DOUBLE-FOCUSING MAGNETIC β-SPECTROMETER FOR ANALYZING THE ENERGY DISTRIBUTION OF ELECTRONS WITH ENERGIES UP TO 35 MeV

V. M. ANTONOV, B. A. KONONOV, V. V. EVSTIGNEEV

(Presented by the joint seminar of the Research Institute of Nuclear Physics and the Physics and Technology Faculty on spectrometry)

At the Tomsk Polytechnic Institute, studies are being carried out on the interaction of electron and bremsstrahlung radiation with matter, in which it is necessary to know the energy spectrum of the radiation before and after passage through the specimen under investigation. Betatrons, which have a narrow energy spectrum of electron radiation [1], are most often used as the radiation source. In a number of cases, for example, in studying the backscattering of electrons or in determining energy losses in very thin layers, the intensity of the electrons passing through the magnetic field of the spectrometer may be small. Therefore, in the general case it is desirable to have a spectrometer with increased luminosity. Described below is a sector-type double-focusing magnetic β-spectrometer, with plane and parallel pole pieces, developed by us for use mainly in experiments on the study of energy losses by electrons in matter. In contrast to the sector spectrometer [1] at our disposal, the present spectrometer has double focusing, a large energy range, and better operating characteristics.

Design of the β-Spectrometer Magnet

Figure 1 shows the particle trajectories in the median plane of the magnet. Focusing in the median plane is provided by a homogeneous magnetic field in the gap; focusing in the field plane is achieved through the decrease of the magnetic field toward larger radii at the entrance and exit of particles from the magnet [2, 3, 4].

Double focusing occurs when the following two conditions are satisfied [3]; in this case first-order focusing is achieved in the median plane:

\[ \tg \varepsilon_2 = \frac{1}{2} \left[ \tg(\Phi-\Psi) + \frac{1}{\Phi-\ctg \gamma} \right]; \tag{1} \]

\[ \frac{1}{l_2} = \frac{1}{2} \left[ \tg(\Phi-\Psi) - \frac{1}{\Phi-\ctg \gamma} \right], \tag{2} \]

where

\[ \operatorname{tg}\Psi=\operatorname{tg}\varepsilon_1+\frac{1}{l_1}; \tag{3} \]

\[ \operatorname{tg}\eta=\operatorname{tg}\varepsilon_1-\frac{1}{l_1}. \tag{4} \]

Here \(\varepsilon_1\) and \(\varepsilon_2\) are, respectively, the angles between the normal to the boundary of the pole piece and the central ray at the entrance to and exit from the magnet. \(l_1\) and \(l_2\) are the distances from the focal points of the spectrometer, respectively, to the entrance and exit faces of the magnet, measured in units of the radius of curvature of the trajectory of the central ray in the region of the uniform magnetic field. \(\Phi\) is the deflection angle of the central ray. The angles \(\varepsilon_1\) and \(\varepsilon_2\) are taken as positive in the case when the trajectories of the particles outside the main magnetic field lie on the same side of the normal to the pole boundary as the center of curvature of the trajectory of the central ray. Equations (1) and (2) are valid under the condition that the height of the magnet gap is small in comparison with the radius of curvature of the particle trajectory in the region of the uniform field.

Fig. 1. Trajectories of particles in the median plane of the magnet of the B-spectrometer.

Since the optimal relation between the luminosity and resolving power of the spectrometer is achieved at \(\Phi\) of about \(90^\circ\) and \(l_1\) of the order of \(1\)–\(1.5\), we chose \(l_1=450\) mm with \(R_0=300\) mm, \(\varepsilon_1=42^\circ\), and \(\Phi=90^\circ\). In this case the corresponding values of \(\varepsilon_2\) and \(l_2\) are \(7^\circ 40'\) and \(1.991\). Thus, the spectrometer possesses the following principal...

...with geometrical characteristics: \(R_0 = 300\) mm, \(l_1 = 450\) mm, \(l_2 = 597\) mm, \(\varepsilon_1 = 42^\circ\), \(\varepsilon_2 = 7^\circ 40'\), \(\Phi = 90^\circ\).

The vacuum chamber of the spectrometer is made from a copper waveguide of rectangular cross section with internal dimensions \(34 \times 72\ \text{mm}^2\) and a wall thickness of 2 mm. On this basis, the height of the interpole gap was chosen to be 42 mm. The width of the pole pieces is 200 mm. The shape of the profile of the pole pieces is shown in Fig. 1.

Fig. 2. Construction of the magnet of the B-spectrometer.

