The characteristic level and the nuclear level density

LEA : 2 .m

Nuclear Physics 9 (1958159) 22-31 ; © North-Holland Publishing Co ., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE C A ACTERIST C LEVEL AND ME NUCLEAR LEVEL DENSITY M . EL-NADI and M . WAFIK Egyptian Atomic Energy Commission, Cairo Received 11 August 1958 Abstract : The suggestions put forward by Bethe and Hurwitz that level densities ought to be measured from a standard energy, such as is given by the semi-empirical formula, and not from nuclear ground states is considered . The semi-empirical formula is corrected, follow ing P . Fong, and this "characteristic level" is determined for nuclei from A --- 10 to A _-__ 250. _''lie experimental data of D . Hughes et al . for the fast neutron cross sections are used to determine the level densities for the corresponding nuclei . These were found to be in agreement with the Fermi-gas model .

1 . Introduction The first attempt to calculate the density of nuclear energy levels was made by Bethe 1). Subsequent studies have been made by other workers 2-5) to refine Bethe's theory. By methods of statistical mechanics Bethe explained the dependence of the energy level density on the mass niunber A and on the excitation energy . It can be generally stated that these theories show an over-all agreement with experiment 11), but are incapable of reproduccing the observed irregularities, associated with odd-even 7) and magic numbers) nucleides . Moreover, some recent measurements of nuclear level densities lead to conflicting conclusions as to the validity of the Fermi gas model on which the above theories are based. In these experiments 10) the excitation energy was measured, as usual, from the ground state. It may seem reasonable to study the possibility of attributing these deviations to fluctuations in the ground state of the nucleus due to even-odd and magic number effects . Eliminating these fluctuations in the ground state energies we may then study the applicability of the Fermi gas model to the nucleus. The introduction of a corrected ground level for nuclei, called the "characteristic level", was suggested by Hurwitz and BeLhe 11) . Fong 12) in his study of the problem of asymmetric fission applied the Hurwitz-Bethe hypothesis to nuclei in the fission product region. This technique has succeeded in showing the applicability of the Fermi gas model to nuclei in this region . We have extended the work of Fong by carrying out the calculations for nearly all mass numbers between 10 and 250, including the fission-product region. The liquid drop model mass formula is corrected 22

THE CHARACTERISTIC LEVEL

23

for each nucleide. The position of the characteristic level is then determined. Analyzing the experimental data of fast neutron capture cross sections and using the above device, the Fermi gas model formula is tested, and the values of its parameters are determined . The results and their possible implications are finally discussed. . Theoretical Considerations We assume a simple model of the nucleus, consisting of A identical noninteracting Fermi-Dirac particles with strong degeneracy. Let si be the individual energy levels and E the energy of excitation of the nucleus. We are interested in calculating the number of distinct ways in Wbich the A particles can be distributed among the allowed individual levels ej so as to give a total energy E'. We shall denote this number by p(A, E'). The exact solution of this problem is given by the well known Darwin-Fowler integra113 ) p(A , E') =

(2ni)-2

ff

exp[%A+PE'+

In (l+e -a-j6 E{)]dacd#,

( 1)

where a and ß are two constants which, ave to be determined from the total number of particles, and total energy E' of the nucleus . We can replace the last term it the ,-xnonent of eq. (1) by an integral and according to Sommera feld ï4) we get ;) 1ln(1+e --ßl La (1+e a-Pc)de i::-- -mA -PEo+(n2/aß 2 )p(Eo) (2 ) = Jo p(s) where p (EO) is the individual level density at the Fermi level so and Eo is the energy of gr( and state of the entire system. Evaluating the integral (1) by the saddle point method we get

(3) p(A , E') = es /(2nD,), where the "entropy" S is the value at the saddle point of the exponent of eq. (1), and D is the determinant formed from the second derivatives of S with respect to a and ß . A simple calculation finally gives for the level density the following expression 2E ' ~ p(A, E') = (48) 'IE'-1 exp

n

__) 36

(4)

where a-1 = p (so) is the individual-nucleon level density, which is linearly related to the mass number A 6-1 --- Const. X A . (5) Refining the above procedure for a case more directly applicable to the nucleus, we shall assume that the system consists of a mixture of protons

