High energy elastic and inelastic p-12C scattering and nuclear deformation

Nuclear Phyiica A271 (1976) 477-494 : © North-Holland Paülithle0 Go ., Muterdaar Not to be reproduced by photoprInt or miao8lm without wrlttm permletioa mom the publlaher

HIGH ENERGY ELASTIC AND INELASTIC p-' = C SCATTERING AND NUCLEAR DEFORMATION Y. ABGRALL, J . LABARSOUQUE and B . MORAND Laboratobe d< Physique 77eéorique, BordeaYx f Received 12 Docember 1975 (Revised 14 June 1976) Abstract : Starting from a microscopic deformed picture of the' =C nucleus, and using angular momentumn projected wave functions, the clastic and inelastic 1 GeV p-scattering differential sow sections have been analysed within the framework of the Cilauiber theory. From a comparison of the resnlta obtained in a tWl (dauber calailation and in the optical limit, it hsa ban shown that the elastic scattering and, to a lesser extent, the transition to the 2 + (4.44 MeV) laud ere oniy weakly affected by the long-range corrdations. In contrast the scattering to the 4 + (14.08 MeV) state inducts c~tcial multiatep contributions which affect the differential ct+oss section both in shape aced magnitude . Similarly the eorcections to the DWIA are essential for the transition to the 3 -(9.64 MeV) state and a satisfactory explanation of the qualitative differences observed betwan the 2+ and 3 - inda :tic cross sections in '=C is given . The great sensitivity of the crow sections to the unclear deformation is shown and a generally good agreement with experiment for both the dedron form factors and 1 GeV p ~ow sections has been obtained for the ground-state intraband transitions using a single oblate intrinsic state . The importance of a eorroct treatmeat of the rotadoaal motion through the aa~ular momentum projection is underlined aced the use ofthe adiabatic appuroxmation is critically examined.

I. Intr~odOCdon The great deal of interest in intermediate and high energy proton collisions with nuclei') is, to a large extent, due to the hope that such hadron-nucleus scattering could be much more fruitful than equivalent experiments at lower energies or than electron-nucleus collisions. Indeed the strong hadronic interaction permits the proton to undergo multiple scattering within the nuclear volume, thus allowing, at least in principle, the investigation of the correlations in the nucleus . The earlier 1 GeV proton scattering experiments of the Brookhaven group s) and the new measurements carried out at Leningrad s) and Saday 4) are now widely known . Their theoretical interpretations have been mostly confined to a variety of approximations based on the Glauber multiple scattering model') or the high energy optical potential 6). A useful review of the results of elastic and inelastic 1 GeV p-nucleus scattering can be found in ref. '). If we confine ourselves to the light nuclei, where ofprimary interest is the fact that f Equipe de Raherche associée au CNRS. Postal addrew : Laboratoire de Physique Théorique, Université de Hordeaus; I. Chemin du Solariums, 33170 Gradignan, Frana. 477

47B

Y. ABGRALL et al.

a good fit to the p-160 elastic cross section can be obtained'' using dynamically uncorrelated wave functions, while the p-12C scattering has resisted such a simple interpretation ' -1 ~. On the other hand it is well known that whereas 160 is predicted to be spherical, most of the Hartree-Fuck calculations using either density independent 11, 12) or density dependent 13) effective forces predict 12C to be an oblate deformed nucleus. Such a picture is consistent with the appearance of an (abbreviated) 0+-2+ -4+ rotational band and yields a satisfactory agreement with the observed enhanced B(E2) values as well as with the e-scattering data. Notice however that HF calculations using strongly density dependent forces like the SIII Skyrme force, give a shallow spherical minimum so that the question whether there is any definitive evidence that 12C is or is not deformed has been raised recently ta,1a), We shall come back to this point later. Several different kinds of model assumptions and approximations have bean done to estimate the effects of the deformation on the elastic p-scattering cross section. These include the use of a deformed well 1 s ' 16), an a-particle model 1' - 19~,) and a liquid drop model 2° -24). In all cases the collective motion is described in the adiabatic approximation 2s). These different nuclear models which all embody collective aspects give qualitatively similar results in better agreement with experiment and confirm especially the influence of the deformation on the ground-state single particle density. However the sensitivity of the elastic cross section to the long-range two-body oorrelations seems rather weak 1' ~~ . Inelastic 1 GeV p-scattering to discrete nuclear levels has not been as thoroughly investigated as elastic scattering and, apart from a few exceptions, all calculations assume a direct transition from the ground to excited state. In such a picture, inincluding the distorted-wave impulse approximation (DWIA) or its eikonal form (the so-called optical limit of the Glauber theory), only the one-body transition densities are nceded, apart from the NN scattering amplitude. Parametrized densities fitting electron scattering data give in 12C a reasonable agreement for the excitation of the 2+(4 .44 MeV) and the 0+(7.65 MeV) levels 1 " 26), at least for not too large momentum transfers. The same analysis gives however a very poor agrcement for the 3- (9.64 MeV) level. There, the maximum in the cross section is badly reproduced and there is no sign of the predicted ditiraction minimum, apart from a slight change in slope. The use of ono-body transition densities deduced from microscopic wave fimdions does not remove the discrepancy with experiment s, 27) It seems, and we get in this work further evidence, that with the 3 - level excitation in 12C we are confronted with a serious breakdown of the simple DWIA picture based on a direct transition to the final state. Corrections to the DWIA due to the dynamics) short-range pair oornlations have been estimated 29, 24, 26, 28, 29~l but ~ a nucleus like 12C which exhibits lowlying strong collective states, the long-range deformation is probably more importaat. The effect ofsuch collective motion in the excitation of the 2+ (4.44 MeV) by 1 GeV

