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THESI‘ um“, ,

Michigan State
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This is to certify that the

dissertation entitled

Experimental Studies of Isobaric Quintets
presented by
Arno G. Ledebuhr
has been accepted towards fulfillment

of the requirements for

Ph 1D - degree in Phys 1C5

Fffl/Zobexm

 

Major professor

Date 24 1982

MS U is an Affirmatiw Action/Equal Opportunity Institution 0-12771

 

 

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EXPERIMENTAL STUDIES OF ISOBARIC QUINTETS

By

Arno George Ledebuhr

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

DOCTOR OF PHILOSOPHY

Department of Physics

1982

ABSTRACT

EXPERIMENTAL STUDIES OF ISOBARIC QUINTETS

BY

Arno George Ledebuhr

The mass excesses of the lowest T=2 states in the
nuclei 2"Al, 8Be, and 8Li have been determined.

The highly proton—rich nucleus 2"Si has been produced
via the 2"Mg(3He,3n) reaction. A cryogenic helium jet and
a recoil time-of-flight mass analyzer system were constructed
for use in observing short-lived beta-delayed particle
emitters. The half-life of 2"‘31 was found to be 103(42) ms,
and the energy of the protons de-exciting the T=2 state in
the daughter, 2"A1, has been measured as 3912.7(37) keV.
From detailed consideration of masses in the A=24 completed
isobaric quintet, it is concluded that this quintet con-
stitutes a test of the quadratic isobaric multiplet mass
equation (IMME) as precise as the mass 9 quartet and that
there is, in this case, no significant departure from the
equation.

Measurements of the Q values for the reactions
1°Be(p,t)°Be(T=2) and 1°Be(p,3He)°Li(T=2) have been carried

out using an Enge split-pole magnetic spectrograph. Data

were recorded on photographic plates and analyzed using
the MSU plate-scanning system. The Q values determined
were -27484.3(14) keV and -26821.3(57) keV for the 8Be and
8Li experiments respectively. These measurements have
removed the deviation from the quadratic IMME that existed
previous to these results.

An analysis of all A=4n isobaric quintets (five com-
plete) for A544 has been carried out. Predictions for
missing members have been made using the quadratic IMME.

In no case has any significant deviation from the quadratic
form of the isobaric multiplet mass equation been found.
This leaves only the mass 9 ground state quartet where a
known deviation is present and where sources of significant

experimental error have probably been eliminated.

To Sharon,
with all my love

ii

ACKNOWLEDGMENTS

I would first like to thank my advisor, Dr. R.G.H.
(Hamish) Robertson for suggesting this thesis project and
for helping and guiding me during my graduate school career.
Hamish has been a real friend and mentor.

I would also like to thank all the staff at the Cyclo-
tron Laboratory with whom I have worked -— both past and
present. They have made working (and sometimes living) at
the lab more enjoyable for me.

To my fellow graduate students -— my nuclear beer
companions -— I would like to say that your support, our
friendship and good times will always be remembered.

Lastly, I would like to thank my wife, Sharon, for her
love and companionship, for her patience through the many

lonely nights and for all her help in preparing this thesis.

iii

TABLE OF CONTENTS

LIST OF TABLES O O O O O O O O O O O O O O O O O 0

LIST OF FIGURES. . . . . . . . . . . . . . . . . .

INTRODUCTION 0 O O O O O O O O O O O O O O O O O 0

CHAPTER ONE

CHAPTER

2.1

2.8
CHAPTER
3.1
3.2
3.3
3.4

Isotopic spin and charge-independence . .
Derivation of the IMME. . . . . . . . . .
Violations of the IMME. . . . . . . . . .
TWO

Introduction. . . . . . . . . . . . . . .
Beamline and beam transport . . . . . . .
Helium jet target chamber . . . . . . . .
Target holder and target. . . . . . . . .
Capillary-skimmer cone interface. . . . .
Foil wheel and particle detection system.
Experimental results and analysis . . . .
2"Si half-life measurement. . . . . . . .
THREE

Introduction. . . . . . . . . . . . . . .
Experimental procedure. . . . . . . . . .
Analysis. . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . .

iv

Page
vi

viii

19
21
24
28
28
31
39

56

61
67
71
81

CHAPTER FOUR

4.1
4.2
4.3
4.4
4.5
APPENDIX
A.1

A.2

A.3

APPENDIX

APPENDIX

A=24 quintet. . . . . . . . . . . . . . .
=8 quintet . . . . . . . . . . . . . . .
Quintet summary . . . . . . . . . . . . .

The D and c coefficients. . . . . . . . .
Conclusions . . . . . . . . . . . . . . .

A

Proton and alpha recoil masses. . . . . .

Beta-decay broadening of the recoil mass
distribution. 0 O O O O O I O O O O O O O

The effect of beta-recoil on the proton
energy. 0 O O O O O O O O O O O O O O O O

B O O O O O O I I O O O O I O O O O O O O

C O O O O O O O O O O O O O O O O O O O 0

REFERENCES 0 O O O O O O O O I O O O I O O O O O O

Page

91
98
100
122

129

131

133

138
142
146
154

LIST OF TABLES

Calipration excitation energies for the 8Li
and Be experiments. . . . . . . . . . . . . .

1"N excitation energies. . . . . . . . . . . .
1977 and 1981 mass excesses. . . . . . . . . .
1977 and 1981 Q values . . . . . . . . . . . .

Contributing uncertainties to the Q value
measurements 0 O O O O O O O O O O O O O O O 0

Reaction 0 values used in the local mass
adjustment 0 O O O O O O O O O O O O O O O O 0

Mass excesses obtained using the local mass
adjustment, and a comparison to the 1977 mass
table (wa77) O O O I O O O O O O I O O O O O I

Summary of the properties of the A=24 isobaric
quintet and coefficients of the IMME fits. . .

Summary of the properties of the A=8 isobaric
quintet and coefficients of the IMME fits. . .

Summary of the properties of the A=12 isobaric
quintet and coefficients of the IMME fits. . .

Summary of the properties of the A=12 first
excited (2+) isobaric quintet and coefficients
Of the IMME fits O O O O O O O O O O O O I O 0

Summary of the properties of the A=16 isobaric
quintet and coefficients of the IMME fits. . .

Summary of the properties of the A=16 first
excited (2+) isobaric quintet and coefficients
Of the IMME fits O O I O O O O O O O O O I O 0

Summary of the properties of the A=26 isobaric
quintet and coefficients of the IMME fits. . .

vi

Page

78
79
82
83

87

95

95

97

101

104

105

106

107

108

Table Page
4-10 Summary of the properties of the A=20 first

excited (2+) isobaric quintet and coefficients

Of the IMME fits . . . O O O . . C . . C . . . 109

4-11 Summary of the properties of the A=28 isobaric
quintet and coefficients of the IMME fits. . . 110

4-12 Summary of the properties of the A=32 isobaric
quintet and coefficients of the IMME fits. . . 111

4-13 Summary of the properties of the A=36 isobaric
quintet and coefficients of the IMME fits. . . 112

4-14 Summary of the properties of the A=4fl isobaric
quintet and coefficients of the IMME fits. . . 113

4-15 Summary of the properties of the A=44 isobaric
quintet and coefficients of the IMME fits. . . 114

vii

Figure

1-1

1-2

2-10

2-11

2-12

2-13

LIST OF FIGURES

The A=8 isobaric (T=2) quintet . . . . . . .
Isospin-mixing in quartets and quintets. . .

Illustration of the B+-delayed proton decay
Of 21.8]... O O O O O O O O O O O O O O O O O O

Beamline layout to target chamber. . . . . .
Cross-section of the target chamber. . . . .
Top view of the target chamber . . . . . . .
Three target holder configurations . . . . .

Layout of the recoil time-of-flight mass
analyzer system. . . . . . . . . . . . . . .

Front view of skimmer, catcher wheel and the
detector system layout . . . . . . . . . . .

Cross section of detector region (not to
scale). Activity was deposited on the right
side of the catcher foil . . . . . . . . . .

The model of the electrostatic lens, showing
equipotentials and electron trajectories . .

Block diagram of typical electronics set-up.

ALICE calculation of production cross sections

versus beam energy for reactions of 3He on

zuMg O O O O O O O I O O C O O O O O O O O O

A section of triple coincidence data with
proton energy (vertical axis) versus recoil
mass (horizontal axis) . . . . . . . . . . .

Spectra of B+-de1ayed proton decays of 2531

and 21Mg (top and middle, respectively) and
the a-decay of 2°Na (bottom) from (3077) . .

viii

Page

16

22
23
26
27
29

30

33

34

38

40

41

42

Figure Page

2-14 Integral mass spectrum computed from particle
energies and time of flight from (R077). . . . 47

2-15 Proton energy spectrum obtained in coincidence
with the MCP for recoil masses from approxi-
mately 22 to 24 u. These data represent a
658 mC bombardment . . . . . . . . . . . . . . 48

2-16 Recoil mass spectra for proton energies within
£28 keV of 3911 keV (top). 4669 keV (middle),
and 4889 keV (bottom). These ener ies corre-
spond to proton groups from 2“Si, 1Mg, and

25Si decay, respectively . . . . . . . . . . . 49
2-17 Three-dimensional plot of a bivariate Gaussian

distribution 0 O O O O O O O O O O O O O I O O 51
2-18 Correlation between proton energy and recoil

time of flight. Momentum vectors when the
beta and proton are: a) parallel, b) anti-
parallel O O O O O O O O O O O I O O O O O O O 52
2-19 Portion of the ramp spectrum from the 25Si
4g89 kev peak. 0 O O O O O O O O O O O O O O 0 57

2-28 Decay of 2"Si. The half-life derived is
1g3(42) ms 0 O O O O O O O O O O O O O O O O O 59

3-1 Residuals for the quadratic IMME fit to the
=8 quintet (R078) . . . . . . . . . . . . . . 62
3-2 Cross-sectional view of Enge split-pole

magnetic spectrograph, showing electrostatic
deflector (stippled area) used to separate
particle types . . . . . . . . . . . . . . . . 65

3-3 Dispersion matching in the spectrograph. . . . 68
3-4 Scan of plate L68. Spectrum of deuterons from
(p,d), top, and tritons from (p,t), bottom, at
9 degrees. . . . . . . . . . . . . . . . . . . 72
3-5 Scan of plate L59.3 Spectrum of helium 3
particles from (p, He) at 9 degrees. . . . . . 73
3-6 Deviations of the 8Be Q values from their
unweighted average . . . . . . . . . . . . . . 84

ix

Figure Page
3-7 Deviations of the 8Li Q values from their

unweighted average . . . . . . . . . . . . . . 85
4-1 Graphical representation of the local mass

adjustment. Solid lines are Q value inputs,

dotted lines are measured proton energies and

calibration excitation energies. . . . . . . . 96
4-2 Residuals for a quadratic IMME fit to the A=24

qUintet O O O O O O O O O O O O O O O O O O O O 99
4-3 Residuals for a quadratic IMME fit to the A=8

quintet. . . . . . . . . . . . . . . . . . . . 102
4-4 Residuals for a quadratic IMME fit to the A=28

qu-ntet O O I C O O O O O O O O O O O O O O O O 115
4-5 Residuals for a quadratic IMME fit to the A=32

quintet C O O O O O O O I O O O O O O O O O O O 116
4-6 Residuals for a quadratic IMME fit to the A=36

quintet O O I O O O O O O O O O O O O O O O I l 117
4-7 The d coefficient of the IMME with e=8 plotted

versus mass number of the quintet. . . . . . . 118
4-8 The e coefficient of the IMME with d=8 plotted

versus mass number of the quintet. . . . . . . 119
4—9 The d coefficient of the quartic IMME (e#8)

plotted versus mass number of the quintet. . . 120
4-18 The e coefficient of the quartic IMME (d#8)

plotted versus mass number of the quintet. . . 121
4-11 The b and c coefficients of the quadratic IMME

for isobaric quintets plotted versus mass

number of the quintet. . . . . . . . . . . . . 123
4-12 The quantity R plotted versus mass number of

the quintet. For a uniform sphere, R=l. . . . 126
4-13 Coulomb radii determined from the b and c

coefficients of the quadratic IMME plotted

versus mass number of the quintet. . . . . . . 127
A-l Fermi and Gamow-Teller recoil energy

distributions 0 O O O O O O O O O O O O O O O O 134

INTRODUCTION

The isobaric multiplet mass equation (IMME) is a result
of first—order perturbation theory, with the assumption that
only two-body forces are responsible for charge-dependent
effects in nuclei. The equation predicts that the mass
excesses AM of analog states of an isobaric multiplet can be
determined by a three-parameter quadratic equation

AM = a + sz + cTé.

Deviations from the quadratic form of the IMME could be
expected if there were charge-dependent many-body nuclear
forces, isospin mixing or shifts in unbound levels. These
deviations are usually parametrized as cubic or quartic
terms in T2 (dT;,eT;). If an isobaric quintet (T=2
multiplet) is used to test this equation, both additional
terms can be determined, whereas only one can be determined
in a quartet (T=3/2 multiplet).

In a review article by Benenson and Kashy (Be79), an
analysis of 22 complete isobaric quartets found that only in
one case - the ground state A=9 quartet - was there a
significant deviation from the quadratic IMME (d coefficient
of 5.8(16) keV). However, mass 9 is also the most

accurately measured quartet and one cannot conclude that

2
this deviation is exceptional without obtaining results of
comparable accuracy in other multiplets. Isobaric quintets
offer the prospect of improved tests of the IMME.

This dissertation describes measurements which yield a
test of the IMME in the A=24 quintet of the same level of
precision as the mass 9 quartet. New measurements on
the A=8 quintet have removed the slight deviation that
existed previous to this work.

The thesis is organized as follows: Chapter One
contains a discussion of the theory of the isobaric
multiplet mass equation: Chapter Two describes the
measurement of the 2"Al T=2 level and includes the details
of the design of a cryogenic helium jet and recoil time-of-
flight mass analyzer system constructed for this
measurement: Chapter Three contains the details of the
measurements of the lowest T=2 levels in 8Be and 8Li;
Chapter Four discusses the A=24 and 8 quintets along with
a summary of the remaining A=4n quintets for A544. There

are now five completed quintets.

CHAPTER ONE

1.1 Isotopic spin and charge-independence.

Isotopic spin has its origins in the observations of
low energy proton scattering which revealed that,
neglecting Coulomb effects, the proton-proton (p—p) and
proton-neutron (p-n) forces are very similar (Br36,Ca36).
These observations led to the hypothesis of the
charge-independence of the nuclear force and started the
development of isospin as an important quantum number.

The hypothesis of charge-independence is that,
neglecting the electromagnetic effects, the force between
p-p, p-n and n-n pairs will be identical when these pairs
are in the same space and spin states. Because the p-n
system is not made of identical particles, there are
space and spin configurations available to this system
that are not allowed for the p-p and n-n system.

It is only those configurations (where the identity
of the nucleons does not matter) for which the forces
may be compared.

Heisenberg postulated (He32) (shortly after the
discovery of the neutron) that the specifically nuclear
forces between a pair of protons (p-p) would be the same as

that between a pair of neutrons (n-n). The hypothesis, of

4

the equivalence of the p-p and n-n force, is known as
charge-symmetry. This is a less restrictive statement than
the one of charge-independence because charge-independence
implies charge-symmetry but not vice versa.

The charge-dependent nature of the nuclear force is
considered to be due solely to the effects of the
electromagnetic interaction. These effects are classified
as direct and indirect (He69). The direct part includes the
Coulomb force, vacuum polarization effects, the neutron-
proton mass difference, and the magnetic forces (such as the
spin-orbit force). These direct effects involve no meson
exchange between the nucleons and are present even in the
absence of hadronic forces.

The indirect charge-dependent effects are present only
when there are nuclear forces. These effects, as described
by Henley (He69,He81), include the mass difference between
the neutral and charged mesons, mass differences of baryons
(during two-pion exchange for example), radiative
corrections to meson-nucleon coupling constants, meson
mixing caused by the electromagnetic interaction, and forces
that arise from meson plus photon exchanges.

In isospin formalism, the proton and neutron
are opposite projections of the nucleon in isotopic spin
space. This is not a projection in physical space as with
normal spin but rather a fictitious space used as a
convenience. Because of the two-state nature of the

nucleon, the same formalism as with spin 1/2 can be applied.

5
The presence of the isospin quantum number allows for the
generalization of the Pauli principle so that the wave
function for a system of nucleons must be antisymmetric
under interchange of any two nucleons.

The operator t is defined as the isospin operator for
an individual nucleon with three cartesian components of tx,
ty, and t2. The projection of the t operator is defined as
tz. The neutron is assigned an isospin projection of
tz=+l/2 and the proton a projection of tz=-1/2. This is the
original (and still-standard) convention that is used in
nuclear physics. In particle physics the opposite sign
convention is used. The standard nuclear physics notation
will be used here.

T is the total isospin operator for the nucleus. This
is defined as

A
a = Z t. ”-1)
1
i=1
and has a projection given by,
A (1-2)
T2 = §E3tz(i) = (N-Z)/2 -
i=1

The charge-independent part of the nuclear Hamiltonian
HN commutes with the total isospin operator,

[HN.T2] = o (1-3)
In the isospin representation, the wave functions of the
nuclear Hamiltonian are eigenstates of T and have no Tz
dependence. That is

HNIao,TTz>=E(°)(ao,T) Iao,TTz> (1-4)

6
Where do represents all the other quantum numbers (aside
from T and T2) that are necessary to specify the nuclear
wave function (such as spin and parity). The energy of
these states E(°)(ao,T) is independent of T2 so there are 2T+l
degenerate states with this energy and with quantum numbers
do and T.

These 2T+l states, all having the same wave functions,
will occur in 2T+1 different nuclei. They all have the same
total number of nucleons and, therefore, have become known
as an isobaric multiplet. The A=8 isobaric quintet (T=2) is
shown in Figure 1-1.

