MATHEMATICAL MODEL OF PARTICLE CAPTURE IN A SYNCHROTRON

V. V. Tsygankov

In a synchrotron with weak focusing, the change in the accelerated charge up to the final energy from cycle to cycle is caused mainly by changes in such parameters as the switching-on times of injection and of the accelerating radio-frequency (RF) voltage, the injection energy, and the initial frequency of the generator. The construction of a mathematical model of capture is the initial stage in designing a system for automatic optimization of capture conditions.

Multiturn injection of particles into a synchrotron is considered under the following assumptions: the collective interaction of particles during injection is not taken into account; the magnetic-field falloff index \(n\), constant over the radius and azimuth of the electromagnet sector and identical in all sectors; the field inside the straight sections is absent; the amplitude of the accelerating RF voltage is established over a time that may be neglected in comparison with the period of radial-phase oscillations.

The injection process is divided into two stages [1]. In the first stage of injection, which lasts until the RF field is switched on, the particles are injected onto an instantaneous orbit contracting toward the center because of the growth of the magnetic field. In accordance with the course of the first stage of injection, the particle distribution function \(g_{\min}(u,\gamma)\) is calculated with respect to the coordinates \(u\) and \(\gamma\) (where \(u\) is the distance from the inner plate of the inflector to the instantaneous orbit, and \(\gamma\) is the angle between the direction of particle emergence from the inflector and the tangent to the instantaneous orbit at the place where the inflector is installed). In contrast to the particle distribution function with respect to betatron-oscillation amplitudes \(\psi(A)\) [1], the function \(g_{\min}(u,\gamma)\) makes it possible to take into account in the capture ratios \(n\) the change in the injection time and the energy of the injected particles.

The solution of the equation of betatron oscillations at the azimuth of the inflector \(\sigma\), for race-track type accelerators, has the form [1]:

\[ x_k = F_c(\sigma)\cdot \cos(4\mu k+\alpha(\sigma)), \tag{1} \]

where

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

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

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

\[ c=\cos \frac{\pi}{2}\chi,\qquad s=\sin \frac{\pi}{2}\chi,\qquad \cos\mu=c-ps, \]

\[ p=\frac{l\chi}{2R},\qquad \sigma=\frac{\chi S}{R},\qquad \chi=\sqrt{1-n}, \]

\(S\) — distance from the edge of the sector to the inflector in the direction of particle motion,
\(R\) — radius of the equilibrium orbit,
\(l\) — length of the straight section,
\(\gamma\) — angle between the direction of emergence of the particle and the tangent to the equilibrium orbit in azimuth,
\(x_0\) — initial deviation of the particle from the instantaneous orbit,
\(x_k\) — value of \(x\) on the \(k\)-th revolution \((k=0,1,2,\ldots)\).

Carrying out the operations (2) and substituting the expressions for \(F_c(\sigma)\) and \(\alpha(\sigma)\) into (1), we obtain:

\[ x_k=x_0 B(k)+\frac{\gamma R}{\chi}A(k), \tag{3} \]

where

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

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

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

In the region \(u>0\) the condition for bypassing the inflector has the form:

\[ (g+u)B(k)+\frac{\gamma R}{\chi}A(k)\leq \Delta R_{\mathrm{ob}}\,k+u, \tag{5} \]

where \(\Delta R_{\mathrm{ob}}\) — contraction of the instantaneous orbit per revolution,
\(g\) — distance from the inner plate of the inflector to the particle,
\(u\) — distance from the inner plate to the instantaneous orbit at the moment when the particle emerges from the inflector.

From (5) follow two inequalities:

\[ 1)\quad g_1(k)\leq \frac{1-B(k)}{B(k)}\,u-\frac{A(k)}{B(k)}\,\frac{R}{\chi}\gamma+\frac{\Delta R_{\mathrm{ob}}\,k}{B(k)}, \tag{6} \]

if \(B(k)>0\), and

\[ 2)\quad g_2(k)\geq \frac{1-B(k)}{B(k)}\,u-\frac{A(k)}{B(k)}\,\frac{R}{\chi}\gamma+\frac{\Delta R_{\mathrm{ob}}\,k}{B(k)}, \tag{7} \]

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

In (6) and (7), for fixed values of \(n,\Delta R_{\mathrm{ob}},\gamma,u\) and among \(k=1,2,3,\ldots\), we find such a value \(k=k'\) at which \(g_1(k')\) is minimal, and such a value \(k=k''\) at which \(g_2(k'')\) is maximal. Then the region of the inflector from which, on emerging, the particles will be captured in the first injection stage is equal to:

\[ 0\leq g_2(k'')\leq g_1(k')\leq g_1, \tag{8} \]

where \(g_1\) is the width of the inflector.

