High energy photonuclear experiments using beam hardening

NUCLEAR

INSTRUMENTS

AND METHODS

94

HIGH ENERGY PHOTONUCLEAR

(t971) 5 5 ~ - 5 5 5 ; % N O R T H - H O L L A N D

EXPERIMENTS

PUBLISH|NGCO.

USING BEAM HARDENING

H. H. TH1ES, B. W. THOMAS, P. E. SIMMONDS, D. M. CRAWFORD and K. BOTTCHER

Unil,ersity of Western Australia, Nedlands, Western Australia, 6009 Australia Received 18 February 1971 The effect of beam hardening in high energy photon difference type photonuclear experiments is calculated. With the gamma inlensities available from linear accelerators an order of magnitude improvement in the accuracy of derived cross sections may be approached. Practical considerations are discussed and demonstrated by a numerical sample calculation. and the normalised yield by

1. Introduction

The resolution of photon-difference type bremsstrahlung experiments is inherently p o o r at energies above the photo-giant resonancel). With the g a m m a intensities available from linear accelerators it is practical to substantially increase experimental resolution above the giant resonance by " b e a m hardening", i.e. by passing the bremsstrahlung beam through a long low A absorber, before it strikes the external target. In photonuclear work, cross sections are derived from yield data by an involution process. It is then not a trivial problem to estimate, how - via the involution process beam hardening will affect the resolution of the cross section. In this paper we derive explicit expressions, which permit in a simple manner, to calculate as function of p h o t o n energy the improvement that can be gained by beana hardening. Practical considerations are discussed and demonstrated by a sample experiment, using for the calculation a known cross section and appropriate absorbers of different lengths.

p(Eo,hv)rr(hv)d(hv ).

y(E0) =

In the photon-difference type experiment y(Eo) is measured at peak bremsstrahlung energies E 0 = E 0 1 , Eo2 ..... Eok. The corresponding discrete set of k yield ordinates ),(Eoz),y(Eo2) ..... y(Eok) can then be used to calculate, in approximation, the discrete set of cross section ordinates a(hvz),a(hv2) ..... a(hvk), where now

hvi = ½(Eoi + Eo(i-i )).

f

y(Eoy) = E p(Eoz,hvi)a(hvi)A(hvi)'

(5)

i-1

which m a y be written as .r

f=

1,2 ..... k,

(6)

i--1

2. Calculation of effect of beam hardening on

where

photo-nuclear cross section

In absence of a beam hardener the photo-nuclear yield Y(Eo) and cross section a(hv) are related through the bremsstrahlung spectral distribution function P(Eo,hv) by the integral equation

y r -- y(Eo

),

P(Eo,hv)a(hv)d(hv),

A, =

= o(h,,,),

and

Pfi

=

p(Eoy,hvl)=

y

e(E°y'hvi)

(7)

Y, P(Eoc,hvi)Ai

Eo

I

(4)

We use the following approximation. We substitute for the integral equation (3) the finite difference equation

)'y = Y, pyiaiAi,

Y(Eo) =

(3)

Eth

(1)

i-I

,,JEt h

Defining

where Eth is the threshold energy of the reaction investigated. For convenience we define the normalised effective bremsstrahlung distribution function p(Eo,hv ) by

f = E e(Eor,h,',)A,,

(8)

i=1

(7) becomes

p(Eo,hv ) =

P(E°'hv)

,

(2)

Pfi = (1/gf)Pfi.

l e o P(Eo,h")d(hv)

(9)

The k equations (6) m a y then be conveniently written in matrix f o r m

Eth

551

552

H . H . T H I E S et al.

which k equations may be written conveniently in matrix form thus

1/g~ LP~

_.1 L

L:Tr7 o1i1,o : .~2

Pkk

1/,q2

P21

P22

0

z

F o ' I A 1- ]

l/Ok__ I P k l

Pkk

(10)

i

LakAkl

e_U2

62A2

×

or

y = (1/o).F.(~A).

(11)

We define the inverse of P by C = P-~. Solving for (aA) and (~xAx) one obtains

(aA) = C ' g ' y ,

(12)

and hence

08)

|

e-"

L6~3kI

or

y = (1/~).P.(e-").~(3).

(19)

Solving for (~A) and 6:A r we obtain

,f

~:A: = Z C:m,y,.

{yi})2

=

i

;

.

