Nuclear level densities with collective rotations included

Nuclear Physics A222 (1974) 493 - 511; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEAR LEVEL DENSI’I’IJZS WITH COLLECTIVE ROTATIONS INCLUDED T. D0SSING The Niels Bohr institute, Uniuersity of Copenhagen, Cope~agen, Denmark

A. S. JENSEN t Nordim, Copeahagea, Denmark Received 14 November 1973 (Revised 11 January 1974) Abstract:

The level density is calculated from the single particle energies in a Woods-Saxon potential with pairing included in the BCS approximation. The collective rotations are included by addition of a rotational band on top of each of the intrinsic levels. The nuclei investigated have mass numbers in the region 100 5 A 5 253. At the ground state deformation and at the neutron separation energy for the nucleus in question we compare calculated and observed level densities. The dependence on the parameters in the model are investigated. Considering the uncertainties in these parameters the calculated results are believed accurate to within a factor of 3. The rotations contribute typically a factor of 40. They must be included for deformed and not for spherical nuclei. We underestimate systematically the level density by a factor of 4 with fluctuations around the average value by a factor of 3. The nuclei lighter than 13*Ba are an exception. We obtain around a factor 100 too few levels in the calculation.

1. Introduction An analytic expression for the nuclear level density was first given by Bethe ‘). For many years only minor variations of this original form were used. The assumptions and approximations behind such a simple formula can be found in Bohr and Mottelson “). The development of the computer has made it possible to evaluate level densities with fewer approximations if we are willing to give up the benefit of a simple analytic formula. The mathematics framework of such a model was given a few years ago 3-5) an d recently Moretto “) extended it to take the spin dependence of the level density into account. The basic parameters are the single particle energies which must be evaluated first e.g. from a Nilsson or a Woods-Saxon model. The effects of pairing are included in the BCS approximation. For a recent review of nuclear level densities see Huizenga and Moretto ‘). The level densities obtained directly from the single particle spectrum have already been applied in many cases ‘- 14). In spite of this, until recently there existed no t Present

address:

Institute

of Physics, University 493

of Aarhus,

DK-8000 Arhus C, Denmark.

494

T. DQSSING

AND

A. S. JENSEN

systematic study of how reliable this crucial quantity is when calculated in this way. Therefore the present investigation was initiated. Since experimental values of the level density are available at the neutron separation energy for a number of nuclei it is natural to start with them. Our prime interests are the actinides and the rare earth nuclei but an extension to lighter nuclei is also of interest. We have therefore studied nuclei with mass numbers between 100 and 253. The aim of this work is to investigate the reliability of the level density calculated directly from the single particle energies. This is only done as a function of mass number for an excitation energy equal to the neutron separation energy, which varies by a few MeV through the nuclei considered. Thus the energy dependence is not studied. For such low excitation energies the shell structure is expected to show up strongly, thereby putting our single particle model to a very severe test. Most but not all of the nuclei we study are deformed in the model as well as in nature. They are in many cases known to have collective features like rotational excited states. As suggested by Ericson ‘*) these should also be counted when we want to make the mentioned comparison. The importance of such contributions was discussed recently by Bjornholm, Bohr and Mottelson 16) and as we shall see their conclusions are confirmed by our calculations. Recently also Huizenga et al. 17) calculated the level density for 16 doubly even rare earth nuclei at the neutron separation energy. Their procedure is the same as ours except for the determination of the pairing strength. They use, however, a Nilsson potential where we apply a Woods-Saxon potential. This difference may be important. As Huizenga et al. 17), we count only the rotational bands built on top of each of the intrinsic states. Thereby all kinds of vibrations are ignored. When the energy of the vibrational state is large compared to the energy of the rotational state the main contribution is in this way taken into account. For the spherical nuclei the collective states are completely neglected in this approximation. The mentioned comparison between the absolute values of the calculated and measured level densities is important. When performed for many nuclei it will give information about the underlying shell model we have used and the importance of the collective states we have ignored. It might be difficult to untangle these two things but other sources of information about the shell model are available. Thus we know alre’ady something about the allowed parameter range for the average potential. Therefore after a brief description in sect. 2 of the theory and notation used we study in sect. 3 the dependence of the calculated level density on deformation, pairing strength and the basic single particle potential parameters. This gives us the accuracy of the calculations, and consequently we can find out if effects not included in the present model are necessary to explain the experimental data. In sect. 4 we present and discuss the numerical results for a number of nuclei and finally sect. 5 gives a summary and the conclusions.

LEVEL DENSITIES

495

2. Theory 2.1. THE SINGLE PARTICLE POTENTIAL

The average single particle potential is an axial symmetric &formed Woods-Saxon potential of constant skin thickness la). Since we want to calculate the level density for different nuclei the dependence of the potential parameters on N and 2 is important. The functional form of the radius, depth and diffuseness parameters is taken from Myers I’). Th e ra d ius of the spin-orbit force ‘*) is taken equal to the radius of the density distribution given by Myers “). The strength of the spin-orbit force is then the only parameter left. It is chosen to give the best agreement between observed and calculated single particle levels around the Fermi energy for “*Pb. Such a potential has several merits. First it is bound to give average properties correctly. This is a consequence of the Thomas-Fermi calculations used to determine the potential ’ “). Secondly it has been used extensively for other purpose with considerable success 20--22). The ground state deformation can be obtained by the shell correction method ““). This was done by G&z et al. *‘) for nuclei with mass number between 140 and 208 and by Haring “‘) for nuclei above *‘*Pb. The lightest nuclei considered here 100 5 A 5 140 were all assumed spherical. 2.2. THE LEVEL DENSITY

The single particle energies si and the z-components CZjof their angular momenta determine the intrinsic level density as function of the excitation energy E and the total projection K of the angular momentum. The saddle point approximation leads as described by Moretto “) to the following equations E+E,

= ; ( 7 siC1-f;:+(si-~k)lEi]-dk2/Gk), K Nk

=

T

21Gk

0)

=CCs2ifi-, k i

(2)

(l-.h*(&i-~k)lEi),

(3)

=

I

C

fi*iEi 3

(44)

where E, is the energy for T = l/p = 0 and K = 0 and

vi+ = tanh [+~(E~+Y&)] -t-tanh [+p(Ei-yQi)J, 2fi-= tanh [-@(E;+@i)]-tanh Ej =i J(Ei-~)’

+ 4’.

