N U C L E A R I N S T R U M E N T S AND M E T H O D S 40 ( I966) 2 6 1 -2 6 6 ; © N O R T H - H O L L A N D P U B L I S H I N G CO.

CALCULATED GAMMA RAY PHOTOFRACTIONS

FOR WELL-TYPE SCINTILLATION DETECTORS

B. J. SNYDER* and G. F. KNOLL

Umverstty of Mwhigan

Recewed 30 July 1965

Monte Carlo calculations for the photofractlons of well-type gamma ray scintillation crystals are described. Leakage of secondary gamma rays, bremsstrahlung and electrons is con- sldered Calculated photofractlons are tabulated for two common- ly used well crystals. Comparisons are made between photo-

fractions for NaI, CsI and CaI2 sclntfilahon crystals. Absorption and scattering w~thln a homogeneous cyhndr~cal source have been considered and results are given for aqueous solutions of monoenergetlc sources.

1. Introduction Scintdlatton counting IS a widely used technique in

the determination of absolute gamma ray emission rates, primarily because of the reproducibility of the detector efficiency. Sodium iodide crystals with care- fully controlled dimensions and compositions are readily available to most laboratories and can be used as scintillation detectors of accurately known efficiency. The perturbing effect of scattering from surrounding media may be eliminated by counting only those interactions which deposit the full gamma ray energy m the crystal. The number of these events is pro- portlonal to the area under the photopeak, Ap, of the pulse height spectrum from the detector. By letting the factor f represent the fractional attenuation of any absorber between the source and the crystal, the abso- lute source strength N O is given by

No = Ap/(feAv), (1)

where eAp = fraction of source gammas totally absorbed m the crystal (absolute peak efficiency). Conventionally, the absolute peak efficiency is written as the product

eAp= p'eAT, (2)

where eAX = fraction of source gammas that interact at least once in the crystal (absolute total efficiency) and p = fraction of mteracting gammas that are totally absorbed (photofractlon), including all secondary particles.

Here the effects of varying the source-to-crystal geometry are reflected mainly m eAT and the photo- fraction has been shown to be only mddly dependent on such variation. Values of the absolute total efficiency have been previously calculated for crystals with the two most common shapes: the solid right circular cylinder 1) and the circular cyhnder with coaxial

* Now at Westinghouse Astronuclear Laboratory, Pittsburgh, Pa.

cylindrical well2). Photofractions for solid crystals have been given by numerous authors, e.g. Heath1), Zerby et al.3'4), Miller and Snow 5) and Jarczyk et al.6). Calculated photofraction values for well-type crystals, however, have not been previously available and are the subject of the present work.

2. Calculational procedure

Monte Carlo methods were used to simulate on a digital computer the sequence of events which occur when a gamma ray is incident upon a well-type scintil- lation crystal. Each source gamma interacting in the crystal generates a sequence of secondary gammas and charged particles. The original interaction, together with the fate of all secondaries, constitutes a single "history". Termination of each history occurs when either the total energy of the incident gamma ray is absorbed, or any secondary gamma (with energy > 10 keV) or charged particle escapes from the crystal. The possibdlty of leakage of each secondary electron is taken into account, as are the bremsstrahlung photons generated along the electron track. The number of histories for which the total gamma energy is deposited within the crystal is then divided by the total number L of interacting source gammas to give the photofractlon, p. The predicted standard deviation of the photo- fraction obtained in this way, is given by

a = [ p ( 1 - p ) / l ] ~. (3)

2.1. SOURCE GEOMETRIES

Gamma rays incident on the crystal can be considered either from isotropic sources or as finite beams in which all rays are parallel to the axis of symmetry. The isotropic sources are typical of small volume radio- isotope gamma emitters and include points (on or off the crystal axis), circular discs centered on the axis, or homogeneous cylinders located along the axis. Self-

