Theory of Moxon-Rae type detectors II

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INSTRUMENTS

AND

METHODS

I30

0975) 4 4 3 - 4 4 7 ;

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NORTH-HOLLAND

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THEORY OF M O X O N - R A E TYPE DETECTORS II S. S. M A L 1 K a n d C. F. M A J K R Z A K

Physics Department, University of Rhode Island, Kingston, R.L 02881, U.S.A. Received 1 July 1975 Theoretical development o f the M o x o n - R a e detector's detection m e c h a n i s m has been extended to include photoelectric effect and pair production for (7,e) processes in addition to the previously considered C o m p t o n scattering. F u r t h e r the converters n o w include AI and Pb along with C that was treated in the earlier theory. Quantitative calculations o f the electron production probability, f(i)(E,t) and the electron transmission probability, f(2)(E,t)

have been m a d e for C, A1 a n d Pb converters for p h o t o n energy E.~ up to 10 MeV. A c o m p a r i s o n o f our f ( 1 ) ( E , t ) f ( 2 ) ( E , t ) values with the recently published M o n t e Carlo calculations o f the response o f M o x o n - R a e detectors indicates that the two sets o f values agree rather well. Various parameters useful for similar calculations, employing other converters, are also included.

i[. Introduction In a previous work by Malik 1) (henceforth referred 1:o as I), a systematic theory of the 7-detection mechanism of a simple Moxon Rae type detector had been advanced. It was shown that the approximate linearity of the 7-detection efficiency as a function of photon energy, E~, results from the combined probabilities for the production of electrons in and, transmission of electrons through, the converter. It was also shown that the probability for the production of pulses by the electrons through the thin scintillator can, with rea,;onable justification, be taken independent of the energy of the electron traversing through the scintillator. Further considerations regarding the specific application of the M o x o n - R a e detector to the measurement of the radiative capture of neutrons were also treated. The response probability, F(E, t), of the detector proper depends primarily on the electron production probability, f ( l ) ( E , t ) and the electron transmission

.f(2)(E,

probability, f(2)( E, t). f(3) (E, t) probability for producin9 ,~,'cintillations was taken to be constant. The converter employed for the discussion of these probabilities was a cylindrical carbon converter. (7, e) processes considered consisted only of Compton scattering. Multiple and plural scattering of electrons in graphite were treated using Moli~re 2) theory. In an independent work lyengar et al. 3) used Monte Carlo technique to calculate the response of the Moxon Rae detectors employing different converters. Their results show that the efficiency, e, per MeV photon energy is ~ 1% for all the converters and that a better linearity is achieved with higher Z converters. The results of the calculations performed by these authors are equivalent to the values of [f~l)(E, t) 443

t)]. Both calculations are based upon probabilistic considerations. The purpose of this work is to extend the theory given in 1 by including photoelectric effect and pair production [in the (y, e) processes] along with Compton scattering. This necessitates modifications in our previous treatment of f(1)(E,t) and f(2)(E,t). In addition we include AI and Pb converters as well. The results of e per MeV in I were given in a relative sense. In this work calculations off(1)(E, t)f(2)(E,t) are performed to obtain quantitative results for all the three converters. Tables and graphs of certain parameters have also been included so that these can be used for calculations of e employing different converters. The results of our calculations compare rather well with those obtained by Monte Carlo technique a).

2. Theory For the sake of completeness we rewrite the mathematical definition of the response probability F(E, t) as

F(E, t) = f(1)(E, t)f(2)(E, t)f(a)(e, t).

(1)

The f ' s in eq. (1) have already been defined. E and t refer to the 7-ray energy and the thickness of the converter. By takingf(3)(E, t) = Ao as before we define

F(E, t)

e- - - f(')(E, t)ftZ'(E, t). f(a)(E, t)

(2)

It is e that we shall evaluate and compare with Monte Carlo calculations 3). 2.1. f(1)(E, t), THE ELECTRONPRODUCTIONPROBABILITY By following the procedure discussed in l, we write

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S. S. M A L I K A N D C. F. M A J K R Z A K

f(~)(E,t)

=

[ 1 - expOmnj%(E)/~r(E).

Eq. (3) is analogous to eq. (6) in I. n is the number of converter atoms per cm 3. %(E) is to be defined as

ap(E) = a~,p+ ~p+np.

