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5108 KzlProleocaPreaICIRClDatoDmJndd

MAGNETIC DIPOLE MOMENT OF THE SHORT-LIVED
RADIOISOTOPE 55Ni MEASURED BY BETA-N MR
SPECTROSCOPY
By

Jill Susan Berryman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Chemistry

2009

ABSTRACT

MAGNETIC DIPOLE MOMENT OF THE SHORT-LIVED RADIOISOTOPE
55Ni MEASURED BY BETA-NMR SPECTROSCOPY

By

Jill Susan Berryman

The double Shell closure at N = Z = 28 in 56Ni has been investigated through
the measurement of the magnetic moment of a nucleus one neutron removed from
this core. Nuclear moments are fundamental, measurable properties that provide in-
formation on the structure of nuclei. The magnetic moments of doubly closed nuclei
:l: 1 nucleon are of particular importance, since the properties of each of these nuclei
are determined by the orbit occupied by that last nucleon. Any deviation from theory
indicates the presence of higher order effects such as configuration mixing, meson
exchange currents, isobar excitation, and/ or even a breakdown of the shell closure.

The 56N i core has been shown to be soft, attributed to the strong proton-neutron
interaction, in comparison to the 48Ca core. The small magnetic dipole moment of
57Cu, with T2 = —1/2 and residing one proton outside 56Ni, suggests the double
shell closure at proton and neutron numbers 28 is broken. However, the experimental
ground state magnetic moments of the Tz = +1 / 2 nuclides 57Ni and 55Co agree well
with shell model predictions, albeit with a “soft” 56Ni core. The ground state magnetic
moment of 55Ni, also with T2 = —1/2 but with one neutron removed from the 56Ni
core can provide critical insight on the nature of the 56N i core, and can be a basis to
understand how the structure of doubly-magic nuclei may change away from stability.

The nuclear magnetic moment of the ground state of 55Ni (1"r = 3/ 2-, T1/2 = 204
ms) has been deduced, in this work to be |p(55Ni)| = (0976 :h 0.026) ,u N using the

fl-NMR technique. A polarized beam of 55Ni was produced by fragmentation of a

58Ni primary beam at energy 160 MeV/nucleon in a Be target. The A1900 and RF
Fragment separators were used to eliminate all other beam contaminants. Results of
a shell model calculation using the GXPF 1 interaction in the full fp shell model space
was found to reproduce the experimental value and support a softness of the 56Ni
core.

Together with the known magnetic moment of the mirror partner 55C0, the
isoscalar spin expectation value (2 oz) = 0.91 :l: 0.07 shows a similar trend to that
established in the 3d shell. Effective 9 factors for the A = 28 system obtained from a
fit to isoscalar magnetic moments, isovector moments, and M1 decay matrix elements
were applied to matrix elements for A = 55 calculated with the GXPFl interaction to
obtain (2 oz) for A = 55. The results of the calculation show the best agreement with
the experimental value for both [4 and (2 oz) and imply that a universal operator
can be applied to both the sd and fp shells.

for en'c

iv

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, Paul Mantica, for all he
has done for me on this long trek known as graduate school. Paul, your wisdom
amazes me and I would not be in this position without your guidance. Thank you
for always keeping the big picture in mind and reminding me to do the same. Thank
you for making me a better scientific writer and speaker. Thanks for giving me the
opportunity to travel all over the world. Thank you for the group bonding time
over lunches, dinners, parties at your house, and picnics at your cottage (thanks to
Stacy for those as well!). I will never forget some of your favorite phrases including:
“Graduate school is a stepping stone, not a career,” and “You have to love what you
do,” (in response to my persistent question on how to be successful).

Another person that deserves recognition is physicist Kei Minamisono. Kei knows
just about everything there is to know about fl-NMR and was always willing to answer
questions, lend a hand in the lab, and discuss my work. Thank you, Kei! I would like
to thank the rest of the beta group members that overlapped with me: Andrew, Josh,
and Heather. Andrew, we did not overlap for long, but I will never forget your kind
and helpful emails before I even arrived at MSU, and your encouraging words during
my first year as a graduate student. Josh, for all four of my years here I looked up to
you for help on everything from my first committee meeting to my second year oral
exam to writing this dissertation. Thanks for doing everything first and then being
willing to tell me what to do and what not to do! Heather, thanks for being a great
friend, travel buddy, and someone that I could always bounce an idea off. You always
helped me look to the positive side of things!

I would like to thank my committee members including David Morrissey, Michael
Thoennessen, and Remco Zegers. Thank you for your guidance and for making me

a better scientist. Thanks to collaborator Warren Rogers who proposed the 55Ni

experiment the first time around and for his help during the experiment. Thank you
Alex Brown and Ian Towner for your helpful theory discussions after the measurement
was complete. Thanks to Andrew Stuchbery for insight on the polarization simulation,
especially the gamma-ray deorientation calculations. I would also like to acknowledge
Michigan State University, the Department of Chemistry, NSCL, and the National
Science Foundation Graduate Research Fellowship program for financial support.

There are many people at NSCL that made my thesis experiment a success. I
have greatly appreciated the friendly attitude of all the staff. Everyone is more than
willing to talk with you, answer questions, and help in any way they can. Al Zeller
was always willing to answer my magnet questions. Thanks to John Yurkon for his
discussions on magnetic shielding and for letting me borrow all kinds of mu metal.
Thanks to Craig Snow for helping me with the mechanical design of my equipment.
Thanks to Jim Wagner for making sure I had everything I needed in the S2 vault,
including the new platform! Brad Powell, thanks for putting in the water lines for
our dipole magnet, in both the South High Bay and then the S2 vault. A lot of credit
goes to the operations department for making sure my experiment ran smoothly.
Special thanks to the A1900 group, including Tom Ginter, Thomas Baumann, and
Marc Hausmann. I know that my experiment required A LOT of settings and I thank
you for your patience and hard work! Thanks to those special operators who gave me
great beam and ordered Big Ten Burrito during the midnight shifts: Carl Cormany
and Dave Schaub. Daniel Bazin deserves many thanks for his willingness to tune the
Radiofrequency Fragment Separator anytime day or night, and the great job that he
did! Thanks to Geoff Grinyer for helping with the experiment and for the helpful
discussions afterward.

I have grown close to a number of people here that I will dearly miss. Thanks for
the great times on DALMAC, guys and gal (Jon Babbage, Thomas Baumann, Jon
Bonofiglio, Renan Fontus, Cindy Fontus, Doug Miller, Dave Miller, Dave Sanderson,

Mathias Steiner, Chisom Wilson, John Yurkon, Andrew Ratkiewicz, and Phil VOSS).

vi

What a great ride with a great group of people! Jon Bonofiglio, thanks for getting up
at 5:30 AM for spinning class two days a week and for swimming on Fridays! Thanks
to my other spinning friends who I have grown close to during the wee hours of the
morning: Tom Mitchell (the greatest spinning instructor ever), Sarah AcMoody, and
Robin Usborne. Phil Voss, I must thank you for organizing Happy Hour every Friday,
and for being a great softball coach! Thanks to Rhiannon Meharchand, Krista Cruse,
Michelle Mosby, and Heather Crawford for the girly outings and for throwing me my
only wedding shower!

Special thanks to my family, including ALL the Pinters and Berrymans. My mom
and dad have always been the most supportive parents a girl could have. Without
you, I would not have been able to do a lot of things. Dad, thanks for making me work
hard my whole life at the greenhouse. Hard work comes naturally now because of you.
Thanks for teaching me I could do anything the boys could do (and more), including
fix heaters, put in sprinkler systems, drive tractors and dump trucks, pour cement,
and haul flats. Mom, thanks for instilling the importance of education into all your
kids. I will never forget how you made us do those math workbooks in the summer
which I hated, but now I guess I thank you for it. Thanks for never restricting and
always encouraging my reading list, no matter how strange the topic of the book. The
overwhelming support that you have both always given means more than you know.

Four years ago if someone had told me that I would get married during graduate
school, I would have thought they were nuts. Marriage was not in the plans during
graduate school or ever! Now, I cannot imagine life without him. Eric, I never could
have gotten through these four years without your constant love and support. I never
thought I would find someone that completes me the way you do. As I write this we
have been together for 3.75 years, married for eight months, and I am giddy over the
fact that we get to spend the rest of our lives together. I love you more and more

everyday.

vii

Contents

List of Tables ................................. x
List of Figures ................................ xiii
1 Introduction ................................ 1
1.1 Electromagnetic Interaction ....................... 2
1.1.1 Electric multipole expansion ................... 2

1.1.2 Magnetic multipole expansion .................. 7

1.2 Magnitude of the nuclear magnetic moment .............. 10
1.2.1 Single-particle model ....................... 14

1.2.2 Effective nucleon 9 factors: microscopic treatment ....... 16

1.2.3 Effective nucleon 9 factors: empirical fit to data ........ 17

1.3 Analysis of mirror moments ....................... 18
1.3.1 Isoscalar spin expectation value ................. 18

1.3.2 Buck-Perez mirror analysis .................... 19

1.4 Nuclear moments and nuclear structure ................. 22
1.4.1 Magnetic moments near closed shells .............. 22

1.4.2 Evidence of 56Ni as a doubly-magic nucleus .......... 23

1.4.3 Magnetic moments around 56Ni ................. 24

1.4.4 Proposed p(55Ni) measurement ................. 26

1.5 Organization of Dissertation ....................... 28

2 Technique ................................. 29
2.1 Nuclear spin polarization ......................... 30
2.2 ,3 Decay .................................. 38
2.2.1 Electron interactions ....................... 40

2.2.2 fi-decay angular distribution ................... 41

2.3 Measuring Spin polarization ....................... 42
2.4 Nuclear magnetic resonance of fi-emitting nuclei ............ 44
2.4.1 Spin-lattice relaxation ...................... 47

2.4.2 Line broadening .......................... 49

3 Experimental Setup ........................... 51
3.1 Nuclide Production ............................ 51
3.2 fl-NMR Apparatus ............................ 55
3.2.1 Overview ............................. 55

3.2.2 Radiofrequency system ...................... 59

viii

3.2.3 Electronics ............................. 64

3.2.4 Calibrations ............................ 70

4 Experimental Results .......................... 79
4.1 Fragment Production ........................... 79
4.2 Particle Identification ........................... 79
4.3 6 energy spectra ............................. 85
4.4 Spin polarization measurement ..................... 91
4.5 N MR measurement ............................ 93

5 Discussion ................................. 97
5.1 Polarization of 55 Ni compared to simulation .............. 97
5.1.1 Momentum distribution reproduction .............. 97

5.1.2 Optical Potential ......................... 99

5.1.3 Results of simulation ....................... 102

5.1.4 Extension to nucleon pickup reactions .............. 103

5.2 Magnetic Moment of 55Ni and the 56Ni closed shell .......... 107
5.2.1 Single-particle waveftmction and effective 9 factors ....... 108

5.2.2 Shell model in full fp shell and ghee ............... 108

5.2.3 Shell model in full fp shell and gefl‘ ............... 110

5.2.4 Isoscalar spin expectation value at T = 1 / 2, A = 55 ...... 111

5.2.5 Buck-Perez analysis ........................ 114

6 Conclusions and Outlook ........................ 115
Appendices .................................. 118
A fi-decay Asymmetry Parameter Calculation ............. 118
Bibliography ................................. 121

List of Tables

1.1
3.1
3.2
4.1
5.1
5.2
5.3
A.1

Theoretical expectations for the magnetic moments of 55Ni. ..... 28
A1900 Bp values for 55Ni fragments. .................. 54
Plastic scintillator energy calibration data ............... 74
Fraction of components of the secondary beam ............. 83
Input parameters for Monte Carlo simulation. ............. 103
Towner corrections to the calculated effective magnetic moment operator109

Magnetic moments of 55Ni,55Co and (2 oz) for A = 55 ........ 110
Experimentally determined values of (a) ................. 119

List of Figures

1.1 Definition of variables used in electric multipole expansion. ...... 3
1.2 Definition of variables used in magnetic multipole expansion ...... 7
1.3 Classical magnetic moment representation ............... 10
1.4 Schmidt diagram for odd proton nuclei .................. 15
1.5 Schmidt diagram for odd neutron nuclei. ................ 16
1.6 Spin expectation values for T = 1/2 mirror nuclei ............ 20
1.7 Buck-Perez plot of nuclear 9 factors of mirror pairs. .......... 21
1.8 Experimental p, for nuclei near closed shells. .............. 23
1.9 Experimental values of E(2i") and B(E2;0'1!' —+ 2?) for the Ni isotopes. 25
1.10 Experimental p of nuclei one nucleon away from 56Ni .......... 26
1.11 p of the odd-mass Cu isotopes compared to theory. .......... 27
2.1 Population distribution of a spin polarized system ............ 31
2.2 Illustration of spin polarization produced in a nucleon removal reaction 33
2.3 Schematic of near- and far-side reactions ................. 34
2.4 Illustration of spin polarization produced in a nucleon pickup reaction 36
2.5 Spin polarization on the chart of the nuclides. ............. 39
2.6 Transmission curve for monoenergetic electrons ............ 40
2.7 Zeeman levels of the 55 Ni nucleus .................... 45
2.8 Schematic description of the fi-N MR technique for an I = 3/ 2 nucleus. 46
3.1 Schematic of the coupled cyclotron facility ................ 52
3.2 Schematic of the primary beam at 2° ................... 53

3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
4.1
4.2
4.3
4.4
4.5

Mechanical drawing of the Radio-frequency Fragment Separator. . . . 54

Photo of the ,B-NMR apparatus ...................... 55
Schematic drawing of the fl-NMR apparatus ............... 56
,B-decay scheme for 55Ni .......................... 57
Schematic drawing of detector system. ................. 57
Photo of the rf coil. ........................... 59
Schematic drawing of the LCR resonance system. ........... 60
Inductance of the rf coil ......................... 62
DC character of the 1f coil. ....................... 63
Resonance Q-curve at frequency 1100 kHz ............... 64
Plastic scintillator electronics diagram. ................. 65
Silicon detectors electronics diagram. .................. 66
Master gate (MG) electronics diagram. ................. 67
Electronics diagram for the radiofrequency system. .......... 68
Dipole magnet pulsing sequence ...................... 69
rf pulsing sequence ............................. 70
Dipole magnet calibration ......................... 71
Dependence of the 6 energy spectra on the strength of H0 ...... 73
Energy spectrum from 137Cs taken with B1 .............. 75
Energy calibration of plastic scintillator detectors Bl—B4. ....... 75
a-decay spectrum of 228Th for silicon detector 1. ........... 76
a-decay spectrum of 228Th for Silicon detector 2. ........... 77
rf calibration with 60Co source. ..................... 78
PID with no wedge at A1900 intermediate image. ........... 81
PID with 405 mg/cm2 Al wedge at A1900 intermediate image ..... 82

Yield distribution of 55Ni as measured at the focal plane of the A1900. 82
Vertical beam position as a function of tof after the RF FS ....... 84

Decay scheme of 55N i and the primary contaminant 54Co ....... 85

xii

4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

Energy loss in silicon detector 1 upstream of the fl-NMR apparatus . 86

55Ni 6 energy spectra with rf on/off. .................. 87
55Ni ,6 energy spectra with rf on/off and higher thresholds ....... 88
55Ni 5 energy spectra with H0 on/off .................. 89
2d energy spectra .............................. 90
2d energy spectra with higher thresholds ................. 91
Two-dimensional background spectra ................... 92
Spin polarization of 55Ni at central fragment momentum ........ 93
Spin polarization of 55N i weighted average ................ 93
NMR resonance spectrum . ._ ...................... 94
NMR resonance spectrum with broad scan ................ 95
Simulated momentum distribution compared to data. ......... 99
Variable definitions for mean deflection angle calculation ........ 100
Spin polarization for 9Be(58Ni,55Ni) with simulation. ......... 102
Parallel momentum / nucleon distribution for nucleon pickup. ..... 105
Polarization plot for 9Be(36Ar,37K) ................... 106
Polarization plot for 9Be(368,34AI). ................... 107
Running sum of Towner corrections ................... 109
Isoscalar spin expectation value with A = 55 result. .......... 113
Buck-Perez plot of nuclear 9 factors of mirror pairs. .......... 114

xiii

Chapter 1

Introduction

The nuclear magnetic dipole moment ([1) is a fundamental property of the nucleus
that can provide detailed information on nuclear structure. Every nucleus with an odd
number of protons and / or neutrons, by virtue of Spin, has a magnetic dipole moment.
The magnetic dipole moment arises from the electromagnetic interaction, which is well
understood. The magnetic dipole operator [I is a one-body operator and the magnetic
dipole moment is the expectation value of #2. Experimental magnetic moments can
be directly compared to predictions of nuclear models, and provide a stringent test of
these models. Deviation of experimental values from model predictions might indicate
the presence of configuration mixing among other orbits, or the need for different or
better parametrized residual interactions. The sensitivity of p to the orbital and spin
components of the nuclear wavefunction yields key information on shell evolution and
shell closures (magicity).

In the extreme single-particle model, the properties of a nucleus with one proton
(or neutron) outside a closed shell are determined solely by the properties of the orbit
occupied by the last odd nucleon. Thus, the magnetic dipole moments of nuclei near
closed shells are particularly important. The simple structure can give critical insight
into the shell structure, and provide a better understanding of how shell closures may

change for nuclei away from the valley of stability.

The magnetic dipole moment, in addition to the other multipole moments, can be
calculated with an expansion of the electromagnetic Hamiltonian. The electromag-
netic interaction is well understood and an analytical form of the Hamiltonian exists.
The more poorly understood strong nuclear interaction can be probed via the elec-
tromagnetic interaction by studying the multipole moments. Such analysis has the
advantage that electromagnetic fields can be thought of as arising from the motion of
the nucleons under the influence of the strong force, and this measurement does not

distort the object of interest.

1.1 Electromagnetic Interaction

The external effects of any distribution of charges and currents (e.g., a nucleus) vary
with distance in a characteristic fashion. An electromagnetic multipole moment as-
sociated with each characteristic spatial dependence is assigned to the charge and
current distribution. For example, the 1/1‘2 electric field arises from the net charge,
which is assigned as the zeroth or monopole moment. The 1/1'3 electric field arises
from the first or dipole moment, the 1/1'4 electric field arises from the second or
quadrupole moment, and so forth. The magnetic multipole moments behave simi-
larly. The electric and magnetic multipole moments can be calculated in the nuclear
regime by treating the multipole moments in operator form and calculating their ex—
pectation values for various nuclear states. These predictions can then be directly

compared with the experimental values measured in the laboratory.

1.1.1 Electric multipole expansion

A systematic expansion for the electric potential of an arbitrary localized charge
distribution, p, has been developed by Griffiths [1]. Figure 1.1 defines the appropriate
geometric variables. The potential at point P, some distance 7“ from the origin, from

some object with a charge density distribution p(7"") that is distance F’ from the

 

Figure 1.1: Definition of geometric variables used in electric multipole expansion.

origin, is given by the expression

 

v<a= / 1..._,.,p< r'w. (1.1)

47r€o

Using the law of cosmos,

2
(F—F’)2 =r2 + (1")2 —2rr’cos6 = 7‘2 [1+ (3) —2 (é) 0069’]: (1-2)

F— F' = n/l + e (1.3)

s (g) (T7, — 2cos0’) . (1.4)

Note that e is much less than 1 for points well outside the charge distribution, and

where

the reciprocal of Eq. 1.3 can be written as a binomial expansion:

1 1 _12_ 1 32 5
__=_1 /_. __ __ 1.
F—F’ T( + e") :(1 2e+8621663+~,> ( 5)

or, in terms of r, r’, and 0’:

 

I I I 2 I 2
1) (r— — 2cos6') +53 (1) (T— — 2c080’)
r 8 r r

+(3;)3(5cos30’—3cosd’)/2+---:l. (1-6)

After like powers of (r’/r) are collected in Eq. 1.6, the resulting coefficients can

be seen to be the Legendre polynomials, Pn(cos 0). Thus Eq. 1.6 can be written as:

 

1 = % i (;)nPn(cosl9'), (17)

-o —o’
7‘ — 7'
n=0

where 6' is the angle between F and 7"”. Substituting back into Eq. 1.1,

 

1 °° 1 n _.
V(f‘) = mgrmfl) /(T’) Pn(cosl9’)p(r')d'r’, (1.8)

or, more explicitly,

 

_ 1 __1_ ~I I i/ I I —.I I
V(F) — 47760 [T/ph‘ )dT + 1‘2 1‘ cosOp(r )dT

+53 [(702 (3- cos2 9' - %) P079617, + ' ' ”] (1'9)

Eq. 1.9 gives the multipole expansion of V in powers of 1/7'. The first term is the
monopole contribution that goes like 1/7'; the second is the dipole that goes like 1/7‘2;

the third is the quadrupole; the fourth is the octopole; and so on.

From Eq. 1.9, the monopole term can be written as

Vmanh"): 1 Q (1-10)

47T60 ‘72-,

 

where Q = f pdr is the total charge of the configuration.

The dipole term from the expansion is

1 1 ..
Vdip(r")=;17r€—O;-2- r'cosd’p(r’)d'r’. (1.11)

Since 0’ is the angle between F’ and I", and
r’cosQ’ =E-F’, (1.12)
so that the dipole potential can be re—written as

_ 1 1 :0 -0/ -0/ I
Vd,p(F) — 47760 r21" fr p(r )dT. (1.13)

The integral in Eq. 1.13 does not depend on F and is the dipole moment of the

distribution,

,7; / F'p(F’)dT’. (1.14)
The dipole contribution to the potential then simplifies to

—o
I

1 P
. = —. 1-
dep<fl 47110 1'2 ( 15)

‘31)

 

Similarly, the quadrupole term in the multipole expansion can be written as

113

47re —3
0 2’" 231:1

unad<fl = fiijij: (1'16)

where
Qij E [[3r'ir’j — (r’)25,-j]p(17’)d'r’. (1.17)

6,5 is the Kronecker delta, and Qij is the intrinsic quadrupole moment of the charge
distribution.

