EXCITATION OF STRETCHED AND NEARLY STRETCHED PARTICLE-HOLE STATES

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EXCITATION OF STRETCHED AND NEARLY STRETCHED PARTICLE-HOLE STATES

G. Emery

To cite this version:

G. Emery. EXCITATION OF STRETCHED AND NEARLY STRETCHED PARTICLE- HOLE STATES. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-389-C4-404.

�10.1051/jphyscol:1984430�. �jpa-00224096�

JOURNAL DE PHYSIQUE

Colloque C4, s u p p l r h e n t a u n03, Tome 45, m a r s 1984 page C4-389

EXCITATION OF STRETCHED AND NEARLY STRETCHED PARTICLE-HOLE STATES

G.T. Emery

Indiana University CycZotron F a c i l i t y and Physics Department, BZoomington, Indiana 47405, U.S.A.

~e/sume/ - Nous resumons sommairement une comparaison d g t a i l l g e e n t r e l ' e x c i t a t i o n ( p , p l ) d e s ' e t a t s de moment a n g u l a i r e 61ev6 d a n s 2 8 ~ i , pour Ep = 80-180 MeV, a v e c plusieurs c a l c u l s DWIA, du p o i n t de vue d e l ' e x t r a c t i o n d e s f o r c e s s p e c t r o s c o p i q u e s de c e s t r a n s i t i o n s . Quelques c a l c u l s & c e n t s de s t r u c t u r e n u c l c a i r e r e l i c s B c e s f o r c e s s o n t d i s c u t c s .

A b s t r a c t - A d e t a i l e d comparison of t h e ( p , p ' ) e x c i t a t i o n of h i g h - s p i n s t a t e s i n 2 8 S i , f o r Ep = 80 t o 180 MeV, w i t h DWIA c a l c u l a t i o n s i s

summarized, from t h e p o i n t of view of e x t r a c t i n g t h e s p e c t r o s c o p i c s t r e n g t h s o f t h e t r a n s i t i o n s . Some r e c e n t n u c l e a r s t r u c t u r e c a l c u l a t i o n s r e l a t i n g t o t h e s e s t r e n g t h s a r e d i s c u s s e d .

F i r s t I want t o r e v i e w w i t h you t h e q u e s t i o n of how w e l l c u r r e n t DWIA c a l c u l a t i o n s f i t t h e e x p e r i m e n t a l d a t a f o r i n e l a s t i c p r o t o n s c a t t e r i n g i n t h e e n e r g y r a n g e 80 t o

180 MeV, and t h e r e l a t e d q u e s t i o n of how r e l i a b l y t h e t r a n s i t i o n s t r e n g t h s c a n be e x t r a c t e d . Then I w i l l d i s c u s s whether o u r c u r r e n t u n d e r s t a n d i n g of n u c l e a r s t r u c t u r e i s a d e q u a t e f o r t h e d e s c r i p t i o n of t h e s t r e n g t h s t h a t seem t o be d e r i v e d from t h e d a t a .

Three high-spin s t a t e s i n 2 8 S i w i l l be d i s c u s s e d i n d e t a i l : t h e 6 - , T=l s t a t e a t 14.35 MeV, t h e 6-, T=O s t a t e a t 11.58 MeV, and t h e 5-, T=O s t a t e a t 9.70 MeV. The 6- s t a t e s have maximum a n g u l a r momentum f o r one-particle--one-hole s t a t e s a t 1 Mw e x c i t a t i o n , and a r e t h e r e f o r e c a l l e d " s t r e t c h e d " ; e x c i t a t i o n of t h e s e s t a t e s i n v o l v e s t r a n s f e r of one u n i t of i n t r i n s i c s p i n , a s w e l l a s a maximal f l i p of o r b i t a l a n g u l a r momentum. The 5- s t a t e i s " n e a r l y s t r e t c h e d " and i s l a r g e l y a n o n - s p i n - f l i p t r a n s i t i o n .

