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dissertation entitled

The Mirror Charge Exchange
Reaction 13C(13N,13C)13N

presented by

Mathias Steiner

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Physics

 

 

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Major professor

Date 10/27/95

 

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MSU to An Nfirmetive ActioNEmel Opportunity Inetitulon
Wm:

— _.._——_._____,__ W, ._

THE MIRROR CHARGE EXCHANGE
REACTION 130(13N,13C)13N

by

MATHIAS STEINER
A DISSERTATION

Submitted to
Michigan State University
in partial fulfilment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1995

ABSTRACT

THE MIRROR CHARGE—EXCHANGE REACTION 13C(13N,13C)13N

By

Mathias Steiner

Differential cross sections have been measured near 0° for the dominant channels
in the mirror reaction l3C(13N,13C)13N at E/A = 57 and 105 MeV. The cross sections
of the peaks in the excitation spectrum are discussed in terms of the Gamow-Teller
and Fermi transition strengths in the target and the projectile. The cross section per
unit Gamow-Teller strength is found to be enhanced relative to that for unit Fermi
strength when compared with previous results from (p,n) reactions. The use of heavy

ions as probes for weak—interaction strengths in radioactive nuclei is disussed.

The present work represents the first use of mirror symmetry to study heavy—ion
charge exchange as well as the first application of the developing radioactive nuclear

beam field to this area.

Meinen Eltern gewidmet

iii

ACKNOWLEDGEMENTS

Before I start to thank those individuals who have made a difference during the last
four years, I want to thank Michigan State University, the NSCL, and the National
Science Foundation for providing support and the opportunity to do my Ph.D. work

in this country.

When I first arrived at the Cyclotron, I was hoping to be able to work for Brad
Sherrill, who had been warmly recommended to me by several physicists at GSI in
Darmstadt, Germany. In retrospect, he really didn’t get much of a chance to say
no... Brad gave me plenty of freedom in my work, but he was always there to help,
advise, or console as needed. Not to mention the fact that it was he who really got

the charge—exchange experiments off the ground. Thank you for being a great boss!

Speaking of bosses, Dave Morrissey was very supportive throughout. Be it postscript
problems, nuclear electronics, or the recalibration of the beamline magnets, Dave
would always help out. His help as well as his, ah, frank conversational style are very

much appreciated.

There is no better tonic for a young physicist’s frayed nerves than a chat with

Walt Benenson, nuclear experimentalist extraordinaire and elder statesman of the

A1200 group.

Thanks are also due to those who contributed to my thesis experiments: Ed,
Nigel, Bob, Sherry, and Brian in the first test run, and later on Daniel, Jim, Mikey,
Maggie, Ned, Toshiyuki, Raman, and Chris. Everything had to work just so for the

charge—exchange runs, and (almost) everything did. Thanks a lot, folks!

Many thanks also to the cyclotron support staff, which is truly deserving of its

name. Thanks to Jeff for a great tuning job, and to Tim for a ton of beam. I needed

that.

The original idea for the mirror charge—exchange experiments came from George
Bertsch, for which I am grateful. As far as the theory behind the experiments is
concerned, I was pretty much in the dark for a considerable length of time. I want to
thank Sam Austin, Carlos Bertulani, Alex Brown, Scott Pratt, Jack Rapaport, and
John Watson for helpful discussions on charge exchange. Anantaraman and Jim Carr

helped me understand the double-folding model.

Without any doubt, those who shared an office with me deserve special mention:
Don, my choice for “best tantrum”, and Brian, who inflicted his infernal radio pro-
grams on the rest of us. Oh well. Ever since Brian left, things have gotten a lot
quieter (except when “Car Talk” is on). It has further been my pleasure to share the
office with Sally and Jing, and with Mikey (as long as he doesn’t explain how “life

stinks” , and why).

Further thanks are due to Ron, Jac, Thomas, and Renan for three great DALMAC
tours; to Michael T. for a memorable teaching experience; to Phil, for culinary and
cultural support; to Jon, for his unwavering support in times of need (love machine
breakdown); to Damian, for comic relief; and to Dave, Nancy, and Michael for a

glimpse of what it’s like.

This work was made possible, in part, by the Peanut Barrel, El Azteco, Velocipede
Peddler, Meridian Auto Parts, and the Sears tool department. Without these fine

establishments, life would not have been the same.

Finally, I want to thank Charlotte for always being there for me.

Contents

LIST OF TABLES

LIST OF FIGURES

1

2

Introduction

Charge—Exchange Reactions

2.1 The (p,n) Reaction ............................
2.2 Fermi and GT Transitions ........................
2.3 The 13C Controversy ...........................
2.4 Heavy Ions as CT Probes ........................
2.5 Other Charge—Exchange Probes .....................

The Experiments

3.1 Secondary—Beam Production .......................

3.2 Cross—Section Measurements .......................

3.3 Ion Optics .................................
3.3.1 A1200 in Dispersion—Matched Mode ..............
3.3.2 Optical Modes ...........................
3.3.3 Dispersion and Energy Calibration ...............
3.3.4 Horizontal Matrix Elements ...................
3.3.5 Vertical Matrix Elements .....................
3.3.6 Achromatism and Image Aberrations ..............

3.3.7 Acceptance Correction ......................

Data Analysis
4.1 Background Determination ........................

4.2 Drifts ...................................
4.3 Energy Spectra ..............................

vi

viii

ix

7
13
15
20
27

28
29
33
35
39
40
44
46
48
51
53

60
60
65
66

4.4 Angular Distributions ........................... 7O

5 Results 77
6 Conclusion and Outlook 82
A The 13C Targets 86
LIST OF REFERENCES 89

vii

List of Tables

3.1

4.1

4.2
4.3

5.1

Experimental parameters for the Charge—Exchange experiments

Integrated cross sections determined from fitting the excitation energy
spectra. ..................................

Fit parameters for the angular distributions ..............

Results from the mirror charge—exchange experiments. ........

Unit cross sections obtained in the mirror charge—exchange experi-
ments. ..................................

viii

List of Figures

2.1 Schematic description of charge—exchange scattering. ......... 5
2.2 Angular distribution of elastic (n,p) scattering at 200 MeV. ..... 6
2.3 Comparison of CT matrix elements for sd—shell nuclei with the predic-

tions from a shell—model Hamiltonian. ................. 11
2.4 Percent of sum rule bound observed by integrated (p,n) cross section

for a variety of targets. ......................... 12
2.5 Zero—degree cross—section spectra for the 14C(p,n)”N reaction at the

indicated bombarding energies. ..................... 14
2.6 The quantity R(Ep) determined from many target nuclei. ...... 16
2.7 Cross section spectrum for 13C(p,n) at 0° and 120 MeV. ....... 17
2.8 Schematic illustration of the nuclear configurations in 13C and 13N for

the states under discussion. ....................... 18

2.9 Comparison of the sensitivity function S(q) (solid line) and the transi-
tion densities for transitions to the lowest six 1+ states of 26Al in the

26Mg(12C,12N)26A1 reaction at E/A = 70 MeV (dashed lines). . . . . 23
2.10 Cross section measurement near zero degrees for the 12C(lr‘lC,12N)12B

reaction at E/ A = 135 MeV. ...................... 24
3.1 AE-TOF particle—identification spectra obtained with the A1200. . . 31
3.2 Energy-calibrated position spectra for the (”N ,13C ) reaction prod-

ucts. .................................... 34
3.3 Principle of the energy-loss mode. ................... 37
3.4 The A1200 Fragment Separator. .................... 38
3.5 A1200 in Medium—Acceptance Mode. ................. 41
3.6 A1200 in Reaction Mode. ........................ 43
3.7 Bp3 vs. x; calibration of the second dipole stage. ........... 45
3.8 Focusing of the second A1200 dipole stage. ............. -. 47
3.9 Determination of the (0“?) matrix element. .............. 49
3.10 Spectra used to determine the vertical first—order matrix elements. . 50
3.11 Checking the dispersion—matched condition. .............. 52

3.12 A plot of 6f vs. Xf for products of the 12C(14N,13C)13N reaction.

3.13 One-dimensional acceptance in the 0; and <25; coordinates. ......

3.14 Acceptance as a function of scattering angle 03cm“. .........

3.15 Acceptance correction of an angular distribution. ...........

4.1
4.2
4.3
4.4
4.5

4.6
4.7
5.1

Background correction for the 57 MeV experiment. ..........
Background correction for the 105 MeV experiment. .........
The slope parameters obtained for the seven energy bins. ......
Energy specta and their interpretation. ................

Schematic illustration of the target and projectile transitions in the
l3C(13N,13C)13N reaction. ........................

Scattering—angle analysis. ........................
Angular distributions for the 13N secondary beam. ..........

Energy dependence of aaT/a‘ip for a number of charge—exchange reac-
tions performed on 13C targets. .....................

81

Chapter 1

Introduction

Recent developments in the rapidly growing field of radioactive nuclear beams (RN B)
have opened up new areas of research, and RNBs can be expected to help answer a
number of important questions in nuclear physics. Experiments with a wide variety of
radioactive nuclei, many of them of astrophysical importance, have become possible
and exciting new phenomena are being observed. For example, halo nuclei are cur-
rently among the most intensely studied objects in nuclear—structure physics [Rii94].

Radioactive beams also make available nuclei with special symmetries such as doubly

magic 100Sn [LeA94, ScF94].

One new capability provided by RNBS are nuclear reaction studies with pairs of
mirror nuclei, since only one member of a mirror pair can be stable and a radioactive
beam is necessary to provide the partner of the stable target nucleus. Due to their
special symmetry, mirror pairs are a useful tool for the study of the strong force
in a nuclear environment and provide a unique opportunity for attaining the long—
term goal of understanding charge exchange in heavy ions. In this work, the relative
importance of spin—flip and non—spin—flip contributions in heavy—ion charge exchange
(HICEX) is investigated, and the usefulness of the reaction as a method of determining

Gamow-Teller (GT) transition strengths in nuclei is explored.

{\D

Weak—interaction strengths are of fundamental interest for our understanding of
the nucleus, and GT matrix elements for a large number of radioactive nuclei are im-
portant parameters in nuclear astrophysics. The physics and terminology of charge—
exchange reactions are introduced in chapter 2. A large part of this chapter is devoted
to the (p,n) reaction, which constitutes the majority of charge—exchange work done
to date. It has been established over the course of the last 15 years that the zero—
degree (p,n) cross sections observed for a given target nucleus are closely proportional
to the corresponding weak-interaction transition strengths, a phenomenon labelled
“specific proportionality.” Heavy—ion charge exchange is presented in the same frame-
work, and the differences to nucleon—induced CEX reactions are discussed. Relatively
few HICEX experiments have been performed to date, and the work described here
is the first such study using radioactive beams as well as the first nuclear reaction

performed with a pair of mirror nuclei.

Due to the relatively poor emittance of the secondary beam, these experiments are
difficult to perform, and it is one of the goals of this study to investigate the general
feasibility of high—resolution nuclear—structure experiments with radioactive beams.
Chapter 3 describes how this was achieved. Detailed experimental parameters are
given, and considerable space is devoted to the ion—optical conditioning in the A1200
fragment separator. Ion optics is a topic of increasing importance in RNB physics,

but concrete experimental examples are still scarce in the literature.

In chapter 4, the analysis of the data is described in detail, and the interpretation
Of the spectra is discussed. The final results are presented in chapter 5 and compared

to other charge—exchange results.

The 13C(13N,13C)13N energy spectra are easily interpreted in terms of weak—

iIlteraction transitions in both the target and the projectile. Other amplitudes ap-

pear to be of little importance at forward angles. The symmetry of the mirror pair
greatly aids in analysing the data and allows the extraction of the so—called Fermi and
Gamow-Teller “unit cross sections”, in close analogy to the parameters used in (p,n)
work. Mirror symmetry further aids in extracting the relative strength of Fermi and
GT contributions from the data. The unit cross sections observed in the present work
are found to be significantly larger than other HICEX results, which illustrates the
need to carefully calibrate the relationship between yield and GT matrix elements.
A controversy regarding the exact values of the transition strengths in 13C prevents
a rigorous quantitative analysis at this time, but the observed cross sections are fully

compatible with the assumption of specific proportionality.

Even though some difficulties still persist in the precise understanding of the un-
derlying physical processes, the results of the present experiments indicate that heavy—
ion charge exchange has the potential to become a powerful tool for GT strength

measurements in radioactive nuclei.

Chapter 2

Charge—Exchange Reactions

Charge—exchange reactions play an important role in our current understanding of the
nuclear force. This is illustrated by the two—nucleon system. Only one of the three
possible systems —— the deuteron — is bound, but neither the dineutron nor the dipro-
ton is. While the similarity of the n—n and p—p systems established the approximate
charge—indipendence of the nuclear force, deuteron binding provided an indicator of
the spin—dependence of the nuclear force. Most of the detailed information on the
nucleon—nucleon interaction, however, was obtained through scattering experiments.
One of the most important and interesting results is shown in Fig. 2.2, which shows
the angular distribution of the 1H(n,p)n reaction at an incident neutron energy of
200 MeV. While the peak at forward angles (0°) is a general feature of elastic scat-
tering from a potential, the sharp peak at backward angles (180°) is evidence for 0°
(n,p) chargeeexchange scattering, as shown schematically in Fig. 2.1. Together with
data from (p,p) and (n,n) scattering experiments over a wide range of energies, the

charge—exchange results led to the formulation of the one-pion exchange potential

[dSF74]:

1 2 3 3
VOPEP = fimnC2(T1'Tz)[(01'02)+ (1+ “— + I" >312] , (2-1)

 

 

      

(a) no charge exchange (b) charge exchange

Figure 2.1: [dSF74] Schematic description of charge—exchange scattering; scattering
of the original proton through the angle 0 looks like backward scattering by an angle
7r — 0 if charge is exchanged.

where a and T are the spin and isospin operators, respectively, 812 is the tensor

operator,
312 = $13071 '1‘)(02'1‘)—(01'02)l, (2-2)

and gz/hc is a coupling constant. Equation 2.1 provides the link to weak—interaction
physics, as it contains the < 01' > operator. The importance of charge—exchange
reactions for measurements of B—decay transition strengths is discussed in the next

section.

