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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS -1963
D.
XV
intermediate
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11
ORN -P330
(To be submitted for publication in Nuclear Physics)
- Driep
MASTER
y
isog.
r
Excitation of the First Level of 12c by 156 MeV Protons*
R. M. Haybron!
Michigan State University 520,
East Lansing, Michigan
30 S000

and
Oak Ridge National Laboratory
Oak Ridge, Tennessee
form as a preprint.
per is being distributed in Microcarů
deuce that it has been published, the pa-
card. Since the U.S.A.E.C. bas bo evir
prior to the issuance date of this Micro-
in the open literature and least
This paper was submitted for publication.
faonths
and
H. McManus TT
Michigan State University
East Lansing, Michigan
Abstract: The inelastic cross-section and polarization for the exci-
tation of the 4.43 MeV of ac were calculated in the distorted wave
impulse approximation using the Tamm-Dancoff and RPA wavefunctions
of Gillet and also of Goswami and Pal. Comparison with experiment
favors the RPA wavefunctions of Gillet.

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KELPASITI 90%! CRIT
IN NUCLER SCIARCE ABSTRACTS
*.
*Research sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation and Michigan State University.
Ton loan to the Oak Ridge National Laboratory from Michigan State
University
111963-64 Guggenheim Fellow at NORDITA from Michigan State University
OCT 1 6 1964

-2-
1.
Introduction
The structure of the levels of various light, even-even nuclei hus
been a topic of considerable interest in recent years in theory and
experiment. It has been found that many levels cannot be described by
single particle excitations between shell model orbitals. In particular,
levels which could be collective (where the parity change in the tran-
sition from the of ground state to the excited state J is (-1)) usually
display matrix elements which are from two to ten times the single particle
result. To reproduce such enhancements, one must generalize the wave-
functions involved to include several nearby orbits. This is currently
done either in the Tamm-Dancoff (1.D.) approximation where the collective
nature of the excited state only is taken into account or in the Random-
Phase Approximation (R.P.A.) where the collective nature of both te
ground and excited states is allowed for.
The scattering of protons by nuclei for bombarding energies above
100 MeV can be described by the distorted wave impulse approximation
(DWIA)!). Here the inelastic scattering operator is expressed in terms of
the free two-nucleon t-matrix. Such a description allows the calculation
of the absolute normalization of the inelastic cross-section which can
provide a stringent test on the wavefunctions used. In addition, the
polarization of the scattered proton can in principle provide information
about the radial variation of the effective coupling scheme.
We shall be concerned here with the excitation of the 2", 4.43 MeV
level of c. The inelastic cross-section and polarization will be
calculated at 156 MeV for wavefunctions obtained by V. Gillet?), and
Goswami and Pal»). Both the T.D. and R.P.A. wavefunctions will be used.
a here with t
-3-
A. The Inelastic Transition Amplitude
The scattering amplitude for the transition from the 0", T = 0
ground state to an excited state J,M(T = 0) is (using the results in ref. 1)
P2 = 2, Sveti
Savulm")vet, d'Ubonusxf
(A-2)
m'n'J
The functions x1*) and x'-) are distorted waves. U(n') and U(m') are
spin functions of the scattered prcton and o is its spin operator.
V(r,o) is the effective inelastic scattering potential given by
V(79) = < vg MF Letragöz) 818 - Fy) I *O"() >
(A-2)
where yoº and V, are the initial and final states of the target nucleus.
ry and are the position and spin of the gth target nucleon. tidios)
is the free two-nucleon scattering amplitude which depends on the bom-
barding energy, momentum transfer and iso-spin transfer as well the spins
of the nucleons. The calculation of (A-2) is treated in ref. 1.
Calculations
The cross-section and polarization implied by (A-1) was calculated
using the T.D. and R.P.A. Wavefunctions of Gillet and also those of
Goswami' and Pal. These are shown in figs. 3 through 4. The distorted
waves were calculated using an optical potential recently obtained by
G. R. Satchler") from the 180 MeV data of Johansson, et al). The cross-
section and polarization for the Gillet R.P.A. vector were also calcu-
lated for the optical potential quoted by Johansson and are shown in
figs. 1 and 2.
The calculations above were performed using the full two-nucleon
amplitude. In order to determine the relative importance of the various
portions of the two-nucleon interaction, the tensor portion of the inter-
action was set equal to zero 80 that t = A +
in the notation
A
of ref. l. The interaction was also represented by t = A for comparison.
Results
The R.P.A. wavefunctions of Gillet can be seen to give results
closest to the data for the cross-section. It is apparent in fig. 1
that the 3-j and L-S coupling limits must be enhanced considerably to
reproduce the strength of the transition.
The T.D. approximation wherein ground state correlations are absent
can be seen to produce an enhancement of between 2 and 3 over the single
particle result and gives a result which is close to the L-S coupled
result.
The introduction of ground state correlations produces an
additional enhancement which yields the best agreement with experiment.
The polarization data ) seems to favor L-S coupling although the
Gillet vector result is quite satisfactory. It should be pointed out
that the distorted waves contribution to the polarization could be
substantially altered by changes in the spin-orbit portion of the optical
potential. Neither V, nor Vs yield good fits to the elastic polari-
zation; a better set of spin-orbit parameters could improve the Gillet
vector result, and studies are in progress to obtain these.
The ratio of the nuclear spin-flip matrix element to the non-spin
flip matrix element, called , is equal to .75 for 3-j coupling and 0.
for L-S coupling. Figures 2, 5, and 6 indicate that ^ is small for the
m
Gillet wavefunctions. Comparison of the values of a derived from these
.5.
wavefunctions (^ is a function of o for wavefunctions containing more
than one configuration) to those obtained from p-y correlations) indi-
cate a Güssable agreement. These measurements were analyzed with the
neglect of the spin orbit portion of the optical potential: better
agreement might be obtained if this were taken into account.
Conclusions
The scarcity of data available at present precludes any detailed
discussion of the shape of the cross-section and polarization as a
quantitative check on the wavefunctions of the levels involved. However,
the results presented here show that the Gillet wavefunctions reproduce
the essential features of the high energy proton data. Further, the
refinement and extension of the high energy data could provide infor.
mation of value in structure calculations such as those mentioned here.
The theoretical curves quoted here contain uncertainties which could
be as large as 20% (aside from possible errors in the two-nucleon ampli-
tude which we do not consider here). The optical potential used here
has been extrapolated from 180 MeV. Although the errors generated by
this procedure should be small, they are uncertain and could conceivably
produce deviations on the order of 10%. In addition, finite range effects
(due to effects of refraction) have been estimated to be on the order of
10% at this energy”).
The aforementioned sources of error are not essential and can be
treated as the inelastic scattering data becomes more extensive. As is
evident here the inelastic data should be accompanied by elastic data at
the correspondong energy to provide for the accurate determination of
distortion effects. It would, of course, be desirable to have elastic
-6-
and inelastic measurements at several energies between about 70 and
300 MeV to facilitate the precise investigation of the wavefunctions.
Wavefunctions such as used here are available for many levels of
12c, 100, and 40ca. These levels are being treated as described here
and the results will be reported later.
Acknowledgements
The authors gratefully acknowledge the aid and encouragement of
G. R. Satchler (ORNL) and express their gratitude to R. M. Drisko
(University of Pittsburgh) for modifying the distorted wave codes for
use here:
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-7-
References
1. R. M. Haybron and H. McManus (submitted to the Physical Review).
2. V. Gillet, Thesis, Paris (1962).
3. A. Goswami and M. K. Pal, Nuclear Phys. 35, 233 (1962); Nuclear Phys.
44, 294 (1963).
4. G. R. Satchler (to be published).
5. A. Johansson, U. Svanberg, and P. E. Hodgson, Arkov for Fysik 19,
541 (1961).
6. J. C. Jacmart, et al., "Direct Interactions and Nuclear Reaction
Mechanisms", Gordon and Breach, Science Publishers, New York, 1963;
edited by E. Clemental and C. Villi.
7. R. Alphonce, A. Johansson, and G. Tibell, Nuclear Phys. 4, 672 (1957).
8. See ref. 1 and also A. B. Clegg, CERN International Conference on
High Energy Physics and Nuclear Structure (1963).
9. R. M. Haybron (to be published).
ZL.
2.
-8-
Figure Captions
Figure 1 - The experimental points shown probably contain errors on the
order of 25% (see ref. 6).
Figure 2 - The polarization produced by the T.D. wavefunction is
essentially identical to the R.P.A. result.
Figure 3 - The cross-section data shown here is the same as in fig. l.
Note that with the error estimate previously stated, either
curve can fit only one point.
Figure 4 - The experimental data has not been shown here although the
R.P.A. result yields qualitative agreement.
Figure 5 - The inelastic cross-section using the Gillet R.P.A. wave-
functions showing the importance of the various parts of the
two-nucleon amplitude.
Figure 6 - Polarizations corresponding to the cross-sections shown in
fig. 5.
Les
:,n
r
. "..**.
..
?
i
.
.
,
UNCLASSIFIED
ORNL-DWG 64-5393

