Direct calculation of the total efficiency of cylindrical scintillation detectors for extended circular sources

~

Radiat. Phys. Chem. Vol. 48, No. 1, pp. 23-27, 1996 Copyright © 1996ElsevierScienceLtd Printed in Great Britain.All fights reserved 0969-806X/96 $15.00+ 0.00

Pergamon 0969-806X(95)OOO47-X

DIRECT

CALCULATION

CYLINDRICAL

OF THE TOTAL

SCINTILLATION

EXTENDED

CIRCULAR

EFFICIENCY

DETECTORS

OF

FOR

SOURCESt

Y O U N I S S. SELIM and M A H M O U D I. ABBAS Physics Department, Faculty of Science, Alexandria University, Alexandria, Egypt (Received 11 January 1995)

Abstract--By the use of spherical coordinates, direct expressions for the total efficiency of a cylindrical NaI(TI) detector arising from a non-axial point (p > R) and coaxial-circular disk (S > R) of isotropic radiating sources, are deduced. Results are compared with previous treatments. Copyright © 1996 Elsevier Science Ltd.

INTRODUCTION

The computed values of f2s are in accordance with those of Masket et al. (1956). By the use of spherical coordinates, a direct mathematical expression for fl~ (contour A A ' C ' B ' B C A ) is derived as (see Fig. 1):

By the use of Monte Carlo calculations only the work of Beam et al. (1978) have treated the total efficiency of a cylindrical detector for the case of an off-axis point source with displacement greater than the detector's radius. In this case the solid angle subtended by the cylinder (2R x L) can be resolved into two components: the solid angle due to the nearest circular face [~t and the solid angle due to the lateral side of the cylinder D~.

n,=2(,/ (h + Lh+L ) 2 + p2 _ R 2 h

+2

MATHEMATICALVIEWPOINT

4~mxsin 0 dO -

4~,~ sin 0 dO

(6)

1

Treatment of Of has been given in a previous work (Selim and Abbas, 1994). By the same spherical trigonometry technique, the solid angle subtended at a point by the lateral surface of a right circular cylinder is derived as: ~.---2

fo

[G(h)-G(h+L)]d~,forh>O

(1)

G ( L ) d~, for h = 0

(2)

where

-R 0~ = tan -I p h '

0

h
(3) ~ax

where, =

h

,

(7)

p2 _ R 2 + (h + L) 2 tan 2 0"~

~m"=c°s-' "

[G(h)+ G(L - h)l d~, for = cos-Z(R/p)

(4)

=

x ( x 2 + p2 _ Rp cos ~) (x 2 + p2 sin 2 ~)(x 2 + p2 + R 2 _ 2Rp cos ~),/2

=cos-'

2p(h+L)t--~n0 p2

}

R2+h2tan20~

-~ht--ffn~

/

(8)

Systematic calculations of ~f, f~s and ~tot~l, of cylindrical detectors (2" x 2" and 3" x 3") for a point source at h = 1 cm from the surface as a function of the displacement p were calculated and represented in Fig. 2(a) and (b). Also, making use of the previous technique leads us to a direct mathematical expression for the total efficiency of a cylindrical scintillator using an off-axis radiating point source. The location of the point

and

6(x)

0~ = tan -l ~/p2 _ R 2

and

D~ = n - 2

~ = 2~ - 2

02 = tan -l x/p2 _ g2 h+L '

0, = t a n - i P - R h+L'

(5)

tPublished as part of the Second Radiation Physics Conference, Menoufia, Egypt, 20-24 November 1994. 23

24

Younis S. Selim and Mahmoud I. Abbas

ie

(1) the upper surface and emerge from the cylinder lateral side, d

Y

h

P where Y = p cos ~b + x / R : - p2 sin: ~b

Ic

(9)

(2) the upper surface and emerges from the cylinder base,

I

i

L d =- cos 0

(lO)

(3) the side of the cylinder and emerges from the opposite side, " . . . . . . . . . . .

B !

2R' d.c.-cos 0

C'

where R '=/

x/

AB - 2 ffp ~ p 2 . R2 R

R 2 _ p2

fl

1+/~

with

C'C" =-~-(p - R)

fl = tan s 0 sin s t~

Fig. I. Coordinate system and notations for calculating D.,. source is specified by quantities (p, h) and the direction of incidence of a ? ray is defined by the angles O and ~b at the point of entrance of considered surface. The effective rays enter the lateral side of the cylindrical detector as well as the upper surface, this ray traverses a thickness d till it emerges out from the crystal. We have four different cases for d. The effective ray may enter

(a)

(11)

(4) the cylinder side and emerges from the cylinder base, h+L d=--

x'

sin 0

cos 0

P

R"

where x' =

(12)

(b) Nal

(TI) 2" x 2"

Nal (TI) 3" x 3"

h=lcm +

h=Icm + omega face omega side al

omega face omega side

41

I

I

I

~ a

n

I

I

2

3

4

5

~

'

6

p (cm)

7

8

I

I

9

i

I

I

I

I

dl

I

I

I

|

'~V"~'I

10

l

2

3

4

5

6

7

8

9

p (cm) Fig. 2. Variation of the solid angle with the displacement p.

