This work was supported by the National Science Foundation Grant NSF PHY 81-14339 and by Indiana Universit

Bacher, A.D.

Drake, T.E.

Emery, G.T.

Jones, W.P.

Miller, D.W.

Olmer, C.

Schwandt, P.

Sobie, R.J.

Yen, S.

English

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ENERGY DEPENDENCE OF INELASTIC PROTON SCATTERING TO ONE-PARTICLE ONE-HOLE STATES IN 28~i 
C. Olmer, A.D. Bacher, G.T. Emery, W.P. Jones, D.W. Miller, and P. Schwandt 
Indiana University Cyclotron Facility, Bloomington, Indiana 47405 
S. Yen, T.E. Drake, and R.J. Sobie 
University of Toronto, Toronto, Ontario, Canada 
The study of iritermediate-energy proton inelastic studies since: (1) the states usually appear as 
scattering to one-particle one-hole states has strong, isolated peaks in the spectrum, (2) 28~i is a 
attracted considerable attention over the last few self-conjugate nucleus so that differences between 
years. One primary purpose of these studies has been proton and neutron transition densities are not 
to exploit the structural simplicity of these excited expected to be a problem, and (3) all three states have 
states as a possible means of spearating effects due to also been studied by pion inelastic scattering1 and 
nuclear structure and those due to the reaction the 5', T=O and 6-, T-1 states>have been studied by 
mechanism. The present work has concentrated on an electron inelastic scattering2, so that this is one of 
examination of the energy dependence of the proton the few cases for which p-n-e complementary information 
inelastic excitation of three high-spin, particle-hole exists. 
states in 28~i: the 5-, T=O state at 9.70 MeV, the 6-, The distorted-wave impulse approximation (DWIA) 
T=O state at 11.58 MeV and the 6-, T=1 state at 14.35 description of inelastic scattering to these states 
MeV. A region of the inelastic scattering spectrum incorporates information concerning the nuclear wave 
measured at an incident energy of 180 MeV is displayed functions of the states, the effective N-N interaction 
in Fig. 1. These high-spin states in 28~i have been and p-2 8 ~ i  elas tic scattering distorted waves. The 
particularly useful in proton inelastic scattering elastic scattering data measured at the various 
energies are displayed in Fig. 2. It is extremely 
180 MeV 
30° 
I ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~  
5- 
I Si (p, p') 
a and the analyzing power A for the elastic scattering 
over a wide range of both proton energy Ep and momentum 
important in a study of the energy dependence of 
inelastic scattering to measure both the cross section 
1800 transfer q. In the present study, a standard 
cn 
t 
C 
a Woods-Saxon optical-model description of the elastic 
0 
1200 mattering was employed, and the resulting energy 
dependence of the deduced optical model parameters was 
600 investigated. By requiring a smooth dependence on Ep 
for these parameters (including the total reaction 
n cross section), a significant reduction in the 
" 
14 13 12 I I 10 9 
Excitation (MeV) ambiguity of parameters was possible. The use of a 
non-standard optical potential (Woods-Saxon plus a 
squared Woods-Saxon term), which was found3 to be 
Figure 1. Inelastic proton spectrum for the scattering 
of 180 MeV protons from silicon. important for reproducing proton elastic scattering 
p + Si ELASTIC SCATTERING 
+ 100 
MeV 
J 1  , , , 80 , , .2 0 MeV 
0 200 400 600 800 
Figure 2. Momentum transfer dependence 
of the cross sections and analyzing 
powers for proton elastic scattering by 
silicon. The curves are the results of 
simultaneous optical-model fits to the 
cross section and analyzing power data at 
each energy. 
from light nuclei, did not significantly affect the described by an RPA wave f~nction,~ which has a 
quality of the optical model fit at 180 MeV. This dominant f7/2d3/2-l term. 
potential resulted in inelastic scattering Several different effective interactions have been 
analyzing-power predictions of similar shape and considered in this study. These include: 
magnitude, and cross-section predictions of similar (1) the Love-Franey interaction6 (LF), which has 
