On the nuclear isovector spin response in the quasielastic peak region

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The spin-isospin response function in infinite nu clear matter is expected to show collective effects even at momentum transfers as high as q � 2 fm-1 .This prediction is based on nuclear matter calculations using the so-called ring approximation and a ph in teraction constructed from pion and rho meson ex changes together with a short range component [l].At moderate momenta this interaction [1,2] turns out to be attractive in the spin longitudinal channel and repulsive in the spin transverse channel and, thus, the corresponding responses or structure functions S L and S T should be enhanced and quenched respec tively, compared to their noninteracting values.
In nuclear matter calculations the finiteness of the nucleus and the surface effects are supposed to be ap-proximately taken into account through the introduc tion of an effective adjustable Fermi momentum kf!l' < kp.The validity of this approach is questionable for light nuclei and also if the response is explored through a strongly interacting probe, such as protons, since then it is the surface response which is involved rather than the volume one (as appears for in stance with electron or neutrino probes).
In order to eliminate this type of uncertainty we apply in the present work a simple semiclassical ap proach to account, in an accurate way, for the fini teness of the system.In a second step we generalise the model of ref.
[I] and investigate the influence of the nonlocality of the Hartree-F ock field on the response function.We analyse consistently (e, e') scattering and the Los Alamos (p, p') experiment [3] on the ratio of the spin longitudinal and trans- The 1 p-lh nuclear response function in ring ap proximation for spin longitudinal and spin transverse excitation operatorsaq exp(iqr) and a X q exp(iqr), respectively, for finite nuclei has been shown to have the following form to lowest order in ti [4]: .( 1) Here v L, T is the ph interaction for which we take the form given in refs.[1,2]; fl(O} is the free (non-inter acting) response in the local momentum approxima tion, i.e. in the nuclear matter expression k F is re placed by k F (R) = [2m(e F -V(r))/1i2J112; V(R) is the local mean field for which we use the parameteri satiOn of ref. [ 5] and R c is the classical turning point determined by e F = V(Rc)• with e F the Fermi energy.
In order to demonstrate the accuracy of the semi dassical approximation we calculate [4]  MeV, R0 = l.2Al/3 fm; a= 0.5 fm.Our result for s0(q, w) =-Im nC0>(q, w)/1T is shown in fig. 1 to- gether with an exact quantum mechanical continuum calculation [ 4] :h .Our approach gives practically the exact result for the structureless part of the nuclear response whereas for the low energy part our result still represents very well the average of the exact cal culation as expected from our numerous investigations on similar problem [6].As yet there does not exist an analogous comparison between semiclassical and exact results for the interacting response in the ring approx imation.However, as explained in ref. [4], the ap proximations used to arrive at (I) are exactly of the same nature as in the simple case (tr-+ 0 limit, local momentum approximation) and therefore we conjec ture that the same type of accuracy is obtained in the interacting as in the noninteracting cases (the same ar guments hold if a nonlocal mean field is introduced, see below).
In a first step we apply the model of ref.
[I] to finite nuclei using our serniclassical theory, that is eq. (1).As in ref.
[I] we take the bare nucleon mass and exactly the same force, i.e. g' = 0.7 for the Landau Migdal parameter.In figs.
2, 3a we show the results of this calculation for the transverse structure functions of ( e, e') scattering on 12c and 48Ca for the two mo mentum transfers, q = 2.02 fm-1 and q = 1.67 fm-1, respectively.We have added to the lp-l h response .0250. 0. 100.0 200.0 1\w(MeV) Fig. 2. Transverse nuclear response in ring approximation at q = 2.02 fm-1 for 12c in a local potential V(R) [5] with and without 2p-2h contributions.Also is given the nuclear mat ter result for k F = 1.15 fm-1 including the 2p-2h contribu tion.The data are taken from ref. (7].The relation between R F and S T is given in ref. (1 ].

