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Abstract
Isospin Character of Transitions to the 2+ and 3 states of90,92,94,96£r
Brian J. Lund
Yale University
May 1996
The elastic and inelastic scattering cross sections of 35.4 MeV alpha-particles by
9o,92,94,96zr ^ave been measured to investigate the isospin character of transitions to the 2\
and 3[ states. At this energy, interference between the nuclear and Coulomb scattering
amplitudes produces structure in the inelastic cross sections which enables the extraction
of the ratio of neutron to proton multipole matrix elements, MJMp, from the data. The
experimental determination of the ratio M JM P can be used to provide a test of nuclear
structure calculations.
There is a significant discrepancy between the M JM P ratios reported from a
previous alpha-scattering measurement and a 6Li scattering measurement for these same
isotopes of zirconium. M JM P had been extracted from those alpha-particle scattering cross
sections using an implicit folding procedure (IFP), while the 6Li scattering had been
analyzed using a deformed optical model potential (DOMP). The M JM P ratios extracted
from the alpha-particle scattering were much larger than those extracted from the 6U
scattering. A partial reanalysis of the previous alpha-particle scattering, using the
deformation lengths deduced therein in a simple relation consistent with the DOMP
analysis of the 6Li scattering, yielded M JM P ratios much smaller than those deduced using
the IFP, but were still significantly larger than those reported from the 6Li scattering.
The present experiment was undertaken in an effort to understand the source of
this remaining discrepancy. Elastic scattering data were measured over a laboratory angle
range of 6° to 46.5°, while inelastic scattering data were measured for laboratory angles of
8° to 46.5°. The data were analyzed using a deformed optical model potential (DOMP)
and a folding model potential. The folding model calculations have been made using
transition densities resulting from a random phase approximation (RPA) calculation as
well as transition densities of the standard collective model form. While the folding model
analysis provides a clearer connection between experimental measurements and the
underlying nuclear structure, most previous experiments have been analyzed using the
DOMP. The elastic and inelastic cross sections measured in the current experiment are
consistent with the previous alpha-scattering measurement. General agreement is found
between folding model calculations for alpha-particle scattering and 6Li scattering when
using the same RPA transition densities, indicating that the two data sets are consistent,
and comparisons between these two experiments are valid. The DOMP analysis of the
( a ,a ’) scattering yields larger M JM f ratios than are extracted from the 6Li inelastic
scattering measurements. The folding analysis of the ( a ,a ’) scattering using standard
collective model form factors produces larger values for M JM P than the DOMP analysis
of the same data. In both the folding and DOMP analysis, deduced M JM f ratios for
92’94’96z r are considerably larger than than their respective ratios of NIZ. The inconsistency
between the DOMP and folding model analyses is demonstrated, as well as the sensitivity
of the folding model analysis to the assumed form for the transition densities.
Isospin Character of Transitions to the 2+ and 3 ~ stateso f 90,92,94,96Zr
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
Brian J. Lund
Dissertation Director: Dr. Peter D. Parker
May 1996
Contents
List of Figures iv
List of Tables vi
Chapter 1: Introduction 1
1.1 General Introduction................................................................................................1
1.2 Zirconium Isotopes.................................................................................................. 3
1.2.1 RPA predictions....................................................................4
1.2.2 Previous measurements.......................................................5
1.3 Present Experiment.................................................................................................. 8
Chapter 2: Theory of the Experiment 17
2.1 Isoscalar Transition Rates..................................................................................... 17
2.2 Coupled-Channels Description of Inelastic Scattering......................................20
2.3 Optical Potential - Folding Models.......................................................................22
2.4 Models for Transition Densities and Potentials.................................................. 23
2.4.1 Collective Model - Bohr-Mottelson form.......................................... 23
2.4.2 Microscopic Models.............................................................................26
2.5 Deformed Optical Model Potential...................................................................... 26
Chapter 3: Description of Experiment 29
3.1 Details of Setup......................................................................................................29
3.2 Ray Tracing.............................................................................................................33
3.3 Electronics..............................................................................................................36
3.4 Experimental Procedure........................................................................................ 37
Chapter 4: Analysis 48
4.1 Extraction of Cross Sections................................................................................ 48
4.2 Deformed Optical Model Analysis...................................................................... 50
Contents ii
4.2.1 Elastic Scattering.................................................................................. 50
4.2.2 Inelastic Scattering................................................................................51
4.3 Folding Model Analysis.............................................................................56
4.3.1 Elastic Scattering..................................................................................56
4.3.2 Inelastic Scattering................................................................................57
4.3.2.a RPA transition densities..................................................... 57
4.2.2.b Bohr-Mottelson transition densities................................. 58
4.4 Investigation of the Phase Shifts..................................................................... 60
Chapter 5: Discussion and Conclusions 99
References 105
iii
List of Figures
1.1 M JM P ratios for several nuclei....................................................................................... 16
2.1 Coordinates in the folding model integrals................................................................... 28
3.1 Schematid drawing of target chamber........................................................................... 41
3.2 Schematic drawing of Yale Split Pole Spectrograph................................................... 42
3.3 Schematic drawing of focal plane detector................................................................... 43
3.4 Illustration of ray tracing r......................................................................................... 44
3.5 Example of use of ray tracing to create an angle spectrum........................................46
3.6 Electronics diagram......................................................................................................... 47
4.1 Illustration of process used to create angle gated spectra.......................................... 70
4.2 Inelastic spectra for “ Z r^ a 'y ^ Z r’ at 0Bpec = 23 .0°.....................................................71
4.3 Inelastic spectra for “ Z r^ a 'j^ Z r* at 0Bpec = 35.0°.....................................................72
4.4 Inelastic spectra for 94Z r(a,a ')94Zr* at 0Bpec = 35.0°.....................................................73
4.5 Inelastic spectra for “ Z^cqa'^Zr* at 0Bpec = 20 .0°.....................................................74
4.6 Illustration of process of angle binning........................................................................ 75
4.7 Elastic scattering cross sections compared to curves of [Ry 87]...............................76
4.8 Cross sections for excitation of 2\ states compared to [Ry 87]...............................77
4.9 Cross sections for excitation of 3j states compared to [Ry 8 7 ]...............................78
4.10 Inelastic cross sections and fits of Rychel et al. [Ry 87]............................................ 79
4.11 Optical model fits to elastic scattering..........................................................................80
4.12 DOMP fits to2j cross sections......................................................................................81
4.13 Coulomb and hadronic contributions to inelastic cross sections................................82
4.14 DOMP fits to 3j cross sections.....................................................................................83
4.15 Folding model fits to elastic scattering..........................................................................84
4.16 RPA and BM folding model predictions for 2j cross sections.................................. 85
4.17 RPA folding model predictions for 6Li excitation of 2\ states.................................. 86
iv
4.18 RPA and BM folding model predictions for 3[ cross sections...................................87
4.19 RPA folding model predictions for 6Li excitation of 3[ states....................................88*4.20 RPA and BM transition densities for excitation of 2\ states......................................89
4.21 Transition potentials for excitation of 2\ states........................................................... 90
4.22 RPA and BM transition densities for excitation of 3 j states......................................91
4.23 Transition potentials for excitation of 3\ states........................................................... 92
4.24 Ratio of BM/RPA transition potentials for excitation of 2\ states...........................93
4.25 Ratio o f BM/RPA transition potentials for excitation of 3" states...........................94
4.26 Folding fits to 2\ cross sections using BM transition densities................................ 95
4.27 Folding fits to 2>\ cross sections using BM transition densities................................ 96
4.28 Folding fits matching phase of 2\ cross sections.........................................................97
4.29 Folding fits matching phase of 3[ cross sections.........................................................98
v
List of Tables
1.1 Summary of predictions of RPA calculations for Zr isotopes..................................... 9
1.2 Contributions of neutrons and protons to Mn>p from RPA calculations.....................10
1.3 Comparison of btN reported for 90Z r ............................................................................11
1.4 Comparison of btN reported for 92Z r ............................................................................12
1.5 Comparison of btN reported for 94Z r ............................................................................13
1.6 Comparison of b * reported for 96Z r ............................................................................14
1.7 Comparison of MJMP and B(E£) f from previous work...........................................15
3.1 Thickness and composition of zirconium targets........................................................ 39
3.2 Contributions to uncertainties in measured cross sections.........................................40
4.1 Optical model parameters from fits to elastic scattering............................................63
4.2 Comparison of b (N and B(EQ f from DOMP analyses............................................. 64
4.3 Parameters for ground state density distributions....................................................... 65
4.4 Strengths of alpha-nucleon effective interaction..............................................i.........66
4.5 Summary of predictions of RPA calculations..............................................................67
4.6 Comparison of DOMP and BM folding results...........................................................68
4.7 Diffuseness parameters required to match phase of cross sections...........................69
5.1 Comparison of (a,a') and (6Li,6Li') r e s u l ts ..............................................................104
Chapter 1: Introduction
1.1 General Introduction
The experimental determination of the isospin character of nuclear transitions can
be used to provide a test of nuclear structure calculations. A measure of this character is
given by the ratio of the neutron to proton multipole transition matrix elements, M JM P,
where
- f g ? r>(r)r''2dr. (1.1)
Here, g nt {p)(r) is the neutron (proton) transition density. For low-lying “isoscalar”
collective mass excitations, the neutron and proton distributions oscillate in phase with
each other, and their transition densities are expected to have the same shape. Therefore
the ratio of the transition matrix elements for this kind of transition is expected to have the
simple value M JM P « NIZ. Deviations from this value reflect the underlying nuclear
structure, and depend, for example, on the specific shell structure of the ground and
excited states of the isotope in question. An accurate nuclear structure theory must
therefore be able to reproduce experimentally measured values o tM JM p.
The proton multipole transition matrix element, Mp, can be accurately determined
from inelastic scattering of electrons or photons, Coulomb excitation, or y-decay
measurements. This is because the electromagnetic interaction is known. The
determination of the neutron multipole transition transition matrix element, M„, requires
the scattering of hadronic probes, whose reaction mechanism is less well understood.
Also, measurements of M„ invariably involve a combination of both M„ and Mp, and thus a
separate determination of Mp is often necessary.
Most information on the isospin character of transitions has come from the
comparison of hadronic scattering data with electromagnetic (EM) transition rates
determined from electron scattering, Coulomb excitation, or y-decay. The earliest such
1
work combined the results of alpha-particle or proton inelastic scattering experiments with
the EM transition rates in order to obtain M„ and Mp. Examples include the scattering of
protons off of 90Zr, 120Sn, 144Sm, and 208Pb [Ga 82]; 58Ni, 40Ca, 58Ni, and 208Pb [Hi 88].
More recently, comparisons of hadron scattering experiments have been used to determine
M n and Mp. For example, comparisons of proton and neutron inelastic scattering have
been used to determine these values in 90'92>94Zr [Wa 88, Wa 90]. Also, the scattering of
Jt+ and it' has been used on several nuclei, including 40Ca and 118Sn [U1 85], 208Pb [SM 86],
48’50Ti, 52Cr and 54,56Fe [Oa 87], and 180 [SM 88],
In principle, it is also possible to use the inelastic scattering of a single probe to
deduce the isospin character of a transition. This can be accomplished if the experimental
parameters can be chosen such that the interference between the Coulomb and nuclear
scattering amplitudes produces structure in the differential cross sections which is sensitive
to their relative amplitudes. This method of Coulomb-nuclear interference (CNI) has been
used with heavy-ion projectiles to investigate the isospin character of transitions to bound
states as well as the giant quadrupole resonsance (GQR). CNI observed in the scattering
of 170 has been used to investigate the isospin character of the transition to the giant
quadrupole resonance (GQR) in 118Sn [Ho 90], 124Sn [Ho 91b], as well as the transitions
to low-lying 2+ and 3‘ states of 204-206-208pb [Ho 91a]. CNI in the scattering of alpha-
particles [Ry 87] and 6Li [Ho 92, Ho 93a] has been used to investigate the .isospin
character of transitions to low-lying states in several zirconium isotopes.
Plotted in Figure 1.1 are values of MJMP for the first 2* and 3' states of several
nuclei recorded from various sources. Also indicated in these plots are the N!Z values for
these nuclei. This figure is not meant to be exhaustive, but is rather to indicate the trend
of M JM P values.
Typically hadron scattering data are analyzed using the deformed optical model
potential (DOMP) (Chapter 2.; also see [Sa 83]) in order to extract a hadronic
deformation length, 6 Bernstein [Be 69] pointed out the conditions under which the
2
6 / extracted from, for example, alpha-particle scattering could be expected to be
physically meaningful when compared to those extracted from EM data. It has been
customary to assume that the &tN extracted in this manner is a property solely of the target
nucleus, and therefore independent of the probe. Madsen, Brown and Anderson noted
that the deformation length, b tN, extracted from inelastic scattering data is a function not
only of the nuclear structure matrix elements M„ and Mp, but also of the probe itself [Ma
75a, Ma 75b]. In addition, Beene, Horen and Satchler [Be 93] have demonstrated the
inconsistencies between the deformed potential and folding models in the analysis of
inelastic hadron scattering of low-lying collective excitations. These discrepancies were
found to increase as the angular momentum transfer £ of the transition increases.
However, as noted by Satchler [Sa 83], this method incorporates the appropriate physics
in a qualitative way while enjoying the advantage of simplicity.
Hadron scattering has also been analyzed using the folding model (Chapter 2; and
see [Sa 83]). In this model, the potential results from folding an effective interaction
between the projectile and a target nucleon with the density distributions of the target
nucleus. This model is dependent on the choice for the effective interaction, as well as the
models used for the ground state and transition densities; however, it has the advantage of
being able to relate parameters extracted from the analysis of elastic and inelastic
scattering to the underlying nuclear density distributions in an unambiguous manner. This
property allows one to compare, in a direct manner, the results of scattering experiments
regardless of the probe used.
1.2 Zirconium Isotopes
Zirconium offers an excellent opportunity to study the effects of valence neutrons
on the isospin character of nuclear transitions, due to the fact that there are four even-even
stable isotopes of this element. In 90Zr, the neutrons close the N=4 major shell, while the
protons close the 2pl/2 subshell (although the ground state of 90Zr is known to include a
3
significant 1 g\n admixture [see, e.g., Ba 83]). 92,94Zr are expected to be dominated by
2d\j2 and 2d^2 neutron configurations, respectively, while 96Zr closes the 2d5a neutron
subshell.
Closure of the major neutron shell is expected to have major consequences for the
transition matrix elements. From simple shell model arguments, one would expect M„ = 0
for 90Zr. Core polarization effects will modify this somewhat, allowing a non-zero value
for A/„, but the shell closure is still expected to inhibit participation of the neutrons in
collective excitations. Indeed, the schematic model of Brown and Madsen, in which
collective core excitations are connected to the shell model space using perturbation
theory [Br 75], predicts MJMP < N/Z for low-lying collective excitations to 2+ states in a
closed neutron shell nucleus [Ma 84]. This value is predicted to rise rapidly to (or slightly
exceed) the collective value M JM P = N/Z as neutrons are added [Ma 84].
1.2.1 RPA predictions
A random-phase approximation (RPA) calculation for 90,92,94,96Zr, done in
conjunction with a 6Li scattering experiment, demonstrate this trend [Ho 92, Ho 93a].
Table 1.1 summarizes the predictions of the RPA calculations, while Table 1.2 lists some
the strongest particle-hole (lp -lh ) contributions to Mn and Mp for the 2\ and 3[ states. In
these calculations, the valence space was defined as the N = 3 major shell for protons {p$a,
ha, Pm, g9n) and the N = 4 major shell for neutrons (ds/2, sm , h im , dm , gia)-
For the 2\ states, the contribution of the valence protons to Mp was found to be
small and relatively indepentdent of neutron number. The valence neutron contribution,
however, rises rapidly from M [vd) = 0 for 90Zr to Af > M for 92>94>96Zr [Ho 93a].
The major valence neutron contributions arise from 2d\n recoupling and 3slf2-2d5j2 p-h
transitions. The 24 2/2 matrix element is nearly the same for 92Zr and 94Zr due to the
similarity of the neutron configurations, but decreases in 96Zr due to the filling of the 2d5/2
orbitals. The increasing contribution from the 3sia-2d'sn transition leads to a larger M JM P
4
ratio for 94Zr compared to 92Zr, but is not enough to offset the decreasing contribution of
the 2 d 2n recoupling in 96Zr.
For the 3[ states, the valence protons transitions are quite strong, and there are
relatively fewer possible valence neutron transitions. Although M ^ is found to increase
with the addition of neutron, the net result is to drive the ratio M JM P upward toward N/Z,
but at a much slower rate than for the 2\ states [Ho 93a].
