Monte Carlo simulation of the decay of neutron resonances to determine resonance spins

2.C: 2.D]

Nuclear Physics A209 (1973) 252--270; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

MONTE CARLO SIMULATION OF THE DECAY OF NEUTRON RESONANCES TO DETERMINE RESONANCE SPINS ARTHUR I. NAMENSON Naval Research Laboratory, Washington, DC Received 20 November 1972 (Revised 26 March 1973) Abstract: The results for two types of experiments which measure the spins of neutron resonances is

predicted by means of a Monte Carlo calculation which simulates the 7-cascades following n e u t r o n capture. The results are compared with experiment wherever possible. For the method which compares the ratios ofT-lines depopulating low-lying levels, the calculation is performed for 143Nd, 14SNd, la7Os, and ~89Os. For the method which compares coincidence to singles counting rates, the calculation is performed for the four nuclei mentioned above plus ~85Re, 87Re, and 177Hf. The validity of the approximations and assumptions used in the calculation are discussed. Also discussed are proposed improvements in the experiments made on the basis of both experimental data and the calculation. All necessary derivations are given in the appendix. 1. I n t r o d u c t i o n

A knowledge of the spins of n e u t r o n resonances is essential both to multi-level cross-section fits a n d to the u n d e r s t a n d i n g of certain nuclear properties such as the dependence of nuclear level densities, radiative widths a n d s-wave strength functions on spin. U n t i l the recent development o f several indirect methods which make use of the statistical properties of the nucleus to deduce resonance spins, the experimental d e t e r m i n a t i o n of these spins was extremely difficult. I n this paper the usefulness of two o f these indirect methods will be examined by use o f a M o n t e Carlo simulation o f the radiative decay of n e u t r o n resonances. The first m e t h o d determines the multiplicity, or average n u m b e r of 7-rays emitted when the nucleus decays from the resonance to the low-lying levels. The theory behind this is that of the two spins which can occur in s-wave capture, the resonances of the higher spin will have a higher multiplicity since it is generally more difficult for them to reach the low-energy states which are usually of low spin. It follows from this that the average y-energy for resonances of higher spin will be smaller since the total binding must be shared by more transitions. The first work using this technique was performed by Coceva et al. t) who d e m o n s t r a t e d it to be valid for a n u m b e r of nuclei. By using two N a I detectors, they determined the spin of each resonance in a give n isotope by c o m p a r i n g the n u m b e r of coincidence counts to the n u m b e r of single 252

NEUTRON RESONANCE DECAY

253

counts. The resonances of higher multiplicity should produce more coincidences. To make the ratio of these two counting rates even more spin dependent, they took advantage of the different spectrum shapes for resonances of different spins. The threshold for the multiples counts was about 0.3 MeV and the threshold for the singles was taken much higher, at about 2.5 MeV. The second method of spin determination deals with the fact that the relative populations of low-lying levels will depend on the spin of the resonance. Early work on this technique was performed by Poenitz and Tatarczuk 2) and by Wetzel and Thomas [ref. 3)]. A still earlier discussion of the subject was given by Huizenga and Vandenbosch 4). All these papers give a good discussion of the basic idea behind the method. The Monte Carlo method of Giacobbe et al. 5) provided important insights into the ?-decay of neutron resonances, especially with respect to the coincidence method of Coceva et al. 1). However, the experimental separations of the two spin groups [parameter d in ref. 1)] were almost always larger than the calculated results. It was desirable for several reasons to perform an extension of the calculation of Giacobbe et al. First, we wished to see how the calculation would apply to our modification of the experiment of Coceva et al. 1). A complete description of our equipment and method is given elsewhere 6, 7). The basic differences are our use of four NaI detectors rather than two, our use of triple coincidences as well as doubles and our use of a window condition on the multiples rather than just a base line. (Using a window takes still further advantage of the dependence of the shape of the T-ray spectra on the spin of the resonance.) A second motivation was to obtain an estimate of the effect expected when one looks at the relative populations of low-lying levels. Once the T-cascade~ have been generated in order to solve the first problem, it is very little extra effort to also examine how the low-lying levels should be populated. Still a third motivation was to investigate the effect for more nuclei - in particular the isotopes which we studied with our own apparatus. Finally, it was desirable to investigate some of the possible causes for systematic errors in the calculation - especially those introduced by questionable assumptions about the level densities in the nucleus, about the detectors and about the average dependence of the intensity of T-transitions on their energy. 2. Method of calculation

2.1. sI MULATING ?,-CASCADES To simulate the T-cascades from the capture states, the level density model of Gilbert and Cameron 8) was used. The spin cut-off factors in that article were used as approximations to the actual parameters. The spin cut-off factors in their article were given at a certain energy E x. Below this energy the factors were taken to be constant while above this energy they were considered to vary as U + where U is the excitation energy corrected for pairing effects. Where they were not listed, the factors for neighboring nuclei were taken. There is some evidence that in practice, spin cut-off factors are smaller than the ones estimated 9) but fortunately, the calculation is quite insen-

