A continuous on-line method for fission yield measurements with the combined GJRT-SISAK technique

Nuclear Instruments and Methods 188 (1981) 375 387 North-Holland Publishing Company

375

A CONTINUOUS ON-L1NE METHOD FOR FISSION YIELD MEASUREMENTS WITH THE COMBINED G J R T - S I S A K TECHNIQUE Tor BJORNSTAD Department of Nuclear Chemistry, University of Oslo, Blindern, Oslo 3, Norway Received 20 March 1981

The principles of a method for fission yield measurements with the combined GJRT-SISAK technique are outlined. Only the simple case where parent effects can be neglected is considered in detail. The article includes derivation of a relation between the production rate and the counting rate of a nuclide. The expression contains various efficiency parameters and the system dependent delay probability function p(t). The way to experimentally determine these parameters and the delay probability curve, as well as the fitting of an analytical function to the curve are shown. The method is general in the sense that the main principles are independent upon the nuclide under study, fissile target material and bombarding particle used.

I. Introduction Since the first observation o f the fission process in 1939 [1] much effort has been expelled in characterizing and describing it with the aim o f a full understanding o f the mechanisms working. To day a large amount of detailed data exists on mass and charge distribution, on angular and energy spectra o f the emitted particles. But no single theory has yet succeeded in unifying a substantial portion o f the information available on fission. There is a constant need o f new and in some cases more accurate data. To arrive at our present state o f knowledge about fission, radiochemistry has been a necessary and highly powerful tool. But the chemical separation procedures used have mostly been manually operated, and therefore rather time consuming, limiting the number o f nuclides possible to study to those lying close to stability. However, in the later years a number of new separation techniques has been invented. Mass (A) separators, even capable o f achieving a separation in the proton number (Z), and automatically fast operating semicontinuous, or fully continuous chemical separation systems have been constructed. These systems can be connected online to the production unit to form semicontinuous or fully continuous on-line production-separation systems. These new techniques largely extend the number o f nuclides accessible for study to include still more short-lived species. The combined G J R T - S I S A K technique is one o f these new techniques. It offers the possibility o f separating nuclides with half-lives down to 0 . 5 - 1 . 0 s.

2. A brief recapitulation of the combined GJRT-SISAK technique The GJRT (Gas Jet Recoil Transportation)-technique has been known for some years. It offers the possibility o f transporting nuclear reaction products from the production site to the separation or direct measurement, by stopping and catching the species recoiling out o f the target in a gas or a gas mixture, which continuously sweeps the target. Some o f the large number o f reported GJRT-systems are found in refs. [ 2 - 7 ] . The SISAK (Short-lived Isotopes Studied by A K U F V E technique) system is a fully continuous multistage solvent extraction system based on H-centrifuges [8,9] for phase separation, the so-called AKUFVE technique [10]. It is intended to deliver radiochemical pure samples of, in principle, any element of interest. The SISAK technique is described in refs. [ 1 1 - 1 4 ] . 0 0 2 9 - 5 5 4 X / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © 1981 North-Holland

3 76

T. Bjdrnstad / Fission yield measurements

In early 1975 a GJRT system (N2/C2H4 gas mixture) was successfully connected to the SISAK system at the Mainz TRIGA reactor [15]. This reactor can be operated in the pulsed mode, thus making time behaviour measurements of the mass flow in the experianental system possible. Such measurements are the basis of the fission yield measurement method described in the present article.

3. Principles of the method

3.1. Derivation of a formula for the production rate of a nuclide The distribution in transport time for products transported from production to detector position is expressed by the delay probability function usually denoted by p(t)dt and first introduced by Winsberg [16]. It expresses the probability of finding a delay between t and t + dt. This function is a helpful notation in the process of converting the number of recorded counts of a nuclide to the production rate. In the following such a mathematical expression will be derived for the simple case where there is no parent effect. A nuclide with decay constant X is formed in the target. Let the production rate be Rp. During the interval t' to t' + dt' the number of atoms formed is dNx = R pdt'.

(1)

The recoil yield, i.e. the fraction of the total amount produced of a nuclide which recoils out of the target and into the jet gas, is denoted ~. The fraction of the recoiled nuclides which reaches the end of the capillary (decay not included) is denoted 7? and represents the GJRT efficiency. The parameter r/depends on the target and target chamber arrangement as well as on the length of the capillary tube. Accordingly the number of atoms reaching the end of the tube neglecting decay) is dN~E = R pOTdt'.

