Indian J. P hys. 69A (5), 5 2 5-535 (1995)
I J P A
— an international journal
Measurement and analysis of isobaric contributions in pre-equilibrium nuclear reactions
A V Mohan Rao
Inter University Consortium for Deportment of Atomic Energy Facilities, Block-LB, Plot 8. Sector III, Salt Lake, Calcutta-700 091. Indio Received2 7 April 1995. accepted II July 1995
Abstract : Excitation functions of the reactions l69Tm f(o,p3n), (a,an), (a,a2n) and (a,a4n)] are investigated upto 60 MeV using foil stock activation technique and Ge(Li) gamma ray spectroscopy method. Isobaric contributions from the decaying parent and grand parent nuclei to the final product nucleus of interest, is discussed and separated out wherever necessary, for which a mathematical formulation from first principle is developed. The measured cross sections are compared with the updated version of hybrid model using initial exciton configuration n 0 = 4 (ApQ/i). While the model predictions give on approximate agreement at higher energies in (a,pxn) reactions, they are equally bad for all (a, uxn) type of reactions for which some direct reaction contributions are indicated.
Keywords : Pre-equilibrium nuclear reactions, isobaric contributions, excitation function PACS No. : 25.55.-e 1
1. Introduction
In recent years, a great deal of efforts have been made for the development of a variety of pre-equilibrium models in order to explain high energy continuum of the ejectiles in the intermediate energy-nuclear reactions. Existing pre-equilibrium models are originally designed to describe the emission of nucleons. They calculate particle emission rates by applying reciprocity to the ejectiles under consideration, rather than the system as a whole.
This does not pose much of a problem as long as only nucleons are considered, since their
existence inside the nuclei are generally accepted and since there is reasonable theoretical
guidance to follow their behaviour (single particle state densities, average potential, Pauli
restrictions
etc).When it comes to clusters, however, no information about pre-formation
and behaviour inside the nucleus is available on an
a p rio ribasis. Thus pre-equilibrium
models to describe cluster emission have introduced parameters or assumptions to fill the
gap. Most notably, the pie-formation parameter and the assumptions employed for a cluster
526
A V Mohan Raoo f s in g le p a rticle state d e n sity h a v e p ro v id ed p o w er fu l to o ls to rep ro d u ce e x p erim en ta l data.
S e v e ra l s u c h a p p ro a ch es [ 1 - 3 ] h a v e b een s u g g e s te d u sin g th is c o n c e p t. C lin e [4 ] su g g ested a c ru d e m o d e l o f sta tistica l c o a le s c e n c e w h ic h w a s su b s eq u e n tly im p ro v ed and corrected by the g ro u p at B ra tisla v a [5 ]. M a ch n er [61 f o llo w e d th is ap p roach and in terp reted its intrinsic c lu ste r p re-fo rm a tio n p r o b a b ilitie s in term s o f th e c o a le s c e n c e m o d e l o r ig in a lly d escrib ed for h ig h e n e rg y h e a v y ion rea ctio n s.