[+D(Ei-yQi)J,

@a)

(w (61

The summation index i runs over the doubIy degenerate single particle energies and k over neutrons and protons. The number of nucleons of the two kinds are iVnCUtlOn

T. D0SSING

496 and Nprotons9respectively.

AND

A. S. JENSEN

The pairing constants are in the same way denoted by Gk.

They are given by l/Gk = gk In (2S/A),

(7)

2 = PI&L

(8)

+S =

451/A*,

(9)

where A is the mass number and gk is the “smooth” single particle level density at the Fermi energy obtained as described in ref. ‘“). It is different for neutrons and protons. The six equations in eqs. (l)-(4) can for given E, K, Nk and Gk be solved for /?, 1,, A,, A,, A, and y. If no finite positive solutions for A, and A, can be found one or both of eqs. (4) is omitted and the corresponding A is assumed equal to zero. The level density itself is given by the familiar expression

(10) where both S and D are given as functions of the solutions of eqs. (l)-(4), see ref. “). Because of the pairing this procedure only works for doubly even nuclei. If one or both of the nucleon numbers are odd we still solve eqs. (l)-(4) with the real odd Nk values in eq. (3). The energy argument in eq. (10) is, however, changed in the following way P,@, X) = ~oe@-A~, K) = p,,(E-A,,

K) = L(E-&-A,,

0

(11)

where A,, and A,, are the pairing gaps in the ground state, i.e. for T = 0, and K = 0. If the pairing has not disappeared at the excitation energy considered, this method has only little theoretical justification. In order to treat the odd-N case in a more consistent way one could follow the suggestions of Soloviev ‘“). At energies with no pairing this is for a uniform distribution of single particle energies equivalent to eq. (11). If all levels arise from intrinsic excitations the level density as function of energy and angular momentum I is approximately given by ‘92*“) 21+1 &&E, I) = ----$- ,QY K = I++),

(12)

where

(13) This derivation assumes that K is the projection of I on the external axis or in other words that the system is rotationally symmetric. Therefore we denoted the resulting level density in eq. (12) by psym.

LEVEL DENSITIES

497

For deformed nuclei the rotational states should also be taken into account. This is done 16) by assuming a rotational band on top of each intrinsic state of the expression in eq. (10). This leads to the level density (14) where j, is the moment of inertia around an axis perpendicular to the symmetry axis. It is a function of energy and deformation. In this case K is the projection of the angular momentum on the body fixed symmetry axis. Since this level density has contributions both from internal and collective degrees of freedom we call it Punif

*

are interested in states with a definite parity x we can obtain the corresponding level density to a very good approximation by If

we

P(K 6 n) = _)P(K 0,

(15)

where p(E, I) can be either P,,,, or pUoic. Very convincing arguments for this can be found in the review article by Ericson 24). When the rotational energy is small compared to the nuclear temperature, i.e. for small I-values, we find ’ “) from eqs. (12) and (14)

the cases considered here a2 w 40. The inclusion of rotational states in this way introduces too many degrees of freedom. For high excitation energies double counting might therefore become a problem. The transition energy where this occurs can be estimated from ref. 16). For the rare earths and actinide nuclei it turns out to be around 50 MeV of excitation. This is an order of magnitude greater than the neutron separation energy. Thus we expect the rotations should be counted fully at this energy. In

3. Model dependence of the results The approximations and assumptions behind the level density formalism in sect. 2 are of interest. For the excitation energies considered here the error introduced by the method is in the case of no pairing less than 25 % as can be verified by direct counting 25). With the inclusion of pairing these errors may increase but they are still believed to be smaller than 50 %. The level density is determined by the uniform pairing constant p in eq. (8) and the single particle energies Ei. In turn Ei depends on the deformation and the parameters of the Woods-Saxon potential. It is interesting to study the dependence of the level density on these parameters of the calculation.

l’. DOSSING

498

Woods-Saxon

50 58

62 66 68 72 74 78 82 94

120 140 150 162 168 178 184 196 208 240

49.62 49.36 49.24

48.81 48.62 48.55 48.31 48.05 47.76 47.46

AND A. S. JENSEN

TABLEt parameters Is) for different nuclei

6.10 6.43 6.59 6.76 6.85 6.98 7.06 7.21 7.36 7.73

5.57 5.90 6.05 6.24 6.33 6.46 6.54 6.70 6.85 7.22

60.38 60.65 60.76 61.19 61.38 61.45 61.63 61.95 62.24 62.54

6.15 6.48 6.64 6.82 6.90 7.04 7.12 7.28 7.43 7.79

5.34 5.66 5.81 5.97 6.05 6.18 6.26 6.40 6.54 6.90

The proton number of the nucleus is Z and the mass number A. Subscripts n and p denote neutron and proton respectively. The quantities V,, R, and I&.,. are depth (in MeV) and radius (in fm) of the central potential and the radius (in fm) of the spin-orbit potential. The strength rVo of the spin-orbit potential is 12 MeV. The diffuseness parameters are the same for all nuclei for both neutrons and protons. For the central potential we use 0.66 fm and for the spin-orbit potential 0.55 fm. The pairing constant in eq. (8) is p = 12 MeV.