p-`=C SCATTERING

479

protons has been approached from dif%rent ways using the macroscopic liquid drop model 2°-24), the a-particle model's and the deformed shell model in the limit of small deformations ' 6). It appears that the two-step processes contributions are negligible up to q2 x 3.5 fm -2 but become substantial above 2°,24). The 3 -(9.64 MeV) has also been treated as a liquid drop one-phonon vibrational state Io.23, 24) In such a description the coupling with other channels can shift the minimum prodided in DWIA but in no case removes it and the agreement with experiment still remains very poor . Recently the dispersive effects have been estimated in a model independent way by inserting into a coupled channel formali~ the form factors deduced from the electron scattering data 4s). In the present work our purpose is to investigate in a clean way the effects of the deformation on the elastic and inelastic 1 GeV p-i2C scattering within the framework of the Glauber multiple scattering theory. The main points and improvements over previous calculations are the following : (i) The full Glauber series which takes into account multi-step processes to all orders has been summed. A comparison with the optical limit (or DWIA) will permit one to appreciate the effeds of the multi-step contributions. (ü) Use is made of microscopic nuclear models sufficiently simple to allow the computation of the full Glauber amplitude but sophisticated enough to permit a good description of the properties of the low-lying states and a rather accurate evaluation of the long-range correlations. The deformed shell model and the microsoopic a-cluster model of Brink 3°) do the job perfectly. (iü) The rotational motion is treated within the Peierls-Yoeooz projection procedure 31) and not in the adiabatic (or strong coupling) approximation 2') as usually done. Indeed the importance of such a correct treatment of the rotational motion on the evaluation of spectroscopic properties and electron form fedora has already been emphasized 32-34) (iv) The present analysis includes not only the elastic scattering and the 2*(4.44 MeV) excitation but also the higher transitions to the 3 -(9.64 MeV) and 4*(14.08 MeV) levels for which the reaction mechanism involves, as shown later, crucial multi-step processes contributions. In sect . 2, we assemble some tools for a Glauber analysis. The nuclear models used in this work are introduced in sect . 3. In the same section theuse ofthe adiabatic approximation in the description of 1 GeV p-scattering is critically examined. The influence of both the multi-step processes and the nuclear deformation is then discussed and the prodidions of the theory are compared to the data (sect. 4). The concluding remarks and a review of the main results of our analysis are given in sect. 5.

Y. AHCiRALL et al.

480

2. The GMober theory of high energy scattering We shall make use of the Glauber multiple scattering formalism which has proven remarkably accurate, even in regions well outside its supposed angular range of validity . The scattering amplitude for transition of the target nucleus from the ground state ~0~ to a final state ~n~ is given by the well-known expression') () where etx,.ocb~

P:ô~(ri . . . r,,)~ {1- r(6-s1)}fit (2) >> J Here k is the momentum of the incident proton, q the four-momentum transfer and b the impact parameter of the scattered proton taken in a plane perpendicular to the beam direction ; s~ is the projection onto this plane of the position rJ of thejth target nucleon. The elementary profile function T is the Fourier transform of the pair amplitudef, -- =

_ d2~~ . r(~ = 2iak bi(a~. J The (average) nucleon-nucleon amplitude will be taken to be of the standard spin independent diffradive form') .Î(4) = 4 (1 - iP) ~P ( -~ß~4~~ with Q = 44.0 mb, p = -0.275 and ß2 = 5.45 (GeV/c)-2 . The computationof the full Glauber amplitude, eqs. (1) and (2), is a rather complex task which is not realized withopt approximations except in a few exceptions (including this work). In most of the calculations done so far only the first terms of the expansions of the phase function X" °(~ in powers of T are taken into account. Indeed assuming oqual p- and n-densities one can write') for the elastic scattering, which we shall consider first, Xoo(~ = iA[roo+~Îrôo+~rô0+ . . .]

where

-~iA(A-1) C(r lr2)I'(6-sl)l'(6-ss~rldr2[1+21'00+ . . .]+ . . ., J

roc

= J Poo(r)r(b- ~.