254

A.I. NAMENSON

sitive to this parameter. Weisskopf estimates were used for the y-intensities and only El, M 1 and E2 transitions were considered. Since dropping the E2 transitions would have only a small effect, we considered higher-order multipolarities entirely negligible. At each step in the cascade, the probability distribution for y-rays to unknown levels in the "continuum" above the highest known level and to known levels was calculated and a random number, generated by the power residue method, was used to determine the next step. Once the cascade reached a known level, the low-lying decay scheme was used to complete the cascade to the ground state. Isomeric levels were treated as final states. 2.2. CALCULATION OF OBSERVABLE RESULTS Calculating experimental effects for the relative populations of low-lying levels was simple and straightforward. An average y-spectrum was generated for the decay of resonances of each spin. Lines depopulating the low-lying levels of interest were included. F o r each spin, the ratio of two such lines observed in the actual experiments was calculated. Then the ratio for the higher spin was divided by that for the lower spin to give the reported results. Predicting experimental results for the coincidence experiment was much more complicated. At each cascade, the number of single, double, triple and quadruple coincidences had to be calculated. This was done analytically to save computer time. Treating four detectors and higher-order coincidences greatly increased the complexity o f the problem over the case of only two detectors. Calculating the singles was not as simple as one might imagine. In our equipment, the simultaneous occurrence of a coincidence event with a singles event would inhibit the registering of the singles event. Difficulties in calculating both singles and coincidences were due to an inexact knowledge of the efficiencies of our detectors and their response to y-rays of different energies. Related to this is the summation of two or more y-rays when they strike the same detector. In practice, for both the high and low thresholds, the y-detectors were assumed to have a constant efficiency of 6 % for all y-rays satisfying the threshold (or window) condition. The efficiency for lines above the higher threshold cancels out in computing the final effect and the important assumption here is its constancy. To predict experimental results one takes the total number of double (triple or quadruple) coincidences for an average resonance of a given spin and normalizes it to the total number of singles for that spin. The percentage by which the normalized ratio for the higher spin exceeds that for the lower spin is our effect. Mathematical derivations for the general case of any number of detectors and any order of coincidences are given in the appendix together with the actual equations used and the approximations made. The results of sample calculations using differing approximations will be presented in sect. 4. 2.3. ESTIMATING THE EFFECTS OF DIFFERENT EFFICIENCIES AND THRESHOLDS As is pointed out in the appendix, using a Poisson distribution for the distribution

N E U T R O N RESONANCE DECAY

255

of the number of transitions per cascade satisfying the threshold or window conditions leads to extremely simple analytic expressions and reasonably good approximations especially for double coincidences. These expressions were used for examining the v a r i a t i o n o f t h e effect w i t h d e t e c t o r efficiency a n d t h r e s h o l d s . O n l y d e t e c t o r efficiencies a n d a v e r a g e m u l t i p l i c i t i e s e n t e r i n t o t h e s e e q u a t i o n s . T o o b t a i n t h e m u l t i p l i c i t i e s for different thresholds, the integrals of the average y-spectra for the two different spins were used.

3. Results T a b l e 1 s h o w s t h e c a l c u l a t e d r e s u l t s f o r t h e c o i n c i d e n c e e x p e r i m e n t o n t a r g e t isot o p e s o f 1 4 3 N d , 1 4 5 N d , 1 8 5 R d , 187Re, 189Os, 187Os a n d 177Hr. T h e e x p e r i m e n t a l c o n d i t i o n s w e r e 0 . 5 5 0 M e V f o r t h e m u l t i p l e s t h r e s h o l d a n d 3.00 M e V f o r t h e singles t h r e s h o l d . W h e n e v e r t h e w i n d o w c o n d i t i o n w a s u s e d f o r m u l t i p l e s , t h e l o w e r level is 0 . 5 5 0 M e V a n d t h e u p p e r level 3.00 M e V . T h e ~ 7 7 H f n u c l e u s is s h o w n f o r c o m p a r i s o n w i t h t h e c a l c u l a t i o n a n d e x p e r i m e n t o f C o c e v a e t al. 1). T h e o t h e r n u c l e i a r e all n u c l e i TABLE l Determination of resonance spins by the coincidence method Target isotope

Possible res. spins

Window

143Nd

3-, 4 -

145Nd

3-,4-

laSRe 187Re 189Os

2 +,3 + 2+,3 + I-,2-

187Os

0 - , 1-

177Hf

3-,4-

no yes no yes no no no yes no yes no

Calculated effect (%)

Experimental effect ( ~ )

doubles

triples

quadruples

doubles

292:10 46£12 212:8 32± 7 -- 7 i 4 --10i14 7± 5 13± 7 0±10 -- 4 2 : 7 11± 4

43±15 702:20 342:9 492:10 -- 9-[_ 4 --11£16 11± 7 17ill 22:11 -- 3 ± 7 145_ 6

552:20 812:30 452:11 622:12 --102:6 --16+15 10-~ 6 187211 62:11 -- 1 2 : 7 16± 6

342:5 ") 4 9 ± 6 a) 16=}_4a) 34:t_7 2) 22:2 b)

triples

4 7 ± 9 ") 58-'--12 ")

2 2 i 7 c)

13 a)

Experimentally, for each resonance, one computes the ratio of the number of coincidences to singles. The experimental percent effect is the percent by which the average ratio for the resonances of the higher spin (J = I+½) exceeds the average ratio for the resonances of the lower spin (J = 1--½). The calculated percent effect is the predicted value of the experimental one. In all cases, the thresholds are 0.550 MeV for multiples and 3.00 MeV for singles. When the window condition is used for multiples, the window is 0.550 MeV to 3.00 MeV. a) Ref. 7), and A. I. Namenson, A. Stolovy, T. F. Godlove, and G. L. Smith, unpublished data. b) Ref. lo), and A. I. Namenson, A. Stolovy, T. F. Godlove, and G. L. Smith, unpublished data. c) A. I. Namenson, A. Stolovy, T. F. Godlove, and G. L. Smith, to be submitted for publication. a) Ref. 1). Note that experimental conditions such as thresholds and detector efficiencies were not the same as those used in our calculation. Also note that the experimental effect, " d " of Coceva et al. was defined slightly differently from our definition. If d' is the effect using our definition, the relation between d and d" is d" = d/(l --½d). If d << 1 there is little difference between the two definitions. The experimental value presented here was converted to our definition using the formula just given.

