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Use of rectangular geometry for radon dosimetry
Nuclear Tracks, Vol. 12, Nos I-6, pp. 693-696, Int. J. Radiat. Appl. Instrum., Part D Printed in Great Britain.
1986.
0191-278X/86 $3.00+.00 Pergamon Journals Ltd.
USE OF RECTANGULARGEOMETRY FOR RADON DOSlMETRY A. Waheed*, A.W. S i d d i q u i * * , H.A.Khan+, A. Sadiq++, S.A. Sheikh+, M.A. Sial+, and A.U. Khan* *
Physics Department, Gomal U n i v e r s i t y , D.l.Khan, Pakistan.
** Mathematics Department, Quaid-e-Azam U n i v e r s i t y , Islamabad, Pakistan. +
N.E.D.,PINSTECH, P.O.Nilore,Rawalpindi,Pakistan.
++ N.P.D.,PINSTECH, P.O.Nilore,Rawalpindi,Pakistan.
ABSTRACT A method is presented to c a l c u l a t e the track density d i s t r i b u t i o n of Radon Alpha p a r t i c l e s in the p l a s t i c track detectors placed along the wall of a rectangular dosimeter. Volume of a sphere of radius 'R' has been found in a rectangular box of dimensions, a x a x b , as a f u n c t i o n of the distance '7' of the centre of the sphere from the edge of the box. The d i s t r i b u t i o n so obtained has been compared with the experimental track density d i s t r i b u t i o n of Radon Alpha p a r t i c l e s found in CN85 p l a s t i c track detectors placed along the wall of a rectangular box of 6 x 6 x 8 cm3 in an a r t i f i c i a l mine in the laboratory. Conclusions regarding the s u i t a b i l i t y of rectangular geometry as compared to the c y l i n d r i c a l geometry have been drawn. KEYWORDS Rectangular geometry; radon dosimetry; uranium e x p l o r a t i o n ; track density d i s t r i b u t i o n ; range of alpha p a r t i c l e s ; c y l i n d r i c a l geometry. INTRODUCTION Detection of radon emanation by measuring the concentration of alpha p a r t i c l e s serves as a tool f o r U/Th e x p l o r a t i o n and in personnel dosimetry(Khan and Akber, 1976). For such kind of measurements however a study of the e f f e c t s of size and geometry of dosimeters is of much importance. The c a l c u l a t i o n
of t h e o r e t i c a l d i s t r i b u t i o n s and t h e i r comparison with the
experimental d i s t r i b u t i o n s may i n d i c a t e that which dimensions and shape of the dosimeters are best suited f o r such experiemtns. In t h i s paper therefore we have given t h e o r e t i c a l track density d i s t r i b u t i o n of radon alpha p a r t i c l e s f o r rectangular geometry and a comparison has been made with the experiemtnal d i s t r i b u t i o n found in rectangular geometry. The d i s t r i b u t i o n s found in rectangular dosimeters have been compared the d i s t r i b u t i o n s obtained f o r c y l i n d r i c a l geometry(Sadiq and others, 1981). THEORETICAL MODEL The track density is taken proportional to the volume of a sphere of radius 'R', where 'R I is comparable to the range of radon alpha p a r t i c l e s . The sphere is taken to be centred at the point
P(a, { , q ) in a box of
a x a x b
cm3, and the sphere moves in such a way that i t s
centre does not leave the (Y,Z) plane. We consider three regions of i n t e r e s t such that
693
X=O,
A. WAHEED e t al.
694
0 ~ ~ ~ a and R~ a ~ b , taking a cubic box with(a x a x a)cm3 in the f i r s t instance.We now calculate the required volume in the following three regions accordingly. Ca~eI: ~ : O, 0~< j" ~< a The volume of the portion of the sphere enclosed in the box is given by,
a /R2-(y - ~ )2 /RZ_(y_ ~ )2_z2 Vo: J J J dx dy dz
T
Theevaluation of this integral gives,
Vo= 7~'/4[R2a-(a - ~ )3/3 - ~3/3 ] Case II: ~ 0
(A)
b
, 0 ~ ~ a- /a 2- ~2
Some portion of the sphere will cross the plane Z=a. The volume above Z=a is given by, R 2~ Cos-l[(a-~)/a]
V'=I/2
; J J (r2Sin 8) dO dg . -q Calculations finally give, V'= ~ (~ /a) [a3-(a-~ )3 ] The volume above Z=O is given by , a f~ + /R2-(y - ~ )2 V= J O0
[ /R2_(y_ ~ )2_(z_ D )2 ] dy dz
Fig.1 Motion of the sphere of radius 'R' for X=O, ~ =a/2 and the corresponding volumes Vo, V1, V2 enclosed in the box.
