Neutron resonance parameters and radiative capture cross section of Gd from 3 eV to 750 keV

2.A.1 I

Nuclear Physics A146 (1970) 337--358; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

NEUTRON RESONANCE PARAMETERS AND RADIATIVE CAPTURE C R O S S S E C T I O N OF Gd F R O M 3 eV T O 750 keV s. J. FRIESENHAHN, M.P. FRICKE, D. G. COSTELLO, W. M. LOPEZ t and A. D. CARLSON Gulf General Atomic Incorporated, San Diego, California 92112, USA tt

Received 9 March 1970

Abstract: Resonance parameters for neutron energies from 3 eV to 200 eV and average capture cross sections for energies up to 20 keV have been measured for the separated isotopes ~ G d and l*7Gd with neutron time-of-flight techniques and a large liquid scintillator. In addition, the average capture cross section of natural gadolinium was measured from 1 keV to 750 keV. The -4 for s-wave strength functions obtained are .9. .17+o-~ovm-* . o.~, . . for . ~SSGd . . and 29+°'9×10 -o.5 aS7Gd' and these values are significantly higher than those predicted from the collective-model generalisation of the optical model. The total radiation widths for resolved resonances are found to have a much larger variance than that predicted by the statistical model, provided the widths are assumed to be spin-independent.The average capture cross section at higher energies is also found to deviate from conventional statistical-model predictions. E[ NUCLEAR. REACTIONS ls5, lS7Gd(n,7), E = 3 eV-20 keV; Gd(n,?), E = 1-750 keV; I measured a(E). ~6, ~5SGd deduced resonances, resonanceparameters. Natural, enriched targets.

1. Introd.etion Since the level spacings of l S S G d + n a n d 1 5 7 G d + n are small near the n e u t r o n s e p a r a t i o n energy, these odd isotopes of g a d o l i n i u m provide a fertile field for the study of r e s o n a n c e - p a r a m e t e r systematics. I n a d d i t i o n to establishing the average s-wave reduced n e u t r o n width a n d level spacing, parameters for a large n u m b e r of resonances can provide i n f o r m a t i o n o n the fluctuation of the total r a d i a t i o n width. The average capture cross section measured at higher energies yields i n f o r m a t i o n on the variation o f the total r a d i a t i o n width with the excitation energy o f the c o m p o u n d nucleus. Both the resonance parameters a n d the average cross section determine the s-wave strength f u n c t i o n for gadolinium. This element lies i n the m i n i m u m of the strength f u n c t i o n n e a r A = 160, where optical-model calculations tend to p r o d u c e values lower t h a n those observed. F r o m a practical s t a n d p o i n t the very high capture cross section of g a d o l i n i u m makes it a n excellent choice for reactor control applications, a n d a n accurate knowledge of the resonance capture integral is of considerable interest in this regard. The gadol i n i u m isotopes do n o t lend themselves to direct m e a s u r e m e n t s of the resonance integrals by activation methods, a n d hence these integrals m u s t be o b t a i n e d from measurements of the resonance parameters a n d average capture cross sections. t Present address: University of California at San Diego, La Jolla, California 92037, USA. tt Work supported jointly by the US Atomic Energy Commission and Gulf General Atomic Incorporated. 337 May 1 9 7 0

338

s.J. FR1ESENBAHNet

al.

2. Experimental techniques Time-of-flight measurements were made with short bursts of photoneutrons from bremsstrahlung produced by 30 MeV electrons from the Gulf General Atomic Linear Accelerator. The electrons impinged on a tungsten-steel alloy target surrounded by

' PATH

...........-- - " " ~ JPOLYETHYLENE MODERATOR

~\~s o~

\

\t

15°

TOP VIEW

IINIUM CAN

TITANIUM WINDOW

R-COOLING

ELECTRON BEAM

.I i I I I J ~ J 6 8 I0 .E : cm

Fig. 1. Cross-section view of uranium-shielded electron target for neutron production. a cylinder of 2aSU (see fig. 1). The uranium moderates the neutrons by inelastic scattering and also shields the detector from the bremsstrahlung. The neutrons were slowed further in a 2.5 cm slab of polyethylene placed perpendicular to the neutron flight path.

155, 157Gd

339

A 20 m flight path was used for the measurements up to 20 keV neutron energy and a 230 m flight path for the measurements at higher energies. Gamma rays from neutron capture were detected at the end of the 20 m flight path with a 4000 1 liquid scintillator, and with a 600 1 liquid scintillator at the end of the 230 m flight path. Further details of these flight-path facilities and the liquid scintillators are given in refs. ~' z). 2.1. MEASUREMENTS BELOW 20 keV The 1 s SGd and 157Gd resonance parameters were determined from shape and area analysis of capture, self-indication and transmission data. The first two types of data extend to 200 eV neutron energy, and the transmission data extend to 20 eV energy. Average capture cross sections were measured up to 20 keV. The capture samples were placed at the centre of the 4000 1 Scintillator, and the transmission and selfindication filters were placed in the collimated neutron beam immediately ahead of the scintillator. Two BF 3 flux monitors were positioned just ahead of the filter in the penumbra of the neutron beam, and a 20 atm, 1.27 cm diameter 3He neutron detector was pIaced just after the scintillator for use in transmission measurements and to determine the flux shape (i.e. the incident-neutron energy spectrum) for a capture experiment. The flux-measuring techniques are discussed in ref. 3). For 1 s 5Gd ' the flux was normalised by using the saturated-resonance technique 4) for the 2.568 eV resonance, and this normalisation was checked b y saturated resonances at 6.3, 19.92 and 21.02 eV. These additional normalisations at different neutron energies also serve as a check on the flux-shape determination. The normalisation for l SYGd was obtained from the calculated area of the 2.825 eV resonance using the resonance parameters given in ref. 5). The relative normalisations for ~SSGd and 157Gd agreed well with the ratio of y-ray spectrum fractions observed for the two isotopes. The discriminator used with the 4000 1 scintillator accepted events depositing 4 to 10 MeV of energy in the liquid. In addition, a coincidence requirement assured that at least 0.5 MeV was deposited in each of the two optically isolated halves of the scintillator. This latter requirement selects the multi-gamma cascades which are usually produced by capture and eliminates many single-gamma events that are usually associated with background. While this coincidence requirement increases the signalto-background ratio by an order of magnitude or more, it might also introduce a sensitivity to the v-ray cascade mode, which in general varies from resonance to resonance. This possibility was examined by comparing y-ray pulse-height distributions for different resonances and also by comparing values of 29Fn deduced both from transmission data and from capture data for a thin sample. Within the precision of the comparisons (2-3 ~ ) no evidence of a sensitivity to the cascade mode was found. The pulse-height distributions used in these comparisons were obtained by lowering the sum-signal discriminator level to 1 MeV and storing the time-of-flight versus pulseheight information in two-parameter form. These data were later sorted to form pulseheight distributions corresponding t o capture at the yarious resonances.