256

A.I. NAMENSON

for which we have done m e a s u r e m e n t s using either the coincidences method, the lowenergy t r a n s i t i o n method, or both. Experimental results are shown wherever they exist. Table 2 is a similar table for the low-energy t r a n s i t i o n m e t h o d for the N d a n d Os isotopes. The m e t h o d o f d e t e r m i n i n g the experimental n u m b e r s a n d their errors is o u t l i n e d in ref. v). TABLE2 Ratios of intensities of lines depopulating low-lying levels Target isotope

Possible res. spins

14aNd

3-, 4-

145Nd

3-, 4-

187Os

0-, 1-

Relevant decay scheme

I -, 2-

Calculated ratio

Experimental ratio

4+ 2+ 0+

1314 696 0

618/696

1.574-0.08

1.46±0.06 ~)

I

4+ 2+ 0+

-r ~ i'

1041 452 0

589/452

1.474-0.10

1 . 3 8 4 - 0 . 0 4 a)

633 479 155 0

324/155

1.98±0.30

1.654-0.23b)

633/155

0.85i0.07

0.97!0.02 b)

548 187 0

360/187

1.75 4-0.13

1.80 !0.14 b)

2+

4+ 2+ 0+ 1a9Os

Lines used in ratio (keV)

4+ 2+ 0+

~

Experimentally, for each resonance, one determines the ratio of the intensities of the indicated lines. The average value of the ratios for the resonances of the higher spin (J = I-t-½) when divided by the average value for the resonances of the lower spin (J = I--½) yields still another ratio which is reported as the experimental ratio. The calculated ratios are the predicted values of the experimental ones. a) Ref. 6). b) A. I. Namenson, A. Stolovy, T. F. Godlove and G. L. Smith, preliminary data. W i t h i n the quoted errors, the experiment a n d theory agree fairly well. N o systematic differences are a p p a r e n t as they were in the work of Coceva et aL 1), b u t such discrepancies might appear if there were better statistics o n b o t h our M o n t e C a r l o calc u l a t i o n a n d experiments. Some interesting features appear in table 1. First, one sees that the size o f the effect increases when one looks at the triple a n d q u a d r u p l e coincidences. Experimentally, the quadruples were so few i n n u m b e r that their m e a s u r e m e n t gave little i n f o r m a t i o n . The triple coincidences did give a decided effect when the w i n d o w c o n d i t i o n was applied to the N d isotopes, b u t i n the other cases fluctuations due to experimental u n certainties a n d P o r t e r - T h o m a s statistics made it impossible to distinguish the two spin groups. Nevertheless, with better detector etticiencies, better resolving times a n d m o r e intense beams, the triples might yield some very good results. A more interest-

NEUTRON RESONANCE DECAY

257

ing feature is the improvement in the effect for both calculation and experiment when the window condition is imposed on the multiples events. In the calculated results, this improvement would not be significant for any one case, but with all the cases taken together, the window definitely seems to improve the situation. The experimental results on the N d isotopes are more definitive on this point. It should be noted, however, that experimentally some of this improvement is lost due to the widening of the Porter-Thomas fluctuations. In both the calculation and experiment for lSSRe we see that some nuclei give almost zero effect and cannot be studied by this method. The reason for its failure is the large number of closely spaced low-lying levels in is 5Re having a number of different spins. In such a case, the resonances of spin 3 + can decay with about the same ease as the resonance of spin 2 +. Other nuclei having small effects are lSTRe and lS7Os. We do not yet have conclusive experimental results on lSVOs, while for lSTRe the possible existence of intermediate structure ~o) makes the measurement impossible. The calculated results for as 9Os are not really statistically significant but preliminary experimental data taken in our laboratory shows that when one imposes the window condition, a noticeable effect appears. The calculations for i77Hf agree with those of Coceva et aL 1) in spite of the different efficiencies and thresholds used, indicating the insensitivity of the calculation to variations in these parameters. The results of table 2 are refinements over those obtained by the simpler methods of Poenitz and Tatarczuk z) and Wetzel and Thomas 3). For example, in lS9Os the predicted effect from the simpler methods for a four step cascade would be 1.97 while the effect predicted here is 1.75__0.13, in better agreement with both our experimental results and those of Wetzel and Thomas 3). A possible exception might be the case of the value predicted for the ratio of the intensity of the 633 keV line to that of the 155 keV line in ~S7Os. The calculated value was 0.85___0.07 while the experimental value was 0.97__+0.02 and the value from the simple calculation would be 1.00. It should be pointed out however, that the measurement was extremely difficult and the experimental results are by no means final.

4. Sources o f error

In order to assess the reliability of a calculation such as this it is necessary to discuss its sensitivity to the assumptions and approximations used in arriving at the final answers. The calculation was therefore performed with other assumptions to see if the results were significantly changed. Since such calculations were often more time consuming and expensive than the original ones, test cases were made on two nuclei - 143Nd and i45Nd. These nuclei were chosen because they were well investigated experimentally and gave large effects for both the coincidence experiments and for the ratios of transitions from low-lying levels. In what follows, the major sources of possible error are discussed one by one.

258

A.I. NAMENSON

4.1. ERRORS IN THE COINCIDENCE EXPERIMENT DUE TO INACCURACIES IN THE DESCRIPTION OF THRESHOLD SETTINGS AND IN THE DETECTOR EFFICIENCIES To estimate these errors, eq. (9) in the appendix was used to approximate variations with respect to efficiencies and thresholds. The thresholds influenced the calculation through the multiplicities. Since the multiplicities as a function of threshold were printed out at the end of each Monte Carlo calculation no additional Monte Carlo runs were necessary. Calculations were made on 145Nd" To conform to experimental conditions, whenever a window condition was used for coincidence events, the threshold for singles events was taken to be the upper level of the window. Small variations of about 30 keV about the 0.550 MeV lower threshold had little effect on the calculation. Likewise variations of about 200 keV about the singles threshold had only a slight effect on the calculation. (In general, for the simple base line condition, the effect went down with the lowering of the singles threshold while for the window condition the effect went up.) Since only an unreasonably inaccurate description of the thresholds could affect the calculation, this is not a likely source of error. It is interesting to note in passing that according to calculation, a significant improvement in the experiment could have been made if we were not restricted to having the upper level of the window equal to the singles threshold. Varying the efficiency of the detectors from 0 to 10 % had practically no effect on the calculation for the case where a window condition was imposed on the multiples events and only a slight effect for the simple base line condition. Using the base line condition for the case of 145Nd, the effect increased slightly with increasing efficiency. In the limit of zero efficiency about 2 % would have to be subtracted from the reported results while for l0 % efficiency 2 % would have to be added to the results. This amount is small compared to the quoted errors. We can conclude that even if the assumed efficiencies were considerably off, the calculation would be unaffected. 4.2. ERRORS IN THE COINCIDENCE EXPERIMENT DUE TO THE SIMPLIFIED REPRESENTATION OF THE RESPONSE OF THE NaI DETECTORS As mentioned in the appendix, the detectors were assumed to have a constant efficiency for all y-rays satisfying the threshold or window conditions and zero efficiency for all others. To check the errors which this assumption might introduce, separate calculations on l ' a N d and 145Nd were run using the efficiencies and peak-to-total ratios of Young et aL ~). The portion of the spectrum outside the peak was very roughly approximated by a constant, adjusted on size so that the integrated spectrum equalled the efficiency of the detector for that energy. Such an approximation was adequate for showing how the results might be altered with a more accurate approxi~ mation. The size of the effects for the window condition was reduced to the point where it was not significantly larger than for the base line condition. This could be due to statistical fluctuations, but there is a reason why such a result would be expected. Accounting for the fact that the y-rays, on the average, deposit less than their full energy in the detectors has an effect which is similar to raising the level of the

