Strong-Focusing Magnetic Analyzer of Charged Particles at 700 MeV/c

B. N. Kalinin, G. A. Saruev, V. M. Kuznetsov

(Presented by the scientific-technical seminar of the High-Energy Laboratory of the Nuclear Physics Research Institute)

The developed magnetic analyzer with double focusing will be used to analyze the energy and angular distributions in reactions of photoproduction of mesons and protons at the synchrotron of the Nuclear Physics Research Institute. The scintillation counters and spark chambers currently used do not make it possible to separate the background and have a limited recorded momentum. In developing the magnetic analyzer, the following considerations were taken into account:

1) the maximum analyzed momentum is up to 700 MeV/c;
2) to provide the largest possible solid angle;
3) to provide the minimum distance between the target and the detector;
4) to use the existing SP-57 electromagnet;
5) the design of the electromagnet must allow measurements to be made over the widest possible range of angles;
6) the momentum resolution is no worse than 2–3% per cm.

In analyzers with a homogeneous field, focusing by the angles of charged-particle emission from the target is accomplished by selecting the corresponding entrance and exit boundaries. Because the focusing forces act over relatively small sections of the target–detector trajectory (the fringe fields at the entrance and exit of the analyzer), the focal distances are rather large and the solid angle is small. Thus, for example, the analyzer of the Institute of Physics and Technology of the Academy of Sciences of the Ukrainian SSR for 500 MeV/c [1], with a homogeneous field, has a target–detector distance of 430 cm and a solid angle of \(2.9 \cdot 10^{-3}\) sr.

Analyzers with an inhomogeneous field use the principle of strong focusing, the essence of which is that the successive arrangement of two or more sectors with \(n > 1\) and \(n < 0\), under certain conditions, leads to a resultant focusing effect in the horizontal and vertical directions. Since in this case the focusing forces act over the entire section of the trajectory located in the magnetic field, the focal distances are considerably smaller and the solid angle is increased in comparison with an analyzer with a homogeneous field. For example, the analyzer in Frascati (Italy) [2] with strong focusing at 900 MeV/c has a target–detector distance of 280 cm and a solid angle of \(5.6 \cdot 10^{-3}\) sr. In view of the small area of the experimental hall of the Nuclear Physics Research Institute synchrotron and the desire to obtain the maximum solid angle at minimum cost, a magnetic analyzer with strong focusing based on a serially produced ...

of the SP-57 electromagnet, weighing 45 tons. Since the SP-57 electromagnet has an Ш-shaped construction, measurements of the angular distribution of reaction products are possible at angles to the direction of the photon beam from 0° to 27°, from 80° to 100°, and from 153° to 180°. For the remaining angles, the electromagnet stands will be located in the path of the bremsstrahlung beam. To expand the angular range, it was decided to modernize the electromagnet from an Ш-shaped construction to a C-shaped one. Such modernization makes it possible to register reaction products at angles of 0°–27°, 37°–143°, and 152°–180°. To form a strong-focusing field in the air gap (height 380 mm) of the SP-57 electromagnet, two pairs of special pole pieces were installed (Fig. 1).

Fig. 1. Pole pieces of the magnetic analyzer and the mechanism for moving the Hall probe.

Choice of a Strong-Focusing Field

The principle of strong focusing makes it possible to obtain focusing simultaneously in the horizontal and vertical planes by using a system of magnetic sectors with alternating signs of the magnetic-field falloff index \(n\). In our case, two magnetic sectors are used: in the first of them \(n \gg 1\), and in the second \(-n \ll 0\).

The magnetic analyzer is characterized by the following parameters:

1) energy resolution,
2) entrance focal distance,
3) exit focal distance,
4) horizontal magnification coefficient,
5) vertical magnification coefficient,
6) coefficient of longitudinal magnification for the horizontal plane,
7) coefficient of longitudinal magnification for the vertical plane,
8) solid angle,
9) interval of registered energies.