shape and slightly different magnitude as compared to been separately derived at 100, 140 and 185 
the standard optical model treatment. A detailed MeV, 
description of the elastic scattering analysis will be (2) The Geramb-Bauhof f interaction7 (GB), which is 
presented elsewhere. a density-dependent interaction based on the 
The wave functions of the three states have been Paris N-N interaction and is available at 
assumed to be of harmonic oscillator form, and the isolated energies (also the ~amada-~ohnston~ 
oscillator parameters derived from (e ,e ' ) studies2 of interaction (HJ), which is available for 
the 5-, T=O and 6', T=l transitions have been used central and spin-orbit components only), and 
(b11.91 and bz1.74, respectively). The 6', T=O state (3) the Picklesimer-Walker interaction9 (PW), 
is assumed to have the same oscillator parameter as the which explicitly incorporates both energy and 
6', T=1 state. Both 6- states are characterized by the momentum dependence. 
f7l2dgI2-l configuration, whereas the 5-, T=O state is The LWIA calculations employing the LF and PW 
2 3 
in te rac t ions  used the code ~ ~ 8 1 ~ ~  and those employing 
the  GB i n t e r a c t i o n  used the code DWBA80.11 A l l  
ca lcu la t ions  t rea ted  the exchange term i n  an exact 
manner. 
Several d i f f e r e n t  types of comparisons of the 
model predict ions with the data  have been examined, 
including: (1)  the dependence of the maximum cross  
sec t ion  %x on Ep, (2)  the  dependence of the 
d i f f e r e n t i a l  cross  sec t ion  a on Ep and q ,  and (3)  the 
dependence of the analyzing power A on Ep and q. A 
complete discussion of the r e s u l t s  of these comparisons 
i s  presented elsewhere. l 
The simplest possible  comparison of the MJIA 
predict ions with the data  involves the energy 
dependence of amax, s ince i t  ignores the dependence on 
the  momentum t rans fe r .  The experimental values of hx 
f o r  the 5', T=0 and 6-, T=O t r a n s i t i o n s  a r e  displayed 
i n  Fig. 3, together with predict ions based on the 
various in te rac t ions .  (An add i t iona l  s e t  of 
ca lcu la t ions  is displayed, using an e a r l i e r  form13 of 
t h e  Love in te rac t ion  derived a t  135 MeV. Also, note 
t h a t  the LF r e s u l t s  a r e  displayed only f o r  energies  of 
100 MeV and above). The individual  contr ibut ions of 
d i r e c t  and exchange a r e  displayed, and these vary 
dramatical ly  f o r  the various in te rac t ions .  These terms 
i n t e r f e r e  construct ively f o r  some in te rac t ions  and 
des t ruc t ive ly  f o r  others.  For some in te rac t ions ,  very 
d i f f e r e n t  d i r e c t  and exchange terms combine t o  produce 
very s imi la r  predict ions.  Experimentally, the energy 
dependences of %x f o r  these two t r a n s i t i o n s  a re  very 
s imi la r .  Only the LF ca lcu la t ions  seem t o  reproduce 
t h i s  observation, whereas the GB and PW ca lcu la t ions  
p red ic t  s i g n i f i c a n t l y  d i f f e r e n t  dependences on proton 
energy. Clear ly  an ana lys i s  over a broader range of 
energy is necessary i n  order t o  ve r i fy  these 
conclusions. 
I 1  I I  I I I  1 1 1  1 1 1 1  1 
2%i (p, p8) 28Si * 5; T = 0 9.70 MeV 
L O M  -FRAMEY LOVE (TELL1 GERAYB- PICKLESIMER- DATA 
BWnQF WALKER 
8.0 ' t :  . 
.;. 
6.0 
- DIRECT + EXCHANGE 
. . .. . . DIRECT 
- - - - -  EXCHANGE 
0.1 LH) 100 Is0 200 100 150 200 I00 I50 200 100 150 200 100 1% 200 
Ep (MeV) 
si (p, p') '%i* 6; T=0 11.577 MeV 
10.0 
. \ 
. . 
Figure 3. Energy dependence of the maximum cross  
sec t ions  f o r  proton i n e l a s t i c  e x c i t a t i o n  of the 5-, T=O 
and 6-, T=0 s t a t e s  i n  28~i. For the da ta ,  a smooth 
curve has been drawn through the four experimental 
measurements f o r  each t rans i t ion .  
A f u r t h e r  comparison can be made by examining the 
renormalization fac to rs ,  shown i n  Fig. 4, which a r e  
needed i n  order f o r  these t h e o r e t i c a l  values of a,, t o  
reproduce the data. For the 5', T=0 t r a n s i t i o n ,  all 
renormalization f a c t o r s  f o r  a l l  the  indicated 
in te rac t ions  have a s imi la r  value and a r e l a t i v e l y  f l a t  