340

$. J. ~FRIESENHAHNe t al.

2.2. MEASUREMENTS ABOVE 20 keV

"['he natural gadolinium average capture cross section was measured at the 230 m flight-path facility. The 600 1 scintillator was also operated to accept 4 to 10 MeV events, but no coincidence requirement was imposed on the signals from this smaller detector. Determination of the average cross section actually involved the use of both flight-path facilities. The probability for capture in the natural gadolinium sample was first measured to 20 keV neutron energy with the short flight path using essentially the same techniques employed in the isotopic capture cross-section measurements discussed in subsect. 2.1. The capture yield was then measured from 1-750 keV using the long flight path, and the results were converted to a capture probability by normalisation to the short flight-path data in the energy region 3 to 10 keV. Gas proportional counters filled with 3He were used to monitor the incident neutron flux at the 230 m flight-path facility. The helium counters were calibrated against the l°B(n, ~)7Li cross section from 1-80 keV and against the n + p scattering cross section from 80 to 750 keV. Details of these flux measurements are given in ref. 2). 2.3. SAMPLES

Two batches of material enriched in 155Gd and 157Gd were obtained in the form of oxide powders from the Isotopes Division of the Oak Ridge National Laboratory. Table 1 lists the enrichments quoted by the supplier. The impurities in both batches of material were negligible, and no resonances attributable to other elements were TABLE 1 Isotopic abundances of enriched samples ~SSGd sample Isotope 152 154 155 156 157 158 160

Atom percent < 0.05 0.61 91.77 5.14 1.13 0.94 0.40

Precision

:t:0.05 ±0.10 ±0.10 ~0.05 ::E0.05 ~0.05

~S7Gd sample Isotope 152 154 155 156 157 158 160

Atom percent < 0.03 0.09 1.54 4.31 88.63 4.61 0.82

Precision

:k0.03 --0.05 :k0.05 :k0.10 7E0.05 7_0.05

155,157Gd

341

observed. E a c h batch was divided into two capture samples and one transmission sample, and the materials were pressed into thin-walled aluminium cans. The thicknesses measured with an X - r a y densitometer were f o u n d to be uniform to +_4 %. Trial calculations o f self-indication and transmission areas indicated that this uniformity was more than adequate to yield reliable results. The absorption o f water can be a problem with oxide samples, and hence the offresonance neutron transmission o f the samples was measured to determine the attenuation due to oxygen, aluminium and absorbed water. The measurement indicated that the a m o u n t o f absorbed water was quite small and, since this experimentally determined b a c k g r o u n d cross section is included in the resonance-parameter calculation, no appreciable error in the parameters is introduced. The sample used for the natural gadolinium measurements was obtained f r o m the Nuclear C o r p o r a t i o n o f America. There was some evidence o f a very small ytterbium contamination, but no corrections for this were necessary. The thicknesses o f all the samples are listed in table 2. TABLE 2 Sample thicknesses Sample 1 2 3 4 5 6 7 8 9 ") b) ~) a) •e) f)

Enriched isotope

Thickness (103 atom/b)

155 155 155 157 155 157 graphite graphite gadolinium

0.518 0.449 1.584 1.297 1.475 1.206 4.88 1.66 3.03

Use ") ") b) b) *) ~) a) c) r)

Thin capture sample for resonance parameters. Thick capture sample for resonance parameters and average capture cross sections. Transmission/self-indication filter. Approximate scattering equivalent of samples 3 and 4. Approximate scattering equivalent of samples 1 and 2. Natural gadolinium average capture cross section.

3. Results 3.1. RESONANCE PARAMETERS The count rates measured at the 20 m flight-path facility were corrected for dead times and backgrounds and converted to capture probabilities or transmissions. Probable errors for these observables were estimated including contributions f r o m counting statistics, possible systematic errors in the backgrounds, uncertainties in the flux shape and normalisation, and uncertainty in the incident neutron energy. Flux

342

s.S. FRIESENHAIIN et al.

uncertainties affect only capture data and were always less than 3 ~ . Maximum systematic errors in the backgrounds were estimated to be 20 ~ . The uncertainties in the resonance-energy determinations, and hence in the boundaries of the area summations, were taken to be one-half the energy width of the time channel nearest to the centre of the resonance. A multi-sample area analysis of the data between 3 and 200 eV was performed using a version of the computer program T A C A S I 6), and a shape analysis was performed on the data below 25 eV with the computer program SHAPE. The latter code is based on the formulations contained in T A C A S I with the addition of the resonance energy 1,0

I I I~

I I I I I I I

I I I

w n~

o_

IJll

Llll

I II

I I i i

ii

tIIIt tiii iiI

>-

m o no_

II1~

it

o

11.53 eV

•

I

T I i I I I I I I [ PI

1!.99 eV

Ir

I t I I I I

i

I r I

I I 1 I

t

I I

0.25/_tse¢ TPME CHANNELS

Fig. 2. Experimental and calculated capture probabilities for the 11.53 and 11.99 eV resonances in 155Gd. The brackets denote the limits on experimental points, and the circles denote calculated values.

as a free parameter. Both of these search codes obtain least-squares estimates of single-level resonance parameters and their uncertainties by a direct fit to the observed quantities for a given resonance. Doppler and resolution broadening a r e calculated; the contributions of nearby resonances are included; and multiple-scattering effects are obtained by Monte Carlo techniques. Except for resonances with known spin, the spin-statistical factor was taken to be g = 0.5. The effect o f this approximation is always small because the gadolinium resonances are weak

(r. << r,).

