AN EXPERIMENTALTEST-OF THE some *
MULT!PLET MASS EQUATION

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
GEORGE FREDERICK TRENTELMAN
1970

 

.....

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LIBRARY

l
T 2 Michigan State
University‘

 

This is to certify that the

thesis entitled

AN EXPERIMENTAL TEST OF THE
ISOBARIC MULTIPLET MASS EQUATION

presented by
George Frederick Trentelman

has been accepted towards fulfillment
of the requirements for

Ph.D Physics
degree in

43 _.
L ”(i ayl’c a “41’ ’.

Major professo

Date August 11, 1970

0-169

 

 

ABSTRACT

AN EXPERIMENTAL TEST OF THE

ISOBARIC MULTIPLET MASS EQUATION
by George Frederick Trentelman

The ground state mass excesses of 9C, 13O, and 21Mg have been
determined through measurement of the Q—values of the 12C(3He,6He)9C,
160(3He,6He)130, 24Mg(3lle,6lie)21Mg reactions. These mass excesses
represent the T2 = — 3/2 members of the T a 3/2 quartets for

130 and 21

A B 9, 13, and 21 respectively, and for Mg represent
measurements of improved accuracy. These experimental values for
the quartet members are used to test the isobaric multiplet mass

equation (IMME)
M(a,T,Tz) . a(a,T) + b(a,T)Tz + c(a,T)T§

The measurements were made using 68—70 MeV 3He beams, and a
split pole magnetic spectrograph as an energy analyzer. An energy
calibration procedure for the spectrograph and beam analysis system
with proton and 3He beams has been developed, and has proven val-
uable in making precision Q-value measurements at high bombarding
energies.

The mass excess of 9C has been measured as 28.9lli.009 MeV,

that of 13O as 23.103i.014 MeV, and that of 21Mg as 10.912i.018 MeV.

George Frederick Trentelman

In addition, Coulomb radii for a uniformly charged sphere model for
A = 9, 13, 17, 21, 25, and 37 have been extracted from the coefficients

of the IMME.

AN EXPERIMENTAL TEST OF THE ISOBARIC

MULTIPLET MASS EQUATION

by

George Frederick Trentelman

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1970

ACKNOWLEDGEMENTS

I would like to thank the faculty and staff of the Cyclotron
Laboratory for their continued support and many extra efforts that
make this work possible.

In particular, many thanks go to Dr. Barry Freedom for his help
in work, and to Professor Edwin Kashy for his guidance and help in
this work and all that preceded it.

Many thanks, also, to Mrs. Jan Amos for typing this manuscript.

TABLE OF CONTENTS

List of Tables

List of Figures

1.

2.

3.

4.

Introduction

Experimental Procedure

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

Introduction

Spectrograph Calibration

3He Beam and 6He Energies

Beam System

Particle Detection and Identification
Kinematic Compensation

Cycling Effects

Dispersion Matching

Constant Radius of Curvature

Target Thickness Measurements

Data Analysis

Experimental Uncertainties

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

Uncertainties of Spectrograph Field Calibrations
Extrapolation to Defined Radius of Curvature
Peak Location

Scattering Angle Fluctuations

Uncertainty in Scattering Angle Measurement

3He Beam Energy Uncertainties

Uncertainty in Target Energy Loss

Stray Magnetic Fields

ii

iv

13

14

l6

17

19

19

20

38

42

45

45

47

50

51

52

53

54

54

iii
Experimental Results
Discussion
Appendix A1

Derivation of the Isobaric Multiplet Mass Equation

List of References

55

60

68

71

2.1

2.2

2.3

2.4

2.5

3.1

4.1

5.1
5.2
6.1

6.2

LIST OF TABLES
Some useful reactions for calibration in momentum
match

Possible reaction pairs for momentum match energy
calibration

Spectrograph calibrating reactions for proton
beams

Spectrograph calibrating reactions for 3He beams
Targets used for (3He,6He) reactions for E(BEAM)=68.0 MeV
Mass excesses of (3He,6He) reaction members

Uncertainties of outgoing 6He energy for GL=ll.OO°i.O3°
and E(BEAM)-68.5 MeV

Experimentally determined (3He,6He) Q-values
Average Q-values and mass excesses for the Tzs—3/2 Nuclei
Emperically determined coefficients for the mass equation

Coulomb radii from the IMME

iv

23

23

24

25

40

44

51
56
58
65

66

1.1

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.103

2.10b

2.11

2.12

5.1

6.1

LIST OF FIGURES

Level diagram of mirror pairs
12C(3He,6He)9C angular distribution
Momentum match

Momentum match spectra
Spectrograph calibrations
1H(3He,3He)1H spectrum
Experimental area

Electronics

Particle identification spectrum
DiSpersion match

a-source collimator

a-source collimator

Position calibration spectrum
Target data

(3He,6He) position spectrum

Mass deviations from IMME

26

27

28

29

3O

31

32

33

34

35

36

37

41

59

67

 

‘l‘\[.[ I

1. INTRODUCTION

The purpose of this work is to determine the ground state mass
excesses of 9C, 13O, and 21Mg. These ground state masses represent
the neutron deficient T2 = — 3/2 members of the T - 3/2 isobaric
quartets for A - 9, 13, and 21 respectively, and with the exception
of 90 are the least well determined members of these quartets. Since
the isobaric multiplet concept appears to be a very sound one, as
evidenced by the similar level structure of mirror nuclei and by T a 1
states in even A, T a 1 triplets, it is of interest to extend the
study of these multiplets to the T = 3/2 quartets.

The basic contention of the isobaric multiplet concept is that
the multiplet members have essentially identical nuclear prOperties,
and differ only through charge dependent effects. The T - 1/2 mirror
nuclei are good examples of this. The level schemes for the T - 1/2
mirror pairs 170 and 17F as well as 13C and 13N are shown in figure 1.1.
Here the level schemes, with the exception of the first excited states
are very similar, and the ground state energy differences between
members of each pair may be attributed to the Coulomb energy difference
(plus, of course, the proton neutron mass difference). The descrepencies
in the first excited states have been attributed to shell effects (No 69),
and do not represent a breakdown of Coulomb systematics.

Further verification of the multiplet concept has come from the

discovery of isobaric analog states. These states were first observed

through the (p,n) reaction by Anderson and Wong (An 62) in a study of

2
nuclei from A a 48 to A - 93. In each case there was a strong neutron
peak whose (p,n) Q-value corresponded exactly to the Coulomb energy
difference between the target nucleus and the residual nucleus having
the same A, but with one more proton and one less neutron (T differing
by 1). The strong yield from these states indicates a similarity
between its structure and that of its analog.

Other analog states have been observed as compound nucleus
resonances formed, for example, by Target + p-) excited nucleus —>
Target + p reaction. Here, the center of mass energy of the incident
proton must be exactly that required to excite the compound nucleus
(Target + p) to one of its analog levels. Normally, these would be

the Tta : 1/2 levels. In some cases, however, forbidden analog

rget

states (Tta : 3/2) are observed again as resonances, found at high

rget
excitation energies and characterized by narrow decay widths. An

example of this is the forbidden T - 3/2 level in the system:

2851 + p —§ng* .3831 + p (Te 69 and Ka 70).

Here the lowest T - 3/2 level is the analog of the 2981 ground state.
Finding such strong resonant behavior at high excitation again
indicates a similarity between this state and the relatively simple
structure of the 2981 ground state.

In the T - 3/2 quartets (and multiplets of higher order) the
comparison of the nuclear level schemes becomes more difficult. First,

the T - 3/2 levels in the T2 - t 1/2 members analagous to the T2 - t

3/2
ground states lie at high excitations (rvlS MeV in the A a 9 multiplet)
and in some cases are unbound. This makes energy levels above these

states difficult to define. Secondly, the neutron deficient T2 = — 3/2

3
(ground state) members lie far from beta stability and are difficult to
measure experimentally. Excited states of these members have yet to be
determined.
The isospin formalism does, however, provide an expression that
connects members of these multiplets in a direct manner. This is the

isobaric mass multiplet equation (IMME).
M(o(,T,Tz) a a(o<,r) + b(ot,'r) 'rz + c(at,T) Ti.

M is the mass of a nucleus, T its isospin, and T2 its isospin
projection. In essence, it is the expansion of a nuclear mass in terms
of its isospin characteristics, and is particularly applicable to
isobaric multiplets of Tbl where the constants a,b, and c may be
determined emperically. If the isobaric multiplet assumption that all
members are identical in nuclear characteristics and differ only by
charge effects is valid, then such an expansion of the multiplet masses
in T2 is valid.

