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Spins of low-energy neutron resonances in 175Lu, 189Os and 187Os
t
2-c
I
Nuclear Physics A237 (1975) 45--63; @ Nor~h-~o~~a~ F~~is~i~g Co., Amsterdam Not to be reproduced
by pbotoprint
SPINS OF LOW-EARLY
ot microfilm without written permission
from the publisher
NEUTRON ~SONAN~~S
IN 1’5Lu, ‘*‘OS AND ‘*‘OS A. I. NAMENSON, A. STOLOVY and G. L. SMITH Naval Research Laboratory, Washington, DC 20375 Received 21 August 1974 Abstract: Resonances up to 300 eV in 175Lu, r8pOs, and Is70s are reported together with spin assignments and probabilities of correctness. Those spins reported with better than 90 % confidence consist of 49, 17 and 10 resonances in 175Lu, ‘8pOs, and’ ‘*‘OS respectively. In Is70s, 39 of the observed resonances have not been reported previously. Ratios of low-energy y-ray intensities and y-ray multiplicity were used for spin determinations. A method developed previously for analyzing these data has been extended to include analysis of uncertainties in relative populations, average reduced neutron widths, and strength functions of the two spin groups. The data in ~ssOs is consistent with a 2J;I level-density dependence, but in the other two isotopes this rule is violated. We suggest spin cut-off factors of about 3 in this mass region. Small differences in <2gFao> and in the strength functions for the two spin groups are not considered very significant.
E 1
NUCLEAR REACTIONS 175Lu, 187, lEsOs(n, y), E = 2.6-300 eV; measured Er, L,. 176Lu, lsOOs resonances deduced .FFgFa, spin cut-off factors. 1s80s resonances deduce‘d .F. Enriched targets.
1. Introduction The experimental simplicity of recently developed indirect methods of determining the spins of neutron resonances by observing the prompt y-rays following neutron capture have lead to much new information. Of the number of such techniques discussed in an article by Bollinger ‘) we will be concerned with the multiplicity method of Coceva er al. “) and with the method of observing low-energy y-rays. Calculations for the multiplicity method were first done by Giacobbe et al. “> and expanded by ~amenson “1. The basis for the other technique was proposed by Huizenga and Vandenbosch “) and later work was done by Poenitz 6), Poenitz and Tarterczuk 7), Wetzel and Thomas “) and others. In both techniques one attempts to generate ratios which will be dependent on the spins of the resonances. A calculation comparing the two techniques “) showed that in almost all cases, the low-energy y-ray method gives better results but the multiplicity method often gives supporting evidence. The new information on resonance spins has renewed interest in some old questions. As long ago as 1958, it was suggested by Feshbach “) that the strength function for neutron capture might be spin dependent if there was a term in the optical model potential of the form d . I where cr is the spin of the neutron and Z is the spin of the target nucleus. There has been some evidence that the strength function may in fact 45
46
A. I. NAMENSON
be spin dependent carefully r3).
lo- 12) but the evidence
et al.
is far from conclusive
when examined
Another interesting question is that of the relative populations of the resonances of the two possible spin groups formed in s-wave capture. This is related to the value of the spin cut-off factor in the level-density formula 14). Previous estimates of this factor (which were admittedly very rough) gave values of about five or six at the neutron separation energy. However, recent measurements show significant departures from a 2Jf 1 rule for some nuclei and spin cut-off factors of the order of three or four ’ “). In this connection ” 5Lu seemed like an interesting case. Examining the eleven spins of a previous measurement of Wasson and Chrien ’ “) one would notice a significant departure from a 2J+ 1 rule, even with this small sample of resonances. It is appropriate to mention here that in the indirect methods of measuring spins which depend on statistical arguments about the decay of an average resonance there is always the possibility of mis-assigning a spin even with good experimental statistics. This is because of the variations in the experimentally determined ratios due to Porter-Thomas fluctuations. In a previous paper ’ ‘), we suggested that probabilities of correctness should accompany spin assignments. It will be shown here that in quoting standard deviations for computed average properties of the resonances (such as strength functions and relative populations) for each spin group, it is important to include the uncertainty in spin assignments.
2. Experimental 2.1. THE GENERAL
techniques
TECHNIQUES
method and the low-energy y-ray For the isotope ‘s’Os, both the multiplicity method method were used. For the two isotopes rs90s and 175Lu, the multiplicity gave inconclusive results and only the low-energy y-ray method was used. Instead of using a single pair of y-lines in the low-energy method, the information content of the experiments on the latter two isotopes was greatly increased by using three low-energy y-lines in l8 90s and nine low-energy y-lines in ” 5Lu. 2.2. APPARATUS
AND
DATA
ACQUISITION
The apparatus and data acquisition system in this experiment has been described before “* la). Th e e q ui‘p ment and methods will only be briefly outlined here. A timeof-flight technique was used to measure the energy of the neutrons which were produced at the Naval Research Laboratory Linac in 60 ns pulses with a peak current of 2 A. For the multiplicity experiment on ‘*‘OS, four NaI detectors were used, with low- and high-energy thresholds of 0.55 and 3.0 MeV respectively. For the low-energy y-ray method, data were accumulated as two-parameter information consisting of time-of-flight by y-ray energy. The time-of-flight information was in the form of 1024 channels while the y-energy information was either 512, 1024 or 4096 channels depending on the experiment. An 80 cm3 Ge(Li) detector was used to detect the
47
SPINS IN lY5Lu, ‘87*1890s
y-rays and measure their energy. For the time-of-bight spectrum, a “crunching scheme” described in ref. ’ ‘) enabled us to have a wide range of flight times without sacrificing good resolution at the short flight times. The experiment on 175L~ was performed with a 22 m flight path while all the other experiments were performed at 10m. 2.3. SAMPLES
All samples were packed into polyethylene discs. The respective masses of the 175Lu, la90s and i870s samples here 56.6 g, 5.55 g and 0.30 g; their respective thicknesses were 912, lg.5 and 59.2 mg/cm*; and their respective enrichments were 99.9 “/_ 87.3 7; and 45.8 %. 3. Treatment of data 3.1. THE NUCLEUS
l’IJLu
Fig. 1 shows the y-ray spectrum integrated over all flight times for the reaction ‘75Lu(n, y) 176Lu . The lines of interest are indicated together with the spins, parities and energies of the levels from which they originate. To identify these levels, we used the decay scheme of Balodis et al. 19). The large number of y-transitions and the extremely complicated decay scheme presented difficulties. It was impossible to resolve most of the lines from nearby ones which were sometimes less than 1 keV away. The nine lines indicated above 153 keV were intense enough not to be significantly perturbed by the presence of nearby lines. The intensities of the 201 and 284 keV lines originating from levels of spins 5- and 5’ respectively, were strongly dependent on the spin of the resonance. The intensities of the 192 and 262 keV lines, which originate from the same level of spin l-, showed a spin dependence opposite to and weaker than that of the 201 and 284 keV lines. Since these former two lines originate from the same level, their intensities were simply added together and the sum was treated
L
-._._-
150
__-__-..
200
250 ENERGY
.-
(keV1
300
._‘_.__
__
350
Fig. 1. Gamma-ray spectrum for 175Lu(n, y)x76L~ integrated over all flight times. The energy of each significant g-ray is indicated together with the spin, parity and energy of its level of origin. All energies are in keV.
48
A. I. NAMENSON
et al.
cd
0’
xi
F
‘75L,
N”i
100
90
l/8 ,uS/CH
80
70
60 50 NEUTRON
40 30 20 ENERGY (eV)
IO
5
2,5
Fig. 2. Time-of-flight spectrum for 175Lu(n, y)“%u integrated over all y-ray energies. Vertical lines denote the boundaries of six different regions each having a different channel width in [&channel. Whenever the vertical scaIe factor changes as one proceeds from left to right, a new vertical scale is indicated. Because the channel widths are different for different regions, the counting rate is, in general, discontinuous as one goes from one region to an adjacent one.
0
,.~
260
250
300
350
400 ENERGY
-_-.
4% CkeV)
500
,A
550.
600
650
Fig. 3. Gamma-ray spectrum for ‘8g0s(n, 1~)r~OOsintegrated over all flight times. The energy of each significant y-ray resulting from neutron capture in ls90s is indicated together with the spin, parity, and energy of its level of origin. All energies are in keV. Changes in the vertical scale as one proceeds from left to right are indicated.
as one line. The remaining five of the nine lines showed no dependence on the spin of the resonance, Their intensities were added together to provide our normalization. For each resonance, then, we had three different spin-dependent ratios. It was noticed, however, that there was a correlation in the intensities of the 201 and 284 keV lines
SPINS IN 175Lu, la'.lssOs
49
after the dependence on the spin was removed. Because treating a situation of partial correlation is extremely complex, a linear combination of these ratios was taken which would maximize the dependence on spin as compared to the fluctuations due to experimental uncertainties and Porter-Thomas statistics. This left us with two sets of independent numbers from which the spin determinations were made. Fig. 2 shows the time-of-flight spectrum integrated over all y-ray energies. even
3.2. THE NUCLEUS
‘*90s
Fig. 3 shows the y-ray spectrum integrated over all flight times for the reaction ‘89Os(n, p) 1900s . The lines of interest are indicated in a similar manner to those of fig. 1. Because they were both normalized to the 187 keV line, which had some spin dependence, the intensities of the 361 and 569 keV lines were combined in a manner similar to that described in subsect. 3.1. The level structure of the daughter nucleus l”‘Os and the y-transitions may be found in the work of Mariscotti et al. ““). The la90s isotope was enriched to only 87.3 %, and it was necessary to guarantee that all the analyzed resonances were due to capture by 18’Os. This was done by choosing energy regions corresponding to the 187 keV line and background regions on either side of it. The time-of-fight spectrum for each region was generated and by suitable subtractjon of the background spectra from the 187 keV spectrum on a channel by channel basis, we obtained the background corrected time-of-flight
175
-
150
140
a0
120
110
102
t
114 pS/CH
118 #XH
IS0 90
130
70 NEUTHON
ENERGY
(et’)
Fig. 4. Time-of-flirt spectrum corresponding to the background corrected I87 keV y-ray line in 1s90s(n, y) lsOOs. The explanations of the six different regions and changes in vertical scale factor are the same as for fig. 2. A solid horizontal line above the bottom of the graph represents zero.
