- Email: [email protected]
Level density enhancement in a statistical treatment of superfluid nuclei
. EIiB
2s
26 September 19%
cm
d
PHYSICS
__
ELSEVIER
LETTERS B
Physics Letters B 385 (1996) 12-16
Level density enhancement in a statistical treatment of superfluid nuclei P. Schuck a, K. Taruishi b a CNRS-IN2P3, Universite’ Joseph Fouriel: lnstitut des Sciences Nucliaires, 53 Avenue des Martyrs, F-38026 Grenoble-Cidex, France ’ Numazu College of Technology, 3600 Ooka, Numazu Shizuoka, 410 Japan
Received 2 February 1996; revised manuscript received 24 June 1996 Editor: W. Haxton
Abstract The single particle level density of superfluid nuclei is calculated in Thomas-Fermi approximation and from a Strutinsky averaged quanta1 approach. The two results, in very reasonable agreement, show a strong enhancement of the level density on both sides of the gap. The importance of this result for various aspects of the physics of the nucleus is pointed out.
A precise knowledge of nuclear level densities is extremely important for many questions in nuclear physics as well as for related atrosphysical questions and neutron stars. The subject is, however, a difficult one and much research is still devoted to this problem (see, e.g. [ 1] ) . In this work we will be concerned with the single particle level density around the Fermi energy in superfluid nuclei. This is a key quantity which for example also triggers the total nuclear density of states [ 21. In a homogeneous superconductor it is well known [ 31 that the quasiparticle level density gA (E) diverges on each side of the Fermi energy EF at a distance IAl which is the gap parameter. In the level density exists therefore a window 2(Aj wide. This is schematically shown in Fig. 1. The question we want to address here is to what extent the divergencies in gA (E) of the infinite system seen in Fig. 1 survive in a finite nucleus. To this purpose we show in Fig. 2 the quantum mechanical quasiparticle level density for “%n calculated with the Gogny force [ 41. We indeed see that with respect to the pure Hartree-Fock level density a gap in the 0370-2693/96/$12.00
spectrum roughly 211 Z 4 MeV wide opens. We also note, the levels close to &F being pushed aside, that the local level density on both sides of the gap is increased. However, due to the scarcity of the levels in finite nuclei a direct quantification of this level density increase is difficult. As usual in nuclear physics one therefore applies some averaging. Two methods are commonly in use: the Strutinsky method [ 51 and the Thomas-Fermi approximation [ 61. In non-superfluid nuclei both methods are known to give results for the level density (and other quantities) in very close agreement with one another [ 5-71. In superfluid nuclei the application of both methods is a little more delicate. The Strutinsky method, because a blind averaging of the levels in Fig. 2 over the shell distance certainly also would wash out the gap of 2 1A 1.Concerning the Thomas-Fermi method it is not clear whether such a crude approximation can accurately enough account for such subtle effects like pairing. We will, however, see that both methods are totally adequate and that in addition they both, once more, yield very similar results.
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00872-6
P. Schuck, K. Taruishi/Physics
E(R,p)
2161
g(E)
Letters B 385 (1996) 12-16
13
= &RP)
-1u2+
IAUCp)12
(3)
is the quasiparticle energy and A (R, p) is the Mgner transform of the gap obeying the gap equation A(R,p)
= -
&zQs
k) A(RYk) 2E(R k)
(4)
Here u(p) is the effective nucleon-nucleon force (Gogny Dl [ lo] ) in momentum space in the S = 0, T = 1 channel. The single particle energies / s(R,p)
EF
E-
Fig. 1. Schematic representation of quasiparticle level density in
homogeneous superconducting (superfluid) (nuclear) matter with the square root singularities on both sides of the Fermi energy.
= $m -k VHF(R,p)
(5)
should in principle also be calculated self-consistently but here for our purpose it will be sufficient to make the effective mass approximation
(6) BCS
-
I
I
-20
-10
"6SIl
I
I
2A
0
I
10
---t E we”)
Fig. 2. Quantum
mechanical quasiparticle level density for ‘16Sn in the supertluid (BCS) and the normal states (I-IF). Note the pushing aside of the levels around EF( E = 0) and the opening of the gap in the superfluid case (numerical values are from Ref.
[41).
