ELSEVIER Nuclear Physics A 592 (1995) 1-8

NUCLEAR PHYSICS A

Parity dependence of the nuclear level density at high excitation

B.V. Rao 1, H.M. Agrawal Department of Physics, G.B. Pant University, Pantnagar 263 145, India

Received 10 November 1994; revised 15 March 1995

Abstract

The basic underlying assumption p( l+ 1, J)=p(1 , J) in the level density function p(U, J, rr) has been checked on the basis of high quality data available on individual resonance parameters (Eo, Fn, J'~) for s- and p-wave neutrons in contrast to the earlier analysis where information about p-wave resonance parameters was meagre. The missing level estimator based on the partial integration over a Porter-Thomas distribution of neutron reduced widths and the Dyson-Mehta /I 3 statistic for the level spacing have been used to ascertain that the s- and p-wave resonance level spacings D(0) and D(1) are not in error because of spurious and missing levels. The present work does not validate the tacit assumption p(l + 1, J ) = p(l, J) and confirms that the level density depends upon parity at high excitation. The possible implications of the parity dependence of the level density on the results of statistical model calculations of nuclear reaction cross sections as well as on pre-compound emission have been emphasized.

1. Introduction

The level density of a compound nucleus at high excitation energy has been a subject

of study for many years [1-4]. In the theoretical description of various characteristics of

excited nuclei, the level density function p(U, J, 7r) has the following form:

o(u, s, U)p(V),

t This work forms part of the Master's thesis (1993) submitted by Mr. B.V. Rao, Physics Department, G.B. Pant University, Pantnagar, India.

0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0375-9474(95)00165-4

2 B.V. Rao, H.M. Agrawal/Nuclear Physics A 592 (1995) 1-8

where the spin and parity components are given by

2 J + l ( - ( J + ½ ) z I Fj(J , U) = 2o.2(U ) exp 2o.2(U ) ] , (1.1)

F = 1 7-

The quantity o'(U) is a spin cutoff parameter. The above equation (1.1) is a consequence of the tacit assumption that level density is

independent of parity, i.e. p(l + 1, J ) - -p( l , J). In many high resolution neutron transmission and capture measurements the assumption of parity independence of the level density has been found [5-10] to be in jeopardy for several nuclei. Moreover, Combinatorial calculations [11-14] show a clear deviation from the equiprobability of positive and negative parity states even at excitation energies as high as the neutron separation energy.

The assumption of an equal parity distribution was justified by Ericson [15] who showed that the contribution to the excitation of even a relatively small number of single particle levels of opposite parity leads very quickly to an equal parity distribution of the nuclear levels. Later, Blokhin and Ignatyuk [16] predicted that in some cases at lower energies one parity may dominate even if the effect should monotonically decrease with increasing energy. The results of Ignatyuk were confirmed by Monte Carlo calculations performed by Angustymak [17]. Maino [18] studied the parity distribution of nuclear excited states within the framework of a statistical formalism and an extension of the method of Blokhin and Ignatyuk [16]. Grimes [19] indicated that the Fermi gas model with the inclusion of both pairing and shell effects fails to predict the parity ratios (F~ = p(Tr = + )/p(~r = - )) even in situations where the level density itself is correctly described. Recently, Mengoni [20] explored the effect of the parity distribution of excited levels on the neutron capture reactions on medium mass nuclei. They followed a semi-empirical approach and found the parity distribution to be well approximated by the relation

= 1 tanh(lotU) (1.2) F~(U)

where U is the excitation energy and a is a free parameter. More recently, Cerf [21] studied the parity distribution of excited nuclear levels in the framework of the equidistant-spacing model and showed sequences of positive and negative parity domi- nated regions in the level density. Parallel to these theoretical pursuits, recent accurate high resolution neutron total and capture measurements confirmed the strong parity dependence of the level density at high excitation for Cr isotopes [22], 89y [23], 67Zn [24] and 2°Tpb [25] nuclides.

Gyulassy and Perkins [26] checked the validity of commonly adopted assumptions, i.e. p(l + I, J) = p(l, J) and p(l, J) ~ 2J + I, on the basis of neutron resonance parameters compiled in a report [27]. They showed the validity of the 2J + 1 depend- ence but could not confirm the parity independence of the level density because of uncertainties in the experimental data.

B. V. Rao, H.M. Agrawal / Nuclear Physics A 592 (1995) 1-8 3

Much new improved data on the s- and p-wave level spacings D(0) and D(1) based on neutron total and capture cross sections measured with a high resolution and good accuracy have now accumulated, and it seems worthwhile to check the aspect of the parity dependence of the level density in view of its great significance in nuclear reaction theories [28-30].