(4)

r and tc in eq. (4) stand for the photoelectric and pair production cross section respectively, ~,~ and the reasons underlying its introduction were discussed in I. In order to calculate J'(~)(E, t) quantitatively %,v must be numerically evaluated. It is, however, more useful to evaluate (a~,v/a~) per electron as a function of incident photon energy, since it can be readily applied to any converter. This is because all Compton cross sections are Z times the cross section per electron. Because of its general usefulness, o-~,p/cr~ is tabulated in table l and shown graphically in fig. 1. In order to evaluate % (which is defined in the same manner as c%,v) we take the angular dependence of for tmpolarized radiation 4) as dz

--

d~2

c c -

sin 2 0

TABLE I

(3)

(5)

(l -ficosO) 4'

where 0 ~ angle between the dilx:ction of incident photon k and the momentum of the electron p in the (pk ) plane. fl = vie with v the electron velocity. Again, because of extensive tables being available for r as a function of E.,, and Z, more general use can be made of the quantity %/z. This has been done for C,

Values o f

(re,~/ae

per electron as a function o f photon energy E v in MeV.

Ev (MeV)

¢;e,p/(r

E~ (MeV)

cre,p/c~

0, I 0.3 0.5 0.7 1,0 1.5

0,47 0.54 0.60 0,64 0,69 0.75

2.0 3,0 5,0 6.0 8.0 10,0

0,79 0,84 0.89 0,91 0.93 0.94

Sn and Pb as a function of E~.. The results are given in table 2 and are graphically represented in fig. 2. An inspection of table 2 and fig. 2 shows that for E~ ~0.8 MeV, the ratio rv/r is essentially independent of Z. This is to be expected since the K-shell binding energy in all cases becomes negligible compared to E~.. Calculation of ~% is quite tedious because of the number of parameters involved. However, an approximate procedure can be adopted by evaluating

;% =

f

;c(E+)cos(Oo)dE+,

(6)

J

1.0

0,8

1.o[

0.6

0,4

'7 Pb

J i 0.2

O, 1

&2 I

-~ 1O0

.........

t 000

500

1500

E~ (keV) 0

2

4 E.) (M~V)

6

8

t0

Fig. 1. Ratio, ~e,]~/cre, for C o m p t o n scattering per electron as a function of the g a m m a - r a y energy, E , , in MeV.

Fig. 2. Ratio rp[~: for photoelectric effect per atom for carbon, a l u m i n u m , th~ and Pb a t o m s as a function o f the g a m m a - r a y energy, E.,, in MeV. T h e values o f r p / r , for C, are calculated only [br Ev = 0.1.0.15 and 0.2 MeV.

MOXON-RAE

TYPE

TABLE 2 "Values o f To/r as a f u n c t i o n o f p h o t o n e n e r g y E ~ in M e V for Al, Sn a n d Pb.

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DETECTORS

a wide range of energies. For a given electron or positron energy we write, following the integration procedure of I,

f(2)(E+, t) = R°(E+)/t,

E~, (MeV)

"gp#" C

AI

Sn

Pb

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5

0.57 0.66 -

0.56 0.66 0.72 0.77 0.81 0.83 0.85 0.92 -

0.54 0.64 0.72 0.77 0.80 0.83 0.87 0.91 0.95

0.43 0.58 0.67 0.73 0.78 0.81 0.87 0.90 0.95

where R°(E+) is the range of the (positrons) electrons for energy (E+)E_ parallel to the axis of the converter. The admixture of energies in pair production is accounted for by weigthing f ( Z ) ( E , t ) in terms of the fractional cross sections giving rise to different energy electrons, i.e.

f(2~(E,t) = _1 [~rc+z RO(E)+ t L ~

+~ o where x(E+)dE+ is the differential cross section for the emission of a position in the energy range between E+ and E+ + d E + . 0 o =mocZ/(xl/3E+) is the most probable angle for the emission of a positron of energy E+ 5). The accuracy of ~p/~: obtained using the above procedure is not high enough to warrant any numerical tabulation. This completes our discussion and the procedure for evaluation off(1)(E, t). t) THE ELECTRON TRANSMISSION PROBABILITY The discussion and evaluation of f(2)(E, t) can be carried out in the same manner as in I. We cannot, however, continue to assume that all the electrons have an energy E = E~ - although for Compton and photoelectrons the assumption of E ~ E~ remains valid but the pair electrons have a continuous distribution over 2.2. f(2)(E,

80

,~(E+){R°(e+)+R°(E_)}d~+

v

40 Pb

12rcnZ2

20 4

6

8

(8/

(9)

where Co is a constant to be determined by applying the equation for 0 to 3.0, 5.0 and 7.0 MeV results for Pb targets. We find an average value of C o = 2.03. Kulchitsky and Latyshev also conclude that the mean angle of scattering is approximately twice as small as that theoretically calculated V). We thus take Co = 2.0. 2 = transport mean free path given by

~, 60

2

.