One restriction on the multipole moments of nuclei comes from the symmetry of
the nucleus, which is in turn directly related to the parity of the nuclear state. Each
electromagnetic multipole moment has a parity, determined by the behavior of the
multipole operator when F is inverted to —7"‘. The parity of electric moments is (—1)L,
where L is the order of the moment (L = 0 for monopole, L = 1 for dipole, L = 2 for
quadrupole, etc.). If the multipole operator has odd parity, then the integrand in the
expectation value is an odd function of the coordinates and must vanish identically.
Thus, all odd-parity static multipole moments must vanish - electric dipole, electric
octupole, etc.

The nuclear monopole electric moment is the net nuclear charge Z e, where e is
the elementary charge. The nuclear electric dipole moment (EDM) is expected to
vanish, as stated above. However, since a nonvanishing electric dipole moment would
violate both parity and time-reversal symmetry, various searches for a non-zero EDM
are underway. The neutron is a good candidate for this study since it is electrically
neutral. The current best upper limit for the neutron EDM is 2.9 x 10’26 e-cm [2],
consistent with zero. On the other hand, the result does not rule out the possibility
of a small symmetry-violating contribution as expected in the Standard Model.

The nuclear electric quadrupole moment provides a measure of the “shape” of the
nucleus. The existence of a nonvanishing electric quadrupole moment implies that the
charge distribution of the state is not spherical and the nucleus is axially deformed.
Usually nuclei near closed shells are Spherical in shape and have small quadrupole
moments. In contrast, nuclei in the middle of a major shell are often deformed and

their quadrupole moments have large absolute values.

The next higher order non-vanishing electric multipole is hexadecapole (which
was not derived above). In general, if an electric hexadecapole moment is present,
an electric quadrupole moment will be present as well. Thus, it is not easy to sepa-
rate contributions from the quadrupole moment from those due to the hexadecapole
moment in the observed results. Furthermore, non-spherical nuclei tend to be domi-
nated by the lowest order deformation, that is, the quadrupole. Thus, of the electric
multipole moments, the one that is most accessible to evaluate nuclear structure is
the quadrupole moment, which provides direct information on the charge distribution

inside the nucleus.

1.1.2 Magnetic multipole expansion

A formula for the magnetic vector potential from a localized current distribution, .7,
can be obtained along the same lines as the electric potential for a localized charge

distribution. Figure 1.2 defines the appropriate geometric variables. The vector po-

 

 

2‘ .1 S
I_ C",
dr —
Figure 1.2: Definition of geometric variables used in magnetic vector multipole ex-
pansion.

tential, AU"), at F arising from a current, 2', that is distance 7“” from the origin, is given

by the expression
J (7‘ ’ )

T—TI

 

[1(5): 4—7‘: ,T'd (1.18)

The integral of the current density, J ( ’,) over the volume, 117’, for line and surface

currents, is equivalent to

 

 

" #0 2‘ I I-‘Oi 1 "I
=_ =_ 1.1
AC 47,]- -,dz 4,, /. -,dz ( 9)

The vector potential of a current loop can then be written as

 

 

u_oi d~ #_oi
A(F)— _ T_ [2’ —: rn+1 Inf ’))"Pn(cos9’ dl’, (1.20)

or, more explicitly:

" __ ”Of _1__ “I if I "I
A(f) — —41r[rfdl +7"2 rcosfl’dl
1 12§ 2 _l *r .
+73%“) (2cos 0’ 2)dl + ]. (1.21)

The first term in Eq. 1.21 is the monopole term, the second the dipole term, the third
the quadrupole term, and so on.
The magnetic monopole term is always zero, because the integral is the total

vector displacement around a closed loop:

fa?" = o. (1.22)
The dipole term is written as
7T7‘2

A5,,(P) = fl?— jf r'cosa’di’ = 11%ng W117. (1.23)
4wr2 4

The integral can be rewritten in a more illuminating way using a property of the

Vector area, a.Note that if the loop 18 flat, (1 is the ordinary area enclosed, with the

direction assigned by the usual right hand rule

if (19'- F’)df = —-7§'>< / d5. (1.24)

Then,

-' #0 I7 X 72'
Adipm = 571 (1-25)

where [Z is the magnetic dipole moment:

[1‘ s 1' f as =15. (1.26)

The parity of the magnetic multipole operator of order L is (—1)L+1. As a result,
even-order magnetic multipole moments must vanish for the same reason as odd-
order electric multipole moments. In addition, the higher-order nonvanishing magnetic
terms are small compared to the magnetic dipole moment, as was the case for terms
higher than the electric quadrupole moment. Therefore, the only magnetic multipole
moment discussed in terms of the underlying nuclear structure is the magnetic dipole
moment, and the terms magnetic dipole moment and magnetic moment are often
used interchangeably. When the classical definition of [.1 in Eq. 1.26 is taken over into
quantum mechanics, the magnetic dipole moment for nuclei can be calculated from
the nuclear wavefunctions, as will be derived in the following sections. Comparison
of calculated values with experiment gives direct and detailed tests of the predicted
nuclear structure.

The electric quadrupole moment and magnetic dipole moment are both fundamen-
tal in the understanding of nuclear structure. The magnetic moment is sensitive to the

single-particle nature of the valence nucleon, and gives direct confirmation of the nu-
clear wavefunction. The quadrupole moment is sensitive to the collective behavior of
the nucleus, and gives direct information on its Shape (deformation). Both quantities

can be directly compared with the predicted values in different nuclear models and

can help explain changes in shell structure away from stability. The study of nuclear
moments near closed shells is especially important as the nucleus can be approxi-
mated as an inert core plus an unpaired nucleon (or hole) and the moments can be
calculated rather easily. The focus of this dissertation is an examination of the nature
of a shell closure removed from the stability line. The nuclear magnetic moment was
used as the probe for such study. What follows in the next section is a review of the

magnetic moments of nuclei.

1.2 Magnitude of the nuclear magnetic moment

The magnetic moment from the motion of an arbitrary charge can be calculated using
classical kinematics. Consider an electron moving in a circular orbit with velocity 12,

about a point at a constant radius r, as shown in Fig. 1.3.

‘l

m, charge e
velocity v

 

Figure 1.3: Schematic representation of an electron with mass m moving in a circular
orbit with radius 1'.

The magnetic dipole moment was defined in Eq. 1.26 as the product of the current
i and the area formed by the electron path, 51'. The area of the circle is 71'7‘2, with the

direction of the magnetic moment pointing out of the loop. The current 2' is the

10

electron charge divided by the time to make a loop, or e / (27r1‘ /v) Thus,

(1.) = id = (26:7) (m2) = .6171. (1.27)

Recall that the angular momentum of the electron with mass, m, moving in a circle

is l = mvr, so that the magnetic moment is simply related to l by the expression

(Isaak-2am (28>

where lfl is the classical angular momentum. In quantum mechanics, the magnetic
moment corresponds to the projection of II on the rotation axis. Thus, the classical
Eq. 1.28 can be “converted” to a quantum mechanical definition by replacing f with
the expectation value relative to the axis where it has maximum projection, mlfi. with
m; = +1. Thus,

ch

=— .2
[.l. 2ml (19)

where now I is the angular momentum quantum number of the orbit.

The constants in Eq. 1.29, (eh/ 2m), are called the Bohr magneton, 113, and has
the value 9.274 x 10‘24 J / tesla. Nucleons behave differently than electrons in terms
of their magnetic moments, as will be discussed. None the less, a similar quantity
can be obtained for the motion of nucleons which is useful for the discussion of
nuclear magnetic moments. Substituting the proton mass for the electron mass in the
expression (eh/2m) yields the nuclear magneton p N = 5.051 x 10’27 J /T. Eq. 1.29

can then be rewritten as

It = 91sz (130)

where the extra factor 9) is the g factor associated with the orbital angular momentum
l. g) is 1 for a proton and 0 for a neutron to reflect the fact that protons are charged,
and contribute to the orbital component of the magnetic dipole moment. Neutrons

are electrically neutral and their motion does not contribute to the orbital component

11

of [1,.
An equivalent expression to Eq. 1.30 can be written to describe the intrinsic or.

spin magnetic moments of the fermions as

Ii = QSSHN (131)

where s = % for protons, neutrons, and electrons. The quantity gs is the spin 9 factor
and can be calculated by solving a relativistic quantum mechanical equation. For a
spin % point particle, the Dirac equation gives 9,, = 2. The experimentally measured
value for the electron is gs = 2.0023 and the small difference between this value
and the Dirac expectation comes from including higher order corrections of quantum
electrodynamics. On the other hand, the experimental values for free nucleons are far

from the expected value for a point particle:

93”“ = 5.5856912i0.0000022 (1.32)

936““ = —3.8260837:i:0.0000018 (1.33)

The proton value is much larger than 2 and the uncharged neutron has a nonzero mag-
netic moment. The observed deviation from expected values is attributed to the inter-
nal structure of the nucleons with internal charged particles in motion (i.e., quarks)
that result in currents giving the observed Spin magnetic moments. It is noted that
gs for the proton is greater than its expected value by about 3.6, while gs for the
neutron is less than its expected value (zero) by roughly the same amount. The dif—
ference has been ascribed to clouds of 7r mesons that surround the nucleons, with
positive and neutral mesons in the proton’s cloud, and negative and neutral mesons
in the neutron’s cloud [5].

The derived spin and orbital components of the magnetic moment of individual

nucleons (protons and neutrons) can be used to determine the magnetic moments of

12

nuclei. The nuclear magnetic dipole moment has contributions from all of the orbital
and spin angular momenta, and the magnetic dipole operator for a nucleus can be

expressed as a sum of two terms [5]

A A
,1 = Z gg’fil") + Z g‘skléu‘), (1.34)
k=1 k=l

where LU“) and S0“) are the orbital and spin angular momentum operators for the
kth nucleon, summed over all A nucleons in the nucleus. 92k) and ggk) are known as
the orbital and spin nucleon 9 factors as defined previously. The magnetic moment is
obtained by taking the expectation value of the z-component of [1' from Eq. 1.34 for
the nuclear substate in which M = J. Thus, for a nucleus described by total angular

momentum quantum number J and magnetic substate M, the wavefunction is 1/) J M,

and the magnetic moment 11 is given by

II = [21211403214114

5 MM = J|(fi)!J,M = J) (135)
A A

= (J, M = J] E 992(k) + Z g‘S")§(")|J, M = J) (1.36)
k=l k=1

where the integration is over the coordinates (position and spin) of all A nucleons.

Then, an equivalent overall expression to Eqs. 1.30 and 1.31 for a nucleus is

l1 = gIImv (137)

where g] is the nuclear 9 factor, sometimes referred to as the gyromagnetic ratio and

written as 'y, and I is the nuclear spin.

13

1.2.1 Single-particle model

The extreme single-particle model is the simplest form of the shell model. It describes
a nucleus in which a single unpaired nucleon moves in a central potential formed as
a result of the other nucleons in the nucleus. In the single-particle limit, the contri-
butions of all of the paired nucleons exactly cancel so that only the single unpaired

nucleon contributes to the overall nuclear magnetic moment and Eq. 1.36 reduces to

[13.19. = (13m = 11911-4 gsglj, m = 1') (138)

where | j, m) is the single-particle wavefunction of the unpaired nucleon. Evaluating
the expectation value of the vector sum in Eq. 1.38 and given that 3 = f+ S', where s is
1 / 2 for protons and neutrons, the single-particle expression can be simplified further

to

 

_. 93—91 ._ 1
[134,—] [glzh 2l+1[forj—li2. (1.39)

Recall that the free nucleon g-factors, gfree, are given as

1 for proton 5.587 for proton
.91, free = 93, free = (1-40)
0 for neutron —3.826 for neutron
and the further assumption is made that the structure of a bound nucleon inside a
nucleus is the same as in its free state (91 = gl,free;gs = gs,free)- The magnetic dipole
moment of an odd-mass nucleus is thus completely determined by the l and j values
of the unpaired nucleon in the extreme single particle model.

The magnetic moment for a single nucleon in orbital nlj that is calculated using
the free nucleon 9 factors is known as the single particle, or Schmidt, value. One way
to illustrate the connection between the single particle expectation and experiment
is to plot experimentally-measured [1 against j, along with the calculated Schmidt

values. Such diagrams are shown in Figs. 1.4 and 1.5 for odd p and odd n nuclei,

14

respectively. The Schmidt values are successful in predicting the general trend of

 

   
 

Nb”.
6 "' Tc”
[“1150
5 _ 11,113
4 — 55 hm 3909.

    

n O

27
-7 Eu”)! ./All31
.th” Rel‘L Sbm

 

 

 

j=1/2 3/2 5/2 7/2 9/2

Figure 1.4: Experimental magnetic moments plotted with Schmidt limits for odd
proton nuclei. Figure taken from [3].

the magnetic moments of odd-mass nuclei, but the experimental values are generally
smaller than the Schmidt values. One limitation of the single particle theory is the
assumption gs = gs’fi-ee. The presence of other nucleons, however, introduces meson
exchange currents (MEC) that produce an electromagnetic field when the two nu-
cleons interact. In addition, the deviation from the Schmidt values grows as more
nucleons are added Within a shell, a trend which can be understood by introducing
Configuration mixing (core polarization) among the single particle states. The basic
Shell model assumes that the odd nucleon is in a Single-particle state, while even small

Configuration admixtures can appreciably change the magnetic moment. The single

15

particle model should be taken as a starting point, with corrections added to account

for its limitations.

1.2.2 Effective nucleon 9 factors: microscopic treatment

The limitations of the single—particle model may be compensated for by introducing

corrections using perturbation theory. When such corrections are applied, the 9 factors

are no longer given by the free nucleon values, but renormalized due to the presence

of other nucleons in the nucleus. The renormalized 9 factors are called “effective” 9

factors and have the form glyeg = g, + 69) and gs’eff = 9s + 695, i.e., the free nucleon

value and a correction to it.

Such treatment has been done in Refs. [5,6] starting from an expanded description

of the magnetic moment operator:

 

 

 
  

 

 

 

 

fieff = gl,efl'<2l> + gs,eff<23) + gp,efle21 (23)]: (1-41)
2 - .
0.1» 1= I - 1’2
SO77
Ba!” 135
1 3:31.737 B“ .an
A —Hg‘” \333
133 171
ff 0 _\_V ;, 2:111 1.1151(ng c1!05 Sm!”
Si 111
1Cdm1-e123 :: H 8“” Cd\/ /Yb”3 /Nd“5
/ /Mg25 \ 141 83
A1? ‘25 Ti”, —" Nd!“ Sm Kt
-1 " \Snus OBe’ 97 \Mo95 / Ca” :
Sn!" Sn!” M0 Ti 9 f"
n! O" J = I + 1/2
'2 " 0He3 I f I I
[:1 /2 3/2 5/2 7/2 9/2

Figure 1.5: Experimental magnetic moments plotted with Schmidt limits for odd

neutron nuclei. Figure taken from [3].

16

where gx,eff = g; + 6935, with :1: = 1,3, or p, and gp denotes a tensor term. The
tensor term contains a spherical harmonic of rank 2 (Y2) coupled to a spin operator
to form a spherical tensor of multipolarity 1. The corrections, 593;, are computed in
perturbation theory for the closed-shell :i:1 configuration.

There have been a number of sources for corrections. Core polarization (CP) is a
correction to the single-particle wavefunction that occurs when there is an excitation
in the closed-shell core made by a particle in orbital (l — .9) coupling to a hole in
orbital (l + s). MEC corrections applied to the magnetic moment operator arise from
nucleons interacting via the exchange of charged mesons. The isobar correction arises
from population of the A-isobar resonance, which de-excites by the electromagnetic
field (the isobar current). Relativistic corrections to the one-body moment operator
to order (p/M )3, where p is a typical nucleon momentum and M is its mass, also can
contribute to the magnetic moment. The perturbation treatment has proven effective
in reproducing experimental magnetic moments over a large mass region, but at the
same time, it is a simple model. There are more complex shell model calculations
that use sophisticated wavefunctions that take into account configuration mixing to
obtain effective 9 factors. One such shell model calculation is described in the next

section.

1.2.3 Effective nucleon 9 factors: empirical fit to data

One way to obtain effective 9 factors is to use shell model wavefunctions with empir-
ical fits to experimental magnetic moment data. The shell model calculation starts
with a simple form of the magnetic moment operator )1 = 93(5) +9) (I) where g) and 93
are the free nucleon 9 factors defined previously. An effective interaction is generally
used to determine the two-body matrix elements for the wavefunction. The wave-
function, therefore, is more sophisticated than the pure single-particle wavefunctions
considered in the earlier section. For example, the GXPF 1 interaction has proven to

be a successful interaction for use in the fp shell [4]. The magnetic moment is first cal-

17

culated with gfree. Higher-order corrections such as configuration mixing over many
major oscillations, MEC, isobar excitations, and / or other effects should be considered
to improve the agreement between the experimental 11 and that obtained in the shell
model calculations with gfi-ee. These higher-order corrections are all represented in
effective nucleon g-factors, yea, that are derived empirically by a least-square fit of
the magnetic moment operator to experimental [1 in a limited region of nuclei. In
general, the values gefl' = 0.69free have been found to be in reasonable agreement with

data [8].

1.3 Analysis of mirror moments

It has been shown in the previous section that the magnetic moment of a nucleus can
be calculated and compared to experimental values to learn about nuclear structure.
Going one step further, the simultaneous consideration of the magnetic dipole mo—
ments of mirror nuclei can provide a framework to test present day nuclear structure
models. A pair of mirror nuclei have the numbers of protons and neutrons inter-
changed. For example, the mirror partner of 33Ni27 is 33C028. Analyses have been
carried out on the magnetic moments of mirror pairs and certain regularities have
been observed. Two approaches described below amplify the different aspects of stud-

ies with mirror nuclei.

1.3.1 Isoscalar spin expectation value

Examination of the specific contribution from nuclear spins to the magnetic moment
can provide insight into shell structure and configuration mixing. The magnetic mo-

ment can be expressed as the sum of the expectation values of isoscalar (2 no) and

18

isovector (Z 112) components, assuming isospin is a good quantum number, as

u = (:40) + (2314) (1.42)
= <21}: + (#10; I‘n)02> + <2 'er12 + (1142» - #nlazl>, (1.43)

where l and a are the orbital and spin angular-momentum operators of the nucleon,

 

 

respectively, 1' is the isospin operator, up = 2.793 p N and an = —1.913;IN are the
magnetic moments of free proton and neutron, respectively, and the sum is over all
nucleons. The isovector (Z 112) part depends on the isospin, T2, and changes its Sign
for Tz = :l:T. The isoscalar spin expectation value (2 oz) can be extracted from the

sum of mirror pair magnetic moments as

<2”) = “(T2 = 2:11-25:22-” _ 1’ (144)
where the total spin is I = (le) + (Z 0;) /2.

Sugimoto [9] and later Hanna and Hugg [10] analyzed data on magnetic moments
of mirror nuclei, and found regularities in the spin expectation values as a function of
mass. All of the ground state magnetic moments of T = 1/2 mirror nuclei have been
measured in the 3d shell and a systematic trend has been established, as shown in
Fig. 1.6. The values of (2 oz) are close to the single-particle value at the beginning
and end of a major shell, and decrease approximately linearly with mass number in
the region in between, reflecting Core polarization effects. All values for the T = 1 / 2

mirror nuclei in the sd shell lie within the single particle model limits.

1.3.2 Buck-Perez mirror analysis

Buck and Perez et al. analyzed the magnetic moments of mirror nuclei in a different
framework. They studied the relationship between gyromagnetic ratios for odd proton,

7p, and odd neutron, 7”, nuclei and the strengths of the fl~decay transitions of mirror

19

 

 

 

 

 

 

 

 

‘8 19 1d 2s 1,: 2p
1.000 « —1’2 3’2 5’2 __1/2 7/2 3/2
. O
O
O Q 1
O . o
0.500 4 , .
O
0 Experiment
/\ _ . .
b 0.000 - Single-particle value
N O
V o
O
‘— O
O
-o.5oo - 1p1/2
1d
312 —
( .5,
-1.000 . . , , 1 .
0 10 20 30 40 50 60 70
Mass Number

Figure 1.6: Isoscalar spin expectation values for T = 1 / 2 mirror nuclei. The single-
particle limits for each orbital are shown by the black horizontal lines.

nuclei [11—13]. Assuming the contributions from the even nucleon to the 2 components
of the total spin 8' and total angular momentum J of the mirror pair are small, they

derived the following relation:

(7p + A712) = a(’Yn + A’Yn) + [3, (1-45)

with a = (ggJ — 9f)/(g§“ — g?) and ,6 = g? —- org,” with the free-nucleon values of the
9 factors gf’ = 1.0, g? = 0.0, g? = 5.586, g? = —3.826. The A'Ypm are corrections
to the gyromagnetic ratios because the even terms, Seven and Javen, are small. Such
treatment is different from the analysis of the isoscalar spin expectation value as a is
a ratio of 9 factors for the proton and neutron. Therefore, any “effective” quenching
of the 9 factors is cancelled. A plot of 7,, against 7,, revealed a linear trend, as shown
in Fig. 1.7 for all known T = 1/2 mirror pairs. The single-particle values for 7,, and
'77, lie close to this line, but the interesting feature is that the points representing

measured magnetic moments deviate from those estimates simply by sliding along

20

 

 

 

 

 

5L; . Data
: —Linearfit
E
4:"
3:-
>-°' I
2.-
15-
05-
C .
4'4. I 1 .1 .Ll.441 .
4 e e 4 o 1
7n

Figure 1.7: Nuclear 9 factors of T = 1 / 2 mirror pairs plotted as the odd proton nucleus
9 factor 7,, versus the odd neutron nucleus 9 factor 71,, also known as a Buck-Perez
plot. The squares are the experimental data and the solid line is a linear fit to the
data.
the same line [11].

The total spin is related to the Gamow-Teller matrix element for the cross-over

,B-decay obtained from the ft value for T = 1 / 2 mirror pairs and thus the following

relations are also true:

 

 

(n+NW=fi+sgfim (Mm
and
G” - g”
(“In + 47..) = 91+ 3 R ’ 24- (1.47)

In Eqs. 1.46 and 1.47, R is the ratio of the axial-vector coupling constant, CA, to the

vector coupling constant, CV, and rm is a variable related to the ,B—decay ft value

The free-nucleon value for R is R = [C A / CV] = 1.26.