261

262 B. J . S N Y D E R A N D G . F. K N O L L

3 (" READ DATA~

/ I' ELECTRON STOPPED ' IN CRYSTAL 'N~-- ~ ' ~ -- ]YES --~

'YES 1 [ GENERATE BREMSSTRAHLUNG

GENERATE BREMSSTRAHLUNG] LPHOTONS ALONG ELECTRON PATH ] PHOTONS ALONG L ELECTRON PATH 1 l

~ANY B R E M S S T R A H L U N G ~ ' qr ' \ ESCAPE CRYSTAL ~ , ' v E S "~

SANY BREMSSTRAHLUNG \ ~ - - IN 0 ' . ESCAPE CRYSTAL ~ @

- - ~ '~ SECOND ELECTRON -~ ~' , INO k. STOPPED IN CRYSTAL ~ ' 4 0 ~ ' ' '

f SECONDARY GAMMA N - - l y e 9. I ~ E N E R G Y C 0 01 MeV /)YES ~ @

IN o [ GENERATEBREMSSTRAHL~NG ' ~ E C O N D A R Y GAMMA j LPHOTONS ALONG/ELECTRON DATAj

2 ("-ANY BREMSSTRAHLUNG ~ _~@

E S C A P E CRYSTAL ,'YES ~" T iNO

FOLLOW HRST 1 IANN,.,LAT'ON PHOTON~

STOP ¢.. -

absorption and scattering have been considered for the cylinder source (sect. 3.2). Beams of parallel rays incident normal to the well bottom can be used to simulate collimated sources.

f E ( R QUIRED N U M B E R O F ~ ~..fflSTOmES FOLLOWEn /YES

_ _ I N O

~ ~ S O SELECT GAMMA URCE PARAMETERSJ

l S Z ~ M A ,NC,DENT NO ~ . U P O N CRYSTAL j

~ YES

~o;~PUTE TOTAL CROSS CTION AND PATH LENGTH~ where

COMPUTE COORDINATES

(POINT INSIDE C RYSTA~FI~ NO ~-SECONDARY GA~IMA ~ N O - - ~ P : p+ l ~ f f @

~ YES

(s~CO~A v GAMM~YESI

I / I I DETERMINE TYPE OF INTERACTION !

L [ coMPTON SCATTERING ] i P~IR PRODUCTION EVENT ]

I I _ C~COMPUTE DIRECTION AND] ] COMPUTE DIRECTION AND : ENERGY OF SCATTERED ] ~ENERGY OF PAIR E L E C T R O N S L GAMMA A~D ELECTRON I

( ELECTRON STOPPED - IN CRYSTAL e N O - - ~ J )

2.2. GAMMA INTERACTION PROCESSES

The general flow diagram for the Monte Carlo computation is shown in fig 1. The appropriate source geometry is sampled to select a single gamma ray defined by seven variables: rectangular coordinates (x, y, z); direction cosines (u, v, w); and energy (E). If the selected source gamma is incident upon the crystal, the total distance, p, traveled in the crystal medium to the first interaction is calculated from

p = - l n ~ / ~ , (4)

PItOTOELECTRIC EVENT

i J o oEA 2Yo ] 1 I

ELECTRON STOPPED IN CRYSTAL NO

IYES

GENERATE BREMSSTRAH LUNG PItOTONS ALONG ELECTttON PATH

t ANY BREMSSTRAttLUNG

ESCAPE CRYSTAL ~YES-- r

[NO

I f

[FOLLO - ' -- ANNItIILATION / L TOTAL ENEItG',' (~ITHIN 0 01 \le~, ) @- ~ N O ~ ABSORBED IN CRYSTAI

GAMMA YEb

A A + l

PRINT RESULTS

REQUIRED NUMBER OF HISTORIES FOLLOWED T O -4 ~: - ~ -

I YES

~J

CALCULATE PHOTOFRACTION A II - AND STANDARD DEVIATION ,

Fig 1. Diagram for Monte Carlo calculation of photofractlons

C A L C U L A T E D G A M M A R A Y P H O T O F R A C T I O N S 263

r = random number obtained from a uniform distri- bution on the interval 0-1,

bl = total gamma ray cross section 7) (without coherent scattering) at energy E.

