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Electromagnetic Dissociation of 8B and the Rate
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presented by
Barry Samuel Davids

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6/01 cJCIRG/DateDuepss—sz

ELECTROMAGNETIC DISSOCIATION OF 8B AND THE RATE OF THE
7Be(p,7)8B REACTION IN THE SUN

By

Barry Samuel Davids

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2000

ABSTRACT

ELECTROMAGNETIC DISSOCIATION OF 8B AND THE RATE OF THE
7M 12,7)813 REACTION IN THE SUN

By

Barry Samuel Davids

In an effort to better determine the 7Be( 12,7)8B reaction rate, we have performed
inclusive and exclusive measurements of the Coulomb dissociation of 8B. The former
was a study of longitudinal momentum distributions of 7Be fragments emitted in the
Coulomb breakup of intermediate energy 8B beams on Pb and Ag targets. Analy-
sis of these data yielded the E2 contribution to the breakup cross section. In the
exclusive measurement, we determined the cross section for the Coulomb breakup
of 8B on Pb at low relative energies in order to infer the astrophysical 5' factor for

the 7Be(p,'7)8B reaction. Interpreting the measurements with Ist-order perturbation

theory, we obtained SEQ/5'51 = 4.7 fig x 10'4 at Ere; = 0.6 MeV, and 517(0) 2 17.8
it: eV b. Semiclassical Ist-order perturbation theory and fully quantum mechanical
continuum-discretized coupled channels analyses yield nearly identical results for the
E1 strength relevant to solar neutrino flux calculations, suggesting that theoretical
reaction mechanism uncertainties need not limit the precision of Coulomb breakup

determinations of the 7Be(p,7)SB 5' factor. A recommended value of 517(0) based on

a weighted average of this and other measurements is presented.

To my family

iii

ACKNOWLEDGMENTS

I would like to express my deepest thanks to my principal collaborators, Sam
M. Austin, Daniel Bazin, Henning Esbensen, and Brad Sherrill. Their knowledge,
intelligence, experience, patience, and equanimity made this work possible.

Henning Esbensen did all of the perturbation theory calculations presented in this
thesis that were not done by me, and the coupled channels calculations were carried
out by Jeff Tostevin and Ian Thompson. Takashi Nakamura contributed prodigiously
to the 2nd experiment.

Thanks go to all those who helped make these experiments successful: Don An-
thony, Tom Aumann, Thomas Baumann, Bertram Blank, Jac Caggiano, Marielle
Chartier, Ryan Clement, Cary Davids, Patrick Hui, Pat Lofy, Chris Powell, Heiko
Scheit, Mathias Steiner, Peter Thirolf, John Yurkon, Al Zeller, and the staff of the
NSCL.

CONTENTS

LIST OF TABLES
LIST OF FIGURES

1 Introduction

2 Experimental Procedures
2.1 Inclusive Measurements ..........................
2.2 Exclusive Measurement ..........................

3 Experimental Results and Analysis
3.1 Longitudinal Momentum Distributions .................

3.2 Theoretical Methods

3.3 Breakup Energy Spectrum ........................

4 Discussion

5 Summary

LIST OF REFERENCES

\J’

vi

vii

21
21
34
38

45
52

54

LIST OF TABLES

2.1 Total number of 8B nuclei on target ................... 7
3.1 Integrated Coulomb dissociation cross sections ............. 36

4.1 Comparison of theoretical E2 strength predictions with present results 46
4.2 Recent 317(0) determinations ...................... 49

vi

LIST OF FIGURES

1.1

2.1
2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

Fluxes of neutrinos (cm"2 3“) produced by different nuclear reactions
in the Sun predicted by the standard solar model of ref. [1]. The energy
ranges in which several neutrino detectors are sensitive are shown on
the top of the figure. ...........................

Schematic view of the S800 spectrometer at the NSCL. ........
Typical Si p-i-n diode versus time-of-flight spectrum illustrating the
secondary beam composition. ......................
Typical ionization chamber energy loss versus stopping scintillator to-
tal energy spectrum. ...........................
Schematic view of the experimental setup for the exclusive measure-
ment showing the detectors, typical trajectories, and contours of con—
stant magnetic field produced by the dipole magnet ...........
Spectrum of the differences between the times for voltage pulses to
reach each end of the delay line in a MWDC. Individual wire peaks are
plainly visible ................................
Spectrum of the sums of the times for voltage pulses induced by elec-
tron clouds to reach each end of the delay line in a MWDC. The max-
imum drift time corresponds to a distance of 4 mm ...........
Spectrum of the signals from one set of MWDC cathode field-shaping
wires versus those from the other set. The two groups correspond to
ions passing on the left of a given anode wire and ions passing on the
right .....................................
Typical proton scintillator bar energy loss versus time-of-flight spec-
trum. The events with small scintillator signals represent crosstalk
from adjacent scintillator bars .......................
Typical 7Be scintillator bar energy loss versus time—of-flight spectrum.
The events with small scintillator signals are due to light produced in
adjacent scintillator bars ..........................
Geometric efficiency for detecting protons and 7Be fragments in co—
incidence from the Coulomb dissociation of 83 MeV/ nucleon 8B with
impact parameters 2 30 frn. The relative errors shown are statistical
uncertainties from the simulation and theoretical uncertainties from
the size of the E2 component, added in quadrature. ..........

12

14

14

15

16

17

20

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Measured longitudinal momentum distributions of 7Be fragments from
the Coulomb dissociation of 44 MeV/nucleon 8B on Pb with several
maximum 7'Be scattering angle cuts. Also shown are lst-order pertur-
bation theory calculations convoluted with the experimental resolution.
See the text for details ...........................
Laboratory frame longitudinal momentum distributions of 7Be frag-
ments with maximum scattering angles of 2.5°, 2.0°, and 1.5° emitted
in the Coulomb dissociation of 44 MeV/nucleon 8B on Ag. The solid
curves represent an incoherent sum of the perturbative Coulomb disso—
ciation calculations and a nuclear component simulated by a gaussian
distribution (a = 35 MeV/c). The dashed curve is the perturbative
Coulomb dissociation calculation for the smallest angle cut .......
Product of Coulomb dissociation probability and impact parameter
squared for the breakup of 44 MeV/nucleon 8B on Pb with several
maximum 7Be scattering angle cuts as a function of impact parameter.
This is a measure of the relative 7Be detection probability, revealing
the impact parameter sensitivity of the various angle cuts. ......
Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb dissociation of 44 MeV/nucleon 8B on Ag
for several maximum 7Be scattering angle cuts as a function of impact
parameter. This is a measure of the relative 7Be detection probability,
revealing the impact parameter sensitivity of the various angle cuts. .
Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb breakup of 81 MeV/ nucleon 8B on Pb with
several maximum scattering angle cuts as a function of impact pa-
rameter. This is a measure of the relative 7Be detection probability,
revealing the impact parameter sensitivity of the various angle cuts. .
Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb dissociation of 81 MeV/nucleon 8B on Ag
with 7B6 scattering angles 3 125° as a function of impact parameter.
This is a measure of the relative 7Be detection probability, revealing
the impact parameter sensitivity of this angle cut ............
Laboratory frame longitudinal momentum distribution of 7Be frag-
ments with maximum scattering angles of 125° emitted in the Coulomb
dissociation of 81 MeV / nucleon 8B on Ag. The curve is a lst-order per-

22

24

26

27

28

29

turbation theory calculation convoluted with the experimental resolution. 30

Laboratory frame longitudinal momentum distribution of 7Be frag-
ments emitted in the Coulomb dissociation of 81 MeV/ nucleon 8B on
Pb with maximum scattering angles of 25°, 1.5°, and 1°. The solid
curves are continuum-discretized coupled channels calculations that
include both Coulomb and nuclear interactions, convoluted with the
experimental resolution. The dashed curve is a DWBA calculation for

@nlax : 205° ooooooooooooooooooooooooooooooooo

viii

3.9

3.10

3.11

3.12

3.13

4.1

4.2

Measured longitudinal momentum distribution of protons from the
Coulomb dissociation of 83 MeV/nucleon 8B on Pb with 8B scatter-
ing angles 3 1.77°. Only relative errors are shown. Also depicted are
Ist-order perturbation theory calculations with different E2 strengths,
convoluted with the experimental resolution ...............
Central region of the 35° angle cut of the 7Be longitudinal momentum
distribution from the breakup of 44 MeV/nucleon 8B on Pb. The
curves are calculations performed with different E2 matrix elements,
expressed in terms of the E2 amplitude of the model of [24], normalized
to the center of the distribution ......................
Contributions of E1, E2, and M1 transitions to the cross section for
the Coulomb dissociation of 83 MeV/ nucleon 8B on Pb with 8B scat—
tering angles S 1.77° in Ist-order perturbation theory. The M 1 cross
section is calculated by folding the M 1 5 factor measured in ref. [8]
with the virtual photon spectrum. The E1 and E2 cross sections are
calculated using the model of ref. [24], scaling the E2 matrix elements
by the factor 0.7 required to reproduce the measured 7Be longitudinal
momentum distributions ..........................
Fraction of the calculated cross section for the Coulomb dissociation of
83 MeV/nucleon 8B on Pb with 8B scattering angles 3 1.77° (b 2 30 fm)
accounted for by El transitions in lst-order perturbation theory. As
the energy falls below 130 keV, E2 transitions become increasingly
important ..................................
Measured differential cross section for the Coulomb dissociation of 83
MeV/nucleon 8B on Pb with 8B scattering angles 5 1.77°. Only rela-
tive errors are shown. Also depicted are continuum-discretized coupled
channels and two lst-order perturbation theory calculations, convo—
luted with the experimental resolution. The point at 64 keV has been