50
A. I. NAMENSON
5x103;
LyL_-..
et nf.
._.._~ ~~ 200 250 300 ENERGY (kc+.‘)
150
i_ 350
1
Fig. 5. Gamma-ray spectrum for “s70s(n, y)‘*‘Os integrated over all flight times. The energy of each significant y-ray resulting from neutron capture in 18’Os is indicated together with the spin, parity and energy of its level of origin. All energies are in keV. One y-ray resulting from neutron capture in ‘*90s is also indicated.
‘-3&j-
100
90
80
250
225
70
200
60
~_.-- _.._.._~ _
175
50 NEUTRON
150
140
130
40 ENERGY
30
20
IO
id’)
Fig 6. Time-of-flight spectrum corresponding to the background corrected 155 keV line in 18’Os (n, y)‘880s. The explanations of the six different regions and changes in vertical scale factor are the same as for fig. 2. The solid horizontal lines above the bottom of the graph represent zero. Note that the position of this line is not necessarily the same for the different regions.
SPINS IN 175Lu, 187. ‘890s
I
8’
01
100
51
90
00
70
60
NEURON
40 ENERGY
30
26
20
‘lb
(eV)
Fig. 7. Comparison of two time-of-flight spectra for r8’0s(n, y) ‘880s. The upper curve is a portion of the background corrected spectrum for the 155 keV line shown in fig. 6. The lower curve is the time-of-flight spectrum integrated over all y-rays. The energy scale on the bottom refers to both spectra. The explanations of the different regions and of the changes in vertical scale are the same as in fig. 2. Resonances not resulting from capture in ‘s’Os are labeled. A label of simply OS means that the isotopic origin is unknown.
NEUTRON
Fig.
8.
ENERGY
(eV)
A cumulative plot for the resonances in rsiOs.
A. I. NAMENSON
52
et al.
spectrum for the 187 keV line, shown in fig. 4. Because the 187 keV line can only arise from neutron capture in ls90s, the spectrum of fig. 4 is effectively that of an isotopitally pure sample. 3.3. THE NUCLEUS
la70s
Fig. 5 shows the y-ray spectrum integrated over all flight times for the reaction ‘870s(n, y)‘**Os. Information about the lines of interest are indicated in a similar manner to that of figs. 1 and 3 (described in subsects. 3.1 and 3.2). The small sample size, the low enrichment and the very low intensity of the 324 keV line made this an extremely difficult measurement. Some useful information was obtained, however, by combining a multiplicity measurement with the ratio of the intensity of the 324 keV line to that of the 155 keV line. These two sets of data were treated as independent. We were able to identify 44 resonances as arising from neutron capture in 18’Os by obtaining a background corrected time-of-flight spectrum for the 155 keV line, using the method described in subsect. 3.2. Fig. 6 shows the resulting time-of-flight spectrum. To illustrate the results of our effective isotopic purification, in the upper half of fig. 7 we have repeated a portion of fig. 6, and directIy beneath it we have shown a time-of-flight spectrum for the identical neutron energy range, integrated over all y-ray energies. Intense resonances at 6.71,9.00 and 10.2 eV (from neutron capture by ls90s) in the lower spectrum are completely absent from the upper spectrum. A small resonance in rs70s at 39 eV is almost masked in the lower spectrum by a strong impurity resonance at 38 eV. However, in the upper spectrum, the 38 eV resonance has disappeared and the 39 eV resonance stands out clearly. At 50.3 eV there are resonances in both is90s and 1870s. Without a method of effectively isotopically purifying the sample, it would have been difficult to identify the 50.3 eV resonance in rs70s, For ‘*‘OS a similar method was used by Jackson and Bollinger 22), but with NaI detectors it was difficult to “purify” the sample as completely as was done here. Fig. 8 is a cumulative plot of the resonances in “‘OS. It shows that above 140 eV, resonances are missed due to poor resolution and low counting rates. 4. Analysis 4.1. MATHEMATICAL
METHOD
The analysis of ref. i7) will b e extended to include a more rigorous method of combining independent experiments and calculating average properties for the resonances in each spin group. In addition, it will be shown how to account for uncertainties in spin assignments in quoting standard deviations and variances for these average properties. Suppose that in m independent experiments, measurements were performed on n resonances. Each resonance can belong to one of L mutually exclusive and exhaustive categories (for example, spin). From the jth experiment, we obtain the number xij (for exampfe, a ratio of two y-ray intensities) for the ith resonance, and this number
SPINS IN l’sLu,
287*‘890s
53
has an experimental standard deviation of a,,. In the jth experiment, for the resonances in category I the values Xii will fluctuate about some average Ujr . The standard deviation for this fluctuation will be a combination of crXijand a standard deviation resulting from Porter-Thomas statistics, apTj. (We have tacitly assumed that these deviations do not depend on the category 1. This is only approximately true.) Because of the many ways of populating low-lying levels, the fluctuations of Xij will be approximately Gaussian. Taking ul to be the apriori probability that a resonance belongs in category E,where L
we may generalize eqs. (l)-(3) of ref. “1 to the case of m independent experiments and L categories by replacing terms of the form exp [-(xi
by
- ~2)~/24]/J%a,
fi exp [-(xii-Uj,)2)/JZ-~ij], j=l
where 0;
=
2 Oql
+
2 $Tj
*
(2)
We finally obtain, for the probability to be maximized,
where dx represents Ili,j dxij, x stands for the set of numbers Xii and similarly for o,., a, ~~~and a. The generalization of eqs. (7) and (8) of ref. I’) gives
for the probability that the ith resonance is in category I. Sets of variables a, gpr and ocare to be found by the maximum likelihood method. When the appiofi probabilities al are determined experimentally, they represent the relative number of resonances in each category. Eliminating one of the variables CI~using eq. (I), we are left with mL-t-m+L- 1 independent variables. We can denote a general variable by zp where 1 s ~16 mLtm+L1. The maximum likelihood relation is then d In P/Sz, = 0
for all p,
(5)
with an error matrix given by (h-l),,
= d2 In
P~az,az,.
(6)
The off-diagonal elements of f2represent covariances between different variables while
54
et al.
A. I. NAMENSON
the diagonal elements represent variances. In practice these off-diagonal elements are very small, and we are only concerned with the diagonal ones, lj~;S = (h-l),,
pjaSEl.
= a2 la
(7) More complete discussions of the maximum likelihood method can be found in many books on statistics [see for example ref. ““)I. It must be pointed out before continuing that because levels of the same spin repel each other, the spacings of resonances can also contribute information about their spins. Eqs. (l)-(4) d o not involve level spacings and so do not account for this factor. Including level spacings in this calculation would make it enormously complicated, and quoted results would be significantly changed only for the calculated variances on the relative populations of the different categories, CI~.These variances are obviously sensitive to the distribution Iaw for resonance spacings, since the more reguIar the spacings of resonances, the smaller the uncertainty due to a sampling of only a finite number of resonances. However, it is still useful to calculate the variance in relative populations on the present basis because: (i) the results will be valid for those cases where large numbers of resonances are missing from the analysis; (ii) the results will always represent an upper limit to the variance; (iii) the form of the results is highly instructive and we may go from them to more exact solutions by simple intuitive arguments. 4.2. FORMULAS
RESULTING
FROM
THE MAXIMUM
LIKELIHOOD
ANALYSIS
To simplify the solutions we introduce the following definitions:
wij=
lla~jj,
@I
Wij$ = Pii Wcj *
(9)
From eq. (S), ajl
=
i
i=l
011= n,/n,
Wijl Xii/ i
i=l
Wi,, 9
(10)
where n, = k pit. i=1
Eq. (7) gives us the variances in the above averages. Because the general expressions are so complicated we will restrict ourselves to the case of interest here, where there are only two possible categories (namely spins in s-wave capture),
5s
SPINS IN ‘75Lu, I*‘* ‘s90s 4.3. AVERAGE
REDUCED
NEUTRON
WIDTHS
AND
STRENGTH
FUNCTIONS
All the resonances in a given category wilt have reduced neutron widths which fluctuate about a certain average according to a x2 distribution with Y = I. Instead of the reduced neutron widths, it is more convenient to use the quantities ZgT,Oiwhere i refers to the number of resonance. We assume that the experimental errors in these numbers are small compared to the Porter-Thomas fluctuations. The average for the Zth category is <2#:),. In order to simplify the following equations, we introduce the following substitutions:
& = 2gr,D,,
ZE = (2grZ),.