21

When the value of 7,3 is known for a particular nucleus, the value of '77”, can
be deduced for that nucleus using Eqs. 1.46 and 1.47. A second prediction can be
made when 7pm is known for the mirror partner using Eq. 1.45. The Buck-Perez
extrapolation is a valid prediction for nuclei with unknown magnetic moments, and an
important tool for future measurements. Specifically, in the fp shell, many magnetic
moments of T2 = +1 / 2 nuclei are known and can be used to predict ,u for the unknown

T7, = —1/2 mirror partners.

1.4 Nuclear moments and nuclear structure

Comparison of both the magnetic moment and isoscalar spin expectation value to
model predictions provides a test of the shell closure and shell evolution. The prop-
erties of nuclei near double shell closures, in particular, are of interest as these nuclei
generally have very simple structures compared to their neighbors. The magnetic mo-
ment is one such property that is sensitive to which orbits are occupied by the valence

particles, and is therefore essential in the investigation of double shell closures.

1.4.1 Magnetic moments near closed shells

The character of stable nuclei with magic numbers of both protons and neutrons,
such as 160 and 40Ca, has been well established. The radioactive doubly magic nuclei,
however, have revealed interesting surprises. An extreme example is that of 2SO, which
was expected to be bound based on its doubly-magic character (proton and neutron
numbers Z = 8 and N = 20, respectively), but has been shown to be unbound [14].
The study of 6 unstable 56N i, residing three neutrons away from the lightest stable
nickel isotope, may provide insight into changes in the structure of doubly-magic nuclei
as one moves away from stability. All eight magnetic moments of the eight neighbors
to the doubly-closed shell 16O and 40Ca (:i: 1 nucleon) nuclei are experimentally

known [15—22] and agree well with the values obtained assuming an inert core :1:

22

 

6.5
° Experiment

5-5 ‘ 6 — Schmidt

4.5 r
150 40Ca
3.5 - 55m

2.5 -

MIN)

1.5 (
0.5 4 o
-0.5 ~

-1.5 - o

t — _ _

 

 

 

 

 

-25 r A . . . .
15N 150 17F 170 39K 3908 4130 410355C0 55Ni 57Cu 57Ni

Figure 1.8: Experimental magnetic moment values compared to Schmidt limits for
nuclei around 16O, 400a, and 56Ni.

1 nucleon (single-particle value), as shown in Fig. 1.8. The agreement reflects the
inertness of the 160 and 40Ca cores. The nucleus 56N i is the first self-mirror nucleus,
with closed-shell neutron and proton numbers (N = Z = 28), that is radioactive.
The three known magnetic moments of neighbors to 56Ni [23—25] do not agree with
single-particle values. The discrepancy indicates the necessity of corrections to the
simple picture of a 56N i closed shell. The 56Ni core is better described by the lowest
order configuration of nucleons plus a sizable mixture of other configurations, in other

words, the 56Ni core is not very inert, that is “soft”.

1.4.2 Evidence of 56N i as a doubly-magic nucleus

The softness of the 56Ni core also appears in contradiction to the behavior of the
first excited 2+ state and the reduced transition matrix element, B(E2; 0;” —> 21!"),
in the even-A Ni isotopic chain. The energy of the 21+ state in 56Ni is E(2i") = 2701

keV and lies significantly higher in energy than E(2'1*') in the neighboring even-even

23

nuclei, which is indicative of a shell closure. On the other hand, B(E2;0’1f —+ 2?)
in 56N i has been deduced from a variety of experimental methods with a variety of
results. A lifetime measurement using the Doppler-shift attenuation method yielded
B(E2;0i!' —-> 2?) = 385(160)e2 fm4 [26] and Kraus et 01. performed a proton scat-
tering experiment that gave a higher value of B(E2;0'lf ——> 2]”) = 600(120) 82
fm4 [27]. Two intermediate-energy Coulomb excitation measurments by Yanagisawa
et (11., yielding B(E2; 01" —-> 2:”) = 580(70) 62 fm4 [28], and Yurkewicz et 01., where
B(E2; 0? —> 2?) = 494(119) 82 fm4 was deduced [29], support the higher value ob-
tained from proton scattering. A summary of the experimental results for E(2i!') and
B(E2; 0?" —> 21!”) values for the Ni isotopes are depicted in Fig. 1.9. While there is a
range of experimental values for B(E2;0'1!" —> 2:") at 56Ni, the overall trend is that
the B(E2; 0? -—> 2?") for 56Ni is not reduced, within error, with respect to neighbor-
ing even-even Ni isotopes, as expected for a good core. Yet, the high E(2f) value
for 56Ni is indicative of a good core. The disparate nature between the E(2if) and
B(E2; 01" —-> 21*) in 56Ni was explained by a large scale shell model calculation with
the quantum Monte Carlo diagonalization method in the full fp shell [31]. The calcu—
lation reproduced the experimentally-observed E(2'1!‘) and B (E2; Oil' —+ 2?) using the
FPD6 interaction, wherein the probability of the N = Z = 28 closed shell component
in the wavefunction of the 56N i ground state is only 49%, compared to an 86% closed

shell component in the wavefunction of the 48Ca ground state.

1.4.3 Magnetic moments around 56Ni

The four nuclei that lie one nucleon away from 56Ni are 55Ni (neutron hole in the 1 f7 )2
shell), 57Cu (proton particle in the 2123/2 shell), 57Ni (neutron particle in the 2173/2
shell), and 55Co (proton hole in the 1f7/2 shell). The measured magnetic moments
of 55Co [25] and 57Ni [24], isospin projection T2 = 1 / 2 nuclei, are well reproduced by
Shell model calculations [4], and support 56Ni as being a soft core (see Fig. 1.10). The

magnetic moment of the T2 = —1/2 nucleus 57Cu was measured to be [0(57Cu)| =

24

 

2800 2:23 I Raman et. al.

‘ Yurkewicz et. al.
D Yanagisawa et. al.

2400 O Kraus et. al.
' Schulz et. al.

2600

2200

2000

15(2‘1’ )

1800

1600

1400 A '
1200 -

 

1000 2:28

900

I-I-i

800 ' r
700
600 ‘

500 (E

h.‘
—.—

B(E2)

 

400 l!

300

 

200

 

52 54 56 50 60 62 64 66 68 70
A

Figure 1.9: Experimental E(2‘1") and B(E2; 01* ——> 2?) for the Ni isotopes. Values
taken from [26—30].

25

O 2.00(5) ”N _
Q 2.49 "N ,.--"Tz — -1/2

 

 

 

 

 

 

 

 

 

 

0 Experiment
0 Theory I 3'79 l"N,“ .. N = Z
I Schmidt value = +1
.' u ..
[El .-""'o-0.798(1)11
. 56 . , N
O-O.999 "N SSNI 2 NI28 57Nl O-0.802 ”N
Z I-1.91 0N , - I -1-91 PN
§5Co
0 4.822(3) "N
‘ O 4.746 "N
I 5.79 "N
b
N

Figure 1.10: Experimental magnetic moment values of nuclei one nucleon away from
56Ni compared to shell model calculation using GXPFl interaction [4]. Note the

discrepency in the value for 57Cu as discussed in the text.

(2.00 :i: 0.05) [I N [23]. The same shell-model calculation for 57Cu gives ”(57Cu)=
2.45 ,u N [4], suggesting a major shell breaking at 56Ni. All of the other odd-mass Cu
isotopes have magnetic moments which agree well with theoretical predictions (see

Fig. 1.11). The one [I value not yet attained is that for the one neutron hole nucleus

55Ni.

1.4.4 Proposed [1(55Ni) measurement

The measurement of 11(55Ni) can provide important information on the N = Z =
28 doubly-magic shell closure, as 55Ni is one neutron removed from the core. The
anomolous 11(57Cu) leaves the open question of whether the one neutron hole in
1f7/2 in 55Ni, also with T2 = -1 / 2, shows the same deviation from shell model as
the one proton particle does for 57Cu.

The theoretical calculations discussed in the previous sections were carried out for

11(55Ni) and the results are shown in Table 1.1. The calculation with gfree refers to the

26

 

Z=29
2.8

0-0-9

0—0—0

0 Experiment
0 A Theory

Ill1lllllll1lilriiillj

 

 

1.8 llllllllllllllllllllllllllll
58 60 62 64 66 68 70

Mass number

31

Figure 1.11: Magnetic moments of the Odd-mass Cu isotopes compared to theory.
Theory is a shell model calculation using the GXPF1 interaction and effective nucleon
g—factors. All values were taken from Ref. [32].

shell model calculation with GXPF1 and free nucleon 9 factors, while the calculation
with ggfoments is the same shell model calculation with effective 9 factors obtained
from an empirical fit to data (section 1.2.3). The calculation with ggértmbafion refers
to the microscopic treatment that added corrections to the magnetic moment opera-
tor through perturbation theory (section 1.2.2). The Buck-Perez predictions (section
1 .3.2) were determined from the known ft value for 55Ni, and in a separate prediction
from the known p(55Co).

A new value for ”(55N i) can also be combined with the known magnetic moment
of its mirror partner 55Co to deduce the isoscalar spin expectation value for the mass
= 55 system. All of the ground state magnetic moments of T = 1 / 2 mirror nuclei
have been measured in the sd shell and a systematic trend of (Z 0;) as a function

0f mass has been established. In the f p shell, however, only three mirror pairs have

been measured, masses A = 41, 43, and 57, and no systematic behavior has been

27

Table 1.1: Theoretical expectations for the magnetic moments of 55Ni.

 

 

 

 

 

 

Theory I1(55Ni) ,uN
with gfree -0.809“
with ggfomnts. -0999“
with gg’gmbatm —1.072b
Buck-Perez (dependence on ft value) —0.872 :h 0.081c
Buck-Perez (linear trend of experimental 9 factors) —0.945 i 0.0396
Single-particle value -l.913
“From Ref. [4]

bPrivate Communication with IS. Towner
cFrom Ref. [13]

established. It is essential to measure more mirror magnetic moment pairs in this
region in order to explore nuclear structure in the f p shell and beyond.

The magnetic moment of 55N i was measured to address questions regarding the
56Ni core. Comparison of both the deduced p(55Ni) and (2 oz) for the T = 1/2,
A255 system with theory provides important information on the structure of doubly-

magic nuclei as one moves further from stability.

1.5 Organization of Dissertation

An introduction to the nuclear magnetic dipole moment was presented in this chapter,
as well as a motivation for the measurement of [11(55N i). In Chapter 2, the experimen—
tal technique of nuclear magnetic resonance of fi-emitting nuclei (fl-N MR), used to

complete the 11(55Ni) measurement, is described. Chapter 3 contains details of the
experimental setup including production of the spin polarized 5E’Ni fragments and the
IB-N MR apparatus. The results of the experiment are given in Chapter 4, followed by
a discussion of these results in Chapter 5. Chapter 6 concludes with a summary of the

present experiment, and outlook on the future of magnetic moment measurements at

NSCL.

28

Chapter 2

Technique

Methods for measuring the magnetic dipole moment of the nuclear ground state de-
pend on the interaction between the magnetic moment and a magnetic field. One
of the earliest methods is the technique of nuclear magnetic resonance (N MR), pi-
oneered in 1946 by Purcell [33]. NMR measurements rely on an external magnetic
field to break the degeneracy of the magnetic substates, and spins being distributed
among those substates according to Boltzmann’s law, with the lower levels slightly
more populated than the upper levels. When an oscillating magnetic field is applied
perpendicular to the external field, the resonance absorption of electromagnetic energy
can occur if there is any population difference. Conventional NMR methods typically
require approximately 1017 nuclei for an observable resonance due to the very small
population imbalance of the magnetic substates at room temperature. This technique
is also restricted to stable or long-lived nuclear states due to the time required to
make such an N MR measurement, on the order of several minutes.
A variation of the N MR technique has been applied to fl-emitting nuclei to measure
the ground state nuclear moments for Short-lived nuclei on the order of 10‘2 to 103 s.
The so—called ,B-N MR technique requires an external magnetic field of order 10‘1 T
and the observation of the angular distribuition of 6 particles from a spin-polarized

“11016113, and will be described in more detail in the following sections.

29

Collinear laser spectroscopy has been employed to measure nuclear moments as
well. Lasers are used to scan the hyperfine structure of atomic transitions and the
relevant energy splittings are determined from the Observed resonance frequencies.
The magnetic moment is deduced from the strength of the hyperfine interaction,
which is obtained from the energy splitting. Further, lasers can be used to spin polarize
nuclei via optical pumping with circularly polarized light. The resulting spin-polarized
ensemble can then be measured with ,B-N MR to determine nuclear moments. N SCL
is developing a laser system as a promising avenue for future measurements of nuclear
spin, charge radii, and nuclear moments.

In the work described in this dissertation, the ,B-N MR method was applied to de-
duce the magnetic moment of 55Ni. 55Ni is suited to the ,B-NMR technique since 55Ni
decays via 6+ emission with a half life of 204 ms. The remainder of this chapter will
describe in detail the necessary components for the fi-NMR measurement including:
1) production of spin-polarized nuclei, 2) fi-decay angular distribution from a spin-
polarized nucleus, 3) measurement of the spin polarization, and 4) measurement of

the magnetic moment with the B-N MR technique.

2.1 Nuclear spin polarization

Nuclear spin polarization is a necessary condition for many types of physics experi-
ments, including fl-N MR spectroscopy. Spin polarization occurs when the population
for a given magnetic substate, m, is not equal to the population for the opposite
substate —m, and a linear distribution among the m states is present (see Fig. 2.1).
Spin polarization is generally discussed in terms of the statistical tensor, p, which
characterizes the orientation of a particular state [34]. The spin polarization for a

given spin value I is defined as the ratio of the statistical tensor p1(I) to its value for

30

maximum spin polarization pinafu). Thus, with

 

 

pm): {2 fig),— (21)
and
max _ ‘I
the spin polarization is
p10) _ mP<m> = g
pi’mU) _ 2,; I " < I >’ (2'3)

where P(m) is the normalized population for substate m [2m P(m) = 1]. Thus, spin
polarization is a measure of the orientation of the total angular momentum relative
to a fixed axis (z).

Several methods are commonly used to produce spin polarized nuclei for fl-NMR
studies. Low-temperature nuclear orientation uses a strong external magnetic field to
break the degeneracy of the magnetic substates. The population of the states follows
the Boltzmann distribution law, as described previously. The splitting of the state
should be of order kT for a measurable polarization effect, namely gpNHO = kT,
where H0 is the strength of the magnetic field, 10 is the Boltzmann constant, and T
is the temperature. The condition is that HO/T = 2.8 x 103 T/K for a state with

gnN = 11110. A successful measurement then requires temperatures as low as 0.002

P(m)

m

Figure 2.1: Population distribution of a spin polarized system with respect to magnetic
substate for a nucleus with I = 3/ 2.

31

K when an external field of 5 T is used. Such conditions have been achieved but are
not well suited to projectile fragments.

Spin polarized nuclei can also be produced by laser Optical pumping as mentioned
previously. Optical pumping relies on the fact that if the electronic spins can be
oriented, the hyperfine coupling will cause the nuclear spin to be oriented as well.
Circularly polarized light is used to excite atomic transitions in an atom to a single
F—spin atomic sublevel. The nuclear spin then follows the orientation of the elec-
tron spins, and a nuclear spin polarization is produced. This technique requires very
low and well defined velocities that have only recently been obtained for projectile
fragments [35].

Another method to produce spin polarization for fl-N MR is via nuclear reactions.
When reaction products are collected away from the incident beam axis, the outgoing
particle and residual nucleus from a reaction will be spin polarized. 'Ifansfer reactions
such as (d,p) are particularly effective, as well as other types of reactions. At NSCL,
rare isotopes are produced by intermediate-energy heavy-ion reactions, in which spin
polarization at small angles has been observed. Spin polarization of projectile—like
residues from intermediate-energy heavy—ion reactions was first reported at the Insti-
tute of Physical and Chemical Research (RIKEN) of Japan in the peripheral reaction
197Au(14N,12B) at a primary beam energr of 40 MeV/ nucleon [36]. The spin polar-
ization, as a function of momentum, was observed to follow an S—shaped curve, with
zero polarization at the peak of the yield distribution, and maximum polarization (as
large as 20%) at the wings of the momentum distribution. A qualitative description
of the polarization mechanism was found in a classical kinematic model that consid-
ers conservation of linear and angular momenta and assumes peripheral interactions
between the fast projectile and target. Figure 2.2 presents a schematic of the ex-
pected polarization and yield for the nucleon removal process for fragmentation of a
projectile on a heavy target.

A systematic study of spin polarization following few-nucleon removal from light

32

projectiles as a function of beam energy and target nucleus was completed by Okuno
et al. [37]. This study demonstrated that the relation between the outgoing fragment
momentum and the Sign of spin polarization depended on the mean deflection angle
édef- N ear-side reactions occur for high-Z targets, where the Coulomb deflection dom-
inates the internuclear potential between projectile and target (see Fig. 2.3). Near-side
reactions give the polarization dependence shown in Fig. 2.2. The nucleon-nucleon po-
tential governs removal reactions on low—Z targets. Far-side reactions prevail in this
case, in which the path of the fragment is toward the target, and the sign of the
observed polarization is reversed.

The spin polarization has a near-zero value at the peak of the fragment yield
curve for both near- and far—side dominated reactions, since [grief] is large. This
behavior can be qualitatively understood from the projectile rest-frame diagram in
Fig. 2.2. The removed nucleons have momentum K. The 2 component of the induced

angular momentum of the projectile-like species is z = —Xky + Ykm, where X, Y

 

 

Removal

 

Polarization
O

 

Yield

 

 

 

 

Pin K =(kx,ky,kz)

 

 

 

0
Relative Momentum

 

Figure 2.2: Illustration of nuclear spin polarization produced in a nucleon removal
reaction at intermediate energies, for a high Z target. The yield and polarization
curves are given relative to the incident projectile momentum. The removal schematic
is given in the projectile-like rest frame (see text for definition of terms).

33

1 YA

  
    
       

Far-side

 

 

vx

 

 
   

,5,
:4 -,_.

I o I}; ‘4. .
PI'OJBCtlle ,. ’..~
1 f 1':

311‘?“ 21’

   
 

Figure 2.3: Schematic of near- and far-side reactions.

34

are the localized Cartesian coordinates of the removed nucleon(s), and km, log are the
momentum components of the removed nucleons in the reaction plane. If the nucleon
removal occurs uniformly in the overlap region, X ~ R0 (the radius of the projectile),
Y ~ 0, then .3 = —Xloy . Zero polarization will therefore result when the fragment
momentum equals the projectile momentum, since Icy = 0 in the projectile rest frame
under these conditions.

If nucleon removal is not uniform in the overlap region, Y 94 0 and the term ka
can contribute to 2. Such a contribution will only be observed experimentally when
[grief] is small. The final scattering angle of the fragment is 0 L = O-def + A6, where A0
is the change in angle caused by the transverse momentum component of the removed
nucleons, A6 = tan‘1(—kx /p) Here, p is the total momentum of the projectile-like
fragment. In reactions where lé-ldefl ~ 0, it is the transverse momentum component of
the removed nucleon(s) that “kicks” the fragments to small angles, and the resulting
polarization is negative since km > 0 to give a positive A9 and Y < 0 for non-uniform
nucleon removal as illustrated in Fig. 2.2.

As stated earlier, the nuclei must be spin polarized to perform a fl-NMR mea-
surement. There is a strong dependence of polarization on the momentum (yield
distribution) of the fragment nucleus, therefore it is crucial to know the magnitude
of polarization prior to the experiment. While fragmentation reactions provide one
means of producing spin-polarized exotic nuclei, these nuclei tend to be produced at
low rates. A useful figure of merit for fl—NMR measurements is P2Y, where P repre-
sents polarization and Y is yield, since the optimization of polarization with yield is
critical. Improvements in yield will come with the development of new radioactive ion
beam (RIB) facilities, but while yields remain small, the ability to accurately predict
the expected polarization is required for experimental success. A Monte Carlo code
was developed [37] based on the ideas discussed above to simulate the spin polarization
generated in nucleon removal reactions at intermediate energies. The general behav-

ior of spin polarization as a function of projectile-like momentum was reproduced,

35

 

 

 

 

 

 

 

Polarization

 

Yield

 

 

 

 

 
 

J g l

 

0
Relative Momentum

 

 

 

Figure 2.4: Illustration of nuclear spin polarization produced in a nucleon pickup
reaction at intermediate energies, for a high Z target. The yield and polarization
curves are given relative to the incident projectile momentum. The pickup schematic
is given in the projectile-like rest frame.

although a scaling factor of 0.25 was needed to match the magnitude of polarization
observed experimentally. Previous work at N SCL improved the quantitative perfor-
mance of this Monte Carlo approach. The progress made in this area will be discussed
later in detail in section 5.1.

While nucleon removal at intermediate energies has been shown to produce spin
polarization, certain nuclei are more easily produced via other reactions, such as
nucleon pickup. Spin polarization in nucleon pickup reactions at intermediate energies
was first demonstrated at N SCL [38]. Positive spin polarization was determined for
37K nuclei collected at small angles from the reaction of 150 MeV/nucleon 36Ar
projectiles with a 9Be target. Figure 2.4 illustrates the features of Spin polarization
and yield from nucleon pickup reactions. The key to understanding the observed spin
polarization in the pickup process is the knowledge that the picked-up nucleon must
have an average momentum equal to the Fermi momentum oriented parallel to the

beam direction. Souliotis et al. [39] showed this to be the case based on the observed

36

shifts in the centroids of the longitudinal momentum distributions for one- and two-
nucleon pickup products. The average projectile-like momentum (p) was found to
satisfy the relation (p) = (pp) + (pt), where (pp) is the average momentum of the
projectile and (pt) is the average momentum of the picked-up nucleon, which is equal
to the Fermi momentum.