(5)

The (ground-state) one- andtwo-body density fondions are obtained by successive

P-' = C

SCATTERING

481

integrations Pö~rirz) _ ~ Poô~r~ " . . r,~dr3 . . . dr,

(7a)

Poo(ri) --_ Pôô~ri) _ ~ P~o~rirz~r

(~)

The two-body cornlation function in eq . (5) is defined as usual as C(rirz) = P~ô~rirz)-Poo(ri)Poo(rz)~ If the target nucleons are supposed to be completely independent, the multidimensional integral in eq . (2) factorises and one obtains the following expression, which also corresponds to the first line in eq. (5), Since T = O(A - }), a reasonable approximation of eq. (9), at least for not too light nuclei, is the so-called optical limit phase shift') (10) Xô ~~~ = iAI'oo. A dit%rent form of the profile function (2), written now in terms of transition densities, can also be obtained ifone assumes factorization of the N-particle densities, e.g. Pôo~ri " . . r,,) ~

~ Po~ri)P.(rz) . . . ppo(r,,), ~ .w . . .p

Pôo~rirz) ^ ~Po.(ri)Po(rz) = Poo(ri)Poo(rz)+ ~ Po.(ri)Po(rz~ 0

(llb)

which when substituted in eq. (2) yields after some simple algebra (assuming all particles identical and p,(r) = Poo(r))~ e~xoocb) _ (1(12) 1`oo~+~A(A -1 x1- roo~ -z ~ l'a,Po + . ., .fo

where the summation deals with a complete set of intermediate states and with

In fact there is a complete correspondence between the two expansions of the full Glauber amplitude, one using correlation functions, eq. (~, and the other using transition densities, «l. (12) [on this point see also ref. z~]. Indeed it is clear that the one-step process in which there is only the intermediate nuclear ground state between successive scattering, e.g. the first term in eq. (12), is equivalent to the assumption (9) of uneorrelated nucleons. Similarly the two-step prooeases oontribu-

482

Y . ABC3RALL et ol.

tion, e.g. the second term in .eq. (12), is easily shown, through eq . (8) and (llb), to be equivalent to the two-body correlation term in (5). Under the same assumption of factorization of the N-particle transition densities, e.g. P:ô~ri ~ . . r,t) ~

~ P~,(ri)P~rs) ~ ~.~ Ppo(r.t~ w.t. . .p

(14b)

P,~,ô~rirs) ^ ~ P,~(ri)P.~o(rsx one gets for inelastic scattering etz~o(b) _s =,!(1_roo~-lr o+~A(A -1x1- roo~

(14a)

~ Ir~~ .o+ . . ..

~to

(1~

Retaining only the first term in eq . (1~ leads to the sculled optical limit The oomsponding scattering amplitude F,o(q) is nothing but an eikonal form of the DWIA 1 " se) amplitude based on the assumption of a direct transition to the final state but including the distortion before and after the inelastic collision. We recall that the optical limit expressions for elastic and inelastic scattering, respectively eqa (10) and (16), depend on the model wave functions only through the (bare) onabody densities poor) and p o(r). It is worthwhile to stress that deviations between the full Glauber amplitude and its optical limit would reflect the effects of the nuclear oorrelations or equivalently of the muiti-step processes. 3. The nrdear model wave fonctions

The 0+(g .s.), 2+(4.44 Mew and 4+ (14.08 Mew levels in 1 ZC will be generated from the rotation of an intrinsic deformed state. If one assumes that this intrinsic state is a Slater determinant, the orbitals are then determined 'by means of the HF method and the rotational band is obtained by projecting out states of good angular momentum ~. Such a projected HF theory has beenwidely used with an incontestable success in the light nuclei as.3s). In a 4n nucleus like 1~C the LS coupling states of maximum [444] symmetry play a prominent role in the description of the low-lying levels and the single particle orbitals can be reasonably taken as the (n~~t:) eigenfunctions of a deformed harmonic oscillator with axial symmetry and characteristic lengths bx = b, = bl and b:. The ground-state band intrinsic state is then the following S = T = 0 spin saturated configuration ~4s) z (000~(10Ô~(Olb~, (1~ is e). which has already been considered in a similar context in refs. ". ~

p-' =C 3CATTERINC3

483

Later on the discussion will be easier in terms of the radius parameter bo = (bibs)} and the field deformation b as introduced by Nilsson 36) : (18)