This equation may be derived from first order perturbation theory,
this is shown in Appendix Al, but fundamentally it comes from two
assumptions. (W1 57)

l. The Coulomb energy is a perturbation on the nuclear

energy (valid for light nuclei) and may be expressed
as:
A 82
vC a g r13 (1/2 - tzi) (1/2 — tzj)
(as a two body force). The T2 dependence comes from
this. In addition, the proton-neutron mass difference

may be expressed as:

4

V(Am)-(M +M)A+(M -M)T.
n p n p z

2. The specifically nuclear properties of multiplet members

are identical and may be characterized by a common set
of quantum numbers (designated by d.in the equation).

Since the equation is quadratic in T2, knowledge of isobaric
quartets are required to test it. Since this expression engulfs the
basic concepts of isobaric multiplets, one of the goals of this
experiment is to test it rigorously with well determined T - 3/2
quartets.

The IMME is a rather insensitive probe of particular charge
dependent phenomena in itself. The fact that very good IMME fits to
the data of isobaric quartets may be obtained, for example, does not
necessarily mean that the assumptions from which it may be derived are
necessarily true. The reason for this lies in the fact that the
quadratic nature of the equation enables it to absorb many other
phenomena as perturbations with accuracy sufficient to fit existing
data well. This is discussed in detail by Janecke, Garvey, and by
Wilkinson (Ja 69), (Ga 69), (W1 64). In particular, the spin-orbit
interaction may be encompassed into isotensors of rank zero, one, and

two:

1: iii: 3“*3’ M‘ P ..{.
V50 n. T+leh 3]

v- P .
+[ 3f (£9‘+£3i) + 313(t31‘£33‘>]

+ [(‘M'ZVXi—giffl‘ .. 1.2 :23] a: gig
.3

5
The derivation of this is found in (Ga 69) where it is pointed
out that this effect can be of the order of 50 KeV.
If information on the specifics of charge dependent interactions

are to be deduced from the IMME, the isobaric multiplets of T )1 3/2

over as wide a range of A as possible must be measured very accurately.

This might then allow trends of A dependence to be determined, and
allow deviations between the equation predictions and experimental
measurements to become apparent. In fact, the most accurately
measured T - 3/2 multiplet, (A - 9), has shown indications of a non-
zero term prOportional to T3. This indicates that charge dependent
perturbation terms above the first order may be needed. (Ge 68) and
(Ca 69).

It is of interest then to determine the masses of the T = 3/2
multiplet members as accurately as possible to test the IMME.

The procedure adopted here is to measure the masses of 9C, 13

21
and Mg by experimentally determining the Q-values of the reactions:

0,

12C(3He,6He)9C, 160(3He,6He)130 and 24Mg(3He,6He)21Mg. This required
the design of new measurement techniques which are described in what

follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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2. EXPERIMENTAL PROCEDURE
2.1 Introduction

A direct measurement of the (3He,6He) reaction Q—values consists
of four major parts:
1. An accurate determination of the incident
3He beam energy
2. An equally accurate determination of the
outgoing 6He energy
3. A measurement of the laboratory scattering
angle at which the reaction is observed
4. A measurement of the effect of target thick-
ness on incident and outgoing particles.
The determination of these parameters and knowledge of the reaction
kinematics is sufficient to determine the Q-value. The mass of the

6

residual nucleus may then be calculated since the masses of 3He, He

and the target nuclei are known.

6

Two particular problems arise in measuring (3He, He) Q—values.

First, the cross sections are small (3¢‘b/sr or less) and this means

6

low yield of He for the 150 to 300 nanoamp beam intensities used.

Second, the 3He and 6He energy losses in the required target materials

6He in

are substantial. As an example, E(loss) - 90.2 Kev for 37 MeV
a 432/4g/cm2 810 target, and E(loss) - 32.8 KeV for 68 MeV 3He in the
same target. The uncertainty involved in measuring these target energy

losses becomes a major contributor to the total experimental uncertainty

8
of the Q-value for target thickness greater than 500,ug/cm2. The

6He count

limits on target thickness and beam intensity restrict the
rate for a given solid angle. The result is that stringent require-
ments are placed on the stability of eXperimental equipment such as
electronics, beam transport elements, targets (deterioration), and
particularly beam energy. These must remain stable for periods of

time sufficient to obtain enough statistics for meaningful analysis.

Two procedures remain available for maximizing the count rate:
observing the reaction at a scattering angle where the cross section
is a maximum, and using the largest possible solid angle. In this
experiment the Optimum scattering angle was determined by taking a
12C(3He,6He)9C angular distribution between 9L = 9° to 19°. This
angular distribution was taken using a 760/“g/cm2 carbon foil, and
consequently some energy resolution was sacrificed. This distribution
is displayed in figure 2.1. As shown, the distribution peaks at
9 L - ll.0°, and most of the data was taken here.

The solid angle was maximized by using a double focussing split-
pole magnetic spectrograph as an energy analyzer. The spectrograph
compensates for the kinematic energy Spread of the reaction products
(l32KeV/deg for the 12C(3He,6He)9C reaction at 9L =- 10°) by focussing
the kinematically spread particles on the focal plane (En 58 and Sp 67).
This allowed a solid angle of 1.2 msr to be used.

In addition, magnetic analysis of the reaction products was an
aid in obtaining clean 6He spectra. In particular, the magnetic
rigidity of the elastically scattered 3He was less than that of the

3

6He for the beam energies used, and for this reason the elastic He

9
peak and its low energy tail was prohibited from reaching the focal plane
detector while 6He data was being taken. This reduced background in the

Spectra and made clean particle identification possible.

2.2 Spectrograph Calibration

An accurate energy calibration of the spectrograph is fundamental
to this experiment since the spectrograph is used to determine the beam
energies, and the energies of the outgoing 6He. Since the only direct
information from the spectrograph is the magnitude of the magnetic
field in the flat field region between the large pole tips as measured
by an NMR probe and the position on the focal plane where the analyzed
reaction products are incident, a method of determining particle
energies with respect to these two parameters was devised. This
calibration technique consists basically of three parts:

1. A calibration involving a reaction with accurately known beam
energy and reaction product magnetic rigidities (BfD. This serves to
define an effective radius of curvature through the Spectrograph,
(>(effective), relative to a focal plane position.

2. A calibration of the field behavior relative to the NMR reading
for reaction products passing through the spectrograph along €(effective).
This calibrates the spectrograph at excitations required for measuring
3He beam and 6He energies.

3. Calibration of the field behavior at higher excitations using
3He beams.

A momentum matching null method using a proton beam was used to

establish an accurately known beam energy and reaction product magnetic

8

of the Q-value for target thickness greater than SOC/ug/cmz. The
limits on target thickness and beam intensity restrict the 6He count
rate for a given solid angle. The result is that stringent require—
ments are placed on the stability of experimental equipment such as
electronics, beam transport elements, targets (deterioration), and
particularly beam energy. These must remain stable for periods of
time sufficient to obtain enough statistics for meaningful analysis.

Two procedures remain available for maximizing the count rate:
observing the reaction at a scattering angle where the cross section
is a maximum, and using the largest possible solid angle. In this
experiment the Optimum scattering angle was determined by taking a
12C(3He,6He)9C angular distribution between 9L = 9° to 19°. This
angular distribution was taken using a 760/“g/cm2 carbon foil, and
consequently some energy resolution was sacrificed. This distribution
is displayed in figure 2.1. AS Shown, the distribution peaks at
9 L - ll.0°, and most of the data was taken here.

The solid angle was maximized by using a double focussing split-
pole magnetic Spectrograph as an energy analyzer. The spectrograph
compensates for the kinematic energy Spread of the reaction products
(l32KeV/deg for the 12C(3He,6He)9C reaction at Eh I'l0°) by focussing
the kinematically spread particles on the focal plane (En 58 and Sp 67).
This allowed a solid angle of 1.2 msr to be used.

In addition, magnetic analysis of the reaction products was an
aid in obtaining clean 6He spectra. In particular, the magnetic
rigidity of the elastically scattered 3He was less than that of the

3

6He for the beam energies used, and for this reason the elastic He

9
peak and its low energy tail was prohibited from reaching the focal plane
detector while 6He data was being taken. This reduced background in the

spectra and made clean particle identification possible.

2.2 Spectrograph Calibration

An accurate energy calibration of the spectrograph is fundamental
to this experiment since the spectrograph is used to determine the beam
energies, and the energies of the outgoing 6He. Since the only direct
information from the spectrograph is the magnitude of the magnetic
field in the flat field region between the large pole tips as measured
by an NMR probe and the position on the focal plane where the analyzed
reaction products are incident, a method of determining particle
energies with respect to these two parameters was devised. This
calibration technique consists basically of three parts:

1. A calibration involving a reaction with accurately known beam
energy and reaction product magnetic rigidities (BfD. This serves to
define an effective radius of curvature through the spectrograph,
(>(effective), relative to a focal plane position.

2. A calibration of the field behavior relative to the NMR reading
for reaction products passing through the spectrograph along €(effective).
This calibrates the Spectrograph at excitations required for measuring
3He beam and 6He energies.