Let us start with the Thomas-Fermi (TF) approximation. This is well documented [ 6,8,9] and therefore we can be rather short. To lowest order in fi the density matrix and the pairing tensor are given in phase space as follows:
KtR,p) = where
A(R,p) W(R, P>
(2)
where PF (R) = d2m( p - V(R) ) is the local Fermi momentum calculated from a realistic phenomenological single particle potential V(R) (we took the one from Shlomo, see Ref. [ 111) and m* is the effective mass corresponding to the Gogny force in local momentum approximation [7]. In Ref. [ 121 it was shown that the gap equation (4) also allows for a very accurate analytical approximation which we want to use here. Our model based on the Gogny force which is known to have good pairing properties is therefore entirely analytical. Before turning to the level density let us make some tests of the Thomas-Fermi approximation in order to obtain a feeling about its validity. In [ 91 it has already been shown that the correlation energy EC = Tr(AK) ‘i=* s
$$i&(R,p)r(R,~)
compares relatively well with the quanta1 results. However, an overestimation of 20 to 30% per cent over the true average can be noted. Since pairing is a surface phenomenon [9,12], it is not astonishing that this error is of the same order as, e.g., the one of the surface energy in TF approximation [ 131. The correlation energy is a global quantity and therefore eventually rather insensitive to approximations. Let us therefore perform one further test which are the
P. Schuck, K. Tan&hi/Physics
14
Letters B 385 (1996) 12-16
matrix elements of the gap itself. Semiclassically these can be obtained in the following way: Ai = Tr(Ali)(il)
= 2.1 $$$A(R,P)B(RTP) “. (7)
where Ii) are the HF states. In the TF approximation the pi ( R, R) are given by [ 141 PARP)
d3 Rd3p
J’
--) WE-~RP))/~
~ (2&)3
x&E-dR,p))
(8)
Using (8) in (7) we then obtain the semiclassical gap as a function of energy which can be compared with the quantum Ai’s. The same procedure of course applies to the pairing tensor: ~~ --+ K(E). We compare in Fig. 3 the semiclassical functions A(E) and K(E) together with the quanta1 values for the case of li6Sn [4]. For the comparison we have to remember that ’ %n is a mid-shell nucleus and therefore pairing is maximum. At both ends of the shell pairing is zero. The TF solution of the gap can at best represent an average over a whole shell which is lower (approximately by a factor of two) than the maximum value. We see from Fig. 3 that this is exactly what is happening. Globally we can be very satisfied with the TF solution and therefore one can be convinced that the TF approximation also works in the superfluid case in the same way as it does in the normal fluid case [ 6,7]. At this point we would like to mention that in this work the continuum is treated by a substraction procedure as in [ 111. Let us now come to the main objective of this work which is the level density. In the superfluid case it is given by [ 151 g*(E)
=CiS(E-
sign(E)Ei)
(9)
where Ei are the quantum mechanical quasiparticle energies and E is counted from the Fermi energy aF as origin. In TF approximation the corresponding level density is then obtained by: gy = 2
s
d3 Rd3p wS(E
- skn(E)E(R,p))
(10)
The phase space integral can be reduced to a one dimensional R-integration in the case of spherical symmetry to be considered here. In ( 10) again use is made
Fig. 3. Quanta1 and semiclassical values for the gap A and the pairing tensor K. The discrete values of KQ~ are connected by a continuous line to guide the eye. Also indicated are the HF single particle energies ayF for ‘%n [4].