I f this Interaction is found to occur within the crystal boundaries, the type of event is chosen by random sampling f rom the possible processes in p r o p o m o n to their individual probabllities7). For gamma rays the only important interaction processes in the scintillation medium are photoelectric absorption, Compton scatter- ang and pair production. Rayleigh (coherent) scattering has been ignored, as this process occurs only for the lowest energies where photoelectric absorption pre- dominates and only a shght change in photon mo- mentum results.

I f a photoelectric event IS sampled, the entire gamma energy IS transferred to a single electron. The electron i.s assumed to be emitted in a direction distributed uniformly in azimuth and at an average polar angle 8) relative to the incident gamma ray direction. Fluo- rescent radiation and/or Auger electrons emitted in filling the vacancy in the tuner shell following a photo- electric absorption are neglected. Treatment of the photoelectron and subsequent bremsstrahlung are discussed in sections 2.3 and 2.4, respectively.

When a Compton event is chosen, the differential Kleln-Nlshlna cross-section for free electrons is sampled by a technique of Kahn 9) to obtain the polal angle of scattering relative to the incident gamma direction. The usual kinematic relationships are then used to obtain the energies of the scattered gamma and electron. Azimuthal symmetry is assumed. Transfor- mations are made to relate all secondary gamma and electron directions back to the coordinate system fixed in the crystal using the method of Kleineckel°). The scattered gamma and electron are further followed through the crystal, as indicated in fig. 1.

I f a pair production event occurs, the positron and negatron are assumed to be emitted at a mean polar angle given by Bethe and Ashkin 11) as

0 = moc2/Ee, (5) where mo c2 = electron rest mass energy, Ee = electron kinetic energy (gamma energy minus

2mocZ). The azimuthal angles for the pair are uniformly dis- tributed, 180 ° apart. The pair electrons are treated identically in slowing down and one is chosen at random to be the positron for purposes of generating annihila- tion radiation. Two 0.511 MeV anmhilation photons are assumed to originate at the end of the positron path

and are followed in the same manner as primary gamma rays.

2.3. ELECTRON LEAKAGE

An approximate method has been used to take into account the possibility of escape through all the crystal surfaces of photo-, Compton-, and pair-electrons. For typical sizes of scintillation crystals, the fraction of all such electrons which escape is quite small and a more rigorous electron transport calculation is not warranted. The probabihty of escape is assumed to be a function only of the ratio p/Ro, where p IS the extrapolated distance to the crystal surface and R o is the electron ,,range,,3, 12). A random sample is obtained from the transmission curve of fig. 2 for each electron path. An electron is assumed to reach the crystal surface when

r < P(p/Ro) , (6) where r = random number, P(p/Ro) = transmission probability (fig. 2), R0 = electron range corresponding to the initial

electron energy 3' 12). The albedo data of Berger t4) have been used to approxi- mately account for that small fraction of electrons reflected from the crystal canning with only a slight loss in energy. All other electrons reaching the surface are assumed to escape from the crystal. Escape of any electron causes the history to be terminated, since the total source energy can no longer be absorbed in the crystal. I f eq. (6) is not satisfied, the electron is con-

I 0 I I I I

. \ "~ 0 8 ISOTROPIC ELECTRON

>- i -

0.6 o c,_ z 0

Z

O 4

0.2

O 0 [ I I 0 2 0.4 0 6

I 0 8 ) .0

NORMALIZED RANGE, F / R °

Fig 2. Electron transmission in NaI, ref.13).

264 B. J. S N Y D E R AND G. F. K N O L L

sldered to be stopped within the boundaries and all its energy deposited in the crystal.

The transmission probability for electrons (fig. 2) has been calculated with the Monte Carlo program of WalnlO 13) for an lsotroplc, monoenergetxc electron source within Nal and includes the effect of back- scattered electrons. Plotting the transmission probabih- ty vs the normahzed range, p/Ro, yields a curve nearly independent of electron energy over the region of interestl<15). An lsotroplc, internal electron source was used in calculating these data in order to simulate the nearly random directions of incidence on the crystal boundaries.

