ANALOGIES BETWEEN NUCLEAR SYSTEMS AND LIQUID 3He

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ANALOGIES BETWEEN NUCLEAR SYSTEMS AND LIQUID 3He

S. Stringari

To cite this version:

S. Stringari. ANALOGIES BETWEEN NUCLEAR SYSTEMS AND LIQUID 3He. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-93-C2-100. �10.1051/jphyscol:1987215�. �jpa-00226480�

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JOURNAL DE PHYSIQUE

Colloque C2, suppl6ment au n o 6, Tome 48, juin 1987

ANALOGIES BETWEEN NUCLEAR SYSTEMS AND LIQUID 3 ~ e

S. STRINGARI

Dipartimento di Fisica, Universita di Trento, I-38050 Povo, (Trento) Italy

Abstract - Several azalogies between nuclear systems and liquid 31ie are presented and discussed. The usefulness of the mean field approach in describing several static and dynamic properties of quantum liquids is emphasized. Using a Skyrme force to account for the interaction in liquid 3 ~ e we give predictions for the equation of state in bulk matter, fcr relevant surface properties

(surface profile, clusters) an? for the liquid-gas transition.

1. Introduction.

A quantum liquid is a homogeneous condensed system for which the thermal wave length (k2/2mk~)ll2 is larger than the average distance between particles. Only if the interaction is weak and the mass of constituents sufficiently light the system remains liquid in the quantum limit. Otherwise it undergoes a liquid-solid phase transition. The most famous examples of quantum liquids are 3 ~ e and 4 ~ e . ibiore recently other systems (polarized hydrogen [l] and polarized >He [2]) have been the object of experimental and theoretical research. In particular in the case of polarized hydrogen the interaction is so weak that the system is not able to liquefy and is e::pected to remain a gas at T=O.

Atomic nuclei represent another important example of quantum liquids. Due to the Coulomb repulsion between protons only small drops of such a liquid are available. Asynmetric nuclear matter is also the object of important theoretical investigations due to its relevance in astrophysical problems. It is interesting to discuss the anaiogies between atomic nuclei and liquid 3 ~ e , both systems obeying the k'ermi- Dirac statistics. In table 1 we presznt a schematic list of relevant features which are well establish5d either experimentally and/or theo- retically. The comparison is here limited to t2ie properties which are relevant for the discussions contained in the present work. The different values for the Fermi energy EF are due to the present unknowledge of the effective mass in the n32clear case. In 3 ~ e they are determined from measurements of magnetic susceptibility and of specific heat. The natural units of lengths and ener ies in the two systems are related by : lfm = ~ o - ~ A and lMeV = l.16*10q0K.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987215

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Table 1.

YUCLEI LIQUID 3He

nucleons

CONSTITUENTS

(fermions) atoms of 3 ~ e DEGENERATE LIMIT

closed shell nuclei (normal phase) 3mK 6 T < < TF unclosed shell nuclei (superfluidity) T 6 ^3mK

SATURATION PROPERTIES

ro = 1.1 fm (average distance) ro = 2.4 A E/A = -15 MeV (binding energy) E/N = -2.5K

9/k = 200 MeV ( ? ) (incompressibility) l/k = 12.1 K A - ~ EF = 30-40 MeV (Fermi energy) EF = 0.5-1.5K

SURFACE PROPERTIES

G = 1.2 ~ e ~ f m - ~ (surface tension) G = 0.11 K A - ~ t = 2.2 fm (thickness 90% to 10%) t = ?

neutron excess POLARIZATION EFFECTS magnetization

( isospin ) (spin)

COLLECTIVE PHENOMENA giant resonances (collisionless regime)

high energy ion collisions

zero sound (hydrodynamic regime) first sound

THERMODYNAMICS liquid-gas transition, critical point, specific heat ...