(16)
If there are n’ resonances with measured neutron widths,
expresses the probability which is to be maximized with respect to Z. The symbols [ and Z stand for sets of numbers. We introduce the probability
and using the maximum Iikelihood method, (19) Once again, in the expression for the variance, we consider only two possible categories and obtain
The s-wave strength function and its variance are obtained from the relations
var [S,(l)]
= [var ((2gr,O>,)J (i
pi;/2dE)‘.
i=l
4.4. INTERPRETATION
OF THE EQUATIONS
In giving intuitive explanations for the equations presented above, only the most important points will be considered. Equations similar to (3), (IO), and (11) have been discussed before I’). Eq. (12) states that the relative population of any category is simply the average probability of a resonance belonging to that category. The quantity PINplays the role of the number of resonances in category I. The probability
56
A. I. NAMENSON
et al.
pi1 in eq. (18) is the probability that the ith resonance is in category I when there is a significant difference in the average <2grt), for the different spin groups. In such a case, the neutron widths would contribute information about the most likely spin. Note that if the averages (2gFt), are not very different for the different spins, we would have Pfl “piI. This is actually the case in almost all practical situations. Eq. (19) for <2grt), is just a weighted average over all CIwhere the weighting factors are the probabilities p;. Eqs. (13)-(15) and (20) show how the uncertainties in spin measurements act to increase the variances through the sums of terms containing the factors pilpiz. If all the spins were known with certainty, the terms pilpil would disappear. In such a case, eq. (15) would become the variance in a binominal distribution of order n, and eq. (20) would be the variance in the average over n’ resonances which follow a Porter-Thomas distribution. In fact, it is useful and interesting to express eqs. (15) and (20) in approximate form, n
var(al) W RI Ez/n+ C Pi1 pi*ln2,
(23)
i=l
var ((2gTz)J
227 = o’(Z,) z -
iF;1 Pi;(Si-z*)2
(24)
+ (i&)*
&l
*
We can see that in both cases, the variance splits into the sum of two terms. The first term is the variance due to the finite number of resonances and the second term is the variance due to the uncertainty in spin assignments. For the approximation to be valid, the second term should be less than the first. The summation z:rpIr in eq. (24) plays the role of the number of resonances with measured neutron widths which belong in category 1. 4.5. VARIANCES
IN THE RELATIVE POPULATIONS
OF DIFFERENT
GROUPS
It is instructive to see how eq. (23) could have been arrived at by intuition. Referring to eq. (12), we see that if all resonances were categorized with certainty, nl would be experimentally determined by a simple counting procedure. When resonances are not known with certainty, its must be calculated from eq. (12), and from statistical arguments, the experimentally determined value of 12~ would have a variance of ”
var (4
= igp”(l
-Pill,
(25)
and for only two categories
var(4 = f
i=l
Pi1 Pi*.
(26)
The variance in c(~consists of two independent variances: one due to the finite value
SPINS
IN ‘75Lu,
1*7*1*90b
57
of n,, and one due to the uncertainty in determining IP, itself, var (gl) = var due to finite number of resonances+
var (ur) ___. n2
(27)
Only the first term of eq. (27) is affected by the specifics of the distribution of level spacings. When these spacings convey no information about spin, the first term is the we11known variance for a binominal distribution and eq. (23) would follow. For a Wigner distribution of Ievel spacings, it can be shown that
Long range ordering effects of the kind suggested by Dyson, and Mehta and Dyson ““) could slightly affect the first term of eq. (28) but this is not expected to be a significant alteration. Note that for eq. (28) to be valid, the second term must again be smaller than the first. 4.6.
PRACTICAL
SOLUTION
OF EQUATIONS
can be solved by introducing initial guesses for the parameters Eqs. (4), (lo)-(12) aji, oPrj and a, and then using a computer to solve by the method of iteration. After obtaining solutions for these quantities and for the probabilities pil, one can use eqs. (18)-(22) t o soIve for the averages,, the strength functions &,(l), the variances in these two quantities, and the probabiIities pi1 . ff there is a realdifference in the averages ,, then the neutron widths contribute significant information about the spin, and eqs. (18) and (19) should be included in the iterative process. This last refinement was not done in practice. 5. Results 5.1. THE NUCLEUS
175Lu
Table 1 shows the resonances in ’ ” Lu, the spin assignments and their probabilities of correctness. There is complete agreement with a previous measurement of eieven of these spins 16). For iIIustration purposes, an optimum Iinear combination of the two independent ratios described in subsect. 3.1 (using the same method described there) was taken to yield the combined ratios in fig. 9. We can see that there are only a few resonances whose spins are ambiguous. The histogram in this figure results from dividing the vertical scale into equal intervals and obtaining the number of resonances in each interval. However, since some of the resonances have rather wide error bars, there is the problem of exactly where to place them. This was solved by noting that the error bars represent a Gaussian probability distribution for the location of the resonances, and so we can calcuiate for each interval the expectation value for the number of resonances in it. We see that the ratios clearly divide into two distinct groups, and that there are about two resonances located in the middle of the two
TABLE1 Spins of neutron resonances in 1T5Lu Neutron energy (eV>
Spin
2.6 4.7 5.2 11.2 14.0 15.3 20.5 23.5 28.1 30.2 31.1 36.5 40.7 49.2 50.2 53.5 56.7 60.9 69.6 85.3 88.1 96.3 99.7 100.9 102.8 107.3 112.7 115.1 118.6
4 4 3 3 3 4 3 3 4 4 3 3 3 3 4 4 3 4 4 A
Measurements Measurements
Probability (%I
Neutron energy feV
Spin
Probability (%I
127.4 129.4 137.8 143.0 146.2 148.8 150.9 155.4 158.7 164.0 169.2 171.5 175.8 180.9 185.4 193.3 196.8 202.6 204.4 217.6 223.9 227.9 229.6 236.9 244.5 251.5 256.4 274.0
3 A
100 99 86 98 99 63 9s 99 100 99 98 94 97 99 99 99 94 51 81 94 94 9s 99 5s 97 69 99 99
100 100 100 100 96 100 100 JO0 100 100 100 97 100 99 99 99 93 99 98 99 70 100 99 82 100 99 99 100 99
4 A 4 4 3 3 3
3 I:1 4 4 3 3 4 3 3 4 3 3 ;I (4) 4 3 4 A 1:1 3 4
with confidence limits between 70 % and 89 % are enclosed in parentheses. with confidence limits below 70 % are enclosed in square brackets.
SPIN
-.0
.___ 50
IO0
3
/ - ..-.- _.... 200 150
ENERGY (eV)
250 .._-.
1.. __i _! 246 NUMBER
Fig. 9. Combined ratio for 175Lu. On the left is a plot of the combined ratio versus resonance energy. The solid lines are the weighted averages for the respective spin groups and the dashed lines show the extent of the Porter-Thomas fluctuations. The error bars on the points refer to experimentai uncertainties. Resonances marked by triangles are those whose spins have been identified with better than 90 % confidence. On the right is a histogram of the ratio versus the number of resonances.
59
SPINS IN 175Lu, ls’~l*sOs
groups. Considering the magnitude of the Porter-Thomas fluctuations and the experimental errors, it is not unreasonable to find two such ambiguous ratios out of 57 resonances. The ratios of fig. 9 could have served for calculating the probabilities in table 1 without introducing much change. 5.2. THE NUCLEUS
‘89Os
Table 2 shows the resonances of rsgOs in a manner similar to table 1 for I7 5Lu. There is complete agreement with the 12 spin measurements of Wetzel and Thomas “). TABLE 2
Spins of neutron resonances in 1890s Neutron energy (ev)
87 8 93.1 104.3 109 110 118 124 128 131 137 141 145 150 160 162 165 167 178 187
100 100 100 99 100 97 100 67 90 78 100 100 95 97 87 96 63 99 19
6.63 8.95 10.3 18.7 22.1 27.4 28.3 30.5 39.1 43.3 50.3 54.9 60.7 63.9 65.4 66.0 72.2 75.3 76.2 Parentheses
Neutron energy (ev)
Probability (%)
Spin
Spin
(1) (1)
r:1
(1) PI (2)
r:1 I21 Dl Ul 121 [:I (2) HI [I I Dl
Probability (%) 83 72 97 56 75 51 70 91 65 57 63 66 53 90 51 78 63 63 69
and square brackets have the same meaning as in table 1. ~,
0.4SPIN
1
2
.
,-.
00
;.
‘_
SPIN
I
.
(8
”
-0,l 0
50
100 ENERGY
150 (eV)
246 NUMBER
Fig. 10. Combined ratio for 1890s. The explanation is similar to that of fig. 9 except that resonances marked by triangles are those whose spins have been identified with better than 85 % confidence. Open circles indicate better than 70 % confidence. Porter-Thomas fluctuations are not shown since they are negligible compared to the experimental uncertainties.
A. I. NAMENSON
60
ei at.