2.4. BREMSSTRAHLUNG ESCAPE

All electrons produced in gamma ray interactions are assumed to generate bremsstrahlung photons according to the average dlfferentml spectra of Zerby and Moran16). Based on these dafferentml spectra, the mean number of photons released with energy greater than 0.04 mo c2 has been calculated by Zerby 3) and used m the present calculation to represent the average number of photons emitted whale the electron is stopped. All bremsstrahlung with energy less than 0.04 mo c2 are assumed to be always absorbed in the crystal. The actual number of photons generated is obtained in a random manner by assuming that the probabdlty for emission is given by a Poisson distribution consistent with the calculated average. For each bremsstrahlung photon released, a separate selection for energy is made from the distribution above 0.04 mo c2 according to the Monte Carlo procedure given an ref. 3). The brems- strahlung photons are assumed to have an lsotropac angular distribution and are distributed in a random manner along the electron path. They are treated in the same manner as pramary gamma rays, except that further electron or bremsstrahlung leakage are not considered. Escape from the crystal of any bremsstrah- lung with energy greater than 0.04 mo ca causes termi- nation of the history, since total absorptmn of the source gamma energy as no longer possible

2.5. SECONDARY GAMMA RAY ESCAPE

The secondary photons generated by primary Compton interactions and pmr positron anmhdataon are further followed through the crystal, using the same techniques described for the primary gamma rays The coordinates of the next interaction point are calculated and tested to determine whether this point hes within the crystal boundaraes. If at is outside the crystal, the secondary photon has escaped and the history as terminated. The interaction point calculatmn is comph-

cated by the posslbdlty of intersection of the photon path with the well volume. In those cases, at is assumed that the gamma ray is unattenuated through the canning and reflector on the well surface and that the well volume is vacuum. Thus, if the gamma ray cuts through the well and re-enters the crystal, the total distance traveled is increased by the length of the path through the wet[ The case of absorption and scattering in the volume source w~thin the well is considered in sect 3.2

C

A B C D 7F8 1 75 2 00 0 75 1 50 8F8 2 00 2 00 1.125 1 50

Fig 3 Well crystal dimensions (inch).

3. Calculated results

3.1. PHOTOFRACT1ONS FOR WELL CRYSTALS

Two widely used sizes of wet[ crystals* were included in the calculation of NaI(T1) photofractlon values. As shown m fig. 3, the primary difference between the two crystals is the diameter of the well Results are glven in table 1 The calculatmns were restricted to point isotropm sources 0 2 cm from the well bottom. The effect of various lsotroplc source geometries (1 e. points, disks, cylinders) within the wet[ was investigated and gave a maximum difference of < 6 °o from these values with most geometries showing considerably closer agreement. Slmdar mild dependence of photo-

* Avadable from Harshaw Chemical Company, Cleveland, Ohio.

C A L C U L A T E D GAMMA RAY P H O T O F R A C T I O N S 265

T A B L E 1

Well crystal photofractJons isotroplc point source on axis

Energy (MeV)

0 279 0412 0 662 117 2.75 4.45 6 00 8 00

10 00

fractions crystals I)

7F8

0.8590 _ 0 0078 0.6460 zk 0 0107 0.4305 + 00111 0 2775 + 0 0100 0.1420 + 0 0078 0.0785 + 0 0060 0.0445 + 0 0046 0.0235 + 0 0034 0.0185 _ 00030

on geometry is also observed

8F8

0.8365 _+ 0 0083 0.6225 ± 0 0108 0.4520 + 0 0111 0 2650 _+ 0 0099 0 1210 _+ 0 0073 0 0815 _+ 0 0061 0 0515 _+ 0 0049 00215 + 00032 0.0125 + 0 0025

for solid

The calculations were made for 2000 histories and the indicated standard deviations were obtained f rom eq. (3). When used with the eAT values of ref. 2), these data permit calculation o f peak efficiencies for the well crystals shown. Since no tabulation of calculated results can cover all possible experimental arrangements that may arise in the laboratory, the specific calcu- lations were limited to these two commonly-used crystals. To permit extension to other cases, the com- puter program, designated BURP-5, has been made available through the Code Center of Argonne Na- ttonal Laboratory , Argonne, Illinois.