2. Theoretical approaches to the ground state.

In the recent years sophisticate microscopic calculations of the ground state of liquid 3 ~ e have become available (see [3] and references therein). Such calculations use an interatomic potential (Van der Waals, Aziz.. . ⁾ with correlated wave functions. The results for the binding energy, saturation density and structure function are in rather good agreement with experiments, the calculations being less re- liable at high pressures. Calculations of similar quality are not pre- sently available in atomic nuclei, due to major difficulties introdu- ced by the tensor interaction and by the presence of inhomogenities in the nuclear density.

Methods based on the Brueckner theor have been also employed (see [4 1 and references therein) in liquid 'He, starting from the pionie- ring work by Brueckner-Gammel [5]. Such methods have been also develo- ped in nuclear physics [ 6 ] . More recently nuclear theorists have found it convenient to use effective interactions of phenomenological type in order to provide a systematic description of the ground state properties of atomic nuclei in the framework of the Hartree-Fock theory.

I n particular the Skyrme interaction [ 7 ] has become very poular due to its simplicity and to the fact that it provides quite a good descrip-

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tion of many measured properties (energies, density profiles..) in the whole nuclear table. Ons of the aims of this work is to show that a similar method can be conveniently used in liquid 3 ~ e too, especially in the description of surface properties.

3. Theoretical approaches to collective excitations.

Both in liquid 3 ~ e and in nuclear systems collective modes can propagate in the collisionless regime. This is the result of the mean field interaction which provides the restoring force responsible for the propagation of sound.

Landau first proposed in 1956 the existence of a sound in the collisionless regime (zero sound) in liquid 3 ~ e . His predictions were confirmed experimentally several years later (see [8] for an exaustive discussion on zero sound). Landau derived his equations for zero sound in the framework of the Vlasov theory. The resulting picture of this collective mode is quite suggestive: the Fermi sphere vibrates taking deformations of different multipolarities. In particular the occurrence of 1=2 deformations in momentum space distinguishes zero sound from first sound and makes it rather similar to an elastic type vibration [g]. The transition between zero and first sound is governed by the collisions and is quite wsll described by the visco-elastic model [g].

Nuclear systems exhibit a large variety of collective modes (giant resonances). They can be classified accor6ing to the multipolarity of the deformations taking place in coordinate space (monopole, dipole, quadrupole ....) and to the spin and/or isospin nature (for a relati- vely recent experimental review see, for example, [10]). In particular the isoscalar quadrupole resonance presents significant analogies with zero sound, the restoring force associated with such a mode being com- pletely provided by an elastic cffect [10]. Many properties of nuclear giant resonances have been explained and prsdicted in the framework of the random phase approximation (RPA) [12]. Many of such calculations use the same effective interaction (Skyrme) as in the static Hartree- Fock (HF) case. Thouless first pointed out the importance of using the same effective interaction in the HF as well as in the RPA calculation (see, for example, ref[l3]). This is a crucial requirement if one wan- ts the conservation laws associated with the symmetries of the problem to be satisfied by the RPA. Time dependent Hartree-Fock (TDHF) calculations have been also become popular in nuclear physics [14]. It is n o w clear that the TDHF theory reproduces, in the limit of small oscillations , the equations of the RPA. Of great incerest is also the link between the TDHF thcory and the Vlasov equations. One can show E151 that the latter can be derived starting from the TDHP equations by taking the long wave length limit or, equivalently, by introducing a semiclassical approximation in the equations of motion.

From the above discussions it emerges that zero sound and giant resonances are solutions of the same equations (TDHF equations) with different symmetries (translational and spherical respectively). It is then interesting to explore in liquid 3:1e the possibility of using effective interactions in the HF-TDHF scheme similarly to what is cur- rently done in nuclear physics. With such a procedure we expect to provide a consistent description of several static and dynamic properties in bulk 3 ~ e as well as in inhomogeneous 3 ~ e (surface, clusters).

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4 . The Skyrme effective interaction.