There are no analytical methods for investigating the function \(g(k)\) of the discrete argument \(k\); therefore the operation of finding \(k'\) and \(k''\) must be performed numerically.

At the first instant after the RF field is switched on, the particles are distributed over the phases \(\varphi_0\) of the accelerating voltage. The region of phases \((\varphi_1,\varphi_2)\),

bounded by the separatrix, is the locus of points of synchronous orbits, around each of which the particles are distributed over the positions of the instantaneous orbit in accordance with the first stage of injection. After a quarter period of radial-phase oscillations, a synchronous orbit having initial phase \(\varphi_0\) will be at a distance \(y \sim \varphi(\varphi_s)\) from the equilibrium orbit; therefore the region of integration with respect to \(u\) for the given synchronous orbit is \((0,u_1,-y)\) (\(\varphi_s\) is the equilibrium phase, \(u_1\) is the distance from the inner plate of the inflector to the center of the working region). If the energy of the injected particles has changed by an amount \(+\Delta E\), then the azimuthal size of the bunch in the plane \((\varphi,\dot\varphi)\) will be \(y_{02}-y_{01}=\varphi(\Delta E)\), and the lower limit of integration with respect to \(y\) is

\[ y_0=\frac{R_s\Delta E}{\beta^2(1-n)E}. \]

In the case when the region of integration with respect to \(u\) does not depend on \(\gamma\), the charge captured into acceleration, taking into account the two stages of injection, is equal to

\[ Q=\int_{\gamma_1}^{\gamma_2}\int_0^{y_{01}}\int_{\varphi_{01}}^{\varphi_{02}}\int_{y_0}^{y_1}\int_0^{u_1-y}\int_0^{g_{\min}(u,\gamma)} q(\gamma,y_0,\varphi_0,y,u,g)\times \]
\[ \times\, dg\,du\,dy\,d\varphi_0\,dy_0\,d\gamma, \tag{9} \]

where \(q\) is the distribution function of the injected particles with respect to the angle \(\gamma\), the energy \(y_0\), the initial phase \(\varphi_0\) of the rf voltage, the amplitudes of the radial-phase oscillations \(y\sim\varphi(\varphi_s)\) without taking betatron oscillations into account, the duration of the injection pulse \(u\), and the inflector width \(g\);

\(y_1\) is the radial half-size of the separatrix;

\[ g_{\min}(u,\gamma)=g_1(\kappa')= \frac{1-B(\kappa')}{B(\kappa')}\,u -\frac{A(\kappa')}{B(\kappa')}\,\frac{R}{\chi}\,\gamma +\frac{\Delta R_{\mathrm{ob}}\cdot \kappa'}{B(\kappa')}. \tag{10} \]

Next we shall assume that the relation between the coordinates \(\varphi_0\) and \(y\) is linear; that is, if the distribution of particles in the phases \(\varphi_0\) is uniform, then the distribution of particles in \(y\) (without taking betatron oscillations into account) may also be regarded as uniform. Such an assumption can be made if, in expression [1]

\[ \varepsilon_{1,2}= \frac{2(1-\varphi_s\operatorname{ctg}\varphi_s)^{1/2}}{\pi}\, \frac{y}{y_1} \left[ 1+\kappa_{1,2}\left(\frac{y}{y_1}\right)^2 \right], \tag{10} \]

which establishes the relation between the amplitudes of the radial and phase oscillations, one neglects the term
\(\kappa_{1,2}\left(\dfrac{y}{y_1}\right)^2\)
(\(\kappa_{1,2}=0.2\div0.25;\ 0\leq y\leq y_1\)), which “gives a small correction” [1]. For the case of a uniform charge distribution,

\[ q(\gamma,y_0,\varphi_0,y,u,g)= \frac{I}{ g_1\cdot\left|\dfrac{dR}{dt}\right|\cdot y_1\cdot 2\pi y_{01}(\gamma_2-\gamma_1) }, \tag{11} \]

where \(I\) is the injector current.