NEUTRON RESONANCE DECAY

259

thresholds - especially the singles threshold. This widening of the window would reduce the computed result for that condition. In practice accounting more accurately for the response of the NaI detectors actually worsened the agreement with experiment. If, however, we also accounted for the fact that two y-rays can sometimes strike the same detector and sum to a larger energy, there would be a compensating effect of effectively lowering the thresholds. As is pointed out in the appendix, it is estimated that for our detectors, the two effects approximately cancel. It should be noted that such may not necessarily be the case for other systems using smaller N a I detectors and approximations about the response of the N a I detectors must be made with caution. 4.3. T H E E X I S T E N C E O F U N D I S C O V E R E D LEVELS I N T H E L O W - L Y I N G D E C A Y SCHEME

Although this was considered the most unlikely of all sources of error a test on 143Nd was re-run with the 2 + and 3- levels at 1.51 and 1.56 MeV in 144Nd deliberately omitted. These levels were chosen because they were omitted in earlier compilations of the Nuclear Data Sheets 11). The results of this calculation showed a change in the answer which was small compared to the quoted uncertainty. 4.4. I N A C C U R A C I E S I N T H E L E V E L D E N S I T Y F O R M U L A S O F G I L B E R T A N D CAMERON

The method of Gilbert and Cameron s) was to fit a specified form to the level densities at the neutron binding energy and at the low-lying levels. For nuclei such as the Re nuclei we would expect the Gilbert and Cameron 8) formalism to give a good fit to the level densities 13) but for nuclei near doubly closed shells the method of Gilbert and Cameron is not as reliable 13). While no nuclei of the latter type are included in our calculation, it is still important to investigate how changes in the level density scheme would affect the calculation. The influence of an erroneous level density formula was checked several ways. First variations of the order of 25 ~o were introduced into the parameters such as a and ~r in the formalism of Gilbert and Cameron 8). These variations produced negligible changes in the results of calculations on 143Nd and 145Nd. As a check on how changing the form of the level density formulas might affect the calculation, a constant temperature model was applied to the calculation on 14 SNd" The level densities at the binding energy and at 2.0 MeV were fixed at the values derived from the Gilbert and Cameron formulation. As can be seen in the eighth row in table 3, this also produced no significant variations. Only drastic changes in the level density formulas produced changes in the computed results. For example, using a constant temperature formula and reducing the level density at the binding energy by a factor of ten reduced the effect to essentially nothing. The same was true when the correct level density at the binding energy was used but the level density at 2.0 MeV was reduced by a factor of eight. The predicted result for the ratio of the 589 to 452 keV lines was

1.774-0.10

324- 8

344- 7

94-14

--104- 9

25-t-13

434-10

554-10

354- 9

404-11

284- 7

344-13

344- 9

triples

124-21

--15::l:: 8

244-16

344-13

45-4-11

quadruples

1.485_0.13

1.41-t-0.13

1.684-0.15

1.664-0.06

1.774-0.06

1.62:k0.06

2.13-t-0.06

1.664-0.06

1.724-0.15

1.47::k0.10

Ratio of 589 to 452 keV lines

For both isotopes the calculation was done for the simple base line condition applied to the multiples. The possible resonance spins in both nuclei are 3- and 4 - . The gaps in the table represent situations where the indicated calculation was not performed for economy of computer time. The first row of this table is a repetition of values shown in table 1 for comparison purposes.

84-10

574- 7

1.69 4-0.06

534- 7

244- 4

Const. temp. model with level density at ~ 2 MeV reduced by a factor of 8

44± 7

36± 7

2.154-0.07

1.654-0.06

294-10

--54-10

234- 5

Same as above with pygmy El resonance at 5.5 MeV and 2 ~ of the intensity of giant dipole

31 4- 7

674-10

1.764-0.15

Same as above but with level density at binding energy reduced by a factor of 10

234- 5

E1 transitions only. Extrapolation of giant dipole resonance. E a law below 3 MeV

634- 7

31 4-10

214- 8

22±10

345_ 5

E1 transitions only, with E 5 law

334- 7

714-17

1.574-0.08

doubles

304- 7

254- 5

El transitions only, with E a law

634-16

554-20

quadruples

145Nd Coincidence experiment effect (in ~ )

Const. temp. model for level density. Weisskopf estimates for ),-rays

46±11

E2 transitions 5 times Weisskopf estimate. El and MI unchanged

43-/-15

triples

Ratio of 618 to 696 keV lines

Extreme single-particle assumption

29±10

Weisskopf estimates using El, M 1 and E2 transitions

doubles

Coincidence experiment effect (in ~ )

143Nd

TABLE 3 Influence of the level density formula and the energy dependence of),-transitions on the computer results

©

>

.>

8

1"4

NEUTRON RESONANCE DECAY

261

hardly affected, however. The results of these last two calculations are shown in the last two rows of table 3. In the region between the low-lying levels and the neutron binding energy, the level densities predicted by the Gilbert and Cameron method would have to be off by an order of magnitude before they could seriously change the computed answers. Since the formulas fit at the low- and high-energy ends of this region, order of magnitude errors in the level density would involve some highly singular behaviour. Since our calculation is involved with spins, a more serious objection might be errors in the dependence of the level density on spins. It has been pointed out by Gilat [ref. 14)] that at very high spins, the Gilbert and Cameron a) formulas do not work. However, in this calculation such large spins are extremely unlikely to arise and this possibility was not considered a source of error.

