Exact Analytic Results for the Solid Angle in Systems With Axial Symmetry

7/27/2019 Exact Analytic Results for the Solid Angle in Systems With Axial Symmetry

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Journal of Applied Mathematics and Physics ( Z A M P ) 0044-2275/89/040608-05 1.50 + 0.20Vol. 40, Ju ly 1989 9 1989 Birkh~iuser Ve rlag, Basel

Exact analytic results for the sol id angle in systems withaxia l symmetryB y V a l e r y A . S h e l y u t o , D . I . M e n d e l e e v I n s t i tu t e o f M e t r o l o g y ,M o s k o v s k y p r o s p e c t 19 , L e n i n g r a d 1 98 00 5, U S S R

For r ad ionuc l i de ac t i v i t y measu remen t by t he me thod o f l im i t ed so l i d ang l e i t i sneces sa ry t o kno w the r a t i o o f num ber o f pa rt i c le s wh ich r each t he de t ec to r t o t henum ber o f pa rt i c le s emi t t ed by t he r ad ionuc l i de sou rce . F o r t he po in t sou rce th i s r a t iowh ich i s ca l led geo m etr ic fac to r (G) i s ev ident ly equ a l to f~ /4n w here f~ i s the so l id angleunde r wh ich t he de t ec to r su r f ace (o r d i aph ragm l imi t i ng r ad i a t i on ) i s s een f rom thesou rce .In t he gene ra l c a se fo r t he f i a t sou rce o f f in i te r ad iu s w h ich i s cop l ana r t o t hed i aph ragm su r f ace geome t r i c f a c to r i s g iven by t he fo l l owing i n t eg ra l ( s ee f ig . 1 )

G (1 )where i t is supp osed t ha t eve ry i n fi n it e s ima l su r f ace e l eme n t emi t s sphe r i ca ll y symm et r i -ca l. I n t eg ra t i on i n eq . ( 1 ) goes ove r sou rce su r face So and d i aph ragm su r f ace Z0 ; R isrunn ing d i s t ance be tween ds a n d de, where de i s t he i n f i n i t e s ima l d i aph ragm su r f acee l emen t ; and h is d is t ance be tween t he sou rce and d i aph ragm . T he I / R 2 f ac to r i n eq . ( 1 )co r r e sp onds t o t he i nve r se squa re l aw and t he r a t i o h /R is equa l t o t he cos ine o f t he ang l ebe tween vec to r R an d n o rm a l t o t he d i aph ragm su r f ace. The i n t eg ra l s simi la r t o tha t i neq . ( 1 ) a r e t yp i ca l f o r many p rob l ems i n ma thema t i ca l phys i c s . Fo r example , t hez - c o m p o n e n t o f th e i n t e r a c t i o n f o r c e b e t w e e n t w o h o m o g e n e o u s l y c h a r g e d p l a te s i sg iven up t o numer i ca l f a c to r by eq . ( 1 ) . Hence , a l l r e su l t s ob t a ined be low fo r t hegeome t r i c a l f a c to r ( 1 ) may be app l i ed i n t h i s c a se a s we l l a s i n many o the r p rob l ems .W ide ly u sed m easu rem en t sy s t em cons is t s o f c i r cu l a r sou rce o f r ad iu s a and coax i a lc i r cu l a r d i aph ragm o f r ad iu s b . I n t he cy l i nd r i ca l coo rd ina t e sy s t em one ea s i l y ob t a in sfo r t he geom e t r i c a l f a c to r ( 1 ) exp l i c it exp re s s ion

h [2~dr f f d rr f ; dpp[h~+ r 2 + p 2 - 2 r p c o s q ~]-3 /2 , ( 2 )G = ~oo jowhere a l l conven t i ons a r e ev iden t .Ty pe (2 ) i n t eg ra ls a r e u sua l l y ca l cu l a t ed by expan d ing t he i n t eg rand i n t he s e rie s w i thh e l p o f L e g e n d r e p o l y n o m e s o r s o m e o t h e r f u n c t i o n s r es p e c ti n g th e s y m m e t r y o f th ep rob l e m ( see i. e. r e f . [ 1 -3 ] and a l so m any o the r pape r s ) . Thes e s e ri e s a r e we l l conv e rgen twhen t he d i s t ance h be tween t he p l a t e s is much m ore t h an t he cha rac t e r i s t ic s iz es a andb o f t he p l a t e s . They a r e poo r ly conve rgen t when t he d i s t ance h i s o f t he s ame o rde r a sthe p l a t e s s iz es . M ore ove r i n t he ca se o f van i sh ing r a t i o h/a---, 0 and a ~ b the in tegra l so f t ype (2 ) c an no t be ca l cu l a ted , a s i s shown be low, w i th t he he lp o f the i n t eg rands s e ri esexpans ion . Th i s i s connec t ed w i th t he nonana ly t i c (ln(h/a)) d e p e n d e n c e o f t h e i n te g r a -t i on r e su l t on t he sma l l pa r am e te r . T he re fo re , t he exac t r e su l t o f t he i n t eg ra t ion , wh ich


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Vol. 40, 1 9 8 9 Exact analyt ic resul ts for the sol id angle 609

Figure 1Source--target configurat ion relevant to eq. (1) .