The region of integration with respect to \(\varphi_0\) for \(\Delta E\ne0\) can be determined if, in expression [2],

\[ \varepsilon_{y_0}(y=y_1)= \frac{\varphi_{02}-\varphi_{01}}{\pi} = \frac{2(1-\varphi_s\cdot \operatorname{ctg}\varphi_s)^{1/2}}{\pi} \left( \sqrt{1-\left(\frac{y_0}{y_1}\right)^2} + \right. \]
\[ \left. +\kappa_{1,2} \left[ \sqrt{1-\left(\frac{y_0}{y_1}\right)^2} \right]^3 \right); \tag{12} \]

neglect the term

\[ \kappa_{1,2}\left[\sqrt{1-\left(\frac{y_0}{y_1}\right)^2}\right]^3,\qquad 0\leq y_0\leq y_1. \]

Substituting (10), (11), (12) into (9), it is easy to calculate the integrals (9).

In addition to the above assumptions concerning the form of the functions \(\varepsilon_{1,2}\) and \(\varepsilon_{y_0}\), which are not of a fundamental character, formula (9) does not take into account the change in the function \(g_{\min}(u,\gamma)\) in the region of instantaneous orbits close to the synchronous one. Indeed, if the instantaneous orbit coincides with the synchronous orbit, which is at a distance \(y\) from the equilibrium one, then \(g_{\min}(u,\gamma)\), since the period of synchronous oscillations is much greater than the period of betatron oscillations. For each instantaneous orbit the maximum amplitude of betatron oscillations (for an accelerator without straight sections) is equal to

\[ A=\sqrt{[u+g_{\min}(u,\gamma)]^2+\left(\frac{\gamma R}{\kappa}\right)^2}. \]

Fig. 1. Distribution function in \(u\) and \(\gamma\) and its change at the 2nd stage of injection

If \(A<u_1-y\), then the particle distribution in \(u\) is found according to (10); if \(A>u_1-y\), then the distribution function is equal to:

\[ g'_{\min}(u,\gamma)=g_{\min}(u,\gamma)-\{A-(u_1-y)\}. \tag{13} \]

Thus, the region of integration with respect to \(u\) is divided into two: \((0,u')\) and \((u'',u''\simeq u_1-y)\); \(u'\) is found from the condition that the expression in braces is equal to zero, and \(u''\) from the condition \(g_{\min}(u,\gamma)=0\). Figure 1 shows a graph of the function \(g_{\min}(u,\gamma)\) (10), (13), calculated for the parameters of the 1.5 GeV TPI synchrotron. It is seen from Fig. 1 that the error of formula (9), which is due to the fact that the change in \(g_{\min}(u,\gamma)\) in the region of instantaneous orbits close to the synchronous ones is not taken into account, is of the same order for a parallel beam with \(\gamma=0\) and \(\gamma\neq0\). The capture coefficient

in the second stage of injection for a parallel monoenergetic beam, calculated by formula (9), is equal to:

\[ \eta_2=\frac{Q}{Q_1}=\frac{\varphi_2-\varphi_1}{2\pi}\cdot\frac{1}{2} \quad(\text{for } y_1=u_1), \]

\[ \eta_2=\frac{\varphi_2-\varphi_1}{2\pi}\cdot\frac{3}{4} \quad\left(\text{for } y_1=\frac{1}{2}u_1\right), \tag{14} \]

where \(Q_1\) is the charge captured in the first stage of injection,
\(\varphi_2-\varphi_1=f(y_1)\) is the azimuthal size of the separatrix.

Fig. 2. Dependence of the captured charge on the energy spread
\(y_{01}\sim \Delta E_{\max}\)

Taking into account the change in \(g_{\min}(u,\gamma)\) (for \(\gamma=0\)),

\[ \eta_2'=\frac{\varphi_2-\varphi_1}{2\pi} \left( \frac{1}{2}-\frac{g_1}{u_1} +\frac{3}{2}\frac{g_1^2}{u_1^2} \right) \quad(\text{for } y_1=u_1) \]

and

\[ \eta_2'=\frac{\varphi_2-\varphi_1}{2\pi} \left( \frac{3}{4}-\frac{g_1}{u_1} \right) \quad\left(\text{for } y_1=\frac{1}{2}u_1\right). \tag{15} \]

Since \(g_1\ll u_1\), the error of formula (9) is insignificant.