C12 (pp') C12*
Eo=156 MeV
2+, T = 0 AT 4.43 MeV
GILLET
a R.P.A. (VS)
6 T.D. (VS)
C R.P.A. (Vy)
I EXPERIMENT
--L-S COUPLING (VS)
..... ; -; COUPLING (VS)
CROSS-SECTION (mb/steradian)
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 1.
.'
-
UNCLASSIFIED
ORNL-DWG 64-5394

POLARIZATION
-0.2
c'(e,po) C12*
Ep = 156 MeV
2+, T = 0 AT 4.43 MeV
GILLET
R.P.A. (VS)
......... R.P.A. (VH)
.
I EXPERIMENT
---L-S COUPLING (VS)
-0.4
-0.6
-0.8
0
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 2.
UNCLASSIFIED
ORNL-DWG 64-5396

C12 (ppi) C12*
Eo = 156 Mev
2+, T = 0 AT 4.43 MeV
GOSWAMI AND PAL
O R.P.A. (VS)
b T.D. (VS)
I EXPERIMENT
CROSS-SECTION (mb/steradian)
L
0
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 3.
UNCLASSIFIED
ORNL-DWG 64-5396

POLARIZATION
___
_
_
_
.
C12 (0,01) C12*
Ep = 156 MeV
27, T = 0 AT 4.43
GOSWAMI AND PAL
R.P.A. (VS)
--- T.D. (VG)
-0.4
-0.6
-0.8
0
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 4.
UNCLASSIFIED
ORNL-DWG 64-5397

C12 (ppi) C12*
Ep = 156 Mev
2+,T=O AT 4.43 MeV
a TWO - NUCLEON
AMPLITUDE
b A AND C ONLY
(DOTTED)
CA ONLY
-.-
-
.
-
.-
CROSS-SECTION (mb/steradian)
-
-
-
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 5.
UNCLASSIFIED
ORNL - DWG 64-5398

0.8
0.6
POLARIZATION
C12 (2.p.) C12*
Eo = 156 Mev
2+, T= 0 AT 4.43 MeV
a TWO-NUCLEON
AMPLITUDE
b A AND CONLY (DOTTED)
CA ONLY
-0.6
-0.8
0
60
10 20 30 40 50
CENTER-OF-MASS SCATTERING ANGLE (deg)
Fig. 6.
END

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DATE FILMED
9/15/65
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