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This is to certify that the
thesis entitled
The g-Factor of the 19/2- State in
1155b and High Spin States in 180w I

presented by

Steven Ray Faber

has been accepted towards fulfillment
of the requirements for

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115

THE g—FACTOR OF THE 19/2- STATE in Sb AND

HIGH SPIN STRUCTURE OF 180W

By

- Steven Ray Faber

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements~

for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

1979

ABSTRACT

THE g—FACTOR OF THE 19/2_ STATE IN ll58b AND

HIGH SPIN STRUCTURE OF l80w

By
Steven Ray Faber

A magnet for measuring g-factors by the time differen-
tial perturbed angular distribution (TDPAD) technique has
been constructed. It was used to measure the g-factor of
the 19/2- state in 115Sb by means of the llSIn(a,Uny)llssb
reaction. The measured value confirmed the suspected
{nd5/2 ® Vhll/2 ® vd3/2} configuration. The effect of core
polarization is discussed in relation to the measured value
(g = 0.287(“)). The g-factor of the 11/2- state was
obtained as a by-product and was found to be l.O6(lO).

The g—factor of the 12+ state in 202Pb was also
measured using a liquid Hg target in a Hg(a,xn)199-2ouPb
experiment with a A8 MeV a beam. The g-factor was found
to be g = -0.lh(2), consistent with the suspected (113/2);2
configuration. The 21/2+ state of 203Pb was shown to have

a positive g-factor contrary to what was expected.

Steven Ray Faber

The high spin structure of 180W was studied by means
of the l80Hf(a,1#n)180w reaction. Three h-quasiparticle

isomers were discovered (K1T = 1H“, 15+ and 16+) consistent
with configurations and spin assignments from calculated
Nilsson orbitals. Four rotational bands have been identi-
fied based on KTr = 2', 8’, 5(-), and 16+ isomers. The
configurations of these are determined from branching ratios
and angular distribution data. The perturbed nature of the
ground band and K7r = 2- band is discussed in terms of

Coriolis interactions.

ACKNOWLEDGMENTS

I would like to thank most of all my Advisor, Fred
Bernthal, for all his help and keen insight in suggesting
these research projects, as well as for his friendship
and support during my stay.

Special thanks to T. L. Khoo, L. E. Young, F. Walker,

C. Dors, and R. Warner, for their help in experiments and
for teaching me much of what I have learned. Thanks

also to office mates, w. Bentley and P. Deason for the
same.

Thanks to C. Morgan and B. Jeltema for their programming
help and M. Edmiston and B. Melin for their expertise; and
all four for their help in "special effects" at the lab.

Thanks to all members of Nuclear Beer, especially
Patricia who managed our wedding at the same time I managed
this thesis.

Acknowledgments also go to P. Miller, H. Hilbert, N.
Mercer, B. Welton, and H. Blosser for their help in obtaining
and constructing equipment including help with the cyclotron.

I also wish to thank P. Warstler for typing this thesis

and the N.S.F. for financial support.

ii

TABLE OF CONTENTS

Chapter Page

LIST OF TABLES. . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . vii
PART 1. g-Factor . . . . . . . . . . . . . . . . . 1
I. INTRODUCTION. . . . . . . . . . . . . . . . . 2
II. THEORY. . . . . . . . . . . . . . . . . . . . u

2.1. Shell Model . . . . . . . . . . . . . . A

2.2. Core Polarization and Configuration
Mixing. . . . . . . . . . . . . . . . . 9

2.3. Higher Order Effects. . . . . . . . . . 12
2.“. Quarks and Mesonic Effects. . . . . . . 13
2.5. g-Factors and the Rotational Model. . . 15
III. EXPERIMENTAL. . . . . . . . . ... . . . . . . 19
3.1. Methods for Measuring g-Factors . . . . 19

3.2. The Time Differential Perturbed
Angular Distribution Method (TDPAD) . . 22

3.2.1. Nuclear Reactions and
Alignment . . . . . . . . . . . 22

3.2.2. Angular Distributions . . . . . 23

3.2.3. Asymmetry Ratio, R(t)
Formation . . . . . . . . . . 28

3.3. Magnet Construction . . . . . . . . . . 30

3.3.1. Experimental Design and
Considerations. . . . . . . . . 30

iii

Chapter Page
3.3.2. Data Taking Method and
Electronics . . . . . . . . . . 32

3.3.3. Design and Construction of
the g-Factor Magnet . . . . . . 32

3.3.A. Target Chamber Design . . . . . AO
IV. RESULTS OF MEASUREMENTS. . . . . . . . . . . . A3
A.l. 210PO Test. . . . . . . . . . . . . . . A3

A.2. The g-Factors of the 19/2' and
11/2' States in 115Sb . . . . . . . . . A6

A.2.1. Introduction. . . . . . . . . . A6
A.2.2. Experiment. . . . . . . . . . . A7
A.2.3. Results . . . . . . . . . . . . 5A
A.2.A. Discussion. . . . . . . . . . . 58

A.3. g-Factor of the 12+ State in
2O2Pb O O I O O O O O O O 0 O O O O I O 61

A.A. Summary and Conclusion. . . . . . . . . 70

PART 2 - HIGH SPIN STATES IN l80w FROM THE
180Hf(a,Any)180W REACTION. . . . . . . . . 72

V. INTRODUCTION. . . . . . . . . . . . . . . . . 73
VI. THEORY. . . . . . . . . . . . . . . . . . . . 75
6.1. Deformed Nuclei . . . . . . . . . . . . 75
6.2. The Hamiltonian . . . . . . . . . . . . 76
6.3. Nilsson Model . . . . . . . . . . . . . 8O
6.A. Pairing Correlations. . . . . . . . . . 83
6.5. Collective Phenomena. . . . . . . . . . 8A

VII 180w EXPERIMENTAL DETERMINATIONS. . . . . . . 87

iV

Chapter

.A.
.5.

~J —q -< «a -q
O)

Page

Singles Spectra . . . . . . . . . . . . 87
Angular Distributions . . . . . . . . . 88
Gamma-ray Lifetimes . . . . . . . . . . .98
Delayed Coincidence Experiment. . . . . 101

Prompt Coincidence“ . . . . . . . . . . 103

VIII 180w EXPERIMENTAL RESULTS . . . . . . . . . . 106

8.1.

00000000

8.7.
REFERENCES.

APPENDIX.

The Ground Band and Back-
bending . . . . . . . . . . . . . . . . 106

The KTr = 2' Band. . . . . . . . . . . . 109
The K" = 5(‘) Band. . . . . . . . . . . 115
The K" =~8’ Band. . . . . . . . . . . . 117

High Spin A-Quasiparticle
Isomers . . . . . . . . . . . . . . . . 119

8.5.1. The 1A’ Isomer. . . . . . . . . 119
8.5.2. The 15+ Isomer. . . . . . . . . 121
8.5.3. The 16+ Band. . . . . . . . . . 125

g-Factors from Branching
Ratios. . . . . . . . . . . . . . . . . 126

8.6.1. Method. . . . . . . . . . . . . 126
8.6.2.
8.6.3. K"

W
:a
ll

8‘ Band. . . . . . . . . . 128
5(’) Band. . . . . . . . . 13o

8.6.u. K“ 16+ Band.. . . . . . . . . 132
Summary . . . . . . . . . . . . . . . . 132

135
1A0

7-2
8-1
8-2

LIST OF TABLES

g-Factors of Elementary Particles.
Results Of the 1158b g-Factor
Analysis

Summary Of Measured g—Factors.

Some Possible Multiquasiparticle
180

States in W . .

y-Rays in the 18OHf(a,Any)l8Ow

Reaction . . . . . . . . . . . .
180

Delayed y's in W.
K" = 8' Band
Summary of Isomers Discovered

and Proposed Configurations.

Vi

Page

55
71

85

9O
99

~129

. 13A

3-2a

3-2b

3-5

A-2

LIST OF FIGURES

Core polarization excitations near

closed shells

Core polarization and spin alignment.

Angular distribution (quadrupole)
and spin alignment in-beam.
Electronics for a g-factor experi—
ment, separate timing . . . .
Electronics for a g-factor experi—
ment, combined timing

The g-factor magnet

Magnet calibration: field as a
function of current

Magnetic field profile as a function
of distance from center along median
plane

Target chamber and target

210PO TAC and partial decay
scheme. . . . . . .

Partial decay scheme for ll58b.

Vii

Page

10

10

2A

33

33
35

38

39

A2

AA
A8

Figure Page

A-3 Time spectra for the 1217 and 1300

keV lines in 115

Sb. Detector 1
(~135°) are [:1 points, detector 2
(135°) A, and sum +. Points + and

C] have been shifted up half a decade

for clarity . . . . . . . . . . . . . . . 50

A-A R(t) for the 1217 and 1300 keV
lines in 1158b. . . . . . . . . . . . . . 52
A-5 Configuration for the 19/2-

isomer and the ground state of
115Sb. The arrows represent core
polarization excitations. Filled

shell-model orbits are indicated

with an "x" . . . . . . . . . . . . . . . 59
A—6 Partial 202Pb decay scheme. . . . . . . . 62
A-7 Delayed V's from A8 MeV
. o's on Hg . . . . . . . . . . . . . . . . 65
A-8 202Pb TAC spectra and ratio (R(t))

for the 179 keV transitions.A indi-
cate -135°,[j indicate +135° detector

angle . . . . . . . . . . . . . . . . . . 66
A-9 203Pb partial level scheme and TAG

spectra. A = -135° spectrumglj = +135°

spectrum. . . . . . . . . . . . . . . . . 68

Viii

Figure Page

6-1 Vector diagram of a particle

coupled to a deformed rotating

core. . . . . . . . . . . . . . . . . . . 77
6-2 Nilsson diagram showing quasi-

particle energies . . . . . . . . . . . . 82
7-1 y-rays from the 180Hf(a,An)l8OW

reaction (A8 MeV a) . . . . . . . . . . . 89
7-2 Delayed spectra showing the

decay of the 2.3 usec isomer. . . . . . . 102

7-3 Level scheme for 180W - . - . - - . . . . 10A
8—1 Backbending plot for 180w . . 107
8-2 The 902 keV coincidence gate. . . . . . . 110
8-3 Angular distribution of the

122.8 keV 5' + A’ transition. . . . . . . 112

8-A AE/2I 1g 212 plot of the

K" = 2' bands of 178,180w . 113
8-5 AE E I for the K" = 8', 5(')

bands . . . . . . . . . . . . . . . . . . 116
8-6 AE/2I Xi 212 plot of the

K1T = 5(-), 8', and 16+ rota-

tional bands. . . . . . . . . . . . . . . 118
8-7 Sum of delayed gates set on

members of the KTr = 8' rotational

band. . . . . . . . . . . . . . . . . . . 122

ix

Figure Page

8-8 Half lives of the 125 and

158 keV transitions . . . . . . . . . . . 123

PART 1

g-FACTORS

I. INTRODUCTION

One of the goals of nuclear physics is to learn about
the structure of nuclei and determine an appropriate
model to explain the observed energy levels and transi-
tions. Ideally all the properties of an excited state
should be predictable from the wavefunction of the state,
the wavefunction being a not too complicated mixture of
convenient basis states. The energy and static moments
of a level practically determine the wavefunction and
many properties of the system such as the decay transi-
tion rates. The measurement of the magnetic dipole moment
of nuclear energy levels is and'has been of value in
determining the validity of nuclear models such as the
shell model, since it is very sensitive to impurity con-
figurations and can be accurately measured in many cases.
For this reason many measurements have been done near
closed shells where impurity admixtures are low. There
are many instances, however, where the magnetic moment
measurement of a level would be of value in determining
its configuration, particularly if one needs only to omit
certain possibilities from a list of possible configura-
tions. This makes magnetic moment or g-factor (U = g?)

measurements of value even on deformed nuclei. There

are many new isomers that have been found by the MSU and
Purdue collaboration in w, Pt, Hg and Pb nuclei that
were accessible through in-beam (O,xn) reactions [He77],
[Sa77], [Pt75].

The method of perturbed angular distributions is
particularly suited for in-beam measurements, so this
method was chosen for use with the MSU cyclotron. A mag-
net for measuring g-factors was constructed and the results
of some measurements are discussed in the following sec-

tions.

II. THEORY

2.1. Shell Model

 

In the simplest model, the single particle shell
model, the nucleus is viewed as a collection of paired
nucleons with only the last unpaired nucleon contributing
to the magnetic moment of the nucleus. The spin and
orbital angular momentum of each pair Of nucleons is
coupled to zero. The odd particle has spin angular
momentum S and orbital angular momentum E that couple
to total angular momentum 3. The respective angular
momentum components :£, as of the total magnetic moment
ET are related to the angular momentum by the following
equations that define the g factors 32 = glf, as = gsS,
and HT = g3. g-Factors are given in units of the nuclear

magnetons

... ...—eh (1)

Classically for a particle charge e, mass m, velocity v,

traveling in an orbit radius r, the current i = 5% where

T is the period of rotation

 

The magnetic moment

_ eV 2 evr
“ 1A c2flr “r 2c
[L] = IPXEI - rmv
_ eL
the magnitude of u - :fiau It follows then for a quantum
mechanical system that
mi

The magnetic moment is defined in terms of the largest

magnetic substate so we obtain the defining equation

u
_ N
“1,-th

where g = 1 for the orbital component of the magnetic
moment. Elementary particles also have a spin component
associated with them giving Us. Table 2-1 gives the g-
factors for the proton, neutron and electron, and the
theoretical predictions by relativistic Dirac theory.
Note that the electron and positron are accurately pre-
dicted but the magnetic moment of the proton and neutron
are quite different than expected. This is due presumably
because the proton and neutron are not fundamental par-
ticles but are composed of quarks or at least mesons.
Quark and meson exchange effects are discussed later.

Now let us consider some Odd A nucleus and assume
that our nucleons are nicely paired in their respective
shell model orbits. The magnetic-moment of the nucleus
taken as a first approximation to be that due to the free
odd nucleon, the paired up "core" nucleons contributing
nothing. The total magnetic moment of the nucleus will
be the vector sum of the orbital moment pg and the spin
moment Us of the particle.

It is useful to have an equation for the sum of
two magnetic moment vectors; If :1 = glgl, and 32 =
g232, DJ = g3 = ($1 + 32)--%—-where 3 = 31 + 32. DJ is

IJI
the sum of the projection of El and U2 on J.

+ -)- + + 3):]:
u = F} = gl(Jl J)J + g2(J2.
J ‘ J2 J2

 

 

Table 2-1. g-factors of Elementary Particles.

 

 

 

Proton Neutron Electron
gm 1 0 -1
Experimental gs 5.586 -3.826 -2

Theory gS 2 0 -2

 

 

2

-> +
Substituting J2 = J12 + 52 + 2J1 ' 32 and simplifying

we Obtain the generalized Landé formula

 

 

gl+g2 + (gl-g2)‘jl(‘jl+l) + (EZ‘gl)32(J2+l)

= )
gJ ‘2 2J(J+1) (3

If we substitute 1 = 11, s = 32, gl = 81, $2 = gs:

we obtain an expression for gJ the g-factor of the nucleus

of spin J = A i %. This is the Schmidt estimate [Sc37].

e -E
E = a, i S 2 . (A)
2£+1

 

If one plots u as a function Of J, J = A i %, one ob-
tains the so called Schmidt lines. Most Of the magnetic
moments Of the ground state of odd A nuclei fall between
these lines suggesting a "quenched" gS value is more
appropriate for a nucleon in a nucleus over that for a
free nucleon. The general trend is predicted well,
however, and the Schmidt formula is useful in predicting
g-factors of certain configurations roughly, especially

near closed shells where perturbations are smaller.

2.2. Core Polarization and Configuration Mixing

In most cases the largest error in the Single particle
shell model estimate for magnetic moments is the assump-
tion that the single particle has no effect on the core.
The odd particle can "polarize" the core, however, or
induce excitation in the core of the type (511’32)Ic = 1
where 31 = A + %, 32 = 2 - %. In other words there is a
nonzero matrix element connecting the ground core state
Ic = O and an excited Ic = 1 core state. Figure 2-1a
shows shell model levels near closed shells, and pos-
sible core excitations. Note that near the N = 8 and
N = 20, 160, “00a closed shells the orbitals available
are not of the same 2.and therefore no core polarization

208Pb closed

can occur, but near the N = 50, 80, Sn +
shells the orbitals are available and core polarization
can occur. It is for this reason that the heavier ele-
ments deviate more from the Schmidt lines than the lighter
elements near 160 + uoCa. The sign of the core polariza-
tion magnetic moment correction is such that it accounts
for the inward deviation from the Schmidt lines. A
simplified explanation of core polarization can be given
which predicts the lowering of the magnetic moment.