(21)

The chemical yield from the moment of mixing the jet gas with the first solution to the moment of measuring is denoted u. The number of atoms in the measuring position at an arbitrary time t" when both decay and delay probability are included, is then given by dN C =Rp~TW dt' exp[ X(t"

t')] p(t" - t').

(3)

Assuming continuous production at a constant production rate Rv, the number of atoms in the counting position at an arbitrary time t" wilt be t"

N~'"--

JdXc=Rv~wf

e × p [ - X C ' - t')]

p(t"-

t') dt'.

(4)

0

It is experimentally shown that 7? is time independent. Introducing the desintegration rate D c D g ' = XN~',

(5)

and the counting rate R c Rg ' = eD~' ' = eXN~',

(6)

where e is the counting efficiency of the nuclide in concideration, gives t It

Rg' =Rp~er~uX f

exp[-X(t" - t')] p ( t " - t')dr' ,

(7)

o

The number of counts recorded in the time period t" to t"+ dt" is t 't

t!

dS = R C dt .

(8)

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377

The total recorded number of counts in a counting period which equals the production period from 0 to t is

S = , f dS =Rp~er~vX 0

f 0

t')] p ( t " - t') dt' at".

exp[-X(t"

(9)

0

If R c is plotted as a function of time the curve would qualitatively look like the one in fig. 1. After a time tz the counting rate R e becomes approximately constant and remains constant throughout the counting period

I.I I I I I

o

I I

tz

TIME

Fig. 1. Qualitative picture of the counting rate, Rc, as a function of the time from start of the counting, indicating the time t z where d R c / d t <~ z.

(assuming that Rp is constant during the counting period). The time tz is determined by the half-life of the nuclide and the shape of the delay probability curve, and is defined mathematically by

(R~' - R c )tz/ R C tz <~z,

(10)

where the value o f z may be taken as small as desirable. This is equivalent to (for t " > tz) tz

t r~

?

exp[-X(t" - t')] p ( t " - t ' ) d t ' - f

O

exp[-X(tz - t')] p(tz - t')dt'

0

<~z,

(11)

tz

j

exp [-X(tz - t')] p(tz -

t')

dt'

0

which gives t"

f

t.

exp[-X(t" - t')] p ( t " - t') dr' <~z --J" exp[-X(tz - t')] p(tz

tz

t') dt'.

(12)

0

If an analytical expression for p(t) is available, tz can be found by numerical integration of the two integerals. Likewise, the constant value of the integral on the right hand side of the inequality (12), C, can be calculated. If the counting is started a time tz after start of production, the following equation for the totally recorded number of counts is valid:

S = Rp~erluXCtc,

(13)

where tc is the counting time. Then

R p = S/~ewXCtc.

(14)

An approximate value of tz may be found by experimentally recording a curve like fig. 1.

3.2. Determination of fission cross sections Knowing the amount of target material (N is number of atoms) and the flux (4)) of the bombarding particles, the cross section (o) is deffmed by Rp -- o~N,

(15)

T. B/Ornstad / Fission yieM measurements

378

Aqueous c,

solution

N!x. Detector Static Gas jet from target

'

mixer

Aq u e o u s solution

Fission products (waste)

Aqueous solution

Fig. 2. Example of an arbitrary flow diagram for the chemical separation system. The arrows indicate the flow directions of the liquids and gases. C1 ..... C4 are units consisting of one static mixer for thorough mixing of the aqueous and organic phases, and one H-centrifuge [9] for fast and efficient separation of the two phases. The jet gas m a y be CO2, N2/C2H4-mixture, N2/KCl-mixture or N2/KF-mixture [7]. The specially indicated static mixer serves to mix the jet gas with an aqueous solution (normally heated to - 7 0 ° C ) for dissolution of the clusters. In the degassing unit the jet gas and volatile reaction products like the noble gases and partly the halogens, are removed. S denotes the standard nuclide with known cross section o, and X denotes the nuclide whose cross section is to be determined.

o = Rp/(oN.