The momentum of the picked-up nucleon will be antiparallel to the incoming pro—
jectile momentum in the rest frame of the projectile-like species. The 2 component
of orbital angular momentum induced by the nucleon pickup process is 12 = RAp,
where Ap is the momentum difference between the projectile and the picked-up nu-
cleon, assuming a peripheral interaction where the nucleon is picked up to a localized
position on the projectile given by R in Fig. 2.4. I; and thus the spin polarization
will be zero when the momentum of the picked-up nucleon matches the momentum
of the incoming projectile (Ap = 0). This zero crossing occurs at the projectile-like
momentum p = [(Ap + 1) /Ap]pp, where Ap and pp are the mass number and momen-
tum of the projectile, respectively. A linear increase in I; is expected with a decrease
in the momentum of the outgoing pickup product. Groh et al. [38] found that proton
pickup reactions follow the trend shown in Fig. 2.4, except for the low momentum
side of the momentum distribution. At low momentum values of the pickup products,
the momentum matching conditions for pickup are no longer satisfied, and the spin
polarization is observed to rapidly approach zero.

'I‘urzO et al. showed that neutron pickup reactions at intermediate energies behave
in a similar manner [40]. Turzo et al. extended the Monte Carlo simulation of Ref. [37]
to include nucleon pickup and the momentum considerations discussed by Groh et
al. [38]. Qualitative agreement of the observed spin polarization as a function of
the projectile-like product was found, as was the case with nucleon removal reactions.
However, the scaling factor of 0.25 was again needed to reproduce the magnitude of the
observed spin polarization. The requirement of scaling factors of the same magnitude

for both nucleon removal and nucleon pickup suggest that the same quantitative

37

correction factors should apply to both and has been demonstrated in Ref. [41]. As
noted earlier, additional components have been added to the Monte Carlo simulation
to achieve better quantitative agreement with data, and will be described in section
5.1.

The kinematic model proposed by Asahi et al. has been successfully employed
to quantitatively explain spin polarization at intermediate energies for both nucleon
removal and pickup reactions [41]. Eirther, polarization produced during fragmenta-
tion and nucleon pickup reactions has proven to be an important means for extending
magnetic moment measurements further from stability. Success has been realized in

improving the reach for such measurements, as shown in Fig. 2.5.

2.2 fl Decay

Another requirement of the ,B-N MR technique is that the nucleus of interest decays
via the spontaneous emission of an electron (6") or a positron (6+), a process known
as 6 decay, and that the asymmetry parameter A3 associated with this decay not be
zero. During fl‘ decay, a neutron is transformed into a proton, while in ,8‘!‘ decay, a
proton is transformed into a neutron. The general form of 6 decay of a parent nucleus

AZ can be written as:

AZN —->A (Z + 1)1'[}_1 + e“ + De + Qfi_ fi’decay (2.4)

AZN —>A (Z —1)1"(}+1 + e+ + ue + Qg+ fl+decay (2.5)

where the Q value describes the energy released during the nuclear reaction:

0,- MHZ] — sz +01 (2.6)

M[AZ] — (M[A(Z — 1)] + 2mec2) (2.7)

Qfi+

38

.qompgaflom 5% 9:05on mo madman 25?? fits @250th mpCoEoSmmmE €0an oEmmwaE ”Wm mnswwm
N o

msgccoou HERMES .
asxofi coo—ozz 4
.925: .8232 9

ES 82605 553258 5% 53> 85358
BSEEBBE ucmEoE uuwcmmz

 

an

           
 

tofi rw> D

0555 2 as :2 %

28> no 950; z 8: to: m
6.an I

39

 

 

 

 

 

 

 

— l
“0
[E I Det.
/ .
Source __ R t
T 9

Figure 2.6: Transmission experiment for monoenergetic electrons, adapted from Ref.
[57]. I is the detected number of electrons through an absorber thickness t, whereas
Io is the number detected without the absorber. Re is the extrapolated range.

2.2.1 Electron interactions

When the electron that is emitted from fi-decaying nucleus passes through an ab-
sorbing material, such as a detector, the electron does not follow a straight path.
Large deviations in the electron path are possible because its mass is equal to that
of the orbital electrons with which it is interacting, and a large fraction of its en-
ergy can be lost in a single encounter [57]. In addition, electron-nuclear interactions
can abruptly change the electron direction. The transmission curve for monoenergic
electrons is shown in Fig. 2.6. Even small values of the absorber thickness lead to
the loss of some electrons from the detected beam because scattering of the electron
effectively removes it from the flux striking the detector. Therefore, the plot begins
to drop immediately and gradually approaches zero for large absorber thicknesses.
Those electrons that penetrate the greatest absorber thickness will be the ones whose
initial direction has changed least in their path through the absorber. Range is there—
fore not a clearly defined concept for electrons because the electron total path length
is considerably greater than the distance of penetration along the initial velocity vec-
tor. Normally, the electron range is taken from an extrapolation of the linear portion
of the transmission curve to zero and represents the absorber thickness required to
ensure that almost no electrons can penetrate the entire thickness.

The continuous distribution of energy from a ,B-emitting nucleus causes the trans-

40

mission curve to differ from that of monoenergetic electrons. The low-energy 6 par-
ticles are rapidly absorbed even in small thicknesses of the absorber, so that the
initial slope on the attenuation curve is much greater. The transmission curve for
fl-emitting nuclei is nearly exponential in shape, although the behavior is only an
empirical approximation.

The tracks of positrons in an absorber are similar to those of normal negative
electrons, and their energy loss and range are about the same for equal initial energies.
Coulomb forces are present for both positive and negative charges, and whether the
interaction involves a repulsive or attractive force between the incident particle and
orbital electron, the impulse and energy transfer for particles of equal mass are about

the same.

2.2.2 fi-decay angular distribution

6 decay is governed by the parity-violating weak force, and the direction of emitted
6 particles can be anisotropic under certain conditions. The angular distribution of

6 particles emitted from a polarized nucleus is given [42,43] as

W(6) = 1 + Achos 6, (2.8)

where 6 denotes the emission angle with respect to the axis of polarization, P, as

defined previously. The asymmetry parameter A5 for allowed 6 transitions is

 

2 / J
A = flp 4p 71—16” (2 9)
5 1+p2 '
where
1 for J —> J’ = J — 1
A = 1/(J+ 1) for J —+ J’ = J (2-10)

—J/(J+1) for J—>J’=J+1.

41

p is the mixing ratio defined by the constant p = (C A (0)) / (CV(1)) where CV and C A
are the Fermi and the Gamow-Teller coupling constants, respectively, and (1) and (a)
are the corresponding nuclear matrix elements. The upper and lower signs correspond
to 6+ decay and fl- decay, respectively.

The B—N MR measurement requires such angular anisotropy of the emitted fl parti-
cles, and the anisotropy also permits measurement of spin polarization. If the nucleus
of interest has some spin polarization, then the 6 particles will be emitted asymmetri-
cally as given by Eq. 2.8, under the condition that A 3 76 0. When the spin polarization
of the nucleus of interest is zero, the 6 particles are emitted isotropically. Thus, the
angular distribution can be used as a probe for measuring both spin polarization and

the magnetic moment, as outlined in the following two sections.

2.3 Measuring spin polarization

The magnitude of spin polarization may be deduced from the results of a successful
fl-NMR measurement, as will be described in more detail in the following section.
However, it is useful to know the spin polarization for the nucleus of interest prior to
the start of a ,B-NMR measurement. The spin polarization depends on the fragment
momentum, as described in the previous section. Therefore, it is desireable to optimize
spin polarization as a function of momentum according to the figure of merit, P2Y,
before the fl-N MR measurement. Also, a spin polarization measurement that deduces
the magnitude of spin polarization as well as direction is ideal, to compare to the
magnitude and direction of the NMR effect observed in the fi-N MR measurement.
A technique has been developed at NSCL to measure polarization using a pulsed
external magnetic field. The technique does not require advanced knowledge of the
nuclide’s magnetic moment [44]. The 6 angular distribution will be anisotropic if
the implanted nuclei have some spin polarization when the external magnetic field is

on. When the magnetic field is off, quadrupolar interactions between the implanted

42

 

nucleus and electrons in the lattice will generally dominate the local field interaction at
the location of the impurity in a face-centered cubic host material. These quadrupolar
interactions will, in effect, depolarize the nuclear spin system and lead to an isotropic
fl angular distribution. The angular distribution shows maximum deviation at angles
0° and 180° relative to the spin polarization axis, as shown in Eq. 2.8. Therefore, the

double ratio
_ [W(0°)/W(18Oo)lfield on
R _ [W(0°)/W(180°)lfieid off, (211)

will deviate from unity when the implanted nuclei are spin polarized, while for un-
polarized nuclei R will be unity. Substituting [W(0°) = W(180°)]fie]d ofi' in Eq. 2.11,

the spin polarization can be deduced from R as

1 + ABP
R ~ 1 + 2A3P. (2.13)

Thus, the spin polarization can be extracted from the experimentally measured quan-
tity R. However, R will also reflect any instrumental asymmetries, for example, the
effect of the external magnetic field on the photomultiplier tubes used to detect the
[3 particles. A normalization for the double ratio can be provided by producing the
secondary beam at 0° along the incident beam direction to correct for this asymmetry.
With the primary beam at 0°, the implanted beam has no spin polarization [45], and
6 emission will be isotropic. The system asymmetry can be removed from the data by
taking a ratio of the double ratios for the polarized (beam angle 2°) and unpolarized
(beam angle 0°) sources. The pulsed magnetic field method for measuring spin po—
larization provides a means of maximizing P2Y for magnetic moment measurements

that use the fi-N MR technique, which is described in the following section.

43

2.4 Nuclear magnetic resonance of fi-emitting nuclei

Nuclear magnetic resonance (N MR) is a branch of Spectroscopy, and therefore deals
with the energy levels of a system and transitions between these levels, either by
absorption or emission of photons. fl-N MR is a type of radiation-detecting NMR. The
sensitivity of fl-NMR is about fourteen orders of magnitude greater than conventional
NMR. While details of conventional NMR experiments will not be discussed, the

017 nuclei

detection step involves measurement of a small electrical signal. More than 1
are needed to obtain a large enough signal above noise. The ,B-N MR technique involves
the detection of 6 particles emitted from radioactive nuclei, which produce a large
electrical signal in the detectors. It has been found that only about 103 nuclei are
needed in the fl-N MR technique. The details of the fl-NMR technique are described
in the remainder of this section.

As mentioned previously, NMR spectroscopy depends on the interaction between

the magnetic dipole moment [I = gpNI and an external magnetic field ITO, which is

defined along the z-axis. The Hamiltonian describing the interaction is given by

H = -[£- H}, (2.14)

—g/1NH0m where m = 1,1 — 1, - -- — I. (2.15)

The interaction induces a splitting in energy known as Zeeman splitting between the
formerly degenerate magnetic sublevels (see Fig. 2.7).

When the frequency of the oscillating magnetic field in a resonance experiment
corresponds to the separation of neighboring levels, transitions between adjacent sub—
states (selection rule Am = :i:1) are induced by this field. Provided that the static
field H0 is sufficiently uniform and that no electric field gradients are present in the
vicinity of the nuclei being studied, the separation between all neighboring levels will
be the same, and transitions induced between adjacent levels will have a common

resonance frequency. The energy levels Em and their separation AB in such a case

44

 

 

+7/2
Figure 2.7: Zeeman levels of the 55N i nucleus in the presence of an external magnetic

field.

are given by

Em = —gpNH0mwherem=I,I—1,---—I, and (2.16)

AE :2 gpNH0=hVL (2.17)

where ”L is the Larmor precession frequency. The value of ”L for 9 ~ 1 and H0 ~ 0.5
generally falls in the radiofrequency (rf) region.

The populations among Zeeman sublevels will be asymmetric after any of the
various techniques described previously have been applied. The fi-angular distribution
from a polarized nucleus shows a maximum deviation at angles 0° and 180° relative to
the spin polarization axis (see Eq. 2.8). Therefore, when the number of ,8 particles are
monitored at these angles, an anisotropy is observed as long as the nucleus maintains
Spin polarization and A 3 aé 0. Given an initially spin-polarized collection of nuclei, an
alternating magnetic field H1 of the proper radiofrequency VL applied perpendicular

to H0 induces transitions between the substates. The H1 drives transitions with Am =

45

 

n‘off rfon at VL

 

Em Em

 

Figure 2.8: Schematic description of the B-NMR technique for an I = 3/ 2 nucleus.

:tl, causing redistribution of magnetic substate population. If enough rf power is
applied, the populations may be equalized and the polarization destroyed (see Fig.
2.8). The fl-angular distribution is then isotropic.

The rf can be applied continuously for a period of time, known as continuous
wave (CW) excitation, or it can be applied in short pulses. The 55Ni measurement
described in this thesis used the CW technique. The rf was scanned using a frequency
modulated (FM) signal, while the external magnetic field was held constant. The FM
signal allowed for an efficient scan of a frequency region in a short period of time. A
wide band FM scan was especially important during the initial search for a resonance.
The FM rf was applied in a repetitive fashion. The beam was always on and the
detectors were always counting. The NMR effect was monitored as the double ratio

[W(0°)/W(180°)lrfofl'

R = lW<0°>/W<180°)l.fon' “'18)

46

When the frequency is off resonance, R is unity, as there is no difference between
having the If off or on. At the Larmor frequency, the rf off condition results in an
asymmetric distribution of 6 particles, while during the If on phase the 6 particles
are emitted isotropically if polarization is entirely destroyed, as shown in Fig. 2.8. As
stated before, at the Larmor frequency, the double ratio reduces to Eq. 2.12. AfiP
is monitored as a function of applied frequency to determine the Larmor frequency.

The 9 factor is then extracted from the resonance frequency

hI/L = QIJNHO- (2.19)

The p can be deduced from the nuclear 9 factor if the nuclear spin is known (Eq. 1.37).
The uncertainty in ,u is evaluated from the width of the FM signal, which appears as
an uncertainty on ”D The uncertainty in H0 is usually small. Hg was measured in
this work with a proton resonance probe to a precision of 1:104. Other experimental
uncertainties are specific to the nucleus under study and the solid lattice into which
it is implanted. In general, for ,B-NMR experiments, these uncertainties are much
smaller than the error in the FM signal, which is typically around 5% for FM: 21:25
kHz and ”L = 1 MHz. The origin of these two uncertainties are described briefly in
the following sections, and will be discussed in the context of the 55N1 measurement

in the next chapter.

2.4.1 Spin-lattice relaxation

The process of spin—lattice relaxation is the means by which a spin polarized system
comes into thermal equilibrium with the surrounding lattice. To conserve energy in
the equilibration process, any nuclear Zeeman transition induced by influence of the
lattice is accompanied by a compensating change to the lattice. Although there are
many contributions to this relaxation, it is generally convenient to use a characteristic

time constant to describe the total process, called the spin-lattice relaxation time T1.

47

A successful fi-N MR experiment clearly requires the spin-lattice relaxation time to
exceed the nuclear lifetime, so that the spin polarization is maintained until the
nucleus decays.

The spin-lattice relaxation time depends sensitively on the nuclear implantation
site and any local radiation damage caused by the implantation process. If the neigh-
boring nuclei in the lattice have non-zero spins, they will change the local magnetic
field that the nucleus of interest experiences, and contribute to the relaxation process.
In addition, the interactions of the nucleus with electrons in the lattice also cause re-
laxation. The contribution and nature of the interaction of the nucleus with electrons
differs depending on whether the lattice is a metal or insulator.

In metals, the interaction between the nuclear moment and the magnetic field
produced by conduction electrons is the dominant spin-lattice relaxation mechanism.
The interaction process can be viewed as a scattering process, in which a conduction
electron scatters from an initial to a final state, while the nucleus undergoes transition
from one magnetic substate to another. The interaction is governed by a potential
V which describes the “scattering”. Calculations of this type yield what is called the

Korringa relation [46], and leads to the following approximate result,

AH 2 n 73
T1 (71—) "4am? (2'20)

 

where 76 and 7,, are the spin 9 factors for the electron and the nucleus, respectively,
and (9}?) is the Knight shift [47]. The Knight shift arises from the difference in
the magnetic field produced by the conduction electrons (AH) and the external field
(H). Note that T1 is proportional to the inverse of the lattice temperature T. Low
temperatures can be employed to lengthen T1 and extend the time window to evaluate
nuclear spin polarization.

The absence of conduction electrons in insulators makes the character of coupling

between nuclei and electrons different, and there is no simple relationship between

48

Ewe-«r

T1 and T, as was the case for metals. One source for relaxation in insulators arises
from the coupling of the nucleus to the magnetic field produced by the electrons
precessing under the influence of H0. Additionally, a nucleus can indirectly couple
with its neighbors via the distortions in the electron shells produced by their magnetic
moments, but for rare isotope experiments, the dilute nature of the impurity makes
such effect unlikely. In general, typical T1 for a metal in metal is on the order of ms,
while a metal in insulator has a T1 greater than seconds. It should be noted that the
spin relaxation time does not contribute to the error on p, but remains an important

experimental consideration.

2.4.2 Line broadening

Resonance line broadening is characterized by a spread in UL for nuclei residing at var-
ious sites within the implantation crystal. Provided that the external magnetic field is
homogeneous, broadening arises from the local environmental effects surrounding the
nuclei. Sources contributing to the overall shape of the resonance line are numerous
and can make the shape quite complicated. A common method used to take into ac-
count the distribution of interactions with different strengths, directions, and symme-
tries is the two—site model [48]. It is assumed that a fraction f of the nuclei experience
the full local field of undisturbed substitutional sites whereas the rest (1 - f) is not
oriented. The latter fraction accounts for nuclei which experience hyperfine fields of
different strengths but no preferred direction in space as may be present for instance
in a nonmetallic material. For this model the anisotropy is simply Reff = f R(Vhf),
where R(uhf) is the anisotropy expected for the undisturbed substitutional frequency
Vhf (or a narrow distribution around it) and R‘Eff is the experimental anisotropy.

In metals, the dominent broadening effect comes from the existence of couplings
between neighboring spins, known as dipolar broadening, and is on the order of a few
kHz or less. The interaction between two nuclear spins depends on the magnitude and

orientation of their magnetic moments and also on their separation [49]. In addition,

49

spin-lattice relaxation processes place a lifetime limit on the Zeeman levels, which
effectively broadens the line by the order of th.

The effects described above for metals are small for insulators if the insulator is
a perfect crystal. Imperfections in the crystal create, at the position of a nucleus,
quadrupole gradients. These quadrupolar effects vary not only in orientation but also
in magnitude from site to Site and have a considerable influence on the shape of the
resonance line. The imperfections in the crystal can be created by dislocations, strains,
vacancies, foreign atoms, and /or radiation damage, and the amount of broadening
depends on the goodness of the crystal. Most of these effects are expected to be small
for fl-NMR spectroscopy on rare isotopes.

In this chapter, the necessary components of a B-N MR measurement were in-
troduced including the production of nuclear spin polarization, the fi-decay angular
distribution, a method to measure spin polarization, and a description of the fi—N MR
technique. In the next chapter, detailed information is given of the experimental setup

that was necessary for carrying out the techniques described previously.

50

""I

Chapter 3

Experimental Setup

In the previous chapter, the techniques required for a successful magnetic moment
measurement using fi-NMR were described. This chapter will detail the experimental

systems required for beam production and the fl-NMR measurement on 55Ni at NSCL.

3. 1 Nuclide Production

At N SCL, radioactive ion beams are produced by projectile fragmentation, in which
a high-energy projectile impinges a stationary target. A large number of fragments,
both stable and radioactive, emerge from the target with velocities near the projectile
velocity. The purpose of the experiment described in this dissertation was to measure
the magnetic moment of 55Ni. 55Ni was produced by neutron removal reaction from
a 58Ni projectile on a 9Be target. A solid sample of 58Ni was vaporized and partially
ionized in a room temperature electron cyclotron resonance (ECR) ion source (see
Fig. 3.1). The 58Ni11+ primary beam was accelerated to 13.7 MeV/nucleon in the
K500 cyclotron, and then injected into the K1200 cyclotron. In the K1200 cyclotron,
the 58Ni primary beam was further stripped with a thin carbon foil to a charge state
of 27+ and accelerated to 160 MeV/nucleon. After exiting the K1200, the primary

beam impinged upon on a 610 mg/cm2 9Be target, resulting in many fragmentation

51

    

production
10 m target

Figure 3.1: Schematic representation of ion source, K500, K1200, and A1900 at NSCL
coupled cyclotron facility.

products including 55Ni produced by three neutron removal. The A1900 [50] was
used to separate the 55Ni from other reaction products. The first half of the A1900
separated fragments based on magnetic rigidity (momentum/ charge), a wedge-shaped
degrader at the intermediate image (dispersive plane) induced a velocity shift propor-
tional to the nuclear charge, and finally the second half of the spectrometer separated
the desired fragment back into a single Spot at the focal plane for transmission to the
experimental areas.

The primary beam was set at 2° with respect to the target to break the symmetry
of the fragmentation reaction and observe polarization, as shown in Fig. 3.2. Two
dipole bending magnets, labeled Z002DH and Z008DS, were used to set the beam
angle. A viewer, labeled Z013, located upstream of the target, was used to check
the beam position. When the beam was at 0°, the beam spot was located at the
center of Z013. At 2°, the beam Spot was located to the left of the center position.
Polarization of 55Ni was measured at three different momentum settings (-1%, 0%,
and +1% relative to the peak of the 55Ni momentum distribution) of the A1900.
The full momentum acceptance of the A1900 was kept at Ap/p = 1% via slits at

the intermediate image. The magnetic rigidity values of the first two dipole magnets

52

 

 

21 0' 2'

Y x

 

 

 

 

 

 

 

Beam spot images

\
/

A1900

r7

55Ni fragments

0‘ 2°

 

Be target
Viewer - 2013

- ] Dipole bending magnet
' ZOOBDS

Dipole bending magnet
“ ZOOZDH

 

Primary beam
58Ni

Figure 3.2: Schematic drawing of the placement of the primary beam at a 2° angle

on the target.

53

 

 

Figure 3.3: Mechanical drawing of the Radio-frequency Fragment Separator. The
beam enters the port on the left and a large time-dependent electric field can be
applied on the perpendicular axis in phase with the arrival of various particles.

