Radiation Chemistry

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38 A Significant Correction Factor in Gamma Ray Dosimetry

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ARI BRNJOLFSSON U. S. Army Natick Laboratories, Natick, Mass. 01760

Softening of the gamma rays as they penetrate light mate­ rials may cause very large differences in the radiation doses absorbed in the samples and in the dosimeters. This is illustrated in the present paper by calculating the dose in 14 dosimeters and several other materials placed at distances of 0, 1, 2, and 4 relaxation lengths from a point isotropic Co source embedded in a large water container. These calcu­ lations show for instance, that the doses in water, Lucite, Fricke dosimeter, lithium fluoride, poly(vinyl chloride) and 0.4M ceric sulfate solution at zero distance from the source are in the ratios: 100: 96: 100: 83: 92: 99; at a distance cor­ responding to µ · r = 1 the dose ratios are 100: 95: 100: 85: 124: 169; and at a distance corresponding to µ ∙ r = 4 the similar ratios are: 100: 93: 101: 87: 162: 251. 60

t

t

A b s o r b e d dose i n a s a m p l e i r r a d i a t e d b y g a m m a rays is u s u a l l y determ i n e d b y m e a s u r i n g t h e a b s o r b e d dose i n a d o s i m e t e r ; f o r instance, a F r i c k e d o s i m e t e r p l a c e d i n the p o s i t i o n of t h e s a m p l e . T h i s a b s o r b e d dose i n the dosimeter is, h o w e v e r , g e n e r a l l y different f r o m that i n t h e sample. T o a r r i v e at t h e a b s o r b e d dose i n the s a m p l e , corrections m u s t b e m a d e f o r t h e difference.

T h e s e corrections are p a r t l y c a u s e d b y

g a m m a electron n o n - e q u i l i b r i u m at the b o u n d a r y , transfer of energy of e x c i t e d states across t h e b o u n d a r y , a n d p a r t l y c a u s e d b y differences i n mass energy transfer coefficients w h i c h are f u n c t i o n s of t h e a t o m i c n u m ­ b e r a n d t h e g a m m a r a y energy. T h e corrections c a u s e d b y b o u n d a r i e s w i l l n o t b e c o n s i d e r e d i n this p a p e r , b u t o n l y t h e corrections c a u s e d b y mass e n e r g y transfer coefficients. I n r a d i a t i o n d o s i m e t r y t h e energy a b s o r b e d p e r ml. of s a m p l e is u s u a l l y the q u a n t i t y of interest. T o a r r i v e at t h e energy a b s o r b e d p e r m l . 550 Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

38.

BRNJOLFSSON

Gamma

of the sample, the dose D

d

Ray

551

Dosimetry

i n the dosimeter—i.e.,

the F r i e k e dosimeter,

is first m u l t i p l i e d b y the r a t i o — , i.e., the ratio of d e n s i t y p of the s a m p l e Pd s o l u t i o n to the d e n s i t y p of the dosimeter s o l u t i o n . S e c o n d l y , the dose D is m u l t i p l i e d b y the ratio — · — , i.e., the ratio of the mass energy Pb μα s

d

d

transfer coefficients.

T h e s e t w o corrections factors are u s u a l l y a p p l i e d .

T h e t h i r d c o r r e c t i o n factor, w h i c h is the r a t i o of the a d s o r b e d dose b u i l d u p factors i n the s a m p l e a n d the dosimeter, is u s u a l l y i g n o r e d , b u t is s h o w n i n this p a p e r to be v e r y i m p o r t a n t . T h e a b s o r b e d dose b u i l d u p Downloaded by CORNELL UNIV on August 24, 2016 | http://pubs.acs.org Publication Date: January 1, 1968 | doi: 10.1021/ba-1968-0081.ch038

factor is d e f i n e d i n this p a p e r analogous to the dose b u i l d u p factor, a n o t a t i o n u s e d w h e n the u n i t r o e n t g e n was s t i l l the u n i t of r a d i a t i o n dose. T h i s p a p e r shows the m a g n i t u d e of this t h i r d c o r r e c t i o n factor, w h i c h is c a u s e d b y differences i n g a m m a - r a y a t t e n u a t i o n coefficients a n d s o f t e n i n g of the g a m m a - r a y s p e c t r u m . A s a n i l l u s t r a t i v e e x a m p l e , the dose i n d i f ­ ferent dosimeters is c a l c u l a t e d as a f u n c t i o n of the distance f r o m a p o i n t i s o t r o p i c cobalt-60 source i n w a t e r .