For a nucleus with T=1/2, there are two degenerate
states with Tz=+1/2 and -1/2. These are called mirror
pairs. A state with a T that is larger than the T2 of the
nucleus (e.g., a T=1 state in a Tz=fl nucleus) is known as an
analog state, because there exist states in neighboring
nuclei that are analogous to it.

The total Hamiltonian for a system of nucleons contains
change-dependent terms which do not commute with the isospin
operator and, therefore, break the degeneracy of the
analog states among a given isobaric multiplet. As'
mentioned earlier, these terms are considered to be
electromagnetic in origin.

Aside from the neutron-proton mass difference, the
major contributor to the breaking of the degeneracy among
the analog states is the static Coulomb potential.

Because the strength of the electromagnetic interaction is

gull};

 

 

 

 

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8
only 1/137 that of the nuclear force, it can be considered
as a perturbation on the nuclear force. Perturbation theory
can then be used to calculate the splitting between the
analog states.

Wigner (W157) was the first to show the effect of the
Coulomb potential on the mass differences of the analog
states. His derivation involved a simple application of
the Wigner-Eckart theorem in first-order perturbation
theory and produced the well-known isobaric multiplet mass
equation (IMME):

AM = a + sz + cT; (1-5)

The crucial decomposition of the Coulomb operator
(in isospin representation) into its irreducible parts was
first accomplished by MacDonald (MaSS). This explicit
decomposition into an isoscalar, isovector and isotensor
part allowed for the simple application of the Wigner-

Eckart theorem (Wi57).

1.2 Derivation of the IMME.
The total Hamiltonian can be written as the sum of the

charge-independent nuclear part H and the charge-dependent

NI
electromagnetic part BC. This latter term can be considered
to be due principally to the static Coulomb interaction.
H = HN + HC (1-6)

In first-order perturbation theory, the correction to

the unperturbed energies Iflm(ao,T) is

E(1)(ao,T) = (aoTTleCIOLoTTz) (1-7)

The Coulomb potential in isospin representation is

given by

l- i l-t '
H _ 22 (2 9‘) (2 2”) <1-a)
- e
C ri.
i<j 3
With the stated convention that tz=1/2 for neutrons and
tz=-l/2 for protons, the (1/2-tz(i))(1/2-tz(j)) product will

then be nonzero only for proton pairs. Multiplying these

terms together yields

HC = e22 —1—[-‘1I-%(tz(i)+tz(j))+ tz(i)tz(j)] (1-9)

r..
i<j 13

+ O + O
Inserting the isoscalar t(l)3t(3) into equation

1-9, (Ma55) results in
+ . + . ‘t (i)+t (j)
_ 1 l t(1)-t(j) _ z z _
Hc'ezz:?"{[Z+ 3 ] l: 2 ]+ (110)

i<j ij
. . _ Em-Eu‘)
[tz(1)tz(3) 3 ]}

The first term in brackets is the zeroth component (in

 

 

 

cartesian form) of a zero-rank irreducible tensor operator
Hfil The second and third bracketed terms are the zeroth
components (in cartesian form) of rank 1 and rank 2
irreducible tensor operators Hg’ and Hg) respectively.

These terms are listed separately as follows,

+ I
1 1 + t(1)’

10

 

H30) = ez

bl

‘t’m. _
3 y.-.

 

 

3

-—— [tz(i)tz(j)-t(i)'t(j)] ; AT = 0, :1, :2

l<j
. t (i)+t (j)
H31’=eZZE%‘[z 2z ]:AT=0.:1 (1'11)
i<j
i<j

The Coulomb operator can now be expressed as

HC = 115°) +1151) +1132) (1-12)

The AT's following the explicit forms of the tensor

operators, Equation l-ll,
operator can make between
These connections are not
perturbation theory where
are considered. However,
these operators may cause

isospin mixing.

show the connections that each
states of differing isospin.
important for first-order

only diagonal matrix elements
in second-order perturbation,

additional energy shifts due to

The fact that the Coulomb operator is reducible into

the zeroth components of these tensors assures that only

states with the same Tz values will overlap (i.e., states in

the same nucleus).

Application of the Wigner-Eckart theorem yields for the

first-order correction to

the unperturbed energies

11

2
3(1) (ac/r) = éoTTZIHCIaOTTZ>=Z@oTTlegk) IaoTTz>
k=0

(1-13)
2
= E <TkTZOITTz> {mom [HUG | law)
k=0
The Clebsch-Gordan coefficients are
(TOTZOITTZ> = 1
T2
<T1TZOITTZ> = ——
/T(T+1) (1-14)

3T;-T(T+1)

 

 

<T2T 0|TT>
z z

/(2T-1)T(T+l)(2T+3)

Inserting the Clebsch-Gordan coefficients into Equation 1—13

yields
E(‘)(ao,'r) = <10TI|H(°)II0L0T> +
T -
+ ——-z— (aoTIIHmllaoT> + (1 15)
/T(T+1)

3T - T(T+1)
+ 2 (ml IH‘Z’I laoT>
/(2T-1)T(T+1)(2T+3)

 

 

Rewriting Equation 1-15 in terms of powers of T2 results in

12

E(1)(ao,T)

 

[(aowlmomm) - T<T+n semifinal] +

/(2T-1)T(T+1)(2T+3)

(1-16)

+ [QOTLleJ 100T):l T +
/TTT:TT z

+

 

[ a gaowjlmmaa) ]T.

/(2T-1)T(T+1) (2T+3) z

The 1%”(ao,T) is the mass correction to the
unperturbed states and is the well-known isobaric multiplet
mass equation (Equation 1-5).
AM = a + sz + CT; (1—17)
The neutron-proton mass difference may be accounted for
by simply including a sum of the neutron and proton masses

in the Hamiltonian.

H = MnN + M z (1-18)

P

where N+Z=A and Mn is the neutron mass and M

AM

p is the proton
mass. The mass difference effect is made more transparent
when Equation l-18 is written in isospin formalism

A

- ' l - .1__ -
HAM ' E [Mn(2+tz(l)) + Mp(2 tz(l))] (1-19)

i=1
Upon rearrangement this becomes

Mn+M A A
HAM = __22 Z . (Mn-Mp)thm
i=1

.=1

(1-2fl)

which further reduces to

M +M (1-21)
_ n _
HAm - (TB)A + (Mn Mp)Tz

13

These terms can then be absorbed into the a and b
coefficients of the IMME.

The spin-orbit interaction has been shown by Garvey
and Hecht (Ga69,He66) to also be expressible in terms of
irreducible tensor operators of rank zero, one and two. So
it too will preserve the form of the IMME.

In general, any two-body force will have the form

(Ga69)
Vij = V1[tz(i) +tz(j)] + V2[tz(i)tz(j)] (1-22)

which may be decomposed into a linear combination of
irreducible scalar, vector and tensor operators. Hence, the
IMME will result from the use of any two-body force in

first-order perturbation theory.

1.3 Violations of the IMME.

In second-order perturbation theory, the correction to

the energies for the static Coulomb potential is given as

(GA69,BA73)

 

. 2
E‘ZNOLO ,T)=ZZ 131' é°TTz|HC|aiT z>|
T' i.

ONO-01.1.) ‘ E(0N('1i,T')
(1-23)
(When T'=T, then i%0.)

where T'=T, Til, T12.

14
With the application of the Wigner-Eckart theorem, the
explicit Tz dependence of this second-order correction can
be extracted. Janecke and Garvey (Ja69,Ga69) have Shown
that the major effect of this higher-order correction is to
alter the a, b, and c coefficients. These changes can then
be absorbed back into the standard form of the IMME
(Equation 1—17). There are, however, terms with a T; and
T; dependence generated by this expression which result in

the "quartic" form of the IMME
.__ 2 3 x. _
AM a + sz + cTz + de + e'I‘z (1 24)

These d and e terms, though small relative to a, b,

and c, will have the form

a T HQ) a.T' a.T' HE) G T
d :2 < 0 ILWLIHT) >-<I-3(:5(0L:)T')H 0) (1-25)
1
(When T'=T, then i#o.)
where T'=T, Til.

and

 

T HP) .TV 2
e g2 (00 l' Ila]. > (1'26)

5- EFNa,,T) - Em(ai,T')

(When T'=T, then i#0.)

where T'=T, Til and T12.

15

In order to distinguish between the effects of a
charge-dependent, many-body force (which presumably would
manifest itself in first-order perturbation theory) over the
higher-order effects of a two-body force, detailed
shell-model calculations will be required.

Isospin mixing is expected to cause deviations from the
IMME. Isospin-forbidden resonance reactions have been used
successfully to discover a number of higher T levels in
lower Tz nuclei (T=2 states in Tz=0 nuclei). So isospin
admixtures are known to exist and could therefore be
responsible for generating relative shifts between multiplet
members. Figure 1-2 illustrates the effect of isospin-
mixing in both quartets and quintets. For quartets, it is
apparent that mixing with lower T levels would primarily
produce only changes in the c coefficient. In quintets,
however, mixing between T=0 and T=2 levels in the Tz=fl
nucleus (R076) would, by symmetry, produce an even-order
eT;.

In the case of the A=8 multiplet, it is known that this
state is admixed with a T=D state (from the observation of
the T=2 state in the Tz=0 nucleus 8Be, as an isospin
forbidden resonance in the 6Li + d reaction). A generation
of an e coefficient is seen to occur, both from the
qualitative argument shown in Figure 1—2 and from the
results of second-order perturbation theory Equation 1-26.
Assuming only one T=fl perturbing state with the same spin

and parity as the T=2 state, the e coefficient has the form

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02:25. ZEmOm. mo whommuw

17

1 (zlln‘z’llo)2
= __ 1-27
8 20 E(°)(2) - 3W0) ( )

where the ilacomes from evaluation of the Clebsch-Gordan
coefficients.

For an e coefficient of 1 keV and an off-diagonal
matrix element of order 169 keV, the perturbing state must
lie within 520 keV of the T=2 state. As of yet this state
has not been observed but evaluation of the sign of the e
coefficient will determine its position with respect to the
T=2 state. Actual observation of this T=fl state would then
allow a determination of the size of the off-diagonal
Coulomb matrix element.

It has also been suggested that changes in the wave
functions between members of a multiplet could cause a
deviation from the quadratic IMME. For example, the
expansion of the radial wave function due to the effects of
the Coulomb repulsion in the proton-rich member of a
multiplet could cause a change in the calculated expection
value, and hence a change in the relative separation of the
levels. However, calculations have shown (Be7fl,Be77) that
the major change occurs in the b and c coefficients and only
small d and/or e coefficients are generated. The change to
the b and c coefficients is absorbed back into the IMME
without causing a significant deviation.

For a complete quintet the five coefficients of the

quartic IMME are completely specified:

18

a = M0

b =f;[8(M1-M_l)-(M2-M_2)1

c =-:—[16(M1+M_1)-(M2+M_2)-36M0] (1-28)
d =332—[(M2-M.2)-2(M1-M.1)1

e = 71;. [(M2+M_2)-4(M1+M_1)+6M0].
From these expressions it is apparent that the coefficients
are more sensitive to the inner members of the multiplet
and therefore accurate determination of these are necessary

for a precise test of the IMME.

CHAPTER TWO

2.1 Introduction

The use of the helium jet technique for the rapid
transport of activities from a region of high background
radioactivity to a much lower one has been quite successful.
In most applications, room temperature helium gas has been
used. This necessitates the use of some sort of macroscopic
impurity or cluster in the gas to obtain useful transport
efficiencies and in fact, quite impressive and even
reproducible results (5@% over ten meters) have been
obtained with some impurities (Ma76).

The high transport efficiencies are understandable in
terms of Bernouilli's equation for laminar flow. A radial
pressure gradient exists that forces the clusters which
carry the activity to remain in the center of the capillary
over long distances.

Though the impurity—laden helium jet has shown its
ability to transport activity over long distances and with
high efficiency, these impurities may limit or even
eliminate its use in certain applications. These include
interfacing the helium jet to certain types of ion sources
for mass spectroscopy, precise studies of B-spectra,

high-resolution Studies of heavy particle emission and

19

20

fundamental weak interaction studies which rely on the line
shapes of B-delayed heavy particle lines.

It was found in 1974 by Aysto et a1. (Ay74) that for
"pure" helium cooled to liquid nitrogen temperatures (77°K)
that a significant improvement in transport efficiency was
possible relative to room temperature impurity-free helium
jets (Ay73).

A model presented by Robertson et al. (R077) explains
this behavior in terms of the thermal diffusion of active
atoms through the carrier gas. This model assumes the loss
of active atoms is due to collisions with the capillary wall
which occur during their transport through the capillary.
The increased efficiency is shown to be due in equal measure
to the decrease in temperature (which decreases the rate of
thermal diffision to the capillary wall) and to the
increased molecular flow rate (which shortens the time the
active atoms remain in the capillary).

In a paper by Robertson et a1. (R077), the description
of a cryogenic (liquid-nitrogen cooled) helium jet coupled
to a recoil time-of-flight mass analyzer for use in
observing short-lived B-delayed particle emitters was
presented. This apparatus was used at Princeton University
in an attempt to observe 2“Si. Results tentatively
suggested that protons from the decay of this nucleus had
been observed and that the mass of the T=2 state in the
daughter nucleus 2“Al was consistent with the prediction

from the quadratic form of the isobaric multiplet mass

21
equation. With these promising results, an improved
apparatus was contructed here at Michigan State University
with the intent of observing 218i and other Tz=-2 nuclei.

It is a feature of this apparatus that mass
identification and background reduction can be obtained with
only a single surface barrier Si detector (rather than a
counter telescope) to detect the protons. The improved
resolution and linearity, as well as the simultaneous
recording of strong calibration groups, make possible a very
precise measurement of the mass of the T=2 state in 2"Al.

In addition, the first direct measurement of the half-life

of 2"Si has been made. Figure 2-1 illustrates the

B+-de1ayed proton decay of 2"Si.

2.2 Beamline and beam transport

For these experiments a beam of 70 Mev 3He ions from
the Michigan State University Cyclotron produced 2|‘Si via
the 2"Mg(3He,3n) reaction. Figure 2-2 shows the beamline
layout to the cryogenic helium-jet target chamber (located
in vault 1). Also shown is the capillary feedthrough to the
recoil time-0f-flight apparatus (located in vault 5).

Beam position and focus were verified with a
scintillator placed fl.3 m in front of the target chamber.
The beam passed through a 1.3 cm diameter aperture in each
side of the target chamber and stopped in a water Faraday
cup placed fl.4 m past the target chamber. This Faraday cup

was electrically isolated from the target chamber and the

22

 

 

 

 

23 3/2+
12

 

+
24AI 4

13

24gigure 2-1. Illustration of the B+-delayed proton decay of

23

 

 

 

 

 

 

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24
rest of the beamline with an 8 cm section of insulating beam
pipe. A current integrator gave both the total charge
and the beam current. The beamline elements were adjusted
to yield the maximum current from the Faraday cup.
Depending on the tuning of the cyclotron, this current
usually lay between 1 and 3 uA of aHe++. Such currents did
not lead to failures of the 5.25 mg/cm2 Havar windows of the
target chamber (H175). However, on one occasion, the
windows ruptured after just three hours of 4-6 uA of beam

current 0

2.3 Helium jet target chamber

The helium jet target chamber consisted of a large (18
cm outer diameter x 35.5 cm long and 2.54 cm thick) aluminum
tube, sealed at both ends with 1.3 cm thick aluminum cover
plates. All flanges were sealed with indium wire.

The chamber was connected to the beamline by two
stainless steel (5 cm outer diameter x 23 cm long and 1.7 mm
thick) tubes, a 7.6 cm long section of which was turned down
to a thickness of only 51 mm. This was done to minimize the
heat loss through these tubes. The chamber was placed in a
large (24 cm inner diameter x 46 cm inside depth) stainless
steel dewar (Mv76) which was filled periodically with liquid
nitrogen (LNZ). A LNZ level controller (Nv76a) was used to
keep the dewar full. A sensor placed 220 cm from the top of
the dewar detected when the level of LN2 went below this

sensor. A solenoid valve then pressurized a feed dewar

25
which delivered LN2 to the target chamber dewar for
approximately three minutes. An insulating styrofoam cover
was fit over the top of the dewar and target chamber and the
stainless steel beamline connecting tubes were wrapped with
a flexible styrofoam insulation. Figure 2-3 shows a
cross-section of the target chamber.

Commercially "pure" He (99.995%) was sent at room
temperature via 1/4 inch "poly flo" tubing to the vicinity
of the target chamber where it connects to a 1.3 cm diameter
copper tube of 2.5 m in length. As Figure 2-3 shows, this
tubing was coiled about the target chamber and also placed
in the LN2 dewar. Thus, the He gas was cooled to g77°K
before entering the target chamber. The gas entered through
a porous stainless steel diffuser with 2 pm openings (Nu77).
This prevented turbulence created by incoming gas which
could significantly reduce the transport of activities
(R077).

The target was mounted on a target holder which
projected into the chamber as shown in Figure 2-4. The
recoiling activities from the reaction of the beam in the
target were thermalized (slowed down) in the cold He gas.

In order to prevent the range of the recoils in the gas from
exceeding the width of the target holder, the gas pressure
was kept at approximately .7 atmospheres. Because of the
increased gas density at 77°K, this pressure was sufficient
to limit the range of the recoils to under 1 cm. The

activities were then swept out of the chamber along with the

26

/lnsulating cover

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Havar
7flj/fl/Ill window
0 <——— Beam
He __. Target
933 ‘ holder
E c
/ /
:. He :
”‘1“ gas ./
E 3 : Porous
: g / stainless
Coiled /;'3 2/ steel
tubing : g , diffuser
/ /
Z r
r j
1 LN2 :
"' w
" z
: : ~— Dewar
. /
V/j

 

Figure 2-3. Cross-section of the target chamber.

27

Capillary feed
to
recoil mass analyzer

1‘

 

 

é—Beam

 

 

 

 

Target
chamber

Figure 2-4. T0p view of the target chamber.

