r PhysicsA537 (1992) 486-500

o0and

CENTELLES, X. VI&AS and M. BARRANCO

Received 25 July 1991

(Revised 30 October 1991)

Won

0375-9474/92/$05.00 @ 1992 - Elsevier Science Publishers B.V. All rights reserved

YSICS

S.MARCOS

Deparianienio de Fisica ~Vodérna, Universidad de Caniabria, E-39006 Santander, Spain

R.J . LOMBARD

Dit,ision de Physique Théorique, Itisiiiiii de Pt~ysèue Nucléaire, F-91406 Orsay, France

panament EnL Faculiai de Fisica, Universitai de Barceléna, Diagonal 647, E-08028Barcelona, Spain

Abstract. Extended Thomas-Fermi calculations up to second order in h have been performed forrelativistic non-linear aw models and compared with the corresponding Hartree calculations.In several respects, the relativistic phenomenology quite resembles the one previously foundin the non-relativistic context using Skyrme forces .

he description of bulk nuclear properties by means of relativistic mean field ap-proaches is retaining an increasing amount of attention . The Thomas-Fermi andartree approximations have been extensively used 1-3) [see especially ref. 1 ), which

summarizes earlier works] . More elaborated techniques have also been investigated,i .e ., Dirac-Hartree-Fock 4) or Dirac-Brueckner-Hartree-Fock 5-7)

ecently, the relativistic Thomas-Fermi approximation has been extended up tosecond order in h [ref. 8 ) 1 . Numerical tests have been performed in the case ofthe relativistic harmonic oscillator . The results are satisfactory and suggest the useof the semiclassical method on a broader scale and in a more realistic context 9 ) .Also, semiclassical expansions have been derived within the relativistic Hartree-Fockcontext 10 ) although, to our knowledge, they have not been applied yet to a practicalcase.The question remains of the comparison between semiclassical and quantal results

in a relativistic framework . It is the purpose of the present work to investigate anddiscuss this point. In the non-relativistic case, such comparisons have underlined thereliability of semiclassical methods 11,12 ) . Due to the highly non-linear equations to besolved, the problem is somewhat more intricate in the relativistic models, and thusmer.', s to be undertaken .

M. Centelles et at / Seiniclassical approxîînations

487

In order to establish useful comparisons, we have considered models reproducingsufficiently well the bulk nuclear-matter data . The first one is an extension of theaw model proposed by Boguta and Bodmer 1 3), in which non-linear contributionsare included through cubic and quartic terms in the scalar field. Among severalpossibilities 2,3), we have chosen two recent sets of parameters 14 ), denoted Sand SRK3M7, for their ability to fit accepted nuclear-matter data . They ditheir effective-mass and surface properties.As an alternative, the non-linear ato model proposed by Zimany and Moszkowski '5 )

has been considered . Its non-linearity is generated by a derivative coupling between thescalar and the fermion fields. In contrast to the previous model, this non-linearity doesnot introduce extra free parameters . For both models, the results of the semiclassicalapproach will be compared with those obtained in the Hartree approximation .

This paper is organized as follows . The lagrangians and energy densities are shortlyrecalled in sect . 2 . The results are displayed and discussed in sect . 3. Finally, theconclusions are drawn in sect . 4 .

with

where

2. Lagrangians and energy densities

The effective mass in* is related to the scalar field 0 by

3~r through

5

For the sake of self-containment, we recall briefly the starting expressions. Thenotation follows that of Serot and Walecka '), and will not be explicited here.The extended Walecka lagrangian advocaied by Boguta and Bodmer (BB) 13) can

be written as

,CBB lb03 C'043

202 )

2v£ =

(W - gV~') - ln* 1 y/ + 1 (040040 - M,

!Fm~,F" + !in

pV",2

4

2 V

(2)

Fg~, = i),g E, - a,,, Vm -(3)

in* = in - g4-

(4)

The Zimanyi-Moszkowski (ZM) lagrangian 15 ) is simply Czm = C, but in this casethe non-linearity is contained in the connection between the effective mass and thescalar field, which reads

in* in

(5)+ goltnThus it requires no extra terms, and consequently this lagrangian is dealing with fewerfree parameters.At low density, the two kinds of non-linearity are of the same nature, and they would

be equivalent up to first order in 0 [ref. 15) 1 . They are clearly different asO gets large.

