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F9II9I GIF SUR YVETTE CEDEX .

CEA-CONF _ . 9101

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CEA-DPh-N-S— 2468B

ELECTRON SCATTERING AND NUCLEAR SHAPES

Goutte, D. CEA CEN Sacloy, 91-Gif-sor-Yvette (France). Service de Physique Nucleoire Haute Energie

Communication présentée à : International conference onnnuclear shapes Aghio Pelagic (Greece) 28 Jun - 4 Jul 1987

ELECTRON SCATTERING AND NUCLEAR SHAPES

D. Goutte

DPh.N.HE

CBN. Saclay

91191 Gif-sur-Yvette

I INTRODUCTION

The large aaount of spectroscopic data now available all over

the periodic table has shown that nuaber of nuclei seeas to be

significantly deforaed. This interpretation is, most of the time, based

on the comparison of the data with a aodel and not on a direct

measurement. Moreover, most of these data are quantities averaged over

the spatial coordinates and contain no information any more on the

radial behavior of the nuclear wave functions.

On the opposit, as we will show in this contribution, the

electron scattering experiments provide detailled information on these

radial features through the precise measurement of the charge and

transition charge densities.

ÏÏ THE TOOL : ELECTRON SCATTERING

The principle of such experiments (figure 1) is very simple :

an incoming electron is scattered by a nucleus of a given target and

is detected in a spectrometer. In such an interaction, the exchange of

a virtual photon, present the unique advantage to be perfectly known.

It becomes so possible to probe fine details of the nucleus without

dependence of any ambiguous reaction aechanisme.

Sptctmtcr

vtactrwi

( T.t)

Nuctauc

Figure 1

S e h i i t t i c p r i n c i p l e of an e l e c t r o n s c a t t e r i n g e x p e r i m e n t .

To show how the measured quantities, the cross-sections, are related to the densities we want to extract, let us use the Plane Wave Born Approximation [1] and consider the simple case of the elastic scattering on an even-even nucleus (J11* 0*ground state) : the cross section is related to the charge form factor F* (q) by the simple relation:

dcr , 1 a * ^ a * o t t T — ' ̂ (q) i2

ree

where o"W o { t and fV t c are just kinematical factors (the Mott cross section and the recoil term)

This fora factor i s related to the charge density p(r) by:

FMq) * J^ P(r) j 0 (qr) r*dr

where j0(qr) is the zero order Bessel function. The charge density is simply obtained by reversing this

formula. In fact, a direct inversion is not possible because the data cannot be extended up to an infinit momentum transfer. However, for high enough momentum transfer, the procedures to extract the density are perfectly under control and the uncertainty due to the lack of data

can be precisely determined.

b

c o

• * *

u « M

in </> o

10 " Z 6 -

~ 1 0 " * -

10

- \

-26 \ 208

Pb fee) -

- » ~ -" -

-30 -

-32

— 8\ —

-34 - -

-36

1 1

1

Experiment

Theory V y

-38 1 i i i i i A

10- 3 0 -

10

10 " * -

10 " 3 6 -

0 1.0 2.0 3.0 4.0

Momentum transfer q ( fm*1)

P i l u r i 2

Ty p i c a l b e h a v i o r of cl a s t i c e l e c t r o n s c a t t e r i n g cross s e c t i o n s

These experim. .„s have been possible only recently because they

require very sophisticated equipment. High energy is needed in order to

have sufficiently small wave lenght. However cross sections are

decreasing very rapidly at high momentum transfer. Cross sections of

10" 3 8cm" 2.sr' 1 are typical for q * '•fm-1 (figure 2). So it is imperative

to have very high intensity electron beams (I > 10MA) and magnetic

spectrometers of large solid angle (dfl > 5<»sr).'In addition, for

inelastic electron scattering experiments, a severe requirement is an

energy resolution ÛE/E better than 10'" in order to isolate the

interesting states.

? I ï 10-'

-T 1 1 r 7 < G « { « » E-500M.V

Pifure 3

Procedure to extract • deneity

» 6 9

r (fm) 10

Actual experimental facilities have been designed to fulfill

these requirements and to permit a full exploitation of the power of

electron scattering. The size of the experimental equipments has

increased significantly with respect to standard nuclear physics. To

give an idea, the spectrometers used at Saclay for the experiment I

will describe here, together with its shielding, have a weight of 600

tons.

