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” "' ' ~—

 

 

 

 

This is to certify that the

dissertation entitled

SPECTROSCOPY OF EXOTIC NUCLEI
WITH THE 8800 SPECTROGRAPH

presented by

JOSEPH ARTHUR CAGGIANO

has been accepted towards fulﬁllment
of the requirements for

Ph.D. Physics

degree in

 

 

Mia/1M

 

V

Major professor

Date FEBRUARY 15, 1999

 

MSU is an Afﬁrmative Action /Equal Opportunity Institution 0- 12771

PLACE IN REI’URN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/98 WWW“

SPECTROSCOPY OF EXOTIC NUCLEI WITH THE S800 SPECTROGRAPH

By

Joseph Arthur Caggiano

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

1999

SPECTROSL

The high '
National Supercor
1996. The purpox
of this thesis. the
compute transfer r
to confirm that the

Binary mu

exotic nuclei I”Li.
at a neutron separ;
hing state ti as ins

The masse

,.
and “P were mea:

excess of the 33 A]

whammy 1:

reaction has used ‘

and its ﬁrst excited

ABSTRACT

SPECTROSCOPY OF EXOTIC NUCLEI WITH THE 8800 SPECTROGRAPH

By

Joseph Arthur Caggiano

The high resolution, broad range 8800 magnetic spectrograph was built at the
National Superconducting Cyclotron Laboratory (N SCL) and commissioned in September
1996. The purpose for building the $800 was to allow the study of exotic nuclei. As part
of this thesis, the magnetic ﬁelds of the spectrograph dipoles were measured and used to
compute transfer maps to allow the high resolution of the spectrograph to be achieved and
to conﬁrm that the spectrograph met design criteria.

Binary multinucleon transfer reactions were used to spectroscopically study the

exotic nuclei 10Li, 23Al, and 27F. The 9Be(9Be,gB)10Li reaction populated a p-wave state

at a neutron separation energy of -500(60) keV in the 10Li nucleus. Evidence for a lower

lying state was insufﬁcient to draw conclusions about the state.

The masses and ﬁrst excited states of the astrophysically signiﬁcant nuclei 23A1
and 27F were measured. The 24Mg(7Li,8He)23Al reaction was used to measure the mass
excess of the 23Al ground state, M.E.=-6773(28)keV, and its ﬁrst excited state location,
Ex=550(20)keV, both in agreement with past measurements. The 28Si(7Li,8He)27P

reaction was used to measure the mass excess of the 27F ground state, M.E.=-670(4l)keV,

and its ﬁrst excited state location, Ex=1.l99(19)keV, which is the ﬁrst experimental report

 

of this State. The sign

be discussed.

of this state. The signiﬁcance of these measurements to the astrophysical rp process will

be discussed.

To my family, both human and animal

 

First and
throughout m." i
\i’alt Benenson
teacher and “‘38
Beers. and Juliu~

Al Zellet
phase.

lmust tha
sauy technical ass

lR'ould lil
dons help with the

Daniel Ba;
detectors and data

lwould lii
Ship. The}. have i
debmfgratitude n-

luant to 1}
gene of baseball.

I would like

Acknowledgements

First and foremost, 1 would like to thank my advisor, Brad Sherrill, for his support
throughout my graduate career. His presence, stamina and abilities have been inspiring.
Walt Benenson has also been an inspiration during my graduate career; he is a great
teacher and was always there when I needed him. I would like to thank Martin Berz, Tim
Beers, and Julius Kovacs for serving on my advising committee.

A1 Zeller provided assistance inumerable times during the S800 construction
phase.

I must thank the Operations and Support staff at the NSCL for providing the neces-
sary technical assistance and beams that made this dissertation possible.

I would like to thank Renan Fontus, Phil Johnson, and Len Morris for their tremen-
dous help with the mapper project.

Daniel Bazin has provided immeasurable assistance, from mapping to conﬁguring
detectors and data acquisition, and to providing transfer maps. Thanks, Daniel.

I would like to thank Jon Kruse, Chris Powell, and Barry Davids for their friend-
ship. They have been true ﬁ’iends, always there to listen, discuss or help. I owe Jon a
debt of gratitude for introducing me to the game of golf.

I want to thank Pat Lofy for his willingness to discuss and understanding of the
game of baseball.

I would like to thank Mathias Steiner, Thomas Baumann and Renan Fontus for two

very enjoyable DALMAC bicycle trips.

 

1 thank m
and Eric.

Finally. I
the big picture at

spent in this God-

I thank my family for being supportive beyond belief. Thanks Mom, Dad, Scott,
and Eric.

Finally, I would like to thank Njema Frazier, Ph.D. She always reminded me of
the big picture and kept me sane when no one else could. She made the 6+ years I’ve

spent in this God-forsaken state worthwhile.

vi

 

List ofFigures . .
List of Tables ..

I Introduction . .

1.1 Mass measurt
1.2 Mass measuru

l3 Outline... .

2 The S800 Spec
2.] Magnetic spe
2.2 Characteristic

2.2.l Beamlint
2.2.2 Detectors
23 Optics of the
2.3.] Notation
23.2 Optical p
23.2.] Dete
23.2.2 Aber

3 Dipole Magnet.

3.] Motivation fo

3.2 Data Acquisit

.2.1 General ,
3.2.2 it

Table of Contents

List of Figures ................................................ xi
List of Tables .............................................. xvii
1 Introduction ................................................ 1
1.1 Mass measurement of exotic nuclei .................................... l
1.2 Mass measurement using binary transfer reactions ........................ 4
1.3 Outline ........................................................... 5

2 The S800 Spectrograph ....................................... 7
2.1 Magnetic spectrographs ............................................. 7
2.2 Characteristics of the S800 ........................................... 8
2.2.1 Beamline and magnets ........................................... 8
2.2.2 Detectors .................................................... 13

2.3 Optics of the S800 ................................................. 13
2.3.1 Notation ..................................................... 13
2.3.2 Optical properties of the S800 .................................... 15
2.3.2.1 Detector resolution considerations ............................. 17

2.3.2.2 Aberration Correction ....................................... 17

3 Dipole Magnetic Field Measurements .......................... 19
3.1 Motivation for the measurements ..................................... 19
3.2 Data Acquisition .................................................. 21
3.2.1 General ...................................................... 21
3.2.2 Mapper hardware and electronics ................................. 23
3.2.2.1 Aluminum frame and cart .................................... 23

3.2.2.2 Induction coils ............................................. 29
3.2.2.3 Analog-to-digital conversion ................................. 29

vii

 

3.2.2.4 El
3.2.2.5 Pt

33 Post pm“
33.] RNd
3.3.2 Dem“
333 AcettU
33.4 \‘oltag
33.5 Field c
33.6 Positit'
3,3,7 Smom
3.4 Results . . —
3.4.1 Field e
3.4.2 Effeeti
3.4.3 Edge :1
3.4.4 Alignn

3.5 Conclusion

4 Spectroscop\

4.1 Motivation
4.1.1 Halo nt
4.1.2 Lithiun'
4.13 Lithium

4-2 EXperimenta
4.2.] Reactim
4.2.2 8800 . .

Detector

4.2.4 Electron

4.2.5 Calibm,

3.2.2.4 Electronic noise suppression .................................. 30

3.2.2.5 Position and time measurement ............................... 31

3.3 Post processing and data conversion ................................... 32
3.3.1 Raw data ..................................................... 32
3.3.2 Deadtimecorrection ................ 32
3.3.3 Accounting for missed ﬁeld data .................................. 34
3.3.4 Voltage offset correction and sweep alignment ....................... 34
3.3.5 Field calibration ............................................... 35
3.3.6 Position calibration ............................................ 36
3.3.7 Smoothing ................................................... 40
3.4 Results .......................................................... 42
3.4.1 Field curvature and gradients ..................................... 51
3.4.2 Effective length ............................................... 54
3.4.3 Edge angles .................................................. 55
3.4.4 Alignment of both dipoles ....................................... 56
3.5 Conclusions ...................................................... 56
4 Spectroscopy of 10Li .................................. , ...... 59
4.1 Motivation ....................................................... 59
4.1.1 Halo nuclei ................................................... 59
4.1.2 Lithium-11 ................................................... 60
4.1.3 Lithium-10 ................................................... 62
4.2 Experimental setup ................................................ 69
4.2.1 Reaction and targets ............................................ 69
4.2.2 S800 ........................................................ 69
4.2.3 Detectors .................................................... 70
4.2.4 Electronics ................................................... 71
4.2.5 Calibrations .................................................. 75
4.2.5.1 Tracking detector position calibrations .......................... 75
4.2.5.2 Elastic and inelastic scattering ................................ 79
4.2.5.3 Spectrograph bending radius calibration ........................ 79

viii

4.2.5.4
4.2.5.5 '

4.3 Results .
43.] ”Be
4.3.2 13B e
433 “it
4.3.3.1 F
4.4 Discussitii

4.5 Conclusio

 

5 Spectroscop)

5.1 Motit'ation

‘ ,a\«;- ’
“"E37 4 , x:
.,- :Mk' “1"" a r
‘ ..*- ....

 

5.1.1 Explw
‘ 5.1.2 Stellar
J 5.13 WhyS“
5.1.4 Why s“-
5.2 Experiment
53 Results . . . .
5.4 Discussion .
5.4.1 nelson
5.4.2 Astrophy
5.5 Conclusions .
tSummary .....
A Mapper Post~Prt
.41 lntroduction . .
42 How to use the .

Sample i"put 171.
. - Output ﬁles . . .