In the limit b = 0 (spherical field) the wave function (1 ~ is nothing but the (zu) _ (04) SU(3) deformed intrinsic state and its J~ = 0+, 2 + , 4+ projected states the usual LS spherical shell-model states . Variational dynamical calculations have shown az) that we deviate markedly from the SU(3) scheme, with typical values of a ~ -o.s. The deformed intrinsic state I~) of course does not correspond to a true physical state. As mentioned above, one has to project out states of definite J to yield the rotational states a'). Explicitly SYJar(ri . . . r,~ = NJ~r 1 . . r,~. (19) ßnz J dQ~~o(ß)R(Q~ri ~ Here NJ~ is a normalization constant and R(1?) a rotation operator where D stands for three Euler angles. The profile function (2) then becomes A

f

(2J,+ lx?1r + 1) CJrMrI ~ D~IJeMr) = Nrrar~,r,er, (8nz)z ~~ai~~~ei,ô(a) X

A

<~rrl ~ OJI~O)r J=1

with O~ = 1-l'(b-s1) and Ian) = R(1?)I~) . The generating intrinsic state I~) being a Slater determinant, the computation of the A-body . operator mean values in eq . (20) is considerably reducod. Imdeod, due to the factorization into one-particle subspaces, one obtains if ua denotes the orbitals, <~rrl ~ O~~n) = det II<~(i)Io~l~(i))Ih

(21)

where R(Q) _ ~~ 1 R(ß) and un(/) = R~Q~a~l). Owing to axial symmetry one of the Euler angles can be eliminated in the HillWheeler integral (19). Consequently the computation of the Glauber scattering amplitude (1) from projected wave functions requires a five-dimensional integration over four Euler angles and the length b of the impact parameter (the angular integration over the direction of b in the impact parameter plane being easily per_ formed analytically). In order to get numerical results only approximate treatments of the projxtion has been done so far. One suitable for small deformation 16), the other for large deformations 1'). In the latter case one assumes that the overlap between I~ni and I~r:~> ~ negligible unless the direction of the symmetry axes are the same. So that

484

Y. ABC3RALL et al.

for a given operator T it becomes which of course greatly simplifies the evaluation of the profile function (20). In fact it is rather straightforward to show that such an approximation isequivalent to the usual adiabatic (or strong coupling) approximation where one assumes the factorization of collective and internal motion . The adiabatic approximation is known in t2C to be very good for the evaluation of the elastic electron scattering form factor but booomes more and more inaccurate for inelastic transitions to higher rotational J-states a3' "). In proton scattering things are almost identical. Indeed the 1 GeV elastic proton cross sections computed in the adiabatic approximation (22) are nearly indistinguishable from those deduced using projected wave functions and eq. (20). Similarly one sees in fig. 1 that in both calculations the shape of the inelastic 2+ and 4+ differential moss suctions are essentially the same . However in the adiabatic approximation the absolute values are underestimated by 30 ~ for the 2+ transition and by more than a factor of two for the 4+ excitation . From what preoodes it is clear that in a light deformod nuclei such as t~C, the correct projection procedure, although much more complex, is however necessary for the evaluation of the inelastic p-scattering cross sections and for a quantitative comparison with experiment.

to'

~ ocw

Fig. 1 . Comparbon between the inelatic scattering a~oa ~ectiom deduced in a fall C3lanber cilwlation wing the projected wave fandioni (fall line) or the adiabatic appraatimation (dashed line).

p-l'C SCATTERING

48S

A minor approximation done in the course of this work concerns the c.m . motion. Indeed the intrinsic state ~~~ has neither rotational nor translational invariance so that one has to project out states of definite total linear momentum as well as good angular momentum J. When b = 0 (spherical field) the c.m . motion is simply a is oscillator state and the c.m . correction reduces to the usual factor R(q) _ exp (-gZb~o l4A) in the amplitude F,o(q). This is no longer true for a deformed field. However our experience with electron scattering has shown that this form of c.m . correction gives form factors which differ by no more than a few percent from those deduced from a correct linear momentum projection procedure. A similar conclusion likely holds for p-scattering so that the c.m. correction factor R(q) will be usod throughout this work. 4. Resalta and dis~tirssion

We have calculated the differential cross sections for 1 GeV p-tsC scattering using the projected wave functions for different values of the field deformation, namely S = 0, -0.15, -0.40 and -0.80. For each deformation, the radius parameter bo is chosen so as to reproduce the experimental rms charge radius sy .3s), 2.46 fm, after the usual c.m . and proton finite size corrections . T~a~ 1 Multipole moments and deformation parameters Q, of the intrinsic states as a funtion of the field deformation b d

Qo (~_)

Hso (~')

~s

0 -0.13 -0.40 -0.80

-10.7 -13.4 -17.0 -21 .4

0 +6.1 +16.3 +33.2

-0.24 -0 .31 -0 .39 -0 .49

.)