3. Calibration of the field behavior at higher excitations using
3He beams.

A momentum matching null method using a proton beam was used to

establish an accurately known beam energy and reaction product magnetic

10

rigidity (Tr 70). This technique requires that the products of two
reactions such as 12C(p,p)12C (elastic) and 12C(p,d)11C (ground state)
be detected simultaneously at the spectrograph focal plane, and be
induced by a beam of energy sufficient to give both reaction products
equal outgoing magnetic rigidities. This beam energy is, of course,
determined by the outgoing particle type, the scattering angle at
which both are observed, and the Q-values of the reactions. For this
pair of reactions with a 12C(p,d)11C Q-value of -16.4953 f .0011 MeV
(Ma 66) the beam energy is Ep - 33.691 I .0022 MeV at €9L . 15.0°.
Furthermore, this beam energy is unique for a given pair of reactions,
and the outgoing particle rigidity (Bf’p - Bf’d - 332.256 Kg-in for
this example) is unique as well. This situation is shown in Figure 2.2,
where outgoing rigidities of the protons and deuterons are plotted
as a function of beam energy. The point at which the curves cross
determines the unique beam energy and magnetic rigidity value. It is
of interest to note that the momentum match Bf> value is quite
insensitive to 9L. Tables 2.1 and 2.2 list other reaction pairs
and their Q-values suitable for momentum matching at other beam
energies and magnetic rigidities. Figure 2.3 shows Spectra of
momentum matched protons and deuterons from the reactions
12C(p,p')12C(4.4398) and 12C(p,d)11C g.s. The Spectra were taken
simultaneously.

The accurately established Bf> value and proton beam energy
provide the vehicle for further calibrations. These reaction products
may now be placed at a point on the spectrograph focal plane where

all subsequent reactions are to be placed, and used to define an

11

effective radius of curvature for particles incident at that point.

The relation is

 

P(effective) = B9 (momentum match)
BNMR

where

BNMR = NMR frequency (MHz)
4.2577 MHz/KG

 

is the field value given by the NMR probe. The establishment of a
unique beam energy and magnetic rigidity is independent of any measure-
ment other than determining that both reactions are incident at the
same focal plane position. This criterion was accomplished by adjusting
the beam energy until the centroids of the proton and deuteron peaks
coincided as seen via the position sensitive detector at the focal
plane. The ability to discern when the centroids were coincident
represents the experimental limit of the calibration, and previous
work indicates that determining centroid differences to 0.1 mm or
better is not unreasonable. Once a match had been established, it
was checked by moving the peaks across the counter by changing the
field. For a real match the centroids must remain coincident. A
relative shift of centroids in this process indicates a spurious
match, possibly caused by DC biasing of the detector pulses, or zero
level Shift in the ADC's.

For these proton reactions the scattering angle was measured
using the reaction 1H(p,p)1H in a formvar foil of negligible thick—
ness. The high kinematic spread of the scattered protons (298 KeV/deg

at 9L - 15.0') allowed measurement of the angle to 10.05. Since

12
the momentum matched beam energy is sensitive to the angle by the
amount dE(beam) - 3O KeV/deg this required the original positioning
567—1337" I
of the spectrograph to be accurate to only i O.5°. The precise value
of the scattering angle was then used to calculate the beam energy to
4 KeV or better.

Once a beam energy and p(effective) were defined by the momentum
match, and the scattering angle measured, the spectrograph behavior
at other field excitations were calibrated. This was accomplished by
observing other reactions at the focal plane position defined by
p(effective). These reactions are listed in Table 2.3. The magnetic
rigidity of the outgoing particles of these reactions were calculated
knowing the beam energy (the momentum match beam), the scattering
angle, the target thickness and the reaction Q-values. These cal-
culated rigidities were then compared to the rigidity measured in the

Spectrograph when the reaction products were placed at p(effective),

namely:
Bp(experimenta1) = BNMR* p(effective).

The comparison of the calculated Bp values to those determined
experimentally calibrates the spectrograph behavior at p(effective)
for a given NMR reading. Several judicious points at various field
excitations provided a calibration over the range of interest. The
ratio Bp(ca1cu1ated was plotted against spectrograph NMR readings,

Bp(experimental)
and this calibration, is shown in Figure 2.4.

 

The upper curve represents two independent calibrations on
separate days using the 12C(p,p)12C elastic and 12C(p,d)11C g.s.

match and indicates the reproducibility of this data. The lower

13
curve is a third calibration using 12C(p,p')12C(4.4398) and 12C(p,d)llC
(g.s.) The beam energy and Bo value required for this second pair
(Ep I 28.927 MeV, Bpp I Bod I 282.822 Kg—in at 0L I ll.0°) defined a
different value of p(effective) for the subsequent calibration and
Q-value measurements. The reproducibility is again indicated by the
fact that the curves are very nearly parallel. For both curves,
several points were taken for each reaction on separate cycles of the

magnet.
2.3 3He Beam and 6He Energies

The 3He beam energy for each run was determined by measuring
the magnetic rigidity of 3He elastic scattering from 12C and 16O.
The procedure required adjusting the spectrograph field such that
these elastic peaks were incident at the focal position defined by
the momentum match. The rigidities of the elastically scattered 3He
were then calculated from the Bp(calculated) curve (Figure 2.4),

Bp(experimental)
the NMR frequency, and the value of p(effective). The relation is:

 

Bp(3He) = E(NMR) * p(effective) * Bp(calculated)
4.2577 Bp(experimental)

 

This rigidity of the 3He (hence the momentum and energy) plus knowledge
of the scattering angle is sufficient to determine the beam energy.
When 3He beams were used, the scattering angle was checked by

measuring the reaction 1H(3He,3He)1H on a formvar foil. The out-

3

going He energy was determined in the manner described above, and

the angle determined from the reaction kinematics. The high kinematic

3

Spread of the He (1490 KeV/deg for E I 68.5 MeV and 0L I ll.0°) makes

14
the angle calculation very insensitive to the 3He beam energy. A
Spectrograph entrance aperature of t0.08° was used for the angle
measurements to reduce the energy spread acceptance and maintain
reasonable peak shape. Figure 2.5 shows a sample position spectrum
of the 1H(3He,3He)1H reaction.

Additional information on the spectrograph calibration was
obtained by measuring the 3He induced reaction shown in Table 2.4.
Since the 3He beam energy and scattering angle were known, and
p(effective) previously defined, these reactions provided additional
Bp(calculated) data. For this reason, these points are also

Bp(experimental)
displayed on the calibration curve (Figure 2.4).

 

The outgoing 6He energies from the 12C(3He,6He)QC, l60(3He,6He)13O
and 24Mg(3He,6He)21Mg reactions were measured in the same manner as
the elastic 3He. However, since the beam energy and scattering angle

6 6He) Q-values to

had been measured, the He energies allowed the (3He,
be calculated.
During each 3He run, data was taken for all calibrating reactions

6He) reactions on each cycle of the spectrograph field.

as well as (3He,
In effect each such cycle provided an independent measurement of the

desired Q-values.
2.4 Beam System

Beams for this experiment were prepared in the Michigan State
University Sector-focus Cyclotron and delivered via the analysis and
transport system (Ma 67). A general view of the experimental area is

given in Figure 2.6. The Slit boxes on the analysis system (81, 52, S3)

15
were typically set to deliver a beam of maximum energy spread of
:20 Rev at 70 MeV and maximum radial divergence of :2 milliradians.
Magnets M3 and M4 provided the initial energy analysis, and M5 is
a switching magnet. The direction of the beam incident on target
was defined by two sets of slits S4 and $5. Slits S4 are current
reading and are permanently located between the scattering chamber
and last quadrupole, 85 inches from the center of the chamber. These
Slits were set at :0.30 inches from the beam line center.

Slits 85 are mounted on the target frame, and a 1.030 inch
opening defined the scattering chamber center and spectrograph object
point. These slits were also current reading. After the beam had
been aligned using M5 and M6, the SS slits were lowered out of the
beam and the targets were bombarded. Continuous monitoring of the
slits S4, and the availability of monitoring SS at any time assured
constancy of incident beam direction. In addition, the switching
magnet M5 was continuously monitored with an NMR.

The slits labled S6 are located at the entrance to the spectro-
graph 10.67 inches from the target, and define the solid angle.

Three solid angle defining slits were used for this experiment;
1.2 msr subtending 2° in the scattering plane and 2° vertically,
0.30 msr subtending 1° in the scattering and 1° vertically, and
0.05 msr subtending 0.l6° in the scattering plane and 1° vertically.

Most of the proton calibration work was done with the 1.0°
slit since the cross sections were substantial, and minimum kinematic
spread in the peaks was desired. This was necessary since the

calibration work was done without moving the detecting apparatus to

16
compensate for kinematic Spread. For the (3He,6He) reaction the large
solid angle 2.0° X 2.0° slit was used.