In table 1 we give for different nuclei the standard parameter set. It is our starting point taken from other calculations ‘“) and we are only going to study variations around these values. In the calculations we applied the usual Ai scaling of the single particle levels ‘O). The scaling interval is chosen such that the errors introduced in this way are less than 30 %. The defo~ation dependence and the influence of pairing are illustrated in figs. la and lb. As a function of the deformation parameters c and h [ref. ““)I we show the level density in eq. (14). It is given in units of the observed level density. The shell corrections are also shown in the same figures. The calculations were performed both with and without pairing for the nucleus 242Pu. The level density without pairing varies with deformation by up to a factor of 100. There is a perfect correlation between the shell correction and the level density. They have maxima and minima at the same deformations. This is because both reflect the single particle level density at the Fermi energy. With the inclusion of pairing this correlation can be reversed. For small excitation energies the quasiparticle level density is low when the single particle level density is high and vice versa. For large excitation energies the pairing has disappeared and the single particle level density is again decisive. The transition occurs for energies between 5 and 15 MeV. At the neutron separation energy the situation is therefore complicated (see fig. 1). The competition between single particle and quasipa~icle level density smooths the total level density as pairing is included (see fig. 1). The variation with deformation is less than a factor of 10. The absolute magnitude also decreases by a factor between

LEVEL DENSITIES

499

242P”

qunlf qobs i-

r;..3--

too -

10"

h+-

+-+--.+--+

16' -

lo2 -

o--o

without

+-+

with

pairing

palring

c-o

without

+-+

with

0.0

0.1

pairing

pairing

2.5 -

SE ,” if 10 -

-2.5 51\ 0

\+/+ / 1.0

+ /

1.1

12

,\,I, 1.3

1.4

-5.0 j-

>

+ 1.5

c

-- 0.2

- 0.1

0.2

h

Fig. 1. (a) At the top is shown the calculated level density punif in units of the observed value for 24zPu at the neutron separation energy. It is shown as a function of the deformation parameter c. The spherical deformation corresponds to (c, h) = (1.0,O.O). Results both with and without the inclusion of pairing are shown. The lower part of the figure is the shell correction energy. (b) The same as a function of the deformation parameter h [see ref. ‘O)].

10 and lo4 depending on deformation. Thus both pairing and deformation must be treated carefully. The valuep = 12 in eq. (8) of the uniform pairing constant was used previously ‘“) in another connection. It leads to the neutron and proton gap given in figs. 2a and 2b with the corresponding “experimental” values derived from the odd-even mass differences 2”). The calculated values are on average slightly smaller than the experimental, but in general the agreement is good. The large deviations, which also occur, can be traced back to the derivation of the “experimental” numbers. They are extracted from the observed binding energies using a deformation independent liquid drop energy formula. This is dangerous in regions of the periodic table where the deformation changes appreciably between neighbouring nuclei. Thus one should be careful with such a comparison. It shows, however, that p = 12.0 MeV is a reasonable value which perhaps should be increased slightly.

T. DQSSING

500

AND A. S. JENSEN

An A 1.5 -

50

60

60

70

90

100

120

110

130

150

140

160

N

Fig. 2a.

I

40

I

50

WA 60

,

@a&

@a

80

90

70

I

100

*

2

Fig. 2b. Fig. 2. (a) Calculated and “experimental” neutron pairing gaps as a function of the neutron number N. The points are obtained with the uniform pairing constant p = 12.0 MeV at the ground state deformation of the nucleus considered. The oscillating solid curve is the average d, given in ref. 26) extracted from the odd-even mass differences. At the shaded regions the deformations change unusually fast. (b) The same for protons.

LEVEL DENSITIES

501

TABLE2 Level density dependence Nucleus

2izBi

(G h)

-f%=l+ PI=12

1.o, 0.0

on different parameters -Pr6=1.25

-Po=l.oo

Pr6=1.16

Pllr0.66

0.55

1.5

1.7

0.85

1.2

2;;Pb

0.59

1.6

0.98

0.76

1.1

2!$jPb

1.0

5.5

2.3

0.92

2.8

‘;;Pb

1.0

4.2

2.1

0.94

2.4

2;;Tl

0.99

3.0

3.1

0.93

2.0

a;?1

0.88

2.0

1.0

0.85

1.7

‘;;Hg

0.74

1.6

0.80

0.91

1.5

0.76

2.3

16

1.1

2.3

1;:W

0.84

2.1

‘f:Re

0.98

2.2

75 lssRe

1.07,0.26

5.5 22

1.0

1.7

1.4

2.4

‘;zW

0.85

1.9

6.0

1.1

1.8

184w 74

0.79

1.4

5.0

0.95

1.6

ll32w 74

0.87

1.3

4.6

0.74

1.1

l;$Ta

1.0

1.5

14

0.60

1.1

‘::Ta

1.1

1.7

18

0.64

1.2

1;;Hf

0.90

1.6

3.3

0.43

0.81

‘;;Hf

0.93

1.7

4.4

0.44

0.88

In the first two columns are given the nucleus and the deformation used [for definitions see ref. 20)]. In the following columns are given level densities from eq. (14) obtained by varying one parameter at a time. They are all given in units of the level density with the standard parameter set from table 1. The radius parameter r. is deiined in ref. lg). Both diffuseness parameters for central and spin-orbit potential were increased by a factor 1.5 in column 5. The results by varying spin-orbit strength K and uniform pairing gap parameterp are shown in columns 3 and 6. The last column show results for a 15% increase of the effective mass. It is usually equal to the nucleon mass.