For calculating the optical characteristics of the analyzer, the following notation is introduced: \(n_1\) and \(n_2\) are the field falloff indices in the first and second sectors, \(R_1 = R_2 = R_0\) is the radius of the equilibrium orbit, \(\varphi_1\) and \(\varphi_2\) are the angles of rotation in the effective magnetic field of the sectors, \(s\) is the angle between

to the main trajectory and the perpendicular to the entrance boundary of the magnetic field; \(t\) is the angle between the main trajectory and the perpendicular to the exit boundary of the magnetic field; \(S\) is the entrance focal distance, \(T\) is the exit focal distance. We give the basic calculated relations for determining the optical characteristics of the analyzer [3]:

Condition for vertical focusing:

\[ \left(S^{D}_{b_1}+U_{b_1}\right)\left(T^{D}_{b_2}+U_{b_2}\right)=I_b , \tag{1} \]

where

\[ S^{D}_{b_1}= \frac{\sqrt{n_1}\,S'_{b_1}}{1-S'_{b_1}\tan s}; \qquad T^{D}_{b_2}= \frac{\sqrt{|n_2|}\,T'_{b_2}}{1-T'_{b_2}\tan t}; \]

\[ U_{b_1}=-\Phi_{b_1}-C_b\left(\Phi_{b_1}^{2}+1\right)q_b; \qquad U_{b_2}=\Psi_{b_2}-\left(\Psi_{b_2}^{2}-1\right)/q_b; \]

\[ I_b=C_b\left(\Phi_{b_1}^{2}+1\right)q_b^{2}; \qquad C_b=\sqrt{\frac{|n_2|}{n_1}}; \qquad q_b=-\Phi_{b_1}C_b+\Psi_{b_2}; \]

\[ \Phi_{b_1}=\cot \sqrt{n_1}\,\varphi_1; \qquad \Psi_{b_2}=\coth \sqrt{|n_2|}\,\varphi_2; \]

\[ S'_{b_1}=S_{b_1}R_0; \qquad T'_{b_2}=T_{b_2}R_0 . \]

Condition for horizontal focusing:

\[ \left(S^{D}_{r_1}+U_{r_1}\right)\left(T^{D}_{r_2}+U_{r_2}\right)=I_r, \tag{2} \]

where

\[ S^{D}_{r_1}= \frac{\sqrt{n_1-1}\,S'_{r_1}}{1+S'_{r_1}\tan s}; \qquad T^{D}_{r_2}= \frac{\sqrt{|1-n_2|}\,T'_{r_2}}{1+T'_{r_2}\tan t}; \]

\[ U_{r_1}=\Psi_{r_1}-C_r\left(\Psi_{r_1}^{2}-1\right)/q_r; \qquad U_{r_2}=-\Phi_{r_2}-\left(\Phi_{r_2}^{2}+1\right)/q_r; \]

\[ I_r=C_r\left(\Psi_{r_1}^{2}-1\right)\left(\Phi_{r_2}+1\right)/q_r^{2}; \qquad C_r=\sqrt{\frac{|1-n_2|}{n_1-1}}; \]

\[ q_r=C_r\Psi_{r_1}-\Phi_{r_2}; \qquad \Psi_{r_1}=\coth \sqrt{n_1-1}\,\varphi_1; \]

\[ \Phi_{r_2}=\cot \sqrt{|1-n_2|}\,\varphi_2; \qquad S'_{r_1}=S_{r_1}R_0; \qquad T'_{r_2}=T_{r_2}/R_0 . \]

Under the condition of double focusing,

\[ S_{b_1}=S_{r_1}=S \quad \text{and} \quad T_{r_2}=T_{b_2}=T . \]

Solving the system (1) and (2), one can obtain the entrance focal distance of the system.

