- Email: [email protected]
Neutron capture in 238U
Nuclear
A121 (1968)
to be reproduced
by photoprint
NEUTRON
@
North-Holland
CAPTURE
IN 238U
D. L. PRICEt, R. E. CHRIEN, 0. A. WASSON, M. R. BHAT, M. BEER tt, M. A. LONEt?? and R. GRAVES Physics
Department,
Brookhacrrr Received
Co., Amsterdam
or microfilm without written permission from the publisher
National Laboratory:, 9 September
tt
Upton, New York
1968
Abstract: They-ray spectrum following slow neutron capture in 23RU has been measured as a function of incident neutron energy at the Brookhaven HFBR fast neutron chopper. The decay scheme proposed by previous (d, p) and high resolution thermal (n, y) measurements has been substantially verified. Partial radiation widths for s-wave resonances up to 117 eV have been deduced. By using a simple multi-level analysis, the spectra between resonances from 21 eV down to thermal have been fitted. The behavior of the strong transitions to low-lying I = 1 states is explained using a direct capture model, while the transitions to low-lying collective states can be explained by level-level interference. The distribution of El transition strengths is not consistent with a x2 distribution of one degree of freedom.
fi
NUCLEAR REACTIONS Z3*LJ(n, y), E = 0.002-120 eV; measured ~J(E; E,,) z39U deduced levels, J, r,,(E?), relative signs of [r,,ry(E,)]t. Enriched target. --
1. Introduction The nuclei of the actinide series have in general non-spherical ground states due to coherent effects between nucleons outside the closed-shell configuration. The higherenergy states can be caused by vibrational or rotational collective excitcltions in addition to intrinsic excitations. The unified model ‘) has been applied with some success to these nuclei, and the odd-A nuclei are particularly interesting in that the mtrinsic excitations often correlate well with Nilsson’s calculations of nucleon binding states in spheroidal nuclei ‘). This paper describes a recent study of the energy levels in 23gU observing the gamma rays following neutron capture in 238U. Previous work on the energy levels of 23gU, using both neutron capture and (d, p) reaction techniques, has been reported 3-6) . However, the investigation reported here is the first to have studied neutron capture in conjunction with high-resolution y-ray detectors as a function of incident neutron energy. The germanium detectors used have a resolution for y-ray energy as low as 6 keV. Since the average level spacing near the ground state of 2 3gU is about 30 keV, this resolution is necessary if closely spaced t Present address: tt State University ttt t
Solid State Science Division, Argonne National Laboratory, Argonne, Illinois. of New York at Stony Brook; present address: Brookhaven National Laboratory, Upton, New York. State University of New York at Stony Brook; present address: Physics Division, Chalk River Laboratories, Chalk River, Ontario, Canada. This work was supported by the U.S. Atomic Energy Commission. 630
N-CAPTURE IN SZSU
631
levels are to be distinguished. The use of energy analysis by means of the time-of-flight technique makes it possible to study the intensities of the transitions to a given final state as they vary over the capturing resonances and the energy regions between the resonances. This has a number of advantages over a thermal capture study. First, the fluctuation in intensity expected from a Porter-Thomas distribution of partial radiation widths 7) m a y make some y-rays appear too weakly in thermal capture to be observed. Second, the variation in relative intensity between transitions to closely spaced levels makes it easier to distinguish them. Third, the possibility of averaging a partial radiation width to a given final state over a number of resonances makes it easier to estimate the multipole order involved, often a crucial factor in identifying the final state. Fourth, it is possible to obtain information about the spins and parities of the capturing states. These points are illustrated in the following sections. The study of the variation in intensity of a given transition as a function of neutron energy has one further feature of considerable interest. The intensity in the regions between resonances may be expected to display interference between contributions from nearby resonances with the same symmetry properties. By studying this interference it is possible to deduce the relative signs of the resonance amplitudes, in addition to the magnitudes which are derived from the intensities at the resonances. Failure to obtain a set of resonance amplitudes which fits the data in this manner must be an indication either of contribution from unknown resonances, for example bound states, or of the fact that direct processes are involved. The present investigation has revealed one case where direct capture is almost certainly involved. This has been briefly reported in a letter s) and is discussed in more detail in this paper.
2. Experimental procedure The experiment was performed at the fast chopper of the Brookhaven High Flux Beam Reactor 9). Briefly, a straight-slot rotor placed at the exit of a beam tube produces periodic bursts of epithermal and resonance energy neutrons. These neutrons are well collimated (8.0 cm F W H M at the target) and fall on the sample placed 22 m away. The y-rays produced by capture events are detected in a germanium diode detector placed near the sample, and the pulses are fed to an on-line computer where they are processed and stored on magnetic tape 10). The events are analysed with respect to time-of-arrival relative to the time the burst is at the rotor and to pulse-height in the detector. These are related to the neutron and y-ray energies, respectively. The sample was a slab of uranium depleted i n / 3 s U , 0.6 cm thick, 50 cm 2 cross section. Three series of runs were made: a) at a rotor speed of 6000 rpm, with a germanium detector of volume 30 cc, resolution 14 keV F W H M , run time 54 h. b) at a rotor speed of 1500 rpm, with a 4 cc detector, 9 keV F W H M , run time 51 h. c) at a rotor speed of 1500 rpm, with a 4 cc detector, 6 keV F W H M , run time 89 h.
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N-CAPTURE I N 23SU
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The series at the higher rotor speed had the best time-of-flight resolution and provided data for the resonances up to 117 eV; most of the information on partial radiation widths was derived from these runs. However, this detector had comparatively poor resolution. Series (b) was made to provide information for thermal capture; thermal is here used to mean neutron energies up to about 0.2 eV; in other words, energies small compared to those at the resonances. The lower rotor speed increases the incident flux for these neutrons. The object of this series was to provide an intercalibration so that the partial widths measured in the first series could be expressed in absolute magnitudes via the thermal measurements of ref. a). When the first two series of runs were evaluated, it became clear that there was considerable interest in the region between thermal and the first and second resonances which could profitably be studied with higher resolution. A further series of runs was, therefore, made at the lower rotor speed with a new detector which had a resolution of 6 keV at 4 MeV. Part of the time-of-flight spectrum for the first series is shown in fig. 1. This includes all v-rays depositing more than 1.96 MeV in the detector. Seven resonances can be resolved at this rotor speed. The highest neutron energy peak (shortest time-of-flight) actually contains two unresolved resonances. All the resonances represent s-wave capture; the small peak at 10 eV has previously been considered as a p-wave capture on intensity considerations 11), but will be shown below to be s-wave. A v-ray spectrum is obtained for each resonance by scanning the magnetic tape on a CDC-6600 computer 1o). A spectrum is formed for those events whose time-offlight falls between two limits set on either side of the resonance peak. The double arrows in fig. 1 show the limits for the scans of the seven peaks; two " b a c k g r o u n d " scans were also made between the resonances. The " b a c k g r o u n d " scans include a significant contribution from off-resonance capture in 2a8U as well as a background component. Similar scans were made in the thermal region for series (b). For series (c) scans were made for the 6.7, 10.2, 21 and 37 eV resonances, and in various regions between the resonances, discussed in sect. 5. Figs. 2 and 3 show two regions of such scans for the three strongest resonances in the first series of runs. The v-ray peaks are indicated by the energy in keV. In general, each v-ray produces three peaks in the detector corresponding to full capture, single escape, or double escape of the photons following a pair production event; these are separated by intervals of 511 keV, and are indicated by the letters F, S and D in the figures. The relative sizes of these peaks depend on the detector volume and the energy of the v-ray. In some cases, two y-rays cannot be resolved, which happens when their energies are close together or separated by nearly 511 or 1022 keV. Compton processes produce a continuum background under the peaks. This rises towards lower energy making it harder to resolve peaks in this region. This limited the present study to y-ray lines between about 3200 keV and 4802 keV, the value of the neutron binding energy in 239U.
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3. Interpretation of observed transitions Sheline e t al. 3) have proposed a level scheme for 239U for the region within 1000 keV of the ground level, based on their thermal capture and (d, p) measurements. This is shown in fig. 4, in which the thin bars designate levels proposed by Sheline
636
D.L.
P R I C E et al.
