Nuclear Physics A411 (1983) 199-208 0 North-Holland Publishing Company

/?-DECAY PHENOMENOLOGY OF NUCLEAR FISSION PRODUCTS

J. A. BEHR and P. VOGEL

California institute of Technology, Pasadma, CA 9112.5, USA

Received 4 July 1983

Abstract: Two data sets representing the two principal experimental techniques used to study the B-decay of short-lived fission fragments are analyzed. The average endpoint and electron energies for nuclei with similar Q-values are the same within statistical accuracy in the two sets. However, the spread of the endpoints is considerably larger in the data set based on integral y-energies (ANR set) compared with the set based on standard y-spectroscopy (ENDF set). The averaged j-strength function for the ANR set follows a “universal” trend, decreasing monotonically and rapidly with increasing B-endpoint energy. On the other hand, the B-strength function in the ENDF set is less steep, and at higher excitation energy it is considerably smaller than in the ANR set. These differences imply a systematic bias in one of these two data sets.

1. Iotroduction

The /?-decay properties of short-lived, neutron-rich fission products are of interest for several reasons. Such p-decays and subsequent y-transitions represent a considerable fraction (N 7 %) of the total energy release of the nuclear fission process, so their study has practical applications. There are also important nuclear physics reasons for interest in such #I-decays. The quantitative understanding of the antineutrino spectrum associated with them is the necessary ingredient of interpretation of any reactor neutrino experiment I). Moreover, study of the /I- decays with high Q-values gives one a unique opportunity to observe the behavior of the /I-strength function over a large interval of excitation energy.

The experimental study of high-Q j&decays has special problems associated with the correspondingly short lifetimes and with the large number of levels, and thus y- rays, in the final nucleus. Traditionally, the decay schemes have been reconstructed from data on y-singles or y-coincidences. About 107 fission fragments with Q > 3 MeV have been analyzed in this way, and the corresponding set of discrete excitation energies Ei and branching ratios b(Ei) has been collected in a standardized form in the Evaluated Nuclear Data File ‘). Its most recent form, ENDF/B-V, is our primary source of spectroscopic information.

A different method has been developed by Aleklett, Nyman and Rudstam 3). It is based on the summation of all y-energies in coincidence with an electron. No

199

200 3. A. Behr, P. Vogel / /l-&cay phenomenology

attempt is made to identify the individual discrete levels in the daughter nucleus. Instead, one obtains the continuous p-feed function b(E), such that b(E)dE is the B- decay probability for the interval dE in the vicinity of E. Altogether 62 fission products with Q > 3 MeV were studied in this way in ref. 3).

It has been suggested 4, that the traditional method is susceptible to potentially serious systematic errors related to the combined effect of limited statistics and large number of final states. As a result, the branching ratios for highly excited states could be systematically underestimated, and consequently the branching ratios for the low-lying states could be systematically overestimated. The method of ref. 3, should be immune to this “pandemonium” effect. The suggestion of Hardy et

al. 4, has been refuted ‘) for the concrete case of 145Gd. Its basic point about the danger of the traditional way of reconstruction of complex decay schemes remains, however, valid.

In this work we analyze the full set of data of refs. 2*3). The purpose of the analysis is twofold. First, we want to test whether the two data sets are mutually consistent. Our test is not based on comparison of the few nuclei present in both sets as done, for example, by Reich and Bunting 6, [or as performed in a limited way in ref. ‘)I. Instead, we define several “global” parameters (average energy, average energy spread, smoothed p-strength function) describing the average characteristics of the complex p-decay schemes. We lind substantial di~eren~s between the above-mentioned two data sets in some of these global characteristics, and similarities in others. Generally, the differences are consistent with the “pandemonium” effect in at least some nuclei of the ENDF/B-V set.

Second, we want to examine the global characteristics themselves. We find that they are reasonably well defined and depend in a smooth way on the available excitation energy, that is, on the Q-value. We find that the Q-dependence of these quantities based on the data set of ANR [ref. “)I agrees in a qualitative way with the current ideas about the behavior of the beta strength function in nuclei. These quantities may be used as a basis for prediction of /I-decay properties of nuclei not yet studied experimentally, but needed for the calculation of other things, for example, of the antineutrino spectrum associated with fission. The global characteristics can also serve as criteria for judging nuclear models designed to describe the beta decay strength over a large interval of excitation energy.