Assume a proton is traveling around a nuclear core

(Figure 2-1b), spin direction as shown. The core pro-

tons will want to align antiparallel and the neutrons

parallel Since like nucleons prefer a singlet state and

IO

 

 

 

 

Pl proton neutron
2
A 3 A '2
f1 X h 2 X '2
2

........... 32 ----d-----|26
_é._d.§. ‘ .

 

 

 

 

i 52' —é———h" —6—ii§-
'2'
icy;

Figure 2-1a. Core polarization excitations near closed
shells.

   

Figure 2-1b. Core polarization and spin alignment.

ll

unlike nucleons a triplet state. The result is the
generation of an oppositely oriented core magnetic moment
from both core neutrons and protons since the direction
of K is opposite the spin for a neutron.

Core polarization is Just another name for configura-
tion mixing though, since the admixture of these core
excited states to the wavefunction as mentioned before
is core polarization. Arima and Horie have done calcula-
tions to estimate the first order core polarization cor-
rection dulst [Ar5A],[No58]. The value depends on the
spin and occupation numbers of the orbitals involved as
well as the energy gap between orbitals, the radial
integrals of the wavefunctions and the strength of trip-
let and singlet nucleon-nucleon interaction. Some simple
assumptions have been made such as Ith = 1.5IVSI that

allow one to easily estimate with reasonable accuracy

the Gulst [Ar5A].

12

2.3. Higher Order Effects

 

If one considers some single particle wavefunction
w as a mixture of w° shell model states and a1 amounts
of some impurity states mi, one could write the wave—

function as

w =V1'ZO‘1 $233,150) + i O‘21‘1’1U’Ic9 (5)

[M076] where wi can contain the appropriate Ic = 1

Gplst components. We could then write
_ 2 2
u - (l-ilail )usp + Gulst + Edi “i

where “Sp is the single particle estimate and “i is the
magnetic moment of impurity state i. If we consider the
ui's as being due to random excitations Of the core caus-
ing an average magnetic moment “R = J-gR we can write the
last term as “R E a12. This random magnetic moment “R
is also referred to as the rotational magnetic moment
or rotational g-factor gR and is expected to Obey the

following relation gR z %. We then have

l3

2
A ' “sp + 6u1st ' (Asp-“R)§“1 (6)

The third term is called the higher order correction
6pHO' The effect of GuHO is the same as if the g2 factor
in the Schmidt formula were reduced for the proton and
increased for the neutron. Calculations of this are
complicated and Show that it is not always small.
Sometimes GUHO tends to cancel sulst' Some theories
[M16A] take care of GuHO by way of "renormalized opera-
tors" which assume some Ggl caused by a nucleon-nucleon

interaction time [Be68].

2.A. Quarks and Mesonic Effects

It isn't too surprising that the magnetic moments of
isolated neutrons and protons don't match those predicted
for Dirac particles, since these nucleons are considered
to be composed of quarks and, more assurably, contain

mesons as exchange particles as these equations demonstrate:

‘H ‘Hv
I

p+1r

Since the n meson has no spin the magnetic moment is
due to the orbital part (ATr = 1). The Trmeson being

lighter than p or n will give rise to a higher moment

1A

(formula 1) SO by simple calculation a dissociation
probability of N30% will reproduce the observed free
nucleon moments. This value is consistent to the expected
theoretical dissociation probability calculated from the
pion-nucleon interaction [M156].

A more elegant and simple theory claims that nucleons
are made of three quarks, a combination of "up" quarks
charge + g and "down" quarks charge - %. A proton is
+++ and a neutron is +++. Quarks are spin 1/2 fermions
with a Dirac g-factor. Assuming the ++ and ++ pairs
couple to spin 1, we find by using the Landé formula
(Equation 3) and the definition of nuclear magneton

(Equation 1) that the g-factors are:

M
=2n ....
8p mg 8n

(ADI-F:

Mn
M—.
q

Since the mass of the quarks haven't been measured, the

g-factors of the nucleons can't be predicted, but the

8 8

ratio -2-= -l.5 is close to the measured -2££££21 = -1.A6.
gn 1 . n(free)

Also if one assumes Mq z 3'Mn the g-factors are predicted

quite accurately.

15

2.5. geFactors and the Rotational Model

In the rotational model the nucleus is viewed as
being deformed and rotating. The deformation thus intro-
duced remove the degeneracies of the shell model orbits
yielding orbitals designated by a new quantum number K,
the projection of the spin on the symmetry axis. (See
Section 6.1 and Figure 6-1 for a more extensive explana-
tion.) Particles residing in these orbitals in excited
states couple together under the influence of pairing
forces giving rise to multi-quasiparticle states. These
intrinsic states are frequently the basis for rotational
bands that is a series of excited levels resulting from
increasing rotational angular momentum added to the system
in this particular particle configuration. These quasi-
particle states are also Often isomeric, allowing a measure-
ment of the g-factor which can be of use in determining
the exact configuration of the state.

To obtain an estimate of the g-factor Of such a
state theoretically we Would start with the basic defin-

ing equation [H059]

fi—TTTLI‘ <7>

16

That is the effective magnetic moment is the expectation
value Of the magnetic moment Operator parallel to the
spin I [B152]. For two quasiparticle states, following
the arguments of Hooke and Khoo [Kh73]:

Hop = ES 81 + 32 £1 + 85 82 + gz £2 + ERR (8)

where 81’ S2, £1, £2, gsl, ggl, etc. are the spin
angular momentum of particle 1 and 2, orbital angular
momentum of 1 and 2, and the corresponding g-factors.
R and gR are the rotational angular momentum and its g

factor. Assuming no interaction between the two quasi-

particles,

$2=K7-‘-;-,andforI=§2=K.

u = T:T{gs<szl> + g£<£zl> : [g82<sz2> + g£<122>] + gR}.
(9)

In the asymptotic limit (of high deformation) <SZ> =

2 and <£Z> = A (see Figure 6-1). So

17

I
u = 3:3{ss(21122) + gg(AliA2) + ER} (10)

we can also write the magnetic moment in terms of the

rotational and intrinsic g-factors gR and gK

_ I
u ‘ I$I{Kgx + 3R} (11)

Comparing Equations 10 and 11 we see

KgK = gs(21:22) + g£(AliA2) (12)

Since

K = |Oltfl2| = [(21+A1) i (22+A2)| (13)

and for singlet spin states (by definition)

18

so gK = gz. For singlet two-quasiparticle states then

0 for two neutrons

gK

1 for two protons (1A)

For triplet spin states 21 i 22 = 1 from Equation (13)

gK = %{gs + g£(K-l)} (15)

These equations apply in the asymptotic limit of infinite
deformation only but are a good approximation for real
highly deformed nuclei. For intermediate cases one can
use formulas incorporating parameters of the Nilsson

wavefunction to obtain the g-factor [Kh72].

III. EXPERIMENTAL

3.1. Methods for Measuring geFactors

To measure the magnetic moment of a nucleus or g-
factor one must measure the magnitude of the Zeeman splitt-
ing of the nuclear or hyperfine states. The splitting
between two magnetic substates in the simple case of a
nucleus in a magnetic field will be AB = gHuN which gives
the g-factor directly. In some experiments the multiplicity
of substates can be determined which allows the nuclear
spin I to be determined and thus the magnetic moment
u = gI. Atomic beam experiments similar to the original
Stern-Gerlach experiment have been done to determine
spins and moments of many stable elements. Nuclear mag-
netic resonance (NMR) is the application of rf radiation
of the energy equal to the difference in energy of the
magnetic substates, which induces transitions between
them. NMR is frequently used in conjunction with atomic
beams and optical spectroscopy to increase the accuracy.
Optical spectroscopy can be used to View directly the
splitting of the hyperfine structure or to detect NMR
resonances. MOssbauer spectroscopy looks directly at

the nuclear states by recoil free resonance emission-

19

2O

absorption Of Doppler shifted low energy Y-rays. A
moving source provides the Doppler shifted y's which are
passed through a cooled absorber where the y's are
selectively absorbed. A plot of absorption vs. source
velocity gives a pattern similar to an optical inter-
feromery experiment. The splitting between absorption
peaks gives the energy difference between substates and
thus the g-factor.

The above methods are suitable for determining g-
factors of stable isotopes or unstable ones with a long
half life that can be obtained in sizable quantities.

To determine g-factors of short lived states, for example
nuclear isomers populated in in-beam nuclear reactions,
one must use the radiation depopulating the isomer as a
probe of what's happening. Luckily, y's and 8's are
emitted with an asymmetric angular distribution with
respect to the nuclear spin axis and this can be used to
determine the orientation Of the nucleus. NMR can be used
then exnloiting the change in the angular distribution to
indicate occurrance of substate transitions. There is
another way to obtain g—factors and that is to measure

the rate of precession of the nucleus in a magnetic field.
The Larmour frequency (precession frequency) ML is equal

to the NMR rf resonant frequency hvrf = fimL = AEsubstateS

= SHUN-
Since the angular distribution Of y-rays depopulating

21

some isomer will follow the precession of the nuclear
spin-magnetic moment axis,.one can obtain ML from the
frequency Of precession of the angular distribution.

One can't observe the precession of a single nucleus only,
since once a nucleus deexcites, it no longer exists in
that state and has given up its signature for detection,
the y-ray. So in order to plot distribution position XE:
time one must have a group of nuclei all aligned the same
or nearly the same relative to each other.

There are several methods for obtaining nuclear spin
alignment. One method is to cool the source to a point
where thermal energy.is lower than the nuclear and elec-
tron spin-spin interactions allowing alignment to occur.
Another method makes use of the alignment provided in an
in-beam nuclear reaction, the perturbed angular distribu-
tion technique (PAD). In this method the arrival of a
beam pulse at the target creates a group of oriented
nuclei which precess in an external magnetic field. The
precession frequency is determined from an oscillation in
the count rate as a function of time in a detector ob-

serving radiation deexciting the state of interest.

22

3.2. The Time Differential Perturbed Angular Distribution

Method (TDPAD)

 

3.2.1. Nuclear Reactions and Alignment

In a typical in-beam y-ray experiment a thin m1
mg/cm2 foil is bombarded by the cyclotron beam. Fusion
of the projectile with the target nucleus yields a com-
pound nucleus Of high energy. The compound nucleus
very rapidly emits neutrons (Of low angular momentum due
to the centrifugal barrier). The recoiling nucleus comes
to rest in about a picosecond for heavier nuclei, where-
after most of the.diScrete y-rays are emitted. The com-
pound nucleus is formed with the spin axis perpendicular
to the beam (Figure 3-1) similar to what one would expect
from hitting a billiard ball off center with a projectile.
The neutrons cause little disturbance Of the alignment
because Of their low angular momentum and the stopping
process with its many collisions over a very short time
doesn't precess the nucleus very far either. The cas-
cade of v-rays can cause some loss of orientation but
most spin dealignment comes from nanosecond time scale
interactions between the nuclear moment and the electrons

and field gradients in the lattice of the stopping material.

23

3.2.2. Angular Distributions

According to the theory worked out by Frauenfelder
and Steffen [St68] the angular distribution of y-rays
from aligned (Gaussian distribution about m = 0,beam

being 2 axis) states is

21
W(8) = 12; AkPk(cose) (16)

even

where Pk are the Legendre polynomials and Ak are the ex-
pansion coefficients, and A the multipolarity of the y—
2)

ray. A2 is positive for stretched quadrupole (A
and negative for stretched dipole transitions. The result
is a peanut shaped distribution about the beam axis
(Figure 3-1). In the time differential perturbed angular
distribution method a magnetic field is applied perpen-
dicular to the beam, Figure 3-1, causing the nuclei to
precess at angular frequency ”L around H. The oscillation
of count rate as a function of time in a y detector placed
at a fixed angle is used to determine ML. Figure A-l
shows a typical time spectrum of an isomer decaying in a
magnetic field. It is an exponential decay modulated by

a sine function. The equation for the time spectrum

would be.

2A

.Emwnucfi newscmwao swam can Aoaomsupmsvv cofiusnfiuumwo unasmc< .Hum cusmfim

ohm. .. N \
m .25

> ‘ ea/s. fiawo

 

ohm...
_ ..3

 

/

N ...
:8 I
a

x 9&5 .2 3.8280

 

25

N = Noe-AtWCG) (17)

where 8 = 60 - th, and A is the decay constant of the
isomer. Taking the expansion for W to k = 2 at 90 = 1350,

we have

1

N = Noe‘mtl + h- A2 + 134—112 sin(2th)] (18)

In the time integral method one Obtains the angular dis-
tribution by recording count rate as a function of angle
with field on and off. The shift in the angular distribu-'
tion 9 observed would be 6 = er where T is the mean life-
time of the state. This method would be of use when

T is too short for a differential measurement. Equations
17 and 18 assume the alignment is retained throughout the
observed lifetime. This is not always the case, however.
Many times it is difficult to maintain alignment for a
sufficiently long time. The electric quadrupole effect
plays a large part in causing a relaxation of the align-
ment [Ab53]. Electric field gradients created by beam-
induced lattice—defects surrounding the nucleus interact
with its quadrupole moment causing a disorientation of

the spin [Be70]. Nuclei coming to rest in interstitial

26

lattice sites experience a similar effect due to inherent
field gradients at these sites as well as nuclei stopping
in a substitutional position in some lattices of non-
cubic symmetry. The relaxation is not spatially symmetric
with time due to the presence of the magnetic field. Two
relaxation times are defined as commonly used in NMR
theory; T1 the longitudinal relaxation time, involves
magnetic substate (spin orientation) changes with respect
to the magnetic field, and T2 the transverse relaxation
time involves a randomization of the alignment in the x-y
plane. Clearly a non-homogeneity in the magnetic field
could cause differenCes in precession frequencies between
nuclei causing this T2 type randomization. Due to these
inhomogenieties in field strength (including quadrupole
interactions) T2 is less than Tl' After a time governed
by T2 the angular distribution in the x-y plane would be
isotropic whereas in planes containing the z axis it would
still be non-isotropic until after a time governed by T1‘
For the TDPAD method where detectors are placed in the
x-y plane, T2 would be the limiting factor.

Gabriel treats the PAC for the general case including

relaxation and introduces the formulas [Ga69]

27

 

_ Aw q qq q
w(9’°) ' 2k+l k2 Ak Yk Gkk Yk (19)
.q
kk

GEE is the perturbation factor introduced, Ak is the usual

anisotropy coefficient as before, YE are the spherical
harmonic functions, k goes from 0 to 2A or 21 as in the
previous eqution, and q the "multipole orientation" goes
from -k to +k and is an even function. MEE is a second
order frequency perturbation which might be important
only when erc z 1, where TC is the correlation time of
the time dependent perturbation. The magnitude of MEE
cannot be estimated without knowing the correlation mechan-
ism, but the effect of this term has been shown to be
suitably insignificant in most experiments. Expanding

this equation to k = 2, e = 90° and O = 135° (typical)

and taking only the real part of GEE:

W = 1 + E-A2e + % A2 e cos (2wL) (20)

Note the exponential decay of the first and second A2
term. The first decays with constant A20 depending only
on T1’ whereas the second term decays with constant

122 dependent on both T1 and T2. Most experiments are
not precise enough to measure T1 and T2 accurately, but

a detector placed out of the x-y plane would be able to

measure Tl'

3.2.3. Asymmetry Ratio, R(t), Formation

In a TDPAD experiment it is useful to get rid of the
exponential decay factor from the time spectrum and ob-
tain a pure sine function that can be fit to obtain the
frequency. To do this one can use two detectors in the

experiment placed 90° apart. The time spectra for each
detector will be out of phase by 1/2 a period, enabling

a ratio

_ N(A5°)-N(-AS°)
R(t) ' N(A5°)+N(-A5°) (21)

to be formed, resulting in first order in a regular sine

29

function. Taking Equation (19) to be k = 2, e = 1350

and substituting in Equation (21), one sees

77 A2 sin(2th)

 

R(t) = (22)
1
If one expands Equation (19) to k = A, one obtains
8 sin 2w t
R(t) = 2 L (23)

 

.1 + 80 - BA cosAth

where

-A t -A t
_ l 20 9 A0
80 - A A2e + 6F Aue
-A t -A t
_ 3 22 5 A2
B2 H-A2e + I6 Aue
-A t
BA _ 35 Aue AA

Even if AA is not insignificant the effect on R(t) as

seen from the equation is pretty small.