A limited investigation was made of the total absorp- tion characteristics of CsI(T1) and CaIz(Eu ) for a well crystal corresponding to the 8F8 dimensions. In fig 4, the results are presented in the form of the ratio of the calculated photofract ion to that of NaI(T1) at the same energy. The Monte Carlo program contains electron and bremsstrahlung data for Na I only and some error ~s introduced in the results for other materials at energies above 1 MeV. It is apparent that a CsI crystal w111 exhibit about 1.5-2.4 times the photofract ion o f the same size NaI crystal over the energy range of most labora tory interest. The compar ison given here is based on equal crystal dimensions and the greater density of CsI (4.510 g/cm 3) over Na I (3.667 g/cm 3) accounts for a major part of this advantage. Only small Calz(Eu ) crystals (1-2 c m 2 x 3 to 6 m m thick) have been produced to date17), but in order to provide a comparison, a hypothetical CaI 2 crystal with 8F8 dimensions was considered. Less than a 10% advantage for CaI 2 over Nal is indicated over the energy range considered.

3.2. ABSORPTION AND SCATTERING IN VOLUME SOURCES

For some gamma ray sources distributed over a finite volume, self-absorption and scattering effects may

significantly reduce the effective detector efficiencies. In a review of the current literature, ref. 18) was found to give the only calculated results in which source interactions were considered and is hmited to sohd crystal detectors. An lnvestiganon of this effect on total efficiencies of well crystals is described below.

The absolute peak efficiency for an ideal source defined as one with zero scattering and absorption cross sections, has been designated as eAp. Also e'Ap is defined as the absolute peak efficiency for the same source geometry assuming non-zero cross sections. Analogous defimtions are made for the photofract lons

26

• 24

>

~ 2.0 -J w

~) 1.8

~ 1,6 u- o

12

I I I I

Csl

Cal 2

'O 1.0

0 " 1 " I I I I 0.4 0 6 0.8 | 0 2.0

ENERGY" MeV

m

m

m

o

4,0

Fig 4 Ratio of CsI or CaI2 photofractlon to Nal for 8F8 well crystal

p and p' and absolute total efficlencies, SAT and E:tA1. By definition e'AX = fraction of source gamma rays that escape the

source without scattering or absorpt ion and interact at least once in the crystal.

!

8 A T z V ~ s .Q Vs

where V~ = source volume,

y = path length in the source extrapolated in the direction o f emission of a source gamma ray,

x = path length m the crystal, • (E) = total cross section of crystal, #(E) = total cross section of source, 0 -- mean solid angle subtended by the crystal at

the source. If the photofract ion p ' is calculated for only those

266 B. J. SNYDER AND G F. KNOLL

rays which escape f r o m the source u n a t t e n u a t e d in

energy, then the abso lu te peak efficiency is g iven by

~'AP = p' '~'~T, (8) where

p ' = t~hotofract ion given by the ra t io A ' / I ' ,

A' = n u m b e r o f g a m m a rays tha t depos i t the to ta l source energy m the crystal ,

I ' = n u m b e r o f g a m m a rays in te rac t ing at least once in the crystal .

Sca t te r ing and a b s o r p t i o n wi th in the source will affect

b o t h the p h o t o f r a c t i o n and the to t a l efficiency, bu t m

m a n y cases the change m p h o t o f r a c t i o n wdl be negli-

gibly smal l I f we ignore the possibi l i ty o f g a m m a rays

sca t tered in the crystal re -en te r ing the well and inter-

ac t ing m the d i s t r ibu ted source , then the a p p r o x i m a t i o n

that p ' = p is, at wors t , equ iva l en t to cons ide r ing the

p h o t o f r a c t i o n fol all po in t s wi th in the source to be

equa l As discussed in sect. 3.1, the la t te r app rox i -

m a t i o n is genera l ly g o o d to wi th in a few per cent for

sources wi th in the well and we h a v e consequen t l y

hm~ted c o n s i d e r a t i o n o f source se l f -absorp t ion to its

effect on eAT. Eq. (7) was numer ica l ly eva lua t ed by G a u s s i a n q u a d r a t u r e for the case o f a h o m o g e n e o u s

a q u e o u s so lu t ion comple t e ly fil l ing the well o f an 8F8

size N a I crys ta l Resul ts , t oge the r wi th those for the

equ iva l en t t r a n s p a r e n t source, a re g iven in tab le 2.

