Magnetic analysis of compton electrons produced by thin-target bremsstrahlung of electrons at 3.5 MeV

NUCLEAR INSTRUMENTS AND METHODS 72 (1969) 277-284; © NORTH-HOLLAND PUBLISHING CO.

M A G N E T I C ANALYSIS OF C O M P T O N E L E C T R O N S P R O D U C E D BY THIN-TARGET BREMSSTRAHLUNG

OF E L E C T R O N S AT 3.5 M e V

R. R. BOURGOIGN1E Laboratorium voor Natuurkunde, Rijksuniuersiteit Ghent, Belgium

Received 25 February 1969 A magnetic Compton electron spectrometer, to be used in the energy range 0.2 to 4.0 MeV, is described. The instrument is calibrated and used to measure thin-target bremsstrahlung of aluminium and gold foils, the incident electron beam baying an energy of 3.5 MeV. Experimental results are compared to theory and existing results. 1. Introduction Compton scattering transforms a photon spectrum into an electron spectrum. When this spectrum is magnetically analyzed it may give information about the original spectrum. A selective measurement of the kinetic energy of the Compton electrons, ejected in a small solid angle about the forward direction will give a spectrum essentially built tip by electrons with maximum energy. It follows that the measured Compton spectrum is a good image of the photonspectrum although it is deformed by an energy dependent efficiency factor and shifted to lower energies.

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Use of beryllium as radiator material has several advantages in the considered energy range: photoeffect may be neglected, pair production is small and may be corrected and electron scattering in the radiator foil is minimized. The Klein-Nishina cross-section formulas for Compton scattering are reliable and wel[-knownl'2). If we want to derive the photon spectrum from the measured Compton spectrum we must know the response function of the spectrometer, or in other words, the line shape produced by a monoenergetic photon beam. It has been proved possible by Latyshev 3) and later by Motz 4-6) to establish a relationship between the intensity of the monoenergetic photon beam and the measured line shape. Furthermore, once such a relationship is established a continuous photon spectrum can be determined in energy and intensity. The way in which we have analyzed our results is in principle similar to the method developed by Motz, although the spectrometer has other characteristics and the calibration procedure is different. 2. The spectrometer 2. I. DESCRIPTION

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Fig. 2. Cross-section of the sector magnet,

magnet has a 90 + sector shape. A vacuum chamber, V, having three portholes, a, b and c, is put between the pole faces of the electromagnet. To the portholes, a and b, metallic bellows, T 1 and T2, are attached so that the tubes or " a r m s " , they carry at their other ends, can be turned in the horizontal plane over an angle of 15°, with a precision of about 0.3 +. A defining baffle, K, is located in the porthole, a. The radiator, R, a beryllium foil, can be arbitrarily pushed into or drawn out of the photon beam without disturbing the vacuum. Before impinging on the radiator, the collimated photon beam enters the spectrometer forearm through a thin mylar window, m~. The undesirable electrons ejected from this window are removed by an auxiliary sweep-magnet located just behind the window. Care is taken that the magnetic influence of the magnet is negligible at the position of the radiator. The forward oriented Compton electrons generated in the radiator are accepted by the defining baffle, K, their deviation in the fringing field of the spectrometer magnet being compensated by bending the forearm of the spectrometer round in the horizontal plane. Only a small fraction of the Compton electrons, i.e., those of which the velocity vector is contained in a small solid angle defined by the baffle opening, reaches the spectrometer chamber, V. If the diameter of the radiated part of the beryllium foil is smaller than 6 mm, the cross section of the collimated photon beam is small enough not to be intercepted by the baffle, K, even if the forearm is turned over an angle of 12 ° . The photon beam then passes through the thin mylar window, m3, which seals off the opening, c, of the vacuum chamber.

After being selected in energy the Compton electrons are focussed onto the 6 mm slit at S. Here, the vacuum is sealed off by a thin aluminized mylar foil, m4, against which a NE-102 plastic scintillator, P, is pressed. This scintillator is 11 mm thick, 18 m m wide and is coated with reflecting paint. The scintillator is stuck onto an EMI 6260B photomultiplier, FM. After preamplification the pulses are counted by a single-channel pulseheight analyzer. The magnetic field, or rather, the momentum Bp is measured with the aid of a Hall-generator (Siemens FA 24) mounted on the vacuum chamber between the pole faces. The Hall-current I, (-~ 400 mA) is measured with a digital voltmeter over a standard resistor of 1 f2. The Hall-voltage, V, is measured with a compensator. 2.2. THE ELECTROMAGNET Fig. 2 shows a cross-section of the electromagnet. This sector magnet has an angle of 90 ° and a radius of curvature, p, of 17.8 cm. It is made double focussing by milling the pole faces to an angle of 3°13 ' . The mean gap distance between the pole faces, e, can be adjusted by enlarging the yoke. First order theory, as presented by Penner v) and Millman 8) was used to calculate the magnetic field configuration. The radial field distribution in the plane of symmetry is measured by examining the magnetic field as function of the distance to the optical axis. This is shown in fig. 3, for e equal to 4.42 cm. From these and similar measurements, the dependence of the fieldindex, n, on the gap width, e, is derived and is shown in fig. 4. Field measurements are done with a movable Hallgenerator of which the location is mechanically very