4.2.5.4 Scattering angle calibration ................................... 80

4.2.5.5 Ray reconstruction ......................................... 82

4.3 Results .......................................................... 83
4.3.1 10Be and 7H6 energy spectra ..................................... 86
4.3.2 l3Benergyspectrum ................ 92
4.3.3 10Li energy spectrum .......................................... 95
4.3.3.1 Fitting procedure .......................................... 98

4.4 Discussion ...................................................... 103
4.5 Conclusions ..................................................... 105
5 Spectroscopy of 27F and 23Al ................................ 107
5.1 Motivation ...................................................... 107
5.1.1 Explosive hydrogen burning and the rp process ..................... 107
5.1.2 Stellar Reaction Rates ......................................... 109
5.1.3 Why study 27P? .............................................. 113
5.1.4 Why study 23A1? ............................................. 115

5.2 Experiment ..................................................... 116
5.3 Results .................................................. ' ....... 120
5.4 Discussion ...................................................... 129
5.4.1 The Isobaric Multiplet Mass Equation (IMME) ..................... 129
5.4.2 Astrophysical implications ...................................... 132

5.5 Conclusions ..................................................... 135
6 Summary ................................................ 137
A Mapper Post-Processing Software ............................ 140
A.1 Introduction .................................................... 140
A.2 How to use the software ........................................... 140
A.3 Sample input ﬁle ................................................ 146
AA Output ﬁles ..................................................... 147

r—v

'1. LL11; It

-

 

lﬂﬂ If...” 1-

B Inverse trz

Bibliography

B Inverse transfer maps ...................................... 150

Bibliography ............................................... 155

 

3.1

33

3.6

 

Schematic of
from the ohie
downstream

Schematic 0‘
after leasing

The dipole
magnets. T
is mountedt
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installed int

Dipole ma;
Resp the ca
3511 rides a

Schematic
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can that m
shown 011:
out of the}

2.1

2.2

3.1

3.2

3.3

3.4

3.5

3.6

List of Figures

Schematic of the 8800 spectrograph. The analysis line consists of those elements
from the object position to the target chamber; the spectrograph itself is the portion
downstream from the target chamber. .................................. 9

Schematic of optical coordinate system, before entering system of magnets M and
after leaving it. The xz plane at y=0 is deﬁned as the midplane .............. 10

The dipole mapper assembly, shown together with the magnet steel in both
magnets. The cart with the coils rides up and down along the guiding track, which
is mounted on an aluminum plate. Only one mapper assembly was used; the mapper
was installed in DI, the magnet was mapped, then the mapper was removed and
installed into D2. ................................................. 24

Dipole mapper cart assembly, top (left) and side (right) views. The pinch rollers
keep the cart on the guiding track, and the wheels keep the cart on the same plane
as it rides along the aluminum plate. .................................. 25

Schematic of the mapper showing the coordinate system used. The thick black lines
are the magnet steel, the thin lines are the mapper plate. The gray box represents the
cart that moves up and down along the plate. The three cylindrical coordinates are
shown originating from a typical coil position on the cart (2 is

out of the page). .......................................... - ........ 26

Schematic of the mapper DAQ system. The PC supplies commands to the motors
and power to the on board electronics and controls event timing. The optosensor on
the cart reads holes in the position strip and sends a square waveform back to the
PC for position and timing information. ............................... 27

Timing diagram for one pass through the magnet. The gate signal is generated when
the cart passes a certain point and tells the PC to start or stop acquiring data. As the
optosensor passes the teeth in the position strip, it generates a square wave. Field
measurement is broken up into 3—tooth (0.090”) samples. Thi and T10 are the times
that the waveform is spent in the high or low state, respectively. T(l) and T(2) are
the live times for the ADC’s. Tstanﬁ) is the time it takes to start the ABC’s
following detection of the ﬁrst tooth in the sample. All the times mentioned are
measured by counting l ys clock ticks using a stopwatch circuit. ........... 28

Raw data from one pass through the D1 magnet at 1.6 Tesla. The top panel is the
time it takes the cart to travel 0.762 mm at a radius of 2.7 m for each ﬁeld sample.
The bottom panel is the raw voltage output from the ADC. The cart oscillation
causes the oscillations in the measured voltage, better seen in the right half of

the ﬁgure. ....................................................... 33

xi

.,. i
‘.' ”FA“.

 

(“—T _

3.7

3.9

3.10

3.14

3.17

3.13

Positior
thick 11
location
at the ce

Hall prt
polynon

Left: Ex
Right: I
probe me

Field me
The magi
Thelines
edge. . . .

Dipole m.
coils was

A D2 field
are separa'

Field megs

Sharp conti
saturation 1

DI (left) a,
the ﬁeld Se
grows With
ﬁeld Streng.

Mailneti‘c ﬁ.
3‘ B=0.28 T

Magnetic n.-
at 8:] .6 T65

6 effect 01
ShOulders in
Peaks in the s

campaign. D6
remain within

Measured Tad,‘

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

Position of Hall probe and NMR probes for magnetic ﬁeld measurements. The
thick lines are the locations of the Hall probe measurements; the dots are the
location of the NMR probe measurements. The origin of the coordinate system is
at the center of the entrance to the D1 magnet. .......................... 37

Hall probe data for D2 at 1.6 Tesla. The line through the data is a Sth order
polynomial ﬁt. The ﬁt residuals are all < 0.1 Gauss. ..................... 38

Left: Excitation functions for D1 and D2 determined from NMR measurements.
Right: Field strength as a function of starting radius for D2 determined by Hall
probe measurements. The ﬁelds range from 0.28 T (top) to 1.6 T (bottom). . . . 39

Field measurement of small permanent magnets placed on the D1 magnet steel.
The magnets are 1cm x 1cm x 2cm long. The top left panel shows all the magnets.
The lines through the centers of the circular contours are 5mm from the magnet steel
edge. ........................................................... 40

Dipole magnetic ﬁeld data for both spectrograph dipoles. The current through the
coils was ~350 A for both, giving ﬁelds of 1.48 Tesla for both D1 and D2. . . . 44

A D2 ﬁeld map at 0.28 Tesla, plotted above 99% of the ﬁeld strength. The contours
are separated by one Gauss, indicating a very flat central ﬁeld. ............. 45

Field measurements for D2 at 1.6 Tesla. All contours are separated by 5 Gauss. In
sharp contrast to Figure 3.12, the ﬁeld exhibits a bowing behavior characteristic of
saturation effects. ......................................... . ........ 46

D1 (left) and D2 (right) magnetic ﬁeld strength along the reference trajectory for
the ﬁeld settings from Tables 3.1 and 3.2. The curvature in the ﬂat ﬁeld region
grows with ﬁeld strength. The negative ﬁeld just outside the coil also grows with
ﬁeld strength. .................................................... 48

Magnetic ﬁeld proﬁle and derivatives along the reference trajectory for a D2 map
at B=0.28 Tesla. The ﬁeld proﬁle is very flat in the center. ................ 49

Magnetic ﬁeld proﬁle and derivatives along the reference trajectory for a D2 map
at B=1.6 Tesla. The ﬁeld is much more curved in the center than are lower ﬁelds.
The effect of the coil is much more prominent than in Figure 3.13; the outermost
shoulders in the ﬁrst derivative are a larger fraction of the peak. The outermost
peaks in the second derivative also demonstrate this effect. ................ 50

Azimuthal ﬁeld curvature for some ﬁeld settings used during the mapping
campaign. Despite the high curvature at the largest ﬁeld settings, the ﬁeld gradients

remain within design speciﬁcations (1 Gauss/cm). ....................... 52

Measured radial ﬁeld gradients in D1 with (+) the trim coils off and (O) the trim

xii

3.19

3.20

4.1

43

4.4

coils 011'

Average
Despite 1
dipoles.

Effecti \ e
integratec
(from Tal‘

Contour p
ﬁeld settin
that 50.0(

correct pO~

The first e
reaction at
spectrum 11

The “Rm .1
state or a co

Results from
65540 k€\

T“’0 measur
Siildence for;

SChematic of"
through the de
An induced 1
information i.
P081110n info”.
back Of the def
ere for clariti

SChematic of S:

0.”!3' One Cathod

Simplify (he din.

3.19

3.20

3.21

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

coils on at their optimized setting. .................................... 53

Average radial gradients measured for D1 and D2 during the mapping campaign.
Despite the apparently erratic pattern, the general pattern is reproduced for both
dipoles. The error bars are 115%. The design limit is l Gauss/cm. ......... 53

Effective length for both dipoles from ﬁeld data, calculated by dividing the
integrated ﬁeld along the reference trajectory by the appropriate NMR reading
(from Tables 3.1 and 3.2). .......................................... 54

Contour plot of the overlapping ﬁeld region of two maps at the same current and
ﬁeld settings. The contours are separated by 20 Gauss, and the region contoured is
that 50.0 Gauss. The overlap of the contours is on the few Gauss level, indicating
correct position and ﬁeld calibration for the two dipoles. .................. 57

The ﬁrst experimental observation of 10Li [W175] using the 9Be(9Be,gB)lOLi
reaction at 121 MeV. The resolution and statistics are poor in the 10Li energy
spectrum (bottom panel). ........................................... 64

The llB(Jt’,p)lOLi reaction [Am90l. It is difﬁcult to tell whether the peak is a single
state or a composite ................................................ 65

Results from the Young, et al. experiment [Yo94]. A prominent peak is observed at
-Sn=540 keV. .................................................... 66

Two measurements by the Bohlen group [8097]. The measurements show
evidence for states at -Sn= 240 and 530 keV. .......................... 67

Schematic of the CRDC operation. A particle track ionizes the gas as it passes
through the detector. The electrons drift to the anode wire where they are collected.
An induced image charge on the cathode pads provides horizontal position
information, while drift time of the electrons to the anode wire provide vertical
position information. (Note the cathode pad locations are really on the front and
back of the detector; they are shown schematically on the bottom

here for clarity.) .................................................. 73

Schematic of S800 focal plane detector electronics. Only one CRDC is shown and
only one cathode pad is shown from the overall electronics to
simplify the diagram. .............................................. 74

Data taken with the position mask placed in front of the S801 CRDC. The top panel
is the raw data taken with a small cut in the dispersive angle, theta, and the bottom

panel is a picture of the prominently illuminated holes. ................... 76

Detector y-position (T AC) drifting over time during this experiment. The circles
are the centers of the distribution and the (+) are the widths. Error bars are not

xiii

 

v I ‘w. n'“!:'.ﬁT

4.9

4.10

4.11

4.12

4.13

4.16

4.17

4.13

plotted an

0913
Spectr g s
scattered

Elastic sc;
carbon “1
scattering
9.64 Me\'.
7.1 MeV.
knossn kin
accurate st

Measured
arises mair

Spectrum (
and excitet
calculated 1
MeV exci
l2C1”Be.' 11

TOP3 Spect
toP Panel 51
the lOWQr P
MeV States

F115 [0 the 7]
shows a

linproyed ﬁ
fullCIiOn~ Tl
has a Small ;

TOp; Energ
"UCleuS_ BO

energy calibi

R8301”

IlOn f
the l

Same Fit

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

plotted and ﬂuctuations are due to low statistics. ........................ 78

Spectrograph bending radius calibration using elastically
scattered 9Be for the 10Li experiment. ................................. 80

Elastic scattering of 98c beam at 40 MeV/u from a carbon target consisting of
carbon with H20 contamination. The solid lines are known kinematics for
scattering off carbon nuclei populating states, from right to left, g.s., 4.44, 7 .65,and
9.64 MeV. The dashed lines are scattering from 16O and populating g.s., 6.1, and
7.1 MeV. The dot-dashed line is elastic scattering off lH nuclei. Aligning the
known kinematics with the spectrum provides a very

accurate scattering angle calibration. .................................. 81

Measured energy loss vs. time-of-ﬂight spectra. Contamination in the 8B gate
arises mainly from scattered beam. ................................... 85

Spectrum of 1'C ejectiles from 12C(9Be,”C)loBe reaction populating the ground
and excited states in 10Be. The solid lines overlaid on the spectrum are the
calculated kinematics for known states in 10Be. The dashed lines are the 11C 2.0
MeV excited state corresponding to the reactions (from right to left)
12C(9Be,' 1C(2.0))IOBe(g.s.) and 12C(913e,l lC(2.0))IOBe(3.368). ............ 88

Top: Spectrum of 11C ions in the S800 focal plane populating states in 10Be. The
top panel shows the entire focal plane extent. Bottom: Five Gaussians were ﬁt to
the lower portion of the spectrum to determine the location of the g.s. and 3.368
MeV states in 10Be. ............................................... 89

Same as ﬁgure 4.12, but using the 9Be(9Be,l lC) reaction to make 7He ........ 90

Fits to the 7He ground state using two different resolution functions. The top panel
shows a purely Gaussian resolution function, whereas the bottom shows an
improved ﬁt (<0 MeV) by including some Lorentzian shape in the resolution
function. The mixture doesn’t affect the peak location (430130 keV) of the ﬁt, but
has a small affect on the extracted width (120130 versus 150130 keV). ...... 91

Top: Energy spectrum of SB ions in the focal plane populating states in the 13B
nucleus. Bottom: Multiple Gaussian ﬁts used for
energy calibration of the focal plane ................................... 94

Resolution function used for the (gBe,8B) reaction (solid line). All three peaks have
the same FWHM. ................................................. 95

Rigidity calibration for this experiment. The line through the data is a linear ﬁt.
The points surrounded by squares are calibration points from the 12C(9Be,l lC)lOBe
reaction; all others are from the 12C(9Be,gB)l3 B reaction .................. 97

xiv

 

‘0 ‘9 bid 1'

 

4.19

4.20

5.1

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for all thre

Two diffe!
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The rp-pro
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Subsetcd‘t
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netw ork ts

Benenson‘
al~l.2 hie

PTCVIOUS 51
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Chapter4).