28
He gas via a flared polyethylene capillary tube (1.8 mm
inner diameter) and transported 2.4 m to the recoil

time-of-flight mass analyzer.

2.4 Target holder and target

It was found by Robertson et al. (R077) that the target
holder design was critical to the transport efficiency. Any
turbulent flow behind the target reduced the transport of
activities dramatically. Three target holder geometries
were tested in this apparatus. Figure 2-5 shows views of
the three configurations. Configurations A and C gave
relatively similar results, but configuration B, which was a
hoped-for improvement to configuration A, gave poor results.
Configuration C was a later design which was used for the
majority of the data collected. Transport efficiencies were
on the order of a few percent.

The target used for these experiments was 5 mg/cm2 of
ang (99%) enriched that had been evaporated on a 2.3 mg/cm2

copper backing.

2.5 Capillary-skimmer cone interface

The activity exited the capillary two to three
millimeters from a skimmer cone. A 149 l/s Roots blower
(R038) backed by an 80 l/s mechanical pump (Ki65) removed
a large fraction of the He gas. Figure 2-6 shows this
arrangement, along with a layout of the recoil time-of-

flight mass analyzer system. The skimmer is a 98°

29

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30

Skimmer
Motor cone .
adapter Catcher Capillary
wheel holder
D.C. Ferro- Roots
motor fluidic blower

 

 

 

 

\

I i
ll}:

 

Servo shaft seal
wheel \ \ T

 

 

 

 

 

 

 

 

 

 

 

 

- [
L... Si a... /|| Feller]. gage,
Light pipe [ r“
‘ L1 :1

 

 

 

 

 

 

 

Turbo
1
Roots
blower
Photo
tube

Figure 2-6. Layout of the recoil time-of—flight mass
analyzer system.

31

cone with a 1.6 mm opening. The capillary is flared and
seals against the o-ring on the end of a glass guide tube.
This tube supports and also aligns the capillary with the
skimmer opening.

During assembly the capillary is first flared on
only one end (by holding vertically downward over a
flame) and then fed through both the target holder and
the shielding wall. After it is all the way past the end
of the glass guide tube's seal, it is cut to length and
flared. It is possible to move the recoil apparatus toward
or away from the shielding wall where the capillary exits,
allowing any slack to be taken up and the capillary put

under tension.

2.6 Foil wheel and particle detection system.

The main chamber was pumped by a 150 l/s Roots blower
(He6l) backed by a 16.5 l/s mechanical pump (Va78) and a 25
l/s mechanical pump (We77) connected in parallel.

The transported atoms passed through the skimmer and
into the main chamber, where they were deposited on a 10-12
ug/cm2 thick, (6.4 mm diameter) Formvar catcher foil.

A vertically aligned foil wheel held six catcher foils.
The foil wheel was stepped at a rate of 5 Hz with a high-
torque hollow-rotor D.C. motor (Mi77). The wheel was
connected to the motor by a Ferrofluidic shaft seal (Fe77).
This seal allows for high shaft accelerations while

maintaining vacuum integrity. Mounted on the opposite end

32
of the motor shaft, were a potentiometer (De78) and a small
six-blade wheel. The potentiometer was used for recording
the wheel position. The six-blade wheel (half the diameter
of the catcher wheel) was part of the servo system for
controlling the motor's motion. A linear filament light and
a photodiode were mounted of Opposite sides of the small
blade wheel. The photodiode was part of a servo circuit
which responded when the leading edge of the blade
interrupted the light coming to the diode. This servo
circuit controlled the motor which could keep the catcher
foils locked into place between steps. The stepping rate
was controlled external to the motor circuitry through a
square wave generator.

The activity was allowed to collect for 266 ms, minus
the approximate 20 ms stepping time. The wheel was then
moved to its next position, where the newly-deposited
activity was placed between the particle detectors (see
Figure 2-7). After B-decay to particle-unstable states,
both the particle emitted (proton or alpha) and the
recoiling ion were observed in coincidence. Protons (or
alphas) passed through the thin catcher foil and were
detected using a single 15% mmz, 368 um deep Si detector
(Or78). The residual nucleus is ejected from the surface of
the foil. Figure 2-8 illustrates this arrangement.

The recoil ions have enough energy (168 keV for 23Mg)
to pass through a thin converter foil, placed opposite the

Si detector. Secondary electrons from the converter were

33

Catcher foils

Skimmer h

 

 

 

 

 

 

 

 

- <——— 3:33"
\
O
_’1

Si \‘__,27

detector

(above Light

plane of \ pipe

diagram)

Figure 2-7. Front View of skimmer, catcher wheel and the
detector system layout.

34

MSUX-79-O7O

 

Catcher foil
Scintillator (NEIOZ)

/ MCP
/' V

 

 

 

£3 .
Sal DISK—P
"hec~“~\\‘
Si detector Converter
foil -——"“’
e—_Light pipe

 

 

 

 

 

Figure 2-8. Cross section of detector region (not to
scale). Activity was deposited on the right side of the
catcher foil.

35
observed by a pair of Microchannel Plates (MCP) (Ga76). The
58.8 mm-diameter converter foil served also to isolate the
MCP from the .14 torr pressure in the main detector chamber.
This allowed the operation of the channel plates in a clean,

'7 torr provided by a 1,588 l/s

high vacuum of 18-6 - l8
turbomolecular pump (Sa77).

The converter foil was shaped as a section of a sphere
with an 88 mm radius of curvature. When placed 88 mm from
the catcher foil, this curvature reduced the time spread of
the recoil ions. Both the Si detector and converter foil
subtended a solid angle of 2.6% of 4n. The converter foil
consisted of a 238 ug/cm2 layer of Formvar on a 295%
transmission curved wire screen. The screen was made with
8.885 inch diameter 00pper wire layed onto a 88 mm radius
convex wooden form. Onto the Formvar surface a 218 ug/cm2
layer of gold and a gl8 ug/cm2 layer of Cal were evaporated.
The layer of gold was used to prevent the converter foil
from charging up. The CsI, with its higher secondary
emission coeficient (Fa77,Bu77), was used to improve the
detection efficiency of the recoils.

It may easily be shown (see Appendix A for a derivation

of this result) that the mass of the recoil (daughter)

nucleus Mrec is given by:

/2E m t
Mrec = d (2‘1)
where t is the time of flight of the recoil mass, Ep the
laboratory energy of the proton (or a), m the mass of the

P
proton (or a) and d is the flight path (distance traveled by

36
Mrec during the time t). The recoil mass is pr0portional to
t, not t2 as in the usual time-0f—f1ight system.

A start signal from the proton or alpha in the Si
detector and a stop signal from the MCP gave a value for the
time-of—flight of the recoil ion. This time, combined with
the particle energy in the Si detector, was used in the
above formula to derive the recoil mass.

The B-decay recoil determines the mass resolution in
these measurements. For example, for 25Si, which was
produced in a competing reaction, this B—recoil limited the
mass resolution to 7%, a value nevertheless adequate for
mass identification. Appendix A also contains a discussion
of this effect.

In initial experiments, it was found that the double
coincidence data (between Si detector and MCP) showed two
different recoil mass groups for the same particle energy.
The second (slower) group was delayed by g9 ns and was
attributed to recoil ions directly striking the MCP but
failing to produce secondary electrons in the converter
foil. In order to remove these ghost groups, which were
producing a background in the region of interest, the design
of an electrostatic lens was undertaken. It was hoped that
a design could be found which would stop the recoils from
striking the channel plates while still allowing the
electrons to do so.

Lens designs were first modeled using the method of

successive over-relaxation to solve Laplace's equation. See

37

Appendix B for a discussion of this method. A program was
developed which allowed the testing of many different
geometric configurations. This program also calculated
particle trajectories through the lens.

A gridded lens design, illustrated in Figure 2-9, that
focused the electrons to the central region of the channel
plates was chosen. This design required only the central
portion of the MCP to be used. The outer regions were
masked off, and a small disk placed in front of the
converter foil prevented the recoil ions from hitting the
channel plates. A loss of only 7% in solid angle resulted
from this arrangement, and a significant reduction in the
background was achieved.

As in the previous apparatus (R077), an annular plastic
scintillator was used to observe the initial B-decay of the
parent nuclei. This scintillator was placed opposite the Si
detector and subtended a solid angle of g37% of 4n. A
plastic light pipe connected the scintillator to a photo-
tube (Am77). Recoil ions traveled through a conical hole
(28° half angle) in the scintillator to the converter foil.
A cross-section of this scintillator is visible in Figure
2-8.

Many different activities are produced by the beam in
the target and several delayed particle emitters were
observed in our experiments. To make use of the mass
identification capability, data was recorded in the event

mode and, upon subsequent playback, proton and alpha energy

38

MSUX-TQ- 072

6.66 cm Diameter\)

 

 

 

 

 

 

U
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Jill

 

 

 

 

 

MCP

 

 

 

 

 

Mesh
+3kV

ff

 

 

 

 

 

 

\

Converter foil

 

 

 

 

Side View of the Cylindrical Lens

Figure 2-9. The model of the electrostatic lens, showing
equipotentials and electron trajectories.

39

spectra for different recoil mass bands were extracted.

Eight parameters for each event were written on
magnetic tape, proton energy, recoil time of flight, energy
and time information from the plastic scintillator, MCP
pulse height, leading-edge to crossover time for pulses from
the silicon detector (used in discriminating against
pileup), position of the foil wheel and, lastly, a ramp
initiated by the end of a foil wheel step and strobed by
pulses from the silicon detector (used in the measurement of
half-lives).

Figure 2-18 shows a block diagram of a typical

electronics set-up for these experiments.

2.7 Experimental results and analysis.

The principal delayed-particle activities observed in
this experiment are 2°Na (T8=445 ms), 21Mg (T%=228 ms), 2551
(T%=125 ms) and 2"Si (T%=183 ms, this work). Figure 2—11
shows the results of a calculation using the compound
nuclear evaporation code ALICE (B176). The cross sectionss
for the production of each activity are plotted versus beam
energy. After losing energy in the target chamber windows
and gas, the beam energy was on the maximum of the
production cross section. The B-delayed activity 2°Mg was
not observed in these experiments.

Figure 2-12 shows a section of triple coincidence data
(in which coincidences between the Si detector, MCP and

scintillator are required), with proton energy on one axis

40

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Figure 2-11. ALICE calculation of production cross sections
versus beam energy for reactions of 3He on 24Mg.

42

 

 

 

 

 

 

 

 

 

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dence data with

proton energy (vertical axis) versus recoil mass (horizontal

lnCl

Figure 2-12. A section of triple co

. ).

3X18

43
and recoil mass (i.e., the atomic mass of the nuclide formed
following proton emission) on the other. The groups
centered on 24 u at channel numbers 322 and 424 are the 4089
and 5402.2 keV protons lines from 25Si decay. The groups
centered on mass 2% u at channel numbers 304 and 367 are the
396% and 4669 keV protons lines from the decay of 21Mg. The
box centered on 23 u at channel number 308 shows the
position of the 3912.7 keV protons from the decay of 2“Si.
The low statistics of the 2"Si group results from the triple
coincidence requirement.

Triple coincidence data (between the Si detector, MCP
and scintillator), though useful in reducing the background
and improving the mass resolution for the verification of
weak groups, were not directly used in obtaining the proton
energy because the beta-induced recoil of the daughter has a
component of its velocity toward the Si detector. Thus the
particles detected in the Si detector would be on the
average shifted up in energy. Therefore, only double
coincidence data (between Si detector and MCP) were used to
obtain the proton energy. Even for the double coincidence
data there are in principle small energy shifts, which will
be discussed later.

In order to convert the energy and time-of-flight data
to these two-dimensional energy versus recoil mass spectra
the expression,

Mrec = G/Ep-Eo (t-to) ‘ (2_2)

was used. Where EO and to are energy and time offsets

44

respectively and G is a scaling parameter. An analysis of
the energy and time-of-flight one-dimensional spectra
provided initial values for these offset parameters.
However, to remove skewing of the 2-D spectra it was
necessary to optimize these offsets. The strong groups in
25Si and 21Mg (made in competing reactions) defined the mass
scale. An assignment of a recoil mass of 24u to the 4089
KeV and 5462.2 keV groups from 2531 and zen to the 4669 keV
and 6225 keV lines from 21Mg was made. The 6225 keV peak
was just at the edge of the silicon detector's range for
these particles. Both with and without this group the
needed parameters could be found.

A program was written that used a portion of the energy
vs. recoil mass data around each peak. For a given Eo and
to a point-by-point construction of a recoil spectrum, i.e.,
projection along recoil axis) and a calculation of the
centroid, via a simple average, was done for each peak. By
using the known recoil mass, a scale factor G was
calculated. This procedure was repeated for different Eo and
to values until the closest agreement between the scale
factors for each group was obtained. The average of these
four quantities was used in equation 2-2. This procedure
successfully provided undistorted two-dimensional energy
versus recoil mass spectra as shown in Figure 2-12.

Spectra from B-delayed protons from 2SSi and 21Mg, and
B-delayed alphas from 2°Na, are shown in Figure 2-13. These

were Obtained with the Princeton apparatus (R077) by setting

45

 

 

 

 

 

 

 

 

I03 I j I |
258i
(p)
I02
'0 i E
M ’ ~ 1! 3'
,0. w I I n Ii I in I
2| 3
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Lil (p) ‘
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I03
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IO .
L I : i!
. will i i I
1 § ‘3' 52 E

 

ENERGY, MeV

Fi ure 2-13. Spectra of B+-delayed proton decays of 2581
and 1Mg (top and middle, respectively) and the a-decay of
0Na (bottom) from (R077).

46
2 u wide windows centered on the appropriate mass regions
(24 u, 20 u and 8 u) in double—coincidence (particle-recoil)
data.

A projection of two-dimensional double-coincidence data
along the recoil axis (as seen with the Princeton apparatus)
forms the integral mass spectrum seen in Figure 2-14.
Recoils (160) from the B-delayed alpha decay of 2°Na will
appear at only half of their actual mass (8 u). Appendix A
also contains a discussion of this effect.

Figure 2-15 shows a proton energy spectrum gated on a
recoil mass of 23 u with a window width of 2 u. The energy
resolution in this spectrum is approximately 25 keV full
width at half maximum (FWHM). In addition to prominent
peaks from 25Si Which extend partially into this band, a new
peak, not seen in lower-energy bombardments (Se73), is
present at an energy of 3912.7 keV. Mass spectra gated on
the new peak, the 4669-keV line from 21Mg and the 4989-keV
line from 25Si are shown in Figure 2-16. The peak at recoil
mass 23 u (to the right of the peak from broad lines in the
21Mg spectrum) identified it as originating from 2"Si
B-decay. This transition has been observed by Aysto et al.
(Ay79) at high mass resolution, with a proton energy of
3914(9) keV, in good agreement with the present value. The
energy is in the vicinity of the IMME prediction for the
ground-state proton decay of the T=2 state in 21‘Al.

The proton energy was obtained from the double-

coincidence (Si detector and MCP) spectra using the strong

47

 

 

 

 

I05 I l I | l
:Li
3 20No
no4
.J
uJ
2:
<2:
I I03
L)
g; ZIMg 253i
2
ualo
5
:5
C)
L)
IO -
1 L 1 L l
0 IO 20 3O

RECOIL MASS, omu

Figure 2-14. Integral mass spectrum computed from particle
energies and time of flight from (R077).

48

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49

100 "flaw

 

l

80
80

1
l

I
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70

 

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30- d

 

20’ ZHSI ‘

 

 

0 a”
111111111—11111I

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C)

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HOO- -

COUNTS PER CHANNEL

I
l

200

 

0‘ 1'11111]I]I[1

800
258:
800- ‘

sons -
200— 4

 

 

 

O'l‘lllTI'lflrl'
18 20 24 28 32
RECOIL MAssud

Figure 2-16. Recoil mass spectra for proton energies within
:20 keV of 3911 keV (top), 4669 keV (middle), and 4089 keV
ébottom . These energies correspond to proton groups from

4Si, 2 Mg, and 258i decay, respectively.

50

lines from 25Si decay as calibrants. The energies of these
lines are determined mainly by the measurement of the
excitation energy of the lowest T=3/2 state in 25Al carried
out by Rogers et al. (Ro77a). This result, 7961(2) kev,
coupled with the 2“Mg(p,y)25Al Q-value (Wa77), 2271.3(8)
keV, and the excitation energy (En78) of the first excited
state in 2"Mg, 1368.59(4) keV, leads to lab proton energies
of 4089.0(22) and 5402.2(22) keV. Peak positions were
obtained by fitting bivariate Gaussian distributions to the
two—dimensional (recoil mass vs. proton energy) spectra,
using the method of Maximum Likelihood (Me75). Figure 2-17
shows an example of a bivariate Gaussian distribution. The
use of such distributions with non-zero correlation
coefficients gives much improved results over simple
projection of recoil bands onto the energy axis because much
of the proton line broadening caused by B-recoil can be
removed, since it appears as a correlation between proton
energy and recoil time of flight. In this way the proton
energy resolution was improved from 25 keV to 15 keV.

An illustration of two extreme cases of the correlation
between proton energy and recoil time of flight is given in
Figure 2-18. This shows the momentum vectors when the
B—decay and subsequent proton decay are (a) parallel and (b)
anti-parallel. For case (a) the excited 2"Al (following
B-decay of 2“Si) is left recoiling toward the MCP detector.
After proton emission the measured proton energy will be

lower and the 23Mg will take less time to reach the MCP.

51

 

Figure 2-17. Three-dimensional plot of a bivariate Gaussia

distribution.

O
"Si
<E-—- Chab
3 “Al"

<e—-—.<s—a o—e»
fl p 23Mg
Opp: 0°
a

52

O
“s;
<4 ———>
"Al" 3

<E———d 03- #—-—e>
' P ”Mg 13
9pp=180°
b

Figure 2-18. Correlation between proton energy and recoil

time of flight.

Momentum vectors when the beta and proton

are: a) parallel, b) anti-parallel.