I0

T o t a l efficiency of c y l i n d r i c a l scintillation d e t e c t o r s

(a)

25

(b) NaI (TI) 3" x 3" h=lcm 0.50 MeV • 1.0 MeV × 5.0 MeV • l 0 MeV h = l 0 cm • 0.50 MeV sk 1.0 MeV a 5.0 MeV + 10 MeV

N a l (T1) 3" x 3" h -0.1cm 0.50 MeV • 1.0 MeV × 5.0 MeV • 10 MeV

h=5cm

hffilcm

0.I

¢3

0.I h : 10cm

[-

0.0]

0.01 I 0

I

I

I

I

I

I

I

I

I

!

2

3

4

5

6

7

8

9 10

t

0.001

0

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

10

p (cm)

Fig. 3. V a r i i a t i o n o f the t o t a l efficiency w i t h the d i s p l a c m e e n t / 7 .

p (cm)

26

Younis S. Selim and M a h m o u d

I. Abbas

(a)

(b)

I

Nal (TI) 3" x 3" h =O.Icm 0.50 MeV 1.0 MeV 5.0 MeV @ 10 MeV h=5cm • 0.50 MeV Ik 1.0 M e V u 5.0MeV

h = 0 . 1 cm ~ ~

~

=

~'0.,

~

o.]

~

0.01

0.01 0

i

2

3

4

5

6

$ (cm)

7

8

9

Nai (TI) 3" x 3" h=lcm 0.50 MeV • 1.0 M e V x 5.0 MeV @ 10 MeV h = 10 cm • 0.50 M e V •k 1.0 M e V ~ 5.0 MeV

10

~

~

m

i

L

I

I

I

i

,

I

~

i'

1

2

3

4

5

6

7

8

9

10

S (cm)

Fig. 4. Variation of the total efficiency with the source radius S.

Total efficiency of cylindrical scintillation detectors Table 1. Non-axial point to bare 2" x 2" Nal(TI) detector source mCs-0.662 MeV Beam et al. (1978) Present work p h (em) (cm) Exp. Theo. ~Tx 10 ~ Ratio 0.0 45.0 1.000 1.000 0.55500 1.0000 22.5 39.0 1.136 1.057 0.63089 1.1367 31.8 31.8 1.201 1.123 0.66282 1.1947 39.0 22.5 1.276 1.172 0.68356 1.2316 45.0 0.0 1.320 1.220 0,73519 1.3247 0.0 15.0 1.000 1.000 4.27900 1.0000 7.5 13.0 1.051 1.034 4.69800 1.0979 10.6 10.6 1.151 1.099 4.99720 1.1679 13.0 7.5 1.244 1.215 5.32330 1.2441 15.0 0.0 1.557 1.414 5.93250 1.3864 Considering the various cases we ultimately get the final expression of ET as a sum of eight (or nine) integrals of the type

27

heights from the surface as a function of the displacement p were calculated and represented in Fig. 3. The total efficiency of a cylindrical NaI(Tl) detector arising from a radiating isotropic and coaxial-circular disk is derived as:

~ET)D= ~S

ETp dp

(14)

where S is the circular disk source radius. Also a computer program ( S G R H ) has been set for those expressions and typical running time for each point is about 4 min. Systematic calculations of (e T)D of a cylindrical 3" x 3" NaI(Tl) detector for coaxial-disk source at different heights from the surface with different radii S(S ~ R) were calculated and given in Fig. 4. CONCLUSION

fo."+' f f ~

(1 -e-Ud)dd~'sinO dO

(13)

according to h less (or greater) than (p - R ) L / 2 R ; with the careful choice of the corresponding O and ~. This is compared to only four integrals for the case of p < R (Abbas et al., 1994). g is the total attenuation coefficient of the scintillator at corresponding energy (Berger and Hubbell, 1993). A computer program ( R H O G R H ) is set to calculate ET and typical running time for each point is about 2 rain. Some values of ET of bare 2" x 2" NaI(T1) cylindrical crystal for non-axial point source were calculated and compared with those obtained by Beam et al. (1978) (see Table 1). Systematic calculations ofET o f a cylindrical 3" x 3" NaI(TI) detector for a point source at different

In this paper, the solid angle subtended by the lateral surface of a right circular cylinder is obtained in a compact mathematical expression. In addition, the corresponding exact values of the total efficiency are deduced in a direct rapid way. REFERENCES

Abbas M. I., Abou-Taleb W. M. A. and Selim Y. S. (1994)

Radiat, Phys. Chem. Berger M. J. and Hubbell J. H. (1993) CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton, Fla. Beam G. B., Wielopolski L., Gardner R. P. and Verghese K. (1978) Nucl. Instrum. Methods 154, 501. Masket A. V., Macldin R. L. and Schmitt H. W. (1956) Tables of Solid Angle and Activations. ORNL-2170, USAEC. Selim Y. S. and Abbas M. I. (1994) Radiat. Phys. Chem. 44, I.