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SEMI-CLASSICAL CALCULATION OF THE
SPIN-ISOSPIN RESPONSE FUNCTIONS
G. Chanfray
To cite this version:
JOURNAL DE PHYSIQUE
Colloque
C 2 ,supplement
au n o 6, Tome
48,
juin 1987
SEMI-CLASSICAL CALCULATION OF THE SPIN-ISOSPIN RESPONSE FUNCTIONS
G . CHANFRAY
Institut de Physique Nucleaire (and IN2P3), Universitb
Claude Bernard Lyon-1. 43, bd du 11 novembre 1918,
F-69622 Villeurbanne Cedex, France
Nou5 p r h ~ e ~ l t o n ~
une apptoche bemi-~Babgque qui ua au-delk de 8'approximation de
Thomab-Fezmi
POUTle cdcul deb ze'ponbeb nucl6aireb. Notze dozmdibme ebt appliquh
a u xte'ponbeb de bpin-ibobpin. Lozbque ~e.?h?b-~i
bunt bonde'eb
p3azdeb hadzonb, noub montzonb que
4eb cozzectionb 5u.rdaciqueb en
h 2
conduibent
h
une tzhb nette diminution du zappo~t
longitudindltzan~vezbe.
ABSTRACT
We pzebent a bemi-c!abbica.i c a k d a t i o n
06the nucleaz zebponbe bunctionb beyond the
Thomab-Fe~mi app~oximation.
Weapply ouz 6ozma4ism to the spin-ibospin tebponbeb and
&how that the [email protected]
peaked
+I*
cozzectionb conbide~ab!g decteabe the m t i o kngitudi-
ndlttanbvezbe a5 obtained th~ough
hadfonic ptobeb.
The quasi-elastic nuclear responses a t intermediate momentum transfer (q
-
2-3 m,) are mainly sensitive t o average volume properties. Consequently we expect that a phase space approach, as Thomas-Fermi (TF) theory, w i l l provide a correct description of the excitation mechanism i n the conti_nuum. This feature has been explicitely proven i n the case of the single particle response f o r which the TF result compares extremely well t o i t s quantum mechanical analog[I].
Moreover, a realistic TF-RPA calculation [2, 31 gives a satisfactory description of both longitudinal and transverse structure functions measured i n (e,e9) scattering experiments [ 4 ] .When applied t o the problematic o f spin-isospin responses, the TF theory [3], as the nuclear m a t t e r approach [5], predicts marked collective effects involving the whole nuclear interior. The transverse response R T should be quenched even a t momenta o f order 2 F m
'
since the short range repulsion (usually characterized by a Landau-Migdal parameter g') s t i l l dominates over the a t t r a c t i v e rho exchange. The experimental response [4], known f r o m (e,e') data is compatible w i t h this collective behaviour. On the other hand, the longitudinal response RL, sensitive t o the long-range pion exchange potential, should exhibit collective e f f e c t s of opposite nature as a r e f l e c t of the nuclear pionic f i e l d enhancement [5]. Thus a net contrast between RL and R T is expected at momentum transfer around 2 ~ m - ' .These theoretical ideas have strongly motivated a ($,pt) experiment which has been performed i n Los alamos [6]. The resulting data were f i r s t rather deceiving as they showed no contrast and seemed t o contradict the existence o f the contrast. We w i l l show that those data do not actually rule out the expected collective nature of the spin-isospin responses. The reason lies i n the fact that protons essentially probe the nuclear surface where specific effects depending on the gradient o f the density show up. Those effects, which cannot be simulated by simply lowering the density i n the TF scheme, w i l l be incorporated through a Wigner-Kirkwood expansion o f the response up t o order Ti2. The surface correction terms acquire an important weight when the nucleus is probed by hadrons and then strongly mask the collective reshaping of the T.F (1.e.h0) response.
C2-66 JOURNAL DE PHYSIQUE
OUTILINE OF THE SEMI-CLASSICAL METHOD
The response of the nucleus t o any external excitation can be obtain5d Jrom the p o l a r i z a t i o n p r o p a g a t o r n ,q' ; w) depicted schematically on fig. ~tB(;ere the end point operators A(?) and
~ ( 6 )
w i l l be e i t l e r tJe longitudinal spin-isospin operatora.
L ( q ) = 0.q T or a component of the t ~ a n s - vTrse s i n isospin o p e r a t o r T (q ) =
(a x
8)
F.