Figure 3 shows schematically the procedure followd to extract

the transition density. The measured spectrum (A) is analysed with the

help of an line-shape fitting méthode taking into account all the

radiative processes affecting the electron before, during and after the

scattering. The surface of the analysed peak is directly related to the

cross-sec*"on and becones one point in the form factor distribution (B). The whole form factor can then be transformed into a density (C) in a completely model independent way.

1 EXPERIMENTAL RESULTS

We will examine here two different regions of the periodic table. The first one i*» the samarium isotopes region where we get typical transitions from spherical to prolate nuclei. The second one is the germanium isotopes region which exhibit strong triaxial and very soft features.

A. THE SAMARIUM ISOTOPES.

24

20

S 16 X

> * 3L

î 8

4 -

- Sm J n=2f • is; - r \ WM 152

— \y E3 150 Vy • 148

, ,

, '

^ * ^ > i i i i 1 1 1 1 1 .

5 6 r Ifm)

10

?1 fur» 4

Exper imenta l t r a n s i t i o n charge d « n * i t l < «

for the f l n t J «2*in tht f l v « i m r l u t l i o t o p u

We have studied, at Saclay five samarium isotopes (»»*-»»»-i5o. l5i.i5»sa ) which are typical.of a vibrational to rotational transition when one goes from the spherical l 4*Sm to the well deformed 1 5*Sm. For all these nuclei, we have extracted the ground state charge densities and the transition charge densities for several excited states, but let us focus here only on the first excited 2*states in the five isotopes (figure k).

The general shape of these five densities is almost the same; all these excitations occure mostly at the surface of the nucleus which is the typical behavior of a collective state. By looking in more detail, one can observe that the transition radius increases with the number of neutrons much more than what would be expected only by the increasing of the nucleus size. This feature confirms that the deformation is increasing from the l 4*Sm to the l 5*Sm.

Another stricking feature of these data is the variation of the amplitude of the transition density. This amplitude reflects the collective nature of the transition and, as expected, this increases with the number of neutrons.

If we consider now the inner part of these densities, the variation between the l 4 4Sm and the other isotopes is quite important : this first nucleus exhibits a much simpler structure with no negative lobe in the inner part of the density. We can explain this effect since the l t t aSm which has a magic number of neutrons, only protons are active while in the other isotopes neutron can play a role as well. Such an effect could also be due simply to higher order terms in the transition operator. In the spherical 1**Sm the ground state is build only with s pairs while the 2* has s and d pairs. Thus the transition from ground state to the 2* consist only in s -» d transitions while in the heavier isotopes d pairs appears in the ground state and d -* d transitions have to be taken into account in the transition operator.

To clarify this point, an experiment on another N«82 neutrons nucleus, the l 4 o C e , is planned at Saclay.

B. THE GERMANIUM ISOTOPES.

The isotopes of germa- a exhibit properties which are at the

limit of our current theoretical understanding of transition regions.

For instance, the first excited 0* state decreases in energy as we move

froï 6 8 G e to 7 2Ge and then increases as we move to 7.6Ge. Such behavior

is not easy to interpret in traditional nuclear model and suggests

shape coexistence : two neutron transfer reactions confirm this

suspicion [2]. In order to clarify this point we have measured the

charge and transition charge densities of the two first 2* states in

7°-76Ge by inelastic electron scattering [3],

The extracted transition densities are shown in figure 5- One

can see that 2\ transition charge densities exhibit very similar shapes

for the four isotopes. For the second 2* state the shapes of the

densities corresponding to the heavier isotopes are again quite similar

while the one of 7 0Ge is completely different. In the next section, we

will see how such a strange behavior can be interpreted.

E x p e r i m e n t a l t r a n s i t i o n charge d e n s i t i e s for the two firat 2 i t i t e i

The excitation energies of the second 2* level as well as the

B(E2) values strongly suggest that these levels are of the same nature

for the whole isotopical chain. This experiment shows clearly that this

is wrong and that the second 2* state in 7 0 G e is of a different nature

than the others.