-20.1

0 +0 .04 +0 .11 +0 .21

+21 .3

-0 .43

+0.12

I~4

The nas charge radius is kept oonataat and equal to its measured value, 2.46 fm . The best fit to the eand 1 GeV proton scxtta~ing data is obtained for b - -0.40. 9 The rosults of ref. as) are listed here.

The corresponding multipole moments and deformation parameters ßs and ß4 of the intrinsic state are given in table 1 . As usual Qo = (~)#~QZOi~

(23a)

H4o = ~Qaoi~

(23b)

_ 4~ ~Qtoi

486

Y . AH(3RALL et al.

with the following mean values P

4 .1 . INFLUENCE OF THE MULTI-STEP PROCESSES

Before a precise confrontation with experiment we would like first to discuss the important problem of the influence of the nuclear cornlations on the scattering cross sections. As said above this can be achieved from a comparison (done in figs. 2-s) of the cross sections deduced in a full Glauber calculation, eq. (2), with those computed in the optical limit, eqs. (10) and (1~, where the nuclear structure enters only through the ono-body densities. From fig. 2 it is clear that the elastic scattering as well as the low transfer behaviour of the 2+ excitation are only weakly sensitive to the oornlations.

l0'

20"

ecw

30"

Fig. 2 . Comparison between the 1 .04 GeV ~' sC di8'erential c .m. cross sediom deduced in a fill Gauber calculation aced in the optical limit . The deformed shell-imadd projected wave fundiona are used for a fidd deformation ô = -0.40 . Experimental data are taken from ref. `).

10' 20' 9 Cw 30' Fig . 3. Comparison between the 1 .04 C3eV p-isC indaatic scattering differentW c.m. cross sediona to the 4*(14.08 MeV) level deduced in a fill Gauber calculation and in the optical limit. The deformed shdl-modd projected wave functions are used for a fidd deformation a - -0.40. The full Gauber results in the SU(3) case (a = 0) are also shown . The teary Saday data's correspond to an angular resolution of 2 .4° lab aced an error of t 16 ~ dne to the absolute normalization moat be added. The point at 4.7° represents an upper limit.

p-'=C SCATTERING

487

In contrast their influence on the 2 + inelastic scattering becomes important for B°.m. > 15° and permits a significant improvement with experiment both in the magnitude of the dit%runtial cross suction and the position of the secondary minimum. Since the cross sections are known to be little influenced by the Pauli oornlations' ~ ~, most of the difference between the full Glauber calctilation and its optical limit comes from the long-range oornlations due to the deformation of the nucleus or, in other words, from multi-step processes through the rotational states summed to all orders in the amplitude (2). ïn that respect the excitation of the 4+ state is still much more interesting. As seen in fig. ~3, large discrepancies between the two calculations are now obvious both in the shape and the magnitude of the dif%rential cross suctions, especially in the forward direction and for scattering angles larger than 20°. The importance in the excitation of the 4+ state of cascades like 0+ (g.s.) -" 2+ (4.44 Mew ~ 4+(14.08 Mew is in fad readily understandable . Indeed the direct transition amplitude 0+ -. 4+ which is roughly proportiona1 23) to ß4 is rather weak in t ~C (sce table 1) and the multi-stop contributions are of the same order (at least in ßi) . This is still much more clear in the SU(3) case (8 = 0) where the direct transition to the 4+ state is now strictly forbidden since po4 as well as ß4

lo

lo-~

1.0

~ ~ (fa'' )3A

Fig. 4. Same as fig . 3 at TP = 80S MeV . The theoretical carves aro evaluated with a = 43 .2 mb, p = -0.40 and ßs = 4.S (GeV/c) - = . The pntiminn+~+ saclay data are taken from ref. ~ .

Fig. S. Comparison between the 1 .04 GeV ~'=C inebitic scattering differential c .m . crass sections to the 3 - (9 .64 MeV) level deduced in a full Glauber calwlation and in the optical limit. The model wave function is used in the adiabatic approximation for a clustering ratio d/bo = 1.1 (a oorrod projection procedure is expected to inaease the theoretical rosalts by mughly SO %a. Experimental data are taken from ref: `).