35

40

41

excluded from the fits because E2 transitions are dominant at this energy. 43

Difference between the cross section for Coulomb breakup of 83 MeV / nucleon

8B on Pb for 8B scattering angles of 1.77° and less predicted by DWBA
(lst—order) and CDCC (all orders) calculations using the same struc-
ture model, expressed as a fraction of the DWBA prediction. Only
Coulomb matrix elements were included in these calculations. No sig-
nificant energy dependence of the higher-order electromagnetic effects
is evident. .................................
Inferred zero-energy astrophysical 5 factors for the 7Be( p,7)8B reaction
from recent direct and indirect measurements. .............

ix

Chapter 1

Introduction

The Sun is a source of energy upon which all life on Earth relies. The origin of
this energy was a mystery until the late 19303, when discoveries in nuclear physics
made it possible for Hans Bethe to suggest that a series of thermonuclear reactions
fusing hydrogen into helium caused the sun to shine. Nearly 30 years elapsed before
this hypothesis was supported by the detection of neutrinos produced in these nuclear
reactions here on Earth. The light created deep in the solar interior follows a tortuous
path, scattering many times before emerging from the surface some 10 million years
after it was produced. For this reason, it carries little information about the center of
the Sun. On the other hand, the neutrinos copiously produced in the nuclear reactions
powering the Sun stream outward freely, rarely interacting with the matter in the solar
interior. By detecting these neutrinos on Earth, we directly probe conditions deep in
the Sun.

As shown in Fig. 1.1, the flux of neutrinos emanating from the solar interior
consists predominantly of low energy electron neutrinos from the p + p —> d + 6+ +
l/e reaction [1]. The higher energy 8B neutrinos, though they constitute less than
10'4 of the total solar neutrino flux, are the best studied. Their flux, direction,

and energy spectrum have been measured in large chlorine radiochemical and water

Cerenkov detectors. Less than half the number of 8B neutrinos expected on the basis
of standard solar models and standard electroweak particle physics has been observed
in terrestrial detectors [2], a situation that has come to be known as the 8B solar
neutrino problem. This discrepancy between theory and experiment appears to be
resolved best by invoking oscillations of 14. into other neutrino flavors. By measuring
the ratio of charged current to neutral current interactions, the heavy water detector
SN 0 will stringently test neutrino oscillation hypotheses. In order to calculate the
theoretical solar neutrino flux and interpret the results of measurements at SNO and
other neutrino detectors, the rate of the radiative capture reaction that produces 8B
in the Sun, 7Be(p,‘y)8B, must be known to a precision of 5% [1]. Thus astrophysics,
nuclear physics, and particle physics meet in addressing the solar neutrino problem.

The cross section for the 7Be(p,7)8B reaction has been measured directly in sev-
eral experiments [3, 4, 5, 6, 7, 8, 9]. Although the shape of the excitation function is
well determined, there is a large spread in the absolute normalizations of these mea-
surements. The cross section must be known at the very low relative energies (~ 20
keV) relevant to 8B production in the Sun, but these low energies are experimentally
inaccessible because the high Coulomb barrier causes the cross section to plummet
with decreasing energy. The strategy adopted is to measure the cross section at the
lowest possible energy, and then extrapolate downward using theory. In order to
extrapolate to low energies reliably, the dominant energy dependences in the cross
section can be factored out, leaving a quantity known as the astrophysical 5' factor,

which varies much more slowly with energy. The 5 factor is defined by
5(E) = E 0(E) €.T])[27T2122€2/(hv)], (1.1)

where the Z,- are the charges, v the relative velocity, and E the center-of—mass energy
of the nuclei involved. Conventionally, the value of the 5 factor for the 7Be(p,'7)8B

reaction, 517, is extrapolated from the data at accessible energies to zero energy.

SuperK. SNO

 

 

 

 

 

 

 

 

.Gallium :Chmrine l——‘:
101- . ' r . . -. '
Bahcall
10” m
10"}
N
:3 10' r
H
Eh
10' r
0 ’Be ’Be
.E 10';
33 .
5 10 f
0 t
z 10. f
10‘ i/’
10'
0.;
' . 1 . . J .
10 in 0.3 1 3 10

Neutrino Energy (MeV)

Figure 1.1: Fluxes of neutrinos (cm’2 3“) produced by different nuclear reactions in
the Sun predicted by the standard solar model of ref. [1]. The energy ranges in which

several neutrino detectors are sensitive are shown on the top of the figure.

In light of the disagreements among the radiative capture measurements of the
7Be(p,"/)8B cross section, and the fact that direct measurements at very low energies
are impractical, indirect techniques have been developed to infer this radiative cap~
ture cross section. Such techniques are subject to different systematic uncertainties.
For photons of a given multipolarity, the detailed balance theorem relates the cross
section for radiative capture to that for the corresponding inverse reaction, photodis-
sociation. In the case of the 7Be(p,~y)8B reaction, the 8B nucleus is radioactive with
a half life of 770 ms and is not a viable photodissociation target. However, when an
energetic beam of 8B nuclei passes through a heavy target, the time-dependent elec-
tromagnetic field of the high Z target nuclei acts as a source of virtual photons capable
of dissociating the incident 8B projectiles into 7Be + p [10]. This process, known as
Coulomb dissociation, is Coulomb excitation to the continuum. The semiclassical
formalism of Coulomb excitation has been extended to Coulomb dissociation at in-
termediate and high energies [11, 12, 13]. The advantages of high energy Coulomb
dissociation over direct radiative capture measurements include thicker targets and
larger cross sections, and thereby the possibility of reaching lower relative energies
with an appreciable yield.

Coulomb dissociation has been used to infer 517(0) [14, 15, 16], but the method
is not without complications. First, several electromagnetic multipoles contribute in
Coulomb dissociation, e.g., E1, E2, and M1, while the radiative capture reaction
is mainly driven by a single electromagnetic multipole transition at solar energies,
E1. Second, even though the electromagnetic interaction dominates, the effects of
nuclear absorption and diffraction must be considered. Finally, one must consider
the effects of higher-order electromagnetic transitions that can destroy the simple
correspondence between radiative capture and Coulomb dissociation. An important

experimental challenge is to identify and understand these complications in order

to firmly establish Coulomb dissociation as a viable alternative to direct radiative
capture measurements. In this thesis, I will show that these complications can be dealt
with in a satisfactory manner by a judicious choice of the experimental conditions
and by applying tested nuclear structure and reaction theories.

The first challenge, that of disentangling the contributions of different electro-
magnetic multipoles to extract one in particular, can be met by carefully studying
the angular distribution of the breakup fragments. In the Coulomb dissociation of
intermediate energy 8B, E2 is the principal unwanted electromagnetic multipole. By
carrying out an inclusive measurement of the 7Be fragments emitted in the Coulomb
dissociation of 44 and 81 MeV/ nucleon 8B, we determined the E2 contribution to the
breakup cross section.

Having measured the E2 strength in the Coulomb breakup, we were in a posi-
tion to study the breakup energy spectrum, da/dEreg, in order to determine the E1
strength at low relative energies and thereby infer 517(0). This was done in a exclusive
measurement with an 83 MeV/ nucleon 8B beam. In the analysis of this experiment,
we used the E2 strength determined in the inclusive measurement, and investigated
the influence of nuclear and higher-order electromagnetic processes. We discovered
that these complications could be minimized, and the theoretical uncertainties made
small enough that our Coulomb breakup measurement is of comparable precision to
the direct radiative capture measurements. In this thesis, I shall describe the inclusive
and exclusive measurements, and interpret them using both Ist-order perturbation
theory and a continuum-discretized coupled channels approach. Finally I will com-

pare the inferred value of 517(0) with recent direct and indirect measurements.

Chapter 2

Experimental Procedures

2.1 Inclusive Measurements

We bombarded a 1.9 g cm‘2 Be production target with 100 and 125 MeV/ nucleon
l2C beams from the K1200 cyclotron at the National Superconducting Cyclotron
Laboratory (NSCL). Typical l2C beam intensities were 10-50 pnA. Fragmentation
reactions in the Be target yielded secondary beams of 44 and 81 MeV/nucleon 8B,
after magnetic analysis in the A1200 fragment separator [17]. A 200 mg cm’2 (CH2)n
achromatic energy degrader aided in the purification of the secondary beams. Slits
limited the momentum spread of the secondary beams to :l: 0.25%. A 17 mg cm“2
plastic scintillator just downstream of the A1200 focal plane provided time-of—flight
and secondary beam intensity information. Typical 8B beam intensities ranged from
(4-20) x 103 s". Table 2.1 shows the total number of 8B nuclei that struck each
target.