We will use an effective interaction of Skyrme type to describe several properties of liquid 3 ~ e in the framework of the HF-TDHF for- malism. The force gives rise to the following expression for the binding energy under the assumption that the ground state is time rever- sal invariant (no magnetization, no currents) [16]:

where !: and T; are the diagonal and kinetic energy respectively. The term proportional to Z fixes the effective mass of 3 ~ e at zero temperature :

With the value Pc ^=4.^l*~ o - ~ A - ~ eq. (2) reproduces the experimental values of Greywall [l71 at all pressures with high precision. The term in b represents an attractive force, while the one in c a repulsive density d e p e n d e n t o n e . T h e t e r m i n d i s repulsive and acting at the surface. The value of the parameters b,c and 8 have been fixed to reproduce experimentally known bulk quantities(binding energy, density, corn ressibility at saturation) using the Thomas Fermi relation= =

19 f/3 with 4 = 3 / 5 ( 3 ~ ~ ) ~ / ~ . The fit yields b = - 6 8 3 . 0 ~ ~ ~ , c=1.405057

~ o ~ K A ~ ( ~ + ) , and &' =2 .l. 'Equation (1) then becomes predictive for the T=O equation of state P= y2d/df(~/~) far from saturation. Fig.1 and 2 show that the Skyrme force well re-

produces the experimental data (cro j'

s s e d p o i n t s ) f o r the equation o f state and for the incompressibility k-l=fd~/d$' up to solidification,

Fig.1. Density vs pressure in 3 ~ e . Fig.2. Incompressibility of 3 ~ e . improving in a significant way the harmonic approximation (dashed line) obtained expanding E/N around the saturation density P O ^:E/N = Eo/N +1/2kfO3* (P-P~?.

The remaining parameter d can be fixed to reproduce the experimental value of surface tension (see ref.[l8] for an exhaustive discus-

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sion of the surface properties of quantum liquids). One gets [l91 d = 2 2 2 2 ~ ~ ~ . The surface term is essential to account for the renormaliza- tion effects induced by the zero point motion of the surface. It plays a more imprtant role in liquid helium than in atomic nuclei.

Once the parameters of the force have been fixed one can proceed to calculate other properties of inhomogeneous 3 ~ e . To this purpose one can follow two different methods. The more microscopic one is to write Hartree-Fock type equations for the single particle wave functions entering the density ( 9 (r)=jiy:(r)yi(r)) and the kinetic energy density ( 't; (r)=zid/dry?(r)d/dryi (r) ) using the standard variational approach:

In practice this method can be easily carried out only for small spherical systems (clusters, see sect. 5) where the resulting Scrodinger equations can be solved with standard iteration procedures. The second method c o n s i s t s of using a semiclassical approximation for the kinetic energy density of the form

which includes Weizsacker corrections ( p =1/18, 1( =1/3) and then of solving the resulting Euler equations for the diagonal density (ac- tually this is the method we have used to fix the value of the parameter d). With the latter procedure, which neglets quantum shell effects, one can quite easily investigate the surface profile of semi- infinite matter . The surface thickness (90% to 10%) is predicted to be 8.4A in agreement with the recent results obtain'ed applying the va riational Monte Carlo approach with the Aziz potential to large 3 ~ e

clusters [20] (see also sect.5).

5. 3 ~ e clusters.

The Hartree-Fock type equations (3) can .be solved for closed shell clusters. In this case the spherical symmetry of the self consistent potential is ensured and eq.(3) becomes a differential equation in the

N E/N ro(N)

I K I [AI

2 0 +O. 08 3.99 4 0 -0.19 3.13 7 0 -0.50 2.90 112 -0.75 2.78

168 _-0.95 2.69 - ^1.0

240 -1.11 2.64 3 3 0 -1.25 2.61

00 -2.49 2.44 - ^0.5

Table 2. Results for 3 ~ e clusters.

radial variable. The

results (Stringari and Fig.3. Profile of the N=168 cluster.

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Treiner, to be published) are reported in table 2 and fig.3,4. Some interesting features of the energy systematics emerge from this analysis [19]:

a) The lack of binding for clusters with less than NZ30 atoms.

b) The occurrence of magic numbers given by the harmonic oscillator rule: N = 40, 70, 112, 168 ...

c) The existence of single particle energy gaps of the order of .2-.5K as can be shown from fig.4 where the energies relative to the last oc- _- cupied and first unoccupied single particle levels are reported. In the same figure the full line is the prediction for the chemical potential obtained using the semiclassical approach described in sect.4.