Calculations

of Absorbed Dose

T h e g a m m a - r a y energy i n rads p e r second a b s o r b e d i n a n i n f i n i t e s i ­ m a l v o l u m e d x d y d z at the p o i n t P ( x , y , z ) is g i v e n b y d = 1.60209 · 10-8 f J ο

E

= 1.60209 · 10-8 Γ Jo

m

Ε

a

—

x

E

.

. J^l

dE

^(E) ρ

dl(E) dE

.

p

d

E

(

1

)

.

where d = Ε = φ(Ε) =

dose rate i n rads/sec. = 100 e r g / g r a m sec. p h o t o n energy i n M e v . p h o t o n flux d e n s i t y = the t o t a l n u m b e r of photons of energy less t h a n Ε w h i c h enter a sphere of cross-sectional area 1 c m . p e r sec. at the c o n s i d e r e d p o i n t P . φ ( Ε ) is i n units of c m . " sec." ( t o t a l n u m b e r of p h o t o n s p e r c m . per s e c ) . 2

2

=

1

2

p h o t o n flux d e n s i t y s p e c t r u m = n u m b e r of photons i n the energy i n t e r v a l Ε to Ε + dE w h i c h enter a sphere of cross-sectional area 1 c m . p e r sec. at t h e c o n s i d e r e d point P. 2

^/IF~ d

* * * °^ M e v - c m . " sec." ( n u m b e r of p h o t o n s p e r M e v . p e r c m . p e r sec. ). s

n

u n

t s

1

2

1

2

( E) —- = p

mass energy transfer coefficient i n c m . / g r a m of the d o s i m eter at Ρ f o r p h o t o n s i n the e n e r g y i n t e r v a l Ε to Ε + d E . ρ is the d e n s i t y i n g r a m / c c . 2

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

552

RADIATION CHEMISTRY

7(E)

=

1

the energy flux d e n s i t y or intensity—i.e., the total energy of a l l the photons w i t h energy less t h a n Ε that cross a sphere of cross-sectional area of 1 c m . p e r sec. at the p o i n t P. 1(E) is i n units of M e v . · c m . " · sec." ( e n e r g y i n M e v . per c m . per s e c ) . 2

2

1

2

dl(E)

the energy flux d e n s i t y s p e c t r u m or i n t e n s i t y s p e c t r u m — i.e., t o t a l energy of the p h o t o n s i n the energy i n t e r v a l Ε to Ε + dE that cross a sphere of cross-sectional area of 1

dE

c m . p e r sec. at the c o n s i d e r e d p o i n t Ρ · = do a£ ^ · Ε is i n units of c m . " sec." ( e n e r g y i n M e v . p e r M e v . per c m . " per s e c ) . 2

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2

1

2

T h e A b s o r p t i o n Coefficient. I n E q u a t i o n 1 the mass energy transfer coefficient — Ρ

should be

used

a n d not

the

mass

energy

absorption

coefficient — g i v e n b y Ρ tl Ρ

=

L- + ^£ + — Ρ Ρ Ρ

(2)

where τ = Ρ σ &

=

p h o t o e l e c t r i c mass a t t e n u a t i o n coefficient i n c m . / g r a m . 2

1.

σ

the a b s o r p t i o n c o m p o n e n t of the total C o m p t o n

cross section i n c m . / g r a m , E is the average e n e r g y g i v e n to the electrons i n the C o m p t o n process w i t h t o t a l cross 2

e

section — i n c m . " / g r a m f o r i n c o m i n g photons of energy 2

h

— = Ρ

,

p

the cross section for the p a i r p r o d u c t i o n i n c m . / g r a m . 2

T h e mass e n e r g y transfer coefficient is s i m i l a r l y g i v e n b y P'k

Ρ

_j_ Ç[a _|_

Ρ

Ρ

K

&

Ρ

(3)

where Ta

Ρ

ρ

( 0 Χ

(4)

(5)

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

38.