The exit focal distance of the system is equal to

\[ T'= \frac{ \Phi_{r_2}\Psi_{r_1}S'\cdot \sqrt{|1-n_2|(n_1-1)}+ } { \sqrt{|1-n_2|} \left[ \Psi_{r_1}S'\sqrt{(n_1-1)|1-n_2|}+ \right. } \]

\[ \left. +\Phi_{r_2}\sqrt{|1-n_2|} +S'(U_1-1) +\Psi_{r_1}\sqrt{n_1-1} \right] \Big/ \left[ +\sqrt{|1-n_2|} -\Phi_{r_2}S'(n_1-1) -\Phi_{r_2}\Phi_{r_1}(n_1-1) \right] . \tag{3} \]

The coefficient of horizontal magnification of the system

\[ \chi_{r_{12}}=\chi_{r_1}\cdot \chi_{r_2}, \tag{4} \]

where \(\chi_{r_1}\) and \(\chi_{r_2}\) are the coefficients of horizontal magnification of sectors 1 and 2, respectively:

\[ \chi_{r_1}= \frac{1}{\cosh \delta^{D}_{m}} \left\{ \cosh\left[\sqrt{n_1-1}\,\varphi_1+\delta^{D}_{m}\right] +\sqrt{n_1-1}\,T'_{r_1}\, \sinh\left[\sqrt{n_1-1}\,\varphi_1+\delta^{D}_{m}\right] \right\}, \tag{5} \]

\[ x_{r_2}=\left\{\cos\left[\sqrt{|1-n_2|}\,\varphi_2+\delta^{D}_{m}\right] -\sqrt{|1-n_2|}\times \right. \]
\[ \left. {}\times T'\sin\left[\sqrt{|1-n_2|}\,\varphi_2+\delta^{D}_{m}\right]\right\}, \tag{6} \]

where

\[ \delta^{D}_{m}=\operatorname{arcth}\left(\frac{\operatorname{tg}s}{\sqrt{n_1-1}}\right); \]
\[ T'_{r_1}= \frac{S'\Psi_{r_1}\sqrt{n_1-1}+1+S'\operatorname{tg}s} {(n_1-1)S'+\Psi_{r_1}\sqrt{n_1-1}(1-S'\operatorname{tg}s)} \tag{7} \]

is the exit focal length for the first sector. The coefficient of vertical magnification of the system is \(x_{b_{12}}=x_{b_1}\cdot x_{b_2}\),

where \(x_{b_1},x_{b_2}\) are the coefficients of vertical magnification, respectively, of sectors 1 and 2:

\[ x_{b_1}=\frac{1}{\cos\delta_m}\left[\cos\left(\sqrt{n_1}\varphi_1+\delta_m\right) -T'_{b_1}\sqrt{n_1}\sin\left(\sqrt{n_1}\varphi_1+\delta_m\right)\right], \tag{8} \]

\[ x_{b_2}=\operatorname{ch}\sqrt{|n_2|}\cdot\varphi_2 +T'\sqrt{|n_2|}\cdot\operatorname{sh}\sqrt{|n_2|}\cdot\varphi_2. \tag{9} \]

where

\[ \delta_m=\operatorname{arctg}\left(\frac{\operatorname{tg}s}{\sqrt{n_1}}\right); \qquad T'_{b_1}= \frac{\Phi_{b_1}\sqrt{n_1}S'+1-S'\operatorname{tg}s} {n_1S'-\Phi_{b_1}\sqrt{n_1}(1-S'\operatorname{tg}s)} \]

is the focal length for the first sector, and \(T'\) is the distance from the exit boundary of the magnetic field to the detector. The energy resolution of the analyzer is found from the expression

\[ \xi= \frac{1} {R_0\left\{\dfrac{1-x_{r_2}}{|1-n_2|} -\dfrac{1-x_{r_1}}{n_1-1}\cdot x_{r_2}\right\}} \ \mathrm{l/cm}. \tag{10} \]

To calculate the solid angle of the analyzer it is necessary to determine the horizontal and vertical acceptance angles \((\alpha_0,\gamma_0)\)

\[ \Omega=4\alpha_0\gamma_0. \tag{11} \]

These angles can be obtained from the equations of motion of charged particles in an inhomogeneous magnetic field:

\[ \frac{d^2r}{d\varphi^2}+(1+n)r=0 \quad \text{for the horizontal plane,} \tag{12} \]

\[ \frac{d^2z}{d\varphi^2}+nz=0 \quad \text{for the vertical plane.} \tag{13} \]