155'157Gd

343

For most of the resonances below 25 eV, shape analysis yielded more accurate resuits than mu!ti-sample area analysis. Above this energy the radiation widths were most accurately determined by the self-indication ratio and the capture area. The interference radii determined by Mughabghab and Chrien 7) were employed in the resonanceanalysis. Contributions from nearby resonances in the even isotopes were taken into account by using the parameters obtained by Karschavina et al. 8). The relative detection efficiency for 158Gd(n ' 7) was determined from 7-ray pulseheight measurements described earlier, and the relative detection efficiencies for the other even isotopes were estimated from binding energies. Because of the high enrichment of the samples, the contributions from the even isotopes were usually small. The results of the resonance-parameter analysis are listed in tables 3 and 4 for i s SGd and 157Gd ' respectively. Below 3 eV the tables contain the parameters recommended in ref. 5) which were obtained from a shape analysis of transmission data. These values have smaller uncertainties than could be readily achieved in the present work using capture techniques. Previous workers have apparently missed the very weak resonance at 3.616 eV. The pulse-height distribution due to capture in this resonance tends to confirm it as a resonance of ~55Gd rather than one arising from an impurity. In spite of the close spacing of the 11.53 and 11.99 eV resonances in 155Gd ' apparently reliable radiation widths were obtained from both using the shape-analysis code. These widths would have been difficult or impossible to obtain using multisample area analysis. The observed and calculated shapes for these two levels are compared in fig. 2. The radiation width of the 23.65 eV resonance in 155Gd is surprisingly large. This large value was obtained from multi-sample area analysis and was also confirmed by shape analysis. On the other hand, the 78.8 eV resonance in the same isotope has a small radiation width. These variations in radiation width are discussed in subsection 3.2. There appears to be a pair of resonances in ~55Gd at 33.14 and 33.50 eV with approximately the same values of 2gF n; however, a very strong resonance in the 156Gd contaminant hinders the analysis of this pair. Our data for 157Gd confirm the low value of the radiation width obtained by Karschavina et al. 8) for the 16.85 eV resonance. The existence of the very weak resonance at 16.25 eV is also confirmed. This low value of the radiation width for the 16285 eV resonance is in disagreement with the higher value reported by Mughabghab and Chrien 7), and this disagreement may be due to the omission of the 16.25 eV resonance from their analysis. The 157G.d resonance at 120.9 eV is wide enough to allow a check on the multi-sample area analysis using the shape code. The agreement was excellent. For resonances in tables 3 and 4 for which both 2gFn and/'~ were determined, we also list the "experimental covariance" of (2gFn, F~) deduced in the least-squares fit to the observed quantities for each resonance. These experimental covariances are included in the error analysis of the resonance integrals discussed in subsect. 3.6.

344

S. I. FRIESENHAHNet al. TABLE 3 15SGd resonance parameters a)

Eo

zlEo

(eV)

(eV)

0.0268 2.008 2.568 3.616 6.300 7.750 10.01 11.53 11.99 14.51 17.77 19.92 21.02 23.65 27.57 29.58 30.10 31.72 33.14 33.50 34.83 35.45 37.13 39.00 43.92 46.10 46.82 47.70 51.36 52.10 53.01 53.68 56.17 59.32 62.74 64.00 65.15 69.50 76.90 77.70 78.80 80.05 80.80 84.05 84.95 90.50

0.0002 0.01 0.013 0.006 0.030 0.007 0.012 0.013 0.01 0.01 0.02 0.02 0.04 0.04 0.05 0.06 0.06 0.06 0.07 0.07 0.07 0.07 0.08 0.08 0.10 0.10 0.10 0.11 0.07 0.08 0.08 0.08 0.08 0.09 0.09 0.10 0.10 0.10 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.13

F'~,

A/'~,

2gT~

A2g/'~

(meV)

(meV)

(meV)

(meV)

108 110 111 130 107.5 127 115 125 112 111 120 108 101 167

1 1 1 17 9.8 4 20 23 11 10 25 16 6 14

90 105 73

22 11 57

147

18

136

9

84

I2

72 61

10 55

82

11

151

36

47

23

0.130 0.280 2.18 0.033 2.6 1.4 0.23 0.44 1.2 2.4 0.50 5.0 19 3.9 0.84 5.4 12 1.3 1.1 1.3 4.6 2.3 5.6 1.4 12 2.6 6.5 0.49 14 14 1.8 9.6 2.7 8.1 9.8 0.36 0.84 7.9 2.1 0.97 5.7 0.39 1.9 6.9 2.3 1.6

a) Parameters for resonance energies below 3 eV are from ref. s).

0.002 0.003 0.02 0.002 0.17 0.027 0.015 0.029 0.051 0.12 0.033 0.43 1.2 0.099 0.023 0.42 0.85 0.04 0.5 0.4 0.17 0.12 0.17 0.44 0.4 0.18 0.2 0.036 0.7 0.52 0.06 0.45 0.11 0.22 0.4 0.017 0.042 0.29 0.076 0.053 0.46 0.14 0.2 0.23 0.12 0.062

Covariance (/'~,, 2g/"n) × 106 (eV) ~

0.03 --1.5 0.31 0.14 0.42 --0.12 --0.62 0.34 --0.54 --4.8 --0.46 0.41 0.38 1.1 0.76

--1.5 --1.6 --1.3

0.49 --2.3 --4.6 --2.8

--6.9

--9.9

155, 157Gd

345

TABLE 3 (continued)

A/to

r,

2g_F~

A2g1~

Covariance

(Pr, 2gTD × lO6 (eV)

(eV)