4.19

4.20

4.21

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Ener y spectrum of 8B ions from both the reactions 12C(9Be,gB)l3B (top) and
9Be( Be, B)'OLi (bottom), shown for comparison. ....................... 99

The loLi energy spectrum gathered in this experiment. The three different ﬁts are
of similar quality, but the single p-wave ﬁts the data best because it has the fewest
degrees of freedom. The background values were held constant

for all three fits. ................................................. 102

Two different reactions measuring the 10Li energy spectrum. The similarities
between the two are striking. ....................................... 105

The rp-process network for speciﬁc temperature and density conditions, reprinted
from [Ch92]. ................................................... 109

Subset of the chart of the nuclides schematically showing the rp-process reactions
affecting the abundance of 26Si. This is just a small subset of the total reaction
network (see Figure 5.1). .......................................... 114

Benenson’s spectroscopic measurement of 27P [Be77]. There appears to be a state
at ~1.2 MeV, but wasn’t reported. ................................... 115

Previous spectroscopic measurement of 23Al. The first excited state is not resolved
from the ground state [W188]. ...................................... 116

Primary beam spot at the focal plane, used to check the dispersion matching of the
spectrograph for the experiment. The line through the data points is a single
Gaussian ﬁt; the error bars are statistical. The beam was transmitted to the focal
plane from the object position with the spectrograph in dispersion matched mode.
The FW HM of the peak corresponds to an energy

resolution of E/AE~4000 ........................................... 119

Particle identiﬁcation spectrum for this experiment. .................... 120

All kinematic spectra for this experiment. The fractional energy is relative to the
central energy and deﬁned as (E-EO)/E0. The resolution is better at lower angles
because of the reduced kinematic broadening effects at lower scattering angles (see
Chapter 4). ..................................................... 123

8He energy spectra for the listed reactions at E(7Li)=50.1 MeV/u and
91ab=0‘ 12°. .................................................... 124

8He energy calibration for this experiment. The line through the data is a linear ﬁt.
Errors on the points are statistical uncertainties (translated into energy) and mass
errors of the calibration nuclei summed in quadrature. The mass errors are taken
from the 1995 Mass Evaluation Update [Au95]. ........................ 125

XV

D

.. 'r 16' L1!

 

‘11..-_l

g).
_.
m

ﬁll
h.
'1!

The en
The en:

New 16\
measure
state loe
respectii
of the pr
seventies

Mass exe
data is a t‘
errors in _'

Mass exec
data is a q
The two st
Benenson‘

Astrophysi

Astroph} 5.“
calculations

 

5.10

5.11

5.12

5.13

5.14

5.15

5.16

The energy spectrum of the 24Mg(7Li,8He)23Al reaction .................. 126
The energy spectrum of the 28Si(7Li,8He)27P reaction .................... 126

New level schemes for 23Al and 27P following this experiment. The thick lines are
measurements reported here; the thin lines are previous measurements. The excited
state locations are given relative to the ground state of the nucleus from their
respective measurements. The proton separation energies are a weighted average
of the present ground state mass measurements and those of Benenson et al. in the
seventies [8e75, Be77]. ........................................... 128

Mass excesses for the A=23, J”=5/2+, T==3/2 isobaric quartet. The line through the
data is a quadratic ﬁt; the box lists the coefﬁcients from the ﬁt with
errors in MeV. .................................................. 130

Mass excesses of the A=27 J“=l/2+ T=3/2 isobaric quartet. The line through the
data is a quadratic ﬁt; the box shows the ﬁt coefﬁcients and their errors in MeV.
The two solid squares are this measurement (lower) and

Benenson’s measurement. ......................................... 131

Astrophysical reaction rates using data from past and present experiments. . . 134

Astrophysical reaction rates due to past and present experiments. Note that the
calculations of Herndl er a1. [He95] agree with the present experiment. ..... 135

xvi

. .-,'I‘ Ptjﬂj

.L' 7 "

 

33

3.4

4.1

'II
by

1.1
Bi

32

Featurl
Parame
Parame
Mappin
Mappin;
Entrance
Entrance
Previous
Reaction 1
sections a
measurem
Flilﬁd para
E-‘Perimen
Eﬂkﬁnmni

T... lg(p..,)2i ‘

Reso
as ”MES P2

Slipn~ P r

2.1

2.2

2.3

3.1

3.2

3.3

3.4

4.1

4.2

4.3

5.1

5.2

5.3

5.4

5.5

A.1

B.1

B.2

List of Tables

Features of the $800 spectrograph. ........................................................................ 11
Parameters of the D1 and D2 magnets. .................................................................. 11
Parameters of the Ql/Q2 quadrupole doublet ........................................................ 12
Mapping parameters used during D1 production maps. ........................................ 43
Mapping parameters used during D2 production maps. ........................................ 43
Entrance and exit angles for the D1 magnet. ......................................................... 55
Entrance and exit angles for the D2 magnet. ......................................................... 56
Previous measurements of 10Li states below -Sn=1.0 MeV .................................. 63

Reaction cross sections from the 10Li experiment. The uncertainties on the cross
sections are purely statistical. A 10-50% systematic error from the beam current

measurement is not included .................................................................................. 84
Fitted parameters for 10L1 energy spectrum. ......................................... . ............... 101
Experimental observations of states in 23Al. ....................................................... 127
Experimental observations of states in 27F. ......................................................... 127
IMME coefﬁcients for the A=23 and A=27, T=3/2 isobaric multiplets. ............. 132
Resonance arameters the levels that dominate resonant proton capture in the

22Mg(p,y)2 A1 reaction. ....................................................................................... 133
Resonance parameters the levels that dominate resonant proton capture in the

26Si(p,y)27P reaction. ........................................................................................... 133
Smoothing parameters and their meanings .......................................................... 143
Inverse map coefﬁcients for the 10Li experiment. ............................................... 151
Inverse map coefﬁcients for the rp-process nuclei experiment. .......................... 153

xvii

 

t :1

Chapte

[nu-odu

1.1 Mass me
As the in

tail techniques art
lei'els are two of
lowest energy stat
meat high precisi
The impor

The understandin c

Chapter 1

Introduction

1.1 Mass measurement of exotic nuclei

As the interest in nuclei farther from stability increases, new and better experimen-
tal techniques are being developed to study them. The mass of a nucleus and its energy
levels are two of its most fundamental properties. Measurement of atomic masses and the
lowest energy states provide invaluable nuclear structure information. As in any measure-
ment, high precision is an important and challenging aspect of the measurement.

The importance of mass measurements can be illustrated through a few examples.
The understanding of stellar processes, which are powered by nuclear reactions, requires
knowledge of the network of nuclear reactions and their rates, and they in turn depend on
each reaction Q-value and, hence, the nuclear masses. Nucleosynthesis and elemental
abundance are largely governed by nuclear binding energies and are often very sensitive to
small mass or excited state uncertainties. High precision mass and energy level measure-
ments are therefore required to constrain the complex models used to understand many
topics ranging from stellar evolution to nucleosynthesis and elemental abundances.

The nuclear shell model uses experimentally determined masses and energy levels

as two of 11.3 mOS
are extremely “5C
and 40Ca illustrat
nucleus is apme
There are
Direct mass mea~
tion based on its
State masses. 111g
neously. Often ‘
relates the mass it
ciple it is possibl
Direct measurem
the elilk‘rimental
Smd‘i‘mg nuclei 1‘;
There are

S a remnant

as two of its most important pieces of data, so precisely known masses and energy levels
are extremely useful. For example, the relatively small masses of nuclei such as 4He, 160
and 40Ca illustrate shell closure effects and suggest that a shell model description of the
nucleus is appropriate.

There are many ways to measure a nuclear mass and low-lying energy states.
Direct mass measurements, in which the nucleus of interest is detected following separa-
tion based on its mass-to-charge ratio, has been used in many cases to determine ground
state masses. High precision is possible if other well-known masses are detected simulta-

neously. Often the separator is a magnetic system and the equation 1%) = YE = E

q (1)
relates the mass to the magnetic ﬁeld, the ion’s speed and its cyclotron frequency. In prin-
ciple it is possible to measure all of these quantities absolutely and with high precision.
Direct measurements are limited to nuclei that live long enough to be transmitted through
the experimental apparatus, typically T1 /2 2 1 us. This is a distinct disadvantage when
studying nuclei far from stability where half-lives can be 10'6 - 10'21 seconds.

There are many direct mass measurement techniques, some of which rely on mea-
suring a resonant frequency for circulation in a ﬁeld or a flight time. Atom traps are exam-
ples of mass measurement through frequency measurement. Atoms stored in an
electromagnetic trap reside in a potential minimum characterized by the ﬁelds and the
ion’s mass to charge ratio. Measuring their characteristic circulation frequency provides a
mass measurement. Penning trap mass measurements rely on this technique and have
been used to measure masses at a precision of 10'7 [Mi97]. However, high resolution Pen-

ning trap mass measurements are limited to ions with half-lives tens of seconds or greater.