P

i s v a l i d a t a r b i t r a r y v a l u e s o f t h e p a r a m e t e r s a n d m a y b e e f fe c t iv e l y u s e d i n n u m e r i c a lc a l c u l a t io n s , is p o o r l y n e e d e d .

D i r e c t i n t e g r a t i o n i n e q . ( 2 ) l e a d s t o t h e e l l i p t i c i n t e g r a l s o n t h e e a r l y s t a g e , t h u se m b a r r a s s i n g f o l lo w i n g c a l c u l a t io n s . T o c i r c u m v e n t t h i s d i ff ic u l ty w e in t r o d u c e a d d i -t i o n a l i n t e g r a t io n o v e r n e w v a r i a b l e z r e m o v i n g t h u s t h e f ra c t i o n a l p o w e r i n t h ed e n o m i n a t o r i n e q. ( 2) :

G = ~ z S o O h d z d o d r r d p p [ z 2 + h 2 + r 2 + p 2 2 r p c o s c p ] - 1

- 2 S o Oh d z d r 2 d p 2 [ ( z2 + h 2 + r 2 + p2)2 _ 4 r2p2] 1/2T h e i n t e g r a n d h e r e d e p e n d s o n l y o n t h e v a r i a b l e s r 2 a n d p 2 a n d t h e f o l l o w i n g c a l c u l a t i o ni s s t r a i g h t f o r w a r d .

I n t e g r a t i o n o v e r p 2 g i v e sZ '~ A + r 2 - b 2

G = ~ f o ~ f l d r 2 { 1 - x / ( A + r 2 + b Z ) Z _ 4 b ~ } , ( 4 )w h e r e A = z 2 + h 2 . T h i s r e s u l t i s i n t e r e s t i n g i n it s o w n r i g h t , b e c a u s e i t is p o s s i b l e t oo b t a i n t h e g e o m e t r i c f a c t o r G a f o r th e p o i n t s o u r c e d i s p l a c e d f r o m t h e a x is o n d i s t a n c ea f r o m i t . F o r t h i s o n e h a s t o m a k e i n e q . ( 4 ) t h e s u b s t i t u t i o n

j ia dr2z~ S o 1 .T h e f o l l o w i n g i n t e g r a t i o n o v e r z i s e a s il y p e r f o r m e d w i t h t h e h e l p o f t a b l e s [ 4] a n d t h eg e o m e t r i c f a c t o r f o r t h e d i s p l a c e d p o i n t s o u r c e t u r n s o u t t o b e

G ~ = ~ 2 ~ 9 ~ - - ~ " K ( q) + - ~ - ~ . I I ( n , q ) , ( 5 )w h e r e K ( q ) , H ( n , q ) a r e t h e c o m p l e t e e l li p t ic i n t e g r a l s o f t h e f ir s t a n d t h i r d k i n d [ 4] ,


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610 V .A . Shelyuto ZA MPr e s p e c t i v e l y , a n d

4 a b ( a + b ) 2q 2 _ h 2 + ( a + b ) 2 ' n - h 2 + ( a + b ) 2 ' (6 )E x p r e s s i o n ( 5 ) l e a d s t o p h y s i c a l ly e v i d e n t r e s u l ts w h e n h - -, 0 a n d o t h e r p a r a m e t e r sh a v e s o m e s p e c if i c v a l u e s . I n th i s c a s e G a = 1 / 2, w h e n a < b b e c a u s e t h e n t h e d i a p h r a g m

f u l ly c o v e r s th e s o u r c e a n d a ll r a d i a t i o n w h i c h is e m i t te d i n t h e u p p e r h a l f s p a c e r e a c hi t; G a = 1 /4 , w h e n a = b a n d t h e s o u r c e i s o n t h e e d g e o f t h e d i a p h r a g m ; G a = 0 , w h e na > b a n d t h e s o u r c e is e x t r a n e o u s t o t h e d i a p h r a g m . T o o b t a i n t h e s e r e su l t s w e h a v eu s e d t h e f o l l o w i n g l i m i t :

l i m x / l _ n . H (n , q ) 7z a + bh ~ 0 - 2 l a - b I" ( 7 )F o r t h e p o i n t s o u r c e d i s p la c e d f r o m t h e a x is o n t h e d i st a n c e e q u a l t o t h e d i a p h r a g m

r a d i u s ( a = b ) o n e o b t a i n s f r o m e q . ( 4 ) t h e a n c i e n t r e s u l t [ 4 ]q '

' - - . K ( q ) , ( 8 )G , = ~ - 2 nw h e r e q ' = ~ / 1 - q2 i s t h e c o m p l e m e n t e l l i p t i c i n t e g r a l m o d u l e .