BRNJOLFSSON

Gamma

Ray

553

Dosimetry

where δ =

average e n e r g y e m i t t e d as fluorescent r a d i a t i o n p e r p h o t o n a b s o r b e d i n the p h o t o e l e c t r i c process.

=

is the c o r r e c t i o n f o r e s c a p i n g r a d i a t i o n f r o m the a n n i h i l a t i o n of t h e p o s i t r o n .

v

δ is m a i n l y d e t e r m i n e d b y t h e fluorescence y i e l d a> i n the K - s h e l l . a> is, a c c o r d i n g to H a g e d o o r n a n d W a p s t r a (4) g i v e n b y k

k

W k

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1

— ( - 6 . 4 · ΙΟ" + 3.4 · ΙΟ" · Ζ - 1.03 · 1 0 " Z ) ; 2

0>

2

6

3

(6)

4

k

where Ζ =

atomic number n u m b e r K - s h e l l vacancies n u m b e r K - s h e l l x-rays

W k

o> as a f u n c t i o n of the a t o m i c n u m b e r ( Ζ ) is s h o w n i n T a b l e I. k

T a b l e I.

Fluorescent

Fluorescent Yield ωκ '100K ·100 r

Atomic Number 2 Element 10 14 16 20 26 29 30 40 50 56 58 60

Ο Ne Si S Ca Fe Cu Zn Zr Sn Ba Ce Nd

K» + K in %

Au

0.18 0.57 2.7 4.9 12 29 39 43 70 83 88 89 90

Yield

Absorptions Coefficient Electron Binding in cm. /gram in Energy in Kev. _ Water at the Gamma Energy K-Shell L-Shell 2

.532 .867 1.839 2.472 4.038 7.112 8.972 9.659 17.998 29.200 37.441 40.444 43.568

33,000 7,200 800 320 72 13.5 6.8 5.4 0.76 0.157 0.075 0.062 0.053

.019 .118 .193 .400 .842 1.100 1.196 2.532 4.465 5.987 6.549 7.126

I n l i g h t elements δ is a l w a y s s m a l l , because most of t h e energy is t a k e n u p b y the A u g e r electrons a n d — c a n t h e n b e r e p l a c e d b v —. A s Ρ ' Ρ the a t o m i c n u m b e r increases, t h e fluorescent r a d i a t i o n increases. A p o r ­ t i o n of the fluorescent r a d i a t i o n , e s p e c i a l l y f r o m the L - s h e l l or the h i g h e r shells, is often a b s o r b e d w i t h i n t h e dosimeter; f o r instance, t h e 1,000 e.v. x-rays f r o m the L - s h e l l i n c o p p e r penetrate o n l y 2 · 10~ c m . of w a t e r . 4

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

554

RADIATION CHEMISTRY

1

T h e r e f o r e , i n these calculations w e h a v e n e g l e c t e d this fluorescent r a d i a ­ t i o n a n d u s e d — i n s t e a d of — i n E q u a t i o n 3. T h i s a p p r o x i m a t i o n is Ρ ρ a d e q u a t e f o r samples a n d dosimeters c o n t a i n i n g a t o m i c n u m b e r Ζ <

30.

B u t for samples c o n t a i n i n g h i g h a t o m i c number—e.g., eerie sulfate s o l u ­ t i o n s — t h i s a p p r o x i m a t i o n i n the calculations leads to a n a b s o r b e d dose w h i c h is s l i g h t l y too h i g h . I n case of

6 0

C o r a d i a t i o n , the p a i r p r o d u c t i o n

is n e g l i g i b l e i n l i g h t elements, w h i l e i n c e r i u m , the heaviest c o n s i d e r e d here, it is

0.8%.