T h ere are a fe w c o m p lic a tio n s e n c o u n te r ed in th e e x p e rim en ta l stu d y o f (oc, Jm za) ty p e o f rea c tio n s u sin g a c tiv a tio n te ch n iq u e , su ch a s iso b a r ic p recu rsor c o n tr ib u tio n to the actu al resid u al n u cle u s w h o s e a c tiv ity is b e in g stu d ied to d e te rm in e the c r o s s s e c tio n . Su ch s itu a tio n is o fte n e n c o u n te r e d in th e a c tiv a tio n m e th o d fo r th e d e te r m in a tio n o f c ro ss s e c tio n s o f [ ( a , (x + 4 ) n ) ] , [ a , p (* + 3 )n ] and (oc, a n ) r ea c tio n s le a d in g to th e resid u al n u clei w h ic h arc iso b a rs o f o n e a n o th er and form a P d e c a y in g c h a in . T h u s, th e g ro w th and d eca y o f la st r e s id u a l n u c le u s is in f lu e n c e d b y th e c o n tr ib u tio n s d o m in g fr o m its iso b a r ic p recu rsors. In su ch situ a tio n , the sim p le c r o s s s e c tio n fo rm u la [7 -E ] ca n n o t b e e m p lo y e d to d e te r m in e th e c r o s s s e c tio n o f resid u a l n u c le u s, s in c e it d o e s npt ta k e in to a c c o u n t the c o n tr ib u tio n s fro m th e d e c a y in g p a ren t a n d /o r gran d p a ren t n u c le i (t h e s e in c lu d e the co n tr ib u tio n s d u rin g th eir g ro w th th rou gh ou t th e p erio d o f irradiation and later d u e to their d e c a y ). In su ch c a s e s , th e c ro s s s e c tio n s o f the parent n u cle u s is first d ete rm in e d u sin g the e x p r e s s io n g iv e n in S e c t io n 2 .1 ; and later th e c r o s s s e c tio n o f the d a u g h te r n u c le u s is c a lc u la te d . A m a th em a tica l fo r m u la tio n from first p r in c ip le is d e v e lo p e d to sep a ra te out e x a c tly p o in t by p o in t, the iso b a r ic p recu rsor c o n tr ib u tio n s from th e m ea su red in c lu s iv e e x c ita tio n fu n ctio n w h ic h are g iv en in S e c tio n 2 .2 .
In th e w ork rep orted h ere, a sy s te m a tic r e in v e stig a tio n o f ,69T m ( a ,p 3 n ) , ,69T m ( a , / . c u n ) ; x = 1, 2 , 4 is m a d e . T o th e b est o f the k n o w le d g e o f th e au th or, o n e o f th ese rea c tio n s n a m ely ,69T m ( a , p 3 n ) reaction w a s earlier stu d ied by S au e t a l [1 0 ] e m p lo y in g a N a l d e te c to r h a v in g 8% r e s o lu tio n , u p to a b ea m e n e r g y o f 5 0 M e V . S in c e in th eir stu dy, th ey h a v e not taken in to a c c o u n t th e iso b a r ic co n trib u tio n o f the d e c a y in g parent n u cleu s, th e o v e r w h e lm in g la r g e c o n tr ib u tio n o f l69Lu fo r m e d in l69T m ( a ,4 n ) r e a c tio n is p r ed o m in a n t and th ey h a v e in fact, m ea su red the l69T m (a ,4 n ) reaction o n ly . T h eir o b v io u s c o n c lu s io n w a s th at th e l69T m ( a ,p 3 n ) r ea c tio n c o u ld n o t b e s tu d ie d b y th e a c tiv a tio n te c h n iq u e b e c a u s e it w o u ld b e m a sk ed by the c o m b in a tio n o f l69T m (a ,4 n ) rea ctio n . In this s c e n a r io , it is fe lt w o r th w h ile ( i) to carry -out th e s e s tu d ie s w ith im p r o v e d a c c u r a c y , e x t e n d in g th e m e a s u r e m e n ts to h ig h e r e n e r g ie s b y e m p lo y in g h ig h r e s o lu tio n H P G e d e te c to r h a v in g 2 K e V r e s o lu tio n fo r 1 3 3 2 k e V o f ^ o an d , ( ii) to e m p lo y ap p rop riate fo r m u la fo r c r o s s s e c tio n d e te r m in a tio n w h e r e v e r r eq u ire d ( i ii ) a n d to c o m p a r e the m ea su red c r o s s s e c tio n w ith u p d ated p r e -eq u ilib r iu m h yb rid m o d e l u s in g n 0 = 4 (4 p 0 h ) to g iv e a m e a n in g fu l p ictu re o f reaction m e ch a n ism .
2. Experim ental
E x c it a t io n fu n c tio n o f l69T m
[(a,
p 3 n ),(a,
otn),(a,
a 2 n ) a n d(a,
a 4 n )J " fe a c tio n s are m e a s u r e d u p to 6 0 M e V . S e l f s u p p o r tin g T h u liu m f o i l s o f t h ic k n e s s e s 2 9 m g /c m 2targets together with several aluminium degraders of varying thickness are used in the stack.
