Energy spectra of delayed neutrons from separated fission products

fZ.A.I:Z.Jl

nuclei

Physics A230 (1974) 153 -X72; @ ~ortb-H~llff~

Not to be reproduced by photopriRt or miaof3m

ENERGY

SPECTRA

without written permission from the publisher

OF DELAYED

FROM SEPARATED

Publishing Co., ~rn~t~r~arn

NEURONS

FISSION PRODUCTS

(I). The precursors 85As, “Br, 134Sn, x3sSb, “36Te and 137I S. SHALEVt

and G. RUDSTAM

Received I8 February 1974 (Revised 30 May 1974) This is the first of a series of articles on delayed neutron sp~~rorne~, in which neutron spectra are analysed for mass-separated isotopes. The theoretical basis is given for calculations of level densities, neutron energies and neutron emission probabilities. Delayed neutron spectra from the precursors *‘Br and 137I are used to obtain a normalized set of level density parameters, which are then used to analyse the neutron spectra from 85As, 134Sn, 13%b and 136Te.

Abstract:

The present work is the first in a series of articles dealing with various aspects of delayed neutron spe~tr~~etry on mass-separa~d fission products. The experimental t~h~iques have been described elsewhere “). In this article the theoretical approach to pr~icting delayed neutron spectra is described, while experiments and predicted spectra are presented for six selected delayed neutron emitters. In for~~oming articles in the series, other groups of emitters will be analysed. The delayed neutron precursors 85As, 87Br, 134Sn, 13%b, 136Te and 137Xare of particular interest. Following b-decay the neutron emitters contain either 51 or 83 neutrons, and neutron e~ssion forms a nucleus with a major closed neutron shell. ~ousequently the neutron binding energy is ex~ptionally low, and the level density parameter also has a greatly reduced value. As a result, the neutrons are emitted from low-lying and relatively widely spaced levels, with a high probabiIity that the neutron energy distribution will consist of discrete, well-separated lines. Preliminary measurements on ’ 37I and “As have indicated that the energy spectra do, indeed, show line structure ‘*“). In the present work we attempt to correlate the expe~mental energy spectra with theoretical predictions of spectral shape, level densities and neutron emission probabilities. t Permanent address: Department logy, Haifa, Israel.

of Nuclear Engineering, Technion - Israel Institute of Techno153

154

S. SHALEV

2. Predictions

AND G. RUDSTAM

of delayed neutron spectra

The basic process of delayed neutron emission was postulated by Bohr and Wheeler “); since then progressively refined calculations have been made of neutron emission probabilities 5-7). The first attempt to predict the energy distribution of delayed neutrons from fission products was made by Batchelor and McKHyder “). Their prediction for the energy distribution of group I neutrons (assumed to be essentially *‘Br) was rather poor, partly due to the inaccurate Q-values available at that time. A more detailed analysis was carried out by Jahnsen, Pappas and Tunaal 9), using a spin-dependent level density expression. The correlation between predicted and experimental spectral shapes was poor, and no attempt was made to compare predicted level densities with the experimental energy distribution. Gauvin and de TourreiI lo) also employed spin and parity seiection as well as a more precise form for the level density term. Both spectral shapes and P, values were predicted for a number of precursors as well as for the spectrum of group 2 delayed neutrons. While the latter showed quite good correlation with the experimental spectrum, it must be noted that this was to a large extent coincidental. For instance, the dominant peak at 400-500 keV in the group 2 spectrum was predicted by these authors to be due to “Br while the low energy spectrum was assigned to r3’J. It will be shown in the present’work that the reverse is the case. More recently Takahashi ‘I) has made a detaiied study of delayed neutron emission using Qb and B,, values obtained both from the Garvey 12) and Myers-Swiatecki 13) formulae. Although both a modified Lorentz and a Gaussian form were used for the strength function, the corretation between the predicted values of P,, and the experimental values r4) was poor. No comparison was made between predicted level spacings and the clearly observed structure in the measured neutron spectrum. Pappas and Sverdrup ’ “) also made calculations of neutron emission probabilities and energy spectra, using two versions of the B-strength function and two different mass formulae to obtain values of QR and B, [refk ’ ‘7 * ‘)I. Poor agreement was found

between calculated and experimental I’, values, with differences of more than an order of magnitude in many cases. Likewise the comparison of energy spectra was inconclusive, with some tentative conclusions concerning the shape of the P-strength function, but no attempt to explain the discrete line structure. In the present work we attempt a more comprehensive approach to the prediction of delayed neutron spectra. For each precursor investigated experimentally, we obtain a spectral shape or envelope, an average spacing between lines or peaks (if such are observed) and a maximum observed neutron energy E,(max). Calculations are then performed to obtain the spectrum, the average density of neutron-emitting states, and the neutron emission probability P,. ideally, we would like to find a logical and consistent set of parameters (level density parameter a, Q-values, binding energies, neutron and gamma widths, etc.) such that the experimental and calculated data are in reasonable agreement for as many precursors as possible.