of the analytical expressions for E( R, p) (Eq. (6) ) and A (R, p) [ 121. However, before discussing the results, we first outline the Strutinsky averaging procedure of the quanta1 level density, which, as was said already, has to be applied with some care. The point is that, in order not to erase effects from pairing, one only must average the underlying lzon superfluid level density. We therefore write
gA(E) = Xi
dwS(w
- &i)
./ x 6( E - sign(E)
J/W)
and replace the individual eraged values ] 5,6] : S(W -&i)
~
tf(~)
(11) peaks by the Strutinsky
av-
P. Schuck, K. Taruishi / Physics Letters B 385 (1996) 12-l 6
where f(x) is the Strutinsky smearing function and y the width parameter. At this point this smearing procedure may seem somewhat ad hoc and needs explanation. For A = 0 we get back from ( 11) the usual Strutinsky averaged non-superfluid level density gst(E) [5]. If we replace in (11) the Ai’s by an average value A., we have g~(J3
=
s
dqst
I% y ‘d_
+0(-E
x’ = (Ei +
@
strut d
(13)
Evidently (13) is also valid in the frequently employed BCS version where the gap is taken as a constant (in the “A-shell”). Expression (13) is a very natural result in the sense that it contains only the average underlying normal fluid level density. It has indeed been shown previously [ 161 that in order to obtain the average for quantities which go beyond the ordinary Hartree-Fock approach (like, e.g. the incIusion of particle-hole correlations) it is sufficient to enter the averaged single particle level density. It is in this way that ( 13) has to be understood. We certainly can take (13) as it stands for our comparison with the TF result. However, we still want to apply a slight further refinement of the averaging procedure which seems more attractive esthetically but is of no consequence for the general conclusions. To this end we keep the individual Ai’s as in ( 11) and only perform the averaging over the A’s in a subsequent step. From ( 11)) ( 12) we straightforwardly obtain =
(MeV-1)
( OJ>
x 6(E-sign(E)d(w--,LL)~+A&)
&j(E)
%E)
IE’
{B(E-
Fig. 4. Strutinsky averaged quantal quasipaaicle level density for “6Sn (broken line) together with the corresponding Thomas-Fermi approximation (full line). In the quantum calculation the continuum has not been taken into account whereas the correct treatment of the continuum [ 111 gives rise to the typical maximum seen at E 2 10 MeV in the Thomas Fermi approximation.
that the poles in (14) must be washed out into a smooth function. However, once the smoothing is achieved the parameter a must not be increased further). We therefore introduce condition
EN
F;-(E) = &
dE’B(E’I
Ai) J&
(l5)
E-a
which after integration yields an analytic expression. The final result for the average quantal level density then is
Ai>f(Xi(E))
Ai>f(x+(E))) - Af)/r
--) E &le”)
EF)
+fiWW(~'(E))}
(14)
where again E and &i are counted from &F as origin. In ( 14) we see the typical square root singularities on each side of the gap in the spectrum. In “%I we have 24 bound levels [4] and corresponding 24 Ai-values (see Fig. 3). The Ai values have a spread of g 2 MeV with a maximum difference of 0.3-0.4 MeV for adjacent states. In order to get a continuous function we therefore propose to do a further very small averaging of each pole in (14) over a width a Z 0.3 MeV (the value of a is rather uniquely determined by the
(16)
which is a continuous function of E and therefore can easily be compared with the TF result. This is done in Fig. 4. We see that even in the superfluid case TF and (average) quantal results are in quite reasonable agreement. The TF gap in the level density is somewhat wider which is to be expected from what we said before on the correlation energy. Certainly ficorrections [ 171 again would bring the semiclassical and Strutinsky values in perfect agreement. The remarkable feature seen from Fig. 4 is that the divergency of the level density in infinite matter still has a pronounced analogue in Fermi liquid drops as
16
P. Schuck, K. Taruishi/Physics
small as nuclei. Indeed the nuclear level density is enhanced by more than a factor of two on both sides of the gap. This feature likely also influences the total nuclear density of states. It may be that something of this sort is seen in the 239Pu (n, 2n) reaction analysed in Ref. [ 181. Work in this direction is in progress. A factor two increase of the level density certainly also has strong effects on many other nuclear processes such as fusion and transfer processes, etc. That pairing increases the level density is not unknown in nuclear physics. However, to the best of our knowledge, discussions of this point are very rare in the literature (the only places which we are aware of are Refs. [ 191) whereas the other mechanism of level density enhancement, namely particle-hole correlations, has received much attention in the past (see, e.g. [20] ). From our study it becomes clear that in this respect particle-particle correlations, at least in superfluid nuclei, are equally important (for particle-particle correlations in non-superfluid nuclei see Ref. [ 211) . We consider it as the merit of this work to have revived this important point and demonstrated (for the first time) that quantitative account is accessible in the ThomasFermi approximation. Of course, in Fermi liquid drops as small as nuclei where the pair coherence length is of the order or larger than the nuclear size, the BCS approach itself may be questioned and quantum fluctuations should be included. They are, however, not expected to play a dramatic role for the level density unless one is close to the non-superconducting phase transition [ 221. This may also be deduced from the fact that in a medium to heavy nucleus there are 2 3-5 levels/MeV (see Fig. 4) around&F where the gap is of the order of 1 MeV (Fig. 3). So at least one criterion for the validity of the BCS approach is met. A study of the quasiparticle level density of rotating hot nuclei (as they are produced in heavy ion reactions) with inclusion of quanta1 and thermal fluctuations along similar lines as in the present work is certainly an interesting subject for the future. Our TF approach may also present some interest in the description of small superconducting metallic clusters (for exhaustive refs. to the literature on this subject, see [ 231 and [ 241) . A much extended version of this work is in preparation. We are grateful to Prof. Soloviev for pointing out to us Ref. [ 191. Discussion with S. Goriely, M. Pearson and P. Ring are acknowledged. The quanta1 results have kindly been provided by J.F. Berger.