Fig. 10 shows the ratios used in obtaining the spin values, and their probabilities of correctness. The histogram is similar to that of fig. 9 described in subsect. 5.1. The separation of the two groups is poorer than the case of “‘Lu, but there is still a definite separation. 5.3. THE
NUCLEUS
f870s
Table 3 for “‘0s is similar to the preceding two tables. Fig. 11 (for illustration purposes only) shows the results of taking an optimum linear combination of the ratios from the multiplicity experiment and the low-energy y-ray experiment as described before. In the histogram of fig. 11, the separation between the two groups is not very well defined and therefore the spin determinations were poorer than in the other cases. We agree with the measurement of Wetzel and Thomas s) for the 9.5 eV resonances, but disagree with their measurement for the 12.6 eV resonance. In the present measurement the multiplicity method corroborates the information from the low-energy y-ray method. It is suggested that the wide error bars in ref. “) do not exclude the possibility that the ratios for the 9.5 and 12.6 eV resonances are approximately the same. In any event, it is dangerous to make spin assignments with only two resonances, since one would have no estimate of Porter-Thomas fluctuations. TABLE 3 Spins of ls70s Neutron energy
Spin
(eV) 9.48 12.6 19.9 26.1 39.3 40.2 43.2 47.6 49.8 50.3 61.9 63.5 64.7 78.0 90.0 92.5 99.1 104.8 107.2 108.7 123.0 124.5
1 1 1 1:1 &
G, (1) (f.l) :,
Probability
W)
99 100 90 98 62 100 80 100 86 86 84 98 51
126.8 138.2 141.2 145.3 155.0 164.6 168.5 176.8 179.0 189.5 201.9 207.8 214.5 225.2 228.3 237.4 251.3 258.8 268.9 278.9 291.1 298.5
Ill
51
:1
91 62 61 96 58 99 55 60
WI ;1 I&
101
Neutron energy
(%I
Parentheses and square brackets have the same meaning as in table 1.
Spin
Probability (%)
(0) (1) LO1 (01 101
101 [Ol (1)
101 t01 t01 t01 (1) [Ol t01 111 ill t01 [Ol [Ol [Ol
101
81 II 55 71 57 65 52 70 58 51 52 58 72 54 52 62 57 55 55 55 55 52
SPINS
IN 175Lu,
0.4
I
‘=-OS 03
SPIN I
o 0.2 F 2 o.i* 00 -0,l
61
1sr.'890~
1
. : ;r.
.
’ SPIN 0
50
/
‘*;
0
150 200 100 ENERGY (eV)
Fig. f 1. Combined ratio for ’ 870s. The expfanation
6. Discussion
2 4
6
is simiIar to that for fig 10.
of results
Some of the average resonance properties are shown in table 4. The relative populations were computed from eq. (12). To compute the uncertainty in the relative populations, eq. (28) for the Wigner distribution was used for l7 sLu while for 1870s (where few spins were well determined), eq. (15) for a binomiai distribution was used. For ““OS, one average was computed for the resonances up to 75.3 eV, where the spins were welt determined, and its uncertainty was computed from eq. (28). For resonances above this energy, another average was computed and the uncertainty obtained from eq. (15). The two averages were then combined by the method of weighted averages with the reciprocal variances as weighting factors. A glance at the relative populations of the two spin groups in table 4 shows that the populations are consistent with a 2Jf 1 law for rs90s. Nevertheless, it is suggested that a generally better approximation to the level density in this mass region would be a formula with a spin cut-off factor of c z 3.5 for 175Lu and rs70s; the spin cutoff factors are 3.5:::; and 0.94,0:: respectively, where the limits represent 68 y: confidence. The spin cut-off factor for ig70s is unusuaily small, but even if the relative popuIations were more than one standard deviation off, we would still have o&y a 1.8 “/: chance that cr 2 3.5. Similar investigations I’> of neutron capture in 17?Hf and 179Hf indicate spin cut-off factors of about 3.5 at the neutron separation energy. It should be pointed out that even for ‘890s thereis still a IO 7: chance that 0 5 3.5. Turning to the averages (2grt>, and the strength functions S,(J) we see that the difference between the two spin groups for l7 5Lu are not really significant. For 18’Os the difference in (2grz), is not considered significant, but the difference in the strength functionsis somewhat suggestive; there is only a 5 % probability that this difference could happen by chance. However, other measurements on nuclei in this mass region have shown no spin dependence of the strength functions “*l’). Most of the uncertainty in the averages (2grz), and in the strength functions S,(J) are due to the small number of resonances with known neutron widths; the measurement of a large number of neutron widths in these nuclei would reduce these uncertainties. Resonance parameters for rs70s are not available at present.
A. I. NAMENSON
62
Spin dependenceof
TABLE4 average properties of resonances for 175Lu, ‘*‘OS and lasOs
Isotope
1’ SLU 1’ SLU 1’ SLLl I 890s 1890s ‘890s ‘S’OS 18’Os 19’0s
ef nl.
Relative population CX) spin 4 spin 3 difference spin 2 spin 1 difference spin I spin 0 difference
48% 4 52* 4 68& 7 32% 7
<2sr”“> “) (meV) 0.68hO.39 1.63hO.73 -0.95f0.83 9.0 *4.1 1.8 &1.6 7.2 +4.4
S&r) x 104
0.38rtO.22 1.4910.67 -1.11+0.70 7.6213.47 0.84&0.07 6.78k3.47
44*14 56&14
Note that the errors for the relative populations of two different spins in the same isotope are completely coherent, since the two relative populations must sum to 100 %. “) Values of 2gFma were taken from ref. 26).
It is of some interest to perform a Wald-Wolfowitz 25) calculation on the isotopes 175Lu and 1890s to test the number of “runs” of one spin or the other to check that there were no systematic variations in the measurements other than spin. (A run is an unbroken sequence of resonances all having the same spin.) The estimate of ref. 25) could be applied to spins if the level spacings followed a Poisson distribution. However, for a Wigner distribution, if the two groups have roughly equal popuIations, a Monte Carlo calculation showed that the number of runs should be 1.4 times this estimate. For ls90s the calculation was performed for resonances up to 66.0 eV, and excellent agreement was obtained. For “‘Lu the calculation was performed for all resonances, and the number of runs was smaller than the above estimate by slightly under two standard deviations. There are, therefore, no obvious systematics in the data. 7. Conclusions and summary In examining the nuclei studied here and elsewhere, we can con&de that in the mass region A x 180, a spin cut-off factor of about 3.5 gives a better estimate of the relative populations of the different spin groups than does a 2J+ 1 law. The differences in the averages {2grz>, are not statistically significant, but the differences in the strength functions are suggestive in the case of lagOs. It is strongly suggested here that in other measurements of this kind, probabilities of correctness accompany spin measurements. When this is done, one would find that many seemingly clear assignments actually had a 5 % to 10 % chance of being incorrect. In calculating average properties for a group of resonances, these probabilities are necessary to properly estimate uncertainties. Finally, since the calculations are easily computerized, the analysis can be done more rapidly and less subjectively than by less quantitative methods.
SPINS IN 175Lu, ‘87~1890s
63
References I) I.,.M. Bollinger, in Experimental neatran resonance spectroscopy, ed. J. A. Harvey (Academic 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)
Press, New York, 1970) p. 317 C. Coceva, F. Corvi, P. Giacobbe and G. Carraro, Nucl. Phys. All7 (1968) 586 P. Giacobbe, M. Stefanon and G. Dellacasa, CNEN Technical Report RT/Fi (68) 20 A. I. Namenson, Nucl. Phys. A209 (1973) 252 J. R. Huizenga and R. Vandenbosch, Phys. Rev. 120 (1960) 1305 W. P. Poenitz, 2. Phys. 197 (1966) 262 W. P. Poe&z and J. R. Tatarczuk, Nucl. Phys. All7 (1968) 586 K. J. Wetzel and G. E. Thomas, Phys. Rev. Cl (1970) 1501 H. Feshbach, Ann. Rev. Nucl. Sci. 8 (1958) 49 J. JuIien, G. Bianchi, C. Corge, V. D. Huynh, G. Le Pottevin, J. Morgenstern, F. Netter and C. Samour, Phys. Lett. 10 (1964) 86 J. Julien, CEA-R-3385 (1968) C. M. Newstead, J. Delaroche and B. Cauvin, in Statisticalproperties of nuclei, ed. J. B. Garg (Plenum Press, New York, 1972) p. 367 H. V. Muryadan and Yu. V. Adamchuk, Nucl. Phys. 68 (1965) 549 A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 C. Coceva, F. Corvi, P. Giacobbe and M. Stefanon, in Stati~t~calproperties ofnuclei, ed. J. B. Garg (Plenum Press, New York, 1972) p. 447 0. A. Wasson and R. E. Chrien, Phys. Rev. C2 (1970) 675 A, Stolovy, A. 1. Namenson, J. C. Ritter, T. F. Godlove and G. L. Smith, Phys. Rev. C5 (1972) 2030 T. F. Godiove, G. L. Smith and A. I. Namenson, Nucl. Instr. 95 (1971) 595 M. K. Balodis et al., Nucl. Phys. Al94 (1972) 305 M. A. Mariscotti, W. R. Kane and G. T. Emery, Brookhaven National Lab. Rep. BNL-11426 (1967) E. Bashandy and S G. Hanna, Nucl. Phys. 84 (1966) 577 H. E. Jackson and L. M. Bolfinger, Phys. Rev. 124 (1961) 1142 H. Cramer, Mathematical methods of statistics (Princeton University Press, Princeton, 1946) F J. Dyson, J. Math. Phys. 3 (1962) 140, 157, 166; M. L. Mehta and F. J. Dyson, J. Math. Phys. 4 (1963) 701, 713 G. D. James, AERE-R 6633 (1971) S. G. Mughabghab and D. 1. Garber, Neutron cross sections, Brookhaven National Lab. Report BNL-325, 3rd ed., vol. 1 (1973)
2-c
I
Nuclear Physics A237 (1975) 45--63; @ Nor~h-~o~~a~ F~~is~i~g Co., Amsterdam Not to be reproduced
by pbotoprint
SPINS OF LOW-EARLY
ot microfilm without written permission
from the publisher
NEUTRON ~SONAN~~S
IN 1’5Lu, ‘*‘OS AND ‘*‘OS A. I. NAMENSON, A. STOLOVY and G. L. SMITH Naval Research Laboratory, Washington, DC 20375 Received 21 August 1974 Abstract: Resonances up to 300 eV in 175Lu, r8pOs, and Is70s are reported together with spin assignments and probabilities of correctness. Those spins reported with better than 90 % confidence consist of 49, 17 and 10 resonances in 175Lu, ‘8pOs, and’ ‘*‘OS respectively. In Is70s, 39 of the observed resonances have not been reported previously. Ratios of low-energy y-ray intensities and y-ray multiplicity were used for spin determinations. A method developed previously for analyzing these data has been extended to include analysis of uncertainties in relative populations, average reduced neutron widths, and strength functions of the two spin groups. The data in ~ssOs is consistent with a 2J;I level-density dependence, but in the other two isotopes this rule is violated. We suggest spin cut-off factors of about 3 in this mass region. Small differences in <2gFao> and in the strength functions for the two spin groups are not considered very significant.
