Correlation effects on the level density of a hot nucleus

Volume 179, number 4

PHYSICS LETTERS B

30 October 1986

CORRELATION EFFECTS ON THE LEVEL DENSITY OF A HOT NUCLEUS R a m e r W. HASSE ~ GSI Datmstadt, P 0 Bo ~"110541, D-61 O0 Darm ~tadt. Fed Rep German v

and P SCHUCK Instttut des Sctences ~ucl('atre~, F-38026 Grenoble Cedex, France

Received 3 July 1986

The previously introduced model for correlation effects on the level dens~ty parameter, a, is generahzed to fimte temperatures It ISfound that the increase of a due to the correlations at zero temperature vamshes w~thm a certain range around 7"~ 4 MeV, m agreement w~th recent experimental studies

In a recent letter, Nebbla et al [ 1 ] reported on the me a su rem en t o f the nuclear level density p a r a m e t e r a for A ~ 160 systems at excitation energies o f 100 to 400 MeV. Together with the old low excltataon energy data [2] which are consistent with a / A - ~ 1/8 MeV [3], they found a surprising drop o f the level density parameter at high temperature, T ~ 6 MeV, to a/A ~ 1/ 13 MeV. The transition occurs at about T = 4 MeV The same effect has also been found in the copper region [4]. Excltatton energy as converted to temperature with the help o f the Lang and LeCouteur formula [ 5 ], E*=aT 2- T.

(1)

This effect cannot be explained with a pure mean field model. Suraud et al. [6] calculated the level density of a hot nucleus plus confining gas system and subtracted the level denstty o f the gas. They found that the p aram et er a / A stays constant up to about T ~ 5 MeV, then increases by about 6% for T ~ 8 MeV. A prehmanary account o f the t e m p e r a tu r e dependence o f the correlations was also gaven in ref. [6] wtth the result that they stay rather constant up to temperatures of T ~ 4 MeV. The f o r m e r result was On leave from Kernforschungszentrum Karlsruhe, D-7500 Karlsruhe, Fed Rep Germany 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

consistent with conclusions from H a r t r e e - F o c k [7] and Extended T h o m a s - F e r m i [ 8 ] calculations on hot systems that the mean field changes little an this range of temperatures Dean and Mosel [9], on the other hand, calculated the entropy S o f a nuclear system at excttatlon energy E* thereby taking the contanuum mto account via the Levxnson theorem. By relating the level density parameter to the entropy, S 2 = 4 a E * , they found large devtatlons from the Lang and LeCouteur formula (1). Th ei r results:t gave evidence to the opposite, h o w ev er small, trend, namely that a / A remains constant = 1/13.5 MeV up to T = 5 MeV and then shghtly decreases by 8% at T = 10 MeV. The strong decrease o f the level density at hagh temperatures, hence, must have a different physical origin which hes beyond the m e a n - f i e l d ' a p p r o x l m a tlon. In a recent paper [lO], we observed that the level density increases drastically, at zero temperature, if ground-state correlation and polarlzatton effects are taken properly into account. The energy dependence of the correlation energy I?(R,P, E)

:~ Note that the numbers given m the text ofref [9] are different from those derived from the figures, i e for E*/A = (2, 3, 5) MeV the text gives A/a = ( 13, 15, 17) MeV whereas from the figures one derlves (131 139, 14 9)MeV, respectlvely 313

Volume 179, number 4

PHYSICS LETTERS B

roughly cancels the m o m e n t u m dependence of the H a r t r e e - F o c k potential in the vicinity of the Fermi energy, ev, thereby gavmg rise to a big b u m p an the effective mass m* Wtth 17bemg the real part o f the correlation and polarlzatton graphs of the mass operator, m semlclasslcal a p p r o x t m a t t o n the slngle-parttcle energy becomes

E=_P:/2m+ VHv(R, P) + ff'(R, P, E) ,

(3)

and the slngle-parttcle level density at the Ferret energy of an N = Z = A/2 system becomes [ 10 ]