Letters B 385 (1996) 12-16
References S. Goriely, Universit6 Libre de Bruxelles, preprint; G. Puddu, 2. Phys. A 353 (1995) 247; N. Canosa, R. Rossignoli and P. Ring, Phys. Rev. C 50 ( 1994) 2850. r21 A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 1, Benjamin (N.Y.) 1969. [31 D.R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger, 1986. [41 J.F. Berger, IN2P3, Ecole Joliot Curie de Physique Nucltaire 1991 and private communication. A.S. Jensen, H.C. Pauli, t51 M. Brack, J. Damgaard, V.M. Strutinsky and C.Y Wong, Rev. Mod. Phys. 44 (1972) 320. [61 P. Ring and P. Schuck, The Nuclear Many Body Problem, Springer 1980. [71 P. Schuck, R.W. Hasse, J. Jaenicke, C. Grbgoire, B. Remaud, E S6bille and E. Suraud, Progr. Part. Nucl. Phys. 22 (1989) 181. [81 R. Bengtsson and P. Schuck, Phys. Lett. 89B (1980) 231; R.A. Broglia, F. De Blasio, G. Lazzari, M. Lazzari and P.M. Pizzochero, Phys. Rev. D 50 (1994) 4781. I91 H. Kucharek, P. Ring, P. Schuck, R. Bengtsson and M. Girod, Phys. L&t. B 216 (1989) 249. [lOI J. Dechargk and D. Gogny, Phys. Rev. C 21 (1980) 1568. t111 S. Shlomo, Nucl. Phys. A 539 (1992) 17. r121 H. Kucharek, I? Ring and P Schuck, 2. Physik A 334 (1989) 119. 1131 M. Durand, P. Schuck and X. Vinas, Z. Phys. A 346 (1993) 87; P. Schuck and X. Vinas, Phys. Lett. B 302 (1993) 1. Chaos in Classical and Quantum [I41 J.M.C. Gutzwiller, Mechanics, Interdisciplinary Applied Mathematics, Springer 1990. [I51 H. Stolz, Supraleitung, WTB, Akademie-Verlag, Berlin 1979. [I61 P. Schuck, R.W. Hasse, J. Jaenicke, C. Gregoire, B. Remaud, E SCbille and E. Suraud, Progr. Part. Nucl. Phys. 22 (1989) 181. [I71 K. Taruishi and P. Schuck, Z. Phys. A 342 (1992) 397. [181 V.M. Maslov, Z. Phys. A 347 (1994) 211. 670; [I91 EH. Bakke, Nucl. Phys. 9 (1958/1959) V.G. Soloviev, Figs. 4.10, 4.13 in Theory of complex nuclei, Pergamon Press, 1976; C.W. Reich and M.E. Bunker, Rev. Mod. Phys. 43 (1971) 348. [20] R.W. Hasse and P. Schuck, Phys. Lett. B 179 (1986) 313; P.F. Bortignon and C.H. Dasso, Phys. L&t. B 189 (1987) 381. [21] N. Vinh Mau and D. Vautherin, Nucl. Phys. A 445 (1985) 245. [22] J.L. Egido, P. Ring, S. Iwasaki and H.J. Mang, Phys. Lett. 154B (1985) 1. [23] J.A.A.J. Perenboom, P. Wyder and E Meier, Phys. Rep. 78 (1981) 173. [24] B. Mtihlschlegel, Physica B 203 ( 1994) 444. [I]
2s
26 September 19%
cm
d
PHYSICS
__
ELSEVIER
LETTERS B
Physics Letters B 385 (1996) 12-16
Level density enhancement in a statistical treatment of superfluid nuclei P. Schuck a, K. Taruishi b a CNRS-IN2P3, Universite’ Joseph Fouriel: lnstitut des Sciences Nucliaires, 53 Avenue des Martyrs, F-38026 Grenoble-Cidex, France ’ Numazu College of Technology, 3600 Ooka, Numazu Shizuoka, 410 Japan
Received 2 February 1996; revised manuscript received 24 June 1996 Editor: W. Haxton
Abstract The single particle level density of superfluid nuclei is calculated in Thomas-Fermi approximation and from a Strutinsky averaged quanta1 approach. The two results, in very reasonable agreement, show a strong enhancement of the level density on both sides of the gap. The importance of this result for various aspects of the physics of the nucleus is pointed out.