(lSa)

In cases with unknown ~ and/or N, the cross section Ox for nuclide x may be calculated relative to another nuclide with know cross section o s from the relation

Ox/Os = (R p)x/(R p)s, Ox = [(Rp)x/(Rp) s] Os,

(l 6 ) (16a)

Two cases are considered: I) x and s isotopes, then the efficiencies 77 and v are the same for the two nuclides, and so are the delay probability functions p(t). By introducing eq. (14) and letting (tc) x = (tc)s one gets

SxGesXsCsos Ss~xexXxCx

Ox

(17)

II) x and s non-isotropic:

ox

-

Sx ~ses~sVsXsCsos SsGex~x Vx~x C=

(1 8)

Here the constants Cx and Cs contain different delay probability functions. The two nuclides are then separated and measured in the same experiment, e.g. according to the schematic separation scheme in fig. 2. This approach requires the evaluation of a delay probability function for two elements.

3.3. Determination of efficieney parameters of the system The recoil yield ~ depends on parameters like the mass, ionic and nuclear charge of the recoiling nucleus, the absorber material type and thickness and the energy of the bombarding particle. The value for ~ for a nuclide with unknown formation cross section can in general probably be estimated to better than 30% by extrapolation

T. BjOrnstad / Fission yield measurements

379

from known or measurable ~-values for nuclides in the neighbourhood to the nuclide of interest (a corresponding error will then, of course, be introduced in the final result for the reaction cross section). However, in fission, for instance induced by thermal neutrons in 23Su, the recoil ranges vary in average from 6 mg/cm 2 to 13 mg/cm 2 for mass numbers from 156 to 77 [17]. A typical target in our experiments at the TRIGA reactor in Mainz contains from 0.1 mg to 0.5 mg, corresponding to 0.127 mg/cm 2 and 0.637 mg/cm 2, respectively, which is a factor 1 0 100 lower than the ranges. Accordingly, the recoil efficiency ~ is close to 50% (the other half recoiling into the target backing) for these irradiations, and there is no need to worry about uncertain estimates. For the sake of generality ~ is, however, all the way kept as a parameter in the formalism. So far there is no evidence that the r/-values are element dependent when using the N2/C2H4-jet. Accordingly the GJRT efficiency can be found by measuring the activity of a suitable nuclide on a catcher foil mounted close to the target relative to the activity of the nuclide collected at the exit end of the capillary. Of course the chemical yield (separation efficiency) v is element dependent. In addition, for a given chemical separation scheme, v is sensitive for variations in temperature and liquid flow rates. It is therefore essential to keep constant running conditions of the system during the measurements. The efficiency v is composed of the chemical yields of each of the separation steps in the system: v = V D G V c , U c 2 ....

(19)

where UDG expresses the fraction of the cluster-bound nuclides in the jet gas that is dissolved in the liquid, the remaining part being swept off together with the jet gas. By measuring the counting rate of the nuclide emerging from the end of the capillary, RCE , and the counting rate of the nuclide in the two liquid phases after the first centrifuge, Rorg , and R a q l , the efficiency VDG can be calculated by Vl)G = (Rorgl + Raql)/RcE.

(20)

The extraction efficiencies VCl , vc2 , ... are found by measuring the counting rate of the nuclide in both phases after extraction, Rorg and Raq. For the extraction from the aqueous into the organic phase the efficiency is expressed by VCI = Rorgl/(Rorgl + Raql ),

(21)

and for the back-extraction from the organic into the aqueous phase we have /~C2 = Raq2/(Rorg2 + Raq2) = Raq2/Rorgl,

(22)

and so on throughout the extraction steps. In cases where difficulties may arise in measuring rt and u separately the product r/v may be obtained assuming that Rp, ~, e, k and C are known for a long-lived isotope of the element in question. According to eq. (13) $2 - St flu = (t2 - tl) Rp~e~kC '

(23)

where S~ and S: are the totally recorded number of counts after the counting times t~ and t2 respectively (start counting at a time to >~ t z ) . The total counting efficiency e is composed of two parts - the absolute intensity of the detected characteristic radiation,/(abs), and a part including the counting geometry and the internal detector efficiency, eD. e = l(abs)eD.

(24)

In summary the following measurements are necessary for evaluation of a fission cross section: Determination of a delay probability function, determination of the GJRT efficiency and the chemical yield of the separation system and the determination of the counting efficiency of the detection system. The following sections show how to determine the delay probability curve, and the fitting of an analytical function to the curve.