From this integral we get the following expression, V= -~- [R2a-(a-~)3/3 - j3/3]+ g /4[(a-~) /R2- D2-(a-~)2 + g /4[(R2_D 2) [ Sin-l( (a- I )/ /R-2~-~2 )+ Sin-l(
+~/ R2_ ~2_ ~2
]
~ / / ~ _ g2 ) ~ ]
+ 1/6 [ [ 3R2(a- ~ ) - ( a - | ) 3 / 3 ]Sin-lEg / /R2-(a - ~ )2)2] + I/6[(3R 2 ~ _ ~3/3)Sin-1 ~ g / /R2_ ~2))]+ g/12[ E(a-~ ) /R2- D2-(a- I )2 ] +E ~ /R2- g2 _ ~2 +Esin-1 ( ~ /
~+Sin-lE(a_ ~)// R2_ ~2 ].(3a2+ D2)
/R2_~2 )] (3a2+~2) ]+4R3/~[ tan-IER /R2_g2_(a_ ~ )2 /(a- I ) D ~
+ tan -I E (R /R 2- q2 _ ~2) / y ~ ] ]
(B)
The required volume VI is then given by, Vt : V - V'
(C)
USE OF RECTANGULAR GEOMETRY
Case l l I :
695
a- /a2-(a - ~ )2 ~ D ~ /a2_(a - ~ )2 , ~> 0
In this case the whole rectangular plane (Y,Z) will lie inside the sphere. The volume V2 of the sphere inside the box will therefore be given by, a-( a-~ V2 = - ~ /
-~/
/a 2 - y2 - z2 dy dz . Solution of this integral gives V2 in this form:
V2= (a-D)/2[ E (a-J) /a2-(a-~)2-(a-~)2
] /2 + Ea2-(a-~)2]/2. Sin-IE(a-J )/ /a2-(a-~) 2 ~ +
[ ~/a2-(a-~) 2- 12 ] /2 + E(a~(a-D)2)/2 ] . Sin-lE ~ /
/a2-(a-D) 2
+ ~/2 [ E (a-~) /a 2 - B2-(a- ~)2 ] /2 +E(a2-B2)/2 ] • Sin -I ( ~ / (a2 - D2) ] + 1/6 [3 a2(a- ~ )-(a- ~ )3 Sin-IE (a-B)/
/a2-(a - I ) 2
3] + 1/6 [(3 a2~- ~3)
Sin -I E(a-~) /
/a 2_ ~2 ~]+ (a-~)/12[ (a- ~ ) /a2_(a_D)2_(a- ~ )2 +
Sin -I E ( a - | ) /
/a2-(a-g) 2 ] E 3 a2+(a-~)2 ~ + ~ /a2-(a-Q) 2- { 2
+ Sin -I E ~ /
/a2-(a-Q) 2 ] E (3 a2 + (a-~) 2 ] +4 a3E a2-(a-g) 2 ] / (a -~)
tan -I E ( a /a2-(a-~)2-(a - ~ )2 ) /(a- J ) (a-~) ]+E4 a3(a2-(a-g)2 ) /(a-9) tan -I E(a /a2-(a-~) 2 - {2 )/(a- ~ )( a - ~) ] ]
+ 1/6 [(3 a2{ - ~3)Sin-iE~ / /a 2- ~2 ]]
+ B/12[ I /a2-D2- ~2 + Sin-lE ~/(a2_B2)] (3 a2 - ~2) +(4 a3/B) tan -I E ( a /a2-~ 2- ~2 ) / y ~ ] ]
(D)
In case of a x a x b geometry the regions of interest will be, (i) O~ ~ ~ b-R, (2) b-R ~ q ~ b- /R2- ~2 , (3) b- /R2- [ 2 ~ D ~ / R2- ~2 The effective volumes are given by Eqs: (B), (C) and (D) by changing the limits accordingly.