The results for the ener- I

gy systematics as well as for -

the density profile of the

clusters quite well agree wi- ^{3 ~ e}

th the theoretical predicti- -

ons of ref.[20] (dot points -

i n fig. 3). In particular a --4 -

value of the same order is -r;;

p r e d i c t e d f o r t h e m i n i m u m n u U

mber of atoms required to fo; - '

rm a bound cluster. It is in-

.

^-

terestirlg t o note that the -

-t , N

properties of 3 ~ e clusters I I I

significantly differ from the ¹⁰⁰ ²⁰⁰ ³⁰⁰ o n e s of 4 ~ e clusters where Fig.4. Single particle energies.

binding is always found eit-

her in the present density functional approach and in the calculations o f ref.[20]. F u r t h e r m o r e n o magic numbers are predicted in 4 ~ e clusters [21].

The significant differences predicted for 3 ~ e and 4 ~ e clusters are expected to stimulate experimental research in this area, at present limited to the study of ionized helium clusters and of helium bubbles in metals.

6 . 3 ~ e a t temperature d i f f e r e n t from z e r o .

Hartree-Fock calculations at T f 0 have been carried out in recent years in order to investigate the thermodynamic properties of nuclear matter (see for example ref.[22]). It is interesting to check the va-

lidity of such a method in situations, like 3 ~ e , where the theoretical predictions can be directly compared to experiment.

A first difficulty in applying the Skyrme force to thermodynamic calculations is that it predicts a T-independent effective mass. This i s c o n t r a d i c t e d by experiments [l71 which reveal a sizeble T- dependence in m* at low te~ilperature (T 6 0.5K). In order to explore the liquid-gas phase tiansition at higher temperatures we have then found [23 l it moxe physical to use the bare mass instead of the T=O effective mzss (eq. (2)). The free energy per particle has then the

follow in^ simplified form:

where fni is the free energy of the non interacting Fermi gas. The pa- rametres b, c and I( have been chosen to reproduce the correct satura-

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tion properties of bulk j ~ e . Equation (5) predicts the existence of instability regions dP/i?f < O associated with the occurrence of a li- quid-gas phase transition. The isotherms predicted by eq.[5] are re-

Fig.5. Tsotherns in 3 ~ e . Fig. 6. Vapor pressure in 3 ~ e . ported in fig. 5, while in fig.6 we have reported the results for the vapor pressure [23]. The agreement with experimental data (dot points in fig. 5, full line in fig. 6 ) is quite good except in the region of

the critical temperature which is predicted too high ( T ~ ~ ~ ~ ~ ~ ~ = " . ~ K , TCeXp=3. 3K ) .

We have shown that from a critical and detailed analysis of the properties of atomic nuclei and liquid 3 ~ e some useful developnents of the Hartree-Fock and time dependent Hartree-Fock formalisln can be proposed for liquid 3~<e. Using an effzctive interaction of Skyrme type we have been able to provide a quantitatively correct description of several properties of bulk 3 ~ e (equation of state) as well as of inhomogeneous 3 ~ e (clusters, surface) at T=O and TfO (liquid-gas phase transition). When possible we have checked the quality of the theoretical predictions comparing with experimental data as well as with other theoretical calc~~lations. The method can be extended L241 to investigate magnetic properties of 3 ~ e and the liquid-gas phase transition in polarized 3 ~ e where interesting phenomena such as metamagnetism (or quasi metamagnetism) and liquid overpolarization are predicted. It can be also employed to study the interaction between 3iIe and *He atoms in

the bulk as well as at the surface [25].

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Acknowledgements.

Part o f the work p r e s e n t ~ d in this paper has been done together with MiBarranco, F-Dalfovo, A-Polls and J-Treiner. It is a pleasure to acknowledge their collaboration.

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