1.2.2 Previous measurements
Previous scattering experiments on the zirconium isotopes have generally been
analyzed using a deformed optical potential having a Woods-Saxon form with the
distorted-waves Born approximation (DWBA) formulation of the scattering amplitude in
order to extract nuclear deformation lengths or parameters. 90Zr has been extensively
studied due to its closed major neutron shell, which allows for relatively simple shell
model calculations. Most of this work is from proton inelastic scattering, but also
included (d,d’), (e,e’), (n,n’), (3He,3He’), ( a ,a ’) and (6Li,6Li’). In comparison, little work
has been done on 96Zr. Tables 1.3-1.6 summarize the nuclear deformation lengths
deduced from previous scattering measurements on 90'92>94>962;r. In cases where only a
nuclear deformation paramter, p, is given, the deformation length is calculated by using a
radius parameter equal to the average radius parameter of the real and imaginary parts of
the scattering potential, i.e., 6 = fir0A m , where r0 = (roT + r0i) / 2.
In contrast, there are very few MJMP ratios reported for the zirconium isotopes.
Rychel et al. [Ry 87] measured the scattering of 35.4 MeV alpha-particles off of
90,92,94,96zr jn orcjer to determine whether CNI effects could be used in order to determine
electromagnetic and isoscalar transitions rates. They analyzed their data using an implicit
folding procedure [Wa 82] with density dependence corrections [Sr 84] to determine
M JM P for transitions to low-lying 2+ and 3' states. The M JM P ratios deduced for the 2\
and 3[ are listed in Table 1.7. Except for 90Zr, they were found to be surprisingly large,
up to M JM P = 4.69 for the 2\ state of 96Zr, and in general significantly larger than the
5
6
value M JM P » NIZ expected for a collective excitation. In contrast, the largest previously
measured value is for the 2\ state of 48Ca: MJMP = 2.55 from 500 MeV (p,p) [Ba 87]
and M JM P = 2.34 from 200 MeV and 318 MeV (p,p’) [Fe 94]. N/Z = 1.40 for both 48Ca
and 96Zr.
Wang and Rapaport [Wa 88, Wa 90] deduced M JMP ratios for transitions to low-
lying 2+ and 3' states in 90i92-94z r from the simultaneous analysis of proton and neutron
inelastic scattering. Their ratios for the 2\ and 3j states are listed in Table 1.7, are
significantly smaller than those reported by Rychel etal. from alpha-particle scattering.
The only other previously reported values for M JM P in the zirconium isotopes are
from 500 MeV (p,p’) [Ba 87] and 800 MeV (p,p’) [Ga 82] on 90Zr. These results are
included in the footnotes to Table 1.7.
In an attempt to resolve the large discrepancies in the reported M JM P ratios for
the transitions to the 2\ and the 3j states, Horen et al. [Ho 92, Ho 93a] made a
measurement of the elastic and inelastic scattering of 70 MeV 6Li ions by the even
isotopes of zirconium. Their results are also listed in Table 1.7. The data were analyzed
using a deformed optical model potential to determine a hadronic deformation length. The
M JM P ratios deduced in this manner were considerably smaller (1/2 - 1/3 times) than
those reported from the alpha-particle scattering experiment of Rychel et al. [Ra 87], and
were in much better agreement with those reported from the proton-neutron scattering
[Wa 88, Wa 90]. The 6Li scattering results also supported the predictions of a random
phase approximation (RPA) nuclear structure calculation [Ho 93a]. The RPA wave
functions were used to construct transition densities which were then used in a folding
model to provide the transition potentials used to calculate the inelastic cross sections.
The implicit folding procedure [Wa 82] used by Rychel et al. is based on an
inverse application of Satchler's theorem which relates the moments of a folded potential
to the moments of the underlying density distribution. For an optical potential given by
the folding integral (see Chapter 2 for details)
U ( r ) = f p(r)v0(r.r ')r '2dr', (1.2)
and a 2*-pole transition potential
Gt(r)=fgt(r)vt(r’rr)r'2dr'’ (1-3)
the moments of the potentials are related to the moments of the density distributions by
J(U ) = J (p )J (v ) , (1.4)
= (1.5)
where J ( f ) is the volume integral, and M t ( f ) is the 21 moment of the function f ( r ) . In
the above equations, r is the ground state density distribution, g is the transition density,
and v is the effective interaction. Using eqs. (1.4) and (1.5), one can determine the
moments of the transition potential
M t (gt ) = AM t (Ge) /J (U ) . (1.6)
The basic assumption of the implicit folding procedure is that the deformed optical
potential model may be used for the transition potential, i.e.
„ dUG‘(r) = - b‘ ~fr (L7)
where b tu is a deformation length determined by matching the measured inelastic cross
sections. Using this form, one can derive the final result of the implicit folding procedure
[Wa 82], namely
(L 8 )
While this result appears to be independent of the forms of U(r) or ge(r), it in fact
depends sensitively on the assumption of the deformed potential [Ho 93c].
Because of the objections to the implicit folding procedure [Ho 93c], Horen et al.
[Ho 92, Ho 93a] performed a partial reanalysis of the alpha-scattering results of Rychel et
al. In this reanalysis, the nuclear deformation parameters reported in [Ry 87] were used in
a simple schematic relation (Chapter 2, eq 2.11). This procedure is consistent with the
DOMP analysis of the 6Li scattering data. The M JM P ratios deduced in this manner from
the alpha-scattering results are listed in Table 1.7; they are smaller than those deduced
using the implicit folding procedure but are still significantly larger than those deduced
from the 6Li scattering. However, the analysis of those two experiments used different
values for the electromagnetic transisition strengths, B(Ef) f , so direct comparisons of the
two experiments could not be made. Listed in parentheses in the fourth column of Table
1.7 are M JM P ratios calculated using the nuclear deformations exctracted from the alpha-
particle scattering as reported in [Ry 87], but using the B(E^) j values adopted for the
analysis of the 6Li scattering [Ho 92, Ho 93a]. It appears that part, but not all, of the
difference in the reported ratios could be attributed to the difference in the B(E^) t used,
once the M JM P were extracted in a consistent manner.
123 Present experiment
In an effort to determine whether the remaining discrepancy was caused by
inconsistencies in data, or by the interpretation of the DOMP model results, we undertook
a new independent measurement of the 35.4 MeV alpha-particle scattering from the
90’92’94’96Zr isotopes. The data were analyzed using a deformed optical model potential
(DOMP) in order to compare with the previous experiments [Ry 87, Ho 92, Ho 93a].
The data were also analyzed using a folding model in conjunction with the same
RPA transition densities used for the analysis of the 6Li-scattering data. This process
gives us a method with which to determine the consistency of the alpha-scattering and 6Li-
scattering data. Folding model calculations have also been made using transition densities
given by the standard Bohr-Mottelson collective model form, in order to investigate the
sensitivity of the extracted M JM P ratios to the shape adopted for the transition potentials.
In addition, we have been able to make a direct comparison of the DOMP and folding
models.
8
TABLE 1.1. Summary of the predictions of the RPA calculations. The RPA calculations were ___________ constrained to reproduce the B(El) t values of column 3.
Isotope Ex(MeV)
B (E /)t( e V )
Ex(MeV)
RPA calculation*
Mp (e fm2)
Mn/Mp N/Z
2; states
90Zr 2.186 0.063b 2.51 25.1 0.84 1.2592Zr 0.935 0.083b 1.40 28.9 1.49 1.3094Zr 0.918 0.066bc 1.55 25.9 1.69 1.3596Zr 1.751 0.055b 2.02 23.3 1.66 1.40
states
s© O N *-* 2.748 0.071d 2.73 267 0.75 1.2592Zr 2.340 0.067** 2.64 257 0.87 1.3094Zr 2.057 0.087** 2.35 295 1.06 1.3596Zr 1.897 0.120* 1.96 346 1.22 1.40
8 References [Ho 93a, Br 88] b Reference [Ra 87]c A recent remeasurement [Ho 93b] found B(E2) f = 0.060 ± 0.004 e2b2 d Reference [Sp 89]e References [Ho 92, Ho 93a]; a recent remeasurement [Ho 93d] found B(E3) t = 0.180 ± 0.018 e2b3
10
T A R 1 F 1n o L C L"£‘' Partial listing of contributions to the neutron and proton multiple matrix elements, M„. and M p (in units of fm1), from the RPA calculations for excitations of the 2 ( and 3 f states in90.92,M.9*2r
lh -lpNeutrons
"’Zr 9:Zr * Z r 96Zr lh -lpProtons
wZr 92Zr 94Zr ,6Zr
2di n -2din 11.50 12.362 r
IASstate
li>9/2*l£9/i 4.75 3.55 3.38 3.94.33 6.35 5.43 4.40 2pj/2-2pi,j 7.74 2.96 2.88 3.86
l?9/2*2^5/2 7.58 5.25 2.88 1.32 t / l / l ' l / 1!!/? 2.31 4.67 4.27 3.49t/7 /2 'lA ||/2 2.39 3.50 2.96 2.34 l / s / i ‘2pi/i 2.15 2.06 1.89 1.862d5,2-3s1/2 3.29 6.15 10.15 l/s/2* l/>9/2 1.40 2.87 2.63 2.15I f 1.61 2.40 2.05 1.66 l^J/2*l£9/2 1.03 2.11 1.95 1.60I p m - l f m 0.74 1.11 0.94 0.76 2pj/j-2p3/j 0.41 0.43 0.36 0.31ldj/2-lg in 0.64 0.96 0.81 0.64 T-Pin' l fm 0.70 1.44 1.33 1.0924j/2-2g9/3 0.69 1.15 1.29 \d}n-\gTn 0.70 1.42 1.30 1.06l h ur2- lh n/2 0.33 0.65 1.19 l£9/2'2d3/2 0.47 0.80 0.67 0.50l-Pxn-lf in 0.37 0.56 0.48 0.39 I f in~ I / 5/2 0.23 0.31 0.26 0.192 4j/j-2 d in 0.44 0.77 0.98 l£9/2'l*lJ/2 0.42 0.79 0.65 0.47l?9/2*l87/2 0.30 0.40 0.34 0.28 2p 1/2 *2 / 5,7 0.25 0.53 0.49 0.42l?7/2'l?7/2 0.14 0.26 0.47 2s 1 /2 *2d 5 /2 0.22 0.44 0.40 0.33Ig i n ' l d i n 0.21 0.37 0.50 ^fin'7-Pin 0.20 0.33 0.26 0.1924 j /j -3 j 1/2 0.18 0.36 0.72 l/ t /2 *2 / 7/2 0.16 0.32 0.29 0.24Igrn' ldj / i 0.14 0.25 0.44 2p j/2 * I / 5/2 0.13 0.20 0.17 0.14
2di n - \h n n 41.15 89.563,"
129.93state
2pj/2*l?9/2 123.35 120.27 113.80 111.7639.52 33.07 37.09 42.58 t/7/2* 1^9/2 16.34 15.80 20.66 26.42
l£9/J*l./lS/J 25.97 22.53 26.69 32.58 1/ 7/2 *!' 13/2 12.67 12.20 16.35 21.58t / 7/2 - l 'u / j 13.42 11.76 13.93 16.97 2pi /2* 1&7/2 8.89 8.78 11.71 15.452Pl/2-1^7/J 9.66 8.26 9.49 11.14 l/3/2*l?7/2 9.02 8.71 11.46 14.85I f i n ' l h i n 9.05 7.92 9.40 11.48 2p 1/2 *21/ 5,1 8.64 8.53 11.21 14.587-P\n-7din 12.84 7.73 5.56 3.61 l/s/2*1^9/2 8.51 8.22 9.28 10.27l/j/2*lg7/2 8.69 7.45 8.57 10.09 l/s /2*!111/2 7.93 7.67 10.33 13.71lS 9/l'2 /l/2 8.45 7.29 8.54 10.31 2pj/2*2d3/2 7.40 7.16 9.34 12.002pm'7di / i 6.98 6.04 7.03 8.38 2p3/2*2d2 7.29 7.06 9.39 12.312ds/z-2fin 4.92 11.20 18.60 * l'/5/2*l^In/2 5.29 5.09 6.82 9.00Id 3/2-lA11/2 5.54 4.79 5.60 6.67 l/7 /I-2 rf,/l 4.31 4.14 5.49 7.17I?9/2'3Pj /2 5.27 4.57 5.38 6.54 2p 1/2 *2^9/2 3.34 3.25 4.39 5.847P)n~7di/2 7.28 4.53 3.36 8.39 2p 3/2*187/2 3.32 3.22 4.23 5.49
a Table VI of [Ho 93a].
TABLE 1.3. Comparison of nuclear deformation lengths, 6 / , reported for the 2\ and 3[ states of 90Zr. ____________Lengths are in fm._______________________________________________________________
( a ,a ’) ( a ,a ’) ( a ,a ’) ( a ,a ’) (P>P’) (P»P’) (p>p’)
31 MeV 35.4 MeV 35.4 MeV 65 MeV 12.7 MeV 25 MeV 65 MeV[Ma 68] [La 86] [Ry 87] [Bi 66] [Di 68] [Bi 83] [Fu 81]
2 ; 0.40 0.42 0.41 0.40 0.42 0.45 0.36
3 ; 0.84 0.77 0.81 0.84 0.96 1.02 0.79
(P>P’) (P>P’) (3He,3He’) (d,d’) (n,n’) (e.e’) (6Li,6Li’)800 MeV 800 MeV 25 MeV 15 MeV 8, 24 MeV 70 MeV[Ga 82] [Ba 83] [Ru 68] [Ba 75] [Wa 90] [Si 75] [Ho92,93a]
2 ; 0.47 0.45 0.69 0.25 0.44 0.42 0.40
3i 0.89 0.81 1.09 0.52 0.86 0.82 0.69
Unweighted average (excluding (d,d’)) for 2\ state: b 2N = 0.44 ± 0.08 fm Unweighted average (excluding (d,d’)) for 3[ state: b 3N = 0.86 ± 0.10 fm
12
TABLE 1.4. Comparison of nuclear deformation lengths, b * , reported for the 2\ and 3[ ___________ states of * l x . Lengths are in fin.____________________________________
(a,ct’) 35.4 MeV
[Si 86]
( a ,a ’) 35.4 MeV
[Ry 87]
(a ,a ’) 65 MeV
[Bi 66,69]
(P>P’) 12.7 MeV
[Di 68]
(P,P’) 19.4 MeV
[St 66]
(P.P’) 104 MeV[Ka 84]
2; 0.74 0.73 0.74 0.71 0.69 0.54
3; 0.93 0.89 1.04 0.96 0.82
(P>P’) (3He,3He’) (M’) (d,d’) (n,n’) (6Li,6Li’)
800 MeV 25 MeV 20 MeV 15 MeV 70 MeV[Ba 83] [Ru 68] [FI 70] [Ba 75] [Wa 90] [Ho 92,93a]
2{ 0.59 0.70 0.66 0.67 0.66 0.563[ 0.94 1.12 0.83 0.89 0.88 0.74
Unweighted average for 2\ states: b7N = 0.67 ± 0.07 Unweighted average for 3[ states: b 3N = 0.91 ± 0.10
13
TABLE 1.5. Comparison of nuclear deformation lengths, 6 ^ , reported for the 2\ and 3[ states of 94Zr. Lengths are in fin.