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those r e p o r t e d b y S t o r m et al.

Energy F l u x Density Spectrum. point isotropic

6 0

Co

u s e d i n this r e p o r t

are

(10). ^(JP

has b e e n c a l c u l a t e d for a

source e m b e d d e d i n a large w a t e r c o n t a i n e r

Goldstein and Wilkins

(3).

(The

by

n o m e n c l a t u r e i n this p a p e r is that

r e c o m m e n d e d b y the I n t e r n a t i o n a l C o m m i s s i o n o n R a d i o l o g i c a l U n i t s a n d M e a s u r e m e n t s (6),

w h i c h differs f r o m that u s e d b y G o l d s t e i n a n d

W i l k i n s w h o u s e d I f o r the same q u a n t i t y as ^

i n this p a p e r . ) C o r r e ­

s p o n d i n g energy b u i l d u p factors i n w a t e r w e r e m e a s u r e d b y G . R . W h i t e (12),

V a n D i l l a a n d H i n e (2),

(8).

T h e s e e x p e r i m e n t a l b u i l d u p factors w e r e f o u n d to agree w i t h the

B i b e r g a l et al. (1 ), a n d b y Sehested et

al.

t h e o r e t i c a l l y c a l c u l a t e d ones w i t h i n e x p e r i m e n t a l a n d c a l c u l a t e d a c c u r a c y of 1 0 % .

W e i s s a n d B e r n s t e i n (11)

s t u d i e d the energy s p e c t r u m b e l o w

150 K e v . a n d f o u n d agreement w i t h Spencer's a n d Fano's

calculated

values ( 9 ) , w h o s e c a l c u l a t i o n s w e r e the basis for the r e p o r t b y G o l d s t e i n a n d W i l k i n s (3).

A l l this indicates that the intensity s p e c t r u m

reported b y Goldstein a n d W i l k i n s for a point isotropic

6 0

C o source i n

w a t e r is f a i r l y correct a n d it w i l l , therefore, be u s e d i n E q u a t i o n 1.

The

spectra are s h o w n i n F i g u r e 1. Calculation of Equation 1. G o l d s t e i n a n d W i l k i n s list o n l y a f e w points o n the spectral curves.

W e h a v e g r a p h i c a l l y i n t e r p o l a t e d these

points so that s m a l l intervals c o u l d b e u s e d i n the n u m e r i c a l i n t e g r a t i o n of E q u a t i o n 1. F u r t h e r , a n e x t r a p o l a t i o n of the s p e c t r a l values b e y o n d the lowest v a l u e r e p o r t e d b y G o l d s t e i n a n d W i l k i n s w a s d o n e b y assum­ i n g that at the l o w energies the s p e c t r a l d i s t r i b u t i o n is s i m i l a r to that for a p r i m a r y p h o t o n energy of 1 M e v . E q u a t i o n 1 was i n t e g r a t e d n u m e r i c a l l y , because neither ^ c a n be expressed a c c u r a t e l y w i t h s i m p l e f u n c t i o n s .

nor

^

T h e w i d t h s of the

energy intervals u s e d i n the i n t e g r a t i o n w e r e 0.01 M e v . f r o m 0.025 M e v . to 0.175 M e v . ; 0.0125 M e v . for photons of 0.1750 M e v . to 0.1875 M e v . ; 0.025 M e v . f o r photons of 0.1875 M e v . to 1.2125 M e v . ; a n d 0.0375 M e v .

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

38.

B R N j OLFSSON

Gamma

Ray

555

Dosimetry

f o r p h o t o n s of 1.2125 M e v . t o 1.2500 M e v . F o r t h e p r i m a r y p h o t o n s f r o m 6 0

C o 1.17 M e v . a n d 1.33 M e v . , a n average energy of 1.25 M e v . w a s u s e d . G o l d s t e i n a n d W i l k i n s (3) d o n o t list t h e p h o t o n i n t e n s i t y s p e c t r u m d i r e c t l y b u t t h e v a l u e of A

dI (E)