488

AL Centelles et al. / Seinielassical approlciî?iaticrs

orenamely

where

old-faced symbols indicate vectors in a three-dimensional space.he energy density is derived in a standard way from the lagrangian . The quantum

structure is expressed as usual by expanding the nucleon field on a single-particle basis(p,, . As in most applications, we restrict the index a to run only over occupied shellmodel orbitals and neglect any contribution from antiparticles . In the static case, withinthe Hartree approximation it yields

with

he two above lagrangians are merely dealing with symmetric nuclear matter. Iner to describe actual nuclei, it is necessary to introduce proton-neutron asymmetry

acts, This is done by adding the p-meson contribution and the electromagnetic field,

for the

eBB=e+403 +'c104

(8)3

0

4

0 1

lagrangian. The first term is given by

+ Pin* + gVO + 11

(9)9P'r3bo + ei(l + T3)AOj(p,, + ef,2Q

~t0

where in* = in - gso, and

Z - 1 Q, - G" + imp2b11

bp

1 gpIF71,ir - bl y/4

2

2

eT71, -21 0 + TA A"V/2

00 )2 + in,202.] -

Similarly, we have ezm = e for the ZM model, now with in* given by eq. (5) (withreplaced by o,,) .The corresponding semiclassical energy densities have a very similar structure, except

that the nucleon variables are now be proton and neutron densities . We define

e2 =

esc = eo + e .) + g,, VOP + 129PbO(PP - Pn) + eAopp + ef,

83 + kF38F

M*4WL.11 kFFj

n LF-+'0Finq

q

3

[(IVV0 )2 + M2V2]

2]

)2 .()

_ 1 [(Vbo) 2 + M22V

flbo'

- 1 (VAO2

2

(10)

(kFq-) M* ) (

(kFq ,3 M* ) (

Pq )2+ B2q(kFqgM*)(

(12)

(13)

K Centelles et aL / Semiclassical approximations

489

where the functions Biq are

Blq(kFq-) M*

X

2

-BF+ 2kFIn

(14)[24kF36F (

m* )lq

FC16kB2q (kFq, m*)

in*

In

(15)2

in*F

q

B3q(kFqgM* ) =

!-F - (2+ !L) In

(16)247r2e2

kF

k2

inF (F

q

Here q denotes the charge state of each nucleon, p = pp -+ p,, is the particle density,2

)1/3

2

*2.kFq = (37r Pq

is the Fermi momentum and CFq = VkFq + inThe eo term corresponds to the usual Thomas-Fermi approach, and the e2 contri-

h2

f 8butions are the relativistic

corrections derived in re . ) . These terms are formallydefined as the semiclassical expectation value of the energy operator when it is ex-panded up to second order in h, i.e.,

with m* = m(l + gO.1m)-' .In this approach, the ground-state densities Pq and the meson and photon fields are

obtained by solving the Euler-Lagrange (EL) equations derived from the variationalprinciple applied to eqs . (19) or (20) . These EL equations read

* Some typing errors have been detected in the semiclassical particle and energy densities inref. 8) . We are grateful to Prof. M.K . Weigel and Dr . D. von Eiff for pointing us these mistakes.The correct equations xvill be given elsewhere 22) .

eo + e2 = (- ia - V + flin*),c . (17)

Similarly, the semiclassical kinetic energy density would be

10 + t2 = (-ia - V + gin - in), (18)

The reader can find the explicit expressions of to and 12 in ref. 8 )* .The semiclassical energy density for the two considered models is given by

escBB = esc + 31b030+

41C04

0" (19)

with in* = in - goo, and by

ez"m Sc= e , (20)