Surprisingly, despite the myriad of nuclear reaction experiments performed over
the past half—century, the 1H(n,p)n reaction remains the only scattering experiment
with a mirror system to date. The reason for this is the fact that only one member of a
mirror pair can be stable and that a radioactive beam or target is therefore necessary
to perform these experiments. The increasing availability of radioactive beams finally
made mirror charge exchange a possibility. While nucleon—nucleon scattering is one of
the most basic tools of probing the nuclear force, charge exchange with “real” mirror
nuclei presents a much more intricate problem. Some of the complications arising

from the extended nature of the projectile will be addressed in section 2.4.

The special symmetry of the mirror pair greatly aids in analysing and interpret-

 

np scattering

_e
0

U1

(daldmm lmb sr"!

 

 

 

 

0 60 120 180

Figure 2.2: [BeE87] Angular distribution of elastic (n,p) scattering at 200 MeV. The
curve is calculated from phase shifts. Note the sharp peak at 180° scattering.

ing the resulting spectra and extracting information on the underlying physics of the
charge—exchange process. Furthermore, the zero Q value of the mirror reaction en-
sures that the transition densities will be probed over the entire range of momentum

transfer including the important region around q = 0.

The goals of this study are to learn more about the mechanisms of heavy—ion
charge exchange and to investigate the feasibility of using these reactions to mea-
sure weak-interaction strengths in radioactive nuclei. The strength and location of
Gamow—Teller resonances in many f p—shell nuclei are important and highly sensitive
parameters in astrophysical calculations describing the evolution of type II supernova
progenitors [Au893]. Measurements of GT strengths in radioactive nuclei in the iron
region are very difficult with presently available methods, and charge—exchange reac-
tions with radioactive nuclei have the potential to become powerful tools for determi-
nation of these parameters. Finding workable experimental solutions to this problem
is all the more important because shell-model calculations in this mass region have

been shown to be unreliable [Au893].

2.1 The (p,n) Reaction

Over the past 15 years, nucleon—induced charge—exchange reactions have received
much attention. The discovery of both the isobaric—analog state [AnW61] and the
Gamow—Teller resonance [DOG75] were achieved using (p,n) charge—exchange scat-
tering. Perhaps the most important application of the (p,n) reaction in the en-
ergy regime of 100 MeV and above is the measurement of GT strengths in nuclei for
excitation—energy regions that are inaccessible to fl—decay. A close correspondence
between zero-degree (p,n) cross section and fl—decay transition strength has been

found [GOG80] and has been thoroughly investigated. A short summary of these re-

sults will be given in the following. For comprehensive reviews, the reader is directed

to Refs. [TaG87, BeE87].

Proportionality between fl—decay transition strengths and (p,n) cross sections is
supported by standard reaction theory as well as experimental observation. The
correspondence is due to the similarity of the operators involved in each type of
reaction. The central isovector terms in the effective nucleon—nucleon interaction

that mediate low momentum—transfer spin—flip (AS = 1) transitions,

2: VaT(r;p)Ui '0‘], 1*, - Tp, (2.3)

and non—spin—flip (AS = 0) transitions,

2: V7(rip)7i ' Tpa (2.4)

are similar to the corresponding operators
GA 2 0113i:t and Gv 2 ti” (2.5)
i i

for Gamow—Teller and Fermi fl—decay, respectively [TaG87]. The sum is over target

nucleons, and the index p indicates the projectile.

The model that is most commonly used to describe the (p,n) reaction is the
distorted—wave impulse approximation (DWIA). It factorizes the process into a nuclear—
structure and a nuclear—reactions part and employs the free nucleon—nucleon inter-
action for the charge—exchange mechanism itself. At zero momentum transfer, the

DWIA cross section for a mixed Fermi or GT transition is of the form [TaG87]:

d_0’ 0 _ ll 2kf D 2 D 2 9
(mm )— (5.7,) k. (NMIJMI B<GT>+N. IJTI mm). 1-6)

where p is the reduced mass and k; and k; are the initial and final wave vectors of the

incident nucleon. J is the volume integral of the interaction strength and has units

of MeV fm3. The distortion factor ND is a number that accounts for the distortion of
the incoming plane waves by the nuclear potential. It is given by the ratio of cross

sections calculated for plane waves and waves distorted by an optical potential:

da(DW)/dQ

ND =
d0(PW)/dQ 9:00

 

(2.7)

and can be calculated reliably in DWIA, i.e. with little dependence on the details of

the model.

An simplified, empirical form of Eq. 1.7 is most useful for comparisons of experi-

mental results:

0 = &0(Epv A)FO(CIa u”13(0), (28)

where a = F or GT and 0 denotes the forward cross section at 0°. The proportionality
constant 6 is called the “unit cross section” and can be dependent on the target mass
and the bombarding energy. The effect of a non—zero Q value is taken into account
through the factor F(q, (.0), which describes the shape of the cross section distribution
as a function of momentum transfer q and energy loss (.0 1 and goes to unity as q and
w go to zero. This factor can be calculated in reaction theory, and for small values of
q and (.0, simple parametrizations have been shown to describe the experimental data

well without reliance on particular models [TaG87]

The factor 6‘ has been the subject of extensive empirical studies [GOG80, TaG87,
RaW87]. The relationship between (p,n) cross section and fl—decay for different
transitions starting from the same parent state is commonly referred to as “specific
proportionality”, while the term “general proportionality” describes comparisons of
cross sections for transitions originating from different target nuclei. Specific pro-
portionality is fulfilled for the majority of nuclei, and once (‘7 has been determined

from transitions with known ,B—decay strength, (p,n) cross sections can be used with

 

10.: describes the effects of excitation energy: w : Ex — Q33, where Q85 is the Q value for the
reaction to the ground state.

10

confidence in most cases to determine the unknown B(GT) values. However, there is
no smooth target dependence, and proportionality constants inferred or extrapolated

from different target nuclei have to be regarded as uncertain at the 20% to 50% level

[TaG87].

Another problem that up to now has not been solved is the phenomenon known
as quenching of GT strength. This refers to the fact that the summed GT strength,
as observed in the (p,n) reaction and normalized to known fi—decay measurements,
falls approximately 40% short of the predicted total GT strength. This behaviour

has been observed for a Wide range of nuclei and appears to be universal.

The total Fermi strength is concentrated in the transition to the isobaric analog
state and is easily calculated as B(F) = N - Z. In contrast to this, the Gamow—Teller
strength is fragmented over a number of nuclear states, and the corresponding sum

rule is less rigorous:

B(GT)B_ — B(GT).+ = 3(N — Z). (2.9)

This only fixes the difference in total strength for fl" and 3+ decay, but since the 5+
strength function is quite weak, the sum rule can be viewed as a close lower bound on
the fi" strength and is called the sum—rule bound. In 11 decay, for example, B(GT) = 3
is indeed observed. The quenching of Gamow—Teller strength is illustrated in Fig.
2.3, which shows a comparison between GT amplitudes obtained from shell—model

calculations and those measured from fl-decay.

While ,B—decay is limited to a small energy range and permits comparison to
theory only for selected transitions, the (p,n) reaction covers tens of MeV of excitation
energy and allows comparison of individual transitions as well as overall integrated
GT strength. Fig. 2.4 illustrates the experimental situation: The fraction of the

sum—rule bound observed in (p,n) reactions at intermediate energies is plotted for a

11

 

1.0 s . v . - . - a -
O .
e
0.8 ' .
.8 C
v t
E e
1: 0.6 . ' -
a:
an 1
m
’E: 0.4 ' *
e
a, 1
0.2 » ' l
l
0.0 *

 

 

 

0.0 0.2 0.4 0.6 0.8 A 1.0
B(GT) WBT ("Free-nucleon")

Figure 2.3: [ChB93] Comparison of GT matrix elements for sd—shell nuclei with the
predictions from a shell—model Hamiltonian [?]. The matrix elements are normalized
so that 1 corresponds to the maximum permitted by the shell model space. For
perfect agreement, the points should lie on the diagonal line. They actually cluster
around a line with a slope of 0.77, showing the quenching of GT strength. Note that
this corresponds to a B(GT) quenching factor of 0.772 = 0.59.

12

 

 

 

 

 

 

 

 

 

 

ll FRACTION of GT-SUMRULE OBSERVED in (p,n)
IOO'I
80h?
o]. 50" ”ix % :I
x 0‘3
40~ 15 I {
20’ :9 39
0 9.6.4.234 9? ".2. '4: ":5 20??“ 21:8 _

 

 

A

Figure 2.3: [Gaa85] Percent of sum rule bound observed by integrated (p,n) cross
section for a variety of targets.

13

variety of targets [Gaa85]. The difference between the cross—hatched area and the
full data points is due to the difficulty of resolving individual peaks for heavy targets.
The transition from 100% of the sum rule bound observed in the 1H(n,p)n case to
60% quenching for heavier nuclei is remarkably smooth and invites speculation that

the quenching is an in—medium effect that reaches saturation around A = 20.

Subnuclear degrees of freedom, such as pions and the delta isobar excitation, may
act to quench the strength in the spectroscopic region. Another possible explanation
is configuration mixing, which may act to shift the strength to much higher energies,

where experimental observation would be very difficult. be very difficult.

2.2 Fermi and GT Transitions

Transitions with AS = 0 are analogous to Fermi fi—decay, and the transition to the
isobaric analog state (IAS) is proportional to the Fermi transition strength B(F) =
N —- Z, where N and Z are the neutron and proton number of the target residue. Since
B(F) is easily calculated and concentrated on one single state, the value of &p can
usually be determined with case. If a relationship between &p and 661‘ can be found,
the IAS (p,n) cross section can be used to calibrate the unit cross sections. This
works especially well for 0+ to 0+ transitions, which are purely Fermi in nature. Fig.
2.5 shows the 0° l4C(p,n)”N cross sections at incident proton energies between 80
and 650 MeV. The spectra have been normalized to the strength of the 1+ GT peak,
and the dashed line indicates the relative strength of the 0° IAS peak. The ratio of

Gamow—Teller to Fermi unit cross section can be parametrized as

&GT(Ep, A) __ 2 ‘
&F(Ep7 A) — [R(Ep, Ail , (2-10)

14

14’C(p,n)MN 9 == 0°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. T I l 0 I l l T r 1 i 1
{80 MeV 120 MeV 160 MeV 200 MeV 500 Mall 650 MeV
:- 1+ 4
IAS j
. .. \ L, ‘ __ A :
.0 .HL . . -—-»—A.— - a
5

0 5 0 5 0 5 0 5 0
Excitation Energy (MeV)

Figure 2.5: [R3895] Zero—degree cross—section spectra for the l4C(p,n)”N reaction at
the indicated bombarding energies. The spectra have been normalized arbitrarily.

15

and may be interpreted as the ratio

2 1‘10le N631” .
IJUI2 N? ( )
as introduced in Eqs. 2.6 and 2.7. Between energies of approximately 20 and 200 MeV,

the following parametrization yields good results:

Ep

R(Ep) = I)?

(2.12)

with E0 = 55.0i0.4 MeV, with little or no A—dependence observed. This relationship
is illustrated in Fig. 2.6, where the value of R2 is plotted versus the incident proton
energy for a variety of even—A and odd—A targets. Aside from the lowest—energy
data, the agreement between the data and the empirical relationship is excellent. In
this energy region, the change in relative strength is due to a decrease of 61: with

increasing energy, while 6GT stays approximately constant [BeE87].

2.3 The 13C Controversy

Knowing the relative strengths of Fermi and GT cross sections is especially important
for transitions that can have either AS = 0 or AS = 1, such as the transition to the
ground state in the 13C(p,n)13N reaction. The J" of the initial and final state is %—,
and the reaction can take place either via the Fermi or the Gamow—Teller interaction.
The corresponding cross section then contains both contributions, which cannot be
disentangled without relying on further information. If Eq. 2.7 holds, knowledge of
B(F) and B(GT) may be used to extract the values of 6. However, there are some
cases, most notably the odd—A nuclei 13C and 15N, where the assumptions of specific
proportionality have been called into question. Fig. 2.7 shows the excitation spectrum

of the 13C(p,n)13N reaction at Ep = 120 MeV. For the ground state and the excited

state at 15.1 MeV, which is the isobaric analog of the 13B ground state, GT strengths

16

 

 

 

 

4 "IIII‘VIIIII1IIIIV‘UTIIWTT
.
. i
p ct
p .4
—4
3- even—A ‘
+- I
b .1
P —
2_ ..
i- d
I d
D u
I d
1— —
. ‘ .
I- ‘.’ 4
b d
an 'r' d
Is: 0'ititlttttl%%ttltttt'%ttt
a: - ' ' ' 4
n A
d
C .
3 - odd-A *
l' '1
r III
C .
2— ‘-
I- .4,
I- d
b a:
h .4
’ .4
b .'
n...”’ 1
D ” q
p
h.‘ -+
0 ’1llllLlLllllllllllJlllll

 

 

50 100 150 200 250
E, (MeV)

0

Figure 2.6: [TaG87] The quantity R(Ep) determined from many target nuclei. The
data points have been segregated into even—A and odd—A cases.

17

 

40 Trlr'WfTTrITTTIrrFII

 

 

 

 

 

 

 

%

E 30 - ‘3C(p,n)’°N 1

L.

m - ROMV ~

B e

g 20 _ A

Q

0

9

s: 10 - a

6) r- a
O I. 1 1 l l 1+1J_1 l 1 +1 4 l 1 1 I L]

15 10 5 0

excitation energy Ex (MeV)

Figure 2.7: [TaG87] Cross section spectrum for 13C(p,n) at 0° and 120 MeV. The
vertical bars represent the GT transition strengths for analogous beta decays. The
Fermi strength is indicated by the dashed vertical line.

3.51 MeV

g.s.

1

J1‘ = 312'

lpl/2
lp3/2

181/2

 

 

 

 

 

protons neutrons

1p 1/2
1p3/2

Isl/2

 

 

 

 

 

13M

8

—_ .

J“ = 312'

A

{,0_ 19112

 

me

000— 1p3/2
lsUZ

3.68 MeV

 

 

 

 

lpUZ

 

 

1p3/2
lsll2

g.s.

 

 

 

1

3C

Figure 2.8: Schematic illustration of the nuclear configurations in 13C and 13N for the
states under discussion. The ground state configurations are shown at the bottom.
The states near 3.5 MeV are strongly excited in both l3C(p,n) and 13N(n,p) charge

exchange (see Fig. 2.7).

19

are available from ,B-decay measurements. The Fermi and Gamow—Teller strengths
for these states are indicated by the dashed and solid vertical lines, respectively. Fig.

2.8 illustrates the configurations involved in the transitions discussed here.

