Thick target yield

International Journal of Applled Radiation and Iaotopm, 1974, Vol. 25, pp. 225-227. Pergamon Pr¢~ Printed in Northern Ireland

Thick Target Yield N. N. K R A S N O V Physico-Energetic, Imdtute, Obnimk, U.S.S.R. (29 October 1973) The present paper is related to a series of published works (l-a) in which the question of units of measurement for the radioisotope rate of formation in reactions with charged particles is discussed. In this paper a rigorous definition of the so-called thick target yield Y is given, and the fundamental relation between the radioisotope activity and Y is derived (formula (2)). It is shown that in this formula the quantity Y is a physical constant which is related to the reaction cross-section by the relation (13) or (15), and to the so-called production rate R by the relation (3). It is seen from the data given that the thick target yield is a convenient characteristic of the radioisotope formation rate. LE R E N D E M E N T DE CIBLE EPAISSE Cet expos6 est 6crit en rapport avec la publication d'une s6rie d'articles(l-s) oiX on discute le probl6me des unit6s de mesure du taux de production des radio-isotopes d a m les r6actions avec les particules charg6es. D a m cet expos6 la d6fmition cxacte de la valeur dite rendement de cible ~paisse (Y) est donn6e. La d~duction de la relation principale de l'activit6 de radio-isotope et de la valeur Y (la formule 2) est pr6sent6e. On montre que la valeur Ydans cette formule est une constante physique li6e avec la section de r6action par la relation (13) ou (15) et avec la valeur R dite taux de production par la relation (3). Les donn6es pr6sent6es montrcnt que le rendement de cible 6paisse est une caract6ristique convenable du taux de production de radioisotopes. B b I X O ~ H3OTOIIA ~ J I H TOJICTOH MHIIIEHH Hacroama~ pa6oTa BHn0~HeHa B CBHSHC ony6JInl~0BaHaeM pm~a cTaTe~, (x-a) B KoT0pI~x o6cym~aeTc~ Bonpoc 06 O~HHHHaX H3mepeHHa cRopocTH o6pasOBaHHR pa~HonaoT0nOS B pea~nnax c 8apnmeHn~MH qaCTH~aMH. B ;~annol~ paSoTe ;~aeTcR cTporoe onpeReneHne Be~MqHHH nasHBaeMol~ BhPXOROM HSOTOHa R/In TOJICT0~ MHIneHH ( Y ) . IIpeACTaBaeH BHBO~ OOHOBHOPO C00THoIneHHff MeH~J" aHTHBHOCTblO p a ~ n o n s o T o n a ~ Be/IHqliHO~ ]7 (0opMyna (2)). HoRasano, ~TO B 9TO~ 0opMyne BeJIHqHHa Y HBJIHeTCH (~HBHqeCHO~ ROHCTaHTO~, RoTopaH CBaaana c ceqeHHeM p e a K ~ n n COOTHOmeHHeM (13) H.rln (15), a c s e n n q n n o l t R , Haa~mae~ol~ cRopocTI~IO Hp0HSBO~OTBa, C00THOIIIeHIIeM (~). I/I8 IIpe~CTaB~IeHHHX ~aHHHX BH~HO~ qTO BMXO~ HSOTOHa ~ I R TO~ICTO~ MHmeHH HB//HeTCH y ~ 0 6 n 0 ~ xapaHTepHCTHRO~ CHOp0CTH o6pasoBanHa pa~0HSOTOIIOB.

AUSBEUTE D I C K E R T R E F F P L A T T E N Die vorliegende Arbeit ist im Zusammenhang mit Ver6ffentlichung einer Reihe yon Artikelntx-a) geschrieben, in denen eine Frage fiber Messeinheiten der Radioisotopenerzeugungsrate bei den Reaktionen mit geladenen Teilchen besprochen wird. In dieser Arbeit ist eine strenge Definition der Gr6sse, einer sogenannten Ausbeute dicker Treffplatten (Y), gegeben. Eine Ableitung der Grundbeziehung zwischen der Aktivitat des Radioisotops und der Gr6sse Y (die Formel 2) ist beschrieben. Es ist gezeigt, dass in dieser Formel die Gr6sse Y eine physikalische Komtante ist, die mit dem Reaktionsquerschnitt durch die Bczichung (13) oder (15) mad mit dem Wert R, einer sogenannten Erzeugungsrate, durch die Beziehung (3) verbunden ist. Die angegebenen Daten zeigen, dass die Ausbeute dicker Treffplatten eine geeignete Charakteristik ffir die Radioisotopenerzeugungsrate ist. INTRODUCTION TaT, Pgocxss o f f o r m a t i o n o f r a d i o a c t i v e isotopes in reactions w i t h c h a r g e d particles is c h a r a c t e r i z e d , like a n y o t h e r n u c l e a r reaction, b y a r e a c t i o n cross-section. B u t in p r a c t i c e

the a p p l i c a t i o n o f c h a r g e d particles for isotope p r o d u c t i o n o r a c t i v a t i o n analysis meets w i t h t h e difficulty t h a t t h e use o f cross-section values is n o t a c o n v e n i e n t w a y in w h i c h to assess the q u a n t i t y o f radioisotopes formed. 223