Figure 2 shows the construction of the spectrometer magnet. The median plane of the magnet is arranged horizontally. The weight of the magnet is about 500 kg, of which 90 kg is accounted for by the two magnetizing coils located on the poles. The coils are made of a copper busbar with a cross section of \(5\ \text{mm}^2\) and are connected in series; each coil has 1250 turns and a resistance of 6 ohms. The parallelism of the pole pieces is maintained to an accuracy no worse than \(\pm 50\ \mu\). Magnetic measurements carried out using nuclear magnetic resonance showed that the static inhomogeneity of the magnetic field within the magnetic track is no more than \(\pm 0.1\%\).

Figure 3 shows the magnetic characteristic of the spectrometer. Here the current in the magnetizing coils is plotted along the abscissa axis, and the magnetic-field strength in the gap along the ordinate axis. At a current of 10 A, which corresponds to a magnetizing force of 25,000 A, the magnetic-field strength...

of the field in the gap is 4100 Oe. At this magnetic-field strength the spectrometer makes it possible to analyze the energy distribution of electrons whose maximum energy is 37 MeV. The power consumed by the magnet in this case is 1.2 kW. The magnet is supplied either by an electric-machine amplifier with a stability of the supply current no worse than $\pm 0.01\%$ or by a storage battery.

Fig. 3. Magnetic characteristic of the β-spectrometer.

Fig. 4. Curve of the decrease of the magnetic field along the calculated trajectory of the central ray upon entry into the main magnetic field.

Figure 4 shows the curve of the decrease of the magnetic field along the theoretical trajectory of the central ray as it enters the main magnetic field. The distance along the trajectory, in cm, is plotted on the abscissa axis; the magnetic-field intensity, in relative units, is plotted on the ordinate axis.

Resolving power of the β-spectrometer

The resolving power of a spectrometer of the type described is given by the relation:

\[ \frac{\Delta P}{P}=\frac{Ay+B\alpha^2}{C}. \tag{5} \]

Here $A$, $B$, $C$ are coefficients determined by the geometrical parameters of the spectrometer, whose explicit values are given in [2]; $y$ is the width of the source in units of the radius of curvature of the trajectory of the central ray; $\alpha$ is one half of the aperture angle of the beam in the median plane of the magnetic field.

It follows from expression (5) that the same value of the resolving power can be obtained for different values of $y$ and $\alpha$. In order for the spectrometer to operate most effectively, it is necessary that, for a given value of the resolving power, the product $y\alpha$ be maximal, since this quantity is proportional to the number of particles that have passed through the main magnetic field of the spectrometer. It is not difficult to show that the quantities $\alpha'$ and $y'$, corresponding

...the half-angle of aperture in the median plane and the source width at which, for a given resolving power, the product \(y\alpha\) is maximal, i.e., the greatest number of particles passes through the magnetic field of the spectrometer, are determined by the expressions:

\[ \alpha'=\left(\frac{C}{3B}\cdot\frac{\Delta P}{P}\right)^{1/2}; \qquad y'=\frac{2C}{3A}\cdot\frac{\Delta P}{P}, \tag{6} \]

where, in the case when the source width is greater than the dimensions of the entrance slit, \(y\) should be understood as the width of the latter.

Table 1

For our spectrometer, relations (5), (6) have the form:

\[ \frac{\Delta P}{P}=\frac{0{,}85y+1{,}88\alpha^2}{3{,}26}; \tag{5,a} \]

\[ \alpha'=0{,}76\left(\frac{\Delta P}{P}\right)^{1/2}; \]

\[ y'=2{,}56\frac{\Delta P}{P}. \tag{6,a} \]

In this case the magnification of the spectrometer in the median plane is \(h_y=-0{,}85\), and the magnification of the spectrometer in the field plane is \(h_z=-2{,}1\).

Table 1 gives the values of \(\alpha'\) and \(y'\) for several values of the resolving power of the B-spectrometer.

The height of the entrance slit in the vertical direction was taken to be \(9\) mm. The maximum solid angle used is \(0{,}047\%\) of \(4\pi\); the maximum entrance angle in the median plane \(2\alpha\) is equal to \(4^\circ 30'\).

As an example of the relation between the efficiency of the spectrometer and its resolving power, the following values may be cited. At resolving powers of \(0{,}1\%\) and \(0{,}01\%\), the optimum product of the source width by the half entrance angle in the median plane is, respectively, \(18{,}5\cdot10^{-3}\) rad mm and \(0{,}58\cdot10^{-3}\) rad mm; in this case the luminosity of the apparatus is \(0{,}03\%\) and \(0{,}009\%\) of \(4\pi\).