(Bpl) and second two dipole magnets (Bpg) for each of the three momentum settings

are summarized in Table 3.1.

Table 3.1: A1900 Bp values for the various momentum settings for 55N i fragments.

 

 

Momentum (%) Bpl (Tm) Bpg (Tm)
-1 3.15860 2.67580
0 3.22240 2.76520
+1 3.19050 2.72080

 

 

 

 

 

A high beam purity is required for observation of maximum NMR effect due to
the continuous nature of the ,6 energy spectrum. The Radio-frequency Fragment Sep-
arator (RFFS) [51] was used in conjunction with the A1900 for further purification.
A mechanical drawing of the RFF S is Shown in Fig. 3.3. The RFFS applied a sinu-
soidal voltage of ~ 100 kVpp across two copper plates that caused a phase dependent
transverse deflection of the beam. The RFFS frequency was operated at the cyclotron
frequency of 24.39780 MHz with an adjustable phase difference. The RFFS deflected

particles based on time—of—flight, as ions with different velocities arrived at different

54

Scintillator

Dipole magnet dEtGCtOFS

     
 

 

~ rf amplifier

Figure 3.4: Photo of the ,B-NMR apparatus.

times with respect to the phase of the applied voltage and experienced different trans-
verse angular deflections. An adjustable vertical Slit system located 5.38 111 after the
end of the RFFS allowed for selective removal of unwanted fragments and provided

a beam purity of >99% for 55Ni.

3.2 fi-NMR Apparatus

3.2.1 Overview

Upon exiting the RFFS, the 55Ni fragments were sent to the fi—NMR apparatus [52],
pictured in Fig. 3.4. A schematic drawing of the important components is shown
in Fig. 3.5. The fragments first passed through a circular collimator, 1.5 cm in
diameter, before being implanted into a NaCl Single crystal located at the center of
the apparatus. The fi-N MR apparatus consisted of a large room-temperature dipole
magnet with its poles perpendicular to the beam direction with a gap of 10 cm.
The magnet induced the required Zeeman hyperfine splitting of the spin-polarized

nuclear ground state. The B particles from 55N1 were detected with a set of plastic

55

 
 

H0 magnet
pole lace

    
 

Plastic detector
UP

     
 

Collimator
NaCl single crystal

Plastic detector
55 ' DOWN
N1 beam
from RFFS

Ho magnet
pole face

Figure 3.5: Schematic drawing of the fi-NMR apparatus. The copper cooling rod was
not used as the measurement was performed at room temperature.
scintillator detector telescopes located between the poles of the magnet. One telescope
was located at 0° and one at 180°, relative to the direction of H0. Each telescope
contained a thin AE scintillator (4.4 cm x 4.4 cm x 3 mm), and a thick E scintillator
(5.1 cm x 5.1 cm x 2 cm). Each scintillator was coupled to an acrylic light guide and
a photomultiplier tube (PMT). The thick detector acted as a total energy detector
for 0 particles up to 4 MeV. 6 particles from 55Ni have an endpoint energy of 7.7
MeV with a mean energy of 3.6 MeV (see Fig. 3.6). Only a fraction of the 6 particles
were completely stopped in the thick detector due to the high endpoint energy and
scattering effects through the scintillator and surrounding material. The ,6 detectors
were labeled as B1 (thick detector on top), B2 (thin detector on top), B3 (thin detector
on bottom), and B4 (thick detector on bottom), as Shown in Fig. 3.7.

The dipole magnet had a fringe field that affected the performance of the PMTS.
The light guides were bent at an angle of 45° to place the PMTs close to the yoke steel

(see Fig. 3.7), where the fringe field was smallest. Even when the PMTs were placed

56

 

7/2'

204.7 ms
55Ni ~I
28

03+ = 8692 keV

 

- ~ 0
7/2—¢100/o

55
2700

Figure 3.6: fi-decay scheme for 55Ni.

17.53 h

 

 

Top view Yoke a“,

 

Plastic scintillator

 

 

 

beam -—->
NaCl
‘1” catcher
y
Catcher insid “'1'
pole gap J
a] Light guide

 

 

 

I PMT

Figure 3.7: Schematic drawing of detector system.

57

 

next to the yoke, there remained a fringe field that ranged from 1 to 6 gauss for a
magnet current up to 190 A (~0.45—T holding field). A number of different shielding
configurations were tested, but none reduced the fringe field Significantly. The final
configuration included a 0.4—mm thick 11 metal Sheet rolled into a cylinder and placed
around the PMT.

Two silicon surface barrier detectors were used for fragment identification. Silicon
detector number 1 (thickness 150 pm) was placed 34 cm upstream of the catcher, and
was attached to an air-activated drive, providing the ability to insert and remove the
detector from the line of the beam without breaking vacuum. Silicon detector number
2 (thickness 300 pm) was placed 12 cm downstream from the catcher and served as
a veto detector for fragments that traveled through the NaCl.

Two identical rf coils in a Helmholtz—like geometry were placed within the magnet
and between the ,6 detectors, with the field direction perpendicular to both the direc-
tion of the beam and the static magnetic field. Details of the rf system are given in
the next section. A 2.5-cm diameter, 2-mm thick disc-shaped N aCl single crystal was
mounted on an insulated holder, between the pair of If coils. The crystal was placed
at an angle of 45° relative to the normal beam axis to minimize the energy loss of the
,6 particles emitted at 0° and 180°. A 1.5-mm thick Al degrader was placed in front
of the collimator to lower the energy of the incoming 55Ni ions to cause the ions to
stop in the center of the NaCl crystal. The LISE++ code [53] was used to calculate
the appropriate thickness of the degrader. NaCl was chosen as a catcher because it
is known to hold polarization for Cu ions with a long T1 [23] given that Ni ions have
similar atomic radii to Cu ions, however, the T1 for Ni ions in NaCl is unknown. A
photo of the If coil, crystal, collimator, and silicon detector 2, all of which under

vacuum during the measurement, is shown in Fig. 3.8.

58

WATT '—

Collimator NaCl catcher

55Ni beam

 

rf coil Si detector 2

Figure 3.8: Photo of the If coil with crystal, collimator, and silicon detector 2. All of
the pictured components are under vacuum during the measurement.

3.2.2 Radiofrequency system

The transverse H1 field used to destroy the polarization was created by the Helmholz-
like coils that made up part of an LCR resonance circuit. L is the inductance of the
7f coil that produces the H1, 0 is the capacitance, and R is the resistance. The LCR

resonance condition for frequency f is given by

f— 1
‘2m/7E'

 

(3.1)

Several variable capacitors were used with fixed L and R to tune the resonance circuit
and achieve impedance matching to the If amplifier. Such operation ensured a sufli-
ciently large value of H1 for all transition frequencies within a frequency modulated
(FM) scan. Transition frequencies were sequentially applied to the LCR resonance cir-
cuit by selecting one of the variable capacitors using fast relay switches. The selected
capacitor was tuned to the specific capacitance that satisfied the LCR resonance
condition for a particular frequency. The basic scheme is shown in Fig. 3.9.

The rf system used one of three function generators to generate the FM rf signal.
A pulse pattern generator, REPIC model RPV071, triggered the function generators.

The rf signal was sent to a 250 W rf amplifier. The amplified signal was then applied

59

Function

Generators Variable

FGl—I son Capacitors
FGZ
_l—J

, RFcoll
FG3

Amplifier

Fast Relay
Switches

Figure 3.9: Schematic drawing of the LCR resonance system.

to the rf coil, which was part of the LCR resonance circuit. A 50 Q resistor fulfilled the
impedance matching condition between the amplifier and the If coil. One (or more) of
six variable capacitors were used to complete the LCR circuit. The primary capacitor
used during the NMR measurement was a 4000 pF variable capacitor. A remotely-
controlled stepper motor was used to tune this capacitor. The generated resonance
curve is also called a Q curve, and an example is shown later in this section. After a
fixed rf irradiation time, the frequency from a second function generator was sent to
the same LCR resonance circuit. A different capacitor was then selected by the fast-
switching relay system. Only one function generator was used for the majority of the
N MR measurements presented in this thesis, as only a single central frequency with
F M was applied for any one rf measurement. However, for a portion of the experiment,
a new multiple-frequency NMR technique was tested. Three function generators were
used to scan a larger frequency region for the initial resonance search. Additional
details on the If system are available in Ref. [54].

The operating parameters for the If depended on the conditions of the NMR
measurement. The theoretical predictions for p(55N i) given in section 1.4.4 suggested
a search region for 11(55N i) between -0.6 [1. N and -1.2 IIN. With an external magnetic
field of H0 = 0.4551 T, the 11 search region corresponded to a frequency range of 588
kHz to 1176 kHz, where g = p/I in Eq. 2.16 and I = 7/2 for 55Ni. The inductance

of the coil and capacitance were chosen according to Eq. 3.1 to match the desired

60

frequency region. With a fixed inductance of ~ 15 pH, the capacitance ranged from
4500 pF for the lowest frequency to 1000 pF for the highest frequency. The capacitors
available in the If system included: one 3900 pF fixed capacitor, two 4000 pF variable
capacitors, two 1500 pF variable capacitors, and one 1000 pF variable capacitor.

The 1f coil support was made of the polyimide—based polymer Vespel®, made by
DuPontTM, and rated to 260°C. The copper wire used to wind the coil was 20 AWG
(round) with a polyimid insulation called Allex®, ordered from Superior Essex®.
The wire insulation was rated to 240°C. Such temperature ratings were suflicient to
withstand the heat generated by the voltage drop across the coil. The inductance
of the coil was measured as a function of turn number as shown in Fig. 3.10a. The
measurement was made with an LCR meter (Electro Science Industries - model 253).
The total turn number is the sum of turns for the two coils. The relationship between
the inductance and total number of turns, N, followed a L ~ N 2 dependence, as
shown in Fig. 3.10b. Therefore, to achieve a coil with an inductance of ~15 [1H, a
14/14 turn coil was used. The measured inductance of the 14/14 turn coil was 14.3
pH.

The strength of H1 needed to destroy the initial polarization is given by the

expression

 

_1 27rAf ”2
H1—;( At ) , (3.2)

where A f is the frequency window, At is the 11' time, and 'y = fipN. A F M of :l:50 kHz
used for the initial wide frequency scan and an If time of 10 ms required an H1 field
of ~8 G. A FM of i25 kHz for the narrower scan and an If time of 10 ms required
an H1 field of ~5 G. Temperature tests proved the If system could withstand the
prolonged application of an H1 of 8 G, as the temperature was observed to saturate
at 155°C.

The DC character of the coil, 0:, was determined by measuring the magnetic field

of the coil as a function of applied current. a is needed to determine H1 of the coil at

61

 

Inductance ( (1H)

 

 

0 . . , ,
0 10 20 30 40

Total number of turns, N

 

 

35- b)
30- o

25‘ 9

20-

15‘ 9
O

10- 9

00.
5-

Inductance (pH)
0

 

 

0 1 fl ,

0 500 1000 1 500
N2

 

 

 

Figure 3.10: Inductance of the If coil as a function of a) total turn number, N and
b) N2.

62

 

L

   

y = 3.2727x - 0.06
R2 = 0.9986

Field (G)
N (A) 4?- 01 O7 ‘1 co (0

 

 

O r I r l l 1
0 0.5 1 1.5 2 2.5 3

Applied Current (A)

 

 

 

Figure 3.11: DC character of the rf coil determined by measuring the magnetic field
of the coil as a ftmction of applied current. The a value is equal to the slope of the
line.

a given frequency. Current was applied to the If coil using a Tenma Laboratory DC
Power Supply (72-6152) fiom O to 2.5 A. The magnetic field was monitored at the
center of the coil using a F W Bell Gauss/Teslameter (model 5080) and the results
are shown in Fig. 3.11. The a value is the slope of the line, (1 =3.3 G/A. The H1 is

then calculated as

H1 = gfig, (3.3)
where V is the voltage across the coil, f is the applied frequency, and L the inductance
of the coil. The voltage is determined from the peak—to—peak value on the resonance
Q—curve, as shown in Fig. 3.12. The example Q-curve was recorded at frequency 1100
kHz with FM :1: 50 kHz. The input voltage from the function generator (PC) was
V,” =100 mVpp and the generated voltage was Vout=870 V. The calculated H1 in this

case was 7.2 G which matches the required H1 given by Eq. 3.2.

63

    

Frequency 1100 kHz
FM t 50 kHz

H1=7.2G

.sl....l....l..1.l.t

 

 

 

 

 

Figure 3.12: Resonance Q—curve at frequency 1100 kHz with FM :1: 50 kHz and input
voltage Vin =100 mVpp. The H1 is calculated from Eq. 3.3.

3.2.3 Electronics
Readout electronics

The plastic scintillator detectors were used to detect the 6 particles emitted in the
decay of 55Ni. A schematic diagram of the electronics for each of the four plastic
scintillators is shown in Fig. 3.13. Each scintillator was coupled to an acrylic light
guide, which was coupled to a PMT. The signal from the PMT was shaped, amplified,
and separated into a fast signal and a slow signal. The slow signal was sent to a
VME analog-to—digital converter (ADC, CAEN mod. V785) where the energy was
determined from the maximum voltage peak. The fast timing signal was sent to
a constant fraction discriminator (CFD, Tennelec TC 455). One of the fast timing
signals from the CFD was converted from NIM type to ECL and used in the VME
scaler module (CAEN sealer C3820) for rate monitoring. Another CF D time signal
was used for establishing the logic of the master gate (MG).

A schematic diagram of the electronics for the silicon detectors used for particle

64

 

Plastic

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Scintillator
Bl
[ Signals for B2, B3, and B4
are obtained from individual
‘ PMT scintillators in a similar fashion
7
Shaper/Amp
slow signal I fag signal
ADC
gate—7 ADC CFD
f
NIM to ECL m
converter
f Bl
Scaler

(see MG diagram)

Figure 3.13: Plastic scintillator electronics diagram.

identification is shown in Fig. 3.14. Silicon detector 1 (Ortec SN 27-259B, model TB-
020—300—150) was located upstream of the fl-NMR apparatus and was used for particle
identification. The energy signal was taken from the Slow output of the amplifier and
digitized in VME. The fast timing signal of Silicon 1 was compared with the cyclotron
rf to generate a time-of-flight (tof) measurement of the incoming beam. Silicon de-
tector 2 (Ortec SN 36—153D, model TB-020—300-300) was located downstream of the
NaCl crystal, and was used for particle identification before the crystal was put in
place. After the NaCl crystal was in place, the detector was used as a veto detector
for fragments that passed through the NaCl crystal. Signals from both detectors were
processed with Tennelec (S / N 2104) preamplifiers, and then amplified (Tennelec TC
241 S). The slow signal was sent to the ADC and the fast signal was sent to the CFD
for timing purposes. One of the CF D timing signals for both silicon detectors was sent
to a logical OR to become part of the master gate (MG). Another CFD timing signal
for both detectors was converted from NIM type to ECL type and sent to the scaler

for rate monitoring. As noted above, a third timing signal from the silicon detector 1

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Silicon Silicon
detector 1 detector 2
Preamp Preamp
.. 1
Amp Amp
fast ' I
slow signal l I srgna slow signal 1 fast signal
ADC l
gate—'7 ADC CFD 32;... ADC CFD
TAC start [
I r i t l
NIM '10 ECL Si 1 to MG TAC ADC Delay Si 2 to MG
converter (see MG A (see MG
1 diagram) , diagram)
rate divider 1
$3!“ I I NIM to ECL
converter
Cydfftm" TAC stop l
Scaler

 

 

 

 

Figure 3.14: Silicon detectors electronics diagram.

66

B334

B1 82 w
w Si(OR)

er.

MG computer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

not busy
F—
..... 6
MG Live
l '1
scaler ADC Trigger Latch/
gate Busy Latch

Figure 3.15: Master gate (MG) electronics diagram.

CFD was used as a start for the time-to—amplitude converter (TAC, Ortec 566). The
TAC stop came from the K1200 cyclotron If. The TAC output represented the beam
tof, and was digitized in VME.

A logical AND was made between B1 and B2, as well as B3 and B4 before being
sent to the MG. The coincidence condition was implemented to reduce readout dead
time and reduce background events and was used to trigger the readout of all other
detectors during the data acquisiton (see Fig. 3.15). The MG was created from the log-
ical OR of scintillator coincidences and the signal from the silicon detectors to trigger
during particle identification. The MG made a logical AND with a computer-not—busy
signal to provide the master live signal. Master live “opened” the data acquisition gate

for ADC conversion.

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

*NIM channel outputs:

 

 

 

RF electronics diagram

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

to RF amplifier
[ A —[ I/O di. 1
2) on _.-.[
- 3 FAN
Double Balazced MIXfI‘ (i I |09lC) Ho on/off
41:63,] F6 3] 3) off 9% r/o ch. 2
‘l i U0 ch. 3
‘ 4) beam —>l:l-[
NIM to TTL FAN
converter (Inverted I091C) 9 Beam pulse
“ 5) rf sum
NIM rf sum 5
IAN/serial : 6) count —>[:]-—-> [lo d1. 4
FAN (split)
7) if 12* attenuator 99am signal 1 to double balanced mixer
NIM ff 1' fr 2’ rf 3 8) rf 2 .p attenuator +gate signal 2 in double balanced mixer
control 9) ff 31-» attenuator +9302 signal 3 to double balanced mixer
970973"! ‘—
ECL to NIM ‘
mnverter" 11) sw 1 FAN NIM to m. rf box sw 1
1 12) SW FAN NIMtoTTL rfboxst
14) 5W if box sw 4
15) sw FAN rf box sw 5
PC RPVO71 16) sw FAN rf box sw 6

 

Figure 3.16: Electronics diagram for the radiofrequency system.

rf electronics

Function generators (FG, Agilent function/ arbitrary waveform generator, 20 MHz

model 33220A) were used to produce the If. Timing control of the If was accomplished
with a VME pulse-pattern generator (see Fig. 3.16). The REPIC model RPV—071

pulse-pattern generator had 32 channel output with 65k / channel data memory. A bit

pattern was loaded into the memory of the RPV-071 through the VME bus. The

pattern was output-synchronized with an external clock signal. Each output was used

to trigger and /or gate devices. These devices are listed on the right Side of Fig. 3.16.

The on, off, beam, and count signals from RPV-071 were sent to an I/O register

(CAEN mod. V977) for recording in the data stream and for software gating. The on

and off signals were used for if pulsing. The on signal was also sent through TTL

68

 

 

Beam

 

 

 

 

 

 

 

 

 

 

Magnet
V One magnet cycle 7
60 6
Count
: Time

 

Figure 3.17: Schematic representation of the external magnetic field pulsing sequence
during the polarization measurement. The field was pulsed on and off every 60 s. The
beam was continously implanted.

to a temperature sensor at the input of the dipole magnet coil to control magnet
pulsing. The rf pulsing sequence generated by RPV-071 was sent to the function
generator. From the function generator, the signal went to the if amplifier (model
BBSOD3FOQ, 58 dB, 250 W), and then to the rf box. The sw1-6 signals shown in
Fig. 3.16 represent the capacitor switch signals, which were generated by RPV-071
and sent to the rf box.

The timing sequences programmed to the RPV-071 module for both the polariza-
tion measurement and NMR measurement are shown in Figs. 3.17 and 3.18, respec-
tively. In both measurements, the beam was implanted continuously and fl counting
was performed for the entire measurement. During the polarization measurement,
the external magnetic field was pulsed on and off every 60 s. The frequency of the
internal clock on the RPV-071 module was 500 Hz, and thus the minimum length of
the pulse was 1 / (500 Hz)=2 ms. The maximum length of the pulse or one cycle of
timing program was (65k data point) / (500 Hz)=l30 8.

During the NMR measurement, the PM was realized in a “sawtooth” function with
a 10 ms rf sweep time. The if was applied continuously for 30 s on and then 30 S
off. The RPV-071 clock frequency was 2000 Hz, so the minimum length of the pulse
was 0.5 ms. The maximum length of the pulse or one cycle of timing program was

32.5 s. The RPV-071 module was controlled with a graphical user interface developed

69

 

Beam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rf
E One rf cycle
8 ]FM width 30s
8 ...
1.1. One rf sweep
10 ms
Count
: Time

 

Figure 3.18: Schematic representation of 11' pulsing sequence during the NMR mea-
surement. The if was pulsed on and off every 30 s, with 20 ms sweep time. The beam
was continously implanted.
using Tcl/Tk (scripting language/ graphical user interface took kit) [55] based on the
NSCLDAQ VME Tcl extension [56].

High voltage (HV) was supplied to each 6 detector PMT and the silicon detector
preamps through a CAEN SY3527 High Voltage Power Supply (HVPS). Individual

software controls for voltage ramp rate and maximum voltage were available for each

device connected to the CAEN HVPS.

3.2.4 Calibrations

External magnetic field

Precise knowledge of the external magnetic field is necessary to reduce systematic
uncertainty in the g-factor result. The 9 factor is calculated from UL and H0 (Eq.
2.19). The contribution to the overall error on the 9 factor from the magnetic field
can be less than the FM. The dipole magnet field was calibrated by measuring the
static field at the center of the rf coil as a function of applied current. The magnetic
field was measured using the Metrolab PT2025 precision NMR Teslameter with a

number 3 solid sample 1H probe (range 0.17 to 0.52 T). Current was supplied to

70

 

5000
H0 = 24.5 I +67.3

 

4500

4000

3500

Field (6)

3000

2500

2000

IIIUIIIIIIIIIVTIIIITIIIllllll—Il

)-
-
b
P
p
-
l
r
-

 

 

 

Set current (A)

Figure 3.19: Dipole magnet calibration.

the magnet with a Power Ten Inc DC power supply (SN 1010740). The calibration
was done with all experimental devices in place, except for the N aCl crystal and
silicon detector 2, which would have been in the way of the probe. The calibration of
the holding field as a function of applied current is shown in Fig. 3.19. The magnet
calibration was done approximately a month before the experiment began to establish
the dependence of the field with the applied current. However, for the experiment,
it is only important that the field is known precisely at the set current, and that
the field is stable for the duration of the fi-N MR measurement. The field was also
monitored immediately before and after the experiment, for one hour to obtain an
estimate of the systematic error on H0. The fl-N MR measurement was conducted
with a current of 180 A, corresponding to 4477.3 G using the calibration. Over the
course of a month period, however, the field shifted. When the field was measured
at 180 A prior to the experiment, the field ranged from 4490.6 G to 4494.3 G over
one hour. Immediately after the experiment, the field was monitored again for one
hour at 180 A and ranged from 4490.0 G to 4491.5 G. The fluctuation in magnetic

field mainly came from two sources: inherent instability of the power supply and

71

temperature. The power supply has a quoted stability of i0.05% of the set point per
8 hours after warm-up. Variations in the temperature of the room affect the power
supply and in turn the magnetic field. A value of 4490.5 G was chosen with an error
of 5.0 G at 180 A to take into account the small fluctuations in field over the course
of the experiment.