53
For case (b) the excited 2"Al is left recoiling toward the
Si detector, so the measured proton energy is shifted up and
the 23Mg will take longer to reach the MCP. According to
equation 2-1 the calculated recoil mass for case (a) will be
smaller than that calculated for case (b). This correlation
is observed as a tilting of the mass groups shown in Figure
2-12.

The shapes of the bivariate Gaussian distributions were
fixed using the strong 2sSi lines as standards. In general,
the recoil distributions are a function of 8+ end point
energy, but the end points for 21‘Si and 25Si are so similar
(within 260 keV) that the same recoil distribution
parameters were used for the two isotopes. There is a
background underlying the 2"Si peak which was fitted making
various assumptions about its dependence on recoil mass and
proton energy. The result for the centroid of the 2"Si peak
was insensitive to these assumptions, and the final form of
the background chosen was linearly varying in the recoil
dimension and constant in the energy dimension. Integration
of the maximum likelihood function provided an estimate for
the uncertainty in the centroid position. Appendix C
contains a discussion of the method of Maximum Likelihood
and its application in this analysis.

A possible systematic effect arises from the B+Aproton
angular distribution. Even though neither the positron (in
double-coincidence data) nor the neutrino are detected,

there are still two effects which can lead to centroid

54
shifts of the proton lines:
1) Kinematic terms in the B+-p correlation
function with a dependence on the cosine of the
angle between the positron momentum and the

8p):
2) Rejection (loss) of proton events for which the

proton momentum (cose

associated positron passed through the Si

detector (i.e., events with 6 €30 ).

8P

Both of these effects were analyzed approximately using
the formalism of Holstein (H074) and were found to cause a
centroid shift of the order of +6.4 keV in the 2“Si peak.
The second effect was found to be the dominating cause of
the centroid shift. This small shift is, furthermore,
almost exactly cancelled by a commensurate shift in the 2581
calibration lines which arise from a quite pure Fermi
transition of similar energy (Se73). Thus, no correction is
needed. Appendix A contains a description of this calcula-
tion.

A linear energy calibration was used to extract the
energy of the 2"Si proton group. The latter group lies
sufficiently close to the 4089 keV line that only small
effects would be expected from neglecting higher-order
terms; however, this assumption was tested by extracting the
energy of the 4669(4)-keV group (Se73) from 21Mg decay for
several different experimental arrangements, involving
different analog-to-digital converter gains, amplifier gains

and amplifier manufacturers.

55

25.
$1

The 21Mg group lies almost midway between the two
calibration lines, so it was a more severe test of the
linearity assumption (see Figure 2-13). Centroids were
extracted from a projection along the energy axis as in
Figure 2-15 using the peak-fitting routine SAMPO (R069).

The standard deviation of the data indicated that the actual
uncertainties were slightly larger than the statistical ones
(which were determined from the fits). An average linearity
error was thus added in quadrature with the statistical
error. It is assumed that departures from linearity will be
in random directions from run to run. The size of this
error was chosen to bring the reduced chi-squared for a set
of four 21Mg energy measurements to approximately unity.

The value obtained was 1.5 keV.

The uncertainties present in the measurement of the
laboratory proton energy from decays of the T=2 level in
2“A1 are the aforementioned linearity error (1.5 keV), the
statistical error in fitting the proton peak (2.5 keV), and
the uncertainty in the calibration lines (2.2 keV). A sum
of these in quadrature (i.e., ‘20: ) yields an uncertainty
of 3.7 keV. The value for the picton energy from the decay
of the T=2 level in 2l+Al is:

3912.7(37) keV.
A weighted average of this value with the result of Aysto et
al. (3914(9) keV), is

3912.8(36) keV

where the common 2.2 keV calibration error is removed before

56
calculating this weighted average. It is then added back
(in quadrature) to obtain the above value.
Chapter Four contains a discussion of the effect of

these results on the IMME fit to the A=24 quintet.

2.8 2"Si Half-life Measurement

The half-life of 2"Si was extracted from the data using
the well-known half-lives of 2°Na (T%=446(3) ms), 21Mg (Tg
=228(3) ms) and 25Si (T%=125(3) ms) as calibrations. Figure
2-19 shows a portion of the ramp data for the 4089 keV peak.
The width of the ramp spectrum determined the total interval
for detecting activity. This interval was divided into four
equal sections, corresponding to equal counting times. The
end points of each of these sections served as gates for the
energy and recoil data. Four two-dimensional spectra of
particle energy vs. recoil mass were generated for each
activity. These data were again fit with the bivariate
Gaussian distribution, but now the only free parameter was
the peak height. These heights would of course be
proportional to peak area and, therefore, the decay rate in
the given interval.

In order to calculate the half-life it was necessary to
determine the actual time that the four intervals
corresponded to. The foil wheel was stepped at a rate of 5
Hz, so the total interval was about 230 ms. However,
because the stepping time was only approximately measured

(526 ms), the exact length of time that the catcher foils

57

 

 

 

 

 

 

Ramp Spectrum for 25Si

100 -
a:
“J
m
2
D ..
z
..n
m
é
‘<1O -
I
C)

1 J l J g I

0 1O 20 3O 4O 50

Figure 2-19.
4089 keV peak.

COUNTS PER CHANNEL

Portion of the ramp spectrum from the 258i

 

58
were between the detectors was unknown. This difficulty was
resolved by using the known activities to define a time
scale.
The half-lives were first calculated for the
calibration activities assuming a known total time interval.
These values were compared to known half-lives using the

expression 2-3.

 

2
3 T5heo- 'r 5"?
2 = 1 l
X Z otheo (2-3 )
i=1 i

e .
TXP 18 the

Where Ttheo is the calculated value,
experimental value and Oitheois the calulated uncertainty.
The three calibration half-lives were known experimentally
to equal precision and therefore these uncertainties were
not included in the X2. First a total time interval was
assumed, and the half-lives were calculated and compared to
their known values. The total interval was then varied
until the best agreement (smallest X2) between the
calculated and known half-lives was achieved. The best
total time interval was found to be 189(1) ms. This
represents a stepping time of only '3.11 ms.

In Figure 2-20 a plot of the extracted peak height as

210.

a function of time is shown for the decay of Si. This

measurement yields a half-life for the B-decay of 2|'Si of,

T8 = 103(42) ms.

59

25 l I I saw—no.3"
218: DECAY

 

 

 

10—

COUNTS

 

 

 

 

., J T

0 50 10iO 1510 200
TIME [ms]

Figure 2-20. Decay of 24Si. The half-life derived is
103(42) ms.

 

60
This result is in good agreement with the predicted value of
115 ms, based on a shell model calculation by Robertson and
Wildenthal (R073). It is also in good agreement with the

. + g .. ..
estimate of 100_“: ms made by Aysto et al (Ay79).

CHAPTER THREE

3.1 Introduction

The A=8 isobaric quintet is of special interest in that
it is the only completed quintet which shows a slight devia-
tion from the quadratic form of the IMME. Figure 3-1 shows
the residuals (Mexp-MIMME) of the quadratic IMME fit for the
A=8 isobaric quintet. These values were derived following
the remeasurement of the 6He mass excess (R078). It can be
seen that a small but significant departure from the IMME
exists. Isospin mixing in the Tz=fl member of the multiplet
(aBe) has been suggested as a possible cause for part of
this deviation (R076). For quintets, as previously
discussed, mixing with T=fl or T=l states immediately causes
a T; dependence in the IMME (see Figure'l-2). To understand
these effects further, a remeasurement of the T=2 levels in
8Be and 8Li was undertaken at the Michigan State University
Cyclotron Laboratory.

The first Observation of the lowest T=2 level in 8Be
was made by Black et a1. (8169). This was Observed as a
narrow resonance using the 6Li(d,p)7Li and 6Li(d,a)"He
reactions. A more precise measurement was subsequently

carried out by Robertson et a1. (R075). This work verified

61

62

 

 

 

 

 

 

 

50. I I I I MSU'X-BZ-MQ
A=8 ISOBARIC QUINTET
3, 25.- ‘
LLI
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Figure 3-1. Residuals for the quadratic IMME fit to the
A=8 quintet (R078).

63
the assignment of the resonance as a 6+ T=2 state (the
angular distribution displayed a prominent L=fl shape). The
1°Be(p,t)aBe reaction was used for that work and serves as
the basis for our new measurement. A measurement of the
radiative width of this level by Noe et al. (N076) using the
6Li(d,Y)eBe reaction provided a value for the excitation
energy of comparable precision.

The only measurement of the lowest T=2 level in 8Li was
made by Robertson et a1. (R075) using the 10Be(p,3He)°Li
reaction.

In the previous measurements by Robertson et al.
(R075), the reactions were analyzed using an Enge split-pole
magnetic spectrograph, and the reaction products were
observed using a position-sensitive Si detector placed in
the focal plane of the spectrograph. The spectrograph field
was set to place the reactions of interest onto the active
region of the detector and then was successively changed to
place the calibration reactions on the detector. This
required an accurate magnetic calibration of the
spectrograph (Tr7l).

An alternative approach, developed by Nolen, Hamilton,
Kashy and Proctor (N074), is to use nuclear emulsions in
the focal plane of the spectrograph. The advantage in this
method is that the necessary calibration lines can be
recorded simultaneously with the reaction of interest. This
helps reduce some of the systematic errors that are

associated with sequential measurements, such as changes in

64
the beam energy, spectrograph saturation and hysteresis
effects as well as changes in the target thickness resulting
from deterioration or a build up of contamination. In
addition, this approach enables a linearized least squares
fit to be performed to determine all the unknowns in the
experiment (assuming sufficient calibration lines are
available). These unknowns are the beam energy, the
reaction angle, and the spectrograph focal plane
calibration.

Photographic emulsions are more linear and more stable
than counters and they provide better position resolution
than that Which is obtainable with electronic detectors (at
best 6.25 mm). There are, however, several drawbacks to
using emulsions. These drawbacks include poor particle
identification properties, poor saturation characteristics
(overexposure) and the inability to display the data in real
time as is possible with electronic detectors.

Though there usually is some observable difference in
the track density between different types of ionizing
particles, this difference cannot be relied on in most
cases to observe a low cross section reaction in a
high background environment. A partial way around this
particle identification difficulty was worked out by
Robertson et al. (R078a) through the development of an
electrostatic deflector Which fitted in the gap of the Enge
split-pole spectrograph. Figure 3-2 shows the location of

the deflector in the spectrograph.

65

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The deflector plates were oriented parallel to the pole
faces and provided a vertical deflection of the particles.
The shape of this deflector was designed so that, to good
approximation, the deflection of the particles is constant
along most of the focal plane. For particles of the same
magnetic rigidity, the vertical deflection is pr0portional
to m/q where m is the mass and q is the ionization state of
the particle. In general, the deflection at the high-rho
end of the spectrograph is approximately 16.6(Vq/E) mm,
where V is the applied voltage in kV, and E is the kinetic
energy of the particle in Mev (R078a). This deflection
resulted in tracks in the emulsion that were vertically
separated into bands along the plate according to the m/q of
the reaction products. For this method to be of use, small
vertical beam spots are required. So care must be taken to
focus the beam onto the target. The deflector would of
course not be of use if particles of the same m/q from some
other reaction fall on the peak of interest.

Figure 2—2 shows the beamline arrangement to the
spectrograph (located in vault 3). The dipole magnets M3
and M4 served to analyze the beam energy and the dipole M5
directed the beam to the spectrograph. The quadrupoles
(labeled Q7, 08, etc.) focused the beam on the target.

The beamline to the spectrograph was designed (8171) so
that the beam could be brought to a dispersed focus at the
target of the spectrograph. By matching the dispersion of

the beam to the dispersion produced by the spectrograph, the

67
line widths in the focal plane could be significantly
reduced (C059). A depiction of a dispersion matched
condition is shown in Figure 3-3. Rays leaving the target
from the E+AE side have a larger radius of curvature (as a
result of their higher energy) but because they are
physically displaced as shown they move towards the central
ray of energy E. The E-AE rays with their smaller radius of
curvature will also move towards the central ray because
they leave the target from the opposite side of the central

E ray.

3.2 Experimental Procedure.

A 45 Mev beam of protons from the MSU cyclotron
bombarded a 113 ug/cm2 target of 94% enriched 10BeO on a
1mg/cm2 Pt backing. This was the same target as was used in
the previous work by Robertson et al. (R075). The target
material was produced in the Oak Ridge high-flux isotope

)1°Be reaction (G074).

reactor via the 13C(n,a
The proton beam was used to populate the T=2 states in
8Be and 8Li via the (p,t) and (p,3He) reactions
respectively. Tritons, deuterons, and helium 3 reaction
products were recorded on 20—inch photographic plates (I174)
mounted in the focal plane of the Enge split-pole
spectrograph. Deuterons were recorded simultaneously with
the tritons and were used as calibrations in the 1°Be(p,t)

experiment. In the 1°Be(p,3He) experiment only the 3He

spectrum was used. Two sets of exposures were made at

68

Spectrograph

 

_ Target

 

 

 

E+AE E-AE
E
Beam

Figure 3-3. Dispersion matching in the spectrograph.

69
laboratory angles of 8, 9, and 16 degrees. The
electrostatic deflector was run at a negative 26 kV
potential on the upper plate (deflecting the positively
charged ions upward).

The spectrograph focal plane is angled at 45 degrees
relative to the particle trajectories (see Figure 3-2).
Particles entering the photographic emulsions do so at 45
degrees with respect to the surface. After development, the
particle paths appear as small tracks under the microscope.
The brightness and density of the tracks depend on the
ionization properties of the particles. Because the
ionization of a charged particle increases as it slows down,
absorbers were placed in front of the emulsion to lower the
particle energy. A 6.662-inch (56-micron) Al foil was used
with both the helium 3 and triton plates while an additional
6.62-inch (6.5-mm) acetate absorber was used with the
tritons. This enhanced the visibility of the particle
tracks.

The plates were developed in a dilute solution of D-l9
developer at 4°C. They were scanned using the automatic
plate scanner developed at MSU by Robertson and Nolen
(Ro73a). This scanner uses an analog approach to the
problem of track identification rather than the more common
digital one (Er79). Tracks are imaged onto a three-segment
diode array through a rectangular aperture (6.6 um x 25 um)
oriented paralled to the tracks. Bright-field illumination

is provided by a high-intensity xenon arc lamp filtered by

70
three infrared filters and a Kodak blue-green gelatin
filter.

Automatic focusing is maintained with a motor drive
connected to the fine focusing knob of the microscope. This
is controlled via a pressure-sensing arrangement that keeps
the objective lens at a constant distance from the emulsion.

The scanning was done in 6.661-inch (25.4-um) steps
with vertical bands that were 4 mm wide. Only a l6-inch
segment of the plates could be scanned at one time, so it
was necessary to scan the 26-inch plates in two segments.

The plate holder moves in a rectangular grid pattern
with tracks passing across the diode array vertically. As
the tracks pass across the aperture the light reaching the
diode array is reduced. This signal is inverted and sent to
the scanner electronics. When these signals are viewed on
an oscilloscope a track event registers as a peak in the
trace. For valid track identification several criteria must
be satisfied simultaneuosly. These include the signal
height, its duration, rise time and centering on the diode
array. If all these requirements are satisfied a valid
track count is recorded at the current position in the scan.
These counts form a histogram of the band of tracks along
the plate. The reliability of the system has been elicited
in a number of comparisons with hand scanned spectra (R081).

Data for both the 8Be and 8Li experiments were recorded
sequentially during the same runs. Two plates were placed

in the focal plane one above the other. The spectrograph

71

angle and the field were set for a given reaction and one
plate was exposed. Then the field was changed for the other
reaction and the plate holder moved so the other plate could
be exposed. Running these experiments together enabled the
use of the beam energies determined in the triton exposures
to be used in the helium 3 experiment. This was necessary
because the observation of only the (p,3He) reaction in the
8Li experiment did not provide the needed decoupling of the
beam energy and the spectrograph angle. Therefore, an
average beam energy was used for the analysis of the 3He
data. The Q value showed little sensitivity to the beam

energy so this procedure was not a problem.

3.3 Analysis

Spectra from the plates are shown in Figures 3—4 and
3-5. Higher energy particles are at lower channel number.
The resolution obtained is 25-35 keV FWHM. Figure 3-4
contains both the deuteron band and the triton band (that
were collected simultaneously during the exposure). The
deuteron band contains some additional peaks from reactions
in the platinum backing. These are the unlabeled peaks at
low channel number. The labeled peaks indicate the
calibrations used in determining the energy of the T=2
state. The ordering is according to increasing channel
number.

The peaks in the deuteron spectra numbered 1, 5, 6, and

14 show a reduction in counts near their centers. This is

72

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Figure 3-5.

degrees.

74

caused by the high track density in the center of these
groups. The track number becomes so large that a large
fraction of the tracks overlap and the scanner electronics
can no longer distinguish the individual tracks. The
overexposure of these lines was necessary in order to obtain
enough statistics in the T=2 state.

A specialized peak fitting routine was written to
determine the centroids of the "burned-in" peaks. This
program performs a peak fit with a symmetric three-parameter
gaussian with a background that could include constant,
linear or quadratic terms. The code allowed the removal of
the central portion of the peak from the fit. The method of
maximum likelihood was used to find the best fit and to
determine the centroid uncertainty.

The use of the maximum likelihood method was necessi-
tated by the discovery that a chi-squared minimization
method did not produce realistic centroid uncertainties for
fits to the burned-in peaks. This was determined by testing
with normal peaks that had their center regions removed. As
more channels were left out of the fit, the resulting
centroid uncertainties decreased. Because less information
should not produce greater certainty in the results, a
different approach to the calculation of the centroid
uncertainties was tried. Integrating the likelihood
distribution with respect to the centroid provided a
reliable estimate of the centroid uncertainty. See

Appendix C for a discussion of the method of maximum

75
likelihood.