Vacuum system of the \(\beta\)-spectrometer

Figure 5 shows a diagram of the vacuum chamber of the apparatus. The vacuum seals are made of rubber and must provide a pressure down to \(10^{-6}\) mm Hg, which is necessary for the direct connection of the vacuum volumes of the betatron and the spectrometer when working with very thin targets. The electron beam emitted by the accelerator is first collimated by diaphragm (1) and strikes the target (2), located at the first focus of the spectrometer. The electrons emerging from the target are collimated by the entrance diaphragm (3) and, after passing through the exit diaphragm (4), are recorded by the detector (5). It is possible to move the diaphragm in the direction perpendicular to the trajectory of the central ray by a distance of up to 50 mm with an accuracy of \(\pm 0{,}04\) mm, and to change the width of the diaphragm slits from 0 to 10 mm with an accuracy of \(\pm 0{,}03\) mm without breaking the vacuum. The vacuum feedthroughs of all dia-

The diaphragms are identical; the slit height of the first and second diaphragms is 9 mm, and that of the exit diaphragm is 27 mm. Each diaphragm consists of two aluminum bars 50 mm thick, the angle of aperture between them in the median plane being \(2^\circ\).

Fig. 5. Schematic representation of the vacuum system of the apparatus: 1—preliminary diaphragm; 2—specimen under investigation; 3—entrance diaphragm; 4—exit diaphragm; 5—radiation detector; 6—radiation detector; 7—entrance chamber; 8—central chamber; 9—exit chamber.

Recording of the spectrum is carried out by means of two scintillation counters (5), (6) using FEU-12B. The scintillators of the counters are located in the vacuum volume and are coupled to the FEU by a light guide passing through a vacuum seal. The light guide and the FEU are mounted in a metal cylinder, which can be moved on a sliding vacuum seal over a distance of up to 70 mm. The scintillator of the detector located behind the entrance diaphragm has a slot through which the main beam passes; the part of the beam traveling in the direction of the pole tips falls on the edge of the scintillator. The presence of two detectors is necessary in order to eliminate the error arising in determining the shape of the spectrum due to fluctuations in the radiation intensity of the betatron. If, as the intensity value,

...not the readings of detector (5), located at the exit from the spectrometer, but the ratio of the readings of the detectors described above, then this error is eliminated [5].

The targets are placed in a special holder rigidly connected to a metal rod introduced into the entrance chamber of the apparatus (7) through a vacuum seal in the vertical direction. The rod can be moved over a distance of up to 110 mm.

To take into account the influence of fringe fields, it is necessary to direct the beam into the region of the main magnetic field at an angle approximately 3°–5° greater than the calculated one; moreover, the new optical axis of the spectrometer must intersect the calculated one at a distance from the pole boundary approximately equal to the height of the interpolar space [6]. The bellows joints of the central chamber of the apparatus (8) with the entrance and exit chambers make it possible to rotate the latter relative to points located from the entrance and exit boundaries of the magnet at distances chosen from design considerations and equal, respectively, to 168 and 148 mm. Rotations are possible through angles of ±6° from a line displaced from the calculated optical axis by 3.5° toward the center of curvature of the trajectory of the central ray.

Since the diaphragms and detectors can be moved in a direction perpendicular to the trajectory of the central ray, it is possible to direct the beam into the magnet gap at any angle and to any point lying within a sufficiently wide interval. In view of the fact that the position of the second focus of the spectrometer can change as a result of the indicated adjustment, provision is made for moving the exit chamber of the spectrometer (9), together with the detector and diaphragm, over a distance of ±80 mm from the calculated position of the second focus. The magnet and the vacuum system are located on a massive steel plate, and a special mechanical system ensures precise displacement of the individual units of the β-spectrometer vacuum chamber relative to one another.

Conclusion

The magnet of the spectrometer described differs advantageously from the designs of magnets of other double-focusing spectrometers whose focal points are located outside the region of the main magnetic field. Thus, in contrast to a sector spectrometer with double focusing, whose magnet air gap is profiled according to a parabola [7], the design of the present spectrometer makes it possible to machine the pole pieces easily and with a high degree of accuracy; in contrast to a 180° spectrometer with an air gap in the form of two intersecting surfaces [8], the magnet of the present spectrometer has a considerably smaller weight at the same working radius. In addition to operation with the electron beam of a betatron, this spectrometer is planned for use in experiments in which it is necessary to study the energy spectrum of bremsstrahlung radiation. The spectrometer can also be successfully used for analysis of the β- and γ-spectra of radioactive isotopes. The developed β-spectrometer has acceptable calculated characteristics; its design makes it possible to carry out precise adjustment and provides the possibility of conducting a variety of physical investigations.

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