A t l a r g e momentum t r a n s f e r t h e s e s t r e t c h e d and n e a r l y s t r e t c h e d p a r t i c l e - h o l e s t a t e s a r e among t h e s t r o n g e s t i n t h e i n e l a s t i c s c a t t e r i n g spectrum. The s i l i c o n ( p , p l ) s p e c t r u m f o r a momentum t r a n s f e r of about 350 MeV/c i s shown i n Fig. 1. As a n o t h e r example, F i g . 2 shows 2 0 8 ~ b ( p , p ' ) a t a b o u t 480 MeV/c. The high-spin p a r t i c l e - h o l e s t a t e s have c r o s s s e c t i o n s comparable t o t h a t f o r t h e v e r y c o l l e c t i v e 3- s t a t e .

S i n c e we s h a l l be d i s c u s s i n g t h e

a p p r o p r i a t e n e s s of t h e DWIA and t h e O 14 1 3 12 1 1 9 i n g r e d i e n t s t h a t a r e c o n v e n t i o n a l l y

u s e d i n i t , i t i s o n l y f a i r t o make Excitation (MeV) a n e x p e r i m e n t a l c a v e a t . These

prominent p a r t i c l e - h o l e e x c i t a t i o n s Fig. 1. I n e l a s t i c s c a t t e r i n g s p e c t r u m f o r o c c u r a t f a i r l y h i g h e n e r g y , where 134-MeV P r o t o n s on s i l i c o n a t e l a b = 35".

t h e d e n s i t y of s t a t e s of lower a n g u l a r From / I / .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984430

C4-390 JOURNAL DE PHYSIQUE

Excitotion (MeV) momentum is large, and there is always a

8.0 7.0 6.0 5.0 4.0 3.0 background. Resolution is limited and the background is not flat, leading to some am- biguity in extracting the experimental cross sections for all but the strongest states. Furthermore, the particle-hole peaks are not prominent in all cases, leading to questions of fragmentation of the strength.

Comparison of results from scattering of different projectiles has been very helpful since different probes have different excitation matrix elements, the hadronic probes have different reaction-mechanism uncertainties, and the experimental difficulties are also somewhat different Channel for the different reactions.

A detailed investigation of how well the Fig. 2. Inelastic scattering DWIA does for the description of the exci- spectrum for 135-MeV protons on tation of these three high-spin states in 2 0 8 ~ b at elab = 50'. From /2/. 2 8 ~ i has been carried out by Prof.

Catherine Olmer at Indiana and her collaborators /I/. Cross sections and analyzing powers were measured at 80, 100, 135, and 180 MeV. Elastic scattering measurements were done at the same energies, and phenomenological optical models were constructed. It was found that, with good data covering a wide range of angles (we went out to at least 550 MeV/c), with the constraint of a smooth variation of the parameters with energy, and with the important further constraint that the derived total reaction cross sections varied smoothly with energy and were in approximate agreement with the experimental systematics, that the optical model parameters were then well determined.

For inelastic proton scattering we compare the data to calculations made using the distorted-wave impulse approximation (DWIA) / 3 , 4 / , and sometimes to variants and improvements of DWIA. In addition to the distorted waves there are two other crucial inputs to DWIA calculations: the structure of the states, as exemplified by the transition form factors, and the effective interaction. Transition form factors have been determined from (e,et) for the 5-, T=O /6/ and 6-, T=l /7/ transitions, and are consistent with L=5 harmonic oscillator densities, with oscillator parameters of 1.91 and 1.74 fm, respectively. For the 6- states we assume in the calculations that the ground state of 2 8 ~ i has a filled d5/2 shell, while for the 5- transition we adopt the open-shell RPA transition density of Yen 181.

There are now available several choices for the effective interaction t-matrix to use in DWIA. These are all derived from nucleon-nucleon scattering data, but with different procedures and different selections and weightings of the free scattering data. Results will be shown which use interactions due to Love and Franey (LF) / 9 / , a free-space interaction derived by Geramb and collaborators /lo/ from the Paris potential /11/, a density-dependent interaction, also from the Paris potential via Hamburg /12/, an energy-independent free interaction due to Picklesimer and Walker /13/, and another density-dependent interaction from Geramb, based on the

Hamada-Johnston potential /14/.