The losses of particles due to the energy spread can be estimated by the formula obtained from (9) for \(\gamma=0,\ y_1=u_1,\ y_{01}=u_1x\):

\[ \frac{Q(y_{01}\ne 0)}{Q(y_{01}=0)}= \]

\[ = \frac{ 2\left[ \frac{5}{16}x\sqrt{1-x^2} +\left(\frac{1}{3}-\frac{x}{8}\right)\sqrt{(1-x^2)^3} -\frac{1}{3} +\frac{5}{16}\arcsin x \right] }{x}. \tag{16} \]

In the square bracket \(0\le x\le 1\). Figure 2 presents the following plots of the function \(\dfrac{Q(y_{01}\ne 0)}{Q(y_{01}=0)}\): the plot denoted by numeral 1 is the plot calculated-

... without taking it into account, and \(1'\)—taking into account the variation of \(g_{\min}(u)\) at the second stage of injection for the parameters of the \(1.5\,\Gamma_{38}\) TPI synchrotron \((g_1=1.5\ \mathrm{cm},\ u_1=10\ \mathrm{cm})\) and \(y_1=u_1\); the number 2 denotes the graph without taking into account, and \(2'\)—taking into account the variation of \(g_{\min}(u)\) for the parameters of the \(10\,\Gamma_{38}\) JINR synchrophasotron, Dubna \((g_1=3\ \mathrm{cm},\ u_1=50\ \mathrm{cm})\), and \(y_1=\frac{1}{2}u_1\).

Let us consider the capture of a parallel monoenergetic beam under nonoptimal capture conditions. Introduce the notation:

\(R_1\) — radius of the instantaneous orbit at the moment when the injection pulse is switched on;

\(R_2\) — radius of the instantaneous orbit at the moment when the RF field is switched on;

\(R'_2\) — radius of the instantaneous orbit at the moment when the injection pulse ends;

\(R_{\mathrm{inf}}\) — radius at which the inflector is installed;

\[ R_1-R'_2=u_i=\left|\frac{dR}{dt}\right|\Delta\tau_i, \]
\(\Delta\tau_i\) — duration of the injection pulse;

\(\overline{R}_s\) — mean radius of the accelerator chamber;

\(R_s\) — radius of the equilibrium orbit;

\[ \Delta u_i=\left|\frac{dR}{dt}\right|\Delta\tau_i, \]
\(\Delta\tau_i\) — error in the injection time;

\[ \Delta u_{\mathrm{RF}}=\left|\frac{dR}{dt}\right|\Delta\tau_{\mathrm{RF}}, \]
\(\Delta\tau_{\mathrm{RF}}\) — error in the time at which the RF field is switched on;

\[ \Delta u_\omega= \frac{R_s}{\beta^2(1-n)F}\cdot\frac{\Delta\omega}{\omega_s}, \]
\(\Delta\omega\) — drift of the initial frequency of the generator;

\[ F=1-\frac{1}{(2\pi R_s+L)\,[\,n+\beta^2(1-n)\,]}, \qquad L=4l; \]

\[ \Delta u_E= \frac{R_s}{\beta^2(1-n)E}\,\Delta E, \]
\(\Delta E\) — drift of the injection energy.

Variations of the above parameters are considered from cycle to cycle. In Fig. 3 a particular case is presented: \(R_1<R_{\mathrm{inf}}\), \(R_s=\overline{R}_s>R_2\). To each position of the instantaneous orbit and of the inflector there corresponds a region \(g_{\min}(u)\), upon leaving which the particles will not collide with the inflector. If the instantaneous orbit of the electrons is returned to its former place after they have made \(k'\) revolutions, then we obtain the graph of the function \(g'_{\min}(u)\), drawn as a solid line. Since in fact the instantaneous orbit moves, the graph of the function \(g_{\min}(u)\) is shown by a dashed line. From Fig. 3 it is seen that the region of integration with respect to \(u\) is \([\,(R_{\mathrm{inf}}-R_1),\,u_1-y\,]\), and with respect to \(y\) is \([\,R_s-R_2,\,y_1\,]\); the azimuthal size of the bunch is:

\[ (\varphi_{02}-\varphi_{01})=(\varphi_2-\varphi_1) \sqrt{\,1-\left(\frac{R_s-R_2}{y_1}\right)\,}. \]