24

11H . EL-NABI AND Ai . WAFIX

and neutrons . This combinatorial problem can be solved by the DarwinFowler method in a similar way and the result is 3 E2)* exp . (6) p(A' E) ° s 21v°E (36 Eq. (6) may take the form

2E p(A, E) = c exp [-n ( 38 where c is a slowly varying function of E and A. To dedace 6 from experiment, we write eq. (7) as follows : 6 = âE [n/(hl p--ln c)] 2 . Eq. (6) can also be written in the familiar form p (A , E) = c exp [2 (aE)I]

where the parameters a and c are given by c --- j-[ß/(216E5)]},

3. Analysis and

a = ot2/66.

r ha

(7) (8) 8

esults

By using the continuum theory of nuclear reactions 15) it has been shown by Bethe 16) that the average radiative neutron capture cross section
THE CHARACTERISTIC LEVEL

25

level" which varies smoothly from one isotope to another. They pointed out that this characteristic level could be represented by a level calculated from the liquid drop mass formula excluding the even-odd energy term . Consequently for the purpose of calculation of the excitation energy values one should determine the position of the characteristic level, using the well known Fermi-Weimäcker semi-theoretical mass formula. This formula takes the form M(A, Z) _ MA+BA(Z-ZA)2+öA (11) where MA == 1 .01464A+0 014AI-0 .041905ZA BA - 0.041905fZA (12) ZA = AI (1 .980670+0 .014962AI)

odd odd-mass ~A =: even. Mass values obtained from this formula, compared to mass-spectrometric ones, have shown errors of the order of 10 MeV. The Fermi-Weizsäcker formula is only an approximation and it does not take into account the existence of nuclear shell structure . To account for the local deviations, the general procedure is to introduce some empirical corrections to MA and ZA (BA and 6A are assumed to be correct) . Moreover, a discontinuous term ,u analogous to the 6A term may be introduced to account for the abrupt variations at the shell edges . The corrected mass formula may be expressed as follows :

.0

0.036

1 0 -1

13 M (A, Z) = MA+dMA+BA(Z-ZA ®ZA)2+ÖA+,u . To determine A ZA we compare the ZA values given by eq. (12) with the positions of the bottoms of the isobaric curves of the experimental betadecay energies 19) . This comparison has been applied to isobaric nucleides of nearly all odd mass numbers between 11 and 243 . The deviations A ZA may be positive or negative as seen in fig. 1. The charge corrections A ZA are plotted, and a smooth curve may be drawn as shown in fig. 2. To determine AMA we compare the MA values given by eq. (12) with the mass-spectroscopically determined masses of stable isotopes. The data used are those of Wapstij 2° i . The mass corrections AMA with their estimated errors are plotted aga -- ist the mass number A, and a smooth curve may be passed through most of the points except the magic number nucleides . The AMA term shows deviations at closed nuclear shells. Thus the introduction of ,u may serve to account for the irregularities associated with magic

26

M. EL-NADI AND M. WAFIK

I

I I

I

I

I .°r,

I

~ O

I

I

,

I I I I I O N e

w Q b

A b

9 4

b b b b

1_I m

1 d

I

I ~

I

n b N

O

THE CHARACTERISTIC LEVEL

27

0

1 r

r VZ

ia

M. EL-NADI AND M. WAFIK

numbers. The values of ,u were found to be between --0.25 and -2.0 MeV. The isobaric curves involving a magic number nucleide cannot generally be represented by smooth parabolas, as can be seen in fig. 1. Better fitting may be obtained if the mass of the magic nucleide is corrected for the P term. Excluding the 6A and ,u terms eq. (13) varies smoothly with A and Z. It becomes free from odd-even effects and irregularities associated with the magic number nucleides, and so may be used as the definition of the characteristic energy level of any nucleus . The characteristic levels of all nuclei may be assumed to coincide with the ground state of odd nuclei and is thus given by MC(Al Z) = MA+®MA+BA(Z-ZA'®Z A ) 2+0.036/Al.