The ESR at GSI also uses frequency measurement to deduce mass [Mi97]. Frag-

..-- c ‘C’- a; ‘-

 

HF ..
‘WTcments. The (

mentatic
when? ms

is U550. to

me t‘requs’“
ratio. OR
good calibre”
the number of
ited to relatiyci
Time of
produce and scp
argues have be.
GA.\'1L[.\1i97}. T
beamline. The m

velocity) are meast

teal time measuren'

11110311“)4 mass res

Adyantages to this 1

s ..
Inultaneously it ith I

lll‘ . '
65 of the isotopes 0

Nuclear mass '

or .
>> 01 one of the pam

In a
the Case of alpha dc.

mentation is used to produce secondary beams which are transported to a storage ring
where they are cooled and circulate for a few to a few hundred turns. A small metal probe
is used to detect the ions as they pass, and the time spectrum is Fourier transformed into
the frequency domain. The frequency spectrum translates direCtly into charge to mass
ratio. Circulating ions with known masses with the ions of unknown mass provides a
good calibration. Typical mass precisions of 6rn/m~104-10'7 are possible depending on
the number of turns and other factors. Because of the cooling time, this technique is lim-
ited to relatively long-lived isotopes.

Time of ﬂight techniques have been extended to exotic nuclei through the ability to
produce and separate isotopes simultaneously with isotopes of known mass. These tech-
niques have been used at such facilities as TOFI at Los Alamos and SPEG at
GAN1L[Mi97]. The nuclei are produced in a nuclear reaction and continue forward into a
beamline. The magnetic rigidity (Bo) and ﬂight time over a known distance (i.e., the
velocity) are measured and the radio provides a measure of the mass-to-charge ratio. Typ-
ical time measurement resolution of 200 ps and typical ﬂight times of 600 ns translates
into 3x10'4 mass resolution if the magnetic rigidity of the ion can be measured to ~1x10'4.
Advantages to this technique are that other nuclei with known masses can be detected
simultaneously with the unknown masses which provides a good calibration, and the half-
lives of the isotopes only have to be of the order or microseconds or longer.

Nuclear mass and excited state measurements are also possible through Q-value
measurements. The Q-value of a nuclear decay or reaction can be used to determine the
mass of one of the participants provided the masses of the others are known. For example,

in the case of alpha decay, the Q-value is Q = M P — M D - Ma where Mp , MD, and Ma

 

are the masses of
ofthe daughter nt
measured. the par
1.2 Mass meat
A method
multinucleon tran
nuclei and throug
value of the reacti
mass of atom A.
mass of the fourth
determined by me;
The adyant
Micipants. thus r
ing to a mass prc.
masses of nuclei it
nuclei; unbound R.

resrdual nucleus car

i
.1. Examples of rea
nch nuclei are the 1

.p'
tulle the (1p) [Dah’

are the masses of the parent and daughter atoms and the mass of the 4He atom. If the mass
of the daughter nucleus is known and the energy of the emitted alpha particle (Q-value) is
measured, the parent mass can be inferred.

1.2 Mass measurement using binary transfer reactions

A method that has been in use for several decades now is the binary, single- or
multinucleon transfer reaction. The method involves striking a target with a beam of
nuclei and through nucleon transfer, populating states in the unknown nucleus. The Q-
value of the reaction A(X,Y)B is given by Q=m(A)+m(X)-m(B)—m(Y), where m(A) is the
mass of atom A. Knowing the mass of any three and measuring the Q-value gives the
mass of the fourth nucleus. The mass of either the residual (B) or the ejectile (Y) can be
determined by measuring the momentum of the ejectile.

The advantage to this method is that nuclei with well known masses can be used as
participants, thus reducing the measured error. Typical mass errors can be 1-50 keV lead-
ing to a mass precision of 6m/msl.0x10’6. The method can be used to measure the
masses of nuclei with lifetimes much less than a microsecond and is not limited to bound
nuclei; unbound resonances can be measured with this technique because the mass of the
residual nucleus can be measured by measuring the ejectile’s energy (see Chapters 4 and
5). Examples of reactions that have been used successfully in the past for reaching proton
rich nuclei are the (p,d), (p,t) [Mi97], (7Li,8He) [W188], and (3He,8Li) [Be77] reactions
while the (t,p) [Da85], (t,3He) [St69], (13C,12N) [13097], and (711813) [Yo94] reactions
have been used for reaching neutron rich nuclei.

In principle there is no limit to the number of transferred nucleons or the Q-value.

However, the cross sections become prohibitively small much beyond 3-nucleon transfer

l”

 

f 1E'_i.
I "Em“ ruin". “I"‘EM' A.Y'.'-"' ‘17“—

or Q-yalues mt
using the 40C at
sr[l{099l and
ofthe 3800. .~‘
special targets
sibie to manu‘
Atten‘
precision of
turn. The 1r
Smements it
Snppression
1(6):" are pg
“Permian
latge solid
35500 an E
1.3 Out
‘.\

VET}. den
sis, and

{“3 “res.

or Q-values much less than -40 MeV. For example, a recent 36Ca mass measurement
using the 40Ca(4He,8He)36Ca reaction (Q-value=-57.6 MeV) has a cross section of ~1 nb/
sr [K099] and was only possible through the large solid angle and high energy resolution
of the S800. Another disadvantage is that accessing nuclei farther from stability requires
special targets of rare or radioactive isotopes which may be difﬁcult or practically impos-
sible to manufacture.

Attempting to measure the mass of a residual nucleus with a transfer reaction to a
precision of slxlO‘6 crucially depends on precise measurement of the ejectile’s momen-
tum. The magnetic spectrograph is a particularly attractive option for doing mass mea-
surements in the case of low cross section reactions because they provide high background
suppression, large acceptance, and high energy resolution. Energy resolutions of 1-50
keV are possible leading to mass precision of ~10'7. The 8800 spectrograph is a new
experimental device which meets all these criteria. Its combination of high resolution,
large solid angle and momentum acceptance, and good background suppression makes the
8800 an excellent device for transfer reaction mass measurements.

1.3 Outline

Magnetic spectrographs can be thought of as nuclear microscopes: they look in a
very detailed way at the microscopic properties of a nucleus, such as masses, energy lev-
els, and angular momentum. Chapter 2 describes the motivation and the key features of
the new magnetic spectrograph, the S800, built at Michigan State University.

Chapter 3 describes a task necessary for obtaining the high resolution required in
the spectrograph: magnetic ﬁeld measurement. The spectrograph consists of large mag-

netic quadrupoles and dipoles, and the goal was to measure precisely and accurately the

 

magnetic ﬁt
acquisition
in this thesi:
Chat
trograph to
”)Li. which
detailed baci
nucleus with
irnental metl
be outlined.
Chap
play Signiﬁcc
explaining W

Presented anc

magnetic ﬁeld proﬁles of each element in the spectrograph. The general techniques, data
acquisition and data analysis will be outlined. The spectrograph dipoles will be discussed
in this thesis. Quadrupole mapping was described in Bo Zhang’s dissertation [Zh97].

Chapter 4 is the ﬁrst of two chapters on mass measurements made with the spec-
trograph to study nuclei far from stability. Chapter 4 is devoted to the unbound nucleus,
10Li, which is important for applying three body models to the halo nucleus 11Li. A
detailed background is given on the status of 10Li measurements. The desire to study this
nucleus with a spectrograph is described, and then the measurement techniques and exper-
imental methods necessary to perform a spectroscopic measurement of states in 10Li will
be outlined.

Chapter 5 describes spectroscopic measurements of two proton rich nuclei that
play signiﬁcant roles in the astrophysical rp process. Background information is presented
explaining why the measurement is necessary, the experiment is described, and the results

presented and discussed. Finally, chapter 6 summarizes the dissertation.

.".
sun. ‘

 

Chapter 2

The 8800

 

 

 

2.1 Magnetic 51
Magnetic 81‘
in nuclear physics.
study nuclear struct
primary adyanta g
terns are back groun
suppression is a yir
cross section meast
interesting eyents.
Particle. neutron. o

imicularly import

4

Chapter 2

The 8800 Spectrograph

2.1 Magnetic spectrographs

Magnetic spectrographs have become an invaluable tool for precise measurements
in nuclear physics. They are generally used as the equivalent of a nuclear microscope to
study nuclear structure. For a thorough review of magnetic spectrographs, see [En79]. The
primary advantages of spectrographs as compared to other types of particle detector sys-
tems are background suppression and very precise momentum determination. Background
suppression is a virtue in modern heavy ion physics experiments because it facilitates low
cross section measurements in the presence of background that would otherwise hide the
interesting events. Spectrographs are good for coincidence experiments with charged
particle, neutron, or gamma ray detectors placed at the target. These characteristics are
particularly important as nuclear physicists strive to learn more about very proton- and
neutron-rich nuclei.

A magnetic spectrograph is an optical system for charged particles. It has an object
and a focal plane, and it contains elements which bend and focus ions like lenses and

prisms do for light. Magnetic spectrographs can be used to measure the momentum of an

 

ion using the full
speed ‘1 passing 1
lar path with rad'.‘

qi-B. and thus 81“

 

mass and 3! is the 1
particles with iden
magnet.
Spectrogra
spectrograph can 1
tie to design the 1
the spectrograph
nuclear physics e
Products. which n
2.2 Character
2.2.l Beamline
The $801
AE=lO.000~ p 91
and solid angle a

Figure 2.

ion using the following principle. A particle with charge, q, and mass, m, traveling at
speed, v, passing through a uniform magnetic ﬁeld with strength, B, will travel in a circu-
lar path with radius p given by balancing the centripetal and magnetic forces, mvz/p=
qu, and thus Bp=p/q, with p=mv. Relativistically, the mass is ‘ym, where m is the rest
mass and y is the Lorentz transformation factor. Thus, for a given magnetic ﬁeld setting,
particles with identical momentum to charge ratios are deﬂected the same amount by the
magnet.

Spectrographs can also measure the angles of emission of the particles. First, the
spectrograph can be set to cover a limited angular range, say 1011°. However, it is possi-
ble to design the optics of the system the angles of each individual beam particle within
the spectrograph acceptance is measured (see section 2.3.2). Measurement of angles in
nuclear physics experiments can be used to determine angular distributions of reaction
products, which may be used to deduce the spin and parity of the state they are populating.
2.2 Characteristics of the S800
2.2.] Beamline and magnets

The 8800 was designed to have high energy and momentum resolution (E/
AE=10,000, p/Ap=20,000), good angular resolution (AO=2 mrad), and large momentum
and solid angle acceptance (9:20 msr, Ap=6%).

Figure 2.1 shows a schematic ﬁgure of the 8800. The S800 consists of an analysis
beamline (from the object to the target chamber), and a large spectrograph. The analysis
line is very similar to the A1200 fragment separator at MSU [Sh92], but with better reso-
lution and larger solid angle acceptance. Table 2.1 summarizes the features of the S800

spectrograph.

I. "- 'w' 1’33“” 'ﬁ

 

The
rupoles. and
object used 1
focus in it an
position. but

the beams ir

object
i
[“1
r“. L

 

Figure 2.1-

ments fTOm 1h

dCMTlSITEam f]

The analysis line consists of four 22.5° dipoles, ﬁve quadrupole triplets, four sex-
tupoles, and one quadrupole doublet. The location labeled “object” in Figure 2.1 is the
object used in high resolution, primary beam experiments. The intermediate image has a
focus in x and y and a dispersion of 1.5 cm/%. There is also a focus in x and y at the target
position, but the dispersion is very high, ~12 cm/%. When making or transferring radioac-

tive beams in the analysis line, the production target is the object position.