92.50 93.00 94.10 95.86 96.55 98.33 100.3 101.4 102.2 104.4 105.9 107.1 109.6 112.4 113.8 116.6 118.7 123.4 124.5 126.1 128.6

0.15 0.15 0.15 0.15 0.15 0.16 0.1 0.1 0.1 0.1 0.1 0,1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0,2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2, 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3

129.8 130.9 133.1 133.9 134.8 137.8 140.4 141.4 145.7 146.9 148.2 149.6 150.2 152.2 154.0 156.3 160.1 161.6

168.3 170.3 171.4 173.6 175.5 178.0 180.4 183.3

(meV)

(meV)

84 67 116

10 12 94

159

65

152

131

110

29

(meV)

(meV)

2.7 3.9 0.68 4.8 4.5 13 1.5 3.4 1.3 6,3 4.6 7.7 3.5 10 25 15 2.5 27 7.6 15 1.4 3.2 35 3.1 3,4 1.1 16 3.1 1.3 7.7 5.3 12 25 31 6.2 1.4 10 12 25 22 10 12 38 2.6 8.0 11 9

0.29 0.36 0.045 0.33 0.31 0.39 0.16 0.3 0.2 1.6 0.38 0.64 0.3 2.4 4.4 2.6 0.2 4.3 0.91 2.1 0.17 0.53 6.1 0.36 0.5 0.18 1.5 0.34 0.21 0.7 0.58 1.4 7.2 11 0.54 0.2 0.84 1.3 3.2 2.4 1.5 1.8

8 0.29 0.69 1.1 0.8

(e¥) 2

--230 --25 --210 --180 --230

100

1200

346

S. J. FRIESENHAHN e t aL TABLE 4

157Gd resonance parameters a) Eo

zlEo

F7

d/'^/

2#/~n

A2g_P~

Covariance

( r 7, 2ar~) x lO~ (eV)

(eV)

(meV)

(meV)

(me¥)

(meV)

0.0314 2.825 16.25 16.85 20.56 21.66 23.33 25.40 40.15 44.20 48.72 58.31 66.57 81.39 82.24 87.20 96.59 100.2 104.9 107.4 110.5 115.4 120.9 138.2 139.3 143.7 148.4 156.6 164.9 168.3 169.5 171.4 178.7 184.0

0.002 0.015 0.2 0.11 0.04 0.03 0.02 0.05 0.09 0.1 0.11 0.08 0.1 0.12 0.13 0.14 0.15 0.1 0.1 0.1 0.1 0.2 0.09 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3

107 97

1 1

77 88 147 121 79

5 5 65 31 23

91 87 104 67

8 4 5 12

113 103 79 66

33 31 19 9

87 113 91

10 56 6

88

10

87 69

70 31

0.59 0.43 0.3 18 17 0.45 0.59 2.3 0.94 12 30 37 11 15 7.7 12 14 26 38 5.8 60 24 198 60 5.3 72 14 24 30 2.5 4.2 29 19 21

0.01 0.003 0.1 1.1 1.3 0.06 0.05 0.088 0.036 0.55 1.8 2.2 0.72 0.89 0.51 0.79 1.1 5.8 8.8 0.61 13 5.7 10 18 1.4 18 1.5 8.6 12 0.39 0.61 4.5 2.1 2.5

(eV) "~

------

0.5 0.37 0.28 0.055 0.12

-----

0.26 0.18 0.41 0.66

-- 2.2 -- 6.4 -- 0.84 -- 3.4 --26 -- 2.3

-- 3.5 --56 --32

a) Parameters for resonance energies below 3 eV are f r o m ref. s).

3.2. A V E R A G E N E U T R O N W I D T H S A N D L E V E L S P A C I N G S

T h e integral n e u t r o n - w i d t h distributions o f z s 5G d + n a n d ~ s 7 G d + n are c o m p a r e d in fig. 3 to curves calculated for two spin p o p u l a t i o n s o b e y i n g a P o r t e r - T h o m a s distribution. There a p p e a r s to be a slight excess o f small widths in the ~5 5Gd + n distribution t h a t c o u l d be due to p-wave resonances. Such a d e v i a t i o n is n o t a p p a r e n t for 1 s 7 G d + n, a l t h o u g h the statistical uncertainties are a p p r e c i a b l y larger in this case because o f the smaller sample o f resonances. T h e average r e d u c e d n e u t r o n widths are given in table 5.

155,157Gd 100

--

,

'

I'.010

I

i

'

.'

I

'

i 0.I

I

'

347 '

'

I

,

r 1.0

'

Zjo

t

t

i .--#

I

i

P

10

r2/rn Fig. 3. C o m p a r i s o n s o f integral n e u t r o n w i d t h distributions for t55Gd a n d ~STGd with t h e distrib u t i o n calculated for two i n d e p e n d e n t spin p o p u l a t i o n s . TABLE 5 A v e r a g e p a r a m e t e r s for G d isotopes Isotope

_Fr(meV)

P~°(meV)

D (eV)

So × 10"

152

54.0~2.2

154

64.6:56.7

155

107.4±2.6

7 ~+,;6 -'v-2.2

1g 7 +1.9

. . . . --2.0

+3.0 4.0 --1.2

a)

4- ' ~1 +2.5 --1.2

lq 1 +2.8 .... -2.3

2.1 +1.3 --0.6

a)

0.78 +°"12

I q9+0.17

-5 17+0.30

--0.12

.... --0.15

$1 × 104

"~'--0.46

156

80.4±5.4

157

I02.6:~2.0

R a+2.8

~'V--l.8

2 ~+o.72 . . . . -0.52

b)

.... - 0 . 1 4

2 ~+0.56

Comments

3 7 +1"¢

e)

" --1.2

49.9 +3.4

1.6 +0.6

a)

5.6 +0.8

2.3 +0.9 --0.5 2.3 +0.8 -0.8

b)

--5.3

-0.7

--0.4

~ 1+1.4 -'~-1.2

c)

158

89.4~4.7

15.5 +5.7

84.4 +5.6

--7.4.