A precise knowledge of nuclear level densities is extremely important for many questions in nuclear physics as well as for related atrosphysical questions and neutron stars. The subject is, however, a difficult one and much research is still devoted to this problem (see, e.g. [ 1] ) . In this work we will be concerned with the single particle level density around the Fermi energy in superfluid nuclei. This is a key quantity which for example also triggers the total nuclear density of states [ 21. In a homogeneous superconductor it is well known [ 31 that the quasiparticle level density gA (E) diverges on each side of the Fermi energy EF at a distance IAl which is the gap parameter. In the level density exists therefore a window 2(Aj wide. This is schematically shown in Fig. 1. The question we want to address here is to what extent the divergencies in gA (E) of the infinite system seen in Fig. 1 survive in a finite nucleus. To this purpose we show in Fig. 2 the quantum mechanical quasiparticle level density for “%n calculated with the Gogny force [ 41. We indeed see that with respect to the pure Hartree-Fock level density a gap in the 0370-2693/96/$12.00
spectrum roughly 211 Z 4 MeV wide opens. We also note, the levels close to &F being pushed aside, that the local level density on both sides of the gap is increased. However, due to the scarcity of the levels in finite nuclei a direct quantification of this level density increase is difficult. As usual in nuclear physics one therefore applies some averaging. Two methods are commonly in use: the Strutinsky method [ 51 and the Thomas-Fermi approximation [ 61. In non-superfluid nuclei both methods are known to give results for the level density (and other quantities) in very close agreement with one another [ 5-71. In superfluid nuclei the application of both methods is a little more delicate. The Strutinsky method, because a blind averaging of the levels in Fig. 2 over the shell distance certainly also would wash out the gap of 2 1A 1.Concerning the Thomas-Fermi method it is not clear whether such a crude approximation can accurately enough account for such subtle effects like pairing. We will, however, see that both methods are totally adequate and that in addition they both, once more, yield very similar results.
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00872-6
P. Schuck, K. Taruishi/Physics
E(R,p)
2161
g(E)
Letters B 385 (1996) 12-16
13
= &RP)
-1u2+
IAUCp)12
(3)
is the quasiparticle energy and A (R, p) is the Mgner transform of the gap obeying the gap equation A(R,p)
= -
&zQs
k) A(RYk) 2E(R k)
(4)
Here u(p) is the effective nucleon-nucleon force (Gogny Dl [ lo] ) in momentum space in the S = 0, T = 1 channel. The single particle energies / s(R,p)
EF
E-
Fig. 1. Schematic representation of quasiparticle level density in
homogeneous superconducting (superfluid) (nuclear) matter with the square root singularities on both sides of the Fermi energy.
= $m -k VHF(R,p)
(5)
should in principle also be calculated self-consistently but here for our purpose it will be sufficient to make the effective mass approximation
(6) BCS
-
I
I
-20
-10
"6SIl
I
I
2A
0
I
10
---t E we”)
Fig. 2. Quantum
mechanical quasiparticle level density for ‘16Sn in the supertluid (BCS) and the normal states (I-IF). Note the pushing aside of the levels around EF( E = 0) and the opening of the gap in the superfluid case (numerical values are from Ref.
[41).