DELAYED

NEUTRON

ENERGY

SPECTRA

(I)

155

3. The theoretical method

The process of delayed neutron emission is treated in two stages. First, P-decay of the precursor leads to a population density of excited levels in the emitter nuclide. The second step is to consider neutron emission from those states with energy greater than the neutron binding energy, subject to competition from photon emission. In general the neutron transition may lead to the ground state or excited levels in the product nuclide. 3.1. THE LEVEL

DENSITY

Low-lying nuclear excited levels are generally well separated and exhibit a relatively simple structure. For higher excitation energies the level spacing decreases rapidly, and a statistical approach is used to describe the level structure. An excellent survey of the theoretical models and experimental methods used has been given by Huizenga and Moretto I’). For our calculations we have employed the so-called “backshifted” zero-order Fermi gas model to express the level density for a single angular momentum J in the form ,@,

J”) =

(2J+- 1) 48$a+u3

exp

.

U5/4

The true excitation energy is replaced by an effective energy U = E - P, where P is the total pairing energy 21). While some authors have treated P as an adjustable parameter 22), we have used the values of Truran, Cameron and Hilf ““). Eq. (1) has been derived on the assumption of an equal distribution of levels of both parities. While this may be generally correct 26), there would appear to be instances when such an assumption is considerably in error 2oY“‘). The spin cut-off factor c2 characterizes the angular momentum distribution of the level density. According to Malyshev ““) this parameter is energy dependent, I72 = O.O889(a U)+A”,

(2)

and values of B for an excitation energy equal to the neutron binding energy are given in table 1. Pairing energies and values for the level density parameter were taken from Truran et al. 23), and b’m d’m g energies from the compilation of Garvey et al. I’). In view of these values, and the experimental result that r~is approximately 4 f 1 in the vicinity of the binding energy 2 ‘), we have used a value of 4 for this parameter throughout all our calculations, Values of the level density parameter a obtained from experimental data 22) demonstrate an overall dependence of the form a = $A, although much lower values are found in the vicinity of major closed shells 22*24, 28,29). However, the reliability of such data is in some doubt, especially for measurements based on counting $’ levels from neutron resonances, a technique affected by finite energy resolution, pwave contamination and the necessity for prior knowledge of the spin and parity

S. SHALEV

156

AND G. RUDSTAM TABLE 1

Data for calculation Precursor

“As *‘Br 13%n lJsSb 136Te 1WI “) Ref. 12).

dependence least-squares

of level density

Pb)

B,“) (McV)

CMeV)

4.10 5.46 3.43 3.86 4.02 4.45

1.449 1.331 0 1.128 0 1.152

“) Ref. 23).

‘) Eq. (3).

od)

UC)

(normalized) (MeV-‘)

9.58 9.89 10.96 11.65 12.24 12.99

2.94 3.34 3.78 3.63 4.06 3.93

7.94 8.25 8.22 8.91 9.50 10.25

“) Eq. (2).

of the total level density. Truran, Cameron fit of the data to obtain the relationship (1

a

(MeV-‘)

= 0.139+0.0102(S-0.33

and Hilf 23) performed

a

D).

A where S is the total shell energy and D is the “distance” to the nearest closed shell. Values of a obtained from eq. (3) are given in table 1. In view of the wide range of values available in the literature for the level density parameter a, and the necessity for a set of values for use with eq. (I), we have adopted the following approach. We accept the shape of the a(A) distribution given in eq. (3) but not its absolute value. For the precursors “Br and t3’I we were able to determine c2from the experimental neutron energy distribution, with the results 8.25 MeV-’ and 10.25 MeV-’ respectively (see subsects. 4.3 and 5.3 below). Normalized values of a for other nuclides were obtained by eq. (3) and are given in table I. 3.2. THE P-STRENGTH

from

these values,

using

ratios

predicted

FUNCTION

The strength function is defined as the product of the level density p(E) and the average of the squares of the nuclear matrix elements,

S, = cons1 2 /&f,,12pJ,(E). Many authors ‘-I “) have assumed that I&.?(’ is independent of energy, and hence S, cc p(E). Attempts by Takahashi ‘l) and Pappas and Sverdrup t j) to use different forms for the strength function have not yielded convincing improvements in predicting P, values or neutron energy spectra. It would appear that, on the neutron deficient side of stability, the strength function may indeed be energy independent, or nearly so, judging by the results of P-strength studies 30) or delayed proton measurements 32). However, the strength function is peaked around the isobaric analog

DELAYED NEUTRON ENERGY SPECTRA (I)

157

state 33) which is located at an energy Xower than the ground state of the 8’ emitter, and higher than the ground state of the ,8- emitter. Consequently the shape of the “tail” of the strength function over the energy range (Qa -IS,) will be quite different in the two cases, and we cannot accept the contention 3i) that the shape of the & strength function for p” decay is applicable to the problem of jl- decay. Ex~~ri~e~~a~ ~easu~e~e~ts have been made of ~-streng~ functions for a wide range of nuclei 34), including s7Br and 13?1. Unfortunately the data do not extend to energies higher than the neutron binding energy, and so cannot be used in the prediction of delayed neutron spectra. A general feature is, however, that the pstrength increases strongly with increasing excitation energy of the daughter. In predicting delayed neutron spectra there are a large number of parameters either adjustable or known to a poor degree of accuracy. These include the level density parameter a, Q8 and 23, values, and the energies, spins and parities of many of the nuclear Xevelsinvolved. While it is true that the energy dependence of the strength f~nct~o~l has an important inthience orz the shape of the predicted neutron spectrum, this is also true for the other parameters. Until such time as more reliabfe data become available, it would a,ppear to be of limited value to investigate the effect of different strength functions, and consequently we have carried out our calculations with the assumption that IMi2 is a constant.