1.8 +0.8 -0.5

a)

160

97.72-4,1

47

159

+25 -20

2.9 +I,7

a)

--

--3.7

+25 -13

-O.S

Colnmelqts: a) Values for even isotopes derived f r o m analysis o f K a r s c h a v i n a e t a L b) Values derived f r o m resolved p a r a m e t e r s listed in tables 3 a n d 4. o) Values derived f r o m average capture cross sections.

s).

348

$. J. FRIESENHAHN e t I00

-

aL

-

15

Z

I0

I

1

--

,

r

0.01

i

OA

r

i

.

r

|

IL_

1,0

Fig. 4. Comparison of integral tevel spacing distributions for lSSGd and xSTGd with the Wigner distribution calculated for two independent spin populations.

?C ~

i ~ I1"

,

I J

50

. 30 "

o<

/I/I/ 1/

~o

~o

~o

1"57Gd / ~

~o

,oo

,~o , ; o

~'d:;;~ /

,;o

,;o ~oo

NEUTRON ENERGY (eV)

Fig. 5. Level-density plots for XSSGd and lsTl~d, The dashed lines indicate 68 % confidence limits on the observed level density.

lss, lsvOd

349

The integral level-spacing distributions are illustrated in fig. 4. The calculated curves are for two independent spin populations obeying the Wigner distribution assuming that zhe mean spacing is inversely proportional to 2 J + 1. The 1s SGd + n data exhibit a slight excess of small spacings whereas the 157Gd+n data do not. This tends to confirm the existence of a population whose spacings are not correlated with the swave spacing. Level-density plots for 155Gd+n and 157Gd+n are shown in fig. 5. The 68 confidence limits on the average level spacing for the entire sample are indicated in the figure. We surmise from the plots that the number of missed levels is small in the energy interval studied. The average observed level spacings are also listed in table 5. 3.3. STP,.ENGTI-I F U N C T I O N S

Since two spin states are populated in s-wave absorption by the odd isotopes of gadolinium, the biases in the strength functions due to the finite number of resonances in the samples cannot easily be calculated analytically. Hence, the level-spacing probability distributions were obtained by generating spacings from two superimposed Wigner distributions corresponding to the two possible total angular-momentum values. The level-spacing probability distributions thus obtained, when combined with analytical calculations of the neutron-width probability distributions, yield the strength-function bias factors. Due to the large number of levels examined in this work, however, the bias factors for both isotopes differed little from unity. The values of the bias-corrected strength functions for both isotopes are listed in table 5 along with the average reduced widths and level spacings. Also listed are values of the average parameters for the even isotopes calculated from the resonance parameters given by Karschavina et ~1. s). Since there is only one spin population for the even isotopes, the analytic formalism of Slavinskas and Kennett 9) could be used to calculate their bias factors. The average parameters found in the present work are in good agreement, within quoted uncertainties, with those reported by Mughabghab and Chrien 7). Optical-model calculations have tended to underestimate the s-wave strength function near mass-160, where a minimum in So occurs from a splitting of the 4s size resonance by collective effects lo). Previous analyses have shown that this discrepancy is not easily removed by variations in the surface absorption potential 11) nor by assuming a spin-spin term in the effective interaction 7). We have examined two other possible sources of error: (i) the use of "real coupling" for the nonspherical potential and (ii) the assumption of a simple 0+-2 + coupling scheme to describe the general features of the collective effect. It has been well established that the proper collective-model generalisation of the spherical optical-model potential is that obtained with "complex coupling", for which both real and imaginary parts of the potential are deformed 12). Coupled-channel calculations 13) of neutron transmission coefficients were carried out for rnass-158 assuming a 0 + ground state and 2 + rotational state. A typical quadrupole deformation

$. 3", FRIESENHAHN et al.

350

parameter (~2 = 0.35) and excitation energy of the 2 + level (80 keY) were obtained from ref. 1,). The transmission coefficients were calculated with the optical parameters of ref. ~o) at an incident energy of 40 keV, and strength functions at 1 eV were then deduced by the methods of ref. 15). The s-wave strength function obtained with complex coupling, 1.5 x 10 -4, was found to be somewhat lower than the value of 1.9 x 10-* obtained with real coupling. Measured 7) values of So for the even isotopes lS6Gd and lSSGd are about 30 lower than those for the odd isotopes of G d (although the quoted uncertainties overlap), and one might suspect that a more realistic coupling scheme is required for the odd nuclei. Coupled-claannel calculations were made for 155Gd including three members of the ground state rotational band: 3 - ground state, 5 - state at 60 keV and ~-- state at 146 keV; other parameters in the calculation were equivalent to those used in the 0 +-2 + calculation. (The additional states of ~5 SGd below 0.26 MeV have positive parity, and the coupling between the ground state and these levels should be much weaker than that for members of the ground state band.) However, this more realistic coupling scheme produced a lower value still for the s-wave strength function (about 1.4 x 10- * versus 1.5 x 10-4). Similar results were obtained for ~57Gd. It thus appears that an interesting discrepancy remains between the prediction of the generalised optical model, So ~ 1.4x10 -~, and the experimental values for 155Gd and 157Gd of So ~ 2.3 x 10 . 4 (from both the present work and ref. 7)). Our measurement (subsect. 3.5) of the p-wave strength function for ~5 SGd is also somewhat higher than that calculated in the manner indicated above. The calculated value is $I ~ 2 x 10 -4, and S 1 varies only 10 ~o w i t h j = l+½ for a spin-orbit well depth of 6 MeV. However, the present uncertainty in the experimental p-wave value precludes a definitive comparison. 3.4. RADIATION

WIDTHS

FOR

RESOLVED

RESONANCES

The least-squares values of the average total radiation widths are listed in table 5 along with the standard deviation of the mean calculated from the variance of the sample. Due to the very large number of levels presumably available for radiative decay of medium and heavy weight nuclei, it is usually assumed that the total radiation widths for different resonances of a given isotope and spin state have a )~2 distribution with a very large number of degrees of freedom. Thus for all practical purposes the total radiation width is constant. If any meaningful test of this assumption is to be made it is essential that the probable experimental errors associated with the individual radiation-width determinations be well understood. In evaluating the probable errors we have assumed pessimistic values to ensure that the resonance-parameter uncertainties represent conservative estimates. This conservatism arises in part by adding the magnitudes of all the systematic experimental errors, and an appreciable over-estimate of the errors can be expected. If the total radiation widths were constant, one might expect their experimentally 2 -2 determined distribution to have a coefficient of variation V~2p~--- (AF~k)/F~, where