380

T. BjOrnstad / Fission yield measurements

4. Methods to determine the delay probability curve 4.1. Previous work

Various methods may be used for determination of transport times in on-line systems. Grover [18] has modified some of the formulas describing molecular flow to fit the transport of rapidly decaying radioactive gases. Oron and Amiel [19] and Winsberg [20] have described a method used at the separator SOLIS [21] in order to determine mass transport tilnes. By means of a fast neutron beam shutter the irradiation time of the target is well defined, and the growth of the activity on a stationary collector during irradiation and the decay of the same activity after irradiation are measured. From the resulting data the delay properties of the system expressed by the delay probability function p(t) may be found as shown in ref 20. Rudstam [ 2 2 - 2 4 ] has carried out thorough theoretical investigations of the delay properties of on-line mass separation systems, taking into consideration various transport mechanisms like diffusion and desorption processes. Hageb6 et al. [25] and Ravn et al. [26] describe an experimental method for determination of the integrated delay function applicable at the ISOLDE facility [27,28]. After reaching saturation in the production of a nuclide at the collector position, the proton beam is deflected off the target and the activity of consecutive samples measured using sampling times much shorter than the half-life of the nuclide collected. Ref. 26 describes how to derive the delay probability function for certain elements released from molten targets using this approach. 4.2. Delay measurements in the combined G JR T - S I S A K system

As shortly mentioned in the introduction a G J R T - S I S A K system is at present installed at the Mainz TR1GA reactor. This reactor can be operated in the pulsed mode with a pulse width of - 3 0 ms (fwhm) and a corresponding neutron flux of ~5 X 1013 n • cm -2 • s -1. Thus an activity, well defined in time, can be produced in the target. This pulse starts the counting equipment, and the activity at the measuring position is recorded as a function of time either by multianalysis or multiscaling. In a typical experiment counting intervals as short as 0.5 s are necessary in order to allow the construction of a delay curve. In such short periods, too poor statistics are obtained by gating at single y-ray peaks. Accordingly gross T-ray counting is applied. In order to reveal the "true" shape of the delay curve, measured values must be corrected for dead-time and decay, qualitatively illustrated in fig. 3. Correction methods will be discussed in the following. 4.3. Corrections to the recorded delay curve 4.3.1. Dead-time correction

The number of counts S" and the effective counting time r are recorded simultaneously by the electronic equipment during the 0.5 s clock-time counting intervals. The counting rate is assumed constant within one interval. The dead-time corrected number of counts is then (25)

S' = S" 0.5/r.

ILl I'-

....

Uncorrecled

---

Corrected

for

deadtime ~

Corrected

deodtime

for

and

ay Z O I

TIME

Fig. 3. Qualitative illustration of the mass delay curve indicating the effect of the necessary corrections that have to be applied to the measured curve.

T. B/Ornstad / Fission yield measurements

381

4.3.2. Decay correction When applying gross 7-ray counting the activity is normally composed of several decaying nuclides. It is therefore essential to record a decay curve of the activity used. The dead-time corrected delay curve must then be corrected according to this decay curve. The decay corrections may be done either semigraphically or computerized. Below a few procedures are outlined for the simple case where growing-in effects can be neglected. It is practical to deffme three index labels: i is the time index starting from the first point in the peak, / defines the nuclide, and p defines the experimental point number (p = 1 is the first one). Fig. 4 illustrates the def'mition of the indexes i and p. The decay measurements are carried out by first letting an activity pulse pass the actual chemical separation. At a well defined time the liquid flow is stopped, and gross 7-ray multiscaling measurements performed on a liquid volume element. The counting intervals are preferably identical to those used for recording the delay curve. As to the exact starting point of the measurements, two cases must be separately considered: Case 1. It is desirable that the midpoint in the first counting interval corresponds to i = 1 in the delay curve. But the countrate at i = 1 in the normal counting position is ~ 0 . In order to assure sufficient activity for the measurement of the decay, the detection position must therefore be moved "closer" to the separation apparatus. However, the counting is started at the time corresponding to i = 1. The recorded decay curve is then qualitatively similar to the one illustrated in fig. 5a. Case 2. If practical hinderances make the method in case 1 inapplicable, the decay measurements can be done in the normal counting position, and started at a time corresponding, for instance, to the delay curve maximum (at i = 5 in fig. 4). The resulting decay curve will then qualitatively look like fig. 5b. When extrapolating the curve back to i = 1, care must be taken for possible short-lived activity contributing significantly only to the first part of the decay curve.