I
. 10 >  the 2p-2h contribution through the ansatz given in ref. [ 1] which accounts for the filling of the dip.This procedure is justified by the absence of structure in the 2p-2h part which means that a nuclear matter approach and a semiclassical calculation most likely would show very little difference.We have also incor porated the effect of the �-resonance and thus used for the response function II (O) = II� ) + II � ) [1].In a first step we take a universal Landau-Migdal param eter g' and later we allow for a difference in the values gN N, gN .A , g� .A .In the latter case the expression for the response function reads (in a self evident notation): As previously stated the above calculations have been performed with the bare nucleon mass.A correct theory should include the renormalisation of the mass coming from the nonlocality of the mean field, giving a typical value m*/m � 0. 7.For high momentum transfers it is, however, more accurate to take into ac count the full nonlocality of the mean field.We achieve this by calculating the nonlocal average poten tial vH F (k F , p) in nuclear matter using the Gogny force [9] and pass again to finite nuclei employing the local momentum approximation.Although the Gogny force for the mean field may seem somewhat inconsistent with the one used for the ph residual in teraction [1 ] we think that this is not too much of a drawback since the Gogny force leads to m*/m = 0. 67 for k F = 1.35 fm-1 which is, as said above, in the range of values any reasonable force should give.The way to calculate the imaginary part of II (O) to lowest order in 1t in the presence of a nonlocal mean field has been shown by Rosenfelder [10].
The introduction of the mass renormalisation shifts the end point of Im II (O) towards higher energies.It also shifts its peak position to the right and brings the peak down (the integral over Im II ( (}) has to be con served).The nonlocality of the mean field therefore has a similar effect as the residual ph interaction: it quenches and hardens the response.When the 2p-2h contribution is added the pure Hartree-Fock response is still considerably higher than the experimental one 293 as shown in fig.3b indicating the presence of some collectivity.It is, however, clear that with the intro duction of the nonlocality of the mean field less col lectivity is needed in order to reproduce the experi ment, i.e. the Landau-Migdal parameters have to be lowered.An acceptable fit to the experiments is ob tained with g� A fixed to the classical Lorentz-Lorenz value currently used for 1T mesic atoms, g� A = 1 /3 [11].The values g!,m = gN A = 0.6 then yield the curve shown in fig.3b which is in reasonable agree ment with the experimental point of the transverse structure function of48ca at q = 1.67 fm-1.
We now turn to the longitudinal response in order to interpret the results of the Los Alamos experiment [3] on the ratio of the longitudinal and transverse spin responses probed through proton scattering.In this case the plane wave expression to the excitation operator is invalid since strong absorption prevents the protons to penetrate the nuclear.interior.The in clusion of this necessitates the use of optical model wave functions.For high energy protons, however, we can use the eikonal approximation leading to [12] with the optical potential U(r) = V(r) + i W(r), W(r) = -t "fl p( r ) (p/m)a(E) , (3)   where p(r) = 2ki(r)/31T2 and a is the total nucleon nucleon cross section.Supposing that the nucleon nucleon interaction is short ranged the excitation operator for the response function then reads: e i (p -p ' ) r-+ t/J p (r)l/;;.(r).
Treating the potentials to lowest order in 1i (i.e.local ly as a constant) we obtain (neglecting the change coming from the real part of U): where the absorption factor C(R) is given by (a similar expression has been given by Bertsch and Scholten [13]) 1.
X J dxexp (-a We here have assumed that the inelastic scattering is forward peaked and that the incoming and outgoing proton momenta can approximately be set equal.The function C(R) for 500 MeV protons on 2 0 8 P b is dis played in fig. 4 for a = 30 mb.Finally we show in fig. 5 the corresponding re sults for SL and ST at q = 1.75 fm-1 calculated ac cording to eq. ( 5) with S L , T = -7T-l Im nL , T• No absolute measurements of S L and S T have been reported in ref. [3], thus not allowing for an in dividual comparison.Carey et al. [3] compare the ratio of the longitudinal and transverse responses of not quite the case due to the tensor force: (SJS T)D > 1 and we have applied a small reduction of !:::!.7% [14] for our theoretical ratio in order to account for this effect.We also have incorporated the isoscalar contribution through the formula (3) of ref. [15].
For a fixed value of g ' the ratio is not very sensi tive to the introduction of the effective mass.How We also calculated the ratio of the longitudinal to transverse isovector spin response measured in a re cent (p, p') experiment including absorption effects.
The ratio is found to be about a factor of two too high at low excitation energies.

Fig. 1 .
Fig. 1 .Semiclassical and exact structure function at two momentum transfers (for details, see text).The exact results have been folded with a lorentzian of 6 MeV width.
the nuclear structure function for noninteracting nucleons in a Woods-Saxon potential with the following values of the parameters (in the usual notation): V0 = -50

Fig. 3 .
Fig. 3. Transverse nuclear response at q = 1.67 fm-1 for 4Sea in two different models (the relation between R T and S T is given in ref. [1] ): (a) For a local potential V(R) in ring ap proximation, with and without 2p-2h contributions.In addi tion the free response including 2p-2h contributions is shown (b) The same as (a) but with a nonlocal potential V(R, P).Experimental points are taken from ref. (8] U.

* 2
We are grateful to M. Bernheim for furnishing his data prior to publication.
ever, as pointed out above, with an effective mass the fit to inelastic electron scattering requires lower g ' values implying more collectivity in the longitudinal response.This leads in fact to a slight enhancement of the ratio SJS T as shown in fig.6.The experimental points lie below the theoretical curve, in particular those 30 and 50 MeV.The dis crepancy with experiment of our results in this work may be attributed to an incorrect treatment of the exchange terms of the ph interaction but also to the fact that for finite nuclei the surface longitudinal and transverse responses do not decouple.Similar results as in this work have been obtained by Bertsch and Esbensen[16], however, without inclusion of the nonlocality of the mean field within the framework of the slab geometry.In summary we have calculated in a serniclassical formalism the isovector spin responses for finite nu clei.We have found that with a local mean field and with the bare nucleon mass very satisfying fits to the ( e, e') data are obtained using by now standard ph interactions[ 1].The introduction of a nonlocal mean field necessitates a reduction of the g' -values leading to somewhat less collectivity.The resulting agreement with experiment is less satisfactory but still acceptable.The fact that the use of the bare mass allows for superior fits to experiments is intriguing.It could be that a correct treatment of the exchange term of the ph force partially cancels the exchange term of the mean field[17] and we intend to investi gate this point.