The 8B beams were transported through a second beam analysis line to the target
position of the S800 spectrometer, as shown schematically in Fig. 2.1. This analysis

line dispersed the secondary beams according to their momenta, resulting in a 5 cm

x 1 cm beam spot on the targets. A ladder held 27 mg cm‘2 Ag and 28 mg cm’2 Pb

Table 2.1: Total number of 8B nuclei on target

Target Beam Energy 8B on Target
(MeV/nucleon) (106)

 

 

 

 

 

Ag 44 360
Ag 81 1 070
Pb 44 840
Pb 81 2 980

targets. A 300 pm Si p-i-n diode detector mounted on a ladder 18 cm upstream of the
targets was intermittently raised into the path of the beam. Energy loss signals from
this detector, in conjunction with timing signals from the plastic scintillator at the exit
of the A1200, yielded both the transmission and composition of the secondary beams.
Times-of-flight were measured for the ~ 70 m flight path between the scintillator at
the exit of the A1200 and the S800 focal plane. Fig. 2.2 shows a typical plot of the
signals in the p-i-n diode detector versus time-of-flight.

The secondary beams were not monoisotopic; 7Be was the principal contaminant,
and was 5-8 times more intense than the 8B component of the beam. Two other nuclei,
6Li and 9C, were also present in the beam. As the energies of these contaminants
differed substantially from that of the 8B ions, their different times-of—flight provided
reliable particle identification.

We used the S800 spectrometer [18] to detect the 7Be fragments emitted in the
Coulomb dissociation of 8B nuclei on the Ag and Pb targets. The spectrometer was
set at 0°, and was operated in a dispersion—matched energy loss mode so that the
0.5% spread in the momentum of the 8B beams did not limit the final momentum

resolution, which was dominated by differential energy loss in the target. The large

Obfect Quadrupole D oubl et [\fDetectors

 
  

+ ' ' i, Q Intermediate Image D1po 1e - D2
Production
Target %
Sextupole Dipoles )‘\
@im %
Quadrupole—yam
Triplet Dipole- D1
Target Chamber/
I-—-|
Quadrupole 2 meters
Doublet

Figure 2.1: Schematic View of the S800 spectrometer at the NSCL.

300

 

250 - i

200; : ‘

Si p-i-n Energy Loss (arbitrary units)

 

 

 

-, 8B
150 — 5
100 — i '7
" Be
6Li
50 l l l
250 300 350 400 450

‘__ Time-of-Flight (arbitrary units)

Figure 2.2: Typical Si p-i-n diode versus time-of—flight spectrum illustrating the sec-

ondary beam composition.

angular acceptance (20 msr) and momentum acceptance (6%) of the 5800 allowed
us to capture essentially the entire momentum distribution at a single magnetic field
setting.

The standard complement of detectors at the focal plane of the S800 spectrometer
comprises two position-sensitive cathode readout drift chambers (CRDCS), a 41 cm
deep, 16 segment ionization chamber, and three plastic scintillators. The CRDCs
are separated by 1 meter to give good angular resolution. Ref. [19] describes these
detectors in detail. The ionization chamber recorded the energy losses, and the first, 5
cm thick scintillator measured the total energies of particles reaching the focal plane.
This information was sufficient to identify the 7Be breakup fragments unambiguously,
as illustrated in Fig. 2.3; the time-of-flight data provided a check. As the nuclei of
interest were stopped in the first scintillator, the other two were not used. The particle
identification was confirmed through comparisons with calibration beams of 7'Be that
had the same magnetic rigidity as the 8B beams. The higher magnetic rigidity of
the detected 7Be fragments compared to the 8B beams made the focal plane particle
identification particularly clean.

The positions and angles of the 7Be fragments were measured in the CRDCs. The
position resolution obtained was approximately 0.3 mm (10), yielding an intrinsic an-
gular resolution of about 2 mrad. We employed the ion optics code COSY INFINITY
[20] to reconstruct the trajectories of the 7Be fragments from their measured positions
in the CRDCS, and the magnetic fields of the spectrometer, which were continuously
monitored by nuclear magnetic resonance probes throughout the experiment. We
calculated the 7Be lab momenta and scattering angles on an event-by-event basis,
allowing reconstruction of the longitudinal momentum distributions.

Corrections to the momentum distributions were made for two different effects.

First, the overall efficiency of the CRDCs was less than unity due to a high threshold

 

200

 

180 -

Breakup 7Be

140 -

 

 

 

 

Ion Chamber Energy Loss (arbitrary units)

0 V l 100 I 200 ' 300 400
Scintillator Total Energy (arbitrary units)

Figure 2.3: Typical ionization chamber energy loss versus stopping scintillator total

energy spectrum.

on the anode wire constant fraction discriminator. This was a small correction in the
case of the low energy beam (< 3%), but larger for the high energy beam (< 15%).
The second was a momentum-dependent correction for the angular acceptance of the
S800, which was important for events having large deviations from the central mo—
mentum and large projections of the scattering angle in the dispersive direction of the
spectrometer. These corrections affected only the tails of the measured momentum
distributions, and amounted to less than 5% of the measured cross sections, even for
the largest scattering angles. Corrections were made on the basis of the data them—
selves by observations of the acceptance limits. Uncertainties equal to half the size of
the corrections were assigned to the data points in the momentum distributions that
required correction. During some runs, the magnetic field of the spectrometer was
varied to move the center of the distribution away from the center of the focal plane

detectors in order to measure the tails of the momentum distributions precisely. The

10

final momentum distributions represent the sums of measurements made at several

different magnetic field settings.

2.2 Exclusive Measurement

The 83 MeV/ nucleon 8B beam used in the exclusive measurement was produced with
a 125 MeV / nucleon primary l2C beam in the same manner described above. Typical
8B beam intensities were 104 s", and the momentum spread in the beam was limited
to i 0.25% by slits in the A1200 fragment separator. A total of 4 x 109 8B nuclei
were incident on the target. A thin plastic scintillator was placed at the exit of
the A1200 fragment separator for beam intensity, transmission, and time-of-flight
measurements. The 8B nuclei were dissociated in a 47 mg cm'2 Pb target located in
front of a room temperature 1.5 T dipole magnet. Four position-sensitive multiwire
drift chambers (MWDCs) [21] recorded the positions of the 7Be and p fragments
produced in the breakup after they passed through the magnetic field. Two MWDCs
measured each breakup fragment, allowing the determination of both position and
angle. A 16 element array of 4 cm thick plastic scintillator bars was placed behind
the MWDCs. A 25 mm x 60 mm stainless steel plate located directly in front of
the first 7Be MWDC absorbed nearly all of the direct beam. The composition of the
secondary beam in the exclusive measurement was roughly 20% 6Li, 55% 7Be, 20%
8B, and 5% 9C. Fig. 2.4 shows a schematic drawing of the experimental setup.

The multiwire drift chambers used in this experiment have active areas of 112 mm
x 112 mm, and use delay-line readout to measure the positions of particle tracks. The
chambers were filled with P30 (70% argon, 30% methane) at a pressure of 700 torr.
Each MWDC has two orthogonal wire planes, providing both x and y positions.
The positions are deduced as follows. The signals from time-to—digital converters

connected to each end of the delay line are subtracted, yielding a spectrum such as

11

 

Plastic Scintillators

._ CID
2° Era-DD

Multiwire Drift Chambers 5" x

1.5 — 3'. '

 

 

N ,’
1.0 — I
Dipole st . .. g!
Magnet "1' g :3 ° B (T)

I

 

0.5 —

 

Beam Direction

f I I l
0.0 0.2 0.4 0.6 0.8 1.0

x (m)

 

 

 

Figure 2.4: Schematic view of the experimental setup for the exclusive measurement
showing the detectors, typical trajectories, and contours of constant magnetic field

produced by the dipole magnet.

12

that shown in Fig. 2.5. The peaks in this spectrum correspond to individual anode
wires, which are separated by 8 mm. Drift time information is used to interpolate
between anode wires using current pulses induced on the cathode field-shaping wires.
The drift time is obtained from the sum of the delay line time signals; an example
of such a spectrum is shown in Fig. 2.6. This spectrum does not show a uniform
distribution of drift times. By integrating such spectra with respect to time and
fitting the result with a polynomial, one obtains calibrated curves of drift distance as
a function of drift time. These time signals do not reveal on which side of an anode
wire an ion passed. The charge collected on the cathode field-shaping wires resolves
this ambiguity. Alternate cathode wires are bussed together, ‘so that ions nearer to
the left side of anode wires produce larger signals in one set of cathode wires than in
the other. Fig. 2.7 depicts a typical spectrum of the signals from one set of cathode
wires versus the signals from the other set. All of the ions passing on the left of an
anode wire are in one group, and those passing on the right fall in the other group.
The principles of operation and geometry of these detectors are described in detail
in ref. [21]. Position resolutions of 0.4 mm (10) were obtained for protons and 7B6
fragments.

Particle identification was achieved through measurements of energy loss in the
plastic scintillator array and time-of-flight between the exit of the A1200 and the
scintillator array. The geometric average of signals from photomultiplier tubes on the
top and bottom of each scintillator bar served as a measure of particle energy loss.
Since the scintillator array was not sufficiently thick to stop the breakup fragments,
direct total energy measurements were not possible, and the time-of-flight measure—
ment was crucial for ion identification. Calibration beams of 7Be and p having the
same magnetic rigidity as the 8B beam were used to confirm the particle identification.

The protons and 7Be struck widely separated scintillator bars, allowing optimization

13

 

100 I I I I I I

_‘ 30.. ..
O

c:

E 60— -
3 l

«‘3 40- —
:3

O

U

 

 

 

 

 

 

 

 

 

 

 

 

20— 1‘ —
O l. -1 L i 1 i

600 700 800 .9001 10001100 1200 1300
MWDC Anode Wire Delay Line Time Difference (channel)

Figure 2.5: Spectrum of the differences between the times for voltage pulses to reach

each end of the delay line in a MWDC. Individual wire peaks are plainly visible.