A check with an optical surveyor's telesc0pe verified that the
centers of all three slits coincided. Also, as a check that no
spurious effects arose in the placement of the reaction peaks at the
focal plane due to slit changes, many of the calibrating reactions
were observed with each Of the three slits (at constant spectrograph
field). No centroid shift was found to be significantly greater than

the statistical error of the centroid itself.
2.5 Particle Detection and Identification

Particle detection and identification at the spectrograph focal
plane was accomplished with a 300 u position sensitive Silicon surface
barrier detector (Da 69 and Jo 70). The energy loss signals position
signals were fed into separate Ortec Model 109 preamplifiers and from
there into two Tennelec Model TC-200 amplifiers. These amplifiers
were used in AC coupled, double differentiating mode to prevent DC
biasing Of the pulses. This is particularly important since the

position information from the detector is computed as:
X(position) = XAE pulse/AE pulse

Thus, any constant added to one of the pulses (DC bias or raised
zero level) would yield erroneous position information. The zero
levels Of the ADC's were also checked with a precision pulser before
each run to eliminate offset zero levels. The amplifier Operating

time constants were .8 usec. The amplified pulses were then sent

17
to two Northern Model NS-629 ADC's and hence to the Laboratory's
XDS Sigma-7 computer where the data was analyzed on-line. Figure 2.7
is a diagram of the detector and electronics setup. The data taking
routine TOOTSIE (Ba 69) displays the data according to the energy loss
in and position along the detector. An example of this display is
shown in Figure 2.8. The particles are incident on the detector at
an angle of about 45° giving the detector an effective thickness of
~425 u. Particles were identified via their differential energy loss
in the detector. The detector was biased at -150 volts and cooled

with alcohol at dry ice temperature.
2.6 Kinematic Compensation

Kinematic compensation requires that the energy spread across the
Spectrograph entrance aperature Of the reaction products due to the
reaction kinematics be refocussed at the detector, and not be observed
in the peak width. The design of the spectrograph provides for a
fixed focal plane for paraxial rays emerging from its Object point in
the scattering chamber. With a finite angular acceptance in the
scattering plane, however, the energy difference between rays accepted
at angles 0L + 6 and 0L -6 due to kinematic spreading is not negligible
(this energy spread would be 264 Rev for the 1.2 msr aperature and the
12C(3He,6He)9C reaction at 10°).

The effect is that the ray entering at O - 6 has a higher energy,

L
and traces a larger radius of curvature than the paraxial rays. The
ray entering at O + 6, being at a lower energy, traces a correspondingly

shorter radius of curvature. The net effect is that these rays

18
intersect one another (focus) closer to the spectrograph than the
paraxial rays do, and the focal point moves toward the spectrograph
and becomes a circle of least confusion Of rays. The equation used

for calculating this shift (A) of the focal point is (Sp 67):

/2

l2 sinO] / [(Ko — KO)(Ti/To)1 c030]

l
A I [DMHOKO(Ti/To)

where: A I focal plane shift for a given reaction at radius of curvature p
D I spectrograph dispersion at p
p I radius Of curvature

l2 / MR

Mi I mass of incident particle

1
K9 ' (MiMo)

M I mass of detected reaction product
MR I mass of residual nucleus

. “Mo/MR
energy of incident particle

PS
I

0-3
I

T I energy Of detected reaction product

0 I mean laboratory scattering angle
More complete discussions of this phenomenon may be found in references
(Sp 67) and (En 58). For the (3He,6He) reactions measured here this
focal shift was three to six inches at p = 32 inches.

The apparatus necessary to achieve this compensation consisted

Of a 50-inch plate or detector holder mounted on two motor Operated
drive screws in the spectrograph camera chamber. This allowed the
detector holder to be driven horizontally toward the spectrograph
exit port. The detector holder may also be driven vertically allowing

the detector to be placed at the position of Optimum count rate.

19
The relativistic kinematics calculations and computations Of
various spectrograph parameters were made with the computer program
SPECTKINE. In addition to the reaction kinematics it provides NMR
frequency and magnet decapot settings as well as calculations of the

kinematic shift described above.
2.7 Cycling Effects

It has been Observed that the spectrograph field behavior is
sensitive to the field recycling procedure used (Sn 67). In particular,
the calibration data showed that cycling at different speeds caused
apparent changes in the effective field strength relative to a constant
reading Of the NMR frequency. This effect was made negligible by
using a cycling time of 40 minutes for a field change of 0.0 to 0.9
to 0.0 Of its maximum value. All data was taken on the 0.0 to 0.9
half of the cycle with the field always rising to its desired value.

Once the prOper cycling procedure had been ascertained for the
spectrograph, there was very little field drift as monitored by the
NMR. Such fluctuations amounted to about :0.7 KHz (at a frequency of
=55 MHz). This introduced an uncertainty in the outgoing 6He energy

of only :0.6 KeV.

2.8 Dispersion Matching

Partial dispersion matching was used to reduce the effects of
beam intensity shift within the energy limits set by the beam trans-
port-analysis system (especially important in three to six hour runs).

A perfect match would mean that the energy dispersion Of the beam

20

produced by the cyclotron and analysis system would be exactly compensated
by the dispersion produced in the Spectrograph. The net result of the
ideal Situation would mean a finite width (and energy dispersed) beam

spot at the spectrograph target would be focussed to a point by the
spectrograph. This is shown graphically in Figure 2.9. The effect of

a total or partial dispersion match is reduced image width for a given
reaction. In addition, the peak centroid is made less sensitive to
fluctuation in the energy intensity pattern of the incident beam.

The dispersion matched condition was created by choosing quadrupole
lens settings that gave the dispersed beam a width on target commensurate
with that required by the dispersion characteristics of the spectrograph.
The amount Of dispersion match actually used in this experiment ranged
from 50-752 of total match. The energy dispersion required by the
spectrograph for perfect matching is:

933-: Do

AE ZMHE

where A§_is the energy dispersion of outgoing particles across the target,
AB

0 is the radius of curvature at which matching is achieved, MH is the

horizontal magnification, and D the dispersion of the spectrograph at

p and E is the mean energy of the outgoing particles.
2.9 Constant Radius of Curvature

The success of accurate energy measurements in the spectrograph
depends upon defining an effective radius of curvature (or a correspond-

ing focal plane position), and placing all pertinent reaction products

21

there. Since the information regarding the peak positions was Obtained
through detector pulses analyzed by computer, and consequently appeared
to change with changes in amplifier gains or ADC conversion gains, some
method of providing a fiducial mark at the focal plane was required.
This requirement was filled by mounting a collimated 241Am alpha source
on the counter bench unit one inch from the counter. The counter could
then be lowered vertically in its bench to the level of the source.
The resulting alpha peaks then provided a positive position indicator on
the counter with a corresponding position channel number. When the
amplifier gains were changed, it was a simple matter to lower the
detector and recalibrate. Figure 2.10 Shows photographs Of the focal
plane apparatus, alpha source, and detector. Figure 2.11 shows the
three resulting calibration alpha peaks.

The alpha source also provided an energy calibration for the
counter since the alpha group energies are well known (5.48 MeV for
the dominant group). Thus particle energy loss in the detector and
electronic noise level could be calibrated without recourse to a pulser.

Since it is also critical that the detector not be moved relative
to the alpha source between the proton calibration and the taking of
(3He,6He) data, each run consisted of both a proton calibration and
6He data taking without removing the detector or its mount from the
camera chamber.

The alpha source, however, moved with the detector when the focal
plane apparatus was moved to compensate for kinematic spreading. To
insure that the radius of curvature defined by the proton calibration

intersected the same point on the focal plane despite this movement,

22

12C were Observed at this point as the detector

the elastic 3He from
was moved to various positions. The result was a calibration for the
drive screws in which the 3He peak centroid always maintained its

proper position. The f0.08° entrance slit was used for this calibration

to minimize the peak widths and maintain good peak shape.

23

 

 

 

 

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38

2.10 Target Thickness Measurements

The 3He and 6He energy loss in the targets used in this experiment
represent a significant correction to the measured outgoing 6He energies,
and therefore, careful measurement of all targets was required. The
targets were measured in air with an alpha source gauge, and the energy
losses for various particles calculated using the published tables
(Wia 66). The appropriate 6He loss was taken to be that of a 3He at
half the 6He energy.

Briefly, the measuring technique requires passing 5 MeV a-particles
from an 241Am source through the target (located a fixed distance from
the source) and detecting them with a solid state detector mounted on
a micrometer. When the total thickness of air and target material
between the source and counter is not sufficient to stop the alphas,
the count rate measured at the detector will be essentially constant.
As the counter is moved away from the source, however, the alphas
begin stopping before reaching the detector. In this region, the count
rate falls rapidly to zero as the counter is moved further away, and
the micrometer reading gives this distance quantitatively. The target
is then removed and the measurement repeated with only air between the
counter and source. This produces a curve similar to the first, but
displaced from it a distance equal to the air equivalent thickness of
the target material for the alphas. Figure 2.11 shows the data of
such a measurement for a 12C foil.