The effect on the level density from an increase of p to 14.0 MeV can be seen in table 2. It decreases by a factor 0.5-0.9 corresponding to a decrease of the ground state energy of about 0.3 MeV. This gives an idea of the uncertainty due to the uncertainty in the pairing strength. On top of this there is a discontinuity in the level density at the critical energy where the pairing gap becomes zero. This energy is in general fairly close to the neutron separation energy and the discontinuity introduces therefore an uncertainty which can be up to a factor of 2. Another source of uncertainty is the Woods-Saxon parameters. From the Fermi gas expression of the level density one finds the following estimates

]n p(R+AR) z 2Jz P(R) ln

Phd- A%d ~ J;~ p(meff)

!A

N

18

Rw

‘2

R’

A meffN g -,Ameff -rv

meff

ln,ff

(17)

502

T. D0SSING

AND

A. S. JENSEN

where R is the radius and meffthe effective nucleon mass. For the numerical estimate in eq. (17) we used a = &A, A = 180 and E = 6.5 MeV. Eq. (17) gives an order of magnitude estimate of the change in the level density with variations of R and m,,. Other parameters are, however, also interesting although it is hard to give analytic expressions for them. In table 2 we give instead some numerical results. A change in R of around 8 % increases the level density by a factor 1.3-5.5, whereas eq. (17) yields a factor of 4. The uncertainty in R is at most 4 %. A 15 % increase of meff multiplies the level density by 0.8 to 2.8 while eq. (17) again yields a factor of 4. An increase of the strength of the spin orbit coupling by 17 %, which is around the uncertainty in rc, produces less than a 30 ‘A change of the level density. The diffuseness increase of 50 ‘A, which is around 3 times the uncertainty, changes the level density of the spherical nuclei by up to a factor of 3 and the deformed nuclei by up to a factor of 22. With our present knowledge of the Woods-Saxon parameters it is tempting to estimate from table 2 the uncertainty in the calculated level densities. At this point it is important to realize that changes in the Woods-Saxon parameters may lead to changes in the ground state deformation. For the cases considered here the nucleus will in general adjust itself to a shape which minimizes the total level density (see also fig. 1). In other words if we for a given deformation find a large increase in level density by a change of Woods-Saxon parameters, it is very likely that it no longer is the ground state. It is also important to notice that the uncertainty in the Strutinsky shell correction approach only enters in the determination of the ground state deformation. This is obvious from eqs. (l)-(4) which only contain single particle energies as parameters. The formalism described in sect. 2 is in principle only valid for doubly even nuclei. In order to test the procedure applied for odd nuclei we tried different things. We considered the odd nucleus as a quasiparticle and an even nucleus which is the one obtained by either adding or subtracting a nucleon, or an average of these two. The differences between the three possibilities are in general less than 15 %. Finally for the moment of inertia j, appearing in eq. (4) the rigid body value was used. This is not correct but because of the small spins involved here this is not important. It would at the most change the calculated results by a few per cent. In this section we have shown the effect on the level density caused by different possible variations of the basic parameters of the model. From other sources we know how large the uncertainties in these parameters are. We know therefore how large deviations from measured values we can explain by these uncertainties. If more is needed something not included in the model calculation must be responsible. In this connection it is of course interesting to compare the results obtained in the Woods-Saxon model with those from other single particle potentials. For the purpose indicated above this is however not necessary because our parameter range is supposed to give limits within which all other reasonable potentials should fall. The ordinary Nilsson model gives an overall higher level density than that of a

LEVEL DENSITIES

503

folded Yukawa potential i3). The latter potential is more similar to the Woods-Saxon potential applied here. It is therefore expected that the ordinary Nilsson model with ho: = 41/A* MeV will lead to an increase of the level density over the results obtained here I’). 4. Discussion of the results The neutron separation energies are from ref. “). The observed level densities are taken from Lynn ‘“) for all nuclei except for the actinide nuclei which are from a preliminary compilation of Bjarrnholm and Lynn 29). The ground state deformations are calculated by G&z et al. “) and by HHring “). Their single particle potential differs slightly from the one used here. In principle the ground state deformations should therefore be redetermined. This was done in some cases and essentially the results are unchanged. For convenience we used the same deformations for groups of nuclei although it should change a little from nucleus to nucleus. On average the deformations from refs. 21*22) were used. Nuclei with mass number in the range 100 5 A 5 140 were taken as spherical. The error introduced by using a deformation slightly different from that of the ground state is discussed in sect. 3. The estimated upper limit is a factor of 2. From eqs. (12) and (14) the level densities are now evaluated for many nuclei with mass number between 100 and 253. The results are given in table 3 together with deformations, neutron separation energies and angular momenta. Because of the different spins and excitation energies it is difficult to see the structure in the level densities. Following Lynn 28) we reduce therefore both calculated and observed level densities to a spin I = 0 and an excitation energy U, = 6.5 MeV by the expression

p(UN,0) = I@*)

(5)’-&

exp(d4JU,-J3},

where I is the spin of the target nucleus and p(E*) is the level density either the observed or the corresponding theoretical quantity. The excitation energy E* is related to U by doubly even nuclei 221 UC@2 for odd-A nuclei (19) doubly odd nuclei, ( 0 where d = 12/JA accounts for systematic odd-even differences ‘). For the constant a we used the Fermi gas expression a = $$A MeV- ‘. It may seem strange to reduce the calculated level density instead of evaluating it directly but we choose to do it this way because then the ratio between observed and calculated level density remains unchanged. For such a reduction to be useful it must be relatively insensitive to the actual parameter a and 2 entering in eq. (18). The maximum change in p( V, , 0) is a factor