a.

ty-

se

,<

1

~- R~ Z

O U =1

3

5

7

TIME

n.-

Decay un-

p =5

correcled

~

,~R s

i

It

~

. . . . ected

15 ~7

T9

D

LABEL

b.

n~

I-<[

9 11 13

Decoy curve

"cQse 2"

uZI =1'~'-O -~ '~'Rd

0C~ i=

t

4 6 8 10 12 14~ TIME

LABEL

I-

'=1 3 5 7 9 11 13 15 17 19 TIME

D-

LABEL

Fig. 4. Qualitative picture of the mass delay curve illustrating the definition of the indices i and p. Fig. 5. Qualitative illustration of the decay of the activity used in the measurement of the mass delay curve. (a) Start measurement at i = 1 (case 1). (b) Start measurement at i = 5 (case 2).

T. B]Ornstad / Fission yield measurements

382

4.3.2.1. Semigraphical correction procedure Case 1. The measured counting rate (cp 0.5 s) in the points 2, 3 4 .... , p of the delay curve are denoted R2,Raa, R44, ..., Rip. The same counting rates corrected back to i = 1 are R 2, R al, R~4 ..., RPt. The counting rates for the points in the decay curve corresponding to the times labelled 1 ... i are denoted Ro~ ... R~. Then the following relation is valid: R{ =~-~id R ~ .

(26)

The Rd-values are taken directly from the decay curve, and the corrected delay curve is readily derived. Case 2. Suppose that short-lived activity which would contribute mainly to the first points in the delay curve < 5 in the example) is absent. Then the extrapolation of the decay curve back to i = 1 may be done rather safely, and eq. (26) can be applied. 4.3.2.2. Computerized correction procedures Case 1. Let 4 symbolize the counting rate of nuclide j at the time labeled i and related to the point p in a delay curve which is smoothed with an analytical function. The measured total counting rate at any time labelled i, is then expressed as i R~ ~ / ~ = 8 1 +82 + -.. + ~ = 'j~l r q , (27) where R,p and K,e are the experimental and fitted total counting rates respectively. This expression may be written as

/ /~f = ~x exp(-kx ti) + ~2 exp(-X2 ti) + ... + ~ exp(-Xjti) = ~

/-1

~ exp(-X/ti).

(28)

By exchanging the index p with the index d, this expression represents the decay curve. A computer fit of eq. (28) to the decay curve gives the best values of the parameters r d, ... r~j and X, ... Xi. The corrected counting rate to be found is expressed by /. /~P = ~ t + ~2 + -.. + & = /--~14 "

(29)

From the eqs. (28) and (29) and the following relations valid for all j,

& & _r l

(30)

the following expression for K'Ip is derived: /

G =

Rf,

/"

(31)

rdl/ exp(--Xjti) j=l

which is equivalent to

(31a) In both these equations the parameter values on the right hand side are known for all p and j. Case 2. Let x be the time label of the fkst experimentally recorded point (i = x) in the decay curve (.v = 5 in fig. 5b). By exchanging the index 1 with x in the eqs. (28), (29) and (30), and the time t i by t i - G, and taking into

T. B]Ornstad / Fission yield measurements

383

account that r~/= rxdjexp(X/tx),

(32)

an expression identical to eq. (31) can be derived. Note that all the decaying components contributing to the recorded delay-curve are here supposed to be clearly detected at the time labelled x.