et al. related their (n, 7) and (d, p) results by identifying the strong 7-line at 4059.4
keV with a strong (d, p) transition to a level measured to be 742.3 keV about the ground state. This leads to a value of 4801.7 keV for the neutron binding energy. The energies of the 7-rays measured in this work were derived by assigning the Sheline et al. value to the 4059 line and measuring energies relative to this line. In all cases
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Fig. 4. Level scheme proposed by Sheline et al. (thin bars) compared with measurements and assignments of this experiment (thick bars). Levels are labelled by energy in keV above the ground state. The letters d, n in parentheses signify whether the Sheline et al. measurement is from (d, p) or (n, 7) studies, respectively. where these lines could be identified with lines measured by Sheline e t al. in either the (n, 7) or the (d, p) studies, the energies were found consistent to within the errors quoted. Since absolute energy measurements were not a primary aim of this work, their values will be used in such cases. In cases of lines not observed by them, the line energies are derived by measuring relative to the 4059 keV line, and the level energies by taking their value for the binding energy. The levels are shown grouped in rotational bands associated with a particular Nilsson orbital 2). There is a well-defined gap from 400 to 600 keV due to the closure o f the 152-neutron shell. The capturing state for all resonances measured in the pres-
N-CAPTURE IN 238U
637
ent study is ½+. The E2 transition to the ~}+ ground state is not observed, but the transition to the 133 keV level is seen. This was seen in the (d, p) work of Sheline et al. and identified with the base of the ½+ [631 ] band, but was not observed in thermal capture, presumably a consequence of the Porter-Thomas distribution of partial radiation widths 7). It was, however, seen by Bergqvist in fast neutron capture work 6). Also observed are transitions to levels at 143 and 191 keV, probably the next higher levels, 3+ and ~z+, in the ½+ [631 ] band which are found in thermal capture. These three levels must be fed by M1 and E2 transitions because there are no negative parity Nilsson orbits with low spin r.mmber in this region. Above the 152-neutron gap, the two levels seen in thermal capture around 700 keV are not observed. The level at 742 keV corresponds to the dominant line of the large y-ray peak at 4059 keV. Sheline et al. observe structure on the low-energy side of this peak and take it to be the next higher level in the ½- [761 ] band, identified with a level observed in the (d, p) work at 754 keV. However, in the resonances this line only appears weakly as structure in the low energy wing of the 4059 keV line (see, for example, fig. 7). On the other hand, there is a strong line at 4068 keV which is not seen in thermal, but shows very strongly in the higher resonances, as can be seen from figs. 3 and 7. This corresponds to a level at 733 keV, and it seems more probable that this is the base of the ½- [761 ] band, and that the level at 754 keV is a positive parity state. The same could be true of the weak level observed in the resonances, but not in thermal capture, at 799 keV. Possible states could be found in the ½+ [620] or ~+ [622] bands. The only other lines deserving special attention belong to the strong doublet at 3981 and 3992 keV, corresponding to levels at 811 and 820 keV. These were not seen in the (d, p) measurements and Sheline et al. tentatively interpret this doublet as an octupole band built on the ½+ [631] orbital. The fact that such a doublet occurs in 238U at 679 and 724 keY above the base level 12) would seem to support this interpretation. However, there are also two negative parity orbitals in this region which provide an alternative explanation. 4. Evaluation of partial radiation widths Partial radiation widths derived from the resonance measurements are given in table 1. These were evaluated as follows. The intensity of the line in the resonance was divided by the intensity of the total ~-ray yield above an energy of 1.96 MeV. This ratio is proportional to the ratio of partial to total radiation width. If the lower energy cut is such that a large number of 7-rays are included in the region above it, the proportionality constant should be the same for all resonances, and should also hold for the thermal measurements. It was, therefore, evaluated by comparing the ratios measured for the thermal run with the values of intensity per thermal capture given by ref. 3). Partial widths were then derived by taking a common value of 26 meV for the total radiation widths of all resonances 11). Corrections were made for background caused by neutron absorption in the detector and for incompletely resolved lines in
Level energy (keV)
133 143 191 733 742 754 799 811 820 928 959 984 1149 1162 1190 1218 1234 1262 1355 1441 1483 1508 1520
Gamma-ray energy (keV)
4669 4659 4610 4068 4059 4048 4003 3991 3982 3873 3843 3818 3653 3639 3612 3584 3567 3540 3446 3361 3318 3293 3282
0 -t-0.03 0.11 4-0.03 0.07 4-0.02 0.03 4-0.03 0.63 4-0.03 0.09 4-0.03 0 4-0.02 0.76 4-0.06 1.13 4-0.07 0.0264-0.040 0.13 4-0.05 0.26 4-0.05 0.10 4-0.02 0.12 4-0.03 0.05 4-0.03 0.54 4-0.04 0.07 -t-0.03 0.10 ±0.03 0.08 4-0.08 0.14 4-0.08 0.0314-0.058 0.0054-0.06 0.13 4-0.06
6.7
TABLE 1
21
04-0.03 0.15 4-0.04 04-0.03 0.06 4-0.05 04-0.03 0 4-0.02 0.01 5:0.23 0.26 4-0.05 0.29 4-0.23 0.93 4-0.03 04-0.03 0.15 4-0.04 04-0.03 0.10 4-0.05 0.53 4-0.26 0.23 4-0.03 0.91 +0.27 0.59 4-0.04 --0.07 4-t-0.25 0.0554-0.037 0.29 4-0.27 0.27 4-0.04 0.42 4-0.27 0.45 4-0.04 0.45 4-0.30 0.15 4-0.05 0.092-t-0.29 0.15 ±0.04 0.15 4-0.30 0.11 4-0.04 0.66 4-0.32 0.14 4-0.04 0.74 4-0.32 0.30 ±0.05 0.49 4-0.32 0.09 4-0.07 0.26 4-0.32 0.12 4-0.05 --0.0354-0.346 --0.0554-0.069 0.19 4-0.36 0.20 4-0.05 --0.0554-0.36 0.20 4-0.08 0.16 4-0.38 0.14 4-0.04
10.2 0.27 4-0.09 0 4-0.02 0 5:0.02 0.24 ±0.03 0.10 4-0.03 0.08 4-0.05 0 4-0.02 0.15 4-0.03 0.32 4-0.03 0.32 4-0.05 0.19 4-0.07 0.0754-0.059 0.07 -t-0.04 0.17 4-0.05 0.25 4-0.05 0.55 ±0.05 0.38 4-0.05 0.20 4-0.08 0.0404-0.078 0.20 4-0.06 0.18 4-0.09 0.18 4-0.09 0.15 4-0.09
37
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Resonance energy (eV)
Partial radiation widths for 2~sU(n, y)'z39U, in units of meV
0.09 ±0.13 0.18 I 0 . 1 8 0 4-0.03 0.15 4-0.12 0.65 4-0.09 0.31 4-0.19 0 4-0.03 0.37 4-0.22 0.24 4-0.11 0.40 4-0.13 0.041+0.159 0.33 4-0.12 0.12 4-0.18 0.25 4-0.19 0.26 ±0.19 0.33 4-0.19 0.36 4-0.13 0.29 4-0.20 0.0924-0.213 0.24 4-0.23 --0.07 5_0.23 0.16 4-0.24 0.12 4-0.24
81
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103+117 0.08 0.08 0.009 0.18 0.51 0.11 0.04 0.31 0.48 0.17 0.18 0.21 0.17 0.15 0.17 0.30 0.32 0.17 0.05 0.10 0.08 0.09 0.08
Average
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N-CAPTURE IN ~ U
639
the cases mentioned in sect. 2. The intensities for the strong lines at 3982, 3991, 4059 and 4068 keV were derived by Gaussian fitting, except for the 10.2 eV resonance. Where possible, intensity information was derived from both full-capture and doubleescape peaks. For the values for the 6.7, 21 and 37 eV resonances, the results for the third series of runs were averaged in with those of the first and second. The errors quoted a re standard deviations, and there are possibly systematic errors beyond these. However, whenever comparison between separately derived numbers was possible (Gaussian fits v e r s u s numerical calculation, first v e r s u s third series) the agreement was consistent w~th the quoted errors. I
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Fig. 5. G a m m a - r a y energy spectra a r o u n d 4 M e V f r o m scans o f 10.2 eV resonance a n d b a c k g r o u n d regions for r u n s at 6000 r p m r o t o r speed.
The last column represents an average over the eight resonances which should give an indication of the multipolarity of the transition. The top two rows represent the M1 transitions to the low-lying levels discussed in the last section. The widths for these transitions are seen to be anomalously large, averaging nearly 0.1 meV. This is about a hundred times the width estimated with a single-particle model, allowing for the appropriate level density at the capturing state. The corresponding estimate for E1 transitions is 0.2 meV, which can be seen to be typical of the values found for those transitions further down the table. It appears, therefore, that the M1 transitions are enhanced by two orders of magnitude, presumably by some kind of collective motion, involving the excitation of 2p-lh states, as proposed by Bergqvist t 3).
640
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N-CAPTURE IN ~ZSU
641
It can be seen that the widths for the 10.2 eV resonance, where they can be measured, have a similar behavior to those of the other resonances. For example, in the higherenergy 7-ray region, the only peaks appearing are the two doublets around 4 MeV which dominate the other resonance spectra. These are illustrated in fig. 5. This supports the conclusion that the 10.2 eV resonance is due to s-wave capture, not p-wave as stated in the literature ~i). Uz38 (n,y) U239 I I 1 1 1
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F i g . 7. E q u i v a l e n t s p e c t r a t o t h o s e o f fig. 6 f o r 21 e V a n d 37 e V r e s o n a n c e s .
5. Analysis of partial cross sections
Several authors have proposed a direct reaction mechanism for the radiative capture of neutrons 14-~7). The best evidence for such a process can be found in the interference between the slowly varying reaction amplitude characteristic of the direct reaction and the sharply peaked resonance amplitude characteristic of the reaction via the compound nucleus. Such interference has been found is) for the ground state transition in 59C0(n, 7)6°C0, and the strongest lines in these measurements were investigated in the same manner.
642
D.L. PRICEet al.
O n e striking effect noticed in the first m e a s u r e m e n t s was the sharp variations in the strong doublets near 4 M e V between t h e r m a l c ap t u r e and the lowest r eso n an ce at 6.7 eV. F o r this reason, a third series o f runs was m a d e with a high-resolution germ a n i u m detector at 1500 r p m r o t o r speed (see sect. 2). Th e variations were b r o u g h t I
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20
Fig. 8. Ratio of partial to total capture cross section for 4059 and 3982 keV gamma rays plotted on a logarithmic scale. The double arrows at the bottom represent the energy limits of the scans from which the points above were derived. The positions of the resonances are indicated by the vertical arrows. The dashed lines ( - - - - ) represent the fits obtained from the single-level formula; the dot-and-dash curves (- . . . . . . ) those from the multi-level interference formula, and the solid curve for the 4059 keV line includes a direct amplitude.
out dramatically in the scans for these lines in the three lowest resonances and various n e u t r o n energy regions off resonance, including the region near t h e r m a l energy. T h e spectra for the two doublets f r o m five o f these scans is sh o w n in figs. 6 and 7. T h e difference between the t o p two spectra in fig. 6, f o r t h e r m a l an d the 6.7 eV resonance, should be noted.
643
N-CAPTURE IN ~88U
The ratio of partial to total capture cross section was measured over these scans for the stronger lines. The method was the same as that used for the evaluation of partial widths described in sect. 4; normalization was made with the thermal intensity measurements of ref. 3). N o corrections for neutron energy resolution have been made. IxlO-I~:
I
I
I
I
I
6.7
D
I
I
I
I
L
10.2
:
21.0:
i~;~
3 991 y RAY
I xlO-e--
k
.~
2
'~:
=:
"
4
I x l O -2 -~ DO= 0
/
b i xlO-~,
/
/
4 0 6 8 y RAY
// /
0°=7.o/',
I x l O -4
IxlO-~
I
2
I 4
6
I
I
I
I0 12 14 ENERGY, eV
f
16
I
18
I
20
Fig. 9. R a t i o o f partial to total c a p t u r e cross section for 4068 a n d 3991 keV g a m m a rays p l o t t e d o n a logarithmic scale. T h e lines represent t h e fits o b t a i n e d f r o m the multilevel interference f o r m u l a . F o r t h e 4068 keV line, t h e d a s h e d curve represents multilevel interference only, t h e solid curve includes a direct amplitude.
The results in the neutron energy region up to 21 eV for four lines are given in figs. 8 and 9, in which the points represent the ratios measured in this way. The form of the partial cross section can be derived from R-matrix theory 19). In the approximation F~ << D, a small ratio of width to level spacing, it can be shown
D. L. PRICE et al.
644
that the partial cross section can be written O.nrf =
~ 0
IDO+ (rool~(r~,,)~2.