W h e n used w~th the p h o t o f r a c t i o n d a t a o f tab le 1,

a p p r o x i m a t e va lues o f the peak efficiency e'Ae can be o b t a i n e d fo r this g e o m e t r y The c o m p u t e r p r o g r a m for

so lu t ion of eq. (7) for any wel l -crys ta l ~s des igna ted B U R P - 2 and is also ava i lab le f r o m A N L C o d e Center .

T h e a u t h o r s are i ndeb t ed to D r . C. Z e r b y a n d K.

W a i n i o fo r d a t a used in the ca l cu l a t i ons a n d to Dr . G.

G y o r e y fo r sugges t ing this i nves t iga t ion .

R e f e r e n c e s

1) S. H Vegors, L L Marsden and R L Heath, AEC Report IDO-16370, 1958

2) B J Snyder and G L Gyorey, Nucleomcs 23 (1965) 80 3) C. D Zerby, m Methods m Computational Physics 1 (Ed B.

Alder, Academic Press, New York, 1962) p. 90 4) C D Zerby and H S Moran, Nucl Instr and Meth 14 (1961)

115

TABLE 2 Absolute total efficlenoes for volume source m 8F8 Nal well

crystal

Energy (MeV)

001 0015 0 02 0 O3 0 04 0.05 0.06 0 08 010 015 0 20 0 30 0 40 0 5O

~tAq

0 0697 0 2388 0 4158 0 5908 0 6539 0 6774 0 6883 0 7003 0 7051 0 6703 0 5698 0 4081 0 3284 0 2849

~" kT

0 8760 0 8760 0 8758 0 8742 0 8756 0 8749 0 8737 0 8705 0 8652 0 8046 0 6725 0 4710 0 3750 0 3207

Energy (MeV)

0 60 0 80 1 oo ! 50 2 oo 3 oo 4 oo 5 oo 6 oo 8 00

10 oo 15 oo 20 oo 30 00

~ ~.q ~ A T

0 2589 0 2890 0 2261 0 2491 0 2049 0 2234 0 1728 0 1855 0 1567 0 1667 0 1429 0 1505 0 1385 0 1444 0 1365 0 1429 0 1371 0 1429 0 1404 0 1455 0 1452 0 1499 0 1581 0 1624 0 1689 0 1731 0 1864 0 1907

I

e' XT: attenuation in aqueous source solution considered e,x~, no attenuation in source.

5) W F Miller and W J Snow, AEC Report ANL-6318 (1961) 6) L Jarczyk, H Kroepfel, J Lang, R. Muller and W Wolff,

Nucl Instr and Meth 17 (1962) 310 7) G W Grodsteln, U S NBS Circular-583 (1957) and R T

McGmnes, Suppl to NBS-583 (1959) 8) C M Davlsson and R D Evans, Rev Mod. Phys. 24 (1952)

79. 9) H Kahn, AEC Reports AECU 3259 (1954) and RM-1237-

AEC 10) D C Klelnecke, Civil Defense Research Project report no.

182-102, University of California, Berkeley, California (1962)

t l) H A. Bethe and J Ashkm, in Experimental Nuclear Physics, 1 (Ed E Segre, Wiley, New York, 1953)

12) A T Nelms, U S NBS, Circular-577 (1956) and Suppl. (1958)

13) K Walmo, University of Michigan, private communication (1965)

14) M Berger, in Methods m Computat¢onal Physics, 1 (Ed B. Alder, Academic Press, New York, 1962) p 135

15) H H Sehger, Phys Rev 100 (1955)1034 1~) C D Zerby and H S Moran, AEC Report ORNL-2454

(1958) 17) R. Hofstadter et a l , Rev Scl Instr 35 (1964) 246 is) M L Verheljke, Int J. Appl Radiation and Isotopes 15

(1964) 559.