Knowing the air pressure and temperature allows a calculation

of the density of, and hence the alpha energy loss through the air

39

(or equivalently, through the target). This in turn allows the target
thickness in mg/cm2 to be calculated. Several measurements were made
over the surface of each target foil to obtain an average thickness
value. It may be noted also that since target thickness is measured
relative to an equivalent amount of air, precise knowledge of the
alpha energy used is not critical if the stopping power of the target
elements are commensurate with that of air.

Once the target thickness and desired energy losses were estab-
lished, they were applied as corrections to the outgoing particles

as follows:

(outgoing) + AE (incident)]

... a E
AELOSS(outgoing) l/2[A LOSS

LOSS

This assumes that on the average the desired reaction will occur at
the target center, and consequently both the incident beam particles
and reaction product will be reduced in energy by half the target.

The average energy loss for the outgoing particles in the target was
then introduced as an effective excitation energy in the corresponding
reaction kinematic calculations.

The uncertainty in the energy losses was estimated by making
several separate measurements of each target. Table 2.5 lists all
targets used for this experiment, their thickness in mg/cmz, the
average energy loss of 6He (as described above) and the uncertainty

in the energy loss.

40

Table 2.5 Targets Used For (3He,6He) Reactions

for E(BEAM) = 68.0 MeV

 

Targets Thickness(mg/cm E(6He), Ave. Loss(KeV)
120 #1 0.153 25.413

810, F-80 0.227 34.114
"Glass” 5102 0.491 71.018

24Mg Foil 0.656 83.5113

810 + C 0.432 66.517

120 "F" 0.117 19.013

 

41

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3. DATA ANALYSIS

Many of the parameters involved in obtaining the (3He,6He)
Q-values were subject to experimental uncertainties that were
difficult to determine quantitatively. Because of this, and because
the goal of this experiment was precise measurement, several inde-
pendent measurements were made. Beam energies and scattering angles
were varied, and for the 9C and 130 reasurements, several different
targets were used. In addition, two different detector geometries
were used; the first with particles incident at 45° in which the
detector operated in the fig mode, and the second with particles
incident at 53° and the dgtector operating in the E mode for the
12C(3He,6He)9C and 160(3He,6He)130 reactions. Rotating the detector
8° relative to the incident particles also increased the effective
energy dispersion acorss it by about 20%. Each of these independent
runs was preceded by its own proton calibration run. In all, three
such independent runs were made, and several measurements of the
Q—values were taken during each run. These measurements made during
each run were, however, subject to any systematic uncertainties in
the preceding proton calibration run, in the detector geometry and in
target thickness measurements.

Each Q-value determination was assigned an uncertainty consisting
of all known parameter fluctuations summed in quadrature. For a
particular measurement this would include beam energy fluctuation,
scattering angle fluctuations, statistical error of the peak centroid,

uncertainties involved in correcting peak centroids not falling

42

43
exactly on the focal point defined by p(effective), and any observed
magnetic field fluctuations. A weighted average of all Q—value measure-
ments for a reaction was then taken, with the weighting factors taken
as the square of the uncertainty of each point. Since these uncertain—
ties were very nearly the same for each point, the weighted average
was numerically equal to an unweighed average for each reaction. In
addition, the uncertainty of the average was computed using the same
weighting factors according to the formula (Be 57)

02 = {2 (W181)2 }1/2 / 2 w o a error of average

1 i 1 W1 . weighting factor

Si = error of point

The systematic uncertainties such as those assigned to the
calibration procedure, and to target thickness measurements were then
summed in quadrature, and added to the uncertainty of the average Q-value.

To check the validity of this assignment of uncertainties, the
standard deviation of the distribution of individual measurements was
also computed for each reaction. (For 9C there were eight measure-

21

ments, nine for 13O and four for Mg). In each case the standard

deviation of the distribution and the total experimental uncertainty

described previously were very nearly equal.

9 21

Finally, in computing the mass excesses of C, 13O, and Mg, the

uncertainties of the other masses involved, 3He,6He, 12C, 160 and 21Mg
were also summed with the Q-value uncertainty in quadrature. The

6He mass excess (Ma 66)

uncertainty of t 4 KeV associated with the
was not negligible in computing the total error of 9C and 13O. The

masses of 3He,6He, 12C, 160, and 24Mg are given in Table 3.1.

44

Table 3.1 Mass Excesses of (3He,6He) Reaction Members (Ma 66)

 

Element Mass Excess(MeV)
120 0.0
160 4.7365i.0003
24Mg -13.9333i.0017
3He 14.9313:.0002
6

He 17.598 i.004

 

4. EXPERIMENTAL UNCERTAINTIES
4.1 Uncertainties of Spectrograph Field Calibrations

Uncertainty in the field calibration curve (Figure 2.4)
(chalc curve) is the consequence of two separate phenomena; first
Bpexp 12 11
the uncertainty in the C(p,d) C Q-value since this reaction is
used for the momentum match, and second the uncertainty of fit of

the calibration curve to the calibrating reactions. These points

will be dealt with separately.
Uncertainty Due to Momentum Match

As pointed out in the section on spectrograph calibration, the
momentum match defines both a proton beam energy and an effective
radius of curvature. Since it is such a point of definition, any
calibration curve must necessarily pass through it regardless of
where other related calibration points may lie. This momentum match
beam energy has a minimum uncertainty of t 2.2 KeV due to the 1.1 KeV
uncertainty in the 12(p,d)11C Q-value. Since the calibration curve is
required to pass through the point itself and since this beam energy
is used in calculating the other calibration points, the effect of
this 2.2 KeV uncertainty is to make any curve through the 2.2 Kev
error bar acceptable as long as such a curve is parallel translation
of the curve best fitting all of the calibration data. In other words,

the location but not the shape of the calibration curve may be altered

45

46
12 11
by the uncertainty of the C(p,d) C Q—value. This effect represents
a systematic uncertainty in any outgoing particle rigidity (and hence
outgoing energy) based on this curve; outgoing 3He for beam energy
and angle measurements, and outgoing 6He, for example.
Quantitatively, the effect is the following. Since a translation

of the calibration curve is reflected directly in outgoing particle

 

 

energies:
AE(outg9ing)§= 2.2 x 10“3 MeV
E(outgoing) 33.691 MeV

where 33.691 MeV is the momentum match beam energy at 0L = 15.0°.

This reduces to:
AE(outgoing) = 6.7 x 10—5 x E(outgoing)

for a 3He beam E(BEAM) = 68.520 MeV producing a 6He of E(6He) - 36.069
the individual uncertainties would be AE(BEAM) B 4.7 KeV and AE(6He) -
2.5 KeV. However, since any such systematic uncertainty must work in
the same direction for both measurements, the uncertainty in the

calculated Q-value is again their difference:
AQ a AE(BEAM) - AE(6He) = t 2.2 KeV for the parameters given above.
Uncertainty Associated With Calibration Curve

The uncertainty associated with the fit of the calibration curve
to the calibrating reactions is basically an estimate. To make this
curve as reliable as possible, an attempt was made to use calibrating

reactions whose Q-values carried minimal experimental uncertainties.

47
The best of these (in this respect) are the 12C(p,p')12C (4.4398)
(Ch 67) and the 7Li(p,d)6Li (g.s.) (Ma 66) reactions, as well as the
momentum match point which falls between them. Fortunately these
reactions span a rigidity region that includes the points required
for determining the 3He beam energy and the 6He energies from 9C and
130. The 3He induced calibrating reactions 12C(3He,3He')12C(4.4398)
and 12C(3He,4He)11C (using a 3He beam energy determined via the proton
calibration curve) then provided a check on the consistency of the
curve shape over a greater rigidity range, particularly to that
rigidity value required by 21Mg. Using the quoted uncertainties of
the calibration reaction Q-values, and considering the consistency
of the curve shape over several sets of experimental measurements it
is estimated that the uncertainty associated with the fit of the
calibration curve is not greater than 0.5 x 10-4 of the outgoing
particle energy. This uncertainty was applied directly to the out—

going 6He energies and hence to the calculated Q-value. Quantitatively

it amounted to about t 2 KeV.
AQ(fit) §_0.5 x 10.4 x E(outgoing 6He) = 2.0 Kev
4.2 Extrapolation to Defined Radius of Curvature

The essence of these Q—value measurements was choosing a Spectro-
graph field that places the desired reaction at a very carefully
calibrated point on the focal plane. Since in practice this was
almost impossible to do exactly for the (3He,6He) reaction (primarily

because the low count rates did not always allow a great deal of time

48

for such fine tuning), some method of extrapolating from the actual
particle group position to the desired position was required. The
approach was the following: knowing the field and where the particles
are, what would the field be if the particle group were where it
belonged? Such a problem depends basically on two factors; the
characteristics of the detector and the dispersion of the spectro—
graph. Since the dispersion is essentially constant, particularly
over the length of the focal plane subtended by the detector (30 mm)
(Sp 67), the relation between a peak centroid in some position channel
C1 (or equivalently at some place on the counter analyzed as channel
Cl) with NMR frequency F1 and an equivalent centroid in channel C2
with frequency F2 is

F1 - F2 = constant

(Cl - C2) (F1 + F2)
2

 

This is actually a statement of constant dispersion and should hold
for a given detector geometry. In principle this constant may be

12C(3He,3He)12C and placing

calculated by taking some reaction such as
the peak in two different channels by changing the field and using
the formula above. This constant and the formula would then allow
determination of the field value that any other peak would have at
p(effective).