(1)
E;t-E +½iF a
Lane and Thomas 2o) show that the above expression is obtained by dropping the off-diagonal elements of the level matrix. The superscript zero indicates values at 1 eV, D Orepresents the amplitude of the direct capture process. It can be seen that the interference between this and a single resonance produces a cross section asymmetric about the resonance, whereas the total capture cross section is symmetric if a large number of v-rays are involved. Three types of theoretical curves were compared with the data: a) A pure single-level formula represented by the sum of the squares of individual resonance terms in eq. (1). Since all the resonances affecting the region considered have the same spin, parity, and neutron orbital angular momentum, this is not physically meaningful. It is nevertheless useful as a base against which to judge the interference fits. b) The multilevel interference formula without direct capture obtained by putting D O = 0 in eq. (1). Such interference has already been observed 21) in partial capture cross sections. Since the experiment only measures the magnitudes and not the signs of the product of the reduced width amplitudes (Fx~,f)½(F°,)~, the signs of these terms were varied by trial and error to give the best fit to the data. c) Direct capture was then included to improve the fit if necessary. The total cross section derived from the neutron and total radiation widths for the lowest eight resonances 11) gives a discrepancy of 0.37 b relative to the measured 2.73-1-0.04 b for the thermal capture cross section. This is presumably due to the influence of bound levels. The discrepancy was removed by postulating a level with a large negative energy and a suitably chosen neutron width. This makes the total cross section correct at thermal energy. The effect on the partial cross sections is unpredictable; initially it was assumed to be zero. The theoretical curves are shown on figs. 8 and 9. The region around the 10.2 eV resonance is omitted because the small neutron width causes a sharp oscillation in this region. For the 4059 keV line, strong constructive interference is observed below the resonance. No amount of multilevel interference could account for the behaviour of the cross section ratio, even in the extreme assumption that the signs of all resonance amplitudes up to 117 eV were such as to produce constructive interference. However, a fit could be obtained (see fig. 8) by postulating a direct capture amplitude D O = 5 x 10-5, equivalent to a potential capture cross section of 1.6 mb at 1 eV. The 3982 and 3991 keV lines could be fitted by multilevel interference alone (see fig. 9). In the case of the 4068 keV line, the best multilevel fit without direct capture gave a curve which was consistent with the data, but which in the region below 6.7 eV was near the upper limit on the ratios, the measured value being zero because the line could not be seen in these low energy scans.
N-CAPTURE I N 238U
645
The curve with a direct amplitude D O = 3 × 10 - s gives the measured data a higher probability. Fits were also made for six weaker lines around 3.5 MeV. In these cases multilevel curves could be found which fitted the data, without a direct amplitude. However, this conclusion is weakened by the poorer statistics for these lines. A typical fit for these lines is shown in fig. 10. Here the region around 10 eV has been included and shows the sharp oscillation caused by the 10.2 eV resonance. I
I
I
5584 kelV I0.2
1
% x
21
T 6.7 I--
0
I
5
I
I0
15
I
20
E. (eV) Fig. 10. R a t i o o f partial to t o t a l c a p t u r e cross section for 3584 keV g a m m a ray, plotted o n a linear scale. T h e oscillation a r o u n d 10 eV is d u e to t h e 10.2 eV resonance. T h e curve represents t h e fit obtained f r o m t h e multilevel interference f o r m u l a .
The combinations of signs of the reduced width amplitudes which gave the best fits to the data are shown in table 2. Signs are not given where either the partial width was not measurable or the statistics on the cross section ratio data are too poor to resolve the signs for the higher resonances. An alternative explanation for the discrepancies from the multilevel fits which were attributed above to direct capture is the possibility of capture into bound levels. Such a level has already been postulated above for the thermal cross section discrepancy. However, the discrepancy is relatively small, and the level must be rather weak; it would be unlikely, although possible, for it to contribute strongly to the 4059 keV line partial capture. Further, the usual Porter-Thomas distribution would cause such a level to contribute randomly to the different lines. Direct capture on the other hand would be expected to favour certain transitions, in particular, those proceeding between initial and final states with strong l = 0 and l = 1 single-particle character,
--
.
66
81
+
+
37
1 0 3 + 1 1 7 (sum)
+
4068
21
6.7 (ref)
(eV)
R es on an ce energy
.
+
+
+
+
+
4095
.
. --
+
+
--
+
3991
--
--
+
--
+
3982
3639
+ + +
3653
+ + +
Line energy (keV)
+
+
+
3612
Signs o f re duc e d w i d t h a m p l i t u d e s relative to 6.7 eV r e s o n a n c e
TABLE 2
+
+
--
+
3584
+
_
+
3567
3540
,...,
t-
N-CAPTURE IN 2~8U
647
respectively. The analysis described above shows evidence for direct capture to two lines, strong in one case and tentative in the other, which have been identified with the ½- and :}- members of the ½-[671] orbital. In the case of the 4059 keV line, the level is strongly populated in the (d, p) reaction. These are strong candidates for the direct capture process. At the same time, there is no evidence of direct capture in the case of the two transitions to states at 811 and 820 keV which are not seen in the (d, p) reaction and which have been associated with collective vibrations. 'Thus, while accurate analysis of further lines would be helpful to resolve the question absolutely, the evidence for direct capture is convincing. The potential capture cross section of 1.6 mb deduced for the 4059 k e y line is somewhat lower than the values of 10 and 20 mb calculated by Lane and Lynn 16). However, the calculation was based on hard sphere scattering which is clearly unrealistic, especially for nuclei in the deformed region. Furthermore, the experimental value is open to uncertainty in that it is difficult to be sure that the choice of signs for the resonance amplitudes is unique. Somewhat poorer fits, but still within statistical error, could be found with higher values for the direct amplitude by changing the signs of the more distant resonances.
6. The statistical properties of partial radiative widths The behaviour of the partial radiative widths in 238U has been the subject of much interest, especially since the original suggestions of Hughes et al. 4) that the width distribution in this nucleide was narrower than expected. In analogy with the particle widths, the radiative widths are usually assumed to follow a Z2 distribution with one degree of freedom, implying a Gaussian distribution of width amplitudes. Early data, taken at Brookhaven 4), Argonne s) and Saclay 25) with low resolution NaI radiation detectors indicated that partial widths for 7-rays near 4 MeV showed relatively little fluctuation from resonance to resonance. These results were generally ascribed to the inability to completely resolve the radiation to discrete final states. With the superior resolution of the Ge(Li) detector, it is now possible to believe that the majority of final states are resolved. This conclusion is supported by the data of the present experiment, which indicate that in the resonances only a few more lines than reported by Sheline et al. 3) at thermal neutron energy are observed. It is therefore of considerable interest to study the distribution of widths obtained in the present experiment. For this purpose we consider the set of 18 final states populated by 5 resonances, omitting the weak 10.2 eV resonance and the incompletely resolved 103 and 117 eV resonances as well as the known M1 and E2 transitions. The set of 90 widths (18 final states, 5 initial states) is normalized to the 4.059 MeV transition by applying an E 3 factor assumed for electric dipole transitions: E
-3
648
D . L . PRICE et
aL
The use of the above relation also assumes that the mean intensities, over all resonances, are the same in populating all final states, so that fractional degrees of freedom are not introduced. With this assumption, we can examine the distribution of partial radiative widths. The cumulative probability distribution for the 90 widths is shown plotted in fig. 11 along with the theoretical curves for v = 1, 2 and 4. F r o m this figure, it is readily seen that the best fit is obtained with v ~ 4. To study the possible distortions in the distribution introduced by experimental errors and limited sample size, a Monte Carlo 90 80 E" 7o At 6 0 co J
~
l
~
,
,
i
\% \ \ ,\ ~
J
~
.....
\\\ '~ ~
,
I
I
i
~:,
--~o2 --,,--4
-
<1"7 i > =0.31 meV
-
u_ 4 0 o
rr 3 0 - w rn :~ 20
N
-(
1
i 0.20
I
i 0.4-0
i
1 0.60
~
,'=---~-~ 0.80 1.00
-r---~-: 1.20
FTi (meV) Fig. 1 l. The cumulative distribution for 90 partial radiative widths o f the present experiment. The widths are reduced to an effective energy o f 4.059 MeV by assuming a cubic dependence o f width
on 7-ray energy. All known MI and E2 transitions have been deleted. analysis employing the program C O R N R O W 2 2 , 2 3 ) w a s carried out to study the distribution of veff, the effective number of degrees of freedom, for the physical sample, assuming parent populations corresponding to v = 1 and 2. For v = 1 a value of 1 +0.55 I~eff = xl."r.~_0.25A~ +0"33 a n d f o r v = 2, Veff . . . . ") . 0.40 w e r e obtained, where the e r r o r s indicate 10 to 90 per cent confidence limits. The experimental value, ~ 4, was determined to be clearly discrepant with both v = 1 and 2. To proceed further, a m a x i m u m likelihood analysis, based on the methods developed at Argonne National Laboratory 24.28), was employed. Briefly, the procedure is to extract the value of v~ff from the physical sample by the method of maximum likelihood, and then to compare the value of v~ff so obtained with values of v~, from mathematical samples chosen to approximate the physical sample. The method is explained in detail in an appendix to this paper. In fig. 12 are shown the distribution of values Vraathematleal>Vphysiea I (obtained from 100 Monte Carlo samples) as a function of the assumed v for the parent population.
649
N - C A P T U R E I N 2aaU
I t.O-
I
I
I
/ ,
m
o
I
I
/"
W~TH ERRORS
o.6-
¢rl
rY (3_
I
o.8-
>_ F-
I
WITHOUT E R R O R S ~ / ' ~ ~
__ A
I
0.4--
0.2--
I
I
Jr"
1.0
2.0
3.0
¢3
~1
/
,
I
I
I
I
I
I
4.0
5.0 v,
6.0
7.0
8.0
9.0
I0.0
Fig. 12. T h e c u m u l a t i v e distribution o f Vetf values obtained f r o m a M o n t e Carlo s i m u l a t i o n o f the physical set o f widths, o b t a i n e d f r o m a m a x i m u m likelihood m e t h o d . T h e values are for s a m p l e s w h i c h are d r a w n f r o m p a r e n t p o p u l a t i o n s o f v indicated o n the x axis.
I
15
i
Vo=4.00 vp=3.80
14 E 13
2O ,2 m
,0i
~
7
~
6
u.l
5
4 3
2I
2.0
3.0
I
I
4.0
5.0
u m
Fig. 13. A typical distribution o f Vet r values o f 100 s i m u l a t e d sets o f physical widths d r a w n f r o m a p a r e n t p o p u l a t i o n with v = 4. T h e ~ett for t h e physical s a m p l e is also s h o w n .
650
D . L . PRICE et al.