Alpha irradiation has been made with 60 MeV alpha particles at Variable Energy Cyclotron Centre, Calcutta, India. Beam current of the order o f 200 nA is maintained in the stack. The experimental technique and procedure adopted in the present work is the same as mentioned in our earlier work [11-14]. Spectroscopic information [15] about various residual nuclei is given in Table 1. The energy and efficiency calibration of the detector are
M easurem ent an d analysis o f isobaric contributions etc 527Tabic 1. Decay characterisiics of the nuclides investigated
Reaction Residual
nucleus
Q-value (MeV)
Tm E y
(keV) h
(keV)
l69Tm(a,p3n) lWYb -30 0 32.02d no 18 0
130 11.4
177 21.7
lft9Tm(a.an) l6*Tm -8.0 93.10d 447 2 2 .0
720 11.0
741 11 3
816 46.3
l69Tm(a,a2n) '"T m -14.87 9.25d 208 41.0
532 1.6
,69Tm(afa4n) ,MTm -30.63 30.06hr 243 35.0
297 24 7
1397 9.3
performed with a standard calibrated *^2Eu source obtained from Radio Chemistry Division o f VECC. The over all error in the present measurements ranges between 8-12% which includes photo-peak area, detection efficiency, uniformity of foil thickness and that of monitor reaction cross section.
2.J. Cross section determination :
The number of observed decays y(t) is related to the total number of decays z(t) during the measuring time t by
y ( 0
Z (,) “ (e{£ « ) /«(abs)J’
where e(En) is the detector efficiency and In (abs) is the absolute y-ray abundance (yield) per decay. The corresponding reaction yield N0. is given for simple decays (simple decays corresponds to direct production o f radio isotopes by the nuclear reactions) by
N . = Z(r) e 1
I ,
4528
A V Mohan Raow h e r e A = It\2/ T is th e d e c a y co n sta n t and t\ and t 2 are th e tim e o f irrad iation a n d w a itin g tim e r e s p e c tiv e ly . N 0 is related to the c ro s s se c tio n a b y th e rela tio n ,
N 0 * < y(N a v & ) /• (1 )
w h e r e N m is th e nu m b er o f a to m s /c m 3 o f th e target m a teria l, Sx is th e th ic k n e s s o f th e fo il in c m s , / is the flu x o f the in c id en t p a rticle (In teg ra ted ).
2 .2 . F o r m a lis m f o r is o b a r ic c o n tr ib u tio n :
A s m e n tio n e d in the in tro d u ctio n , the a b o v e e x p r e s s io n is v a lid fo r stra ig h t-fo rw a rd c a se s.
B u t m ore o ften th e n u clear rea ctio n s in w h ic h tw o g e n e tic a lly related p rod u ct n u c le i (isob ars or iso m e r s ) are fo rm ed w ith co m p a r a b le h a lf liv e s and o n e o f th em , th e d a u g h ter n u cle u s, b e in g c o n tin u o u s ly fed by th e m o th er n u c le u s b y r a d io a c tiv e A ccay. In su c h c a s e s , the e x p r e s sio n g iv e n a b o v e c a n n o t b e e m p lo y e d to d ete rm in e th e c r o s s se c tio n o f th e d au gh ter n u c le u s , as it d o e s not ta k e in to a c co u n t th e co n trib u tio n fro m fe e d in g p recu rso r n u cle u s.
T h e r efo r e, th e c r o s s s e c tio n 0 \ o f the m o th er n u c le u s is first d e te rm in e d u s in g eq . (1 ). T h e c ro s s s e c tio n o 2 ° f the d au gh ter n u cle u s is d eterm in ed u sin g the e x p r e s s io n g iv e n b e lo w :
P 9 y r
= N T<t> ° i * 2
A2 - A, ( l - e '* -'" ) A./u’ - e ■ Ajfnv+A) 2IW _ ?(/u- + 4 )
At
g l + f f 2
* 2 .
x [ e - 1* * - +
A 2 - A,
( l - e - X2“ )
( 2 )
w h er e s u ffix 1 sta n d s fo r m o th er and 2 stan d s for d au gh ter n u cleu s.