In addition to three-parameter symmetric gaussians, an
attempt was made to fit the peaks with functions containing
more parameters. This was done in the hope that the
slightly asymmetric nature of the peaks would be better fit
with a distribution that could include this asymmetry.
Four- and five-parameter functions were tried but the extra
parameters produced unrealistic peak shapes. So only fits
with three parameters were used in the final analysis.
Because of the inability to obtain realistic peak shapes
with the four- and five-parameter functions, it was not
possible to determine if a systematic error was present when
the fitting was done with the three-parameter functions.
However, because of the large number of calibration lines on
each plate, as well as the number of plates used in the
analysis, the deviations should average out. Furthermore,
if a constant centroid shift were introduced, the relative
peak separations would not change and the shift would have
no effect. No additional uncertainty will be introduced to
account for this possible effect.

DOALL, a code developed at MSU by Hamilton and Vance
(Ha74), was used to determine the Q values. This program
performs an energy calibration on a given spectrum. It
utilizes a linear chi-squared minimization procedure on
designated calibration lines with accurately-known reaction
excitation energies to calculate changes in the beam energy

(AE) and spectrograph angle (A9) while simultaneously

76

calculating the parameters for the relation between the
effective radius of curvature p and the particle position D,
along the focal plane of the spectrograph. Measurement of
the spectrograph magnetic field, combined with the effective
radius of curvature, enables the determination of the
particle energy. This code then calculates accurate
excitation energies and Q values from the fit parameters.

Nominal values for the magnetic rigidity, (Bpi)O for a
given line i are provided from calibration of the beam
analyzing system. The analyzing system determined the beam
energy to 3166 keV. The spectrograph angle was known
nominally to 6.2 degrees. DOALL calculates corrections to
the nominal values of the beam energy Eo and the scattering
angle 90 and applies these to an actual value of B01 using
the first order expansion,

i

. i 1
BO = (Bpl)o + ELEM AB + 34133; AG (3-1)

8E 30

The relation between D and D is usually expressed in
terms of an expansion around an arbitrary point and takes
into account the nonlinear relationship between these
quantities.
pi = p0 + a(Di-DO) + 8(Di--DO)2 (3-2)
This can then be related to the magnetic rigidity as

Bpi = Boo + Ba(Di-DO) + BB(Di--DO)2 (3-3)

The po,<1, and B are the unknowns of the spectrograph

calibration. Combination of equations 3-1 and 3-3 results

77

in an expression for each calibration line which contains

five unknowns.

_ 3 (B01)

1 - - -—
(Bo ’0 ‘ as 30

A6 + BpO +

(3-4)

+ aB(Di - D0) + eswi - DO) 2

With five or more appropriately-chosen calibration lines it
is then possible to determine these unknowns. Once the
chi-squared minimization procedure determines the best
values for these unknowns the energy and Q value of the
reaction of interest can be obtained (N073).

Table 3—1 contains the excitation energies used in the
analysis (Aj79,Aj86,Aj81). Recent high precision
measurements (Bi81,Ba82) of states in 1“N were combined in
quadrature with the compilation of Ajzenberg-Selove (Aj81).
These are shown in Table 3-2. The Q values used in the
analysis were those calculated from the 1981 mass tables of
Wapstra and B08 (Wa81).

The weighting for each calibration line included both
the centroid uncertainty from the peak fitting routine taken
in quadrature with the excitation energy uncertainty. The
excitation energy was first converted to channel number via
the first order conversion between energy and channel number
along the focal plane. In the compressed spectra the
energy/channel number conversions for 3He, d, and t were
5.6, 6.9, and 4.5 keV/channel respectively.

The target thickness and uniformity were determined

using the alpha particle backscatter technique (Fo77a,Ro82).

78

Table 3-1 Calibration excitation energies for the
8Li and 8Be experiments.

Nuclide Excitation Peak Numbers
Energy(keV) in Spectra

8Li Experiment

1“N 6.6 1
2312.96(3) 4

3948.12(l8) 6

5165.88(15) 8

5691.43(12) 16

5834.21(16) 11

6263.5(6) 12

7628.94(13) 16

7966.6(6) 17

8487.7(12) 18

9172.44(36) 19

8L1 6.6 2
986.8(1) 3

2255.(3) 5

16822.2(55) 23

1°8 6.6 7
718.32(9) 9

l746.16(l7) 13

2154.3(5) 15

5163.9(6) 26

8Be Experiment

150 6.6 5
5183.(l) 16

5246.9(3) 11

6176.3(17) 14

9Be 6.6 l

2429.4(13) 2

14392.6(18) 12

11c 6.6 6
2666.6(5) 9

1“o 6.6 3
6596(16) 8

8Be 27494.6(26) 13

1°C 6.6 4

3356.8(9) 7

Table 3—2

Excitation
Energy (keV)
(Aj81)

2312.87(7)
3947.8(4)
5165.87(18)
5689.6(11)
5834.23(21)
6263.5(6)
7627.9(14)
7966.6(6)
8487.7(12)
9176.7(16)

79

Excitation
Energy (keV)
(Bi81)

2312.96(3)
3948.2(2)
5165.9(3)
5691.55(13)
5834.3(3)

7629.4(3)

9172.5(3)

1+ 0 o o
1 N exc1tation energies.

Excitation
Energy (keV)
(Ba82)

5696.5(4)
5834.6(4)
7628.85(14)

Weighted
Average (keV)

2312.96(3)
2948.12(18)
5165.88(15)
5691.43(12)
5834.21(16)
6263.5(6)
7628.94(13)
7966.6(6)
8487.7(12)
9172.44(36)

80
A 3 MeV alpha beam, from the single-ended Van-de-Graaff at
Los Alamos National Laboratory, was used in the measurement.
Alpha particles were backscattered off the platinum target
backing. The energy shift and width of the high energy side
of the Pt backscatter peak (with particles passing through
the BeO layer) showed that the target was quite uniform
(though the platinum backing appeared not to be). The
measured target thickness of 113 ug/cm2 was in excellent
agreement with a previous measurement of 114 ug/cm2 by
Robertson et al. (R075).

To account for energy loss in the target, corrections
were made to the excitation energies corresponding to the
calculated energy loss for each reaction in the target. The
program SPECTKINE IV, developed at MSU by Trentelman,
Robertson, Gleitsmann and Robinson (Tr76b) contained the
calibration of the beam transport system and calculated the
particle energies for the reactions. The energy loss for
each reaction was then calculated using the stopping power
tables from Saclay (W166). An average beam energy was used
in this calculation, and the particle energies were
determined for a scattering angle of 9 degrees. Energy loss
corrections for the deuterons and tritons were between 6 to
4 keV while those for 3He were between 6 to 13.2 keV.

The 1981 mass tables have not as yet been incorporated
into the library of the MSU computer system. Therefore, to
take into account the new 0 values, it was necessary to

enter corrections to the excitations energies used in the

81

calibration. For a given reaction, the ground state as well
as all the excited states were corrected with the negative
of the Q value change (1981-1977).

Table 3-3 lists the ground state mass excesses that are
needed in the calibration reactions (Wa77,Wa81). There
currently exist some discrepancies between the adOpted
values and some recent high resolution mass measurements.

In particular the masses of 1“O and 10C are in doubt.
Unpublished measurements by Barker and Nolen (Ba82a), differ
by several keV from the 1977 and 1981 mass tables (see Table
3-3). An additional and independent measurement of the
10C-“O mass difference (see footnote in Table 3-3), also by
Barker and Nolen, is consistent with the mass difference
extracted from their individual mass measurements. Table
3-4 lists the Q values based on the 1981 and 1977 mass
tables for all the ground state reactions in the
experiments. The Q value changes using the Barker and Nolen
mass excesses are also listed. An analysis of the 8Be data
using these alternate values produced a change in the 8Be
T=2 Q value of less than 6.13 keV. These masses were not
used in the 8Li measurement, and so had no effect on the Q

value in that case.

3.4 Results
The deviation of the Q values (about their unweighted
average) derived from individual plate exposers, from the

8Be and 8Li experiments are shown in Figures 3-6 and 3-7.

Table 3-3

Mass Excess
1977

18 7289.634(23)
H 13135.84(4)
3H 14949.94(5)
3He 14931.32(5)
Li 26946.9(9)
88e 494l.76(16)
Be 11348.6(4)
1°8e 12667.6(6)

1°8 12651.7(5)

c 15762.9(7)
11c 16656.6(11)
12c 6.6

1“N 2863.444(23)
o 8668.3(5)
150 2855.4(7)

o

-4737.62(4)

1977 and

82

1981 mass excesses.

Mass Excess
1981

7289.628(23)
13135.823(42)
14949.916(52)
14931.31l(52)
26945.312(787)

4941.696(165)
11347.666(397)
12666.869(412)
12656.642(393)
15766.472(394)
16649.287(435)

6.6

2863.436(24)

8666.563(81)

2855.456(636)

-4737.648(47)

Difference

-6.666
-6.617
-6.636
-6.669
-l.588
-6.664
-6.466
-6.731
-1.658
~2.428
-6.713
-6.614
-1.737

6.656

(1981-1977) (Nolen-1977)

a
-2.7l(47)

6.61(29)

a
6.35(51)

Calculated 10C-“O mass difference relative to the 1977 mass

table is 3.66(42) keV.

keV (Ba82a).

Measured mass difference is 3.44(27)

83

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85

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Figure 3—7. Deviations of the Li Q values from their

unweighted average.

86

In the 8Li data, one 9-degree run was not included because
of an incorrect setting of the spectrograph field which
placed the T=2 state off the end of the plate. The
uncertainty for each value is from a sum (in quadrature) of
the centroid uncertainty (from the peak fitting routine) and
the uncertainty in the excitation energy (calculated in
DOALL from the variance-covariance matrix). The difference
between a weighted and unweighted average of these data is
also shown in the figures. The standard deviation of the
data points (internal error) is shown as dotted lines in
these figures. The weighted average falls well within the
internal spread of the data. The unweighted average of the
Q values is adopted for the final result. For the
1°Be(p,t)°Be reaction the Q value is

~27484.3(l4) keV.
For the 1°Be(p,3He)8Li reaction the Q value is

-26821.3(57) keV.

Table 3-5 contains a list of the major uncertainties in
the measurements. All the uncertainties were taken in
quadrature to yield the errors quoted above.

For the 8Be experiment there are four ground state Q
values that could change:

Q =12C(Ptd)llc

1
02=1°Be(p,d)9Be
03=160(p.t)1“o

Qu=12C(ptt)1 0C0

Future corrections to our 0 value will be approximately

87

Table 3-5 Contributing uncertainties to the Q value

measurements 0

A 16% uncertainty in 10BeO

A possible 16 ug/cm2 carbon
layer

25 keV shift in beam energy for
8Li experiment

16% uncertainty of the relative
shift between d and t from
deflector

Uncertainty in appending two
halves of the scans together;
1/16 channel shift to all
peaks ?- 1266 channels

Barker and Nolen Q values
(Table 3-4)

Uncorrelated Q value
uncertainty from %

2 so (T=2) 265

i BQi 1
(Partials from Equations
3-5 and 3-6. Q value

uncertainties from

Internal errors
(See Figures 3-6 and 3-7)

88e(T=2)
AQ(keV)

6.36

6.23

0.12

8Li('r=2)
AQ(keV)

6.42

6.45

6.32

Total sum in quadrature

1.41

5.67

88

AQ=(6.59)AQI+(6.1634)AQZ+(6.625)A03+(1.6217)AQH (3—5)

where AQ's are defined as QNew-QOld°
For the 8Li experiment the qround state 0 values
subject to change are
Ql=12C(p,3He)1°B
Qz=1°Be(p,3He)°Li.

Possible corrections to our value will be approximately

Equations 3-5 and 3-6 are based on the approximation
that small Q value changes may be taken as independent.
These expressions give the change in our result as a
function of calibration Q value changes. The coefficients
are partial derivatives calculated by making 1 keV changes
to the calibration O values and taking the difference of the
averages for all the runs before and after the change. This
sum of partial derivatives works quite well and gives
reasonable results even when several values are changed at
once.

The Q values from the measurement of Robertson et al.
(R075) are -27487.6(26) and -26864.1(54) keV for the
reactions 1°Be(p,t)BBe(T=2) and 1°Be(p,3He)°Li(T=2)
respectively. Comparison of our values with these previous

results shows a fairly good agreement between the 8Be

89

measurements. For the 8Li 0 value, however, our new value
is 17.2 keV below the older measurement. The calibration Q
values used in the Robertson analysis have only changed by a
few keV and are not the cause of this discrepency. Because
of the introduction of the additional calibration lines as
well as the non-sequential aspect of the new measurements,
it may be argued that only the new values should be adopted.

The mass excess for the T=2 state in 8Be (based on our
new Q value of -27484.3(14) keV for the reaction
1°8e(p,t)°8e) is

32436.4(15) keV.

The resonance energy for the reaction 6Li(d,Y)BBe(T=2)
by Noe et al. is Ed=6962.8(36) keV. This measurement is
based on an accelerator energy calibration of Ep=14236.75(26)
keV from the resonance reaction 12C(p,Y)13N(T=3/2) (Aj81).
This calibration has not changed, so just using the Ed
resonance energy and the 1981 mass table to recalculate the
Q value, the T=2 8Be mass excess becomes 32435.7(23) keV.

A weighted average of the mass excesses obtained from
the resonance experiment of N08 et al. and from our (p,t)
work is 32432.6(13) keV. Because these two values are two
standard deviations apart, the approach of the particle data
group (Ba76) has been adopted. Namely, the uncertainties of
the measurements have been scaled to bring the reduced
chi-squared to unity. The resulting change is only in the
size of the error and does not affect the value of the

average. The adopted mass excess for the T=2 state in 8Be

90

is then
32432.6(24) keV.

The 8Li T=2 mass excess (based on the Q value from our
new measurement -26821.3(57) keV and the 1981 mass tables)
is

31785.9(57) keV.
Chapter Four contains a discussion of the effect of

these new results on the IMME fit to the A=8 quintet.

CHAPTER FOUR

4.1 A=24 Quintet

In the A=24 quintet, the mass of the Tz=2 nucleus 21‘Ne
was measured by Silbert and Jarmie (Si61) using the
22Ne(t,p)2”Ne reaction. The most accurate measurement of
the lowest T=2 state in 2“Na was made by Start et al. (St73)
via the 22Ne(3He,py) reaction. The lowest T=2 level in 21'Mg
was first observed in the 26Mg(p,t) reaction by Garvey,
Cerny and Pehl (Ga69). It has subsequently been studied as
an isospin-forbidden resonance in the 23Na(p,y) reaction by
Riess et al. (Ri67), by Szucs, Underwood, Alexander, and
Anyas-Weiss ($273), and by Heggie and Bolotin (He77).

The T=2 level in 2“Al was first observed by Aysto et
a1. (Ay79) using the 2"Mg(3He,3n)2“Si reaction. The 2"Si
activity was transported via helium jet to an on-line mass
separator system. Mass-24 activity was deposited in front
of a AE-E telesCOpe and the energy of protons from the T=2
level in 2"A1 (POPUIated by the beta-decay of 21+Si) were
measured. This level has subsequently been measured using
the same reaction (see Chapter Two and Le86). The Observa-
tion of 2"Si in the 28Si("He,8He) reaction by Tribble et al.
(Tr86) brings this quintet to completion.

In the A=24 multiplet some of the mass excesses are

91

92

derived from measurements of nuclear reaction Q-values, some
from excitation energies and some from proton resonance or
decay energies. Considered independently, each of these T=2
state mass excesses includes in its uncertainty the
uncertainty in a ground state mass excess. However, mass
differences in a local group of nuclides are frequently
known more precisely (by direct measurement) than are the
absolute mass excesses themselves. It is these mass
differences which are relevant in testing the IMME because
the higher-order terms are all expressible as mass
differences. In other words, the uncertainties in the
ground state masses are strongly correlated, and a proper
evaluation of the uncertainties in the coefficients of the
IMME requires consideration of these correlations.

We therefore return to the original measurements which
link the nuclei of interest. Those measurements constitute
a practically uncorrelated body of data, although there are
small correlations which arise from the use of common
calibration lines, e.g., the 6129.17 keV line of 16O. The
uncertainties in these calibrations are small enough that
their influence can be neglected. We may confine our
attention to masses which affect the T=2 states in 2"A1,
2“Mg and 2“Na because the uncertainties in the masses of
2"Si and 2"Ne, 22 keV and 16 keV, respectively, are so large
Furthermore, the masses of the T=2 states in 2“Al and 2"Na
are, for all practical purposes, correlated only with the

masses of 23Mg and 2"Na (respectively), while in 21’Mg the

93

gamma decay energy to the ground state and the proton
resonance energy in 23Na(p,y) are known to comparable
precision. In earlier mass evaluations these two
independent measures of the mass <15 the 2"Mg T=2 state were
consistent, but the 1977 Wapstra-Bos tabulation (Wa77) leads
to values obtained in the two types of experiment
(Ri67,Sz73,He77) which differ by 5.6(24) keV. The origin of
this discrepancy is difficult to identify exactly because of
the global nature of mass adjustments; however, it appears
to lie outside of the local group of masses needed for the
present purposes, a group interrelated by several precise
and consistent experimental measurements.

This problem in the middle of the s-d shell has been
noted by Wapstra and Bos (Wa77). Wapstra (private
communication) has carried out an adjustment in Which the
direct mass spectrosc0pic data is omitted and the results
show reduced closure errors. Furthermore, Wapstra and Bos
have published tables of mass differences which have been
calculated taking correct account of the correlations
between the masses concerned. However, these tabulations do
not list the correlations between mass differences, nor do
they include the linkage through the T=2 state of 2"Mg,
i.e., the proton resonance energy and the Y-decay energy,
which significantly constrains the 23Na-“Mg mass
difference. For these reasons we have adopted a local mass
adjustment.

The ground state masses required in the analysis are

94

23Mg, 23Na, 2“Mg and 2“Na. In addition, the mass of the

T=2 state in 2"Mg should be included on the same footing
as the ground state masses because it links 2"Mg and 23Na.