There is one correction to DWIA that is rather well established: the effective interaction is in principle density dependent, and the effects of this density dependence for a number of isoscalar low-spin natural-parity transitions have been shown to be important by Kelly and his collaborators /5/. Including the density dependence in a local-density approximation seems to give a definite improvement.

With that one caveat, one has had the impression that the DWIA works quite well for medium energy (p,pV), even for energies as low as 100 MeV. One would like to know if it is possible to calibrate the DWIA so that it can be used to determine the strengths of the nuclear excitations.

In Fig. 3 (top) we see a comparison with the data of the peak cross section predictions for the 5- transition in 2 8 Si, and we see that the way the direct and knock-on exchange amplitudes combine can be very different for the different interactions. For this excitation the predicted total results do not vary very much. For the 6- transitions the variations are somewhat larger: the 6-, T=l transition is shown in the lower part of Fig. 3. Even when the predicted peak cross sections are similar, results for other observables may differ more widely.

The differential cross sections for these high-spin states have a characteristic bell-shaped form quite different from the diffraction-like shapes seen for states of

LOVE ( T E L L 1 PARIS (FREEI PARIS (DD) PICKLESIMER- DATA WALKER

\:

.

1 4 . 0

, 2.0

;:.<.""'

'.. :\.. _.

DIRECT+ EXCHANGE

- - _ _ . _ - EXCHANGE

Ep (MeV)

I I I I I I I I I I I I I I I

28 Si (p, p ' ) 2 8 ~ i * 6; T = I 14.35 MeV

I

6 . 0 1 LOVE-FRANEY LOVE (TELL)

....

4.0

. ..;---

PARIS (FREEI PARIS (DO) PICKLESIMER-

--

. .

1

... . . . .

EXCHANGE

Ep (MeV)

Fig. 3. Energy dependence of the peak cross sections for (p,p') excitation of the 5-, T=O and 6-, T=l states in The separate contributions of direct and exchange processes are indicated for the DWIA calculations. For the data (extreme right) a smooth curve has been drawn through the experimental points for each transition.

C4-392 JOURNAL DE PHYSIQUE

lower spin. The data is shown in Figs. 4-6. The middle and bottom parts of these

1 1 1 1 1 I I l l , 1 1 , , , 1 , 1

~i ( ~ i ; p ~ " ~ i * 5 ; T = O 9.70 MeV LOVE - FRANEY

100 MeV 134 MeV 180 MeV

I I I I I / 1 1 , 1 , , , , , , , , 1

Si ( ~ p ' ) ' ~ S i * 5-, T = O 9 . 7 0 MeV PARIS (FREE)

100 MeV 134 MeV 180 MeV

Fig. 4. Cross sections for (p,pl) excitation of the 5-, T=O state of 28~i. The curves are from DWIA calculations using optical potentials appropriate for the various energies. Those for the Love-Franey interaction (top) and the free Paris interaction (middle) have been multiplied by 0.667, while those for the

density-dependent Paris interaction (bottom) have been multiplied by 0.625.

1.0

I I l I I I / I / I I I I I l I I

- 28 Si (ii;p/)28~i* 5; T = O 9.70 MeV PARIS (DO) _-

E p = 8 0 MeV 1 0 0 MeV 134 MeV 180 MeV

4

I I I I I l I , / l J l I I l I I I I l l

28 Si (E pr) " ~ i * 6-, T = 0 11.58 MeV LOVE - FRANEY

- -

ED= 8 0 MeV 100 MeV 134 MeV 180 MeV

.4

Figure 5. Cross sections for (p,pt) excitation of the 6 - , T=O state in 2 8 ~ i . The curves are from DWIA calculations using optical potentials appropriate for the various energies. Those from the Love-Franey interaction (top) have been multiplied by 0.15, those for the free Paris interaction (middle) by 0.10, and those for the density-dependent Paris interaction (bottom) by 0.114.