 

[ETECTOR

  
  
   
 
   

focal plane

SE XI’ um

INTERUEUATE

 

 

dispersive
image and
target

 

 

 

Figure 2.1: Schematic of the S800 spectrograph. The analysis line consists of those ele-
ments from the object position to the target chamber; the spectrograph itself is the portion
downstream from the target chamber.

 

 

Yi Yf
X1 I Xf I
Angles:
y x
id, i4
2 2

Figure 2.2: Schematic of optical coordinate system, before entering system of magnets M
and after leaving it. The xz plane at y=0 is deﬁned as the midplane.

 

 

 

 

10

 

 

 

Tabl
We
Dip
TE
D1;
D1;
Dip
Dip
Elie

\
Ma.

 

Trir

 

Mai
\

Table 2.1: Features of the 8800 spectrograph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Energy, momentum resolution AE/E=104, Ap/p=5x10'5
Energy, momentum range 11.6%, 5.8%

Solid angle 20 msr
Angular resolution 5 2 mrad
Horizontal dispersion (x16) 9.5 cm/%
Vertical dispersion (yl¢) 0.9 mm/mrad
Horizontal magniﬁcation (xlx) 0.74

Focal plane size 55 cm (hor.) x 15 cm (ver.)
Detector separation 1 meter

Focal plane tilt 285°
Maximum rigidity 4.0 T—m
Maximum dipole ﬁeld 1.6 Tesla

Dipole bend radius 2.8 m

Dipole bend angle 75°
Horizontal detector resolution 0.3 mm

Vertical detector resolution 0.3 mm

 

 

 

 

Table 2.2: Parameters of the D1 and D2 magnets.

 

 

 

 

 

 

 

 

 

 

 

Weight 70 Tons each

Dipole gap 15 cm

Maximum central ﬁeld 1.6 Tesla

Dipole bend radius 2.8 m

Dipole bend angle 75°

Dipole entrance angle Dl=0°; D2 = 30°

Dipole exit angle D1=-30°; D2=0°

Effective length (nominal) 3.7 111

Maximum current 450 A

Trim coil setting 43.75% of main current
(1% of main coil A-Tums)

Maximum trim coil current 250 A

 

 

 

 

ll

 

 

 

The beamli
tion [Zh97]. The tr‘
irnum ﬁeld gradien
tons each.

Table 2.2 s
magnets. The dipoli
mum central ﬁelds .
We the absolute ﬁt

and bend angles are

 

Table 2.3: Parameters of the Q1/Q2 quadrupole doublet.

 

 

 

 

 

 

 

 

 

Weight 5 Tons

Pole tip diameter Q1=24 cm, Q2~=42 cm
Maximum ﬁeld gradient Q1=19.7 T/m, Q2=7.5 T/m
Maximum current Ql=85 A, Q2: 100 A
Length of pole tips Q1=30cm, Q2=30 cm
Effective length (nominal) 40 cm for both
Sextupole location Q2

Maximum sextupole current 9 A

 

 

 

The beamline quadrupoles triplets are described in detail in Bo Zhang’s disserta-
tion [Zh97]. The triplets have a 6 inch bore for large acceptance. They are capable of max-
imum ﬁeld gradients of 35 T/m and maximum currents of 100 Amps. The triplets weigh 5
tons each.

Table 2.2 summarizes the features of the superconducting spectrograph dipole
magnets. The dipoles each weigh 70 Tons and have 6”x36” gaps. The dipoles have maxi-
mum central ﬁelds of 1.6 Tesla and NMR probes are placed in a ﬂat ﬁeld region to mea-
sure the absolute ﬁeld setting of the dipoles during operation. The bend radii are 2.8 m,
and bend angles are 75°. There are 30° edge angles on the dipole steel which defocus ions
in the x direction while focussing them in the y direction, i.e., they act as quadrupoles.
The dipole power supplies deliver up to 450 Amps to the coil and are stable to 1 part in
105. Trim coils are present on the inner and outer radii of the dipoles to homogenize the
ﬁeld near the edges of the magnet. In the event of cryogenic failure, a coil protection

switch takes the power supply out of the circuit and reroutes the current from the coils

12

 

through a dissip
into this resistor
Table 2.;
design is describ
heat load of 2 \\
with no damage
tion to maximizt
to measure the ﬁ
There is
rect the dominar
10“"D'lng excite
to a known shap
in the focal plan
2.2.2 Detector
The steam
bet fOr energy l
This detector 5}.
Seated in an 31m
1101] requira
men

diw ..
Missed later 1.]

through a dissipating resistor. At maximum current, 1.1 M] of stored energy is dumped
into this resistor with a decay constant of much less than a minute.

Table 2.3 summarizes the features of the spectrograph quadrupole doublet. Its
design is described extensively in [Zh97]. It is a superconducting. magnet with a cryogenic
heat load of 2 Watts. There is no protection circuit on the quadrupoles as they can quench
with no damage to the coils. The doublet focuses in the y direction ﬁrst, then the x direc-
tion to maximize the acceptance of the spectrograph. Hall probes are ﬁxed to the pole tips
to measure the ﬁeld gradient during operation.

There is a sextupole coil conﬁguration around the bore tube of Q2. This is to cor-
rect the dominant (XIOZ) spectrograph aberration in speciﬁc experiments such as studies of
low-lying excited states via inelastic scattering. This will reduce the curvature of the lines
to a known shape corresponding to the (xkbz) aberration, and a beam blocker can be placed
in the focal plane to block the dominant elastic scattering.

2.2.2 Detectors

The standard S800 detector system consists of two position detectors, an ion cham-
ber for energy loss measurement, and three scintillators for total energy measurement.
This detector system and electronics will be described in detail in Chapter 4, and are pre-
sented in an annual report contribution by J. Yurkon and collaborators [Yu98]. The resolu-
tion requirements and their impact on spectrograph energy and angle resolution will be
discussed later in this chapter.

2.3 Optics of the 8800
2.3.1 Notation

The ions are guided through the beamline by the magnetic elements. The optical

13

 

coordinates of t1
jectory passes th
by the 1’0:qu
dipoles. B is the
Figure 3.
the distance alor.
pendicular to z a
dipoles bend the
defined in this co
angle in the non-
The Coordinate ]
an“ Particle. 1 =
AS the 10:
the action of the
the ﬁnal COOFdin;
about how the ior

matrix can be (let

that ' ,
Tlx eqUHIIOD i,‘

coordinates of the ions are described in relation to a central, reference trajectory. This tra-
jectory passes through the center of the magnets and has the reference momentum, deﬁned
by the p0=qu0 for the dipole elements, where p0 is the central bend radius for the
dipoles, B is the dipole ﬁeld, and q is the ionic charge.

Figure 2.2 illustrates the coordinate system used. The coordinate z is deﬁned as
the distance along the reference path at any point in the motion. The x coordinate is per-
pendicular to 2 at all points along the path, and deﬁnes the dispersive direction, i.e., the
dipoles bend the ions in the xz-plane (labeled horizontal by convention). Two angles are
deﬁned in this coordinate system. Theta is the angle in the dispersive plane, and phi is the

angle in the non-dispersive plane. The momentum coordinate is deﬁned as 5 = 11 - .

P0
The coordinate l is the distance traveled along the central trajectory in relation to the refer-

ence particle, l = 11 — l .
0

As the ions pass through the magnetic elements, their coordinates change due to
the action of the magnetic ﬁelds on them. A simple transfer matrix can be'used to relate
the ﬁnal coordinates to the initial coordinates. The transfer matrix contains information
about how the ions travel through the ﬁelds in the system. If the ﬁelds are known, then the
matrix can be determined by solving the Lorentz differential equation. To ﬁrst order, the
matrix equation is:

x, “(xixi (x16) (XIy) (XI¢) (XII) (xiai RI

9, (Glx) (9|9)(9|)')(9l¢) (9|1)(9l5) 9,-

y, = (YIX)(yl9)(y|y)(yl<l>)(YI|)(y|5) y.- (21)
¢; (¢Ix)(¢|9)(¢|y)(¢lt>)(¢II)(¢|6> «i '
1; (IIX) (ll9) (lly) (||¢) (Ill) (llé) I.-

5n _(5|X) (ble) (5W) (5W) (5“) W5), 92

~

 

 

 

 

 

 

The transfer matrix equations take on a polynomial form when expressed as a Tay-

14

.25”! H ‘
ﬁnd

 

I

loreXPEmSion a

where xf l5 the
For examplt th

equation for the

.r, = (X;
where the i and
principle the eq
having a smalle
zero. For examr
zero in Eq. 2.1.
23.2 Optical;

The ﬁrst
and parallel-to-p
51ml terms dom
Scattering angle :
PﬂSion has unit
trograph is 9.5 e

x, 0m.

6 ﬁrSI(

lor expansion about the central ray for only one ﬁnal variable. The form is:

l ai+j+k+l+m+n

x,
iljlk!1!m!n!axiaejayka¢lalmaan (12>

 

(xflxiejyk¢llm6n) =

where xf is the ﬁnal coordinate and the derivatives are with respect to initial coordinates.

2
la—ﬁ. The

For example, the (xIBZ) term has i,k,|,m,n=0 and j=2 and is given by (x | 92) = 569

equation for the transformation of the x coordinate:

x, = (xlx)x,- + (x|8)9,-+ (xla)o,.+ (x|6¢)9,¢, + (xloo)o,.a,.+ (x|92)9,-2+ (2. 3)
where the i and f subscripts indicate the initial and ﬁnal coordinates, respectively, and in
principle the equation extends up to inﬁnite order, with each successive order hopefully
having a smaller affect. If a system has midplane symmetry, all terms with (k+1) odd are
zero. For example, terms like (xly), (XI¢), (xly3), (ylx), and (CNS) are not allowed and are
zero in Eq. 2.1.

2.3.2 Optical properties of the S800

The ﬁrst order optics of the S800 are point to point in the x direction, i.e. (x|9)=0,
and parallel-to-point in the y direction, i.e. (yly)=0. In addition, the (yltp) and (xlb) (disper-
sion) terms dominate so the y position measurement is an accurate measurement of the
scattering angle and the x position is an accurate measurement of the momentum. The dis-
persion has units of distance/(relative momentum in %). The dispersion of the spec-
trograph is 9.5 cm/%. Hence, two rays having the same initial positions and angles but
differing in momentum by 1% will be separated by 9.5 cm at the focal plane.