The de-excitation mode of levels above the neutron binding energy in the emitter nuclide, through either photon or neutron emission, is determined by the neutron r~(~~) and y-widths r,. Pappas 9y1“) has used the approach that r, is e~ec~vely independent of excitation energy, while the neutron width is given by the BlattWeisskopf expression 40). Takahashi I’) has calculated neutron widths using transmission factors extrapolated from the tables of Auerbach and Perey 41). In our ca~cu~ations~we assume a sharp cut-off energy E,*(t), such that

The actual values of ~~(~} are determined by the shape of the experimental neutron spectra. For light masses, with p- and f-wave neutron emission, the cut-off energies were usually selected at 30 keV and 400 keV respectively. For heavy masses with s-, d- and g-wave neutron emission, the cut-o2T energies were 0 keV, 350 keV and 1000 keV in most cases. 3.4. THE NEUTRON ~~~~RUM Zse enve?fopeof the

energy ~s~bntio~

of delayed neutrons is giveu by:

158

S. SHALEV

AND

G. RUDSTAM

Within this envelope the spacing of levels with spin and parity J” is given by the reciprocal of p(E, J”), according to eq. (1). The Fermi functionf(2-t 1, Q, -E) has been shown to vary according to (Qa - E)5 [ref. ‘“)I, an approximation that has been used by previous authors “). In the present work we have also adopted this simplification, and used eq. (6) to compute the envelope of the spectrum. 3.5. NEUTRON

EMISSION

PROBABILITY

This parameter is defined as the probability that a given J-decay of the precursor will lead to the emission of a neutron and, with the assumptions and simplifications used in this work, is given by

A computer program has been written to calculate P,, using step-wise integration with 10 keV intervals. The function is constrained to linearly approach zero at the energy U = P. 4. Data collection and analysis A detailed description of the experimental techniques has been given elsewhere ‘). Basically, a sample of 235U is located in a region of high neutron flux, and the fission products are extracted, accelerated and formed into isobaric ion beams by means of the OSIRIS isotope-separator on-line facility. The neutron spectrometer is a cylindrical 3He ionization chamber, with energy resolution less than 35 keV FWHM for neutrons with energy up to 1 MeV. Moving tape systems are available ‘) for transporting samples of the selected isotope to the spectrometer, or for removing the longer-lived decay products. The spectrometer yields a pulse-height spectrum which has to be converted into the incident neutron energy spectrum. Such a conversion requires knowledge of the response function of the detector and its energy dependence, and the techniques for obtaining these data have already been described I735). A computer program has been developed for unfolding pulse-height spectra 36), as well as for correcting for distortion due to y-ray pile-up. 5. Experimental

results

All the neutron emitters investigated in the present work have a single neutron outside a closed shell, so that following neutron emission the final nucleus has the form (2” +m, N*) where Z* and N” have the magic numbers 28,50 or 82. The value

DELAYED

NEUTRON

ENERGY

SPECTRA

(I)

159

TABLE 2 Summary of Qa and B, vabres (in MeV) Garvey et al. “)

Myers and Swiatecki “)

9.05 4.10

8.95 3.64

6.68 5.46

6.43 4.65

6.07 3.43

7.01 2.87

7.52 3.86

7.95 3.23

3.28

7.57 3.49

7.7 3.3

4.47 4.02

5.13 3.57

3.72

4.11 3.84

5.0 3.7

5.79 4.45

6.10 3.91

5.88 4.17

5.7 4.0

QP

=As

B. s7Br

Qfl

13%n

&

QS

‘=Sb

B,

Q@

136Te

B,

QS

1371

B,

“) Ref

. r2) .

“) Ref. 13).

‘) Ref. 38).

Wapstra and Gove “)

6.50 5.511 10.008

Zeldes et nl. 0)

Seeger ‘)

9.16 4.48

8.0 4.2

6.64 5.41

5.4 5.2

6.3 2.9

5.40 3.86 f0.02 d, Ref. 16).

“) Ref . I’)

.

of m is even for the precursors *5As, 87Br, l3 ?Sb and I3 71, and odd for 134Sn and I3 6Te. The precursors 1331n (m = 0)and 84Ge also belong to this group, but we have not succeeded in obtaining their neutron spectra at this stage. For the precursors “Br and 1371, accurate values of the neutron binding energy are given in the Wapstra and Gove tables 38). Consequently we have used the observed level spacings in these cases to deduce values for the level density parameter ra for emitters in the light and heavy mass regions respectively, and hence obtain a complete set of normalized values for all the isotopes investigated (see table 1). A summary of Qp and B,, values for all the cases investigated is given in table 2. 5.1. THE PRECURSOR

s7Br

The precursor ’ 7Br was collected at the on-line sample position, with a 1 mm thick lead absorber between the sample and the neutron spectrometer. The tape was moved 0.X ~rn~rnin to remove long-lived activities. During a run time of 11.5 h a total of 37000 fast neutrons (above 100 keV) were detected, of which 3.4 % were due to background. The experimental pulse-height spectrum, shown in the upper part of fig. 1, extends up to 1.22rfiO.06 MeV, at which energy there is no significant increase over background. The energy spectrum, obtained by oorrecting for detection efficiency and sp~trometer response function, is shown as a histogram in the lower part of the same figure. The ordinate gives the fraction of fast neutrons (above 40 keV) per keV energy interval, while each step in the histogramrepresents 6.15 keV. Prom-