The argument centers on a comparison between the zero—degree cross sections
measured for the lowest excitations in the reaction and B(GT) values obtained from

shell—model calculations. The discussion is summarized as follows:

Using the measured zero—degree 13C(p,n)13N cross section at 160 MeV for the
transition to the 13N ground state, and normalizing the Fermi and GT contributions
according to Eq. 2.10, Goodman et al. [GoB85] found a 6GT value of 12.7 mb/sr,
which is large compared to e.g. the 6GT = 8.1mb/sr found for the 14C(p,n)”N
reaction at the same proton energy. From the measured cross section and their 6GT
value, the authors obtained B(GT) = 0.83 :l: 0.03 for the %_ to 3. transition to the
state at 3.68 MeV excitation energy in 13N. The shell—model value for this state is
B(GT)=2.38 [LeK80], which lead them to suggest that the B(GT) strength for this
transition is quenched by a factor of 0.36, which is a larger effect by far than the

usual quenching factor of approximately 0.60.

Further corroboration for the high 6GT value was originally found in the fact
that the cross section for the transition to the g: T = g state at 15.1MeV, where
the B(GT) is known from fl—decay measurements, corresponds to 6GT = 13.8 mb/sr.
Later on, that measurement was shown to be in error due to an unexpectedly high
amount of 12C in the 13C target, and subsequent zero—degree cross—section mea-
Surements of the 13N(n,p)l3C and l3C(p,n)13N reaction at 200 MeV yielded values of
6821’?) = 7.24 :l: 0.33 mb/sr and 683'?) = 9.81 :l: 0.8 mb/sr [MiA91]. It should be noted,
however, that the erroneous value had no direct bearing on the 5 to %- B(GT)

measurement of Ref. [GoB85].

20

Watson et al. [WaP85] contended that the 6GT value used in Ref. [GOB85] was
inflated by an unusually high GT cross section in the 13C(p,n)13N groundwstate peak.
They argued that this is due to renormalization of the Gamow-Teller operator in the
nuclear medium, which has a tensor correction that is known to assume a maximum
value for %— to %— transitions and causes an increase in the observed (p,n) cross sec-
tions [Wat94]. Their recalibration yielded B(GT) = 1.38 for the {I to %_ transition,
which represents a. much more ordinary quenching of 0.58. The corresponding unit

cross section is 6GT = 8.4 mb/sr, quite comparable to other measurements at this

energy.

However, there are arguments to support the results of Goodman et a1. as well:
The assignment of the Fermi and GT portions of the ground state cross section was
repeated at a later date using polarization measurements [Rap95] and confirmed the

earlier result.

In the present work, the value B(GT) = 0.83 i 0.03 is used, since it represents the
conventional approach and has not conclusively been shown to be in error. It is also

the most recently available value in the literature [RaW87].

2.4 Heavy Ions as CT Probes

In recent years, heavy—ion charge exchange (HICEX) reactions have begun to at-
tract interest as probes for GT strength, and experiments have been carried out to

investigate the feasibility of the method [WiA86, WiA87, IcI94, Au594].

The experimental methods available for measurements of GT strength in radioac-
tive nuclei are limited. One obvious choice is to employ the (p,n) reaction in inverse
kinematics, using radioactive beams on a hydrogen target. If good energy resolution

is to be obtained, however, these experiments will be difficult, since they involve the

21

H

detection of rather low energy neutrons, and experiments in the [3+ direction are not

possible, as there are no neutron targets.

HICEX offers several advantages over nucleon—induced charge—exchange reactions.
Since it only involves.charged particles, the energy resolution can in principle be very
good. HICEX further offers favorable kinematics, the capability to sample both the
6+ and the 6’ direction, and the additional advantage of quantum selectivity for
certain projectile—ejectile choices. In the (“’Li,6 He) reaction, for example, final states

can be chosen which limit the transitions to AS = 1, AT = 1 [WiA87].

The problems concerning the suitability of the HICEX reaction for spin—isospin
spectroscopy are twofold: Firstly, the importance of contributions from sequential
stripping and pickup of nucleons, such as [(6Li,7Li)(7Li,°He)], has to be investigated.
Such studies — both experimental and theoretical — have been carried out for the
(6Li,°He) [WiA87] and the (12C,12N) [WiA86, AnW9l, LeW89] reaction at varying
energies. The results of these studies indicate that for the (6Li,°He) reaction, the
one—step process dominates above approximately E/ A = 25 MeV, while two-step
processes are important in the (12C,12N) reaction up to E/A = 50 MeV. Extrap-
olations from the detailed study by Lenske et al. [LeW89] lead to an estimated
30% two—step contamination in the forward cross section of the the (12C,12N) re-
action near E/ A = 60 MeV. The proportions might be expected to be similar for
the l3C(13N,13C)13N reaction, since the GT strengths involved in both systems are

comparable in magnitude.

The second, more involved question regards the effects of the finite size and struc—
ture of the projectile and has several facets. One issue is that the internal structure
of the projectile allows additional amplitudes, not proportional to GT strength, to

contribute to the cross sections. In contrast to (p,n) charge exchange, orbital angular

22

momentum as well as spin angular momentum can be transferred in the reaction,
and different values of projectile and target orbital angular momentum can couple
together to give the same angular—momentum transfer. For example, in the 0+ to 1+
transition of the (12C,12C) reaction, the L = 0 cross sections contains “cross terms”,
where Lp = 0 couples with L, = 2 and vice versa [AnW91]. However, computations

performed to investigate these amplitudes have found the contributions to be small

[AnW91].

In this context, the (3He,t) reaction deserves special mention, as all nucleons in
the A=3 system are in the IS state (L = 0), and no orbital angular momentum can
be transferred to the projectile. This reaction is a valuable tool in its own right,
combining the advantage of a charged ejectile with the relative simplicity of the (p,n)

reaction, but also serves to bridge the gap between nucleon—induced and heavy—ion

charge exchange [JéiB94, AkD94].

A further complication, also connected with the size of the participants, concerns
the strong absorption in heavy—ion reactions, which ensures that only the surface
regions of the projectile and target transition densities are sampled in the reaction.
This limitation to a narrow spatial region implies — by the uncertainty principle
— that the reaction is sensitive to a large range of the momentum transfer q in
the transition density. This should make the measured cross sections an unreliable
measure of GT strength, which is equal to the corresponding transition density at q z
0. The issue has been studied in a theoretical experiment for the 26Mg(12C,12N)2°Al
reaction [AnW91], where the same wave functions were used to determine the B(GT)
and cross section values, and a close proportionality between GT strength and cross
section was found. This surprising behaviour was investigated by Osterfeld et
al. [OsA92] using an eikonal model calculation. It was found that while the cross

section did indeed involve an integral over all momentum transfers in the transition

23

 

(arbitrary units)

t

9..

 

 

 

 

 

 

 

500
0.0 0.5 1.0 1.5 2.0 2.5
q’ Um")

Figure 2.9: [AUS94] Comparison of the sensitivity function S(q) (solid line) and
the transition densities for transitions to the lowest six 1+ states of 26Al in the

26Mg(12C,12N)2°Al reaction at E/A = 70 MeV (dashed lines).

24

 

    
 
 

 

 

 

 

lolirlrlilll llilllllt‘:
— —— o<r<4.5 rm um I
" '''' 0<r<4.6 {In L-Z .
- - ' - - 4.6<r<1? (to L-O ..
100 E7. 12 12 t 12 ”—5
C ' C( C, 2N) B :
A 3 ' E/A = 135 MeV:
L. _ ‘. 1 _
< P I . I. 23 (1") 8'8'_
p ‘ I l e
C: : . ' ' 2
:3 I 'l i . ‘ 3
b h i:.' ' . ‘E‘ '1
'O _‘e..e.'l ‘ ..I. ‘\ \ '-
ii 1 V l
— — " ' f °
10 2 : 1' i . ‘. '0. .
Z I- - I -. . :,
I- -| I - .
c |. -]
A -| \.
I O
.. : .. a;
10-3 thlllllllllll‘lllf’
0 2 4 6 8

0cm. (deg)

Figure 2.10: [IcI94] Cross section measurement near zero degrees for the
12C(nC,”N)12B reaction at E/A = 135 MeV. The lines show results of a DWBA

calculation, described in the text.

25

density, these were weighed by a sensitivity function which limited contributions to a
small range of q. This is illustrated in Fig. 2.9, which shows six different transitions
densities, all normalized to the same peak value, together with the sensitivity function,
peaked near q = 0.35 fm’l. The observed specific proportionality is due to the fact
that the shapes of the transition densities is nearly identical in the region selected by
the sensitivity function. This illustrates an important advantage of mirror symmetry:
Due to the structural similarity of the nuclei involved, only one set of transition

densities enters the equations.

The strong absorption in heavy-ion reactions also influences the cross section
directly by shielding large parts of the nuclear volumes from each other. This effect
as well as the influence of the Q value on the reaction cross section are illustrated
in the following example: Anantaraman et al. [AnW91] studied the l2C(12C,”N)12B
reaction at E/ A = 70 MeV as part of an effort to measure GT strengths in 56Mn. A
forward cross section of 1.4 mb/sr was measured for the 0+ to 1+ transition to the
ground state in 12B. By a careful analysis using the distorted—wave approximation, the
L = 0 portion of the cross section was determined as 6(L = 0, 0 = 0) = 0.70 mb/sr,
with approximately a 10% error in overall normalization. Because of the large Q value
of the reaction (Q33 2 —30.71MeV), the extrapolation to zero momentum transfer
requires a large correction factor: F(q,w) = 6.81, which makes the L = 0 portion

cross section at zero degrees and zero momentum transfer
0'(L = 0,0 = 0, Q = 0) = 4.78 mb/sr. (2.13)

When the same reaction was studied at E/ A = 135 MeV by Ichihara et al. [Ic194] and
a similar analysis performed, the measured forward cross section was 0’ = 5.9 mb/sr

(i20%), with an L = 0 contribution of 6(L = 0, 6 = 0) = 4.5mb/sr. The extrapola-

26

tion to Q20, yielded
0(L = 0, 0 = 0, Q = 0) = 14.8 mb/sr, (2.14)

much higher than the 70 MeV result. In an effort to understand this discrepancy, a
distorted—wave Born approximation (DWBA) calculation was carried out, in which
the nuclear form factor was divided into an inner part (0 < r < 4.5fm) and an outer
part (4.5 < r < 17fm), and the contributions of the inner and outer regions were
studied separately. The result of this study is shown in Fig. 2.10. It is apparent
that the contribution of the inner region of the form factor is nearly as large as
that of the outer region near zero degrees. This behaviour is due to the increased
nuclear transparency at higher energies. While both contributions are approximately
2.0 mb/sr, their coherent sum gives an L = 0 cross section of 4.8 mb/sr, indicating
constructive interference. The 2 0 contribution from the outer part of the form
factor only is 6.2 mb/sr, which is compatible with the 70 MeV measurement, where

contributions from the inner region of the form factor are expected to be negligible.

While a consistent, microscopic understanding of the HICEX mechanism is still
lacking, the studies cited above indicate that the reaction has the potential to become
an important tool for GT strength measurements. One obvious disadvantage is the
requirement to calibrate each reaction individually, as the overall proportionality
factor appears to be even more variable than for the (p,n) reaction. However, HICEX
is at present the most promising method for measuring the GT strength function in
radioactive nuclei for all but the lightest isotopes. Especially in the astrophysically
important region near the Fe peak, charge—exchange experiments with radioactive

beams and nuclear targets will likely be the method of choice.

 

 

k5

27

2.5 Other Charge—Exchange Probes

There are other experimental techniques, such as pion scattering or experiments with
electromagnetic probes, that are relevant to the measurement of weak—interaction
strength, but which are not discussed in the present work. For more information on

these, the reader is referred to the review by Rapaport and Sugarbaker [Ra595].

Chapter 3

The Experiments

The mirror charge—exchange experiments were performed with the A1200 fragment
separator operated in a dispersion—matched mode using an 14N primary beam. The
secondary 13‘N beam was produced in a beryllium target located at the A1200 target
position and was focused onto a 13C reaction target, manufactured from 13C-—labelled
polyethylene powder (see Appendix A). Detailed experimental parameters are given
in table 3.1. The final l3C reaction products were registered in a detector setup
at the focal plane, which consisted of two position monitors, a position—sensitive
silicon detector (PSD), and a thick plastic scintillator, providing measurements of
two-dimensional position and angle, energy loss, total energy, and time—of—flight with
respect to the cyclotron RF. Fig. 3.4 shows the setup schematically. The main

challenges in performing the charge—exchange experiments were:

Achieving good energy resolution despite the energy spread of the secondary

beam

e Minimizing background while obtaining a sufficiently high secondary—beam rate

Obtaining reliable cross—section measurements

Measuring the angular distribution by ray tracing

28

29

Table 3.1: Experimental parameters for the Charge—Exchange experiments

 

 

 

NSCL Experiment Number 92065 93043
Primary Beam 14N, E/A=70 MeV 14N, E/A=120 MeV
Intensity (approx.) 50 pnA; 3X10“s‘l 15 pnA; 1x10113-l
Production Target 9Be, 570 mg/cm2 9B6, 1100 mg/cm2
13N Energy E/A=57 MeV E/A=105 MeV
Intensity (approx.) 5x1063‘1 6x1053-1
Reaction Target 13CH2, 9.0 mg/cm2 13CH2, 18.0 mg/cm2
Energy Spread (fwhm) 15 MeV 27 MeV
Second Stage Dispersion 3.6 C???- 4.1 (:77?

13N Resolution (fwhm) 1.0 MeV, E/AE=750 1.5 MeV, E/AE=900
13C Resolution 2.0 MeV 2.4 MeV
l3N‘S‘i/IQ’N7+ Ratio (1.0:l:0.1) x 10‘5 (1.5:l:0.2) x 10“6

 

 

The following sections address these issues and may be used as a reference for fu-
ture radioactive—beam experiments. The two experiments were sufficiently similar in
design and execution that figures from both experiments are shown and discussed

interchangeably.

3.1 Secondary—Beam Production

The optimum production target is thick enough to produce the desired secondary
beam in sufficient intensity, but at the same time thin enough not to cause broadening
of the ions’ momentum distribution too far beyond the acceptance of the spectrometer.
While moderate gains in usable secondary—beam intensity are possible with very thick
targets, the introduction of a wider variety of isotopes into the acceptance window
adds difficulties that must be taken into account. Target thicknesses for the mirror

charge—exchange experiments are listed in Table 3.1.