( a ,a ’) ( a ,a ’) ( a ,a ’) (P,P’) (P>P’)35.4 MeV 35.4 MeV 65 MeV 12.7 MeV 19.4 MeV
[Si 8 6 ] [Ry 87] [Bi 66,69] [Di 6 8 ] [St 6 6 ]2\ 0 .6 8 0.63 0.64 0.71 0.70
3[ 1.03 1 .02 1.08 0.98 1.08
(3He,3He’) m (n,n’) (6Li,6Li’)25 MeV 20 MeV 8 MeV 70 MeV[Ru 6 8 ] [FI 70] [Wa 90] [Ho 92,93a]
2; 0.60 0.50 0.65 0.53
3[ 1.26 0.93 0.94 0.84
Unweighted average for 2\ states: 6 ^ = 0.62 ± 0.07 finUnweighted average for 3[ states: 63* = 1 .02 ± 0 .1 1 fm
14
TABLE 1.6. Comparison of nuclear deformation lengths, b tN, reported for the 2\ and 3[ ____________states of 96Zr. Lengths are in fm._____________________________________
( a ,a 1) (a ,a ') ( t ,f ) (6Li,6Li’)35.4 MeV 35.4 MeV 20 MeV 70 MeV
[La 8 6 ]_________[Ry 87]_________ [FI 70] [Ho 92,93a]2\ 0.64 0.64 0.38 0.473[ 1.19 1.23 1.05 1.05
Unweighted average for 2\ states: b 2N = 0.53 ± 0.08 fm Unweighted average for 3[ states: b 3N = 1.13 ± 0.08 fm
TABLE 1.7. Comparison of M JMP and B(E£) T deduced from the 35.4 MeV (a ,a ') data of Rychel et al., the (p,p') and (n,n') data of
Wang and Rapaport [Wa 8 8 , Wa 90] and the 70 MeV (6Li, 6Li') data of Horen et al. [Ho 92,Ho 93a]. Also shown are the
__________ M JM P deduced from the reanalysis of the 35.4 Mev (a ,a 1) by Horen et al. [Ho 92, Ho 93a] (see text)_________________
Reanalysis of Wang and
Rychel et al. ( a ,a ')* Rychel et al.b Rapaport0 Horen et al. (5Li, 6Li')d
Nucleus Mn/Mp B(E*) t
( e V )
Mn/Mp M„/Mp Mn/Mp B (E f)T e
( e V )
90Zrf’8
2\ states
1 .2 2 ±0 .1 2 0.06210.006 0.89 0.8510.06 0.85+0.10 0.06310.00592Zr 2.91±0.19 0.06910.006 2.31 (2.02) 1.05+0.07 1.3010.10 0.08310.00694Zr 3.02±0.22 0.05010.005 2.48 (2.03) 1.5010.22 1.5010.15 0.06610.01496Zr 4.69±0.64 0.027+0.007 3.86 (2.43) 1.50+0.15 0.05510.022
90Zrtg
3" states
1.80±0.31 0.066410.0073 0.94 (0.86) 0.92+0.13 0.60+0.08 0.07192Zr 2.17±0.45 0.0556+0.0077 1.44 (1.23) 1.20+0.13 0.85+0.10 0.06794Zr 2.3610.51 0.0794+0.0118 1.53 (1.42) 1.95+0.20 0.90+0.10 0.08796Zr 2.6710.47 0.10410.011 1.77 (1.57) 1 .1 0 +0 .1 0 0 .1 2 0
a Reference [Ry 87]b Reference [Ho 92, Ho 93a]; the value in parentheses was calculated using the B(E£) f from the last column.0 From the simultaneous analysis of (p,p') and (n,n'); Reference [Wa 88, Wa 90] d Reference [Ho 92, Ho 93a]e The B(Ef) f were fixed in the analysis of the 6Li scattering data [Ho 92, Ho 93a]. f From 90Zr(p,p’)90Z r\ Ep = 500 MeV [Ba 87]: Mn/Mp= 1.47 (2,+ state) and Mn/Mp = 1.12 (3 ‘ state).8 From 90Zr(p,p’)90Z r\ Ep = 800 MeV [Ga 82]: M„/Mp = 1.12±0.16 (2,+ state) and Mn/Mp = 1.06±0.05 (3[ state).
16
(a) 2\ States
2.5
5 2a% 1.5
(b)
0.5
2.5
a- 2
i 1.5
1
0.5
A - g O Q
B O “
-f-
K1>-‘ K i ^ . 4 k . ^ . i n i n o s > - ‘ O O O O ^ O O A M O M0 ' z n n n ' 2 ’Z ’z % „« B 9 8 5 B 8 f 6 " , , B “ , s P £ PB B
G>O00*0O'
3j States
§ = 5 - o - “
h-» G> in in o\ l-B Gi00 O O .U 00 Os 00 © 1- N OO 2 nn os
n n65 ft 2»■* 2 ON
C/3OC/3
00
B B cr
Figure 1.1 Some MJMP values reported for (a) 2\ and (b) 3J" states for several nucleii. Circles are reported from (p,p’) [Ba 87, Fe 94, Ga 82], triangles are (jij^jr) [Hi 92, Mo 87, Oa 87, SM 88], squares are ( a ,a ’) [Gi 75] and diamonds are (170 ,170 ’) [Ho 91b]. The horizontal lines are M JM P = N/Z.
Chapter 2: Theory of the Experiment
2.1 Isoscalar transition rates
A measure of the isospin nature of a nuclear transition is given by the ratio of the
neutron to proton multipole transition matrix elements, M JM P. An isoscalar transition is
one in which thje protons and neutrons respond equally, and in phase, to an external probe.
In isovector transitions, the proton and neutron response is opposite, or out of phase.
Excitation of the well known isovector giant dipole resonance, in which the protons and
neutrons vibrate against one another, is an example of an isovector transition. Transitions
to low lying \jibrational states, which are surface oscillations of the nuclear density
distribution, involve the proton and neutron distributions equally. These are therefore
expected to be isoscalar, or mass, transitions.
The neutron and proton reduced multipole transition matrix elements are given by
[Be 81a, Sa 89]
**„(,) (2.1)
is the neutron (proton) transition density. The reduced electric transition
lated to the proton transition matrix element [Sa 87]
where g f p)(r)
probability is re
We can
B (E £ )1 = e2M 2p = e2[fg; ( r ) r u2dr (2.2)
define an isoscalar, or mass, transition density as the sum of the neutron
and proton transition densities [Sa 87, Sa 89]
.isSi = g t + g \ (2.3)
The corresponding isovector transition density is the difference between the neutron and
proton transition densities. From (2.1) and (2.3) we find that the isoscalar (mass)
multipole transition matrix element and the corresponding reduced isoscalar transition
probability are given by [Ho 93a]
17
18
B M t = M . + M .
(2.4)
(2.5)
The ratio M JM P may now be expressed in terms of the electric and isoscalar
transition strengths. From equations (2.5) and (2.2) we see that
If we assume t
then
KM .
B M tB { E t ) \ j e 7
1/2
-1 . (2 .6)
lat the neutron and proton transition densities have the same geometry,
g ; ( r ) = M > ; g ( r ) , * / ( r ) - Z 6 ' * ( r ) . (2.7)
An isoscalar (mass) deformation length can be defined in terms of the neutron and proton
deformation len
Using (2.7) and
Iths
(2.8)
'2.8), the mass and proton transition matrix elements may be expressed as
< - M , af g ( r y ' 2dr, (2.9)
K - Z b ' f g ( r ) r M dr. (2.10)
The ratio of the neutron and proton multipole transition matrix elements is now given,
using eqs. (2.9) and (2.10) to determine Bb (^) t and B(E^) f in eq. (2.6), by
M n A bjis
M p Z b f— 1 . (2.11)
For a simple homogeneous isoscalar mass vibration, the neutron and proton
deformation parameters would be equal ( 6 / = 6 / ) [Be 81b, Ma 84]. In this situation, the
ratio of neutron to proton matrix elements is simply MJMP = N!Z. However, the detailed
shell structure for a given nucleus can produce a difference in the neutron and proton
19
deformation lengths, in which case MJMP will differ from this simple collective model
result. This will especially be true for those nuclei having closed major neutron or proton
shells [Be 81b, Ma 84].
The filling of a neutron shell will hinder the vibration of the neutrons in the
nucleus, and therjefore the neutron distribution will have a smaller deformation length than
the proton distribution [Be 81b, Ma 84]. This will push M JM P toward values that are less
than N/Z. The opposite is true for the filling of a proton shell. Closure of a neutron shell
thus drives M JM P to zero, while a closed proton shell drives the ratio toward infinity.
However] transitions among the valence nucleons may induce vibrational
excitations of the core distribution of the nucleus. This is known as "core polarization."
[Br 75, Be 81a, Be 81b] Because the nucleon-nucleon force is most effective between
unlike nucleons, excitations of valence neutrons will have a stronger effect on core
protons, and vice versa. Thus, while the closure of a neutron shell is expected to inhibit
the participation of valence neutrons in a given excitation, driving the neutron deformation
length to zero, the valence protons will excite the core neutrons, which will tend to
increase the neutron deformation length.
Although shell closure and core polarization effects tend to compensate for each
other, the vibrational excitations need not have equal proton and neutron deformation
parameters, even in open shell nuclei. When the underlying shell structure o f the filling
shells is strongly dominated by one type of nucleon, this type is favored in the ratio M JM P
[Ma 84]. This effect is most notable, however, in single closed shell nuclei. Using an
extension of the schematic model to open shell nuclei, Madsen and Brown [Ma 84] have
shown that there is a large jump toward equality of the deformation lengths with the
change of only two nucleons away from a magic number.
20
2.2 Coupled-Channels Description of Inelastic Scattering
The description of elastic and inelastic scattering of a projectile a on a target
nucleus A involves the solution of the time independent Schroedinger equation [eg. see Sa
90]
[£ - T/j'P = 0, (2.12)
where the Hamiltonian is given by
H = H . + H + K + V.A a (2.13)
HA>a are the Hamiltonians describing the target and projectile, K is the relative kinetic
energy, and V is the interaction potential. When the interaction V is particularly strong, a
perturbative treatment, such as the DWBA may be inadequate to accurately describe the
scattering [Fe 92]. An example is the excitation of low-lying collective states in the target
nucleus, where the amplitude for the transition to the collective state may be nearly as
large as the elastic scattering amplitude. In such cases, the strongly coupled channels must
be treated on an equal basis with the elastic scattering. In coupled-channels calculations,
several coupled states are treated exactly. Other, weakly coupled states are treated in a
phenomenological manner using an optical potential.
Coupledjchannels theory has been fully described in several references [Ta65,
Sa83, Fe92, Au70]. A description of the theory for the special case of a spinless projectile
which is not excited by the interaction is given below, in order to demonstrate how
information about the target nucleus is obtained from the scattering. This situation is
realized in the present alpha-particle scattering experiment.
The wave function W is written as a sum over a number of strongly coupled
channels, which are further expanded in partial waves
where
' aJMl(2.14)
4 w ( * J = \jM )ilY?‘ ( r ) ^ A(IAmA;xA) (2.15)mAm,
is the coupling of the orbital angular momentum with the target nucleus having spin IA.
The index a indicates the channel in which the target nucleus is in a particular state with
energy EA and spin IA. The variable x A represents the internal space coordinates of the
target, while r is the target-projectile separation.
Inserting (2.14) into the Schroedinger equation results in a series of coupled radial
equations for the functions u(r):
21
h2 ( d 2 £(£ + l ) \ . , . 'E " E a + 2 ^ [ d S " ~ 7 ~ ) ■
= J (aJMl |a JM£')ua Jt. (r)
uaji(r)(2.16)
a t
where the primes indicate that the combination a £' should be different from a£. The
matrix elements indicate integration over the angles and internal coordinates of the target
nucleus, and thus remain a function of r, the radial separation of the target and projectile
centers of mass. The potential of (2.13) has been separated into a diagonal part, V0, and a
non-diagonal part, Vc, which induces transtions in the target nucleus. The left side of the
equation describes elastic scattering in the channel a ; the diagonal matrix element of V is
the optical potential. The right side includes the coupling to other, inelastic channels. It is
these matrix elements which are important for the extraction of nuclear structure
information.
The coupling potential can be written in the general form [Ta65]
(2-17)k
where Q operates only on the internal space coordinates of the target and projectile
nucleus. The matrix elements of Vc are then
<a«|Vc | a « '> - (218)
The coefficient A is a geometrical factor. The matrix element of Q, taken between states
of the target nucleus, contains the information about the nuclear dynamics. It is through
this term that a scattering experiment allows us to extract information about a nucleus.
The coupled channels equations are solved for the radial functions u with the usual
boundary conditions that at large distances only the elastic channel has an incoming wave.
The other channels have only outgoing spherical waves.
The usual method for coupled-channels analysis is to choose a form for the optical
potential. The elastic scattering is then fit to fix any free parameters of the potential. A
model is chosen for the transition (or coupling) potential and the nuclear structure,
perhaps leaving one or more free parameters which are adjusted to best fit the inelastic
scattering.
2.3 Optical Potential - Folding Models
If the effective nucleon-nucleon interaction is known, then the optical potential
describing elastic scattering is given by folding the effective potential v(r) with the
projectile and target ground state density distributions [Sa 83]:
VpiT) “ J J P A ( O P . ( O K O < M rA. (2-19)
where the ground state density distributions are of the form
P . M - W S ^ r - r a K ) . (2.20)I
Figure 2.1 illustrates the coordinates used in the integral. Wa is the wave function
describing the ground state of the target. This folded potential is identified as the diagonal
matrix element of V in equation 2.16.
If a form for the projectile-nucleon effective interaction is known, then the
potential reduces to the single folding form [Sa 83]
V p ( r ) = f P a ( r A ) v ( | r " r A | ) ^ r A • ( 2 -2 1 )
22
Folding model transition potentials are obtained by replacing the ground state
density distribution in (2.20) or (2.21) by the transition densities [Sa 83]
S « - W 2 6 < r - r , ) k > - (2-22)I
The folding model relates the properties of the scattering potentials to the
properties of the density distributions in an unambiguous manner. There may still remain
some uncertainty as to the form of the interaction, v(r), and the density distributions, p(r).
However, once a model has been chosen, parameters extracted from fits to the scattering
data may be easily understood in terms of the distributions. The folding model also allows
one to put in constraints or known physical properties into the scattering problem. For
example, the density distributions may come from Hartree-Fock or shell model
calculations, or they may be of a phenomenological form (e.g. Fermi shape).
The effective interaction v used in the folding integrals is usually real. Therefore
the imaginary potential must be treated in a phenomenological manner. The usual method
is to assume that the imaginary potential has the same shape (but not necessarily the same
strength) as the real potential [Sa 83, Sa 87].
2.4 Models for Transitions Densities and Potentials
2.4.1 Collective model - Bohr-Mottelson form
The collective model for nuclear structure is described in several places [e.g. Bo
75]. We start with spherically symetric density distribution, and introduce a set of
multipole deformation parameters axji, which become the dynamical variables of the
model. The standard method is to start with a density distribution p(r,R), with a surface at
r = R . One example of this is the Fermi shape
p(r,/?0)= p0(l + ex)" \ x = (r - R0)/a . (2.23)
23
The surface of the distribution is then deformed according to the standard prescription [Sa
83, Sa 87]
24
(2.24)
For the case of a statically deformed, rotating nucleus (rotational model), the are the
deformation parameters of the nucleus in the frame of the body-fixed axes. If we are
considering harmonic vibrations of a spherical nucleus (vibrational model), they are
phonon creation and annihilation operators of angular momentum X and z-projection p.
We assume that we can make a multipole expansion of the density distributions [Sa
83]
p M .< p ) = 2)p<«(r )y<*(0 ’<P )’. (2.25)
P a .(0 = f P(r>e ><P)^(0 ,<P)dr
The multipole moments ptm play the role of the transition densities needed for the folding
model potentials. An expression for the multipole densities is found by making a Taylor
expansion of the density distribution, which to first order in the deformation parameters is
given by [Sa 83]
p (r ,R)= p (r ,R 0) + dp(r,xa ), „ v <*p(r,J?0) , (2.26)
df,- R4 a‘-Y‘- ^ rwhere xa stands for the dynamical variables a. The multipoles are given by
dp(r)(2-27)
For density distributions which are dependent on (r - R0), such as the Fermi shape
(2.23), we have d/dR0 = - d /d r . Note that the transition densities in this model are given
simply by the derivative of the ground state density distribution.
In order to see how the above density distribution gives rise to a coupling potential
of the form of equation (2.17), the effective interaction of the single folding model (2.21)
is expanded in a series of multipoles about the center of mass of the target nucleus [Sa 83,
Ma 75c],
25
When this expression is inserted into the folding integral (2.21), and the integral over the
angle coordinates of the target nucleus is performed, we obtain a multipole expansion for
the potential,
U r (r) - „(r) + ' 2 u b ,(r ,xA) Y T ( f ) - , (2.29)tm
where U0(r) is the optical potential of (2.19), and the Utm are given by
Utm(r ,xA) = « (r >rA)rAdrA . (2.30)
Equation (2.27) has been used for ptm. This expression is now of the form (2.17), where
we identify Qkfl s c t X(l, and
vx(r) s Rof d M ~ ViC’rA ) r 2AdrA . (2.31)
In the vibrational model, the reduced matrix element of eq. (2.18) for the excitation of one
2*-pole phonon in an even-even nucleus is [Bo 75]
(n, - 1 , r - 4 > J M - (2.32)
where the Kronecker delta insures conservation of angular momentum. The parameter (3
is the amplitude of the oscillation. The strength of the coupling for exciting the target
nucleus is proportional to 6 / = ft tRQ, the deformation length of the nucleus. This is
clearly the same as the isoscalar deformation length of eq. (2.8).