The ﬁrst order optics determine the momentum and angular resolution of the sys—
tem as well as the measurement of certain other quantities. The momentum resolution of

the system is deﬁned as the reciprocal of the momentum deviation at which two peaks are

15

 

 

where (‘ll‘l is‘
s}'Stem add [ht
correction i5 ‘0

One Pt"
spread. Primdf
spreads of 0.1‘
trograph is nor
pensate for this
require that the
target position t
ﬁrst order. by x‘:
sic dispersion ir
focal plane of tl‘
denote the bean
311le 6) is the (

T1131 the ﬁnal Spc

separated by their widths:

 

_ (XI5)
R - (xlx)x0+Ax

(2. 4)
where (xlx) is the magniﬁcation and x0 is the object spot size. Aberrations present in the
system add the Ax contribution to the denominator of Eq. 2.4. The goal of aberration
correction is to make that contribution zero; this is the theoretical resolution limit.

One possible limitation to the resolution of the $800 is the intrinsic beam energy
spread. Primary beams from the K1200 cyclotron at the NSCL have intrinsic energy
spreads of 0.1% while secondary beams have 3% or more energy spread. The spec-
trograph is normally operated in dispersion-matched, energy-loss mode [N089] to com-
pensate for this ﬁnite beam energy spread. The optical conditions for dispersion matching
require that the ﬁnal image have zero width, in the following manner. The spot size at the
target position (dispersive image) between the beamline and the spectrograph is given, to
ﬁrst order, by xb=(x|x)bx0+(x|6)bb where x0 is the object beam spot size andb is the intrin-
sic dispersion in the beam, assuming a ﬁrst order focus where (xl0)=0. The spot size at the
focal plane of the spectrograph is then given by xs=(x|x)sxb+(x|6)so. The b and s subscripts
denote the beamline and spectrograph, respectively. The term (xlx) is the magniﬁcation M
and (xlo) is the dispersion D, and 6 is the intrinsic beam momentum spread. The condition
that the ﬁnal spot size is negligibly small, i.e. xS=O implies the condition Msxb = -D56. The

initial spot size is assumed to be negligibly small so that x0=0 and xb=Dbb. Thus, the dis-

persion matching condition becomes:

_DS

Db = M3

(2. 5)

 

An excellent diagnostic for this condition in the S800 is a plot of the beam’s time—of-ﬁight

16

 

waves are“

“.4 1".

versus the X P07
intrinsic mome
position has no
2.3.2.1 Detect
The rest
design. In order
horizontal resolt
The measured a.
tion measuremei
lutions listed in
error of 0.3 mm
Aberratic
ﬁed by a high or.
”0“ 1* 82)=0.ooe
the detector re“
B‘(tﬂmram 0,4 It:
sortable eStlmate

acce

ptable prec 1 S

versus the x position at the focal plane (momentum). The TOF will vary according to the
intrinsic momentum spread in the beam. The system is dispersion matched if the x
position has no dependence on the TOF.
2.3.2.1 Detector resolution considerations

The resolution requirements of the spectrograph were considered in the detector
design. In order to achieve the momentum resolution requirements given in Table 2.1, the

horizontal resolution of the detectors should be better than 0.005% * 95mm/%= 0.475mm.

xr‘xz
D

 

The measured angles are given by 0 = atan( ) , where X] and x2 are the two posi-
tion measurements and D is the detector separation. The speciﬁed detector position reso-
lutions listed in Table 2.1 correspond to a momentum precision of 0.003% and an angle
error of 0.3 mrad.

Aberration correction requirements must be considered. Any angle error is magni-
ﬁed by a high order aberration in the system. For example, consider the dominant aberra-
tion (x|62)=0.006 mm/mradz. Given the 0.4 mrad angle measurement error as speciﬁed by
the detector resolution, this translates into a worst case (i.e. if all events came with
0=60mrad) 0.4 mm error from the aberration, which is considered marginal. A more rea-
sonable estimate would be if all events were evenly spread over the focal plane, giving an
acceptable precision of 0.15 mm. Similar arguments can be made for the detector resolu—
tion considering the angular resolution requirements. A detailed study of these effects was
made prior to construction of the S800 and is given in [N089].
2.3.2.2 Aberration Correction

In a magnetic spectrograph with large solid angle acceptance such as the 8800,

higher order aberrations signiﬁcantly affect image quality, in both momentum and angle.

17

“"71

5;”;1..‘.‘

 

The dominant
focal plane. I
affecting the CI
rations such as
dominant angu

Both h:
tions in the SM
tions involx'e l
multipoles not
duced. and the
determined that
angular and em
thart building In
accurately and \
THIEly Calculate
tain constraintS

measurem CI] [5 Vt

The dominant aberrations are the angular aberrations because they blur the image in the
focal plane. These are the (xl0ncpm) terms where n+m22. The largest optical aberrations
affecting the energy resolution are the (xl02), (xloz), and the (xl03) terms. Chromatic aber-
rations such as (xl06) and (x152) are also substantial. Angular resolution is degraded by the
dominant angular aberration (ylytp).

Both hardware and software methods were evaluated for the correction of aberra-
tions in the 8800 [N089]. Because of the large solid angle in the 8800, hardware correc-
tions involve building large multipole magnets which can be costly. In addition, the
multipoles would not be manufactured perfectly, and other aberrations could be intro-
duced, and the need for software corrections still might be necessary. A detailed study
determined that software correction for the 8800 would be sufﬁcient to achieve the desired
angular and energy resolution [Be93]. Soﬁware correction is much more cost effective
than building multipoles but comes at a price; the ﬁelds of the magnets need to be known
accurately and with great precision. As speciﬁed in the article, aberrationsimust be accu-
rately calculated up to 7th order to reach the design energy resolution which places cer-
tain constraints on the ﬁeld knowledge. Chapter 3 will describe why and how ﬁeld

measurements were made.

18

 

C113.

Dipol

3.1 Mom:
The 34‘

through it. I“
known. then an
the Lorentz fort
final momenta at
expressed as a n:
Each element in i
motion changes as
the total transfonna
Knowledge 1
llt‘ti‘atislated into mr;

and scattering ang]

bar and

C‘.

the emerging

and 330

K

.165 are measu-

 

Chapter 3

Dipole Magnetic Field Measurements

3.1 Motivation for the measurements

The 8800 spectrograph makes use of magnetic elements to direct the ions that pass
through it. If the magnetic ﬁelds of all the magnetic elements at all points in space are
known, then an ion traveling through the ﬁelds can be tracked analytically, as they obey
the Lorentz force law dp/dt=q(E + va). Integrating the equations of motion provides the
ﬁnal momenta and positions for a given set of initial conditions. This relationship can be
expressed as a matrix transformation of the initial coordinates to the ﬁnal coordinates.
Each element in the beamline has its own transfer matrix that describes how an ion’s
motion changes as it travels through the element. For a sequence of consecutive elements,
the total transformation is just the product of transfer matrices.

Knowledge of these transfer matrices allows measurement of ﬁnal coordinates to
be translated into more meaningful nuclear physics quantities such as momentum, energy,
and scattering angle. Speciﬁcally, in the S800, nuclear reactions occur in the target cham-
ber and the emerging ions from the reaction are detected in the focal plane. Their positions

and angles are measured in the two transverse directions. If the transfer matrix is known,

19

I - il' —-V.._.__ ,,_‘_ _

then it can be
it left the targt
quantities in tl
as ray-reconsti
The sp

to four dift‘erei

where the SUE
chamber. respl
in the dispersi
describe the it
be negligible.
negligible. Tl

then it can be inverted to arrive at initial coordinates, such as the energy the ion had when
it left the target and angle through which it was scattered. This process of reconstructing
quantities in the target from the measured ﬁnal coordinates in the focal plane is referred to
as ray-reconstruction.

The spectrograph measures four quantities in the focal plane, and these are related

to four different quantities in the target chamber by a matrix equation:

 

 

 

xf 0i
9 .
f = S y' (3.1)
Yf ‘l’i
.‘l’o .5L

 

where the subscripts f and i denote the measurements in the focal plane and the target
chamber, respectively. The coordinates (x, 0, y, o) are the two transverse positions, angles
in the dispersive and non-dispersive directions, respectively. The coordinates (0, y, (p, 6)
describe the initial coordinates in the target chamber. The target spot size x0 is assumed to
be negligible, or equivalently that the incoherent width of the dispersion matched beam is
negligible. Theta, phi and y carry the same meaning as in the focal plane; 5 is the
fractional kinetic energy difference from the central ray, b=(E-EO)/E0. The knowledge of
the quantities in the target chamber now requires that this matrix equation be inverted. The
inversion procedure is complicated and described in detail in [Be93]. The result is a
matrix, R, which is not true inverse of S, but can be used to deduce energy and angles in

the target chamber:

20

This ”“96
Kn

magnetic ﬁ‘
men! in C0:
speciﬁcation
ei‘eQV'her e i
system. This 1
nates than in J
tem requires it
The prc‘

along the beam
ofdescribing tht
tighter constrain
and depend on th«
This cha pt

3.2 Data Acqui

31.1 General

 

 

 

 

0i xf
- 9
y' = R f (3. 2)
(l), Yf
Pi. Elf.

This inversion process includes the ﬁrst order optics and all the optical aberrations as well.

Knowledge of this transfer matrix requires precise and accurate knowledge of the
magnetic ﬁeld data in the elements. The ﬁeld data was intended to be used in the MF ele-
ment in COSY [Be95]. If the system has midplane symmetry as the 8800 does, then only
speciﬁcation of the midplane ﬁelds is necessary for complete description of the ﬁelds
everywhere in the space of the magnet gap. COSY requires ﬁelds in a Cartesian coordinate
system. This is easily understood: Maxwell’s equations are far simpler in cartesian coordi-
nates than in any other coordinate system. Thus, taking data in any other coordinate sys-
tem requires interpolation onto a cartesian grid.

The precision and accuracy goal of the mapping process was 1 Gauss at all points
along the beam path. However, it was shown that using the Gaussian image charge method
of describing the data in COSY, 10 Gauss of random noise is allowable [De92]. There are
tighter constraints on systematic errors as COSY may interpret them as ﬁeld structures,
and depend on the exact form of the error, but they are certainly 510 Gauss.

This chapter describes the techniques used for mapping.
3.2 Data Acquisition

3.2.1 General

The goal of the mapping project was to measure the ﬁelds in the dipole gaps to 1

Gauss accuracy, i.e., measure the ﬁelds as a function of position, B(x,y,z). The ﬁeld mea-

21

suremer
induced
detennir
are made
ing teeth
lute posi
strong. pt
Tl
coordinat:
central be:
perpendiu
dipole.
The
(one for ear
Control boa

dlPOles.

The
”'0 Passes t;

mat the eﬂtin

surement technique was to pass a coil of wire through the magnetic ﬁeld and measure the
induced voltage, e=-d<I>/dt, where (I) is the magnetic ﬂux through the coil. The ﬁeld is
determined along the path by time integral of the voltage. Field calibration measurements
are made with NMR and Hall probes. The relative position measurement is done by read-
ing teeth on a ﬁxed strip along the coils’ path and will be described in section 3.3.5. Abso-
lute position calibration (relative to the magnet steel) is achieved by mapping small,
strong, permanent magnets placed on the steel edges and is described in section 3.3.6.