(j _ M2)VV0 gV'O' (21)

(A 2- in - 1 -P )bo = 2 9P (PP Pn), (22)

AAO = - epp, (23)

g-,Vo + - - -2igflbo + eAo + -% 2B, q Apq B2qAM * 0Pq )2

p~'

-2 'OBiq JB2q C9B3q_ - = 0,*(Vpq .,VM*)

( )

(VM* )2jUq (24)

0M am* apq

490

11 . Centelles et al. / Semiclassical approximations

for q protons and neutrons, an

"ere

OCOA ON

- In")0. = -g4p,, + 6,

In* + OB2q

OBIq( UN

W) Pq )2

IUD3

Qq+ 2"" (Vpq - Vn* ) + L ' . (MlIq2I

(26)Upq

Um

is the semiclassical scalar density . The quantities 6 and ~ are defined as b = b02 + C0300'~

~ = 1 for the BB lagrangian, and à = 0, ~ =

for the ZM lagrangian.

The sets of coupling constants, masses and nuclear matter properties correspondingto the three forces mentioned in the preceding sections are collected in table 1 .The p-meson coupling constant has been adjusted to approximately reproduce theexpaimental binding energy Of 208Pb in the Hartree approximation. To compare theenergies calculated in the present work with the experimental values, one has to add

TAUE IParameters and nuclear matter properties (energy per particle EIA, particledensity po, incompressibility K and effective mass Wlin at saturation) of

the three forces considered in the text .

K (MeV)

300

300

225WN

0.55 0.75 0.855

3. esults an iscussion

C7 97

i = s, v, p. T = bl (g,3tn), Z = clg,4 . The masses aregiven in MeV.

(25)

T X 103

F X 1011 .618

-2.2973 .2923.978

In 939 939 938in s 500 500 420Inv 783 783 783flip 763 763 763

ES4 (MeV) -16.0 -16.0 -16.0po (fIn- 3 ) 0.15 0.15 0.16

SRK3M5 SRK3M7 zM

Q' I 381792 231239 161200CIF, 261687 131497 51100

C~" 26.645 85 .645 11 .395

M. Centelles -I aL / Seiniclassical approximations

491

to them the centre-of-mass correction -Ekin/A, where Ekin is the corresponding totalkinetic energy, as it is usually done in non-relativistic calculations '6) .

The different approximations will be labelled in the following way. Semiclassicalresults comprise two cases : ho-order Thomas-Fermi (TRO ), and extended Thomas-Fermi calculations which include h2 corrections (TFh2 ) . The quantal results have beeenobtained from self-consistent Hartree calculations .

In the non-relativistic case, a simple my of incorporating shell effects into thesemiclassical calculation consists in adding them perturbatively, i.e., the Hartme-Fock(HF) equations are iterated only once employing the self-consistent TFh2 potentialsand then use is made of the wave functions so obtained to calculate the densities andenergies . The results of this so-called expectation value method 1"

7) are found to be

in good agreement with the quantal ones for spherical closed shell nuclei . The sametechnique can be applied to the relativistic case: the Dirac-Hartme equations for thenucleons are solved only once with the self-consistent TFh2 fields and the resultingquantal densities am used to evaluate the quantum fields by solving their correspondingKlein-Gordon equations. Finally, the energy is obtained by means of these densitiesand fields. We will denote this approach by H* .The semiclassical densities and fields are obtained from the set of coupled differen-

tial equations (2l)-(26), which we solve numerically using the imaginary time-stepmethod, as outlined in appendix B of ref. 12), until self-consistency is achieved . Aword of caution has to be said about the convergence of the numerical solution of theKlein-Gordon equation obeyed by the semiclassical scalar field, eq . (25) . When theh2-order corrections are included, we have not been able to find stable solutions forparametrizations with tn*lm < 0.60. Rather than a failure of the numerical methodwe have employed, we consider it an intrinsic instability of the equation itself. Wehave checked that the term B3q (kFq ,,m*)Ain* arising in the scalar density, eq. (26),originates the numerical instability. This term comes from the variation with respect to

of the term B3q (kFq , in*) (iVM* )2 which appears in the energy density e~-, eq . (13) .