H THEORETICAL DESCRIPTIONS

In order to reproduce the properties of these low-lying excitations several theoretical approaches are possible. We will examine here a Microscopic one, treating the individual degrees of __,_ freedoa of the nucléons and a Macroscopic one, taking into account the collective degrees of freedom of the nucleus.

A Microscopic approach : HFB calculations.

As an example of such an approach, we will compare here our experimental data with the results of very sophisticated Hartree Pock Bogoliubov calculations performed by N.Girod and B.Grammaticos [4] using Gogny's Dl force. In this kind of approach the internal as well as the collective aspects of nuclear motion are described in a unified and consistent way. These calculations are completely microscopic and the only ingredient of the formalism is the effective nucleon-nucleon force.

In a first step, the potential energy surface E(&,"r), characterising the nucleus, is obtained by a constrained HFB method.

For the description of the nuclear dynamics, the complete resolutionof the Griffin-Hill-Wheeler equation is approximated by a Bohr-like Hamiltonian of the following form :

MC y2 2 BOO(0O|32) Q,\*2 BO2«3O0.) *, & • B 2 2(3 03 2) (5** 2. k G k < l W

where the collective potential V a s is deduced from the HFB energy E(0,"v) by substraction of the zero point energy. The other ingredient of the potential are the vibrational mass parameters B x ^(A.M^O.2) and the rotational moments of inertia Ix , IY and I z calculated in the cranking approximation, starting from the HFB quasi-particles wave functions. A numerical resolution of this Bohr Hamiltonian, using Kumar's code gives the collective wave functions of the system. The transition charge densities can then be obtained from averaging an

appropriate transition operator with the collective wave functions of the initial and final states. *

1) SatMriua isotopes.

l 5»s«

l* 2Sa

^"Sa

«8 Sm

144 Sa

Figura 6

P o t e n t i a l enarglei i.t t h r u and two d i a a n a l o n a and e o l l a e t i v * vivt

function» calculait by M.fiirod for th* five m i r l u i i a o t o p t a .

Figure 6 shows how such nuclei appears in this framework. One can see that the minimum of the potential energy surface V 0 7 is changing continuously from spherical prolate when one goes from l 4*Sm to 1 5*Sm. Figure 7 exhibit the experimental 2* transition charge densities compared to such aicroscopic calculations by N.Girod. The general trend of all these densities lâf well reproduced. However the amplitude of the transitions are in good agreement only for the cost deformed isotopes. For the more spherical ones, the theory predict too much strength at the surface.

i i ' ' * ' i t 0. 2. 4. S. 8. 10. 12

» (fm**)

T i f il r e 7

Experimenta l 2* t r a n s i t i o n d c m i t i c i c o i p i r t d to the HPB c i l c u l t t l m i ,

2) Germanium region. The same type of calculation as the one performed in the

samarium isotopes was performed here for the germanium case [5]. The figure 8 shows a quite gond agreement for the first 2" states. The general shape is well reproduced, however, the disagreement in the amplitude -is important and the gap between 7 0 - 7 2 G e on one hand and 7*.76 G e o n t h e o ther cannot be reproduced. That seems to be a general feature for the Bogoliubov calculations because in such approaches the number of particles is conserved only in average, which means that a

calculation of 7 2 G e for instance mixes some contribution from 6 eGe, 7°Ge ,7*Ge , etc... It is then obvious that some rapid changes between two adjacent isotopes will be washed out by the calculation. For the second 2* states (figure 8), again the three heavier isotopes are approximatively well described but the strange 2* in 7 0Ge is completely missed. We reach here, with suctr soft nuclei, the actual limit of these kind of microscopical calculations.

M- J >

i ' : . i; t .«m

I V _ / 3 7 Mlml

" G . ? ;

oos / i \\

i \\ on l ii

V\ 00)

-M

r i f u r i 8

F i r s t a n d n c o n d 2* n p i r i i t n t i l t r a n s i t i o n d c n t l t l t t c o a p a r c d

t o t h « • i c r o s e o p l e c i l c u l t t l o n i o f • . C i r o d t t « 1 .

B Macroscopic approach : Interacting Boson Model.

It is well established that the Interacting Boson Model of A.Arima. F.Iachello and I.Talni is able to reproduce a very large amount of spectroscopic data in several regions of deformed nuclei. However, the_model was not originally able to take into account any radial degrees of freedom. It was shown recently that such an extension was possible by the introduction of phenomenological boson structure functions [6](figure 9).