Let us start with the Thomas-Fermi (TF) approximation. This is well documented [ 6,8,9] and therefore we can be rather short. To lowest order in fi the density matrix and the pairing tensor are given in phase space as follows:
KtR,p) = where
A(R,p) W(R, P>
(2)
where PF (R) = d2m( p - V(R) ) is the local Fermi momentum calculated from a realistic phenomenological single particle potential V(R) (we took the one from Shlomo, see Ref. [ 111) and m* is the effective mass corresponding to the Gogny force in local momentum approximation [7]. In Ref. [ 121 it was shown that the gap equation (4) also allows for a very accurate analytical approximation which we want to use here. Our model based on the Gogny force which is known to have good pairing properties is therefore entirely analytical. Before turning to the level density let us make some tests of the Thomas-Fermi approximation in order to obtain a feeling about its validity. In [ 91 it has already been shown that the correlation energy EC = Tr(AK) ‘i=* s
$$i&(R,p)r(R,~)
compares relatively well with the quanta1 results. However, an overestimation of 20 to 30% per cent over the true average can be noted. Since pairing is a surface phenomenon [9,12], it is not astonishing that this error is of the same order as, e.g., the one of the surface energy in TF approximation [ 131. The correlation energy is a global quantity and therefore eventually rather insensitive to approximations. Let us therefore perform one further test which are the
P. Schuck, K. Tan&hi/Physics
14
Letters B 385 (1996) 12-16
matrix elements of the gap itself. Semiclassically these can be obtained in the following way: Ai = Tr(Ali)(il)
= 2.1 $$$A(R,P)B(RTP) “. (7)
where Ii) are the HF states. In the TF approximation the pi ( R, R) are given by [ 141 PARP)
d3 Rd3p
J’
--) WE-~RP))/~
~ (2&)3
x&E-dR,p))
(8)
Using (8) in (7) we then obtain the semiclassical gap as a function of energy which can be compared with the quantum Ai’s. The same procedure of course applies to the pairing tensor: ~~ --+ K(E). We compare in Fig. 3 the semiclassical functions A(E) and K(E) together with the quanta1 values for the case of li6Sn [4]. For the comparison we have to remember that ’ %n is a mid-shell nucleus and therefore pairing is maximum. At both ends of the shell pairing is zero. The TF solution of the gap can at best represent an average over a whole shell which is lower (approximately by a factor of two) than the maximum value. We see from Fig. 3 that this is exactly what is happening. Globally we can be very satisfied with the TF solution and therefore one can be convinced that the TF approximation also works in the superfluid case in the same way as it does in the normal fluid case [ 6,7]. At this point we would like to mention that in this work the continuum is treated by a substraction procedure as in [ 111. Let us now come to the main objective of this work which is the level density. In the superfluid case it is given by [ 151 g*(E)
=CiS(E-
sign(E)Ei)
(9)
where Ei are the quantum mechanical quasiparticle energies and E is counted from the Fermi energy aF as origin. In TF approximation the corresponding level density is then obtained by: gy = 2
s
d3 Rd3p wS(E
- skn(E)E(R,p))
(10)
The phase space integral can be reduced to a one dimensional R-integration in the case of spherical symmetry to be considered here. In ( 10) again use is made
Fig. 3. Quanta1 and semiclassical values for the gap A and the pairing tensor K. The discrete values of KQ~ are connected by a continuous line to guide the eye. Also indicated are the HF single particle energies ayF for ‘%n [4].