To bypass this problem, which we encounter with the SRK3M5 force On*lm =0.55), we have splitted the term B3q in eq. (13) by retaining only a fraction f of itin the self-consistent calculation, with f < 1 . The total energy is computed addingperturbatively the remaining (I -f )B3q part not taken into account in the minimizationprocedure. We have checked that the EL equations can be solved if one uses f , 0.5and that the perturbative correction is only a few per cent of the total energy in theworst case (40Ca, SRK3M5 force) .The calculated energies, and proton and neutron r.m.s. radii are displayed in tables

2 and 3 for 40Ca and 208Pb, respectively. For small values of the effective mass theTRO solution is less bound than the Hartree one, but becomes more bound for largevalues of in* . When the relativistic h2 corrections are added, thc TFh2 energies alwayslie below the Hartree ones. In general, the semiclassical radii are smaller or largerthan the corresponding Hartree values depending on whether the binding energies are,,respectively., larger or smaller than the Hartree ones.

492

TABLE 2

Total energy (in MeV), and proton and neutron r.m.s . radii (in fm) of 40Ca calculatedwith the different approaches studied in this paper (the label H corresponds to the Hartreeapproximation) . The listed energies have not been corrected for the centre-of-mass motion .

AL Centelles et al. / Seiniclassical appro,~:iînations

The discrepancy Knween TRO and Hartree results varies almost linearly as a functionof the effective mass, both energies would be roughly the same for iWlin ;ZZ 0.65 . Thesituation is illustrated in fig . 1, where the difference of the semiclassical energies tothe Hartree ones (Es' - EH ) has been plotted as a function of m*1in for the ho andh2 aporoximations. Resuft~ are presented for the three forces studied in this paperand also for the parametrization of ref. "s ) which has in*lm = 0.68 . Carrying outcalculations with parametrizations which have an incompressibility K = 400 MeV andin*lnz = 0.55 or 0.75 [ref. 14)1, we have checked that this behaviour as a function ofnz*lm is not significantly affected by the value of the nuclear incompressibiliiy.

If we compare the TRO and TFh 2 results with the Hartree ones, we can see thatin practiczIly all cases the second-order correciions reduce the differences betweenTRO and Hartree energies in a sensitive way . For the radii, the improvement is not soobvious nor systematic. Had we considered a force with nz*lm ;Z:~ 0.65, the TRO resuitswould have been in closer agreement with the Hartree values than the TFh 2 ones . Forinstance, this happens with the parametrization of ref, 18) (see fig . I ) . It does notmean the TRO approximation to be better than the TR2 one. Indeed, semiclassicaland quantal calculations must differ in the shell energy . This is a subtle quantity that

TABLE 3Same as table 2 for 208 Pb

SRK3M5

E rn rp

SRK3

E

7

rn E

ZM

rn

TRO -1563.8 5.76 5.60 -1697.3 5.60 5.47 -1739.0 5.48 5.46TFh2 -1641 .2 5.67 5.51 -1679.8 5.59 5.45 -1677.1 5.49 5.46

H -1620.1 5.65 5.47 -1620.1 5.63 5.44 -1619.1 5.53 5.49H* -1615.5 5.71 5.55 -1617.2 5.64 5.47 -1618.0 5.53 5.50

SRK3M5

E rn rp

SRK3M7

E rn rp E

ZM

rn rp

TRO -292.3 3.40 3.46 -355.2 3.19 3.22 -345.0 3.17 3.23

TR2 -318.8 3.28 3.33 -348.1 3.17 3.21 -321 .8 3.20 3.26

H -304.9 3.35 3.42 -337.0 3.25 3.30 -317.0 3.27 3.33

H* -304.8 3.33 3.40 -336.6 3.24 3.26 -316.9 3.25 3.33

M. Centelles et at / Seiniclassical approximations

493

75

25

-75

-125 '

I

.-I

.