To account for multiple and plural scattering we have adopted the following procedure. For C the values of the mean scattering angle, 0, have been previously calculated in I. For Pb there exist experimental measurement by Slawsky and Crane 6) for three values of electron energy, i.e. 3.0, 5.0 and 7.0 MeV and by Kulchitsky and Latyshev 7) for electrons of 2.25 MeV. A comparison of these experimental values was made by Slawsky and Crane with the case of complete diffusion in accordance with the theory of Bethe et al.8). These authors concluded that the experimental widths for Pb were narrower than the calculated values 6' 7). Since the effective thickness of Pb converter for all electron energies is > 200 mg/cm 2, the case of thick target 9) and complete diffusion should be applicable provided the results of the theory are adjusted in accordance with the experimental observation6). This enables us to express the most probable angle as Co 0 = (x/2) ~,

xx

0

(7)

10

E (MeV) Fig. 3. V a l u e s o f the a v e r a g e d m e a n s c a t t e r i n g a n g l e , in degrees, for c a r b o n a n d P b c o n v e r t e r s as a f u n c t i o n o f e l e c t r o n e n e r g y , E, in MeV.

2

e4 W 2

(2ap~

(W 2 -- m 2 C4)2 In \hZ+/l"

In eq. (10) W = total energy of the electron, p - electron momentum,

(10)

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S.S.

MALIK

AND

a = B o h r radius, e, m o c 2 are the electron charge (e.s.u.) and the electron rest m a s s energy respectively. X is the target thickness in cm. The value o f X is, however, not unique but varies between 0 and a length equal to the range of the electron in the particular target. H e n c e the values o f 0 are averaged, as in I, by using the f o r m u l a

O(E) =

Odx.

(11)

0

Fig. 3 gives the values of 0(E) as a function o r E for graphite and Pb. i

i

i

-I~ b

i

zx

o

12

o

&

o

C. F. M A J K R Z A K TABLE 3 V a l u e s o f efficiency, e, p e r M e V in u n i t s o f 10 a as a f u n c t i o n o f p h o t o n e n e r g y E~¢ ( M e V ) f o r C, AI, a n d P b c o n v e r t e r s .

E~. (MeV)

a:/MeV C AI Pb 2.58 c m 2.3 c m 0.64 cm P r . C a l . a M . C . h P r . C a l . a M . C . b Pr. C a l . a M . C . b

0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0

4.3 6.8 8.3 8.3 8.2 8.1 7.2 6.5 6.0

o

"~ - - P r e s e n t calculations o - - M o n t e Carlo method

4 i

o

I

I

I

I

I

I

I

c

I

I

i

AI

0

e-8

0

o

10.3 12.0 10.7 11.0 ll.9 12.4 13.0 13.9 14.6

9.3 11.4 10.6 11.8 12.4 12.7 13.0 13.0 12.9

For a converter with arbitrary Z we shall assume that for a given electron energy O(E) is proportional to Z ( s i n c e ) . ~ o c Z ) in accordance with Bethe et al.S). This permits a linear interpolation and thus the values of O(E), for a given converter and given E, can be easily obtained from the graph. In terms of the actual range R(E) we can now write

o o

o

l

E 0) o

O

+ I

0

3.8 7.2 8.3 8.2 8.1 7.4 7.4 6.7 6.7

a Pr. Cal. - present calculations. b M . C . - M o n t e C a r l o c a l c u l a t i o n s . AI c o n v e r t e r t h i c k n e s s u s e d in M . C . is 2.24 c m .

8

kid

3.8 7.2 8.0 8.4 8.2 8.3 8.1 8.1 8. l

o

O

~_ 4

3.9 6.9 8.2 S.0 7.3 6.9 6.9 6.2 5.8

t

I

1

I

I

e

o

o

8

] f E - 2m0c2

70

~c('E+)

[R(E+ )cosO(E+ ) +

1.