With the production targets in place, the first step in the experiments was to

determine the magnetic rigidity that would produce the maximum number of 13N nu-

30

clei. This was achieved by varying the magnetic rigidity with the entire spectrometer

set to the same Bp and observing the 13N rate.

The top of Fig. 3.1 shows an energy—loss/time—of—flight spectrum recorded at the
optimum Bp setting, with several isotopes identified. This is a very typical parti-
cle identification spectrum, with the N=Z nuclei all arriving at the focal plane with
nearly identical time of flight. Note the absence of unstable 8Be, which provides a
convenient clue to particle identification. While the presence of nuclei with mass
number smaller than 14 is attributed to nuclear fragmentation, 14N itself should not
be detected, as the magnetic rigidity of the primary beam after the target does not
fall within the acceptance of the spectrometer. Due to the high number of incident
particles, however, even a small fraction of the primary beam scattering off an aper-
ture and producing secondary reaction products could produce significant amounts of
contamination. Scattered primary beam is indeed evident in nearly all fragmentation
experiments over a wide range of magnetic rigidities that are smaller than the Bp of
the beam. In order to minimize the amount of 14N reaching the reaction target, the
primary beam was put through the production target at an angle of about 1° hori-
zontally. This was achieved by steering the beam towards the positive x coordinate
with a horizontal dipole magnet, located approximately 2.7m before the production
target, and then steering it back on the center of the target with the quadrupole ele-
ments of the beam line, which act as bending magnets for beams that are off—center.
The E/ A = 57 MeV experiment was performed using the “Low—Acceptance” target
position, and the primary—beam particles not reacting in the target were dumped on
water—cooled slits located approximately 1800 mm downstream of the target position.
In the 105 MeV experiment, however, the production target was moved upstream to
the so—called “Mid—Acceptance” position in order to take advantage of a newly in-

stalled quadrupole doublet, located immediately upstream of the Mid—Acceptance

31

 

AE

 

 

 

.‘ , 5:7,, a, 13Ns.
.7’ i A. .®(‘3N.13C)

  

 

 

 

Tlme of Fllght

Figure 3.1: AE-TOF particle—identification spectra obtained with the entire A1200
set to the magnetic rigidity of the 13N fragments (top) and with the magnetic rigid-
ity of the second dipole stage scaled by a factor of 7/6 to collect the 13C reaction
products (bottom). The bottom spectrum was recorded with a primary—beam inten-
sityh five orders of magnitude higher than for the top spectrum. The large amount
of background is clearly visible. The desired 13C particles are marked by the oval.

32

position. A horizontal beam spot size of approximately 0.5 mm across was achieved,
compared with about 1 mm in the earlier experiment. Instead of using the slit drives,
which cannot be adjusted to less than 3 mm full Opening, water—cooled aperture plates
were used, located only 95 mm downstream of the Mid position. Two different round
apertures were used during the experiment, I / 32” andI / 16” in opening diameter. The
short distance from beam spot to aperture made the secondary—beam properties very

sensitive to slight changes in the position of the primary beam.

The importance of angling the primary beam to prevent it from entering the
spectrometer should be emphasized; an early test run begun without this technique

[StBQl] suffered from background so intense that it rendered the 13C group invisible.

With the magnetic rigidity of the second dipole stage increased by a factor of 7/6
to accept the (13N,13C) reaction products, the spectrum at the bottom of Fig 3.1 was
obtained. The primary—beam intensity was approximately five orders of magnitudes
higher than for the previous spectrum. 14N and 13N appear in the same position as
before, but in their 6+ charge state. The 13C reaction products, which have very
nearly the same velocity as the 13N secondary beam, appear at the same time—of—
flight, but at lower energy loss, due to their lower nuclear charge. They are visible
inside the oval. Other particle groups come in two different varieties: Those that
have the same general shape and size as the original isotopic clusters and are due to
nuclear reactions of contaminants, and those that are smeared out diagonally over
relatively large areas. The latter are attributed to particles scattering off beam line
walls or apertures and their secondary reaction products. Note that one such group
is intruding into the 13C cluster from above. In chapter 4, the amount of background
particles in the (13N,13C) spectra will be shown to be correlated with the amount of

14N present.

33

3.2 Cross—Section Measurements

Two quantities have to be known in order to determine the reaction cross sections: the
number of 13C atoms per unit area in the reaction target and the number of incident
13N ions. The number of target atoms was obtained from the measured weight and
area of the target, together with the chemical composition of the target material.
Fig. 3.2 shows the energy—calibrated position spectra collected simultaneously for
the 13C(13N,13C)13N reaction products and the 13N6+ secondary beam at E/A =
105 MeV. The incident 13N intensity was determined by monitoring the intensity of
the l3N6+ peak and dividing it by the 6+ / 7+ charge state ratio, 6. In the 57 MeV
experiments, 6 was determined by adjusting the magnetic rigidity of the second dipole
stage to select 13N7+ and 13N64”, respectively, counting the number of particles arriving
at the focal plane, and monitoring the beam current using four “beam monitors”.
These are small silicon detectors mounted around the production target that measure

scattered beam particles and provide an excellent relative measure of beam current.

Since these detectors were unavailable in the 105 MeV experiment, the charge»
state ratio was determined by a direct measurement. Two parallel—plate avalanche
counters (PPACs) located at the dispersive image plane were used to monitor the
incoming 13N7+ beam, and the 13N6+ leaving the reaction target were collected at
the focal plane. Since the count rate observed in the PPACS ceases to be proportional
to the number of incoming particles at rates above approximately 5000 particles
per second, the PPAC response was calibrated by sending the l4N7+ primary beam
through the A1200 at varying intensities and counting ions in the PPACS as well as

in the PSD — which is assumed to have 100% efficiency.

With either method, the accuracy obtained for e is estimated to be no worse than

10%. The results are listed in Table 3.1.

34

 

.4

I l I I I I I I I T I I I I I
18C(13N, 13C)13N
1 05 MeV/ nucleon

80

60

COUNTS/CHANNEL
3

20

II I] f! II [I II’Ilgt‘ill [I
II ll '1 ll 1' II II [I ll.Ll l

 

+
-i-

db-
di- _
db.

 

d.
l l

 

 

5000
4000
1 3 N 6+
3000

2000

llelllllllllllllll

1000

COUNTS/CHANNEL

 

 

UTTIIYTTTIITIIIIIUIIUIUU I .

() I L_ I l I l I l l l l l I 1.Al

o 10 20
Excrrmou ENERGY [MeV]

 

Figure 3.2: Energy—calibrated position spectra for the (”N ,13C ) reaction products
(top) and the 13N6+ charge state (bottom) at E/ A = 105 MeV. The l3N6+ peak
provides a convenient method of monitoring the secondary-beam intensity and also
supplies the absolute calibration of the energy scale.

35

3.3 Ion Optics

The present section is a quick review of the basics of charged—particle Optics. For a

comprehensive introduction, see e.g. [W0187].

It is convential in ion optics to define the position of a charged particle as a
vector in a six—dimensional phase space, with the coordinates given with respect to
a reference particle, thought to move along the optical axis of the device with the
desired momentum. The coordinates used are x, the horizontal distance of the particle
from the optical axis; 0, the angle between the particle trajectory, projected onto the
horizontal plane, and the optical axis; y and 45, the vertical position and angle; 1,
defining the distance between the ion and the reference particle along the optical
axis; and 6, the deviation of the ion’s momentum from that of the reference particle.
Units used conventionally for the six coordinates are cm and mrad for the position
and angle coordinates, and % in Ap/p for momentum. The coordinate z, with units
of m, describes the reference position along the optical axis. The l coordinate is not

of interest in the present case, but is carried along for reasons of convention.

The optics of a beam transport system are expressed as a matrix equation in
the form of a Taylor expansion. If the final (image) coordinate Xf, for example, is

a function of the initial (object) coordinates, Xf = f(xi,01,yi,q§i,6), the Taylor series

reads:
0f 6f182f 1 62f
=— i —6i —— X2 —— i i .I
X‘ 6xX+80 + +2162xx'+2!6x60X0+ (3)
It is conventional to write e.g. the partial derivative $2,; as (xldqfi). The first—order

terms usually dominate the overall optical properties, and the first-order equation is
given by an equation of the form x} 2 Air}. A is called the transfer matrix. The full

equation reads:

36

 

 

 

 

 

 

(x\ ((xlx) (x)6) (xly) (>46) (xll) (x)6)\ (x;
6 (6(x) (6)6) (61y) (6)6) (6(1) (6(6) 6
y = (yIX) (H9) (yly) (y|¢) (Ylll (yl5) y
6 (6(x) (6)6) (6Iy) (6)6) (6)1) (6(6) 6
1 (le) (1(6) (lly) (1(6) (1(1) (1(6) 1

K6). \(6lx) (6(6) (6(y) (6)6) (6)1) (6(6)) l6),

In most instances, dipole and quadrupole elements are adjusted to provide the
desired first—order imaging characteristics, while sextupoles and other higher—order
lenses are used to minimize second—order effects. Some aberrations are excluded ab
initio by taking advantage of certain symmetries in the design of the beam transport
system. For example, symmetry around the horizontal plane defined by y = 0 (“mid-
plane symmetry”) requires that two rays with the same overall coordinates and the
same magnitude but opposite signs in (253 be transported to the same final coordinate

Xf. Generally, midplane symmetry results in the condition (xlqufin) = 0 if m + 11 odd.

Also, many of the first—order terms are automatically equal to zero because of
symmetry considerations; vertical and horizontal coordinates, for instance, are de-
coupled from each other, and in any system where the bend plane is horizontal, only
the x coordinate can have first-order momentum dependence. The first-order matrix

to be considered here has the following form:

((XIX) (Xl9) 0 0 0 (Xl5)\
(0|x) (ale) 0 0 0 0

0 0 (YIY) (W25) 0 0

0 0 (65M (¢|¢) 0 0

0 0 0 0 1 0
\ 0 0 0 0 0 1 )

 

 

Conservation of phase space density imposes the additional condition that the

determinants of the horizontal and vertical submatrices be 1.

37

Dis ersion-Matched Ener -Loss S ctrometer

 

am 39 lm
W Second Dipole Stage

First Dipole Stage
zip/F196

 

 

 

 

 

AMP-1%
Reaction Target

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

Mid Ac/rmm/Ii‘ lmaya
_.L_.

| Production Target Focal Plane

 

 

 

Figure 3.3: Principle of the energy—loss mode. The solid lines illustrate the principle
of dispersion matching: Particles leaving the object (production target) with different
magnetic rigidity are focused to a point at the final achromatic image (focal plane). If
energy is lost in the reaction target, for instance due to excitation of a target nucleus,
the particle will be displaced at the focal plane, as indicated by the dashed line. This

displacement is independent of the original momentum.

38

The A1200 Radioactive-Beam Facilig

  

D' . l l!l
E I I' I l D' . I E2
2PPACs
\ Reaction Target
In).

“N Beam from

2 PPA Cs/CRDCs (x, y)
PSD (AE, x)
Plastic (TOF, Eta )

 

 

 

First Dipole Stage Second Dipole Stage ————4

 

 

Figure 3.4: The A1200 Fragment Separator.

39

3.3.1 A1200 in Dispersion—Matched Mode

Radioactive beams produced by projectile fragmentation generally have large emit-
tances. It is still possible, however, to perform high—resolution experiments with
beams of poorly defined momentum by using a fragment separator in dispersion—

matched mode, also known as energy—loss mode [ShM91].

The principle is illustrated in Fig. 3.3: The secondary beam particles leave the
production target with a wide range of momenta and are focused by an initial dipole
stage onto an image plane in the mid—section of the spectrometer. The beam spot
at this dispersive image is large in the horizontal (bend) coordinate, and position
is correlated with momentum, as indicated by the lines in the figure [(xl6) 7A 0].
The second dipole stage is tuned to cancel the dispersive effect of the first, which
leads to the condition (xl6) = 0. Furthermore, the final angle 0f must not depend on
momentum, which would degrade the angular resolution: (0|6) = 0. If both conditions
are met simultaneously, the system is said to be achromatic. In this situation, the
only effect of the spectrometer — to first order - is to impose a cut in magnetic rigidity
(Bp) on the beam, eliminating the primary beam and most of the unwanted reaction
products. If a reaction target is placed at the dispersive image, and the magnetic
rigidity of the second stage of the spectrometer is set to accomodate the reaction
products, these will again be focused to a point at the final focal plane, provided
the overall tune of the system is preserved. Energy lost in the reaction due to a
nuclear excitation, however, will be missing from the ejectile’s kinetic energy, causing
it to be displaced at the focal plane, as indicated by the dashed line in Fig. 3.3.
Excitation energy is determined from the focal—plane position spectrum after it has

been calibrated (see Sec. 3.3.3).

More rigorously, the position Xf at the focal plane as a function of the coordinates

40

at the reaction target is (to first order)
Xi = (XIX) Xr + (Xl5) (58 + 5R), (32)

where focusing [(xl0) = 0] is assumed and (SB and 63 are the momentum coordinate
of the beam and that induced by the reaction, respectively. The matrix elements

describe the properties of the second dipole stage. Dispersion matching implies
(XIX) Xr + (Xl5) 53 = 0, (33)

so that the position x; at the focal plane is independent of 63 and a measure of (SR
only. Going back one step, the position xr at the dispersive image in terms of the

coordinates at the production target is given by
x, = (xlx)1xi+ (XI6)16B, (3.4)

where the superscript denotes matrix elements describing the first stage of the system.

Assuming negligible initial beam—spot size, equation 3.3 becomes
((xlx)(x|6)1+(x|6)) 63 = o. (3.5)

This illustrates the fact that dispersion matching does not require equal and oppo-
site dispersions (xlo): Because the beam spot at the dispersive image is large, the

magnification of the second dipole stage is important as well.