Because the transition densities are given by the derivative of the ground state
density distribution, once the elastic scatter cross sections have been fit, any parameters in
the transition densities, as well as those of the effective interaction v, are fixed. The only
free parameter left in the inelastic cross section is the nuclear deformation, b tN.
26
2.4.2 Microscopic models
Wave functions generated by a nuclear structure calculations (for example, random
phase approximation (RPA) or shell model calculations) may be used to calculate the
transition densities, eq. (2.22), for use in the folding integrals for the transition potentials
[Sa 83, Ma 75c]. These structure calculations are generally required to reproduce some
known property of the nuclear states in question, such as the energies or the transition
strengths. Cross sections calculated using these folded potentials, when compared to
measured cross sections, provide a means by which inelastic scattering may be used to
assess the accuracy of given model of nuclear structure.
2.5 Deformed Optical Model Potential
The deformed optical model assumes that due to the short range of the nuclear
potential, the scattering potential has the same shape as the underlying nuclear density
distribution, and is statically deformed and rotating, or undergoing shape oscillations in the
same manner [Sa 83]. This assumption is strictly true only for the special case of folding
with a zero range interaction and a point projectile.
In the deformed optical model (DOMP), we start with an optical potential with a
surface at r = R , and deform the surface according to the standard prescription, eq.
(2.24). A Taylor expansion is made in the same manner as for the density distribution
(eqs. (2.26) and (2.27). which leads to a multipole expansion for the potential,
U (r,R ) = U(r,R0) + ^ U tm(r,xa )Ytm(rT
dU dU • (2-33)Utm=R° d R ^ atn>=~R ° dr a *m
The last form is for the special case of a potential which depends only on the distance from
the surface of the potential, (r-R). An example of such a form is the commonly used
Woods-Saxon potential.
27
The potential (2.33) is once again of the form (2.17). The reduced matrix
elements are given by eq. (2.32). The strength of the coupling is once again proportional
to a deformation length, b tN = (5*R0. However, this is the deformation length of the
potential, U(r,R). Unlike the folding model, we cannot simply identify b eN of the potential
with 6 /5 of the density distribution.
It is common to assume that b tN = 6 /5 in the DOMP, i.e., the deformation length
of the potential is identical to the deformation length of the density. This is not strictly
true, and we may expect the excitation of a given transition by different probes (e.g. a , p,
n, e', etc.) to yield different values for the deformation parameters when the DOMP is
used [Be 81a, Be 81b]. This probe dependence has been examined [Be 81a, Be 81b] for
0+-*2+ transitions, and the values of MJMP deduced using different probe combinations
was found to generally agree to -15%.
Beene, Horen and Satchler [Be 93] have examined the inconsistencies between the
use of deformed optical potentials and the folding model potentials for the analysis of
inelastic scattering. They have shown that the discrepancy in the deformation parameters
(and thus the transition rates) extracted using the two methods can be quite large. These
discrepancies increase as the multipolarity I of the transition increases. They also note
that while the transition potential of the DOMP is independent of the multipolarity £ , a
realistic nuclear force having a finite range will generate transition potentials having a
strong dependence on £.
The deformed optical model potential (DOMP) has the advantage of simplicity,
while incorporating the appropriate physics in a qualitative manner [Sa 83]. However, it
has the disadvantage in that it loses the unambiguous connection between the density
deformation and the potential deformation of the folding model.
28
a)
projectile a target A
b)
Figure 2.1 Coordinates in the folding model integrals for a) the double folding model,
and b) the single folding model.
Chapter 3: Description of Experiment
3.1 Details of Setup
The experiment was performed over a period of two weeks at the A.W.Wright
Nuclear Structure Laboratory at Yale University. A 35.4 Mev beam of alpha-particles
was produced by the Yale ESTU tandem accelerator. Beam currents ranged from 0.3 nA
for small-angle, elastic scattering measurements, up to 200 nA for large-angle elastic and
inelastic scattering measurements.
Isotopically enriched targets of 90’92’94’96Zr were used for the cross section
measurements. The characteristics of the targets are shown in Table 3.1. These targets
are the same ones used in the 6Li scattering experiment [Ho 92, Ho 93a]. The thickness of
these targets was remeasured by recording the energy loss of the 8.784 MeV and 6.288
MeV alpha-particles emitted by a 228Th source. Energy losses for various thicknesses of
zirconium were calculated using the program STOPX, and were then compared to the
measured energy losses to determine the actual target thickness. These new
measurements indicated that the zirconium targets were roughly 10% thinner than
previously reported [Ho92, Ho93a]. This accounts for the renormalization that was
necessary in the analysis of the 6Li scattering cross sections [Ho 92, Ho 93a]. Note that
the 96Zr target contained a small tungsten contaminant from use in a previous experiment;
this was corrected for in data analysis. In addition, a Mylar target was used to help
identify elastic scattering from carbon and oxygen contaminants in the zirconium targets.
A schematic drawing of the target chamber is shown in Figure 3.1. The beam
dump used to measure the beam current during this experiment was rather small (3/8"
diameter x 1/2" deep). The small size allowed measurements to be made at angles in to 4
degrees. This beam dump was biased at +300 volts in order to suppress electrons from
escaping and leading to erroneous beam current measurements. In order to ensure that we
were making the accurate beam current measurements necessary to extract absolute cross
29
30
sections, the performance of the small beam dump was tested against a large cup which
covered 99.5% of 4jt, with electron suppression provided by a +300 V bias and by
magnets mounted at the entrance to the cup. The cup could be rotated in and out of the
beam, while the beam dump remained fixed. Beam currents from 0.5 nA to 50 nA were
checked, using current readings from the image Faraday cup of the accelerator’s analyzing
magnet to provide a normalization. In all cases the small beam dump measured the same
current to within ~1% as the large cup.
Two 1” square permanent magnets were placed on either side of the target in order
to suppress electrons liberated from the target by the incident beam from reaching the
beam dump. A field of -900 gauss was produced at the target center, and fell off rapidly
outside the dimensions of the magnets. The performance of this system was tested by
comparing beam currents measured with a target in place and with the targets completely
removed from the incident beam. The analyzing magnet image Faraday cup was used as
above to provide a normalization. No measurable difference was observed between beam
currents measured (to <1%) with or without the targets in place.
The calibration of the Beam Current Integrator (BCI) was checked by comparing it
with the reading of a Keithley picoammeter normally used to measure the currents in
several Faraday cups along the ESTU beam line. Current for these measurements was
supplied by connecting a battery in series with a large resistor.
The combination of target thickness, beam current integration, and aperture solid
angle was calibrated by comparison between our measured elastic scattering cross sections
and optical model predictions (see Fig. 4.11 and section 4.2.1). These comparisons
resulted in overall renormalizations of < 5% (Table 4.1) relative to the nominal measured
values.
The scattered alpha-particles were momentum analyzed in an Enge Split Pole
Magnetic Spectrometer [Sp 67, En 79]. Figure 3.2 shows a diagram of the spectrometer,
along with some of its performance characteristics. The magnetic field of the
31
spectrometer was set at 1.1 Tesla, well below the maximum value of 1.63 T. Note that,
by design, the path of a particle along the central ray of the spectrometer crosses the focal
plane at an angle of 45 degrees. In addition, the position of a particle crossing the focal
plane of the spectrometer was calibrated in terms of the radius of curvature, p, of the path
of the particle as it passes through the magnet. This was done by detecting the 8.78 MeV
alpha-particle emitted by a 228Th source in the spectrometer focal plane detector
(described below) using several magnetic field strengths. The radius of curvature, p, was
calculated using the standard non-relativistic formula for a charged particle passing
through a magnetic field. For an alpha-particle, this reduces to
p(mm) =1440 E f ( M e V ) /B ( K g ) . (3.1)
A blocker plate, located just before the entrance of the focal plane detector, was then
positioned to reduce the count rate of the 8.78 MeV alpha-particles by 1/2. The position
of the blocker gave a measure of the position, in mm, at which the alpha-particles crossed
the focal plane with respect to some arbitrary reference point. Empirically it was found
that the position was given by
X(mm) = 2.79 p (mm) + C (3.2)
where C is an arbitrary constant use to define the reference point.
Slit plates (each with an array of five vertical apertures with the centers of adjacent
apertures separated by 1°) were mounted in the target chamber so that they could be
positioned in front of the entrance to the spectrometer. These apertures were used (a) to
accurately define a reduced solid angle, smaller than that of the spectrometer and (b) to
admit particles simultaneously which had been scattered through five discrete angles.
The slit plates were machined out of 0.076 mm thick tungsten. The dimensions of
each of the apertures were measured to a precision of ±0.01 mm using an optical
comparator. The aperture widths were nominally 0.80 mm, corresponding to an angular
width of « 0.3° in the scattering plane. The out-of-plane acceptance angle was different
for each of the plates, varying from ±0.3° to ±1.2° (corresponding to nominal aperture
heights of 1.5 to 6.1 mm). The taller apertures were used for measurements at large
scattering angles, while the shorter slits were used at smaller angles in order to minimize
the deviation of the (out-of-plane) scattering angle over the rectangular aperture from the
in-plane scattering angle. The thickness of the tungsten plates was -1/10 the width of the
individual slits, which helped to reduce the amount of slit scattering observed in the
experiment. A 35 MeV alpha-particle is not stopped in this thickness of tungsten, but
loses about 12 MeV in passing through the plate. However, the 23 MeV alpha-particles
were bent through too small a radius in the magnetic fields used during this experiment to
hit the focal plane detector. Thus, only those particles which passed through the slit
openings were detected.
The focal plane detector is similar to those described in [Er 76, Fu 79]. It consists
of a gas filled ionization chamber followed by a 6.35 mm thick plastic scintillator (figure
3.3). The ionization chamber was filled with isobutane at a pressure of 150 torr. The
cathode signal gives a measure of the energy lost by any particle in the gas. Particles
passing through the gas chamber are generally stopped in the plastic scintillator. The
anode signals from the photo tubes at the ends of the scintillator are summed to provide a
measure of the energy lost by the particle in the plastic. The combination of the gas
counter cathode and the plastic scintillator enables E-AE particle identification.
The gas counter also has two position sensitive wires (fig. 3.3). Each of these
consist of a 0.051 mm diameter wire surrounded by a series of small copper “C”-shaped
pickups with 1 mm segmentation. The central wires were biased at +1300 V (front wire)
and +1400 V (rear wire). Electrons from ionized isobutane create an avalanche as they
are attracted to the wires, which induces a pulse in the copper pick-up near the site of the
avalanche. These pulses are read out through a series of lumped delay line chips. The
signals travel to both ends of the series, and the relative delay between the signals at each
end is a measure of the position of the particle as it passes the wire.
32
33
3.2 Ray Tracing
The use of two position sensitive wires enables us to trace the path of a particle as
it traverses the detector. This allows us (i) to measure the angle at which it crossed the
focal plane, thereby determining the scattering angle of the particle, and (ii) to transform
the spectra measured at the position of the front wire to the focal plane of the
spectrometer. Because of the ray tracing capability, the front wire of the detector need
not be placed at the focal plane of the spectrometer in order to obtain good resolution.
Figure 3.4a illustrates how ray-tracing is performed. In order to carry out the ray
tracing, the front and rear wires must be carefully calibrated. This was done by using
elastic scattering of 35.4 MeV alpha-particles off the 90Zr target at 0 = 10° using various
values for the spectrometer magnetic field. The entrance aperture to the spectrometer was
closed down to ±10 mrad to provide a narrow beam of particles through the magnet. The
position along each wire was fitted to a quadratic function of the recorded channel
number,
where i = F,R refers to either the front or rear wire, and C is the channel number. The
constant terms, a0R and a 0F, are somewhat arbitrary, and are used to determine the point of
reference from which positions along each wire are determined. The position along the
focal plane, X, was determined from the X(mm) vs. p(mm) calibration, with p (radius of
curvature through the magnet, given by eq. 3.1) being calculated from the known alpha-
particle energy and magnetic field settings. The front and rear wires were calibrated
separately by placing each wire in turn as close to the location of the focal plane as
possible.
For a given event, the channels recorded for the front and rear wire were
converted in software into position along the wire in millimeters. The angle of incidence
on the front wire for each individual particle path was then calculated using the relation
(3.3)
(3.4)
where the separation of the front and rear wires was H = 102 mm. XR(F) refers to the
position along the rear (front) wire. Note that the angle is defined relative to a normal to
the front wire, and therefore the large angle ray of Figure 3.2 will have an incident angle
on the front wire less than 45°. Once the incident angle was determined, the position at
which the ray crossed the focal plane could then be calculated by the relation
X foc= X F -D tan(Q ), (3.5)
where D is the distance from the front wire to the focal plane of the spectrometer. The
spectrometer is designed so that a particle passing through the center of the spectrometer
should intersect the focal plane at 45°. Therefore, the constant terms of the software
calibrations (a0 of eq. (3.3)) were adjusted so that when the spectrometer entrance was
closed down to ±10 mrad, an incident angle of 45°, as seen in the calculated angle
spectrum, was observed.
A blocker was place in front of the detector to cut off the elastically scattered
particles so that higher beam currents could then be used in order to enhance the count
rate of inelastically scattered alpha-particles. The best location for the blocker, in order to
most efficiently block off elastically scattered particles while not blocking any of the
inelastically scattered particles, is right at the focal plane of the spectrometer (Fig. 3.4).
However, this required the front wire to be placed about 10 cm behind the focal plane.
The detector could not be moved this far from the focal plane. Instead it was moved as far
as possible in order to place the blocker as close to the focal plane as possible.
Figure 3.4(b) illustrates the typical situation for inelastic scattering measurements
using a five-slit plate. Rays from each of the five slits pass through a point on the focal
plane, which is located ~4 cm from the front wire. This diagram is for inelastically
scattered alpha-particles having an energy of E = 33 MeV, which is fairly typical for states
in the zirconium isotopes located just under 2 MeV in excitation energy. The slits are
separated by 1° in the horizontal scattering plane, while each slit subtends an angle of 6©
= 0.3°. The Split-Pole Spectrometer has a horizontal magnification of Mx = 0.39
34
35
(Fig. 3.2), which means that the rays will be separated by A0fp = 2.6° at the focal plane,
and will subtend an angle of 6©fp = 0.8°. With the wire and focal plane separations
indicated in Figure 3.4(b) we find that ray (1), which intersects the focal plane at 0 fp =
39.8°, covers ~3.5 mm at the rear wire, while ray (5), which intersects the focal plane at
©fp = 50.2°, covers ~4.8 mm. Rays (1) and (2) are separated by ~11.5 mm, while rays (4)
and (5) are separated by ~15 mm. These sizes and separations are to be compared with
the 1 mm segmentation of the “C” position pick-ups on the wires. (Note that the 1 mm
segmentation should enable the resolution of tracks differing by -0.4° for particle paths
intersecting the focal plane at angles near 45°. This corresponds to scattering angles at the
target differing by ~0.16°.)
Figure 3.4(b) is drawn for the ideal case of a perfect focus. If we assume that none
of the observed energy resolution of 6E ~ 130 KeV is due to the focal plane detector
(obviously an incorrect idealization), then using eqs. (3.1) and (3.2), we find that scattered
particles having energy E = 33 MeV will produce a spot of ~4 mm along the focal plane.
Over this small distance (the active areas of the position wires are on the order of 700 mm
long) the angles of the particle paths cannot vary much. The net effect will be to “smear”
the rays of Fig. 3.4(b) over a space of 4 mm along the focal plane, but will have negligible
effect on the relative positions (Xf - Xr) at which particle tracks pass the front and rear
wires. Thus a finite beam size will not affect our ability to accurately construct an angle
spectrum, or equally, to determine the scattering angle by identifying the slit through
which the particle passed.
Knowledge of the scattering angle, through the measurement of the angle of the
particle passing through the detector, allows us to measure the scattering yield at several
angles simultaneously. This greatly reduces the amount of beam time necessary to
perform the experiment. During most runs, one of the plates with five slits was located in
the target chamber just in front of the entrance to the spectrometer. Figure 3.5 shows the
raw front and rear wire spectra for elastic scattering off 90Zr at a spectrometer angle of
26.0°, along with the resulting angle spectrum. The front wire was near the focal plane.
The spectrum from the rear wire, which was 10.2 cm behind the front wire, shows how
the rays from different slits separate after passing throught the focal point. The angle
spectrum has been gated on the front wire elastic peak. The scale of the angle spectrum is
such that the channel number is roughly ten times the angle of incidence to the front wire.