155, tSTGd

351

the AF,..~are the estimated probable errors for the total radiation widths F~k and where /~, is the average radiation width. The brackets denote an average over the ensemble of resonances. The observed coefficient of variation for the set of measured widths is V~s = ((F,,,k-F~)2)/P~. This quantity is a measure of the real variance of the radiation-width distribution broadened by the experimental errors. (Hereafter, we refer to V 2 simply as a "variance".) To increase the number of radiation widths for analysis we have normalised the widths of the two isotopes to their respective means and combined them into one set of 52 values of F~. When the variances of this set are calculated, the expected variance Ve~p is larger than the observed variance V,obs, a a result anticipated from the conservative error estimates. Since the major portion of the expected variance is due to the few resonances with large estimated uncertainties, we reduced the sample size systematically by excluding resonances with the largest values of AF~/F~k. When this is done the expected variance quickly becomes significantly smaller than that observed. When the sample has been reduced to 37 resonances by limiting AF~k/F~ to < 0.25, the observed variance is V~s = 5.5x 10 . 2 and the expected variance is Vfxp = 1.7 x 10 -2. I f one assumes that the expected and true variances pertain to normally distributed quantities, an estimate of the true variance can be obtained from 2

2

where V(~o is the actual variance of the radiation widths and C is the correlation coefficient. This leads to an estimate of Vt~u~ 2 ~ 3.5 x 10 -z, which corresponds to 57 degrees of freedom for a Z2 distribution. This estimate remains fairly constant as the sample size is reduced further, which lends credence to the hypothesis that an observable Vt~.~ 2 exists. A histogram containing the set of 37 radiation widths is shown in fig. 6 along with Zz distributions with 20, 57 and 100 degrees of freedom. Before accepting Vt~.e 2 as a measure of the variance of the total radiation widths, we calculated correlation coefficients with other experimental quantities to test for unsuspected experimental aberafions. Correlation coefficients were calculated for the full set of 52 resonances as well as all the subsets. No significant correlation was found between [F~k--/=~[ and 2gF. or the resonance energy Eo. A slight correlation (C = 0.34+0.10) was found with AF.~k, as would be expected. i f some of the radiation widths were those of p-wave resonances, any difference between p- and s-wave radiation widths might be important. However, the fact that most of the radiation widths were determined for relatively strong resonances at low energy tends to make p-wave contributions to the distribution of radiation widths unlikely, Of the resonances analysed for a radiation width, the 3.616 eV resonance in ~5 SGd is the most likely candidate for a p-wave assignment on the basis of its very small neutron width. The radiation width of this resonance happens, however, to fall near the average value. 'Unless the apparent true variance Vtr~ 2 is increased by the fact that two spin states are populated by s-wave capture in the odd isotopes, i.e. unless the total radiation

352

s. 3. FRIESENHAHN e¢

a[.

width is assumed to be spin-dependent, one must conclude that this variance is vastly greater than that expected from the statistical model. The density of states at the binding energy in the compound nuclei lS6Gd and 15SGd that are populated by E1 transitions from the capture state (J~ = 1- or 2 - ) is roughly 1.6 times the observed level density. Thus a constant-temperature density formula with a value 16) of T = 0.49 MeV yields approximately 3.9 x I0 s levels available for primary transitions in 156Gd and l a x 105 levels in ~SaGd. For a Porter-Thomas distribution of partial radiation widths that vary with p r a y energy as E~, and for the constant-temperature 3.2

I

I

I

I

2.8 N =IO 0 2.4

2.0 = 57

L,•1.2 =20

el

t, 0.4

-o

0.4

ry, 7

,.G

2.0

Fig. 6. Distribution of 37 radiation widths a b o u t the mean value for the combination of resonances for 155Gd and lSTGd. The s m o o t h curves denote Z z distributions with 20, 57 and 100 degrees o f freedom.

level density, one can then estimate ~7) the variance of the distribution of total radiation widths expected for each isotope. Using an average appropriate for the ensemble of widths from both isotopes, we obtain a variance corresponding to V 2 ~ 10 -4. This statistical-model estimate of V 2 is a factor of 350 less than the value deduced experimentally. There is relatively little experimental information on the effective number of degrees of freedom for decay of compound nuclei by radiative transitions. Recent results obtained by Glass et ~l. l a) for the distribution of radiation widths of 23SU+ n indicate

155, 157Gd

353

44 degrees of freedom, or a V z which is about 100 times larger than the statisticalmodel estimate. Large variances in the total radiation widths have been observed t 9) for capture by 59Co and were also found 2o) for 232Th. The experimental value of V 2 for 23ZTh is about 17.5 times larger than the statistical-model estimate, and for S9Co the experimental value is 10 times larger ~1). Large variations in the total radiation width have been interpreted 21) as arising from an anomalously large number of primary gamma transitions to low-lying levels. While the thermal capture 7-ray spectrum z2) for natural gadolinium shows no highenergy resolved lines of exceptional intensity, there may be some evidence that the overall spectrum is somewhat harder than that expected from the statistical model. The data given in ref. 22) were compared to a Monte Carlo calculation of the secondary y-ray spectrum obtained with a simple treatment of the transition probabilities. From a constant-temperature level density p and the Weisskopf estimate 23) of the average partial radiation width (F~i) for dipole transitions, the probability for gamma emission at each step in the cascade was taken to be P(E~) o: (F~,)Pnn,l oc E ¢ x exp ( - E y / T ). For a nuclear-temperature of T = 0.49 MeV, the calculated spectrum falls off considerably faster than the data with increasing gamma-ray energy E~ above 3 MeV. The calculated 7-ray intensity near E~ = 5 MeV is more than a factor of three smaller than the measurement. However, this indication must be strongly qualified, since we have not considered here the effects of E1 and M1 resonances 24) and~ especially, the effect of spin-parity selection rules together with the particular scheme of low-lying levels. The latter effect might change even the gross features of the y-ray spectrum for resonance capture from those of the thermal spectrum 25). 3.5. A V E R A G E C A P T U R E CROSS SECTIONS