5. Examples of experimental determination of delay probability curves A number of delay curves has been determined experimentally for various chemical systems with variation of parameters like the measuring position, speed of the different liquids, length of tubes and the shape and volume of the measuring cell. As an example delay curves measured by short-lived isotopes of lanthanum isolated from fission products of 23Su are presented below. The chemical separation scheme is described in ref. 29. To facilitate the understanding, the procedure is outlined step by step: a) The delay curve is measured in the following way: The counting cell is a tank volume of ~100 ml. A plug flow (see section 6.1) delay line with a volume of 28.3 ml, corresponding to 4.3 s delay, is inserted in front of the cell. The whole experimental sequence is controlled by an electronic timer, and the time sequence for the various events is shown in fig. 6. The data are accumulated in the gross "r-ray multiscaling mode in the energy region 100-1000 keV, and with counting intervals of 0.5 s; b) The main contributors to the number of counts in the delay curve are known to be X44La (42.1 s) and 146La (8.5 s) with a small contribution also from 14SLa (25.2 s) [29]. Accordingly, the dead-time corrected counting number also have to be corrected for decay. The decay curve is accumulated according to case 1. The delay line is disconnected (which in practice means that the detector is closer to the separation apparatus). When the decay measurements start, the liquid has to be at rest. This is achieved by a valve operated by the electronic timer. The measuremenys is started at a time corresponding to the first channel in the delay peak. The whole time sequence is shown in fig. 6. c) By applying the decay data in the semigraphical correction procedure described in section 4.3.2.1 the correct form of the delay curve is found, and illustrated in fig. 7. The results from a similar measurement, keeping all the parameters constant except for the counting cell which is changed into a short piece of tubing (5 cm, plug flow cell, see section 6.1), approximately a "volume element source", are given in fig. 8.

.Start

of

Start

. n- pulse

Delay

4.3s

//

I

Start

DELAY

measurement

~ o o.1

MASS

4.3s

sequence

18.1

0.5 of

~ I

.

4.3s Start meaf,ure-~ ment .... =-~....

sequence

'..

n-pulse

~

DECAY CURVE

°°°°o

Liquid flow ~ ~ ~ " //

I

0.5

. ~ . I , ,~.

17.6 TIME

18.1

AXIS(s)

Fig. 6. The time sequence for the various events when accumulating a mass delay curve, and the corresponding decay curve.

T. Bjdrnstad / tqssion yield measurements

384

I .... ! 0 0 0 0 L-

....

-

I ....

I ' ' ' ' I ' ' ' '1

....

•

:

IT,

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Decay curve of La isotopes ~sed for me(Isurm9 the muss

%%', "',,..% "oo .

delay curve

e~N ~

5000~--

1 ' ' ' ' I ' ' ' ' I ' ' '

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•

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experimental

mass

01:10 0 0

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curve

o. c "o

10000

c : o

CoPra , 5r:,

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Uncorrected delay

o >-

ejo

Lr') C~

o

e

i,~,,~

)-

Oecaycorrected exper,mentat mass delay curve

(~,

~c u

z

ILl

500

nl

z

'm, :'

o

9 P

I,--

oo

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8

o o.o _ o

"3 :'

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~ ....

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50

60 TIME

> "c o o

nro L .... " ~ ?~, ~:/Y ,~, ~, ~ 70 80 90 I00

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cos ~ ....

b ', 'oo,

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"°'o.c

5000

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t

(s)

0

" o~-

I , , , , I ....

20

I ....

o

o

9-a,~r~

30

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TIME (s)

Fig. 7. Mass d e l a y c u r v e ( a n d t h e c o r r e s p o n d i n g d e c a y c u r v e ) m e a s u r e d o n a t a n k - v o l u m e cell ( - 1 0 0 m l ) by s h o r t - l i v e d La i s o t o p e s i s o l a t e d f r o m t h e r m a l n - i n d u c e d f i s s i o n p r o d u c t s o f 23 s U. T h e d a s h e d c u r v e s are " h a n d - f i t s " to t h e e x p e r i m e n t a l p o i n t s . l:ig. 8. Mass d e l a y c u r v e ( a n d t h e c o r r e s p o n d i n g d e c a y c u r v e ) m e a s u r e d d i r e c t l y o n a s m a l l p i e c e o f t u b i n g ( - 0 . 7 5 lived La i s o t o p e s , t h u s a p p r o x i m a t i n g a v o l u m e e l e m e n t s o u r c e ( s e e also t e x t to fig. 7 ) .