Complications arise, however, due to nonlinearities in the detector
response. This situation effectively makes the dispersion appear non—

linear. In order to calibrate for this effect, the detector was care—

fully mapped with one of the calibrating reactions (usually elastic

49

3He from 12C) during each run. This essentially yielded a set of
dispersion "constants" for various areas of the counter. This infor-
mation permitted subsequent mounting of the detector so as to utilize
its most linear region. Doing so minimized nonlinearities to the
point where an average value of the dispersion constant could be
calculated over the region of interest. Since the dispersion is
viewed via the detector and related electronics, it appears to depend
upon the amplifier gains. The constant was, therefore, redetermined
after any gain change.

The values of the dispersion constant for any given setup would
vary by approximately 10% over the detector range of interest. Sub-
sequently any such extrapolation of the field was given an uncertainty
of 10% of the extrapolated value. Such extrapolations were generally
in the range of 5-15 KHz, and all but two were less than 25 KHz. At
an extrapolation of 25 KHz the uncertainty introduced is i 2.5 KHz
which translates into an uncertainty of i 3.5 KeV in the outgoing
6He energy (=37 MeV).

The energy uncertainty was calculated as follows: for a given
frequency interval AF at some average frequency F determining outgoing
particle energy E(6He)

ZAFBAE
F E

AB 8 ZAF E

Therefore, for dispersion constant 8 AF with uncertainty
FAC

50
6(AF ) = 10%.A§_ the Q-value uncertainty becomes

FAC FAC

AQ(6) = 10% x AF x 2 x E = 0.20 x AF x E(6He).
F

4.3 Peak Location

The uncertainty of the location of a peak centroid due to its
shape and statistics was accounted for by calculating the statistical
uncertainty of the centroid. For a peak with calculated centroid at C,

the expression for C is

C = 2 n c / Z n, where n1 is the number of
counts in channel ci,
and its uncertainty is
. 1/2
6C = [2 ni(ci-C) / N(N-1)] where N is the total number
i
of counts in the peak
Translating this to energy units is accomplished as follows: for a
centroid with uncertainty 6C located at a detector position where the
dispersion constant - AF , 6C becomes an effective field uncertainty:

FAC

6F = 6C x AF x F
FAC

6
and hence an uncertainty in the outgoing He energy:

E(6He) = 6C x (AF ) x F x 2 x E(6He)
FAC

In the majority of cases the centroid uncertainty ranged between

0.2 - 0.7 channels. For the typical value AF - 1.5 x 10"2 KHz

 

FAC MHz channel

51
this yielded uncertainties in the outgoing 6He energies of 2.0 - 7.0 KeV.
These uncertainties are reflected directly in the calculated Q-values:

AQ(6C) = ac x (AF ) x F x 2 x E(6He)
FAC

4.4 Scattering Angle Fluctuations

6He energies due to uncertainties in

The uncertainty of outgoing
the scattering angle 0L consists of two parts. The first is a random
contribution to each point due to possible fluctuation in beam direction
while a particular data point was being taken. From consideration
concerning the geometry of the beam delivery system and the general
consistency of the calibration data this effect was estimated to yield
an uncertainty in 0L of 1 .03°. This is reflected in the outgoing
6He energy through the value of the kinematic energy spread for each

reaction. Table 4.1 shows these values and the energy uncertainty

for each reaction for OL = 11.00° 1 .03° and E(BEAM) = 68.5 MeV.

Table 4.1. Uncertainty of Outgoing 6He Energy for 0L = 11.00° 1 .03°

and E(BEAM) = 68.5 MeV

Reaction 12C(3He,6He)9C 160(3He,6He)l30 24Mg(3He,6He)21Mg

 

Kinematic Spread 150 KeV/deg 108 KeV/deg 70 KeV/deg

E(6He) for 00=:.03° t 4.5 KeV : 3.3 KeV : 2.1 KeV

 

52

These uncertainties are reflected directly in the calculated Q-values:
AQ(0L) = AE(6He)
4.5 Uncertainty of Scattering Angle Measurements

The second uncertainty in the lab scattering angle is that attached
to the actual measurement of the angle. Since the 1H(3He,3He)1H reaction
used to measure BL is extremely sensitive to angle because of the kine-
matics of the reaction (1480 KeV/deg at 0L 8 ll.0°, E(BEAM) - 68.530 MeV)
it is quite insensitive to the beam energy. The angle measurement was
therefore assumed good to i .03°, since this meant determining the
outgoing 3He energy to only i 45 Rev. The angle uncertainty affects
the outgoing 6He energy two ways; first through the beam energy, since
the reaction 12C(3He,3He)12C and 0L are used to determine the beam energy,
and second, directly through the measured 6He energy. Fortunately, these
two effects tend to cancel one another since the Q-value reflects a
difference of incident and outgoing particle energies. This is illustrated

in the following example:

Suppose the elastics from 12C(3He,3He)12C are measured and have
energy E(3He) - 67.892 MeV for OL = 11.0° this gives E(BEAM) = 68.530 MeV.
For the same outgoing 3He energy but 0L = 10.0°, E(BEAM) = 68.420. Now,
for the reaction 12C(3He,6He)9C, these two sets of parameters would

yield:
E(6He) = 36.069 MeV for E(BEAM) - 68.530, 0L = 11.0°

and E(6He) = 36.116 MeV for E(BEAM) = 68.420, 0L = 10.0°

53
6 _ . 12 3 6 9
Therefore, AE( He) — 47 ReV/deg for C( He, He) C. The same calculation

for the other reactions gives:
AE(6He) = 6 KeV/deg for l60(3He,6He)130
AE(6He) = 45 KeV/deg for 24Mg(3He,6He)21Mg

Assuming OL = i .03°, the uncertainty reflected in the final outgoing

6He energies and hence the calculated Q-values becomes:

AQ(A0L) = 1 1.4 KeV for 90
AQ(A0L) = 1 0.2 KeV for 130
AQ(A0L) = 1 1.4 KeV for 21Mg

Since such an error in angle determination would be systematic
over an entire run, these uncertainties are applied to.the average

Q—value of each reaction, rather than to each particular measurement.

3

4.6 He Beam Energy Uncertainties

The systematic uncertainties associated with absolute measure-
ment of the 3He beam energies are so much a function of the calibration
procedure and scattering angle determination, that the values used for
3He beams are effectively defined by these procedures. Systematic
errors in its value has therefore been absorbed into these other
uncertainties.

Estimates of the beam energy fluctuations during a run were

obtained from the scatter of the 3He calibration reaction points

54
over the course of a run. For example, nine individual measurements
of the 12C(3He,3He)12C elastics over the course of a two day run varied
by a maximum of only 8 Rev, with the other calibration reactions show-
ing similar scatter commensurate with their sensitivity to beam energy.
This remarkable stability of beam energy was of prime importance to
the experiment. The largest such fluctuation for any of the runs was

AE(BEAM) 8 i 10 KeV.
4.7 Uncertainty in Target Energy Loss

This problem is discussed in the section on target energy loss
measurements, and uncertainty values are given there. In the case of
the 24Mg(3He,6He)21Mg measurement where only one target was used for
all measurements, the uncertainty is treated as systematic and applied
to the average of the Q-value measurements. For the 9C and 130 measure-
ments several targets were used, and the uncertainties applied to the

Q—value measurements for the corresponding targets.
4.8 Stray Magnetic Fields

To guard against spurious effects on incident beam direction due
to stray fields of the cyclotron, bending magnets, and particularly
the spectrograph whose field was often changed, as well as effects
due to the earth's magnetic field, all exposed areas of the beam

transport system were wrapped with soft iron for magnetic shielding.

5. EXPERIMENTAL RESULTS

Table 5.1 lists all individual measurements of the Q-values
obtained along with the experimental parameters pertinent to each.
The column labled "total error” represents all known experimental
uncertainties for that point summed in quadrature as though that
were the only measurement made. The column labled ”partial error”
represents the sum (unquadrature) of random errors associated with
that particular point.