C o n s i d e r n o w t h e g e n e r a l c a s e o f s p r e a d s o u r c e in e q. (4 ) . I n t e g r a t i o n o v e r v a r i a b ler 2 i s t r ivia l :

h f ~ d z { A + a 2 x / ( A + a 2 b Z ) 2 _ 4 a 2 b 2 } .G = ~ o o Jo A + b 2 - + (9)F i n a l i n t e g r a t i o n h e r e i s s i m i l a r t o t h e o n e p e r f o r m e d i n e q . ( 5 ) a n d w i t h t h e h e l p o f

t a b l e s [ 4 ] w e o b t a i nG = ~ o . (a 2 + b 2 ) - ( a - b ) 2 x / 1 - n . I I ( n , q ) - 4 a b x / 1 - n . D (q ) , ( I 0 )

w h e r e D ( q ) = ( l / q 2) 9 [ K ( q ) - E ( q ) ] is t h e l i n e a r c o m b i n a t i o n o f e ll i p ti c i n t e g r a l s o f th ef i r s t a n d s e c o n d k i n d .

E q . ( 1 0 ) g iv e s al l p r e v i o u s l y k n o w n s p e ci fi c c a s es . T h u s f o r l a r g e d i s ta n c e b e t w e e nt h e so u r c e a n d t h e d i a p h r a g m w e o b t a i n G ~-Y~o/4rchL W h e n h g o e s t o z e r o t h e l i m i td e p e n d s o n t h e r e l a t io n b e t w e e n a a n d b : G ~ Z 0 / 2 S o i f a > b a n d G ~ 1 /2 i f a < b . F o rt h e p o i n t s o u r c e ( a / b ~ 0 ) p l a c e d o n t h e s y m m e t r y ax i s

h

G e o m e t r i c a l f a c t o r h a s t h e s im p l e s t f o r m w h e n t h e r a d i i o f t h e s o u r c e a n d t h ed i a p h r a g m a r e e q u a l :

2 q 'G = 8 9 D ( q ) . ( l l )7E

I t s a s y m p t o t e i s e q u a l t o ( 1 2 ) w h e n h / a ~ 0G ~ ! - ~ h - 2a ' ( l n h + 3 1 n 2 - 1 ) ( 1 2 )

a n d c o n t a i n s t h e l o g a r i t h m i c t e r m m e n t i o n e d a b o v e a n d h e n c e c a n n o t b e o b t a i n e d f r o me q . ( 2 ) b y e x p a n d i n g t h e i n t e g r a n d i n t h e s e r i e s o v e r t h e r a t i o h / a . I t s h o u l d b em e n t i o n e d t h a t t h is a s y m p t o t e w i t h r a t h e r g o o d a c c u r a c y d e sc r ib e s t h e b e h a v i o u r o f t h e


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Vol . 40 , 1 9 8 9 Exact analy t ic resu l ts fo r the so l id ang le 6 i lgeome t r i c f a c to r f o r t he sma l l va lue s o f h / a . F o r e x a m p l e , a l r e a d y w h e n h / a = 0.2 theva lues g iven by eqs . (11) an d (12) d i f fe r by less than 0 .1% .

In t eg ra l r ep re sen t a t i on (3 ) f o r G i s ve ry conven ien t and a l lows on e t o so lve t he morecom pl i ca t ed p rob l e m o f c a l cu l a ti ng t he geo me t r i c f a c to r G~ fo r coax i a l sy s t em cons i s ti ngo f the c i r cu l a r d i ap h rag m and t he cy l i nd r ica l r ad i a t i on sou rce . The r ad ioac t i ve subs t anceis s u p p o s ed t o b e h o m o g e n e o u s l y d i s tr i b u te d o v e r t h e v o l u m e o f t h e s o u r c e a n d t h eabso rp t i on i s supposed t o be absen t . I n t eg ra t i ng t he r i gh t s i de o f eq . ( 3 ) ove r va r i ab l e hf rom the l ower l im i t h i equa l t o t he d i s t ance be tween t he d i aph ragm to t he nea re s t ba seo f t he cy l i nde r up t o t he u ppe r l im i t h 2 equa l t o t he d i s t ance be tween t he d i aph rag m andthe d i s t an t ba se o f t he cy l i nde r one ob t a in s