30

3.3. Magnet Construction

3.3.1. Experimental Design and Considerations

The main factors to consider in attempting to measure
a g-factor of some isomer with this method are: the life-
time of the isomer, the magnitude of the expected g-
factor, the relaxation time (suitable host lattice), the
magnetic field strengths, the maximum beam pulsing inter-
val, and the population intensity of the isomer. For
good accuracy one would like to observe at least one full
cycle of the sine function which is half the Larmour period

80

fin 656
NT 1 “NE 01" NT(nsec) 1m (2“)

where NT is the number of lifetimes the radiation is-
servable. This means that in general a high magnetic
field and high g—factor are more desirable although there
is an upper limit to the observable Larmour frequency set
by the time resolution of the system, A—lO nsec. It is
also important that the alignment be preserved over a
significant portion of the observable lifetime of the

radiation. One must choose a suitable target or target

31

backing to host the recoils such as a metal with a cubic
lattice structure like Pb. Use of a target heater may

be necessary to allow the fast annealing of lattice dis-
locations in the case of some high melting pOint materials,
in order to extend the relaxation time. The beam pulsing
period may be an important factor. At the MSU cyclotron
the system used to provide microscopic pulses (those that
don't have a finer structure) of varied duration is an
electrostatic beam sweeper. A sinusoidal voltage is ap-
plied to a set of deflecting plates that sweeps the beam
past a slit. The frequency of the sweeper is some lower
harmonic of the cyclotron frequency allowing one to
separate up to one out of 11 natural beam pulses (period
~50 nsec). The maximum pulse period attainable is about

1 see. For isomers with lifetimes greater than about 200
nsec the radiation produced by previous beam bursts has

a time overlap with radiation from later beam bursts, thus
affecting the time spectrum. The effect of this overlay
should be taken into account in the analysis, although

for a simple system the frequency of the oscillation of
the R(t) spectrum won't be affected, only the amplitude.
For the case when the half life is not much less than the
beam pulsing interval a stroboscopic resonance experiment
can be done [Ch68]. It is still important that the
relaxation time be greater than the half life or the

spectrum would be washed out by isotropic background.

32

The time required to take data with sufficient statistics
is determined by the intensity of the line(s) depopulat-
ing the isomer in the delayed spectrum, the magnitude of

A2 and ML, and the relaxation time.

3.3.2. Data Taking Method and Electronics

There are a few different ways to take data depending
on the data taking programs available. One method is to
do two separate timing experiments, one with each detector
(Figure 3-2a). (One can then use the program TOOTSIE
setting bands on the y-rays and collecting the time spectra
(see Appendix for notes on TOOTSIE). Another method is
to OR the two detectors and strobe 3 ADC'S whenever either
detector fires, two of which contain the information for
the event. This method was used in conjunction with a
program written by Brian Jeltema (GFACTOR) on the PDP-
ll/A5. The PDP is count rate limited to N5 K cps rate
with the Camac ADC's so one must gate out or scale down
the prompt events in order to take data efficiently

(Figure 3-2b).

3.3.3. Design and Construction of the g-Factor Magnet

To do the TDPAD experiments the target must be held in
a magnetic field perpendicular to the beam. A magnet

had to be constructed that would contain a Vacuum-target

 

 

 

 

 

 

33

W
Fraction

 

 

 

 

 

Disc.

 

 

 

rt l
filter
‘Tlming ‘
- - Filter
Amplifier Amp.
SCA
A . . SCA
oincidence on,

 

 

 

3

DC

lGoteeDelay'
Gen.

 

F—

 

 

Energy

Figure 3-2a.

gate

 

 

 

 

ADC
Time

 

 

 

stop

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rt ‘
filter
Constant TiminT \
Fraction Filter -
Amplifier
Disc. Amp.
.1 L; r...
S C A
TAG r-sfig1LCoincidence
e+Delayx
Gen.
ADC ADC
Time gate Energy

 

 

 

Electronics for a g-factor experiment,
separate timing.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r rt
fitter filter
1—1
Constant ‘ Timing A I I
Amplifier FPOCti Filter Amplifier
Disc "—" “"9-
.——. . —-i
O R —
start
Gate+Deloy C'A _ r_t_’
Gen. 0" TAC ”OP
, golte _ [ I , 1
ADC ADC ADC
Energy Time Energy
Figure 3-2b. Electronics for a g-factor experiment, com-

bined timing

3A

chamber between its pole tips. The pole tips and thus

the area of the magnetic field were kept small so the

beam would be bent as little as possible before reaching
the target. A high magnetic field was desired so the

pole tip gap was made fairly small. The maximum field
attainable with a non-superconducting iron magnet with
reasonable currents is around the saturation field for
iron, 22 kgauss. It has been found that the optimum pole
tip shape is a truncated cone with A5° slope [K165].

This is true when the coils are placed around the sloping
portion of the pole tip, since the field lines get pinched
together in the coil-area, thus assisting in achieving

high flux density (field) at the tips. The magnet that

was used was a small beam line magnet that was modified

by adding yoke extensions and new pole tips. The A5° pole
tip design was used although the coils were left at the top
of the poles rather than the middle. The pole tips were
machined from castings generously supplied by the Bay City
Foundry maker of the iron for the MSU K=500 superconducting
cyclotron magnet. A diagram of the magnet is given in Figure
3—3. The top pole tip is an inch smaller in diameter than
shown due to an error by the shOp. The coils consist of A
layers top and bottom of 36 turns each l/A" square hollow con-
ductor copper encased in epoxy. For greater cooling the
water supply was remade to deliver parallel feeding

of all 8 coils. Field calculations were done before

35

 

 

 

 

 

 

 

dee\Hew
G-fcctor e _;
Sim
magnet / \
O O
O O O O
O O

  

 

 

 

 

 

 

 

  

 

Top View

Figure 3-3. The g-factor magnet.

36

construction. It is possible to do a magnetic circuit
calculation by hand. A magnetic circuit is analogous to
a DC electrical circuit; EMF is equivalent to magneto
motive force = NI (ampere turns), resistance is reluctance
R = §%%-where A is cross sectional area, dz length, n

is the permeability, and current is the flux 6 [K165].
The equivalent Ohm's Law is NI = Re. In this case,
however, resistance varies with flux density (field)
since u varies with field. One can start with a desired
field strength at the gap and estimated flux loss to
fringing fields and determine total flux, the reluctance
of each section of the magnet and determine total NI
needed. It is difficult to estimate the fringing field
due to the sloped pole tips, however. Another method is
to start with the NI available and calculate the final
field. An iterative calculation is necessary here, how—
ever, since the reluctance depends on the flux density.
A program called TRIM adapted by Dave Johnson for the
Z-7 computer was used to estimate the field by this
iterative method. A grid is made of the magnet dimen-
sions assuming cylindrical symmetry including material
and coil data. The program estimated that at 308 amps

a 22.8 kgauss field could be achieved and at 500 amps
2A.? kgauss. A four inch diffusion pump setup from an
old Veeco leak detector was used for a pumping station.

A "Spectro Magnetic" DC power supply shared with the

37

neutron time of flight set up was used to power the
magnet. The supply delivered more than enough current

at good regulation. A field calibration test was done
first with a Hall probe and then using a.flip coil which
had been calibrated with NMR. The results are in Figure
3-A. The measured field was about 10% smaller than the
prediction from TRIM. A thermal switch placed on the out-
let side of each coil guarded against overheating. The
maximum continuous current before trip-off was around

A30 amps. The water flow rate was about 3 cu ft/min, the
voltage across the magnet at A00 amps was around 50 volts,
thus dissipating around 20 kwatts. As seen from the
graph, A00 amps is about the saturation point of the mag-
net. The reproducability of the magnetic field with

dial setting was tested and found good to 1%.

A field profile was also done as a function of dis-
tance from the center of the gap along the beam entrance
line using a Hall probe (current = 300 amps). This is
shown in Figure 3—5. The amount of beam bending can be
calculated from this curve. A ray tracing code gives a
bending angle of 7.A° at the target, with A8 MeV alpha
particles. A simple geometrical calculation can be done
by assuming an effective gap radius so as to give the same
area under the curve as that in Figure 3-5. The effective
gap radius turns out to be 3.03 in a 20.5 kgauss field.

The radius of curvature of the beam (non-relativistic)

 

 

 

 

 

 

 

 

 

Dial

 

 

 

 

 

3

 

 

 

 

l- ls ‘4 II I
“no
--.- I ... - m
0'
.-I.. -ltl ..I-r 1| it All
.0
. it [l.t. II... til. . 'IL I-
- .- Sir Ii TI... Iii i

 

 

 

i

400 Amps

T

 

 

field as a function of

 

 

 

38

T

 

300

 

 

 

 

Current

 

 

 

 

 

is
200

 

 

 

 

 

Magnet calibration:

current.

T
IOO

 

 

 

 

 

 

 

 

 

rill i

 

 

 

 

 

 

 

 

 

 

 

 

IO!

.0 5
2 . i

8.: 22... 0:23:

 

 

 

 

 

 

 

 

Figure 3-A.

39

 

 

 

 

 

 

 

 

III I
II
..IuIte I
II I
I J I
I. It:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I‘IIéI III I uIcIII II I III
IAfiA-I ! I IILwII. I
I II II II II II fi II I I II II. #1- I
I‘ 'I I- file I O‘HI ..... I. I ‘Ifi -"
I II I I II f II II.II I I II I I II
IIrI’III I II ssssss I I I fiIII I .I
..
III I II QII I I I I III I I III I II TI I II II I
9; I fi l. I II I I III II II II I III I. II IILII-I
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III I I II I II II I I I I II I II
n
10 I I I III fl .0. I I I I I II III. I I . .I
I I. I. g I I v I IL I! I i ..... 10.: u . f I. . I .I I III II II&
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I l! I. II- v I II 0
I II I I IIL IIIILIII filI III II [I
\\
II I It: I III fiIIL%K III. I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2‘5

0 5
8.: 32“. 0:28.:

Distance from center (in)

Magnetic field profile as a function of

Figure 3-5.

distance from center along median plane.

U0

is

r = lHH/mE(MeV)
(cm) B(kg7h

 

and the angle at the target will be

sin-

I!

0

where d is the effective radius of the gap. This gives
9.l° as a result. When running lower rigidity beams and
higher fields one would expect a greater bending angle,

so this would be a typical minimum bending angle.

3.3.N. Target Chamber Design

 

The target chamber was designed to be as simple as
possible and yet afford suitable access and provide an
exit path for the beam. Since the beam is bent sig-
nificantly, an angled exit port is useful, so two exit-
access ports were made, one at a larger angle than the
other. The whole chamber was cut from a S-l/M" 6060

aluminum pipe. A 2" diameter aluminum entrance pipe was

Ml

used to provide easy positioning of the Ge(Li) detectors
at il35°. The exit ports were 2.87" diam. and fitted
with standard beam line flanges. One port served as a
window through which a scintillator could be viewed.

The targets slide in rails between the pole tips. Two
targets or a target and scintillator can be positioned
in the chamber. These are attached to a brass rod and
movable thorugh a sliding seal in the target port on the
side of the chamber, see Figure 3-6. A collimator is
situated 0.25 in from the target to facilitate focusing
of the beam. In a 20 kgauss field electrons with energy
14.3 MeV or lower from the target won't reach the collima-
tor. The rear of the chamber and the pole tips were
covered with 1/8" thick lead sheet to prevent them from

being activated by the beam.

U2

a’x

“ beam exit .
port

    
 

 

 

 

 

 

 

 

 

5
access port
tar at
target L g \-
ort '-———--“
p L====aAlc===J
[ \ tcollimator
beam

target

 

 

 

 

 

 

Figure 3—6. Taréet chamber and target.

IV. RESULTS OF MEASUREMENTS

u.1. 210Po Test

 

To test out the system the g-factor of a previously

210

4.
well known case, the 8 isomer of Po was measured.

To produce the isomer we used the reaction 208Pb (a,2n)-
210Po. 25 MeV alphas were used on a 2 mil (58 mg/cmz)
natural Pb target. A partial level scheme is given in
Figure fl-l. The 8+ isomer has t1/2 = 112 ns and g =

0.910(5) [Ya70]. The 6+ level has t1/2 = “1 ns and g =
O.93(2) [Ya70],[Na73]. The line we observed was the 1180
keV line. To minimize background from beam hitting the pipe,
a u ft long beam dump was used with a split Faraday cup at
the end in order to allow the beam to be centered in the
beam dump by adjusting the field of the magnet. A field of
18.6(1) kgauss was used. The beam was swept 1 out of h
leaving 261.6 nsec between the beam bursts. The data taking
method in Figure 3-2a was used, Just a normal timing experi-
ment with the y-side gated. Only one detector was used in
this experiment; and it was placed at 135°. The 1180 keV
line was gated and the resulting TAC spectrum showed the

oscillations due to the precessing angular distribution very

clearly (Figure M-l). The data were fit to the function

”3

 

 

 

I...
10 I T I I I I
- 8” e4 l557 Hens fl 2“"30 3
__ 6* I473 4Ins :
_ 4+ I427 |l8l keV
+ 246 "
.. 2 + ll8l ..
f2 3 ”8|
§10 t’ ‘:
o - ..
o - ..
._ 0+ _

 

I
l

 

 

 

 

100 l ' ' l 1 l E" L

0 200 L+00 800
CHANNEL

 

Figure h-l. 210Po TAC and partial decay scheme.

“5

N = Noe't/Ttl +1171;2 + {>- A2 sin(2th-2A6)], (25)

where A6 is the phase shift due to the beam bending and
. other factors, with the least squares program KINFIT
[Na71].. NO, T, A2, wL, and A6 were left as parameters
in the fit. To the period of oscillation = gL-was found

L
to be 38.8(5) nsec. Using the equation

_ fifl
' HuNTw (26)

(Section 3.1) one obtains g = 0.909(13), very close to
the accepted value 0.910(5) [Na7u]. There is a complica-
tion, however, due to the presence of the 38 nsec 6+
isomer. Since the 1180 keV line is being observed, the
nucleus will have precessed under the influence of the

6+ isomer also, introducing a change in the observed

wL and a phase shift (incorporated here in A9). Nagamya
gives a detailed treatment of the PAD pattern equation
with the presence of two isomers [Na70]. He also makes

210Po case and how different experi-

reference to the
ments can yield different g-factors and half lives

depending on the beam pulsing interval T and the initial

R6

population ratio of the two isomers. Using his estimated
value for a = Nl/N2 = .9 the initial population ratio,

and our value T = 262 nsec the apparent w should be

L

0.90 m + 0.10 w2. Substituting the measured

“’L‘ 1

“L = 38.9 nsec and calculating w from Equation (26)

2
assuming g2 = 0.93 [Na7u] one obtains ml = 38.9 nsec
corresponding to g = 0.907 showing a minimal deviation

in this case due to the similarity of the g-factors

g1 and g2 and the small effect of w2 on ”L' This ex-
periment confirmed our faith in the setup and indicated
certain needed improvements in the target positioning
method. Holes were put in the edges of the target frames
as in Figure 3-6 and a set screw was installed in the
target positioning rod to rigidly fix the targets. A
metal sleeve was used on the outside portion of the rod

to fix target and scintillator position relative to the

collimator hole.

u.2. The g-Factors of the 1912' and 11/2' States in 115Sb

H.2.l. Introduction

 

The nucleus 115Sb lies in a shape transitional region.
It is one proton outside the Z = 50 closed shell and
exhibits some vibrational states similar to the neighbor-
ing Sn nuclei, but it also has a rotational band built
on the 9/2+[U04] Nilsson state [Ga75]. The non-rotational

#7

part of the spectrum has been explained by coupling a
d5/2 proton to excited states in the 11“Sn core [Va7l].
Recently an in-beam y-ray study of 115Sb yielded data on
high spin states in that nucleus [Br77]. Three isomers
were seen: a tl/2 = 6.7 nsec, 11/2- isomer at 1300.2
keV, a 156 ns 19/2’ isomer at 2796.3 and a u.o ns 25/2+
isomer at 3059.7 keV corresponding to a (15/2 proton
coupled to the 11“Sn 3', 7' and 10+ states, respectively.
Figure 4—2 shows a partial decay scheme involving those
isomers. The 19/2' isomer is thought to be a three
particle state [wd5/2 Q ”111/2 @ vd3/2], where [mu/2
Q vd3/2] is the 7" state in the 11”Sn core. According to
Bron [Br77] the wavefunction probably contains some ad-
mixture of ["h11/2 Q 14+] since the transition 19/2" to
11/2' in 115Sb is enhanced about a factor of three over
the corresponding 7' to 5' transition in ll“Sn. The
measurement of the g-factor could give an indication
of the extent of the admixture.