2. Average energies and their spread

The average energies are the most important quantities for practical applications. As pointed out above we consider the available decay energy (Q- value) to be the most important parameter. In order to smooth the variations of corresponding quantities in individual nuclei with similar Q-values, we average

J. A. Behr, P. Vogel / /3-decay phenomenology 201

TABLE 1

List of considered nuclei

Q WV Data set ENDFIB-V Data set ANR [ref. ‘)I [ref. ‘)I

3 -3.5

3.54

4 -4.5

4.5-5

5 -5.5

5.54

6 4.5 Ql& QORb QORb* > 3

96y QQy 140,-J 1 > 7

6.5-l ‘6~~ 8’Br 96ye Q’y 1361 1361* , > , 3 7

I -7.5

7.5-8 8 -8.5

“As, “As* 86Br Q3Rb Q’Sr Q’y+ , , , , 7

Q8y 96~:

“K;, “Rb 134Sb 133Sb* 144Cs

8.5-9 9 -9.5

*sBr

s2Br *‘Br* ‘%r Q6Nb lo7Ru, “OAg, 1 1 , “% ’ 1161g, “‘Sn, “‘Sn, 13%b, 133Te*, “‘lBa ’ 144pr* 152pm, 152pm*

%a, ‘slAs, ‘%e, s3Se* *‘Kr, Q3Sr, Q2y 106Rh

133ib 13*1,

106Rhf, llZig, 126S,,, 126SbL,

’ ‘40La

7QGe “‘Kr , ls4Prn , 1 54pm*

“Rb, “Y lo8Rh, losRh , ’ * “81~,

“Zr, ‘O’Te i’SIn* 128&, 128.&e,

1341 1341; 13’~~ 140~~ 139~~ ‘46~~

84Br 84Br*, ’ 89&, 94y, d8N,,, QkN,,*,

lo’& 102Tc*, 13OSb, 13OSb*, 141cs, 142~~, 148~~

**Rb lo4Tc lJgXe ’ 138Cs

“ORh, loRh*, 1201n, 1201n*,

’ 138cse, 3 , 144La

80As, “Kr, “Rb, Q8Sr, “‘Sb, “‘Sb*, 14’Xe

7QGe “Kr, l”Ag, li’A;* 1231~

7 1 13’Xe > 143Xe 139(-J ‘45La

114A ’ lZ31n*, 13Qxe, Lx,

14’Cs, 14’Ba “Ga, “Ge, l16Ag, 116~ * llQAg,1231n

1*qng*: 144La

81Ge 85Se*, ‘lRb, “‘In;, 13*Sb, 132&!b* , 13’1 141xe, 140cs,

143cs

76Ga, “Kr, ‘*%, ‘*‘In*, 13’Te, 13’1 “Zn, “Ga, s’Br, 142cs 81~~ Q3Rb 118~~*

, 9

lz41n 86Br g2Rb, 12’In 18Ga, ’ “Br l*oA lZ61,.,, 134&e

&

“Br “Ga Q4Rb “*In , 9

throughout the paper over all nuclei in the same Q-value bin. The list of nuclei and /I-radioactive isomers in each Q-bin for the two data sets is shown in table 1.

For a given nucleus A, Z the average endpoint energy is

and the average over the Q-bin (normalized to the corresponding Q-value) is

(1)

(2)

The index i in eq. (1) characterizes individual branches; the summation in eq. (2) is

202 J. A. Be&r, P. Vogel I #Ldecuy pheno~enology

over all @-decaying nuclei in the corresponding Q-bin ; Nbin is the number of nuclei and isomers in that bin. (The p-decaying isomers are treated as independent nuclei.)

As a measure of the accuracy of the mean endpoint energy we use the standard deviation of the mean

Note that the distribution (standard deviation of the population) of

L&4 zYQ(4Z) is 6 t' lmes wider than the quantity d@/Q). In an exactly analogous way, one defines the average electron kinetic energy by

replacing Ednd in eq. (1) with the corresponding average electron energy. The allowed j&spectrum shape is assumed throughout.

For the ANR data set 3), we replace the summation in eq. (1) by integration over energy, and b&) by b(E).

The resulting average energies are shown in table 2. (In a few cases there is only one nucleus per bin and the deviations A vanish in that case.) It is important to remember that the deviations d@/Q) have nothing to do with the experimental uncertainties ; they only reflect the spread among different nuclei. In evaluating eqs. (l)-(3) we assumed that the experimental uncertainties are unimportant for our purposes.

TABLE 2

Average endpoint and electron kinetic energies with their standard deviations of the mean (see eqs. (2) and (3) for de~nitions)

Q-bin (MeV) @/Qh,i,+&%'Q) ~~~/Q)~i"~~(~~/Q)

ENDF/B-V ANR ENDF/B-V ANR

3.0-3.5 3.5-4.0 4.0-4s 4.5-5.0 5.0-5.5 5.5-6.0 6.0-6.5 6.5-7.0 7.0-7.5 7.5-8.0 8.0-8.5 8.5-9.0 9.0-9.5

0.58 f 0.08 0.58 f 0.06 0.69 f 0.05 0.64f0.06 0.71 kO.06 0.77 f 0.07 0.75 +0.07 0.65 f 0.06 0.73 +0.05 0.84kO.13 0.87 20.08 0.70