3.3.2 Optical Modes

Fig. 3.5 shows some characteristics of the so—called Medium—Acceptance Mode. Much
like in geometrical optics, rays emerging from a point at the object (production target)
come back to a point at the real image, as indicated by the beam envelope. This so-
called ‘point—to—point focusing’ is employed in the x and y coordinates throughout

the device and is illustrated in the figure for the horizontal direction: images occur

41

 

 

 

 

 

 

6 I l I I I T I I I I l I I l I I I T l
.. l l P I T J
' a
:3" I
v i
a _
a .
3, ‘i
:3 2
8 )
8
0
E I
if -(
d _
a .—
I- I I I I I I I I L J I I I I i I I I I I I I I I d
o 5 10 15 20 25
z [m]

Figure 3.5: A1200 in Medium—Acceptance Mode. The figure shows the result of
a TRANSPORT calculation. The horizontal beam envelope (dotted line) and the
matrix elements (x|9) (dashed line) and (xlé) (solid line) are shown as a function of
position along the optical axis. Units are cm, cm / mrad X 0.1, and cm / %, respectively.
The histogram indicates the location of the magnetic elements of the A1200, beginning
with a quadrupole triplet, followed by a dipole segment, a sextupole, another dipole
segment, and so forth. The zero crossings of (XIO) indicate the target position (2 = 0),
the intermediate images Nos. 1 and 2, and the final image at the focal plane (z =
22.65 m). The size of the enveIOpe is a function of angular size as well as momentum
spead of the beam, and becomes small only at the focal plane, where (xl0) = (x|6) = 0.

42

wherever the (xl0) matrix element is zero, since at these points, the first—order vector
equation for x reduces to Xf = (xlx)xi, with no angular dependence. The overall
dispersion matching of the system is illustrated by the (x|6) line, which goes to zero

after the last dipole magnet.

The Medium—Acceptance mode is the optical mode used most often in the A1200
and provides straightforward raytracing as well as small focal—plane beam spots in

both the x and y coordinates.

In the present experiments, however, a different mode was used, known as Reaction
Mode. In this optical mode, the vertical angular acceptance in the second dipole stage
of the spectrometer is much larger than in Medium—Acceptance Mode. This makes
it ideal for secondary—beam experiments using a reaction target at the dispersive
image No. 2, since the angular distribution of the final reaction products is usually
much wider than that of the secondary beam, which is routinely limited in angle by

apertures at the production target.

The tradeoff is that while the focusing in the horizontal coordinate is point—to—
point throughout the device, it is “point—to—parallel” for the vertical coordinate in
the second dipole stage. This means that trajectories emerging from a point at the
reaction target with different angles are parallel to each other but have different y
positions at the focal plane. Fig. 3.6 illustrates this. The (xlx),(x|0),(y|y), and
(qu5) matrix elements are shown as a function of z for the area around the A1200
focal plane as predicted by a TRANSPORT calculation [BrR77]: The image in x [i.e.
(xl0) = O] coincides with the physical position of the focal—plane detectors as well
as the point where vertical point—to—parallel focusing occurs [(yly) = 0]. In reality,
the x focus was adjusted to occur at the silicon PSD, but the (yly) = 0 point was

off by several cm. TRANSPORT calculations typically provide a good estimate of

43

 

 

 

 

 

 

4 1.. I I I I f I I I\T\T T n I T i I I I I l I331 I I-
Ix ‘ \ j
.. \ \ \\ -
2 :— ‘ ‘ - \ T
.. \.\ \ q
: F \ \\ . :
VI 0 ‘

:a -
0 )- _. _.
8 b -
0 )- .4
d " ..
: Focal Plane 2
-3 T 1
i . I
-8 h I 1 1 1 1 imi 1 1 I 1 1 1 4 I L 1 1 1 i 1 J 1 14

21.0 21.5 22.0 22.5 23.0

2 [m]

Figure 3.6: A1200 in Reaction Mode. The figure shows the matrix elements (xlx)
(solid), (xl0) x 0.01 (dashed), (yly) (dot-dashed), and (y|¢) x 0.01 (dotted) as a
function of position along the optical axis. The range is limited to the area around
the focal plane and the standard units are used. The position of the position—sensitive
silicon detector is indicated by the two arrows and coincides with the x focus as well as
the (yly) = 0 plane. Angles 03 and 43, at the reaction target are related to focal—plane
coordinates through the first-order equations 0f = (0|0)0; and Yf = (y|¢)¢i.

44

the overall optical properties, but individual conditions and matrix elements have
to be determined experimentally. The agreement between calculated and measured
matrix elements varies, but is often not too impressive. Some examples of measured
(calculated) values for the second dipole stage are: (0|0) = —0.65 (—0.44), (ylo) =

-0.022 (—0.013)cm/mrad, and (xl6) = 4.4 (4.5) cm/%.
3.3.3 Dispersion and Energy Calibration

The energy scale was determined using the secondary 13N beam with several settings
for the magnetic rigidity of the second dipole stage. Fig. 3.7 shows the magnetic
rigidity vs. beam spot position at the focal plane as obtained in the 105 MeV experi-
ment. The line represents a linear least—squares fit to the data, which forms the basis
for the energy calibration. Energy, momentum, and magnetic rigidity are linked to

each other through the following equations:

 

szg; pc=\/2mc2E+E2; E=T+mc2,
q

where p, q, and m are the momentum, charge, and mass of the ion, and E and T are
its total and kinetic energy, respectively. The dispersion (xl6) was found to be 3.6 CT}?-
(4.1 %) in the 57 MeV (105 MeV) experiment and determined the slope of the energy
calibraton. The E = 0 point was fixed by measuring the energy loss of the 13N beam
in the reaction target. From this value, the thickness of an equivalent l2C target was
obtained and the difference in energy loss for 13N and 13C was calculated. The zero
of the energy scale was then determined for the 13C(13N,13C)13N reaction products,
with the reaction taking place at half the target thickness. The broadening of the 13C

peaks due to differential energy loss is taken into account during the data analysis.

45

 

2.820 T f r I j I I T I I I I I I I I
I I I

2L815
£L810

£L805

BPS [Tm]

ELBOO

IlIIIIlIITIIlIIIlIIIIlIIII

2&795

 

 

IIIIIIIIIJIIIIIIIIJIIlllllll

 

2.790 I I I l J + I I I l I I I I l L I I I

200 300 400 500 600
x, position [chs.]

Figure 3.7: Bp3 vs. Xf calibration of the second dipole stage. The data points are
the measured position of a 13N secondary beam for different values of magnetic
rigidity. The line is a linear least—squares fit to the data, which was used in the
energy calibration.

46

3.3.4 Horizontal Matrix Elements

A very important goal to be achieved regarding the horizontal coordinates is proper
focusing of the beam on the position—sensitive silicon detector. While raytracing can
in principle provide sufficient resolution even if the focus is not exactly on the detector,
it is usually desirable to avoid this additional complication. Focusing was checked
with the primary beam and a carbon reaction target at image No. 2, employing the
12C(14N,13C)13N reaction. The first dipole stage was tuned to produce a 14N beam
spot of minimal size on the reaction target (% 0.5 mm), in an effort to ensure correct
focusing. Due to the large angular width induced on the beam by the reaction, any
possible remaining correlation of angle and position is washed out. The focusing
condition of the second dipole stage can then be verified by plotting 0f versus Xf,
as shown in Fig. 3.8. The top part of that figure shows the A1200 focal—plane to
be clearly out of focus, Whereas the bottom part exhibits good focusing. The only
adjustment between collecting the two spectra was a 4% increase of the “A061TA”

quadrupole magnet, located right after the second pair of A1200 dipoles.

In order to obtain the scattering angular distribution from the coordinates mea-
sured at the focal plane, some of the matrix elements governing the optics between
the reaction target at the dispersive image and the focal plane have to be deter—
mined. This was achieved by transporting the 13N secondary beam through the
A1200 with the entire device set to the same magnetic rigidity. Position and angle at
the dispersive image, which are measured by two parallel—plate avalanche counters,
were correlated with coordinates at the focal plane and the matrix elements could be

determined.

When measuring these correlations, it is very important not to be misled by

cross—correlations. For example, if the secondary beam is not focused properly on

47

 

 

 

 

 

 

 

Figure 3.8: Focusing of the second A1200 dipole stage. Both spectra show a plot
of Of vs. x; for the 12C(MN,13C)13N reaction products. The 130 phase space is
dominated by the reaction, so that the focusing condition from the reaction target
to the focal plane can easily be checked. The top spectrum shows a strong (xl0)
correlation, with the actual focus occurring downstream of the focal—plane detectors.
The bottom spectrum, obtained after increasing the quadrupole field, shows that the
focusing condition is fulfilled. Scale: 60 mrad by 58 mm, top and bottom.

48

the reaction target, but farther up— or downstream, the vertical position y; at the
reaction target will be correlated with angle (15;. A plot of e.g. y, versus the focal—
plane coordinate y; will then have built—in correlations with (I); which will lead to
erroneous results. This complication is avoided if a small cut in (15,- is imposed on
the y; versus y; plot, ensuring that only the intrinsic dependencies are investigated.
Other coordinates have to be treated likewise. Fig. 3.9 shows the spectrum used to
determine the angular correlation (BIO), with a cut in x; imposed on the data. The

line through the data represents the resulting matrix element (GIG) = —0.65.

3.3.5 Vertical Matrix Elements

Fig. 3.10 shows the correlation spectra used to determine the vertical first—order
matrix elements. The position 2; at which y; is calculated was varied to satisfy the
condition (ny) 2: 0. This corresponds to the intersection of the dashed line and the
optical axis in Fig. 3.6 and is equivalent to the focal point in the the point—to—parallel
mode. The lines through the data show the correlations determined from the spectra.
The parameters that are of importance in calibrating the angular spectra are the
matrix element (ylo) = —0.022 cm/mrad and the coordinate Zf = 78 cm, upstream of

the nominal focal~plane position.

There is the possibility, at least in principle, to exploit the unitarity condition in
determining the matrix elements. For example, under the focusing condition (xl0) =

0, the determinant of the horizontal submatrix becomes

(xIx) 0
(QIX) (9W)

 

 

'2 (xlx) (BIG) 2 1. (3.6)

However, the (xlx) matrix element cannot be determined experimentally, since the
beam spots are only marginally larger than the detector resolution. In the vertical

coordinate, it was similarly found that the matrix element (ylo) could be more reliably

49

 

 

 

 

 

Figure 3.9: Determination of the (BIO) matrix element. This spectrum was obtained
with the 13N secondary beam throughout the A1200 using position monitors both at
the dispersive image and at the focal plane. A cut in horizontal position x; has been
imposed at the location of the reaction target. Scale: 35 mrad by 20m1‘ad.

50

 

 

 

 

 

 

VI

Figure 3.10: Spectra used to determine the vertical first—order matrix elements. The
top spectrum shows a plot of y; vs. y;. The coordinate Zf at which y; was calculated
was varied until the (yly) = 0 condition was reached. The bottom spectrum shows
a plot of (251 vs. yr, recorded for the same Zf. The line through the data indicates
the matrix element (y|¢). The beam was 13N throughout the A1200, with position
monitors at the dispersive image and the focal plane. Scale: 8mm by 14’ mm (top),
20 mrad by 14 mm (bottom).

51

determined from the data than its complement (gbly).

3.3.6 Achromatism and Image Aberrations

While most first—order matrix elements can be determined off—line, some conditions
are required to successully perform the experiments. As discussed in section 3.3.4,
the imaging condition (xlfl) = 0 is convenient, but not absolutely essential, because
raytracing can be used to improve the resolution in the data analysis. The most
important optical property in reaction experiments is achromatism: (xlb) = 0 and
(0|6) = 0, applied to the entire device. The two dipole stages by themselves are, of
course, dispersive. Since momentum measurements of the secondary beam cannot
be performed due to the high incident rate during production runs, any deviation
from (xlb) = 0 will degrade the energy resolution. The achromatic condition can be
checked by sending a limited rate of secondary—beam particles through the device
and using the x, coordinate as a measure of momentum, since xi and 6; are correlated

by the dispersion of the first dipole stage.

Fig. 3.11 shows a X; versus x; plot obtained under those conditions. The data
show that there is no first—order correlation between position at the focal plane and
secondary—beam momentum, i.e. (xlb) = 0. This spectrum was obtained as the result
of several hours of tuning, with quadrupoles being adjusted to match the dispersions
of the two dipole stages and sextupoles eliminating any ’banana’ shapes of the curve.
The hourglass shape of the data indicates higher—order image aberrations, which
could not be eliminated with the available optical elements. An attempt was made to
reproduce and understand this shape with the second—order computer code TURTLE
[Car64], but was unsuccessfull, and the origin and nature of the aberrations are not
fully understood. In an effort to balance count rate and resolution, the reaction target

was cut to a smaller size, as indicated by the two horizontal lines in the figure, trading

52

 

Xi

 

 

 

 

X1

Figure 3.11: By plotting xi vs. Xf for the 13N secondary beam, the dispersion‘
matching condition can be checked. Since the horizontal beam spot size at the disper—
sive image is dominated by the momentum width of the secondary beam, dependence
of position Xf at the focal plane on momentum can be determined. The general ver—
tical trend of the data suggests first—order achromatism, but the overall asymmetric
hourglass shape indicates higher—order aberrations. Scale: 96mm by 21 mm.

53

about a third of the available secondary—beam intensity for a 20% improvement in

resolution.

An example where image aberrations could be corrected with the available mag-
nets is shown in Fig. 3.12, which shows a 6{ versus x; plot for the 12C(MN,13C)13N
reaction (compare with Fig. 3.8). The curved shape of the lines in the spectrum indi—
cates a second—order aberration, which was corrected using the sextupole magnets of
the second dipole stage. It is interesting to note that Fig. 3.12 is already an improve-
ment over the initial situation, where the sextupole settings had incorrect polarity
and the aberrations were so severe that the problem could not even be diagnosed

from the spectrum.

3.3.7 Acceptance Correction

One of the technical difficulties associated with measuring the angular distribution
of the the charge—exchange reaction products is the limited acceptance of the A1200.
While the Reaction Mode offers significant sensitivity at fairly large scattering angles,
the acceptance window is highly asymmetric, with a full width of about 20 mrad in
0; and more than 40 mrad in (15;. In order to convert the measured angular distribu-
tion of the reaction products, the acceptance for each scattering—angle bin has to be

determined and divided out.

Acceptance was calculated with the beam—optics code TURTLE [Car64], which
uses a Monte Carlo approach and the same underlying calculations as TRANSPORT.
Input for the program were the known dimensions of the second A1200 dipole stage
and magnetic—field settings calculated from the magnet currents, together with cor-

rection factors routinely used in applying TRANSPORT calculations to the A1200

magnets.

\

 

 

54

 

 

 

 

 

X1

Figure 3.12: A plot of 0; vs. x; for products of the 12C(MN,13C)13N reaction. The
spectrum is similar to Fig. 3.8, but the curved shape of the data suggests strong image
aberrations, caused by faulty settings of the sextupole elements. Scale: 50 mrad by

47mm.