The mapping coordinate system was chosen to be cylindrical, which is the natural
coordinate system of the dipole magnets. The radial coordinate was perpendicular to the
central beam’s path, the angular coordinate parallel to the beams path, and the z coordinate
perpendicular to both of those and parallel with the magnetic ﬁeld lines in the center of the
dipole.

The data acquisition system consisted of a 386-based personal computer, 5 ADC’s
(one for each induction coil), a scaler board (to count teeth and measure time), and a motor
control board to control the steppor motors that moved the cart back and forth in the
dipoles.

The general data acquisition algorithm was controlled by the PC. The coils make
two passes through the ﬁeld at a ﬁxed radius. Once a lap is completed the coils are moved
a ﬁxed radial distance and another lap is started. It is through this rasterization procedure
that the entire dipole gap is mapped.

The ﬁeld measurements were taken at roughly one million individual points on a
polar grid of 2.3 mm radial pitch by 0.05° angular pitch at 5 different vertical positions.

Each sample contains position and time information for the coil package, and the 5 indi-

22

“dual col.
data during
data on its
3.2.2 Mal
3.2.2.] All
The
frame is 0.5.
central radiu
held in place
stainless stee
plate and
on the plate to
The co
is mounted on ‘
rectangular pic
pinch TOllers, to
forth 01] an arc (
one on each con
in hag-ht and deg

Only One

sequentially, The r

vidual coil voltages. A PC controlled the motion of the coils through the ﬁeld and took
data during the mapping cycle. Once a map was started, the system could be left to take
data on its own until it had completed measurements over the whole gap of the dipole.

3.2.2 Mapper hardware and electronics

3.2.2.1 Aluminum frame and cart

The mapper was designed to ﬁt in the gap of the two dipoles (see Figure 3.1). The
frame is 0.5” thick aluminum plate, 27” across, with a central radius of 2.7 m, to match the
central radius of the dipoles, and an angular extent of 110 degrees. The plate was ﬁrmly
held in place against one side of the dipole gap with a combination of non-magnetic
stainless steel brackets and suction. (A vacuum was applied to a large area between the
plate and the dipole steel, creating approximately 6 tons of force
on the plate to ensure it was ﬂat against the magnet face.)

The coils ride on a G-10 cart, which is designed to ride on an aluminum guide that
is mounted on the large aluminum frame that ﬁts in the dipole gap. The cartiwas a 12”x25”
rectangular piece of G-10, with aluminum hardware attached. The cart had four plastic
pinch rollers, two on each side of the aluminum guide, to enable the cart to roll back and
forth on an arc of ﬁxed radius. The cart was held ﬁxed in the y direction by eight wheels,
one on each corner of the rectangular cart, top and bottom. The top four wheels were ﬁxed
in height and designed to press ﬁrmly against one surface at all times.

Only one mapper assembly was manufactured, so the dipoles were mapped
sequentially. The mapper was ﬁrst installed in D1, and the ﬁelds were measured. The map-

per was then removed from D1 and installed in D2.

23

moto

 

 

-
.."°

 

.. guiding track
,v for cart

i ........ ,» aluminum
““““ plate

motors

Figure 3.1: The dipole mapper assembly, shown together with the magnet steel in both
The cart with the coils rides up and down along the guiding track, which is
mounted on an aluminum plate. Only one mapper assembly was used; the mapper was
installed in DI, the magnet was mapped, then the mapper was removed and installed into

24

Chain attac
point

Pinch roller

wheels

 

Chain attachment
point

 

 

 

 

 

Pinch rollers

 

wheels

 

 

 

 

motor for moving
coils in radial
direction

 

 

 

 

 

 

 

 

radial coil path

Figure 3.2: Dipole mapper cart assembly, top (left) and side (right) views. The pinch roll-
ers keep the cart on the guiding track, and the wheels keep the cart on the same plane as it
rides along the aluminum plate.

25

 

  

 

 

lll[IIIIIIIIIIIIII'IIITIITWIIIIIIrTIIIIIIIIIL

l- ’ u

p 2 (out of page) aluminum. .

: plate . :

P- -i

.- -i

h +

- a

2— .—

r- -l

P -

- -i

b d

T; - a
L- - u
3 1- ................ -
Q) I- -I
E - .
p -

P -

0 .-

.1

D d

E 1

I- d

P -i

_1—. .—

P- . -

I- i i J
lll[111L114]lLllLlllllllllLlllllllllllILLJILIJ

 

 

 

0 1 2 3
x [meters]

Figure 3.3: Schematic of the mapper showing the coordinate system used. The thick black
lines are the magnet steel, the thin lines are the mapper plate. The gray box represents the
cart that moves up and down along the plate. The three cylindrical coordinates are shown
originating from a typical coil position on the cart (2 is out of the page).

26

 

 

Figur
”Title
in th
TOT p.1-

 

Field measurementzj

 

is

V‘—

coil of wire

Ampliﬁer

 

Position measurement: I

 

Optosensor

 

WUDQDDD“:WW

oles

 

 

 

5 Position strip»
(front) .
(Side)

 

Position strip

   

_J—L_l_|_

square wave out

     

detector

 

 

Data
Stream

  
 

 

 

M torContrQl , if I

 

 

 

 

Motion
control and
power

   

  
    

 

 

Figure 3.4: Schematic of the mapper DAQ system. The PC supplies commands to the
motors and power to the on board electronics and controls event timing. The optosensor

on the cart reads holes in the position strip and sends
for position and timing information.

27

a square waveform back to the PC

9.316

Optosensor

AtDClit-‘e (\

OPTOSC‘HSU;

ADCt’l)

ADCtZ)

TStan( l )

gate

stop

 

optosensor

 

 

 

ADC live (x5)

optosensor

ADC(l)

ADC(2)

Tstart(1 )
Tstardz)

..|

 

 

 

 

 

 

 

 

T(l) -

 

 

 

 

—l:

 

 

 

 

T(2) -

 

 

 

 

 

+5V

0V

Figure 3.5: Timing diagram for one pass through the magnet. The gate signal is generated
when the cart passes a certain point and tells the PC to start or stop acquiring data. As the
optosensor passes the teeth in the position strip, it generates a square wave. Field measure-
ment is broken up into 3-tooth (0.090”) samples. Thi and T10 are the times that the wave-
form is spent in the high or low state, respectively. T(l) and T(2) are the live times for the
ADC ’s. Tsm(i) is the time it takes to start the ADC ’3 following detection of the ﬁrst tooth
in the sample. All the times mentioned are measured by counting 1 us clock ticks using a
stopwatch circuit.

28

In tl
(side with ri
the coils sta;
coils were cc
held 6.000:(
paths in the g
path inside at
a plastic chair
(attached to tl
its center of m
3.2.2.2 lndu
The im
O-OOl diametc
$0M of Wire
diameter of 0. 3
ltilool-mS)3 mea.
a ﬁeld gradient
Ulidplane‘ one b!
3.2.2.3 Analog
The ADC

The ADC Chip u

In the gap of the dipole, the springs on the bottom four wheels pressed the cart
(side with rigid wheels) ﬁrmly against the pole-tip steel, which is very ﬂat, ensuring that
the coils stayed in the same vertical plane. In the fringe ﬁeld region (outside the gap), the
coils were constrained by an aluminum plate bolted to the aluminum frame. This plate was
held 6.0002t0.005” apart, and shimmed such that the coils remained coplanar with their
paths in the gap. For example, the midplane coil remained on the midplane throughout its
path inside and outside the magnet gap. The cart was propelled through the dipole gap by
a plastic chain connected to a steppor motor. The plastic chain made a loop between a gear
(attached to the driving motor) and a sprocket. The chain was ﬁxed to the cart at roughly
its center of mass.
3.2.2.2 Induction coils

The induction coils used for ﬁeld measurement consist of very ﬁne (42 gauge or
0.001” diameter) wire spooled around a bobbin. The height of the bobbin was 0.5”, but the
spool of wire was only 0.375” tall. The wire was wrapped very tightly with a ﬁnal
diameter of 0.375”. The wires were copper, and given their diameter and resistance (5-10
kiloohms), means they were miles in length, providing a large voltage signal (~0.7 Volt for
a ﬁeld gradient of 60 Gauss/mm). There were ﬁve of these coils, one on the magnetic
midplane, one below and three above.
3.2.2.3 Analog-to-digital conversion

The ADC’s were printed circuit modules designed for use in a personal computer.
The ADC chip was an Analog Devices model number 1170. The exact workings of the
chip are proprietary, but according to the manufacturer they operate roughly as voltage-to-

frequency converters. The voltage is digitized into 24 bits, with the last four being

29

 

essential'
3.2.2-4 I
Th
during the
the)" “cm 5
such 35 RA:
ADC from 5
integration ll
Sever:
noise ratio. Tl
of the raw Sign
The supply wa
quency comp0n
Cooling 1
the entire PC C351
voltage by a factt
old) also had the e
An simple
hhhg the level of t
Slant (not ampliﬁed

r2 .
110. Consequentli

the ﬁeld setting,

 

essentially unusable. The input range is -5V to 5V.
3.2.2.4 Electronic noise suppression

The voltage induced on the coils is 0—1 Volts for the typical ﬁeld gradients present
during the normal operation of the dipole magnets. Because the ADC ’s resided in the PC,
they were subject to substantial electronic noise from various operations in the computer
such as RAM and hard drive access. Typical peak to peak noise present at the inputs of the
ADC from such sources was 300 mV. Because the ADC ’s were set to have a fairly long
integration time, Tim=16.67 ms, most of the noise was suppressed.

Several steps were taken to reduce the noise level, and thus enhance the signal to
noise ratio. They were shielding, cooling, elimination of noise sources, and ampliﬁcation
of the raw signal. Removing the switching power supply from the PC helped substantially.
The supply was replaced with a different (non—”switching”) supply whose highest fre-
quency component was 60 Hz.

Cooling the ADC ’s also provided some noise relief. This was achieved by cooling
the entire PC case to approximately 5° C. This reduced the ﬂuctuations in the output ADC
voltage by a factor of two. Replacing the AD1170 chip itself (if it was more than 1 year
old) also had the effect of cutting the noise in half.

An simple ampliﬁcation circuit with adjustable gain and offset was designed to
bring the level of the input signal above that of the PC noise. Since the PC noise is con-
stant (not ampliﬁed), simply amplifying the input signal will increase the signal to noise
ratio. Consequently, the signal to noise ratio increased by a factor of 7-30 depending on

the ﬁeld setting.

30

3.2.25 P0sit
The p0
nere obtained
3.2). The strip
0.030". The let
transmissive 0
contained 3 sm
was detected b
strip. The outpi
trigger. The out
and falling edict.
Two str
The azimuthal !
tinuously as the
“1356 measure
frame. AbSOlute-

3330’ for ﬁnal ;

3.2.2.5 Position and time measurement

The positions and times of the ﬁeld measurements necessary for ﬁeld integration
were obtained by reading teeth from a small strip with a comb-like structure (see Figure
3.2). The strip had 0.010”x0.25” rectangular holes regularly spaced at a distance of
0.030”. The length of a three-foot section was measured and was reproduced to 0.002”. A
transmissive optical sensor was used to detect the teeth in the strip. The optosensor
contained a small, collimated infrared photodiode on one side of the strip, and light from it
was detected by a small (also collimated) photo-transistor placed on the other side of the
strip. The output of the transistor (0 or 5 Volts) was supplied to the input of a fast schmitt
trigger. The output of this chip is a square wave (rise time and fall times <2us). The rising
and falling edges of the wavetrain then translate into position information.