F r o m this figure we may read off the best value of Veff and its 10 and 90 percentile limits, and we obtain Veff = 4.6-+ 11.9 .2' Fig. 13 shows a typical distribution of v values obtained from the maximum likelihood analysis of 100 mathematical samples of 90 widths, drawn from a parent population of v = 4, and broadened by a normal error distribution. Also shown in this figure is the maximum likelihood value of v for the physical sample, v~ff = 3.80. The large value obtained is not consistent with the Porter-Thomas distribution, or )~2 distribution with one degree of freedom. A subset of widths was further examined by selecting only the strong transitions at 4.068, 4.059, 3.991 and 3.982 MeV. The subset of 20 widths gave a value oi6 v =~a~. /,+2 _l: in good agreement with the larger set. The large value of v,~f obtained here is difficult to explain in terms of unresolved transitions, since, as noted above, the comparison between thermal and resonance capture indicates that most levels are probably being resolved. Furthermore, it is difficult to visualize other systematic errors which would lead to this result. A theoretical framework to accommodate departures from the Porter-Thomas distribution, without violation of the usual invariance properties of nuclear wave functions, has been advanced by N. Rosenzweig 26). Rosenzweig's expression for the joint probability distribution of the widths, W(r)6{
P ( Z , . . • Z,,) oc
Z,
r2}rdr,
i=1
is derived from considering only those interactions which induce orthogonal transformations in the N compound nuclear states which leave the subspace spanned by ~ n invariant. Here, ~ n is defined as the space spanned by PVqg~ where V is the interaction Hamiltonian, goz as one of n final states, and P as the projection onto the space spanned by the N compound nuclear states. The Porter-Thomas distribution is considered as a special case of the above where the weighting function W ( r ) is a Gaussian, i.e. W = exp(-½Nr2). The relative variance, v, of a single partial width is shown to obey the following inequality: v >__ 2 ( n - 1 ) -
(n+2)
Since the present experiment clearly indicates that v < 1, we conclude that n < 4. Rosenzweig also shows that it follows that at least one of the width correlation coefficients must be negative. The data of the present experiment do not include enough resonances to establish firmly this fact. However, average correlation coefficients between reduced neutron widths and partial radiative widths, and among radiative
N-CAPTURE IN 238U
651
widths, have been calculated. The results are as follows: R -- (corr {F °~, r~,j}> i = +0.058, T -- (corr {F~j, F~j,})j.ej, = +0.06. These values are consistent with zero falling at the 75th and 64th percentile, respectively. The presence of correlations of both signs, as expected from Rosenzweig's analysis, is not ruled out by the data.
7. Conclusions These measurements of resonance neutron capture in 238U(n, ~)239U are on the whole consistent with the level scheme for 2 3 9 U proposed by Sheline et al. in ref. 3), on the basis of rotational bands based on Nilssons orbitals; there is only one point of disagreement. There is evidence of unusually strong M1 transitions. The most significant feature of the experiment has been the most convincing evidence, to date, for a direct, potential contribution to neutron radiative capture. However, single-particle capture effects in the resonances appear to be small. The authors would like to encourage more theoretical work on these lines: (i) calculations of the strength of transitions of different multipolarity in the region of deformed nuclei. (ii) Quantitative examination of the proposal for octupole vibration bands near 800 keV excitation. (iii) A more realistic calculation of the strength of direct capture. The results of the experiment show that the partial radiative widths of 238U are inconsistent with a Porter-Thomas distribution, and are also inconsistent with a X2 distribution with two degrees of freedom. The authors would like to thank Frederick Paffrath for his tireless work maintaining the rotor system, Vito Manzella for electronic assistance, and Jean Domish, Judy Harte, and Isaac W. Cole for help with the computations. The authors would also like to acknowledge the invaluable assistance of David Parks for setting up the m a x i m u m likelihood analysis of sect. 6. The germanium detectors used in this experiment were made by H o b a r t Kraner and Marco Jamini of the B N L Instrumentation Department.
Appendix MAXIMUM LIKELIHOOD METHOD g4--28) The original physical data consist of N partial radiation widths {Fij } and their respective errors {e~j}, where i and j denote a particular final state and resonance, respectively. F r o m these sets a normalized set of widths (xij} and a normalized set of errors {e~j} are formed, where the xij and e~j are related to the Fij and eij by
x~j = F~/u~, eij' = egj/Ui,
1 < i <_ n, 1 <=j =< m,
(1) (2)
652
D . L . PRICE
et al.
where
1 & uz = - -
(3)
L rij,
mj=l
and where m is the number of resonances and n is the number of final states. One assumes that the {x~j} belong to a Xz distribution with 2p degrees of freedom, for which the probability distribution function is pP P ( x i j , Pl =
i-(o)
x ° - l e- - P x ' l d-0i~, 'z -ij
(4)
where F(p) is the gamma function of p = v/2, v being the number of degrees of freedom. Following the maximum likelihood procedure 27), one forms the likelihood function L(x~j, p) which is the joint probability distribution function for the random variable x~:
L(xij, p) = Vl p a x °.-'e -'x'~ ~/ r(p) "
(5)
or
L(r,;u,,pl
PP ( F z J l P - l l e x p ( - p F i J ) u, , Us ,"
=
(6)
The estimator of the parameter p, or the value ofp which maximizes L(xe, p), is found by solving the simultaneous set of equations d In L(r,j/u,, p) = O, dp
(7)
d In L(Fij/u,, p) = O.
(8)
dui
Since the mean value of x~j using eq. (4) is 1, one chooses the estimator of the mean, or the value of u~ that satisfies eq. (8), to be 1 also. Using this and performing the differentiation above, one gets the transcendental equation dlnF(p)_lnp dp
= ~ .1.
In x,j,
(9)
where N = (n)(m), the total number of widths in the physical sample. Taking into account the finite size of the physical sample, eq. (9) can be replaced by the more accurate form 7)
d ln F(p)- ln p = l ~ ln x~j - 0"5 + 1 dp N ,j [Q" ~12Q
1
120Q ~ +
2~2Q61
'
(101
where
Q = rap.
(11)
N-CAPTURE IN Z38U
653
The solution to eq. (10) yields the estimator of the parameter p. The term In x~j in eq. (10) diverges as x~j ~ 0 and is undefined for x o < 0. Since the physical data may contain zero and/or negative widths, the data is truncated at a value determined by the normalized error distribution of the physical sample. If the normalized error distribution has a well defined peak and a smooth tail, the cutoff value is equal to the mean error. If the distribution has a well defined peak but also has a b u m p in the tail due to large errors of certain resonances, then the cutoff value is equal to the median error. The median error, and not the mean error, is chosen in the latter case so as not to overestimate the error and hence the cutoff. The criterion for using the cutoff is that any width lying below the cutoff value is set equal to onehalf of the cutoff value. Following this procedure the normalized widths {x~j} are corrected using the appropriate cutoffs, renormalized over resonances, and are used in eq. (10) to get the estimator of p. After the number of degrees of freedom vp for the physical sample has been calculated, a Monte Carlo technique is employed to test the hypothesis that the physical sample is drawn from a Xz distribution with Vo degrees of freedom. A mathematical sample "identical" to the physical sample is generated from a X2 distribution with v o degrees of freedom, where v o = 1, 2, 3, 4, etc. This is done by first choosing at random N numbers from a Xz distribution. The number of "widths" in this mathematical sample must be equal to the total number of g a m m a widths in the physical sample. These "widths" are normalized in groups of rn widths, corresponding to eq. (1) for the physical sample. To further simulate the physical data, errors are folded into the mathematical distribution of "widths" according to zij = y~j + r/d,
(12)
where y~j is a normalized width, d the mean or median error of the physical normalized error distribution, and r/a r a n d o m number drawn from a normal distribution with mean zero and standard deviation one. The z~j's are normalized over "resonances" and are corrected using the same cutoff value and cutoff criterion as for the physical sample. The corrected widths, after renormalization over resonances, are used in eq. (10) to calculate Vm, the number of degrees of freedom for the mathematical sample. The entire process of forming a mathematical sample simulating the physical sample is iterated 100 times for each Vo. For a given v o, the above hypothesis is tested by plotting the distribution of vm (100 values) and Vp: if vp lies within 10 ~ to 90 ~o of the distribution, then the physical data is consistent with Vo degrees of freedom. The distribution of vm for v o = 4.00 and vp = 3.80 for the 23sU data is plotted in fig. 13. The "best value" of v for the physical data is the v o for which Vp is the median value of the distribution of vm, or when the probability that Vm > Vp is 0.5. Such a value can be obtained by plotting the probability that vm > vp for each value of vp. Also, a "best value" for v can be obtained from truncated distributions without errors folded in. A comparison is made in fig. 12.
654
D.L. PRICE et al.
References 1) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk., 27, no. 16 (1953) 2) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1 no. 8 (1959) 3) R. K. Sheline, W. N. Shelton, T. Udagawa, E. T. Jurney and H. T. Motz, Phys. Rev. 151 (1966) 1011, and references contained therein 4) D. J. Hughes, H. Palevsky, H. Bolotin and R.E. Chrien, in Proc. Int. Conf. on nuclear structure, edited by D. A. Bromley and E. W. Vogt (University of Toronto Press, Toronto, 1960) p. 771 5) H. E. Jackson, Phys. Rev. 134B (1964) 931 6) I. Bergqvist, Nucl. Phys. 74 (1965) 15 7) C. F. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 8) R. E. Chrien, D. L. Price, O. A. Wasson, M. R. Bhat, M. A. Lone and M. Beer, Phys. Lett. 25B (1967) 195 9) R. E. Chrien and M. Reich, Nucl. Instr. 53 (1967) 93 10) M. R. Bhat, B. R. Borrill, R. E. Chrien, S. Rankowitz, B. Soucek and O. A. Wasson, Nucl. Instr. 53 (1967) 108 11) J. R. Stehn et al., Brookhaven Nat. Lab. Rep. 325, Suppl. 2 (1965) Vol. III, 92-238-3 12) E. K. Hyde, I. Perlman and G. T. Seaborg, The nuclear properties of the heavy elements (Prentice-Hall, New Jersey, 1964) p. 150 13) I. Bergqvist, B. Lundberg and N. Starfelt, in Proc. Int. Conf. on nuclear physics with reactor neutrons, edited by F. E. Throw (Argonne National Laboratory Report 6797, 1963) p. 220 14) C. K. Bockelman, Nucl. Phys. 13 (1959) 205 15) H. Morinaga and C. Ishii, Prog. Theor. Phys. (Kyoto) 23 (1960) 161 16) A. M. Lane and J. E. Lynn, Nucl. Phys. 17 (1960) 563 and 586 17) I. Lovas, Z h E T F (USSR) 41 (1961) 1175; JETP (Sov. Phys.) 14 (1962) 838 18) O. A. Wasson, M. R. Bhat, R. E. Chrien, M. A. Lone and M. Beer, Phys. Rev. Lett. 17 (1966) 1220 19) E. P. Wigner and L. Eisenbud, Phys. Rev. 72 (1947) 29 20) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 21) R. E. Cot6 and L. M. Bollinger, Phys. Rev. Lett. 6 (1961) 695 22) M. Beer, Correlation effects of primary resonance neutron capturey-rays in heavy nuclei (Thesis, State University of New York at Stony Brook, 1968, unpublished) 23) M. Beer, M. A. Lone, R. E. Chrien, O. A. Wasson, M. R. Bhat and H. R. Muether, Phys. Rev. Lett. 20 (1968) 340 24) Argonne Nat. Lab., Illinois 60439, Appl. Math. Div. Program Library 867/PHY-186, Dec. 22, 1960 25) C. Corge, V.-D. Huynh, J. Julien, J. Morgenstern and F. Netter, J. Phys. Rad. 22 (Oct. 1961) 722-723 26) N. Rosenzweig, Phys. Rev. Lett. 6 (1963) 123-125 27) R. V. Hogg and A. T. Craig, Introduction to mathematical statistics (Macmillan, New York, 1965) 2nd ed., p. 243 28) L. M. Bollinger, R. E. Cot6, R. T. Carpenter and J. P. Marion, Phys. Rev. 132 (1963) 1640
A121 (1968)
to be reproduced
by photoprint
NEUTRON
@
North-Holland
CAPTURE
IN 238U
D. L. PRICEt, R. E. CHRIEN, 0. A. WASSON, M. R. BHAT, M. BEER tt, M. A. LONEt?? and R. GRAVES Physics
Department,
Brookhacrrr Received
Co., Amsterdam
or microfilm without written permission from the publisher
National Laboratory:, 9 September
tt
Upton, New York
1968
Abstract: They-ray spectrum following slow neutron capture in 23RU has been measured as a function of incident neutron energy at the Brookhaven HFBR fast neutron chopper. The decay scheme proposed by previous (d, p) and high resolution thermal (n, y) measurements has been substantially verified. Partial radiation widths for s-wave resonances up to 117 eV have been deduced. By using a simple multi-level analysis, the spectra between resonances from 21 eV down to thermal have been fitted. The behavior of the strong transitions to low-lying I = 1 states is explained using a direct capture model, while the transitions to low-lying collective states can be explained by level-level interference. The distribution of El transition strengths is not consistent with a x2 distribution of one degree of freedom.