nil) b y s h o r t -

6. Mathematical description of the delay curve 6.1. Comments on the form o f the measured delay curves Two idealized flow patterns, plug flow and back-mix flow, are of particular interest. Plug flow assumes that a volume element of the fluid moves through the vessel with no overtaking or mixing with earlier or later entering volume elements. Back-mix flow assumes that the fluid is perfectly mixed and uniform in composition throughout the vessel. True laminar flow is present when there is no mixing between two adjacent infiniteshnal fluid layers in the radial direction of the tube. The flow velocity varies and is parabolic in its form with the highest velocity in the tube middle and zero velocity at the tube walls. All patterns of flow other than plug and back-mix flow may be called non-ideal flow patterns [30]. In process equipment, even with proper design, some extent of non-ideal flow remains due both to molecular and turbulent diffusion and to the viscous characteristics of real fluids which result in velocity distributions. The dimensionless Reynolds' number [31] may be used to predict the flow velocity profile in a pipe. This number depends upon the four parameters density and viscosity of the fluid, linear velocity of the flow and diameter of the tube. Plug flow is obtained when Reynolds' number ~>2300. For Reynolds' number <2300 there might be incomplete mixingof the fluid in the radial direction, and laminar layers near the tube walls may occur. True plug flow in tubes yields nearly Gaussian distribution curves [32]. Back-mix flow can be obtained in a vessel with thorough mixing, and yields a distribution curve that decreases exponentially in time [30]. Laminar parts in the fluid will bring about a slower decrease in the distribution curve than expected from the two flow patterns mentioned above mainly due to the slow exchange of fluid elements between the laminar and turbulent (plug) flow regions [321. The chemical separation equipment consists of several different units, each of then] influencing the mass flow.

T. B/drnstad / Fission yield measurements

385

The 30 ms broad (fwhm) neutron pulse will not contribute much to the broadening of the delay peak. However, the reaction chamber [15] may be regarded partly as a back-mix vessel. The glas flow in the capillary is presumably laminar. In the degassing unit the liquid is swirling around on the glass walls and a certain degree of laminar flow is created also here. The flow pattern in the static mixers may be composed of both laminar, back-mix and plug flow regions. Aronsson [ 12] has shown that the flow pattern in the centrifuges is composed of back-mix and plug flow regions. With the liquid speeds normally used in the SISAK system, plug flow is attained in the tubes. Essentially fig. 8 reflects the influence of the entire production/separation system on the mass distribution, while the extra broadening and increase in screwness as documented in fig. 7 is due exclusively to the counting cell which must be regarded as a back-mix volume. In summary one can say that the average flow pattern in the combined G J R T - S I S A K system is not ideal, and that the deviation from pure Gaussian distributed delay curves is ascribed to well recognized back-mix and laminar flow regions. 6.2. Fitting o f a n analytical f u n c t i o n to the measured delay curve

A mathematical model of the mass flow behaviour in the SISAK-system, containing exclusively physical parameters, would be valuable with respect to predicting the form of the delay probability curve for a given experimental setup and selected running conditions. Evaluation of such a model would however require much effort in a direction that is outside the frame of this work. In order to evaluate the double integral in eq. (9) by computerized numerical integration, it is sufficient that p(t) can be described by an analytical function. Procedure 1. The measured delay probability curve may be represented by a step function (fig. 9) P(ti

-+

for t i > t > t i _ l ,

t i - 1 ) = qi,

(33)

where qi is a constant for each step number i. The step length on the time axis, t i - - t i 1, can be varied over the time axis so as to make the difference between the integral of the step function and the measured curve as small as desirable. Procedure 2. More convenient would it be to represent the curve by a continuous analytical function. The curve shape suggests a Gaussian distribution with some tail function superimposed on the right side. Tail functions norreally used in fitting 3,-ray peaks [33,34] have been tried with small modifications, using the minimization program MINUIT [35]. However, the fits obtained were not satisfying, especially in the tail region. A substantial improvement is obtained when exchanging the Gaussian part with a log-normal distribution [36]. The tail excess can be properly fitted with a function composed of a "growing-in" term multiplied by a "decay" term. The total composite function finally used in the fitting procedure of the measured delay probability curves is: p ( t ) : Y 0 exp

L-

in

)1

+ 1

+K1K2{1 - exp[-X2(t - t2)]} e x p [ - X l ( t - t2)] ,

(34)

where the parameters to be fitted are Yo = the ordinate value at the peak maximum, p = (t2 - to)/(to - tl), a mea-

v CL

.q M _J <~ rt~ o ix.

_.1 taJ o ti-i

ti

TIME (t)

Fig. 9. Qualitative illustration of a step-function fit to the delay curve.

T. B/6rnstad / Fission yield measurements

386 ]

I

'

l

l

~1000

I

I

I

'

0

I

I

'

'

"

0

:

Experimental

I

l

'

'

'

•

II

values

JlJlllrllr

J Jll~l

frll

flr[llllllf1111~l~l

15000

o ~

Io~

- - :

Computer eqn.