The average isotopic capture cross sections in the low keV region can be calculated from the average resonance parameters; however a direct measurement of the cross sections provides a valuable consistency check on these average parameters and also yields information about the contributions from higher orbital angular momenta. Such information is of particular value in the calculation of infinite-dilution resonance integrals, since appreciable contributions to these quantities are made by capture above the resolved-resonance region. The natural gadolinium cross-section measurements provide a check on the isotopic measurements in the low keV region and also provide a test of the statistical-model calculation of this reaction in the upper keV region. In addition, the data at higher energies are of interest in the use of gadolinium for the control of very hard spectrum reactors. Since a moderately thick capture sample was used in order to obtain acceptable signal-to-background ratios in the data at higher energies, the self-shielding of the unresolved resonance structure must be taken into account. The resonance self-shielding and multiple-scattering effects were calculated with the code SESH 29) which uses Monte Carlo techniques to generate resonance environments from the Wigner and Porter-Thomas distributions using average resonance parameters. The calculated net

s . J . FII.IESENHAIiN et al.

354

correction for multiple scattering and self-shielding was always less than 5 % for natural gadolinium. The self-shielding effect was negligible for the isotopic samples, and hence only multiple-scattering corrections were necessary to obtain the ~55Gd and ~57Gd average capture cross sections. Small corrections ( ~ 2 % at 400 keV) were also applied to the average cross-section data at the higher neutron energies for the increase in detection efficiency with incident energy. These corrections were derived with the assumption that the pulse-height distribution is stretched linearly with the total energy available for g a m m a emission (the neutron separation energy plus the incident energy).

.O

o ~10 Z

}

o

o LO O. £3

I N E U T R O N ENERGY (keV)

Fig. 7. Average capture cross sections for ~SSGdand 157Gd from 1 to 20 keV. The solid curve was calculated using the s- and p-wave strength functions listed in table 6. The isotopic average capture cross sections are shown in fig. 7 along with calculated cross sections based on s- and p-wave strength functions obtained by fitting the measured cross section at 2 keV and 20 keY. In these fits the p-wave strength function was assumed to be spin-independent (a reasonable assumption from the discussion in subsect. 3.3), and width-fluctuation corrections appropriate for isolated levels 26) were made. The strength functions obtained from the fits are shown in table 5 along with those values obtained from the resonance parameters. As can be seen the s-wave strength functions obtained from the two techniques are in good agreement. Since the major portion of the natural gadolinium capture cross section is due to 1 s SGd and ~57Gd ' a good consistency check on all these measurements can be obtained by comparing the measurements for the natural gadolinium average capture cross section to an abundance-weighted average obtained from the measured 155Gd and 157Gd cross sections plus a calculated contribution for the even isotopes. The pwave strength functions of the even isotopes are unknown and hence were chosen to

J_55, 157Gd

355

be 2 x 10-+, in conformity with expectations from the optical model. The excellent agreement between the natural gadolinium capture cross section obtained in this way and that measured between 1 and 20 keV is illustrated in fig. 8. The apparent systematic disagreement below 3 keV can be attributed to systematic errors in tile large ambient-background correction at low energies which must be applied to the data obtained at the 230 m flight-path facility.

o•

z

o

o

_o v-

g o

o

[z

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o

o

[

o o

o

{--

o

o

0.4

t_

i

1

+

NEUTRON

[ i I0 ENERGY ( k e Y )

i.

I

o

I

100

Fig. 8. Natural gadolinium capture cross section. The values measured with the natural sample are denoted by circles, and the solid curve denotes an abundance-weighted sum of measured values for the odd isotopes and calculated values for the even isotopes.

d,•

t.

I

I

I I

I

i

I

I

I

Z

o t--

I

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--

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@

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•

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MODIFIED


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(keV)

Fig. 9. Natural gadolinium capture cross section versus statistical-model calculations described in the text.

Statistical-model calculations were made for the high-energy natural gadolinium cross section, and the results are compared to the data between 10 and 500 keV in fig. 9. These calculations were made in the manner of ref. 27) using Hauser-Feshbach

356

s.J. rRIESENHAgNet aL

theory with corrections for width-fluctuation correlations. The competition from all open inelastic-scattering channels is included explicitly, and the exit-channel penetrability (0~) = 2 n ( F ~ ) / D j is calculated with an average level spacing Dj whose energy and angular-momentum dependences are determined from the formulas of Gilbert and Cameron ~6). The energy dependence of the effective average total radiation width (Fo~) given by the Weisskopf estimate 23) is approximately nil in this energy region, which is below the region of significant effects from the (n, 7n') process. This yields a penetrability (0~) that increases exponentially with increasing neutron energy and produces the solid curve in fig. 9. The dashed curve in fig. 9 was produced by reducing the energy dependence to ( 0 r ) c c (B~+En) 6, where Bn is the neutron separation energy and En the incident energy. As has been found for other heavy nuclei 27), this reduced energy variation of (0~.) (or the radiative strength function) greatly improves the agreement with the measured average capture cross section above 100 keY. To date, however, the source of this anomaly has not been identified. 3.6. RESONANCE INTEGRALS The infinite-dilution resonance integrals were calculated from the parameters reported here and were compared with the values of Mughabghab and Chrien 7). In conformity with other workers the low-energy limit for the calculations was chosen to be 0.5 eV. The values listed in table 6 were caiculated at 0 ° K using the resonance parameters of tables 3 and 4 and strength functions listed in table 5. The upper limit of the calculations was chosen to be 100 keV. The uncertainties quoted for the integrals include the uncertainties in the neutron and radiation widths and the experimental TABLE

6

Resonance integrals of Gd isotopes ~) (from 0.5 e¥ to 100 keY) Isotopes

Resonance integral (b)

Abundance

Contribution to natural Gd (b)

152

622 +13 --11

0.002

1. . .9d+0"o3 . --0.02

154

215 +11 --11

0.0215

4.62 +0.24. -0.24-

155

1538 +1,~ -24

0.1478

227.3 +2.1 -3.5

156

q7 ~.+3,o - - " ~'-- 3 . 0

0.2059

20.0 +0.6 --0.6

157

765 +17 -24-

0.1571

120.2 +2.7 -3,8

158

63 +2.2 --2.3

0.2478

15.6 +o.6 --0.6

160

8.0 -+0.5 -0.6

0.2179

"

natural gadolinium

1.97+°'12 --0.14 390.9 +6,3 -8.9

~) Integrals for even isotopes were calculated from the parameters of Karschavina et al. s).