 

 

 

 

 

40 — -
“E:
5 30 — -
.G
U
3 20 — ‘
t:
:1
O
U 10 .. _.
O I l l
0 1000 2000 3000 4000

MWDC Anode Wire Delay Line Time Sum (channel)

Figure 2.6: Spectrum of the sums of the times for voltage pulses induced by electron
clouds to reach each end of the delay line in a MWDC. The maximum drift time

corresponds to a distance of 4 mm.

14

 

 

 

i 200 - '
5 -.-;I ‘
.c: .
3 :-, ..'
.‘i-I’ "'
3 100 _
d)
8
f“:
a 50 —
3
l l L l J
50 100 150 200 250

Right Cathode Wire Charge (channel)

Figure 2.7: Spectrum of the signals from one set of MWDC cathode field-shaping
wires versus those from the other set. The two groups correspond to ions passing on

the left of a given anode wire and ions passing on the right.

15

60

 

a

:3 50

E Breakup Protons

.5.

g 40‘ . I

E . I I

>3 30-

is" .

G I

m 20-

V)

E

:2 10—

.‘E

E .' . II 'II'. :I'I.
0 I ---

 

 

 

I I I I
150 200 250 300 350 400 450
4— Time-of-Flight (arbitrary units)

Figure 2.8: Typical proton scintillator bar energy loss versus time—of-flight spectrum.
The events with small scintillator signals represent crosstalk from adjacent scintillator

bars.

of the individual bar electronics for the appropriate fragment energy losses. Fig. 2.8
shows the scintillator energy loss versus time-of-flight spectrum for a scintillator bar
that detected protons, while Fig. 2.9 shows that for a bar used to detect 7Be frag—
ments. These spectra are gated, requiring a good position signal in at least one proton
MWDC plane and one 7Be MWDC plane.

We reconstructed the 4-momenta of the breakup fragments from the measured
positions in all 8 MWDC planes and the magnetic field using the ion optics code
COSY INFINITY [22]. The magnetic field was measured with a Hall probe at 2184
points in each of 4 planes in the gap of the dipole magnet to a precision of :1: 2
mG [23]. Second-order Taylor series expansions about a reference trajectory were

employed, and the trajectory reconstruction was checked through the use of proton

16

 

 

 

 

E 200

:1 7

E Breakup Be

I:

'~ 150- "P

g .

.8.

>» 100—

2"

Q)

1::

LL]

('1

2 50-

.9

.59.

E

'E . *— 0 ' :fl' 2'.“

c2 0 I I "T I r I
150 200 250 300 350 400 450

<_Time-of-Flight (arbitrary units)

Figure 2.9: Typical 7Be scintillator bar energy loss versus time-of-flight spectrum. The
events with small scintillator signals are due to light produced in adjacent scintillator

bars.

17

and 7Be calibration beams. The invariant mass method was used to calculate the

relative energy of the fragments according to

Ere, : t/E2 — p2 —- mike2 — mpcz, (2.1)

where E is the total relativistic energy, and p the total momentum in the laboratory

frame. The energy and momentum are defined by
E = 7387713662 + 7pmpc2, (2.2)

and
P = VBemBeVBe + 7pmpr- (2-3)

The relative energy is simply the kinetic energy in the center-of-mass reference frame.
We obtained a relative energy resolution of 55 keV (10) at Ere; = 100 keV; the energy
resolution increased for higher relative energies approximately as m. The small
separation between the lst and 2nd MWDCs, combined with the MWDC position
resolution, caused the angular resolution to limit the relative energy resolution. This
small distance was necessitated by the requirement that the first detector be far
enough away from the magnet that there be adequate separation between the 7Be
fragments and the 8B beam, and by the limitations of an existing vacuum chamber.
Other contributions to the relative energy resolution included energy loss and multiple
scattering in the 47 mg cm‘2 target and the MWDCs, each of which was 30 mg cm“2
thick.

The resolution and efficiency of the experimental apparatus were determined
through a Monte Carlo simulation. The inputs to the simulation included the beam
emittance (6 mm beam spot diameter, :1: 6 mrad in the dispersive direction of the
magnet, :E 9 mrad in the non-dispersive direction) and the measured detector position

resolution. The beam emittance was measured by reducing the magnetic field, caus-

ing the 8B beam to miss the beam blocker and be detected in the MWDCs, while the

18

detector position resolution was determined through the use of a mask. The Monte
Carlo simulation was also used to calculate the small fraction of the 7Be breakup frag—
ments that were intercepted by the beam blocker. The FORTRAN source code of the
Monte Carlo simulation is available at http://www.nscl.msu.edu/~sherrill/davids.

In order to evaluate the geometric efficiency of the setup, we employed a model
for the breakup of 8B that includes both E1 and E2 transition amplitudes, which
have different distributions in 983, the laboratory scattering angle of the excited 8B.
To account for the E1-E2 interference observed in the asymmetry of the longitu-
dinal momentum distribution of 7B6 fragments, we included an anisotropic angular
distribution of the breakup fragments in the excited 8B rest frame. The shape of
this distribution is similar to those shown in Fig. 9 of ref. [24], but was empirically
adjusted to reproduce the longitudinal momentum distribution of protons measured
in this experiment, which will be discussed in section 3.1. The E1 and E2 dissocia—
tion probabilities were taken from the model of ref. [24], after scaling the E2 matrix
elements by the factor 0.7. This quenching of the E2 amplitudes, required for the
best fit of the inclusive data, is discussed in more detail below. We gauged the model-
dependence of the efficiency determination by also computing the efficiency using the
same model without E2 transitions. The difference between the computed efficien-
cies with and without E2 transitions was less than 5% for the angular and relative
energy ranges covered in the experiment. This difference was used as the theoretical
uncertainty in the efficiency determination.

Since both E2 transitions and nuclear absorption and diffraction effects are rela-
tively more important at small impact parameters than at large ones, we imposed a
impact parameter cutoff at 30 fm. For 83 MeV/ nucleon 8B on Pb, this corresponds
classically to 983 = 1.77°. In practice, 933 was determined from the reconstructed

total laboratory momentum vector, and the 10 resolution of this quantity was 4.5

19

 

 

 

 

: q; I I I
3 " -I
2 - Q -
,2 e
o e
'8 {D
L: g- m ..
s .. ° a -
8 4~ “3 § -
8 E
. a} ..
O 3 i m
2' l
0.01 I ' '
0.0 0.5 1.0 1.5 2.0
Relative Energy (MeV)

Figure 2.10: Geometric efficiency for detecting protons and 7Be fragments in coinci-
dence from the Coulomb dissociation of 83 MeV/ nucleon 8B with impact parameters
2 30 frn. The relative errors shown are statistical uncertainties from the simulation

and theoretical uncertainties from the size of the E2 component, added in quadrature.

mrad. The geometric efficiency for detecting 8B breakups with b 2 30 fm is shown
in Fig. 2.10. The efficiency falls off rapidly with increasing relative energy, primarily
due to the small solid angle subtended by the proton MWDCs. As the goal of the
experiment was to determine the Coulomb dissociation cross section at low relative
energies, the experimental arrangement was most sensitive to the events of interest.
The intrinsic detection efficiency, i.e., the probability that all 8 MWDC planes pro-
vided good position signals when the breakup fragments passed through them, was

measured to be 0.414 :1: 0.008 using the scintillator array.

20

Chapter 3

Experimental Results and Analysis

3.1 Longitudinal Momentum Distributions

Measured laboratory frame longitudinal momentum distributions of 7Be fragments
from the Coulomb breakup of 44 MeV/ nucleon 8B on a Pb target are shown in Fig.
3.1. The momentum resolution obtained was 5 MeV/c, and the error bars indicate
the relative uncertainties of the data points, which are dominated by statistical errors.
The systematic uncertainty in the measured cross section due to target thickness and
beam intensity was :1: 9%. This systematic uncertainty is common to all of the 7Be
momentum distribution measurements. Fig. 3.1 also includes the results of lst-order
perturbation theory calculations performed using a modified version of the model of
ref. [24]. Both the overall normalization and the E2 matrix elements of this calculation
have been scaled, the former by 1.22 and the latter by 0.7. We shall return to this
point later.

To investigate any possible dependence of higher-order electromagnetic effects on
target charge, we also made inclusive measurements with an Ag target. The measured
longitudinal momentum distributions of 7Be fragments produced in the dissociation

of 44 MeV/nucleon 8B on Ag are shown in Fig. 3.2 for several different maximum

21

 

‘I

 

4 I I , I I
44 MeV/nucleon

 

 

 

 

 

 

do/dpz (mb/(MeV/c))

 

 

 

1950 12100

 

7Be Longitudinal Momentum (MeV/c)

Figure 3.1: Measured longitudinal momentum distributions of 7Be fragments from
the Coulomb dissociation of 44 MeV/ nucleon 8B on Pb with several maximum 7Be
scattering angle cuts. Also shown are lst-order perturbation theory calculations con-

voluted with the experimental resolution. See the text for details.