The equations for determining the angles \(\alpha_0\) and \(\gamma_0\) have the form:

\[ \begin{aligned} &\alpha_0^2\left\{|1-n_2| \left[S'\Psi_{r_1}+\frac{1+S'\operatorname{tg}s}{\sqrt{n_1-1}}\right]^2 +\left[S'\sqrt{n_1-1}+\Psi_{r_1}(1+S'\operatorname{tg}s)\right]^2\right\} \\ &\quad+\alpha_0\cdot 2z'_M \left\{(1+S'\operatorname{tg}s) \left(\frac{|1-n_2|}{\sqrt{n_1-1}}\Psi_{r_1} +\frac{|1-n_2|}{n_1-1}\operatorname{tg}s +\Psi_{r_1}\sqrt{n_1-1} +\Psi_{r_1}^{2}\operatorname{tg}s\right)\right. \\ &\qquad\left. +S'\left[ |1-n_2|\Psi_{r_1}^{2} +\frac{|1-n_2|}{\sqrt{n_1-1}}\Psi_{r_1}\operatorname{tg}s +(n_1-1)\Psi_{r_1}\sqrt{n_1-1}\operatorname{tg}s \right]\right\} \\ &\quad+r_M'^2\left\{|1-n_2| \left(\Psi_{r_1}+\frac{\operatorname{tg}s}{\sqrt{n_1-1}}\right)^2 +\left(\sqrt{n_1-1}+\Psi_{r_1}\operatorname{tg}s\right)^2\right\} \\ &\quad -r_0'^2\frac{|1-n_2|}{\operatorname{sh}^2\sqrt{n_1-1}\,\varphi_1} =0. \tag{14} \end{aligned} \]

where

\[ r'_{\mathrm{M}}=r_{\mathrm{M}}/R_0,\quad r'_0=r_0/R_0, \]

\(r_{\mathrm{M}}\) is the radial coordinate of the point of emergence of the charged particle from the target,

\(r_0\) is the half-size of the radial working region in the first sector,

\[ \gamma_0^2\left[n_1S'^2+(1-S'\tg s)^2\right] +\gamma_0\,2z'_{\mathrm{M}}\left[n_1S'-(1-S'\tg s)\tg s\right] -n_1\left(z_0'^2-z_{\mathrm{M}}'^2\right)+z_{\mathrm{M}}'^2\tg^2 s=0, \tag{15} \]

where

\[ z'_0=z_0/R_0;\quad z'_{\mathrm{M}}=z_{\mathrm{M}}/R_0, \]

\(z_0\) is the half-size of the air gap,

\(z_{\mathrm{M}}\) is the vertical coordinate of the point of emergence of the charged particle from the target.

For the preliminarily selected values \(n_1=+21.3\) and \(n_2=-16.2\), pole profiles were calculated according to the method of [4], which assumes that 1) the magnetic permeability of the pole material is infinite and 2) there is no pole scattering. Therefore an experimental selection of the pole profile was carried out on a plane model. The plane model was powered by alternating current. Field measurements were made with an induction coil. On the plane model, regions of linear variation of the field with radius were obtained for the first sector, \(n_1=+21.3\), \((R_0+3\ \text{cm})\div(R_0-5.5\ \text{cm})\), and for the second sector, \(n_2=-16.2\), \((R_0+6\ \text{cm})\div(R_0-4.5\ \text{cm})\). The relatively small region of linear field in the direction of increasing air gap is explained by the influence of the field of the round poles, between which the profiled pole tips are installed.

In view of the fact that the stray field of the SP-57 electromagnet, because of the large air gap between the round poles, is fairly extended, magnetic screens are installed at the entrance and exit of the analyzer; these make it possible to form the stray field over a small section of the trajectory and to calculate the optical characteristics of the analyzer.

The distance from the physical boundary of the pole to the effective boundary of the magnetic field in the presence of screens is determined by the expression [1]

\[ \xi=\frac{\delta}{2}-\frac{\delta}{\pi}\arcsin \frac{\delta}{2g\sqrt{1+\dfrac{\delta^2}{4g^2}}} +\frac{g}{\pi}\ln\left(1+\frac{\delta^2}{4g^2}\right), \tag{16} \]

where \(\delta\) is the distance between the screen and the pole,

\(g\) is the half-height of the air gap in the pole.