6a/rc: = go -

h3

d R R : m * ( R , PF)Po,(R,p~),

(4)

where Pv is the local Fermi m o m e n t u m . At zero temperature the polarlzatton graph restricts 2 p - l h excttattons to energies above ev and the correlation graph restricts 1p - 2 h excitations to energies below ev which gives rise to a strong imaginary part of the mass operator which ts quadratic around ev and to a strong linear dependence of the real part wtth a negative slope [ 10]. At higher temperatures, however, the two phase spaces overlap and large cancellattons occur in both graphs which results in a finite imaginary part at ev. Thts induces, for instance, the decrease of the nuclear mean free path [ 1 1 ] and a weakening o f the slope of the correlatton energy which, by virtue of eq (3) d l m m l s h e s the effecttve mass and, by eq (4), also the level denstty This effect is not restricted to the semlclasslcal a p p r o x i m a t i o n but the decrease of the effecttve mass wtth temperature has also been found recently m a Brueckn e r - H a n r e e - F o c k calculation for hot nuclear m a t t e r [12]. In the actual calculattons, we e m p l o y e d our methods of refs. [ 10,11,13 ] and first calculated the tmagmary part of the mass operator W(R, P, E) at finite temperature with a finite-range two-body interaction in the local m o m e n t u m a p p r o x i m a t i o n , 1.e putting the p - h pair at the same position as the extra particle 314

09

i

i

i

t

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i

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i

T= 0 HeY

L~

2-'O7 E

106

(2)

which has the so called on-shell solution Eo~ = E(R, P) or, inverted, Po,=P(R, E) to give the total singleparttcle potenttal V,p( R, P) = VHv( R, P) + iV(R, P, Eo~). The effective mass ts then defined as

m*(R, P ) / m = [ 1 + (rn/P)dV~r,/dP ] l ,

30 October 1986

05L I I I -7S 6 0 - l , 0 - 2 0

I 8

I I J 10 30 SO

I 80

I 100 120

E [MeV] F~g 1 The effecnve mass at the center of a nucleus as funcuon of energy for various temperatures using the Perey-Buek potentml

or hole. Then the real part was obtained by numerical princtpal-value lntegratton. With an underlying Perey-Buck slngle-parttcle potential, cf. ref. [ 10], we obtain the energy dependence of the effective mass for vartous temperatures, see fig. 1. In comparing our results wtth those o f r e f [ 12 ] we note that due to the proper lncluston o f both graphs, here m* is centered about ev = - 8 MeV, whereas in the Brueckner calculation tt is centered at eF+ 16 MeV because it lacks the correlatton graph. The temperature dependence, however, ts very similar Since the prlnctpal-value calculation is very computer-time consuming we extracted a d a m p i n g factor with temperature of the slope of the correlation energy at eF for vartous radlt (or nuclear densities), see fig 2. The influence o f the temperature becomes nonneghgible for T > 2 MeV 10

I

r

I

I

I

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,

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E

]

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I

09 08 07 06

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2

4

6

8

10

12

14

T, Temperature

(MeV)

Fig 2 The correlation damping factor at full and at one-fourth nuclear density as funct]on of temperature

Volume 179, number 4

PHYSICS LETTERS B

30 October 1986

160

,

j

j

,

j

,

j

,

,

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T=0

E

1211

MeV

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06

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--

Gogny-HF Woods-Saxon

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40

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00

i

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A=160

20

A=160

02

00

exp 1969 exp 1986

J 7

J

Radius (fm)

Fig 3 Temperature dependence of the effective mass using the semlclasslcalHartree-Fock potential with the Gogn} force when the particle and hole Fermi functions overlap substantially. Furthermore, due to the peaking of the correlations in the nuclear surface, the effect there is more pronounced than m the bulk. In the next step we employed the semlclassical expression of the more sophisticated nonlocal Hartree-Fock potential based on the Gogny force ++2with the Fermi energy ev = - 7 . 8 MeV and the nuclear radius parameter ro= 1.1 fm adjusted self-consistently to accomodate A = 160 particles. For comparison, also a local Woods-Saxon potential has been taken with the parameters V~- 54.4 MeV, ro= 1.0 fro, ao=0.75 fm, ev = - 10.8 MeV, which were fitted in order to reproduce a/A ~-1/7 MeV at zero temperature and also to contain 160 particles. Then the correlation energy was calculated anew but without the above mentioned local m o m e n t u m approximation and at T = 0. The latter is multiplied by the d a m p i n g factor of fig. 2, added to the H a r t r e e - F o c k potenUal, and m*(R, Pv), Pos(R, PF) and go are obtained from eq. (4) as functions of temperature In fig. 3 we show the final effective mass calculated for different temperatures w~th the above prescription according to eq. (3) using the Gogny force for the nonlocal Hartree-Fock field and the method described in ref. [ 10] to account for correlaUon and In table 1of ref [ 10], it should read B = 32 495 MeV fm4 This now g~vesgood agreement in fig 7 of this reference with the experimental single-particlepotentml depths

i

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i

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1

2

3

~

~

s

7

8

9

10

~1

12

T, Temperature (MeV)