Q = 9 - d h a ( h ) = 5 V o o" ( a 2 + b 2 ) - ( a - n1

I I9 9 D ( q ) ' [ ( 1 - q 2 ) . D ( q ) + E ( q ) ] ( 1 3 )I-l (n, q) 1 - n 3 x fi -~ - ~ h,

wh ere V0 = So(h2 - h~) = So " H is t he vo lum e o f t he sou rce . I n t he ca l cu l a t i on l e ad ing t oeq . (13) the form ulae f r om [4] wh ere explo i ted as well as the in teg ra lf _ q y q Z _ n I I ( n , q ) q K ( q ) , (14)_ . _ _ ,d q II(n, q) = n q n

which i s absen t i n handbooks .Th e mo s t conc i sed fo rm G~ has when r ad i i o f t he d i aph ra gm and t he cy l i nd r ica l

sou rce a r e equa l2

G v 1 .= 2 - - ~ [ A (h 2 ) - - A ( h l ) ] , ( 1 5 )w h e r e

h I + I " E ( q ) ] . ( 16 )( h ) = ~ r " q " D ( q ) q "In the case of inf in i te ly th in s o u r c e ( h 2 ~ h i ) the express ions (13) and (15) co inc idewi th the express ions (10) and (11) respec t ive ly .The sha re o f r ad i a t i on ( r/ l) wh ich goes t h roug h each o f t he cy l inde r ba ses i s g iven by

eq . ( I 5 ) where h i = 0 a nd h 2 = H (H here i s he ight of the cy l indr ica l source)rl~ = ~ - - ~ . q ' . D ( q ) + - - . E ( q ) - . (17)q 'The sha re o f r ad i a t i on go ing t h rough the cy l i nde r s i deways i s equa l t oq 2 = 1 - - 2 r / l = ~ ' q ' - D ( q ) + ~ . E ( q ) - . (18)

I t i s ev ident th a t ~/1 ~ 0 , th -~ 1 wh en a / H ~ O a n d 2 r h ~ 1 , t / 2 ~ 0 i n t h e a l t e r n a t e c a s ew h e n H / a - -, O .Al l exp re s s ions fo r geome t r i c f a c to r f o r d i f f e r en t r ad i a t i on sou rce s ob t a ined abovea re va l i d fo r a rb i t r a ry va lue s o f t he pa rame te r s a , b , h and H . Th i s p r e sen t s t he s e r i ousadvan t age o f t he se fo rm u lae ove r t he ones , wh ich a r e g iven by t he in t eg rand ' s pow erexpans ion , i f th i s expans ion i s a t a l l poss ib le .We shou ld l i ke t o men t ion t ha t t he t r i cks u sed above fo r ana ly t i c c a l cu l a t i on o fin t eg ra l in eq . ( i ) m ay be a l so app l i ed i n o the r ca se s. Fo r examp le , we ea s il y ob t a in t he


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612 V .A . She lyu to ZAM Pv a l u e f o r t h e i n t e g r a l

S o ~ -~ o'R 3 r c x / ~ . q 3 [ ( l + 2 q Z - 3 n ) ' K ( q )- ( 1 + q2 _ 3 n ) - E ( q ) + 3 ( 1 - n ) (n - q 2 ) . n ( n , q ) l ,

w h i c h c o r r e s p o n d s t o i n te r a c t io n e n e r g y o f tw o h o m o g e n e o u s l y c h a r g e d c o a x ia l d is k s ,T h e e x p r e s s i o n f o r t h e i n t e r a c t i o n f o r c e , w h i c h i s p r o p o r t i o n a l t o e q . ( 1 ) i s o b t a i n e d f r o m( 1 9 ) b y d i f f e r e n ti a t in g o v e r t h e p a r a m e t e r h .

R e f e r e n c e s

[1] A. H. Jaffey, Rev. Scient. lnstrum. 25 , 349 (1954).[2] K. A. Petrzhak and M . A. Buck, Sov. Ph ys. JTP 25 , 636 (1955) (in Russian).[3] P. M. Morse and H. Feshbach, Methods of Theoret ical Physics, Part II. McGraw-Hil l Book Co.,New York 1953.[4] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (formu lae 3.152.1, 3.157.2,3.169.1,2). Academic Press, New Y ork 1980.

A b s t r a c t

Exact analyt ical expressions for the sol id angle subtended b y a circular diaphragm are obtained interms of complete elliptic integrals. Point, disk and cylindrical sources of radiation are considered.(Received: November .2, 1988)

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Exact Analytic Results for the Solid Angle in Systems With Axial Symmetry