Expression (16) cannot be solved analytically with respect to \(\delta\). Introducing the notation \(x=\xi/g\) and \(\delta/2g=y\), we rewrite (16) as

\[ x=y-\frac{2y}{\pi}\arcsin\frac{y}{\sqrt{1+y^2}}+\frac{1}{\pi}\ln(1+y^2). \tag{17} \]

Next, from the graph of function (17), for possible values of \(y\), one can, at \(\xi=\mathrm{const}\), determine \(\delta\) for each \(g\).

The distance between the screen and the pole is chosen from the following considerations:

1. The magnetic induction in the screen must be no greater than the induction in the part of the pole facing the screen.

2. The distance between the entrance screen and the entrance focus must be sufficient for placement of the hydrogen target.

On the basis of these requirements, \(\delta_0=6.3\ \text{cm}\) was chosen at the radius of the equilibrium orbit.

Results of Measuring the Magnetic Field of the Analyzer

The program of magnetic measurements included:

1) Measurement of the radial distribution of the magnetic field to determine \(n\).

2) Measurement of the azimuthal distribution of the field to determine the effective magnetic angles and to check the assembly quality.

All field measurements were performed with an E11-3 Hall magnetometer at fields \(B_0 = 900, 3000, 5900, 8000, 8800, 10400, 11400, 13000\) G. The measurement error was \(\pm 1.5\%\). To move the Hall probe in the radial direction, a special mechanism was designed (Fig. 1).

Fig. 2. Dependence of the magnetic-field falloff index and the effective angles of the sectors on the values of the magnetic field on the equilibrium orbit.

Figure 2 presents the dependences of the field index and the effective angles of the sectors on the magnetic field on the equilibrium orbit.

At \(B_0 > 6\) kG, saturation effects of the analyzer poles begin to appear and the focal distances increase. However, taking into account the sufficiently large longitudinal dimensions of the target (3–5 cm), it may be assumed that the focal distances do not change up to fields of 10 kG. Above 10 kG, correction of the positions of the target and detectors is necessary. The measurement results showed that up to fields of 13 kG the radial regions in both sectors remain constant with an error of no more than 2%.

From the results of the magnetic measurements, the main parameters of the analyzer were determined:

1) analyzed momentum — \(43 \div 700\) MeV/\(c\),

2) magnetic field on the equilibrium orbit — 740–13000 G,

3) radius of the equilibrium orbit — 200 cm,

4) height of the air gap — 9.05 cm,

5) field index of the first sector — \(+21.7\),

6) field index of the second sector — 17.6,

7) effective angle of the first sector — \(17^\circ 40'\),

8) effective angle of the second sector — \(16^\circ 30'\),

9) entrance focus — 40.5 cm,

10) exit focus — 103 cm,

11) momentum resolution — \((2.32 \pm 0.2)\%\)/cm,

12) solid angle — \(5 \cdot 10^{-3}\) sterad,

13) momentum acceptance interval — \(\pm 0.16\).

In conclusion, it may be said that at the Nuclear Physics Research Institute, on the basis of the standard SP-57 electromagnet, a strong-focusing magnetic analyzer of charged particles with a maximum momentum of 700 MeV/c has been developed for carrying out experiments on meson photoproduction at the Nuclear Physics Research Institute synchrotron.

REFERENCES

1. N. G. Afanas'ev. Scattering of high-energy electrons by light and heavy nuclei. Dissertation, Kharkov Institute of Physics and Technology, Academy of Sciences of the Ukrainian SSR, Kharkov, 1967.

2. Sacerdoti and Tau. Charged-particle spectrometer. Nucl. Instr., 16, 139 (1962).

3. R. M. Sternheimer. Double focusing of charged particles by a system of two magnets with an inhomogeneous field. Rev. Sci. Instrum., 24 No. 8, 873 (1953).

4. C. C. Iliescu. Ideal profile for soft-, strong-, and FFAG-focusing accelerators and other nuclear instruments, Instrum. and Methods, 21, 145—154 (1963).