160

S. SHALEV AND G. RUDSTAM

1 I

/

-4

I

7

/ I

---_ 400

800

NEIJTRON ENERGY (keV

I 1200

1600

1

Fig. 1. Upper part: experiments p~s5height spectrum of delayed neutrons from the precursor *‘Br. The error bars correspond to + one standard deviation. Lower part: Corrected energy spectrum, the ordinate being the fraction of fast neutrons per keV. Predicted envelopes according to mass data from Garvey et aLiZ) [full line], Myers and Swiatecki13) [dashed line] and Wapstra and Gove3*) [dash-dot line].

inent peaks occur at 130, 183, 253 and 440 keV, with smaller peaks at 315, 400, 534 and 614 keV. Allowed P-decay from the ground state of ’ 7Br ($-) populates $-, $- and 4- levels in 87Kr. For this nuclide the neutron binding energy is given by Wapstra and Gove to great accuracy, although there is no agreement amongst different mass formulae with regard to the Qa value (see table 2). However, neutron emission wil almost certainly not lead to the first excited level in *%r at 1564 keV [ref. *‘)I, and only p- and f-wave transitions to the 86Kr ground state need be considered. The experimental peak spacing of about 50 keV in the energy range 100-300 keV is attributed to p-wave transitions from + - and 3- levels in *‘Kr. From eq. (1) we then obtain a value of 8.25 MeV-l for the level density parameter a, this being used for normalizing the a-values for light isotopes in table 1. From eq. (6) we obtain predicted envelopes for the neutron spectrum, and three cases are shown in the lower part of fig. 2. The predicted curves are normalized to the experimental data for energies above 100 keV, and E:(f) is taken as 400 keV. The best fit is obtained for the Wapstra-Gove data, suggesting a Q, value of about 6.5 MeV. It is interesting to note that the shape of the predicted envelope is not sensitive to

DELAYED

NEUTRON

ENERGY

SPECTRA (I)

161

the value of a. For instance, an increase in a from 8.25 to 10 MeV-r would increase the height of the step at 400 keV by only 7 %. The envelope prediction using the Wapstra-Gove data also gives reasonable values for the predicted level spacings. At 200 keV the calculated spacing is 50 keV, since a was normalized to give this value. At 400 keV the calculated spectrum comprises 65 %. f-wave emission, with an average line spacing of 22 keV, and 35 % p-wave emission with an average spacing at 41 keV. At 600 keV the level spacings are 18 and 33 keV respectively, and in view of the spectrometer resolution one would not expect to observe such line structure. The P, value was calculated from eq. (7) using the Wapstra-Gove values for B, and Q,. The result, 0.16 % is in very poor agreement with Tomlinson’s adopted value of(2.3k0.2) ‘A [ref. ‘“)I. Using the experimental value 44) of 7.06LO.l MeV for Q, yields P,, = (1.35kO.4) %, while a value of 2.3 % is obtained by assuming a Q, value of 7.3 MeV. 5.2. THE PRECURSOR

137I

A preliminary measurement of the delayed neutron energy spectrum has been published previously “). A more accurate measurement was made using the on-line tape system with the neutron spectrometer located directly behind the sample posi-

0

400

600 NEUTRON

ENERGY

17.00

1600

(keV 1

Fig. 2. Upper part: experimental pulse-height spectrum of delayed neutrons from the precursor 13% Lower part: corrected energy spectrum and predicted envelopes (see caption to fig. 1).