0.75 *0.05 0.80&0.04 0.83 kO.04 0.75 +0.07 0.78 *to.04 b.81 ho.03 0.78 f0.04 0.72 rf: 0.07 0.75f0.02 0.81 +O.lO 0.76 f 0.03 0.56 0.64f0.08

0.23 kO.03 0.24 f 0.03 0.29 f 0.03 0.27kO.03 0.30+0.03 0.34*0.03 0.33 * 0.03 0.29 + 0.03 0.33 * 0.03 0.38 + 0.06 0.39 f 0.04 0.32

0.30 f 0.02 0.34 -t-o.03 0.36 kO.02 0.32 +0.03 0.34 f 0.02 0.36 f 0.02 0.35 iO.02 0.32 +0.03 0.33 +0.01 0.37 f 0.05 0.34+0.02 0.25 0.29 * 0.03

The standard deviation is omitted if the Q-bin contains only one nucleus.

J. A. Behr, P. Vogel / /?-decay p~e~ornen~~ogy 203

Inspection of table 2 shows that the two data sets give essentially consistent average energies, as concluded also by Aleklett and Rudstam *). Closer inspection shows that E/Q for the ENDFIB-V data set are slowly increasing with Q, while for the ANR data set they are slowly decreasing. The former tendency would have to stop when i?/Q reaches the unphysical value of 1 (which happens for Q N 16 MeV if one uses linear extrapolation). This small difference in average energies may be related to the di~erent relative numbers of the odd-A and even-tl nuclei in the two data sets.

Perhaps more interesting, because more sensitive to details of the decay scheme, is the spread of endpoints in individual nuclei, averaged as before over the corresponding Q-bins. This quantity is defined as

and is shown together with the standard deviation of the mean (defined in analogy with eq. (3)) in fig. 1. The difference between the two data sets is readily apparent.

Fig. 1. Average energy spread & *~~.~~DF/B-V data *) denoted by full line, ANR data j) denoted by dashed line. Points are connected by lines to guide the eye.

204 J. A. Behr, P. Vogel / p-decay phenomenology

The endpoints in “traditional” data have a consistently narrower distribution in energy. Exactly that tendency is expected when weak branchings to faraway states are underestimated. Let us note that the data of the ANR set 3, are not truly continuous but come in energy bins N 142 keV wide. It is clear that the difference in DE in fig. 1 are much larger than any spreading associated with smearing over such energy bins or with the differences in selection of nuclei mentioned above.

3. /?-strength function

The essential physics of b-decay is only indirectly contained in the directly measured excitation energies and branching ratios. The p-strength function, which takes into account the energy dependence of the phase space factor f(E,Z), represents the quantity of direct physical significance. For a discrete transition the dimensionless B-strength is

(5)

where D = 6260 sec. For continuous /?-feeding one uses instead 3, the B-strength per unit energy (with dimension energy- ‘); in a trivial generalization

WE) S(E) = f(E, Z)T+ ’

(6)

To calculate the analogous strength per unit energy for discrete states, we choose energy bins of width SE and define

S(E) = 6ET~ i f(E, Z)’ D c b(Ei)

where the summation is over all branches with endpoints in the interval 6E around E.

In accordance with our general aim of finding smooth trends, we additionally average the strength (6) and (7) over relatively wide energy intervals AE (thus removing structures in energy dependence, which presumably strongly depend on the considered nucleus), and as before average over all nuclei with similar Q-values. The resulting smooth average strength

S(E) = ~ ~ FfcE, ~)(~~~~, Z)

is found to be typically defined only with poor accuracy.

(8)

As before we use the

J. A. Behr, P. Vogei J @decay p~nomenoio~y 205

standard deviation of the mean

AS(E) = Jy&i din 5 [ ( ;AEf(EflbZ(;;(A. 2) * -s(E)2 + 1 1 (9) as a measure of accuracy of S(E), This poor accuracy means that the strength itself varies considerably from one nucleus to another. Actually, for the largest excitation

energy E,,, - Q, the average strength S(E) is often so poorly defined that its value loses any meaning. In this context one has to remember that the highest excitation energy is associated with the smallest branching ratios most susceptible to experimental diffl~ulties. Nevertheless, some qualitative conclusions are possible.

It is instructive to consider first the theoretical expectations for S(E). Let us assume that essentially all B-decays for the considered nuclei are the allowed Gamow-Teller transitions. We expect ‘) that the main GT strength is concentrated in (typically, two) giant resonances, above and below the isobar analog state (IAS). For the /I- unstable nuclei the excitation energy of IAS is

A T,T+l = &+Q-W,-M,k2, 00)

where empirically the Coulomb energy shift is lo)

A& N (1.43 Z/..@ - 1.0) MeV (11)

with (Z = $(Z. +Z,)). In a series of isotopes of a given Z the quantity AE, changes with A or iV much

more slowly than Q does. Thus for all such isotopes the region E,,, N Q is at a similar position with respect to IAS (and the giant resonance), while the ground states (E,,, m 0) are further removed as A or N increases. Thus we expect that nuclei with different Q-values will follow the same general trend of monotonically decreasing strength with increasing value of the quantity Q-E,,,.