l I I I l I I I I ~ I I ~ 1 1 1 ~ ~

28 Si (K p') " ~ i * 6-, T = O 11.58 MeV PARIS (FREE)

- ^-

Ep = 8 0 MeV 100 MeV 134 MeV 180 MeV

4 2 -

I I I l I I l I l l I I I I I l I I l I

28 Si (E p p / ) " ~ i * 6-, T = O 11.58 MeV PARIS (OD)

- -

Ep = 8 0 MeV 100 MeV 134 MeV 180 MeV

-

b .02- .Ol- ,008 -

,006- ,004 -

.002 -

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I I / I 1 I I I , I I / 1 1 1 1 I

28 Si (E p 0 " ~ i * 6-, T = I 14.35 MeV LOVE - FRANEY

.4 - _{ED = 8 0 MeV 100 MeV 134 MeV 180 MeV}

4 -

Fig. 6. Cross sections for (p,pt) excitation of the 6-, T=l state in 2 8 ~ i . The curves are from DWIA calculations using optical potentials appropriate for the various energies. Those for the Love-Franey interaction (top) have been multiplied by 0.30, while those for the free Paris (middle) and density-dependent Paris interactions (bottom) have been multiplied by 0.50.

I I l I I I l l I I l l l , I I l I I

28 Si ( A P ' ) ' ~ s ~ * 6; T = I 14.35 MeV PARIS (FREE)

Ep = 8 0 MeV 100 MeV 134 MeV 180 MeV ^-

4

I I l I l I I I l l l I l I I I l ~ l ' l

28 Si ( ~ p ' ) ' ~ s i * 6; T = I 14.35 MeV PARIS (OD)

- E,=80 MeV 100 MeV 134 MeV 180 MeV

figures compare the data to calculations using the free and density-dependent Geramb-Paris interactions. For all three transitions the differences are small, which is consistent with the transition form factors being peaked in the surface of the nucleus. Density dependence is not an important effect for these cross

sections, though, as we shall see later, the analyzing powers are somewhat more sensitive.

The various forms of the effective interaction are not equally successful in fitting the shapes of the cross sections, however. The 5- data is compared with the

Love-Franey predictions in the top part of Fig. 4, and it can be seen that the predicted curve is too narrow. The Paris calculations (middle and bottom) fit the width of the distribution better, and the difference comes from a bigger second lobe of the central force. Even though the spin-orbit force gives the biggest

contribution to the peak cross section, we can learn something about one of the weaker components.

For the 6-, T=O state the oscillator parameter chosen may have been a little too large (since there is no (e,e7) transition density available, we chose it the same as for the T=l state). All the predicted distributions may be a little too narrow, but the principle deficiency is that the data show a shift with bombarding energy of the q-value of the peak cross section--it shifts to lower q as the energy gets smaller--that is not given by any of the calculations.

In the case of the 6-, T=l transition the q-value of the peak is given reasonably well by all the calculations, but there seem to be extra counts at low q at the two lowest energies. These could be from an unresolved state, but similar behavior was seen in the RCNP data at 65 MeV, even though the resolution was better 1151. The Love-Franey calculations fit rather well at 180 MeV, but the tensor contribution seems to deviate from the data at lower energies. The Paris calculations give too broad a distribution.

Our group has already noted /8/ that for a high-spin isoscalar natural-parity transition, like the 5- state in 2 8 ~ i , the analyzing power is already large in the plane-wave limit and crosses through zero where the central force term changes sign.

Distortion does not change this characteristic shape. Love and Franey /9/ stressed that this should be a general feature for states of this type. Fig. 7 shows the 5- data together with that for a very prominent peak at 3.5 MeV excitation in the 116~n(p,p7) spectrum (see van der Werf, et al. 1161). We think the tin peak is mainly due to a 9- state at 3.52 MeV 1171, but there may be contributions from nearby 10+ and 8+ states.

A comparison of the 5- analyzing power with calculations based on various

interactions is shown in Fig. 8. All the interactions reproduce this signature at the higher energies, and all deviate more from the data at the lower energies.