Y. AHCiRALL et al.

488

cancel . However, as acen is fig. 3, the cross suction which comes uniquely from the meld-step processes is still apprxiable. The long-range oorrelations are predicted to dominate the behaviour of the 4+ cross section in the forward direction and to be the origin of the bump at 22° while the direct transition contributes mainly to the peak near 12°. The preliminary Saclay ~~ 39 . ~ at 1 GeVand 805 MeV, shown in figs. 3 and 4, coverunfortunately an angular range too restricted to confirm (or contradict) our analysis t . The tzC(p, p~ data to the 3 -(9.64 MeV) at 1 GeV are available too and the measurements of the angular distribution have been extended to angles involving substantial values of the momentum transfer. We shall prove that while the simple optical limit .picture alone is inadequate, the inclusion of meld-step contributions permits a good fit to the data . Of course the 3- state is not a member of the ground-state positive parity band and in the context of p-scattering it has been so far destxibed as an octupole vibrational state. However another description, still more satisfactory, can be done within the microscopic Hrink a~luster model 3~. ïn such a picture the lowest intrinsic state in tzC is the usual equilateral triangle configuration which has neither angular momentum nor well defined parity, an extra richness of the a intrinsic structures. As a consequence, low-lying bands of both parity can be projected, among them the K = 0+ ground-state band and a K = 3 - band whose J = 3 - state is the first member . l:n fact it is well known that as far as the ground-state band is concerned, the deformed shell model of sect. 3 and the a-model strongly overlap 41). Calculations done in the adiabatic approximation (the only one managed for computational reasons in the a-model) confirm that the elastic and inelastic 2+ and 4+ 1 GeV proton cross sections are very similar in both descriptions . Consequently only the 3- excitation which is specific to the a-model will be discussed here . In the equilateral a-configtu~ation of tzC there are two parameters, the size parameter bo ofthe clusters-and their distances d from the c.m . The 3 - angular distribution shown in fig. 5 has barn calculated for a clustering ratio dlbo = 1.1, value which is known to be reasonable from a dynamical point of view and which furthermore gives the best overall fit to the elastic and inelastic electron scattering data 3~). As previously, the absolute value of bo and d are chosen so as to reproduce the experimental rms charge radius . The corresponding multipole deformation parameters of the intrinsic configuration are ßz =

where

-0.39,

ß3 =

+0.06,

ßa =

+0.08,

(26)

f we are indebted to the Saclay-Strasbourg collaboration and particularly to Dr. Aslanides for communicating to us their results at 1 GeV prior to publication .

p ."C SCATTERING

489

As usual the optical limit cross section exhibits a well pronounood diffraction pattern near 20° which is not seen experimentally. The inclusion of multi-step contributions present in the full Glauber integral, eq. (2), nicely corrects this weakness and explains the (puzzling) qualitative difference between the observed 2 + (4.44 MeV) and 3 - (9.64 MeV) inelastic differential cross sections. Notice that the a-model results are deduced in the adiabatic approximation. Now from sect. 3 we know that such a procedure underestimates the cross sections by an amount whiff is exported to be roughly 50 ~ for L = 3. As a consequence a correct projection procedure will bring the theoretical results of fig. 5 into very good agreement with experiment. It is interesting to compare the conclusions of our analysis on the influence of the dispersive effects with those of Viollier 4s) who approaches the same pmblem in a model independent way, using the electron form fedora in a coupled channel formalism which includes the first four levels of tZ C. Although nearly equivalent as far as the sleeting scattering is concerned, the results of both calculations differ for inelastic scattering . Indeed the dispersive effects in the excitation of the 2 + and 3 - states are predicted by Viollier `e) to be negligible, while we found them very importânt, especially, as shown above, for the 3 - state. The discrepancy comes from the coupling betwcen different excited states which is

10 4

p2 T

a

v0 10 5 10' 30' A CM 20' Fig . 6. The p-"C elastic scattering differential c .m . aoss sections at 1 .04 GeV . The theoretical predictions are deduced from a frill Glauber calculation using the deformed shell-model projected wave functions for different values of the field deformation a. The experimental data are taken from ref.') .

10' 20' 30' gcw Fig. 7 . The differential c.m. moss suctions for the excâtation of the 2* (4 .44 MeV) level in'=C by 1 .04 GeV protons . See caption to fig . 6.

Y . AHC~RALL et d.