E 1
NUCLEAR REACTIONS 175Lu, 187, lEsOs(n, y), E = 2.6-300 eV; measured Er, L,. 176Lu, lsOOs resonances deduced .FFgFa, spin cut-off factors. 1s80s resonances deduce‘d .F. Enriched targets.
1. Introduction The experimental simplicity of recently developed indirect methods of determining the spins of neutron resonances by observing the prompt y-rays following neutron capture have lead to much new information. Of the number of such techniques discussed in an article by Bollinger ‘) we will be concerned with the multiplicity method of Coceva er al. “) and with the method of observing low-energy y-rays. Calculations for the multiplicity method were first done by Giacobbe et al. “> and expanded by ~amenson “1. The basis for the other technique was proposed by Huizenga and Vandenbosch “) and later work was done by Poenitz 6), Poenitz and Tarterczuk 7), Wetzel and Thomas “) and others. In both techniques one attempts to generate ratios which will be dependent on the spins of the resonances. A calculation comparing the two techniques “) showed that in almost all cases, the low-energy y-ray method gives better results but the multiplicity method often gives supporting evidence. The new information on resonance spins has renewed interest in some old questions. As long ago as 1958, it was suggested by Feshbach “) that the strength function for neutron capture might be spin dependent if there was a term in the optical model potential of the form d . I where cr is the spin of the neutron and Z is the spin of the target nucleus. There has been some evidence that the strength function may in fact 45
46
A. I. NAMENSON
be spin dependent carefully r3).
lo- 12) but the evidence
et al.
is far from conclusive
when examined
Another interesting question is that of the relative populations of the resonances of the two possible spin groups formed in s-wave capture. This is related to the value of the spin cut-off factor in the level-density formula 14). Previous estimates of this factor (which were admittedly very rough) gave values of about five or six at the neutron separation energy. However, recent measurements show significant departures from a 2Jf 1 rule for some nuclei and spin cut-off factors of the order of three or four ’ “). In this connection ” 5Lu seemed like an interesting case. Examining the eleven spins of a previous measurement of Wasson and Chrien ’ “) one would notice a significant departure from a 2J+ 1 rule, even with this small sample of resonances. It is appropriate to mention here that in the indirect methods of measuring spins which depend on statistical arguments about the decay of an average resonance there is always the possibility of mis-assigning a spin even with good experimental statistics. This is because of the variations in the experimentally determined ratios due to Porter-Thomas fluctuations. In a previous paper ’ ‘), we suggested that probabilities of correctness should accompany spin assignments. It will be shown here that in quoting standard deviations for computed average properties of the resonances (such as strength functions and relative populations) for each spin group, it is important to include the uncertainty in spin assignments.
2. Experimental 2.1. THE GENERAL
techniques
TECHNIQUES
method and the low-energy y-ray For the isotope ‘s’Os, both the multiplicity method method were used. For the two isotopes rs90s and 175Lu, the multiplicity gave inconclusive results and only the low-energy y-ray method was used. Instead of using a single pair of y-lines in the low-energy method, the information content of the experiments on the latter two isotopes was greatly increased by using three low-energy y-lines in l8 90s and nine low-energy y-lines in ” 5Lu. 2.2. APPARATUS
AND
DATA
ACQUISITION
The apparatus and data acquisition system in this experiment has been described before “* la). Th e e q ui‘p ment and methods will only be briefly outlined here. A timeof-flight technique was used to measure the energy of the neutrons which were produced at the Naval Research Laboratory Linac in 60 ns pulses with a peak current of 2 A. For the multiplicity experiment on ‘*‘OS, four NaI detectors were used, with low- and high-energy thresholds of 0.55 and 3.0 MeV respectively. For the low-energy y-ray method, data were accumulated as two-parameter information consisting of time-of-flight by y-ray energy. The time-of-flight information was in the form of 1024 channels while the y-energy information was either 512, 1024 or 4096 channels depending on the experiment. An 80 cm3 Ge(Li) detector was used to detect the
47
SPINS IN lY5Lu, ‘87*1890s
y-rays and measure their energy. For the time-of-bight spectrum, a “crunching scheme” described in ref. ’ ‘) enabled us to have a wide range of flight times without sacrificing good resolution at the short flight times. The experiment on 175L~ was performed with a 22 m flight path while all the other experiments were performed at 10m. 2.3. SAMPLES
All samples were packed into polyethylene discs. The respective masses of the 175Lu, la90s and i870s samples here 56.6 g, 5.55 g and 0.30 g; their respective thicknesses were 912, lg.5 and 59.2 mg/cm*; and their respective enrichments were 99.9 “/_ 87.3 7; and 45.8 %. 3. Treatment of data 3.1. THE NUCLEUS
l’IJLu
Fig. 1 shows the y-ray spectrum integrated over all flight times for the reaction ‘75Lu(n, y) 176Lu . The lines of interest are indicated together with the spins, parities and energies of the levels from which they originate. To identify these levels, we used the decay scheme of Balodis et al. 19). The large number of y-transitions and the extremely complicated decay scheme presented difficulties. It was impossible to resolve most of the lines from nearby ones which were sometimes less than 1 keV away. The nine lines indicated above 153 keV were intense enough not to be significantly perturbed by the presence of nearby lines. The intensities of the 201 and 284 keV lines originating from levels of spins 5- and 5’ respectively, were strongly dependent on the spin of the resonance. The intensities of the 192 and 262 keV lines, which originate from the same level of spin l-, showed a spin dependence opposite to and weaker than that of the 201 and 284 keV lines. Since these former two lines originate from the same level, their intensities were simply added together and the sum was treated
L
-._._-
150
__-__-..
200
250 ENERGY
.-
(keV1
300
._‘_.__
__
350
Fig. 1. Gamma-ray spectrum for 175Lu(n, y)x76L~ integrated over all flight times. The energy of each significant g-ray is indicated together with the spin, parity and energy of its level of origin. All energies are in keV.
48
A. I. NAMENSON
et al.
cd
0’
xi
F
‘75L,
N”i
100
90
l/8 ,uS/CH
80
70
60 50 NEUTRON
40 30 20 ENERGY (eV)
IO
5
2,5
Fig. 2. Time-of-flight spectrum for 175Lu(n, y)“%u integrated over all y-ray energies. Vertical lines denote the boundaries of six different regions each having a different channel width in [&channel. Whenever the vertical scaIe factor changes as one proceeds from left to right, a new vertical scale is indicated. Because the channel widths are different for different regions, the counting rate is, in general, discontinuous as one goes from one region to an adjacent one.
0
,.~
260
250
300
350
400 ENERGY
-_-.
4% CkeV)
500
,A
550.
600
650
Fig. 3. Gamma-ray spectrum for ‘8g0s(n, 1~)r~OOsintegrated over all flight times. The energy of each significant y-ray resulting from neutron capture in ls90s is indicated together with the spin, parity, and energy of its level of origin. All energies are in keV. Changes in the vertical scale as one proceeds from left to right are indicated.
as one line. The remaining five of the nine lines showed no dependence on the spin of the resonance, Their intensities were added together to provide our normalization. For each resonance, then, we had three different spin-dependent ratios. It was noticed, however, that there was a correlation in the intensities of the 201 and 284 keV lines
SPINS IN 175Lu, la'.lssOs
49
after the dependence on the spin was removed. Because treating a situation of partial correlation is extremely complex, a linear combination of these ratios was taken which would maximize the dependence on spin as compared to the fluctuations due to experimental uncertainties and Porter-Thomas statistics. This left us with two sets of independent numbers from which the spin determinations were made. Fig. 2 shows the time-of-flight spectrum integrated over all y-ray energies. even
3.2. THE NUCLEUS
‘*90s
Fig. 3 shows the y-ray spectrum integrated over all flight times for the reaction ‘89Os(n, p) 1900s . The lines of interest are indicated in a similar manner to those of fig. 1. Because they were both normalized to the 187 keV line, which had some spin dependence, the intensities of the 361 and 569 keV lines were combined in a manner similar to that described in subsect. 3.1. The level structure of the daughter nucleus l”‘Os and the y-transitions may be found in the work of Mariscotti et al. ““). The la90s isotope was enriched to only 87.3 %, and it was necessary to guarantee that all the analyzed resonances were due to capture by 18’Os. This was done by choosing energy regions corresponding to the 187 keV line and background regions on either side of it. The time-of-fight spectrum for each region was generated and by suitable subtractjon of the background spectra from the 187 keV spectrum on a channel by channel basis, we obtained the background corrected time-of-flight
175
-
150
140
a0
120
110
102
t
114 pS/CH
118 #XH
IS0 90
130
70 NEUTHON
ENERGY
(et’)
Fig. 4. Time-of-flirt spectrum corresponding to the background corrected I87 keV y-ray line in 1s90s(n, y) lsOOs. The explanations of the six different regions and changes in vertical scale factor are the same as for fig. 2. A solid horizontal line above the bottom of the graph represents zero.