The magnetic field strength chosen for the polarization measurement was 1000 G,
which corresponds to 40 A of current in the magnet. In the polarization measurement,
the magnetic field is pulsed on and ofl. The PMTS are affected by the fringe field
of the magnet, and this effect is field dependent, as shown in Fig. 3.20, where the
energy spectra for B1 and B4 are shown for a 90Sr source as a function of applied
current. Therefore, 40 A (1000 G) was chosen as a field strength for the polarization

measurement, where the field effects were minimal (see Fig. 3.20).

Plastic scintillator detectors

Ideally, the plastic scintillator detectors would be calibrated with [3 particles of varying
energy. However, there are limited off-line or long-lived sources of [3 particles with a
large endpoint energy. One alternative is to calibrate the energy response with the
Compton edge from a 'y-ray source, since the scintillators are able to detect the
scattered electrons. The accuracy of such a calibration is not high, but is sufficient to
check the functionality of the detector.

A Compton scatter results in the creation of a recoil electron and scattered 'y—ray
photon, with the division of energy between the two dependent on the scattering
angle [57]. The energy of the scattered r) ray, E3, in terms of its scattering angle 6

and the initial 7 energy, E7, is given by

 

E’ E7

7 : 1+ (E7/m0c2)(1 — cos 6)’ (34)

where 771062 is the rest mass energy of the electron (0.511 MeV). The Compton edge

72

 

B1 B4

 

   

 

 

   

   

 

 

6000 0A On
E5000 Off
'E
=- 4000 -
.0
I—
33000 -
.9
c 2000 -
D
0
01000 -
o 200 400 600 800 1000 1200 0 200 400 600 000 1000 1200
Energy (arb. units) Energy (arb. units)
6000 .
A5000 5 A @5000
'E
g a 4000
54000 9'
E 33000
g 3000 *2
Ezooo 3 mo
3 0
81000 1000
0 .
0 200 400 600 800 1000 1200 200 400 600 1000 1200
Energy (arb. units) Energy (arb. units)
5000‘.
2‘ E
C C
3 2'!
.o' n'
I— h
1! 3
£2 £2
E C
D 3
O O
0 0
0 200 400 600 300 1000 200 0 200 400 600 800 1000 1200
Energy (arb. units) , Energy (arb. units)
5000.?
35 ' 165A
'7:
3
.o‘
I—
11
B
C
3
O
0
0 200 400 600 600 1000 1200 '0 200 400 600 800 1000 1200
Ener arb. units .
gy( ) Energy (arb. units)

 

 

Figure 3.20: Dependence of the 6 energy spectra for a 908r source on the strength of
the external magnetic field. Energy spectra of thick detectors B1 and B4 are shown
with magnetic field off (dotted line) and magnetic field on (solid line) at 0, 75, 110,
and 165 A.

73

Table 3.2: Peak 7 energy and calculated Compton edge for each source used in the

 

 

plastic scintillator energy calibration.
Source Peak 7 energy (keV) Compton edge (keV)
57Co 122 40
1370s 662 481
6000 1170 964
1330 1120

 

 

 

 

 

represents the maximum energy that can be transferred to an electron in a single
Compton interaction, that is, in a head-on collision in which 0 2 1r. In this case, Eq.

3.4 reduces to
E
E’ g —7 3.5
'7 1 + 4E7’ ( )
and the Compton edge, E0, is the maximum energy transferred to the electron, that
is

EC = E, — El, (3.6)

Three 7 sources were used to energy calibrate the ,6 telescopes: 5700, 137Cs, and
6000. The 7 peak energy and calculated Compton edge for each source are shown in
Table 3.2. An average was taken for the two 6000 values, as the separate Compton
edges are not resolved in the plastic scintillator. The Compton edge was determined
in the spectrum by taking the channel value at the half maximum of the Compton
continuum for each detector. An example of the determination of the Compton edge
from the energy spectrum taken with a 137Cs source with B1 is shown Fig. 3.21. This
channel number is shown as a function of the calculated Compton edge in Fig. 3.22.

The linear trend shows that the detectors are functioning as expected. The abso-
lute values obtained from the calibration may not be accurate, as it was difficult to
determine the location of the Compton edge from the spectra due to the scattering
and energy loss properties of electrons discussed in section 2.2.1. Such properties are
also the reason for the difference in the slope of the line for thick detectors versus

thin detectors.

74

 

 

 

 

 

 

150200250300350H00450500550600
384 Channel

Figure 3.21: Energy spectrum from 137Cs taken with B1 to determine the Compton

edge. The Compton edge was determined in the spectrum by taking the channel value

at the half maximum of the Compton continuum.

 

900
l 0 B1 Linear regression through 84
800 - I 32 y=0.5801x+144.09

  
 

 

700*
600‘
5004
400*
300‘

Linear regression through 83
y = 02241): + 103.59

Channel Number of Compton Edge

200 a

100 ‘

 

 

o r 1 r I r l
0 200 400 600 800 1000 1200
Calculated Compton Edge (keV)

 

 

 

Figure 3.22: Energy calibration of plastic scintillator detectors B1-B4. 57Co, 137Cs,

and 60Co were used as 7 sources and the calibration was done using the Compton
edge of the 7 spectrum.

75

 

 

 

 

 

 

50° :— 559 9V 6.25: MeV Silicon detector 1
Z 6.78 MeV
4oo _— /
I 5.42 MeV
.— \
a 300 _—
C
3 .—
0 -
U _ 8.78 MeV
20° .- 6.05 MeV
C
100 _—
o

 

 

 

400 500 600 700 800 900
Channel number

Figure 3.23: a-decay spectrum of 228Th collected with silicon detector 1.

Silicon detectors

The silicon detectors were tested prior to the experiment to check the energy resolu—
tion. Silicon detector 1, placed upstream of the fi-NMR apparatus on an air activated
drive, was the primary detector used for particle identification at the experimental
end station. Silicon detector 2, placed downstream of the NaCl crystal, was used to
determine if fragments were passing through the crystal. Both detectors were tested
by collecting an a spectrum from a 228Th source, with the results shown in Figs. 3.23
and 3.24. The observed resolution was sufficient for particle identification purposes
in the 55Ni region. The FWHM at 5.69 MeV was observed to be 64.8 keV for silicon
detector 1 and 76.5 keV for silicon detector 2. The silicon detectors were also tested
with the external magnetic field on at both 180 A and 40 A, with little change in

energy resolution.

fi-NMR apparatus

The entire ,B-NMR apparatus was calibrated to ensure that there were no inherent

asymmetries. The same rf sequences executed during the experiment were also per—

76

    
   

 

5.69MeV 3-2 MeV
\

Silicon detector 2
V ‘6/78 MeV

5.42 MeV
\

Counts

600

400

 

 

 

 

200

 

300 400 500 600 700 800
Channel number

Figure 3.24: a—decay spectrum of 228Th collected with silicon detector 2.

formed with a 60C0 7 source at the crystal position. Calibration with a 6 source
would again be ideal, as was the case for the plastic scintillator calibration, but the
selection of [3 sources is limited. The available 6 source, 90Sr, 6‘ decays to 90Y with
a Q value of 546 keV (mean ,6 energy 196 keV). 90Y then ,8‘ decays to 90Zr with a Q
value of 2.3 MeV (mean 6 energy 933 keV). Most 6 particles fiom this source would
be absorbed before making it to the thick detector for a coincidence measurement.
Some 6 particles make it through to the thick detector, but the activity of the source
was not sufficient to complete the necessary calibrations in a reasonable amount of
time. Thus, a 7 source was thought to be the best available option. Prior to the
experiment, the frequency range of 600 kHz to 1600 kHz was scanned, pulsing the
rf on and off every 30 s, with a constant external magnetic field of 4490.5 G. The
double ratio R was determined as given in Eq. 2.18. Two frequency modulations were
applied, the first at :t 50 kHz, and the second at :i: 25 kHz, to reproduce the expected
experimental conditions. Both calibrations are shown in Fig. 3.25 and no significant
asymmetry was apparant.

The H0 on/off double ratio was also determined with the 6000 source. The dipole

77

 

 

1.05 .05

1.04 - FM :50 kHz 1.04 - FM :25 kHz
1.03 - 1.03 -
1.02 - 1.02 - +

 

 

.. Jr+++++t . ,1. + + .r+i..++.
0.99_ '+_l—+ 0.99- TT+ “l“ ++++ITFT

 

 

0.98 - 0.98 -
0.97 - 0.97 -
0.96 - 0.96 -

 

 

 

 

095 .klk“1...| ..l. .l 0.35.1...l ..I . I I I.
I 600 800 1000 1200 1400 1600 600 800 1000 1200 1400 1600

Frequency (kHz) Frequency (kHz)

Figure 3.25: rf calibration prior to the start of the experiment with 60Co source. The
rf is pulsed 30 s on then 30 s off. TWO frequency modulations (FM) were checked:
21:50 kHz and :i:25 kHz.

magnet was pulsed on and oif every 60 s, at a field of 1000 G. The double ratio R
was taken as given in Eq. 2.11. The calibration value of R was found to be 1.0012 :1:

0.0074, consistent with zero asymmetry.

78

Chapter 4

Experimental Results

4. 1 Fragment Production

The experimental systems described in the previous chapter were used in the produc-
tion and identification of 55Ni fragments, and the ,B-NMR measurement. The 55Ni
secondary beam was produced under a variety of conditions to maximize spin polar-
ization and complete the B—N MR measurement. The spin polarization measurements
were completed with the 55Ni secondary beam produced at primary beam angles of
2° and 0°, as well as three separate fragment momenta settings of the A1900. The
NMR measurement was completed with the primary beam at an angle of 2°, and a
single A1900 setting with the momentum corresponding to the peak yield of 55Ni.
This chapter presents the particle identification of the secondary beam, the response
of the ,6 detectors under the various conditions described above, and the results of

both the spin polarization and NMR measurements.

4.2 Particle Identification

Secondary beam particle identification (PID) was performed using the A1900 focal

plane detectors for energy loss and tof information. The PID with no wedge at the

79

intermediate image of the A1900 and a constant value of Bp is shown in Fig. 4.1. The
constant value of Bp = mv/ q yields certain features that are characteristic of the A/ q
of the fragments. The unbound fragment 8Be did not appear in the PID, and the
“hole” where the fragment should appear was used as reference (shown in the lower
part of Fig. 4.1). The energy loss of the 55Ni fragments was determined to be 597
MeV through the 0.5 mm thick A1900 focal plane PIN detector.

A 405 mg/cm2 Al wedge was placed at the intermediate image of the A1900, and
the PID was measured again as shown in Fig. 4.2. Based on the energy loss observed
in the unwedged PID, the 55Ni fragments were identified with the wedge present.

The Bp values were scanned and the rate of 55Ni was measured at each Bp setting
to establish the yield distribution as a function of fragment momentum. The mo—
mentum distribution is especially important for the spin polarization measurement,
as polarization was later measured as a function of fragment momentum at three
settings to establish the variation. The measured momentum distribution is shown
in Fig. 4.3 with a Gaussian fit. The measured momentum distribution agrees with a
calculation that considers conservation of linear momentum, as described in section
5.1.1.

The 55Ni fragments were sent on to the RF FS. Particle identification was per-
formed after filtering at the RFFS diagnostic box, which was located 5.3 m down-
stream of the RFFS exit. The diagnostic box consisted of an adjustable vertical
slit system sandwiched between two retractable parallel-plane avalanche counters
(PPAC), and a retractable telescope of Si PIN detectors. The PPACs are position
sensitive detectors and were used to determine the slit position for selective removal
of unwanted fragments. The vertical position of fragments in the secondary beam
after deflection in the RF FS is shown as a function of arrival time in Fig. 4.4. The tof
was taken as a time difference between a signal in the RF F S PIN detector and the
if frequency of the K1200 cyclotron and thus the faster fragments have longer times

in this figure. The upper vertical slit (G183 tOp) was positioned to +4 mm to select

80

 

7001“ . . . -.-'_

I!”

  

600: ' ~. «13. ..

HQ 4123‘ ggw' . ‘93:: ~

   

. . 8.?»5-‘31‘ «’n‘yu-ugw.’ .1

     
  

Energy Loss (arb. units)

‘5‘: .‘ .
x 3%.}:

   

3
70:—
:
_
z

30 33%;}. .

‘l

Energy Loss (arb. units)

 

_l|ll]_llll_l_.ll

I I I l I l I I I I I I I I I I I I I I | I I I l I I I I I I l I I I I I I I I
10 20 30 40 50 60 7O 80
Time of Flight (arb. units)

Figure 4.1: Energy loss at the A1900 focal plane PIN detector as a function of time—of-
flight with no wedge at the intermediate image. The expanded PID shows the “hole”
where unbound 8B6 is expected, providing a reference for 55N i.

81

 

700

600

500

400

300

Energy Loss (arb. units)

200

 

100

I I I I 'I'I‘.lfll'"I"-l"l'l ‘I I I I I l l I I I 1"l'1'r \.-.l.::I-"-1-l ‘I I I I I I I I

 

llllllllllllllllIllllflllllllll‘lllll

 

10 20 30 40. 50 60 70 80
Time of Flight (arb. units)

Figure 4.2: Energy loss at the A1900 focal plane PIN detector as a function of time-
of—flight with 405 mg/cm2 Al wedge at the intermediate image.

 

9000
8000
7000
6000
5000
4000
3000
2000
1000

, ' Experiment
Gaussian fit

   

Rate [pps/pnA]

IrlIllllllllllllllIIIlllllllllllllllllllll'l

 

 

11-

....i .i....i... ......i
-0.03 -0.02 -0.01 0 0.01 0.02 0.03

 

Momentum (% of peak)

Figure 4.3: Yield distribution of 55Ni as measured at the focal plane of the A1900.

82

Table 4.1: Fraction of components of the secondary beam, relative to 55Ni, before the
RFF S was turned on and after.

 

 

Nuclide Fraction before RF F S Fraction after RFFS
55Ni 1 1
5400 1.33 0
53Fe 0.24 0.01
52Mn 0.02 0.02

 

 

 

 

 

the deflection region that included the fragment of interest and eliminated unwanted
contaminants. Two beam steerers located downstream of the RFFS diagnostic box
and upstream of the experimental endstation were used to recenter the fragments
onto the optical axis of the experimental endstation. The fiactions of each nuclide in
the secondary beam relative to 55Ni, before the RFFS was turned on and after, are
given in Table 4.1. The primary contaminant in the secondary beam from the A1900
was 54Co, as seen from Fig. 4.4a. Implantation of 54Co would present a particular
problem for the ,B-N MR measurement because it has a half-life and fi-endpoint energy
similar to that of 55Ni (see Fig. 4.5).

When the RF F S was on with the slits closed, the 54Co contamination was elim-
inated completely, as shown in Fig. 4.4b. The other contaminants, 53Fe and 52Mn,
were not a problem for success of the measurement due to their low fl-endpoint en-
ergies, although these low-energy contaminants were observed, as will be discussed
in section 4.3. Contributions from these low-energy contaminants were removed from
the 0 energy spectra collected at the ,B-N MR apparatus by making an energy out in
software.

The purified beam was implanted at the center of the fl-NMR apparatus. Final
particle identification was performed at the endstation as well. Energy loss of the
secondary beam was recorded with silicon detector 1, and the tof was taken as a
time difference between a signal in that detector and the rf frequency of the K1200
cyclotron. The PID measured before the RF FS was turned on is shown in Fig. 4.6a.

After the RF F S was turned on and the vertical slits were adjusted, the PID was taken

83

 

Vertical position (arb. units)

 

Time of flight (arb. units)

 

. 55Ni

Vertical position (arb. units)

 

0 100 200 300 400 500 600 700
Time of flight (arb. units)

Figure 4.4: Plot of the vertical beam position as a function of tof after the RFFS.
a) The gray rectangle highlights the region allowed to pass through the vertical slit
system and b) the particle ID resulting from the cut is shown below.

84

7/2' 0"”

204.7 ms 193.3 ms
55Ni l 54CO NI

 

 

 

 

28 N 27
0,3,, = 8692 keV 013+ = 8243 keV
' o + ~ o
1753 h 7/2—‘~100A, Stable 0—‘ 100/0

55 54
27Co 26Fe

Figure 4.5: Decay scheme of 55Ni and the primary contaminant 5400 in the beam
from the A1900 removed by the RFFS.
again as shown in Fig. 4.6b (the poor resolution in the energy loss spectrum was a

result of noise from the EFF S slit motors).

4.3 6 energy spectra

The fl-decay energy spectra for detectors 81-84 are shown in Figures 4.7 through
4.11. The 1-dimensional spectra taken near the beginning of the experiment as the if
was being pulsed on and off is shown in Fig. 4.7. The solid line represents the if on
condition, and the dotted black line represents the if off condition. The dotted line
falls nearly on top of the solid line, which indicates that the rf did not interfere with
the detection of 6 particles or cause the PMTS to behave differently when the 1f was
on versus when it was off. This observation is in agreement with the if on/off source
calibration data discussed in section 3.2.4. The small low energy peak that is visable
around channel 300 in detectors B1 and B4 represents the decay product, 55Co, which
has a Q value of 3.5 MeV (mean 5 energy 570 keV). A majority coincidence register
was added shortly after the experiment started to lower the dead time. The timing
signals from B1 and B2, as well as those from B3 and B4, were taken as a logical
AND. Further, the CFD thresholds on B1 and B4 were raised from -0.7 V to -1.6 V

85

 

Energy loss (arb. units)
§
llIIIUIIIIIIIrIIIIIIIIrIllllll

 

 

 

    

Energy loss (arb. units)
§

 

    
     

250

 

200

 

100 150 300 350

Time of flight (arb. units)

Figure 4.6: Energy loss in silicon detector 1 upstream of the ,B-NMR apparatus as a
function of tof a) before the RFFS was turned on and b) after.

86

 

_ L'fi— n.—

and -1.4 V, respectively, to eliminate the 5500 and other low-energy contamination.
The higher threshold on B1 and B4 was at ~400 channels as shown in Fig. 4.8, which
corresponds to ~440 keV from the energy calibration in section 3.2.4.

Changes in the 5 energy spectra were also checked when the external magnetic
field, Hg, was pulsed on and off at 1000 G (Fig. 4.9). The spectra show no difference
when the external field was on, as compared to when it was off. Again, this observation
is in agreement with the calibration data discussed in section 3.2.4. The particular
spectra shown were taken before the thresholds on B1 and B4 were raised, and thus
the small peak corresponding to 55Co is present in both spectra.

Two-dimensional fl-decay energy spectra were constructed by plotting counts in
the AE detector (B2, B3) as a function of counts in the corresponding E detector
(B1, B4) for both up and down telescopes. The thin AE detectors has a more uniform

response independent of [3 energy, as demonstrated by the calibration in section 3.2.4.

 

 

      
   

 

 

 

 

 

      

 

 

10: B1 _ :3; 1o5
10 104
103 103
102 102
10 - 10
g ( 1000 2000 3000 4000 o 1000 2000 3000 4000
§
U
105 .
104
103
102 .
10 r
1 I I . I I I .
o 1000 2000 3000 4000 c 1000 2000 3000 4000
Channels

Figure 4.7: 55Ni [3 decay energy spectra for thick detectors B1 and B4, and thin
detectors B2 and B3, for both rf on (red) and if off (black).

87

 

 

 

 

 

Counts

 

 

 

 

 

 

 

 

Channels

Figure 4.8: 55Ni ,B-decay energy spectra for thick detectors BI and B4, and thin

detectors B2 and B3, for both if on (red) and 11' off (black). The Bl and B4 thresholds
were raised to reduce low energy contamination.

88

 

 

 

 

 

 

 

 

Counts

 

 

 

 

 

 

Channels

Figure 4.9: 55Ni fl-decay energy spectra for thick detectors BI and B4, and thin
detectors 82 and B3, for both external magnetic field H0 on at 0.45 T (red) and H0
off (black).

89

 

 

 

E

 

 

 

13ml
2 I %
250° :
E ' 5
s i; f;
N
£13400! m400
.0.
i
i . _.
o o 200 400 600 300 10000

Bl channels B4 channels

Figure 4.10: Two-dimensional 55Ni B—decay energy spectra plotted as AE versus E
detector. For these spectra, the external field was held constant at 0.45 T, the rf was
off, and the threshold on BI and B4 was -0.7 V.

All B particles deposit the same amount of energy, more or less, as they travel through
the thin detector because less scattering occurs. The thick E detector has more of an
energy—dependent response since more of the B particles will come to rest in this
detector. The 2.0 cm thickness of the E detector is only sufficient to stop B particles
with a maximum energy of about 4 MeV. Only a fraction of B particles were stopped in
the thick detector due to the high Q value of 55 Ni (Q value of 8.7 MeV, mean B energy
3.6 MeV) and the scattering and energy loss properties of the B particles. Shown in
Fig. 4.10 are the 2—D telescope spectra for both B2 versus B1 and B3 versus B4. The
2-D spectra constructed after the installment of the MG coincidence condition and
with higher thresholds on B1 and B4 are shown in Fig. 4.11.