T. DQSSING

504

AND

A. S. JENSEN

TABLE 3

Comparison Nucleus

Cc.h)

between calculated and experimental level spacings I

‘ZiRU

1.0,o.o

9.67

looTc

1.0,o.o

6.60

4, 5

lz:Mo

1.o, 0.0

5.39

b-

1ozRU

1.o, 0.0

9.22

2,3

103RU I::,

1.o, 0.0

6.23

t

1.o, 0.0

7.00

O, 1

I::,,

1.o, 0.0

7.07

$Pd

1.o, 0.0

9.56

‘i;Pd

1.0,o.o

6.54

!z

‘!$Ag

1.0,o.o

7.27

O, 1

‘$Pd

1.o, 0.0

6.15

& O, 1

43

44

D OtJS

(M?V)

W)

2, 3 24

430

18

470

4.5 x 103

8.6 x lo4

&Is *4 > 93

z% 233

15

16

> 100 19

13

&4 > 140

‘$Ag

1.o, 0.0

6.81

‘$Pd

1.0,o.o

5.76

l:;Cd

1.0,o.o

6.98

3

‘:;Cd

1.o, 0.0

9.40

O, 1

26

&40

12.8hl.3

3

38 1.4x103

1.0 x 103 3.3 x 104

36

1.1 x 103

460

1.2 x 104

26

740

980 35

13.5*1.3 57

Dvm WI

> 55 > 180 16

D “nil (ev)

1.8x103

2.4x lo4 1.1x103 4.3 x 104

70

2.1 x lo3

>6

750

2.1 x lo4

> 45

420

1.1 x lo4

53

1.5 x 10s

&5

‘:;Cd

1.o, 0.0

6.54

4

200

150

640

1.7x 104

‘$Sn

1.o, 0.0

7.74

+

108

&loo

440

1.25 x lo4

O, 1

25

85

2.5 x lo3

l:;Cd

1.o, 0.0

9.04

‘:fLn

1.o, 0.0

7.31

495

15

14

6.5f2

‘$d

1.0,o.o

6.15

B

160

&50

lt:Sn

1.o, 0.0

7.53

+

150

160

‘$jIn

1.o, 0.0

6.78

4,5

‘fj$n

1.o, 0.0

9.57

O, 1

50

lt7Sn 50 ‘$n

1.o, 0.0

6.94

+

180

‘lgSn 50 lzoSn 50 ‘$Sn

1.o, 0.0

9.33

25

1.o, 0.0

6.49

3

180

1.o, 0.0

9.10

O, 1

30

1.o, 0.0

6.18

4 2, 3

200

‘$b

1.0, 0.0

6.81

lz3Sn 50 lz3Te

1.o, 0.0

5.95

t

1.o, 0.0

6.93

t

+b

1.o, 0.0

6.47

394

28

lz4Te

1.o, 0.0

9.42

O, 1

39

1.o, 0.0

6.58

4

1.0,o.o

7.70

if

1.0,o.o

9.11

O, 1

52 ‘$Te

lz5Xe 54 lz6Te 52

470

14 400

65

2.8 x lo4 1.1 x 103

28

800

&30

90

2.2 x 103

*50

940

2.2 x 104

+

110

3.2 x lo3

1.6~10~

4.1 x 104

6.7&2

O, 1

1.04x 103

260

5

&50 f

8

*70

250

8.0 x lo3

2.4 x 10’

7.3 x IO4

33

1.2 x lo3

1200

2.8 x 103

9.8 x lo-’

> 36

140

4.6 x lo3

112

34

1.3 x lo3

514

17

700

> 21

180

7.0 x 103

> 2.5

31

1.1 x IO3

h-25

11

1.0x 103

*4

LEVEL

505

DENSITIES

TABLE 3 (continued) Nucleus

I

(c, h)

D unl1

(Mh)

(eV)

1.o,0.0

6.30

+

1.o, 0.0

6.83

233

13

f0.5

4.8

1.o, 0.0

6.46

334

18

k6

3.3

1.o, 0.0

9.26

0, 1

1.o, 0.0

5.92

4

1.o, 0.0

8.94

132

31

+16

5.1

1.o, 0.0

6.89

3,4

20

f

2.6

110

1.0,o.o

9.11

192

51

f14

6.8

260

1.o, 0.0

6.90

t

(8

f4)x103

260

> 100

82

14

+55

(3.5+0.6)x

lo3

1

1.o, 0.0

8.61

1, 2

200

+150

1.o, 0.0

4.72

t

(10

f4)