The data base given in Table 4-1 forms the input for a
local least-squares mass adjustment. Each mass is linked to
at least two others, and 5 masses are related by 7 mass
differences. This is illustrated graphically in Figure 4-1.
The solid lines show the 7 connections between the 5 masses
and the dotted lines represent the measured proton energies
and excitation energies used in determining the T=2 level in
2"A1.

The normalized x: for the fit is 1.1 (2 degrees of
freedom). To obtain actual numerical values for masses we
arbitrarily assign 2“Mg a mass excess of -13936.6 keV -- all
the derived masses are then based on (and fully correlated
with) this mass. The resulting ground state mass excesses
(along with a comparison to the 1977 mass tables) are listed
in Table 4-2. Table 4-3 contains a summary of the T=2 state
mass excesses (and their excitation energies) as derived
from the local mass adjustment. These masses are used to
derive the coefficients of the IMME which are also shown in
Table 4—3. The covariances are negligible with the
exception of 23Na - 2"Na. It should be borne in mind that

the masses in Tables 4-2 and 4-3 do not include the 2“

M9
mass uncertainty (6.7 keV), and that it is this distinction
Which eliminates the otherwise strong correlation between

them. The coefficient (a) is therefore subject to the

95

Table 4-1 Reaction 0 values used in the local mass
adjustment.

Reaction Q values (keV) References
23Na(p,Y)“Mg 11691.2(11) (Wa77)
23Na(n,v)2“Na 6959.4l(12) (Wa77)
23Na(p,n)23Mg -4839.1(26) (Wa77)
2"Mg(p,d).“Mg -l4367.5(15) (Wa77)
2"I~IzaI(I3—)“Mg 5514.8(26) (Wa77)
2"I~Ig('r=2)-+“1~I.-.-1 + p 3741.2(23) (Ri67,Sz73)a
2"I~Ig('r=2)+“mg + y 15436.3(6) (St73,Sz73)a

Weighted average of these measurements.

Table 4-2 Mass excesses obtained using the local mass
adjustment, and a comparison to the 1977 mass table (Wa77).

Nuclide Local Adjustment 1977 Mass Table
Mass Excess(keV) Mass Excess(keV)
23Na -9527.7(9) -9529.6(8)
23Mg -5476.2(13) -5476.6(15)
2“Na -8415.6(9) -8417.5(8)
2“Mg -13936.6a -13936.6(7)
2"IIg(T=2) 1565.5(6) -

Assigned reference mass excess.

96

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U_~l

97

Table 4-3 Summary of the properties of the A=24 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excessa Excitation References
(keV) (keV)
2“s1 -2 l6782(22) 6 - (Tr86)
29 b
Al -1 5963.4(46) 5955.4(57) (Ay79,This
work)
2“Mg 6 1565.5(6) 15436.1(6) (Wa77)
2“Na 1 -2445.4(13) 5976.2(9) (St73)
2“Ne 2 -5949(16) 6 (Wa77)
-------_--___-_---_------------_------------------_-----;---
----3 ___________ E __________ f ........... 3 _________ f _____ 52---
1565.5(6) -4l71.6(19) 224.7(18) - - 6.9
1565.4(6) -4174.3(25) 225.6(18) -1.2(16) - 1.3
1565.5(6) -4l75.3(26) 223.9(28) - 6.4(9) 1.7

1565.5(6) -4l7l.6(35) 222.1(31) -2.8(21) 1.4(12) -

Mass excesses of the three inner members of the quintet
(A1,Mg,Na) make use of the local mass adjustment. See
tables 4-1 and 4-2 for input values. Outer members (Si,Ne)
do not.
b
Weighted average of these measurements.

98

additional uncertainty contributed by the uncertainty in the
mass of 2"Mg.

Examination of Table 4-3 shows that the quadratic IMME
gives a reasonably good fit to the data, with X$=6.9 per
degree of freedom. A plot of the residuals of the quadratic
fit is shown in Figure 4-2. Addition of either a cubic or
quartic term by itself does not improve the quality of the
fit. The uncertainty in the d coefficient, 1.6 keV in the
cubic fit, may be compared directly with other d

coefficients in both quintets and quartets.

4.2 A=8 Quintet

In the A=8 quintet, the T=2 nucleus 8He was first
observed by Cerny et al. (Ce66), using the 26Mg(“He,8He)22Mg
reaction. More recently the 18O(“He,8He)“O (Ja74) and
6"Ni(“He,3He)5°Ni (Ko75a,Tr77) reactions have been used to
determine the 8He mass excess. Previous to the work
described in Chapter Three, the only measurement of the T=2
state in 8Li was made by Robertson et al. using the
1°Be(p,3He)8Li reaction. The T=2 level in 8Be was discovered
as a narrow resonance by Black et al. (B169), using the
6Li(d,p)7Li and 6Li(d,a)”He reactions. This was later
verified by Robertson et al. using the 10Be(p,t)8Be reaction
(R075). The T=2 state in 8B has only been measured once
using the 11B(3He,6He)°B reaction (R075). The T=2 nucleus
8C was first observed by Robertson et a1. (R074), via the

12C(“He,°He)°C reaction. A measurement by Tribble et al.

99

 

 

 

 

 

 

 

50. 1 I I MSU'X-BZ-l47
A=24 ISOBARIC QUINTET

i, 25.- f -
DJ

2

g . , I
2 o I"

'd I

8
2

-25.— -
'-5CL .1 I .1 I I

-2 -l o l 2
T2

Figure 4-2. Residuals for a quadratic IMME fit to the A=24
quintet.

100

(Tr76) also used this reaction. A later measurement by
Robertson et al. (R076), used the 1L'N(3He,9Li)8C reaction to
determine the 8C mass excess. This later measurement helps

8c

to reduce the correlation between the 8He mass and the
mass that results when using the 12C(“He,°He)8C reaction.

Table 4-4 contains the mass excesses for the A=8
isobaric quintet. The 1981 mass tables (Wa81) have been
used to derive these masses. Because of the large number of
connections between the ground state masses in this region,
it was felt that a local adjustment was not necessary. The
measurements of the T=2 states were, however, analyzed with
this table.

Table 4-4 also contains the coefficients from a fit
using the IMME. With the change in the 8Li mass from its
previous value, the deviation from the quadratic form of the
IMME has been eliminated. This holds even when the new and
old 8Li masses are averaged together. Figure 4-3 shows the
residuals of the quadratic fit to the A=8 quintet.

It is striking that even in this case, where several
members of the quintet (8C, 8B and 8Be) are unbound to

isospin allowed particle decay, that agreement with the

quadratic form of the IMME is still maintained.

4.3 Quintet Summary
Besides the A=8 and 24 isobaric quintets, the A=26, 32,
and 36 quintets have been completed. Of the remaining A=4n

systems for A344 there are four multiplets A=12, 16, 28,

101

Table 4-4 Summary of the properties of the A=8 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
8 a
c -2 35695(23) 6 (Wa81)
e b
B -1 33542.3(79) 16622.1(86) (R075)
C
88e 6 32432.6(24) 27496.3(24) (N076,This
work)
8 d
Li 1 31785.9(57) 16846.6(58) (This work)
8He 2 31598.6(91) 6 (Wa81)
------------_------------------------_--_---------------;---
_----f ___________ E __________ f __________ 5----_-_--f_-_-_§2-__
32432.3(23) -876.7(38) 229.8(25) - - 6.2
32432.4(23) -878.1(64) 229.4(28) 6.6(23) - 6.3
32432.6(24) -876.7(38) 232.3(76) - -6.7(18) 6.3

32432.6(24) -879.5(68) 233.3(72) 1.3(26) -1.2(26) -

Based on a weighted average of (Tr76,Ro76).

Used measured Q value ratio from (R075) and 1981 mass
tables (Wa81) to Obtain these values.
0

Adjusted value from (N076) using (W881) then took a
weighted average with this work. Quoted error has been
scaled according to the prescription of (Ba76).
d

The measurement of (R075) has not been included.

102

MSUX-BZ-MS

8 ISOBARIC QUINTET

 

3’
u .

N

9'
l
l

Mexp ' MIMME (REV)
o
r
_.._

 

 

 

 

 

-25.- -
_ so. 1 J I 1 1
-2 -l O I 2
T2

Figure 4-3. Residuals for a quadratic IMME fit to the A=8
quintet.

103

and 46 which contain four known members (A=44 has only three
members known). Each of these is missing the Tz=-l member
of the quintet. Experimentally, this has been the most
difficult mass to measure, because no good general method
has been found to identify the T=2 state in these Tz=-l
nuclei (Be79).

There are at least 9 excited quartets that have been
completed (Be79). However, there is as of yet no case of
a complete excited quintet. Only for the A=l6 multiplet
are four members known. For the A=12 and 26 multiplets
three members are known.

Tables 4-5 through 4-15 contain a summary of these
quintets. The uncompleted cases are included and use the
quadratic form of the IMME to predict the unknown mass
excesses and excitation energies. The residuals of the
quadratic fit to the IMME have been plotted for the three
completed quintets (26, 32, and 36) and are shown in Figures
4-4 through 4—6.

In all cases the 1981 mass tables (Wa81) were used in
the analysis of the data.

Figures 4-7 through 4-16 contain plots of the d and e
coefficients obtained from IMME fits. For fits with only a
d or e coefficient the uncompleted quintets were included.
It is worth noting, that for a multiplet where the Tz=-1
mass is missing, the derived d coefficient will equal the e
coefficient. For the completed quintets both d and e could

be tested simultaneously. These coefficients are shown in

104

Table 4-5 Summary of the properties of the A=12 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
12 a
o —2 32666(45) 6 (Ke78, 8u86)
12 b
N -1 [29589(16)] [12251(16)] IMME
12c 6 27595.6(24) 27595.6(24) (R078)
C
128 1 26686(19) 12711(19) (Ne7l,As76)
128s 2 25677(15) 6 (A178)
' """"""""" g """"""""""" 3""""“"";3"
- ____ f ........................ S--------__-__-_-__-f_-__--2--
27594.9(24) -l749.6(1ll) 244.7(59) - - 6.2

27595.6(24) -l763.6(266) 244.1(66) 4.6(69) - -

27595.6(24) -l747.3(119) 228.3(292) - 4.6(69) -

A weighted average of these two measurements. The value
from ref. (Bu86) required a small -3 keV correction
(M082a).

Prediction using the quadratic form of the IMME.
c

A weighted average of these two measurements. This
differs from the adopted value of Ex=12756(56) keV in
(Aj86) based on (As76) and (Aj75). The latter measurement
is 9Be(7Li,a)12B which is isospin forbidden to the T=2
level and is therefore unlikely to populate it.

105

Table 4-6 Summary of the properties of the A=12 first

excited (2+) isobaric quintet and coefficients of the IMME
fits.

Nuclide T Mass Excess Excitation References
(keV) (keV)
a
12o -2 [33816(866)] [1746(866)] IMME
12 a
N -1 [31566(346)J [14166(346)] IMME
12c 6 29636(56) 29636(56) (A876)
128 1 28189(l66) 14826(l66) (As76)
1 b
2Be 2 27179(19) 2162(12) (A178,Al78a)
a b c d e X3
29636(56) -l657(214) 216(164) — - -
a

Predictions using the quadratic form of the IMME.

A weighted average of these two measurements.

106

Table 4-7 Summary of the properties of the A=16 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
16 a
Ne -2 24622(51) 6 (Ke78,Bu86)
16 b
F -1 [26793(17)] [16113(l9)] IMME
16o 6 17983(5) 22726(5) (Aj82)
16N 1 15669.8(74) 9928(7) (He68)
C
16c 2 13694.2(36) 6 (Wa81)

a b c d e X3
17981.3(47) -2589.4(161) 222.8(49) - - 6.9
17983.6(56) -2595.3(119) 218.8(65) 3.3(35) - -
17983.6(56) -2582.6(128) 265.4(196) - 3.3(35) -
a

A weighted average of these two measurements. The value
from (Bu86) required a -3 keV correction to the Q value
(M082a).

Prediction using the quadratic form of the IMME.
c

This is based on a weighted average of Q values from ref.
(F077) and (Se78).

107

Table 4-8 Summary of the properties of the A=16 first
excited (2+) isobaric quintet and coefficients of the IMME
fits.

Nuclide Tz Mass Excessa Excitation References
(keV) (keV)
16Ne -2 25712(87) 1696(76) (Ke78)
b
168 -1 [22593(28)] [11913(29)] IMME
16o 6 19785(11) 24522(11) (A176)
C
16N 1 17382.8(74) 11761(7) (He68)
15¢ 2 15466(11) 1766(16) (F077)
a b c d e X3
19771.1(94) -2666.9(156) 226.9(75) - - 6.6

19785.6(116) -2615.6(167) 266.3(113) 13.2(54) - -

19785.6(116) -2563.6(219) 147.7(368) - 13.2(54) -

The ground state masses used are tabulated in Table 4-7.
Prediction using the quadratic form of the IMME.

This could be a l- or 2+ level.

108

Table 4-9 Summary of the properties of the A=26 isobaric
quintet and coefficients of the IMME fits.

Nuclide T Mass Excess Excitation References
(keV) (keV)
20 a
Mg -2 1757(27) 6 (Wa81)
20 b
Na -1 13426(56) 6586(56) (M079)
20 C
Ne 6 9689.4(23) 16735.7(31) (B167,Ku67,
Ad69)
20F 1 6562.4(31) 6519.4(36) (Mi76)
20 d
O 2 3799.5(84) 6 (Wa81)
___-__-__---_------_-----__--------__---_--__-------_---3-__
_--_-3 ........... E-__-_---_-f __________ §__--_----f--__-32-_-
9696.4(22) -3437.6(49) 247.5(32) - - 1.1
9689.5(23) -3434.4(56) 249.4(36) -2.3(19) - 6.8
9689.4(23) -3444.6(68) 259.7(95) — -2.6(l9) 6.4

9689.4(23) -3464.1(335) 279.4(335) 5.3(87) -7.6(85) -

Based on a measurement by ref. (Tr76a).

This value based on the mass difference between the T=2
and ground state (Wa81) mass excesses. This differs from
the value of Ex=6576(56) keV from ref. (M079).
c
Weighted average of these three measurements.
d
Based on weighted average of Q value from (Ja66,Hi62).

109

Table 4-16 Summary of the pr0perties of the A=26 first
excited (2+) isobaric quintet and coefficients of the IMME
fits.

Nuclide Tz Mass Excess Excitation References
(kev) (keV)
a
2°Mg -2 [26466(866)] [2966(866)] IMME
a
2°Na -1 [15536(366)] [8696(366)] IMME
b
2°Ne 6 11383.8(29) 18436.1(35) (Ad68,Ku72)
C
20F 1 8633(166) 8656(166) (Aj78)
2°o 2 5473.2(84) 1673.68(15) (Wa73)
’ """""""" g """"""""""" 3"“""""""'Z"
_ -3 ________ ° ............ f __-_-§2-_
11383.8(29) -3746(266) 396(166) - - -
a

Prediction from the quadratic form of the IMME.
Weighted average of these two measurements.

From a measurement in ref. (Ce64).

110

Table 4-11 Summary of the properties of the A=28 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
28s -2 -4134(l66) 6 (M082)
28 a
P -1 [-1252.8(98)] [5968(11)] IMME
23 b
Si 6 -6269.3(31) 15223(3) (Sn69,Je72)
28A1 1 -16858.l(8) 5992.4(4) (St73)
28Mg 2 -15618.7(21) 6 (Wa81)
..-.-___-._-_-._._......._._-._._-.-....................3...
a b c d e Xv
-6269.4(3l) -4862.7(56) 214.6(26) - - 6.1
-6269.3(31) -4798.6(l37) 266.7(266) 2.5(67) - -
-6269.3(3l) -4788.2(466) 196.9(468) - 2.5(67) —
a
Prediction using quadratic form of the IMME.
b

Updated the value from (Sn69) brought it into agreement
with (Je72). This excitation energy differs from that of
(En78) Ex=15225(3) keV.

111

Table 4-12 Summary of the properties of the A=32 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
32 a
Ar -2 -2176(56) 6 (Bu86,M082)
32c1 -1 -8296.4(54) 5633.1(93) (Ha77)
32$ 6 -13971.1(5) 12645.6(4) (An86)
32 b
p 1 -l9232.6(11) 5673.2(9) (Pi66,Ly67,
Va67,Fo72)
3231 2 -24679.1(12) 6 (Wa81)
----_----------_----------------------_--------__-_-----;---
- .... 3 ........... E __________ E __________ §_-_-__--_f-_--_§2_--
13971.1(5) -5468.8(l9) 267.4(11) - - 6.2
13971.1(5) -5468.8(21) 267.3(26) 6.6(16) - 6.4
13971.1(5) -5468.5(27) 266.8(33) - 6.1(5) 6.4

13971.1(5) -5465.5(56) 265.2(43) -2.6(43) 1.4(23) -

Based on Q value for 32$(fl+,fl-)32Ar = -23846(56) keV.

Weighted average of these four measurements.

112

Table 4-13 Summary of the properties of the A=36 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
36 a
Ca -2 -6446(46) 6 (Wa81)
36K -1 -13168(22) 4258(23) (Ay81a)
36 b
Ar 6 -19379.6(22) 16851.8(22) (Hu76,Ma76a)
C
36c1 1 -25223.2(16) 4298.9(16) (Ma75,Ri75,
Ve76)
353 2 -36664.51(26) 6 (Wa81)
' """""""" g """"""""""" 5"""""""';E"'
_____E __________________ c __________ e v _
19379.8(26) -6644.5(34) 261.1(13) - - 1.9
19386.2(22) -6645.3(38) 263.1(46) -6.8(17) - 3.5
19379.6(22) -6643.2(74) 199.3(91) - 6.3(14) 3.7

19379.6(22) -6618.1(151) 176.4(156) -9.5(56) 7.6(41) -

Based on Q value from (Tr77).

A weighted average of these measurements; uncertainty
scaled according to (Ba76). This excitation energy differs
from that of (En78) Where Ex=16853.4(15) keV.
c

Result is a weighted average of these three measurements.
In ref. (En78) the value adopted was from ref. (Ve76)

where Ex=4299.5(11) keV.

