C A L C U L A T I O N O F T H E F L A W I M A G E I N

T - R A Y F L A W D E T E C T I O N

A. M. K o l T c h u z h k i n a n d V. V. U c h a i k i n UDC 620.179.152.5

A pe r tu rba t i on - theo ry method is descr ibed for calculat ing the image of a smal l flaw. R e - suits calculated for flaws of s eve ra l s imple shapes a r e repor ted .

The flaw image produced by the Y-rad ia t ion f r o m a plane, perpendicular source can be sepa ra ted into two components: the even image, due to unsca t te red radiat ion, and the odd image, produced by the sca t t e r ing of radia t ion in the sample . Calculation of this second component and the de te rmina t ion of its ro le in reveal ing hidden flaws is an impor tan t p rob lem in the theory of x - r a y and y - r a y flaw detection. Below we desc r ibe a pe r tu rba t ion - theo ry method for solving this problem.

w T h e o r y

We consider a p lane -pa ra l l e l l aye r on whose z = 0 boundary monoenerge t ic radia t ion is incident normal ly . The change in the radiat ion intensi ty at point r due to a flaw in volume V is given in f i r s t - o r d e r per tu rba t ion theory by [1]

E~

~ l (r) = -- S d r" f d to S d E *+ (r', ~, E; r) A Q (r' , to, E) . (1) V 4~ 0

Here ff+(r r, co, E; r) is the conjugate function for a de tec tor which m e a s u r e s the intensi ty at point r of the homogeneous medium; and ZXQ(r T, w, E) is the dis t r ibut ion densi ty of the source equivalent to the flaw.

The equivalent sou rce cons is t s of two par ts :

A Q = A Q o + A Q S " (2)

The f i r s t par t is d i rec t ional and is due to unsca t te red radiation:

~ Qo (r', to, E) = ~ ( t o - - % ) a ( E - - Z O ) a ~ ( r ' , E o ) e -~o~"

The second par t has a continuous angular distr ibution:

AQs (r' , to, E) = A ~ ( r ' , E ) O s ( r ' , t o , E) -- Sdto'f~dE'aI~s ( r ' ,E ' ) k ( to" -+ to, E' ~ E ) q g ( r ' , t o ' , E ' ) ,

where ~s is the di f ferent ia l flux of s ca t t e r ed radiation. Multiplying (2) by the conjugate function, and in te- gra t ing over co and E, we find

[21

~ (r', E0) e - J O+ (r', too, E0; r) + S d to S d E q~+ (r', to, E; r) A Qs (r', to, E ) . (3)

At sma l l p = ~/(x - x q z + (y - y q Z for suff icient ly la rge z - z ' , the conjugate function can be wri t ten as

1 O] E o e - L~o (~ - z') r (r', too, e0; r) = ~ [~ (~) + ~s0 (4)

S. M. Kirov T o m s k Polytechnical Insti tute. Trans la ted f r o m Izves t iya Vysshikh Uchebnykh Zaved- enii, Fizika, No. 1, pp. 111-114, January , 1971. Original a r t i c l e submit ted November 27, 1969.

�9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or an), purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

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F o r a de t ec to r which m e a s u r e s the in tens i ty we have

o = (cos E, (cos o) dO, Eo

0

w h e r e k(cos d) is the s c a t t e r i n g c h a r a c t e r i s t i c ; E 1 is the e n e r g y of a photon s c a t t e r e d th rough an angle 8 and having an e n e r g y E 0 be fo re the s ca t t e r i ng , and we have 8 m = 7r/2 if the d e t e c t o r is a t the boundary l a y e r and d m = 7r if the in tens i ty is being ca lcu la ted within the b a r r i e r , f a r f r o m the boundary .

It fo l lows f r o m Eq. (4) tha t the f i r s t t e r m in the s u m (3) has a s i ngu la r i t y as p ~ 0. The second t e r m of this s u m is the r ad ia t ion in tens i ty at r p roduced by a point s o u r c e with a cont inuous angu la r d i s t r ibu t ion a t r ' . As p ~ 0 with z - z ' ~ 0, the second t e r m r e m a i n s finite. If the in t eg ra t ion r ange in (1) is suf f ic ient ly sho r t and inc ludes the s ingu la r i t y p = 0, we can neg lec t the second t e r m in (3). Then we find

d 2 ~p • (r ' ) d r ' . (5) V

To evaluate the f i r s t i n t eg ra l we use a cy l ind r i ca l coord ina te s y s t e m whose OZ axis pas ses th rough point r ; we find

:X Io (r) = - - Eo e-~'o~ / &~o (x, 3% z') dz'. (6)

T h e r e is no p a r t i c u l a r d i f f icul ty in eva lua t ing quant i ty (6), which d e s c r i b e s the f law image produced by the u n s c a t t e r e d radia t ion . The second in t eg ra l in (5) g ives the image produced by t he s c a t t e r e d r a d i a - tion; we wr i t e i t as

A I ~ ( r ) = l~*~ ~ ( r ) = ; A l~ dx 'dy ' (7)

Equat ion (7) d i r e c t l y g ives the r e l a t i onsh ip be tween the even f law image AI0, which c o r r e s p o n d s to the g e o m e t r i c s i ze of the flaw, and the odd ("b lur red") image produced by the s c a t t e r e d radia t ion . I n t e r - es t ingly , this r e l a t ionsh ip conta ins no addi t ional i n fo rma t ion about the flaw. The e x p r e s s i o n fo r q)(x, y, z) is s i m i l a r to that fo r the e l e c t r o s t a t i c potent ia l in the z = cons t plane with a s u r f a c e cha rge dens i ty of

o(x, y)=-- A lo(X, y , z ) .