fi
NUCLEAR REACTIONS Z3*LJ(n, y), E = 0.002-120 eV; measured ~J(E; E,,) z39U deduced levels, J, r,,(E?), relative signs of [r,,ry(E,)]t. Enriched target. --
1. Introduction The nuclei of the actinide series have in general non-spherical ground states due to coherent effects between nucleons outside the closed-shell configuration. The higherenergy states can be caused by vibrational or rotational collective excitcltions in addition to intrinsic excitations. The unified model ‘) has been applied with some success to these nuclei, and the odd-A nuclei are particularly interesting in that the mtrinsic excitations often correlate well with Nilsson’s calculations of nucleon binding states in spheroidal nuclei ‘). This paper describes a recent study of the energy levels in 23gU observing the gamma rays following neutron capture in 238U. Previous work on the energy levels of 23gU, using both neutron capture and (d, p) reaction techniques, has been reported 3-6) . However, the investigation reported here is the first to have studied neutron capture in conjunction with high-resolution y-ray detectors as a function of incident neutron energy. The germanium detectors used have a resolution for y-ray energy as low as 6 keV. Since the average level spacing near the ground state of 2 3gU is about 30 keV, this resolution is necessary if closely spaced t Present address: tt State University ttt t
Solid State Science Division, Argonne National Laboratory, Argonne, Illinois. of New York at Stony Brook; present address: Brookhaven National Laboratory, Upton, New York. State University of New York at Stony Brook; present address: Physics Division, Chalk River Laboratories, Chalk River, Ontario, Canada. This work was supported by the U.S. Atomic Energy Commission. 630
N-CAPTURE IN SZSU
631
levels are to be distinguished. The use of energy analysis by means of the time-of-flight technique makes it possible to study the intensities of the transitions to a given final state as they vary over the capturing resonances and the energy regions between the resonances. This has a number of advantages over a thermal capture study. First, the fluctuation in intensity expected from a Porter-Thomas distribution of partial radiation widths 7) m a y make some y-rays appear too weakly in thermal capture to be observed. Second, the variation in relative intensity between transitions to closely spaced levels makes it easier to distinguish them. Third, the possibility of averaging a partial radiation width to a given final state over a number of resonances makes it easier to estimate the multipole order involved, often a crucial factor in identifying the final state. Fourth, it is possible to obtain information about the spins and parities of the capturing states. These points are illustrated in the following sections. The study of the variation in intensity of a given transition as a function of neutron energy has one further feature of considerable interest. The intensity in the regions between resonances may be expected to display interference between contributions from nearby resonances with the same symmetry properties. By studying this interference it is possible to deduce the relative signs of the resonance amplitudes, in addition to the magnitudes which are derived from the intensities at the resonances. Failure to obtain a set of resonance amplitudes which fits the data in this manner must be an indication either of contribution from unknown resonances, for example bound states, or of the fact that direct processes are involved. The present investigation has revealed one case where direct capture is almost certainly involved. This has been briefly reported in a letter s) and is discussed in more detail in this paper.
2. Experimental procedure The experiment was performed at the fast chopper of the Brookhaven High Flux Beam Reactor 9). Briefly, a straight-slot rotor placed at the exit of a beam tube produces periodic bursts of epithermal and resonance energy neutrons. These neutrons are well collimated (8.0 cm F W H M at the target) and fall on the sample placed 22 m away. The y-rays produced by capture events are detected in a germanium diode detector placed near the sample, and the pulses are fed to an on-line computer where they are processed and stored on magnetic tape 10). The events are analysed with respect to time-of-arrival relative to the time the burst is at the rotor and to pulse-height in the detector. These are related to the neutron and y-ray energies, respectively. The sample was a slab of uranium depleted i n / 3 s U , 0.6 cm thick, 50 cm 2 cross section. Three series of runs were made: a) at a rotor speed of 6000 rpm, with a germanium detector of volume 30 cc, resolution 14 keV F W H M , run time 54 h. b) at a rotor speed of 1500 rpm, with a 4 cc detector, 9 keV F W H M , run time 51 h. c) at a rotor speed of 1500 rpm, with a 4 cc detector, 6 keV F W H M , run time 89 h.
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The series at the higher rotor speed had the best time-of-flight resolution and provided data for the resonances up to 117 eV; most of the information on partial radiation widths was derived from these runs. However, this detector had comparatively poor resolution. Series (b) was made to provide information for thermal capture; thermal is here used to mean neutron energies up to about 0.2 eV; in other words, energies small compared to those at the resonances. The lower rotor speed increases the incident flux for these neutrons. The object of this series was to provide an intercalibration so that the partial widths measured in the first series could be expressed in absolute magnitudes via the thermal measurements of ref. a). When the first two series of runs were evaluated, it became clear that there was considerable interest in the region between thermal and the first and second resonances which could profitably be studied with higher resolution. A further series of runs was, therefore, made at the lower rotor speed with a new detector which had a resolution of 6 keV at 4 MeV. Part of the time-of-flight spectrum for the first series is shown in fig. 1. This includes all v-rays depositing more than 1.96 MeV in the detector. Seven resonances can be resolved at this rotor speed. The highest neutron energy peak (shortest time-of-flight) actually contains two unresolved resonances. All the resonances represent s-wave capture; the small peak at 10 eV has previously been considered as a p-wave capture on intensity considerations 11), but will be shown below to be s-wave. A v-ray spectrum is obtained for each resonance by scanning the magnetic tape on a CDC-6600 computer 1o). A spectrum is formed for those events whose time-offlight falls between two limits set on either side of the resonance peak. The double arrows in fig. 1 show the limits for the scans of the seven peaks; two " b a c k g r o u n d " scans were also made between the resonances. The " b a c k g r o u n d " scans include a significant contribution from off-resonance capture in 2a8U as well as a background component. Similar scans were made in the thermal region for series (b). For series (c) scans were made for the 6.7, 10.2, 21 and 37 eV resonances, and in various regions between the resonances, discussed in sect. 5. Figs. 2 and 3 show two regions of such scans for the three strongest resonances in the first series of runs. The v-ray peaks are indicated by the energy in keV. In general, each v-ray produces three peaks in the detector corresponding to full capture, single escape, or double escape of the photons following a pair production event; these are separated by intervals of 511 keV, and are indicated by the letters F, S and D in the figures. The relative sizes of these peaks depend on the detector volume and the energy of the v-ray. In some cases, two y-rays cannot be resolved, which happens when their energies are close together or separated by nearly 511 or 1022 keV. Compton processes produce a continuum background under the peaks. This rises towards lower energy making it harder to resolve peaks in this region. This limited the present study to y-ray lines between about 3200 keV and 4802 keV, the value of the neutron binding energy in 239U.