of the analytical expressions for E( R, p) (Eq. (6) ) and A (R, p) [ 121. However, before discussing the results, we first outline the Strutinsky averaging procedure of the quanta1 level density, which, as was said already, has to be applied with some care. The point is that, in order not to erase effects from pairing, one only must average the underlying lzon superfluid level density. We therefore write
gA(E) = Xi
dwS(w
- &i)
./ x 6( E - sign(E)
J/W)
and replace the individual eraged values ] 5,6] : S(W -&i)
~
tf(~)
(11) peaks by the Strutinsky
av-
P. Schuck, K. Taruishi / Physics Letters B 385 (1996) 12-l 6
where f(x) is the Strutinsky smearing function and y the width parameter. At this point this smearing procedure may seem somewhat ad hoc and needs explanation. For A = 0 we get back from ( 11) the usual Strutinsky averaged non-superfluid level density gst(E) [5]. If we replace in (11) the Ai’s by an average value A., we have g~(J3
=
s
dqst
I% y ‘d_
+0(-E
x’ = (Ei +
@
strut d
(13)
Evidently (13) is also valid in the frequently employed BCS version where the gap is taken as a constant (in the “A-shell”). Expression (13) is a very natural result in the sense that it contains only the average underlying normal fluid level density. It has indeed been shown previously [ 161 that in order to obtain the average for quantities which go beyond the ordinary Hartree-Fock approach (like, e.g. the incIusion of particle-hole correlations) it is sufficient to enter the averaged single particle level density. It is in this way that ( 13) has to be understood. We certainly can take (13) as it stands for our comparison with the TF result. However, we still want to apply a slight further refinement of the averaging procedure which seems more attractive esthetically but is of no consequence for the general conclusions. To this end we keep the individual Ai’s as in ( 11) and only perform the averaging over the A’s in a subsequent step. From ( 11)) ( 12) we straightforwardly obtain =
(MeV-1)
( OJ>
x 6(E-sign(E)d(w--,LL)~+A&)
&j(E)
%E)
IE’
{B(E-
Fig. 4. Strutinsky averaged quantal quasipaaicle level density for “6Sn (broken line) together with the corresponding Thomas-Fermi approximation (full line). In the quantum calculation the continuum has not been taken into account whereas the correct treatment of the continuum [ 111 gives rise to the typical maximum seen at E 2 10 MeV in the Thomas Fermi approximation.
that the poles in (14) must be washed out into a smooth function. However, once the smoothing is achieved the parameter a must not be increased further). We therefore introduce condition
EN
F;-(E) = &
dE’B(E’I
Ai) J&
(l5)
E-a
which after integration yields an analytic expression. The final result for the average quantal level density then is
Ai>f(Xi(E))
Ai>f(x+(E))) - Af)/r
--) E &le”)
EF)
+fiWW(~'(E))}
(14)
where again E and &i are counted from &F as origin. In ( 14) we see the typical square root singularities on each side of the gap in the spectrum. In “%I we have 24 bound levels [4] and corresponding 24 Ai-values (see Fig. 3). The Ai values have a spread of g 2 MeV with a maximum difference of 0.3-0.4 MeV for adjacent states. In order to get a continuous function we therefore propose to do a further very small averaging of each pole in (14) over a width a Z 0.3 MeV (the value of a is rather uniquely determined by the
(16)
which is a continuous function of E and therefore can easily be compared with the TF result. This is done in Fig. 4. We see that even in the superfluid case TF and (average) quantal results are in quite reasonable agreement. The TF gap in the level density is somewhat wider which is to be expected from what we said before on the correlation energy. Certainly ficorrections [ 171 again would bring the semiclassical and Strutinsky values in perfect agreement. The remarkable feature seen from Fig. 4 is that the divergency of the level density in infinite matter still has a pronounced analogue in Fermi liquid drops as
16
P. Schuck, K. Taruishi/Physics
small as nuclei. Indeed the nuclear level density is enhanced by more than a factor of two on both sides of the gap. This feature likely also influences the total nuclear density of states. It may be that something of this sort is seen in the 239Pu (n, 2n) reaction analysed in Ref. [ 181. Work in this direction is in progress. A factor two increase of the level density certainly also has strong effects on many other nuclear processes such as fusion and transfer processes, etc. That pairing increases the level density is not unknown in nuclear physics. However, to the best of our knowledge, discussions of this point are very rare in the literature (the only places which we are aware of are Refs. [ 191) whereas the other mechanism of level density enhancement, namely particle-hole correlations, has received much attention in the past (see, e.g. [20] ). From our study it becomes clear that in this respect particle-particle correlations, at least in superfluid nuclei, are equally important (for particle-particle correlations in non-superfluid nuclei see Ref. [ 211) . We consider it as the merit of this work to have revived this important point and demonstrated (for the first time) that quantitative account is accessible in the ThomasFermi approximation. Of course, in Fermi liquid drops as small as nuclei where the pair coherence length is of the order or larger than the nuclear size, the BCS approach itself may be questioned and quantum fluctuations should be included. They are, however, not expected to play a dramatic role for the level density unless one is close to the non-superconducting phase transition [ 221. This may also be deduced from the fact that in a medium to heavy nucleus there are 2 3-5 levels/MeV (see Fig. 4) around&F where the gap is of the order of 1 MeV (Fig. 3). So at least one criterion for the validity of the BCS approach is met. A study of the quasiparticle level density of rotating hot nuclei (as they are produced in heavy ion reactions) with inclusion of quanta1 and thermal fluctuations along similar lines as in the present work is certainly an interesting subject for the future. Our TF approach may also present some interest in the description of small superconducting metallic clusters (for exhaustive refs. to the literature on this subject, see [ 231 and [ 241) . A much extended version of this work is in preparation. We are grateful to Prof. Soloviev for pointing out to us Ref. [ 191. Discussion with S. Goriely, M. Pearson and P. Ring are acknowledged. The quanta1 results have kindly been provided by J.F. Berger.