F o r s impl i c i ty , we will use this notat ion below.

w C y l i n d r i c a l F l a w

We t r a n s f o r m to the new v a r i a b l e s r , r ' , and 8, us ing

x' = r 'cos ~%, y' = r 'sin O, r = V x a + y,a.

We wr i t e the i m a g e of a cy l i nd r i ca l f law of r ad ius a produced by the u n s c a t t e r e d r ad ia t ion as ~ (r) /

= ]cr0, r < a; whe re cr 0 is a cons tan t which depends on the height of the f law and the d i f fe rence Ap within the 0, r > a;

flaw. The e x p r e s s i o n fo r ~0 is

a 2~

a(r) = j ' ! l f r= 5~ �9 -,77# 2 = ) 7 / c o s

0 0

In tegra t ing over r ' ,

r~

a, ( r ) - - 2 % t ' Q V r 2 + a 2 - - 2 r a c o s { } - - r + r c o s O h l C ~ - - r c ~ 1 6 2 1 7 6 0

in t roduc ing the v a r i a b l e • = (Tr- 8 ) / 2 and in t eg ra t ing the t e r m conta in ing the l o g a r i t h m by pa r t s , we find

? (r) - 9 ~o [ (a + r) E (k) ~ (a - r) 1((k) ] , k 21/7-d, (8) r + a

87

w h e r e K and E a r e comple te e l l ip t ic i n t eg ra l s of the f i r s t and second Mnds. At r = 0 we have E = K = rr / 2 , and Eq. (8) c o n v e r t s into the equat ion found p rev ious ly fo r the cen t r a l point of a f law [3, 4]: ~0(0) = 2 7ra O-o.

w A x i s y m m e t r i c F l a w

In genera l , the quant i ty ~0(r) fo r the c a s e of an a x i s y m m e t r i c f law is

CO 2r~

(r} = V r~ =: r'==-~ - 2rr~ cos ~ ' 0 0

Expanding the r e c i p r o c a l d i s tance in t e r m s of L e g e n d r e po lynomia l s ,

[r 2 + r "2 - 2rr' cos O] 7= 1 ,7 r< P~(cos~)

(9)

and subs t i tu t ing this expans ion into (9), we find

" r Cn , ~ r ) a ( c ' )d r '+ ;7) ~(r ' )dr ' , n = O 0 r

w h e r e

1 n 9_ C~-=,i'p~(cos~)d{)=2r, P~(O)= 2= ~ n/2 '

0

for odd n.

for even n.

The angu la r i n t eg ra t ion in Eq. (9) can be c a r r i e d out immedia t e ly :

co

~ ( r ) = 4 ~ K ( l e ' ) ~(r ' )r 'dr ' k " - 2 V - ~ (10) r + r" ' r + r" o

The image of an a x i s y m m e t r i c flaw of this shape can be found d i r e c t l y by c a r r y i n g out a n u m e r i c a l i n t e g r a - t ion in (10).

w A n I n f i n i t e l y L o n g F l a w

If the f law is inf ini te ly long in one d i r ec t i on (e.g., e long the OY axis) , and we have a(x , y) - a(x) , we can conve r t the i n t eg ra l f o r ~p to

+ o o

(x) = -- 2 5 ~ (x')lnlx - - x'[clx'. (11) - o o

The r e su l t s of in t eg ra t ing this e x p r e s s i o n fo r two c a s e s of i n t e r e s t a r e as fol lows: = / ~ [x[ < a;

1. F o r a f law of r e c t a n g u l a r c r o s s sec t ion (a slot) , we have a(x) | 0 , Ix[ > a;

(x) = 2% [2a + (x - - a) lnlx -- a I -- (x + a) lnlx + a[].

2. F o r a f law of t r i angu l a r c r o s s sec t ion , we have

and

[ 0 ~ x < O ,

~ ( x ) = % ' 0 < x < a ,

O, x > a ;

~ ( x ) = ~ % +ax--xalnlx], - - ( a ~ - x 2 ) I n [ a - x ] .

The i m a g e s of inf ini te ly long f laws having o ther s imp le c r o s s sec t ions can be found in an ana logous manne r .

8 8

An i m p o r t a n t c a s e in connect ion with the expe r imen ta l s tudy of the ro l e of the s c a t t e r e d r ad ia t ion in

image is the ndrop in the l a y e r t h i cknes s , " a(x) = ~0 , x < 0; In this ca se we have ~(x) = q~(0) t "

producing the - 2o-0x[ln I xl - 1]. t o-0, x > 0.

The au tho r s thank V. G. Bagrov for de r iv ing Eq. (8).

I.

2.

3. 4.

LITERATURE CITED

G. I. Marchuk and V. V. Orlov, in: Neutron Physics [in Russian], Atomizdat, Moscow (1961), p. 30. A. M. Kol'chuzhkin and V. V. Uehaikin, this issue, p. 81. A. M. Kol'chuzhkinand V. V. Uchaildn, Izv. VUZ, Fiz., No. II, 129 (1968). A. M. Kol'ehuzhkin and V. V. Uchaikin, Izv. VUZ, Fiz., No. II, 130 (1968).

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