Fig 4 The inverse nuclear level density parameter -lla versus temperature Experimentaldata (1986) and ( 1969) are taken from refs [1,2] and the full and dashed hnes are calculated with underlying Gogny-Hartree-Fock potential and a Woods-Saxon potential, respectively polarization effects Note the somewhat different behavlour of m*/m at T = 0 from the result in ref. [ 10] which is due to use of the G o g n y - H a r t r e e - F o c k potential (instead of the Percy-Buck potenUal) on one hand and the different form of the effective mass, eq. (3) (instead of the simpler prescription in ref. [10] ), on the other hand. These details, however, should not influence the global conclusions drawn in this letter. The results of fig. 4 show that for temperatures from zero to 12 MeV, the inverse level density parameter A/a calculated from eq (4) using the results of fig. 3 rises from 9.2 to 14 1 MeV or from 6.8 to 12 8 MeV for the nonlocal and the local potential, respectively. Experimentally, the rise IS confined to the narrower range 2.5 MeV < T < 4 5 MeV, whereas theoretically it shows up in the wider range 2 5 MeV < T < 8 MeV. We therefore obtain quahtaUve agreement with experiment. The remaining differences are very likely due to our neglect of collective effects which are particularly i m p o r t a n t for energies a r o u n d the Fermi surface. Nevertheless, the decrease of the e n h a n c e m e n t of the effective mass at ~v and the decrease of the level density parameter a/A with temperature with a transition point at T ~ 4 MeV should basically remain unaltered m a more realistic approach As a final remark we note that one should not be tempted to try to reproduce the experimental 315

Volume 179. number 4

PHYSICS LETTERS B

level d e n s i t y p a r a m e t e r o f 1/8 M e V in a m e a n - f i e l d c a l c u l a t i o n at z e r o t e m p e r a t u r e w i t h o u t c o r r e l a t i o n effects b e c a u s e temperatures.

the

latter

vamsh

only

at

high

We are grateful to Dr. J B N a t o w l t z for b r i n g i n g his w o r k to o u r a t t e n t i o n

References [1] G Nebbta, K Hagel, D Fibres, Z Malka, J B Natowltz. R P Schmltt B Sterhng, G Mouchat~, G Berkowltz, K Strozewskl, G Vlesll, P L Gonthler, B Wllkms, M N Namboor&rl and H Ho, Phys Lett B 176 (1986) 20 [2] J M Alexander and J B Natowltz, Phys Rev 188 (1969) 1842 [3] G Rohr, Z Phys A318 (1984)299, N Vmh Mau and D Vauthenn, Nucl Phys A 445 (1985) 245

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30 October 1986

[4] M Klcmska-Hablor, prl'~ate commumcatlon (1986) [5] J M B Langand J K LeCouteur, Proc Phys Soc London A 67 (1954) 586. J K LeCouteur and D W Ling, Nucl Phys 13 (1959) 32 [6] E Suraud P Schuck and R W Hasse, Phys Left B 164 (1985) 212 [7] P Bonche S Levlt and D Vaulherm, Nucl Phys A 427 (1984) 278 [8] J Bartel, M Brack and M Durand, Nucl Pbys A 445 (1985) 263 [9] D R Dean andU Mosel, Z Plays A322 (1985) 647 [10] R W Hasse and P Schuck, Nucl Phys A 445 (1985) 205, R W Hasse, Proc XX1V Winter Meeting on Nuclear physics) (Bormlo, Italy), to be pubhshed [11] A H Bhn, R W Hasse, B Hdler and P Schuck, PhTys Lett B 161 (1985) 211, B 165 (1985) 454 (E) [t2] A Lejeune, P Grang6, M Martzoloffand J Cugnon, Nucl Phys A453 (1986) 1 [13] R W Hasse and P Schuck, Nucl Phys A 438 (1985) 157