4.5. VARIATIONS FROM THE WEISSKOPF ESTIMATES IN COMPUTING THE INTENSITIES OF ),'-RAYS There is a certain amount of experimental evidence tbr doubting the accuracy of the Weisskopf estimates. For example, it has been suggested by Axel ~5) that for the E1 transitions, instead of using the E 3 rule, one should use an extrapolation of the giant dipole resonance, and around the binding energies of nuclei, an E s rule might be more appropriate. In addition, Earle et al. 16) have noticed that some nuclei seem to exhibit an anomalous bump in their capture ?,-ray spectra at about 5 MeV. Some variations from Weisskopf estimates, assuming only El transitions, were run on the calculations for 143Nd and 145Nd" The variations included an E 3 law, an E 5 law, an extrapolation of the giant dipole resonance and an extrapolation of the giant dipole resonance with a pygmy resonance superimposed to simulate an anomalous bump. For the last two calculations we adopted a procedure of Bergqvist and Starfelt 17) and Starfelt 18). Above 3 MeV the 7-ray strength function was derived from the giant dipole resonance or the giant dipole resonance plus the pygmy resonance. Below 3 MeV the strength function was assumed to be proportional to a constant times E 3, the constant being determined so that there were no discontinuities. The giant dipole resonance was taken to have an energy of 13 MeV and a width of 3.5 MeV while the pygmy resonance was taken to have an energy of 5.5 MeV and a width of 1.5 MeV. The relative size of the pygmy resonance was chosen so that it had an integrated intensity of 2 ~o of the giant dipole resonance. An additional computation was performed using Weisskopf estimates but with the E2 transitions exaggerated by a factor of five. In all these calculations, the base line condition for the coincidence experiment was assumed. The results are presented in the first six rows of table 3. For ~43Nd the difference is not statistically significant while for the coincidence experiment on 145Nd, only the use of an E s law for E1 transitions made a significant difference. However, considering an E 5 rule to hold down to energies below 2 or 3 MeV would be unreasonable. It seems safe to say that no reasonable variation in the

262

A.I. NAMENSON

behavior of the ?-ray intensities as a function of energy would affect the calculation for the coincidence measurement. Calculations for the method of examining the ratios of ?-lines depopulating lowlying levels are more sensitive to the law used for ?-ray intensities. Introducing an E 5 rule produces an enormous difference in the computed result for both isotopes. The more reasonable extrapolation of the giant dipole resonance does not produce results significantly different from the E 3 law. In 1,SNd ' the addition of a pygmy resonance to the extrapolation of the giant dipole resonance does seem to produce a significant change. The results on X*3Nd, while not significant in themselves, support this conclusion, and when the two differences are combined, the probability of a real difference becomes 95 ~ . Intuition would lead one to conclude that such a difference must exist in the presence of a pygmy resonance since such a resonance would enhance the higherenergy transitions and reduce the multiplicity of all resonances. It would follow that there would be less of a tendency to "forget" the spin of the capture state and the populations of the low-lying levels would be more influenced by it. 4.6. A C C O U N T I N G F O R T H E I N F L U E N C E OF T H E SPINS OF T H E I N I T I A L A N D F I N A L STATES ON T H E INTENSITIES OF i~-TRANSITIONS

It was assumed in the calculation that the intensities of ?-transitions depended only on the multiplicity and the parity of the transitions and not on the spins of the initial and final levels connected by the transition. The resonance averaging experiments of Bollinger and Thomas 19) provide experimental evidence that this is a good assumption for the highly excited states. Nevertheless, at the lower end of the "continuum" such an assumption might not hold up as well. For example, in the limit of a singleparticle model, the spins of the states connected by a ?-transition would play an important role in its intensity. As a test case, the calculation on 145Nd was re-run adopting an extreme single-particle model. The spin of the even-A daughter nucleus ~46Nd was considered to be composed entirely of the coupling of two odd nucleons - one of which was the captured neutron. The ?-transitions were considered to be singleparticle transitions involving one of these particles. Because these calculations involved a lot of computer time per cascade, only E1 transitions were considered. Two different assumptions were made about the odd particles. One was that the particles consisted of two neutrons and the other that they consisted of a neutron and a proton. For each of these cases the calculation for the coincidence experiment was performed for both the window and the simple base line conditions. The results of the calculation where the base line condition was imposed and where the odd particles were considered to be a neutron and a proton are shown in the seventh row of table 3 (labeled extreme single-particle assumption). This row may be compared with the third row showing the results for the calculation which makes the usual assumptions but is also limited to only E1 transitions. We see that for the ratio of the 589 to 452 keV lines, the answers are unaffected. For the double coincidences the answers are likewise within statistics. For the triple coincidences the two results differ by just

NEUTRON RESONANCE DECAY

263

slightly over a standard deviation. If one compares the triple coincidence rate of the third row of table 3 with the value of the first row which should be basically the same calculation, one would conclude that the low value in the third row is a statistical fluctuation. The other calculations with the single-particle model, all showed still less deviation from the calculation which ignored any possible spin dependence on the 7-ray transitions other than selection rules. Since even an extreme single-particle approach produces so little change in our results we can conclude that such spin dependences are not a source of error.