[\3
[\D

7Be scattering angle cuts. The agreement with the lst-order perturbation theory cal—
culations done with the model of ref. [24] (shown here only for Om” = l.5°) is not
as good as it is with the Pb target. In particular, the magnitude and width of the
calculations are insufficient to describe the data. Suspecting that nuclear processes
not accounted for in the Coulomb dissociation calculation were responsible for this
discord, we simulated a nuclear contribution to the breakup momentum distribution
with a gaussian distribution having a width (0) of 35 MeV/c. This parametrization
was motivated by the measurement of ref. [25] of the nuclear-induced breakup of 41
MeV/ nucleon 8B on a Be target. Arbitrarily scaling the normalization of the gaus-
sian, we added this simulated nuclear contribution to lst-order perturbation theory
calculations having the same E2 matrix element scaling and overall normalization as
the 44 MeV/ nucleon Pb target calculations in order to minimize X2 for the central
6 points of the momentum distributions. The resulting curves are shown in Fig. 3.2.
The point of this exercise was not to extract the nuclear—induced breakup compo—
nent, but simply to see if the data could be described by such an incoherent sum of
nuclear and Coulomb components. The agreement between these calculations and
the measurements is fairly good. The difference between the Coulomb dissociation
calculations and the data increased with maximum scattering angle, consistent with
an increasing relative importance of nuclear-induced breakup. The breakup of 8B on
Ag can be studied with the CDCC method, but these results are outside the scope of
this thesis, and will be presented elsewhere.

Placing different cuts on the angles of the emitted 7Be fragments allows one to
probe different impact parameters. However, a maximum 7Be scattering angle does
not correspond to a fixed minimum impact parameter, because the breakup energy
and the angle of the emitted proton are not determined in the inclusive measurement.

The sensitivities of the various angular cuts of the 44 MeV/nucleon longitudinal mo—

2.0 , ,

 

    

 

 

Ag Target
15 _ 44 MeV/nucleon

 

 

 

 

1.0

,¢-.
......
o

do/dpz (mb/(MeV/c))

0.5

 

 

 

/ -- ,.. I "A " I. ‘
1950 2000 2050 2100

7Be Longitudinal Momentum (MeV/c)

Figure 3.2: Laboratory frame longitudinal momentum distributions of 7Be fragments
with maximum scattering angles of 25°, 20°, and 1.5° emitted in the Coulomb dis-
sociation of 44 MeV/nucleon 8B on Ag. The solid curves represent an incoherent
sum of the perturbative Coulomb dissociation calculations and a nuclear component
simulated by a gaussian distribution (a = 35 MeV/c). The dashed curve is the

perturbative Coulomb dissociation calculation for the smallest angle cut.

24

 

mentum distributions to different impact parameters are shown in Fig. 3.3 for the
Pb target and Fig. 3.4 for the Ag target. These curves are a measure of the relative
probability that 7Be fragments emitted in Coulomb breakups at various impact pa-
rameters will fall within specified angular cuts. The corresponding relative detection
probability distributions as a function of impact parameter for the 81 MeV/ nucleon
beam are shown in Fig. 3.5 and Fig. 3.6. All of these calculations were performed
using the model of ref. [24], with the E2 matrix elements scaled by 0.7 [26]. A compar-
ison of the figures reveals that the Ag distributions probe smaller impact parameters
than the Pb distributions, indicating that nuclear absorption and diffraction should
play a larger role for the Ag target.

Fig. 3.7 shows the 7B6 longitudinal momentum distribution for the 81 MeV / nucleon
8B beam on Ag with a maximum 7Be scattering angle cut of 1.25°. The curve is a
lst-order perturbation theory calculation done with the model of ref. [24] with E2
matrix elements scaled by 0.7. The overall normalization of this calculation has not
been altered. The perturbative calculation describes the data fairly well, with the
most important discrepancy being the greater width of the measured distribution. It
is possible that nuclear absorption and diffraction not accounted for in the Coulomb
dissociation calculation broaden the measured distribution beyond the predicted ex-
tent.

The inclusive 7Be longitudinal momentum distributions measured at 81 MeV / nucleon
with the Pb target are depicted in Fig. 3.8 for 7Be scattering angle cuts of 25°, 20°,
and 1.5°. Also shown here are CDCC calculations convoluted with the experimental
resolution of 5 MeV/c. The CDCC calculations describe the data reasonably well,
accurately reproducing the slopes of the central regions of the momentum distribu-
tions, particularly for the largest angle cut. These calculations are not fits, but rather

are absolute predictions based on the assumed structure model; the E1 and E2 ma-

25

 

10

 

 

 

 

 

 

 

 

 

 

8:‘._T"""r1rrvr1"‘r:
6' — 1.5 deg -
4: ------ 2.4deg :
'. --- 3.5 deg .
[ X ‘ ‘t - no angle cut
NE 2_ I, “\ «J
“a .' """"" ‘~ m,‘ Pb Target
3 l — 1' ‘\ 44 MeV/nucleon _
.59. 8: I :
a? 6: I
(:0 4: :
2-
0.1 ‘

 

 

0 50 100 150 200
b (fm)

Figure 3.3: Product of Coulomb dissociation probability and impact parameter
squared for the breakup of 44 MeV/ nucleon 8B on Pb with several maximum 7Be
scattering angle cuts as a function of impact parameter. This is a measure of the
relative 7Be detection probability, revealing the impact parameter sensitivity of the

various angle cuts.

26

 

 

 

 

 

 

 

 

 
 
 

- '. — 1.5 deg a
2 b '- . ------ 2.0 deg _
. - - - 2.5 deg
' _ - no angle cut

NA 1 L- , " ~ \ -_

E 3E ,’ “Elli?“ Ag Target 2

g 6; ,' "in...“ 44 MeV/nucleon :

“is 4 : I ,' ‘ :
".0

    

 

 

 

 

8 53’
6E 11: L l I I l l l L l l l
0 50 100 150 200

b (frn)

Figure 3.4: Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb dissociation of 44 MeV/ nucleon 8B on Ag for several maxi-
mum 7Be scattering angle cuts as a function of impact parameter. This is a measure

of the relative 7Be detection probability, revealing the impact parameter sensitivity

of the various angle cuts.

27

 

 

 

 

 

 

 

    
  

 

4: — 1.5 deg j
' . _ ------ 2.5 deg
’ , .-, - - - 3 deg ‘

2 . ,' ”15.5 - no angle cut .
Nag ," \\ Pb Target
X 18 ' 81 MeV/nucleon
e ,
as? 6
ND 4

 

 

 

 

0.1 , l l l l I l l l l I
0 50 100 150 200
b (fm)

Figure 3.5: Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb breakup of 81 MeV/ nucleon 8B on Pb with several maxi-
mum scattering angle cuts as a function of impact parameter. This is a measure of
the relative 7Be detection probability, revealing the impact parameter sensitivity of

the various angle cuts.

28

 

 

 

 

 

 

 

 

 

 

 

 

 

2 I I M I I I I I I I l I T I I I I l I
— 1.25 deg
I F- - no angle cut -:
A 6f Ag Target 2
NE St 81 MeV/nucleon .
“CL 4-
e. 3-
mm 2"
.D
0.1 :-
L
g.
0 50 100 150 200

b (fm)

Figure 3.6: Product of Coulomb dissociation probability and impact parameter
squared for the Coulomb dissociation of 81 MeV/nucleon 8B on Ag with 7Be scat-
tering angles 3 1.25° as a function of impact parameter. This is a measure of the
relative 7Be detection probability, revealing the impact parameter sensitivity of this

angle cut.

 

08 _ r l l l

(amax = 1.25°

   

 

 

 

 

Ag Target
81 MeV/nucleon

 

 

 

0.6 b

0.4

0.2

do/dpz (mb/(MeV/c))

 

1
2700 2750 2800 2850

7Be Longitudinal Momentum (MeV/c)

 

 

0.0

Figure 3.7: Laboratory frame longitudinal momentum distribution of 7Be fragments
with maximum scattering angles of 1.25° emitted in the Coulomb dissociation of 81
MeV/nucleon 8B on Ag. The curve is a lst-order perturbation theory calculation

convoluted with the experimental resolution.

30

 

l l l I f l l l

 

 

 

 

 

 

 

O 9111;111:250 ............... Pb Target

A 3 — [:1 (~) -- 81 MeV/nucleon _
3? <> 9 .
>
d.)
E 2
B .—
ii
a?
E
8 1—

m ....

 

K __

0 -,.s. ' ' ;~
2700 272 2740 2760 2780 2800 2820 2840

 

 

 

7Be Longitudinal Momentum (MeV/c)

Figure 3.8: Laboratory frame longitudinal momentum distribution of 7Be fragments
emitted in the Coulomb dissociation of 81 MeV/nucleon 8B on Pb with maximum
scattering angles of 25°, 1.5°, and 1°. The solid curves are continuum-discretized
coupled channels calculations that include both Coulomb and nuclear interactions,
convoluted with the experimental resolution. The dashed curve is a DWBA calcula-

tion for Om“. = 25°.

trix elements have not been scaled in the CDCC calculations. The dashed curve is
a distorted wave Born approximation (DWBA) calculation for the largest angle cut
that assumes the same structure model and interactions as the CDCC calculation.
The difference between the lst-order DWBA and the CDCC calculations reflects the
influence of higher-order processes, which tend to reduce the effective E2 strength
needed in the lst-order calculation. As is the case for the 81 MeV/ nucleon Ag data,
the calculations predict distributions narrower than were measured. The difference in
magnitude between the calculations and the data for the smaller angle cuts is within

the error due to the angular uncertainty of 025°.