'1.2 o.s
0
I
I
I
1
2
3
I
4 (cm)
I
5
I
I
6
7
Fig.2 Track density distributions.Theoretical and experimental distributions are given by solid and dotted lines respectively. NT 12:I/6-TT
8
A. WAHEED et al.
696
EXPERIMENTAL RESULTSAND CONCLUSIONS CN85 plastic detectors were placed along the walls of rectangular dosimeters stretched from top to the bottom of each cylinder. Dimensions of the dosimeters were (6 x 6 x 8) cm3.With this arrangement exposures were carried out separately for 8 and 16 hours in the laboratory over a simulated mine containing uranium ore. The detectors were then etched in three groups for one and a half, two and three hours in 6N NaOHat 50±3 CO. The best distributions were obtained for 16 hours exposures and 2 hours etching. The normalized track density d i s t r i b u t i o r is displayed in Fig.2 alongwith the calculated distribution for R = 6 cm as range of alpha particles (Khan and others, 1976;1977). Agreement between the distributions is f a i r enough. In a previous paper(Sadiq and others,1981) we presented the distributions for cylindrical geometry. Those distributions with some improvements are reproduced in Fig.3 .
~
1.2 1.0
%
o e
x
x
.~,~~_
~
~,~
-
0.6 v
0.4 0.2
0
I
I
|
0.2
0.~
1.0
i 1.4
I
J
1.8
2.2
Z R Fig.3 Theoretical ( solid line) and the experimental (dotted line) distributions in a cylincrical geometry. From a comparison of the two geometries under consideration i t may be concluded that with i t s constant track density distribution over a f a i r l y large region the rectangular geometry is more r e l i a b l e from the experimental point of view to find the maximum track density with the accuracy. Future investigations with d i f f e r e n t sizes of dosimeters are intended to study the geometry effects in d e t a i l . This work has been carried out in the framework of German-Pakistan agreement on s c i e n t i f i c and technological cooperation. We thank the International Office,KfK,Karlsruhe for support. One of us (HAK) is grateful to Alexander von Humboldt(AVH) Stiftung for financial support and equipment kindly given to the SSNTD laboratory (PINSTECH), Pakistna. REFERENCES Khan HameedA. and Akber Riaz A.(1976). Proc. IX Conf. SSNTD, Pergamon Press,Oxford, pp. 803 Sadiq A. and others(1981). Nucl. Tracks Suppl., 3 , 543 . Khan HameedA. and others (1976). Proc. IX Conf. SSNTD, Pergamon Press, Oxford, pp. 815. Khan Hameed. and others(1977). Nucl. Instr. Methods, 147, 125.
1986.
0191-278X/86 $3.00+.00 Pergamon Journals Ltd.
USE OF RECTANGULARGEOMETRY FOR RADON DOSlMETRY A. Waheed*, A.W. S i d d i q u i * * , H.A.Khan+, A. Sadiq++, S.A. Sheikh+, M.A. Sial+, and A.U. Khan* *
Physics Department, Gomal U n i v e r s i t y , D.l.Khan, Pakistan.
** Mathematics Department, Quaid-e-Azam U n i v e r s i t y , Islamabad, Pakistan. +
N.E.D.,PINSTECH, P.O.Nilore,Rawalpindi,Pakistan.
++ N.P.D.,PINSTECH, P.O.Nilore,Rawalpindi,Pakistan.
ABSTRACT A method is presented to c a l c u l a t e the track density d i s t r i b u t i o n of Radon Alpha p a r t i c l e s in the p l a s t i c track detectors placed along the wall of a rectangular dosimeter. Volume of a sphere of radius 'R' has been found in a rectangular box of dimensions, a x a x b , as a f u n c t i o n of the distance '7' of the centre of the sphere from the edge of the box. The d i s t r i b u t i o n so obtained has been compared with the experimental track density d i s t r i b u t i o n of Radon Alpha p a r t i c l e s found in CN85 p l a s t i c track detectors placed along the wall of a rectangular box of 6 x 6 x 8 cm3 in an a r t i f i c i a l mine in the laboratory. Conclusions regarding the s u i t a b i l i t y of rectangular geometry as compared to the c y l i n d r i c a l geometry have been drawn. KEYWORDS Rectangular geometry; radon dosimetry; uranium e x p l o r a t i o n ; track density d i s t r i b u t i o n ; range of alpha p a r t i c l e s ; c y l i n d r i c a l geometry. INTRODUCTION Detection of radon emanation by measuring the concentration of alpha p a r t i c l e s serves as a tool f o r U/Th e x p l o r a t i o n and in personnel dosimetry(Khan and Akber, 1976). For such kind of measurements however a study of the e f f e c t s of size and geometry of dosimeters is of much importance. The c a l c u l a t i o n
of t h e o r e t i c a l d i s t r i b u t i o n s and t h e i r comparison with the
experimental d i s t r i b u t i o n s may i n d i c a t e that which dimensions and shape of the dosimeters are best suited f o r such experiemtns. In t h i s paper therefore we have given t h e o r e t i c a l track density d i s t r i b u t i o n of radon alpha p a r t i c l e s f o r rectangular geometry and a comparison has been made with the experiemtnal d i s t r i b u t i o n found in rectangular geometry. The d i s t r i b u t i o n s found in rectangular dosimeters have been compared the d i s t r i b u t i o n s obtained f o r c y l i n d r i c a l geometry(Sadiq and others, 1981). THEORETICAL MODEL The track density is taken proportional to the volume of a sphere of radius 'R', where 'R I is comparable to the range of radon alpha p a r t i c l e s . The sphere is taken to be centred at the point
P(a, { , q ) in a box of
a x a x b
cm3, and the sphere moves in such a way that i t s
centre does not leave the (Y,Z) plane. We consider three regions of i n t e r e s t such that