55

9, [mrad]

-60 -4O -20 0 20 40

I I I I I III III ITT’I I I I I I I I I r I T’ITI I I I

 

Irr
11G
O

100
80
60
40
20

0
60

Counts

 

IIIIIIIIIIIIIIIIIIIIIII

    
  

 

40

Counts

20

1L11l11LJlL1IJJ 4

I I I I l I l I I I I I I l

-40 -20 0 20 40 60
III [mrad]

_

 

 

O
EI I II [VTITIIF1TI IITI [I IIIIIIIIIIIIIIIIIIIIIIIII

 

 

l
G
O

 

Figure 3.13: One—dimensional acceptance in the 0; and 45; coordinates. The histogram
at the top shows the 0; spectrum for the 12C(MN,13C)13N reaction. The dashed line
is the acceptance calculated with TURTLE and normalized to the data. The bottom
figure shows the equivalent (bi curves.

56

Fig. 3.13 shows the result of a TURTLE calculation for the 57 MeV experiment
using first— and second—order matrix elements. The histogram in the top part is the
91 distribution for the l("C(14N ,13C)13N reaction products, which have an angular dis-
tribution wide enough to fill the aperture. Acceptance is determined by simulating a
beam with a wide, rectangular-shaped angular distribution, originating at the reac-
tion target. The calculation is done twice, once without apertures and once imposing
realistic constraints on the physical size and shape of the beam pipe, quadrupoles, and
dipoles. The ratio of count rate with and without aperture constraints for any given
angle bin provides the acceptance. The TURTLE simulation, shown by the dashed
line, fits the measured data quite well, which indicates the validity of the method.
The lower portion of the figure shows the equivalent plot for the vertical coordinate.
Obviously, the measured (15.- distribution is off-center with respect to the acceptance,
which is the result of a misalignment of the secondary beam. This may have been
caused by an error in the vertical position of the beam spot on the production target.
Upon entering the quadrupole fields, such a shift will cause the beam to be deflected
towards the optical axis, resulting in an angular misalignment. The angular width of

the 13C reaction products appears to be significantly smaller than the acceptance of

the device.

In order to determine the acceptance for each scattering angle, a two—dimensional
approach was used. Just as in the one—dimensional case, a wide beam was simulated,
once with and once without aperture constraints. A two—dimensional histogram was
developed, ranging from -20 to +20 mrad in 0; and from -60 to +60 mrad in (b;, with
bin sizes of 0.5 and lmrad, respectively. Simulated particles emerging from the
reaction target were recorded in the appropriate bin, and the acceptance for each bin

was determined. The coordinates of the secondary beam, (9? and (25?, were then used

 

 

 

 

 

57

to convert the two—dimensional result to a one—dimensional scattering angle:

 

6...... = ./(6. — 9,0)2 + (<61 — 6?)? (3.7)

The result of this calculation is shown in Fig. 3.14, where the acceptance is plotted
as a function of laboratory scattering angle. Due to the limited horizontal acceptance,
there is a significant loss of particles for any angle larger than about 10 mrad. How-

ever, thanks to the vertical point—to—parallel focusing, there is significant acceptance

beyond 40 mrad scattering angle.

Fig. 3.15 illustrates the process of acceptance correction: The solid line shows a
Gaussian curve with a width similar to that obtained in the experiment (see chap-
ter 5), and the full symbols show how this distribution is distorted by the limited
spectrometer acceptance. During the data analysis, the measured differential cross
section for each angular bin is divided by the corresponding acceptance value in or-
der to obtain a realistic angular distribution. In the present example, this acceptance

correction of course reproduces the original Gaussian, as shown by the open symbols.

 

 

 

 

Acceptance

Figure 3.14: Acceptance as a function of scattering angle 03mm... The curve shows
the result of a TURTLE calculation, taking into account the coordinates of the 13N

secondary beam.

1.0

0.8

0.6

0.4

0.2

0.0

58

 

 

 

 

 

_ I I I I I I I I I I I I T I I I I I I I I I I I I I I I .1
r- -(
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I- —1
P d
- d
.1

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P d
.— —
b c-I
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.— -.
1— _
I— _
1— .-
— —
h -
b u-l
l- -
p—-— ——
l— —l
b _
l- H
t l I I | l ‘

I I I I I L I I I I I I I I I I I I I I I I I
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0)
O

 

 

 

59

 

 

    

 

 

 

b I I I I l l I l I I 1 I I l I I I I i F-
1.00<_ —:
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“if - I
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0 I Q :1
p I O 1
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O

0.01 :— 9 —:
E ’ 5
- 1 1 1 1 I 1 L 1 1 l 1 1 1 1 L 1 m 1 L L 1 1

0 10 20 30 40

am, [mrad]

Figure 3.15: Acceptance correction of an angular distribution. The solid line is a
Gaussian curve with a width similar to that observed experimentally. The full symbols
represent the angular distribution that would be observed in the experiment, while
the open symbols illustrate the effect of acceptance correction.

 

Chapter 4

Data Analysis

While some of the experimental parameters had to be determined on—line, others
were left to the off—line data analysis. The following sections deal with the issues
of determination of the background rate, total and differential cross sections, and

unfolding of the angular distributions.

4.1 Background Determination

In the 57 MeV experiment, background was determined by replacing the 13C target
with a 12C target of the same thickness and observing the 13C ions at the focal plane.
Because the Q value of the l2C(13N,13C)12N reaction is -15.1 MeV, the reaction prod-
ucts are displaced by 15 MeV on the excitation energy scale and do not contribute
to the region of interest. Scattered beam particles, however, and fragmentation reac-
tions of the 14N primary beam are influenced little by the change in target, so that

the observed 13C spectrum should provide a good measure of the background rate.

The top panel of Fig. 4.1 shows the energy—calibrated focalwplane position spec-
trum of all data collected with the 13C target. The counts observed at negative
excitation energy are a clear indicator of background, since this region is energeti-

cally inaccessible to the 13C(13N,13C)13N reaction products. The center panel shows a

60

 

 

 

61

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

80L rI I I I I I I I I I I I I I I I I rr_I:

E E

i ”I '1

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E E a

8 20: _

Ti #11 I I IJ l I I l I I] I 1 LII :

eg I I Ii I I I I I I I I I I I I I I I] E

5:— —:

g 4;— -;

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E. =— (1W1 E

o 1: -
° 0 . “IL. I

- | PE

Mr 1

g 405- I

\ I E

g 20:— _

o ;_ I

U 0‘_| IJIJIII IIIHU ll ..

11 I I l I 14—1 l_l. L4 14 l 1 I I I | Q

-5 0 5 10 15

zxcnmon ENERGY [MeV]

Figure 4.1: Background correction for the 57 MeV experiment. The top panel shows
data collected with the 13C target, the middle panel shows the result of a l2Cv-target
background run, and the bottom panel contains the background—corrected spectrum.

62

13C spectrum collected over a period of six hours with the 12C target. The count rate
tends to increase towards higher excitation energy, but no peak structure is visible.
At the bottom of Fig. 4.1 is the background—subtracted 13C(13N,13C)13N spec-
trum. It was obtained by multiplying the background spectrum by a factor of 6.2 to
account for the difference in the amount of incident secondary—beam particles, and
subtracting it from the 13C(la‘N ,13C) spectrum. While the result of this procedure is
unsatisfactory due to the poor statistics involved, the average number of counts at

negative excitation energy is close to zero, indicating the validity of the approach.

Instead of subtracting the spectra directly, the background was assumed to be
smooth and the spectrum was fit with a first—order polynomial —— shown by the

dashed line in the center panel — which was used in the subtraction process.

A somewhat different approach was used for the 105 MeV experiment. It became
apparent during the run that the background rate was correlated with the amount
of scattered primary 14N beam observed in the particle—identification spectrum. The
ratio of 14N to 13N , which had been very stable during the 57 MeV experiment, ranged
from approximately 0.7 to more than 6. The reason for this was most likely the fact
that the target beam spot and the aperture limiting the angular acceptance of the
A1200 were very close together. While that distance had been approximately 180 cm
in the 57 MeV experiment, beam spot and slits were only 95 mm apart in the 105 MeV
run, and the aperture was accordingly very small. A small change in the position of
the beam spot could, therefore, have a large effect on the amount of scattered primary
beam entering the device. The rate of counts observed in the negative excitation—
energy region was high whenever the 14N contamination was strong. In light of this
behaviour, the approach used in the low—energy exeriment was considered insufficient,

and the following analysis was performed instead:

63

O
(0
IF
0
CD

0 2 4 6 8

 

  
    
  

 

 

    

 

 

 
  
 

 

     

 

 

 

 

15fiftiI""I""I"""""tfj"l""I'fi"I""I'1'i 15
l I.
I II
10' “ 10
m ; E - -?.2 luv 1; E - —3.6 HOV j
a p I} 0 1
I 0 4
g 5? 1r 1 5
o p o I
U i I: '
or 0 I o
I IE I
I I
Bo’fiftItf¢i%::;:%tff3{:t:f c:¢:§:c::}:;::}~:fi%t -: —5
I I
E f- 540
15~ u: *
3 ; j; f-SJIloV E
: 1;" .30
1°." 1' i
I
d
s : r we"
5_ l-ODIQV _j: 1
> 4’ + 110
I I
I I
: L I L l j: l I I l :
o>::::'::::': :t'---:'::::0:::‘,:::‘l‘:‘:,:‘~ 'ctcc‘ 0
I I
40” "E' 440
I I I
p 1 1
m : ‘: :
+’ 30.- 4.,- 130
g I I: I
I II
o 20 <1» 1r*+ 4.20
I I
0 ; r-m luv 1, r- 10.5 Rev 3
105- 1:" 110
I I
I I: I
0 :::::AH4::;:: ‘c: ‘-“““““-‘1**“1“~‘ 0
g 2 4 s a
b O
m g Ratio
+> L
g :
I
o n
o i
: E-iunov
101'
I
I

 

o ----I----I----I--J_.1L+LA

2 4 6 8
Ratio

0

Figure 4.2: Background correction for the 105 MeV experiment. The top panel shows
the excitation—energy spectrum, divided into bins for the background analysis. The
six panels underneath show the normalized count rate as a function of contamination
ratio for the first six energy bins.

64

 

IIIrIIIIIIIIIjIIIIIIIIIIIIITI

4F _

Slope

 

 

 

 

I I I L I I I J L I I I L I l I I I I I I L I I l I I I I
-10 —5 O 5 10 15 20
Excitation Energy

 

Figure 4.3: The slope parameters obtained for the seven energy bins, plotted as a
function of the average energy of the bin. The line is the linear least—squares fit used

in the final background subtraction.

65

o The excitation energy spectrum was divided equally into seven bins, 3.6 MeV

wide each, and covering the region between -9.0 and +162 MeV.

o The available data files were divided into five groups, according to their 14N/13‘N

contamination ratio.

0 For each energy bin, the count rate, normalized to the intensity of the incident

beam, was plotted as a function of contamination ratio, as shown in Fig. 4.2.

o The data were fit by first—order polynomials. The intersection of the curve with
the y axis indicates the 13C rate at zero contamination. Note that this value

is close to zero for the two lowest—energy bins.

o The slope of the curve determines the background rate as a function of con—
tamination. The slope values were plotted as a function of excitation energy, as
shown in Fig. 4.3. In order to obtain a function varying smoothly with energy,

a linear least—squares fit was performed, as indicated by the solid line.

The parameters of the fit, together with the observed numbers of 14N and 13N
ions, was then used to perform a background subtraction similar to that described
above for the low—energy experiment. The background—subtracted 105 MeV spectrum
is shown in Fig. 3.2. Again, the average number of counts for negative excitation

energy is compatible with zero.

4.2 Drifts

Each of the mirror charge—exchange experiments lasted just over a week, and several
of the experimental observables exhibited a slow variation over time. Most notably,
the position of the primary beam spot on the production target moved horizontally by

up to a millimeter, probably due to variations in the cyclotron operating conditions.

66

Since the position at the A1200 focal plane is a real image of the initial beam spot,
this behaviour caused the overall energy resolution to deteriorate significantly. Other
observables, such as the total-energy signals from the plastic scintillator that stopped
the ions, also exhibited slow variations with time, likely as a result of changing ambient

temperature.

During A1200 experiments, data taking is routinely interrupted every two hours,
and a new file is opened. Since the typical time scale of the variations were signifi-
cantly longer than two hours, the problem was solved by determining the average value
of the observables for the 13N6+ ions and recalibrating the position, total—energy, and
other signals relative to these values for each file. This procedure restored the original

resolution almost completely.

4.3 Energy Spectra

The bottom panels of Fig. 4.4 show the background—subtracted, energy—calibrated
position spectra for the 13C(13N,13C)13N reaction products. Three peaks are visible,
at excitation energies of approximately OMeV, 3.5 MeV, and 7MeV, plus a broader
background structure above 10 MeV. The insets show the ”N“ position spectra,
which were used for the absolute calibration of the energy scales. The peaks are

slightly shifted from OMeV because of the differential energy loss in the reaction

target.

To understand the structure, it is instructive to consider Fig. 4.5. On the assump—
tion that the reaction proceeds primarily via single—step charge exchange, possible
candidates for final states in the ejectile are the J " = 1/2' ground state and the 3/2‘
excited state at 3.68 MeV. The neutron~emission threshold of 13C is at 4.95 MeV.

Possible states in the target residue are the ground state, the 3/‘2‘ state at 3.50 MeV,

67

 

 
 

 

 

 

 

 

 

E/A - 57 NOV E/A - 105 MeV

,- ' ' ' ' I ' I ' ' I ' fi' I ' r ' I ' I # r

: Shell “Odd

E- (clltnry nib)

E 1 - 1 .ii i

p' ' ' ' ' I I I ' ' T Ifi? 10°

40: ”c(”x.”c)”u 1f

'3 : "N“ 0% :r 80
3 3°? 1; :
\ t II if ' 6°
.3 m :- 0! ‘9" .L l ‘0
g : . fl 1’ I

L .l i _. : .
8 1o: (‘i‘l‘l f i”

o : II I I . ' - . I . A . . I . . . . I :i -- 1 L - - L 1 A - L #1 I -1 o
—5 0 5 10 15 0 10 20
Excitation Energy [MeV] Excitation Energy [MeV]

Figure 4.4: Energy specta and their interpretation. The histograms in the bottom
panels show the background-corrected 13C excitation—energy spectra, and the full
lines are the fits described in the text. The insets show the ”N“ charge state peaks,
and the top panels contain the results of shell—model calculations.