9

,

B

x

where / X t

0.0632 c m . " is t h e t o t a l a b s o r p t i o n coefficient i n w a t e r a t the p r i m a r y p h o t o n energy E

=

1

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0

W e h a v e , therefore, first c a l c u l a t e d t h e v a l u e d • · 4ΤΓΓ · β χ ρ ί ^ · r) = 2

1.602

· ΙΟ" · 3.7 · 1 0 C · 8

10

/ \ Γ 2 . 5 β χ ρ ( - ^ · r ) · /%(Ε ) 4*r* · e x p U * 0 · [ Λ

2

0

+

m

( ) 7

T h e first t e r m i n t h e b r a c k e t is t h e c o n t r i b u t i o n f r o m t h e p r i m a r y g a m m a rays (1.17 a n d 1.33 M e v . ) at t h e p o i n t P, r c m . f r o m t h e p o i n t source a n d t h e last t e r m is t h e c o n t r i b u t i o n f r o m t h e scattered

gamma

rays at t h e p o i n t P . d is the dose rate i n rads p e r s e c ; r is t h e distance i n water f r o m the point isotropic

6 0

C o source of C curies; p = 0.0632 c m . t

- 1

is t h e t o t a l l i n e a r a b s o r p t i o n coefficient i n w a t e r f o r 1.25 M e v . p h o t o n s ; is t h e scattered g a m m a r a y i n t e n s i t y s p e c t r u m ; a n d

is the

energy transfer coefficient i n t h e dosimeter. I t is a s s u m e d that t h e d o s i m e ­ ter is s m a l l e n o u g h n o t to c h a n g e t h e energy i n t e n s i t y s p e c t r u m i n t h e w a t e r at t h e p o i n t P, a n d that i t is large e n o u g h to m a k e t h e effect of g a m m a electron n o n e q u i l i b r i u m negligible. Definition of Absorbed Dose Buildup Factor. I n t h e analysis of t h e dose v a r i a t i o n , t h e c o n c e p t of dose b u i l d u p f a c t o r is u s e f u l . T h e u s u a l d e f i n i t i o n of dose b u i l d u p factor (3, 5,7) dosimeter.

l i m i t s its use to dose i n a n a i r

T h e present d e f i n i t i o n of a b s o r b e d dose m e a s u r e d i n rads,

b y w h i c h dose i n a n y m a t e r i a l o r i n a n y dosimeter is d e f i n e d ( 6 ) makes t h e p r e v i o u s d e f i n i t i o n of dose b u i l d u p factor t o o restrictive. W e w i l l , therefore, r e p l a c e t h e dose b u i l d u p factor b y d e f i n i n g t h e a b s o r b e d dose b u i l d u p f a c t o r B(r) f o r a g i v e n d o s i m e t e r i n a g i v e n m e d i u m as t h e r a t i o of t h e a c t u a l a b s o r b e d dose i n t h e d o s i m e t e r to t h e a b s o r b e d dose that w o u l d b e m e a s u r e d i n t h e d o s i m e t e r i f there w a s n o scattered r a d i a t i o n . T h e v a l u e o f E q u a t i o n 7 w a s , therefore, d i v i d e d b y t h e a b s o r b e d dose

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

556

RADIATION CHEMISTRY

1

rate f r o m t h e u n s c a t t e r e d p h o t o n s , t h e first t e r m o n t h e r i g h t side i n E q u a t i o n 7. T h i s q u o t i e n t v a l u e w e c a l l B(r),

J ^ e x p U - r ) . dE ^^^-^

/%(Εθ)

B(f)

where J

0

(8)

μ*(Εο)

=

2.5 M e v . p e r o n e d i s i n t e g r a t i o n of ι—ι

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i.e.,

6 0

Co.

ι ιM1I

DIFFERENTIAL ENERGY SPECTRA IN WATER C 0 POINT ISOTROPIC SOURCE 60

Figure

1.