693
X=O,
A. WAHEED e t al.
694
0 ~ ~ ~ a and R~ a ~ b , taking a cubic box with(a x a x a)cm3 in the f i r s t instance.We now calculate the required volume in the following three regions accordingly. Ca~eI: ~ : O, 0~< j" ~< a The volume of the portion of the sphere enclosed in the box is given by,
a /R2-(y - ~ )2 /RZ_(y_ ~ )2_z2 Vo: J J J dx dy dz
T
Theevaluation of this integral gives,
Vo= 7~'/4[R2a-(a - ~ )3/3 - ~3/3 ] Case II: ~ 0
(A)
b
, 0 ~ ~ a- /a 2- ~2
Some portion of the sphere will cross the plane Z=a. The volume above Z=a is given by, R 2~ Cos-l[(a-~)/a]
V'=I/2
; J J (r2Sin 8) dO dg . -q Calculations finally give, V'= ~ (~ /a) [a3-(a-~ )3 ] The volume above Z=O is given by , a f~ + /R2-(y - ~ )2 V= J O0
[ /R2_(y_ ~ )2_(z_ D )2 ] dy dz
Fig.1 Motion of the sphere of radius 'R' for X=O, ~ =a/2 and the corresponding volumes Vo, V1, V2 enclosed in the box.
From this integral we get the following expression, V= -~- [R2a-(a-~)3/3 - j3/3]+ g /4[(a-~) /R2- D2-(a-~)2 + g /4[(R2_D 2) [ Sin-l( (a- I )/ /R-2~-~2 )+ Sin-l(
+~/ R2_ ~2_ ~2
]
~ / / ~ _ g2 ) ~ ]
+ 1/6 [ [ 3R2(a- ~ ) - ( a - | ) 3 / 3 ]Sin-lEg / /R2-(a - ~ )2)2] + I/6[(3R 2 ~ _ ~3/3)Sin-1 ~ g / /R2_ ~2))]+ g/12[ E(a-~ ) /R2- D2-(a- I )2 ] +E ~ /R2- g2 _ ~2 +Esin-1 ( ~ /
~+Sin-lE(a_ ~)// R2_ ~2 ].(3a2+ D2)
/R2_~2 )] (3a2+~2) ]+4R3/~[ tan-IER /R2_g2_(a_ ~ )2 /(a- I ) D ~
+ tan -I E (R /R 2- q2 _ ~2) / y ~ ] ]
(B)
The required volume VI is then given by, Vt : V - V'
(C)
USE OF RECTANGULAR GEOMETRY
Case l l I :
695
a- /a2-(a - ~ )2 ~ D ~ /a2_(a - ~ )2 , ~> 0
In this case the whole rectangular plane (Y,Z) will lie inside the sphere. The volume V2 of the sphere inside the box will therefore be given by, a-( a-~ V2 = - ~ /
-~/
/a 2 - y2 - z2 dy dz . Solution of this integral gives V2 in this form:
V2= (a-D)/2[ E (a-J) /a2-(a-~)2-(a-~)2
] /2 + Ea2-(a-~)2]/2. Sin-IE(a-J )/ /a2-(a-~) 2 ~ +
[ ~/a2-(a-~) 2- 12 ] /2 + E(a~(a-D)2)/2 ] . Sin-lE ~ /
/a2-(a-D) 2
+ ~/2 [ E (a-~) /a 2 - B2-(a- ~)2 ] /2 +E(a2-B2)/2 ] • Sin -I ( ~ / (a2 - D2) ] + 1/6 [3 a2(a- ~ )-(a- ~ )3 Sin-IE (a-B)/
/a2-(a - I ) 2
3] + 1/6 [(3 a2~- ~3)
Sin -I E(a-~) /
/a 2_ ~2 ~]+ (a-~)/12[ (a- ~ ) /a2_(a_D)2_(a- ~ )2 +
Sin -I E ( a - | ) /
/a2-(a-g) 2 ] E 3 a2+(a-~)2 ~ + ~ /a2-(a-Q) 2- { 2
+ Sin -I E ~ /
/a2-(a-Q) 2 ] E (3 a2 + (a-~) 2 ] +4 a3E a2-(a-g) 2 ] / (a -~)
tan -I E ( a /a2-(a-~)2-(a - ~ )2 ) /(a- J ) (a-~) ]+E4 a3(a2-(a-g)2 ) /(a-9) tan -I E(a /a2-(a-~) 2 - {2 )/(a- ~ )( a - ~) ] ]
+ 1/6 [(3 a2{ - ~3)Sin-iE~ / /a 2- ~2 ]]
+ B/12[ I /a2-D2- ~2 + Sin-lE ~/(a2_B2)] (3 a2 - ~2) +(4 a3/B) tan -I E ( a /a2-~ 2- ~2 ) / y ~ ] ]
(D)