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3. Interpretation of observed transitions Sheline e t al. 3) have proposed a level scheme for 239U for the region within 1000 keV of the ground level, based on their thermal capture and (d, p) measurements. This is shown in fig. 4, in which the thin bars designate levels proposed by Sheline
636
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et al. related their (n, 7) and (d, p) results by identifying the strong 7-line at 4059.4
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Fig. 4. Level scheme proposed by Sheline et al. (thin bars) compared with measurements and assignments of this experiment (thick bars). Levels are labelled by energy in keV above the ground state. The letters d, n in parentheses signify whether the Sheline et al. measurement is from (d, p) or (n, 7) studies, respectively. where these lines could be identified with lines measured by Sheline e t al. in either the (n, 7) or the (d, p) studies, the energies were found consistent to within the errors quoted. Since absolute energy measurements were not a primary aim of this work, their values will be used in such cases. In cases of lines not observed by them, the line energies are derived by measuring relative to the 4059 keV line, and the level energies by taking their value for the binding energy. The levels are shown grouped in rotational bands associated with a particular Nilsson orbital 2). There is a well-defined gap from 400 to 600 keV due to the closure o f the 152-neutron shell. The capturing state for all resonances measured in the pres-
N-CAPTURE IN 238U
637
ent study is ½+. The E2 transition to the ~}+ ground state is not observed, but the transition to the 133 keV level is seen. This was seen in the (d, p) work of Sheline et al. and identified with the base of the ½+ [631 ] band, but was not observed in thermal capture, presumably a consequence of the Porter-Thomas distribution of partial radiation widths 7). It was, however, seen by Bergqvist in fast neutron capture work 6). Also observed are transitions to levels at 143 and 191 keV, probably the next higher levels, 3+ and ~z+, in the ½+ [631 ] band which are found in thermal capture. These three levels must be fed by M1 and E2 transitions because there are no negative parity Nilsson orbits with low spin r.mmber in this region. Above the 152-neutron gap, the two levels seen in thermal capture around 700 keV are not observed. The level at 742 keV corresponds to the dominant line of the large y-ray peak at 4059 keV. Sheline et al. observe structure on the low-energy side of this peak and take it to be the next higher level in the ½- [761 ] band, identified with a level observed in the (d, p) work at 754 keV. However, in the resonances this line only appears weakly as structure in the low energy wing of the 4059 keV line (see, for example, fig. 7). On the other hand, there is a strong line at 4068 keV which is not seen in thermal, but shows very strongly in the higher resonances, as can be seen from figs. 3 and 7. This corresponds to a level at 733 keV, and it seems more probable that this is the base of the ½- [761 ] band, and that the level at 754 keV is a positive parity state. The same could be true of the weak level observed in the resonances, but not in thermal capture, at 799 keV. Possible states could be found in the ½+ [620] or ~+ [622] bands. The only other lines deserving special attention belong to the strong doublet at 3981 and 3992 keV, corresponding to levels at 811 and 820 keV. These were not seen in the (d, p) measurements and Sheline et al. tentatively interpret this doublet as an octupole band built on the ½+ [631] orbital. The fact that such a doublet occurs in 238U at 679 and 724 keY above the base level 12) would seem to support this interpretation. However, there are also two negative parity orbitals in this region which provide an alternative explanation. 4. Evaluation of partial radiation widths Partial radiation widths derived from the resonance measurements are given in table 1. These were evaluated as follows. The intensity of the line in the resonance was divided by the intensity of the total ~-ray yield above an energy of 1.96 MeV. This ratio is proportional to the ratio of partial to total radiation width. If the lower energy cut is such that a large number of 7-rays are included in the region above it, the proportionality constant should be the same for all resonances, and should also hold for the thermal measurements. It was, therefore, evaluated by comparing the ratios measured for the thermal run with the values of intensity per thermal capture given by ref. 3). Partial widths were then derived by taking a common value of 26 meV for the total radiation widths of all resonances 11). Corrections were made for background caused by neutron absorption in the detector and for incompletely resolved lines in
Level energy (keV)
133 143 191 733 742 754 799 811 820 928 959 984 1149 1162 1190 1218 1234 1262 1355 1441 1483 1508 1520
Gamma-ray energy (keV)
4669 4659 4610 4068 4059 4048 4003 3991 3982 3873 3843 3818 3653 3639 3612 3584 3567 3540 3446 3361 3318 3293 3282
0 -t-0.03 0.11 4-0.03 0.07 4-0.02 0.03 4-0.03 0.63 4-0.03 0.09 4-0.03 0 4-0.02 0.76 4-0.06 1.13 4-0.07 0.0264-0.040 0.13 4-0.05 0.26 4-0.05 0.10 4-0.02 0.12 4-0.03 0.05 4-0.03 0.54 4-0.04 0.07 -t-0.03 0.10 ±0.03 0.08 4-0.08 0.14 4-0.08 0.0314-0.058 0.0054-0.06 0.13 4-0.06
6.7
TABLE 1
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04-0.03 0.15 4-0.04 04-0.03 0.06 4-0.05 04-0.03 0 4-0.02 0.01 5:0.23 0.26 4-0.05 0.29 4-0.23 0.93 4-0.03 04-0.03 0.15 4-0.04 04-0.03 0.10 4-0.05 0.53 4-0.26 0.23 4-0.03 0.91 +0.27 0.59 4-0.04 --0.07 4-t-0.25 0.0554-0.037 0.29 4-0.27 0.27 4-0.04 0.42 4-0.27 0.45 4-0.04 0.45 4-0.30 0.15 4-0.05 0.092-t-0.29 0.15 ±0.04 0.15 4-0.30 0.11 4-0.04 0.66 4-0.32 0.14 4-0.04 0.74 4-0.32 0.30 ±0.05 0.49 4-0.32 0.09 4-0.07 0.26 4-0.32 0.12 4-0.05 --0.0354-0.346 --0.0554-0.069 0.19 4-0.36 0.20 4-0.05 --0.0554-0.36 0.20 4-0.08 0.16 4-0.38 0.14 4-0.04
10.2 0.27 4-0.09 0 4-0.02 0 5:0.02 0.24 ±0.03 0.10 4-0.03 0.08 4-0.05 0 4-0.02 0.15 4-0.03 0.32 4-0.03 0.32 4-0.05 0.19 4-0.07 0.0754-0.059 0.07 -t-0.04 0.17 4-0.05 0.25 4-0.05 0.55 ±0.05 0.38 4-0.05 0.20 4-0.08 0.0404-0.078 0.20 4-0.06 0.18 4-0.09 0.18 4-0.09 0.15 4-0.09
37
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Resonance energy (eV)
Partial radiation widths for 2~sU(n, y)'z39U, in units of meV
0.09 ±0.13 0.18 I 0 . 1 8 0 4-0.03 0.15 4-0.12 0.65 4-0.09 0.31 4-0.19 0 4-0.03 0.37 4-0.22 0.24 4-0.11 0.40 4-0.13 0.041+0.159 0.33 4-0.12 0.12 4-0.18 0.25 4-0.19 0.26 ±0.19 0.33 4-0.19 0.36 4-0.13 0.29 4-0.20 0.0924-0.213 0.24 4-0.23 --0.07 5_0.23 0.16 4-0.24 0.12 4-0.24
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N-CAPTURE IN ~ U
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the cases mentioned in sect. 2. The intensities for the strong lines at 3982, 3991, 4059 and 4068 keV were derived by Gaussian fitting, except for the 10.2 eV resonance. Where possible, intensity information was derived from both full-capture and doubleescape peaks. For the values for the 6.7, 21 and 37 eV resonances, the results for the third series of runs were averaged in with those of the first and second. The errors quoted a re standard deviations, and there are possibly systematic errors beyond these. However, whenever comparison between separately derived numbers was possible (Gaussian fits v e r s u s numerical calculation, first v e r s u s third series) the agreement was consistent w~th the quoted errors. I
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The last column represents an average over the eight resonances which should give an indication of the multipolarity of the transition. The top two rows represent the M1 transitions to the low-lying levels discussed in the last section. The widths for these transitions are seen to be anomalously large, averaging nearly 0.1 meV. This is about a hundred times the width estimated with a single-particle model, allowing for the appropriate level density at the capturing state. The corresponding estimate for E1 transitions is 0.2 meV, which can be seen to be typical of the values found for those transitions further down the table. It appears, therefore, that the M1 transitions are enhanced by two orders of magnitude, presumably by some kind of collective motion, involving the excitation of 2p-lh states, as proposed by Bergqvist t 3).
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N-CAPTURE IN ~ZSU
641
It can be seen that the widths for the 10.2 eV resonance, where they can be measured, have a similar behavior to those of the other resonances. For example, in the higherenergy 7-ray region, the only peaks appearing are the two doublets around 4 MeV which dominate the other resonance spectra. These are illustrated in fig. 5. This supports the conclusion that the 10.2 eV resonance is due to s-wave capture, not p-wave as stated in the literature ~i). Uz38 (n,y) U239 I I 1 1 1
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5. Analysis of partial cross sections
Several authors have proposed a direct reaction mechanism for the radiative capture of neutrons 14-~7). The best evidence for such a process can be found in the interference between the slowly varying reaction amplitude characteristic of the direct reaction and the sharply peaked resonance amplitude characteristic of the reaction via the compound nucleus. Such interference has been found is) for the ground state transition in 59C0(n, 7)6°C0, and the strongest lines in these measurements were investigated in the same manner.
642
D.L. PRICEet al.
O n e striking effect noticed in the first m e a s u r e m e n t s was the sharp variations in the strong doublets near 4 M e V between t h e r m a l c ap t u r e and the lowest r eso n an ce at 6.7 eV. F o r this reason, a third series o f runs was m a d e with a high-resolution germ a n i u m detector at 1500 r p m r o t o r speed (see sect. 2). Th e variations were b r o u g h t I
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out dramatically in the scans for these lines in the three lowest resonances and various n e u t r o n energy regions off resonance, including the region near t h e r m a l energy. T h e spectra for the two doublets f r o m five o f these scans is sh o w n in figs. 6 and 7. T h e difference between the t o p two spectra in fig. 6, f o r t h e r m a l an d the 6.7 eV resonance, should be noted.
643
N-CAPTURE IN ~88U
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I 4
6
I
I
I
I0 12 14 ENERGY, eV
f
16
I
18
I
20
Fig. 9. R a t i o o f partial to total c a p t u r e cross section for 4068 a n d 3991 keV g a m m a rays p l o t t e d o n a logarithmic scale. T h e lines represent t h e fits o b t a i n e d f r o m the multilevel interference f o r m u l a . F o r t h e 4068 keV line, t h e d a s h e d curve represents multilevel interference only, t h e solid curve includes a direct amplitude.
The results in the neutron energy region up to 21 eV for four lines are given in figs. 8 and 9, in which the points represent the ratios measured in this way. The form of the partial cross section can be derived from R-matrix theory 19). In the approximation F~ << D, a small ratio of width to level spacing, it can be shown
D. L. PRICE et al.
644
that the partial cross section can be written O.nrf =
~ 0
IDO+ (rool~(r~,,)~2.
(1)
E;t-E +½iF a
Lane and Thomas 2o) show that the above expression is obtained by dropping the off-diagonal elements of the level matrix. The superscript zero indicates values at 1 eV, D Orepresents the amplitude of the direct capture process. It can be seen that the interference between this and a single resonance produces a cross section asymmetric about the resonance, whereas the total capture cross section is symmetric if a large number of v-rays are involved. Three types of theoretical curves were compared with the data: a) A pure single-level formula represented by the sum of the squares of individual resonance terms in eq. (1). Since all the resonances affecting the region considered have the same spin, parity, and neutron orbital angular momentum, this is not physically meaningful. It is nevertheless useful as a base against which to judge the interference fits. b) The multilevel interference formula without direct capture obtained by putting D O = 0 in eq. (1). Such interference has already been observed 21) in partial capture cross sections. Since the experiment only measures the magnitudes and not the signs of the product of the reduced width amplitudes (Fx~,f)½(F°,)~, the signs of these terms were varied by trial and error to give the best fit to the data. c) Direct capture was then included to improve the fit if necessary. The total cross section derived from the neutron and total radiation widths for the lowest eight resonances 11) gives a discrepancy of 0.37 b relative to the measured 2.73-1-0.04 b for the thermal capture cross section. This is presumably due to the influence of bound levels. The discrepancy was removed by postulating a level with a large negative energy and a suitably chosen neutron width. This makes the total cross section correct at thermal energy. The effect on the partial cross sections is unpredictable; initially it was assumed to be zero. The theoretical curves are shown on figs. 8 and 9. The region around the 10.2 eV resonance is omitted because the small neutron width causes a sharp oscillation in this region. For the 4059 keV line, strong constructive interference is observed below the resonance. No amount of multilevel interference could account for the behaviour of the cross section ratio, even in the extreme assumption that the signs of all resonance amplitudes up to 117 eV were such as to produce constructive interference. However, a fit could be obtained (see fig. 8) by postulating a direct capture amplitude D O = 5 x 10-5, equivalent to a potential capture cross section of 1.6 mb at 1 eV. The 3982 and 3991 keV lines could be fitted by multilevel interference alone (see fig. 9). In the case of the 4068 keV line, the best multilevel fit without direct capture gave a curve which was consistent with the data, but which in the region below 6.7 eV was near the upper limit on the ratios, the measured value being zero because the line could not be seen in these low energy scans.