These polarization propgqators . . -a r e cb#sidered -as one body oper-ators, w r i t t e n i n momentum space, and are expressible in terms of+t eir Wigner-trans-Fiquie
1 - The$04
tization ptopagato.r
forms (WT)bABIW
(R
,I!? ; w) accordingnae[q
,$
;,I
t o :Ti'
( ~ ? + ~ , L F ; w ) = J ~ ~ R
t
L
e
w
(1)
The semi-classical method consists i n an expansion in power of
fi
of the WT [nPBlW. In practice we w i l l l i m i t ourselves t o order -ti2. The responze t o an external weakly in e r a c t ~ n g probe, transferring t o the n u c l e u p a momentum-energy q ,w and interacting w i t h the nucleus throuqh the one-body operator A(q ) is then obtained as :THE PARTICLE-HOLE (OR MEAN FIELD) RESPONSE FUNCTION
We consider a spherical nucleus w i t h N = Z wherc the nucleons move i n a central local single particle potential V(R). The pure particle-hole response is independent o f the spin-isospin structure of the excitaLion operators and writes very generally :
A
where H est the single particle Hamiltonian and E~ is the F e r m i energy. We expand the r.h.s. of Eq. (3) up t o order 5 ' . The f a m i l ~ a r Thomas-Fermi result [I] is recovered as the f i r s t t e r m (i.e. f1° term) of the expansion. In the momentum range of interest, the surface-peaked f l Z contribution gives only a few per cent correction. We show on fig. 2 the result of the calculation for 4 0 ~ a (Woods-Saxon potential w i t h parameters taken f r o m Myers [7] ) a t momentum q
=
350 MeV/c. By comparison of the dotted curve (TF response) w i t h the dashed curve (fiO + 5' responses) it is seen that then2
contributions is positive a t low energy (w < 100 MeV) and becomes negative a t high energy. The smallness of thefi?
correction i m p l i c i t e l y confirms the accuracy of TF theory, when we are dealing w i t h average volume properties.
THE RPA SPIN-ISOSPIN RESPONSES
+
a
+
The response R (R ) t o a longitudinal (transverse) spin-isospin excitation O.q TL T
a n t i s y m e t r i z a t i o n is assumed t o be incorporated i n t h e Landau-Migdal parameters g'. This procedure has been j u s t i f i e d by S t r o l h e t a!. a t t h e TF l e v e l [Z], i.e. a n t i s y m e t r i z a t i o n yields a simple r e d e f i n i t i o n o f g'. The R P A p o l a r i z a t i o n propagation nLL is obtained as a solution of coupled i n t e g r a l equations w h i c h w r i t e i n a schematic way :
Taking t h e Wigner-transform o f Eq. 4, we get an equivalent set o f equations w h i c h are solved order b y order up t o fi2. F o r doing that, we need t h e WT of t h e mean f i e l d quantities. F o r instance [7PLl.
Iw
has the f o r m ( o m i t t i n g a tensorial piece i r r e l e v a n t f o r spherical nuclei) :A p a r t f r o m t h e inclusion o f A-hole c o n t r i b u t i o n w h i c h is specific t o t h e spin-isospin channel, t h e f i r s t t w o t e r m s (1.e.
a
and.n2a2) o f E q . 5 are independent o f t h e p a r t i c u l a r f o r m o f t h e end p o i n t operatossklntoegrating t h e i r imaginary p a r t s would give t h e response( 3 ) c a l c u l a t e d t o order S2. a ( q , R ;
w)
is f o r m a l l y i d e n t i c a l t o the p o l a r i z a t i o n propagator o f u n c o r r e l a t e d nuclear m a t t & once one has i n t r y d y e d a l o c a l F e r r n i m o m e n t u m kF(R) =[ 2 M ( E ~ - V ( R ) ] 1 / 2 . The 1engt.h~ expression of a (q ,R ; w) w i l l be given i n a f o r t h c o m ~ n g paper. N o t e t h a t t h e last t e r m of Eq. (5) does n o t c o n t r i b u t e t o t h e p-h response. H o w e v e r i t w i l l become i m p o r t a n t i n the r e n o r m a l i z a t i o n procedure (see Eq. (10) below). F r o m Eq. (4) we also see t h a t there is a longitudinal transverse coupling t e r m whose W-T w r i t e s :
One sees t h a t t h e L - T m i x i n g e f f e c t occurs only t o order -I7 and thus disappears i n the T-F a p p r o x i m a t i o n . A f t e r s o m e a l g e b r a w e f i n d f o r t h e R P A l o n g i t u d i n a l r e s p o n s e :
H e r e
goL
is t h e w e l l known T F - R P A expression :w h i c h yields r e n o r m a l i z a t i o n and enhancement o f t h e response. We have d i v i d e d i n Eq. (7) t h e h2 c o r r e c t i o n t e r m i n four d i s t i n c t pieces. The f i r s t one m a y be understood as t h e r e n o r m a l i z a t i o n of t h e h2 part. of t h e m e a n - f i e l d response. It w r i t e s :
C2-68 JOURNAL D E PHYSIQUE
which contains the L T mixing term. I t tends t o attenuate the collective reshaping o f the response. The last two*' terms have the same e f f e c t but are less important f o r our cases o f interest. +Their $xpressions are quite complicated and involvf gradients of a w i t h respects t o q and R as well as gradient of V w i t h respect to q . As for the traisverse response, R T is simply obtained by exchanging {he role of L and T i n Eqs. (7), (8) and (9). The e f f e c t of the h2 terms is reversed i.e the quenching is reduced.