162

S. SHALEV AND G. RUDSTAM

tion. A lead absorber 1 mm thick was used, and the tape system operated at 10 cm,Imin. In a run time of 4 h 10 min a total of 37900 pulses were recorded above 100 keV, of which 1.6 ‘A were due to background. The experimental pulse-height distribution is shown in the upper part of fig. 2, and the energy spectrum after correction is shown in the lower part of the same figure. The coordinates are the same as those in fig. 1. Prominent peaks are observed at 272, 380, 487, 583, 756, 863, 965 and 1140 keV, while smaller but still evident peaks occur at 166, 325,425, 515, 695 and 1063 keV. There is no doubt that with even better energy resolution, some of these peaks would be shown to consist of several componems, especially in the higher energy region. The ground state of 137I is probably -2’ and hence allowed p-decay will populate $+, 3* and 4’ levels in r37Xe. Decay by neutron emission to the ground state (Of) of l3 6Xe will involve neutrons with wave numbers d,, g%and g%respectively. Neutron decay to the first excited level 43) in 136Xe at 1.313 MeV, if energetically possible, will involve s- and d-wave neutrons. The low neutron intensity observed below 300 keV suggests that s-wave emission is weak, and hence we assume that the dominant line structure in the energy range 300-600 keV is due entirely to d-wave neutrons emitted from +’ states in 137Xe. Using the experimental level spacing of l/p(O.3,$‘) = 0.1 MeV, and the accurate value of B,, given by Wapstra and Gove 38), eq. (1) may be used to obtain a value for the level density parameter a of 10.25 MeV- I. (If the weak peaks in the experimental spectrum are also taken into account, the level spacing is about 50 keV and a becomes 11.65 MeV-I. However they could also be due to p-wave neutron emission from +J- states, following first-forbidden /Idecay.) The value of a = 10.25 MeV-l is used to normalize the a-values for heavy isotopes in table 1. The envelope predicted by eq. (6) is very dependent on the values of Q2aand B, used, and three examples are shown in fig. 2. The envelopes are normalized to the experimental data for energies above 350 keV. The curve using the Myers-Swiatecki data extends to too high energies and includes too large a g-wave component, while that based on the Garvey data has too low an end-point. We conclude that the Wapstra-Gove data give the best prediction of the spectral envelope above 350 keV. Presumably a more realistic approach to the y-competition factor for d-wave neutrons would improve the fit in the energy range 250-350 keV. The predicted level spacing at 400 keV is 88 keV, due entirely to d-wave neutrons. At 1000 keV the predicted spectrum consists of 33 % d-wave, with a level spacing of 30 keV, and 67 7; g-wave, with a level spacing of 14 keV. This is not inconsistent with the experimental spectrum. Neutron decay to the 1.3 13 MeV in 136Xe is entirely negligible. Calculations of the P, value using the Wapstra-Gove mass data in eq. (7) give a value of 0.75 %, much lower than Tomlinson’s adopted value of (5.4 + 1.3) % [ref. ’ “>I. The P, value would increase to this figure if a value of 6.05 < QB < 6.3 MeV were adopted. This is supported by experimental measurements of Q, by Adams et aI. 3Q) and by Lund 44), which gave 5.8 and 6.21 +0.36 MeV respectively.

DELAYED 5.3. THE PRECURSOR

NEUTRON

ENERGY

163

SPECTRA (I)

13sSb

The precursor 135Sb was collected at the on-line sample position using a 10 mm thick bismuth absorber and a tape speed of 40-50 cm/min. In a run time of 269 min a total of 43000 fast neutrons (above 100 keV) were detected, and the experimental pulse-height spectrum is shown in the upper part of fig. 3. Very prominent peaks occur at 1040, 1205 and 1450 keV, while at lower energies there would appear to be some structure superimposed on a continuous distribution. The energy spectrum after correction for the spectrometer response function is shown in the lower part of fig. 3 as a histogram, each step representing about 5.48 keV. The ordinate gives the fraction of fast neutrons (above 50 keV) per keV energy interval. In the low energy region there are a large number of peaks with spacings in the range 30-70 keV. The structure becomes less obvious in the energy range 600-1000 keV presumably due to decreasing level spacings and the progressively lower spectrometer energy resolution. However the appearance of three major peaks above 1000 keV clearly demonstrates the presence of a high energy neutron transition from levels with average spacing about 200 keV. By analogy with I3 71, we assume that the ground state of l3 ‘Sb is 3+, and hence allowed p-decay will populate $‘, 3’ and $’ levels in 135Te. Considering the first three levels in 134Te [ref. ““)I, it is seen from table 3 that neutron emission will ‘8 r-

I

13%b

0

I 0

I 400

I

1

800

1200

1600

NEUTRON ENERGY fkev)

Fig.

3.

Upper part: experimental pulse-height spectrum of delayed neutrons from the precursor i35Sb. Lower part: corrected energy spectrum and predicted envelopes (see caption to fig. I).

S. SHALEV AND G. RUDSTAM

I64

TABLE3 Spins and parities of levels involved in the allowed @-decay of 13%b Precursor =%b

Emitter ’ 35Te JZ

Product 134Te energy (MeV)

Neutron wave number

0

1.278 S

1.575 d;

consist of a mixture of s-, d- and g-wave transitions. Neutron emission will undoubtedly lead to even higher levels in 134Te, but such transitions will be low in energy and intensity, and will not be considered here. Spectrum envelopes have been calculated for the Garvey and Myers-Swiatecki data, and are shown in the lower part of fig. 3. They represent the experimental data fairly well except in the low energy region, presumably due to the effect of highly excited levels in r34Te. Values of E,*(d) = 350 keV and E,*(g) = 1000 keV were selected by analogy with the 137I case. An additional calculation, based on the assumption that the 13%b ground state is S+ 2 , gives very similar results, although the high energy (g-wave) contribution is reduced in relative amplitude by about 20 %. Likewise, the shape of the envelope is ahnost independent of the level density parameter. Increasing a from 8.91 to 11.65 MeV-” [the value obtained from eq. (3), see table l] alters the step at 1000 keV by only about 4 %. It is interesting to compare the predicted level spacings with the experimental results. For the Myers-Swiatecki mass data and a value of a = 8.91 MeV-l, table 4 shows the calculated level spacings and relative intensities for a number of different neutron energies. At 200 keV the calculated distribution consists of a series of peaks with average spacing 34 keV, together with a second series (of lower intensity) with average spacing 101 keV. At 500 keV five series of peaks are predicted, but in view of the finite spectrometer resolution one expects that many peaks will overlap. However, the series with average spacings of 357, 70 and 49 keV will presumably cause visible structure superimposed on the continuous distribution. At 800 keV the level spacings are greatly reduced, and little discrete structure is to be expected. Above 1000 keV, g-wave emission is expected to occur. The calculations for 1100 keV show that peaks due to g-wave transitions should have an average spacing of 78 keV, and comprise about 36 % of the total distribution. We conclude that the calculated neutron spectrum agrees well with the experimental distribution, representing both the envelope and the discrete structure remarkably