3. Results and discussion
T h e stu d y o f e x c ita tio n fu n c tio n is g e n e r a lly c o m p lic a te d by th e p o s s ib ilit y o f iso b a r ic p r e cu rs o rs c o n tr ib u tin g to th e c r o s s s e c tio n o f th e fo r m a tio n o f th e r esid u a l n u c le u s o f in terest. T h e a p p lic a b ility o f the m a th e m a tic a l-fo rm u la tio n for th e iso b a r ic co n trib u tio n to the r ea c tio n r e s id u e s o f in terest and their sa lien t featu res are d is c u s s e d b e lo w :
3 .7 . l69T m (a> p3n ) 169Yb r e a c tio n :
In th e p r e s e n t in v e s t ig a t io n , th e e x c ita tio n fu n c tio n for th is r ea c tio n is e x c lu s iv e ly and c a r e fu lly d e te r m in e d e m p lo y in g the a c tiv a tio n te ch n iq u e and the a b o v e fo r m a lism . A t first, th e c r o s s s e c tio n s o f th is r ea ctio n for v a r io u s p r o je c tile e n e r g ie s is c a lc u la ted u s in g eq . ( I ) an d is fo u n d that th e o f th is rea ctio n is o f the order o f 1 0 0 0 m b. T h is is b e c a u se o f the la r g e c o n tr ib u tio n o f *69L u fo r m e d in ( a ,4 n ) r ea c tio n s. T h is in terferin g co n trib u tio n in th e m e a s u r e m e n ts o f ( a ,p 3 n ) c r o s s s e c tio n fr o m th at o f ( a , 4 n ) r e a c tio n is r e a lly a m a jo r
Measurement and analysis ofisobaric contributions etc
529
p r o b le m , e s p e c ia lly in v ie w o f th e fa c t th at the later c r o s s s e c tio n is g e n e r a lly 10 tim es larger than th e fo rm er, w h ic h is in q u estio n . T h e m ajor part o f th is e f fe c t is o f c o u r se , d u e to the C o u lo m b f ie ld h in d ra n ce to the p roton s.
It is p r e c is e ly fo r th is r ea s o n , S a u e t a l [1 0 ] c o u ld n o t s u c c e e d in s tu d y in g the e x c ita tio n fu n c tio n o f l69Tm (0t, ,p 3 n ) rea ctio n a lth o u g h th ey m a d e an u n s u c c e s s fu l attem p t b y m e a s u r in g th e a c tiv ity o f th e p ro d u ct n u c le u s l69Y b (T m = 3 2 d ). T h u s , ta k in g th e e x p e r im e n ta l c r o s s s e c tio n o f ( a ,4 n ) rea ctio n from ou r ea rlier w ork [1 2 ], true c r o s s se c tio n o f ( a ,p 3 n ) r e a c tio n are c a lc u la te d u s in g th e e x p r e s s io n in e q . (2 ). F ig u r e 1 s h o w s th e c o m p a r is o n o f th e p resen t e x p erim en ta l resu lts for 169T m (a ,p 3 n ) rea ctio n w ith th e u p d ated
n E
100
10I E - 1 '** Lu
u .o ih \ Yb JLfi
J J -
Present Exptl Values _ Hybrid Model:
nD - 4 (4 p oh ) (A LIC E/9 0)
20 90 40 50 60 7 0
E j M e V )
Figure 1. Excitation function of 169Ttn(a,p3n)169Yb reaction
h y b rid m o d e l u s in g n Q = 4 (4 p 0 /i). It m ay b e o b s e r v e d that in th e lo w e n e r g y part, the c o m p o u n d n u c le u s p e a k is a b sen t. T h e broad e x c ita tio n fu n ctio n m ay in d ic a te a m ix tu r e o f e q u ilib r iu m an d p r e -eq u ilib r iu m e f fe c t s in th is e n e r g y r eg io n . F or th is rea so n , it^can be o b s e r v e d th at h y b rid m o d e l p r e d ic tio n s are in s e n s itiv e to th e in itia l e x c ito n n u m b er.