An example of a background run is shown in Fig. 4.12. A small fraction of high
energy cosmic rays were observed. Also, some 5500 (Q value 3.5 MeV, mean B energy
570 keV) remains as background due to its half life of 17 hours, and the fact that
the threshold on B1 and B4 was raised to only ~100 channels on B1 and B4, which

90

 

82 channels
B3 channels

   

.' 'ufiaivm " ' " h.-
0 200 400 600 800 1000

Bl channels B4 channels

Figure 4.11: Two-dimensional 55Ni B—decay energy spectra plotted as AE versus E
detector. The external magnetic field was held constant at 0.45 T and the if was off.
Thresholds on B1 and B4 were raised to -1.6 V and -1.4 V, respectively, in order to
reduce low enery contamination.

corresponds to ~440 keV from the enery calibration in section 3.2.4. At this enery
threshold, the majority of the B particles from 5500 were removed as the two strongest
B particles have mean energies of 436 keV (26%) and 649 keV. (46%). However, the
highest enery particles constitute only a small fraction of the continuous B enery
distribution. Additionally, 53Fe and 52Mn are present after the RFFS, and have Q
values of 3.7 MeV (mean B enery 1.1 MeV) and 4.7 MeV (mean B enery 1.2 MeV),
respectively. 53Fe and 52Mn also contribute to the background spectra. The enery

cut taken on the 2-D enery spectra to determine the double ratio did not include

the low enery background.

4.4 Spin polarization measurement

An important first step of the 55N i magnetic moment measurement was to optimize
the spin polarization of the secondary beam. The figure of merit for an NMR mea-

surement is P2Y, where P is the spin polarization and Y is yield. Spin polarization

91

 

 

1000— 30

i-_ 25
600—
.02 fl ' 20
g 2 We
r: C -
(0 IO
.5 -5 i 15
m i'
8 m 400-

 

 

 

 

 

   

Bl channels 84 channels
Figure 4.12: Two-dimensional background spectra. The external magnetic field was
held constant at 0.45 T and the if was off.
measurements were made at Ap/sz and :l:1% relative to the fragment momentum
distribution peak to optimize PZY and identify the best conditions for the magnetic
moment measurement.

The spin polarization was determined for 55 Ni fragments produced from bom-
barding 58Ni on a Be target. The polarization was deduced from the pulsed magnetic
field method, where in this application the external magnetic field was set at 1000 G
when on, and the pulse duration was 60 s. The B asymmetry was determined from
the number of counts in the up and down detectors using Eq. 2.11. The measurement
was completed at a 2° primary beam angle to break the symmetry of the fragmen-
tation reaction and realize spin polarization. A normalization run was also taken at
0°, as discussed in section 2.3. At the momentum corresponding to the peak yield of
55N i, three separate spin polarization measurements were completed at both 2° and
0° at different times throughout the experiment. The deduced spin polarization as a
function of the experimental run time is shown in Fig. 4.13. The two spin polariza-

tion measurements at Ap/ p = :l:1% were completed near the time of the final central

92

 

 

. Data taken at 0.14%
0-03 relative fragment momentum

— Weighted average of data

 

 

 

 

11me relative to first measurement (hours)

Figure 4.13: Spin polarization of 55Ni plotted as a function of time relative to the
first measurement. Data were taken near the central fragment momentum.

 

0.04
0.03
0.02
0.01

an

< 0
-0.01 “—l.—

-002
003

004w
-3 -2 -1 o 1 2 3

Relative momentum (% of peak)

 

 

IIIY WIHIUIIIIIITIIIIII

 

 

Figure 4.14: Spin polarization of 55Ni plotted as a function of percent momentum
relative to the peak of the yield distribution.

momentum measurement. A weighted average was taken of the three points at the

central momentum to obtain the final spin polarization curve, as shown in Fig. 4.14.

4.5 N MR measurement

The maximum polarization was observed for 55Ni fragments at the peak of the yield

distribution. Therefore, the B—NMR measurement was completed under these condi-

93

 

 

  
 

 

[Li H 1.] Tl

 

 

 

-o.oos ‘-
-o.o1; .
- --~ I Experiment
: I — Zero band
'°'°1° 600 000 1000 1200 1400 1600
Frequency (kHz)

Figure 4.15: Asymmetry AflP as a function of applied frequency. Data taken with
FM=:l: 25 kHz is represented by the solid squares with the weighted average of base-
line data represented by the gray band.
tions. The primary beam angle was maintained at 2° to break the symmetry of the
fragmentation reaction and observe spin polarization. The rf was pulsed on and off
every 30 s in the region of 605 kHz to 1455 kHz in steps of 50 kHz. H0 was held
constant at 4490.5 G. The FM was :l:25 kHz, and the if sweep time was 10 ms (see
Fig. 3.18). The H1 field produced under these conditions was ~5 G. Data was taken
at each frequency in Fig. 4.15 for 30 min, and three scans were performed for a total
of 90 minutes per frequency point. A resonance was observed at 955 kHz and was
found 3.50 below the weighted average of the other baseline data points. The per—
mutation calculation based on Gaussian statistics gives a probability of 0.83% for a
random deviation of at least 3.50. Further, the confidence interval for the mean of the
baseline was determined, and compared to the statistical error in AflP at 955 kHz.
At the 95% confidence level, the 955 kHz point lies 30' from the baseline.

Prior to the B-N MR scan shown in Fig. 4.15, a new technique was attempted to test
the capabilities of the if box. As discussed in Chapter 2, the new rf system allowed

for the fast, sequential scan of multiple frequencies. Using a frequency modulation of

94

 

  

 

 

 

 

 

0.01 L
0.005 :— .....
0. I
<1?Q 0 _
E FITI‘I-I-é'imm
4.005 :- E l l I-I-e
- i 3
0.01 '__ I - FM :l:25 kHz
I ""' A FMeff 1:150 kHz
I l — Zero band
.0 015 L l I n .l 1 I 1 a n L n n l I a 1 1 l n n n l
' 800 800 1000 1200 1400 1600
Frequency (kHz)

Figure 4.16: Asymmetry ABP as a function of applied frequency. Data taken with
FM=:l: 25 kHz is represented by the solid squares with the weighted average of base-
line data represented by the gray band. The multiple frequency scan that used three
sequential frequencies of FM=:l: 50 kHz each is represented by the open triangles
with dashed error bars.

:l:50 kHz, with three sequential frequencies from three different function generators, a
frequency region spanning 300 kHz could effectively be monitored. For example, the
first scan region included the three frequencies 630, 730, and 830 kHz with a PM of
:i: 50 kHz each. The 1f sweep time was still 10 ms, but each frequency was applied
for 55 ms in sequence. This sequential application was performed for 30 s, then the
rf was off for 30 s, and the cycle repeated. Thus, the frequency region 580-880 kHz
was scanned in 120 min. The wide modulation scan for the full region 580-1480 kHz
is presented in Fig. 4.16. However, the point that covers the identified resonance with
the effective :l:150 kHz PM did not show the same magnitude of asymmetry as the
21:25 kHz resonance point. It may be because the if condition was not exactly the
same in both measurements. First of all, the frequency modulation was different; one
was i25 kHz and the other was three points each of :l:50 kHz. The rf sweep time
was the same for both measurements at 10 ms. The wider FM of 21:50 kHz required

an H1 of 6 G at the resonance point, according to Eq. 3.2, while an PM of 21:25 kHz

95

only required 4 G. Such difference was accounted for by using an H1 of 8 G for the
i50 kHz scan and 5 G for the 21:25 kHZ scan, but there may have been a problem
when multiple frequencies were introduced. Further, the statistics on the wide FM
data are lower than that for the narrow FM data due to a lower beam intensity at
the time the wide FM data was collected. This experiment was the first time the
multiple frequency scan technique was attempted for an NMR measurement, and the
technique may need more testing before it is fully understood.

The resonance at UL = 955 kHz with FM: i25 kHz was used to deduce the
corresponding 9 factor as |g| = 0.279 :i: 0.007. The magnetic moment was further
extracted as p = gI, with I = 7/ 2 for the 55Ni ground state [58]. The final result is

Ip,(55Nl)| = (0.976 i 0.026)pN.

The uncertainty on p was evaluated from the width of the FM. The p was not cor—
rected for the chemical shift due to the interaction of 55 Ni with electrons in the lattice,
which is not known, but assumed to be small compared to the error on the present
result. The sign of g and thus 11 cannot be determined directly from the measurement.
However, it was assumed negative based on theoretical considerations for a neutron

hole in the 1 f7 /2 shell.

96

 

Chapter 5

Discussion

5.1 Polarization of 55N i compared to simulation

In Chapter 2, the development of a Monte Carlo code that simulates spin polar-
ization produced in nucleon removal and pickup reactions at intermediate energies
was described. The original simulation as described in Ref. [37] was revised to im-
prove the quantitative agreement with experiment [41, 59]. Simulations of the 58N i
fragmentation reaction to produce 55Ni were performed to test the reliability and
predictive power of the Monte Carlo code. Details regarding the reaction observables

are provided in the following sections.

5.1.1 Momentum distribution reproduction

The Monte Carlo simulation was first used to provide predictions to compare the
experimentally-observed momentum distribution to predictions. The momentum of
the outgoing fragment was calculated based on conservation of linear momentum.
The linear momentum (x, y, and 2 components) of the group of removed nucleons

was modeled using a Gaussian distribution centered at zero with a width, 0, given by

97

the Goldhaber formula [60],

 

AF(AP — AF)
(AP _ 1) (5.1)

 

0:00

where A F is the fragment mass, A p is the projectile mass, and 00 is the reduced
width. The reduced width is related to the Fermi momentum of the nucleon motion
inside the projectile 08 = 17%. mm / 5. The 00,835,” deduced from experimental distribu-
tion variances have been observed to depend on the mass number of the fragmenting
projectile nucleus, with a weak dependence on the mass number of the target nucleus
and kinetic enery of the projectile [61]. Therefore, a subsequent phenomenological
parametrization was used to determine the reduced width for 55N i. The parametrizar
tion considers dependence on fragment mass, target mass, and incident projectile
enery, and is applicable over a wide range of masses from A p = 12 — 200. The re-
duced width was shown to have a linear dependence as a function of A p. The reduced

width was calculated as

E 2A
00,6xpt = (1 1‘ 4Tlcab) (70 + —3P) (5.2)

 

where Tlab is the beam enery in MeV/nucleon and EC is the Coulomb enery for

the relevant fragmentation reaction, given by

_ 1.114sz17

. 5.3
Tp+TT ( )

C

In Eq. 5.3, Z P,T are the projectile and target charge numbers, respectively, and T‘P’T

are the uniform distribution nuclear radii given by

7'P,T = M(TP,T)rmsi (5-4)

98

F *‘ elm-vvw

 

9000
93e(58Ni,55Ni) I Data

3000 _ Simulation

7000 I—.
6000
5000

4000

Rate [ppslpnA]

3000

2000

—.

 

1000 l I I 1 l
-o.03 ' -o.02 -0.01 0 0.01 0.02 ‘ 0.03
Relative momentum (% of peak)

Figure 5.1: Simulated momentum distribution compared to data for the reaction

9Be(58Ni,55Ni) at 160 MeV/nucleon. The red squares represent the data and the
blue line represents the results of the simulation.

where the nuclear rms radii are taken from electron scattering measurements [62]. For
a 160 MeV/ nucleon 58Ni beam on a 9Be target,

00 = 112 MeV/c. (5.5)

With this reduced width, the simulation yielded a momentum distribution in good

agreement with experiment, as shown in Fig. 5.1.

5. 1 .2 Optical Potential

The real part of the optical model potential, required to calculate the nucleus-nucleus
interaction, V0, is an input parameter for the mean deflection angle, a parameter
of the spin polarization simulation. The deflection angle 0 (see Fig. 5.2) for a single
interaction is given by

6 = 7r — 2d), (5.6)

99

 

 

 

 

b
at I

Figure 5.2: Variable definitions for mean deflection angle calculation.

with

 

 

q, ___ /°° M; , (5.7)

rmin r2\/1_ 32 _ 91;)
In Eq. 5.7, b is the impact parameter, r is the distance between the centers of the
two interacting objects, U (r) is the potential governing the interaction of the two
objects, rm," is the separation between the centers of the two point-like objects at

the distance of closest approach and the enery, E, is given by

I
E = §mvgm (5.8)

where 1200 is the velocity of the projectile at r = 00 [63].

The projectile is assumed to move away from the target after the scattering event
with momentum equal to the incident momentum, thus E0200) = E(v,-,,c,-de,,t). Eq.
5.7 is general for any spherically symmetric potential.

The potential U (r) is defined as

U“) = UCoulomb(T) + Unuclear(r)- (5'9)

The Coulomb part of the potential is repulsive and is given in Eq. 5.3. The nuclear

100

 

 

 

part of the potential is taken to be the real part of the optical model [64], and is

attractive:
—V0

= 1+ c(r—R)/a' (5.10)

 

Unuclear (7‘)

Here V0 is the depth of the optical model potential, R = 1.2( w + W) where Ap
and At are the masses of the projectile and target respectively, and a is a measure
of the diffuseness of the nuclear surface. V0 and a are parameters fit to experimental
data. There are very limited nucleus-nucleus scattering data available in the litera-
ture, and an exact determination or parametrization of V0 is difficult for any given
projectile-target combination. Typically this is not a problem because in head-on col-
lisions, the nuclear potential does not have a large influence. However, the treatment
of peripheral collisions depends strongly on the optical potential. In the minimum,
a determination of V0 is needed. A parametrization of V0 based on enery and/or
number of nucleons removed would suffice, but unfortunately, such a parametrization
does not presently exist.

In the work described in the following sections, V0 was determined with a folding
model calculation [65]. The model was chosen because it reproduces experimental
scattering data for heavy ions in the enery range of interest. The folding calculation
yields the real part of the optical potential (V0) as a function of the internuclear
radius, the distance between the center of the projectile and target. The internuclear
radius was calculated in the simulation code, based on the relations by Gosset et
al. [66]. For 58Ni at 160 MeV/nucleon on a 9Be target, Khoa calculated the optical
potential for a three nucleon removal reaction to be V0 = 41 MeV which corresponds
to a mean deflection angle of 6def = 0.049. A renormalization of the real folded
potential is usually assumed to account for higher-order effects, with a renormalization
coeflicient N = 1 i 0.2 multiplied by the potential. In the case of the 9Be(58N i,55N i)
reaction, a normalization coefficient of 1.1 (V0 = 45 MeV) was shown to have the best

agreement with data.

101

 

 

0.02
0.01 5
0.01
0.005

 

 

 

Polarization
O

-0.005
-0.01
-0.015

 

-0.02
-0.025

' Experiment
— Simulation

 

IIIIIIIIIIII'IIIIIIITIIIIII llll'llll'l'llllllll

I I l Ll l l l l l [JJlklll I I [ll 0 IL

I -0.02 -0.01 0 0.01 0.02 0.03
(p-pollpo

p

 

 

_b
O
{A

Figure 5.3: Spin polarization as a function fragment momentum p relative to the peak

of the yield distribution pa for the three neutron removal reaction 9Be(58Ni,55Ni) (160
MeV/ nucleon). The red squares are the experimental data points and the grey band
represents the range of the Monte Carlo simulation results within a 10 distribution.
The input parameters used in the simulation are given in Table 5.1.

5.1.3 Results of simulation

The spin polarization measurement for the reaction 9Be(58Ni,55 Ni) at 160 MeV / nucleon
(see Fig. 4.14) is shown along with simulation results in Figure 5.3. The parameters
used in the simulation are given in Table 5.1.

A value of A3 = 0.885 was used to extract polarization for the 55Ni analysis.
Calculation of Ag as outlined in Appendix A gives two values, A5 = +0885 or
A5 = —0.747 depending on the sign of the mixing ratio p, which is not experimen-
tally kmown. The polarization simulation predicts negative polarization at the peak
of the yield distribution. A positive value of AB is needed for the polarization mea-
surement to have the same sign as simulation. The sign of the Gamow-Teller matrix
element should be determined to confirm this assignment of A5. A negative spin

polarization is expected for the three neutron removal reaction based on the previ-

102

Table 5.1: Input parameters used in the Monte Carlo simulation to model the spin

polarization of the nucleon removal reaction 9Be(58Ni,5‘r’Ni), and the nucleon pickup
reactions 9Be(3°Ar,37K) and 9Be(3°S,34Al).

 

 

 

 

 

 

Parameter 9Be(58Ni,55Ni) 9Be(36Ar,37K) 9Be(368,34A1)
A, Z of projectile 58, 28 36, 18 36, 16
A, Z of target 9, 4 9, 4 9, 4
Incident enery (MeV/ nucleon) 160 150 77.5
Distance of closest approach (fm) 5.47 5.44 5.40
Number of events 500000 500000 500000
Angular acceptance (deg) 2 :l: 2.5 2 :i: 2.5 2 :i: 1
Optical potential (MeV) 45 29 32
Mean deflection angle (rad) 0.014 -0.07 -0.49

 

 

ous considerations of conservation of linear momentum. Recall that the definition of

 

polarization is dependent on lz/[L|. [L] = «Lg. + L3 + L2 is a positive value and
I; = —Xky + Ykz. At the peak of the momentum distribution, the fragment momen—
tum is zero, and thus Icy = 0. The fragments accepted into the A1900, as shown in
Fig. 3.2, had an :c-component of linear momentum that was negative. Therefore, the
m-momentum of the removed nucleons, km is positive. As discussed above, Y < 0 for

non-uniform removal as shown in Fig. 2.2; therefore, I; and P must be negative.

5.1.4 Extension to nucleon pickup reactions

A complete quantitative treatment of intermediate enery reactions is important to
the success of the spin polarization simulation code. In addition to nucleon removal
reactions, nucleon pickup reactions at intermediate energies provide a means for pro-
ducing spin polarized nuclei away from stability. The spin polarization mechanism
for both nucleon removal and pickup reactions is believed the same. Therefore, the
simulation code was extended to include nucleon pickup, independent of the efforts
reported in Ref. [40].

The pickup process follows the observations of Souliotis et al. [39], in that the
picked-up nucleon has an average momentum equal to the Fermi momentum (230

MeV/c), oriented parallel to the beam direction. The momentum distribution for the

103

one-neutron pickup reaction 27A1(180,190) at 80 MeV/nucleon is shifted below the
momentum / nucleon of the beam, as observed in Ref. [39], in contrast to the observed
shift for nucleon removal products. The simulated position of the centroid agrees
with the calculation of Ref. [39], where a simple model based on momentum conser-
vation was used (see Fig. 5.4). The agreement demonstrates that angular momenta
considerations are employed correctly in the Monte Carlo code modified for nucleon
pickup.

The width of the momentum distribution is observed experimentally to be small

(around 20 MeV/c), while it is calculated to be zero. The GE from Goldhaber [60] is

2 02 A10190419 - APF)

where A p F = A F — AAt is the mass of the projectile part of the final product and
AAt is the number of nucleons picked up from the target. As discussed in section
5.1.1, the parameter 00 is the reduced width, and is related to the Fermi momentum
of the nucleon motion inside the projectile: 03 = 12%. (mm. / 5. Eq. 5.11 assumes that the
nucleon is picked up from the target with a fixed momentum and direction, and the
picked-up nucleon makes no contribution to the width. Thus, for any pure nucleon
pickup process, A p = A p F and the parallel width is zero. To model the experimental
observations of Ref. [39], a parallel width of 0‘" = 20 MeV/c was used. In addition to
the parallel width, Van Bibber et al. [67] showed that in heavy-fragment studies in
the 100 MeV / nucleon region, the projectile is subject to an orbital deflection due to
its interaction with the target nucleus before fragmentation takes place. The orbital
deflection gives an additional dispersion of the transverse momentum, as given in the

expression:

A (AP — A ) A (A — 1)

2 2 PF PF 2 PF PF

—_ + . .12
0"” 01 AP -1 02 AP(AP — l) (5 )

 

 

The first term in Eq. 5.12 was defined previously (Eq. 5.11, where 00 is replaced

by 01), and the second term contains 0%, the variance of the transverse momentum

104

 

 

Counts (Arb. Units)

 

TIIIIIHIIIIIIIIllllllll'l'lll

 

 

jllllllllll
372 374 378 378 380 382 384 388 388 390 392

Fragment Momentum (MeV/c per nucleon)

Figure 5.4: Parallel momentum/ nucleon distribution calculated with the simulation

code for the reaction 27Al(180,190) at 80 MeV/nucleon. The red squares are the
data [39] and the black line represents the simulation results. The arrow corresponds
to the momentum/ nucleon of the beam. The simulated momentum distribution has
been scaled by the ratio observed in Ref. [39] of experimental centroid / calculated
centroid (0.969/ 0.978), in order to compare to the data.

of the projectile at the time of fragmentation (200 MeV/c as used in Ref. [67]). A
comparison of the simulated momentum distribution is shown in Fig. 5.4 with the
data taken from Ref. [39].

The simulation results for one-nucleon pickup processes discussed in the literature
are shown in Fig. 5.5 and 5.6. Souliotis et al. [39] used the “typical” Fermi momentum
ppermi=230 MeV/c in the momentum distribution calculation. ppenni was calculated
here based on data taken from Moniz et al. [68]. The ppermi ranges from 170 MeV/c
for the lightest targets to 260 MeV/c for heavier targets. The results of the simulation
for a proton pickup 9Be(3°Ar,37K)X, first observed by Groh et al. [38], are given in
Fig. 5.5. The parameters of the simulation are listed in Table 5.1. The momentum
matching conditions [69] for simple surface-to—surface pickup are best met for the
two data points on the high momentum side of the yield distribution, where the
simulation agrees with the data. On the low momentum side of the peak of the

yield curve, the picked-up nucleon has a momentum less than the Fermi momentum,

105

and the momentum matching conditions for direct pickup are poorly satisfied. More
complex transfer mechanisms are therefore required to describe the polarization on

the low momentum side [38].

 

 

 

 

 

 

 

 

0.3 :—
:
0.2 _—
c L.
.9 0.1 _
4.0 n
m -
.g -
9 ° 1"
o b
0- Z
-o.1 —
l:
-02 :—
h l I I I I l I I I I I I L .I. l I I_ I l
-0.01 0.005 0 0.005 0.01
(p-po)/po

Figure 5.5: Polarization as a function of fragment momentum p relative to the pri-

mary beam momentum'po for the one-proton pickup reaction 9Be(3°Ar,37K) (150
MeV/ nucleon). The red squares are the experimental data points from Ref. [38] and
the grey band represents the range of simulation results within a la distribution.