113

Table 4-14 Summary of the properties of the A=46 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
u a
0T1 -2 -8792(l66) 6 (M082)
“0 b b c
Sc -1 [-16164(39)] [4364(39)] IMME
d
“°Ca 6 -22874(15) 11974(15) (Ce68,Ha76,
Ad72,Bo77)
“°K 1 -29151.9(12) 4384.6(3) (St77)
“°Ar 2 -35646.8(l4) 6 (Wa81)

a b c d e x:
22859.7(131) -6494.2(197) 261.8(66) - - 3.8
22874.6(156) —6562.3(261) 239.4(263) -15.6(77) - -
22874.6(156) -6562.2(466) 299.3(563) - -15.6(77) -

e e e
22874.6(156) -6472.4(226) 194.5(76) - - -
a

This mass is predicted to be -9151(9l) keV from the
quadratic IMME (using Ca,K and Ar).

This mass is predicted to be -16267(45) keV with
Ex=4321(45) from the quadratic IMME (using Ca,K,

and Ar).
c

Prediction from the quadratic IMME, with Ti in the fit.
d

Weighted average of these four measurements.
e

Coefficients from the IMME fit, did not include Ti.

114

Table 4-15 Summary of the properties of the A=44 isobaric
quintet and coefficients of the IMME fits.

Nuclide Tz Mass Excess Excitation References
(keV) (keV)
a

““Cr -2 [-13461(36)] 6 IMME

a
““v -1 [-21623(14)J [2836(1666)] IMME
b
““Ti 6 -28211.8(25) 9338(2) ($172)
C

““Sc 1 -35627.3(46) 2787.4(27) (En78)

HCa 2 —4l469.4(l6) 6 (Wa81)
--------------------------------_-_----__---------------_-Z-
----- 3-_---_------E___---_---f---_-_----§---------f------§2_
28211.8(25) -7662.2(89) 186.7(43) - - -

a

Prediction using the quadratic form of the IMME.

A possible doublet with a companion level at Ex=9298(2) keV
(Di78).
c

A weighted average of seven measurements tabluated in ref.
(En78), the adopted value in this ref. was Ex=2787(3) keV.

115

 

 

 

 

 

 

 

 

MSUX-BZ-MS
50. . 1 1 . .
A=20 ISOBARIC
QUINTET
:2 25. - «-
g I
2- 0. 1—4 1
.d +
3‘)
5 -25.— -
_50. l l 1 1 1
-2 -| o I 2
T2

Figure 4-4. Residuals for a quadratic IMME fit to the A=20
quintet.

116

 

 

 

 

 

 

50 ' MSUf-BZ-MB
A=32 ISOBARIC QUINTET
:‘j 25.— + -
“J
2
2
5" 0. +——o———o—o——
'd
8
2
-25.— -
-5CL ..1 1 1 1 1
-2 -l o l 2
T2

Figure 4-5. Residuals for a quadratic IMME fit to the A=32
quintet.

117

 

 

 

 

 

 

50 MSUX-BZ-ISO
1 A=36 ISOBARIC QUINTET
E 25.- ‘
DJ
2
E
2 ° *—’_' ' ”
'a
3‘;
2-25— -
-50. 4 ‘ 1 '
-2 -I o 2
T2

Figure 4-6. Residuals for a quadratic IMME fit to the A=36
quintet.

118

MrSUXI-BZ- I44

 

 

 

 

 

ZCE I ’I 1 l ‘1 T
d COEFFICIENTS FfiOM
'5-" M=0+sz+cTz+de -
Io.- e-ground state'
5. _ + x-excuted statefl
..__I .
‘6 ' + I‘ |
..5._. a
-l 0. - -
'45." 1
_20. J J J I I A J 1 l . 1 1
0 8 I6 24 32 4O 48
A
Figure 4-7. The d coefficient of the IMME with e=0 plotted

versus mass number of the quintet.

119

 

 

 

 

 

 

MSUX-82-l4l
20- I I I U I I 1 T I T 1 I
e COEFFICIENIS FROM
'5- ' M=0+sz+cTz JeeTZ4 "
'0- " e -gr0und state“
5 __ x-exited state
'6‘ i
x 0.
2: + l +
-5.— -
-|0.- -
-l5. - -—
_20. L 1 L J 1 1 1 1 1 I 1
0 8 l6 24 32 40 48

Figure 4-8. The e coefficient of the IMME with déo plotted
versus mass number of the quintet.

120

 

20 MSUX-82-142
d COEFFICIENT FROM

'5. .. M=O+sz+CTzz+des+eTz4 ..

l0. " "

 

d(keV)
o
.._.

 

 

 

 

-l 0. - -
-l 5. - -
_20 1 1 1 I l J I j 1 1 1 1
0 8 I6 24 32 4O 48
A

Figure 4-9. The d coefficient of the quartic IMME (e#0)
plotted versus mass number of the quintet.

121

 

 

 

 

 

 

20 MSUX-BZ-l43
e COEFFICIENTS FROM
us.~ M=a+sz+cTzz+dT23+eTz4 -
IO.- -
A 5.- -
o. - 2 +
*6 f
-5... -
$
-|O.‘- 4
-l5.- ..
-2C> 1 I 1 1 1 1 1 .1 1 1 J. I
0 8 IS 24 32 4O 48
A

Figure 4-10. The e coefficient of the quartic IMME (d#0)
plotted versus mass number of the quintet.

122

the latter two figures (4-9 and 4-15).

It is apparent that for fits using either a d or e
coefficient there is quite good agreement with the quadratic
form of the IMME. Only in the excited A=16 and the ground
state A=40 multiplet are there significant deviations from
the quadratic IMME. Neither of these multiplets is
completed and in particular the measurement of the I‘°Ti mass
excess is still preliminary (M082).

For the completed quintets there is a larger spread in
the coefficients when both d and e are used simultaneously,
but only in the A=36 case are the two coefficients
inconsistent with zero. However, because the reduced chi-
squared for the quadratic fit was significantly smaller
than that for a fit using either a d or e coefficient, there
is no real need for these coefficients. Also, the residuals
for the quadratic fit in this case reveal that both the Tz=
-l and -2 masses are poorly determined relative to the other
members of the multiplet, which may account for the apparent

deviation.

4.4 The b and c coefficients

Examination of the experimentally-determined mass
excesses shows that in most cases the quadratic form of the
IMME is satisfied. Figure 4-11 shows the b and c
coefficients of the quadratic IMME plotted versus mass
number (A) of the quintet. The trend of a decreasing

positive c coefficient and an increasing negative b

123

MSUX-BZ-l38

 

 

 

 

270. I 1 I I I I I W I I 9.
b and c COEFFICIENTS for
250. .. ISOBAR IC QUINTETS x- b .1 3.
o-c
250.- x -7.
+ + x
240.- x -6.
X
230.— I . x -5.
C(keV) + o -b(MeV)
220.- X .4,
x I
2I0.- -3.
x o
zoo.~ ‘ jz.
X
IQOJ' x + “I.
I80. J L 1 1 1 1 I J 1 i O.
4 I2 20 A 28 36 44

Figure 4-11. The b and c coefficients of the quadratic
IMME for isobaric quintets plotted versus mass number of
the quintet.

124

coefficient is the same as that observed in isobaric
quartets (Be79).

A simple model which explains the approximate size of
the b and c coefficients may be used to compare these
coefficients.

Considering the nucleus to be a uniformly-charged

sphere of radius Rc' the total Coulomb energy is

3 2(z--1)e2 (4'1)
Em(ao,T) = (aoTTzIHCIaoTTZ> = E R

 

C

Expressing the proton number Z as A/2-Tz (from Tz=(N-Z)/2
and A=N+Z), this expression becomes

1) .. 9_-_6e_2 a: _ _ “-2)
EI (OtotTI- RC [4 (A 1)Tz+Tzz]

Comparing this quadratic equation in T2 to the quadratic

IMME the coefficients b and c may be expressed as

(4-3)
- 2
b = 0.6e

R

A- -
c( 1) +(MJr1 Mp)

and

 

0.6e2 (4-4)
R

where the neutron-proton mass difference has been added to
the b coefficient (used Mn'MIH in the calculations).
Defining the quantity b* as

b = b - (Mn-Mp) (4-5)

an expression for the relation between b* and c is

*

- b* _ (4-6)
R : C(l-A) — 1

125

where the quantity R is by definition 1 for a uniform
sphere. Figure 4-12 shows the quantity R plotted versus
mass number (A) of the quintet. Deviations are apparent,
but the agreement is quite good for such a simple model.
The Coulomb radius parameter can also be extracted
from this model by setting RC in Equations 4-3 and 4-4

equal to roAyi Solving for r in each equation the Coulomb

O

radius parameters are

 

r = 0.6e2(1-A) (4-7)
ob b*A1/3
and
= 0.6e2 (4-8)

 

r
CC C A1/3

For a uniformly-charged sphere the ro extracted from the b
and c coefficients should be equal. Figure 4-13 shows the
Coulomb radii ro determined from the b and c coefficients of
the quadratic IMME plotted versus mass number (A) of the
quintet. There is quite good agreement between the value of
ro as calculated from the two different coefficients, much

better in fact than for quartets, where the r derived from

o
the c coefficient shows more fluctuations (Be79).

For a given multiplet there are two Coulomb energy
effects that occur when moving from the neutron-rich to the
proton-rich member (Be79). There is the increase in energy
from the Coulomb interaction between the valence proton and

the core. Then there also is the increased energy from the

Coulomb interaction between the valence protons themselves.

126

 

 

l.20 . . . 1 , I I ' IISU’E'azi' 4o
ISOBARIC QUINTETS
I “3" e _
’ = 24:.
R I-A
n.05— _
+ UNIFORM
I SPHERE=I
R LOO I /

.95— 0 I . I I 4
.90- + + I _

.85 I I I l l l I 1 1 1 1

0 8 IS 24 32 4O

 

 

 

Figure 4-12. The quantity R plotted versus mass number of
the quintet. For a uniform sphere, R=1.

127

 

 

 

 

2 O MSUX-BZ-l39
' COULOMB RADII FOR
- ¢ ISOBARIC QUINTETS ~
I.8- x .
" x - (0 from b "
I.6- x o-ro from 0.
r0 (fm) - x '-
I.4’—’ x X '
X x X
._ “ I. I. ‘C ., +. ‘I .
I .2 - ‘
P d
|.o I J1 I J J I l l I l
4 l2 20 28 36 44

A

Figure 4-13. Coulomb radii determined from the b and c
coefficients of the quadratic IMME plotted versus mass
number of the quintet.

128

The first effect is constant for each proton added so it is
expressed in the linear b coefficient. The second effect
increases as each proton is added so this dependence is
found in the nonlinear c coefficient. Hence, the b
coefficient is an average over the core whereas the c
coefficient is more sensitive to the wave functions of the
valence particles.

In the case of quartets where A=4ni1, the number of
protons alternates from even to odd for a given Tz member in
one multiplet to the same Tz member in the next multiplet.
Because of the strong pairing correlation between nucleons,
there is more of an energy difference when a paired proton
is converted to a neutron than when an unpaired one is.

This additional pairing effect would more likely manifest
itself in the b coefficient which displays more of the
average properties of the nucleus. In quintets where A is
always even, the proton number does not alternate from one
Tz member in one multiplet to the same Tz member of the next
multiplet. So this pairing effect is greatly reduced.

It is interesting to note the rather large Coulomb
radius parameter for the A=8 quintet (see Figure 4-13).

The neighboring nuclei 7Li and 9Be have ground state root-
mean-square (rms) Coulomb radii inferred from electron
scattering of 2.39(3) fm and 2.Sfl(9) fm respectively (Ja74).
Taking the radius parameter derived from b (r =l.82 fm), the
rms Coulomb radius obtained for the T=2 states is 2.81 fm.

This is comparabletx>the rms Coulomb radius of 18O and 19F.

129

It is apparent that the T=2 states are significantly larger
than the ground states of the neighboring nuclei in this

region.

4.5 Conclusions

In the case of the A=24 quintet application of a novel
experimental method was successfully applied to yield a
precise value for the mass of 2“Al as well as the first
direct measurement of the half-life of 21'Si. For the A=8
quintet the remeasurement of the mass excesses of 8Be and
8Li removed the deviation from the quadratic IMME that
existed previous to these measurements.

From an examination of all these A=4n quintets, it is
found that agreement with the quadratic form of the isobaric
multiplet mass equation is quite good. There seems to be no
evidence for substantial higher-order charge—dependent
effects in the nuclear interaction, though these effects may
be mostly manifested in the c coefficient as is the case for
isospin-mixing, Coulomb repulsion and second—order
perturbation effects.

It may be conjectured that cases of disagreement from
the quadratic form of the isobaric multiplet mass equation
may be attributed to experimental error. This leaves only
the case of the A=9 ground state quartet where a known
deviation is present and where sources of significant
experimental error have probably been eliminated.

The significance of the b and c coefficients has been

130

examined and it is found that they can provide detailed
structure information of the analog states.

Further experimental study of both isobaric quartets
and quintets is recommended. Mass measurements that
complete the remaining quintets as well as improve the
accuracy in the known members can, in conjuction with
theoretical calculations, provide useful nuclear structure

information.

APPENDIX A

A.1 Proton and alpha recoil masses.

The formula for the recoil mass (M ) can be derived

rec
simply from the conservation of momentum in a two-body
decay. Following the B-decay of the (A,Z) parent activities
to particle unbound levels in the (A.2-1) nuclei, particle
emission populates states in the residual recoil nuclei
(A-l,Z-2 for proton emission and A-4,Z-3 for alpha
emission). For now we neglect the B-induced recoil, since
this just mostly broadens the distribution, ultimately
limiting the mass resolution, and does not cause a
significant change in the centroid (this effect will
examined in Section A.3).
For a two-body decay at rest we have

mpvp = Mrecvrec (A'l)
where the subscript p denotes the emitted particles (proton
or alphas), m, mass and v, velocity. The rec subscript is
for the recoiling nucleus. In our case, we have 23Mg
recoiling from the delayed proton emission from the T=2

state in 2“Al.

Non-relativistically, the energy of the proton is
1
E = —1n v2
p2pp
solving for the momentum of the proton

(A-Z)

131

mpvp = ' 2Epmp (A-3)

Substituting this into the conservation of momentum

equation we find

VZE m
M __B_.L (A—4)

V
rec rec

The velocity of the recoil from the flight path d and the

time of flight t is simply

_ d
Vrec — t (A-S)

Replacing this in the recoil mass equation leads to the

desired result:

VZE _
M = m t (A 6)
rec d

For B-delayed alpha emitters expression for the recoil

mass is
Ma = find—t (A-7)
rec d

If, however, the proton mass is used in this equation we

 

obtain
a
M = ELI: = 3593013: Mrec (A—8)
rec d 2d 2

Because we will be using mp in our calculations rather than
mg, the alpha emitters will appear at one half of their

actual mass.

133
A.2 Beta-decay broadening of the recoil mass distribution.
Neglecting B-recoil, we have been able to obtain a
simple relation for the recoil mass Mrec (following particle
decay) in terms of the experimentally-observable quantities
Ep and t.

Assuming Mre is known, we can instead find the recoil's

c
velocity and hence its time of flight. For 25Si decaying to
the T=3/2 state in 25Al followed by ground-state proton
emission we have a 2"Mg recoil with a velocity of Vrec = .12
cm/ns and a time of flight of 68 ns for our 8 cm flight
path. This is in contrast to the proton velocity of vp =
2.8 cm/ns.

We now consider the approximate effect of B-recoil on

25Si we have both

our mass resolution. For the B-decay of
8+ and v emission. Since we are not detecting either of
these particles, it is tempting to believe that the recoil
effects may just average out. However, as mentioned in the
text, the rejection of 8+ - p coincidences in the Si
detector and 8+ - p angular correlations complicate the
problem. Also, in any beta decay we have the sharing of
energy between the 8+ and the v as well as the angular
correlation between their decay. Figure A—l shows recoil
energy distributions between a Fermi type and Gamow-Teller
type decay (Di80,J063). For the Fermi decay, the B-v
correlation is peaked at small relative angles. We can
estimate an upper limit to the size of the effect by

assuming that the 8+ and v are collinear, and then,

134

 

 

 

 

 

 

G-T
)-
t:
:1
G)
<1
8 \
#~
(I Q-
o. 6'
L _L I J J J I J _L
OI) OJ (12 CM3 DJ! (15 015 (17 (18 O£I LO
ER / 50

Figure A-l. Fermi and Gamow-Teller recoil energy distribu-
tions.

135

furthermore, giving all the energy to the 8+. As before, we
again have a two-body decay, and using momentum conservation
can obtain an expression for the initial recoil velocity Vi
which occurs before proton emission:
pa = Mivi (A-9)
Mi is the mass of nucleus before proton emission, and

is related to the final recoil mass M (after proton

rec
emission) by
M. s M + 1
1 rec (A-1g)
From the known Q-value of this decay, (to the first
T=3/2 level in 25Al) thefyIkinetic energy is T=4325 keV.
This positron is relativistic so we use
Eé = péczi-méc“ .(A-11)

and from conservation of momentum we have for the initial

recoil velocity

 

JEE - mécT 3 (A-12)
Vi = Mic? c = 61(10 cm/ns

 

Because this velocity is so much less than the proton
velocity (vp = 2.8 cm/ns), we will neglect any 8+ - p
correlation effects in this discussion of the mass
resolution. We will consider these effects in relation to
changes in the proton centroid in Section A.3.

In comparison to the final recoil velocity (Vrec = .12
cm/ns) the initial recoil is more significant for this
discussion. Defining the direction toward the center of the

Si detector as 6°, we write the new velocity of the recoil

a
Mrec 8

136

' = + . e
Vrec Vrec V1COS

(A—13)
where 6 is the angle that the beta-decay makes with the
axis.