T h e r e fo r e , e x c e p t to r e c o g n iz e th e in flu e n c e o f p re-eq u ilib riu m el l e e ls in the en erg y region 5 0 - 6 0 M e V , n o d e fin ite c o n c lu s io n can be draw n .
3 .2 . l69T m ( a ,a n ) t6HT m r e a c tio n :
A s a lre a d y sta te d , th e stu d y o f ( a ,o u n ) ty p e o f r ea c tio n s is d iffic u lt and in te re stin g at the s a m e tim e . T h e y h a v e g e n e r a lly lo w c r o s s s e c tio n so that th eir c h a ra cteristic g a m m a rays are m a s k e d by th e c o m p to n b a ck g ro u n d d u e to m u ch stro n g er g a m m a rays c o m in g from (a ,jtn ) ty p e o f r ea c tio n s o n the sa m e target. T h e in terest in th is ty p e o f rea ctio n is cen tered a ro u n d th e m e c h a n is m o f a lp h a e m is s io n . T h e b a sic a ss u m p tio n su ch as th e e x is t e n c e o f 'p refo rm ed ' a lp h a p a r tic le s in n u c le u s a n d th e id e a s o f 'c o a le s c e n c e ' m o d e ls h a v e to be te s te d a g a in s t d e ta ile d e x p e r im e n ta l s tu d ie s o n (ocourn) ty p e o f r e a c tio n . H o w e v e r , as p o in te d o u t p r e v io u s ly , th eir stu d y b y a c tiv a tio n te ch n iq u e , as far as the e x c ita tio n fu n ctio n is c o n c e r n e d , is g e n e r a lly c o m p lic a te d b y th e p o s s ib ility o f tw o is o b a n c p recu rso rs o n e c o m in g fr o m (a ,p jr'n ) a n d th e o th e r (a,0C c"n) r e a c tio n s . B u t in s p e c if ic c a s e s u c h a s
,69Tm(a,otn),68Tm reaction, such isobaric precursor contributions are absent simply because the immediate precursor, l68Yb, happens to be a stable nucleus.
A theoretical comparison is shown in Figure 2 for the reaction Tm(oc,an) with the prediction of updated hybrid model. It can be seen that the theoretical predictions are lower by more than an order of magnitude, compared to the experimental ones. In fact the shape o f the excitation function is completely different with a very broad peak occurring in the energy region where a deep valley is indicated by the theoretical predictions. This is of course not surprising because hybrid model is not designed to deal with alpha particle emission in the pre-equilibrium phase which is quite likely at moderate energies.
530 A V M ohan R ao
Figure 2. Excitation function of **’l9Tm(a,an),^8Tm reaction.
The only plausible explanation left out is the direct interaction effect. The bare, structure-less shape o f the excitation function is itself a pointer to the direct reaction, namely, the inelastic scattering of the incident a-particle followed by a neutron evaporation.
Similar observations were made by Blann and Lanzafame [16] while studying the ex- induced reactions on gold, especially ,97Au(a,an) reaction. In their work, they have studied the recoil ranges o f the residual nuclei ,96Au and found that there is very little momentum transfer to recoiling nucleus (roughly about 10% of the momentum transfer for that a compound nucleus would have involved). On this basis, they have proposed the operation of considerable non-compound mechanism such as direct interaction and/or pre-equilibrium decay.
3.3. l69Tm (a,a2n)I67Tm reaction:
Figure 3 shows the presently measured values o f excitation function together with
theoretical predictions o f hybrid model for the initial exciton number n0 =» A(4p0h). There
are isobaric precursor contribution in the study o f this reaction with the result that the
experimentally measured cross section at any energy is the sum o f the cross-sections for the
three reactions, (a,6n), (a,p5n) and (cx»a2n) on the target ,69Tm.