31

 

 

 

 

 

 

 

 

 

 

 

 

2,5 _ F r I I I I -
0 Data .. .. 83 MeV/u 813 on Pb
........... 121111132 . (988 s 1.77 °
2.0 - --—— Scaled E2 : """""""" I. ~~~~~ ..
R ------ No E2
Q o '..‘
E) % 1'21.
2 1.5 — ' ‘1
S ..
g (I
g: 1.0 — n 0 ‘
3 I
0.5 — , _
00 l l I l I I
340 360 380 400 420 440 460

Proton Longitudinal Momentum (MeV/c)

Figure 3.9: Measured longitudinal momentum distribution of protons from the
Coulomb dissociation of 83 MeV/ nucleon 8B on Pb with 8B scattering angles 3 1.77°.
Only relative errors are shown. Also depicted are lst-order perturbation theory cal-

culations with different E2 strengths, convoluted with the experimental resolution.

Fig. 3.9 depicts the longitudinal momentum distribution of protons measured in
coincidence with 7Be fragments from the Coulomb breakup of 83 MeV/ nucleon 8B
with reconstructed 8B center-of-mass angles of 1.77° and less. The proton momentum
resolution was estimated from the Monte Carlo simulation to be 4 MeV/c (10). Also
shown in the figure are lst-order perturbation theory calculations using the model of
ref. [24], one with the full E2 amplitude, one with the E2 matrix elements scaled by
0.7, and another with no E2 matrix elements. The overall normalization of the full
E2 calculation was 0.90, while the calculation with scaled E2 matrix elements was
multiplied by 0.95, and the calculation without E2 matrix elements was not scaled.

All of the measured longitudinal momentum distributions share a common feature:

32

an asymmetry attributed to interference between E1 and E2 transition amplitudes
in the Coulomb breakup. This effect was first predicted in ref. [27]. An early mea-
surement of the momentum distribution of 7Be fragments from the Coulomb breakup
of 41 MeV/ nucleon 8B on gold [25] provided evidence of this effect, but the statistics
were insufficient to draw any definitive conclusions. By measuring longitudinal mo-
mentum distributions of 7Be nuclei and protons on two targets at two different beam
energies with two different experimental setups, we have conclusively demonstrated
the existence of this asymmetry.

In lst-order perturbation theory, the size of the predicted asymmetry is propor-
tional to the E2 transition amplitude. Fig. 3.10 illustrates this point, depicting the
central region of the 35° 44 MeV/ nucleon 7Be longitudinal momentum distribution
from the Pb target along with three calculations. These calculations were performed
with different E2 amplitudes, and are normalized to the same value at the center
of the distribution. The simple potential model of 8B structure from ref. [24] makes
predictions for the E1 and E2 matrix elements. By arbitrarily scaling the E1 and
E2 matrix elements in a lst-order perturbation theory of the Coulomb breakup, we
fit the central 6 data points of the 35° 7Be longitudinal momentum distribution from
the breakup of 44 MeV/ nucleon 8B on the Pb target in order to minimize the X2
value. Using this procedure, we found that the optimal ratio of E2 and E1 matrix
element scaling factors was 0.7. The same ratio of matrix element scaling factors was
required to best fit the 81 MeV/ nucleon 7Be longitudinal momentum distribution on
Pb in perturbation theory, a calculation that is not shown here (see ref. [28]). In
the exclusive experiment, it was not possible to measure the longitudinal momentum
distributions with a precision comparable to that of the inclusive measurement. Fur-
thermore, any nuclear—induced breakup contribution is relatively more important for

the Ag target than for the Pb target. For these reasons, we used only the inclusive

33

measurements on Pb to deduce the E2 strength. The preliminary findings of the in-
clusive measurement were described previously [28]. In ref. [28], the optimal ratio of
the E2 and E1 matrix element scaling factors was incorrectly reported as the ratio of
the scaling factors for the E2 and E1 strength distributions; the correct value for this
ratio is 0.72 = 0.49. As a consequence, the reported [28] ratio of E2 and E1 S factors
at Ere; = 0.6 MeV should be replaced by 4.7 ”3:3 x 10'”. This result assumes the
validity of lst-order perturbation theory in describing the reaction mechanism, and a
particular 8B structure model. If higher-order electromagnetic effects are important,
a larger intrinsic E2 strength is required to fit the data. Hence we have determined
the effective E2 matrix element which, within a lst—order perturbation theory with a
given E1 matrix element, fits the empirically observed asymmetry in the longitudinal
momentum distributions.

Table 3.1 lists the integrated cross sections obtained in the inclusive longitudi-
nal momentum distribution measurements on both targets. The purpose of these
inclusive measurements was to deduce the E2 strength in the Coulomb breakup.
A determination of low-lying E l strength would be subject to large nuclear struc-
ture uncertainties, since the inclusive measurements are sensitive to electromagnetic
strength over a large range of excitation energies. However, the observed asymmetry
in the longitudinal momentum distributions is a clear signature of E1—E2 interference,

through which these measurements probe the total E2 strength.

3.2 Theoretical Methods

We have performed lst-order perturbation theory calculations of the Coulomb disso—
ciation of 8B using the model of ref. [24], and have used them to interpret both the
inclusive and exclusive measurements. This simple, single-particle potential model

describes 8B as a pg/Q proton coupled to an inert 3/2" 7Be core. The model predicts

34

 

 

 

 

 

 

 

 

 

 

4.0 I I I
44 MeV/nucleon ”if“.

z”: 3.8 F @max = 3'50 I ’ ........ \‘, —
Q 5.13%. \‘
2 3.6 — of} \—
S . if” 0 ‘l ‘\
a . "I I’ i'.‘ ‘1
\/N 3.4 — f0 ‘
2a. 0 Experiment ‘1‘
8 -------- 50% E2 Amplitude '-,

3-2 ‘f.’ — 70% E2 Amplitude -‘

I." 0 ’1’ " ' ‘ ’ 100% E2 Amplimde
3.0 I l I I

 

2010 2020 2030 2040 2050

7Be Longitudinal Momentum (MeV/c)

Figure 3.10: Central region of the 35° angle cut of the 7Be longitudinal momentum
distribution from the breakup of 44 MeV/ nucleon 8B on Pb. The curves are calcu-
lations performed with different E2 matrix elements, expressed in terms of the E2

amplitude of the model of [24], normalized to the center of the distribution.

35

 

Table 3.1: Integrated Coulomb dissociation cross sections

 

Target Beam Energy 7'Be angle cut (°) 0 (mb)
(MeV / nucleon)

Ag 44 1.5 61 (7)
Ag 2.0 97 (10)
Ag 2.5 140 (15)
Pb 1.5 68 (7)
Pb 2.4 156 (16)
Pb 3.5 252 (25)
Ag 81 1.25 67 (7)
Pb 1.5 130 (8)
Pb 2.0 201 (13)
Pb 2.5 266 (17)

 

 

 

 

 

36

that 517(0) 2 19.1 eV b, and that 532/351 :2 9.5)(10‘4 at a relative energy of 0.6
MeV. The use of lst-order perturbation theory is justified theoretically because of

the small value of the electromagnetic strength parameter,

(EA) N Zt€<fllM(E)‘)lli>
XHI ‘ bubA ’

 

(3.1)

which measures the number of photons exchanged in a transition [11]. The inverse
dependence on the beam velocity '0 implies that for intermediate energy beams the
strength parameter will be less than unity, guaranteeing that most processes are
mediated by single photon exchange, and that lst-order perturbation theory is a
reasonable approximation.

We have also investigated the importance of higher-order electromagnetic ef-
fects and nuclear-induced breakup through continuum-discretized coupled channels
(CDCC) calculations. In the CDCC approach [29], the breakup of 8B is assumed to
populate a selected set of spin-parity excitations with proton—7Be relative energies up
to some maximum value. This excitation energy range is subdivided into a number
of intervals, or bins. For each such bin a representative square integrable wave func-
tion is constructed, a superposition of those proton-7Be scattering states internal to
the bin. These bin wave functions form an orthonormal basis for the expansion and
coupled channels solution of the proton + 7Be + target three-body wave function.
This coupled channels solution is carried out here using the code FRESCO [30]. The
subsequent evaluation of the fragment energy and angular distributions, from the
CDCC bin-state inelastic amplitudes, is discussed in detail in ref. [31].

The parameter space used in the CDCC calculations is as follows. Partial waves
up to Lmar = 15000 and radii up to 1000 fm were used for the computation of the
projectile-target relative motion wave functions. The wave functions for each bin and
their coupling potentials were calculated using proton-7B6 separations up to 200 fm.

Excitations up to a proton-7Be relative energy of 10 MeV were considered. In these

37

calculations the TBe intrinsic spin is neglected, assuming that the core behaves as
a spectator. The proton spin dependence is included, however, and all proton-7Be
relative motion excitations consistent with orbital angular momentaf S 3, i.e. relative
motion states 6]. up to f7 /2 , were included. The effects of the g-wave continuum
are small and are neglected. The calculations use potential multipoles A S 2 in the
expansion of the proton- and 7'Be- target interactions. The real potential used to
construct the wave functions for each bin was the same as that used to bind the
8B ground state, a pure p3/2 proton single—particle state. This proton-7Be binding
potential was taken from Esbensen and Bertsch [24], and was used for all spin-parity
channels. The fragment—target nuclear interactions are also included; for the 7Be-
208Pb system we take the (Li) interaction of Cook [32] and for the proton-208Pb

system the global nucleon optical potential of Becchetti and Greenlees [33].