There is a definite difference at the lower energies between the free and

density-dependent Paris interactions, with the density-dependent fitting the data somewhat better. To isolate density dependence one should compare calculations based on the same family of interactions: for these transitions differences between predictions of forces from different families are often much greater than the effects of density dependence. The analyzing power is more sensitive than the cross section to the effects of density dependence because it requires (in the plane-wave limit) interference between two different force components.

For the 6- transitions the situation is more complicated, since in the plane-wave limit the analyzing power is predicted to be small, and distortion is important.

However, one seems to see in the analyzing power data for isovector stretched transitions also a characteristic signature. Fig. 9 shows data for the 6-, T=l state in 2 8 ~ i together with some data for the excitation of its analog in the (p,n) reaction, from the Kent State group /la/. The characteristic rise near 450 MeV/c is evident in the Love-Franey calculations (Fig. lo), but the calculated values do not rise enough to match the data. Calculations using the other interactions do not do as well, at least at the higher energies.

J O U R N A L DE PHYSIQUE

Fig. 7. Analyzing power in the (p,pr) excition of high-spin natural-parity states at 135 MeV. (Left) the 5-, T=O state in (Right) a line in l16sn at 3.52 MeV excitation, probably 9- (see /16/).

I I I

28 S i

- -

5-, 0

- 9.70 MeV

-

p ^4P0

- 0

11

0

- P

0

- ₁₁ --

0

- 0 -

0 n

- II --

I I I

Now let's return to the question of what are the strengths of these transitions.

Before looking at the silicon results in detail I would like to mention results for other high-spin transitions. Stretched transitions in several nuclei have been studied by (p,p') now, and while most have been examined at only one bombarding energy, and compared with calculations using only one effective interaction, the results usually require a normalization factor of between 0.3 and 0.5 (see, e.g., /19/). For example, Comfort et al. /20/ have located two close-lying 8- states in 4 8 ~ a and they are reported to sum up to 0.38 of the strength expected from a filled f7/2 neutron shell. Some of the O R b stretched states found in (p,n) by Anderson et al. / 2 1 / , are shown in Fig. 11. Their calculations require renormalization by factors of 3 ^{0.5. The 0} &I stretched states lie lower in the level schemes than 1 hw states and may thus be less fragmented. The 14- state in 2 0 8 ~ b was

earlier /2,19/ found to need a normalization factor of about 0.5 at 135 MeV, and recent work from Osaka at 65 MeV finds a similar result by comparison with a G-matrix force /22/. In this last work, by the way, two-step contributions were calculated and found not to be important.

I I I

116 Sn -

(9")

-- 3.52 MeV ^-

-- -

-- -

P

--

P

P -

P -

4

P -

$

I I 1

To return to the transitions in silicon, we show in Fig. 12 the normalization factors necessary to adjust the calculated peak cross sections to fit the data. For the 5- transition all the interactions (except perhaps Picklesimer-Walker at 180 MeV) give the same factor, approximately independent of bombarding energy. The value found, between 0.6 and 0.7, is smaller than was found with electron scattering / 6 / , where it was within 10% of unity, but that's not so very disturblng, since while the (e,ef) results determine the longitudinal transition density very well they only place crude bounds on the transverse density, which is relatively more important in (p,pf). It is perhaps somewhat more disturbing that 800-MeV results f o ~ this transition /23/ seem to need no renormalization, but at that bombarding energy again thelongitudinal density is more emphasized. The largest uncertainty in describing this transition may come from uncertainties in the ground-state

properties of 28Si.

I l l

-LOVE-FRANEY

- - _{PARIS DD}

PARIS FREE

-1

. . . .. . .

:

.

-

I'

\ * /

134 MeV *

-

Fig. 8. Momentum-transfer dependence of the analyzing powers for ( p , p ' ) excitation of the 5-, T=O state in 2 8 ~ i . The curves are from the DWIA calculations discussed in the text.

For the 6-, T=O transition the different interactions predict quite different peak cross sections, leading to quite different normalization factors, though none vary strongly with energy. The values found with the Love-Franey interaction are consistent with results from inelastic pion scattering / 2 4 / , but the (n,s') results have been analyzed with only one particular interaction.