Table 5.2 lists the resulting average Q-values, their total
uncertainties and the resulting values for the mass excesses of
90, 130, and 21Mg.

Figure 5.1 displays typical position spectra for the (3He,6He)

reactions.

55

56

 

:mmwao; Ho.HH moo.H~Hm.wo Nae.“ ~NO.H sam.om- cmaammo.mmmvooa mmum
owIE .oam Ho.HH moo.0~am.m6 0H0.“ HHo.H con.0m- cmaxmmo.mmmVooH aa-m
Ha QNH Ho.HH moo.w~am.mo moo.“ HHo.H nam.Hm- umnmmo.mzmvuma swim
He UNH Ho.HH moo.w~am.m6 sac.“ HHo.H sam.Hm- omammo.mmmVoNH maum o~\NH\6
anon wzqm 69.0H woo.HORs.mo 0H0.“ “so.“ mam.n~- wzfluxmmo.mmmvw26~ woai~
anon wzsm 6m.OH moo.noks.w6 mac.“ mac.“ mmm.n~- mzH~A6:6.6mmezqm moaum
om1m .onm 6m.oH moo.HoNs.wo 0H0.“ HHo.w som.om- omHAmmo.m:mv06H noaum
om-m .on om.oH moo.no~s.w6 0H0.“ Nae.“ smm.om- emafimmo.ommVooa HOH-N
omum .oam om.oH woo.nonq.m6 HHo.H Nae.“ oom.om- cmaammo.mmmvooa manm
Ha QNH 6m.oH woo.wo~q.m6 0H0.“ HHo.H Nam.am- Ammo.mzmvoma 60H-N
Ha omH 6m.OH woo.wo~q.mo 0H0.“ HHo.H Hmm.am: oaxmmo.mmmvomfi swim
Ha UNH om.oa woo.wohq.m6 0H0.“ sac.“ sam.Hm- omfimzo.mmmvoma mmum o~\M\q
:mmmao: mm.qa oao.waam.mo mac.“ ONO.“ oam.am- emafimmo.mmmVooa mmaua
Ha UNH mm.sa oao.fisam.mo mac.“ ONO.“ oom.Hm- omAmxo.mmmvu~H Hmaufl oa\M\m
Awmwv A>mzv A>mzv A>mzv
A xwumcm uouum uouum A>mzv umnasz
umwume o Edmm 6mm Hmwuumm Hmuoe o=Hm>IO cofiuommm wax mum:

mosam>lo Ammo.mmmv vmcHEumuma haamucmEHumaxm H.m wanna

57

 

 

o+ofim No.0H OHo.Hom~.m6 HHo.H 6H0.“ oam.omu omaAmmo.ommvoofi omum
o+onm m6.oa oao.woa~.mo HHo.H sac.“ oam.om- omHAmmo.mzmvooH mmnm
o+owm 16.0H o~o.wmma.wo Has.“ 6H0.“ m~q.omu omafimzo.mmmvooa whim
1m: UNH w6.oa oao.wmma.wo ~HO.H mac.“ Nam.Hm- umxmmo.mmmquH amum
:1: UNH m6.oa oao.ao-.m6 HHo.H Nae.“ Nam.am- umfimmo.mmmvoma mmum o~\om\m
anon wzsm Ho.HH moo.w~am.m6 0H0.“ ado.“ mom.-- mzHNAmmo.mmmvwzs~ mmum
anon quN HO.HH moo.w~am.m6 moo.“ mam.“ mmq.-- wzflmawmo mmmeZqN Hmum oa\~H\q
Awmsv A>6zv A>6zv A>6zv
A hwumcm uouum uouum A>ozv umnasz
uomuma e Baum mam Hmfiuumm Hauoe maam>lo coauommm cam mama
wmscaucoo H.m maan

58

Table 5.2 Average Q—values and Mass Excesses for the T2 8 -3/2 Nuclei

 

Element Reaction Q-value (MeV) Mass Excess (MeV)
9c 12C(3He,6He)9C -31.5781.008 28.9111.009
130 160(3He,6He)130 —30.5061.013 23.1031.014
ZlMg 24Mg(3ne,6ne)21ng -27.5121.018 10.9121.018

 

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6. DISCUSSION

Table 6.1 gives the coefficients of IMME for the A = 9, l3, and
21 isobaric quartets. The coefficients a(a,T), b(a,T) and c(a,T)

were obtained from a least squares fit of the form
M=a+bT +cT2
z z

to the mass excesses of the quartet members. The d(a,T) coefficient
is the coefficient of a T2 term when the same data is fit to the

expression

2+dT3

M = a + sz + cTz 2

For both cases the mass excess values for the T2 = - 3/2 members are

2

taken from Table l of (Ga 69). The term represents the quality

of the fit, and the expression used was

1/2

 

2 = §[M(calc) - M(exp)]
oM(exp)

Deviations of the experimentally determined masses and predictions
of the IMME with a, b, and c coefficients of Table 6.1 are displayed
graphically in Figure 6.1. Only the A - 9 quartet shows deviations
greater than the experimental uncertainty of the points.

The addition of higher order terms such as T3 and T: to the IMME
are predicted when the Coulomb potential is expanded as a second order
perturbation. Such a second order treatment then involves mixing of
states of T(perturb) - Til through the off diagonal matrix elements of

the expansion. This has been done in detail in a recent article by

60

61
Janecke (Ja 69), where the IMME is expanded to a quartic in T2. The
conclusion reached here is that terms in T: and T: will be small, not
so much because the perturbation is small, but because the major
effects of such perturbations are absorbed mostly in the T2, and T:
terms.

The size of the d coefficient (of dTi) has been estimated to be
=Zac (Ga 69) where Z is the average charge of the multiplet, is
the fine structure constant, and c the coefficient of Ti. For the
A = 9 quartet this would be =9 KeV and the data indicates a d term
of this magnitude. For A = 13 and 21 such a term is not evident.

The determination of such a cubic term from isobaric quartets
must be done with caution. Since the cubic will fit a quartet
exactly, the d term and its uncertainty may be expressed in terms
of the masses:

d = m(3/2) - M(—3/2) - 3{M(1/2) - M(—l/2)}
6

 

Ad a AM(3/2) + AM(—3/2) + 3{AM(l/2) + AM(—l/2)}
6

 

where M is the mass of the multiplet member whose Tz value is in
paranthesis. From these expressions, it is apparent that the
generally small experimental uncertainties of the T2 = t 1/2

members (= :5 KeV) may contribute as much or more to the uncertainty
in d as the larger uncertainties of the T2 = t 3/2 members. For this
reason, careful determination of all member masses is required.

Another attempt to make some estimate of the size of the d term

 

tr—

 

62
is presented in (He 69), where the Schroedinger equation is solved
directly for a nuclear model. The model consists of three nucleons
outside an inert core, with each of these extra nucleons in a Woods-
Saxon nuclear potential, a Lane symetry potential, and the Coulomb
potential of a uniformly charged sphere with radius equal to that of
the Woods—Saxon well. The coefficients for the T2 and T: terms agree
with experimental values, generally to within 10 - 20%. The predicted
d term, however, is on the order of several hundred eV rather than
the =9 KeV (Zac) estimate. Such values are not inconsistent with
those of A = 13 and 21 of Table 6.1.

Because many charge dependent effects may be absorbed into the
T2 and T: terms, the quadratic form of the IMME fits data well over
a wide range of mass values (Table 2 in Ca 69, for example). This
quality provides confidence that the quadratic IMME may be used to
extrapolate to unknown masses with good accuracy. This has been done,
for example, to determine the mass of 2581 (Wi 70).

Several attempts to extract specific information from a study of
the IMME coefficients (Ja 66 and Nil 64) but were limited to some
extent by the quantity and quality of data available on T91 multiplets.

The uniformly charged Sphere model serves as an example of how
information may be extracted from the coefficients. For a Coulomb
energy of

vC=_3_2(z-1) e2
5 RC

 

the term Z(Z - 1) may be expanded in terms of T2 and T: to yield the

63

IMME coefficients (Ca 69):

b = - §_(A-1 e2 + (Mn — Mp)
5 R A
O
c =‘3 e2
5 ROAI/3

A is the mass number of the multiplet and (Mn-Mp) the neutron-proton
mass difference. The Coulomb radii Ro may be obtained for known b and
c coefficients, and these radii are given in Table 6.2. The validity
of such a model will be reflected in the values of the radii, and in
their consistency. In addition, the radius values should be the same
whether extracted from the b or c coefficient.

The radii from Table 6.2 are reasonable and compare with those
of various models (No 69, for example). This would indicate that such
a Coulomb potential is a major contributor to T2 and Ti dependence.
The fact that the b and c coefficients yield different results, however,
indicates that other effects are also present. An attempt to reduce
this effect by deforming the sphere yields negative results since
such a reduction in the Coulomb energy could not account for the
fact that for A = 37 the ratio Rig) > 1 while it is less than one
for the other multiplets. Estifiétis of the deformation effect also
indicate that changes in the Big) ratio would be small. The failure
of this simple model to reprOESZ; these effects indicates that charge

dependent interactions in addition to the Coulomb interaction are

contributing to the T2 and T: terms.