DELAYED

NEUTRON

ENERGY

165

SPECTRA (I)

TABLE4 Calculated line spacings for neutron emission from f35Te Neutron

energy (kev)

Neutron wave number

Final state JR

Average line spacing Rev)

Relative line intensity (%)

200

s

2+ 4’

101 34

35 6.5

500

S

2” 4+ 0+ 2” 4’

70 24 357 34 49

I4 26 17 31 12

2+ 4’ 0+ 2” 4+

49 17 239 23 35

15 25 23 26 11

2* 4’ 0’ 2” 4+ 0”

3.5 12 162 17 25 78

9 13 17 19 6 36

d

800

s

d

I100

S

d

6

well. An even better comparison would be obtained for a B, value rather lower than given by the Myers-Swiatecki tables, the optimum value being close to 3.0 MeV. Calculations of P, using eq. (7) give 23 % and 50 % for the Garvey and MyersSwiatecki data respectively. Tomlinson’s adopted 14) value is (8+2)%. Very large reductions in Qs are required to obtain a better calculated value. For instance, using the Garvey value of 3.86 MeV for B,, a value of QB = 6.5 kO.15 MeV is necessary in order to obtain an acceptable P, value. 5.4. THE PRECURSOR

85As

The OSIRIS on-line separator does not provide samples of 85As sufficiently intense for neutron spectrometry. Herrmann and his colleagues “) have used a chemical separation loop to extract 85As from thermally fissioned 235XJ, and measured the delayed neutron spectrum with a 3He proportional counter. While the spectrometer used for the 85As measurements had limited energy resolution (about 60-70 keV FWHM for 1 MeV neutrons), it was sufficient to detect discrete line structure in the neutron spectrum. Our normal data reduction techniques were used to correct for detector efficiency and response function, albeit that such calculations must be somewhat inaccurate due to lack of precise data. The corrected energy distribution is shown as a histogram in fig. 4. The ordinate gives the fraction of fast neutrons

166

S. SHALEV AND G. RUDSTAM

40

20

0 0

800

400

NEUTRON

izco

1600

ENERGY (keV)

Fig. 4. Corrected

energy spectrum for the precursor 85Asfrom ref. 3)7 and predicted envelopes according to mass data from Garvey et at. “) Ifull line], Myers and Swiatecki13) [dashed line] and the values Qp = 6.0 MeV, 3, = 4.0 MeV [dash-dot line].

(above 70 keV) per keV energy interval. Pr~minant peaks are evident at 176, 495, 645 and 900 keV, and indications of peaks with fewer &en&y are seen at abont 110, 565, 1110 and 1350 keV. [Recently a new spectrum has been obtained by the same groupie), in which the peaks appear more clearly at energies close to those quoted here. J We assume that the ground state of g5As is $-, by analogy with 87Br_ Allowed /I-decay populates Q-, +j- and +- levels in ssSe with subsequent p- and f-wave neutron transitions to the ground and first excited levels in *“Se. We have adopted the original authors’ figure of 1455 keV excitation energy for the first 2’ level in 84Se [ref. 3)‘j, and not treated higher levels. The f-wave cut-ofF energy E:(f) was set at 450 keV on the basis of the shape of the experimental spectrum, and the levet density parameter cz = 7.94 h&V-’ (see table I). There is little difference jn the predicted neutron energy envelopes calculated with the Carvey et at. and the MyersSwiatecki mass data, and neither appears to represent the experimental spectrum satisfactorily. The third envelope shown in fig. 4 (dash-dot line) was calculated using the arbitrary values of Q@= 6.0 MeV and -tr, = 4.0 MeV, although from the mass data summarized in table 2 it is seen that there is no support for such a low Qp value, Great care must be exercised when comparing a calculated envelope of neutron energies and an experimental energy d~str~bLlt~on_in the case where level densities are very low, and hence peaks widely spaced, there may be little supeticial resemblance between the two curves. Table 5 shows the predicted line spacings and relative peak intensities at two energies in the neutron spectrum, using the data from both Garvey et af. aad Myers and Swiaiecki. It is seen that the calculated Iine spacings are very Large, parti~~1~~~ for the dyed-Swiate~k~ data due to the low binding energy of 3.64 MeV. At a neutron energy of 200 keV, 58 % of all the neutron intensity is concentrated in peaks with average spacings greater than 300 keV, and 48 Y0is concentrated in peaks with average spacings greater than 750 keV. Under these circum-

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167

TABLE5 Calculated Iine spacing for neutron emission from 85Se Neutron energy (keV)

Final state .Tn

200

Of

“) Ref. 12).