P(Eo,hv ) = P(Eo,hv)e -'u'È)',

(15)

gf

f P(Eo:,hv,)e-"'

(16)

~ P(Eoy,hvi)e-mAi i=1

Then in analogous approximation to above, the normalised reduced yield with beam hardener is given by

(21)

i=l

Then the standard deviation of #:A: is defined in analogy to above st {6:3:} =

where p(hv) is the g a m m a absorption coefficient of the beam hardener and 1 is its length. We shall write a short-hand p(hv~)l = u; and now define the normalised effective bremsstrahlung distribution function for the case with beam hardener by

7

f

6:3: = E; e "' C:ig~y -- v

(14)

When beam hardening is used, the bremsstrahlung spectral distribution function becomes

P:i = ~(Eo.r,hvi ) _ P:i _

(20)

and

We shall assume that experimental errors in yz have a Gaussian error distribution with a standard deviation in Yl defined by st{yl}. The standard deviation in a:A: defined by st{a:A:} is then calculated from

st{~:Af}

(~a) = (e')" C - y . y

03)

i=1

I ~:

}

(e":C:flist {yi})2 )

i=i

(22)

and R:, the ratio of st{ 6:z[:} from an experiment with beam hardener to st{arAr } - i.e. the corresponding standard deviation from an experiment without beam hardener - becomes r f

e

"½

_ St {ff:Af} _ !i~_, (e":c:'o'st {y'} )2

....

i~=, ( C :igi st {),,})z

i

We assume that the precision, with which Yi and )7i can be measured, is limited by the stability of the accelerator and the detection system and not by beam intensity - and hence time considerations. With modern equipment the practical limit of accuracy in redticed yield measurements is at present of order 0.1°/,o. We thus assume that

f

Y: = Z P,ficriA~

st{y;} _- st{y;}__(greater or of order 0.001). (24) Yi Y;

i 1 f

= Z (1/,Of)P:i e-"'-GiAi,i=1

f = 1 , 2 ..... k,

(17)

Thus with (6), (17) and (24) we obtain

HIGH ENERGY PHOTONUCLEAR

st {y~} = st {.vA yj

12-

Yi

:

xlO-:' l

(a)'

!

i

J

~, Pise-'%7~As gi= s=l g~ i ~, Pisa,sAs

= st{yi}

553

E X P E R I M E N T S U S I N G BEAM H A R D E N I N G

v

[NevF

(25)

s= l

~Eth-

and hence substituting (25) into (23)

,i

20

2. "~-1(Cfigist{yi}ri)2]½ Rf=

e s~

f

~

Z

(C,rigist{yi}) 2 }

(26)

,

40

hv NeV

60

Fig. la. Normalised effective bremsstrahlung distribution function p(Eo,hp) as function of photon energy hp. Eth = threshold of nuclear reaction aH(tu,) investigated. No beam hardening.

i=l

where i

Pis e-"~a~A s Pi ~

(27)

i

[The error in Rf due to the approximations used is of order ( - 10%) near the giant resonance and nil at high energies.] 3. Numerical

Alternatively, using normalised matrix i.e. with Pi., = (l/gl)Pi~ and

Cis = st {#IA-j-} Rf - m st {~rfA.r }

gsCis,

( p - i)i s =

__

e2US

elements,

,

(28) "

"

~. (<'//St

"

{y/})2)

(29)

i=1

and i

pise_U,asAs s=l

(30)

i

Pi~asAs S=I

In an actual experiment the standard deviations st{y~} and st{y~} vary only very slowly as function of i. On the other hand the coefficients cz~ decrease very rapidly with decreasing i - see e.g. the tabulations of Penfold and LeissZ). Hence in good approximation one may write R¢

st {cT"rA-S} st { a : A : }

-- -

'

- - e"O" i

example

and practical

considerations

Normalized effective bremsstrahlung spectra without and with beam hardener are shown for three different peak bremsstrahlung energies Eo on figs. la and lb, respectively. The effect of the beam hardener is to enhance the high energy region and thus to convert the bremsstrahlung spectrum into one which has a broad maximum below Eo. The effect of beam hardening on the yield - and hence on the precision of the derived corresponding cross section may be understood from the following qualitative consideration. Fig. 2 shows a hypothetical cross section aH(hv) which was used in computations throughout the following numerical example. Fig. 3 shows the normalised reduced yield as function of E o for an experiment without beam hardener and with beam hardener, as produced by spectra shown on figs. l a and lb. The yield without hardener approximates to

12" ,,lo -2

/,~

/

.#.

I

ha.

1

(b)

WE.=

~

27.ev

;Eo= 44NeV

6

(31)

or

f

0 -

pfse-U"~rsAs RZ _ st {e/A-,-} _ e,,S s=, st {afAr}

20

40

hv MeV

60

(32) f

~ pf~c%A~ S=I

Fig. l b.p(Eo,hr) as function of hi, same as fig. l a, but blemsstrahlung beam has been hardened by passing it through 1000 g/cm e of Li H.

554

H . H . THIES et al.

IEth an ( hv)d(hv )

1.0. ~.

a n d bears little r e s e m b l a n c e to aH(hv). T h e yield with h a r d e n e r r e s e m b l e s an(hv) as seen u n d e r r a t h e r low r e s o l u t i o n . In t h e idealised case o f an e x t r e m e l y l o n g

"~'no beam hardener

Rf=0.5.st (aA) ~ % ~ ~ ~ ~ 1 ~ ~ 1

1000 gin/eraa ~ 0

Io0. in

0

2{3

6'0

4'0 hv l~leV

Fig. 4. Relative improvement of cross-section accuracy R~ = st l~rf,If} / st ~crr.l tJ as a function of photon energy In, for various lengths of Li H beam hardener. 20 0

4'0 hv HeV

2O

60

Fig. 2. Hypothetical nuclear cross section O'H(hv) as function of photon energy hr. This cross section was used throughout for numerical calculations.