MAGNETIC

ANALYSIS

OF C O M P T O N

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culations, at least in the case of a sector magnet with a homogeneous field, by replacing the magnet by a fictitious one characterized by the sector angle but with a larger radius of curvature, the so called "effective p". We shall not follow this procedure as in our geometry we may, due to the bellows, compensate for the fringing field by bending the spectrometer arms over some angle e. So we consider a fictitious magnet having the same radius of curvature (17.8 cm) but with a sector angle of 90 ° - 2e. Plotting the fringing field leads to a value of 8.25 ° for e. We later found, experimentally, 9.3" to be a more accurate value. We may now also compute the optimal positions of the conjugated loci, in the symmetric case. Finally we find a magnification factor of 0.93 and the plausible value of 68.7 cm as dispersion.

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2.4. EXPERIMENTALCHECK ON THE FIRST ORDER RESULTS The radiator is replaced by a 137Cs point-source. This isotope has an electron line at 624.2 keV (Kconversion, Bp = 3381.28 Gcm). We may now follow what happens to the electron peak as the spectrometer arms gradually change in orientation. We may also see how the resolution changes and consider the influence of the source or of

precisely determined. For the 4.42 cm gap the fieldindex is 0.50 with a precision of about l/o. o/ No measurable deviations are observed for other settings of the magnetic field along the optical axis. We must stress that in later experiments the largest distance of any path to the central one is kept smaller than 2.8 cm. The variable magnet current is stable up to 1°/oo . 0.50

2. 3. INFLUENCE OF THE FRINGING FIELD

The effects caused by the fringing field are discussed by Judd and BludmanO). These effects are mainly that the optical or geometric axes are bent towards each other and that the paraxial conjugated foci are displaced along the optical axis. Furthermore, the field in the plane of symmetry is is given by :

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B: = Boil --n(A P/Po) + fl(Ap/Po) z +...3, where Ap = P - P o , P0 being the radius of curvature of the central path and p being the radius at the location where the field B_ is being measured. Double focussing occurs when the value of the fieldindex, n, is 0.5. The value offl, a parameter determining the aberrations of the sytem, is also affected by the fringing field. The fringing field is usually dealt with in the cal-

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The photon energy may be correlated with the extrapolated front edge (Bp)c or the peak position (Bp)p of the line shape. The extrapolated front edge corresponds to the maximum energy of the Compton electrons so we should use this value in the calibration. If, however, a complex type of input photon spectrum which cannot be resolved by the spectrometer is to be treated such as a bremsstrahlung spectrum the photon energy is identified with the top of the peak of the line shape. The use of the top of the line shape requires consideration of two phenomena: First the energy variation of the Compton electron with angle of ejection, and second the electron energy loss in the radiator. The first effect is negligible in our geometry but the second shall have to be accounted for. Motz et al. 4) present a relationship between the intensity of a monoenergetic photon beam and the peak counting rate of the line shape. In our geometrical situation we may use this relationship in a formal way:

Fig. 5. Resolution as function of the angle e. the detector slit on the resolution and the position of the K-peak. Up to now the Hall-generator is not yet calibrated so we can just measure the ratio of the Hall-voltage, V, to the Hall-current, /, but not the field B or the momentum Bp itself. Nevertheless, we may, in the considered energy range, accept a linear relation between these quantities, so that

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Q(k) = R(k)/ [N T(t,IQ" f i~f(O) {da(k,O)/dO}dOJ . R(k) is the peak counting rate in counts per minute for Q(k) photons of energy k impinging on the radiator per minute. N is the number of electrons in the foil per c m 2. COUNTS MINUTE

C o + C 1.Bp,

C O and C1 being at present unknown constants. We know that Co is at the most 1% of VII so it follows that

ABp/Bp = A(V/I)/(V/I). As we expect resolutions of the order of 1°/ /o, Bp is almost constant over the width of the K-peak. This means that, at least for the present, we may take as resolution the ratio of the width of the peak at half the height to the VII value of the extrapolated front edge. The experimental results are satisfactory. One of the results is presented in fig. 5, showing the resolution as function of the angle e.