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13C 13"
16.06 - _
Target transitions 3’2, T‘s”
11.88 3,2-
___1.Q.L 1’2-
—-s GT 9'43 an-
8.92 “2'
'IU'IUI'IUUI555III' F
3.51 312'
1a. 0.0 Mov .IIIIIUUIIUIIIUDII' 0.0 MeV 1n-
4,95 120 + n Projectile transitions
312’ 3°68 \
1a. 0.0 Mov 'UUIIICIUUIICIOIIOIU 0.0 Nov 1&-

 

 

 

Figure 4.5: Schematic illustration of the target and projectile transitions in the
13C(13N,13C)13N reaction. Since the ejectile is detected in the reaction, only tran-
sitions to particle—stable states contribute to the observed cross section. There can
be no cross—coupling of Fermi and GT—type transitions.

69

and further negative—parity states above 9 MeV of excitation energy. The transitions
depicted in Fig. 4.5 were obtained from Mildenberger’s analysis of 200 MeV 13C(p,n)

data [MiA91].

Since the one-step reaction can proceed either via a Fermi or via a GT transition,
there can be no F/ GT cross terms. Accordingly, the observed strength above the g.s.

peak is entirely GT in nature, while the ground state is mixed.

For the analysis, only the three lowest resolved states were taken into consider-
ation. The energy spectra were fit with multiple Gaussian curves of fixed positions
and widths, with the peak area as the free parameter. For the 57 MeV experiment,
three peaks were considered sufficient to describe the data: a) Ground state (g.s.):
Both ejectile and target residue are in their ground states, b) Single Excitation (SE):
The target residue is in the 1/2‘ ground state and the ejectile in the 3/2’ state, or
vice versa; c) Double Excitation (D.E.): Both nuclei are in the 3/2' state. For the
105 MeV experiment, higher excitations were also allowed in the fit in order to obtain
a reliable measure for the area under the DE. peak. The results are shown by the

solid lines.

The curves in the top panels of Fig. 4.4 were produced by assuming strict pro-
portionality of the cross section to transition strength. GT strengths obtained from
a shell—model calculation [ChB93] were multiplied by the experimentally determined
unit cross section (see Chapter 5) and convoluted with the observed energy resolution.
The double excitation peak at 7.2 MeV is overpredicted relative to the experiment,
but this a result of the “13C controversy” discussed in Section 2.3: The unit cross
sections were determined assuming B(GT)3/2— = 0.83, while the shell—model calcula-
tion predicts B(GT)3/2— = 2.38. Aside from this built—in deviation, the shape of the

spectra is reproduced remarkably well.

70

Table 4.1: Integrated cross sections determined from fitting the excitation energy
spectra.

 

 

13N Energy E/A=57MeV E/A=105MeV

 

13Cg,_.,,(7 31:1:4 pb 15 i2 pb
13051.10 36 :l:4 [1b 23 i2 pb
”(30.5.0 102 :1:6 pb 59 21:4 pb

 

 

An unusual feature of the energy fits should be pointed out: Upon close inspection,
the E/ A = 57 MeV g.s. peak appears to be shifted up in excitation energy by 200
to 300 keV. While it is conceivable that this shift could be caused by an error in the
determination of the zero of the energy spectrum, the fit to the double—excitation
peak at 7.2 MeV is excellent. It seems rather unlikely that errors were committed in
the determination of both the absolute value and the slope of the energy calibration
in such a way as to cancel each other out near 7 MeV. N o explanation has been found
for this effect, but it should be noted that the area of the g.s. peak appears to be
well reproduced by the fit and did not change appreciably when the peak position
was allowed to vary. The problem was not investigated further. The E/ A = 105 MeV

data do not exhibit this kind of complication.

The integrated reaction cross sections, determined from the number of counts
observed in each peak, are given in Table 4.3. In order to determine the forward
cross sections da/dQ(0°), the angular distributions of the reaction products were

determined for the three excitation—energy peaks in each experiment.

4.4 Angular Distributions

As outlined in Ch. [3], the horizontal and vertical angles 0, and 45, were determined

from measurements of the two—dimensional particle position and angle at the focal

71

plane, after careful calibration of the first—order ion—optical properties of the sec-
ond A1200 dipole stage. The 13N6+ ions were used to define the 03:0 and 97220

coordinates. The scattering angle was then determined from the relation

 

gscatter : \/(0i — 013N)2 + (¢i — ¢13N)29 (41)

and histograms for the number of counts per scattering-angle bin —— in the laboratory
(lab) system —— were obtained. These were converted to units of counts per msr of solid
angle in the center—of—mass (c.m.) system. Due to the symmetry of the reaction, the
relationship between lab and c.m. angle is a very simple one to first approximation:
96m = 2 x glab- The difference in mass between 13C and 13N is 2.22 MeV/c2 and is
ignored, but relativistic effects are more important, and the actual conversion factors

between 0cm, and 010;, are 2.030 (2.053) for E/A = 57 MeV (105 MeV).

Fig. 4.6 illustrates this process. The top panel shows the energy—calibrated posi-
tion distribution of 13C ions at the focal plane for the E/A=57 MeV experiment. A
software cut is used to select the ”Cg, reaction products, as indicated by the two ver-
tical lines. The center panel shows the 0161, histogram for the selected particles. The
bottom panel shows the same data, but converted to units of counts/msr. The axes
for the angle spectra were chosen to be equivalent in order to facilitate comparison,
and the error bars represent the statistical uncertainty only. Once the spectrum is
corrected for the spectrometer acceptance, a realistic angular distribution is obtained.

However, care must be taken with the normalization of the data.

Firstly, the number of counts for which angular coordinates could be obtained are
limited by the efficiency of the focal—plane position monitors. While this detector effi-
ciency was close to 100% for the CRDCs used in the 105 MeV experiment, it was only
approximately 65% for the 57 MeV experiment, where PPACs were used. Secondly,

the software cuts imposed on excitation energy are sufficient to determine the shape

72

Excitation Energy [MeV]

 

 

 

 

 

 
 
   

 

 

 

 

 

 

-5 O 5 10 15
.8 80 j_I j T I I T I r I I I I I I I I I—r 1
a 60 E -:'.
\ =— =
3 4° E a
5 20 Lmfl q
o P 1
O l 1 1 1 I 1 1 1 LI 1 1 1 1 L '
F. 25 :_I 1 I I I I 7 I r r Ifi I T I I I I T I T_—:
9 '2 2
g 20 I. u'C 3.3. ”i
\ 15 E— E/A - 57 Nov "a"
g 10 ;—- a
8 5 :— 1:
O I I I I I I I I I I I I I— I m I m I II'1'_I:
0 10 20 30 40
0“ [mrld]
I I I I I Ij I I I I I I I I I I I I I I I I I I
100 I
B 9111 3
3 10% If a
a E f a
8 : I I f :

In”... ....illlII.I.I.ili

I l .
0 1 2 3 5
on... [(100008]

Figure 4.6: Scattering—angle analysis, shown for the E/A=57 MeV 13C ground state
as an example. The middle panel shows a scattering—angle histogram for the data
selected by the software cut in the top panel. The same data are displayed at the
bottom after conversion to units of counts per c.m. solid angle. No acceptance
correction was performed, and the scales of the x axes in the two lower panels are
equivalent. The error bars represent the statistical uncertainty only.

73

Table 4.2: Fit parameters for the angular distributions

 

 

 

 

 

E/ A = 57 MeV
Maximum rms width (c.m.) X2
”N“ — 052° 24
13C ground state 63:1:6 Insr‘l 1.16° 0.67
13C single excitation 56:1:5 Insr’l 1.43° 1.73
13C double excitation 133:1:8msr‘1 1.42° , 1.29
E/A = 105 MeV
Maximum rms width (c.m.) X2
l3N6+ —— 050° 78
130 ground state 190:1:14msr"1 0.85° 0.96
13C single excitation 294:1:17 msr’l 0.92° 1.36
13C double excitation 533:1:22 msr"1 0.91° 1.00

 

of the angular distribution, but do not accurately reflect the number of counts in the
peak, which has to be determined by peak fitting. Lastly, the widths of the measured
angular distributions are not solely due to the charge—exchange reaction process, but
also in part to the width of the secondary beam, which has to be deconvoluted from

the measured distributions.

The following procedure was used in determining the forward cross section. For
all peaks, the acceptance—corrected angular distribution was determined in units of
counts per msr. The peak value, dN/dfl(0°), was determined by fitting the data
with a Gaussian curve, with the peak width and area as free parameters, and the
centroid fixed at 0°. This procedure is empirically justified by the fact that the data
are rather well reproduced, as indicated by the X2 values in Table 4.4. Furthermore,
the angular distribution for single—step charge exchange obtained from a model cal-
culation in eikonal approximation [Ber93] show a similar functional behaviorfor the
13C(13N,13C)13N cross section at small angles. The angular distributions obtained

for both experiments are displayed in Fig. 4.7. The top panels show the angular

74

 

    

 

 

E/A = 57 MeV E/A = 105 MeV
9m [mrad] 9» [mrad]

,_, o 10 so so 40 o no no so 40 a
.3 ' ' ' ' ' ' I I I I ' I ' I ' ' ' ' I ' q
a 13N0+ 1.00 E

. :3
'n H
3 0.10 5‘
C:
'U
E 0.01 E,
'U . IN 1-1

: . 1’C(”N.“C)”N
3.8. 10

 

AAIAAJ I

/. I—O-I
Juan-J J
p

 

1N ‘ 100

da/dO [mb/sr]
J:
;
[.Is/qm] up/ap

too: 1“.“ DE. i .‘KN‘ DE-
10 I 1 “I

L. I.

‘100

 

 

 

l l 1 *
1 1 \ I i l
L...1....1....1....1§\..i ....1....1.1..IL.- I.. .
0 1 3 8 5 5 0 I. 8 8 4 5
Hon [degrees] 6‘um [degrees]

Figure 4.7: Angular distributions for the 13N secondary beam, obtained from the 6+
charge state, and the three lowest peaks in the excitation—energy spectra.

75

Table 4.3: Results from the mirror chargckexchange experiments. The forward cross
sections da/dQ(0°) were obtained through the procedure described in the text.

 

 

13N Energy E/A=57 MeV E/A=105 MeV
13C“. da/d0(0°) 17:1:3 mb/sr 16 i3 mb/sr
1309.15,: da/dQ(0°) 15:1:3 mb/sr 22 i4 mb/sr
”ODE, d0/dfl(0°) 44 i8 mb/sr 56 i10mb/sr

 

 

 

distributions for the 13N6+ secondary—beam. The solid lines depict the spectrometer
acceptance, calculated with a modified version of the computer code TURTLE (see
chapter 3). The curves are slightly different for the two experiments because of an
improved optical solution employed in the 105 MeV experiment. The dashed lines
show the results of non—linear least—squares Gaussian fits, limited to data at c.m. an-
gles S 1°. While the fits fail to account for the tails in the spectra, they describe the
13N6+ data rather well down to 10% of the maxima, and were considered sufficient
to use in unfolding the measured angular distributions. The angular distributions for
the ground-state, single—excitation, and double—excitation peaks are shown in the six
lower panels. After fitting, the spectra have been calibrated in units of mb/sr, The
dashed lines show the results of Gaussian fits, limited to center—of—mass angles S
24° (2.0°) for the 57 MeV (105 MeV) data. Parameters of the fits are given in Table
4.4. The final results for the forward cross sections were obtained from the following
relation:

3’10") = Fit(0°) x C x 2E. x —i——, (4.2)

d” Z:Osmatter 0'2 — 0'3

 

where Fit(0°) is the peak value of the fit to the angular distribution, C is a factor
that converts the number of counts to units of millibarn, EE“ is the number of counts
obtained from the fit to the excitation energy, and 295mm,, is the number of counts

in the angular—distribution spectrum. The final correction applied accounts for the

76

finite resolution (00) of the secondary 13N beam, which is subtracted in quadrature
from a, the width of the 13C reaction products. The resulting increase in the forward
cross sections is relatively small — typically between 10 and 20% -— which justifies
the approximation of the 13N distributions by a simple Gaussian. The results of this

analysis are listed in Table 4.4.

Chapter 5

Results

As outlined in chapter 2, the relationship of zero—degree cross section to target tran-

sition strength for (p,n) reactions is conventionally written as:
da/dfl(0°) : 6pF(q,w)B(F) + 6GTF(q,w)B(GT), (5.1)

This formula, which connects the charge—exchange cross section with the weak—
interaction strength in the target only, is practical in the nucleonic case. It is not
useful, however, for a description of heavy—ion charge exchange, especially in cases
where the ejectile may be excited as well. Consider the present case: the physics
of the interaction process clearly depends as much on the structure of the projectile

nucleus as it does on the target structure.

Therefore, a more general form of equation 5.1 is proposed, which is physically
better justified and will allow meaningful comparison of CEX data obtained with

different systems:
d0/dfl(0°) = 0EF(q,w)B(F)pB(F)T + aaTF(q,w)B(GT)pB(GT)T, (5.2)

where the transition strengths in both projectile (P) and target (T) are taken into
account, and 0" replaces 6. The latter is only defined for the (p,n) reaction, in which

501‘, since B(F) = 1 and B(GT) = 3. In the present case,
77

case a; = 5;: and GET 2 i

78

Q8, = 0, and the correction factor F (q,w) is estimated from the systematics outlined
in Ref. [TaG87] to be greater than 0.95 (0.98) for the 57 MeV (105 MeV). It will be

dropped from the equations and is not considered further.

Using equation 5.2, the cross section of the three peaks in the energy spectra, and

the structural symmetry of the mirror nuclei, the following equations are derived:

010/ d9(0°)g.s. = UEB(F)f/2— + UETB(GT)f/2- (5-3)
da/dn(0°)s.E, = 205TB(GT)1,2-B(GT)3,2- (5.4)
da/dn(0°)D,E, = agTB(GT)§,,_. (5.5)

There is no mixing of Fermi and GT—type transitions, since the interaction is pre-
sumed to be one—step in nature. Note that the possibility of ejectile excitations gives
rise to three peaks instead of the two obtained in nucleon—induced charge exchange.
From the resulting three equations 5.3—5.5, it is in principle possible to extract the
unknowns 6p and (AMT, as well as B(GT)3/2—, which is not available from B—decay
measurements.1 However, this requires reliance on the assumption of specific propor-

tionality, which has been called in question for 13C.

Given the uncertainty regarding 13C(p,n), it is considered a better choice to use
the (p,n) results of Ref. [RaW87] for and to draw conclusions based on a comparison

between the HICEX and (p,n) studies.