5. Conclusions Both the calculation and experiment show that the statistical methods of examining the spins of resonances can produce sizeable effects. The method of examining lowenergy transitions gave definitive results in all the calculations performed here. This is also borne out by experiment. However, the coincidence method is a simple way of providing useful corroborating information. There are some isotopes which are evidently poor candidates for coincidence-type experiments since the expected effects are quite small. While extreme variations in the density formula or in the dependence of average 7-ray intensities on energy do affect the results for the coincidence experiments, reasonable variations do not. No doubt a more accurate simulation of the detectors would improve the calculation, but it would very likely make the co~t of a calculation comparable to that of the experiment itself. Once again the calculation is not very sensitive to reasonable variations from the detection system considered here, but for any drastic change, the simplifications would have to be reconsidered. For the calculations on the ratios of lines depopulating low-lying levels, even very large changes in the assumptions about the level densities do not change the compated results significantly. However, the calculation does seem to be somewhat sensitive to the assumptions about the average intensities of 7-rays as a function of energy. It would seem that the introduction of a bump at about 5.5 MeV in this function in the form of a pygmy resonance does affect the results. This last source of difficulty is also a source of interest since experiments of this type could conceivably detect such variations which persist over a large number of resonances. However, to make this practical, the calculation would have to be shown to be reliable for a larger number of nuclei than illustrated here and the statistical errors on both the experimental and calculated results would have to be improved.

Appendix A.I. GENERAL PROCEDURE AND NOTATION The situation where one has many detectors is much more complicated than the situation where one has only two detectors. This is especially true when one wishes to

264

A.I. NAMENSON

account for multiple coincidences of higher order than the simple double coincidences. The methods and results of a formalism for handling this situation will be presented here but the derivations will be omitted. First to introduce the notation to be used, suppose that x is a statement. We let P[x] represent the probability that x is true. If x and y are two different statements then P [ x y ] is the probability that the logical A N D of the two statements is true and P [ x + y ] is the probability that the logical OR of the two statements is true. Further, suppose that some condition x can arise in N~ mutually exclusive ways and each way has the same probability, P [ x ]. The statement describing that the condition x has occurred is N x x which obviously has a probability of P [ N x x ) = NxP[x]. If y is a condition which is mutually exclusive to condition r it is evident that P [ N x x + Ny y] = N,~ P[x] + Ny P[y]. Now consider that we have N detectors some of which are identical to others. Suppose we have N 1 detectors of type l, N2 of type 2 and so forth such that P

~ , N i = N, i=1

where p is the number of different types of detectors. Let a~vit stand for the statement that n t detectors of type i failed to respond and correspondingly let ~ t stand for the statement that n i detectors of type i did respond. (It will become apparent later why the symbol a~ is used to express the fact that a detector failed to respond rather than the fact that it did respond.) The probability of an M-tuple coincidence between a particular set of detectors composed of n 1 detectors of type 1, n 2 detectors of type 2 and so forth is P = P [ a lN,-n, aaN~-,= . . . apN~-,~,,~,~ -1 -2

.. ap ], --np

"

(i)

where ~f= ln~ = M. To evaluate this we replace each symbol at with the expression ( 1 - a t ) to give P = P[a~'-"'...

a~P-"~(l--a,)"'...

(l-ap)"P].

(2)

The expression within the brackets in eq. (2) is now expanded as if it were an ordinary algebraic expression to yield a sum of terms each having the form C ( m 1, m2, • • • mp ) a'~'ar~2.., a'~p", where C ( m l . . . mp) is simply a numerical coefficient. This sum represents the logical OR of a set of mutually exclusive conditions and so its probability can be found by simply "multiplying" through by the factor P - that is the probability reduces to a sum of probabilities each having the form C ( m 1 . . . mp)P[a~ 1 . . . a~p"]. If we are interested in all possible M-tuple coincidences which can occur, we must enumerate them by enumerating all possible ways in which the detectors can respond or fail to respond: POSSIBILITIES = (a 1 -]- ff,)N'(a 2 + ~2)N2... (a v + ~v) Np.

(3)

NEUTRON RESONANCE DECAY

265

The right h a n d side o f e q . (3) is once again expanded as if it were an algebraic expression to again yield a polynomial in terms o f the symbols a and a. If we are interested in M-tuple coincidences, we pick out all the terms o f order M in the symbols 6. The summation o f these terms represents all the ways in which an M-tuple coincidence can occur. As before, each symbol 6i in this polynomial is replaced by the expression ( 1 - a l ) and then the whole expression replaces the one in brackets in eq. (2). The evaluation o f the probability proceeds as before according to the rules o f ordinary algebra.

Fig. 1. Schematic for sample calculation of situation where two detectors are identical and one is different. As an example, consider the simple case illustrated in fig. 1 where we have two identical detectors (labeled a) and one different one (labeled b). We wish to express the probability o f a double coincidence: ( a + 6)2(b + b) = aZb + 2ab~ + a2~ + b ~ 2 + ~2~ + 2a6b. The terms o f the second order in the barred symbols are 2a~b + b ~ 2 ~ 2 a ( 1 - a ) ( 1 - b ) + b ( 1 - a )

2,

a n d finally the probability o f doubles is P = P[Za(1-a)(1-b)+b(1-a)

2]

= P[3a2b-2a2-4ab+2a+b] = 3P[a2b] - 2P[a 2 ] - 4P[ab] + 2P[a] +P[b], where p[a2b] is the probability that all the detectors failed to respond, P[ab] is the probability that detector b and a particular detector o f type a failed to respond, etc. The full expansion o f eq. (3) is convenient when one is interested in all orders o f coincidence. I f one is interested in only a particular order (M-tuples) one has from a simple consideration o f the binomial theorem P

M-tuple P O S S I B I L I T I E S = Z all

m

1--[

N~!

i= 1

m i ! ( N i - lrli)!