In the bottom of the figure we see the results for the isovector 6- state. Again the differences are large, and in some cases the energy dependence is not correctly

JOURNAL DE PHYSIQUE

described. Again it is the

Love-Fra~ey results that agree best with values derived from (e,er) /7/

and (n,nr) /24/.

Ratios of the normalization factors for the two 6- transitions are shown in Fig. 13. The energy dependence is well described by the Love-Franey force, which overpredicts the isoscalar/isovector ratio by a factor of about 2. Both calculations using the Paris interaction overpredict this ratio by a factor of about 5.

We must conclude that the extraction

0.2 ^{- -} of normalization factors, or

spectroscopic strengths, from (p,pr)

A~ O excitation of high-spin states is

not yet unambiguous. The

- 0.2 ^- differences found here between

results obtained from the different

-0.4 ^- interactions are disturbing, and we

I I I I I shall probably have to use all the 200 400 tools we can find, including the

MOMENTUM TRANSFER- MeV/c study of states of various types, energy dependence, and the study of more complicated spin observables (which bring in interferences Fig. 9. Analyzing power for between different interaction isovector 6- transitions in 2 8 ~ i + ^p components) to find an effective at 135 MeV. The (p,n) data is from interaction that can be relied on.

the Kent State-MIT-Indiana group /18/.

In the meantime it may still be worthwhile to review some of the ideas about why these states have

the strengths they seem to. The 5- stren th is rather sensitive to the amount of d3/2 in the ground state, since the d3/2-?f 12 configuration is largely spin singlet, while the occupation-favored d5/2"f 2!7 configuration has only a modest spin-singlet component. A more precise determination of the transverse form factor in (e,e') would be a great help here.

The 6- transitions will have their strength reduced by the lack of full occupancy of the d5/2 orbital in the ground state; that alone will reduce the normalization factor to somewhere between about 0.50 and 0.75.

While we cannot be sure exactly what the ratio of normalizations is for the two 6- transitions, it is clear that the simplest models give them the same value, while both the (x,xr) results and the Love-Franey analysis of (p,pr) give isoscalar strength about one half the isovector (and the Paris force analyses of (p,pr) give an even smaller factor, 1/5 or 1/6).

It has been pointed out several times /25-27/ that in stripping experiments, starting from the 2 7 ~ 1 ground state, these two states are populated with approxi- mately equal strength, in agreement with simple expectations. It has sometimes been argued that this necessarily implies approximately equal strength in inelastic scattering /27/. That argument suffers from overly restrictive assumptions. If the wave functions of the states involved (ground states of 2 8 ~ i and z 7 ~ 1 , and the two

6 - states in silicon) are sums of more than one term, so that there are multiple

routes for the transitions, then there is no reason to believe that the spectroscopic strengths for stripping and inelastic scattering should be proportional.

1 I I I I I

100 MeV

0 . 6

.

28 s i ( ~ ~ ' ) ~ ' ~ i * 6; T = l

I I I I I I

8 0 MeV

0 . 6

+

0 . 8 0 . 6 0 . 4

Fig. 10. Analyzing powers for (p,pl) excitation of the 6 - , T=l state in 2 8 ~ i . The curves are from the DWIA calculations discussed in the text.

(p.n) SPECTRA AT 134 MeV

For example, Amusa and Lawson /28/ have done a calculation in which in the ground states particles can be in either d5/2 or s1/2, while for the 6- states one pdrticle is transferred to f7/2. They find a substantial fractionation of the strength for both channels. The stripping strength is not proportional to the inelastic scattering strength. They get a somewhat greater quenching for the inelastic strength to the strongest isoscalar state than for the isovector. The set of basis

Y

0 -5

Fig. 11. Excitation of the O& stretched states in the (p,n) reaction, from /21/.

Spin-parity values are 7+, 7+, 9+. and 13+,

0 respectively.

16 12 8 4 0

I l I I 1 I

134 MeV - LOVE-FRANEY

- - - _{PARIS DO -}

PARIS FREE

"

'...