For the ll/2“ isomer a g-factor measurement would
indicate the extent of possible admixture of the [ads/2 Q

3' (octupole vibration)] with a pure "hll/2 configuration.

0.2.2. Experiment

The g-factors were measured using the time differen—
tial perturbed angular distribution technique (TDPAD).

The reaction was 115In (a,uny)1158b, induced by a “8 MeV

U8

 

 

 

 

 

 

 

 

 

2796-3 [279 4 @578 l9/2' .156 nsec
26383 I 1512'
25l6-9 I 15/2“
I338o2

lane-7
1300-2 V we" 6-7nsec

1300-2

¢ 512’

“58b

Figure 3-2. Partial decay scheme for ll58b.

“9

a particle beam from the MSU cyclotron. The target

was a thick (110 mg/cmz) natural metallic In foil (95.7%
115In). Indium is a good host lattice for the 115Sb
recoils due in part to its tetragonal lattice structure
which maintains the alignment over several precession
cycles. The low melting point of In (156°C) insures a
high mobility of lattice defects induced by the beam

even at room temperature and thus reduces the main cause
of loss of alignment generally experienced with this tech-
nique.

The magnet used produced a 22.7 kG magnetic flux
density with less than 0.5% deviation over the target
area. The field was calibrated to less than 1% error
using a flip coil that had been calibrated with nuclear
magnetic resonance. Two 10% efficient Ge(Li) detectors
were placed at il35° to the beam. The beam was chopped
with an electrostatic deflector so there were 526 ns
between bursts. Six energy gates were set per detector,
one on the 1217 keV line, one on the 1300 keV line, and
two background gates, one on either side of each peak.

Figure u-3 shows the background subtracted time to
amplitude converter (TAC) spectra and the normalized sum
spectra formed by adding the spectra from the two de—
tectors. The 1300 keV transition is a doublet contain-

116

ing some of the 129R keV line from Sb decay. The

amount of this background in each 1300 keV spectrum was

50

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51

determined by fitting each TAC spectrum to the equation
for a time spectrum incorporating the 156 ns half-life as
determined from the 1217 keV sum TAC, plus a background

116

due to the 15 min decay of Sb. The ratio

R(t) =

was formed for each Y ray, where N(1) and N(2) are counts
from detector 1 and 2, respectively (Figure 4-H). The
time-dependent equation for the counts N in a TAC spectrum

in a pulsed beam and magnetic field is [M076]

N(t) = Noe‘mune‘m)’1

x [1+BOistin(2wt—2A6+¢2)
-Bucos(th-uAe+¢u)], (28)
where A is the decay constant of the isomer, T the time
between beam bursts, w‘is the precession frequency, and

A6 the correction for beam bending due to the magnet.

The coefficients

52

 

0.3
l2|7 keV

o.2-' 1

R (t)
.0
o

1

 

R (t)

 

 

 

 

 

 

0.3 1 L' 1 L 1 L 1 L 1
I300 keV
-Q3 1 1 1 1 1 1 1 1 1
O ICC 200 300 400

TIME (nseC)

Figure 1-1.

500

R(t) for the 1217 and 1300 keV lines in 1158b.

B = b er k (29)

in Equation (27) contain an exponential decay term with
constant 1k accounting for the relaxation of the align-
ment. The factors Gk and ¢k in Equation (28) arise from
the overlap in time of decays from preceding beam bursts
and are l and 0, respectively, when t1/2 << T. They are

defined as follows:

G = 1-8 ,
k (l—2BBkcoskwT+B2Bl2)l/2

 

BBksinkwT
¢k = , (3O)
1-BBkcoskwT
-A T
where B = e-AT and Bk = e k .

The factors bk in Equation (29) are combinations of

A2 and A“, the usual anisotropy coefficients,

bo =71“? + BEAM b2 =131‘A2 + TSEALH bu = 83% (31)

The ratio R(t) is then

5H

N(1)-N(2)

R(t) = N(17+N(27

 

stin(2wt-2A0+¢2)

 

 

= . (32)
1+BO-Bucos(uwt-hA6+¢u)
The data were fit to
%A2e-A2tG2(wT)
R(t) - sinE2wt—2A0+¢2(wT)1 (33)

by varying the four parameters, A2, 12, w, and A6 with
a least squares fitting program. The simplifying assump-

tions, Au = 0 and 10 << 12 have been made.

H.2.3. Results

The results of the fit to the TDPAD data are in
Table u-l. The period of oscillation, T2 = W/w, is 100.7
ns, corresponding to a g-factor of +0.287(u) obtained from
the expression g = -fiw/BuN. With the field B "down"
and w in the opposite direction, the g-factor is positive.
Statistical errors were carried in the background sub—
traction of the TAC spectra and the formation of the ratio.

Weighted least squares fits to the data were performed.

55

Table u-l. Results of the 1158b g-Factor Analysis.

 

 

 

 

 

 

 

Y-Ray
Energy 819/2' 811/2-
1217 keV 0.28701)a
1300 keV 0.287(u) 1.06(10)
Parameters Determined by Least Squares Fitb
1217 keV 1300 keV
- 1.: _ = _
Tm - w 100.7(13) Twl 100.7(11)
A9 = 0.13(6) T = - 27.2(6)
“2
A2 = 0.26(3) A2 = o.36(6)
12 = 0.0021(8) Ar = 0.0021(13)
c = 0.019(5)0 c = — 0.056(5)°

 

 

Constants Used in the 1300 keV Fit

 

(l)

t1/2 = 156(3)ns. e = 0.414(3) A0 = 0.13(6) aél) = 0.903
13; = 6.2(ll)ns 01 = 0.7(2) fl = 1.11

 

 

aThe errors in the results reflect all fit uncertainties as
well as magnetic field uncertainties and in the case of
the ll/2‘ g-factor were determined by varying entered
parameters as stated in the text.

bThe errors quoted reflect only statistical errors in the
fit. The tabulated g-factor results also reflect the
errors associated with the constants used for the 1300 keV
fit, as determined from the 1217 keV fit and the 1300 keV
sum spectrum fit.

CC is an offset constant left as a parameter in the fit.
Its only function is to shift the curve vertically to
correct for any normalization errors.

56

The effect of the Au term on the derived g-factor can

be estimated by including it as a constant in the first
equation for R(t). Varying A“ from 0 to -0.1 changed the
g-factor only about +0.l% so no adjustment was made to
the final value. The magnetic field at the nucleus is
not always the same as the applied field, however. In
our case, the Knight shift due to polarization of the s
electrons in the metal target increases the field at the
nucleus (+0.82%), and the diamagnetic effect decreases
the field (-0.6%), resulting in a correction in the measured
g-factor that is small (-0.22%) compared to the estimated
error [M076].

The ratio for the 1300 keV line can also be fit, but
is complicated by the presence of the short-lived isomeric
state that it depopulates. Nagamiya gives a detailed
treatment of this problem [Na74]. The expression we have
employed to calculate the intensity ratio of the line

depopulating isomer 2 under isomer 1 is

R(t) = b2N(t)'l[cosaG§l)lee-Altsin(2wlt+6+¢él)—2A0)

<2)f

-A -
-G§2)f28e 2tCOSGSin(2w2t+5+¢§2)-2A9)+aG2 e A2t

2

x sin(2o2t+¢§2)-2Ae)], (3h)

57
where
-12t

N(t) = Bf]_e->‘1t + (a-B)f2e ,

B = Al/(x2-xl)’

b2 = 312e'xrt/(1+A2),

fi = (1-e-AiT)-la

a = N2/Nl,

a = tan‘1[2(w2-el>/<12-1111. (35)

In the above, Xi and mi are the decay constant and preces-

sion frequency for isomer 1, 1 is the decay constant for

r
loss of alignment, a is the intensity ratio of initial
population, and Géi)(wiT) and oéi)(wiT) are defined as
before. Note 8, b2, and 11 are defined differently in
Equations (34) and (35) than in Equations (28)-(33).

The same assumptions made in Equation (33) apply to Equa-
tions (3”) and (35). In this case, f2 z 0:2) 3 l, and

¢§i> ~ 0. The constants a and 12 are determined by fitting
the sum TAC for the 1300 keV line to N(t) and solving for

a and 12. These were entered as constants together with
fl, A0, 0&1), Al, and e and the parameters w2, ml, A2,

and Ar were obtained from the best fit. The data for

58

the 1300 keV line later than about 230 ns after the beam
burst were left out of the fit since a small amount of
prompt beam leakage associated with the beam sweeper
occurred at that time period. The g-factor of the ll/2‘
isomer obtained from w2 is +1.06(10). The error was
estimated by varying the constants over an appropriate
range and noting the effect on w2. The derived g-factor
is in agreement with the value +0.97(10) determined by

Ketel gt_al. [Ke78]. The effect of the higher lying 4.0

ns 25/2+ state on the measured g—factor was assumed to
be negligible due to its short half life and low popu-
lation, perhaps introducing a small phase shift incor-

porated in A0.

u.2.u. Discussion

 

The expected g-factor for the supposed configura-

tion [ads/2 ® vh11/2 ® vd3/2] for the 19/2 state can be

obtained by combining the measured g-factors for the 111“Sn
7 isomer [Vhll/Z Q vd3/2]7 , g = -0.810(6) with that of

+, g = 1.380(M) to

115
the ground state of SbEnd5/2JS/2
give g19/2_ = 0.305(3) using the additivity relation
(Equation 3). This procedure, however, neglects changes
in core polarization due to different configurations.

Figure H-S shows the suggested configuration of the 19/2'

isomer and that of the ground state. The arrows represent

59

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60

excitations corresponding to the core polarization of the
neutrons by the odd proton. In the 19/2’ configuration the
holes in the g7/2 orbital allow an extra core polarization
over that of the ground state where that orbital is filled.
0n the other hand, the extra particle in the d3/2 orbital
causes a blocking in the amount of d-orbital core polari-
zation. The effects cancel each other, suggesting the
additivity result should be reasonably accurate. The
Arima-Horie formalism [N058] was used to estimate this
effect and the result was g19/2- = 0.293 to 0.313 using
a pairing energy C = 30 to 37 MeV. If there were an
admixture of ("1111/2 ~Q 11+) one would expect the g-factor
to be somewhat larger. The measured value, slightly lower
than predicted, suggests a relatively pure [ads/2 Q vh11/2
Q Vd3/2119/2- configuration.

The g-factors of the 19/2' and ll/2‘ states in 115Sb
have recently been measured independently by other labs.

T. J. Ketel measured the g-factor of the 11/2- state by
TDPAD using a 116Sn(p,2n)llSS‘p reaction where little or no

population of the higher lying 19/2’ state occurred [Ke78].
He obtained a g-factor of 0.97(10) and suggested a 50%
admixture in the configurations "hll/2 and [11d5/2 Q 3'].
Shroy, gt_al. concurrently measured the g—factor for the
19/2' state using a 112Cd(6Li,3n)115Sb reaction and re-
ported it to be g = 0.290(5), in agreement with our measure-

ment but of opposite sign [Sh77]. They have since determined

61

the sign is actually positive but was quoted wrong due to
an error in the phase of the R(t) of a standard used to
calibrate the magnetic field [Sh791. Ketel g§_al. also
published a measurement of the 19/2' state g-factor using
the 113In(o,2n)llssb reaction and obtained g = 0.282(6)
[K079] in agreement with the others.

u.3. g-Factor of the 12* State in 2021311

The MSU-Purdue collaboration has recently obtained
y-ray spectroscopic data in the light Pb region [He77],
[Sa771, [D078]. Many isomers were discovered, some above

the 3.6 hour 9' isomer in 202Pb.

The presence of this
isomer has made it difficult to observe the higher lying
states, but the presumed 20 ns 12+ state, and two higher
lying isomers t1/2 = 150 ns and 100 ns have been found
(Figure 4-6). The g-factors of the 12+ states in the even

l9“:1962198’200’206Pb have all been measured

lead isotopes
[R077], [Yo75], [Na72]. The g-factors were found to be
-o.158(6), -0.157(7), -0.1uu(11), -0.157(6), and -0.155(u),
respectively. They are all amazingly close together
showing little A dependence of the g-factor. The Schmidt
g-factor for an 113/2 neutron hole state ( 113/2)-2

would be gSchmidt = -0.29H (Equation h). The large dif-

ference between the Schmidt and experimental values can

be explained in part by core polarization (Section 2.2).

62

 

 

 

 

 

 

 

 

l6 IOOns

||5|
l50ns

853 (EZ)

12+ ‘V

II. l79(ED
8 88 (£2)

9" V 3.6 111

ZOZPb

Figure U-6. Partial 2O2Pb decay scheme.

63

Using the Arima-Horie formalism [ArSH] and assuming only
core polarization resulting from the 113/2, ill/2 shells
(Figure 2-la), one obtains gcorr = -0.20. Core polariza-
tion from other levels such as the (“hll/Z’ wh9/2) pair
would further reduce the magnitude of the g-factor, but
even with all corrections considered the discrepancy is
considerable [R077]. If one assumes as a first approxima-
tion that all the neutron holes are in the 113/2 orbit

one would expect a decrease in the core polarization

as the mass number decreases since the particle number

in the 113/2 orbit is decreasing. The g-factor would then
be predicted to become more negative for the lower mass

Pb isomer [LEY]. The effect of other core polarizations
(c.p.) may counteract this effect however. A calculation
by Roulet g£_al. [R077] shows the (f7/2,f5/2) excitation
compensating the effect of the (113/2, ill/2) levels.

In this case as the f5/2 shell fills, the higher neutron
number blocks c.p. excitations to effectively counteract
the increased (113/2, 111/2) core polarizations.

The experimental method used is similar to that used
to take the 115Sb data. Gates were set on the 179 keV,
202Pb, 197 keV, 19F, 222 keV, 201Pb, and 258 keV 203%
lines, and a background gate on the high energy side of
each. A “8 MeV alpha beam was used to induce (a,xn)
reactions on a thick l/l6" natural liquid Hg target. It

is estimated from the natural abundance of the Hg isotopes

6H

and the cross sections from the program CS8N [CS8N] that
about 27% of the reaction goes into the production of

202Pb at'u8 MeV. Another peak in 202Pb production occurs

202Pb.

at 30 MeV, where 27% of the cross section produces
Since the target is thick enough to stop the beam, however,
there will be lower energy components of the beam present
that will result in a shift in production toward higher
mass isotopes. Figure h-7 shows a spectrum of delayed
y-rays from this reaction.

About 13 hours of data were collected at a magnetic
field of 22.7 kgauss. One out of eleven beam bursts were
selected by the sweeper, providing 526 ns between beam
bursts. The relaxation time of Pb in liquid Hg is greater
than 100 usec [Qu69]. No relaxation should take place be-
fore the isomers decay. Figure h-8 shows the final back-
ground subtracted TAC spectra and the ratio R(t) for the
179 keV line. Fitting the equation to a simple sine
function (Equation 22) including a phase shift, A6, and
assuming a negative A2 for the 179 keV El transition,
0.28(29).

one obtains A2==-0.10(2) tw = 207(25) ns and A0
The phase shift A6 is close to what one would expect from
beam bending.) The g—factor from Equation (26) is then

- "655-9 -
g ’ 22.7kg-207ns ' “0°1”(2)

 

close to the g-factors of the other measured l2+ isomers.

65

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66

 

 

 

 

 

: I I r
5 “2% .5
1- 179 keV TAC 4,. _
.. s“: _
g 100 ‘3; .-
8 E- fié‘a?” 501:
(.J 3: of o° ::
_ 1:300“ “(fife ..
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10 e“ ‘1‘. 1 ° 1 L r
0 250 500 750 1000
CHANNEL
ZCHZ
02— Pb +_
I79 keV +

 

-0.2- -

 

 

-0.3 1 L 1
0 50 100 150 200

TIME [nsec]

Figure U-8. 202Pb TAC spectra and ratio (R(t)) for the 179
keV transition A indicate -l35°, Elindicate
+135° detector angle.

 

67

There exist, however, two isomers above the 20 ns l2+
isomer as can be seen from the longer tl/2 components in
the 179 keV TAC spectrum and in coincidence data obtained
on this nucleus. The effect of these isomers on the PAD
pattern is not known, but from the size of the 20 ns TAC
peak one can conclude that the prompt feeding of the 12+
level is significant, indicating that the g-factor ex-
tracted mostly reflects the 12+ isomer.

