Monte-Carlo calculation of photofractions and intrinsic efficiencies of cylindrical NaI(Tl) scintillation detectors

NUCLEAR

INSTRUMENTS

AND

METHODS

23

(1963)

13-18;

NORTH-HOLLAND

PUBLISHING

CO.

MONTE-CARLO CALCULATION OF PHOTOFRACTIONS AND INTRINSIC EFFICIENCIES OF CYLINDRICAL NaI(TI) SCINTILLATION DETECTORS C. W E I T K A M P

Institut /~y Neutronenphysik und l~eaktorlechnik, Ifern/orschungszentrum Karlsruhe R e c e i v e d 30 N o v e m b e r 1962

This p a p e r describes a Monte-Carlo calculation of efficiencies a n d p h o t o f r a c t i o n s of N a I ( T I ) detectors for g a m m a - r a y s between 0.2 a n d 10.0 MeV. T h e crystals considered are r i g h t circular cylinders r a n g i n g in size f r o m 3" d i a m e t e r × 2" length to 5" d i a m e t e r × 6 ~ length. T h e results m a y be used to deter-

m i n e t h e o p t i m u m crystal dimensions for g i v e n c r y s t a l v o l u m e and g a m m a - r a y energy. The influence of collimators of different a p e r t u r e s is investigated. E x c e p t two characteristic cases, however, results are g i v e n for uncollimated r a d i a t i o n only.

1. Introduction

account the whole energy range. Moreover, most of the programs existing, especially those calculating the entire response spectrum, take rather a long time, even when run on a fast computer. A rapid Monte-Carlo program therefore has been developed to calculate the photofraction and the efficiency of right cylindrical NaI(T1) detectors for ?-ray energies from 0.2 to 10 MeV. Pair production was treated thoroughly, but no corrections were made for bremsstrahlung and escape of electrons from the crystal. It can be seen easily that, even for the smallest crystal and the highest energy, the error introduced by those effects is below the statistical fluctuations. The calculations were carried out for point sources located on the crystal axis at various distances; the influence of collimators of different apertures was investigated by dividing the crystal face into ten concentric rings. The program was set up for the simultaneous consideration of four crystals of the same diameter but different length, and 6000 Monte-Carlo histories were computed for each set of energy and geometry parameters.

As the photofraction of a gamma-ray scintillation detector is an important parameter in nuclear spectroscopy, m a n y authors have performed calculations of the photofractions for given source and crystal geometry. While the approximations used in older publications I -5) are somewhat coarse (pair production is often considered as complete absorption of the incident photon), more recent papers give a rigorous treatment of the annihilation radiation 6) and sometimes take into account even small effects like bremsstrahlung and electron escape energy losses from the crystal 7"8). None of the papers mentioned, however, allows the photofraction optimization of the dimensions of scintillation detectors as used in capture g a m m a coincidence and angular correlation work. Either the variety of crystals considered is too small or too widely spaced, or the calmflations do not take into 1) D. Maeder, R. Mtiller a n d V. W i n t e r s t e i g e r , H e i r . Phys. A c t a 27 (1954) 3. 6) j . G. Campbell a n d A. J. F. Boyle, A u s t r a l i a n J. Phys. 6 (1953) l ? l . 5) M. J. B e r g e r a n d J. D o g g e t t , R e v . Sci. I n s t r . 27 (1956) 269. 4} M. J. B e r g e r a n d J. D o g g e t t , J. Res. N a t . Bur. Stand. 56 (1956) 355. 5) M. H. WS.chter, W. H. Ellett a n d G. L. Brownell, Rev. Sci. Instr. 28 (1957) 717. 6) W. F. Miller a n d ~,V. J. Snow, R e v . Sci. I n s t r . 31 (1960) 905. 7) C. O. Zerby, TID-7594, P a p e r 8 (1960). 8) W. F. Miller a n d "W. J. Snow, ANL-6318 (1961).

2. Method of Calculation 2.1. PHOTON HISTORIES The principle of the Monte-Carlo method consists in simulating the history of a great number, N, of individual photons of energy E~ on their way through the crystal. Physical quantities that obey 13

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CALCULATION

OF PHOTOFRACTIONS

AND EFFICIENCIES

a statistical distribution are sampled according to the given distribution function. If n T is the number of photons interacting at least once, thus causing a pulse at the detector output, and if n~ photons lose allt their energy within the crystal, the intrinsic efficiency*t is defined as T = n T / N , and the photofraction is P = n~,/nT, nT and n e, and therefore T and P are characterized by two indices i(i = 1... 4) and fl(fl = 1...10) indicating the number of the crystal of the " s e t " and the collimator aperture, respectively. The sampling of the photon histories m a y easily be understood from the schematic diagram of fig. 1 and need not be described in detail. Fig. 2 gives I

OF SCINTILLATION

DETECTORS

15

photon under consideration. In the case of a pair production the separation of the amounts of energy lost in the different crystals is somewhat more difficult than for Compton and photoelectric processes. Therefore four quantities E t . . . E 4 are introduced "gathering" the fraction of energy dissipated by the gamma in the largest crystal only, in the two largest crystals, in the three largest ones and in all four crystals, respectively. 2.2. R A N D O M S A M P L I N G