The angle of incidence is defined from a normal to the front wire. This means a particle
with a larger scattering angle will have a smaller angle of incidence at the detector. Thus,
in this figure, the peak at the highest channels corresponds to particles scattered through
24°; the one at the lowest channel is from particles scattered through 28°.
For angles less than 20 degrees, slit-scattering of the elastically scattered particles
on the slit plate produced a long tail which could not be removed by the blocker or by ray
tracing. This tail was so large that it prevented us from making inelastic scattering
measurement using the slit plate at 0 < 20°. For these angles, a procedure of angle binning
had to be used. First, a short elastic run was made using the slits in order to get an angle
calibration. Then the slits were removed, and the spectrometer entrance aperture was
used to define the vertical opening angle. A series of software gates on an angle spectrum
were then used, based on the angle calibration, to determine the horizontal angle. The
solid angle for the given measurement was then given by the product of the vertical angle
opening of the aperture and the horizontal angle of the software gate. This process is
discussed in detail in Chapter 4. Use of the slits gives a more reliable method of
determining the scattering angles, however, and so as much of the data as possible was
acquired using the slit plates.
33 Electronics
Standard NIM and CAMAC electronics were used for the data acquisition. An
electronics diagram is shown in Figure 3.6. The event strobe consisted of a coincidence
between the anode and cathode of the gas counter and the plastic scintillator. For each
36
event, the front and rear wire positions, the gas counter anode and cathode pulse heights,
and the scintillator pulse heights were recorded. The data were written to tape event by
event for off-line analysis.
The combination of gas cathode-gas anode or plastic scintillator-gas cathode were
used for E-AE particle identification as described in section 3.1. Normally, particle
identification would be used to help filter out background from the wire spectra, by
placing a two dimensional gate around the desired ion species on the E-AE spectra.
However, in the present experiment, only alpha-particles were observed in the focal plane
detector.
The target current was monitored with Brookhaven Instruments Corporation
Model #1000a beam current integrator feeding a CAMAC scalar. In addition, the number
of event strobes during a given run were also scaled. “Live” scalars were inhibited during
the time in which the data acquisition computer was processing an event, and only
counted event strobes or BCI pulses which occured during a time at which the computer
was ready to accept new data. “Raw” scalars counted all event strobes or BCI pulses
which occured during a given run. The ratio of “live” to “raw” counts was used to
determine the percentage of time during a run in which the computer was able to accept
new data. Both raw and live BCI and events were counted in order to measure the live
time of the acquisition system.
3.4 Experim ental Procedure
Measurements were made in 0.5 degree increments from 0|ab = 6.0° to 46.5° for
elastic scattering, and from 0|ab = 8.5° to 46.5° for inelastic scattering. Elastic and
inelastic yields were measured during separate runs, with a blocker inserted in front of the
detector to screen off the elastically scattered particles during inelastic scattering
measurements. While making the inelastic runs, a small amount of the elastic peak was
37
allowed to leak into the detector in order to ensure that none of the inelastically scattered
particles were being blocked.
Most runs were made using the slit plate. After the measurements were made at
one angle setting, the spectrometer was then moved 0.5 degrees in order to get the half
degree increments. The spectrometer was then moved 2.5 degrees further out. Thus, a
series of measurements were made for angle sets of (6°,7°,8°,9°, 10°),
(6.5°,7.5°,8.5°,9.5°,10.5°), (9°,10°,11°,12°,13°), (9.5°,10.5°,11.5°,12.5°,13.5°), etc. For
each run, there were two overlapping angles with previous runs, enabling us to check the
consistency of the results between the various angle sets.
Whenever the angle of the spectrometer was changed, a short run was made using
elastic scattering off the 90Zr target with the spectrometer entrance aperture closed to ±10
mrad. The slit plate was then positioned so that scattering through the central slit was
measured to be incident on the detector at the same angle as scattering through the
spectrometer entrance aperture.
The elastic scattering cross sections were measured with an uncertainty of ~4%,
while the inelastic scattering cross sections were measured to -5-8% . Table 3.2 lists the
most important contributions to the uncertainties of the cross section measurements.
38
39
Table 3.1 Thickness and isotopic composition of the zirconium targets
TargetThickness3(mg/cm2) 90Zr
i
91ZrComposition (%)
92Zr
b
94Zr 96Zr
90Zr 0.855±0.027 97.67 0.96 0.71 0.55 0.1392Zr 0.845±0.025 2.86 1.29 94.57 1.15 0.14
94Zr 0.910±0.027 1.67 0.42 0.76 96.93 0.22
96Zr 0.828±0.025 7.25 1.41 2.24 3.85 85.25
a Remeasured for this experiment (see text). b Ref. [Ho 93a].
40
Table 3.2 Typical contributions to uncertainties of measured cross sections
Instrumental: Target thickness 3%Solid angle 1.8%, slit plate
7%, angle binning
BCIa 1.5%
Statistical Scattering yields 1.4%, elastic scatter3%, inelastic scatter
a BCI = Beam Current Integrator.
Figure 3.1 Schematic drawing of target chamber. The magnets about the target are used to suppress target electrons from reaching the beam dump. BCI refers to the beam current integrator. Also indicated is the position of the slit plate. This slit plate contains five slits, separated by 1“ in the scattering plane.
42
Performance Characteristics of the Yale Split-Pole Spectrometer
Solid Angle: 160 mrad x 80 mrad = 12.8 msr
Orbit radii at full solid angle: 51.1 cm to 92.0 cmFirst order resolution for 1mm target spot: Ap/p = 1/4290 for p = 92 cm
Momentum range: Pm„ /Pm;, = 1.80Maximum field strength: B = 16.3 TeslaMagnifications: Mx = 0.39, My = 2.9
Figure 3.2 Schematic drawing of the Yale Split Pole Spectrograph. The positions of the target and focal plane detector are indicated. The central ray of the
spectrometer intersects the focal plane at a 45° angle, (adapted from
[Ha 93])
43
PhotomultiplierTube
Side View bc-404PlasticScintillator
Figure 3.3 Schematic drawing of the focal plane detector. The AE and E(residual) anodes are usedto identify heavy particles which stop in the gas. For light particles, such as a particles, the cathode is used for the AE signal while the scintillator is used to provide the E signal used in particle identification. Position information is obtained from charge induced on the series of 1 mm wide copper “C”-shaped pick-ups surrounding each wire. This induced charge is collected from each end of the series of delay line chips, and the time difference between the arrival of the signals at each end provides a measure of the position at which the tracks of the incoming particles passed the wire. FW and RW refer to the front and rear wire, respectively.
44
a)i
tanG = { x f - X r) / H X foc = X f - Dtand
Figure 3.4 a) Diagram of the focal plane detector showing how ray tracing is performed. Use of two wires enables the extrapolation of the path back to the focal plane. Distances along the wires and the focal plane are measured from an arbitrary point to the right of the diagram.b) (next page) Illustration showing a typical situation in which a five-slit plate is used to define solid angles for cross section measurements. The blocker is positioned to cut out elastic scattering events, allowing a larger beam current to be used when recording inelastic scattering data. Some of the elastically scattered particles were allowed through to ensure that no inelastic scattering events were being cut off. The detector could not be moved back far enough to put the blocker at the focal plane, in this example 40 mm in front of the front wire; instead, the blocker was located as close to the focal plane as possible. Ray tracing enables the creation of spectra with good resolution, even if the events are measured well beyond the focus. Assuming the slit plate to be positioned so the center slit is set at a laboratory scattering angle of 0sp, then ray (1) corresponds to inelastic scattering through an angle 0sp-2°, ray (2) through 0sp-l°, ray (3) through 0Sp, ray (4) through 0sp+l°, and ray (5) through 0sp+2°. Ray (e) is elastic scattered events allowed to “leak” around the blocker plate. See text for further discussion.
46
channel
Figure 3.5 Example of the use of ray tracing to measure the scattering angle. Elastic scatter of 35.4 MeV alpha-particles off 90Zr with the spectrometer at 26°. A plate with five slits separated by one degree is located before the entrance to the spectrometer, a) raw front wire spectrum. The front wire is near the focal plane, and the elastic peak is focused, b) raw rear wire spectrum. The rear wire is 10.2 cm behind the front wire, and the separation of scattering through each of the five slits is clearly seen beyond the focal point, c) angle spectrum created by gating on the elastic peak on the front wire spectrum. The channel number is equal to ten times the angle of incidence to the front wire. The peak near channel 510 is from the slit at 24° scattering angle, while the peak near channel 490 is from 28°.
47
Figure 3.6 Electronics Diagram for AZr(a,a'). FW = Front Wire, RW = Rear Wire, BPA = Brookhaven cold termination pre-amp, CPA = Canberra pre-amp, TFA = timing filter amplifier, CFD = constant fraction discriminator, TAC = time-to -amplitude converter, SCA = single channel analyzer P/S = Phillips Scientific.
Chapter 4: Analysis
4.1 Extraction of Cross Sections
The data were stored on tape event-by-event and analyzed off-line in order to
extract the elastic and inelastic scattering yields. All elastic measurements, as well as
inelastic measurements for 0|ab ^ 20°, were made using the slit plate to define the solid
angle. For inelastic scattering at 0|ab< 20°, a process of angle binning was used to extract
the scattering yields (see below). An energy resolution of AE~130 keV was obtained. At
this resolution, the 2\ and 3[ states of 96Zr are just resolved.
The elastic scattering data were analyzed using the front and rear wires to create
an angle spectrum, as discussed in Section 3.2. The angle spectrum was gated by the
elastic peak on the raw front wire spectrum. An example of such a spectrum for the
elastic scattering by the 90Zr target with the spectrometer at 26° was shown in figure 3.4.
The number of alpha-particles elastically scattered through a given angle was found by the
summing the number of counts in the appropriate peak of the angle spectrum.
An angle spectrum for the inelastic data was created using the front and rear wires
in a manner similar to the elastic analysis. The angle spectra were gated by the inelastic
peaks on the front wire. However, the inelastic scattering peaks sit on significant
background caused mostly by slit-scattering of the elastically scattered alpha-particles.
The number of inelastically scattered alpha-particles could therefore not be determined
simply by summing the counts in a given peak of the angle spectrum. For those runs in
which the slit plate was used, ray-traced spectra which had been transformed back to the
focal plane in order to provide the best energy resolution were gated by the peaks of the
angle spectrum. The process is illustrated in Figure 4.1. Figures 4.2-4.5 show examples
of such spectra for each of the four targets. The scattering yields were measured by fitting
these spectra with a series of Gaussian peaks.
48
49
The small angle inelastic data (0 < 20°) had to be analyzed using the process of
angle binning. This process is illustrated in Figure 4.6. A short run was first made using
the slit plate in order to calibrate the angle spectrum in terms of the scattering angle at the
target. The slit plate was then moved out of the way in order to record the inelastic
scattering data. The angle calibration was used to create gates in the resulting angle
spectrum which were centered on the desired scattering angle, and subtended a horizontal
(in-plane) angle of 0.25° at the target. The spectrometer entrance aperture was used to
define the vertical (out-of-plane) angle. The solid angle when using angle binning for the
measurement was given by
4 D - ( A 0 ) fc,-(A q p ),^ , (4.1)
where A0fcjn=O.25° was the width of the angle bins. The angle bins were used to create
angle-gated ray-traced spectra as before, which were then fitted using Gaussian peaks in
order to extract the low-angle inelastic scattering yields.
The absolute cross sections were calculated using the target thickness and
compositions, the experimental geometry as determined by the slit plate or angle binning,
and the Faraday cup readings. The data were corrected for the live time of the acquisition
system. In addition, the cross sections were adjusted using the renormalization factors
obtained from the elastic OMP fits (section 4.2.1). The resulting elastic scattering cross
sections were measured with an uncertainty of -4% , while the inelastic cross sections have
an uncertainty of ~5-8%.
The elastic cross sections are shown in Figure 4.7, while the cross sections for
exciting the 2\ and 3j states are shown in Figures 4.8 and 4.9, respectively. Also shown
in these figures are curves calculated using the results of Rychel et al. [Ry 87]. We note
that while the curves of [Ry 87] match our data in the CNI region (0 ^ 20°), there are
significant shifts in the positions of the maxima between the calculated and measured cross
sections at larger angles. The data of [Ry 87] are not available to us for direct
comparison; plots of their inelastic cross sections (taken from [Ry 87]) are shown in
50
Figure 4.10. The fits of [Ry 87] show angle shifts with respect to their data at the larger
angles, similar to the shifts seen in Figures 4.8 and 4.9. From a comparison of figs. 4.8,
4.9 and 4.10, we note that the data from the present experiment is generally consistent
with the cross sections of [Ry 87], although the cross sections for exciting the 3[ state of
90,96Zr are slightly smaller than those of [Ry 87]. The nuclear deformations, B(E£) |
values and M JM P ratios extracted from the present experiment may be compared to the
corresponding results from [Ry 87] in order to determine the consistency of the two
experiments.
4.2 Deformed Optical Model Analysis
In the DOMP model, because of the short range of the nuclear interaction, we
assume that the scattering potential has a shape similar to that of the underlying nuclear
density distribution [Sa 83]. The deformation length of the DOMP potential is assumed to
be the same as that of the deformed nuclear density distribution. While plausible, this
assumption can be shown to be inconsistent with folding model calculations and thus
makes questionable the meaning of deformation lengths deduced from analyses of inelastic
data using the DOMP model [Be 93, Ho 93c]. However, we have analyzed our data using
the DOMP model in order to make comparisons with the earlier analyses which have also
employed this method [Ry 87, Ho 92, Ho 93a].
4.2.1 Elastic scattering
The nuclear portion of the optical potential used to describe the elastic scattering
was taken to be of the standard Woods-Saxon form,
where
/ ( * , . ) - ( l + e ' ) _1, x.t - ( r
R ^ r - ^ A f + i=V,W.
(4 -2)
51
Here, Ap and A, refer to the mass numbers of the projectile and target nuclei, respectively.
The real and imaginary parts of the potential were assumed to have the same shape, i.e.
rv = rw and av = aw. The Coulomb potential was taken to be that of a point charge
interacting with a uniform charge distribution of radius R c = rc {^A^3 + A,V3) fm. In this
work we adopted the value rc=1.2 fm.
The program PTOLEMY [Mac 78, Rh 80] was used to optimize the fit to the
elastic data from each target by varying the four optical model parameters using the
standard %2 criterion and the experimental uncertainties. In addition, an overall
normalization parameter was allowed to vary. This renormalization factor is used to
account for uncertainties in such factors as beam current integration and target thickness
measurements, and was used to adjust the extracted cross sections for all further analysis.
The optical model parameters thus obtained are listed in Table 4.1. There is considerable
ambiguity associated with optical model parameters determined from fits to low energy
alpha-particle scattering [e.g., see Sa 83]; however, it is found that those giving
equivalent fits to the elastic data also yield very similar inelastic cross sections [Ry 87].
This is due to the strongly absorbing nature of alpha particles [Be 69]; the scattering wave
functions, and thus the cross sections, only depend on the shape of the surface region of
the scattering potential. Two scattering potentials having the same form in the surface
region will lead to identical scattering cross sections. We have chosen potential sets with
V between 200 and 300 MeV. The corresponding fits to the elastic data are shown in
Figure 4.11.
4.2.2 Inelastic Scattering
In the DOMP calculations of the inelastic cross sections, the nuclear transition
potential for angular momentum transfer i is assumed to have the form
U ?(r) = S tN ^ j r (4-3)
where UN(r) is the optical potential (eq. 4.2) with parameters determined by the fits to the
elastic data (Table 4.1). This form follows from eqs. (2.33, 2.32, and 2.18). Here we
52
have assumed that the real and imaginary deformation lengths, btN, are equal. The total
transition potential is the sum of the nuclear and Coulomb transition potentials. At large
radii, the Coulomb interaction is completely determined by the reduced electric transition
probability, B(E£) | . For radii less than Rc, the Coulomb interaction was taken to have
the form for a point charge interacting with a deformed, uniformly charged sphere of
radius Rc [Mac 78, Rh 80, Sa 87]:
where Zp is the atomic number of the projectile.
The assumption was made that the deformation length of the DOMP, b tN, was the
same as the mass deformation length, 6 /5, and may be used in equation (2.11) in order to
deduce the value of M JM P. A measure of the proton deformation length was obtained
This expression corresponds to the proton radial transition density being a delta function
at r = Rc ; calcuiations with a more realistic shape indicate that the error made with this
expression is smJll, e.g. less than 5% in b f .