F i g u r e 9

Th« b o s o n s t r u c t u r e f u n c t i o n s i n t r o d u c e d i n t h e IBM.

The transition operator (for an E2 transition for instance) has to be written as :

T<">(r) s 2 e p fap(r) (s +d*d +s)< 2 ) •Xpfi(r) (d+ d)(2> 1 p=ir,i> l J

where ap(r) and 0p(r) are these phenomenological boson densities and e p

fixed effective charges. A transition charge density then becomes :

p(r)« S e p fap(r) <2*l(s +d*d +s) ( 2 ,IO* > • xp0(r) <2M(d +d) ( 2 >IO*>)

where (2*I(s+d*d fs)< 2> 10* > and < 2*I(d + d ) ( 2 '10* > are matrix elements fixed by the choise of the Hamiltonian. That leads to four (o^, 0 f f , ocv

and 0 V ) boson functions to be determined in the general IBM-2 framework and only two (a* a^o^and P* 3«*3V) i n a n IBM-1 approximation.

One can choose two or four experiment»'' transition densities to extract these boson functions and i t becomes then possible to predict any other density with no free parameter. I t i s also possible to adjust

these phenomenological functions on all the available experimental densities. The only difference in these two procedures is that, in the second case, the discrepancies will be averaged all over the data.

1} The samarium isotopes. Let us consider again the 2\ states in all the five isotopes

and compare the data to an interpretation in the IBM framework. If we assure that the boson structure functions can be kept constant for all the isotopes we can adjust the two functions Ot(r) and 0(r) by the five experimental transition densities.

i i i i i i t i i i 0 1 2 3 4 5 6 7 8 9 10 11

R Ifm) Ptfur» 10

Experiaental 2* transition l e n t i t l u for th« fivt i m r l u i isotopci coaparad to «n 1BH-1 adjustaent.

The result of such an adjustment is shown in figure 10. As one can see, the agreement is quite good with respect to the few number of parameters involved. Of course, the densities measured here present only regular variations and it seems to be easy for a phenomenological approach to reproduce such behavior. We will see with the germanium isotopes, that nature is not always so simple but that the IBM framework can still reproduce it.

2) The germanium region.

The region of validity of the model is confined to the medium to heavy mass nuclei because, in this mass region the relevant shell-model space involves many orbitals which give rise to large collectivity. In addition neutron-proton pair? should be unimportant in these nuclei since neutrons and protons are occupying different major shells with generally opposite parities. In the case of the germanium isotopes this assumption does not hold and the neutron-proton pairing may play an important role. We have shown [7] that it was possible to take this into account by adopting the configuration mixing technique introduced by Duval and Barrett [8] . We describe these isotopes as containing two separated configurations which are allowed to mix.

The nature of the second configuration was chosen to be a proton two-particle-two-hole excitation between the 2p J / 2 and lf 5 / 2

orbitals. So both configurations have the same number of bosons. We performed two IBM calculations using the standard IBM-2

Hamiltonian and mixed the resulting wave functions using the following Hamiltonian:

\ i x - * (sJ8;*sns+)<0>* 0 ( d X < 4 ) ( 0 '

where the prime is used to distinguish the "excited" proton-boson. The agreement with the experimental level schemes, the reduced

transition probabilities and the quadrupole moments is quite good. The strong variation of the excitation energy of the first excited 0* state is well reproduced as well as the results of (p.t) and (t.p) transfer reactions.

Here again we repaired that proton-bosons and neutron-bosons have identical radial behaviors. So we !<ad four boson fuctions to extract from the data, two for each configuration. Instead of adjusting these four functions on our eight experimental densities we choose four densities, we extract the boson functions and we calculate the four remaining transition charge densities. This procedure is not different, in principle, from the one used for the other regions but the result might be more striking in this case.

1 — '1 1 1

1

1

K" J 'V

1 1 _L 1

1 F " T

»e» J"

A-r •2?

V roc, j *

A •2r

A -

i i , i .