1.o, 0.0

8.78

B, J$

23

*

1.o, 0.0

5.17

3,4

73

150

2,3

51

x lo3 7

520

1.0x 104 220 150 640 1.8~10~ 230

260

7.7 x 103

79

2.1 x 103

6.9 x lo3

1.3 x lo5

8.0

260

75

1.9x103

+16

56

1.5 x 103

z 1000

1.1 x lo3

1.6~10~

72

iSO

19

33

f13

15

300

5.2f1.4

16

430

394

8.0+0.7

12

280

394

2.4f0.6

1.0,o.o

5.84

1.08, -0.08

5.18

1.08, -0.08

7.82

394

1.14, -0.15

7.57

334

1.18, -0.13

5.90

374

1.14, -0.15

8.14

1.18, -0.13

7.99

3

8.4

430

450 6.9 x lo3

1.18, -0.13

5.59

3

1.21, -0.12

8.27

374

1.3kO.4

1.21, -0.12

6.31

2, 3

0.75fO.15

1.21, -0.12

5.87

3

1.21, -0.12

6.44

2,3

1.21, -0.12

8.53

1,2

1.20, -0.09

7.93

132

1.20, -0.09

6.38

132

1.18, -0.05

8.19

2, 3

1.17,0.02

6.27

3

130

1.17,0.02

7.65

293

11

1.17,0.02

5.72

4

1.17,0.02

6.24

394

6.1f1.2

8.5

300

1.17,0.02

7.77

3.4

3

&1.5

9.5

340

1.17,0.02

6.59

0, 1

6

hl.5

22

790

1.13,0.12

6.85

4

> 15

50

2.2 x 103

1.13,0.12

6.29

334

3.7hO.7

10

440

1.10,0.20

7.07

8, %+

2.3f0.4

11

540

> 10

>4

350 6.9

220

10

330

300

7.9 x 103

1.4hO.4

8.9

280

2.1*0.2

6.8

220

12

& 6

3.950.6 2.1 hO.4 *50 *3 > 70

12 8.8 4.8 110

360 280 160 3.6 x lo3

li

400

300

9.2 x 103

T. DPISSING

506

AND

A. S. JENSEN

TABLE 3 (continued) Nucleus

Z

(c, h)

DOb¶ W9

1.10,0.20

7.63

3,4

1.10,0.20

6.10

&

1.07,0.26

7.39

495

1.07,0.26

5.69

#

> 36

1.07,0.26

7.64

“h. %

>0.2

1.07,0.26

6.06

334

1.07,0.26

6.19

t

1.07, 0.26

7.41

0, 1

1.07, 0.26

5.75

4

1.07, 0.26

6.18

2, 3

1.07,0.26

5.41

3

> 3.9 4.OkO.7

55

15 &4 > 90 3.830.8 150

4.5rt1.2

1, 2

3.1 *to.6

0.88,0.25

6.25

*

0.88,0.25

6.07

192

7.92

192

18

6.51

132

16.8&1.6

0.94,O.lO

5.57

3

0.94,O.lO

6.65

&

0.94,O.lO

8.01

O, 1

5.99

&

2.0 x lo4 860 7.6 x lo3

53

2.4 x lo3

500

2.1 x lo4

28

k4

220

180

2.1

8.251.6

0.88,0.25

0.94,O.lO

450

18

.% 20

0.88, 0.25

4

680

25

2, 3

1. 2

14

710

6.20

7.76

510 9.5 x 103

+80

5.87

6.23

11

18

kl8

Dm. W)

220

4.3

4.4f0.4

1.07,0.26

0.94,O.lO

W)

2.940.7

0.88, 0.25

0.94,O.lO

D un1r

780 3.2 x lo4 1.2x103 60 750

3.7

93

8.8

240

12

350

> 48

69

1.3 x 103

90

*30

27

720

70

128

11

(2.2f0.7)

x lo3

100 20

480

x lo3

600

1.1 x 104

100

f40

(2.4f1.3)

300 2.2 x 103

38

1.1 x 103

380

8.9 x lo3

s 5x104

2.0 x lo3

4.6 x lo4

1.o, 0.0

6.66

O, 1

(2

1.o, 0.0

6.50

O, 1

(l.OhO.3)

1.o, 0.0

6.74

3

&1)X10” x 104

1.o, 0.0

7.37

O, 1

z 8~10~

2.8 x 10’

6.0 x lo4

1.o, 0.0

3.94

*

> 3.5 x 105

2.0 x lo4

4.2 x 10”

1,o, 0.0

4.60

4, 5

230

7.4 x 103

1.20, -0.20

6.79

2, 3

0.41

1.5

39

1.20, -0.15

5.57

192

0.41

1.6

48

1.20, -0.15

4.79

+

16.7

1.20, -0.20

5.74

#z

1.20, -0.15

5.18

132

(3.5f1.2)

7.6&1.5 0.69

x lo3

115

2.8 x IO3

19

530

3.0

91

1.20, -0.15

5.31

*

13

f2

34

990

1.20, -0.15

5.13

*

17

13

39

1.3 x 103

1.20, -0.15

6.14

O, 1

2.5

10.1

360

1.20, -0.15

5.48

2.3

0.69

1.43

55

LEVEL DENSITIES

507

TABLE3 (continued) Nucleus

(c. h)

(hi&) -

I

239u 92

1.20, -0.15

4.80

3

20.8

67

2;;pu

1.20, -0.15

5.66

?z

8.0

22

2fiPu

1.20, -0.15

6.53

0, 1

2.25

a$;Pu

1.20, -0.15

5.24

B

af;PU

1.20, -0.10

6.31

2, 3

13 12 0.65

2$zAm

1.20, -0.10

5.53

2, 3

2fiPU

1.20, -0.10

5.04

t

0.77

2f$Am

1.20, -0.10

5.36

2, 3

2$$m

1.20, -0.10

5.52

t

13.7

2$Cm

1.20, -0.10

6.45

3, 4

1.5

15 42 0.68

7.4 42 3.8 1.8 80 3.3 46 3.3

2.3 x lo3 810 290 1.6x lo3 150 77 3.0 x 10s 140 1.9x 103 150

2f$m

1.20, -0.10

5.16

3

2$m

1.20, -0.10

6.21

4, 5

2$Cm

1.20, -0.10

4.71

3

40

71

3.0 x 10s

2GiCi

1.20, -0.10

4.79

fr

16

35

1.7 x 10s

40 1.6

70 2.5

2.9 x 10” 120

The nucleus and the deformation used are given in columns 1 and 2. The neutron separation energy is S,, and the spins Z given in column 4 correspond to the I = 0 neutron resonances. The observed spacings are given in column 5. They are from refs. 28*2g). In the last two columns are listed the two calculated level spacings from eqs. (12) and (14).