We make a simple velocity addition because the recoils
are nonrelativistic. From this expression we see that, for
B-decays toward the Si detector, the Mi recoils in the
direction of the channel plates with a velocity Vi“ Upon
the subsequent proton decay, the recoil will have this
additional velocity. If, however, the beta is emitted
toward the channel plates, then following proton emission,
the Mrec will have less velocity. Figure 2-18 illustrates
this effect.

Still neglecting any proton velocity shift, we can
express momentum conservation in terms of an angle dependent

recoil mass

= Mr(6)(Vr + Vicose)

m v
P P ec (A-14)

We use the Equation A-3 to obtain

JZE m

Mr(6) = p
( V.cos(6)) (A-lS)
V 1 +
rec

and making use of the expression for Mre

"O

<L‘

rec

c from Equation A-4.

M
rec

Vicos(6) (A—16)
(w- >

rec

 

Mr(9)

This is an expression for the angle-dependent mass

Mr(6) in terms of the mass Mr and velocity Vre and the

6C C

initial B-recoil velocity Vi'

137

The range of allowed angles is only from 19° to 186°.
This is because we reject positrons that enter the Si
detector as they will be coincident with the protons and
hence have too large an energy signal.

If we put into Equation A-16 the above limits on the
angle 6, we have a percentage change in the derived mass of
8.8%. A better estimate of our mass resolution is to
calculate the variance or root-mean square of the deviations
via the expression

0;! = <M§(e)> - (Mr(6)>2 (A-l7)
where the < > denote an integration over angles. We need to
evaluate expressions of the form

62
M 2 ' ede
ng r(6) n31n (A-18)

<Mr(6)> 62

2flsin6d6

 

91

92
f M2 (6)21Tsin6d6
9 r
1 (A-19)

62
Znsinede
62

Substituting Equation A-16 into the above integrals

 

(M; ( 8))

(61 = 19° and 62 = 188°) yields the result for the
fractional standard deviation in the recoil mass of
0M (A—Zfl)

E 0.03
Mrec

 

If we treat the distribution as if it were Gaussian, we
can obtain an estimate for the full-width at half-maximum

(FWHM) for the percentage mass resolution as

138
o

M
rec

 

FWHM 5 2.354 x 100% E 7% (A-21)

This is the "intrinsic" mass resolution limit and has
no dependence on the length of the flight path used. A long
flight path, however, will reduce the effect of a finite
source size which can degrade the mass resolution. Monte
Carlo calculations were used in the design of the Princeton
apparatus to ensure that this effect was kept smaller than
the B-recoil limit (R077). we therefore used the same

length flight path and source size in our apparatus.

A.3 The effect of beta-recoil on the proton energy.

The recoil mass was a crudely-determined quantity and
was used only to filter out nearby masses. In contrast, the
proton energy is a precisely-determined quantity and,
therefore, scrutiny of any possible systematic effects is
essential. In the previous section, the 8+ - p correlation
was neglected; we now consider this, along with initial
B-recoil effects on the proton energy.

As with the recoil mass, Equation A-l3, we can write
the new velocity of the proton as

v' = vp — Vicose (A-22)
where e is the relative angle between the proton and the
positron. Zero degrees is again in the direction of the Si
detector. vp is the velocity without B-recoil and v'p is
the observed velocity. We can write Equation A-22 in terms

of the kinetic energy by squaring it,

139

v52 = v; — 2vpvicose + Vicosze (A-23)
this becomes
2
Ep(0) = Ep [1 - ecose + goosze] (A-24)
h - 2V1 - 1 2 d E (0) = lwn v"2 A
wares—VET,EP—-§mpvp an p 2p . swas

shown in Section A.2, Vi<<vp' and therefore 8 is quite
small.
The angular correlation function between 8+ and the p
can be expressed as (H074,Fr77)
W(0) = 1 + a(E)cos0 + p(E)cosZG (A-25)
It is now only necessary to calculate the average
proton energy to determine if there are any shifts from the

E value.
p e
2

E(0)W(6)2wsin6d0

(Ep(e)> = 61 92 (IX-26)

Jl. W(6)2nsin6d0
91

 

We again have 61: 19° and 02: 180°.

Though it is possible to carry out this integration
exactly, it is more instructive, and just about as accurate,
to take the following approach.

Inserting Equation A-24 into the integral, our

expression becomes

82 2
Jf (-€cose + E cosze)W(6)sin0d0

 

E (e) 61 4 (A-27)
<—E3———>= 1 +
E 92
P / W(0)sin0d6

91

140
We make use of the fact that the coefficients a(E) and p(E)
are small (Fr77) and about the same order of magnitude as a.
With 6 as the leading factor in our integral, we see that
only the first term in the correlation function W(6) will
contribute (i.e., W(e) s 1). All the other terms are second
order in the 8, a(E) and p(E) coefficients, and are

therefore much smaller than the first term. So this

 

 

 

becomes
62
<% (0» le cosesinede
_2__ g 1 _ 91
Ep 62 . (A-28)
Jr Sinede
91
This reduces to
(152(9)): 1 + 2V1 1- coszl9° (A-29)
E v 2(14—cosl9°)
P P
Now recalling Equation A-12, we note that for Eé>>méc"
we have
BBC
1

Substituting this expression into Equation A-29 and

rewriting, we obtain for the proton energy shift

YE M (MD
AEP — (Ep(e)) Ep MiCT l

 

1

/§

 

where
[1 + cos 19°

1 - coszl9°]

In the decay of 25Si, we find a value of AEP = .443 keV

for this shift. For 2"Si this shift is AEp = .467 keV. We

see that the maximum possible relative shift between these

141
two lines is only 24 electron volts. This effect is

insignificant and can therefore be neglected.

APPENDIX B

The design of the electrostatic lens was numerical in
nature since the complexity of the problem would exclude an
analytical approach. The relaxation method is a finite
difference approach to the solution of Laplace's equation

V2¢ = 0. (B-l)

It is based on the property that the value of the
solution (to Laplace's equation) at any point in space is
equal to the average of the solutions over an arbitrary
surface surrounding the point.

Through the use of a Taylor series expansion of a
function, one can obtain an approximation to the derivatives
of this function in terms of nearby values.

d2f(x) _ f(x+Ax) -2f(x) + f(x—Ax)
dxz (Ax)2 (B-z)

 

By substituting these approximations to the derivatives
into Laplace's equation, one obtains the value of the
potential as an average of nearby potentials. For a
two-dimensional Cartesian system with equally-spaced x and y
mesh points, Ax = Ay = h, we have the expression
<I>(x,y) =%[<I>(x+h,y) + <I>(x-h,y) + <I>(x,y+h) + <I>(x,y-h)] (3-3)

Here we see quite clearly that the value of the potential

142

143
can be expressed as an average of the potentials around a
given point.

Since our apparatus has axial symmetry (except for the
square face of the channel plates), a lens with this
symmetry was designed. With axial symmetry our problem
reduces to a two-dimensional one. Laplace's equation in
cylindrical coordinates for an axisymmetric system,

(I) = <I>(p,z), is

3% 13¢ 32¢
W+ET>+5ET=° (M)

The potential with equally-spaced p and z mesh points,
Ap = Az = h! for p > Z is
(B-S)

<I>(p.z) =%[(1+-2%)<)>(p+h.z)+ (1-2—11)-)¢(p-h.z)+ <I>(o.2+h)+<1>(o.z-h)]

and for p = 0 we have

(p(olz) = '16; [44)(1‘112) 'I' ¢(0,Z+h) 'I' @(0,Z-h)] (B-6)

The alternate form of the equation at p = 0 is due to the
behavior of the first derivative term in Laplace's equation.
In this coordinate system as O + 0, we find that the first

derivative term becomes a second derivative term:

B-7
11ml_a_g_,1im_1__a_g_*azq> I )
p+Oo o o+0Ap p 302
Therefore, for p = 0, Laplace's equation becomes
2 2
23¢+3¢=0 (3‘8)

 

 

144
which results in the second form for the potential (Equation
B-6).

This relaxation approach to the solution of boundary
value problems is a powerful yet simple method to apply in
practice. One first chooses a set of grid (or mesh) points.
These are the points at which the potential will be
calculated. The boundaries, as well as fixed potential
regions of this array, are held constant. The potentials
for the remaining points of the potential array are obtained
by systematically applying (in our case) Equations B-S or
B—6. The value of the potential from these equations is
then put into the array. After this has been done for all
the "free" points (those not held fixed), the process is
repeated. These iterations are continued until the
potential values have sufficiently converged.

A faster convergence can be obtained through the
application of successive over-relaxation. This method is
similar to a weighted average and uses the previous
iteration's value of the potentials. After a potential at
some point has been calculated, a correction to this
potential is made using the expression

(B-9)

current previous
¢.. ¢.. - ..

13 13 + (1 w)¢lj

where l < w < 2. This corrected value is then placed in the
array. Choosing w = 1 reduces over-relaxation to
relaxation. Over-relaxation acts to more rapidly diffuse

the correct values of the potentials from the fixed

145

boundaries and regions to the rest of the array.

To test a given lens design an existing Runge-Kutta-
based code was modified to calculate electron trajectories
through the lens (D180). This program used the potential
array in order to calculate the electric fields acting on
the electrons. The program was previoulsy tested with
analytically-solvable cases and found to be in good
agreement (Di80).

In our design studies, a number of possible
configurations were tested until a design with acceptable
focusing characteristics was obtained.

Figure 2-9 shows the model lens upon which the actual
lens was based. Eleven equipotentials and nine particle
trajectories are shown. The particles started from rest
near the surface of the converter foil. The inner and
outer trajectories are at 0.5 cm and 2.54 cm respectively.
A time difference of 3.1 ns was calculated for these two
paths. The mass resolution is still limited by B-recoil
before the particle emission.

In this calculation an array of 81 x 126 points (0,2)
with Az = Ap = .41625 mm was used. The largest change in
the potential was monitored for each iteration. After 250
iterations the largest change was that of 2.2 mV for a 50 mV
potential. The total potential drop across the lens was
3000 V, this change was thus insignificant and shows the

good convergence of the calculation.

APPENDIX C
The principle of maximum likelihood is based on the
assumption that the sequence of observations that actually
occur in an experiment are those with the maximum

probability of occurring. The likelihood function

n
fie) = TTf(xi;C) (C-l)
i=1

is the joint probability density of getting a particular
experimental result, x1, x2, . . . xn, assuming f(xi,c) is
the "true" normalized one parameter distribution function,
where

1I(x;c)dx = 1 (C-2)
The best value of c will maximize the value of the likeli-
hood function. Another useful quantity is the log-

likelihood function, which is just the natural log of the

likelihood function.

n n -
NC) = lutfic) = 1n(TTf(xi:d) = Z ln(f(xi;c)) (C 3)
i=1 °

l=1

To find the maximum of the likelihood function we require
the first derivative offlzzc) with respect to the parameter c

to be zero:

di(C) = 0 (c_4)
do

146

147
In addition, the second derivative must be negative:
ga:5;2§(c)< 0 (C’s)
Since taking the log of a function does not change the
position of the maximum of the function, we can instead work
with the log-likelihood function £(c).

As a point of reference, we can show that for a set of
measurements which are Gaussian-distributed this approach
reduces to one of minimizing Chi-squared for the set of
measurements (Be69).

For example, consider a linear relationship between two
quantities (x,y):

y(x) = a + bx. (C-6)
We wish to obtain the best values for the coefficients a and
b from a series of n measurements of (xi,yi). The
probability for a given measurements will be

’ y -y( .) 2 (c-7)
f(x.;a,b) = 1 exp [-%( 1 o 1)]
l o./2? i

J.

 

 

where y(xi) contains the a and b explicitly.
The log-likelihood function becomes

(C-B)

n n n
. - (x) 2
Ham) = Z? ln(f(xi;a,b)> = —1— 31— .. .1. 2(y1 y 1)

 

VZN 1 1

i=1 ' 1 1 2 i=1 0'

The first term has no a or b dependence and, hence, is
constant for a given set of measurements. We recall that

the Chi-squared is defined as

n
2 y.-y(x.) (C-9)
x 2 EE:( 1 1 72
i=1 °

1.

 

148

(a,b) thus becomes
£(a,b) = constant - %; (c—1g)
From this expression we see that to maximize the
log-likelihood function is equivalent to minimizing the X2.

Underlying all of the least-squares approaches to data
analysis and curvefitting is the minimization of the X2. We
see that this approach is consistent with the method of
maximum likelihood but is valid only for Gaussian-
distributed data. However, in many applications, it is the
Gaussian distribution which is most useful and is a good
approximation to the binomial and Poisson distributions for
good statistics.

From the protons (or alpha) energies and the recoil
time-of-flight we Obtain a two-dimensional plot of energy
vs. recoil mass (see Figure 2-12). Because of the B-recoil,
there is a correlation between the proton energy and the
recoil time-of-flight. On the plot of energy vs. recoil
mass, we observe a tilting of the particle groups.
Therefore, if we simply project these groups onto the energy
axis, their energy resolution will be degraded.

As a result of this correlation, we choose to fit the
data with bivariate Gaussian distributions (BGD). (C-ll)

_ 2 .. -
why) = 1 exp _ 2(1-1-p2) {(xcux)2 _ among) (y 111)+
2noxoy/I:67 x

7%))

 

 

 

149

The ux and u are respectively the x and y values for the

Y
centroid. The Ox and 0y are respectively the x and y
standard deviations for the Gaussians, and p is the
correlation coefficient between x and y. If p = O, and ifvua
require that the joint distribution function be bivariate
normal (Gaussian), then we have that x and y are
uncorrelated. A three-dimensional plot of this can be seen
in Figure 2-17. Many points in the two-dimensional data
contain zero counts, and most have somewhat low statistics.
We therefore need to use the Poisson distribution rather
than the Gaussian distribution to fit our BGD (Equation

C-ll) to the data. The Poisson distribution function is

given as n.
‘u e

fIni;u) - ——ETT-—' (C-l2)
1
We recall that the experimental points must be integers,
since they are either observed or not observed (i.e., one
cannot have half a count). Since we had a two—dimensional
array of integer data, our actual distribution function was

expressed as

 

u.¥je 13
f(n..;u..) = 13 ' (C-13)
13 13 nij’
The likelihood function becomes
N M (c-14)
manila-H.516) =1T ’IT f(ni.;u..)
i=1 j=1 3 13

150

where we have for uij the expression

 

 

(C-lS)
1 ° 2 (' )(j )I J 2 |
.. _ _ 1-a3 _ a2 1-a3 -a5 '-a5
U13 alexp (1'32) {( a). auas (as )I
a74-a3j

The energy is along the i—axis and the recoil mass is
along the j-axis. The last terms a, and as are for the
background. These terms were first fit by looking at
projections (along the recoil axis) of regions around the
peaks of interest. The improved statistics from these
projections reduced the fluctuations of the background and
made for a better fit to it. These parameters were also fit
using the method of maximum likelihood and, once obtained,
were not changed during the BGD fit.

As one can see from Equation C-15, this background was
constant along the energy axis and linearly-varying along
the recoil axis. In the fit of the bivariate Gaussian
distribution several alternative forms for the background
were tested (e.g., constant along the recoil axis and
different a7 and a8 values), none of which had much effect
on the centroids.

The coefficients a2, a, and a6 (the correlation
parameter, the energy width and the recoil width) were
obtained from the 4089 keV 25Si peak. These were then not
allowed to vary when fitting the 5402 keV 2SSi group and the

2"Si group. In addition, for the 2"Si peak, the recoil

151
centroid a5 was determined from the mass scale, previously
defined by the 21Mg and the 25Si lines. Fixing a5 for the
2"Si peak reduced any errors because of statistical
fluctuations and background effects.

These two-dimensional fits were superior to simple
projections of the data, both in an improvement of the
energy resolution and in the reduction of the bias which may
occur when choosing the width of the mass band to be
projected.

The fitting program used was a modification of a
grid-search non-linear least-squares program from Bevington
(Be69). This program originally was designed to find the
minimum of the x2 in a multiparameter space. By taking the
negative of the log-likelihood function, the maximum became
a minimum and the same algorithm applied.

To obtain the errors using the maximum likelihood
formalism, there are three accepted approaches. Two of
these involve a weighted integration of the likelihood
function: these are discussed in references (Me75) and
(Or58). A more straightforward approach, also discussed in
the above references, is to obtain the confidence interval

for the fitting parameters. This is given as

Cz
I(c)dc
C1
C

max (c_16)
. I(c)dc
min

PrIClSCSCZI +

C

152
which is the probability that the interval between the
estimates c1 and c2 includes the "true" value of the
parameter c. The limits cm and c . are where the value

ax min

of fic) .18 effectively zero (Zfic < cmin) = 1(c > cmax) = 0).

and c1 = c . unit probability is 1, hence,

If c2 = c
min

max
dividing by the integral over the entire range of possible
values of c just normalizes the probability to 1.

We recall that for a normalized Gaussian distribution
the area enclosed by one standard deviation is 0.68269 of
the total area of 1.0. By choosing the limits of the
integration such that we enclosed 68.269% of the total area,

we can obtain an estimate of the error in the value of our

parameter.

Co'I'A
f iICIdc
C

+11
.68269 = ° 1

cmax (C'17)
j: ;:(c)dc

min

 

In practice we integrate from co + A2 or C0 - A1

fco+A2 fco
Co or C )°

o-Al

separately (i.e.,

When the likelihood function is symmetric we have

A1 = A2 = A and the usual result of co + A.

When integrating multiparameter likelihood functions

1(c1,c2, . . .,cn), it is necessary to keep 1(c1, . . ,cn)

at a local maximum. As the likelihood function is

153
numerically integrated, that is, for each step made in the
integration variable (e.g., c1), the other parameters
(ci;i¢1) are changed until we are at a maximum. This
ensures that the error in the parameter is not

underestimated.

Ad68

Ad69

Ad70

Ad72

Aj75

Aj78
Aj79

Aj80

Aj81
Aj82

Aj82a

A178

Al78a

Am77

An80

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