50
A. I. NAMENSON
5x103;
LyL_-..
et nf.
._.._~ ~~ 200 250 300 ENERGY (kc+.‘)
150
i_ 350
1
Fig. 5. Gamma-ray spectrum for “s70s(n, y)‘*‘Os integrated over all flight times. The energy of each significant y-ray resulting from neutron capture in 18’Os is indicated together with the spin, parity and energy of its level of origin. All energies are in keV. One y-ray resulting from neutron capture in ‘*90s is also indicated.
‘-3&j-
100
90
80
250
225
70
200
60
~_.-- _.._.._~ _
175
50 NEUTRON
150
140
130
40 ENERGY
30
20
IO
id’)
Fig 6. Time-of-flight spectrum corresponding to the background corrected 155 keV line in 18’Os (n, y)‘880s. The explanations of the six different regions and changes in vertical scale factor are the same as for fig. 2. The solid horizontal lines above the bottom of the graph represent zero. Note that the position of this line is not necessarily the same for the different regions.
SPINS IN 175Lu, 187. ‘890s
I
8’
01
100
51
90
00
70
60
NEURON
40 ENERGY
30
26
20
‘lb
(eV)
Fig. 7. Comparison of two time-of-flight spectra for r8’0s(n, y) ‘880s. The upper curve is a portion of the background corrected spectrum for the 155 keV line shown in fig. 6. The lower curve is the time-of-flight spectrum integrated over all y-rays. The energy scale on the bottom refers to both spectra. The explanations of the different regions and of the changes in vertical scale are the same as in fig. 2. Resonances not resulting from capture in ‘s’Os are labeled. A label of simply OS means that the isotopic origin is unknown.
NEUTRON
Fig.
8.
ENERGY
(eV)
A cumulative plot for the resonances in rsiOs.
A. I. NAMENSON
52
et al.
spectrum for the 187 keV line, shown in fig. 4. Because the 187 keV line can only arise from neutron capture in ls90s, the spectrum of fig. 4 is effectively that of an isotopitally pure sample. 3.3. THE NUCLEUS
la70s
Fig. 5 shows the y-ray spectrum integrated over all flight times for the reaction ‘870s(n, y)‘**Os. Information about the lines of interest are indicated in a similar manner to that of figs. 1 and 3 (described in subsects. 3.1 and 3.2). The small sample size, the low enrichment and the very low intensity of the 324 keV line made this an extremely difficult measurement. Some useful information was obtained, however, by combining a multiplicity measurement with the ratio of the intensity of the 324 keV line to that of the 155 keV line. These two sets of data were treated as independent. We were able to identify 44 resonances as arising from neutron capture in 18’Os by obtaining a background corrected time-of-flight spectrum for the 155 keV line, using the method described in subsect. 3.2. Fig. 6 shows the resulting time-of-flight spectrum. To illustrate the results of our effective isotopic purification, in the upper half of fig. 7 we have repeated a portion of fig. 6, and directIy beneath it we have shown a time-of-flight spectrum for the identical neutron energy range, integrated over all y-ray energies. Intense resonances at 6.71,9.00 and 10.2 eV (from neutron capture by ls90s) in the lower spectrum are completely absent from the upper spectrum. A small resonance in rs70s at 39 eV is almost masked in the lower spectrum by a strong impurity resonance at 38 eV. However, in the upper spectrum, the 38 eV resonance has disappeared and the 39 eV resonance stands out clearly. At 50.3 eV there are resonances in both is90s and 1870s. Without a method of effectively isotopically purifying the sample, it would have been difficult to identify the 50.3 eV resonance in rs70s, For ‘*‘OS a similar method was used by Jackson and Bollinger 22), but with NaI detectors it was difficult to “purify” the sample as completely as was done here. Fig. 8 is a cumulative plot of the resonances in “‘OS. It shows that above 140 eV, resonances are missed due to poor resolution and low counting rates. 4. Analysis 4.1. MATHEMATICAL
METHOD
The analysis of ref. i7) will b e extended to include a more rigorous method of combining independent experiments and calculating average properties for the resonances in each spin group. In addition, it will be shown how to account for uncertainties in spin assignments in quoting standard deviations and variances for these average properties. Suppose that in m independent experiments, measurements were performed on n resonances. Each resonance can belong to one of L mutually exclusive and exhaustive categories (for example, spin). From the jth experiment, we obtain the number xij (for exampfe, a ratio of two y-ray intensities) for the ith resonance, and this number
SPINS IN l’sLu,
287*‘890s
53
has an experimental standard deviation of a,,. In the jth experiment, for the resonances in category I the values Xii will fluctuate about some average Ujr . The standard deviation for this fluctuation will be a combination of crXijand a standard deviation resulting from Porter-Thomas statistics, apTj. (We have tacitly assumed that these deviations do not depend on the category 1. This is only approximately true.) Because of the many ways of populating low-lying levels, the fluctuations of Xij will be approximately Gaussian. Taking ul to be the apriori probability that a resonance belongs in category E,where L
we may generalize eqs. (l)-(3) of ref. “1 to the case of m independent experiments and L categories by replacing terms of the form exp [-(xi
by
- ~2)~/24]/J%a,
fi exp [-(xii-Uj,)2)/JZ-~ij], j=l
where 0;
=
2 Oql
+
2 $Tj
*
(2)
We finally obtain, for the probability to be maximized,
where dx represents Ili,j dxij, x stands for the set of numbers Xii and similarly for o,., a, ~~~and a. The generalization of eqs. (7) and (8) of ref. I’) gives
for the probability that the ith resonance is in category I. Sets of variables a, gpr and ocare to be found by the maximum likelihood method. When the appiofi probabilities al are determined experimentally, they represent the relative number of resonances in each category. Eliminating one of the variables CI~using eq. (I), we are left with mL-t-m+L- 1 independent variables. We can denote a general variable by zp where 1 s ~16 mLtm+L1. The maximum likelihood relation is then d In P/Sz, = 0
for all p,
(5)
with an error matrix given by (h-l),,
= d2 In
P~az,az,.
(6)
The off-diagonal elements of f2represent covariances between different variables while
54
et al.
A. I. NAMENSON
the diagonal elements represent variances. In practice these off-diagonal elements are very small, and we are only concerned with the diagonal ones, lj~;S = (h-l),,
pjaSEl.
= a2 la
(7) More complete discussions of the maximum likelihood method can be found in many books on statistics [see for example ref. ““)I. It must be pointed out before continuing that because levels of the same spin repel each other, the spacings of resonances can also contribute information about their spins. Eqs. (l)-(4) d o not involve level spacings and so do not account for this factor. Including level spacings in this calculation would make it enormously complicated, and quoted results would be significantly changed only for the calculated variances on the relative populations of the different categories, CI~.These variances are obviously sensitive to the distribution Iaw for resonance spacings, since the more reguIar the spacings of resonances, the smaller the uncertainty due to a sampling of only a finite number of resonances. However, it is still useful to calculate the variance in relative populations on the present basis because: (i) the results will be valid for those cases where large numbers of resonances are missing from the analysis; (ii) the results will always represent an upper limit to the variance; (iii) the form of the results is highly instructive and we may go from them to more exact solutions by simple intuitive arguments. 4.2. FORMULAS
RESULTING
FROM
THE MAXIMUM
LIKELIHOOD
ANALYSIS
To simplify the solutions we introduce the following definitions:
wij=
lla~jj,
@I
Wij$ = Pii Wcj *
(9)
From eq. (S), ajl
=
i
i=l
011= n,/n,
Wijl Xii/ i
i=l
Wi,, 9
(10)
where n, = k pit. i=1
Eq. (7) gives us the variances in the above averages. Because the general expressions are so complicated we will restrict ourselves to the case of interest here, where there are only two possible categories (namely spins in s-wave capture),
5s
SPINS IN ‘75Lu, I*‘* ‘s90s 4.3. AVERAGE
REDUCED
NEUTRON
WIDTHS
AND
STRENGTH
FUNCTIONS
All the resonances in a given category wilt have reduced neutron widths which fluctuate about a certain average according to a x2 distribution with Y = I. Instead of the reduced neutron widths, it is more convenient to use the quantities ZgT,Oiwhere i refers to the number of resonance. We assume that the experimental errors in these numbers are small compared to the Porter-Thomas fluctuations. The average for the Zth category is <2#:),. In order to simplify the following equations, we introduce the following substitutions:
& = 2gr,D,,
ZE = (2grZ),.