of two when a is changed by 30 ‘%. One must also require that the reduction does not introduce any significant systematic effects. If these conditions are fulfilled random differences in spin and excitation energy have been removed. Thus the shell structure effects in the level densities are uncovered by this procedure [see also ref. ‘“)I. The level densities reduced in this way by eq. (18) are shown in fig. 3. The observed level density has two narrow regions around mass number 138 and 208, where p varies appreciably. It has three regions 100 5 A 6 130, 150 6 A 6 193 and 230 $ A S 253 of nearly constant p. The closed shells at (N, 2) = (126,82) and N = 82 are usually held responsible for this structure. The experimental errors are for the actinides 30) typically a factor of 2 and (10-40) % for the lighter nuclei ‘s). In some cases they might be even larger as seen from table 3. The solid curves in fig. 3 represent Fermi gas expressions “) of punif for I = 0 and E = UN

PFnif=

&(2j,.b./ti2)-iU~3

exp (2Jaz)

(20)

where the rigid body value of the moment of inertia was used. Three different values of a was tried a = &A (harmonic oscillator) , &A (square well) and an intermediate value &A. It is clear from the figure that an expression like eq. (20) needs a z ,‘$A to reproduce the average behaviour of the experimental level density. The value *A

T. DQSSING

508

s, 100

I

120

I

/

140

I

I

160

AND A. S. JENSEN

I

I

180

ID,

200

L

220

I

I

II

240

Fig. 3. The reduced level density p(U,, 0) described in the text at the energy Vu = 6.5 MeV. It is given as a function of the mass number A in the three cases where p is the experimental values (closed circles) or the calculated values with (crosses) or without (open circles) rotational contributions included. The solid curves are Fermi gas expressions given in eq. (20) with a = &, &4 and AA.

often quoted in the literature “) is obtained by using the expression corresponding to eq. (20) without inclusion of the rotational contributions 2), i.e. = &(2jr.b./h2)-3U{2a+ PsFym

exp {2JaU,}.

(21)

The calculated level densities psym and punif, reduced by eq. (18), are shown in fig. 3 together with the observed values. In the deformed regions 150 5 A S 190 and 230 5 A =<253 Punir reproduces as expected the experimental numbers much better than psym. The observed dips at A = 208 and A = 138 are present in both Psymand Punif. At the bottom of the “‘Pb dip psymis clearly closest to the observed values. At the 13’Ba dip and the transition region around “‘Hg the experimental numbers fall in between the two calculated level densities. The observed shell structure for A 2 145 is approximately reproduced but only when the proper transitions between psymand punif are taken into account. For the assumed spherical nuclei 100 S A S 130 psym is typically a factor 100 too small. Unexpectedly punif is systematically much closer to the observed values. The overall agreement, irrespective of shell structure, can better be seen in fig. 4, where we show the ratio of the calculated level density to the experimental. All systematic odd-even effects are reproduced by the prescription described in sect. 2.

LEVEL DENSITIES

0.1

0.001

i

' 100

+

I 120

140

I 160

J 180

200

220

240

Fig. 4. The ratio of calculated to observed level density at the neutron separation energy as a function of A. The rotational levels are included for the closed points and not for the open points. The squares are the doubly even nuclei, the circles odd-odd nuclei and the crosses are the odd nuclei. The tilted crosses have an odd proton.

In the deformed regions Punif is on average a factor 4 too small and except for a few nuclei we find 0.2 5 P”nif/Pobo5 0.5. The Pb region is reproduced better by P symand a similar tendency is observed around ‘38Ba. The transition from spherical to deformed nuclei in the region 190 g A 5 208 is clearly seen. For the lighter nuclei 100 5 A s 130 psym underestimates the level density by a factor 100. This is surprising in view of the reasonable agreement for the other spherical nuclei around “*Pb. The different uncertainties discussed in sect. 3 give rise to “error-bars” on the calculated level density. We estimate that because of the Woods-Saxon parameters, the BCS approximation and the Strutinsky shell correction method, the calculated level density is only reliable to within a factor 3 up or down. From our present knowledge of the Woods-Saxon parameter we feel they must be wrong by unreasonably large amounts if we are to explain the observed discrepancies. However, in view of our systematic underestimate other explanations are more likely. A momentumdependent potential is perhaps necessary. Table 2 indicates that in this case an effective mass 20-30 y0 larger than the nucleon mass is needed. A more immediate and more natural possibility is that we have ignored all collective levels except the rotations. For the rare earth and actinide nuclei a factor of