(14

E = 931 .162[M(A, Z)+Mn+1-Mc (A+1, Z)] MeV

(15)

The E-values may then be calculated as follows :

where the masses are expressed in amu and the energies in MeV. The individual-nucleon level spacing 6 may now be determined for the particularly selected nucleide pairs, referred to above) by substituting the values of the excitation energies into eq. (10) and using the experimental data of Hughes et al. for the fast neutron capture cross sections. The quantity a [.--_ n2/68] is then plotted for these nucleides against the mass number A, and a straight line may be drawn passing through most of the points. This straight line was found to be t a = 0.03A MeV- 1 .

(16)

To determine the actual values of a for other nucleides, we shall assume the validity of the linear relation (16) as a first approximation . Using eqs. (15), (8) and the measured values of the level densities, we can determine the values of the other parameter c. It is clear from the dependence of c and a on the mass number A, according to the Fermi gas model, that the values of the parameter c are not very sensitive to variations in A . Consequently, the calculated values of c may be smoothed out by a simple curve without producing much error in the values of a. This curve was taken as a straight line in the work of Fong 12) . In the present work better representation (both for the a and c values) as obtained when the c values are represented by a slightly curved line represented by c = 0.082 X exp [0.071A-0.00026A 2]. (17) With these values of c, which we assume to be correct, together with the predetermined level densities and eq. (8) we can calculate the values of the t This is to be compared with the value a = 0 .05A McV-1 obtained by Fong regio=n 71-18 .5 .

12)

for the mass

THE CIiARAC TERISTIC LEVEL

vIvi v

29

30

M. EL-NADI AND M. WAFIK

100

I

I

I

I

150

I

I

I

I

I

200

I

I

I

I

250

Fig. 4

parameter a for all nucleides . These are represented in fig. (4). This figure shows clearly the linear dependence of a on the mass number A which is in conformity with the predictions of the Fermi gas model. eferences 1) 2) 3) 4) 5) 6) 7) 8)

9) 10) 11) 12) 13) 14)

H. A. Bethe, Phys . Fev. 50 (1936) 332 ; Revs . Mod. Phys . 9 (1957) 69 G . Van Lier and G . E . Uhlenbeck, Physica 4 (1937) 531 J . M . B. Lang and K. J . Le Couteur, Pr(,c . Phys. Soc . A 67 (1954) 585 C. Bloch, Phys . Rev. 93 (1954) 1094 N . Rosenzweig, Phys . Rev . 105 (1957) 950 P . C. Gugelot, Phys . Rev. 81 (1951) 51 ; 93 (1954) 425 Feld, Fesbach, Goldberger, Goldstein and Weisskopf, U.S . Atomic Energy Commission, NYO-636 (unpublished) pp . 176, 185 H. W. Newson and R . H . Rohrer, Phys . Rev. 87 (1952) 177 ; Àsaro and I . Perlman, Phys . Rev . 87 (1952) 393 ; P. Staehelin and P . Preiswerk, Helv . Phys. Acta 24 (1952)623 ; P. J . Grant, Proc . Phys . Soc . A 65 (1952) 150 ; B. B. Kinsey, Report to the Brookhaven Conference on Neutron Physics, November 1950 (unpublished) G . Igo and H . E . Wegner, Phys . Rev. 100 (1955) 1364 K . G . Porges, Phys. Rev . 101 (1956) 225 ; Eisberg, Igo and Wegner, Phys . Rev. 100 (1955) 1309 ; G . Igo, Phys. Rev. 106 (1957) 256 H . Hurwitz and H . A . Bethe, Phys . Rev . 81 (1951) 898 P. Fong, Phys . Rev. 102 (1956) 434 R. H. Fouler, Statistical Mechanics (MacMillan, New York, 1936) A . Sommerfeld, Z . Physik 67 (1958) 1

IRE CHARACTERISTIC LEVEL

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15) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 1952) 16) H. A. Bethe, Phys. Rev . 57 (1940) 1125 17) Hughes, Garth and Levin, Phys. Rev. 91 (1953) 1423 15) J . Heidmann and H. A. Bethe, Phys. Rev. 84 (1951) 274 19) Way, Fano, Scott and Thew, Nuclear Data, National Bureau of Standards Circular no . 499 (U.S. Government Printing Office, Washington, D. C ., 1950) and Supplements ; Way, King, McGinnis and Van Lieshout, Nuclear Level Schemes (A = 40 - A = 92), U.S. Atomic Energy Commission 20) A. H. Wapstra, Physica 211 (1955) 367