3. Calibrating the spectrometer 3. I. METHOD With monoenergetic photons incident on a thin radiator foil we obtain a "line shape" by measuring the counting rate vs the momentum Bp, as shown in fig. 6.

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MAGNETIC

ANALYSIS

OF C O M P T O N

da(k,O)/dO is the Klein-Nishina differential crosssection in cmz per electron. f(O) is the probability that an electron leaving the radiator at any point, its path making an angle 0 with the forward direction, passes through the baffle opening, and is consequently detected if the magnetic field setting is appropriate. The correction factor T(t,k) includes corrections accounting for the finite thickness of the radiator and including Coulomb scattering and energy loss in the foil. 3.2. EXPERIMENTAL The radiator is replaced by a Th-B source; this source is characterized by three electron lines: the L, M and X lines. The VII values of the front edges of the peaks are used to calibrate the spectrometer. The values of the constants Co and C1 are determined by least-squares analysis. Finally we find in our case:

Bp = {(V/I)(0.99366-6 × 0.00106)+0.00012}/ /(0.4681 x 10-s), where VII is the value measured for a 6 m m slit and 6 is the width of the source in mm. The measured resolution of the lines is satisfactory. When the Th-B source is replaced by a 13~Cs source the measured K-conversion line lies exactly as predicted. A Fermi-plot shows that the spectrometer may be operated at least down to 150 keV, this limit probably being set by the quality of the source rather than by the spectrometer itself. We may now control how well this calibration in energy predicts the position o f a C o m p t o n electron line. We have therefore analyzed the 1.17 MeV and 1.33 MeV photon lines of 6°Co at the same time using these peaks to calibrate the spectrometer in intensity. Certain precautions have to be taken such as beam collimation and radiation shielding. The radiator is 0.01121 cm thick. The resulting corrected spectrum for the 1.33 MeV line is shown in fig. 7. The indicated errors are standard deviations determined from counting statistics. The extrapolated front edge is expected at B p = 5 1 5 6 Gem so that the calibration in energy is precise up to about 20/00 . We finally calculate that there are (4.44 + 0.40) x 106 photons of 1.33 MeV impinging on the radiator per minute. The photonflux can also be measured with a l~3 I t x 2 t t NaI(T1) Harshaw scintillation detector. The spectrum

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As photon spectrum the bremsstrahlung spectrum in the forward direction of 3.5 MeV electrons impinging on thin targets is used. The electron source is the 4.3 MeV linear accelerator at the Natuurkundig Laboratorium of the Ghent University. We chose gold ( Z = 79, d = 2 3 . 8 2 mg/cm z) and aluminium ( Z = 13, d = 16.89 mg/cm 2) foils as targets because of their theoretical interest. The uncertainties in target thickness are of the order of 1%. The electron current of the accelerator is measured by capturing the electrons after radiating in a Faraday cup. This cup, together with the experimental set up and the specific technical problems arising from the fact that the accelerator is a pulsed machine have been treated elsewhere ~~), 4.2. THEORY The number of bremsstrahlung photons in the forward direction with energy ranging from k to k + d k per electron of the accelerator is given by:

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R.R. BOURGOIGN1E

P'(k)dk = EdZa(Eo,k,O)/(dQdk)]o=o(n,/TB)AOdk, where the factor in brackets is the bremsstrahlung cross-section, and nB the number of nuclei/cm 3 in the target. TB is a correction for multiple scattering and energy loss and A (2 is the solid angle. Let these P'(k) photons now impinge on the Compton radiator then they shall produce N'(Bp) counts if the spectrometer is set up to detect electrons of momentum Bp; so that N'(Bp) is given by

N'(Bp) =

e'(k)S(k,Bp)dk,

(1)

0

S(k, Bp)

being the energy response function of the spectrometer. This function is already determined by the measurement of the line shape of the 1.33 MeV 6°Co peak. Let us represent this line shape by R(Bp) normalized so that the peak value is equal to one. Then the response is given by:

S(k,Bp)dk = [nrTr(k)f lo~f(O){da(k,O)/dO}dO] x x [{dEc/d(Bp)}(dk/dEc)]R(Bp)d(Sp).