Using the measured cross sections, B(F)1/2— = N — Z = l, and the literature
values B(GT)1/2— = 0.200 :1: 0.004 (from fi—decay) and B(GT)3/2— = 0.83 :t 0.03
[from the (p,n) reaction] [RaW87], equations 5.3—5.5 can be solved to yield values
for (If; and 0E”. The results are given in Table 5.1. For comparison, experimental

results obtained for 13C targets with beams of protons and 3He are also included

 

1This analysis results in B(GT)3/2- = 1.2 :t 0.3 (1.0 :1: 0.2) for the E/A = 57 MeV (100 MeV)
experiment, right between the Goodman/ Watson results of 0.83/1.38.

79

Table 5.1: Unit cross sections obtained in the mirror charge—exchange experiments.
The extraction of these parameters is based on results from the 13C(p,n) reaction.

 

 

 

 

 

 

 

13N Energy E/A=57 MeV E/A=105 MeV
of; 15:1:2mb/sr 13 i2mb/sr
GET 55:1:10mb/sr 74:1:13mb/sr
Other Results: (3He,t) E/A=67MeV (p,n) E2160 MeV
of; 5.5i0.5 mb/sr 1.6i0.3 mb/sr
023T 4.1:1:0.2mb/sr 4.2:i:1.2 mb/sr

 

 

[RaW87, JaBQ4].

The specific cross sections found for the present case are surprisingly large, and

GET shows a small increase with increasing energy, while 0;; decreases slightly.

A model calculation [Ber93] for the 13C(13N,13C)13N reaction at E/A=70 MeV
predicts 3.5 mb/sr for the GT part of the forward g.s. cross section. The experimental
value, determined from the E/A = 57 MeV 0“ and relation 5.3, is 2.2 :l: 0.4 mb/sr.
This calculation is based on a relatively simple eikonal approximation and employs
harmonic-oscillator wave functions, so that the expected accuracy is less than a factor
of two. It should be noted that the theoretical prediction implies (“7" = 87.5 mb/sr,
which is larger still than the experimentally obtained values. Given the level of

sophistication of the model, the results are surpisingly close to the experimental ones.

The increase in cross section with bombarding energy is in contrast with (p,n)
charge exchange, where 033T remains unchanged, while of; decreases with energy
[TaG87]. In 12C(12C,12N)12B HICEX, however, an increase in GT cross section
has been observed, with values for of” of 5.4 mb/sr (17 mb/sr) at E/ A = 70 MeV
(135 MeV) [AnW91, IcI94]. As outlined in section 2.4, the increase in cross Section is

attributed to enhanced nuclear transparency at increased bombarding energy.

A test of proportionality that is independent of calibration is obtained by com-

80

bining Eqs. 5.4 and 5.5:

da/dQ(0°)D.E. = B(GT)3/2-
d0/dfl(0°)s_E. 2 B(GT)1/2— .

 

 

(5.6)

The experimentally obtained values for the left—hand side of Eq. (6) are 2.9 :1: 0.5
(2.5 d: 0.7) for the 57 MeV (105 MeV) experiment, in fair agreement with the right—
hand—side value of 2.08 :1: 0.09.

For further comparison, the values of GET/0E as a function of energy for the
three charge—exchange reactions 13C(p,n)13N [TaG87, RaW87, Rap95], 13C(3He,t)13N
[JaB94, AkD94], and 13C(13N,13C)13N are shown in Fig. 5.1. The data indicate that
GET/0;; increases with projectile mass. This increase is attributed to the strong ab-
sorption in heavy—ion reactions, which selects large impact parameters and therefore
emphasizes the longest—range part of the nuclear force. The spin—flip part of the
interaction is mediated by single pion exchange and hence is enhanced with respect
to the non—spin—flip portion, which has the shorter—range characteristics of multiple

pion exchange.

In summary, the results obtained in this study indicate that HICEX can be used
for weak—interaction studies in much the same way as the (p,n) reaction. The ex-
perimental energy spectra can be understood in terms of weak—interaction strengths
in both projectile and target. However, the large specific cross sections obtained for
(13N,13C) mirror charge exchange, as well as the unexpectedly large ratio of GT to
Fermi unit cross section, underscore the need for a reliable calibration procedure.
With only a few systems studied so far, much further experimental work is necessary

to understand the physics involved in the HICEX process.

81

 

_ I I I I I I I I I I a I I I I ' I I I I I I
10.0 _ 1
4

2.0 _ g 5 ‘

 

 

 

In. .
‘Q I
8 1‘0 I , (laN,13C) "
I: Q , ' .
b . .
..' o (P'n)
0.2 l' é ‘
x ($31)
0.1 x . . . I I . . A I L A L 4 I a L a A I x
O 50 100 150 200

Energy per Nucleon [MeV]

Figure 5.1: Energy dependence of GET/a} for a number of charge—exchange reac-
tions performed on 13C targets. The dotted line is the equivalent of the (”MT/(‘71: =
(Ep / 55 MeV)2 curve, scaled appropriately.

Chapter 6

Conclusion and Outlook

With the increasing availability of radioactive nuclear beams, charge—exchange re-
actions with fl-unstable nuclei can now be studied. This is demonstrated by the
present work: Quantitative results and high energy resolution were achieved despite

a secondary beam with limited intensity and poorly defined momentum.

The difficulties associated with these experiments are obvious. Because of the
large emittance of radioactive beams produced by projectile fragmentation, the ion—
optical setting of the analyzing beam line has to be tuned much more carefully than
for typical primary—beam experiments, since the phase—space volume of the radioac-
tive beam provides a large “lever arm” for image aberrations. Using just the A1200
for production and puricifation of the secondary beam and analysis of the reaction
products provides an additional challenge. Further complications are the compara-
tively low count rate, the limited angular resolution, and the high background rate.
A comparison of the quality of the final data of the present work with the results

obtained by Raman et a1 and Ichihara et a1 (Refs. [AnW91, Ic194]) illustrates the

relative difficulty of stable—beam and radioactive—beam experiments.

However, the mirror charge—exchange results are the first of their kind, and ma-

jor improvements both in beam intensity and in experimental equipment can be

82

83

expected at laboratories worldwide. At the NSCL, the coupled—cyclotron upgrade
[MSU94] will provide the former, while the S800 spectrograph [N0189] is the first
high—resolution beam—analysis device specifically built for nuclear experiments with
radioactive beams. Had the 5800 been available as an analysis device for the present
study, the A1200 could have been used to produce and purify the secondary beam,
and a drastic reduction in background rate could have been realized in addition to
improved energy resolution. With a projected energy resolution of E/AE = 5000,
equivalent to a FWHM of 0.27 MeV for the E/ A = 105 MeV experiment, the final
13C resolution would have been 1.9 MeV and thus slightly better than that achieved
in the E/ A = 57 MeV experiment. At the cost of a reduction in rate, thinner reaction

targets could provide excellent overall energy resolution.

The suitability of heavy—ion charge exchange as a tool for weak—interaction strength
measurements is another issue addressed by the present work. The analysis of the
mirror charge—exchange experiments greatly profits from the symmetry of the mirror
system, which aids in the interpretation of the spectra and ensures that only one nu-
clear form factor enters the picture. In this sense, the mirror reaction offers a unique
opportunity for the study of the HICEX process. The present results are compatible
with the assumption that the reaction occurs via single—step charge exchange, the
underlying physics of which is the nucleon—nucleon interaction. The narrow angular
widths of the observed states as well as the good agreement of the shapes of the en-
ergy spectra with the shell—model calculations shown in Fig. 4.4 further suggest that
the forward cross section is indeed dominated by ATzl, AS=0,1 (i.e. Fermi and
Gamow—Teller) transitions. This is analogous to the results obtained theoretically
and experimentally for the (6Li,6He) and (12C,12N) systems, and is another piece of
evidence that HICEX can be instrumental in measuring weak—interaction strengths

in radioactive nuclei.

84

Qualitatively, the 13C(13N,13C)13N excitation spectra are readily interpreted in
terms of the transitions in projectile and target. The quantitative argument for “spe-
cific proportionality” is not quite as strong. While the observed spectra do suggest
that the measured cross section is very likely proportional to the strengths of the Fermi
and GT transitions, the controversy regarding the interpretation of the 13C(p,n) re-
sults renders the conclusions somewhat tentative. Including these uncertainties, it
can be said that specific proportionality is born out by the 13C(13N,13C)13N results
to within ISO—50%.

Clearly, more work remains to be done before HICEX will be as well understood as
the (p,n) reaction. From an experimentalist’s point of view, it would seem desirable
to obtain as much data as possible. The (3He,t) reaction is of interest as a first step
towards nucleon—induced charge exchange, while it does not allow angular—momentum
transfer to the projectile. It is desirable that CEX reactions should be studied over
a wide range of projectiles and targets. Nuclei such as 1“C, which has been studied
extensively with the (p,n) reaction, as well as other “benign” nuclei should be used

as benchmarks to probe the CEX response in a variety of systems.

Mirror charge exchange will also be a valuable tool due to its relative structural
simplicity. Two obvious candidates for further mirror HICEX studies are the (11C,“B)
and (278i,27Al) systems. A charge—exchange experiment using the latter has been
proposed and has received approval for a feasibility study with the S800 spectrograph
[Ste95b]. One of the advantages of the mirror reaction is that it is possible to observe
transitions of both Fermi and the Gamow—Teller type. Together with data from the
(p,n) and (3He,t) reactions, it is to be hoped that a consistent picture will emerge
regarding the relative magnitudes of Fermi and GT unit cross sections. This would

greatly aid in calibrating future HICEX results obtained with radioactive beams.

85

Much work also remains to be done regarding theoretical issues. DWIA calcula-
tions, for expample, are very helpful in understanding the general features and trends
of the data, such as the effects of energy loss in the reaction. But when absolute
cross sections and angular distributions are to be determined, much fine—tuning of
model parameters is usually required in order to provide acceptable fits, as shown
e.g. in Ref. [RaW87]. While empirical formulae describe most of the (p,n) charge-—
exchange results rather well, the difficulties associated with the 15N(p,n) and 13C(p,n)
results indicate that nucleonic charge exchange is not well enough understood on a
microscopic, fundamental level. Until such understanding progresses satisfactorily,

theoretical descriptions of HICEX will be at least problematic.

While more work is clearly necessary in both experiment and theory before HICEX
reactions can be fully understood, the present results allow cautious optimism. Given
the current improvements in beam intensity and spectrometer performance, it can
be expected that radioactive nuclear beams will provide the opportunity to reliably

measure Gamow—Teller strengths in radioactive nuclei.

Appendix A

The 13C Targets

The targets for the charge—exchange experiments had to satisfy a number of require-
ments: A thickness of about 10 to 20 mg / cm2 was required to strike a balance between
count rate and and energy resolution. The targets had to be an inch high to cover
the target frame, about two inches in width because of the beam size at the dis-
persive image, and have maximum uniformity so as not to degrade the spectrometer

resolution.

Since targets of enriched 13C that meet the above specifications are not offered
commercially, they had to be manufactured from available raw materials. First at-
tempts to produce acceptable targets by machine-pressing amorphous 13C powder into
pellets were unsuccessful: Even though the pressure applied to the powder was suffi-
cient to leave imprints in the stainless-steel dies, the targets were not self—supporting.
A first acceptable batch of targets was produced by dissolving styrofoam in acetone,
mixing amorphous 13C powder in and pressing the substance betwe[21 en pieces of
glass. After the acetone had evaporated, 13C constituted about half of the targets by
weight. A first test run [StB91] showed that a higher target purity was desirable and

that a different production method had to be developed.

The targets used in the charge—exchange experiments were made from l3C—labelled

86

87

polyethylene [(13CH2)n, 99% enriched], which was purchased in powder form from the
ISOTEC corporation for a price of $3000 for 1 gram. (It may be helpful to know that
the procedure outlined here did not work at all with deuterated polyethylene supplied
by Cambridge Isotope Laboratories. That material appeared to have a higher melting

point than the polyethylene supplied by ISOTEC.)

In order to ensure an even, well—controlled thickness and to avoid air inclusions in

the final target, the following procedure was employed in making 10 mg/ cm2 targets:

1. 250 mg of the polyethylene (PE) powder are spread on the clean, polished
surface of a steel brick that forms the lower half of a target press and the brick
placed in a vacuum oven and heated to 150 degrees C. The powder is left in
the oven for a period of around 16 hours, during which it melts and forms small

beads on the brick’s surface.

2. The oven is then switched off, the vacuum broken, and the brick taken out.
Since the brick will retain its temperature for several minutes, a tongue depres-
sor or similar instrument can be used to form the PE into a compact pellet,

approximately the size and shape of a large booger.

3. The pellet is again placed on the steel brick, and the brick is placed in the
vaccuum oven overnight. Over the course of 24 or so hours, air bubbles inside
the plastic will be largely pumped out. What little air remains inside the
polyethylene, however, forms large bubbles due to the surrounding vacuum.
Therefore, when the oven is switched off, it should also be let up to air, which
shrinks the remaining air inclusions to a very small size. When everything has

cooled off, the solid pellet of PE is ready for pressing.

4. In order to be able to remove the finished target — which is about 0.1 mm thick

88

— from the surface of the target press, the polyethylene is sandwiched between
two sheets of 2—mil brass shim stock. The brass sheets are first sprayed with a
Teflon release agent and then buffed to remove excess material. A practical size
for the brass sheets is about three by four inches. Two double—thick strips of
shim stock are laid along the length of the sheets as spacers to define the outline
of the target and ensure that the material is not pressed thinner than the desired
thickness of 0.1mm (4 mil). The PE pellet is placed on the bottom sheet, the
top sheet placed over it, and the target press is assembled and tightened, taking
care that the spacers are not displaced. Since spring pressure alone is sufficient
to produce the target, the springs of the target press should be preloaded but
not fully compressed in order to avoid rupturing the target and creating enclosed

air volumes.

5. The target press is placed in the vacuum oven again and left for several hours.
The press must be under vaccum well before the pellet begins to melt in order
to avoid further air inclusions. Once the target has been formed, the vacuum
should be broken and the oven switched off. The press will take an hour or

more to cool off by itself, but can also be cooled with water.

The targets produced by this procedure had all desired characteristics. Even
though tiny bubbles were visible to the eye, the overall inhomogeneity was better
than 10% of thickness, and the deterioration of spectrometer resolution as determined

with a beam of 13N at 105 MeV/nucleon was negligible.

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