N. N. Krasnoo

224

The method is inconvenient because, firstly, the cross-sections are continually varying according as the charged particles slow down in the material being irradiated, and also because often a particular radioisotope is formed as the result of different nuclear reactions simultaneously which have different crosssections and energy thresholds. In refs. 1 and 2 an opinion was expressed as to the expediency of laying down convenient units of measurement for use by expefimentors in measuring the radioisotope formation rate in reactions with charged particles. In this regard it is stated in the cited papers that the existing unit of measurement for the radioisotope formation rate in reactions with charged particles, i.e. the so-called thick target yield, is inaccurate because for short-lived isotopes it depends on the irradiation time. But this is untrue because it pre-supposes that the activity and thick target yield are associated by the following relation: a = ~It

(I)

where A is the activity at the end of irradiation, Y the thick target yield, I the beam current and t the irradiation time. Actually, however, formula (1) is an approximate formula which holds good for t < Tit I (Tx/~--half-life). The exact formula which holds in all cases, is

Here 2 is the isotope decay constant. I f 2t < 1 (t < Tt/2), then formula (2) becomes formula (1). On introducing the notation Y ~- = R

(3)

formula (2) becomes the well-known formula (4): ,4 .= R I (1 -- e-a). (4) The quantity R is usually referred to as the production rate, or as the thick target Yield at saturation. I f 2t ~ 1, then formulae (2) and (4) assume the form IF a =

IR -

~.

(5)

Since the quantities Y and R are interrelated by the simple relation (3), the question raised in (I-3) as to which of these quantities is best, cannot be an object of discussion. The two quantities are both physical constants which are equally acceptable for practical use. Clearly, if the value of Y has been found experimentally, then the value of R can be calculated without difficulty and vice versa. The cause of the misunderstanding of this question is, apparently, the lack of published works in which the thick target yield concept would be correctly defined and showing the validity of formula (2). In this connection, the present paper gives the derivation of formula (2), and from this follows a precise definition of the thick target yield as a physical constant. DERIVATION OF THE FUNDAMENTAL FORMULAE The differential equation describing the process of radioisotope formation and decay in the irradiation of a thin layer AX (when one can assume that ~r = const), is written s o : (4)

dN d'--i = F n ~ A X

-

-

(6)

AN.

Here N is the number of atoms of the isotope formed; F (partlcles]sec) is the particle flux; n (atoms/rag) is the concentration of the atoms from which the radioisotope is formed; a (cm s) is the nuclear reaction cross-section; AX(mg/cm 2) the thickness of the layer; )~ (sec-1) the decay constant of the radioisotope; t (sec) the time. Equation (6) does not have regard to the reduction of the number of atoms of the initial element (n), or of the radioisotope itself (N), which are formed due to nuclear reactions. In actual conditions these effects make a negligible contribution to the overall balance describable by equation (6), so in the usual way these effects are disregarded. ~° Given that the particle flux is constant in time, integration of equation (6) gives the number of radioisotope atoms formed during time t in a thin layer AX: N=FnoAX

(1 - ~ e - ~

.

(7)

225

Th~k ~r&a ~dd

In the thin layer the radioisotope activity (dps) is A A --- N t = F n t r A X (1 -- e-u). (8) This relation is usually used for determining the cross-sections of nuclear charged-particle reactions by the irradiation of fine films and foil. Clearly, in accordance with expression (8) the radioisotope activity distribution by thick-target depth (dA[dx) is dA dx - - Fmr(x) (1 -- e-u). (9) But the total radioisotope activity in a thick target is found by integration of equation (9) : A = (1 -- e-U)n

F(x)cr(x) dx.

(10)

Here x, (mg/cm s) is the particle range on the portion where ~r(x)~ 0, i.e. on the portion where the particle energy is higher than the reaction threshold energy. The coordinate X = 0 corresponds to the surface of the target. The integration limit x, can be replaced at infinity. In this respect the value of the integral remains unchanged, since a ( x ) = 0 ifx > x , . The quantity n is taken outside the integral since the atom concentration in the target is assumed to be uniform. Clearly, formula (10) also precisely defines the value of the isotope activity in a thick target, just as formula (8) applies to a thin target. The beam intensity is usually measured in/~A and so for the particle flux we have:

F(x) = 6"3. I0*'l(x)".