155,157Gd

357

covariance of these two quantities. The uncertainties in the strength functions were assumed to be systematic. For those resonances in which the radiation width was not determined, only the uncertainty in the neutron width was included. The value of the resonance integral for 157Gd(765_24 + 17 b) is in good agreement with that obtained by Mughabghab and Chrien (711 b from 0.5 to 600 eV), when the difference in the upper energy limit is taken into consideration. The value of 1720 b quoted by Mughabghab and Chrien for the 1 s SGd resonance integral from 0.5 to 240 eV has been revised 28) to 1417 b, which is slightly smaller than that obtained from our data for the same energy interval. Since uncertainty estimates were not quoted in ref. 2s), however, the significance of these comparisons is diminished. The present result for lSSGd from 0.5 eV to 100 keV is 1538_24+14 b. 4. Conclusions

The present work has improved the knowledge of gadolinium resonance parameters and their statistical behavior. When the results of this work are combined with the existing data below 3 eV, an accurate representation of the capture cross section can be obtained for most of the energy region of importance in reactor calculations. The assumption of a constant radiation width is not necessarily valid even for nuclei as heavy as gadolinium although, if the present result could be ascribed to a spindependence, the assumption might still hold true for many even nuclei. If fallacious, this assumption could lead to erroneous results in self-shielding calculations which are important in the calculation of reactor control-rod worths and Doppler coefficients. Some conventional optical- and statistical-model calculations have failed rather generally to account for the various properties determined here, and the sources of discrepancy remain largely undetermined. It is particularly evident that the mechanism of neutron radiative capture in the upper keV region deserves some new fundamental investigation. We are grateful to the crew and staff of the G G A LINAC for the exceptional performance to which we have grown accustomed, and to Mr. W. E. Gober for his assistance with the apparatus. It is a pleasure to acknowledge the continued support of our neutron cross-section work by Drs. C. A. Preskitt, J. L. Russell and V. A. or. van Lint. References 1) E. I-Iaddad, R. B. Walton, S. J. Friesenhahn and W. M. Lopez, Nucl. Instr. 31 (1964) 125 2) M. P. Fricke, W. M. Lopez, D. G. Costello, S. J. Friesenhahn and A. D. Carlson, Gulf General Atomic Report No. GA-9275 (1969) unpublished 3) S. J. Friesenhalm, D. A. Gibbs, E. I-raddad, F. I-L FrShner and W. M. Lopez, J. Nucl. Energ. 22 (1968) 191 4) F. H. Fr6hner and E. Fiaddad, General Atomic Report No. GA-5137 (1964) unpublished

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5) M. D. Goldberg, S. F. Mughabghab, N. P. Sucendra, B. A. Magurno and V. M. May, Brookhaven National Laboratory Report No. BNL 325, Supplement No. 2 Second Edition Vol. IIC 6) F. H. Frtihner, General Atomic Report No. GA-6906 (1966) unpublished 7) S. F. Mughabghab and R. E. Chrien, Phys. Rev. 180 (1969) 1131 8) E. N. Karschavina, N. N. Phong and A. B. Popov, 3oint Institute for Nuclear Research (Dubna USSR) Report No. JINR-P3-3882 (1968) 9) D. D. Slavinskas and T. J. Kelmett, Nucl. Phys. 85 (1966) 641 10) B. Buck and F. Perey, Phys. Rev. Lett. 8 (1962) 444 11) A. P. Jain, Phys. Rev. 134 (1964) B1 12) E. R. Flynn and R. H. Bassel, Phys. Rev. Lett. 15 (1965) 168 and refs. therein 13) Taro Tamura, Oak Ridge National Laboratory Report No. ORNL-4152 (1967) 14) Paul I-L Stelson and Lee Grodzins, in Nuclear Data Sheets, compiled by K. Way et al. (Academic Press, Inc., New York, 1965) Vol. 1, No. 1, p. 21 15) H. Feshbach, Ann. Rev. Nucl. Sci. 8 (1958) 49 16) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 17) C. E. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 18) N. W. Glass, A. D. Shelberg, L. D. Tatro and J. H. Warren, Proc. of Conf. on neutron cross sections and technology, Vol. 1, 573, ed. D. T. Goldman 19) M. C. Moxon, Proc. Int. Conf. on study of nuclear structure with neutrons, Antwerp (1965), contributed paper 88 20) M. Ashgar, M. C. Moxon and C. M. Chaffey, Proc. Int. Conf. on study of nuclear structure with neutrons, Antwerp (1965) contributed paper 65 21) J. E. Lynn, The theory of neutron resonance reactions (Clarendon Press, Oxford, 1968) 321 22) T. L. Harper and N. C. Rasmussen, private communication 23) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) 24) N. Starfelt, Nucl. Phys. 53 (1964) 397 25) K. J. Yost, Nucl. Sci and Eng. 32 (1968) 62 26) A. M. Lane and J. E. Lynn, Proc. Phys. Soc. 70 (1957) 557 27) M. P. Fricke and W. M. Lopez, Phys. Lett. 29B (1969) 393 28) S. F. Mughabghab, private communication 29) F. H. Fr/Shner, General Atomic Report No. GA-8380 (1968) unpublished