The w end shape o f the excitation function with sudden exponential rise in the measured cross sections beyond 55 MeV, is clearly seen to be the result o f isobaric contribution from the two interfering reactions (a, 6n) + (a, p5n). However, below 55 MeV
Measurement and
a n a lysis c f isobaric contributions e tc531
Figure 3.
Excitation function of *®^Tm(cx,a2n )^ T m reaction.which is the effective threshold for (a,6n) and (a,p5n) reactions, the theoretical cross sections are that due to l69Tm(a,a2n) reaction. It can be seen that hybrid model predictions are lower by a factor o f ten or even more. Furthermore, it is also interesting to see that the exponentially rising part o f the excitation function is adequately accounted for, by the hybrid model predictions. This is because, this part of excitation function comprises mainly of the cross section of (a,6n) and (a,p5n) reactions of which later is approximately an order of magnitude smaller than the former.
3.4
.
m Tm(a* ctfn) l65Tmreaction :
Figure 4 shows the excitation function for this reaction together with theoretical prediction -o£ hybrid model using
n0=
4(4p0h)configuration. For this reaction also, although in principle the isobaric precursor contributions are possible as shown in the inset of the Figure, in practice these contributions are very small. This is why there is no abrupt rise observed in the experimental measurements in this case as found in the previous ones. The shqpe o f the excitation functions is reminiscent of the direct reaction model.
In the present work, alpha induced reactions on the target element thulium are investigated upto 60 MeV. Excitation functions for the reactions l69Tm(a,p3n), l69Tm(a,cxxn); x = 1,2, 4 are studied in which isobaric contributions are observed. The observed isobaric contributions in the case o f 1*9Tm(a,p3n) reaction is adequately accounted for, by using the formula developed from first principles. The corrected experimental cross sections for both
$9A15V9
532
A V M ohan R aoth e se r e a c tio n s s h o w a fa ir a g r ee m en t w ith h y b rid m o d e l p r e d ic t io n s w ith « 0 = 4 (4p0h) c o n f ig u r a t io n . In th e c a s e o f (oc,cu:n) ty p e o f r e a c t io n s , th e m o d e l p r e d ic tio n s are u n d e r e stim a te d by m o re than an ord er o f m a g n itu d e . A p o s s i b l e r e a s o n fo r th e large e x p e r im e n ta l v a lu e s m a y be the d irect in e la s tic sc a tter in g o f in c id e n t a lp h a fo llo w e d by a neutron e v a p o r a tio n in this ty p e o f rea ctio n s.
E^(MEV)
Figure 4. Excitation function of ,69Tm(a,a4n),6<5Tm reaction.
Acknowledgments *
T h e au th or w is h e s to than k the en tire s t a f f o f Inter U n iv e r s ity C o n so r tiu m and V a ria b le E n e r g y C y e lo tr o n C e n tr e, C a lc u tta fo r th eir e f f e c t iv e c o o p e r a tio n in th e p r e se n t w ork.
H e is th an k fu l to D r. S N C h in ta la p u d i fo r p r o v id in g th e f a c ilit ie s . T h e fin a n c ia l support fr o m th e C o u n c il o f S c i e n t if i c an d I n d u str ia l R e s e a r c h , N e w D e l h i is g r e a tfu lly a c k n o w le d g e d .
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(14) N L Singh, S Mukheijee, A V Mohan Rao, L Chaturvedi and P P Singh J. Phys.G21 399 (1995) (15) CM Lederer and V S Shirley Table o f Isotopes 7th tdn.(New York : Wiley) (1978)
[161 M Blann and F W Lanzafame Nucl. Phys.A142 559 (1970) [17] Batemann Froc Cambridge Phil. Soc.15 423 (1910)
Appendix
Let
Tbe a target containing A^r nuclei per unit area and 0be the flux of a particles incident on it for a period of time
trDuring this period of irradiation, two isobaric nuclei
Band C are produced through (a^m) and (cx.pjt'n) reactions with cross sections
(7/and O
2respectively. Also the isobaric nuclei are radioactive with disintegeration constants
Xjand
Xnrespectively forming a decay chain,
7 -4 B -> C
X T A, A2
with the notation that
Tstands for target nucleus which is stable
(XT= 0),
Band C are mother and daughter nuclei respectively.