6: xlO-2 (Nev) -i

1000 gm/cm 2

(Ii .m

3

0

a~dener ,

0

20

40 E, MeV

60

b e a m h a r d e n e r the r e d u c e d s p e c t r u m w o u l d r e s e m b l e a n e a r m o n o c h r o m a t i c g a m m a line j u s t b e l o w hv = E o, a n d yield and cross section - a p a r t f r o m a c o n s t a n t m u l t i p l i e r - w o u l d be identical. T h e r a t i o R I = st{ #A}/st{rrA} was n u m e r i c a l l y calc u l a t e d for rrn(hv) for v a r i o u s lengths o f Li H b e a m h a r d e n e r . R I as f u n c t i o n ofhv is s h o w n o n fig. 4. F r o m this figure it is seen t h a t w i t h b e a m h a r d e n e r , a b o v e the g i a n t r e s o n a n c e there is a s u b s t a n t i a l i m p r o v e m e n t in e x p e r i m e n t a l a c c u r a c y . F o r 1000 g / c m 2 o r Li H in the b e a m R l b e c o m e s less t h a n 0.1, i.e. the e x p e r i m e n t a l e r r o r in t h e d e r i v e d values o f cr,(hvi) are r e d u c e d by a f a c t o r m o r e t h a n ten o v e r t h o s e w i t h o u t b e a m h a r d e n e r . T h e r e exists, h o w e v e r , a p r a c t i c a l c o n s i d e r a t i o n which limits the m a x i m u m l e n g t h o f the b e a m h a r d e n e r . T h i s l i m i t is set by t h e t o t a l " a t t e n u a t i o n " o f the yield. W e define the t o t a l " a t t e n u a t i o n " by

Fig. 3. Yield y(E0), without beam hardening, and yield v(E~) with 1000 g/cm 2 Li H beam hardening as functions of peak bremsstrahlung energy E0. Here

f

~PY~e-"~GAs Y(E0y) fy

s=l

/ ~Eo=60 MeV

y(Eo) = Y(Eo)..~/IEo=E,~

Y(Eo)dEo

Af =

and

f E Pfs GsAs

_

-

Y(Eof)

(33)

-

Yf

5--1

/ ('~o:~O ~¢v

;(Eo) = Y(Eo)/|Eo=E,~/j

_

Y(E°)dE"

H e n c e w i t h (27) a n d (31) eq. (33) m a y be written

TABLE 1 Relative cross section accuracy improvement Rf and corresponding attenuation As as function of photon energy hv for Li H beam hardening.

hv (MeV) 13 16 27 44 60

310 g/cm 2 Li H

640 g/cm 2 Li H

RI

Af

Rf

0.825 0.676 0.413 0.331 0.313

1/135 1/99 1/68 1/54 1/47

0.696 0.477 0.195 0.147 0.146

1000 g/cm 2 Li H

Af 1/24.7 1/12.3 1/5.2 1/2.6 1/1.8

x x x x x

RI 10:~ 103 10:3 103 103

0.603 0.353 0.105 0.086 0.090

AI 1/6.7 x I/2.1 x 1/4.7x 1/1.3 x 1/6.3 x

108 108 l07 l0 s 104

HIGH ENERGY PHOTONUCLEAR

A: = e-":R]~

EXPERIMENTS

(34)

which is readily evaluated once R: has been calculated. A: is the ratio of count rates with beam hardener in position over those with no hardener. The practical lower limit for A: in a linear accelerator experiment is 10 -3, possibly 10 -4. Numerical values of R: and corresponding values of A: are tabulated in table 1. The relative decrease of g a m m a absorption coefficients /l(hv) as function of hv is greatest for low A nuclei and hence ideally a long liquid hydrogen absorber would give the smallest value of R: for the least total attenuation A:. The density of liquid hydrogen is, however, so low that it is impractical to use it in an experiment as the inverse square law loss

U S I N G BEAM H A R D E N I N G

555

of intensity due to the length of the liquid hydrogen absorber offsets its low attenuation advantage. It appears that the optimum in R: with still sensibly low attenuation and a sensible length of beam hardener is obtained using Li H with its relatively high density (0.82). R: has a very shallow minimum at about 40 MeV in our example. This is typical. The advantage of a continuous decrease of/~(hv) above this region is offset by the fact that with increasing E o the normalised effective bremsstrahlung spectrum y(Eo) resembles less and less a broad "monochromatic line" (see fig. l b). References 1) H. H. Thies, Australian J. Phys. 14, no. 1 (1961) 174. ~) A. S. Penfold and J. E. Leiss, Analysis of photo-cross-sections, Report (University of Illinois, Champaign, Ill., 1958).