Of the other lines observed, only the 258 keV 21/2+

l7/2+ transition in 203Pb showed any hint of precession

in the TAC spectra (Figure H-9). The difference in the
spectra between the two detectors is small but shows
indication of a positive g-factor, assuming a positive A2
for the 258 keV E2 transition. Theoretically the 21/2+
£12 ns isomer in 203Pb is thought to be (2011Pb n+0
113/2-1)21/2+ [Sa77]. Combining the effective g-factor of
an 113/2 neutron hole g = -0.157(6) (from 200Pb) [Yo75] and
the measured g-factor of the 201(Pb u+ 127a keV level (260 ns
t1/2) g = +0.057(20) [Sa63] using additivity (Equation 3)
one obtains g = -0.032(l6). This g-factor is close to
zero, but a measurable positive g-factor would be un-
expected.

Another experiment was attempted with the same reaction
using data taking method in Figure 3-2b, without the scale-
down of prompt events. Timing SCA's were used to try

and cut down the spectrum in uninteresting areas to reduce

68

 

 

 

 

 

 

 

 

 

 

 

 

29/2" 2949 0.48 s
(211/21+ {L93
874
21/2“ ‘I '3??- 4 2 as
258
839
13/2* J 825 5,2 5
‘ 203Pb
3
10 I I I
p _
:_ 203Pb :
: 258 keV TAC rfi' -
U) .- ‘aap ‘: .d
I— fit’u’ °
3 " a:
O C 590'; ....
Q :- “W. 8 °"
_ 1:300 1:0 "
P I I m
D
D O
05‘.“ 1 1 1 1
0 250 500 750 1000
CHANNEL.
Figure U-9. 203Pb partial level scheme and TAG spectra.

A -l35° spectrum; I]

+135° spectrum.

1'?

69

the count rate. The program IIEVENT was then used to col-
lect data on tape. The TAC spectra for the 179, 222, 258,
600, 839, and 888 keV lines in 202Pb, 201Pb, 203Pb, 201Pb,
203Pb, and 202Pb, respectively, were sorted from the 16
tapes collected. The effective counting rate was quite
low (W1 khz), limited by the tape collection method, and
the statistics obtained were quite poor. The 222 and 600
lines had very poor statistics, the 179 and 258 lines were
added to the previous data and the 839 and 888 lines showed
no hint of precession taking place.

To improve these types of experiments a flowing or
replaceable Hg target should be used, since the activity
built up by the long lived 9’ isomers in 202, 20h Pb
increase the background considerably after about an hour.
A 30 MeV beam could also be used so less high spin isomer
population would occur to possibly interfere with the 12+
202Pb precession pattern.

Work was started on developing a flowing Hg target.
The design consisted of a normal target frame backed with
a 1/8in. thick piece of Al into which two polystyrene
capillaries entered, top and bottom. The back plate must
be glued on using a suitable epoxy that sticks to A1
(not 5 min type). The capillaries run through the plastic
beam port window (use of hypodermic needles glued in the
window may be helpful) to the Hg reservoirs. The Hg is

dripped under the action of gravity into a shielded

70

receiving bottle. The most suitable window material for
the target proved to be 1 mil polystyrene, held on with
cyanoacrylate "super glue", although it is not clear
whether it can take prolonged exposure to greater than
one atmosphere pressure for a sustained period. The Hg

reservoirs may have to be evacuated to relieve this problem.

A.A. Conclusion

 

In conclusion, this system works quite well for

measuring g—factors of suitable isomers using an (a,xn)
reaction. The use of 50 MeV protons has been attempted in

l90Pt using the

the measurement of a 10‘ isomer in
193Ir(p,lIn)190Pt reaction. The protons produced an ex-
cessive neutron flux background, especially when the beam
struck parts of the chamber. The Pb lining tends to become
activated quickly producing long-lived activity. Alpha
particles and heavy ions are much nicer in this respect.
When planning an experiment it is important to consider

the target material in light of previous discussions on
relaxation. It may also be advisable to do an experiment
in a goniometer first to prove the feasibility of the
experiment in terms of angular distribution and intensity
before one attempts the use of the more difficult g-

factor apparatus (in cases where little is known of the

reaction). Summary of the measured g-factors is given in

Table u—2.

71

Table 1-2. Summary of Measured g—Factors.

 

 

Nucleus Spin g-Factor tl/2 . Configuration

 

115 -
Sb 19/2 0.28701) 156(3) ns 105/2 9 Vhll/Z 9 1103/2
11/2" l.O6(lO) 6.2(11) ns trill/2 Q [MS/2 Q 3"]

202 +

Pb 12 -0.lu(2) 20 ns (0113/2)’2

2031911 21/2“ Positive? 12 ns 20uPb n+0 113/2-1 ?

 

 

PART 2

HIGH SPIN STATES IN 180w FROM THE

180Hr(a,uny)180w REACTION

72

V. INTRODUCTION

Recently much research has been done by our group and
by others on the Hf and W isotopes. They have been found
to yield interesting information about the "backbending"
phenomenon [Be76] and display the first observable U-quasi-

180

particle high K isomers [Kh75]. The nucleus W being

amongst these isotopes and an isotone of the fruitful
178Hf was chosen for study of the high spin states using
an (a,un) reaction. The MSU facility provided the rare.
combination of equipment to make this project possible;

a cyclotron capable of providing a pulsed beam of U5-50
MeV alphas with good timing and the necessary data taking
equipment and data analyzing computer.

Previous work included decay studies of levels in
180W from 180Re ([Ho67],[Go67] and [Ha68]) which showed
the first 3 members of the ground band and the first two
members of the 2' band. An (o,2n) reaction study showed
the 8' isomer in 180W and other nuclei [Bu66]. An (a,un)
reaction done long ago with NaI (Tl) detectors showed the
ground band to spin l2 [La65]. Some work was done using
the (p,2n) reaction observing y's and conversion electrons
but the data quality and difficulties involved lead to
a questionable level scheme [Gr66]. Recently in beam

Y work done concurrently with this work using the (p,2n)

73

7”

reaction has confirmed the low spin portion of this work

[Ma791. The present study investigates high spin states

by the 180Hf(o,14n)180W reaction using modern high resolu-
tion Ge(Li) gamma-ray detectors and high speed computers

for analysis, enabling more information to be gained than
heretofore possible. The following discussion will in-

clude some of the relevent theory, a description of the

experiments performed, and the results obtained.

VI. THEORY

6.1. Deformed Nuclei

 

The problem of developing a quantitative nuclear
model is much more complicated in many ways than that for
an atom. In the nucleus one doesn't have a central force
problem such as that for the atom. The nuclear force is
a strong short range interaction and the nucleus is more
like a particle in a box problem. One can approximate the
nuclear force in terms of a square well or harmonic os-
cillator potential, fill it with spin 1/2 nucleons with
orbital angular momentum A and obtain a series of possible
energy levels. If spin—orbit coupling is included (M. G. .
Mayer) [Ma50], a nuclear shell model results that predicts
the observed "magic numbers" or closed shells observed
for nuclei. Excited nuclear states near closed shells can
also be explained in terms of these orbitals. When one
observes nuclei with many valence particles, far from closed
shells things change rapidly and the spherical shell model
doesn't work as well. Measurements of quadrupole moments
of these nuclei suggest that they are non—spherical.
Nuclei in the rare-earth region Er, Yb, even through W show

large deformations usually a prolate shape with 8 z 0.3

(B = major axis - minor axis )

average of’tfie axes of the ellipsoid* The energy

75

76

levels of these nuclei can be separated into groups that
correspond to rotational bands similar to those seen in
molecular spectra. The energy levels obey the relation

E = gg-I(I+l) where I is the sum of the particle angular
momentum j and the rotational angular momentum R (see
Figure 6-1). The unpaired nucleons traveling around the
deformed core have intrinsic spin S = i%, orbital angular
momentum I and total angular momentum 3. Consider a nu-
cleus rotating about the 2 axis with angular velocity 0.
The projections of the angular momentum vectors on the sym-
metry axis (3) for I, S, 3, and I are A, 2, 0, and K,
respectively. For an axially symmetric core R is parallel
to Z so 0 = K. These considerations are the basis of a

model explaining deformed nuclei behavior, the Nilsson

model.

6.2. The Hamiltonian

The Hamiltonian for deformed nucleus can be broken up
into a part describing the energy of the rotating nucleons
as a whole (collective motion) and the energy of the
particle configurations in the deformed potential on
which the rotational bands are based (intrinsic motion).

For a single valence particle it can be written:

77

 

 

 

 

 

Figure 6—1. Vector diagram of a particle coupled to a
deformed rotating core.

78

2
Hrot ._. 23.7 (R) = %{(Il-Jl)2 + (I2’32)2}

2
_ h 2_ 32 2
where
h2
Ho = - 27 (I+J_ + I_J+)
. + + +
HC corresponds to the classical Coriolis force F = -wxp,

exerted on a particle momentum p in a rotating system

with angular velocity 0. This force tends to orient the
angular momentum vector along 0. The wavefunction for a
particle moving in a deformed system is characterized by
the quantum number 0 and a linear combination of spheri-

cal states ij [M076].

79

Each rotational band can be characterized by its projec-
tion quantum number K which is a function of the intrinsic
configuration of the nucleus. The Coriolis force mixes
bands with AK = :1. For a single particle the non-diagonal

matrix element is:

 

<K,I|H |Ki1,I> = -A<K|jI-|Kil> \/(I¥K)(IiK+1) (38)

rot

where

 

<K|j$|xil> = ZCjKCjK‘rl \l(jT-K) (Jix+l)
j _
The diagonal rotational energy is:

E = A{I(I+l)-K2 + o

1 a(-)I+1/2(I+§)} (39)

K,1/2

where a is called the decoupling parameter:

2 (no)

a = - (Q = %|J+|Q = - '12:) = §(—)j-l/2(J+%)ICJl/2I

80

The Coriolis mixing is usually small because the 0:“2
coefficients are small. The exceptions are when high j
unique parity spherical orbitals are involved (hll/2’
113/2...). In these Cja = 1 because there are no other
states to be mixed, and high j and low 0 leads to in-

creased Coriolis mixing.

6.3. Nilsson Model

 

The Nilsson model is a description of the splitting
of the spherical shell model states as a function of
nuclear deformation [Ni55]. The Hamiltonian for the

single particle problem is

++ 2
HO + 2Kfiw(£-S - Hi ). (U1)

H0 is the deformed harmonic oscillator potential part of
which depends on the deformation. The parameters K and

u are adjusted to reproduce observed level schemes of
rare earth nuclei. The wavefunctions are linear combina-
tions of cylindrical harmonic oscillator functions. Due
to the lack of spherical symmetry j is not a good quantum
number, but jz = O is. The eigenfunctions for each 9 are
composed of a sum of terms characterized by N, 2, A, and

2, where N is the number of oscillator quanta involved (the

81

other quantities are shown in Figure 6-1). On a Nilsson
orbital diagram the energy of the eigenstates are plotted
as a function of deformation, see [Le68]. Each Nilsson
orbital can hold 2 nucleons and is labeled with the asymp-
totic quantum numbers 0“[NnZA] where nZ is the number of
nodes along the symmetry axis. The energy levels can be
calculated as a function of both quadrupole (82) and hexa-
decapole (Eh) deformation. The quadrupole deformation was
estimated for 180W from the formula in Stelson and Gro-

dzins [St65]

E 82 = B(E2)l/2 [__u11 1 (”2)

The B(E2) was calculated from the lifetime of the 2+

+ o+ transition (see Section 8.3). R0 is the radius of
the nucleus R0 = 1.2 x 10'13 Al/3 cm. The parameters
used and the final result of the Nilsson calculation are
given in Figure 6-2 for both protons and neutrons. The
Fermi surface was placed somewhat arbitrarily half way
between the last "filled" orbital and the nearest "un-
filled". The Fermi surface (A) is given a value of zero
on the energy scale. On the basis of these orbitals one
can predict the energy and spins of possible intrinsic

states for 180W, but first a word about pairing.

 

 

 

 

 

 

 

 

 

82

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Nilsson diagram showing quasiparticle energies.

83

6.4. Pairing Correlations

Nucleons which occupy identical but time reversed
(opposite m3).orbitals interact with each other through a
pairing force. The cumulative effect of this interaction
lowers the energy levels below the values predicted from
the pure intrinsic calculation given above. The inter-
action is similar to that in an electron gas in the theory
of superconductivity by Bardeen, Cooper, and Schrieffer
(BCS) [Ba57]. The nucleus can be viewed as a Fermi gas
also where each orbital has a certain probability of being
occupied at a given time with the average occupation level
being at the Fermi surface. The occupation probability

parameters for holes (vi) and particles (pi) is given by:

2
“K (E —A)
1 = 911 t +1 (113)
V2 91)
K

These are called quasiparticle excitations and Eqp is the

quasiparticle energy. Eqp can be found from the formula:

 

Eqp = {(13Sp - 1)2 + 112 (1111)

where A is the pairing gap parameter [Og7l]. The pairing

84

gap 2A can be obtained from the odd-even mass differences.
The value A is larger for protons than neutrons. The values
used for 180W were obtained from an empirical formula
derived by Nilsson [N169]. The quasiparticle energies and
parameter values are listed in Figure 6-2. For an even-
even nucleus the excitations observed are even quasiparticle
excitations obtained by simply adding the quasiparticle
energies for the orbitals involved. For an odd A nucleus,
the ground state is a single quasiparticle excitation with
higher lying 3-quasiparticle states, etc. A list of 2

and h-quasiparticle states possible for 180W and their
energies is given in Table 6-1. Each of these excitations
could be expected to be the basis of a rotational band

with K isomerism displayed (that is the bandhead may decay
with a long lifetime due to AK # 0 forbidden interband

transition).

6.5. Collective Phenomena

In even—even nuclei the lowest excited states will be
those in the rotational band built on the ground state.
The lowest intrinsic excitations will be two—quasiparticle
levels separated from the ground state by an energy about
2A. There are other excitations possible, however, other
than just the quasiparticle configurations and their rota-
tional bands. Collective vibrations are possible which may

lie lower in energy than the 2—quasiparticle levels. These

85
Table 6—1. Some Possible Multiquasiparticle States in 180W.

 

 

Configuration Energy 3K

 

2-Quasiparticle States

 

 

 

8’ %*[621+]n %7[5111]n 1.52 0

6+ % [5111+]n §7[512+Jn 1.82 0

_ + 9-

7 g [102+]p E-+[511+]p 1.82 1.31

7‘ g [512+]n % [62111n 1.82 -0.33

8‘ g’t10111p 271511+lp 1.81 1
+ 7+ 5+

6 2;[404+]p §;[402+]p 1.8“ 1

1+ §£52l11n % [5111]n 2.03* -0.58

5' %;[5211]n 2716211]n 2.03* 0

- - +

5 % [510+]r1 % [621+]n 2.28 —0.16

u-Quasiparticle States
.1. - -

15 8 n 7 p 3.31 0.63
- - +

11 8 n 6 p 3.36 .5
+ _ -

l6 8 n 8 p 3.36 .5

13‘ 6+n 7‘p 3.61 0.72
+ .. ..

1H 7 n 7 p 3.6“ .5
- + _

11 6 n 8 p 3.66 .5
+ + +

12 6 n 6 p 3.66 .5
+ _. _

15 6 n 8 p 3.66 0.37
- - +

13 7 n 6 p 3.66 0.28

 

*Probably not realistic because the -]2=--[521I]n orbital is at

a lower energy experimentally as evident in 179W.

86

vibrational states may also have rotational bands built on
them. Some of the lowest energy modes seen in even-even

rare-earth deformed region nuclei are the following:

1. 8 vibration - the elongated ends vibrate in and

 

out keeping a circular cross section. It has

zero angular momentum along the symmetry axis.

2. y vibration - Similar to the B vibration but the

 

cross section becomes eliptical. It carries 2

units of angular momentum along the symmetry axis.

3. Octupole vibration - This excitation carries
0-3 units of angular momentum along the symmetry
axis, has odd parity, and oscillates about a 3

lobe shaped cross section.

These vibrations may contain one or more units of
vibrational quanta (phonons). The lowest energy modes
will be single phonon states. Higher energy bands based
on multiple phonon excitations are also possible.

In summary, the energy levels in a deformed nucleus
are made up of rotational bands of I(I+l) spacing that
are built on intrinsic quasiparticle states (derived from
Nilsson type orbitals) or collective vibrations. The
Coriolis force may introduce mixing between bands or cause
anomolies at high spins which are reflected in the level

scheme.

