.

2. A. L. Mints, A. A. Vasil’ev, É. L. Burshtein. Cybernetic accelerator to an energy of 1000 GeV. International Conference on Charged-Particle Accelerators, Dubna, 1963.

3. G. G. Kazanskii, A. P. Tsarenkov. Suppression of phase oscillations of a proton bunch in the 10 GeV synchrophasotron. JINR, Dubna, preprint 2491, 1965.

4. V. P. Tarasenko. Influence of disturbances on discrete systems of automatic optimization (review). Izv. AN SSSR OTN “Technical Cybernetics”, 1963, No. 2.

5. G. E. P. Box and Hunter. Technometrics, 1962, vol. 4, No. 3, pp. 301–318.

6. H. Hansmann F. J. Operat. Res. Soc. Japan, 1964, 6, No. 4, pp. 200–207.

7. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Nauka Publishing House, Moscow, 1965.

8. V. I. Kotov, N. B. Rubin. ZhTF, vol. 28, issue 2, 1958.

9. N. B. Rubin, O. I. Yarkovskii. Features of the particle-capture regime in a synchrophasotron. Dubna, preprint R-649, 1961.