If W ( y ) is the given probability distribution of the quantity y ( a < y <~ b), and r(0 < r ~< 1) are so-called random numbers with the distribution function [ 1

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the meaning of the geometry parameters, n is the number of Monte-Carlo histories already considered, l is the free path of the photon between two successive interactions, q~, ~ and r are the azimuthal and polar angles of the propagation direction and the endpoint of the gamma-ray in the crystal coordinate system, and t/ and ~ represent the scattering angle and azimuth involved in Compton scattering. ~ stands for the energy of the photon in units of m0c2; the mean free path of gamma-rays with energies < 50 keV being 3 x 10 -2 cm or less, photons with e < ~ * - 0 . 1 are assumed to be completely absorbed. The subscript k indicates the number of the interaction undergone by the Because of the finite resolution of t h e d e t e c t o r s y s t e m a p h o t o n m a y c o n t r i b u t e to the p h o t o p e a k of t h e pulse h e i g h t s p e c t r u m e v e n w h e n losing less e n e r g y w i t h i n t h e crystal t h a n E~, s a y E~ E a . T h e choice of E o is not critical. I t a p p e a r s reasonable to t a k e 1"2a -- C. ~/E.~. the c o n s t a n t C being determ i n e d b y s e t t i n g ( E a / E . , ) ( E G / E e ) E e = e~2 keY 10 ° o. In the higher e n e r g y region (E~ ~ 2 MeV) E c has been chosen from t h e resolution m e a s u r e m e n t s of K o c h and Footeg). "~t Often referred to as interaction ratio.

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CALCULATION

OF P H O T O F R A C T I O N S

AND EFFICIENCIES

and a, b, c, d and e are functions of the energy parameter ~k of the incident photon. Starting with the solution v = 1, (2) m a y easily be solved b y the method of Newton to a n y accuracy wanted. 2.3. C R O S S S E C T I O N S

Using the tables of Grodstein 11), Crouthame112) has calculated the cross sections of 0.1 per cent thallium-activated sodium iodide for photoelectric effect, Compton effect and pair production. These data were used in the present calculation and, if necessary, interpolated quadratically. Rayleigh and Thomson scattering of gammas were not taken into account. 3. Results

The results of the calculation for uncollimated radiation are given in figs. 3, 4 and 5. It should be noted that the m i n i m u m of the photofraction does not coincide with the m i n i m u m of the intrinsic 11) G. W. Grodstein, Nxt. Bur. Stand. Circular 583 (1957). 12) C. E. C r o u t h a m e l ed., A p p l i e d G a m m a - R a y S p e c t r o m e t r y (Pergamon Press, Oxford, 1960).

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DETECTORS

17

efficiency and of the total cross section of NaI located at 5 MeV but is shifted to about 3 MeV. This m a y be explained b y the fact that, in this energy range, small Compton scattering angles and therefore small energy losses of the gammas are very frequent; thus, absorption of secondary gammas is not likely, and little energy will be transmitted to the crystal. Pair production, on the other hand, yields secondary radiation of relatively low energy which m a y easily be absorbed within the crystal, and therefore increases the photofraction in the energy region where its cross section becomes relevant. The discontinuity of the P curves shown in figs. 3, 4 and 5 results from the energy dependence of the resolution. The first escape peak was taken to be separated from the full energy peak up to 6.6 MeV only. Efficiency and photofraction are both increasing with increasing size of the crystal. But while the length dependence of T and P is about the same, the photofraction varies rather markedly with the diameter of the crystal, whereas the efficiency does not (cf. fig. 6). To choose the optimum length-to10,

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Fig. 5. I n t r i n s i c efficiency T and p h o t o f r a c t i o n P for u n c o l l i m a t e d r a d i a t i o n of energy E~. The p a r a m e t e r is the crystal length L. Crystal d i a m e t e r D = 5", source distance H = 15 and 30 cm. The a r r a n g e m e n t of each set of curves corresponds to the a r r a n g e m e n t of the L values left of the figure.

18

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plot for E~ = 2 MeV and H = 15 cm. The dotted lines correspond to one set of crystals each, i.e. to one D value, the various points result from the variation of L. For every crystal volume there is one optimum diameter. It may be obtained by determining the (dotted) curve that touches the envelope (solid line) at the given volume. The corresponding D value is the crystal diameter wanted. The collimator influence results in an improvement of the photofraction (see fig. 8a). For crystals of large diameter and small length it may happen, however, that at energies of about 3 MeV the photofraction as a function of the collimator aperture goes through a maximum (fig. 8b). The explanation is again the dominating forward direction of Compton scattering at higher energies where the probability of multiple processes increases rapidly with increasing "effective" thickness of the crystal. Only few of the results of the present calculation

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diameter ratio for a given crystal volume and given energy of the incident gammas, it is useful to plot P against the crystal volume. Fig. 7 shows such a 060

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7. P h o t o f r a c t i o n P f o r 2 M e V y - r a y s vs. c r y s t a l E~, = 2 M e V , H = 15 c m .

Acknowledgement The author would like to express his gratitude to Professor K. Wirtz for the possibility of doing this work in his laboratory, and to Dr. K. H. Beckurts for his advice and for many helpful discussions. He is very much indebted to Dr. W. Michaelis for verifying the calculation as well as for the help given throughout the work, and to Miss A. D6derlein for setting up and running the machine program.

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