510

T. D0SSING

AND

A. S. JENSEN

around 4 is needed. Estimates of vibrational contributions using the expressions in ref. 16) makes this plausible. The nuclei lighter than A = 130 are more difficult to understand. If they are spherical i.e. without rotational states, vibrations of different character should contribute around a factor 100 in order to explain the data. This is much more than we believe is reasonable to expect [c.f. ref. ‘“)I. At this point it is interesting that the magic proton number Z = 50 gives a dip in the calculated level density but the contrary in the observed (see fig. 3). All single particle models with shell closure at Z = 50 will lead to a dip. Therefore collective levels seems to be important in this region since only they can be responsible for this discrepancy. 5. Conclusion The results presented here for the level density are obtained directly from the single particle energies in a Woods-Saxon potential. No new parameters are introduced and no adjustment of old parameters has been attempted. The potential and the liquid drop model employed are taken from previous investigations of different problems. The small amount of direct experimental information available makes it very difficult to test the energy dependence of the calculated level density. Essentially only the A-dependence has therefore been investigated here. The nuclei investigated have mass numbers in the interval 100 5 A 5 253. At the ground state deformation we compare calculated and observed level densities at the neutron separation energy. Shell and pairing effects are taken into account in this approach. To a certain extent the shell structure in the level density are reproduced. Systematic odd-even effects in the present calculation are reproduced. The inclusion of collective rotational states for deformed nuclei is necessary to get reasonably close to experimental values. Otherwise we are off by a factor of around 100. For the region around “*Pb only intrinsic level densities are needed to account reasonably well for the observed values. The nuclei lighter than 13*Ba present a problem, since we obtain a factor 100 too few levels. For almost all nuclei we find level densities between 2 and 5 times too small. Thus we obtain a systematic underestimate of the level density, on average by a factor of 4. The estimated “errors” on the calculated level density is a factor of 3. Thus to explain the discrepancies we need effects which have not been included in the model. In other words intrinsically excited states and collective rotational states are not sufficient to explain the data completely. On the other hand a considerable improvement is obtained by inclusion of the rotational states. The authors wish to thank S. Bjornholm and B. Mottelson for many suggestions, discussions and for their continuous interest in this work. We are indebted to Dr.

LEVEL DENSITIES

511

E. Lynn for the permission to use his preliminary compilation of the observed level densities for the actinides. One of us (A.S.J.) would like to thank NORDITA for the award of a fellowship. References 1) H. A. Bethe, Phys. Rev. 50 (1936); Rev. Mod. Phys. 9 (1937) 69 2) A. Bohr and B. Mottelson, Nuclear structure, vol. 1 (Benjamin, New York) 3) P. Decowski, W. Grochulski, A. Marcinkowski, K. Siwek and Z. Wilhelmi, Nucl. Phys. All0 (1968) 129 4) L. G. Moretto, Nucl. Phys. Al82 (1972) 641 5) A. S. Jensen and Jens Damgaard, Nucl. Phys. A203 (1973) 378 6) L. G. Moretto, Nucl. Phys. A185 (1972) 145 7) J. R. Huizenga and L. G. Moretto, Ann. Rev. Nucl. Sci. 22 (1972) 427 8) P. Decowski, W. Grochulski and A. Marcinkowski, Nucl. Phys. Al94 (1972) 380 9) L. G. Moretto, Nucl. Phys. A180 (1972) 337; Phys. Lett. 35B (1971) 379, 38B (1972) 393, 4OB (1972) 1 10) H. C. Britt, M. Bolsterli, J. R. Nix and J. L. Norton, Phys. Rev. C7 (1973) 801 11) L. G. Moretto, S. G. Thompson, J. Routi and R. C. Gatti, Phys. Lett. 38B (1972) 471; U. Mosel and R. Vandenbosch, Phys. Rev. Lett. 28 (1972) 1726; V. Metag, S. M. Lee, E. Liukkonen, G. Sletten, S. Bjornholm and A. S. Jensen, Nucl. Phys. A213 (1973) 397 12) A. V. Ignatyuk and Y. N. Shubin, Sov. J. Nucl. Phys. 8 (1969) 660; A. V. Ignatyuk, Sov. J. Nucl. Phys. 9 (1969) 208; A. V. Ignatyuk, V. S. Stavinskii and Y. N. Shubin, Sov. J. Nucl. Phys. 11 (1970) 563; A. V. Ignatyuk, G. N. Smirenkin and A. S. Tishin, Sov. J. Nucl. Phys. 15 (1972) 622 13) F. C. Williams, G. Chart and J. R. Huizenga, Nucl. Phys. Al87 (1972) 225; H. Freisleben, H. C. Britt and J. R. Huizenga, Proc. IAEA Symp., New York, 1973 14) A. S. Jensen and Jens Damgaard, Nucl. Phys. A210 (1973) 282; A. S. Jensen and T. Dossing, Proc. IAEA Symp., New York, 1973 15) T. Ericson, Nucl. Phys. 6 (1958) 62 16) S. Bjornholm, A. Bohr and B. Mottelson, Proc. IAEA Symp., New York, 1973 17) J. R. Huizenga, A. N. Behkami, H. C. Britt and H. Freiesleben, private communication (1973) 18) J. Damgaard, H. C. Pauli, V. V. Pashkevich and V. M. Strutinsky, Nucl. Phys. Al35 (1969) 432 19) W. D. Myers, Nucl. Phys. A145 (1970) 387 20) M. Brack, Jens Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky and C. Y. Wong, Rev. Mod. Phys. 44 (1972) 320; H. C. Pauli, Physics Reports 7C (1972) 35 21) U. Gotz, H. C. Pauli, K. Junker and K. Alder, Nucl. Phys. Al92 (1972) 1 22) R. HBring, Diplomarbeit, University of Base1 (1973) 23) V. G. Soloviev, Mat. Fys. Skr. Dan. Vid. Selsk. 1, no 11 (1961) 24) T. Ericson, Adv. of Physics 9 (1960) 425 25) 3. Skak-Nielsen, private communication (1973) 26) P. E. Nemirovsky and Y. V. Adamchuk, Nucl. Phys. 39 (1962) 551 27) A. H. Wapstra and N. B. Gove, Nuclear Data Tables, A9 (1971) 267 28) J. E. Lynn, The theory of neutron resonance reactions, (Clarendon Press, Oxford, 1968) 29) S. Bjornholm and E. Lynn, Rev. Mod. Phys., to be published 30) J. E. Lynn, private communication (1973)