- I __ .

0.55 0.65 0.75M*/M

0.85

Fig . 1 . Difference between the semiclassical and Hartree energies (Esc - EH ) as a function ofnz*lm, in !he TRO and TR2 approximations, for 208 Pb (circles) and 40Ca (triangles) . The lines

connecting the symbols are only to guide the eye.

comes from the difference of two large numbers ., and it is not easy to evaluate it

correctly . We thus consider the agreement between TRO and Hartree results aroundin*lin = 0.65 as fortuitous .To some extend, the above comments also apply to the phenomenology found in

non-relativistic semiclassical calculations with Skyrme forces 11,12 ) . That is, the non-

relativistic TR2 energies lie below the HF ones, and the position of TRO relative to

TR2 and HF strongly depends on the effective mass of the force . For parametrizations

like SV, SIV and S11 16) which have in*lm = 0.38,0.47 and 0.58 respectively, the

TRO binding energies are smaller than the HF ones. As the effective mass increases .

TFhO yields more binding, with binding energies which lie between the HF and TFh 2

ones for SkM* (in*lm = 0.79) and are larger than the TFh2 ones when the effective

mass is close to unity [SVI, ref. 16) or T6, ref. 19 ) 1 .Tables 21 and 3 also show that the perturbative treatment of the shell effects (H*)

starting from the semiclassical TR2 fflields is a good approximation to the full self-

consistent quantal calculation, as it happens in the non-relativistic context ' 7 ) . The

evaluation of H* from the TRO fields leads, in general, to an agreement in the energies

only slightly poorer than when the input are the TFh2 fields, the major discordances

0%) have been found for SRK3M5. However, the results are considerably worse for

the radii .To ascertain the importance of the different terms of the semiclassical functional in

the TR2 case, it is useful to discuss in some detail the different contributions to the

494

TAKE 4Am0wions On MeV) of f2O and f12 order (TO, 7~) tothe kinetic energy of 40Ca. Also displayed are the corre-sponding contributions to the total energy (E0, E.)), as

well as their partial decomposition .

M. Centelles et al. / Seiniclassical appro.,riî?zaliotis

total energy. Tables 4 and 5 show, for 40Ca and 208Pb respectively, the zeroth-order Toand second-or4ler 71, kinetic energy contributions. They correspond to the integrationover the space of to and t2, eq. (18) . which are obtained from the solutions of thef, 2-order EL equations (2l)-(26) . The same tables collect the contributions E0 and E2to the total energy calculated from eo and e~,, as given by eqs . (12)-(16) . It is seenthat the importance of the h2 contributions to the kinetic energy (71,) and to someparts of E2 decreases when in* increases, since the gradients of in* are more importantfor small effective masses . It is interesting to note that around Wlm = 0-75 the -'~,correction vanishes . This is due to the fact that the contributions involving Vin* havethe opposite sigyn of the (17P )2 contribution . Again, this behaviour is not sensitiveto the value of the incompressibility. A similar situation is met in non-relativistic

TAKE 5Same as Mble 4 W 2UPb

SRK3M5 SRK3M7 ZM

TO 2386.1 3570.9 3813.90 230.9 127.4 91 .3

Eo -61956.7 -35773.2 -19284.9P -98.8 5.4 51 .1y[( ,21 87.7 101 .9 89.7P[( p - Vtn*) I -126.2 -79.6 -35.2P [ ( in*) 2 1 -60.3 -16.9 -3.4

SRK3M5 SRK3M7 ZM

TO 430.4 655.1 672.971, 87.8 48.1 33.4

Eo -10807.3 -6557.6 -3361 .3P -36.5 2.2 19.2y[( ,21 32.6 38.5 32.9Pl( P-VM*)] -4&3 -219 -12.5Q 1 ( ln *)21 -218 -&4 -Q

M. Centelles et aL / Seiniclassical approxiinalîons

495

calculations with Skyrme forces: the h2 corrections, which include effective mass anspin-orbit contributions, are negative for small and medium in*, becoming positive forin* /in >, 0.90 (SVI and T6 forces for example) .The way the energy is distributed among the various contributions is displayed in

tebles 6 and 7 for 40Ca and 208Pb, respectively . The dominant contributions are thoseof the a- and oi-mesons, although there is a strong cancellation between them. Thecontribution of the p-mieson is not negligible for 208Pb, all its semiclassical estimateslying close to the quantal result .