64

A detailed study of the coefficients to obtain specific information
on the Coulomb systematics requires good experimental measurements of
high T multiplet members. In addition, the determination of excited
levels in the T2 - - 3/2 members would be valuable for comparison with
structure of the other multiplet members. The experimental procedures
developed in this work will, in addition to making absolute Q-value
measurements at high bombarding energy possible with high precision,
allow study of neutron deficient nuclei and thus aid in the study of

the Coulomb systematics of high T multiplets.

65

 

Table 6.1 Empirically determined coefficients for the mass equation
M = a + sz + cTZZ. The last column indicates the
coefficient of a T23 term assuming the equation to
have the form M = a + bTZ + cTZ2 + dTZ3,
Mass a(a,T) MeV b(a,T) MeV c(a,T) MeV X2 d(a,T) KeV
9 26.343i.004 -l.3185:.003 0.266t.003 4.0 0.0083:.0039
13 19.257i.0027 -2.l802:.0035 0.256i.003 .002 -0.0002:.0035

21 4.8987i.0046 -3.6573i.005 0.240i.0048 1.28 0.0057i.0051

66

Table 6.2 Coulomb Radii from the IMME Coefficients

 

Mass b(a,T) Ro(b) C( ,T) RO(C)
A (MeV) (Fermi) (MeV) Fermi
9 -1.3198 1.581 .2668 1.562

13 —2.1803 1.489 .2568 1.436

17 —2.878b 1.469 .238b 1.412

21 -3.6573 1.411 .240a 1.305

23 -3.960c 1.409 .223c 1.362

25 —4.387b 1.372 .216b 1.368

37 --6.181b 1.340 .174b 1.490

 

Present work
b See (Me 70)

c See (Ga 69)

67

A9

 

 

 

D
11

2|

X2: 0.002

 

 

 

 

Figure 6.1

_ I ' T
.9 : 1 1

3/2 Mass deviations from If

APPEND Di

APPENDIX A1
Derivation of the Isobaric Multiplet Mass Equation (IMME)(Ga 69).
The IMNE:
M(a,T,TZ) = a(a,T) + b(a,T)Tz + c(a,T)T:

may be derived from first order perturbation theory with the follow-
ing assumptions: the Coulomb potential is two-body, and is a
perturbation on the nuclear potential, and all members of an isobaric
multiplet are identical in their charge independent characteristics.
The charge independent characteristics are denoted by a in the

equation. Then for

VC = z e2 (1/2 - ‘21)(1/2 - cz

....—

)
i>j rij j

where tz is the isospin value +1/2 for neutrons and -1/2 for neutrons,

and r. is their relative separation, the expression for VC becomes:

13

2
V = X 1 — 1 2 + + t .
c 1>j§‘“( l4 / (tzi tzj) 21‘23)
ij
and judicious addition of a term tzinzj yields
3
Vc - 2 e2 (1/2) +(tzi'tzj - 2 e2 (121+tzj)
1>J rij "’7¥"’ 1’3 2‘13
2 _ t .t
+ it : (tZith Zi Zj)
>1“““ij “*3

68

69

The Coulomb potential has thus been separated into tensor operators
VC(O), Vc(1), and Vc(2) of rank 0, l, 2, and the perturbing Coulomb

term <aTTz'VC'<TTZ> may be reduced with respect to isospin:
<a,T,Tz'Vc'a,TTZ> = <a,TTzIV(0)‘aTTz>
<0,TTZ'V(1)'0TTZ>
<0,TTz|V(2)|aTTz>
using the Wigner-Ekhart theorem this becomes:
<a,TTZlV(0)'aTTz> a <TOTZOITTZ><a,T||V(0)I|a,T>
<a,TTz|V(1)laTTz> = TszTZ'TTz><a,Tu v(1)fl a,T>
<a,TTle(2)|aTTz> = (TZTZTZII‘TZ><0,TH V(2)u a,T>

where the double bracket represents matrix elements reduced w.r.t.

isospin. The Clebsch-Gordon coefficients are:

<T0TZOITTZ> = 1

 

 

<TlT T ITT > = T2 11
z z [mum/7
2-
<T2TZTZ|TTZ> a 3Tz T(T+1)

[72T-1)T(T+1)(2T+3)]1/2

Since T is a constant for a given multiplet the Coulomb perturbation

then becomes

<a,TTZlVCIa,TTz> = a(a,T) + b(a,'r)'rz + c(a,T)T§

70
The mass difference between the proton and neutron which must be
taken into account between the members of an isobaric multiplet may

be written:

V(Am) = (I~1n-+MP)A + (Mn—Mp)Tz
2

and may therefore be absorbed into the IMME since it is linear in

T .
z

LIST OF REFERENCES

A:

An

Ba

Be

Ce

Ch

Da

En

Ga

He
Ja

Jo

Mar

No

66

62

69

57

68

67

69

58

69

69

69

70

70

66

67

67

69

71
T. Lauritsen and F. Ajzenberg—SeLove, Nucl. Phys. 78
(1966) No. 1 pp 24

J. D. Anderson, C. Wong, and J. W. McLure, Phys. Rev.
126, 2170 (1962)

D. Bayer, Bull. Am. Phys. Soc., (1969)

Yardley Beers, Introduction to the Theory of Error,
Addison-Wesley, Reading, Mass. (1957)

 

J. Cerny, Ann. Rev. of Nucl. Sci., 18 (1968) pp 27

 

C. Chasman, K. W. Jones, R. A. Ristinen, D. E. Alburger,
Phys. Rev. 159 (1967) pp 830

Guide to the Selection and Use of Position Sensitive
Detectors, Nuclear Diodes (1969) ed. by W. W. Daehnick

H. A. Enge, Review of Scientific Instruments, 29 (1958)
pp 885

G. T. Garvey, Nuclear Isospin, Proceedings of the Conference
on Nuclear Isospin at Asilomar, ed. by J. D. Anderson, S. D.
Bloom, J. Cerny, and W. W. True, Academic Press, New York,
703 (1969)

 

E. M. Henley and C. E. Lacy, Phy. Rev. 184 (1969) pp 1228
Joachim Janecke, Nuclear Physics A128 (1969) 632

R. K. Jolly, G. F. Trentelman and E. Kashy to be published
in Nuclear Instruments and Methods

E. Kashy, R. K. Jolly, and G. F. Trentelman, to be published

C. Maples, G. W. Goth and J. Cerny, Nuclear Reaction Q-values
Lawrence Radiation Laboratory, Berkeley, California,
UCRL-l6964 (1966)

G. H. Mackenzie, E. Kashy, M. H. Gordon, and H. G. Blosser,
The Beam Transport System of the Michigan State University
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No. 3, 450 (1967)

J. B. Marion, University of Maryland Technical Report
#0R0—2098—58 (1967)

J. A. Nolen and J. P. Schiffer, Coulomb Energies, University
of Maryland Technical Report No. 70-035 (1969)

Sn

Sp

Te

Tr

Wa

Wia

Wi

Wi

Wi

Me

Ja

67

67

69

70

66

57

64

70

70

66

72

J. L. Snelgrove and E. Kashy, Nucl. Inst. and Meth.
52 (1967) pp 153

J. E. Spencer and H. A. Enge, Nucl. Inst. and Meth. 49
(1967) pp 181

B. Teitelman and G. M. Temmer, Phys. Rev. 177 pp 1656
(1969)

G. F. Trentelman and E. Kashy, to be published in Nuclear
Instruments and Methods

E. K. Warburton, J. W. Olness, D. E. Alburger, Phys. Rev.
140 B1202 (1965)

C. F. Williamson, J. P. Boujot, J. Picard, Tables of

Range and StOpping Powers of Chemical Elements for

Charged Particles of Energy 0.05 to 500 MeV, Centre D'Etudes
Nucleaires De Saclay, CEA—R3042 (1966)

E. P. Wigner, Isospin-A Quantum Number for Nuclei Proceed—
ings of the Robert A. Welch Foundation Conferences on
Chemical Research, I. The Structure of the Nucleus (1957)
Houston, Texas

D. H. Wilkinson, Phys. Rev. Letters, 13, No. 19 (1964)
pp 571, Physics Letters, Vol 12 No. 4, (1964) pp 348

D. H. Wilkinson and D. E. Alburger, Phys. Rev. Letters,
24 (1970) pp 1134

R. Mendelson, G. J. Wozniak, A. D. Bacher, J. M. Loiseaux,
and J. Cerny, submitted to Phys. Rev. Letters (1970)

Joachim Janecke, Phys. Rev. 147 pp 735, (1966)

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