(II)

Z

Here /(HA) is the beam current; z is the relative particle charge (for protons and deuterons z = I). O n substituting into relation (I0) the expression for the particle flux in accordance with equation (ll), and also dividing and multiplying the right-hand side of equation (10) with 2rio, we get:

{I -- e-u~ A=

k

at

zIo

x

fO"

Y = 6.3.

n2 f0 ~'.

ax.

(12)

(IS)

Expression (I2) then simplifiesto _

_

e--U

T i m is also the formula (2) given above and very convenient for practical purl~Ses. Here Y (the thick target yield) is a physical constant. In its physical essence the yield is defined by expression (13), but it is independent of the irradiation time and it is measured without difficulty experimentally and in accordance with the relation (14) by measuring the activity of the isotope, the beam current and the irradiation time. The particle flux varies significantly by target depth (I = I(x)) only for high particle energies ( > 100 McV) when the total interaction cross-sections of particles with target nuclei greatly increase. But at the particle energies usually used in obtaining radioisotopes (1040 MeV), the particle loss duc to nuclear reactions is negligibly small, so therefore the particle flux by target depth remains constant until the particles stop, it means I(x) = I o. In this case expression (13) simplifies to

Y = 6"3 . 10 ls n2 f=, (15) Z Jo In this case the value of Y can be calculated without difficulty by numerical integration of equation (15) if the excitation function o(E) is known. In those cases where the isotope is formed in several nuclear reactions sirnultancously, the expression for Y assumes the following form:

= 6.3. I01,

6"3. ~oX'na

] Iot

Here I o is the value of the beam current incident on the target surface, i.e. for X = O. Putting Y for the following expression which is called the thick target yield, we have

21 n,

06)

Here j is the ordinal number of the reaction leading to the formation of the isotope in question, whilst q is the number of such

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226

reactions. T h e thick target yield in this case is found experimentally just as simply by expression (14) as in the case of one reaction. This is one of the chief advantages of using thick target yield values compared with reaction cross-section values. Usually the term "thick target yield" is applied to a target which entirely consists of a particular element and contains a natural mixture of stable isotopes of this element. In those cases where the element from which the radioisotope is formed, is contained in the target as an impurity, an alloy component or chemical compound, formula (14) becomes

A,, n = Y~,~ ~ It [1 -- e - ~ x, \ at 1 .

(17)

Here A,,~ is the activity of an isotope which is formed from an element i contained in the irradiated material m; ~m is the relative content of the element i in the material re(gig) ; Y is the thick target yield of a target which consists entirely of the element i, i.e. the thick target yield in the generally accepted meaning of the term; x,~ and x~ are the charged-particle paths in the material m and in the element i. From consideration of formulae (14) and (17), it follows that for material m with isotope content of element i, the thick target yield Y~,nis found by the following expression:

Y,1, k,n •

(18)

T h e derivation of formulae (17) and (18) appears in the paper, tS) Formula (18) is analogous to the formula obtained by RIccX and HAHNte~ for activation analysis. Formula (18) is readily verifiable by experimental testing. I n particular, this formula was verified by us, taking as an example the yield of asSr on irradiation of RbCl and of metallic rubidium by deuterons.t~ T h e s~Sr yield on irradiation of RbC1 was 65 per cent compared with metallic rubidium, which agrees exactly with the value obtained by formula (18). Clearly, formula (17) is a convenient one to use for evaluating the sensitivity of activation analysis on charged particles when the thick target yields of isotopes are knownJ s~

DISCUSSION OF THE RESULTS From the above derivation of formula (14) it is clear that the isotope thick target yield used in this formula and determined by expressions (13) and (15) is a physical constant and it does not depend on the irradiation time. I n two extreme cases (2t ~ 1 and ;tt >~ 1) formula (14) is transformable into formula (1) or (5) respectively, and these are also correct if the conditions stated are satisfied. I f the nuclear-reaction excitation function a(E) is known, then the thick target yield can be calculated by formula (15) using numerical integration and the pathenergy curve. Consequently, the thick target yield of an isotope can serve as a universal characteristic of the rate of formation o f radioactive isotopes in reactions with charged particles. T h e thick target yield of an isotope is determined without difficulty experimentally by means of formulae (14) and (1) or (5), so it is a convenient quantity for practical use in producing isotopes and in performing activation analysis on charged particles. Readily obtainable experimental curves Y(E) give full and convenient information for selection of the optimum conditions for isotope production or activation analysis. Thus there are no grounds for seeking any new ways of estimating the rate of formation of radioisotopes in reactions with charged particles, as proposed by the authors of the works, a'2) Especially as data have already been published for more than 30 years on the thick target yields found by formula (1) on condition that ;tt ~ 1. In this connection one is surprised that the authors of the works a'~) should state that the isotope yield is an inconvenient quantity because of the errors which can occur in experimentally determining it, owing to the isotope loss in the process of irradiation arising from overheating of the target. Clearly, from this point of view there is no difference between experimental determination of nuclear reaction cross-sectlons and isotope thick target yields. In the former case a thin target (foil) is irradiated and the cross-section is evaluated by formula (8), whilst in the other case a thick target is irradiated and the isotope yield is evaluated by formula (14). In both cases it is necessary to measure accurately the beam current, the