(2)

The expression in the first square brackets represents an "effective" cross-section that applies to those processes in which a Compton electron with an initial recoil angle 0 is accepted by the spectrometer after emergence from the foil. The expression in the second square brackets, where E¢ is the Compton electron energy in the forward direction arises from a change of variables from photon energy to Compton electron m o m e n t u m in Gcm. As the function R(Bp) is sharply peaked in a small interval all other factors can be removed from the integral in eq. (1), when eq. (2) is introduced into eq. (1). The remaining integral

f ] R(Bp)d(Bp) represents the area under the normalized line shape. The bremsstrahlung intensity is finally given by the expression:

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number of counts/V. is a constant, 3.121 x 1013 electrons/V, of which the value is determined by the choice of the integrating condensor in the current integrator. The function T R is calculated in a way similar as given in l l). 4.3. EXPERIMENTAL RESULTS

At the energy under consideration the theoretical cross-sections for emission of bremsstrahlung in the field of the nucleus are formulated under the assumptions of the Born approximation. As it is well-known, these assumptions are rather severe and depend on the atomic number, Z, of the nucleus. The approximation will be the better, the lower Z material is used. If the screening of the atomic electrons is neglected, the cross-section dZa/dkdOdifferential in photon energy and angle is given by Sauterl2). Comparison of the Sauter formula with the exact Sommerfeld theory at nonrelativistic energies shows that the Born approximation gives good results in the long wave region of the spectrum, while it underestimates the exact cross-section more and more in going toward the high frequency limit where it shows a sharp drop to zero instead of giving a finite value as the exact theory does. The larger deviations towards the high energy tip are easily understandable as the momentum of the outgoing electron goes to zero and such an electron in the Coulomb field of the nucleus does not look like a plane wave. Calculations of screened bremsstrahlung crosssections are performed in Born approximation by Bethe and Heitlert3), but their formula is in differential form in photon energy and in photon and electron emission angles. Schiff 14) integrated this formula over outgoing electron angles and obtained a cross-section that is differential in photon energy and angle. His analytical calculations were done using an approximate screened atom potential but his calculations are restricted to energies that are large compared to the electron rest energy. Starfelt and Koch 15) have performed experiments with electrons of 2.72, 4.84 and 9.66 MeV using beryllium, aluminium and gold as targets. For the two lower electron energies and in the long wave region of the spectrum they find the shape of the spectrum to agree with theory but great differences occur in the upper part of the spectrum. For beryllium and aluminium the experimental points lie below theory but the experimental errors are large (up to 28%). As for gold, the experimental points lie without

MAGNETIC

ANALYSIS

OF

d o u b t above theory for 4.54 MeV and still more so for 2.72 MeV. Figs. 8 and 9 show our experimental results. The theoretical curves for the bremsstrahlung intensities are calculated from the Scbiff and Sauter formules setting the p h o t o n emission angle equal to zero. The indicated errors to the experimental values contain only standard deviations determined from counting statistics. Other sources of incertainties may lead to errors of about the same order. A shift of the energy axis is possible due to uncertainties in the energy calibration of the spectrometer and of the accelerator but this can only be 0.1 MeV at the most. Our results for gold show that even when screening is included the Born approximation does not hold, the largest deviations appearing at the high frequency limit of the spectrum. In the case o f aluminium the shape of the spectrum seems to agree with the Schiff formula better than in the preceding case, but nevertheless the experimental points lie above theory in the high energy region. This is in contradiction to the Starfelt and Koch results.

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I should like to thank Prof. Dr. J. Verhaeghe, Director of the N a t u u r k u n d i g L a b o r a t o r i u m for experimental facilities, Prof. Dr. C. Grosjean for calculating the function, f(0), and Prof. Dr. V. Vanhuyse for stimulating discussions.

References i) A. T. Nelms, NBS Circular 542 (1953). 2) H. E. Johns et al., Can. J. Phys. 30 (1952) 556. a) G. D. Latyshev et al., J. Phys. (USSR) 3 (1940) 251. 4) j. W. Motz et al., Rev. Sci. Instr. 24 (1952) 929. ~) J. W. Motz et al., Phys. Rev. 89 (1952) 968. ~) J. W. Motz et al., Phys. Rev. 96 (1954) 1344. 7) O. Penner, Rev. Sci. Instr. 32 (1961) 150.

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R. R. B O U R G O I G N 1 E

8) B. Millman, l'Onde Electrique 421 (1962). 9) D. L. Judd and S. A. Bludman, Nucl. Instr. 1 (1957) 46. 10) M. Berger and J. Doggett, J. Res. Nat. Bur. Std. 56 (1956) 6355. 11) R. R. Bourgoignie, V. J. Vanhuyse and W. L. Creten, Z. Physik 188 (1965) 303.

12) F. Sauter, Ann. Phys. 20 (1934) 404. la) A coherent summary of the brernsstrahlung cross section formulas is provided by Koch and Motz in Rev. Mod. Phys. 31 (1959) 920. a4) L. I. Schiff, Phys. Rev. 83 (1951) 252. 15) N. Starfelt and A. W. Koch, Phys. Rev. 102 (1956) 1598.