VII. 180W EXPERIMENTAL DETERMINATIONS

7.1. Singles Spectra

The 180W nuclei were produced by bombarding a thin
W1 mg/cm2 foil of 180Hf with 18-50 MeV 0 particles result-
ing in the reaction 180Hf(a,un)180W. The target was a
self supporting metal foil made by sputtering techniques
at Niels Bohr Institute in Copenhagen by G. Sletten
[$172]. In later experiments (all but the first timing
experiments) the target material remaining from the broken
target was remounted on formvar. The spectra were fairly
clean consisting of the (0,1n) reaction, some (0,5n) and
(a,3n) contamination as well as some 18’19P production due
to the oxide existing on the target. The program CS8N
[CS8N] was used to calculate excitation function for the
(a,xn) reactions. It predicted maximum production at
17-18 MeV which were the energies used in all experiments
except for the prompt coincidence experiment where 19.1
MeV a's were used to favor the high-spin states. The
singles spectra were taken in a goniometer at 125° and
counted with and without 75Se and 110mAg sources as internal
standards for energy calibration. An 8% efficient Ge(Li)

detector (Edax) with FWHM of 2.0 keV at 1332 keV was used

87

88

to take the spectra. A copper-cadmium absorber was used

to cut the X-ray intensity. The data were analyzed using
the y-ray peak fitting program SAMPO [R069] and the inter-
active version TVSAMPO as adapted by Clare Morgan. The
relative efficiency of the detector was determined by
analyzing spectra of the standard sources 166mHo and 75Se
using the standard relative efficiencies as determined in
reference [Ge77]. The density of lines is quite high;

about 150 peaks were fit, almost all at an energy under

600 keV. The spectrum is shown in Figure 7-1 and a list

of the peaks, energies, intensities and angular distribu-
tion coefficients (mentioned later) is given in Table 7-1.
The efficiency curve accuracy is poorer than indicated below
80 keV due to the absence of calibration points. The energy

calibration accuracy is around 0.1 keV.

7.2. Angular Distributions

In an in-beam nuclear reaction the product nuclei are
aligned in a plane perpendicular to the beam. The 7-
radiation is emitted in a preferred direction yielding an
angular distribution as mentioned in Section 3-1. The
experiment was performed by moving the detector about the
target in the goniometer arm and counting for an hour and
15 minutes at each angle. Seven angles were done: 90°,

105°, 115°, 125°, 135°, 145°, and 155°. The spectra were

89

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ma
JNZZ<I0

 

(ooom ocom ooor ooom
_ _

 

 

 

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has
rmv. Hm 00 l m
”I”. .L H “.1 .l n
6 8 H N
0 SS .1 ..I.-
E 689 SC.— ll
180. an ..1. as
S r
a
9 .l
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q _ _ _ a
1m 3
CS .W .ro;nnu
19% w m. i _ 1 mm
an m” w 1 .1
f. .W I I 1. 03
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C4 0 TL CA In
0 E a w M m
m ..n .1. ...mod

 

 

 

Table 7-1.

y-Rays in the

90

18

OHf(a,4ny)180W Reaction.

 

 

 

EY (keV) Nucleus IY (rel. int.) A2 A4

71.2 0.27I15)

72.8 0.71(10)

75.0 1.72I12)

76.1 a . 0.77(8)

83.1 0.26I5)

81.8 1.25(9)

87.2 0.26I5)

88.9 0.51(6)

91.9 181w 1.92I12)

93.3 lBOHf 3.12(20)

95.8 179w ' 3.11(19) -0.10(3) -0.10(5)
100.6 182w 0.81(l3) —0.21(7) 0.20(11)
103.6 a 31.0(19) 0.036(13) -0.007(18)
105.0 178w 0.83(l3)

108.1 1.56(15) 0.17(9) -0.03(l6)
109.9 181w 2.13(19)

113.1 181111 1.05(23) -0.3l4(22) 0.06(3)
118.1 l.07(ll)

119.9 l79w 12.1(7)

122.8 a 2.05(11) 0.21(6) 0.17(9)
l25.0* 15.7(9) -0.10(3) 0.03(1)
136.8 1.05(19) -0.5(1) 0.2(7)
137.7 l81w,179w 8.8(5) -0.13(3) 0.09(6)
139.6 l.07(ll)

112.0 1“N 3.79(23) —0.39(9) -0.22(12)
144.6 l79w 1.18(26)

116.9 a 1.58(l3) -0.55(7) 0.12I12)
152.3 2.11(l1)

151.3 l.01(7) 0.18I9) 0.03(15)

 

Table 7-1.

Continued.

91

 

 

 

EY (keV) Nucleus IY (rel. int.) A2 A4
156.1 0 72(6) 0.25(1) -0.05(7)
158.1 a 4.09(23)

159.6 179w 1.ll(8) 0.09(8) -0.07(17)
162.5 l79w l.33(15) —0.18(9) 0.3(2)
163.1 18111 3.03(29) —0.28(8) 0.23(17)
161.2 1.67I26)

165.2 l.71(14) —0.29(9) -0.08(12)
168.1 179111 1.68(11) —0.51(7) 0.03(11)
170.1 0.54I7)

171.3 a 1.2l(9) -0.18(12) -0.07(20)
173.3 ' 1.15(9)

175.1 0.29I7) -0.83(15) -0.32(27)
178.7* a 6.0(3) 0.17(3) 0.08(1)
181.5 l79w 0.52(7)

181.2 185 6.7(1)

185.2 181w 1.90(l4) -0.59(6) -0.12(10)
189.1 l79w 11.0(8)

191.1 a 1.50(19) -0.33(1) 0.05(9)
196.5 a 25.9(15)

197.3 19F 5.6<5)

200.1 l79w 2.l6(15) —0.22(1) -0.16(6)
202.5 1.17(10) -0.28(3) 0.09(1)
201.2 17911 1.50(11) 0.18(6) 0.08(10)
205.1 l79w 1.59(l2) —0.11(1) 0.01(6)
209.8 o.36(12)

213.0 179111 2.l8(17) -0.30(5) 0.01(8)
211.11 179w 6.6(5)

215.3 18°Hf 1.6(4)

219.5 a 13 2(8) -0.17(3) 0.10(5)

 

92

Table 7-1. Continued.

 

 

 

EY (keV) Nucleus IY (rel. int.) A2 A“
222.0* a 179w 5.8(3)
225.2 a 8.1(5) 0.21(1) -0.05(6)
229.1 182N 1.83(l3) 0.25(11) -0.12(15)
230.3 0.80(12)
233.9 a 98.(5) 0.132(18) 0.001(20)
236.9 179w 1.l9(l8)
238.3 2.46(21)
211.8 a 8.5(5) —0.63(5) -0.03(7)
213.1 l79w 0.97(l9) 0.518(8) -0.32(2)
215.0 0.75(2l) -0.51(11) 0.1(2)
250.0 ‘ 1.88(12) 0.21(5) 0.00(ll)
251.2 181N 0.91(8) o.31(18) -0.1(3)
252.9 179w 5.2l(29) -0.16(1) -0.l6(6)
251.7 1.10(12)
255.5 1.90(l4)
257.0 l.11(9) 0.32I7) -0.21(9)
264.0 a 7.9(5)
261.6 l79w 5.1(1)
268.0 0.92I8)
271.7 (181W) a 1.51(lo)
273.3 0.61(7) 0.33(11) -0.3(2)
275.8 2.02I15) 0.49(5) 0.21(8)
276.9 a 7.7(4) 0.26(2) 0.04(3)
279.6 17911 7.1(1) 0.18I3) -0.05(5)
281.1 1.3l(15)
285.3 a 2.13(19) -0.16(1) -0.l9(l)
289.7 0.11(5)
292.7 0.29(7)
291.5 0.93(8) 0.29I9) -0.26(15)

 

93

Table 7-1. Continued.

 

 

 

EY (keV) Nucleus IY (rel. int.) . A2 A4
297.9 0.26(11)
298.7 0.52(10)
301.2* 181w,179w 2.96(18)
303.9 0.80(8)
306.1 a 1.58(10) -0.47(6) 0.11(9)
308.9 179w 8.2(5)
312.7 17911 1.65(28) 0.23(6) —0.07(8)
316.6 a 11.1(6) 0.31(1) 0.02I6)
318.0 a 0.83(7) 0.16(9) -0.50(17)
321.1 0.18(7)
322.6 ' l.77(13) 0.31(10) -0.3(2)
323.9 1.44(11)
325.6 2.61(17) -0.l3(10) -0.1l(20)
328.8 0.75I7)
332.2 a 3.03(18)
339.2 0.51(ll)
310.3 a 3.02(20) 0.18(7) 0.l3(12)
311.7 0.73(7)
313.6 1.51(10)
345.1 0.38(6)
348.6 181w 2.21(16)
350.9 a 100.(6) 0.130(11) 0.000(18)
351.2 2.l8(l8)
355.0 17911 5.1(1) 0.31(8) 0.03(13)
358.1 179w 5.1(3) 0.26(5) -0.12(9)
361.0 a l.01(9) 0.38(6) 0.09(10)
362.8 a l.44(1l) 0.21(5) 0.17(8)
365.6 l81w 2.62(l7)
368.9 a 5.1(3) 0.29(2) 0.01(1)

 

Table 7-1.

Continued.

9“

 

 

 

EY (keV). Nucleus IY (rel. int.) 12 Au
373.2 0.37(9)
375.0 (179w) 1.59(27) 0.35(5) -0.11(8)
376.9 1.28(11) 0.58(18) -0.5(3)
386.1 0.50(6)
390.5 a 58.(3)
100.1 a 8.7(5) 0.30(3) -0.03(1)
101.5 179W 5.1(3) 0.27(8) -0.18(13)
112.2 1.55(12) 0.13(8) 0.06(11)
115.9 a 11.8(8) 0.17(3) 0.00(1)
121.2 0.22(8)
126.5 ' 1.37(11)
129.8 l79w 1.66(12)
139.9 1.80(13)
111.5 179w 1.21(25) 0.29(3) -0.15(5)
150.0 a 95(5) 0.116(11) -0.023(16)
153.2 a 1.9(3) 0.29(6) -0.23(10)
155.3 a 3.17(26)
159.7 l79w 2.88(18) 0.32(6) -0.12(9)
161.1 a 11.1(8) 0.16(3) -0.01(1)
161.1 l79w '3.86(23)
166.8 l79w 0.73(7) 0.1(1)
173.7 1.31(12)
176.7 a 5.3(3)
178.5 179w 3.71(21)
182.0 181Ta 3.50(22)
193.7 0.85(8) 0.39(8) 0.02(13)
196.0 1.21(9) 0.31(7) -0.13(10)
198.8 0.69(8) 0.39(15) -0.09(23)
505.9 a 17.1(9) O.l5(3) -0.0H(5)

 

Table 7-1. Continued.

95.

 

 

 

EY (keV). Nucleus IY (rel. int.) 12 Ad
518.6 0.82(11)
521.3 0.0051(6)
525.7 a 27.1(16) 0.29(3) -0.05(5)
528.0 a 7.1(1)
529.1 a 2.87(26) 0.21(8) -0.12(15)
531.1 0.80(7)
537.1 1.39(12)
511.5 l79w 1.53(13) 0.13(11) -0.01(12)
516.3 2.51(16) 0.35(9) —0.07(5)
519.1 11.1(6) 0.17(3) 0.07(5)
550.3 ' 0.56(5)
558.1 1.11(8)
562.2 0.81(7)
563.6 2.36(16) 0.26(6) 0.26(9)
565.3 179w 2.18(15)
571.3 a 15.5(9) 0.31(9) -0.0l2(55)
571.5 0.51(10)
581.0 0.70)10)
587.6 a 8.9(5) 0.32(1) 0.02(6)
589.6 a 1.8(3) o.36(1) -0 08(6)
591.2 a 9.2(5) 0.13(1) —0.03(5)
596.6 2.13(20) 0.25(15) 0.07(26)
597.8 a 3.81(27)
609.7 a 2.80(18)
631.6 3.61(22) 0.25(5) -0.02(8)
670.7 2.09(15) 0.16(1) -0.13(7)
715.0 1.90(11)
760.0 1.23(9) 0.32(7) 0.39(11)
813.1 a 6.2(3)

 

Table 7-1. Continued.

96

 

 

 

EY (keV). Nucleus IY (rel. int.) A2 Au

835.2 1.08(11)

838.5 1.08(11) 0.21(1) -0.15(8)
813.9 1.61(28)

816.9 3.31(21)

883.5 2.61(17)

902.8 a 20.3(12)

937.1 18F 6.0(3)

1011.2 7.8(5)

1098.9 0.82(13)

 

 

a. Placed in level scheme.

*Multiple peak.

-97..

analyzed in the same manner as the singles spectra with
SAMPO. The 390 keV gamma ray which depopulates a 5.2

msec isomer was used for normalization between angles,
since it would be isotropic. The peak areas were then fit
to the usual angular distribution function (Equation 16)

in this case:

I = IO [1 + A2P2(cos0) + AuPu(cos6)]. (45)

The coefficients A2 and A1 were obtained and used to indi-
cate the multipolarity of the transition and to some extent
the dipole-quadrupole mixing ratios. Due to inherent in-
accuracies and error in multiplet peak areas, accurate A2
and A“ coefficients were not obtainable in many cases.
Angular distributions from lines depopulating isomers

as short lived as 25 nsec were isotropic. The hexagonal
crystal structure of Hf causes rapid dealignment and
isotropy (Section 3-1). Useful angular distribution data
concerning the multipolarity of lines deexciting isomers
was lacking therefore, and for the most part the data served
to confirm the dipole and stretched quadrupole nature of

transitions in the rotational bands.

98

7.3. Gamma-ray Lifetimes

Gamma-ray lifetimes were obtained by performing timing
experiments in which the detection of the gamma-ray is
timed against the arrival of the beam pulse. Two para-
meter data were required (energy, time) as with the g-factor
data, Section 3.3.2. This time the time spectrum was gated
into 10 bands and the respective energy spectra accumu-
lated.

The first timing experiment was done with the beam
sweeper removing six out of seven beam pulses resulting in
338 nsec between bursts. The time bands were set giving
30.7 nsec between each collected energy spectrum. The
spectra were analyzed with SAMPO, sorted and least squares
fitted with a program called FITBAND [FITB]. The results
are given in Table 7-2 for some delayed lines found.

A series of long-lived lines is present, the half-
lifes of which are beyond the range of this experiment.
These form a rotational band as confirmed by coincidence
data. Apparently a long-lived isomer feeds into this
band. In order to determine the half-life a slow beam
pulsing experiment was done. In this experiment a high
voltage square wave is applied to the deflector plates
allowing one to obtain macrOSCOpic beam pulses with
minimum width of 10 nsec and beam off time of 50 nsec.

Using a 20 nsec TAC range and 1.28 nsec between time

99

Table 7-2. Delayed y's in 180W.

 

 

 

Energy ' t1/2 Comments
103* 19.2 ns + 5.2 mseca 8- + 5(-)
108 13 ns unknown
125 8.6(6) ns 15+
158 20.3(6) ns 16+
178 26(5) ns 5(')
196* 150 ns + 2.3 useca 19F, 11‘
219 2.05(20) nsec
222* 2.3 useca + 5 mina 11', 179w 1‘
221 21(7) ns 5(-
231 5.3 msec 8'
238 ‘26(8) ns unplaced high
spin line
211 2.7(1) nsec 11'
263 2.26(7) usec 11'
281 2.82(9) usec 11-
306 2.09(22) usec 11-
332 23(7) nsec 5“)
351 5.3 msec 8-
365 18.3(1) usec 181W 5-
390 5.3 msec 8’
116 2.25(6) nsec 11'
150 5.3 msec 8‘
155 20 nsa
161 2.l7(6) usec 11'
505 2.3 useca 11-
528 2.13(1) usec 11'
518 2.38(8) usec 11'
591 2.25(5) usec 11’

 

 

*Multiple peaks.

a. Evident but not measured in this work.

The 5.2(2) msec

100
Table 7-2. Continued.
Xiiiiri$e§§°80£lafifiijsf3§ $23.12; 12°?iia [37“ 19'2”) “sec
a79].
Summary of Isomer Half Lives Measured
Spin

8’ 5.2(2)*msec [Bu66]

5(-) 21(7) nsec, 19.2(3)* [Ma79]
11— 2.3(2) usec

15* 8.6 (6) ns

16+ 20.3(6) ns

*Taken from stated reference, others from this work.

lOl

bands a half life of 2.3(2) sec was found for the isomer.
Figure 7-2 shows the decay of these lines. The high energy
tailing is a result of excessive count rates. The count-
ing rate was kept at around 15 kHz assuming most of the rate
was in the prompt part of the spectrum, but pileup still
resulted in the delayed spectra causing the skewed peaks.
It became evident that there were short lifetime iso-
mers in the 20 nsec region from the first timing experi-
ment. To more accurately determine these lifetimes an-
other experiment was done, this time using a low energy
photon spectrometer (LEPS) for better low energy resolu-
tion (0.771 keV FWHM at 122 keV 5706). One of five beam
sweeping was used, giving 8 nsec per time band. The
results showed a delayed 125 keV and 158 keV line which
were suspected of feeding the 11' isomer as deduced from
delayed coincidence data. The results are included in

Table 7-2.