Letters B 385 (1996) 12-16
References S. Goriely, Universit6 Libre de Bruxelles, preprint; G. Puddu, 2. Phys. A 353 (1995) 247; N. Canosa, R. Rossignoli and P. Ring, Phys. Rev. C 50 ( 1994) 2850. r21 A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 1, Benjamin (N.Y.) 1969. [31 D.R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger, 1986. [41 J.F. Berger, IN2P3, Ecole Joliot Curie de Physique Nucltaire 1991 and private communication. A.S. Jensen, H.C. Pauli, t51 M. Brack, J. Damgaard, V.M. Strutinsky and C.Y Wong, Rev. Mod. Phys. 44 (1972) 320. [61 P. Ring and P. Schuck, The Nuclear Many Body Problem, Springer 1980. [71 P. Schuck, R.W. Hasse, J. Jaenicke, C. Grbgoire, B. Remaud, E S6bille and E. Suraud, Progr. Part. Nucl. Phys. 22 (1989) 181. [81 R. Bengtsson and P. Schuck, Phys. Lett. 89B (1980) 231; R.A. Broglia, F. De Blasio, G. Lazzari, M. Lazzari and P.M. Pizzochero, Phys. Rev. D 50 (1994) 4781. I91 H. Kucharek, P. Ring, P. Schuck, R. Bengtsson and M. Girod, Phys. L&t. B 216 (1989) 249. [lOI J. Dechargk and D. Gogny, Phys. Rev. C 21 (1980) 1568. t111 S. Shlomo, Nucl. Phys. A 539 (1992) 17. r121 H. Kucharek, I? Ring and P Schuck, 2. Physik A 334 (1989) 119. 1131 M. Durand, P. Schuck and X. Vinas, Z. Phys. A 346 (1993) 87; P. Schuck and X. Vinas, Phys. Lett. B 302 (1993) 1. Chaos in Classical and Quantum [I41 J.M.C. Gutzwiller, Mechanics, Interdisciplinary Applied Mathematics, Springer 1990. [I51 H. Stolz, Supraleitung, WTB, Akademie-Verlag, Berlin 1979. [I61 P. Schuck, R.W. Hasse, J. Jaenicke, C. Gregoire, B. Remaud, E SCbille and E. Suraud, Progr. Part. Nucl. Phys. 22 (1989) 181. [I71 K. Taruishi and P. Schuck, Z. Phys. A 342 (1992) 397. [181 V.M. Maslov, Z. Phys. A 347 (1994) 211. 670; [I91 EH. Bakke, Nucl. Phys. 9 (1958/1959) V.G. Soloviev, Figs. 4.10, 4.13 in Theory of complex nuclei, Pergamon Press, 1976; C.W. Reich and M.E. Bunker, Rev. Mod. Phys. 43 (1971) 348. [20] R.W. Hasse and P. Schuck, Phys. Lett. B 179 (1986) 313; P.F. Bortignon and C.H. Dasso, Phys. L&t. B 189 (1987) 381. [21] N. Vinh Mau and D. Vautherin, Nucl. Phys. A 445 (1985) 245. [22] J.L. Egido, P. Ring, S. Iwasaki and H.J. Mang, Phys. Lett. 154B (1985) 1. [23] J.A.A.J. Perenboom, P. Wyder and E Meier, Phys. Rep. 78 (1981) 173. [24] B. Mtihlschlegel, Physica B 203 ( 1994) 444. [I]