3.3 Breakup Energy Spectrum

In contrast to the E2 component, the size of the M 1 contribution to the cross sec-
tion for Coulomb breakup can be determined from the measurement of the radiative
capture cross section at the 0.64 MeV 1+ resonance [8]. M 1 transitions only play a
role in Coulomb dissociation near this energy, and the magnitude of the contribution
is obtained from the measured resonance parameters [34] and the calculated virtual
photon spectrum [12]. The energy resolution of our exclusive measurement is too
large and the contribution too small to allow us to clearly see this resonance, but it
represents a few percent of the measured cross section.

Since the radiative capture reaction involves protons and 7B6 nuclei in their ground
states, Coulomb breakup that yields excited 7Be nuclei is not relevant to the inverse
radiative capture rate. As our experimental setup did not include any provision for

detecting ’y rays, a correction for the yield to the 1/2‘ excited state of 7Be was made

38

 

on the basis of equation 41 of ref. [35], and the analysis of the data of ref. [36] found
in ref. [37]. The size of this correction ranged from 1% at 200 keV to 9% at 2 MeV.

By placing a 1.77° cut on the reconstructed angle of the dissociated 83 MeV / nucleon
8B projectiles, we have ensured that nuclear diffraction and absorption effects are
small, and that the point-like projectile approximation employed in lst-order pertur-
bation theory is valid. Furthermore, by also excluding relative energies below 130
keV from our analysis, we minimized the role of E2 transitions and considered only
relative energies where E1 was the dominant (> 90%) contribution to the breakup.
Fig. 3.11 shows the theoretical breakup energy spectrum calculated in lst-order per-
turbation theory. The calculation was performed with the model of ref. [24] for the
E1 and E2 components, scaling the E2 matrix elements by 0.7, while the M 1 com-
ponent was calculated as described above. As illustrated in Fig. 3.12, E1 transitions
dominate the breakup cross section from 130 keV to 2 MeV except for a narrow range
surrounding the 0.64 MeV 1+ resonance. E2 transitions contribute significantly at
relative energies under 130 keV, accounting for the sharp fall in the E 1 fraction of
the cross section at low relative energies.

To deduce the E1 strength at low relative energies, we carried out the following
procedure. After fixing the E2/ E 1 ratio using the inclusive data, we convoluted the
calculated E 1, E2, and M1 cross sections with the energy-dependent experimental
resolution, and then scaled the combined E 1 + E2 cross section in order to minimize
X2 for the measured differential cross section between 130 and 400 keV. Recent work
[38] suggests that above 400 keV, nuclear structure uncertainties increase appreciably.
The best-fit normalization factor of the E 1 + E2 calculation for the data between 130
keV and 400 keV was 0.93 i333, resulting in 517(0) 2 17.8 it; eV b, with all sources
of uncertainty added in quadrature. We extrapolated to zero energy using the pre-

scription of Jennings et al. [38]. The quoted error (10) includes energy—dependent

39

 

 

 

 

 

 

 

 

 

 

 

l l I
83 MeV/u 813 on Pb j_j_j_ g

150 — 983 5177 ........... M1 ‘
9 ‘ _ ~ — Sum
Q) 1’ ‘\\
g 100 - ’1 i‘. -

'12
EI
'U 50 _ :2: ‘~~~‘ —
""""""""""""" 1":’*Mll-
0 0 0 5 1.0 1 5 2 0
Relative Energy (MeV)

Figure 3.11: Contributions of E 1, E2, and M1 transitions to the cross section for the
Coulomb dissociation of 83 MeV/ nucleon 8B on Pb with 8B scattering angles 3 1.77°
in lst-order perturbation theory. The M 1 cross section is calculated by folding the M 1
S factor measured in ref. [8] with the virtual photon spectrum. The E1 and E2 cross
sections are calculated using the model of ref. [24], scaling the E2 matrix elements
by the factor 0.7 required to reproduce the measured 7Be longitudinal momentum

distributions.

40

 

 

 

 

 

 

 

 

 

 

 

l .0 I l I T l l l I r I T T l I l I I I I
0.9 — _
E
O
[—
L2 0 8 - —
LL]
0
0.7 _ —
83 MeV/u 8B on Pb
98,, s 1.77 °
06 1 1 1 1 l 1 1 1 1 I 1 1 1 1 I 1 1 1 J
0.0 0.5 1.0 1.5 2.0
Relative Energy (MeV)

Figure 3.12: Fraction of the calculated cross section for the Coulomb dissociation of
83 MeV/nucleon 8B on Pb with 8B scattering angles 3 1.77° (b 2 30 fm) accounted
for by E 1 transitions in lst-order perturbation theory. As the energy falls below 130

keV, E2 transitions become increasingly important.

41

contributions from statistics, momentum and angular acceptance, detector efficiency,
and the 7Be excited state yield correction, added in quadrature with systematic uncer-
tainties from the beam intensity (1%), extrapolation to zero energy (1%), size of the
E2 component (2.5%), target thickness (2.6%), and momentum calibration accuracy
(4.2%).

We also analyzed the measured breakup cross section at higher relative energies,
carrying out the same X2 minimization procedure for the data from 130 keV to 2 MeV.
The data above 2 MeV were excluded from the fit because of a 3+ resonance at 2.2
MeV that was not included in the theoretical calculation, and because the statistics
there are poor. The best-fit normalization factor obtained for the data between
130 keV and 2 MeV with this procedure was 1.00 fgfig. We assign a 5% theoretical
extrapolation uncertainty for this energy range [38]. The result of the perturbation
theory analysis of data from 130 keV to 2 MeV is 517(0) 2 19.1 it: eV b. This result
is consistent with the value extracted from the data up to 400 keV, implying that
the simple potential model of ref. [24] describes the physics well even at large relative
energies, within the uncertainties. Nevertheless, we prefer the value of 517(0) inferred
from the data below 400 keV because of its relative insensitivity to the details of 8B
structure.

Fig. 3.13 shows the differential cross section measured in the exclusive experiment
along with the results of our CDCC calculations, convoluted with the experimental
resolution. The figure also includes the results of the best-fit lst-order perturbation
theory calculations for the two energy ranges described above using the model of ref.
[24], with E2 matrix elements quenched as required to fit the inclusive data. The per-
turbation theory calculations include M 1 transitions and also have been convoluted
with the experimental energy resolution. The CDCC calculations employ a slightly

simplified version of the structure model of ref. [24], and provide a means of gauging

42

 

I l

__ 0 Data

120 - ' ----- Perturbative (best fit to 400 keV) —
- --------- Perturbative (best fit to 2 MeV)

— CDCC

140 j

 
 
 
   
   

100 —
80

60

D...
u..
......
I...
'0

Q..
O

4O 1' .............................

§..
.§
y
.
‘Q
‘~
5
' u
..
‘-
o u.

do/dEreI (mb/MeV)

20

 

I
0.0 0.5 1.0 1.5 2.0

 

Relative Energy (MeV)

Figure 3.13: Measured differential cross section for the Coulomb dissociation of 83
MeV/ nucleon 8B on Pb with 8B scattering angles S 1.77°. Only relative errors are
shown. Also depicted are continuum-discretized coupled channels and two lst-order
perturbation theory calculations, convoluted with the experimental resolution. The
point at 64 keV has been excluded from the fits because E2 transitions are dominant

at this energy.

the importance of nuclear-induced breakup and higher-order electromagnetic effects;
the E1 and E2 reduced transition probabilities predicted by the two structure models
agree at the 1% level. These fully quantum mechanical CDCC calculations include
both nuclear and Coulomb interactions, and have not been renormalized.

The two reaction models describe the data between 130 keV and 2 MeV equally
well, implying that the theoretical uncertainties in the reaction mechanism are smaller
than or comparable to the experimental uncertainties here. In large measure, this is
due to the experimental conditions of the exclusive measurement. By limiting the

angular acceptance as we did, we probed large impact parameters where the E2 and

43

nuclear contributions are small. These CDCC calculations indicate that nuclear-
induced breakup is negligible at relative energies less than 400 keV. Higher-order
electromagnetic effects are also smallest at the largest impact parameters [13, 24]. The
fact that the zero energy .5' factors implicit in the CDCC calculation (18.9 eV b) and
the best-fit lst-order perturbation theory calculation for the data up to 2 MeV (19.1
eV b) agree within 1% gives confidence that lst—order perturbation theory adequately
describes the underlying physics of the breakup reaction under these experimental

conditions, provided the E2 matrix elements are appropriately quenched.

44

Chapter 4

Discussion

The E2 strength deduced from the inclusive momentum distribution measurement
is 10 to 100 times larger than the upper limits reported in other experimental stud-
ies [16, 36]. We studied an observable that directly probes E1-E2 interference, the
asymmetry in the longitudinal momentum distribution. Our experimentally deduced
value for the E2/ E 1 ratio is only slightly smaller than or in good agreement with
recent theoretical calculations [24, 39, 40, 41, 42], and is consistent with the mea-
surement of [43], although this group does not give a value for 332/5131. That the
extracted experimental value should be somewhat smaller than the theoretical values
is consistent with the idea that lst-order perturbation theory overestimates the E2
contribution to the cross section [24]. Table 4.1 shows the E2 strength predictions
given in several recent papers using potential models, microscopic cluster models, the
shell model embedded in the continuum, and R-matrix theory, along with the results
of this work. The concordance of these predictions of the E2 strength made on the
basis of disparate theoretical methods and the result deduced from the measured lon-
gitudinal momentum distribution asymmetries imply that the E2 component must
be accounted for in a proper theoretical description of the Coulomb breakup.