7.1. Delayed Coincidence Experiment

Due to the intensity of the 2.3 usec delayed lines it

was expecfédwthat they made up a rotational band built

on the known K1T 8' 5.2 msec t1/2 isomer [Bu66]. To
place this 2.3 nsec isomer and find higher lying states
feeding it a delayed coincidence experiment was planned
in which both y-y and y-beam timing information were

recorded along with the detector-energies El and E2.

 

 

 

 

 

105
180W 2.1% USEC/BAND BANDl
10"
U)
E 103 II I I
8 ' 28.3311 .
L) «N3 2 7'5 ’9' | I
100 13» co __
" 3 ‘34:? a;
105- L L 1
18014 2.19 USEC/BAND BANDZ
10” ,_
2
g 103 ll ' I ' I
8 I
| l I |
100: - I
105 - l L L L
18011 2.1% USEC/BAND BANDS
10“
33
2103 u .
D l | '
o
L) | |
100- I I I I
10‘. L L J 4
500 1000 1500 2000
CHANNEL

Figure 7-2. Delayed spectra showing the decay of the 2.3
usec isomer.

103

The beam was swept at 1 out of 11 leaving 536 nsec between
beam bursts. The 1-parameter data was recorded event by
event on tape to be sorted later as desired. Two 10%
efficient Ge(Li) detectors were used resolution 2.1 to
2.2 keV FWHM at 1332 keV. lO tapes of data were col-
lected at approximately 3 million events per tape.

Coincidence data were sorted using the program
KKRECOVERY [KKRE] which allows up to 120 2018 channel
spectra to be generated from corresponding gates in about
30 minutes of Xerox 27 computer time per tape.

Clean background-free coincidence data on the delayed
lines were obtained from setting a delayed gate on the
y— beam TAC showing clearly the K1T = 8" band. A delayed
gate on the y-y TAC yielded information on levels feeding
isomers including confirmation of the K1T = 8‘ band place-
ment. These data provided the bulk of information from
which the level scheme was deduced. The level scheme is

shown in Figure 7-3.

7.5. Prompt Coincidence

In looking at the high spin states the intensity of the
lines was very weak. Various tricks were used to attempt
to improve the statistics such as gating on both detectors,
shifting the spectra by computer to match gains, and adding
the resultant spectra. The delayed coincidence data

were limiting since the duty cycle of the beam was reduced

 

‘62. 9

 

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105

11 to 1, resulting in fewer prompt coincidence events per
unit time than could be obtained with a non-swept beam.
To clarify the complicated high spin states feeding the KTr
= 11' isomer a prompt coincidence experiment would be of
value. Delayed gates on the y-y TAC would provide the
necessary delayed-coincidence data to resolve the place-
ment of the nanosecond isomers above the K1T = 11'. For
these reasons a prompt coincidence experiment was done.
The beam energy used was 19.1 MeV compared to 17.0 MeV
used previously. More (a,5n) product (179W) was produced
with the higher energy beam but the higher energy also
favored the population of the high-spin states which were
the subject of investigation. The Appendix shows some

gates that were useful in determining the level scheme.

VIII. l80w EXPERIMENTAL RESULTS

8.1. The Ground Band and Backbending

The ground band has been determined to spin 18 as
shown in the level scheme, Figure 7-3. In a pure rotational
band there is a constant difference between successive
transition energies. This is certainly not the case with

this band. In order to show rotational band deviation more

clearly it is customary to plot 2—2/' where J is the moment
3 ‘h
of inertia vs. (hm)2 = (AE/2)2 where w is the rotational

180W.

frequency. Figure 7-1 shows such a plot for For

a rigid rotor one would obtain a horizontal line. A
vibrator would give a vertical line. The variable moment
of inertia model (VMI) which predicts a smooth increase
in the moment of inertia with increasing rotational fre-
quency would give a straight line with a positive slope
intermediate between the rotor and vibrator models. 180W
shows a sharp upbend around spin 11 (Figure 8-1). Certain
rare-earth nuclei such as 162EP exhibit an even more pro-
nounced backbending of this curve from which the term
"backbending" arises.

Backbending is thought to be caused by a rotation-
alignment effect where a pair of high-J, usually 113/2

neutrons, unpair and align themselves with the spin of

106

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ENERGY (luv)
2 § § § § §
200 e
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0.0: 0.02 0.03 004 0.05 0.06 0.07 0.00 0.09 0:0 0." 0:: ma
(fin-11
180
Figure 8-1. Backbending plot for W.

108

the rotating core due to the Coriolis force [St72]. This
causes an effective increase in the moment of inertia
responsible for the observed curve. In a less simple but
related model proposed many years before the actual dis-
covery of backbending it was explained as a breakdown of
the superfluidity (paired spins) of the nucleus [M060].
Some calculations have been done in the 180W region
by Bernthal 32 al. that describe the backbending behavior
quite well [Be71], [Be76]. A comparison is made between
the amount of decoupling observed in 113/2 bands in cor-
responding odd-A nuclei and the degree of backbending ob-
served in the even-A-nuclei. The amount of decoupling is
calculated by obtaining wavefunctions for the 113/2 bands
from a fit to observed energy levels and calculating the
expectation value of the rotational angular momentum

<R2>. The values for <R2> are compared to the minimum

+ +

value possible for R, R

2
dec

++
dec = I-J (See Figure 6-1). The

difference [<R2> = <R >] tends to be zero for highly
decoupled (particle angular momentum aligned with the core
angular momentum) bands and approaches zero with increas-
ing spin of the nucleus. The amount of decoupling in the
odd-A nucleus predicts quite well (with the exception of
170Yb) the amount of backbending observed in the correspond-

ing even (A+l) nucleus.

109

8.2. The K = 2' Band

The 2' level of the octupole band has been seen in the
study of the radioactive decay of 180Re [H067], [0067].
The 3' level is also seen, but the placement of these
levels was wrong until clarified by conversion electron
data [Ha68] from the decay. The 3' level is strongly
populated in the (d,d') reaction [Gu7l]. The decay of the
180Re ground state {w%+[102+] x v%'[511+]}1- to the 2'
level is an unhindered allowed beta transition consistent
with the observed log §§_value suggesting a single particle
transition of v%'[511+] + w%'[511+] and a configuration of
{fig-[511+] Q 1% [102132- for the 2' level [H667]. This
decay occurs with a 7.1(1) ns tl/2 [0067].

The K" = 2‘ band has been determined to spin 11' in
this work and has subsequently been seen to spin 11' in
the (p,2n) reaction [Ma79]. The 902 keV gate, Figure 8-2,
and gates on transitions within the band confirm its
existence quite conclusively. Only the E2 transition con-
necting the odd spin members to the odd, and even to the
even are seen clearly. The odd-even Ml/E2 transitions
are very weak; only the 122.8 keV 5' + 1‘ transition can
be seen clearly. A good angular distribution for one of
these transitions may indicate the proton-neutron character
of the band as in Section 8.3. The 122.8 keV coincidence

gate shows some uneXplainedlines suggesting it may be a

doublet but the majority of the intensity is from the

110

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111

5' + 1' transition. The angular distribution for this
peak is shown in Figure 8-3 gives A2 = 0.21(6) and A1 =
0.17(9).' Assuming an 02 z .6, 6 = 0.37 or 1.89 requiring
an A1 of 0.05 or 1.6. The first value of 6 = 0.37 is more
compatible with the observed A1' The positive sign ob-
tained for the 6 suggests a proton configuration in keeping
with the previous assignments. The g-factor by branching
ratio method was not attempted due to the questionable
value of such a result obtained from such a highly per-
turbed structure.

The "perturbed" nature of the band is indicated
by the non-rotational energy level spacing of the band.
The odd and even spin members act almost as separate bands
in that they each have different moments of inertia when
plotted on a AE/2I vs 2I2 plot (Figure 8-1). A K # %

rotational band obeys the following formula:

E = AI(I+l) + BI2(12+1). (16)

2
The A term is A = gg-and the B term is a higher order

correction that accounts for the stretching of the nucleus
as a function of rotational frequency and subsequent in-
crease in moment of inertia. On the above type of plot

A is the intercept and B is the slope. A good rotational

112

 

1.5, -
I22.8 keV

1H 1

1.3 ..

 

1.2 _ 1

‘ I

1.0 -

INTENSITY

 

A2: 0.24(6)

 

.A ‘=(D.l7
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0° 10° 20° 30° 10° 50° 80° 70° 80° 90°
ANGLE

Figure 8-3. Angular distribution of the 122.8 keV
5‘ + 1‘ transition.

113

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111

band would be a level line with some intercept A and small
slope B. The K" = 2‘ bands in the light w and Yb nuclei
exhibit this odd-even splitting behavor. This has been
attributed to the effect of mixing of the odd spin members
with an unseen KTr = 0' band (which contains only odd spin
members for symmetry reasons). The energies of these odd
spin levels are then depressed compared to the evens giving
riSe to the apparent higher moment of inertia. This
mixing is greater when the state has some 113/2 2-quasi-
particle component since the Coriolis mixing is greater
for high J orbitals. There is indirect evidence for i13/2
neutron components in the systematics of l7“’176’178’180W
negative parity yrast bands [WALK]. If the 113/2 neutron
bands of the corresponding odd-mass nuclei are plotted
similar to Figure 8-1 the magnitude of the oscillations
follows the same trend as the octupole bands in Figure 8-1
[WALK]. The magnitude of the oscillations also increases
with decreasing mass. This can be explained in terms of
the increasing Coriolis mixing as the Fermi surface
descends to lower 0 Nilsson orbitals with decreasing mass.
(Lower 0 results in increased Coriolis forces.)

The negative A2 of the 113.5 keV 1' + 3‘ and 133.8
keV 5' + 1' transitions suggest neutron character for the

KTr = 2' band in 182

178

W [Je77]. The Ml/E2 transitions are not
seen in W, however, so no such argument can be made

there [D079]. Similar arguments have been made for the

115

170

negative parity yrast bands in Yb however [W379].

+ transition has been shown to have

The 902 keV 2' + 0
mixed E1;M2-E3 character from angular correlation studies
[K071]. The presence of the E3 octupole component sug-
gests a highly collective (octupole vibration) nature. The
strong population of the 3' level in the (d,d') reaction
also suggests collectivity [Gu71].

In conclusion it is probably impossible from this

evidence to assign a particular particle configuration to

this band, as it is probably a mixture of all of these.

8.3. The K1T = 5(') Band

The coincidence data give good evidence of a band
feeding into the 1', 5', and 6' levels of the KTr = 2'
band. It is populated rather weakly but is seen clearly
in the 181Ta(p,2n)180W reaction also [Ma79]. From the way
it feeds the K1T = 2' band one would expect it to be a
K1T = 5', 6: band. Some possible 2-quasiparticle states
include (See Table 6-1) a {;[511+]nI9 %'[512+]n} 6+ state
and a {%[101+]pI33%[102+]p}6+ state. Due to the unexpectedly
low energy of the %'[521+] neutron orbital as seen in 179W
a configuration of {'3'szan Q -12-'[521+]n}5' is also pos-
sible (See Section 8.6.3). If one plots AEy_s_ I
for this band it is a straight line through the origin

for a K = 5 band (Figure 8-5) with the same slope as the

116

 

 

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H
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J . q _ _ _ _ J . _ _ _ (TI \fiw\ 0
($0010. .00.
ICON
ad
03. m0 028:. 0 m0 050» a
2.2. hi. .6 .68
0:00 Immpx 0
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30o.

 

 

 

117

K = 8 band. The E/2I X§.2I2 plot (Figure 8-6) is also
more consistent with a K = 5 assignment but no direct

proof exists since the angular distributions for the inter-
band transition are isotropic due to the 20 ns lifetime of
the bandhead. Timing measurements show a half life of
26(5) nsec from the 178 keV doublet in the LEPS timing
data, and a tl/2 - 21(7) nsec for the 221 line. These

are close to the tl/2 = 19.2(3) nsec measurement from the
(p,2n) reaction [Ma79], but the transitions are very weak
in this reaction (0,1n) and hence yield inaccurate half

life values.

8.1. The K" = 8' Band

As mentioned previously the delayed coincidence experi-
ment provided the proof of the existence of a band built
on the 8' tl/2 = 5.2 msec isomer fed by a higher lying
isomer. Some out of beam coincidence gates and angular
distributions of transitions in the band are given in the
Appendix and show conclusively levels up to 11' in the
band. Delayed gates on the ground band and 390 keV transi-
tions show the presence of the band. The intensities are
also consistent with the band based on the KTr = 8' isomer.
A plot of E/2I vs 2I2 for the band (Section 8.2) gives
a very flat line (Figure 8-6). No backbending or other

perturbation is seen. The moment of inertia is consistent

118

.m0cmn Hmsoaumpos +mH 000 .Im .AIVm 1 ex on» no poaa NHN mm Hm\m< .mIm madman

 

 

NH N
00s 000 000 00¢ 00m 00m 00. 0
q u _ q _ _ _
I0
.0
m :0 0:0. 0 0. IHIWI
0 0 _ .0 . 2,00. 00
10.
+2.90.
mTIImTIIlm. L:

 

 

 

N.

119

with the bands seen in the isotone 178Hf [Kh77]. The
I particle configuration of the band is deduced in Section
8.6.2 from the branching ratios. The t1/2 of the 8'
isomer has not been measured in this work but is given

by Burde 22 a1, as 5.2(2) msec [Bu66].
8.5. High Spin 1-Quasiparticle Isomers

8.5.1. The K1T = 11' Isomer

When it was discovered that the transitions in the
KTr = 8' rotational band were delayed (as seen in the 350
ns range timing experiment) the existence of a high lying
1-quasiparticle state was suspected. A 20 nsec range,
slow beam pulsing experiment was done (Section 7.3) and
the half life of this isomer was found to be t1/2 = 2.3(2)
usec (averaged from several transitions).

The spin of the isomer can be guessed at in a manner
similar to the K = 5 band assignment, by the way it feeds
into the K = 8 band (see level scheme Figure 7-3). Prob-
able Spins are l1i or 13:. Multipolarities cannot be
obtained from the angular distribution, but conversion
coefficients can be obtained from intensity balances in
the delayed spectra. On this basis the conversion co-
efficient for the 222 keV transition can be deduced. The
conversion coefficient for the 528 and 813 keV lines are

quite small and can't be obtained with much accuracy.

120

The 222 keV line is a doublet with the %

delayed (5.2 min) transition in 179W, but the percent con-

+ % (gnd. state)

tribution to the observed line can be calculated from the
published relative intensities in 179W [Be75] and the
observed relative intensities for the 180W reaction (Table
7-1). Using intensities obtained from the 20 nsec range
timing experiment and subtracting an estimated 19% of the
222 keV intensity due to the delayed impurity line from
179W, the conversion coefficient is QT z 0.55(1) (neglect-
ing efficiency calibration errors). The theoretical total

conversion coefficients for various multipolarities are:

.;1
M1 0.51
El 0.050
M2 2.21
E2 0.22

The data predicts an M1 which means the level would be a
spin 13' or 11'.

Using a model dependent argument, a look at possible
1—quasiparticle isomers of that energy (Table 6-1) shows

_ + 7.. 7+ +
a 11 {0% [621+] 0 65 [511110 1P2- [101110 n; [102+3}
configuration at one of the lowest excitation energies,
E = 3.1 MeV in good agreement with the experimental findings

Eisomer = 3.261 MeV.

121

From systematics, a look at the isotone l78Hf shows
the presence of a similar 11' isomer feeding a K1T = 8'
band [Kh77]. From these arguments, a K1r = 11' is assigned

to this isomer.

8.5.2. The K1T = 15+ Isomer

Delayed coincidence gates from the 1 parameter experi-
ment were set on transitions in and feeding the K1T = 8'
band in order to determine transitions feeding the K1T = 11'
isomer.

The resultant sum of these gates is shown in Figure 8—7.
One can see clearly 125 and 158 keV lines feeding the isomer
Both the 125 and 158 keV transitions are seen to be de-
layed in the timing experiments with a t1/2 of 15 to 20
nsec. Half life curves for these transitions taken from
the LEPS timing experiment are shown in Figure 8-8. The
experiment was performed with a band set in the prompt
part of the TAC spectrum starting at the centroid (t = 0).
This enables one to detect the presence of a prompt com-
ponent of a delayed line. Note that both the 125 and 158
keV transitions show no apparent prompt component, indicat-
ing that they directly depopulate an isomer. (See note in
Appendix.) In other words there are two higher lying iso-
mer§’(1iquasiparticle)thatlieabéve the K1T = 11' isomer.