Such a plot is shown in figs. 2 and 3 for Q-bins 1 MeV wide. The plot does not include several points where ,!$ is very poorly defined because ,!? - A%

Several comments should be made: (a) At low excitation energy (E,,, s 2 MeV) the two data sets have mutually

consistent strength (the 3 < Q < 4 bin is an exception, but the ANR set contains only 4 nuclei in that bin). At these low excitation energies the average strength is decreasing with increasing Q-value as expected.

(b) The data of the ANR set all follow the same general trend. This “universal” strength seems to depend exponentially on Q-E,,,. The exponential dependence (but not the universal character) was noted in ref. 3, and follows from the suggestion that S(E) N p(E). One has to remember, however, that the total level density p(E) is not really a relevant quantity for the one-particle transition operator encountered in B-decay.

206 J. A. Behr, P. Vogel / /Wemy phenomenoiogy

IO"

T

2 5 G

Id'

r;i;

Id2

I”

0 I 2 3 4 5 6 7 8 9 IO

Q - E~~~~~eV)

Fig. 2. Averaged p-strength function s(E) for the ANR data set as a function of the endpoint energy Q-E,,,. A few standard deviations d!@) are shown for illustmtion of accuracy. Points belonging to the same Q-bins (1 MeV wide) are connected to guide the eye; each line is labeied by the midpoint of

the corresponding Q-bin in MeV.

(c) The average p-strength function for the ENDF/B-V data is less well defined, suggesting more fluctuations between nuclei with similar Q-values. Generally, the average b-strength function decreases more slowly with Q-E,,,. With the single exception (7 < Q < 8 MeV) the average strength function at E,,, N Q is typically ten times smaller in the ENDF data set when compared with the ANR data set.

4. conclusions

The analysis shows subtle yet definite differences between the two data sets representing the two principal experimental techniques used to study short-lived fission fragments. The differences resemble the “pandemonium” effect 4), that is, a systematic underestimation of weak branches leading to highly excited states in the data set based on standard y-spectroscopy. However, such a resemblance is not a real proof, and it is possible that the ANR experimental procedure is subject to some unknown systematic effects responsible for the differences. Further work to

J. A. Behr, P. Vogel / fi-decay phenomenology 207

lo-3 1 I I I I I I I I I 0 I2 3 4 5 6 7 8 9

Q-E,,, (MeV)

Fig. 3. Averaged /?-strength function S((E) for the ENDF/B-V data set. For notation see caption of tig. 2.

resolve these differences is clearly needed. Until this is done the properties which depend on quantities other than the average /I- and y-energies should be treated with extreme caution.

We have established that the “global” quantities (average energies, their spread, smoothed p-strength) have well-defined values for sets containing many nuclei with similar Q-values. However, the corresponding standard deviations of the population distribution are large so that the-degree of agreement of individual nuclei with these averaged properties has little meaning. Nevertheless, when dealing with larger groups of nuclei one can still use these trends as a testing criterion for experiments or calculations.

This work was supported by the US DOE Grant DE-AT-03-81ER40002. One of the authors, J.A.B., would like to acknowledge the initial support of the Caltech Summer Undergraduate Research Fellowship (SURF) program.

208 J. A. Eehr, P. Vogel / b-decay phenomenology

References

1) P. H. Frampton and P. Vogel, Phys. Reports 82 (1982) 342 2) Fission Product Library of the Evaluated Nuclear Dam File, Version V (available from, and

maintained by, the National Nuclear Data Center at the Brookhaven National Laboratory) 3) K. Aleklett, G. Nyman and G. Rudstam, Nucl. Phys. A246 (1975) 425 4) J. C. Hardy, L. C. Carrar, B. Jonson and P. G. Hansen, Phys. Lett. 71B (1977) 307 5) R. B. Firestone, Phys. Lett. 113B (1982) 129 6) C. W. Reich and R. L. Bunting, Nucl. Sci. Eng. 82 (1982) 132 7) P. Vogel, G. K. Schenter, F. M. Mann and R. E. Schenter, Phys. Rev. 24C (1981) 1543 8) K. Aleklett and G. Rudstam, Nucl. Sci. Eng. 80 (1982) 74 9) C. Gaarde, Nucl. Phys. A3% (1983) 127~

10) J. Janecke, in Isospin in nuclear physics, D. H. Wilkinson, ed. (North-Holland, Amsterdam, 1969) p. 297