The inelastic cross sections were calculated by the program PTOLEMY using
coupled-channels [Sa 83, Mac 78, Rh 80]. The effects of the couplings of the inelastic
channels on the elastic cross sections were found to be small, so the optical model
parameters deduced from fitting the elastic data (listed in Table 4.1) were adequate for use
in the calculations of the inelastic cross sections. A series of calculations were performed
for several combinations of B(EQ f and b eN to obtain the best fits to the inelastic data.
In Figure 4.12, the best fit calculations for exciting the 2\ states are compared
with the data. The corresponding B(E2) \ and 6 * are listed in Table 4.2, where they are
compared with the results of Rychel et al. [Ry 87]. Rychel et al. report their deduced
(4.4)
from the B(Ef) f value by using the expression for a uniform charge distribution of radius
Rc = 1.2A f fm,
53
isoscalar transition strengths, B]S(£) j- We have used these values, and the formulae in
[Ry 87] in order to extract the nuclear deformation parameters listed in Table 4.2. The
overall agreement between these two alpha-particle measurements is considered to be
excellent. Rychel et al. [Ry 87] analyzed their data by means of x2 fitting, whereas we
have determined the fits by visually comparing calculations to the data. In their x2
analysis, Rychel et al. [Ry 87] restricted their analysis to data for which 0c.m. < 30° because
they found that inclusion of the larger angle data broadened their x2 distributions. The
reason for this can most likely be understood from Figure 4.12, where it is clear that at the
larger angles there are significant phase shifts in the oscillations of the angular distributions
between the data and the calculated curves. At angles ^ 25°, the maxima for the
calculations of the 90Zr cross sections are shifted toward smaller angles than our data,
while the maxima for the calculations of the other zirconium isotopes are shifted toward
larger angles than the data. Because our data for each angle were measured sequentially
for all four targets without changing any of the spectrometer or slit-plate settings (e.g. in
the sequence ... 90Zr(25° - 29°), 92Zr(25° - 29°), 94Zr(25° - 29°), 96Zr(25° - 29°), 90Zr(28o -
32°), 92Zr(28° - 32°), etc.), this difference between the 90Zr data and the 92>94>96Zr data is
real and not the result of some systematic errors in the angle settings, as might have been
the case if the data had been measured in the sequence (... 90Zr(25o - 29°), 90Z t{2So - 32°),
90Zr(33o - 37°), ... 92Zr(25° - 29°), 92Zr(28° - 32°), etc.). Furthermore, the angular
dependence of our data is in excellent agreement with the independent measurement of
Rychel et al. [Ry 87] (figs. 4.7 - 4.10). We address the possible explanations for this shift
of phase between the data and calculations in Section 4.4.
In the top section of Figure 4.13 we show calculations for excitation of the 2\
state of 90Zr by pure Coulomb, pure nuclear and combined interactions. The cross section
at the larger angles is dominated by the nuclear component which determines b tN. There
is a strong interference near 0 « 15° which can be used to determine the ratio of the
nuclear and Coulomb amplitudes, 6 * / b f . In the analysis of the present data, as well as
54
that of Rychel et a I. [Ry 87], emphasis was placed upon reproducing the shape and
magnitude of the data in the smaller angle regions. The fact that the calculations also
match the magnitudes of the maximum cross sections at the larger angles, but not the
locations of the maxima, suggests that other phenomena beyond the simple coupling of the
2\ state and the ground state are occurring which affect the relative phase between the
measured and calculated oscillations. We have investigated the inclusion of other
couplings as well as reorientation effects, but these did not produce phase shifts of the
observed magnitudes (see Section 4.4).
In Table 4.2 we also list the h 2 obtained from DOMP analyses of the 6Li data [Ho
92, Ho 93a] in which the values of B(E2) \ were held fixed. As noted earlier [Ho 92, Ho
93a], there is excellent agreement between the alpha-particle and 6Li results for the 2\
state of 90Zr. The agreement for 92,94Zr is not quite as good, as the mean values of the h 2
deduced from the alpha-scattering data are about 20% larger than those from the 6Li
measurements for these two isotopes. In light of the excellent agreement found for the 2\
state of 90Zr, the reason for the differences obtained for the b 2 for 92,94Zr is not clear.
Part of this could be accounted for by the small differences in the corresponding B(E2) \
values, but it is not sufficient to completely account for all of the differences. For
example, if the present 92Zr alpha-scattering cross sections are fitted by fixing B(E2) t =
0.083 e2b2, then the best fit is obtained for 6 * = 0.651 fm, as compared to the value 6 * =
0.557 fm extracted from the 6Li scattering data. If the value B(E2) f = 0.066 e2b2 is
adopted to fit the 94Zr cross sections then we obtain 6 * = 0.594 fin from the present
alpha-scattering data, which is to be compared to b 2 = 0.525 fin extracted from the 6U
scattering. Although it has been commonly expected that the b tN extracted from inelastic
data for different probes using a DOMP analysis would be nearly the same [Be 81a,b],
there is no guarantee that the deficiencies in the DOMP method [Be 93, Ho 93c] will
apply equally to different probes. The fact that the deduced b 2 agree for the 2 \ state of
90Zr but differ for the 2\ states of 92,94Zr might be a reflection of nuclear structure
55
differences in the transition densities that are sampled somewhat differently by the two
probes. Our B(E2) | =0.058 ± 0.010 e2b2 for the 2\ state of 94Zr is in excellent agreement
with a recently measured [Ho 93b] value of 0.060 ± 0.004 e2b2. For the 2 \ state of 96Zr,
the B(E2) f from the DOMP analyses of the alpha-particle scattering [Ra 87 and present
experiment] are about one half the adopted value [Ho 92, Ho 93a], but essentially overlap
within experimental uncertainties. Comparison between the alpha-particle and 6Li results
for this state of 96Zr are difficult because of the poor experimental resolution achieved in
the 6Li measurements [Ho 92, Ho 93a].
Unlike the 2 \ states, the Coulomb scattering amplitude contributes very little to
the 3] inelastic scattering cross sections. This is illustrated in figure 4.13, where the cross
sections for purely nuclear and purely Coulomb scattering are compared to the total
inelastic cross sections for the states in 90Zr. The Coulomb contribution to the 2\ cross
sections is significant even at large angles, but is only a very small part of the 3\ cross
section. Fits to the 3j data therefore measure the nuclear deformation parameter, b " ,
essentially independent of the adopted value for B(E3) f . Accurate independent
measurements of B(E3) f are necessary to deduce the value of M JM P for these states.
However, except for 96Zr [Ho 93d], the values of B(E3) t for the zirconium isotopes are
not well known.
To determine b f from our data, we fixed B(E3) | to the values adopted for the
6Li analysis [Ho 92, Ho 93a], and then searched on b f in order to produce the best fit to
the measured cross sections. The values of B(E3) f are shown in the last column of Table
4.2, while our deduced values for b f are given in the fourth column of the same table. In
Figure 4.14 we show calculations in which we use our deduced b f and the values of
B(E3) f adopted in [Ho 92, Ho 93a]. For 96Zr, we show a calculation using the value
B(E3) f = 0.120 e2b3 (solid curve) used in [Ho 93a], as well as a calculation using the
recently determined [Ho 93b] value B(E3) f = 0.180 ± 0.018 e2b3 (dashed curve). These
two curves are almost indistinguishable, except in the region 10° ^ 0c.m. ^ 15°, even
56
though the values of B(E3) f used to make the calculations differ by a very large amount.
This illustrates the difficulty of using the alpha-scattering data to determine B(E3) f
values for these zirconium isotopes.
While we were unable to precisely determine the value of B(E3) f for the even
zirconium isotopes, we were able to investigate the range of values of B(E3) f that are
consistent with the alpha-scattering data. This was done by using our deduced
deformations b * and performing calculations for several values of B(E3) f which were
then compared to the measured cross sections. In the fifth column of Table 4.2, we list
the upper and lower values of B(E3) | which are supported by our data.
4.3 Folding Model Analysis
A folding model analysis was made using a complex alpha-nucleon effective
interaction having a Gaussian form [Sa 87, Sa 89]. A folded potential was used to analyze
the elastic data in order to be consistent with the use of folded transition potentials to
analyze the inelastic data.
4.3.1 Elastic Scattering
The optical potential used to describe the elastic scattering was obtained by folding
the effective alpha-nucleon interaction with the ground state density distribution of the
target nucleus. The potential was given by the single-folding integral, (2.21). The
effective interaction was taken to have the Gaussian form
where V, W, and t were to be determined from fits to the elastic scattering cross sections.
The form (4.6) assumes that the real and imaginary parts of the optical potential have the
same shape, but not necessarily the same strength.
A two parameter Fermi shape (eq. 2.23) was used for the ground state density
distributions of the zirconium isotopes. Initially, the diffuseness parameter, a, for each of
the zirconium isotopes was set to the same value and the radius was scaled by A
(4.6)
57
Folded potentials were calculated for a variety of ranges, t, of the alpha-nucleon
interaction and least square fits to the data were made using the program PTOLEMY. It
was found that the minimum x2 occurred at a different range (t) for each isotope, and the
values of V and W varied considerably. Based upon the increase in the range as a function
of A at the minimum x2, we decided to investigate whether we could find a single effective
interaction which would simultaneously fit the elastic data for each of the isotopes. To
this end, we scaled the radius of the ground state density as A V3 and searched on the value
of the diffuseness, a, while fixing the range of the effective interaction at t= 1.94 fin, a
value used successfully in earlier descriptions of alpha-particle scattering [Sa 87, Sa 89].
The searches with folded potentials under these conditions resulted in the values of the
diffuseness parameters tabulated in Table 4.3 and the strengths of the potentials given in
Table 4.4. The corresponding fits to the elastic cross sections are shown in Figure 4.15.
As seen in Table 4.4, the depths of the real part, V, of the effective interaction are
consistent to within 5% and the depths of the imaginary part, W, to within 13%. The
diffuseness parameters (Table 4.3) increase monotonically from 90Zr to 96Zr. There do not
exist nuclear structure calculations of the ground state densities of these zirconium
isotopes with which to compare this trend. The noted differences in the depths of the
effective interaction would not change the calculated cross sections significantly.
4.3.2 Inelastic Scattering
The transition potentials for inelastic scattering were calculated using the
generalization of eq.(2.2) in which the ground state density distributions are replaced by
the transition densities [Sa 87, Sa 89]. We performed analyses using transition densities
obtained from a quasi-particle random-phase approximation (RPA) calculation, as well as
transition densities having a standard Bohr-Mottelson collective model form (BM).
a. RPA transition densities
In order to examine the consistency of folding model calculations for alpha-particle
and 6Li particle scattering, transition densities which were obtained from quasi-particle
58
RPA calculations using separable quadrupole or octupole interactions were used in folding
model calculations. The RPA calculations are described in [Ho 93, Br 88]. The
corresponding B(EQ f and M JM P were taken from Table III of [Ho 93], and are
reproduced in Table 4.5. In Figure 4.16, the results of the folding model calculations for
the 2 \ states (solid curves) are compared with the data. Figure 4.17 compares folding
model calculations using the same RPA transition densities with the cross sections for
exciting the 2\ states by the 6U ion scattering [Ho 92, Ho 93a]. Except for the strong
interference region for the 2 \ state of 96Zr, the folding model calculations with RPA
transition densities reproduce the alpha-particle data in a manner comparable to the
folding model calculations for the 6Li scattering.
The folding calculations using the RPA transition densities reproduce the
magnitudes of the 2\ cross sections at the larger angles which suggests that the sum of the
neutron and proton matrix elements, or the isoscalar (or mass) matrix element, is
reasonable. However, except for 90Zr, the calculations fail to reproduce the CNI region
accurately; this is especially noticeable for 96Zr. This indicates that the predicted ratios of
M JM P from the RPA calculations are not correct. Finally, we call attention to the angle
shifts between the oscillations in the calculated and measured 2 \ angular distributions
which are similar to those observed in the DOMP calculations.
Similar calculations (solid curves) for the 3] states are compared with the data in
Figure 4.18 for the alpha-particle scattering, and Figure 4.19 for the 6Li ion scattering [Ho
92, Ho 93a]. Again we find that the results of the folding calculations for the alpha-
particle scattering are similar to those for the 6Li scattering. For both projectiles, the 3]
calculations significantly underestimate the data.
b. Bohr-Mottelson transition densities
Folding model analyses were performed in which the zirconium nuclei were
described using the standard Bohr-Mottelson (BM) model of a collective vibrational
59
nucleus [Bo 75, and see section 2.4]. The resulting transition densities to be used in the
folding integral for the transition potential then follow from eq. (2.27) and are given by
S ,( ') - - 6 (4-7)
where p(r) is the ground state density distribution (section 4.3.1, above), and b tM is the
mass deformation length. Hence, the deformation parameters deduced using the folding
model are the deformations of the density distributions of the target nuclei themselves. In
the DOMP, it was assumed that the measured deformations of the potentials were equal to
the deformations of the underlying nuclear density distributions. While the form for the
density distributions must be adopted from some model, measurement of the inelastic
scattering gives some measure of the properties of this distribution.
b.l. Calculated cross sections using the RPA parameters
Folding model calculations were performed using BM transition densities which
gave the same values for B(EQ f and M JM P as were used in the RPA calculations above
(also see Table 4.5). The results of these calculations are shown as dashed curves in
Figure 4.16 for the 2* states and Figure 4.18 for the 3j states. The calculated cross
sections using the BM transition densities are systematically lower than those using the
RPA transition densities. The reason for this can be seen in Figures 4.20-4.25, where the
BM and RPA transition densities and transition potentials are compared. The tails of the
RPA transition potentials shown in Figures 4.21 and 4.23 are larger than the
corresponding tails for the BM transition potentials in the region over which the
interaction takes place, i.e. 7 s r s 11 fm. This is clearly seen in Figures 4.24 and 4.25,
where the ratio of the BM transition potentials to the RPA transition potentials are
plotted. In order to produce cross sections of the same magnitude as the RPA densities,
the BM densities will require larger deformations, btM, and therefore lead to larger
extracted M JM P ratios. Thus we see that the extracted M JM P ratios are fairly sensitive to
the assumed shapes for the transitions densities.
60
Also shown in Figures 4.21 and 4.23 are the transition potentials corresponding to
the DOMP fits to the 2\ and 3j states, respectively.
b.2 Cross section fits with Bohr-Mottelson transition densities
Because PTOLEMY does not have a search routine for inelastic scattering, folding
model calculations of the inelastic cross sections were performed for various combinations
of the values of b f 1 and B(E-f) | assuming BM transition densities. Calculated cross
sections using best fit values of 6 * and B(E2) f are compared with the data in Figure
4.26. Although the calculations reproduce the small angle data and the magnitude of the
large angle data, phase shifts can be seen analogous to those found in the DOMP
calculations. The best values of 6 / and B(E2) ja re listed in Table 4.6.
The cross sections for exciting the 3\ states mainly determine b " and are only very
weakly dependent upon the B(E3) t- We used a procedure similar to the DOMP analysis
of the 3\ states to deduce the values of b f and to determine the upper and lower limits of
B(E3) f . These values are listed in Table 4.6. In Figure 4.27, calculations of the cross
sections using our deduced b f and the B(E3) t values adopted in [Ho 92, Ho 93a] are
compared to the data. Also shown for the 3\ state of 96Zr are calculations with B(E3) f =
0.120 (solid curve) and 0.180 e2b3 (dashed curve) in order to demonstrate the lack of
sensitivity of the cross sections to the B(E3) f values. The 3\ calculations also exhibit
phase shifts relative to the data similar to those observed in the DOMP calculations.
4.4 Investigation of the Phase Shifts
Several calculations were performed in an attempt to understand the phase shifts
with respect to the data. Note that the DOMP fits to the 6Li inelastic scattering [Ho 92,
Ho 93a] do not show such shifts. In a study of the data of Rychel et al. [Ry 87], Satchler
found that there was about a 1° phase shift between calculations of inelastic cross sections
for exciting the 2\ states in 92,96Zr using the DOMP and folding models [Sa 89]. The
61
present calculations do not show such a shift, and, in fact, the two models give nearly
identical cross sections.
We performed coupled-channels calculations for scattering to the 22 state which
included coupling to the 3] or 2\ states as well as the ground state. In addition, Satchler
investigated the effects of reorientation coupling [Sa 94]. These couplings can cause shifts
of the order of 0.5°, which is not enough to explain the discrepancy between the
calculations and the measurements (Fig. 4.26). To produce the largest shift requires
coupling parameters of similar size as the couplings to the ground state (e.g. &23 of similar
magnitude as 6 20). However, it is not at all clear whether the magnitudes of the coupling
parameters required to effect such a phase shift are realistic.