After a "Period of Irradiation"
t,the products are allowed to decay for a waiting time
‘ V and then the activity of the daughter nucleus C is measured after the time,
utw+
A"starting from
“tw”then,
S t a g e 0 :
Upto the end of irradiation. According to Batemann's [17] equations.
( I )
and _
Nr 4>(
T| f A 2 X,t.
*1 X,!, ]" A 2 [ A, - A 2 A, - A 2 J
a i )
S t a g e 1 : A fte r th e e n d o f irradiation, at an y tim e
B(t)
= (H I)w ith th e c o n d itio n that at r = 0,
B
=Bt
a . Further, the rate o f d e c a y o f C is g iv e n as— =
k xB
- A2Cdt
or — + A , C = A
dt
m534
A V Mohan Raosubstituting the value of
Bfrom equation (III) and multiplying equation (TV) by
ex*\we have
e * * + A ,C «Ai' = X ^ B ^ e '^ e ^
d tor
Integrating the above expression, we get
C e Xl* -
* ■
Bl'° eiX2~Xl)t+
KA2
—Ai
w here AT is a constan t.
At
I . O . C . c , .
then,
c
-* ' V + ,
A2 - A,
or v - r ^ B,< °
A “ L'r<> a . _ 3.
substituting the value of
K, in equation (V),
A| j a2 - A,)/
C e X’'
=
A2 - A,
+ C t,o - A2 - Aior
A2C(0
A l A 2 V A |, . - A , , _ A 2 A l _ - A , ,A, - A. * + A 2 H » * " J, _ 3 " •
A2 - Ai ' A2 - A,
substituting the values of
Bt oand
Cr ,afrom equations (I) and (II);
(V )
(VI)
and
. AV0O|A2 f/ -A ,/.\-A .i A2 ~ Al f 1 t ^
2
A j-A , LI
>A2 l (A| - A2)
1 1
1 *
a , - a 2) J e l 1 *
j ej
1 (VII)
A2 C2 =
N f0 <r2 — e j e ^2l (vm)
The reaction product C can be formed in two ways, (i) by the decay of its isobar
Band (ii) directly in the reaction.
These two modes of formation of C are given by equations (VH) and (VHI).
Therefore, for total activity of C,
Measurement and analysis ofisobaric contributions etc
535
Aj C s A2 Cj + A2 Cj from e q u a tio n s (V II) and (V III)
g 1^2 A ? C = Nt
(A2 — A j) + <t2 (1 - e ' Al,' ) e * A2']
In tegratin g b etw een t w and tw + A and sim p lify in g w e get,
?~A|' ( 1 - e~X''<) - e~Aj' ( 1 - e‘ Aj'')
A2 ' '
0iA2 / .
+_
11
1
5 e ~xiK _ e, - A2('„ +4) '
P Y e r N T<i> 1 * 2 ~ * . 1 > \ * 1 a2
1 *1 + ff2 / , - e ' ^ - c ' A2<,“+ 4 )J + ^ T
* 2 l \ 2 - A,
( e
11 | - C' A2<'»+‘1>jj (IX )
T h e q u an tities on L H S o f equation (IX ) can b e written as A » r
0 0 p
y Y Y P, " , " a .
A y - p h o to p eak area o f ch aracteristic y r a y w ith ab u n d an ce 0 r and the e ffic e n c y o f the d etector b ein g P y .
P n A r w , and N av are iso to p ic abu nd ance, m ass num ber, w e ig h t o f the fo il per unit area and A v a g a d ro N u m ber resp ectiv ely .