Q.

-g

fit

~

(34)

o :

~ - - :

C

o

values

Computer

with

(~orr~ted ~ordecayl

eqn

0. I000C

g

Experimental

fit

(39)

500

>-

Z

5000

,,=, Z

z

I I , , I I I I I I I I I I I , I , I I

20 t1 to

t2

30

TIME (s)

40

20

30

40

50

. . . .

60

I

70

. . . .

I ....

80

I I , , I I

90

100

TIME (s)

Fig. 10. T h e s a m e c u r v e as given in fig. 7 w i t h t h e c o m p u t e r fit (fully d r a w n c u r v e ) b a s e d o n eq. ( 3 4 ) . T h e p a r a m e t e r values f o u n d a r e y o = 1 2 8 5 7 , p = 3 . 5 8 , W = 3 6 . 4 9 , to = 2 9 . 5 4 , K z = 0, X l = X2 = 0. l:ig. 11. T h e s a m e c u r v e as given in fig. 8 w i t h t h e c o m p u t e r fit (fully d r a w n c u r v e ) b a s e d o n eq. ( 3 4 ) . T h e b e s t p a r a m e t e r values f o u n d a r e y o = 9 3 8 , p = 1 . 4 2 , If = 5 . 4 0 , to = 2 4 . 6 8 , K 2 = 9 7 0 , Xl = 0 . 2 4 7 , X2 = 0 . 0 8 5 .

sure of the peak skewness, W = t2 - t l , the full with at half-maximum, to = the time at the peak maximum, K2, )` and ),2 = parameters for the tail function. Here K1 = 0 for t < t2 and 1 for t > t2. The times to, tl and t2 are defined in fig. 11. Initial values ofyo, p, W and t2 are easily found from a plot of the delay curve. Figs. 10 and 11 show the measured delay probability curves from figs. 7 and 8 respectively, properly corrected, and the fits to these curves with eq. (34). The fits are acceptable. Accordingly, by inserting the expression for p(t) given in eq. (34) in the normalized form [by demanding that ~fop(t)dt = 1], eq. (9) can be solved by numerical integration.

7. Concluding remarks The fission yield measurement method outlined here in the present article is, in principle, general with respect to the nuclide under study, type of fissile material and bombarding particle. It can even be employed in high energy spallation and fragmentation reactions provided that the kinetic energy given to the products is high enough for a substantial portion to recoil out of the target. As mentioned above, the enrphasize has been put on a relatively detailed description merely of the principles of the method by assuming all the way that parent effects can be neglected. This restricts the presently derived formalism (section 3) to be applicable solely on the following types of measurements: a) Determination of primary reaction yields of nuclides shielded by a ~-stable nuclide. Examples may be s6mRB (1.02 min) and 11°gAg (24.6 s). b) Determination of primary reaction yields of nuclides shielded by a "long-lived" nuclide (the decay yield of the "long-lived" nuclide during the time from production to separation from the nuclide of interest must be negligible). Examples of nuclides in this category a r e 9 ? m N b ( 5 3 s) and l°SmRh (45 s). c) Determination of cumulative reaction yields for nuclides whose parents are short-lived enough to be considered totally decayed at the production site. Examples are l a~,la2Sn (59 s and 40 s). Extending the method to include also parent effects during the mass transfer is possible by using the necessary growing-in functions in the mathematical formalism, in a treatment parallel to the one given by Rudstam [24], but adjusted to fit the present experimental technique. But since it is hardly feasible to construct a simple, general formalism, this work should be carried out for each actual experiment that might come up.

T. BfOrnstad / Fission yield measurements

387

The data shown in the figs. 7, 8, 10 and 11 are taken f r o m a j o i n t S I S A K e x p e r i m e n t , and the permission to use the data f r o m m y collaborators, Drs. K. Brod6n, N. Kaffrell, I. Rudstad-Haldorsen, G. Skarnemark, E. Stender and N. T r a u t m a n n , is kindly acknowledged. Special thanks are due to Dr. M. Skarestad for having critically read the manuscript, for good advice and fruitful discussions. Financial support has been o b t a i n e d f r o m the Norwegian Research Council for Science and the Huaminities.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [ 13] [ 14] [15] [16] [17] [18] [ 19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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