energy dependence is ,seen f o r  these f a c t o r s  (with the 
exception of the PW i n t e r a c t i o n  a t  high energies).  A s  
a r e s u l t  of these observations, i t  may be concluded 
t h a t  both the LF' and GB in te rac t ions  reproduce 
reasonably well the energy dependence of amax f o r  the 
I I I I 
28 28 .* Si (p,pO) SI 
RENORMALIZATION FACTORS 
LOVE - FRANEY 
0 PICKLESIMER -WALKER 
x GERAMB -BAUHOFF 
Ep (MeV) 
Figure 4. Energy dependence of the renormalization 
factors for proton inelastic excitation of the 5', T=O, 
6', T=O and 6', T=1 states in **~i. 
5' transition. For the 6', T-1 transition, only the LF 
interaction appears to require the same renormalization 
factor over the indicated range of proton energies and, 
moreover, the value of this factor is remarkably close 
to that obtained1,* in (e,eV) and (x,xl) studies. 
Calculations with all interactions indicate that this 
transition is dominated by the tensor component of the 
increase in strength of the tensor component in this 
energy region (100-200 MeV) is reproduced reasonably 
well by the LF interaction, but that the energy 
dependence of the tensor component in both the GB and 
PW interactions does not appear to be quite correct for 
this transition. 
The energy and momentum-transfer dependences of a 
and A for the three transitions have been examined for 
the various interactions. For example, the 
calculations for the 5' transitions using the LF and GB 
interactions are displayed, together with the data, in 
Fig. 5 and 6. The calculations have been renormalized 
by the factors indicated in the appropriate figure 
captions. (Calculations at 80 MeV for the LF 
interaction employed the 100 MeV force). The 
decomposition of the predicted cross section into 
central (C), spin-orbit (LS) and tensor (T) 
contributions is indicated for each of the 
interactions. Calculations using the LF interaction, 
shown in Fig. 5, appear to reproduce the energy 
dependence of %, and qmax (the location of %ax) 
reasonably well, but consistently fail to reproduce the 
width of the distributions. The predicted 
distributions are narrower than experimentally 
observed, and moreover, the calculations do not exhibit 
the observed increase in width of the distributions at 
the lower incident energies. Calculations for the 5'
transition using the GB interaction, shown in Fig. 6, 
also appear to reproduce the energy dependence of %, 
and q,,, and here the calculations reproduce the 
experimental width of the distributions significantly 
better than was observed for the LF interaction. 
In particular, the shape of the distribution at 135 MeV 
is not exhibited in the LF calculation, but is 
well-reproduced in the GB calculation, where it 
interaction. Thus it may be concluded that the results from the large, second maximum of the central 
I I 1 I I 28 I 1 1 28 .* I 1 I I I I 1 I Si (p,p') SI 5- ,  T=O DWIA;  LOVE - FRANEY 
9.702 MeV 
E,=80 MeV 100 MeV 135 MeV 180 MeV 
Figure 5. Momentum transfer dependence of the cross sections for proton inelastic excitation of the 5', T=O state 
at 9.70 MeV in 28~i. The curves are DWIA calculations (multiplied by 0.67) using the Love-Franey effective 
interaction and optical potentials appropriate for the various energies. 
Figure 6. Momentum transfer dependence of the cross sections for proton inelastic excitation of the 5-, T=O state 
at 9.70 MeV in 28~i. The curves are IWIA calculations (multiplied by 0.67) using the Geramb-Bauhoff effective 
interaction and optical potentials appropriate for the various energies. 
1.0 
.8 
I 1 I 1 I I I 1 I I I 1 I 
- 28 I I I I 28 .* Si (p, p') SI 5-, T = O  DWIA; GERAMB -BAUHOFF 
- 
- 9.702 MeV - 
component (very weak in the LF calculation) 
contribution to the large momentum transfer region. 
This broadening of the distributions also results from 
calculations using the HJ interaction. Both the GB and 
HJ interactions are density dependent, and both exhibit 
a double-humped central contribution for proton 
energies between 100 and 180 MeV. Thus it appears that 
density dependence is important for a complete 
understanding of this transition. 
