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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