-,nN'-"'n" ,

(3a)

266

A.I. NAMENSON

where ~ ' = ~ m i = M. It is sometimes more convenient to use the expression * (3b)

A.2. SIMPLIFYING ASSUMPTIONS The first simplifying a p p r o x i m a t i o n is to assume that the probability of a detector n o t responding to a series of v-rays is the p r o d u c t of the probabilities of it n o t responding to each one separately. As an example of a situation where this a s s u m p t i o n is not true, suppose we had a cascade of two 7-rays which were both below a threshold. T h e p r o b a b i l i t y of n o t responding to each v-alone is unity a n d so our a s s u m p t i o n w o u l d lead us to believe that there was a 100 ~o chance of the detector not responding. However, if both v-rays struck the same detector and their energies s u m m e d to m o r e t h a n the threshold, there would be a considerable chance that the detector would respond. Similar considerations lead to the conclusion that even if one or both v-rays were above the threshold, the probability of the detector not responding would be slightly less than the p r o d u c t of the probabilities of it n o t responding to each one separately. F o r the lower energy threshold used for multiples, calculations showed that a c c o u n t i n g for such s u m m a t i o n s would have a negligible effect o n the answers. A somewhat more serious objection to neglecting the s u m m a t i o n of v-lines is with respect to the singles. With our typical singles threshold of 3.0 MeV, almost every cascade had at least two v-lines that s u m m e d to more than 3.0 MeV but which were n o t themselves more t h a n 3.0 MeV. I n addition, the presence of v-lines below the threshold, noticeably increases the probability that v-lines above the threshold will cause a response. A detailed study of some r a n d o m l y selected cascades, indicated, however, that one could compensate for these effe:ts by assuming a c o n s t a n t p r o b a bility of detection for all v-rays over the singles threshold a n d zero probability for those u n d e r it. The remaining error is a few percent in the calculation of singles. However, since the same error is introduced into the singles rate for both spins, o u r final answers are negligibly affected. The ability to neglect the s u m m a t i o n of y-rays in a detector e n o r m o u s l y simplifies A more intuitive but not more convenient expression is the polynomial obtained by eliminating the operators at + from the expression

__1 ( a l a l + a 2 a 2 + +--

+--

. . . a p a p+ -)-

",.M

a N~ I a N2 2 .

..

aNplo~,p~,,

M~

where the symbols are treated like operators obeying boson commutation rules. [a~+, a~] -- [dt +, d j] - - O i j and all other commutators are zero and where al+]0) = di+]0) = 0. In this situation at represents an operator which creates a detector that fails to respond while dt is an operator which creates an "anti-detector"-that is a detector which does respond, aLNlaz~V2.., a~Npro) represents a state where none of the detectors respond while (M!)-~(al +dl +a2+d2 • •. ap+ap)Mis an operator representing all the ways that we can change M detectors into anti-detectors without regard to order.

NEUTRON RESONANCE DECAY

267

the calculation and justifies the a p p r o a c h o f calculating the probability o f detectors not responding. Suppose we pick n detectors, a l . . . a,, and we have m 7-rays 71 - - • 7,,. If r/ij is the probability that the ith detector will respond to t h e j t h y-ray, the probability, P[al a2 • . . a,,], that none o f these detectors will respond is m

tl

P[ai a z . . . a.] = l-I ( 1 - E qiJ). j=l

(4)

i=1

The next simplifying assumption is to approximate the response o f the detectors as being a constant for all 7-rays over the threshold and zero for all those under the threshold. Such an approximation has already been mentioned with regard to the singles. Using the shapes, efficiencies, and peak-to-total ratios of 7-lines in the 12.7 cm diameter by 12.7 cm N a I crystals studied by Y o u n g et al. 11) this turned out to be a valid approximation for the multiples threshold as well. Now, if the detectors which do not respond consist o f n~ detectors o f type al, the corresponding probability is given by P

P[a"1'a~=... a~"] = (i-- Z niq,)',

(5)

i=I

where rh is the absolute efficiency o f detectors of type i and m is the number of y-rays over the threshold (or in the window if a window condition is imposed). Still a third approximation is useful even t h o u g h this one was not accurate enough to use in the final calculations. If we assume that the number of y-rays meeting our threshold or window conditions follows a Poisson distribution, having an average o f / ~ , eq. (5) reduces to the form, P

P[a]'a~ = . ' ' a; ~] = I~ e-m"'""

(6)

i=I

In this last expression, P is equal to the expression in brackets if we make the substitution al = e-m% In the case o f a Poisson distribution one only needs the expansion o f eq. (3) and the substitutions al = e - ' " ' and al = 1 - e -m"'. A.3. ACTUAL CASE Fig. 2 shows the schematic used for analyzing the singles and multiples events. We considered all four detectors to be intrinsically the same. However, one detector was singled out from the others and its output was routed t h r o u g h two different channels. In the "multiples" channel its pulses were treated just as they were for the other detectors but in the "singles" channel its pulses were accepted only if they exceeded the higher threshold. In fig. 2a, representing the singles channel, this detector is labeled b. In fig. 2b, it is not distinguished from the other detectors. In our set-up if another detector responded simultaneously with a count in the "singles" channel, the event was registered in the coincidence spectra but not in the singles. This was electronically simpler, and also enhanced the dependence o f the multiples to singles ratio

268

A.I. NAMENSON

on the multiplicity. However, the presence o f o t h e r detectors n o w affects the singles rate. The analysis for the simple threshold c o n d i t i o n on the multiples is s o m e w h a t different from that for the w i n d o w condition. In b o t h cases, however, we m u s t start with the expansion

f

(a + ~)3 (b + b) = b(a + ~)3 + ~a 3 _{_3~a~Z q_ 3 ~ a 2 + ~ 3 . A

SINGLES

MULTIPLES

Fig. 2. Schematic of actual set-up. Diagram A is how the experiment appears to the singles channel while diagram B is how the experiment appears to the multiples channel. F o r the simple threshold condition, detector b must r e s p o n d while all the o t h e r detectors m u s t fail to r e s p o n d in o r d e r to register a singles event. This is represented by the t e r m / ~ a 3 a n d following our formalism:

e = P[a a] - P[ba 3] = (1 - 3q) "m +,s _ (i - 3q - r/s)"'(1 - 3q)"m, where r/is the efficiency o f the detectors for multiples while qs is the efficiency o f detect o r b for singles. Efficiency n~ is the n u m b e r o f ?-rays in a p a r t i c u l a r cascade which exceed the u p p e r threshold, while n m is the n u m b e r between the u p p e r a n d lower thresholds. In the expression written above, eq. (5) was used to express the n u m e r i c a l value o f P[a 3] while eq. (4) was used for P[ba3]. In practice, the n u m b e r o f ?-rays exceeding the higher threshold c a n n o t be very large since the sum m a y n o t exceed the binding energy o f the nucleus. I n fact, in less t h a n 1 ~o o f all cascades was the n u m b e r o f ?-rays over 3.0 M e V greater than one. A p p r o x i m a t i n g that n, is either zero or one we e n d u p with P = q~ ns(l - 3q) "m, (7a) for the singles with the simple base line condition. F o r a singles to be c o u n t e d when the w i n d o w c o n d i t i o n is applied, d e t e c t o r b m u s t r e s p o n d while no m o r e than one o f the other detectors m a y respond. This c o r r e s p o n d s to the terms in our expansion ~ a 3 + 3 / ~ a 2. I f we m a k e the same a p p r o x i m a t i o n as before that rh is zero or one, we are finally led to the e q u a t i o n P = qs ns3(1 -- 2qm)"~ -- 2(1 --2qm) "m ,

(7b)

NEUTRON RESONANCE DECAY

269

for the singles with the window condition, where qm is the effÉciency of the detectors for the window condition and n m is the number of 7-rays in the window. For the coincidences, we have only to deal with the case of four identical detectors. The results are doubles:

P = 6[(1-2r/)"-Z(1-3q)"+(1-4r/)"];

(8a)

triples:

P = 4[(1-r/)n-3(1-Zq)"+3(1-3q)"-(1-4q)"];

(8b)

quadruples:

P =

t-4(1-q)"+6(1-2rl)"-4(l-3q)"+(1-4q)";

(8c)

where n is the number of y-rays in the cascade satisfying the threshold or window condition depending on which is imposed. Finally, if we approximate the distribution of the number of y-rays per cascade by a Poisson distribution, we are left with the approximate equations: singles, integral condition:

P = qs ns e - 3.n~ ;

singles, window condition:

P = r/S ns(3 e - 2.nm -- 2e- 3.nm) ;

doubles:

P = 6e-Z~"(1--e-,")2;

triples:

P = 4e-,"(1-e-,")3;

quadruples:

P = (l-e-")*.

(9)

3°°1 ,z

I 0

2 4 G B I0 12 TOTAL NUMBER OF 7 RAYS

O" 2 4 6 S lO IZ NUMBER OF 7 RAYS OVER BASE LLNE

n

0 2 4 6 B iO 12 NUMBEROF 7" RAYS IN WINDOW

Fig. 3. Frequency distributions for multiplicities per cascade as compared with a Poisson distribution having the same average. The solid lines represent the actual distribution obtained in a Monte Carlo calculation having 1000 trials while the dashed line represents a Poisson distribution. The curves are for average 3 - resonances in 145Nd.

In practice, the Poisson approximation gave fair results in computing the effects for double coincidences and somewhat worse results for the triples and quadruples. It is nevertheless extremely convenient in estimating such things as the variation of the effect with efficiency and thresholds. Fig. 3 shows graphs comparing the Poisson distribution with an actual distribution resulting from 1000 cascades. We see that the

270

A.I. NAMENSON

approximation

is r a t h e r p o o r i n d e s c r i b i n g t h e f r e q u e n c y d i s t r i b u t i o n f o r t h e t o t a l

m u l t i p l i c i t y p e r c a s c a d e . I t is s o m e w h a t b e t t e r f o r t h e d i s t r i b u t i o n o f t h e n u m b e r o f 7 - r a y s o v e r t h e l o w e r t h r e s h o l d , a n d is f a i r l y g o o d f o r t h e n u m b e r o f y - r a y s in t h e window.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

C. Coceva, F. Corvi, P. Giacobbe and G. Carraro, Nucl. Phys. A l l 7 (1968) 586 W. P. Poenitz and J. R. Tatarczuk, Nucl. Phys. A151 (1970) 569 K. J. Wetzel and G. E. Thomas, Phys. Rev. C1 (1970) 1501 J. R. Huizenga and R. Vandenbosch, Phys. Rev. 120 (1960) 1305 P. Giacobbe, M. Stefanon and G. Dellacasa, CNEN technical report RT/FI (68) 20 T. F. Godlove, G. L. Smith and A. I. Namenson, Nucl. Instr. 95 (1971) 595 A. Stolovy, A. I. Namenson, J. C. Ritter, T. F. Godlove and G. L. Smith, Phys. Rev. C5 (1972) 2030 A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 L. M. Bollinger, G. D. Loper and G. E. Thomas, Bull. Am. Phys. Soc. 17 (1972) 556 A. Stolovy, A. I. Namenson and T. F. Godlove, Phys. Rev. C4 (1971) 1466 F. C. Young, H. T. Heaton, G. W. Phillips, P. D. Forsyth and J. B. Marion, Nucl. Instr. 44 (1966) 109 Nucl. Data Sheets, N R C 59-1-106 S. M. Grimes et al., Phys. Rev. C6 (1972) 236 J. Gilat, Phys. Rev. C1 (1970) 1432 P. Axel, Phys. Rev. 126 (1962) 671 E. D. Earle, M. A. Lone, G. A. Bartholomew, B. J. Allen, G. G. Slaughter and J.A. Harvey, Statistical properties of nuclei, ed. J. B. Garg (Plenum Press, New York, 1972) p. 263 I. Bergqvist and N. StaffeR, Progress in nuclear physics, vol. 11, ed. D. M. Brink and J. H. Mulvey (Pergamon Press, Oxford, 1970) p. 1 N. Starfelt, Nucl. Phys. 53 (1964) 397 L. M. Bollinger and G. E. Thomas, Phys. Rev. C2 (1970) 1951