N-CAPTURE I N 238U
645
The curve with a direct amplitude D O = 3 × 10 - s gives the measured data a higher probability. Fits were also made for six weaker lines around 3.5 MeV. In these cases multilevel curves could be found which fitted the data, without a direct amplitude. However, this conclusion is weakened by the poorer statistics for these lines. A typical fit for these lines is shown in fig. 10. Here the region around 10 eV has been included and shows the sharp oscillation caused by the 10.2 eV resonance. I
I
I
5584 kelV I0.2
1
% x
21
T 6.7 I--
0
I
5
I
I0
15
I
20
E. (eV) Fig. 10. R a t i o o f partial to t o t a l c a p t u r e cross section for 3584 keV g a m m a ray, plotted o n a linear scale. T h e oscillation a r o u n d 10 eV is d u e to t h e 10.2 eV resonance. T h e curve represents t h e fit obtained f r o m t h e multilevel interference f o r m u l a .
The combinations of signs of the reduced width amplitudes which gave the best fits to the data are shown in table 2. Signs are not given where either the partial width was not measurable or the statistics on the cross section ratio data are too poor to resolve the signs for the higher resonances. An alternative explanation for the discrepancies from the multilevel fits which were attributed above to direct capture is the possibility of capture into bound levels. Such a level has already been postulated above for the thermal cross section discrepancy. However, the discrepancy is relatively small, and the level must be rather weak; it would be unlikely, although possible, for it to contribute strongly to the 4059 keV line partial capture. Further, the usual Porter-Thomas distribution would cause such a level to contribute randomly to the different lines. Direct capture on the other hand would be expected to favour certain transitions, in particular, those proceeding between initial and final states with strong l = 0 and l = 1 single-particle character,
--
.
66
81
+
+
37
1 0 3 + 1 1 7 (sum)
+
4068
21
6.7 (ref)
(eV)
R es on an ce energy
.
+
+
+
+
+
4095
.
. --
+
+
--
+
3991
--
--
+
--
+
3982
3639
+ + +
3653
+ + +
Line energy (keV)
+
+
+
3612
Signs o f re duc e d w i d t h a m p l i t u d e s relative to 6.7 eV r e s o n a n c e
TABLE 2
+
+
--
+
3584
+
_
+
3567
3540
,...,
t-
N-CAPTURE IN 2~8U
647
respectively. The analysis described above shows evidence for direct capture to two lines, strong in one case and tentative in the other, which have been identified with the ½- and :}- members of the ½-[671] orbital. In the case of the 4059 keV line, the level is strongly populated in the (d, p) reaction. These are strong candidates for the direct capture process. At the same time, there is no evidence of direct capture in the case of the two transitions to states at 811 and 820 keV which are not seen in the (d, p) reaction and which have been associated with collective vibrations. 'Thus, while accurate analysis of further lines would be helpful to resolve the question absolutely, the evidence for direct capture is convincing. The potential capture cross section of 1.6 mb deduced for the 4059 k e y line is somewhat lower than the values of 10 and 20 mb calculated by Lane and Lynn 16). However, the calculation was based on hard sphere scattering which is clearly unrealistic, especially for nuclei in the deformed region. Furthermore, the experimental value is open to uncertainty in that it is difficult to be sure that the choice of signs for the resonance amplitudes is unique. Somewhat poorer fits, but still within statistical error, could be found with higher values for the direct amplitude by changing the signs of the more distant resonances.
6. The statistical properties of partial radiative widths The behaviour of the partial radiative widths in 238U has been the subject of much interest, especially since the original suggestions of Hughes et al. 4) that the width distribution in this nucleide was narrower than expected. In analogy with the particle widths, the radiative widths are usually assumed to follow a Z2 distribution with one degree of freedom, implying a Gaussian distribution of width amplitudes. Early data, taken at Brookhaven 4), Argonne s) and Saclay 25) with low resolution NaI radiation detectors indicated that partial widths for 7-rays near 4 MeV showed relatively little fluctuation from resonance to resonance. These results were generally ascribed to the inability to completely resolve the radiation to discrete final states. With the superior resolution of the Ge(Li) detector, it is now possible to believe that the majority of final states are resolved. This conclusion is supported by the data of the present experiment, which indicate that in the resonances only a few more lines than reported by Sheline et al. 3) at thermal neutron energy are observed. It is therefore of considerable interest to study the distribution of widths obtained in the present experiment. For this purpose we consider the set of 18 final states populated by 5 resonances, omitting the weak 10.2 eV resonance and the incompletely resolved 103 and 117 eV resonances as well as the known M1 and E2 transitions. The set of 90 widths (18 final states, 5 initial states) is normalized to the 4.059 MeV transition by applying an E 3 factor assumed for electric dipole transitions: E
-3
648
D . L . PRICE et
aL
The use of the above relation also assumes that the mean intensities, over all resonances, are the same in populating all final states, so that fractional degrees of freedom are not introduced. With this assumption, we can examine the distribution of partial radiative widths. The cumulative probability distribution for the 90 widths is shown plotted in fig. 11 along with the theoretical curves for v = 1, 2 and 4. F r o m this figure, it is readily seen that the best fit is obtained with v ~ 4. To study the possible distortions in the distribution introduced by experimental errors and limited sample size, a Monte Carlo 90 80 E" 7o At 6 0 co J
~
l
~
,
,
i
\% \ \ ,\ ~
J
~
.....
\\\ '~ ~
,
I
I
i
~:,
--~o2 --,,--4
-
<1"7 i > =0.31 meV
-
u_ 4 0 o
rr 3 0 - w rn :~ 20
N
-(
1
i 0.20
I
i 0.4-0
i
1 0.60
~
,'=---~-~ 0.80 1.00
-r---~-: 1.20
FTi (meV) Fig. 1 l. The cumulative distribution for 90 partial radiative widths o f the present experiment. The widths are reduced to an effective energy o f 4.059 MeV by assuming a cubic dependence o f width
on 7-ray energy. All known MI and E2 transitions have been deleted. analysis employing the program C O R N R O W 2 2 , 2 3 ) w a s carried out to study the distribution of veff, the effective number of degrees of freedom, for the physical sample, assuming parent populations corresponding to v = 1 and 2. For v = 1 a value of 1 +0.55 I~eff = xl."r.~_0.25A~ +0"33 a n d f o r v = 2, Veff . . . . ") . 0.40 w e r e obtained, where the e r r o r s indicate 10 to 90 per cent confidence limits. The experimental value, ~ 4, was determined to be clearly discrepant with both v = 1 and 2. To proceed further, a m a x i m u m likelihood analysis, based on the methods developed at Argonne National Laboratory 24.28), was employed. Briefly, the procedure is to extract the value of v~ff from the physical sample by the method of maximum likelihood, and then to compare the value of v~ff so obtained with values of v~, from mathematical samples chosen to approximate the physical sample. The method is explained in detail in an appendix to this paper. In fig. 12 are shown the distribution of values Vraathematleal>Vphysiea I (obtained from 100 Monte Carlo samples) as a function of the assumed v for the parent population.
649
N - C A P T U R E I N 2aaU
I t.O-
I
I
I
/ ,
m
o
I
I
/"
W~TH ERRORS
o.6-
¢rl
rY (3_
I
o.8-
>_ F-
I
WITHOUT E R R O R S ~ / ' ~ ~
__ A
I
0.4--
0.2--
I
I
Jr"
1.0
2.0
3.0
¢3
~1
/
,
I
I
I
I
I
I
4.0
5.0 v,
6.0
7.0
8.0
9.0
I0.0
Fig. 12. T h e c u m u l a t i v e distribution o f Vetf values obtained f r o m a M o n t e Carlo s i m u l a t i o n o f the physical set o f widths, o b t a i n e d f r o m a m a x i m u m likelihood m e t h o d . T h e values are for s a m p l e s w h i c h are d r a w n f r o m p a r e n t p o p u l a t i o n s o f v indicated o n the x axis.
I
15
i
Vo=4.00 vp=3.80
14 E 13
2O ,2 m
,0i
~
7
~
6
u.l
5
4 3
2I
2.0
3.0
I
I
4.0
5.0
u m
Fig. 13. A typical distribution o f Vet r values o f 100 s i m u l a t e d sets o f physical widths d r a w n f r o m a p a r e n t p o p u l a t i o n with v = 4. T h e ~ett for t h e physical s a m p l e is also s h o w n .
650
D . L . PRICE et al.
F r o m this figure we may read off the best value of Veff and its 10 and 90 percentile limits, and we obtain Veff = 4.6-+ 11.9 .2' Fig. 13 shows a typical distribution of v values obtained from the maximum likelihood analysis of 100 mathematical samples of 90 widths, drawn from a parent population of v = 4, and broadened by a normal error distribution. Also shown in this figure is the maximum likelihood value of v for the physical sample, v~ff = 3.80. The large value obtained is not consistent with the Porter-Thomas distribution, or )~2 distribution with one degree of freedom. A subset of widths was further examined by selecting only the strong transitions at 4.068, 4.059, 3.991 and 3.982 MeV. The subset of 20 widths gave a value oi6 v =~a~. /,+2 _l: in good agreement with the larger set. The large value of v,~f obtained here is difficult to explain in terms of unresolved transitions, since, as noted above, the comparison between thermal and resonance capture indicates that most levels are probably being resolved. Furthermore, it is difficult to visualize other systematic errors which would lead to this result. A theoretical framework to accommodate departures from the Porter-Thomas distribution, without violation of the usual invariance properties of nuclear wave functions, has been advanced by N. Rosenzweig 26). Rosenzweig's expression for the joint probability distribution of the widths, W(r)6{
P ( Z , . . • Z,,) oc
Z,
r2}rdr,
i=1
is derived from considering only those interactions which induce orthogonal transformations in the N compound nuclear states which leave the subspace spanned by ~ n invariant. Here, ~ n is defined as the space spanned by PVqg~ where V is the interaction Hamiltonian, goz as one of n final states, and P as the projection onto the space spanned by the N compound nuclear states. The Porter-Thomas distribution is considered as a special case of the above where the weighting function W ( r ) is a Gaussian, i.e. W = exp(-½Nr2). The relative variance, v, of a single partial width is shown to obey the following inequality: v >__ 2 ( n - 1 ) -
(n+2)
Since the present experiment clearly indicates that v < 1, we conclude that n < 4. Rosenzweig also shows that it follows that at least one of the width correlation coefficients must be negative. The data of the present experiment do not include enough resonances to establish firmly this fact. However, average correlation coefficients between reduced neutron widths and partial radiative widths, and among radiative
N-CAPTURE IN 238U
651
widths, have been calculated. The results are as follows: R -- (corr {F °~, r~,j}> i = +0.058, T -- (corr {F~j, F~j,})j.ej, = +0.06. These values are consistent with zero falling at the 75th and 64th percentile, respectively. The presence of correlations of both signs, as expected from Rosenzweig's analysis, is not ruled out by the data.