It is worth noting that the good agreement in the total energy between the H* andHartree approximations is not achieved term by term; it results firom compensatorjeffects and is better for parametrizations having larger in* . On the other hand, thedifferences between Hartree and the semiclassical methods (TRO and TR2) in thepartial contributions to the energy are rather large, and also compensatory effectsappear in the final result . The situation is similar in the non-relativistic case if wecompare the different contributions to the total energy evaluated semiclassically withthe corresponding HF contributions (see for example table 10 of ref. 12)) . Indeed,because of the shell effects present in the Hartree or HF results, there is no reason forthese partial quantities to be the same in the quantal and semiclassical approaches .As a representative example, the proton density of 208Pb calculated with SRK3

in the four approaches considered has been plotted in fig . 2 . The results obtainedfor neutron and scalar densities exhibit similar trends . We notice the good agreementbe!ween Hartree and H* densities . The shell effects are rather well incorporated byH* in the internal region of the densities, which become almost identical in bothapproximations in the surface and tail . The semiclassical TRO and TRI 2 densitiesshow no oscillations since they lack shell effects, but average the Hartree results . Thesecond-order corrections improve the TRO densities and in general come closer to theHartree ones, have a better surface and always show a notably better decay than TFh"densities, despite that the fall-off is still too steep with respect to the Hartree densities .To complement this picture, figs . 3 and 4 show the proton densities of 40Ca and

208 Pb, respectively, calculated in the Hartree and TFh2 approximations with the threeforces analyzed in this work . The comparison between Hartree and TR 2 results showsthat the situation is quite similar to the one found in the non-relativistic case, see fig.I of ref. 12 ) . Fig . 5 is the equivalent of fig . 2 for the proton kinetic energy density 'rp ;similar comments to those made for the proton density in fig. 2 apply here .

Finally, the 208 Pb scalar and vector meson potentials V, and V,,, are displayed in fig.6 for all the forces in the Hartree and TFh2 approaches. Notice the change of scalein the vertical axis for each parametrization . Inside the nucleus, the ratio between thevector potentials of any two of these parametrizations is, to a good approximation,equal to the ratio of the corresponding coupling constants Q2. For the scalar potential,this is not so because of the non-linear couplings in eq. (25) .A Foldy-Wouthuysen reduction of the Dirac equation for a nucleon feeling the action

of G and V, transforms it into a Schr6dinger equation whose central potential involves

7

496

ÏL Centelles et al. / Semiciassical 2pproximations

*5

0 ic =

efj ÇA

ÇD

21- AM .r_ uM >

-E.,

'Eh0

CI cmm -0Cu u

ÇA

à :3

7S

m

e

Z; 2

j m

triM

&05

ce 'l

Iz

U.

-Z

oq 01 ~ licI~ ;; rq -= fIlrl-

tr; ri r- ,o 0,la 00 el% "

rq CD trb rli 00

00

"d'

tr; r-: --: (Nle

tn

- -

10>1>

1-0

ÇA

m

u

-Z

Li.

fV r,à r--

in = M

IC

ëI

CD00 V-~

q

CD 0; tri

4D r- V-b

111IN - 1-i ":

-Zo~ --: r- Co

U. - oo qr~ -e

là%1

CD

. ~er~tedles et ad. / Sed»icdassical aper®xifnati®ns

0.®

v®.®

®.®2

Fig . 2 . Proton density of 2°gPb obtained with the Sit

3

3 force in the ~Iartree (solid line),(long-dashed), TFfi2 (shore-dashed) and TFfa° (dotted) approximations .