In case of a x a x b geometry the regions of interest will be, (i) O~ ~ ~ b-R, (2) b-R ~ q ~ b- /R2- ~2 , (3) b- /R2- [ 2 ~ D ~ / R2- ~2 The effective volumes are given by Eqs: (B), (C) and (D) by changing the limits accordingly.
'1.2 o.s
0
I
I
I
1
2
3
I
4 (cm)
I
5
I
I
6
7
Fig.2 Track density distributions.Theoretical and experimental distributions are given by solid and dotted lines respectively. NT 12:I/6-TT
8
A. WAHEED et al.
696
EXPERIMENTAL RESULTSAND CONCLUSIONS CN85 plastic detectors were placed along the walls of rectangular dosimeters stretched from top to the bottom of each cylinder. Dimensions of the dosimeters were (6 x 6 x 8) cm3.With this arrangement exposures were carried out separately for 8 and 16 hours in the laboratory over a simulated mine containing uranium ore. The detectors were then etched in three groups for one and a half, two and three hours in 6N NaOHat 50±3 CO. The best distributions were obtained for 16 hours exposures and 2 hours etching. The normalized track density d i s t r i b u t i o r is displayed in Fig.2 alongwith the calculated distribution for R = 6 cm as range of alpha particles (Khan and others, 1976;1977). Agreement between the distributions is f a i r enough. In a previous paper(Sadiq and others,1981) we presented the distributions for cylindrical geometry. Those distributions with some improvements are reproduced in Fig.3 .
~
1.2 1.0
%
o e
x
x
.~,~~_
~
~,~
-
0.6 v
0.4 0.2
0
I
I
|
0.2
0.~
1.0
i 1.4
I
J
1.8
2.2
Z R Fig.3 Theoretical ( solid line) and the experimental (dotted line) distributions in a cylincrical geometry. From a comparison of the two geometries under consideration i t may be concluded that with i t s constant track density distribution over a f a i r l y large region the rectangular geometry is more r e l i a b l e from the experimental point of view to find the maximum track density with the accuracy. Future investigations with d i f f e r e n t sizes of dosimeters are intended to study the geometry effects in d e t a i l . This work has been carried out in the framework of German-Pakistan agreement on s c i e n t i f i c and technological cooperation. We thank the International Office,KfK,Karlsruhe for support. One of us (HAK) is grateful to Alexander von Humboldt(AVH) Stiftung for financial support and equipment kindly given to the SSNTD laboratory (PINSTECH), Pakistna. REFERENCES Khan HameedA. and Akber Riaz A.(1976). Proc. IX Conf. SSNTD, Pergamon Press,Oxford, pp. 803 Sadiq A. and others(1981). Nucl. Tracks Suppl., 3 , 543 . Khan HameedA. and others (1976). Proc. IX Conf. SSNTD, Pergamon Press, Oxford, pp. 815. Khan Hameed. and others(1977). Nucl. Instr. Methods, 147, 125.