490

indudod in our full Glauber calculation, but neglected in ref. ~ because there exist no experimental data for these transition form factors. This reveals that the virtual excitations between extxted states, such as 0*(g.s.) -~ 2*(4.44) ~ 3'(9.64), should be included in any reliable calculation. 4.2 . DEPENDENCE OF THE CROSS SECTION ON THE NUCLEAR DEFORMATION

Elastic and inelastic 1 (3eV proton scattering cross sations deduced from a full Glauber calculation using the deformed shell-model projected wave fimdions are shown in figs. 6-8 for different values of the field deformation b. The Coulomb interaction is included in the elastic scattering by following the approach of Glauber and Matthiae 4s) in which the nucleus is replaced by a spherit~l Merge distribution. Hence the effect of the deformation on the (weak) Coulomb part of the amplitude is in fact neglected. Since the rms radius is kept constant the forward elastic peak is almost independent of the deformation (fig . ~. As already shown in ref. t'), the position of the first (" )

lo~

E lotie

10-a

,:

l

2 + (4 .441W)

~~ 17s

(e)

4* (14.O611eV)

10'

~

eCM

Fig. 8. The differential e .m . cross sections for the excitation of the 4* (14.08 McV) .level in' = C by 1 .04 GeV protons . See caption to fig. 6. The crosfes correspond to the b ~ -0.40 theoretical results folded to simulate a 2.4° lab angular resolution. The preliminary Saday data are taken from ref: -3~.

t0

20

~t

(1)

W

y

3 - (A6411~)

30 , (fi~)lA

2D

IFI s

3A

Fig. 9. Elastic sect inelastic electron scattering form fedora in' 1C deduced from the projected wave fundiotu in the deformed shell model for b ~ -OAO (curves (a) . (b) and (c)) and in the amodel for d/bo - 1 .1 (cnve d). Corrections for c .m. motion and proton finite size are intro duced . The experimental data are taken from refs. "-as) .

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minimum as well as the height of the second maximum are in better agreement with experiment for an oblate non-spherical field. The best fit to the 12C elastic data proves to be of the same quality as that obtained') for' 60. The sensitivity of the inelastic cross suctions on the deformation is much more visible. From fig. 7 one sees that the inelastic 2+ data are rather well reproduced for d x -0.40, while d = 0 undoubtedly fails. For reasons given in subsed . 4.1, the agreement with experiment is better than in the simple DWIA picture using the electron density ' .'e) . There is still room for improvement near theminima. However a good agreement was not expected there since, as is well known, the Coulomb scattering (not included in our formulation for inelastic transitions) and the correotions to the Glauber amplitude ") are important near the diffraction minima. The 4+ inelastic results shown in fig. 8 drastically depend on 8. The simple 1s4 1 pa LS spherical description of the 4+ state obtained with b = 0 is definitively disqualified while an oblate deformation, slightly larger than b = -0.40, reproduces the right magnitude and the position of the observed peak both at 1 GeV and 805 MeV. Part of the apparent discrepancy in curvature between the predictions and the data is in fact spurious. Indeed the preliminary Saclay data at 1 GeV correspond to an angular resolution of 2.4° lab so that in order to compare comparable things one must also fold the theoretical cross sections. The corresponding results plotted in fig. 8 show a less pronounced minimum . From figs . 6-8 and fig. 4, it follows that a single intrinsic state with an oblate deformation 3 x -0.40 can satisfy simultaneously the observed intraband 0+, 2+ and 4+ high energy p-scattering cross sections. It is very pleasant for the consistency ofthe model to observe that such a value of the deformation precisely gives the best overall fit to the electron scattering form factors as shown in fig. 9. Notice however that a slightly better agreement of the elastic form factor alone is obtained for a spherical field [see also ref. 4')] . The multipole moments of our best fit intrinsic state reported in table 1 are mughly 20 ~ lower than those deduced by Nakada et al. ~) from a rather similar analysis of the electron form fedora . The lack of angular momentum pmjoction in the latter case explains in large part such a difference . S. Coododieg remarks sect some~ary

In the light of what precedes, we would like to come back to the question of the deformation of the 1~C nucleus. Indced the notion of a deformed intrinsic state is sutFciently remote from observables than an unambiguous demonstration of the evidence of deformation is not a trivial problem in the light nuclei where the oollaxive aspects are never so dear as in medium or heavy nuclei . One of the main arguments in favor of such a deformation in the 1~C nucleus is the existence of a rotation like 0+ , 2+, 4+ band and the appearance of a low-lying collodive 3- state. Such a spectrum, as wdl as the observed ß(E2) and the inelastic