Garvey ei ai. “) ~-_______ av. line rel. peak int. spacing (“/,) (keV)

Myers-Swiate~ki’) av. line spacing (keV)

rel. peak int. (%)

Qb = 6.0 MeV; 3, = 4.0 MeV av. line rel. peak spacing int. (%) (keV)

P+ P* Pg P+

985 541 45 185

14 25 49 12

1767 970 75 306

12 22 53 13

x115 613

Of

PaP* f+ P; P$

11 21 26 34 8

1203 660 515 54 220

10 19 24 38 9

771 424 330

2+

684 376 293 33 135

2+ 500

Neutron wave number

35 65 < 0.3 20 35 45 0

b, Ref. 13).

stances, the physical meaning of an envelope calculated from statisti~l theory is far from clear. Calculations of P, using eq. (7) give values of 60 % and 69 % for the Garvey and Myers-Swiatecki data respectively, compared with (20 h4) % according to Tomlinson 14). For th e same neutron binding energies, it would be necessary to reduce QP to 7.0 and 6.3 MeV respectively in order to obtain this value for P,, and there is no obvious justification for such a step. However, it is interesting to note that calculations of both the spectrum shape and the P, value give results closer to the experimental results if a low value of Q, is used. 5.5. THE PRECURSOR

l=Te

The precursor ‘36Te was measured at the off-Iine position using a 1 mm lead absorber and a run time of 1217 min. The total number of fast neutrons (above 100 keV) was 30000. The experimental pulse-height spectrum is shown in the upper part of fig. 5. It comprises a single dominant peak at 429 keV, and a series of smaller peaks at 251,313,466,525,593,692 and 766 keV. Since this series consists of energies quite close to those found in the Is71 sp ectrum the question of cross-contamination of adjacent ion beams presents itself. Measurements were made with the central ion beam adjusted to masses between 136 and 137, and any contamination, if present, was shown to be negligible, The doubly even precursor ‘36Te is not exactly comparable to the other heavy precursors discussed previously, namely 13%b and 137I_ True, neutron emission from the emitter nuclide I3 61 leaves the closed neutron shell N = 82, but there is an odd number of protons and hence a different situation with regard to pairing energy.

S. SHALEV AND G. ~~~STA~

Fig. 5. Upper part: experi~~~ta~ pu~s~~e~~t spectrum of delayed neutrons from the precursor 139e. Lower part: corrected energy spectrum and predicted envelopes according to mass data from Myers and SwiateckiX3) [dashed line] and with &-& = I.0 MeV [dash-dot linef.

Allowed &decay will populate 1’ states in 13?I, followed by d-wave ~nsi~o~s to the $+ ground state af 135I. No information is available on higher excited levels in this nuclide. It has been shown “) that E:(E) is somewhat higher for doubly add emitters, and so we use E:(d) = 400 keV for this case rather than the value of 350 keV used previously. From the mass values given in table 2 it is clear that the Garvey data cannot be used to predict neutron energies greater than 450 keV. Consequently Wehave calculated the envelope of the neutron spectrum for the Myers-Swia~e~ki data, and for the values Qp = 4.57 and B,, = 3.57 MeV. The calculated envelopes are shown in the lower part offtg. 5, normalized to the experimental data above 400 keV. The ordinate gives the fraction of fast neutrons (above 50 keV) per keV energy interval, and each energy step represents about 5.6 keV. Using eq. (l), B, = 3.57 MeV and cb = 9.5 MeVf, the level spacing of 1’ levels in 13’1 at an excitation energy of B, -f-400 keV is found to be 70 keV. This is in excellent agreement with the peak spacing in the experimental spectrum_ It wonid a.ppear to be reasonable to explain the. presence of peaks below 400 keCVas due to a slow change in ri(E,J, rather than the abrupt step used in our calculations. The Garvey et ~2. mass data are rtot accurate for the case of 136Te. The neutron binding energy of 4.02 Me?’ is a~~a~~tly too high, since a level spacing of 41 keV

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169

is predicted for 1* states at B, = 400 keV and the experimental result is greater than this. The Q, value is low, or to be precise the QP - B,, value is too low, as seen by the fact that neutrons are emitted with energies considerably in excess of 450 keV. Furthermore, the P, value calculated on the basis of these mass data is obviously too low (< lo-’ %) in view of the high counting rates obtained by us. The Myers-Swiatecki mass data yield a value for P, of 1.O % assuming E,*(d) =400 keV and a = 9.50 MeV-I. This result is not very dependent on the level density parameter, rising to 1.73 % for a = 12.24 MeV-I. There is little accurate information available on the value P,, for comparison. Amiel and Feldstein 47) p re d’ic t va 1ues of about 0.25 and 1.5 % for the Garvey and Myers-Swiatecki data respectively, using a very simple empirical formula. Schussler et al. 48) obtain an approximate value of 0.5 ‘A on the basis of the ratio between the measured neutron yield and the estimated cumulative fission yield. Takahashi 11) calculated values of 1.50 and 3.02 ‘A using the Myers-Swiatecki mass data. 5.6. THE PRECURSOR

+??n

The precursor 134Sn was measured at the on-line position, using a tape speed of 0.4 cm/set. The total number of fast neutrons (above 100 keV) was 4600. It was found that most of the neutron activity was associated with a half-life of about 0.6

/

I

I

I

,

800

NEUTRON

ENERGY

I

1200

I

I

1600

(keV)

Fig. 6. Upper part: experimental pulse-height spectrum of delayed neutrons from the precursor 134Sn. Lower part: corrected energy spectrum and predicted envelopes according to mass data from Garvey et aE. 12) [full line] and for the values Qs = 4.5 MeV, B, = 3.0 MeV [dashed line].