(16)
If there are n’ resonances with measured neutron widths,
expresses the probability which is to be maximized with respect to Z. The symbols [ and Z stand for sets of numbers. We introduce the probability
and using the maximum Iikelihood method, (19) Once again, in the expression for the variance, we consider only two possible categories and obtain
The s-wave strength function and its variance are obtained from the relations
var [S,(l)]
= [var ((2gr,O>,)J (i
pi;/2dE)‘.
i=l
4.4. INTERPRETATION
OF THE EQUATIONS
In giving intuitive explanations for the equations presented above, only the most important points will be considered. Equations similar to (3), (IO), and (11) have been discussed before I’). Eq. (12) states that the relative population of any category is simply the average probability of a resonance belonging to that category. The quantity PINplays the role of the number of resonances in category I. The probability
56
A. I. NAMENSON
et al.
pi1 in eq. (18) is the probability that the ith resonance is in category I when there is a significant difference in the average <2grt), for the different spin groups. In such a case, the neutron widths would contribute information about the most likely spin. Note that if the averages (2gFt), are not very different for the different spins, we would have Pfl “piI. This is actually the case in almost all practical situations. Eq. (19) for <2grt), is just a weighted average over all CIwhere the weighting factors are the probabilities p;. Eqs. (13)-(15) and (20) show how the uncertainties in spin measurements act to increase the variances through the sums of terms containing the factors pilpiz. If all the spins were known with certainty, the terms pilpil would disappear. In such a case, eq. (15) would become the variance in a binominal distribution of order n, and eq. (20) would be the variance in the average over n’ resonances which follow a Porter-Thomas distribution. In fact, it is useful and interesting to express eqs. (15) and (20) in approximate form, n
var(al) W RI Ez/n+ C Pi1 pi*ln2,
(23)
i=l
var ((2gTz)J
227 = o’(Z,) z -
iF;1 Pi;(Si-z*)2
(24)
+ (i&)*
&l
*
We can see that in both cases, the variance splits into the sum of two terms. The first term is the variance due to the finite number of resonances and the second term is the variance due to the uncertainty in spin assignments. For the approximation to be valid, the second term should be less than the first. The summation z:rpIr in eq. (24) plays the role of the number of resonances with measured neutron widths which belong in category 1. 4.5. VARIANCES
IN THE RELATIVE POPULATIONS
OF DIFFERENT
GROUPS
It is instructive to see how eq. (23) could have been arrived at by intuition. Referring to eq. (12), we see that if all resonances were categorized with certainty, nl would be experimentally determined by a simple counting procedure. When resonances are not known with certainty, its must be calculated from eq. (12), and from statistical arguments, the experimentally determined value of 12~ would have a variance of ”
var (4
= igp”(l
-Pill,
(25)
and for only two categories
var(4 = f
i=l
Pi1 Pi*.
(26)
The variance in c(~consists of two independent variances: one due to the finite value
SPINS
IN ‘75Lu,
1*7*1*90b
57
of n,, and one due to the uncertainty in determining IP, itself, var (gl) = var due to finite number of resonances+
var (ur) ___. n2
(27)
Only the first term of eq. (27) is affected by the specifics of the distribution of level spacings. When these spacings convey no information about spin, the first term is the we11known variance for a binominal distribution and eq. (23) would follow. For a Wigner distribution of Ievel spacings, it can be shown that
Long range ordering effects of the kind suggested by Dyson, and Mehta and Dyson ““) could slightly affect the first term of eq. (28) but this is not expected to be a significant alteration. Note that for eq. (28) to be valid, the second term must again be smaller than the first. 4.6.
PRACTICAL
SOLUTION
OF EQUATIONS
can be solved by introducing initial guesses for the parameters Eqs. (4), (lo)-(12) aji, oPrj and a, and then using a computer to solve by the method of iteration. After obtaining solutions for these quantities and for the probabilities pil, one can use eqs. (18)-(22) t o soIve for the averages
175Lu
Table 1 shows the resonances in ’ ” Lu, the spin assignments and their probabilities of correctness. There is complete agreement with a previous measurement of eieven of these spins 16). For iIIustration purposes, an optimum Iinear combination of the two independent ratios described in subsect. 3.1 (using the same method described there) was taken to yield the combined ratios in fig. 9. We can see that there are only a few resonances whose spins are ambiguous. The histogram in this figure results from dividing the vertical scale into equal intervals and obtaining the number of resonances in each interval. However, since some of the resonances have rather wide error bars, there is the problem of exactly where to place them. This was solved by noting that the error bars represent a Gaussian probability distribution for the location of the resonances, and so we can calcuiate for each interval the expectation value for the number of resonances in it. We see that the ratios clearly divide into two distinct groups, and that there are about two resonances located in the middle of the two
TABLE1 Spins of neutron resonances in 1T5Lu Neutron energy (eV>
Spin
2.6 4.7 5.2 11.2 14.0 15.3 20.5 23.5 28.1 30.2 31.1 36.5 40.7 49.2 50.2 53.5 56.7 60.9 69.6 85.3 88.1 96.3 99.7 100.9 102.8 107.3 112.7 115.1 118.6
4 4 3 3 3 4 3 3 4 4 3 3 3 3 4 4 3 4 4 A
Measurements Measurements
Probability (%I
Neutron energy feV
Spin
Probability (%I
127.4 129.4 137.8 143.0 146.2 148.8 150.9 155.4 158.7 164.0 169.2 171.5 175.8 180.9 185.4 193.3 196.8 202.6 204.4 217.6 223.9 227.9 229.6 236.9 244.5 251.5 256.4 274.0
3 A
100 99 86 98 99 63 9s 99 100 99 98 94 97 99 99 99 94 51 81 94 94 9s 99 5s 97 69 99 99
100 100 100 100 96 100 100 JO0 100 100 100 97 100 99 99 99 93 99 98 99 70 100 99 82 100 99 99 100 99
4 A 4 4 3 3 3
3 I:1 4 4 3 3 4 3 3 4 3 3 ;I (4) 4 3 4 A 1:1 3 4
with confidence limits between 70 % and 89 % are enclosed in parentheses. with confidence limits below 70 % are enclosed in square brackets.
SPIN
-.0
.___ 50
IO0
3
/ - ..-.- _.... 200 150
ENERGY (eV)
250 .._-.
1.. __i _! 246 NUMBER
Fig. 9. Combined ratio for 175Lu. On the left is a plot of the combined ratio versus resonance energy. The solid lines are the weighted averages for the respective spin groups and the dashed lines show the extent of the Porter-Thomas fluctuations. The error bars on the points refer to experimentai uncertainties. Resonances marked by triangles are those whose spins have been identified with better than 90 % confidence. On the right is a histogram of the ratio versus the number of resonances.
59
SPINS IN 175Lu, ls’~l*sOs
groups. Considering the magnitude of the Porter-Thomas fluctuations and the experimental errors, it is not unreasonable to find two such ambiguous ratios out of 57 resonances. The ratios of fig. 9 could have served for calculating the probabilities in table 1 without introducing much change. 5.2. THE NUCLEUS
‘89Os
Table 2 shows the resonances of rsgOs in a manner similar to table 1 for I7 5Lu. There is complete agreement with the 12 spin measurements of Wetzel and Thomas “). TABLE 2
Spins of neutron resonances in 1890s Neutron energy (ev)
87 8 93.1 104.3 109 110 118 124 128 131 137 141 145 150 160 162 165 167 178 187
100 100 100 99 100 97 100 67 90 78 100 100 95 97 87 96 63 99 19
6.63 8.95 10.3 18.7 22.1 27.4 28.3 30.5 39.1 43.3 50.3 54.9 60.7 63.9 65.4 66.0 72.2 75.3 76.2 Parentheses
Neutron energy (ev)
Probability (%)
Spin
Spin
(1) (1)
r:1
(1) PI (2)
r:1 I21 Dl Ul 121 [:I (2) HI [I I Dl
Probability (%) 83 72 97 56 75 51 70 91 65 57 63 66 53 90 51 78 63 63 69
and square brackets have the same meaning as in table 1. ~,
0.4SPIN
1
2
.
,-.
00
;.
‘_
SPIN
I
.
(8
”
-0,l 0
50
100 ENERGY
150 (eV)
246 NUMBER
Fig. 10. Combined ratio for 1890s. The explanation is similar to that of fig. 9 except that resonances marked by triangles are those whose spins have been identified with better than 85 % confidence. Open circles indicate better than 70 % confidence. Porter-Thomas fluctuations are not shown since they are negligible compared to the experimental uncertainties.
A. I. NAMENSON
60
ei at.