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electron form factors are consistent with simple deformed or a~luster models . Furthermore, dynamical HF calculations or variational calculations within the a-cluster model, assuming [444].spatial permutational symmetry and neglecting the spin-orbit interaction, predict a sufficiently deep oblate energy minimum that the concept of a deformed shape seems meaningful. However, one knows that the spin-orbit interaction breaks the spatial symmetry, i.e. the supermultiplet, and admixes the states of lower permutational symmetry [4431], [4422], etc. into the highest symmetric state [444]. Competition is therefore expected between the effects of clustering which favor deformations and those of the spin-orbit interaction which favor a spherical shape 4~. Recently, axial symmetric HF calculations ' 4), including the spin-orbit interaction but without angular momentum projection, have shown that the deformation energy curve is extremely flat and, for most of the forces used, yields a slight minimum at spherical shape. On the other hand, the forces which favor a deformed shape predict rms radii in serious disagreement with experiment . A correct quantitative evaluation of the subtle competition between clustering (deformation) and the spin-orbit effects is however a rather complex task. Further, it drastically depends on the effective interaction. This problem has been already approached by Arima et al. within the hybrid mode1 49). Another possible way would be to consider the shape oscillations in the quadrupole coordinate as suggested by Friar and Negele i4) . To date, the complex structure of the 12C nucleus is not yet well understood and further investigations as well as a better knowledge of the effective forces in light nuclei are needed. Let us now discuss the information the medium energy p -~ 12C data seem to convey about the structure of 12C. Previous analyses of the 1 GeV data had already shown that a rather good description of the elastic and 2 +(4.44 MeV) inelastic scattering is obtained using the densities deduced from a deformed model. Notice however that the same data could be explained on other grounds, such as for instance the RPA vibration of a spherical core s'). In the present analysis, the transitions to the 3 - (9.64 MeV) and 4 +(14.08 MeV) states are considered, as well as the elastic and the 2+ inelastic scattering . An original and important conclusion we reach is that a nice fit for the whole inelastic differential cross sections - both in electron and proton scattering - can be obtained in a deformed scheme. In that respect the 3 - excitation which has been shown to be sensitive to the dispersive effects is rather instructive. Indeed, while the RPA description meets with some difficulty 2'), ~ the nice agreement with experiment in the a-cluster model implies that not only the direct transitions 0+ -. 2+ or 3 - , but also the transition 2+ -. 3 - are rather accurately described in such a picture. If the consistent description of inelastic scattering is an uncontestable success of the deformed scheme, on the other hand, the difficulty to get a simultaneous fit for both e- and p elastic cross section reveals some weaknesses of the deformed wave

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functions which could be related to the effects of the spin-orbit force. Indeed, as discussod above, such a force admixes in thenuclear wave function other components than the [444] states considered here . Due to the collective enhancement, the relation between the ground state and each of the low-lying 2+(4.44 MeV), 4+ (14.08 MeV) and 3'(9 .64 MeV) states is probably dominated by the collective component and rather accurately described in a deformed (or a-cluster) scheme . This could explain the good agnxment for inelastic scattering we obtained. The same argument however no longer holds for elastic scattering which oonsequeatly is more sensitive to other components that collective . Let us now briefly review the main points of our work. Starting from a deformedpicture ofthe 12C nucleus, the elastic and inelastic 1 GeV p-scattering cross sections have been analysed within the framework of the Glauber multiple scattering theory. From a comparison of the results obtained in a full Glauber calculation and in the optical limit, it has been shown that the elastic scattering and to a lesser extent the transition to the 2+(4.44 MeV) state are only weakly affected by the multi-step contributions. In contrast, the inelastic scattering to high-L states within a rotational band (L > 2) do not proeoed only by direct transitions (which are roughly proportional to ß;) but include also crucial multi-step contributions which affect the differential cross sections both in shape and magnitude. The pertinence of such an analysis is confirmed by the behaviour of the 3'(9 .64 MeV) excitation and, for the first time, a satisfactory explanation of the qualitative differences observed between the 2+ and 3' inelastic cross sections has been given. Indeed, in the latter case, the breakdown of the simple DWIA based on a direct transition is manifest and the multi-step contributions explain the lack of the diffraction minimum predicted so far and the slight change in slope seen experimentally. The great sensitivity of the 1 GeV p-' 2C cross sections on the nuclear deformation has been established and a single oblate intrinsic state with a Nilsson deformation S ~ - 0.40 can satisfy simultaneously the observed electron form factors and proton cross sections for the ground-state intraband transitions. We have also demonstrated that in a light deformed nuclei like 12 C, angular momentum projection is ncoeasary for a correct evaluation of the inelastic scattering cross sections. The use of the adiabatic approximation can vitiate noticeably the comparison with experiment especially for the transition to the 4+(14.08 MeV) level. The present analysis, which obviously can be extended to nuclei other than "C, shows that a careful investigation of the intermediato- and high-energy p-nucleus scattering, in particular the inelastic transitions to high-J states, may provide near information and teats of nuclear structure, espadally with regard to the long-range cornlations due to the deformation or clustering .

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