The simulation code was also used to model data from a neutron pickup reaction,
9Be(3°S,34Al) at 77.5 MeV/nucleon obtained in Ref. [40], as shown in Fig. 5.6. Again,
the simulation parameters are given in Table 5.1. These data were reproduced by an
independent simulation of the nucleon pickup in Ref. [40], but required a scaling factor
of 0.25. No scaling factor was applied in the results presented here to reproduce the

polarization from neutron nor proton pickup reactions.

.
C
. -
7..
;1
. .
. .
..
b.-
a
..
O
C
O
-
- .
.'.,
.
..
.:
lb
';
. ;.5
I ‘.
.
7-
,.
2‘.
-.
.
r
..
Y
4.-
...
‘3
. ._
.
:t
...

J. 1
In...
9v"

106

 

n ' 1 ‘ _ ‘
an. mayday???

 

l
I
i .'
”'5" I
I
l
l
l
1
fl '

0.4

0.2

 

IIFIITIIII—FIIII

 

 

 

Polarization

-0.2

-0.4

 

 

 

lll'llllIIl
O

 

‘ I ll lllllllllllllll
'°'° -0.02 0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

(p'po)/p0

Figure 5.6: Polarization as a function of fragment momentum p relative to the mo-
mentum at the peak of the yield distribution pa for the one-neutron pickup reaction
9Be(3°S,34A1) (77.5 MeV/ nucleon). The red squares are the experimental data points
and the blue dashed line is the previous simulation result, both from Ref. [40]. The
grey band represents the range of the present simulation results within a 10' distribu-
tion.

5.2 Magnetic Moment of 55Ni and the 56Ni closed

shell

As given in section 4.5, the magnetic moment of 55Ni was deduced as
p(5°Ni) = (—0.976 i 0.026)/1N.

The new p(55Ni) is compared below to theoretical predictions. The starting point for
the discussion is a simple single-particle wavefunction, where p is then corrected with
an effective operator. The discussion is then expanded to consider a more sophisticated

wavefunction for the 55N i ground state.

107

 

 

 

 

5.2.1 Single-particle wavefunction and effective 9 factors

The new p(°5N i) was first compared to the results of a calculation that used a simple
form of the wavefunction, where 56Ni was assumed to be an inert closed core. The

magnetic moment operator was described in Refs. [5, 6] as:

fieff = gl,eif<l) + gs,eff<3) '1' 9p,eff<lY213l>1 (5'13)

where 93,63 = 9;; + 691-, with a: = l, s, or p, and 9,, denotes a tensor term. Here 9,; is
the free nucleon 9 factor gfree (gs = 5.586, g) = 1 for proton and ya = —3.826, g; = 0
for neutron) and 6gx the correction to it. s and I represent spin and orbital angular
momentum, respectively. The results of the calculation for both a single proton (5°Co)
and single neutron (55Ni) configuration in the 1 f7 /2 shell are shown in Table 5.2.
Details of the calculation and individual corrections can be found in [5-7], and the
corrections were discussed in section 1.2.2. Starting from the single-particle values for
11(55Ni) and 11(5500), the CP corrections overcorrect experimental values (see Fig.
5.7), but the MEC restore the theoretical prediction close to the experimental values.
The isobars and relativistic effects have only small contributions to the correction.
perturbation

The simple theoretical model, labeled as geff , reproduces the experimental

values for 55Ni and the mirror partner 55Co well, as shown in Table 5.2.

5.2.2 Shell model in full fp shell and gfree

Another theoretical approach was taken using a complex wavefunction in a shell
model calculation to gain more insight on the details of the 56Ni core. The shell
model calculation was performed in the full fp shell with the effective interaction
GXPF1 [4], where 40Ca was assumed to be an inert closed core. Here, the 5°Ni
core is soft as the probability of the lowest order closed-shell 7r(1 f7 /2)81/( 1 f7 ”)8
configuration in the ground-state wavefunction is ~60%. The magnetic moment can be

calculated from gfree with a form of the magnetic moment operator [1' = gs (s)+gl (I). In

108

Table 5.2: Contributions to the calculated effective magnetic moment operator for a
1f7/2 neutron in 55Ni and a lf7/2 proton in 55Co.

 

 

 

 

 

Neutron 1f7/2 (55Ni) Proton 1f7/2 (55Co)

91 93 .91) ll .91 93 9p fl
CPa 0.185 1.933 3.339 1.744 -O.183 -2.188 -3.892 -1905
MECb -0.245 -O.614 -O.368 -1.066 0.270 0.693 0.340 1.181
Isobars 0.010 0.288 -O.889 0.117 -0010 -O.288 0.888 -0.117
Relativistic 0.000 0.093 0.000 0.046 -0024 -0151 -0040 -0150
jg“ all °°”°°' -0.049 1.701 2.082 0.841 0.052 -1935 -2704 -0.990
Elgfpmmle 0.000 -3.826 0.000 -1913 1.000 5.587 0.000 5.794
single-particle
value + correc- -0049 2125 2.082 -1072 1.052 3.652 -2704 4.804
tions

 

 

 

“contains both random phase approximation (EPA) and second-order effects
(CP(2nd)).

contains meson exchange corrections as well as a core-polarization correction to the
two-body MEC operator (MEC-OP).

 

 

 

  
 

2 6
1.51 5.5
1 /\.\‘— = 5
A 0.5 .4. "z
5 145500 5 i
-’= o -4 8
m2 i- =31 Exptfiriment ‘1?)
L0 ' V
‘5. -O.5 (wr error) . 3'5 1
-1 we “‘3... . 3
-1.51 2.5
-2 2

 

 

 

Single—particle cp
value

MEC ISObarS Relativistic Experiment

Figure 5.7: Running sum of Towner corrections to the single-particle magnetic moment
for 55Ni (blue diamonds) and 55Co (pink squares).

109

Table 5.3: Magnetic moments of 55Ni,55Co and the isoscalar spin expectation values
of the mass A = 55 system.

 

 

 

 

 

 

 

 

 

#(55Ni)#N 745500) #N (2 dz)

Experiment -0.976 :l: 0.026 4.822 :1: 0.003 [25] 0.91 :i: 0.07
Single-particle value -1.913 5.792 1.00

65mm“ -1072 4.803 0.61
full [7) ghee -0.809 4.629 0.84
full fp ggoments -0999 4.744 0.65
full fp 93,, fit -1071 4.926 0.94
full fp 93;; fit without

-1129 4.868 0.63

isoscalar 6gfd fit term

 

general, good agreement is realized by this treatment for N ~ Z nuclei over the range
A = 47 - 72. The shell model calculation gives p(55Ni)=-0.809 p N with ghee, which
is in fair agreement with the present result as compared with other it calculations in
Ref. [4]. The observed agreement supports the softness of the 5°Ni core. Similar results
were obtained for the probability of the 7r(1f7/2)8u(1f7/2)8 closed shell component
in the wavefunction from a separate shell model calculation [31] that explained the

discrepancy between the systematics of E(2f) and that of B (E2; 0? -» 2?) for 5°Ni.

5.2.3 Shell model in full fp shell and gas"

Effective nucleon 9 factors, ggffments, may be employed in the previously discussed
full fp shell model calculation for better agreement. The ggf‘fments were derived em-
pirically by the least-square fit of the magnetic moment operator to experimental
11(57’65'67Ni) and p(°2"°8i7OZn) [4]. The values 933 = 0.993%, 9,133 = 1.1 and -0.1 for
protons and neutrons, respectively, were obtained. The resulting magnetic moment,
B(55Ni)=-0.999 pN, gives good agreement with the experimental value. The results
of the theoretical calculations are summarized in Table 5.3. It is noted that all of the
theoretical calculations give good agreement with the experimental value, and within

the accuracy of nuclear structure models, there is not a significant difference between

the result of the calculations for p.

110

5.2.4 Isoscalar spin expectation value at T = 1 / 2, A = 55

The known value 11(55C0)= 4.822 :i: 0.003 MN [25] was combined with the present
result for B(55Ni) to extract (2 oz) for the mirror pair at A = 55. Using Eq. (1.44),

(:03) = 0.91s 0.07

was obtained. A peculiar feature is noted in Table 5.3 between calculated )1 and (2 oz)
for A = 55. The shell model calculation with gfree reasonably reproduces the (Z 02),
although the agreement with p is only fair. However, the calculation considering
ggf‘fments gives good agreement for p, but does not agree with experimental (2 0;).
Such discrepancy was already noted in the 3d shell, and can be explained by examining
the isovector and isoscalar components of the M1 operator separately [70,71]. The
magnetic moment is dominated by the isovector term due to the opposite signs and
nearly equal magnitude of the neutron and proton magnetic moments, whereas (2 oz)
is an isoscalar quantity.

The effective 9 factors for the A = 28 system obtained from a fit to isoscalar
magnetic moments, isovector moments, and M1 decay matrix elements [71], 92% fit,
were applied to matrix elements for A = 55 calculated in Ref. [4] with the GXPF1
interaction to see if a similar approach would realize success in the fp shell. This
approach assumes the hole configuration in the 1d5/2 shell for A = 28 is analogous to
that of 1f7/2 for A = 55. Effective 9 factors for A = 28 were obtained as 93" fit = 4.76,
-3.25, gf’d fit = 1.127, -0.089 and (9%)“! fit = 0.41, -0.35 for protons and neutrons,
respectively (9;, = gp/\/8_7r). The calculated B(55Ni)=-1.071 and (2 dz) = 0.935 with
93% fit shows the best agreement with the present result as summarized in Table 5.3.

(2: oz) is known to be quenched relative to the extreme single particle model. Fur-
ther, (Z 02) was shown to be quenched relative to the theoretical (2 0,) calculated
with gfree [70] (dotted line in Fig. 5.8) at the beginning (1d5/2) and the end (1d3/2)

of the A = 17 — 39 region. It is also known that the (Z 0;) around A = 30 are rela-

111

tively well reproduced with gfree, as shown in Fig. 5.8. Optimum M1 operators were
determined semi-empirically for the sd shell nuclei based on the fit to the isoscalar
magnetic moment derived from the sum of the mirror magnetic moments [71]. This
procedure to determine the effective M1 operator can be justified since the effective
operator determined by the magnetic moments (ggflfments is dominated by the large
spin isovector component {g},VMl = (95 — g?) / 2 = 4.706] and thus is not sensitive to
the small isoscalar components, to which (2 0;) is sensitive. Corrections to gfree were
determined for possible pairs of orbits in the 8d shell [71]. The (2: 0;) calculated with
the effective operator better reproduces the experimental result over the 3d shell and
quantitatively reconciles the observed quenching (dashed line in Fig. 5.8). Similarly, in
the fp shell, the (2 oz) for the A = 41 and 43 mirror pairs at the beginning of the f p
shell (1 f7 /2) are quenched relative to values calculated with gfree. The present result
at A = 55 with single hole in the 1f7/2 shell is well reproduced by the (Z az)=0.84
calculated with ghee and is close to the extreme single-particle value. The same trend
can be seen in the sd shell at A = 27 (a hole configuration in the 1d5/2 shell), where
the (Z 02) is well explained by the calculation with gfree and restored close to the
single-particle value relative to neighboring (2 oz).

The 56Ni core could be considered as a good core since (2 oz) for A = 55 is very
close to the single-particle value. However, if the 56N i core is soft as shown from the
satisfactory [1 results from the shell model calculation with the GXPF1 interaction,
then configuration mixing should account for the ~40% of the ground state wavefunc-
tion not attributed to 1r(1f7/2)8V(1f7/2)8. This configuration mixing should appear
as a deviation in (2 oz) from the single-particle value, which was not observed. It
can be shown from the (Z az)=0.628 calculated without isoscalar correction to the
gfd fit, 5113, that a contribution from the large orbital angular momentum (f orbit)
to the 9,“ fit enhances the (Z 0;). The contribution to (Z: 0;) from the large orbital
angular momentum correction cancels the effect from configuration mixing, support-

ing the softness of the 56Ni core and emphasizing the sensitivity of (2 oz) to nuclear

112

 

 

 

 

 

 

 

 

 

 

1 000 - Lilla—M _1d__5/l_ 2.51/2 1f7/2 2p3/2
o e\\\\: .. o l, . Experiment
\.-_ 0‘74 \
.i. 11 1‘ [ ,. - Present
. \... . I I ......
0.500 - \ J {1, 9free
2L 4 -_. from sd fit
""4.) gefi
/\ ii — Single-particle
[3 0.000 - :‘,\ value
V o :i— —%\ .
e... ? ‘.. —
. .
-o.5oo ~ 1p1/2
1d _,
3/2 [
1“5/2
.1000 . . , , . .
0 10 20 30 40 50 60 70
Mass Number

Figure 5.8: Isoscalar spin expectation value for T = 1 / 2 mirror nuclei. The black
diamonds represent previous experimental data while the red square is the present
result. The blue solid line represents a shell model calculation with free nucleon g-
factors. The pink dashed line is the Brown calculation [71] with effective g-factors
that were obtained from a fit to the isoscalar magnetic moment in the 8d shell. The
black horizontal lines are the single-particle values.

structure. Similar enhancement of (Z 02) due to 6115 was found in Fig. 5 of Ref. [71]
for A = 39. The enhancement may be attributed to a large MEC contribution to
6115. Calculations by Arima et al. [72] that included MEC corrections were found to
agree with the empirical value of 6113. However, it is noted that the MEC depends
sensitively on the choice of the meson-nucleon coupling constants (see Ref. [6, 71])
and that calculations by Towner [6] do not show such enhancement, attributed to the
MEC being offset by the relativistic effect. The contribution to (2 0;) from the ten-
sor term ggd fit is small as (Z az)=0.94(0.87) is calculated with(without) the tensor
term. The good agreement between the present result and the (2: 0;) calculated with
92% fit in the 3d shell implies that a universal operator can be applied to both the sd
and fp shells. However, for more detailed discussion, effective M1 operators of the fp
shell nuclei have to be determined from the mirror moments in the fp shell, for which

more experimental data are required.

113

 

 

 

 

 

5 'Data
5? —Linearfit
4?-

37—

C

>9 :
2.—

C
1.—

0:-

-1:'1. I ..I. .11. 1. .14

-4 -3 -2 -1 o 1

7n

Figure 5.9: Nuclear 9 factors of mirror pairs plotted as the odd proton nucleus 9 factor
7,, versus the odd neutron nucleus 9 factor 7", also known as a Buck-Perez plot. The
squares are the experimental data and the solid line is a linear fit to the data.

5.2.5 Buck-Perez analysis

The p(55Ni) result can also be compared to the predictions made by Buck and Perez et
al. based on the systematic linear relationship between ground state 9 factors and the
B-decay transition strengths of mirror nuclei [1 1—13], as introduced in section 1.3.2.
The predicted values are p(55Ni)=-0.872 i 0.081 )1 N based on the dependence of ft
values and [1(55Ni) =-0.945 :t 0.039 p N from the linear trend of experimental 9 factors.
Both predictions agree with the observed ”(5°Ni)=-0.976 :i: 0.026 pN, although the
predictions have large errors.

The experimental 9 factors of the T = 1 / 2 mirror nuclei, including the new A =
55 value, are shown in Fig. 5.9. A linear fit was performed and the new A = 55
value follows the linear trend well. The Buck-Perez extrapolation is a valid prediction
for f p shell nuclei with unknown magnetic moments, an important tool for future

measurements.

114

Chapter 6

Conclusions and Outlook

The magnetic moment of the T = 1 / 2 55Ni nucleus was measured for the first time
with the B—NMR technique. The 55Ni ions were produced at NSCL from a 160
MeV/ nucleon 58Ni beam impinging on a Be target. The resulting secondary beam
was purified using both the A1900 and RF fragment separators. A three neutron re-
moval reaction was employed, yielding a nuclear polarization of [P | ~ 2% at the peak
of the momentum distribution. An N MR resonance was observed at 955 :i: 25 kHz,
with an external magnetic field of 0.4491 :1: 0.0005 T. The deduced magnetic moment
was [11(55N 1)] = (0.976 :l: 0.026) pN. The experimental result agreed with shell model
calculations with the GXPF1 interaction in the full fp shell. Results of the shell model
calculation with free nucleon 9 factors showed reasonable agreement, while effective
9 factors obtained from an empirical fit to neighboring magnetic moments showed
better agreement with experiment. The present [1 supports the softness of the 56Ni
core.

The spin expectation value was extracted together with the known [1(55Co) as
(Z 0;) = 0.91 :i: 0.07. The shell model calculation, with free 9 factors showed reason-
able agreement with (Z 0;) while the effective 9 factors from the empirical fit did
not. The effective 9 factors determined by isoscalar magnetic moments, isovector mo-

ments, and M1 decay matrix elements in the sd shell combined with A = 55 matrix

115

elements are able to explain the present (2 oz). The agreement implies that a univer-
sal operator can be applied to both the sd and fp shells. However, for more detailed
discussion, effective M1 operators of the fp shell nuclei have to be determined from
the mirror moments in the fp shell, for which more experimental data are needed.

Continued studies of magnetic moments of nuclei immediately outside of presumed
doubly-magic cores are important in the ongoing investigation of the resilience of the
magic numbers away from stability. Moving further from stability comes at a cost of
both spin polarization and yield, and for the B—NMR technique the figure of merit is
PZY. The magnitude of spin polarization is expected to decrease as more nucleons
are removed and / or picked up. In addition, the cross sections become lower for the
most exotic nuclei. Greater magnitudes of spin polarization and greater yields are
necessary to optimize the figure of merit P2Y for B—N MR measurements on nuclei
far from stability. A laser polarization beam line is currently being implemented at
NSCL to provide polarized beams by optical pumping. Typically, the magnitude of
spin polarization achieved via optical pumping is much greater than that obtained
from fragmentation reactions.

With the new p(55Ni) result, 57Cu remains the only nucleus d: 1 nucleon away
from 56Ni with a magnetic moment that does not agree with shell model. This leads
to the question of whether the proton outside the 1f7/2 orbit is in some way affecting
the core. The magnetic moments of the T = 1 / 2 nuclei 33V”, gZCr23, ggMn24, géFe25,
and 33C025 are important measurements that would provide insight on the Z = 28
shell closure. The magnetic moments of the mirror partners of 45V, 49Mn, and 51Fe
are known (45Ti, 49Cr, and 51Mn, respectively). Completion of the mirror pair would
allow the spin expectation value for the A = 45,49 and 51 systems to be extracted,
and would provide important information on shell evolution in the fp shell. These
nuclei are difficult to produce at ISOL facilities, due to the chemistry involved in the
extraction. Production of these polarized nuclei via optical pumping at NSCL may

provide an avenue to access these difficult transition metals.

116

Finally, in this dissertation the systematics of only the isospin T = 1 / 2 mirror
nuclei have been discussed. However, there have only been five T = 3/ 2 mirror pairs
measured in the sd shell and none in the fp shell, and systematics have yet to be
established. The magnetic moment of the heaviest bound T7, = -3/ 2 fp shell nucleus
55Cu is another important measurement that not only would contribute to the T =
3/ 2 systematics, but would also provide important information on the Z = 28 shell
closure, as 55Cu is one proton above the 1f7/2 orbit. The magnetic moments of the
Cu isotopes heavier than 57Cu all agree with shell model. It is therefore necessary
to go further from stability within the Cu isotopic chain and determine whether the

Z = 28 shell closure is broken.

117

Appendix A

B—decay Asymmetry Parameter

Calculation

The B—decay asymmetry parameter, AB is given in Ref. [42] by:

= :l:]CA(0)|2/\ - 2CACV(1)(0)\/ J/(J +1)°JJ’
ICV(1)|2 + ICA(0)|2

 

A 3 (A.1)
where CV and C' A are the vector and axial-vector coupling constants, (1) is the Fermi

matrix element, and (a) is the Gamow-Teller matrix element. The :1: refers to Bi

decay, 5JJ’ is the Kronecker delta, and /\ is defined by

l forJ—iJ’=J—1
»\ = 1/(J+ 1) for J——> J’ = J (A2)
—J/(J+ 1) for J—+ J’=J+1.

To simplify the expression A.1, the mixing ratio is defined as:

_ 021(0)
p ‘ 0v<1>’ (A3)

 

118

and A.1 simplifies to

 

:lzpzA — 2p,/J/(J + 1)6 J J,
Afl = 2 . (AA).
1 + p
55Ni B+ decays to 5500 (7/2‘ -—> 7/2‘). Therefore, A = 1/(J + 1) = 2/9 and for
T = 1 / 2 mirror B decays (1)21. The world average for the ratio of coupling constants
C A / CV has been experimentally determined to be —1.2699 :l: 0.0029 [73] from the

B-asymmetry parameter of the free neutron. p then reduces to
p = —1.2699(29)(0). (A5)

The absolute value of the Gamow-Teller matrix element, [(0)] has been experimentally

determined for 55N i as shown in Table A.1.

Table A.1: Experimentally determined values for the Gamow-Teller matrix element

|(0)l-

 

 

 

B(GT) Hail
Reusen et al. [74] 0.466 :1: 0.027 0.538 :1: 0.031“
Ayste et al. [58] 0.508 :i: 0.008
Hornshpk et al. [75] 0.613 :l: 0.017
Weighted mean 0.528 :i: 0.007

 

 

 

 

 

“Extracted from B(GT) = (CA/CV)2(0‘)2.

From Eq. A.5,
p = i0.671(9). (A.6)

Note that the sign of p is determined by the sign of (a), which is not known.
The sign of p has been determined for sd shell T = 1/2 mirror B decays based on
systematics in that shell. In the fp shell, only four mirror T = 1/2 B decays have

been studied, including the A = 55 pair, and such systematics are not established.

119

Solving Eq. A4 for both positive and negative p,

49

(2/9)102 - 21/7799
1 + p2
(2/9)(:t0.671)2 — 2./7/_9(:i:0.671)
1 + (:i:0.671)2

—0.747 i 0.003 for +p (A.7)

+0885 :t 0.005 for -p. (A8)

120

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125

 

   

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