2. Method of analysis

The data (D(0), D(1)) used for input were obtained primarily from the latest BNL reports [31,32]; however, some of the more recent high quality data [22-24, 33-36] which have been measured subsequent to these compilations have also been included in the analysis. A statistical procedure put forward by Gyulassy and Perkins [26] has been used. According to them, the equation which connects the level density within a specific (l) and corresponding (l, J ) sequence is given by

2 l + 1 p(l) = p(l, J ) Wt, ' (2.1)

g

where the statistical weight factor g is defined as g = (2J + 1)/2(21 + 1). There are two types of summations over J, and in the channel formalism

1+1+1/2 g

Wt" = ~ 2l + 1' 1l-1-1/21 (distinct J)

a term that takes care of the number of ways, a particular value of J can be found. Eq. (2.1) reduces to p(1) = 3Wt, 1 p(0), after incorporating the assumption of equal

probability of positive and negative parity states, i.e.

p( l + 1, J ) = p(l, J ) . (2.2)

We define a dimensionless ratio x = 3W1,1 D(1)/D(O) and check if this ratio is consistent with unity, thereby confirming the underlying assumption p(l + 1, J ) = p(l, J ) .

To compare the experimental ratios to unity, an approximate averaging technique for ratios must be defined. We expect the values of x to be asymmetrically distributed about unity. However, we may expect that the logarithms of the ratios are distributed symmetrically about zero, i.e. with the same number of ratios greater than some value R as the number of ratios less than 1/R. Therefore, after defining the new variable z as z = ln(x), an average ~ and standard deviation (s) are calculated. The average ratio will then be defined by

( x ) = exp(2) (2.3)

and the standard deviation by

A += +__ [exp(2,___ s) - ( x ) ] . (2.4)

4 B.V. Rao, H.M. Agrawal/Nuclear Physics A 592 (1995) 1-8

To the extent that the z 's are normally distributed, { (x ) - A-, ( x ) + A +} represents the 68% confidence interval for the ratio x.

In order to check the validity of the commonly employed assumption of parity independence of the level density, the uncertainties in the values of D(0) and D(1) should be minimal. The usual method of staircase plots to determine D(0) and D(1) can not detect any neutron energy independent missing level fraction. Moreover, for the very narrow resonances, there could be little difference in the shapes of s- and p-wave resonances, and parity assignments would be difficult. This missing fraction and misassignment can cause an error in the value of D(0) and D(1). We have used the missing level estimator (M3) and the Dyson-Mehta A 3 statistic (as discussed in the following sections) to ascertain that D(0) and D(1) are not in error because of spurious and missing levels.

2.1. Missing level estimator (M3)

An estimate for the correct number of resonances in a given level sequence can be made by the missing level estimator (M3) [37] which uses some results of the partial integration over a Porter-Thomas distribution of neutron reduced widths. The method

consists of calculating the quantity N E g F n l / ( Y ' . g ~ I )2, starting with the largest value of g in t in the interval and adding one level at a time from larger to smaller until this quantity reaches 1.206. Then the total number of levels in the studied interval is N/0.617. This method is sensitive to the magnitude and number of large neutron widths, which, of course, can be determined with much greater accuracy than the smaller widths.

2.2. A 3 statistic

An additional test for missing resonances and/or resonances for which incorrect (l, J ) assignments have been made is provided by the A 3 test of Dyson and Mehta [38]. The A 3 statistic of Dyson and Mehta is quite sensitive to the strategic location and lack of levels in a pure sequence of levels. This technique was used extensively by the Columbian [39] and ORELA group [22,24,34,40,41] as a sensitive test for detecting the presence of spurious levels or the absence of weak resonances.

Columns 8 and 9 of Table 1 indicate the modified values of D(0) and D(1) following the above techniques.

3. Results and discussion

Table 1 represents the input data (D(0), D(1)), the energy range, modified D(0), D(1) and the ratio x. Modified values of D(0) and D(1) are given only in the cases where missing/spurious levels have been found. The experimental values of x selected by the stated procedure in Section 2 are shown in Fig. 1 as a function of atomic weight.

B. V. Rao, H.M. Agrawal / Nuclear Physics A 592 (1995) 1-8 5

Table 1 s- and p- wave level spacings D(0) and D(1), energy range, corrected values of D(0) and D(1) after taking into account the missing level fraction, and ratio x = 3WI, 1 D(1)/D(O).

S. No Nuclei I ~ D(0) Range D(I) Range Modified x Ref. (keV) (keV) (keV) (keV) O(0) O(1)

(keV) (keV)

1 3°si 0 + 1000 1400 100 1400 0.3 [33] 2 4°Ar 0 + 49 1520 20.5 1520 1.26 [34] 3 49Ti 7 5 240 1,5 240 0.6 [31] 4 5°Ti 0 ÷ 125 590 10 590 0.24 [31] 5 5°Cr 0 + 15 590 4.1 590 11.3 2.98 0.79 [31] 6 52Cr 0 + 45 600 8.5 600 0.57 [22] 7 54Cr 0 ÷ 60 600 9.2 600 0.46 [22] 8 55Mn 5 2.7 210 1.3 210 0,96 [31] 9 54Fe 0 4 13 650 4.4 650 1 [31]