U 2 1 6 I V 0 2 1 6 S W

r ( f m )

P i f u r t 11

C o a p a r i a o n b e t w e e n t h e I BU p r e<J i c 11 on» «nd t h e d a t a . >

We choose the two first 2' states in 7 6 Ge and 7 2 G e to extract the four boson densities. With these boson densities and the matrix elements obtained by the IBM-2 calculations of ref.7, we calculate the transition charge densities for the four remaining 2* states in 7 0 G e and 7*Ge. All the calculated densities are in good agreement with the experimental ones including the second 2* state in 7 0Ge, whose strange behavior is very well reproduced (figure 11).

The hypothesis of two intrinsic states coexisting in these nuclei account for apparent pecular behavior of the first excited 0* level. In this picture che two first 0* states are both ground states from two different configurations. The energy splitting between these two configurations varies from one isotope to the next. The configurations are well separated in 6 t G e , completely mixed in 7 2 G e and again well separated in 7 6Ge; but they have exchanged their role in the

meanwhile and the ground state configuration fro» 6 8Ge i s the "excited" one in 7 6Ge and vice versa. The sudden change in shape between the 2*

E*(MeV) , I o + I o +

/ 2 2 / 2 1

10 +

_ i L ^ ^ A x 2 , + - |

I o + £ X P . — IBM —

6 8 G e 7 0 G e 7 2 G e 7 t G e 7 6 G e Figure 12

Trtniition charfe d t n i l t i n for the flr»t 2 * i t t t n in É , 8 " 7 6 C «

tipirliint (thick l i n e s ) and ISM pre d i c t i o n s (thin l i n e s ) .

states of 7 0Ge and 7 2 G e is no more surprising in terms of

configuration nixi"* : in 7°Ge both 2\ and 2* states belong to the

first configuration and the 2* is the first 2* state of the second one. In 7 2 G e the reduction in the splitting energy between the

configurations oakes that the first two 2* states is the result of the mixing between the first 2' state of each configuration while the 2* is now purely of the first one.

So, if one wants to connect the states between the two isotopes, the 2* in 7 0 G e has to be related to the 2* in 7 2Ge and not to the 2* . In fact, when we observed 2* transition densities in 7o-76 G 6 i ^ s e e a 2*state from the first configuration only in 7 0Ge and one can understand that its shape is not observed"" "In the other isotopes; there is no change in shape but just an inversion between 2* and 2* when one goes from 7 0Ge to 7 2Ge. The figure 12 summarizes this result.

CONCLUSIONS

Inelastic electron scattering was shown to be the tool to provide detailed information on the spatial behavior of collective excitations. The transition charge densities measured by this technique represent a very strong constraint for any theoretical approach trying to reproduce the collective behavior of nuclei. They transition charge densities are also, in certain cases such as the germanium isotopes, the only way to bring out strange features hidden in the corresponding integral properties of the nucleus.

Among all the possible theoretical treatments of collective *

degrees of freedom in deformed nuclei we have shown that the microscopic approach of M.Girod et al. using as only input the Gogny's nucleon-nucleon interaction is able to reproduce reasonably well the measured transition charge densities in most cases. However, the actual limit of these calculations seems to be reached in the very soft germanium isotopes.

We developped also a phénoménologies! approach based on the Interacting Boson Model framework. This was shown to be able to predict very well the measured radial features. In addition, it allows a clean and easy-to-understand classification of the complicated and seemingly erratic behavior of the germanium transition charge densities. In that, it provides us with a valuable interpretative tool for the complicated nuclei of transition regions.

Notes and References

[1] Plane Wave Born Approximation (PWBA) : one consider that the electron is exchanging only one virtual photon with the nucleus and that the incoming as well as the outgoing electron is represented by a plane wave. This wave is in fact distorded by the field of the nucleus and this distorsion is taken into acount in the extraction of a density. [2] M. Vergnes et al.. Phys. Lett. 72B(1978)447 [3] J-P. Bazantay et al., Phys. Rev. Lett. 54(1985)643 [4] M. Girod and B. Grammaticos, Nucl. Phys. A3jO(1979)40. [5] H. Girod, D. Gogny et B. Grammaticos,

6**eet 7*" Sessions d'Etude d'Aussois 1981 et 1983. [6] F. Iachello, Nucl. Phys. A358(1981)89 [7] P. Duval D. Goutte and M. Vergnes, Phys. Lett B124(1983)297 [8] P. Duval and B.R. Barrett, Phys. Lett. 100B{1980)223