Thus,

\[ Q= \frac{I(\varphi_2-\varphi_1)} {g_1\left|\dfrac{dR}{dt}\right|\cdot y_1\cdot 2\pi} \sqrt{\,1-\left(\frac{R_s-R_2}{y_1}\right)^2\,} \int_{R_s-R_2}^{y_1} \int_{R_{\mathrm{inf}}-R_1}^{u_1+(R_{\mathrm{inf}}-R_1)-y} \int_{0}^{g_{\min}(u)} d g\,d u\,d y . \tag{17} \]

Fig. 3. On the placement of the limits of integration with respect to \(u, y, \varphi_0\) under nonoptimal capture conditions

Fig. 4. Functional diagram for modeling the capture relations (18) on an analog computer
(operations: 1 — summation, 2 — extraction of the modulus, 3 — comparison with zero,
4 — multiplication, 5 — comparison of two quantities, 6 — extraction of the root)

In the general case, the dependence of the captured charge on the above-indicated parameters has the form:

\[ \begin{aligned} Q={}& \frac{I\cdot 2(1-\varphi_s\cdot \operatorname{ctg}\varphi_s)^{1/2}} {g_1\left|\dfrac{dR}{dt}\right|y_1\cdot \pi} \sqrt{1-\left(\frac{|R_s-R_2|}{y_1}\right)^2} \Bigg( \int_{|R_s-R_2|}^{|R_{\mathrm{inf}}-R_1|-|R_s-\bar R_s|} \\ &\times \int_{\{(R_1-R_{\mathrm{inf}})\geq 0\}}^{u_1-\{(R_1-R_{\mathrm{inf}})\geq 0\}} + \int_{|R_{\mathrm{inf}}-R_1|-|R_s-\bar R_s|}^{\min\left\{u_1-|R_s-\bar R_s|,\;y_1\right\}} \int_{\{(R_{\mathrm{inf}}-R_1)\geq 0\}}^{u_1-(y+|R_s-\bar R_s|)-\{(R_{\mathrm{inf}}-R_1)\geq 0\}} \Bigg) \\ &\times \int_{0}^{g_{\min}(u)} dg\,du\,dy, \end{aligned} \tag{18} \]

where:

\[ \left. \begin{aligned} R_1-R_{\mathrm{inf}}&=\Delta u_E-\Delta u_{\mathrm{i}},\quad R_s-\bar R_s=\Delta u_{\omega},\quad R_2-\bar R_s=\Delta u_E-\Delta u_{\mathrm{HF}},\\ |R_2-R_s|&=|(R_s-\bar R_s)-(R_2-\bar R_s)|=|\Delta u_E-\Delta u_{\mathrm{HF}}| \end{aligned} \right\}. \tag{19} \]

Substituting (19) into (18), we obtain the dependence of the captured charge on \(\Delta u_{\mathrm{HF}}, \Delta u_{\mathrm{i}}, \Delta u_E, \Delta u_{\omega}\). From (18) it is seen that, for a nonuniform distribution of the captured charge in \(u\), the dependence on \(\Delta u_{\mathrm{i}}\) and \(\Delta u_E\) is not symmetric with respect to zero. From (18) it is easy to show that, in order to compensate the drift of the injection energy, it is desirable to make the moments of injection and of switching on the HF field coupled. Figure 4 presents a block diagram for modeling relation (18) for \(g_{\min}(u)=g_1\).

REFERENCES

1. M. S. Rabinovich. Fundamentals of the theory of the synchrophasotron. Proceedings of the Lebedev Physical Institute, Academy of Sciences of the USSR, vol. 10, 1958.

2. I. S. Danilkin, M. S. Rabinovich. Capture of particles into the synchrophasotron acceleration regime. ZhTF, vol. 28, no. 2, 1958.