170

S. SNALEV AND G. RUDSTAM

1 set, and hence we conclude that *34Sn is the dominant precursor and not 37) ‘34Sb. The experimental pulse-height spectrum is shown in the upper part of fig. 6, and is seen to comprise a single dominant peak at 500 keV and smaller peaks at 320, 435, 760,860 and 1020 keV. The similarity with the ‘36Te spectrum is remarkable. By analogy with 136Te, we assume d-wave neutron emission to the 3’ ground state of *33Sb, and a value of E:(d) = 400 keV. The normalized level density parameter a = 8.22 MeV-*. The corrected neutron energy distribution is shown in the lower part of fig. 6, each energy step representing about 10.8 key. The ordinate is the same as for fig. 5. The predicted neutron energy envelope using the Garvey data extends to energies far above the experimental spectrum, since the neutron “window” is 2.64 MeV. The Myers-Swiatecki data give an even higher figure, 4.14 MeV, for this parameter. The predicted envelope, using Q, = 4.5 MeV and a mean value of B,, = 3.0 MeV is shown in fig. 6, and would seem to provide a lower limit for the QP value. The average level spacing at 500 keV is predicted to be 274 keV on the basis of these mass data, and the P, value is 0.83 “/ In comparison, the P, vaiue is 9.1 x, and 39.0 % for the Garvey and Myers-Swiatecki data respectively. The presence of low energy neutrons in the spectrum is probably due to s-wave neutrons transitions to excited levels in i33Sb. Contamination by t3%b could also cause s-wave or p-wave neutrons, depending on the spins of the 13%b and 133Te ground states. Average level spacings of about 30 keV are predicted for this case, and P, values of about 0.5 7: and 3.5 y0 for the Garvey and Myers-Swiatecki data respectively. 6. Conclusions Discrete, weft-defined peaks are clearly seen in all the neutron spectra measured in the present work. For two cases, the precursors 87Br and 1371, the experimental level spacing could be used to deduce values for the level density parameter, making possible the calculation of the spectrum shape and P, value. For both examples, an acceptable spectrum shape implied Qp values similar to those predicted by the Wapstra and Gove and the Garvey c?tal, mass tables, while reasonable P, values required values of QP considerably higher, and close to the values obtained by direct experimental measurement. This would perhaps suggest the tentative conclusion that not only are the mass tables in error, but that envelope calculations should be modified, perhaps by reducing the energy dependence of the beta strength function. One would expect the precursor 85As to be analogous to 87Br, and I3 ‘Sb to r3’I. In fact, for these cases acceptable envelopes and P, values can only be obtained with Qp values considerably lower than those predicted by the mass formulae. This appears to be the case also for the precursors 134Sn and 13GTe, although no experimental values of P, are available for comparison. The simple approach used in preparing a normalized set of values for the level density parameter is apparently quite successful. There is good agreement between calculated and observed peak spacings for all the cases studied.

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ENERGY

SPECTRA (I)

171

Effective cut-off energies for gamma competition are found to be approximately 1000 keV for g-wave neutron transitions (from 13%b), 350 keV for d-wave (from I3 ‘I), 400 keV for d-wave from doubly odd emitters (13%n and ’ 36Te), and 400-450 for f-wave transitions (““As and s7Br). However, it will be interesting to see whether these tentative values are applicable to neutron emission from a wider range of nuelides, as further experimental data becomes available. A close examination of table 2 shows that the Myers and Swiatecki tables generally predict higher Q, and lower 3, values than the other data sets. Consequently the neutron %indow” is usually higher than the m~imum neutron energy, and the P, value is unduly large. There is little difference in the data from Garvey et al. and Zeldes et al., except for perhaps slightly lower values of B, in the latter set. Seeger predicts exceptionally low values of Q, for *‘As and s 7Br. Unfo~unately, no single data set can be shown to give satisfactory results for the neutron envelope, level density and P, value for all the isotopes under consideration. It is not difficult to suggest points in our treatment which could lead to inaccurate results. The simplified treatment of the P-strength function, the assumption of equal positive- and negative-parity states and the use of an energy-independent level density parameter 19) are among the subjects requiring more detailed treatment. There are, however, even more basic assumptions that are based more on naivety than on physical reasoning. Should on.e treat level densities by a simple statistical model when one is dealing with only tens of levels per MeV or less? Is the level density a smooth function or should one take into account predicted ‘O) oscillations? Fluctuations in the ~-strength may be of importance, causing fme structure even for level spacings much smaller than the detector resolution 51). Indeed, it is surprising that one can obtain even the modest degree of agreement that we have demonstrated, in view of the simple approach employed. The invaluable assistance of Mr. 0. C. Jonsson in carrying out the experimental work is gratefully acknowledged. Our thanks are also due to Mr. L. Jacobsson for running the isotope separator, to Mr. E. Neeman for performing the P, calculations and to Mr. M. Schary for preparing the figures. The work was supported by the Swedish Atomic Research Council.

References 1) 2) 3) 4) 5) 6)

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S. SHALEV AND G. RUDSTAM

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