One possible way to effect such a phase shift is to assume that the potential in the
outgoing channel is different from that for the incoming channel. This would be justified if
the density distributions of the excited states were different from those of the ground
states. To investigate this possibility, we performed folding model calculations in which
we assumed that the density distributions of the excited state was similar to that of the
ground state but with a different diffuseness. (We could have obtained the same result by
adjusting the radius instead.) The diffuseness parameter was adjusted until the folded
potential for the outgoing channel caused the desired shift between the calculated and
measured cross sections. The transition density was again taken to be of the BM form and
the transition potential recalculated assuming the diffuseness for the transition density to
be the average of the diffusenesses of the ground and excited states.
Fits to the inelastic data using this procedure are shown in Figures 4.28 and 4.29
for the 2 \ and 3} states, respectively. The b tN and B(E£) f values are essentially the
same as those listed in Table 4.6 which were obtained from the fits to the data using BM
transition densities. The changes in the diffusenesses that are required to account for the
observed phase shifts are listed in Table 4.7 The changes in diffuseness vary from -10%
for the 2* state of 90Zr to +18% for the 2* state in 96Zr. These would translate into
62
changes of the mean square radius of about 2-4% which is probably somewhat unphysical.
Also, the results suggest that the 2\ and 3] states of 90Zr are smaller in size than the
ground state, which is probably somewhat unphysical. Except for the 3[ state in 90Zr, a
smaller change in diffuseness is'required for the 3] states than is required for the 2 \ states
in order to match the data.
An equally effective means to accomplish a shift of the necessary amount is to
change the strength of the potentials for the exit channel (and transition potential). The
required change is about 50% of the potential for the entrance channel [Sa 94]. Whether
this is due to an accumulation of several processes, e.g., an overall sum of many couplings,
or has a single cause, is not clear.
The phase shifts between the calculated and measured cross sections are seen in
the DOMP fits as well as the folding fits, which suggests that these shifts are not due to
the form of the transition density or effective interaction. The most obvious couplings,
considered above, cannot produce the observed shifts. There may be other couplings (e.g.
coupling to transfer channels) that can produce large shifts, but this would be somewhat
surprising [Sa 94]. We are able reproduce the shifts, but are unable to explain them at this
time.
TABLE 4.1. Woods-Saxon optical model parameters determined from fits to elastic scattering data. ACoulomb radius parameter of rc=1.20 fm was fixed for all cases.
Isotope
V
(MeV)
W
(MeV)r„
(fm)au
(fm) Renorm.8 x2/pt90Zr 220.59 23.989 1.018 0.575 1.02 0.88092Zr 241.51 41.145 0.993 0.600 1.02 0.59694Zr 253.61 74.592 0.959 0.645 1.05 0.77696Zr 279.42 86.879 0.953 0.645 1.005 1.271
a Used to renormalize all cross section for all subsequent analysis.
TABLE 4.2. Comparison of btN and B(E£) f deduced from DOMP analyses of the 35.4 MeV (a ,a ') data of Rychel et al. and the
present work. For the 3[ states, we give a range of values for B(E3) t which is supported by the data (see text). The values from Rychel et al. [Ry 87] were derived using the tables and formulae therein. Also listed are b tN deduced from
the 70 MeV 6Li scattering of Horen et al. [Ho 92, Ho 93a], and the adopted B(El) t used in the DOMP analysis.
Rychel et al. (a ,a ') Present work (a ,a ') Horen et al. (6Li,6Li!)Nucleus 6 / B (E /)t 6 / B(E*)f B(E£) t "
(fm) ( e V ) (fm) («2b4) (fin) ( e V )
2\ states
SO © N 0.408±0.016 0.06210.006 0.40010.020 0.06310.005 0.39610.021 0.06310.00592Zr 0.73110.007 0.06910.006 0.67310.034 0.07510.010 0.55710.024 0.08310.00694Zr 0.633±0.006 0.05010.005 0.63210.032 0.05810.010 0.52510.032 0.06610.01496Zr 0.639±0.003 0.02710.007 0.58910.030 0.02510.005 0.46610.028 0.05510.022
3[ states
90Zr 0.80610.007 0.066410.0073 0.75010.038 0.051 - 0.091 0.68610.034 0.071b92Zr 0.89410.005 0.055610.0077 0.83110.042 0.047 - 0.087 0.74210.040 0.067 b94Zr 1.02010.006 0.079410.0118 0.93210.047 0.067 - 0.107 0.83910.044 0.087 b96Zr 1.22810.011 0.10410.011 1.11110.056 0.060 - 0.180 1.05110.050 0.120 b
a The B(E£) t were fixed in the analysis of the 6Li scattering data [Ho 92, Ho 93a]. b B(E3) t value which produced the largest value of M JM P was reported in [Ho 92, Ho 93a].
TABLE 4.3. Parameters* for a two-parameter Fermi model of the ground-state density
distributions of the zirconium isotopes, where p (r) = p0(l + e('"‘r)/a) \
Isotopec
(fm)a
(fm)RMS radius
(fm)
90Zr 4.90 0.519 4.25892Zr 4.94 0.529 4.30294Zr 4.97 0.539 4.34096Zr 5.01 0.549 4.385
3 Slightly different parameters for the ground state density distribution were used in Ref. [Ho 92, Ho 93a].
VOVO
TABLE 4.4. Strengths of the real and imaginary parts of the alpha-nucleon effective interactionwhich fit the elastic alpha scattering from the zirconium isotopes.
Isotope
V
(MeV)
W
(MeV) x2/pt.
90Zr 49.736i 16.774 2.93092Zr 48.443 18.153 1.29494Zr 48.263 18.081 0.84196Zr 47.348 18.977 1.394
TABLE 4.5. Summary of the predictions of the RPA calculations. The RPA calculations wereconstrained to reproduce the B(El) t values of column 3.
Isotope Ex(MeV)
B(E£) t ( e V )
Ex(MeV)
RPA calculation*
Mp (e fm2)
Mn/Mp
2i states
90Zr 2.186 0.063b 2.51 25.1 0.8492Zr 0.935 0.083b 1.40 28.9 1.4994Zr 0.918 0.066bc 1.55 25.9 1.6996Zr 1.751 0.055b 2.02 23.3 1.66
3T states
90Zr 2.748 0.071d 2.73 267 0.7592Zr 2.340 0.067** 2.64 257 0.8794Zr 2.057 0.087** 2.35 295 1.0696Zr 1.897 0.120* 1.96 346 1.22
a References [Ho 93a, Br 88] b Reference [Ra 87]0 A recent remeasurement [Ho 93b] found B(E2) t = 0.060 ± 0.004 e2b2 d Reference [Sp 89]e References [Ho 92, Ho 93a]; a recent remeasurement [Ho 93d] found B(E3) t = 0.180 ± 0.018 e2b3
TABLE 4.6. Summary of DOMP and BM folding results. For the 3[ states, we give the range of B(E3) f supported by the data (see
text).
Isotope B(E£) t
( e V )
DO M PX N
(ftn)Mn/Mp B(Ef) t
( « V )
BM FoldingX N
(fin)
Mn/Mp N/Z
•O © N
2* states
0.063±0.005 0.400±0.020 0.84±0.12 0.063±0.005 0.440±0.022 1.04±0.13 1.2592Zr 0.075±0.010 0.673±0.034 1.93±0.24 0.080±0.010 0.758±0.038 2.22±0.26 1.3094Zr 0.058±0.010 0.632±0.032 2.21±0.32 0.060±0.010 0.671±0.034 2.39±0.33 1.3596Zr 0.025±0.005 0.589±0.030 3.70±0.53 0.022±0.005 0.621±0.031 4.34±0.67 1.40
90Zr3[states
0.051 - 0.091 0.750±0.038 0.75±0.09a 0.051 - 0.091 0.947±0.047 1.31±0.11a 1.2592Zr 0.047 - 0.087 0.831±0.042 1.07±0.10a 0.047 - 0.087 1.024±0.051 1.68±0.13a 1.3094Zr 0.067 - 0.107 0.932±0.047 l.ll± 0 .1 1 a 0.067 - 0.107 1.124±0.056 1.68±0.13a 1.3596Zr 0.080 - 0.160 1.111±0.056 1.22±0.11a,b 0.060 - 0.180 1.330±0.067 1.82±0.12a,c 1.40
a M JM P calculated assuming the B(E3) t given in the third column of Table 4.5, adopted from [Ho 92, Ho 93a]. These values do not include the uncertainties of theB(E3) f .
b Using B(E3) t = 0.180 ± 0.018 e2b3 [Ho 93d] gives M JM P = 0.81 ± 0.13. c Using B(E3) t = 0.180 ± 0.018 e2b3 [Ho 93d] gives M JM P = 1.30 ± 0.16.
TABLE 4.7. Diffuseness parameters of the BM ground state and excited state density
distributions required to reproduce the phase of the cross sections at large angles. The radii of the distributions are given in Table 4.3, while the strengths of the
effective alpha-nucleon interaction are given in Table 4.4. The diffuseness of the transition density was taken as the average of the diffuseness parameters of the ground and excited state densitiy distributions.______________________________
ground state excited stateIsotope a RMS radius a RMS radius
(fin) (fm) (fm) (fin)
2! states
0.5190.5290.5390.549
4.2584.3024.3404.385
0.4600.5900.6200.650
4.1634.4104.4874.571
37 states
0.5190.5290.5390.549
4.2584.302
4.3404.385
0.4100.5290.560
0.580
4.0904.3024.376
4.439
70
Figure 4.1 Illustration of the process used to create angle-gated spectra for the analysis ofAZ r(a,a’)AZr\ a) Raw Front Wire spectrum (e.g. Fig. 3.5(a)). A gate is placed about the desired peak or peaks (e.g. elastic scattering peak) on this spectrum in order to produce a clean angle spectrum, b) Angle Spectrum (e.g. Fig 3.5(c)) calculated from front and rear wire positions on an event-by-event basis. Only those particle tracks whose front wire position fall within the front wire gate are included Peak #1 is at the smallest scattering angle, while peak #5 is at the largest angle. Gates placed around each of the peaks in the angle spectrum are used to create the angle-gated ray-traced spectra, c) Calculated focal plane spectra created using several of the angle gates from (b) (but not the gate on the front wire spectrum). The particle paths have been ray- traced from the front wire position to the focal plane. The shift in the positions of the peaks from one spectrum to the next is due to kinematics.
co
un
ts/c
ha
nn
el
71
1000 1050 1100 1150 1200 1250channel
Figure 4.2 Inelastic spectra for 90Zr(a,a')90Zr’, taken with the spectrometer at 23.0°.The spectra have been transformed from the front wire to the focal plane (section 3.2). Each of the five spectra have been created by gating on the peaks of the angle spectrum. The spectrum at 21.0° is near a minimum of the cross section for scattering to the 3[ state. The positions of the 2\ (2.186 MeV), 5" (2.319 MeV), and the 3j (2.748 MeV) states of 90Zr have been indicated. Also indicated are the positions of the peaks due to elastic scattering off carbon and oxygen.
co
un
ts/c
ha
nn
el
72
1000 1050 1100 1150 1200 1250 1300channel
Figure 4.3 Inelastic spectra for 92Zr(a,a')92Zr’, taken with the spectrometer at 35.0°.The spectra have been transformed from the front wire to the focal plane (section 3.2). Each of the five spectra have been created by gating on the peaks of the angle spectrum. The spectrum at 36.0° is near a minimum of the cross section for scattering to the 2\ state. The positions of the 2\ (0.935 MeV), 2\ (1.847 MeV), 2\ (2.067 MeV), and 3" (2.340 MeV) states of 92Zr have been indicated. Also indicated is the position of the peak due to elastic scattering off oxygen, which overlaps the 3\ state at 33.0°. The far right of the spectra show the effect of the blocker plate on the elastic peak which was used when collecting inelastic data (sec. 3.2 and fig 3.4).
co
un
ts/c
ha
nn
el
73
1000 1050 1100 1150 1200 1250 1300channel
Figure 4.4 Inelastic spectra for 94Zr(a,a ')94Zr*, taken with the spectrometer at 35.0°.The spectra have been transformed to the focal plan, and gated by the peaks of the angle spectrum. The spectrum at 35.0° is near a minimum of the cross section for scattering to the 2\ state, and near a maximum of the cross section for scattering to the 3[ state. The positions of the 2\ (0.918 MeV), 4* (1.470 MeV), 2\ (1.671 MeV), 3[ (2.057 MeV), and 2\ (2.151 MeV) states of 94Zr have been indicated. Also indicated is the position of the peak due to elastic scattering off oxygen.
co
un
ts/c
ha
nn
el
74
1100 1150 1200 1250 1300 1350channel
Figure 4.5 Inelastic spectra for 96Z r(a,a ')96Zr*, taken with the spectrometer at 20.0°.The spectra have been transformed to the focal plane, and gated by the peaks of the angle spectrum. The spectrum at 20.0° is near a minimum of the cross section for scattering to the 3 state. The positions of the 2* (1.751 MeV) and 3‘ (1.897 MeV) states of 96Zr have been indicated. Also indicated are the positions of the peaks due to elastic scattering off carbon and oxygen. The right sides of the spectra show the effects of using the blocker to cut off elastic scattering when collecting inelastic scattering data.
i
75
a) Calibrate angle spectrum using slit plate
> Angle calibration
Figure 4.6 Illustration of the process of angle binning used to analyze inelastic data for 0iab < 20°.a) The angle spectrum was first calibrated by making a short run using a slit plate. An angle spectrum, gated by the elastic scatter peak on the raw front wire spectrum, was used to calibrate the angle spectrum in terms of the laboratory scattering angle ( 0 vs channel), b) A run was then made without the slit plate. The calibration obtained in (a) was then used to calculate the positions of gates on the resulting angle spectrum, which has no peak structure. The gates (angle bins) were centered on the desired scattering angle, and were calculated to subtend a horizontal angle of 0.25°. Focal plane spectra were then calculated by ray-tracing particle paths falling within each angle gate from the front wire position to the focal plane, similar to fig. 4.1(c).
76
£sb5£b
b
0 „ (d e g ) 0c« (d e g )
Ib \ bb\
0 cy (d e g ) ©cm (d e g )
Figure 4.7 Elastic scattering cross sections. The curves are calculated using the fits of Rychel, et. al. [Ry 87]. The cross sections are plotted relative to the Rutherford cross section.
da/d
O
(mb
/s
r)
da/d
O
(mb
/s
r)
77
0 C (d e g ) 0 c (d e g )
0 c (d e g ) 0 c (d e g )
Figure 4.8 Differential cross sections for exciting the 2 \ states o f 90,92,94’96Zr. The curves are coupled channels calculations using the results of Rychel, et. al. [Ry 87].
do/d
O
(mb
/s
r)
do/d
O
(mb
/s
r)
78
© C M ( d e g ) ©CM (d e g )
0 Cy ( d e g ) 0 a , (d e g )
Figure 4.9 Differential cross sections for exciting the 3[ states of 90’92,94’96Zr. The curves are coupled channels calculations using the results of Rychel, et. al. [Ry 87].
4nU
ft(mb/
n)
79
9°-96Zr(aJa') Ea=35.4Md/
90-96zr(a,a') Ea=35.4MeV
0CM ©CM (deg)
Figure 4.10 Cross sections from Rychel et al. [Ry 87] for the excitation o f low-lying 2+ and 3 states of 90'92-94'96z r by the inelastic scattering o f 35.4 MeV alpha- particles. (adopted from [Ry 87])
80
i b S b
£ i b \ b
0 C, (d e g ) © c m (d e g )
Ib \ b
&b\
0c» (deg) ©OT (deg)
Figure 4.11 Optical model fits to the elastic scattering data for 90'92-94-96z r + a at Eiab =35.4 MeV. The optical model parameters are given in Table 4.1. The cross sections are plotted relative to the Rutherford cross section.
do/d
0 (m
b/
sr)
do
/dO
(m
b/
sr)
81
0 Cy (d e g ) 0cy ( d e g )
©cm (d e g ) ©m (d e g )
Figure 4.12 Differential cross sections for exciting the 2 \ states of 90,92’94’96Zr. The curves are the coupled-channels calculations of the cross sections using the DOMP. The values of b 2 and B(E2) f deduced from the fits are listed in Table 4.2.
o (m
b)
a (m
b)
82
© c (d e g )
0 c ( d e g )
Figure 4.13 Contributions of the Coulomb and nuclear interactions to the total cross sections for the 2\ state (top) and the 3[ state (bottom) of 90Zr. The curves are calculated in the framework of the DOMP. The dashed curve corresponds to the cross section arising from the nuclear potential only, the dot-dashed curve to the Coulomb potential only, and the solid curve to the total cross section.
da/d
Q
(mb
/s
r)
dcr/
dO
(mb
/s
r)
83
0 „ (d e g ) © a (d e g )
0 Cy ( d e g ) ©cm (d e g )
Figure 4.14 Differential cross sections for exciting the 3[ states of 9G,92,94’96Zr. The curves are the coupled-channels calculations of the cross sections using the DOMP. The B(E3) t values used in the calculations are adopted from [Ho 92, Ho 93a] and are shown in the last column of Table 4.2. The values of 6 * deduced from the data are listed in Table 4.2. For 96Zr, calculations using B(E3) t = 0.120 e2b3 (solid curve) and 0.180 e2b3 (dashed curve) are shown in order to demonstrate the weak sensitivity of the cross sections on the value of B(E3) t .