7. Conclusions These measurements of resonance neutron capture in 238U(n, ~)239U are on the whole consistent with the level scheme for 2 3 9 U proposed by Sheline et al. in ref. 3), on the basis of rotational bands based on Nilssons orbitals; there is only one point of disagreement. There is evidence of unusually strong M1 transitions. The most significant feature of the experiment has been the most convincing evidence, to date, for a direct, potential contribution to neutron radiative capture. However, single-particle capture effects in the resonances appear to be small. The authors would like to encourage more theoretical work on these lines: (i) calculations of the strength of transitions of different multipolarity in the region of deformed nuclei. (ii) Quantitative examination of the proposal for octupole vibration bands near 800 keV excitation. (iii) A more realistic calculation of the strength of direct capture. The results of the experiment show that the partial radiative widths of 238U are inconsistent with a Porter-Thomas distribution, and are also inconsistent with a X2 distribution with two degrees of freedom. The authors would like to thank Frederick Paffrath for his tireless work maintaining the rotor system, Vito Manzella for electronic assistance, and Jean Domish, Judy Harte, and Isaac W. Cole for help with the computations. The authors would also like to acknowledge the invaluable assistance of David Parks for setting up the m a x i m u m likelihood analysis of sect. 6. The germanium detectors used in this experiment were made by H o b a r t Kraner and Marco Jamini of the B N L Instrumentation Department.
Appendix MAXIMUM LIKELIHOOD METHOD g4--28) The original physical data consist of N partial radiation widths {Fij } and their respective errors {e~j}, where i and j denote a particular final state and resonance, respectively. F r o m these sets a normalized set of widths (xij} and a normalized set of errors {e~j} are formed, where the xij and e~j are related to the Fij and eij by
x~j = F~/u~, eij' = egj/Ui,
1 < i <_ n, 1 <=j =< m,
(1) (2)
652
D . L . PRICE
et al.
where
1 & uz = - -
(3)
L rij,
mj=l
and where m is the number of resonances and n is the number of final states. One assumes that the {x~j} belong to a Xz distribution with 2p degrees of freedom, for which the probability distribution function is pP P ( x i j , Pl =
i-(o)
x ° - l e- - P x ' l d-0i~, 'z -ij
(4)
where F(p) is the gamma function of p = v/2, v being the number of degrees of freedom. Following the maximum likelihood procedure 27), one forms the likelihood function L(x~j, p) which is the joint probability distribution function for the random variable x~:
L(xij, p) = Vl p a x °.-'e -'x'~ ~/ r(p) "
(5)
or
L(r,;u,,pl
PP ( F z J l P - l l e x p ( - p F i J ) u, , Us ,"
=
(6)
The estimator of the parameter p, or the value ofp which maximizes L(xe, p), is found by solving the simultaneous set of equations d In L(r,j/u,, p) = O, dp
(7)
d In L(Fij/u,, p) = O.
(8)
dui
Since the mean value of x~j using eq. (4) is 1, one chooses the estimator of the mean, or the value of u~ that satisfies eq. (8), to be 1 also. Using this and performing the differentiation above, one gets the transcendental equation dlnF(p)_lnp dp
= ~ .1.
In x,j,
(9)
where N = (n)(m), the total number of widths in the physical sample. Taking into account the finite size of the physical sample, eq. (9) can be replaced by the more accurate form 7)
d ln F(p)- ln p = l ~ ln x~j - 0"5 + 1 dp N ,j [Q" ~12Q
1
120Q ~ +
2~2Q61
'
(101
where
Q = rap.
(11)
N-CAPTURE IN Z38U
653
The solution to eq. (10) yields the estimator of the parameter p. The term In x~j in eq. (10) diverges as x~j ~ 0 and is undefined for x o < 0. Since the physical data may contain zero and/or negative widths, the data is truncated at a value determined by the normalized error distribution of the physical sample. If the normalized error distribution has a well defined peak and a smooth tail, the cutoff value is equal to the mean error. If the distribution has a well defined peak but also has a b u m p in the tail due to large errors of certain resonances, then the cutoff value is equal to the median error. The median error, and not the mean error, is chosen in the latter case so as not to overestimate the error and hence the cutoff. The criterion for using the cutoff is that any width lying below the cutoff value is set equal to onehalf of the cutoff value. Following this procedure the normalized widths {x~j} are corrected using the appropriate cutoffs, renormalized over resonances, and are used in eq. (10) to get the estimator of p. After the number of degrees of freedom vp for the physical sample has been calculated, a Monte Carlo technique is employed to test the hypothesis that the physical sample is drawn from a Xz distribution with Vo degrees of freedom. A mathematical sample "identical" to the physical sample is generated from a X2 distribution with v o degrees of freedom, where v o = 1, 2, 3, 4, etc. This is done by first choosing at random N numbers from a Xz distribution. The number of "widths" in this mathematical sample must be equal to the total number of g a m m a widths in the physical sample. These "widths" are normalized in groups of rn widths, corresponding to eq. (1) for the physical sample. To further simulate the physical data, errors are folded into the mathematical distribution of "widths" according to zij = y~j + r/d,
(12)
where y~j is a normalized width, d the mean or median error of the physical normalized error distribution, and r/a r a n d o m number drawn from a normal distribution with mean zero and standard deviation one. The z~j's are normalized over "resonances" and are corrected using the same cutoff value and cutoff criterion as for the physical sample. The corrected widths, after renormalization over resonances, are used in eq. (10) to calculate Vm, the number of degrees of freedom for the mathematical sample. The entire process of forming a mathematical sample simulating the physical sample is iterated 100 times for each Vo. For a given v o, the above hypothesis is tested by plotting the distribution of vm (100 values) and Vp: if vp lies within 10 ~ to 90 ~o of the distribution, then the physical data is consistent with Vo degrees of freedom. The distribution of vm for v o = 4.00 and vp = 3.80 for the 23sU data is plotted in fig. 13. The "best value" of v for the physical data is the v o for which Vp is the median value of the distribution of vm, or when the probability that Vm > Vp is 0.5. Such a value can be obtained by plotting the probability that vm > vp for each value of vp. Also, a "best value" for v can be obtained from truncated distributions without errors folded in. A comparison is made in fig. 12.
654
D.L. PRICE et al.
References 1) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk., 27, no. 16 (1953) 2) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1 no. 8 (1959) 3) R. K. Sheline, W. N. Shelton, T. Udagawa, E. T. Jurney and H. T. Motz, Phys. Rev. 151 (1966) 1011, and references contained therein 4) D. J. Hughes, H. Palevsky, H. Bolotin and R.E. Chrien, in Proc. Int. Conf. on nuclear structure, edited by D. A. Bromley and E. W. Vogt (University of Toronto Press, Toronto, 1960) p. 771 5) H. E. Jackson, Phys. Rev. 134B (1964) 931 6) I. Bergqvist, Nucl. Phys. 74 (1965) 15 7) C. F. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 8) R. E. Chrien, D. L. Price, O. A. Wasson, M. R. Bhat, M. A. Lone and M. Beer, Phys. Lett. 25B (1967) 195 9) R. E. Chrien and M. Reich, Nucl. Instr. 53 (1967) 93 10) M. R. Bhat, B. R. Borrill, R. E. Chrien, S. Rankowitz, B. Soucek and O. A. Wasson, Nucl. Instr. 53 (1967) 108 11) J. R. Stehn et al., Brookhaven Nat. Lab. Rep. 325, Suppl. 2 (1965) Vol. III, 92-238-3 12) E. K. Hyde, I. Perlman and G. T. Seaborg, The nuclear properties of the heavy elements (Prentice-Hall, New Jersey, 1964) p. 150 13) I. Bergqvist, B. Lundberg and N. Starfelt, in Proc. Int. Conf. on nuclear physics with reactor neutrons, edited by F. E. Throw (Argonne National Laboratory Report 6797, 1963) p. 220 14) C. K. Bockelman, Nucl. Phys. 13 (1959) 205 15) H. Morinaga and C. Ishii, Prog. Theor. Phys. (Kyoto) 23 (1960) 161 16) A. M. Lane and J. E. Lynn, Nucl. Phys. 17 (1960) 563 and 586 17) I. Lovas, Z h E T F (USSR) 41 (1961) 1175; JETP (Sov. Phys.) 14 (1962) 838 18) O. A. Wasson, M. R. Bhat, R. E. Chrien, M. A. Lone and M. Beer, Phys. Rev. Lett. 17 (1966) 1220 19) E. P. Wigner and L. Eisenbud, Phys. Rev. 72 (1947) 29 20) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 21) R. E. Cot6 and L. M. Bollinger, Phys. Rev. Lett. 6 (1961) 695 22) M. Beer, Correlation effects of primary resonance neutron capturey-rays in heavy nuclei (Thesis, State University of New York at Stony Brook, 1968, unpublished) 23) M. Beer, M. A. Lone, R. E. Chrien, O. A. Wasson, M. R. Bhat and H. R. Muether, Phys. Rev. Lett. 20 (1968) 340 24) Argonne Nat. Lab., Illinois 60439, Appl. Math. Div. Program Library 867/PHY-186, Dec. 22, 1960 25) C. Corge, V.-D. Huynh, J. Julien, J. Morgenstern and F. Netter, J. Phys. Rad. 22 (Oct. 1961) 722-723 26) N. Rosenzweig, Phys. Rev. Lett. 6 (1963) 123-125 27) R. V. Hogg and A. T. Craig, Introduction to mathematical statistics (Macmillan, New York, 1965) 2nd ed., p. 243 28) L. M. Bollinger, R. E. Cot6, R. T. Carpenter and J. P. Marion, Phys. Rev. 132 (1963) 1640