WALKER -

+

* +

I I I I I I

180 MeV

- -

- -

- * + -

- 0 . 2 - - 0 . 4 -

EXCITATION ENERQY(MeV)

0.8- 0 . 6 - 0.4-

-

1

I I I I I I

- 0 . 2 - - 0 . 4 -

. .

-

a.

I I I I I I

-

c4-400 JOURNAL DE PHYSIQUE

RENORMALIZATION FACTORS

0.21 _{50 100} I ₁₅₀ I _{200 250} I

I

Ep (MeV) 0.9

0.8 0.7 0.6

Fig. 12. Energy dependence of Fig. 13. Energy depenaence of t h e r e l a t i v e n o r m a l i z a t i o n f a c t o r s r o r s t r e n g t h s of t h e 6- t r a n s i t i o n s i n 2 8 ~ i ,

( p , p ' ) e x c i t a t i o n of high-spin i s o s c a l a r / i s o v e c t o r . (Top) DWIA r e s u l t s s t a t e s i n 2 8 S i . Experimental and d a t a shown s e p a r a t e l y . (Bottom) r a t i o s peak c r o s s s e c t i o n s a r e of DWIA r e s u l t s t o experiment. Values from d i v i d e d by DWIA r e s u l t s f o r (n,n ') /24/ a r e shown on t h e r i g h t .

t h e i n d i c a t e d e f f e c t i v e i n t e r a c t i o n s .

PARIS DD 0 PARIS FREE x LOVE - FRANEY

A PICKLESIMER-WALKER

- C

5-, T = O - - '..., ,,.'

:&d&:g?

- '.A- x..

s t a t e s i n t h e i r c a l c u l a t i o n i s s t i l l n o t l a r g e enough t h a t one would f e e l c o m f o r t a b l e i n making d i r e c t comparisons w i t h t h e d a t a , b u t t h e r e s u l t s a r e c e r t a i n l y encouraging. As an example of t h e m u l t i p l e r o u t e s p o s s i b l e f o r t h e t r a n s i t i o n s , Fig. 14 shows a t r u n c a t e d s e t of t h e wave f u n c t i o n components i n v o l v e d .

Another approach, a l s o r e s t r i c t e d t o v a l e n c e - s h e l l e f f e c t s , i s suggested by a l l t h e e v i d e n c e t h a t 2 8 ~ i i s a deformed nucleus. Zamick /29/ has noted t h a t a J=6

c o n f i g u r a t i o n i n a deformed n u c l e u s i s s p l i t i n t o 7 components, w i t h K v a l u e s r u n n i n g from 0 t o 6 , and t h a t t h e i n e l a s t i c s c a t t e r i n g s t r e n g t h i s d i s t r i b u t e d , w i t h 2/13 of t h e t o t a l g o i n g t o each K component between 1 and 6, and 1/13 t o t h e K=O l e v e l . The s t r i p p i n g i s more c o m p l i c a t e d , s i n c e i t s t a r t s from 27Al., which has some d e f i n i t e K v a l u e o r mixture of K v a l u e s . T r a n s f e r of an f 7 / 2 p r o t o n cannot connect a n i n i t i a l K of 1 / 2 o r 3 / 2 t o K=6, f o r example, and i n g e n e r a l t h e p o ~ u l a t i o n of t h e

Fig. 14. A truncated sample of some of the wave function components in the calculation of Amusa and Lawson /28/, showing some of the multiple routes possible for the transitions.

final K components is unequally weighted and depends on the initial K value(s) and vector coupling coefficients.

Now, I had thought about this problem, and had early decided that Coriolis mixing must play an important role. For a reasonably large eccentricity the deformation splitting between K=O and K=6 is only about 2-3 MeV. On the other hand, the off-diagonal Coriolis matrix elements that admix adjacent K-values are all larger than about 3.5 MeV.

The calculation I have done has a limited basis, with the hole restricted to the d5/2 shell and the particle to f7/2. The effects of deformation and of the Coriolis force have been included. The only distinction between T=O and T=l is that a residual particle-hole interaction was added in the form of the modified surface delta interaction, which gives approximately the splitting between T=O and T=l 1301.