( Ye - YW ), whereas the strength of the spin-orbit force depends on ( Ye ~ ~w ) [ref. ' ) ] .

Fig. 6 shows that in the bulk, the central potential is around -60 eV, deeper for small

m*, shallower for large m* in accordance with the optical-model phenomenology z® ) .

The changes are far more drastic for the spin-orbit potential, which decreases by a

factor of around 3 frown the SRK31~5 to the Z1VI forces . This leads to a small splitting

493

Fig. 3 . Proton density of a°Ca, calculated with the three forces considered in this paper, in thet-iartree (solid line) and TF~2 (dashed) approximations .

w-.1 0.()0.

.11. Centlles et al. / Sep?iiclassical approminations

10

S

r (M)

7

Fig. 4 . Same as figure 3 for '-0s b .

10

le

in the single-particle levels for forces having an effective mass of the order of 0.80, avalue compatible with the one extracted from optical-model analyses 21 ) . It seems tobe quite a general feature of all the aw lagrangians that the experimental splitting canonly be obtained with rather small effective masses, W/in < 0.60 .

Fig . 5 . Same as figure 2 for the proton kinetic energy density.

M. Centelles et aL / Seinielassical approlciî?îaîions

r (fm)

Fig. 6 . Scalar (V,, = in - W) and vector (V. = gv VO ) potentials obtained for 208Pb with thethree parametrizations analyzed in the text . Solid line, Hartree approximation ; dashed line, M12

approximation .

4 . Conclusions

10

10

499

In the present paper, we have carried out extended Thomas-Fermi calculations forrelativistic non-linear aw models using two kinds of lagrangians whose parametershave been fixed to reproduce some bulk nuclear matter properties.Comparing TRO with Hartree results, we have found that TFhO yields an under-

binding when the effective mass of the model is small, and an overbinding when in* islarge . For a value around tn*lm = 0.65, both TFhO and Hartree would roughly yieldthe same binding energy .When the fuil h2 order corrections are taken into account, we have found a numerical

instability in the solution of the semiclassical Klein-Gordon equation for the scalarfield in the case of parametrizations with in* lin < 0.60 . Whether this instability is dueto the particular nature of the aw model or to a failure of the relativistic semiclassicalexpansion of h2 order, is an open question that can only be answered when thisexpansion be applied to other relativistic models.

Second-order corrections in h to the TRO approximation improve the agreementwith Hartree solutions in a sensitive way, always yielding more bound nuclei thanwithin the Hartree approach . The sign of the h2 corrections depends on in*, and theyare found to vanish around m*lm = 0.75 for the models of the type considered here .When studied as a function of the effective mass, the semiclassical relativistic results

500

H Centelles et at / Semiclassical approximations

behave with respect to the Hartree ones as in the non-relativistic case, in spite of the1 2different origin of nz* in both situations . Using the TR

semiclassical mean 51-ld asa starting point, perturbative quantal solutions have been found which are in goodagreement ~-.,:'Lh the exact Hartree results .The TFf1 2 approximadon As two apparent merits over the simple TRO one. First,

it allows to obtain fully variational densities that go exponentially to zero, and second,it takes into account non-local, i.e., effective mass and spin-orbit corrections, up to h 2

order. However, it should be pointed out that the seNclasical expansion of the energydensity functional in terms of the local density, effective mass and their gradientshas likely not converged if one stops the expansion at the h 2 terms. This can beinferred from the compariion of this M expansion with the Wigner-Kirkwood onein the particular case of a set of fermions moving in a relativistic harmonic externalpotential 22 ). To improve on the results we have presented here, it would be necessaryto add f14-order corrections, as it has been done in the non-relativistic case 11,12) .

nfortunately, to work out these f14 corrections seems a very cumbersome task, as itcan be easily realized if one looks at the way the h 2 corrections have been derived 8-10 ) .

This work has been supported in part by the IN2P3-CICYT exchange program andby the DGICYT grant P1389-332.

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