Energy

dl spectra -j= in water at a

alii

distance r corresponding to μ · r — 1; μ · r = 2; and ^ · r = 4 from a point isotropic Co source. The ordinate shows and 4-rrr exp (μ - r) - - J ^ ; the abscissa the photon energy in Mev. ί

χ

60

t

2

ΐ

T h e a b s o r b e d dose b u i l d u p factor B ( r ) i n E q u a t i o n 8 is t h e r a t i o of the a c t u a l dose i n t h e d o s i m e t e r at a p o i n t P , r c m . f r o m a p o i n t iso­ tropic

6 0

C o source i m b e d d e d i n large w a t e r container, to t h e dose that

w o u l d b e m e a s u r e d at t h e same p o i n t i f there w e r e n o scattered r a d i a t i o n . I n this e q u a t i o n 7 is t h e energy e m i t t e d b y the source; I is t h e scattered 0

s

r a d i a t i o n flux at Ρ ; μι is t h e t o t a l a b s o r p t i o n coefficient of w a t e r (0.0632 cm." ); 1

and —

is t h e energy transfer

coefficient i n c m . / g r a m i n t h e 2

dosimeter. T h e i n t e g r a l i n E q u a t i o n 8 w a s c a l c u l a t e d f o r t e n elements c o m m o n i n a p p l i e d dosimeters. T h e s e t e n elements w e r e H , C , O , A l , S i , S, C I , F e ,

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

38.

BRNJOLFSSON

Gamma

Ray

557

Dosimetry

C u , a n d C e . T h e c o r r e s p o n d i n g b u i l d u p factors c a l c u l a t e d a c c o r d i n g to E q u a t i o n 8 are l i s t e d i n T a b l e I I . Table II.

Dose Buildup Factors in Elements at Different Distances in Water from a Point Isotropic C o G 0

Downloaded by CORNELL UNIV on August 24, 2016 | http://pubs.acs.org Publication Date: January 1, 1968 | doi: 10.1021/ba-1968-0081.ch038

Element

He Li Be Β *C Ν *0 F Ne Na Mg *A1 *Si Ρ *s *C1 A

Element

Buildup Factors at μ · r = ΐ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

4

1.958 1.96 1.96 1.97 1.98 1.995 2.02 2.051 2.10 2.17 2.25 2.36 2.494 2.665 2.86 3.106 3.38 3.61

3.101 3.10 3.11 3.13 3.16 3.201 3.27 3.363 3.50 3.69 3.92 4.25 4.627 5.115 5.65 6.367 7.14 7.90

5.618 5.62 5.64 5.68 5.74 5.850 6.01 6.228 6.59 7.20 7.62 8.32 9.179 10.32 11.50 13.24 15.04 17.00

Κ Ca Sc Ti V Cr Mn *Fe Co Ni *Cu Zn Br Zr Rh Sn I *Ce

Buildup Factors at ^ · r = t

19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 53 58

I

2

4

3.94 4.31 4.71 5.19 5.75 6.17 7.00 7.66 8.2 9.0 9.86 10.8 16 23 31 40 45 38.5

8.8 9.9 11.2 12.5 14.0 15.6 17.3 19.24 21.2 23.2 25.43 28 43 62 85 112 130 103.6

19.1 21.5 24.4 27.5 31.0 34.7 39.0 43.29 48.0 52.8 57.7 64 97 138 190 252 295 241.1

Interpolation of the Values of B. T h e energy a b s o r p t i o n coefficient can be approximated b y :

J± = ρ

a

(

£

)

A

(£)

b

z +

-( ) f

z

( 9 )

A

where Ζ =

atomic number

A =

atomic weight

— —^ Λ

a(E) b(E)

=

== a f u n c t i o n of t h e p h o t o n energy E, b u t i n d e p e n d e n t of Ζ and A .

· f(Z) = b(E)

C o m p t o n absorption

photoelectric absorption

=

a f u n c t i o n of the p h o t o n energy Ε b u t i n d e p e n d e n t of Ζ and A .

f(Z) =

a f u n c t i o n of t h e a t o m i c n u m b e r b u t i n d e p e n d e n t of Ε

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

558

RADIATION CHEMISTRY

1

I 1 -Π1 !

1

"""ΤI ΤΓ

200

-

-

o «oo Ο