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irradiation time and the isotope activity at measurement of activity and of radiation the end of irradiation. So if any vaporization dose. Sometimes it is convenient to express of the isotope occurs during irradiation, or t h e isotope yield in units mOi/kWh. In this uncontrollable isotope loss in chemical separa- case the curve for the Y(E) relationship has tion, the results obtained will be incorrect a maximum at which the isotope yield is in either case. Clearly, it is simpler to ensure largest for a given beam power dissipated reliable cooling of a thick target than thin on the target. Such a method of determining foil. It is therefore a simpler procedure to the optimum charged-particle energy by yield determine the isotope yield experimentally, values given in mCi]kWh was first published than to determine the cross-sections of nuclear in the paper of GREEN and MARTIN(I0) in 1960. reactions. T h e isotope yield is converted from units of I t is thus clear that the thick target yields mCi//~Ah into mCi/kWh by the trivial relation: of isotopes, like the reaction cross-sections I000 z too, must be measured under conditions which Y(mCi/kWh) = Y(mCi/pAh) E ( M e V ) " (18) ensure absence of isotope loss. In the production of isotopes on cyclotrons, In conclusion it should be mentioned that as justifiably pointed out by the authors of it is no longer so important always to publish the works, t1'2) the technical isotope yield is isotope yield values in identically the same often less than the virtual thick target yield units, since each investigator can convert owing to isotope losses arising from overheating these quantities into units which suit his own of the target. But this fact cannot be an convenience. It is important to remember obstacle to the thick target yield concept that the thick target yield of an isotope is a being used as a physical constant. On the physical constant, and so in experimental contrary, knowing the virtual isotope yield, measurement of this quantity steps must be it can be compared with the technical process, taken to eliminate errors, as necessary when and thus the quality of the isotope production measuring any physical constant. technology can be assessed correctly. Such a comparison of isotope yields enables the REFERENCES isotope production technology to be raised to 1. SvononA K. The Uses of Cyclotrons in Chemistry, a higher level so that the technical yield becomes Metallurgy and Biology, p. 383. Butterworths, equal to the physical thick target yield. London (1970). T h e dimensionality of the isotope thick 2. SVOBODAK. and SXLV~ST~RD. J. Int. J. appl. target yield is the ratio of the radioactivity Radiat. Isotopes 22, 269 (1971). to the radiation dose. T h e radiation dose 3. LEBowrrz E. Int. J. appl. Radiat. Isotopes 23, 203 is usually measured in units of pAh, and the (1972). activity in pCi or dps. In radioisotope pro4. FmEDLANVERG., K~NNEDYJ. and MXTTERJ. Nuclear and Radiochemistry. Wiley, New York duction the yield is usually given in units (1964). of pCi/pAh, ta) but in activation analysis it 5. KRASNOVN. N. Atomnaja E~rg. 26, 284 (1969). is more convenient to use the units dps//~Ah, ta} 6. l~ccI E. and HAHNR. Anal. Chem. 37, 742 (1965). In calculating Y by the relation (15), the 7. K~SNOV N. N. et al. AtoranajaEnerg. 27,13 (1969). dimension of 2 should be h -1, and in this case 8. K~.SNOV N. N. et al. The Uses of Cyclotrons in the yield dimension is dps/pAh. In order Chemistry, Metallurgy and Biology, p. 341. Butterto obtain the yield in units of pCi/pAh, the worths, London (1970). numerical coefficient in expression (15) must 9. KnASNOV N. N. et al. The Uses of Cyclotrons in be equal to 1-7 x l0 s instead of 6.3 × 1012. Chemistry, Metallurgy and Biology, p. 159. ButterFor experimental determination of the isotope worths, London (1970). thick target yield on the basis of the relation 10. GREEN F. L. and MARTI~J. A. Nucl. 8d. Engng (14), the yield dimension is defined by units of 7~ 787 (1960).