Rut
h
84
5Ib
b
0 C (d e g ) 0 c (d e g )
0 „ (d e g ) ( d e g )
Figure 4.15 Folding model fits to the elastic scattering cross sections for 90,92’94,96Zr+a at E|ab=35.4 MeV, using a gaussian alpha-nucleon effective interaction and Fermi density ditributions. The cross sections are plotted relative to the Rutherford cross section. The parameters of the ground state density distributions are given in Table 4.3, while the strengths of the interaction are given in Table 4.4.
dcr/
dO
(mb
/sr)
da
/dO
(m
b/
sr)
85
©«. (d e g ) ©cu (d e g )
©cu (d e g ) ©a. (d e g )
Figure 4.16 Comparisons of folding model predictions of the cross sections for exciting the 2 \ states of 90,92,94,%Zr. The solid curves are calculations using the RPA transition densities. The dashed curves use BM transition densities that have been constrained to reproduce the B(E2) f values and M JM P ratios predicted by the RPA, as shown in Table 4.5.
86
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Qc.m. (deg) ®c.m. (deg)
Figure 4.17 Cross sections for exciting the 2[ states of 90’92>94-96z r by the inelastic scattering of 70 MeV 6Li ions [Ho 92, Ho 93a]. The curves are folding model calculations using the RPA transition densities, (adopted from [Ho 93a]).
87
0 C (d e g ) 0 c ( d e g )
0 C„ (d e g ) © c ( d e g )
Figure 4.18 Comparisons of folding model predictions of the cross sections for exciting the 31 states of 90'92'94’96Zr. The solid curves are calculations using the RPA transition densities. The dashed curves use BM transition densities than have been constrained to reproduce the B(E3) t values and the M JM P ratios predicted by the RPA, as shown in Table 4.5.
da/d
O
(mb
/sr)
do
/dO
(m
b/
sr)
88
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45 0 50.08c.m. (deg) 0c.m. (deg)
Figure 4.19 Cross sections for exciting the 3[ states of 90*92'94>96Zr by the inelastic scattering of 70 MeV 6Li ions [Ho 92, Ho 93a]. The curves are folding model calculations using the RPA transition densities, (adopted from [Ho 93a]).
89
r ( f m ) r ( f m )
r ( f m ) r ( f m )
Figure 4.20 Comparison of the RPA and BM transition densities for the excitation of the 2 \ states of 90’92,94,96Zr.
90
r ( f m ) r ( f m )
r ( f m ) r ( f m )
Figure 4.21 Comparison of the transition potentials for the excitation of the 2* states of9o,92,94,96^ obtained by folding the alpha-nucleon effective interaction with the transition densities of Figure 4.20. These transition potentials were used to calculate the curves shown in Figure 4.16. Also shown are the DOMP transition potentials.
) ,
g3
(frr
f
91
r ( f m ) ( f m )
r ( f m )
8 10 r ( f m )
Figure 4.22 Comparison of the RPA and BM transition densities for the excitation of the 3j states of 90'92,94’%Zr.
92
r ( f m ) r ( f m )
10 15r ( f m )
>0)
0)q:
10 15r ( f m )
Figure 4.23 Comparison of the transition potentials for the excitation of the 3[ states of 90,92,94,96 0btajned folding the alpha-nucleon effective interaction with the transition densities of Figure 4.22. The transition potentials were used to calculate the curves shown in Figure 4.18. Also shown are the DOMP transition potentials.
93
<Q_ <Q_
GOw>"
r ( f m ) r ( f m )
r ( f m ) r ( f m )
Figure 4.24 Ratio of the BM to RPA real transition potentials for the excitation of the 2j states of 90-92-94>96z ri jn the surface region 7 &n 5 r s 10 fm, the BM transition potentials are smaller than the RPA potentials, indicating that the BM potentials will require a larger deformation length in order to fit the data, and hence predict a larger M JM P ratio than the RPA potentials.
94
<Q_
<S'CO
<Q.
s.
( f m ) r ( f m )
<Q_ <Q_
r ( f m ) r ( f m )
Figure 4.25 Ratio of the BM to RPA real transition potentials of Figure 4.23 for the excitation of the 2>\ states of 90-92-94-96Zr. In the surface region 7 f i n s r s 10 fm, the BM transition potentials are smaller than the RPA potentials, indicating that the BM potentials will require a larger deformation length in order to fit the data, and hence predict a larger M JM P ratio than the RPA potentials.
do/d
O
(mb
/s
r)
da/d
O
(mb
/s
r)
95
0 C (d e g ) 0 c (d e g )
0 c (d e g ) 0 c (d e g )
Figure 4.26 Results of fits to data using folding model calculations with the BM transitions densities for the 2\ states of 90-92-94’96Zr. Searches were made by varying both B(E2) f and b 2 . The resulting parameters along with the extracted values for MJMP are shown in Table 4.6.
dcr/
dQ
(mb
/s
r)
da/d
O
(mb
/s
r)
96
0 C« (d e g ) 0 c (d e g )
0 c (d e g ) 0 c ( d e g )
Figure 4.27 Results of fits to data using folding model calculations with the BM transition densities for the 3[ states of 90-92-94-96Zr. Searches were made by varying 63* while fixing the B(E3) t values. For 96Zr, calculations are shown using B(E3) t = 0.120 e2b3 (solid curve) and 0.180 e2b3 (dashed curve) in order to demonstrative the weak dependence of the cross sections on the value of B(E3) t- The extracted values for b 3 and the deduced M JM P ratios are shown in Table 4.6.
97
0 cy (deg) 0cu (deg)
0CU (deg) ©cu (deg)
Figure 4.28 Results of fits to data using folding model calculations with the BM transitions densities for the 2\ states of 90,92,94’%Zr. The diffuseness parameter of the inelastic channel was varied in order to match the phase of the data at large angle. The diffuseness parameters are shown in Table 4.7.
98
0CB (deg) ©c« (deg)
©c« (deg) ©cu (deg)
Figure 4.29 Results of fits to data using folding model calculations with the BM transition densities for the 3[ states of 90-92-94-96z;ri The diffuseness parameter of the inelastic channels was varied in order to match the phase of the data at large angles. The diffuseness parameters are shown in Table 4.7.
Chapter 5: Discussion and Conclusions
From Table 4.2, we see that the agreement between the DOMP analyses of the
alpha-scattering data of Rychel et al. [Ry 87] and the present data is quite good; the mean
values of the b tN differ by -10% , which is about the uncertainty in the measured cross
sections. The agreement for the 2\ state of 90Zr is much better. The deformation lengths
reported from these two alpha-scattering experiments are also in good agreement with
those values of b tN reported from other scattering experiments (Tables 1.3-6). Our b (N
for the 2\ states (except for 90Zr) are about 20% larger than those deduced from the 6Li
scattering [Ho 92, Ho 93a], and about 10% larger for the 3[ states. It is not clear what
causes this discrepancy. Attempts to fit the 2\ scattering cross sections using the B(E2) f
values of [Ho 92, Ho 93a] yielded b f = 0.651 fm for 92Zr and b f = 0.606 fin for 94Zr,
while the interference region of the 96Zr data could not be reasonably described using the
value B(E2) t = 0.055 e2b2 adopted in [Ho 92, Ho 93a]. Thus the disparity in the values
of b tN for the 2 \ states cannot be attributed solely to the differences in the B(E2) f values
used in the analyses of the alpha-particle and 6Li data. Our B(E2) t values deduced for
90,92Zr (given in Table 4.6) are in excellent agreement with the corresponding values
adopted from Coulomb excitation measurements [Ra 87], and for the 2\ state in 94Zr we
obtain B(E2) f = 0.058 ± 0.010 e2b2 in good agreement with a value 0.060 ± 0.004 e2b2
obtained from a recent lifetime measurement [Ho 93b].
A more meaningful comparison is that between the DOMP and folding model
analysis as shown in Table 4.6. Here it is found that the btN from the folding analyses are
about 5-10% larger than those from the DOMP analyses for the 2 \ states and about 20%
larger for the 3[ states. An examination by Beene, Horen and Satchler indicated that this
is about what would be expected from the non-equivalency of the DOMP and folding
models for a BM type transition density [Be 93]. They have shown that the differences
between the DOMP and folding analyses increase as the multipolarity, t , of the transition
99
100
increases. This is due to the neglect of the explicit t dependence of the shape of the
transition potential by the DOMP [Be 93].
The two methods yield B(E2) f values that are in excellent agreement with each
other. However, this is not unexpected, as both methods use the same form for the
Coulomb interaction, eq. (4.4), which is uniquely determined by the B(Et?) f . The cross
sections for exciting the 3[ do not exhibit enough sensitivity in order to determine
B(E3) | , although the ranges of values which can reasonably describe the data using the
two methods are similar (Table 4.6).
From eq. (2.11), the ratio o iM JM p can be calculated from the deduced quantities
bf and 6 / asAb N
(5.1)
In Table 5.1 we compare the M JM P ratios obtained from the two types of analyses. Since
the B(E^) f values determined by the two methods are nearly the same, the difference in
the M JM P ratios is mostly a reflection of the different nuclear deformations, b eN, which
are extracted using each of the two methods. The two methods give comparable M JM P
values for the 2j states which (except for 90Zr) are at least 50% larger than those reported
in the 6Li work. Furthermore, the MJMP for the 2\ states of 92,94,96Zr are considerably
larger than those for a "pure" isoscalar transition, i.e., N/Z. The larger values of 6 *
deduced in this work from the alpha-particle scattering (as compared to the 6Li scattering)
are mainly responsible for the larger M JM P ratios determined here.
In the case of 96Zr, the much smaller value of the B(E2) f causes an additional
discrepancy between the MJMP ratios. The large M JM P = 4.34 ± 0.67 for 96Zr and the
low B(E2) f = 0.022 ± 0.005 e2b2 suggests that this 2\ state is predominantly a neutron
excitation. This is rather surprising and is contrary to the nuclear structure calculations of
which we are aware. If true, it would suggest some strong interaction between the closed
d5/2 subshell neutrons and the valence protons. On the other hand, these results might
simply be an indication that the 2\ excitation in 96Zr does not have a collective type
101
transition density, in which case utilization of either the DOMP or our folding model
would be incorrect. The measured B(E3) f = 0.180 ± 0.018 e2b3 = 47.1 ± 4.7 W.u.
[Ho 93b] is one of the most enhanced E3 transition to a 3~ state [Sp 89]. In addition, the
measured B(E3:3] -•> 2 [) = (1.27 ± 0.08)xl0'3 W.u. is similar to that observed in the Ba-
Nd and Ra regions where octupole collectivity is strong [Ho 93b], suggesting that
octupole correlations are unusually strong in 96Zr. Also, several calculations have
suggested that 96Zr lies in the region of a shape transition from a spherical nucleus to that
of a deformed rotor [e.g. Ma 90, Bo 85]. These factors may indicate that 96Zr undergoes
rather complex anharmonic vibrations, in which case the RPA formalism would break
down [Ma 90, Ho 93b]. Furthermore, our analyses based on either the use of the DOMP
or the folding potential using Bohr-Mottelson type transition densities, which assume that
the excited states may be described as the harmonic surface oscillations of a spherical
nucleus, would be inadequate. Further experimental information pertaining to the form
factor for this state is needed, which could be obtained by either inelastic electron
scattering or intermediate energy proton scattering.
The folding model value MJMP = 1.31 ± 0.11 extracted from our data for the 3[
state in 90Zr suggests that the transition is nearly isoscalar, contrary to the smaller M JM P =
0.75 ± 0.09 value deduced from the DOMP analysis. We have already noted our
preference for the folding model analysis, due to its firmer physical basis. The trend of
increasing values for M JM P with increasing mass number A for the 3[ states is similar to
that predicted by the RPA calculation [Ho 93a], although the folding model values
indicate that the transitions have M JM P a: N/Z, while the RPA predicts M JM P < NIZ. Use
of the recently measured value B(E3) f = 0.180 ± 0.018 e2b3 [Ho 93d] for the 3[ state of
96Zr would give M JM P = 1.30 ± 0.16 which is in excellent agreement with the value
M JM P = 1.30 extracted from the 6Li scattering data using a folding model with a BM type
transition density [Ho 93d]. Since there is some uncertainty as to the values of B(E3) |
102
for the other isotopes, one should not put too much weight upon the M JM p values for
those 3[ states reported here.
In conclusion, we have measured elastic and inelastic cross sections for scattering
of 35.4 MeV alpha-particles by 90’92,94,%Zr. The data were analyzed using both
microscopic and macroscopic models. In the macroscopic analysis, the elastic scattering
cross sections were fitted using an optical potential of the Woods-Saxon form; the
inelastic cross sections were analyzed using the deformed generalization of the optical
potential (DOMP). In the microscopic analysis an effective alpha-nucleon interaction
having a Gaussian shape was used. The strengths of the interaction were found by fitting
the elastic data; the interaction was then folded with transition densities in order to predict
the inelastic cross sections. Transition densities from RPA calculations as well as a Bohr-
Mottelson collective model form for the transition density were used. Our DOMP
analyses give 6 / and B(EQ \ values in good agreement with an earlier alpha-scattering
study [Ry 87]. The b tN are larger than those deduced from 6Li scattering [Ho 92, Ho
93a]. Comparison of the b tN deduced from DOMP fits versus folding model fits to the
present data clearly show the inconsistency [Be 93, Ho 93c] between the two methods. In
addition, the excellent agreement between the alpha-particle and 6Li ion scattering for the
3[ state of 96Zr using the folding model with BM transition densities suggests that the
DOMP does not offer an accurate method for the extraction of nuclear properties. Similar
analysis of the remaining 6Li scattering data would be interesting in order to further test
this hypothesis. For both the DOMP and folding potential methods, we observed shifts at
large scattering angles between the oscillations in the experimental and calculated inelastic
angular distributions which require further investigation. The rather large values of M JM P
deduced for the 2 \ states of 92’94-96z r certainly raise questions about the validity of using
BM collective type transition densities, as well as currently available nuclear structure
calculations. The sensitivity of the deduced b tN with respect to assumed transition
103
densities has been demonstrated, as has the need for independent precision measurements
of B(E*) t •
104
TABLE 5.1 Summary of M JM P ratios extracted from (a ,a 1) and (6Li,6Li') data.
Isotope
1 1 U11M v
M„/Mp ratios
*-'aJ J wuvu*
N/ZDOMPa (a ,a ')
BM folding3 (a ,a ')
DOMPb(6Li,6Li’)
RPAbcalc.
2i+ states90Zr 0.84±0.12 1.04±0.13 0.85±0.10 0.84 1.2592Zr 1.93±0.24 2.22±0.26 1.30±0.10 1.49 1.3094Zr 2.21±0.32 2.39±0.33 1.50±0.15 1.69 1.3596Zr 3.70±0.53 4.34±0.67 1.50±0.15 1.66 1.40
states
90Zr 0.75±0.09c 1.31±0.11c 0.60±0.08 0.75 1.25
92Zr 1.07±0.10c 1.68±0.13c 0.85±0.10 0.87 1.30
94Zr l.ll± 0 .1 1 c 1.68±0.13c 0.90±0.10 1.06 1.35
96Zr 1.22±0.11c,d 1.82±0.12c,e 1.10±0.10 1.22 1.40
a Present work. b References [Ho 92, Ho 93a].0 M JM P calculated assuming the B(E3) t values adopted in [Ho 92, Ho 93a]. These values do not include the uncertainties of the B(E3) f .d Using B(E3) f = 0.180 ± 0.018 e2b3 [Ho 93d] gives M JM P = 0.81 ± 0.13. e Using B(E3) t = 0.180 ± 0.018 e2b3 [Ho 93d] gives M JM P = 1.30 ± 0.16.
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