We interpret the required quenching of the E2 matrix elements in lst-order per-

45

Table 4.1: Comparison of theoretical E2 strength predictions with present results

 

 

 

 

 

 

Author Reference Method 532/531 (0.6 MeV)
Esbensen and Bertsch [24] potential model 9.5 X10"4
Typel et a1. [39] potential model 8.0 X10_4
Bennaceur et a1. [40] SMEC 7.72 ><10‘4
Descouvement and Baye [41] cluster model 6.2 ><10"4
Barker [42] R—Matrix 8.7 ><10"4
Davids et al. this work experiment 4.7 "333 ><10'4

turbation theory as a manifestation of higher-order dynamical effects. For a fixed
E2 strength, the predicted asymmetry of the longitudinal momentum distribution is
diminished when higher-order effects are considered compared with lst-order pertur-
bation theory [24]. In dynamical calculations of the Coulomb dissociation of 8B that
include higher-order processes [24], the E1 strength is essentially unaltered, while the
E2 strength is reduced with respect to lst-order perturbation theory calculations. As
such dynamical calculations are difficult and time-consuming, we have accounted for
these effects by quenching the E2 matrix elements in the context of a lst-order per-
turbation theory description of the reaction dynamics. The dynamical calculations
include the same physics as do the CDCC calculations presented here. A comparison
between the CDCC calculations and (lst-order) DWBA calculations using the same
structure model indicates that the reduction in E2 strength caused by higher-order
dynamical effects does not exhibit any significant relative energy dependence. Fig. 4.1
shows the result of this comparison. Hence the approach we have adopted, namely,
scaling the E2 matrix elements by the same factor for all relative energies in lst—order

perturbation theory, is justified.

46

 

0.10 I I I I I I T r I I I I I I I I I 1 I

0.02 - ._

:8

0.08-
E 00 1
b o
\ O OO
,3 0.06— 00 ~—
8 O 0000
U O 0000
t,’ 0.04— 00 _
é
a
E

 

 

0.00 1 1 1 1 l 1 1 1 1 I 1 1 1 1 l 1 1 1 4
0.0 0.5 1.0 1.5 2.0

 

Relative Energy (MeV)

Figure 4.1: Difference between the cross section for Coulomb breakup of 83
MeV / nucleon 8B on Pb for 8B scattering angles of 1.77° and less predicted by DWBA
(lst—order) and CDCC (all orders) calculations using the same structure model, ex-
pressed as a fraction of the DWBA prediction. Only Coulomb matrix elements were
included in these calculations. No significant energy dependence of the higher—order

electromagnetic effects is evident.

47

By measuring the Coulomb dissociation cross section at low relative energies and
small scattering angles of the 8B center-of—mass, we have ensured that the contribution
of E2 transitions is small, and that nuclear diffraction effects are negligible. Using our
inclusive measurement of 7Be longitudinal momentum distributions to determine the
relative contributions of the E2 and E1 components, we extracted the E 1 strength at
low relative energies from the exclusive measurement. The value of the astrophysical
zero—energy 5 factor for the 7Be(p,7)8B reaction we infer, 17.8 11:; eV b, is in good
agreement with other recent measurements, and with the recommendation of a recent
workshop on solar nuclear fusion cross sections [44]. Fig. 4.2 and table 4.2 show the
results of recent radiative capture [8, 9], Coulomb breakup [15, 16], and asymptotic
normalization coefficient [45] determinations of 517(0), along with the results of this
work.

The concordance of our measurement and the other Coulomb breakup measure—
ments conceals an underlying difference in interpretation. The analyses of references
[15, 16] have treated the contributions of E2 transitions as negligible, while our data
imply they are not. Since these experiments covered angular ranges larger than
this measurement, they probed smaller impact parameters where E2 transitions are
relatively more important. If E2 transitions are considered, lst-order perturbation
theory calculations imply that the astrophysical 5 factor inferred from the RIKEN
Coulomb breakup measurement should be reduced by 4-15% [46], and that of the
G81 measurement by 15-20%. Such a reduction would bring these measurements into
even better agreement with the present work. If we were to analyze our measured
Coulomb breakup cross section between 130 and 400 keV without considering E2
transitions, the extracted E1 strength would be 5% greater, and the inferred value

of 517(0) would increase to 18.7 :l: 1.3 eV b. The small E2 correction is the result of

restricting the angular range covered in this experiment, making the E2 contribution

48

Table 4.2: Recent 817(0) determinations

 

Author Reference Method 517(0) (eV b)
Filippone et al. reanalysis [8, 9] radiative capture 18.4 :l: 2.2
Hammache et a1. [9] radiative capture 18.5 :1: 1.0
Kikuchi et a1. [15] Coulomb breakup 18.9 i 1.8
Iwasa et al. [16] Coulomb breakup 20.6 :1: 1.0 :l: 1.0
Azhari et al. [45] transfer reaction 17.3 :t 1.8
Davids et al. this work Coulomb breakup 17.8 it;

 

 

 

 

 

to the breakup cross section comparable in magnitude to the statistical uncertainty
of the measurement.

It appears that the three techniques used to infer 317(0), direct radiative capture
measurements, asymptotic normalization coefficient determinations, and Coulomb
breakup, yield consistent results with different systematic uncertainties. In light of
these facts, we take a weighted average of these measurements to obtain a recom-
mended value. In this average, we include the radiative capture measurement of ref.
[8], which was deemed the only reliable measurement at the 1997 workshop on solar
nuclear fusion cross sections [44], and the recent measurement of ref. [9]. Here we
use the reanalysis of the data of ref. [8] presented in ref. [9]. We also include the
asymptotic normalization coefficient result [45] shown in table 4.2, and the present
Coulomb breakup measurement. We do not include the other Coulomb breakup mea-
surements [15, 16] in this average because we lack sufficient information to precisely
correct for the E2 component neglected in the published analyses of these data. The
weighted average we obtain is (517(0)) 2 18.1 :1: 0.7 eV b. This value of 517(0) implies

a reduction of the predicted 8B solar neutrino flux of about 5% from the value used

49

 

24——H . .

 

 

 

 

 

 

 

 

GSI

22 — " —
A Argonne RIKEN
.0 ,. “r Cb
% 20 — _
X MSU
9; o
m" o

18 — c> -

Orsay / Bordeaux " O
16” “ TexasA&Md_ '
———I.I . .
1983 1998 2000
Year

Figure 4.2: Inferred zero-energy astrophysical 5 factors for the 7Be(p,7)8B reaction

from recent direct and indirect measurements.

50

in ref. [1].

51

Chapter 5

Summary

In summary, we have carried out inclusive measurements of the Coulomb dissocia-
tion of 8B on Pb and Ag targets at 44 and 81 MeV/nucleon. Using a high-resolution,
large-acceptance magnetic spectrometer, we measured the distribution of longitudinal
momenta of the emitted 7Be fragments. The longitudinal momentum distributions
reveal E2 strength in the Coulomb breakup in the form of an asymmetry produced
by E l-E2 interference. By comparing the measured longitudinal momentum dis-
tributions with lst-order perturbation theory calculations, we deduced the effective
E2 contribution to the Coulomb breakup. Expressing our result as the ratio of E2
and E l 5 factors at an energy where previous results have been compiled, we found
SEQ/SE] = 4.7 fig x 10'4 at EM 2 0.6 MeV. This result is at least a factor of 10
larger than other experimental determinations, but in reasonably good agreement
with theoretical predictions arrived at through several different methods.

In a separate experiment, we made an exclusive measurement of the Coulomb
dissociation of 83 MeV/ nucleon 8B on a Pb target using a dipole magnet to separate
the beam from the breakup fragments. Measuring the differential Coulomb breakup
cross section at low relative energies and small 8B scattering angles yielded the as-

trophysical 5' factor for the 7Be(p.7)8B reaction with minimal complications from E2

transitions, higher-order electromagnetic effects, and nuclear—induced breakup. Inter-
preting this exclusive measurement in the context of a lst-order perturbation theory
description of the reaction dynamics and a single—particle potential model of 8B struc-
ture, we obtained 317(0) 2 17.8 :13; eV b. We checked the validity of the perturbative
approach through continuum-discretized coupled channels calculations that assume
an essentially identical model of 8B structure. The two reaction theories describe the
data up to relative energies of 2 MeV equally well within the experimental uncertain-
ties, implying that a slightly modified lst-order perturbation theory is adequate for
understanding the Coulomb breakup of 8B at intermediate beam energies and small
angles.

This measurement agrees well with other recent experimental determinations of
517(0), and shows that the uncertainties associated with the Coulomb breakup tech-
nique, unwanted multipolarities, higher-order electromagnetic effects, and nuclear-
induced breakup, can be controlled Well enough to obtain a precise value for the
7'Be(p,')()gB cross section. Direct radiative capture measurements, asymptotic normal-
ization coefficient determinations, and Coulomb breakup measurements yield consis-
tent results for 517(0), despite their different systematic uncertainties, giving confi-
dence that this quantity is now well determined. We recommend a weighted average
of measurements using these 3 different techniques, (517(0)) 2 18.1 :1: 0.7 eV b, for
use in solar modeling. On the basis of this test case, we believe Coulomb breakup

holds promise for indirectly measuring other radiative capture cross sections.

53

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57