(12)

U 2.3. VALUES OF e FOR C, A1 AND Pb CONVERTERS

8

8

i

5 Ey (MeV)

a~.+-c R(E)cosO(E) + (7

+ R(E_)cosO(E_)}dE+

C

8

I

i

i

10

Fig. 4. Efficiency, e, p e r M e V in u n i t s o f l 0 a f o r c a r b o n , a l u m i n u m a n d l e a d c o n v e r t e r s as a f u n c t i o n o f the g a m m a - r a y e n e r g y , E.~, in M e V . T h e r e s u l t s o f b o t h t h e p r e s e n t c a l c u l a t i o n s a n d M o n t e C a r l o c a l c u l a t i o n s a r e s h o w n . T h e s t a t i s t i c a l e r r o r s in t h e M o n t e C a r l o c a l c u l a t i o n s , as q u o t e d b y l y e n g a r et al. a) a r e 5 % f o r E~,>2.0 M e V , 1 5 % at E.~ = 0.5 M e V a n d 1 0 % at E v = 1.0 M e V .

The calculation o f e is now straightforward by using values o f at, T, to, o- and R(E) from Siegbahn lo) and other literature11). 111 table 3 and fig. 4 our values o f a / M e V for C, AI and Pb converters are shown as a function o f E r along with those o b t a i n e d by Monte Carlo techniqueS). 3. Conclusion

This work is an extension of the Moxon Rae detector theory advanced in I. We have incorporated all the three primary interactions namely the C o m p t o n scattering, the photoelectric effect and the pair production for the (y, e) processes. The results for graphite

MOXON-RAE

TYPE DETECTORS

in I were presented in a relative sense. We show that quantitative results can be obtained with relative ease using the extended theory given in this work. A comparison of our results with Monte Carlo ralculations of Iyengar et al.3) shows that there is a rather good agreement between the two sets of values. Our treatment of the contribution to e due to the pair electrons is somewhat overestimated. This is because, Jn our procedure, we have essentially neglected the wide angle emission of the pair electrons. Our values of e for C are everywhere within the statistical error of the Monte Carlo calculations except for E~ = 4.0 and 5.0 MeV. The maximum difference :in the results of the two calculations is for E~.--5.0 MeV. The Monte Carlo value is within 15% of our value. For AI the maximum differences arise for E.,= 8.0 and 10.0MeV. The corresponding Monte Carlo values differ by approximately 17% from the present calculations. For a Pb converter the Monte Carlo results are within 10 % of our values. As seen from fig. 4, present calculations suggest that the value of e/MeV for a Moxon Rae detector, employing an AI converter, is essentially constant for gamma ray energy, E;., between 2 and 10 MeV. This conclusion has to be tempered with the statement that, as the need for more accurate calculations of e arises,

447

better evaluation of pair production contribution should be made. We, however, believe that the theoretical framework advanced in I and extended here provides a sufficiently complete basis for the understanding and evaluation of the properties of Moxon Rae detectors.

References 1) S. S. Malik, Nucl. Instr. and Meth. 125 (1975) 45. "~) G. Moli6re, Z. Naturforsch. 3a (1948) 78. 3) K. V. K. lyengar, B. Lal a n d M. L. Jhingan, Nucl. Instr. a n d Meth. 121 (1974) 33. 4) C. M. D a v i d s o n and R. D. Evans, Rev. Mod. Phys. 24 (1952) 79. 5) B. Lal a n d K. V. K. Iyengar, Nucl. Instr. a n d Meth. 79 (1970) 19. 6) M. M. Slawsky and H. R. Crane, Phys. Rev. 59 (1939) 1203. 7) L. Kulchitsky and G. Latyshev, Phys. Rev. 61 (1942) 254. 8) H. A. Bethe, M. E. Rose a n d L. P. Smith, A m . Phil. Soc. Proc. 78 (1938) 573. 9) H. Frank, Z. Naturforsch. 14a (1959) 247. 10) G. K n o p and W. Paul, in Alpha-, beta- and gamma-ray spectroscopy (ed. K. Siegbahn; N o r t h Holland Publ. Co., A m s t e r d a m , 1965) vol. 1 p. 18; C. M. Davisson, id. vol. 1 p. 827. 11) M. J. Berger a n d S. M. Seltzer, Tables o f energy losses and ranges o f electrons and positrons, N A S A S.P. 3036 (1966); L. K a t z and A. S. Penfold, Rev. Mod. Phys. 24 (1952) 30; E. Storm and H. I. Israel, Nucl. D a t a A7 (1970) 565.