The analyzing-power distributions for the 5-, T=O 
transition are shown in Fig. 7, together with 
predictions for the four interactions. The most 
positive value of A is observed to decrease from -0.8 
at 180 MeV to near zero at 80 MeV, and this general 
trend is apparent in calculations for all the 
interactions. Since A is sensitive to the interference 
of amplitudes, a comparison of the predicted and 
observed analyzing powers provides an independent test 
- - O E A * U ) - W F  
. . , . PICKLESIMR- 
WALKER 
Figure 7. Momentum transfer 
dependence of the analyzing powers for 
proton inelastic excitation of the 5', 
T=O state at 9.70 MeV in **~i. 
of the  i n t e r a c t i o n ,  s epa ra t e  from t h a t  furnished by the  
c r o s s  sec t ion .  Here, both the  GB and LF c a l c u l a t i o n s  
a r e  i n  reasonable  agreement wi th  the  d a t a ,  a l though t h e  
GB i n t e r a c t i o n  appears  t o  be s l i g h t l y  preferred.  
The d e t a i l e d  comparisons of p red ic t ions  based on 
va r ious  a v a i l a b l e  e f f e c t i v e  i n t e r a c t i o n s  wi th  i n e l a s t i c  
s c a t t e r i n g  d a t a  over a broad range of inc iden t  energy 
has only r e c e n t l y  become poss ible .  The importance of 
such comparisons has become c l e a r  from the  present  
s tudy  which has revealed informat ion concerning the  
e f f e c t i v e  i n t e r a c t i o n  which would normally be 
i n a c c e s s i b l e  f o r  experiments and analyses  a t  i s o l a t e d  
energies .  
1 )  C. Olmer, B. Zeidman, D.G. Geesaman, T-S. H. Lee, 
R.E. Segel,  L.W. Swenson, R.L. Boudrie, 
G.S. Blanpied, H.A. Thiessen, C.L, Morris, and 
R.E. Anderson, Phys. Rev. Lett. 3, 612 (1979). 
2) S. Yen, R.J. Sobie, H. Zarek, B.O. Pich, . 
T.E. Drake, C.F. Williamson, S. Kowalski, and 
C.P. Sargent ,  Phys. Let t .  z, 250 (1980); S. Yen, 
t o  be published. 
3) H.O. Meyer, P. Schwandt, G.L. Moake, and 
P.P. Singh, Phys. Rev. C 23, 616 (1981). 
4) C. Olmer, A.D. Bacher, G.T. Emery, W.P. Jones,  
D.W. Mi l l e r ,  P. Schwandt, T.E. Drake, R.J. Sobie, 
and S. Yen, t o  be published. 
5) S. Yen, R.J. Sobie, T.E. Drake, A.D. Bacher, 
G.T. Emery, W.P. Jones,  D.W. Mi l l e r ,  C. Olmer, 
P. Schwand t , W. G. Love, and F. Pet rovich,  Phys . 
Let t .  E, 421 (1981); S. Yen, Ph.D. Thesis ,  
Univers i ty  of Toronto (1983). 
6) W.G. Love and M.A. Franey, Phys. Rev. C 24, 1073 
(1981). 
7) K. Nakano and H.V. von Geramb, P a r i s  P o t e n t i a l ,  A 
Table of E f f e c t i v e  Density and Energy Dependent 
I n t e r a c t i o n s ;  Centra l  Spin-Orbi t  and Tensor 
P o t e n t i a l s ,  p r e p r i n t ,  Un ive r s i t a t  Hamburg (1982); 
M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, 
P. Pines,  and R. de J o u r n e i l ,  Phys. Rev. C 23, 2405 
(1981). 
8) H.V. von Geramb, Table of E f f e c t i v e  Density and 
Energy Dependent I n t e r a c t i o n s  f o r  Nucleons, Pa r t  A: 
Cen t ra l  P o t e n t i a l ;  W. Bauhoff and H.V. von Geramb, 
Pa r t  B: Spin-Orbi t  P o t e n t i a l ,  p r e p r i n t ,  
U n i v e r s i t a t  Hamburg (1980); T. Hamada and 
I.D. Johnston, Nucl. Phys. 2, 382 (1962). 
9) A. Pickles imer  and G. Walker, Phys. Rev. C 17, 237 
(1978). 
10) R. Schaeffer  and J. Raynal (unpublished);  modified 
by J. Comfort. 
11) R. Schaeffer  and J. Raynal (unpublished);  modified 
by W. Bauhoff. 
12) C. Olmer, A.D. Bacher, G.T. Emery, W.P. Jones,  
D.W. Mi l l e r ,  P. Schwandt, T.E. Drake, R.J. Sobie, 
and S. Yen, t o  be publ ished;  C. Olmer, AIP Conf. 
Proc., The I n t e r a c t i o n  between Medium Energy 
Nucleons i n  Nuclei, (McCormickls Creek S t a t e  Park),  
Spencer, Indiana,  t o  be published. 
13) W,G. Love i n  the  The (p,n) Reaction and the  
Nucleon-Nucleon Force, e d i t e d  by C. Goodman e t  a l . ,  
(Plenum, New York, 1980), p. 23. 


Energy Dependence of Inelastic Proton Scattering to One-Particle One-Hole States  in 28-Si

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Energy Dependence of Inelastic Proton Scattering to One-Particle One-Hole States in 28-Si

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