The results for inelastic scattering strength are shown in Fig. 15, and show that typically something like 1/2 of the total strength is collected into one state. In this simplified calculation the difference between isoscalar and isovector is not very large. The results for stripping strength are shown in Fig. 16. The description of 27Al is done in the same simplified way, but the results are in reasonable agreement with this of Dehnhard /31/. Again the T=O results do not differ very much from those for T=l for the strength of the principal transition.

One can see that there are differences in detail between the inelastic scattering and stripping strengths from Fig. 17.

This model is, of course, too simplified to be directly compared to the data in any realistic sense. There are the problems of what the deformations are, and how one handles the differences in deformation between the different states. The basis should be enlarged, bringing in real Nilsson states mixed in II and j. And a more

C4-402 JOURNAL DE PHYSIQUE

realistic residual interaction should he used, with off-diagonal matrix elements.

But I think the model has a certain amount of interest in its exploration of the effects of deformation; deformation is likely to play a role for those transitions, even if it may be better in the end to treat it via indirect (shell-model) means.

The effects discussed so far have all involved the valence shell. There may also be effects associated with 6- strength at 3 & and 5 &, and possibly in the delta region. A recent calculation of Blunden, Castel, and Toki /32/ considers such effects. They find, as was already indicated by the Julich group /33/, that delta-hole admixtures have a much smaller effect for these high-spin states than they do for Gamow-Teller and MI strength. On the other hand, they report that 3 4 h mixing produced by a Landau-Migdal force reduces the isoscalar 6- strength by about 25%, while it has a much smaller effect on the isovector strength. It is perhaps puzzling that the effect is attributed to a stronger repulsive force in the

isoscalar channel, but in fact the strong isoscalar state lies 2.8 MeV below the strong isovector strength.

DEFORMED 2 8 ~ i , 6; d 51%' f 7/2 To summarize, there are

20

I 2

-

(3 IL

y W Z

i

! x"

understanding of the reaction

E mechanism for (p,pl) and will,

I am confident, be able to find an appropriate effective interaction for the DWIA. If Fig. 15. Distribution of inelastic we do not yet know how to fix strength from the simplified deformed the strengths of these

model described in the text, for various elementary nuclear excitations

deformations. precisely from nucleon

inelastic scattering, and how to understand them, we have made considerable progress toward that goal.

-

' 5 -

-

o = -

10-

5 -

-

I

I

I 3 I I

3

T = I o T=O

0 -0.1

perhaps more difficulties than had earlier been thought in extracting believable spectroscopic factors from medium-energy proton inelastic scattering, but there is considerable evidence that the strength of the l a u

excitations in 2 8 ~ i is quenched, and that the

isoscalar 6- state is quenched more than the isovector. Part of the overall 6- quenching F

I

comes from the obvious fact that the d5/2 orbital is not fully occupied in the ground state. Differences in quenching between the

isoscalar and isovector states can reasonably be expected to come from valence-shell fragmentation, and perhaps from 3 Mo mixing, and these mechanisms need not affect the stripping spectroscopic factors in the same way. We are now beginning to have a much more comprehensive

I

-

I

-

I

-

= ~ z i

-0.4

I

3

-

.

a -

-0.5

-0.2 -0.3

DEFORMED "Si, 6; d 5/2' _f 712 S T R I P P l ~ ~ DEFORMED 28~i, 6; d 5/f1 f 712, E = -0.4 INELASTIC STRIPPING

-

-

20

STRENGTH

Fig. 17. Comparison of inelastic Fig. 16. Distribution of stripping strength and stripping strength in the from the simplified deformed model described simplified deformed model, for a in the text, for various deformations. deformation of -0.4

I would like to thank all my collaborators for their cooperation, and especially C.

Olmer and A.D. Bacher for intensive discussions. For their help in the deformed model calculations, I thank P. Bradley and L. van Ausdeln. This work was supported by the U.S. National Science Foundation.

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