Delayed gates set on the 125 and 158 keV transitions

122

.0000 HmcofipMpop Im u 0x 000 no 0000505 :0 now moumw 00mma00 00 Sam

.hIm madman

 

 

 

nmzz<ro
oooN com" 003 com o
d _ A o
.
l m 12:
00
P
0000 ..m 5 «200 023.00 .0 £3 3
. 300. 9
b _ _

 

 

 

ooN

SlNHOU

123

 

   
    
 
  

 

 

 

I04L
: 0 l25 keV
.. o
- TVZ= 8.6 (6) nsec A '58 keV
— 20.3 nsec
- 36.2(22) nsec
2
z '03:
2 _
O - A
C) P
P T I/2 = 20.3(6) nsec
'02 I I 1 I
O 20 4O 60

TIME (nsec)

Figure 8-8. Half lives of the 125 and 158 keV transitions.

124

show a rotational band built on the 158 keV transition.

The level scheme then takes the form as in Figure 7-3.
Spectroscopically the multipolarities of the 125 and 158

keV transitions can't be absolutely determined, but a
coincidence gate on a transition feeding the two isomers
should give the relative electron conversion ratio of the
two lower transitions. That is, the coincidence gate of the
3H0 keV transition should show the 125 and 158 keV transi-
tions with equal intensity except for a small correction
taking into account the differing half lives. Any intensity
difference seen would be due to internal conversion and the
efficiency of the detector. The 125 keV transition is seen
much more strongly than the 158 keV indicating a higher
conversion coefficient for the later.

From the list of possible quasiparticle isomers pre-
dicted (Table 601), a 16+ {v%[62u+] Q vyi-[Sll-H] o nguom
o flg-[slmn or a 15“ {V‘g'E52l-H‘] Q vg-[slun Q ngwozn
Q Tr-g-[5lu+]} are probable. Assuming these are the isomers,
an ordering of 15... above the'l'6+ would give multipolarities
of M1 and M2 for the 158 and 125 keV transitions respectively.
The M2 would be highly converted and the y-intensity would
be low. The data supports the opposite ordering consistent
with a low conversion for the 125 keV El and a moderate

conversion for the 158 keV Ml.

125

8.5.3. The K" = l6+ Band

The rotational band built on the 16+ isomer is plotted
in Figure 8-6 and is consistent with the moments of inertia
seen for the other bands, giving additional evidence in
favor of the adOpted scheme.

The transitions within the band are seen to have highly
anisotropic angular distributions with large positive A2
values (Table 7-1). The mixing ratio calculated from the A2
is large suggesting a high g-factor (proton configuration)
for the band. Indeed, the gk calculated (in Section 8.6.H)
accurately corresponds to the theoretical g-factor for the
proposed 2 proton-2 neutron configuration. The large mixing
ratio may also explain the unseen E2 crossover transitions.

This isomer is also seen in the isotone 178Hf where it
lies at a lower energy than the K1r = 1h“ isomer [Kh77].

The K" 178

in", and 16+ isomers in Hf are yrast traps;

that is they lie lower in energy than corresponding spin
members of the other bands. This may result in a long
lifetime for the isomer if they can decay only by high
multipolarity K forbidden transitions. The KTr = l6+

example in 178Hf is particularly striking in that it has

a half life of 31 years [He68]. In 180W, however, the Fermi
surface has risen above the two proton orbitals involved
(%7[51N+] and %f[uou+]) (See Figure 6-2), causing the energy
of the 16+ isomer to be higher in this nucleus than in

178Hf.

126
8.6. gyFactors From Branching Ratios

8.6.1. Method

G-factors for the bandheads of rotational bands can
be deduced from branching ratios within the band in the
framework of the Nilsson model. The branching ratios
here are the ratio of intensity of the E2 crossover to the
Ml+E2 cascade transition leaving the same level within the
band. Values of the mixing ratio-squared 62 and ng-gfi|
can be obtained if Q0, the quadrupole moment of the nucleus,
is known. These quantities are computed from the following

formulas [A164]:

1. = (1+1)(I-1+K)(l-l-K) [B(I I-2)]5

 

 

l +
a? 2AK2(2I-l) B(T I'15
I (I+I-2)
A = 1
I (I+I-l)
Y

0.933 QQ;E(I+I-l)(MeV)}

(g -g ) = . (n?)
K R «IE-1) a

 

Sometimes 5 can be deduced from conversion electron or

angular distribution data. The quadrupole moment Q0

127

is taken to be that of the ground state of the nucleus
and can be determined from rotational band theory from

the B(E2) of the 2+ + 0+ transition [St65] where
_ -6l -5 -1 -l
B(E2) - u.o8 x 10 [EY(MeV)] [1(3e9)] [1+al (“8)

and T is the mean life of the 2+ and a the total conversion
coefficient. (One must be careful which definition of
B(E2) one uses because they differ from source to source.)

Q0 is then
Q0 = [l6nB(E2)/5]l/2 (M9)

Using r = l.76(u) ns [2173] and the energy 103.6 from this
work, and a from [St65], the value Q0 = 6.“8(2H) was ob-
tained.

The value of gR of the ground band was determined from
the measurement of the g-factor of the 2+ state using
Mossbauer [Z173] g(2+) = .258(17) = gR. The ground band
value for gR was taken to be approximately the same for
the other bands when determining gK for these bands.

Angular distributions can also be used to obtain the

128

mixing ratio 5. They are not usually accurate enough to
yield a good value however. One must have alignment
parameters (a2) which represent the fraction of total
maximum theoretical alignment lost. These can be obtained
from known unmixed transitions (E2 for example) of similar
spin. The corrected A2 coefficient is entered in a quadratic
equation which is solved for 6 (see [M076] p.55). The
appropriate one of the two 6 values obtained can sometimes

be chosen by comparison with the observed Au values.

8.6.2. KTr = 8' Band

 

The K7T = 8' band is thought to be a two neutron state
on the basis of systematics observed through the N = 106
nuclei [Bu66]. The lower mass isotone 178Hf has two K" = 8'
bands, however, a corresponding two neutron KTr = 8' and
a two proton K1T = 8' at a lower energy [Kh77]. These two
bands are mixed about 6H%-36% [He68], [B076], as determined
by gK values from branching ratios. In 180w the two
proton K1T = 8' is not seen. It would lie at a higher
energy in 180w since the Fermi surface for the protons would
be farther away from the %[Slh+] and %[h0h+] proton or-
bitals (Section 6.3). The two neutron singlet state
{-3—[62LH] Q 75-[51N]}8_ would be expected to have gK = 0
(Section 2.5), but the gK values obtained from the data
are gK = O.UOO(l2) or 0.120(12) depending on the sign of

6, (Table 8-1). The angular distributions of the Ml/E2

129

 

 

 

 

Table 8—1. K1T = 8' Band
. l
I of Level '32 gK for +6 gK for -6
10 o.67(20) 0.367(2u) o.lu9(2u)
ll l.u5(27) o.u19(23) 0.097(23)
l2a l.03(2u) o.39u(2u) 0.122(2u)
13 l.03(2u) 0.393(23) 0.123(2u)
lu l.US(25) O.Hl8(23) 0.098(23)
averageb o.u00(l2) 0.120(12)
final gK 0.120(12)
K" = 5(‘) Band
8° 9.3(10) 0.376(27) -o.12(3)
9 11.0(18) 0.69(u) -o.l7(u)
average ' O.530(2h)' -0.150(25)
Final gK -o.150(25)

 

 

aThg 26H geV transition from the I=12' level in a doublet
an may e unre ab e.

bComputed omitting the 12' level data.

0The l2u.9 keV and 271.8 keV lines from the I=6
levels are doublets and hence omitted.

and 7

3R = 0.258(17) Q0 = 6.u8(2u)

130

transition in the KTr = 8' band have highly negative A2
values implying a negative 5, so gK = 0.120(12). This is
slightly greater than the expected gK = 0 indicating a pos-

178Hf case

sible small proton admixture similar to the
but less, around 12%, assuming gK = l for the two proton

band.

8.3.3. K" = 5(‘) Band

The highly negative A2 coefficients of the Ml/E2
transitions in the K = 5 band suggest a negative also.
The branching ratio method yields gK = -O.150(2S) (See
Table 8-1) consistent with a two neutron band. From the

Nilsson orbitals calculated (Figure 642, Table 6-1) one

wouldn't have expected any low lying K 5 isomer since
the %f[52l+] orbital is 1 MeV from the Fermi surface.

Looking at the neighboring nucleus 179W, however, the
%'[52l+] band is seen at an excitation energy of only 222
keV. The actual energy of this orbital must be much closer
to the Fermi surface than calculated. On this basis the
existence of a K = 5 isomer is not surprising and one would
expect it to be {v%+[62u+] Q v%-[52l+]}5-. This is a singlet
neutron state and would be expected to have gK = O. A

value of gK = -O.150(25) would be more representative of a
triplet neutron configuration such as {%'[510+]<3‘%[62u+]}5_-
From formula 15 gK = -O.U6 for this configuration if the

empirical "quenched" value for gs is used where gS = 0.6 gfree'

131

The experimental value is closer to gK = 0 suggesting the
singlet configuration, but there may be some admixture with
this higher energy triplet configuration.

The errors quoted on the gK values do not reflect
possible model dependent errors in determining the 6's.
That is the 6 value extracted assumes pure rotational band
theory is correct with good K quantum number. From the ob-
served energy level spacing and E/2I vs 2I2 plots (Figure
8-6) the assumptions made are probably quite good.

The 6 values calculated from the angular distributions
are not model dependent but harder to obtain to a known

accuracy. Values obtained for the K“ = 8' band are

typically 6 —O.6 or -l.h by solving the quadratic equa-
tion for 6. (Alignment parameters a2 were obtained from
the E2 transitions.) The inaccurate A“ values observed

are not sufficient to choose which 6 is appropriate, but

I

each is close to the value 6 “ -1 obtained from the branch-
ing ratio data.

The same methods applied to the K = 5 band gives
6 = -O.27 or -2.5 for the I = 8 + 7 transition corresponding
to an Au = 0.023 or 0.30. The 6 = -O.27 value is consistent
with the observed Au's and the value 6 ~ -O.3 from branch-

ing ratios.

132

8.6.u. The K" l6+ Band

No E2 crossovers were observed for this band so a
branching ratio calculation could not be done. The angular
distribution of the Ml+E2 transitions, however, are highly
anisotropic allowing a 6 value to be extracted which could
be used to obtain a gK value for this isomer. The calcula-

tion gives (estimating a2 ~ 0.9) an unambiguous:

07
II

0.38 for spin l7 yielding gK 0.51

and

0.57

0‘)
II

0.39 for spin l8 yielding gK

These gK values are close to the theoretical gK = 0.50
for the proposed configuration {v%+ [62%] Q21 [5114+] Q
n- [noun a «3 [51u+]} 16 (Table 6-1).

8.7 Summary and Conclusions

In summary three H-quasiparticle isomers have been dis-
covered in 180W with a rotational band seen on one of them
(16+) The rotational band built on the known K7r = 8'
isomer has been discovered and characterized along with the
K1T = 2' octupole band, both to spin lU'. Knowledge of the
ground band has been extended to spin 18+. A K = 5, 2-
quasiparticle side band has been identified which feeds the

K1T = 2' band. The configurations proposed for the intrinsic

133

states are summarized in Table 8-2.
180W has proven to be a fruitful nucleus for the study
of low lying multi-quasiparticle states. The presence of
the Fermi surface in an island of closely spaced high spin
Nilsson orbitals accounts for this. More information may
be obtained from further study of this nucleus. The
present data shows as yet unplaced transitions feeding the
15+ isomer and 16+ isomer in the prompt and delayed gates

on the 125 and 158 keV transitions. More work might

resolve the problem further.

l3“

 

 

 

Table 8—2. Summary of Isomers Discovered and Proposed
Configurations.
K1T Energy tl/2 Configuration
h-g.p.
+ -
lu" 32614 keV 2.3(2) usec g-[621H1n 73. [5'1an
7+ +
2- [uounp % [u02+]p
+ ..
15+ 3389 8.6(6) ns % [621M]r1 7- [5111+]n
- +
g- [512+]n g- [62u+]n
+ ..
16+ 35117 20.3(6) ns % [62u+]n £- [5111+]n
+ -
7 9
§'[HOH+]p 5 [514+]p
2-g.2.
8'b 1529 5 2(2) mseca 2f[62u+] ZT[514+]
' 22 n 2 n
+ -
5" 16140 19.2(3) nsa 3462an %[52l+3n°
-b a 2' 2+ 0
2 1006 7.H(U) ns 2 [Slu+]p 2 [MO2+]p ‘

+ octupole vibration

+ neutron character?

 

 

a. Halflives taken from References [Bu66],[Ma79], and [0067].

b. Isomers known previously.

REFERENCES

[Ab53]
[Al6u]

[Ar5A]

[Ba57]

[Be68]
[Be70]
[Be7A]

[Be75]

[Be76]

[8152]

[B076]
[Br77]

[Bu66]

[Ch68]

[D078]

[D079]

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[0271]

[P175]

[Qu691
[R069]

[R077]

[Sa63]
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[8037]
[Sh771

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[Va711

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APPENDIX

APPENDIX
A. Notes on Data Acquisition Programs

1. TOOTSIE

TOOTSIE is a program for the 2-7 computer designed
primarily for taking spectrograph data. It enables one to
display E yg. AE for example and set non-linear bands around
groups of counts in the 2—D spectrum. It then can accumu-
late l-D spectra corresponding to these bands.

This program can be used to take y-ray timing data by
setting straight line gates on the vertical (time) axis
and accumulating up to 10 time sliced 20A8 channel y-ray
spectra. To take g-factor data one can make the vertical
axis y—energy, set gates on it and collect time spectra,
but the visual resolution (on the screen) for the vertical
axis is only 256 channels.

A typical g-factor configuration would use this

initial command
DEFINE,EDE,WX,256x6A,102A,10,20A8

The BBB is the operating configuration, WX refers to

1A0

1A1

the ADC's used W(time) horizontal, X(enerEY) vertical,
256x6A in the 2-D array size (256 vertical), 102A is the
size of the 1-D spectra, 10 refers to the number of bands
to be set, and 20A8 is the conversion gain of the XADC.

A little known feature of this program makes it possible
to set accurate gates on a 2K or greater Y-spectrum. The
program retains the resolution equal to the conversion gain
specified in the original command. To set bands one must
then expand the whole screen to one channel, divide it into
say 8 equal parts and set bands surrounding these drawn
in lines. A band set surrounding the 3rd line drawn between
a full scale expansion between channel 100 and 101 will

contain channel 803 in the 2K ADC space.

2. IIEVENT

This program records up to A parameter coincidence
data on magnetic tape event by event. KKRECOVERY is the
program of choice to sort the data quickly but has the
limitation of a fixed gate on the first ADC. IIEVENT
RECOVERY does not have this limitation but is slower.
CDRECOVERY is a special version of KKRECOVERY that sorts
with the last ADC as the fixed gate (but otherwise the
same). Another special version that sorts four parameter
data exists in ABS form (the source is lost), the first
and last ADC's being fixed (the last having no background

subtract).

1A2

B. Notes on Resolving the 125 keV and 158 keV Half Lives

Figure 8-8 shows the resolving of the 125, and 158
keV transitions half lives into three components. The first
data point represents a band set starting in the middle of
the prompt peak. Its purpose was to indicate the presence
of a prompt component and is not expected to contain all
the counts associated with that time bin due to the spread
of the TAC prompt peak and for that reason is left out of
the fits. It is seen that the 158 transition contains
little or no prompt component but the 125 keV line does
have some since the first point is relatively higher than
in the 158 keV case.. This is expected due to some contribu-
tion from the prompt 125 keV line in the K = 5 band.

The data for the 125 keV transition was fit with the
least squares program KINFIT to a complex decay function
(Equation 35) incorporating the 20.3 nsec half life (de-
termined from the 158 keV data with the same program) and
the initial population ratios (determined from the peak
area ratios from the summed timing data) as fixed parameters
and solving for the short half life component plus an
added exponential containing the third long lived component.
The evidence for the existence of an unknown longer-lived
component was strengthened by including another data point
not shown on the figure (Figure 8-8) that represented a
wide band set on the late part of the time spectrum. This
last point was corrected for unequal time band width in

the program.

1A3

C. Some Interesting Coincidence Gates from the A9 MeV

Prompt Coincidence Experiment

 

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