10 56Fe 0 ÷ 17 900 4 900 2.68 0.47 [31] 11 57Fe 1 6 200 2.2 200 5.06 0.98 [31] 12 58Fe 0 4 35 350 8.3 350 0.71 [31] 13 58Ni 0 ÷ 13.7 650 4.1 650 10.39 3.68 1.06 [31] 14 6°Ni 04 16 600 4.3 600 13.5 3.29 0.76 [31] 15 63Cu 3 - _ 1.04 50 0.495 50 0.95 [6] 16 6SCu 3 1.47 51 0.685 51 0.93 [6] 17 64Zn 04 3.44 370 0.68 370 2.87 0.7 [31] 18 66Zn 0 ÷ 4.7 350 0.84 460 0.54 [31] 19 67Zn 5 0.413 24 0.137 24 0.67 [24] 20 68Zn 0 4 5.56 380 1.63 380 0.88 [43] 21 7°Zn 04 6.82 360 2.42 360 1.06 [31] 22 78Se 0 + 1.39 41 0.515 41 1.17 1.32 [31] 23 85Rb 5 7 0.2 18 0.1 1 [31] 24 86Sr 04 1.9 38 0.63 38 1 [36] 25 S7Sr 9 ÷ 0.37 15 0.18 15 0.97 [31] 26 8SSr 0 ÷ 25 850 8.7 850 7.46 0.89 [31] 27 89y ~_ 4.6 26 1.35 26 0.66 [23] 2

28 91Zr 5-+ 0.57 25 0.34 25 0.313 1.1 [31] 2 29 96Zr 04 13 100 4.5 100 11.56 1.04 [31] 30 ll3Cd 1 + 0.021 2.3 0.014 0.5 1.5 [35] 31 ll8Sn 0 4 0.87 5 0.32 5 1.1 [31] 32 12°Sn 0 4 1.4 100 0.55 100 1.18 [31] 33 2°6pb 0 + 37.1 900 5 900 0.4 [32] 34 207 Pb l _ 35.7 860 9.3 860 0.58 [32] 35 209 Bi 9 4.5 265 1.1 265 0.48 [32] 36 238U 0 + 0.021 6 0.0072 6 1.03 [32]

A c lu s t e r o f p o i n t s a r o u n d A = 5 0 - 7 0 l ie b e l o w t h e l ine c o r r e s p o n d i n g to t he ra t io

un i ty . T h e p o p u l a t i o n m e a n fo r a s a m p l e s i ze o f 36 i s o t o p e s ( x ) = v . . . a ,7,7+o.26_o.2o is n o t

u n i t y a n d t h u s c o n f i r m s tha t t h e u n d e r l y i n g a s s u m p t i o n p ( l + 1, J ) = p(l , J ) is n o t

co r r ec t .

6 B.V. Rao, H.M. Agrawal/Nuclear Physics A 592 (1995) 1-8

2-01

1.0

0.5

0-0

• E V E N - E V E N 0 ODD

0

°0 0 O.

~ ~ I I 100 150 200 250

A

Fig. 1. Level spacing ratios obtained from the experimental data vs. atomic weight.

Although it cannot be claimed that no resonances in s- and p-wave level sequences have been missed and correct J~ assignments have been made to all the resonances for all nuclei, we believe that the use of the Dyson-Mehta A 3 statistic and the use of M3 (missing level estimator) as a further step to modify D(0) and D(1) (as described in Sections 2.1 and 2.2) provide a sufficiently reliable set of D(0) and D(1) to conclude that there is a dependence of the level density on parity.

The possible implication of the parity dependence of the level density in the calculations of neutron induced charged particle spectra of 58'6°Ni following the com-

pound and pre-equilibrium reaction mechanism has been pointed out by Graham et al. [42].

The variation of this ratio from unity will have a limited effect on the decay of the first stage compound nucleus for which many decay channels are available; but it could affect the decay of the second stage substantially. Because of the strong parity dependence of the sub-Coulomb barrier decay widths for protons and the widths for gamma rays as well, the cross sections for reactions of the type (n, n'p) are sensitive to this ratio p(Tr = + ) / p ( z r = - ) .

Mengoni et al. [20] investigated the parity effects in the neutron-nucleus interaction at low energy. Reffo [4] also mentioned that the parity dependence may in principle have a strong impact on the results of statistical model calculations as well as on precom- pound emission. In most of the codes (ALICE [28], EXIFON [29] and GNASH [30]) which are used to calculate compound and pre-compound contributions to cross sections for various nuclear reactions, the parity component to the level density function F= is

1 taken equal to : . The question for theorists is how to put this aspect of parity dependence of level density in various codes.

B. V. Rao, H.M. Agrawal /Nuclear Physics A 592 (1995) 1-8 7

4. Conclusions

The present work conf i rms the aspect of parity dependence of level densi ty based on

good quali ty data on s- and p-wave resonance parameters from the latest B N L reports

and recent publicat ions.

It is also pointed out that the inc lus ion of parity dependence in the level density

funct ion p(U, J , 7r) is very important f rom the point of v iew of calculat ing nuclear

cross sections based on var ious statistical and p re -compound models, and it remains a

chal lenging task for theorists.

Acknowledgements

This work has been part ial ly supported by DES, G.B. Pant Univers i ty vide project no.

BS-376.

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