Fiquie
2
-L
ongitudind spin-isospin
tespOn5eb
601 O C ~at
q-
3
50MeVlc. The dotted bne (mean-6ield
responsej and the dot-dashed fine
l R P A
~ e s p o n s e l c o ~ t e s p o n d t o a
Thomas
-Fetmi ca5cdation.
The
dabhed
line (mean-dicedtesponbel
and t h e dull line l R P A tesponsej
ate obtained aXter inclusion
O X
the
o
(MeV)
RESULTS A N D DISCUSSION
For mathematical convenience the calculations have been carried out a t f i n i t e temperature and the zero temperature case is obtained as a limit. In practice T = 1 MeV is sufficient. On fig. 2, we show the results for the longitudinal response of ' O C ~ a t q = 350 MeV/c. The Landau-Migdal parameter is taken t o be g' = 0.7. A t the TF level, we have a n e t c o l l e c t i v i t y as can be seen by comparison of mean field result (dotted line) w i t h the RPA response (dot dashed line). When the 1F2 corrections are included, the mean field response (dashed line) although slightly larger, remains very close t o i t s TF analog but the RPA response (full line) is reduced, w i t h respect to the TF-RPA one, by about t e n per cent a t low energy. Thus, once+' corrections are added, the c o l l e c t i v i t y is a l i t t l e reduced but s t i l l very net.
Unfortunately, the volume longitudinal ysponse is not known experimentally. The only available information comes f r o m the (P,~') Los Alamos experiment [ 6 ] which essentially measures surface responses. Thus, the surface-peaked -t12 t e r m w i l l obviously acquire a much more important relative weight yielding a considerable reduction of the collective effects. I n addition the isoscalar spin responses also enter, I n this experiment Carey and collaborators measured i n f a c t the ratio.
a t a momentum q = 350 MeVlc. The t i l d e mean t h a t we are dealing w i t h surface responses and the distorsion e f f e c t is taken into account through a position dependent weight factor C(R) multiplying the integrand i n Eq. ( 7 ) . C(R)is derived in ref. [3]. In Eq. ( I I ) , we have s p r o x i m a t e d the isoscalar responses RL T ( ~ , ~ ) by the mean field one. The calculated r a t i o
easy w i t h small computer t i m e t o perform calculation for d i f f e r e n t conditions, such as other incident energies or other probes 'He-T. Although our results s t i l l remain above the data, they are quite conclusive and mainly show that the absence of contrast is mainly due t o the peripheral nature o f the probe. We should also mention that improvements i n the theoretical description (such as inclusion of 2p-2h contribution, r e n o ~ m a l i z a t i o n of the isoscalar responses [ 9 3 is likely t o further reduce the calculated r a t i o X. F r o m the above discussion, one can conclude that the access t o the l o ~ g i t u d i n a l volume response require further experiments. Charged pion electroduction (e,e'f
!
1101 could provide additional information i n t h a t direction but such an experiment is feasible only w i t h the next generation o f high duty cycle machines.Fiquie 3
-The iatio
?
601'OCa a5
a dunction
06
the tian56e~ied eneigy
w
at
q =3 5 0
MeVIc. The dashed
curve5 iedei to a
TF-RPA
calcu!a-
tion and the did! cuiveb coi1e5pond
t o t h e
R P A
~ e 5 u l t including
4i2 co.r<ections.The e x p e . r i m e n t a l d a t aate 6iom ie6.
161.
REFERENCES
[I 1 STROTH, U., HASSE, R., SCHUCK, P., J. de Phys., C-6 (T984) 343 [21 STROTH, U. e t al., Phys. Lett., 156-6 (1985) 291
[31 STROTH, U., HASSE, R., SCHUCK, P., Nucl. Phys., A-462 (1987) 45 ALBERICO, W.M. e t al., Nucl. Phys., A-462 (1987) 269
[41 MEZIANI, Z.E. e t al., Phys. Rev. Lett., 54 (1985) 1233
[51 ALBERICO, W.M., ERICSON, M., MOLINARI, A., Nucl. Phys., A-379 (1982) 429 161 CAREY, T.A. e t al., Phys. Rev. Lett., 53 (1984) 144
[71 MYERS, W.D., Nucl. Phys., A-145 (1970) 387
181 ALBERICO, W.M. e t al., Phys. Lett., 183-6 (197) 1 3 5
[91 ORLANDINI, G., TRAINI, M., ERICSON, M., Phys. Lett., 179-6 (1986) 201 [I01 CHANFRAY, G., OELORME, J., Phys. Lett., 129-6 (1983) 167