Fig. 10 shows the ratios used in obtaining the spin values, and their probabilities of correctness. The histogram is similar to that of fig. 9 described in subsect. 5.1. The separation of the two groups is poorer than the case of “‘Lu, but there is still a definite separation. 5.3. THE
NUCLEUS
f870s
Table 3 for “‘0s is similar to the preceding two tables. Fig. 11 (for illustration purposes only) shows the results of taking an optimum linear combination of the ratios from the multiplicity experiment and the low-energy y-ray experiment as described before. In the histogram of fig. 11, the separation between the two groups is not very well defined and therefore the spin determinations were poorer than in the other cases. We agree with the measurement of Wetzel and Thomas s) for the 9.5 eV resonances, but disagree with their measurement for the 12.6 eV resonance. In the present measurement the multiplicity method corroborates the information from the low-energy y-ray method. It is suggested that the wide error bars in ref. “) do not exclude the possibility that the ratios for the 9.5 and 12.6 eV resonances are approximately the same. In any event, it is dangerous to make spin assignments with only two resonances, since one would have no estimate of Porter-Thomas fluctuations. TABLE 3 Spins of ls70s Neutron energy
Spin
(eV) 9.48 12.6 19.9 26.1 39.3 40.2 43.2 47.6 49.8 50.3 61.9 63.5 64.7 78.0 90.0 92.5 99.1 104.8 107.2 108.7 123.0 124.5
1 1 1 1:1 &
G, (1) (f.l) :,
Probability
W)
99 100 90 98 62 100 80 100 86 86 84 98 51
126.8 138.2 141.2 145.3 155.0 164.6 168.5 176.8 179.0 189.5 201.9 207.8 214.5 225.2 228.3 237.4 251.3 258.8 268.9 278.9 291.1 298.5
Ill
51
:1
91 62 61 96 58 99 55 60
WI ;1 I&
101
Neutron energy
(%I
Parentheses and square brackets have the same meaning as in table 1.
Spin
Probability (%)
(0) (1) LO1 (01 101
101 [Ol (1)
101 t01 t01 t01 (1) [Ol t01 111 ill t01 [Ol [Ol [Ol
101
81 II 55 71 57 65 52 70 58 51 52 58 72 54 52 62 57 55 55 55 55 52
SPINS
IN 175Lu,
0.4
I
‘=-OS 03
SPIN I
o 0.2 F 2 o.i* 00 -0,l
61
1sr.'890~
1
. : ;r.
.
’ SPIN 0
50
/
‘*;
0
150 200 100 ENERGY (eV)
Fig. f 1. Combined ratio for ’ 870s. The expfanation
6. Discussion
2 4
6
is simiIar to that for fig 10.
of results
Some of the average resonance properties are shown in table 4. The relative populations were computed from eq. (12). To compute the uncertainty in the relative populations, eq. (28) for the Wigner distribution was used for l7 sLu while for 1870s (where few spins were well determined), eq. (15) for a binomiai distribution was used. For ““OS, one average was computed for the resonances up to 75.3 eV, where the spins were welt determined, and its uncertainty was computed from eq. (28). For resonances above this energy, another average was computed and the uncertainty obtained from eq. (15). The two averages were then combined by the method of weighted averages with the reciprocal variances as weighting factors. A glance at the relative populations of the two spin groups in table 4 shows that the populations are consistent with a 2Jf 1 law for rs90s. Nevertheless, it is suggested that a generally better approximation to the level density in this mass region would be a formula with a spin cut-off factor of c z 3.5 for 175Lu and rs70s; the spin cutoff factors are 3.5:::; and 0.94,0:: respectively, where the limits represent 68 y: confidence. The spin cut-off factor for ig70s is unusuaily small, but even if the relative popuIations were more than one standard deviation off, we would still have o&y a 1.8 “/: chance that cr 2 3.5. Similar investigations I’> of neutron capture in 17?Hf and 179Hf indicate spin cut-off factors of about 3.5 at the neutron separation energy. It should be pointed out that even for ‘890s thereis still a IO 7: chance that 0 5 3.5. Turning to the averages (2grt>, and the strength functions S,(J) we see that the difference between the two spin groups for l7 5Lu are not really significant. For 18’Os the difference in (2grz), is not considered significant, but the difference in the strength functionsis somewhat suggestive; there is only a 5 % probability that this difference could happen by chance. However, other measurements on nuclei in this mass region have shown no spin dependence of the strength functions “*l’). Most of the uncertainty in the averages (2grz), and in the strength functions S,(J) are due to the small number of resonances with known neutron widths; the measurement of a large number of neutron widths in these nuclei would reduce these uncertainties. Resonance parameters for rs70s are not available at present.
A. I. NAMENSON
62
Spin dependenceof
TABLE4 average properties of resonances for 175Lu, ‘*‘OS and lasOs
Isotope
1’ SLU 1’ SLU 1’ SLLl I 890s 1890s ‘890s ‘S’OS 18’Os 19’0s
ef nl.
Relative population CX) spin 4 spin 3 difference spin 2 spin 1 difference spin I spin 0 difference
48% 4 52* 4 68& 7 32% 7
<2sr”“> “) (meV) 0.68hO.39 1.63hO.73 -0.95f0.83 9.0 *4.1 1.8 &1.6 7.2 +4.4
S&r) x 104
0.38rtO.22 1.4910.67 -1.11+0.70 7.6213.47 0.84&0.07 6.78k3.47
44*14 56&14
Note that the errors for the relative populations of two different spins in the same isotope are completely coherent, since the two relative populations must sum to 100 %. “) Values of 2gFma were taken from ref. 26).
It is of some interest to perform a Wald-Wolfowitz 25) calculation on the isotopes 175Lu and 1890s to test the number of “runs” of one spin or the other to check that there were no systematic variations in the measurements other than spin. (A run is an unbroken sequence of resonances all having the same spin.) The estimate of ref. 25) could be applied to spins if the level spacings followed a Poisson distribution. However, for a Wigner distribution, if the two groups have roughly equal popuIations, a Monte Carlo calculation showed that the number of runs should be 1.4 times this estimate. For ls90s the calculation was performed for resonances up to 66.0 eV, and excellent agreement was obtained. For “‘Lu the calculation was performed for all resonances, and the number of runs was smaller than the above estimate by slightly under two standard deviations. There are, therefore, no obvious systematics in the data. 7. Conclusions and summary In examining the nuclei studied here and elsewhere, we can con&de that in the mass region A x 180, a spin cut-off factor of about 3.5 gives a better estimate of the relative populations of the different spin groups than does a 2J+ 1 law. The differences in the averages {2grz>, are not statistically significant, but the differences in the strength functions are suggestive in the case of lagOs. It is strongly suggested here that in other measurements of this kind, probabilities of correctness accompany spin measurements. When this is done, one would find that many seemingly clear assignments actually had a 5 % to 10 % chance of being incorrect. In calculating average properties for a group of resonances, these probabilities are necessary to properly estimate uncertainties. Finally, since the calculations are easily computerized, the analysis can be done more rapidly and less subjectively than by less quantitative methods.
SPINS IN 175Lu, ‘87~1890s
63
References I) I.,.M. Bollinger, in Experimental neatran resonance spectroscopy, ed. J. A. Harvey (Academic 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)
Press, New York, 1970) p. 317 C. Coceva, F. Corvi, P. Giacobbe and G. Carraro, Nucl. Phys. All7 (1968) 586 P. Giacobbe, M. Stefanon and G. Dellacasa, CNEN Technical Report RT/Fi (68) 20 A. I. Namenson, Nucl. Phys. A209 (1973) 252 J. R. Huizenga and R. Vandenbosch, Phys. Rev. 120 (1960) 1305 W. P. Poenitz, 2. Phys. 197 (1966) 262 W. P. Poe&z and J. R. Tatarczuk, Nucl. Phys. All7 (1968) 586 K. J. Wetzel and G. E. Thomas, Phys. Rev. Cl (1970) 1501 H. Feshbach, Ann. Rev. Nucl. Sci. 8 (1958) 49 J. JuIien, G. Bianchi, C. Corge, V. D. Huynh, G. Le Pottevin, J. Morgenstern, F. Netter and C. Samour, Phys. Lett. 10 (1964) 86 J. Julien, CEA-R-3385 (1968) C. M. Newstead, J. Delaroche and B. Cauvin, in Statisticalproperties of nuclei, ed. J. B. Garg (Plenum Press, New York, 1972) p. 367 H. V. Muryadan and Yu. V. Adamchuk, Nucl. Phys. 68 (1965) 549 A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 C. Coceva, F. Corvi, P. Giacobbe and M. Stefanon, in Stati~t~calproperties ofnuclei, ed. J. B. Garg (Plenum Press, New York, 1972) p. 447 0. A. Wasson and R. E. Chrien, Phys. Rev. C2 (1970) 675 A, Stolovy, A. 1. Namenson, J. C. Ritter, T. F. Godlove and G. L. Smith, Phys. Rev. C5 (1972) 2030 T. F. Godiove, G. L. Smith and A. I. Namenson, Nucl. Instr. 95 (1971) 595 M. K. Balodis et al., Nucl. Phys. Al94 (1972) 305 M. A. Mariscotti, W. R. Kane and G. T. Emery, Brookhaven National Lab. Rep. BNL-11426 (1967) E. Bashandy and S G. Hanna, Nucl. Phys. 84 (1966) 577 H. E. Jackson and L. M. Bolfinger, Phys. Rev. 124 (1961) 1142 H. Cramer, Mathematical methods of statistics (Princeton University Press, Princeton, 1946) F J. Dyson, J. Math. Phys. 3 (1962) 140, 157, 166; M. L. Mehta and F. J. Dyson, J. Math. Phys. 4 (1963) 701, 713 G. D. James, AERE-R 6633 (1971) S. G. Mughabghab and D. 1. Garber, Neutron cross sections, Brookhaven National Lab. Report BNL-325, 3rd ed., vol. 1 (1973)