Two strips provide the radial and azimuthal coordinates of the coils at all times.
The azimuthal strip was mounted near the center of the aluminum plate and was read con-
tinuously as the cart passed through the magnet. The radial strip was mounted on the cart.
These measured positions are relative to an arbitrary starting point on the aluminum
frame. Absolute position measurement (reference to the magnet steel, for example) is nec-
essary for ﬁnal ﬁeld analysis. This will be discussed later.

The time measurements for the ﬁeld integration in the azimuthal direction were
made by timing the cart between the teeth on the strip. A stopwatch circuit was established
to measure the time spent in the high (+5V) and low (0 V) portions of the wavetrain from
the optosensor. The stopwatch counted lus clock ticks while the waveform was in a high
or low state. In addition, the integration time from the ADC ’s was measured. The ADC ’3

provide a gate signal to indicate when they are busy taking data, and the scaler board

31

Counts ClOC
3.3 Post
The
corrections
the magnet
chosen OPtic
given severa
are passed t'
calibration dd
to conserve 5
output ﬁle forr
33.] Raw da
Figure 3
magnet. It is cle.
This is caused by
passes thrOUgh th

33.2 Dead timt

  
  

The produ

ment for that sam,
1

counts clock ticks while this gate signal is active (Figure 3.3).
3.3 Post processing and data conversion

The objective of the post processing software was to interpret the raw data, make
corrections where necessary, and provide the magnetic ﬁeld values on a cartesian grid in
the magnet’s coordinate system. As mentioned before, the input requirements of the
chosen optics code COSY demand that the ﬁeld be in cartesian coordinates. The user is
given several options including alignment, calibration, interpolation and smoothing. These
are passed to the program through an input ﬁle which includes the location of the
calibration data ﬁles. The output is a ﬁle written in binary format which is zero suppressed
to conserve storage space. Appendix A gives the relevant details regarding input and
output ﬁle format.

3.3.1 Raw data

Figure 3.6 is a plot of the raw voltage and time data for a typical sweep through the
magnet. It is clear from the ﬁgure that the cart’s speed was not uniform during its motion.
This is caused by the propelling chain stretching like a spring. The cart then bounces as it
passes through the magnet.

3.3.2 Dead time correction

The product of the voltage with the integration time gives the differential ﬁeld ele-
ment for that sample. If the ABC’s could integrate continuously with no dead time, simply
summing the products of these voltages and times would yield the ﬁeld value at the end of
each integration period. This is certainly the preferred method, as it is largely indepen-
dent of the speed of the cart.

Correcting for dead time would be simple if the speed of the cart was constant or

32

slowl)
comps
were I
anothe
testing

tooth. ii

l 2\ ll)”:

l0

limo/0.7mm l/l\'

'nducvd Clnl' nn CUM [Volts]
C

slowly changing. Unfortunately, this was not the case, as is shown in the Figure 3.6. When
comparing two identical sweeps through the magnet it was seen that the ﬁeld differences
were largest in places where the motion of the cart differed greatly from one pass to
another. This large difference was attributed to insufﬁcient position resolution during the
testing phase. During the production mapping phase, the position strip was read at every

tooth, in effect tripling the position resolution.

 

 

    

 

 

 

 

 

 

 

 

 

 

1228103 rIIIIIIIIIIIITIIIﬂjWI ’ IIIIIIIIIIIIITTIIWTTIIIIIIIIIIIIITIIT
r— -l
_ to l0x103 -— —
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h 6— 8 ‘lll-l «uni. A! .
s5 = 3 E - lllhtll‘ltvul/lr l W
25- -I 6— —
:llllLlltllllllllllllLLll: O:lLutJutilutilirrtlliiilittilunl:
:22" 4:. L E I I
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.0 L _ .g _ ..
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PillllllllllllJJJllIlllll- _4Tilrruluulinilunlnulrrulnnlr‘
0 1000 2000 3000 4000 5000 3800 4000 4200 4400
Distance cart has traveled [mm] Distance cart has traveled [mm]

Figure 3.6: Raw data from one pass through the D1 magnet at 1.6 Tesla. The top panel is
the time it takes the cart to travel 0.762 mm at a radius of 2.7 m for each ﬁeld sample. The
bottom panel is the raw voltage output from the ADC. The cart oscillation causes the oscil-
lations in the measured voltage, better seen in the right half of the ﬁgure.

33

 

The i
the voltage l
sample V0113!
norrnalizatior
occurred. thei
there was no t
adding it to th
3.3.3 Accour
To mal
spaced polar c
(radial) and Off,
the dipoles u as
lnlegration t and
pl”assul‘POse
ing resumed at
POSition and Vol

order interpolatit

The \‘Oha:

i .

by more than 10 G

ltw .
as p‘OSSlble to c

3”” (away from th

 

The variation in the speed of the cart also causes a change in the response, since
the voltage induced is pr0portional to the speed. The ﬁrst correction was to multiply the
sample voltage by the ratio of the sample time to the integration time, essentially a speed
normalization. The second correction was to compute precisely where the sample
occurred, then the normalized voltage was interpolated to the center of the sample as if
there was no dead time. This method was superior to computing the missed voltage and
adding it to the sample.

3.3.3 Accounting for missed ﬁeld data

To make post processing easier, the ﬁeld measurement was taken on a regularly
spaced polar coordinate grid. The goal was to have a ﬁeld measurement every 2.3 mm
(radial) and 005° (azimuthal). One unfortunate consequence of the vertical geometry of
the dipoles was cart oscillations. Occasionally the speed of the cart was too fast for an
integration (and subsequent conversion) to ﬁnish before the next regularly-scheduled sam-
ple was supposed to start. In this case, the sample was skipped and marked, then data tak-
ing resumed at the next regular grid point. (This is referred to as a bad sample.) The
position and voltage information from surrounding samples was used to perform a 3rd
order interpolation over the missed data.

3.3.4 Voltage offset correction and sweep alignment

The voltage offset from the electronics must be removed to determine the true
induced voltage. This is necessary and important because it can affect the ﬁeld calibration
by more than 10 Gauss. By subtracting the forward and backward passes from each other,
it was possible to determine this offset by ﬁtting a line to the region of the map inside the

steel (away from the fringing ﬁelds). This process also facilitated the alignment of forward

34

and ba
propor
shifting
structur

3.3.5 I

calibrate
MIR pr
E
to be add
coil. The
Vals with
mll'nonii.
pitch),

Fig
tings used
(ﬁller sligh
Currt3nt is n
a Signiﬁcar
inferred by
ﬁelds pTESen
”’6 Power 5

ﬁelds fmm 1h

and backward passes at the same radius, because the difference of two raw sweeps is
proportional to the ﬁrst derivative if there is a position error. The passes were aligned by
shifting one of the passes relative to the other until the difference held no peak like
structures. The two sweeps then have the same coordinates.

3.3.5 Field calibration

Once the raw data was converted and corrected, the ﬁeld strengths needed to be
calibrated. The ﬁeld measurements were calibrated using a combination of Hall probe and
NMR probe data. Figure 3.7 shows the location of the probes during ﬁeld measurement.

Because the ﬁelds are the integral of the voltage data, an integration constant needs
to be added to the ﬁeld at all points; this is the ﬁeld strength at the starting position of the
coil. The starting ﬁeld strength was measured at the starting point of the coils at 1” inter-
vals with a Hall probe, then the data was ﬁt with a ﬁfth order polynomial (Figure 3.8). The
polynomial was used to determine the ﬁeld strength for each starting position (2.3 mm
pitch).

Figure 3.9 shows Hall probe and NMR probe data for D1 and D2 for all ﬁeld set-
tings used during the mapping campaign. The excitation functions for the two magnets
differ slightly at high ﬁelds. This is due to the location of the power supplies and how the
current is measured. The current through the cables coming from the power supply creates
a signiﬁcant and measurable magnetic ﬁeld around the supply cables. The current is
inferred by measuring the ﬁeld it creates around the supply cable. Any other magnetic
ﬁelds present, such as the ones from the magnets themselves, can affect this measurement.
The power supply for D1 is very close to the magnet steel. At high ﬁeld settings, stray

ﬁelds from the magnets affect this current measurement.

35

NMR 1
locations in th
the NMR prob:
at ﬁve 2 locatit
dead time com
ibrated ﬁeld H;
The ﬁeld Was

the Small ﬂuctu

 

polynomial V31,
more reliable C;
Central Vaer_ ‘3

NMR measurements were used to determine the absolute ﬁeld strength at known
locations in the magnet polar coordinates (r,B,z), placed to coincide with the location of
the NMR probe during S800 experiments (midplane only). The measurements were made
at ﬁve 2 locations, one for each coil, and were accurate to better than 0.1 Gauss. Once the
dead time corrections were applied and the ﬁeld integrated along the coils path, the uncal-
ibrated ﬁeld F(r,9,z) is known at the calibration points.

The ﬁeld was ﬁt as a function of radius at the NMR angular location to eliminate
the small ﬂuctuation error in the ﬁeld. The calibration value was then taken from the
polynomial value at the NMR location rather than the raw data value. This provides a
more reliable calibration and ensures that the measurements ﬂuctuate around the correct
central value. A scaling factor can then be determined based on the set of NMR and Hall

probe measurements, as follows:

 

S = Bnmr(rnmr’ enmr) _ Bhall(rnmf’ 65‘3”) (3. 3)
f‘(rnmr’ 9nmr) _ F( rnmr’ Gstart) '

and the calibrated ﬁelds are then given by

B(r, 8) = Bha”(r, 0m”) + S - (F(r,9) — F(r,65,an)) (3. 4)
3.3.6 Position calibration
Absolute position calibration was achieved by mapping small, strong, permanent magnets
located on the steel edge. The magnets are 1 cm x 1 cm wide and 2 cm long and have a
pole tip strength of ~ 0.5 Tesla. The magnets were placed in holes in the mapper plate and
ﬁxed ﬁrmly to the magnet steel edge by their magnetic attraction. The small permanent
magnets are mapped individually, then the residual ﬁeld from the magnetization of the
spectrograph magnet steel is subtracted to reveal just the ﬁeld due to the magnets. This

procedure is also used to align the induction coils to each other as well

36

 

mctcrs

Figure 3.;

The tthk “n‘

M‘s NMR
e entrance

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

JilrllI—IIIIIIIIIIIIIIIITTTIIIIIIIUIIIIIIIUIT
3 (2.1382, 6.3587“?
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a ; :
/1 ~ —:
E 5 i E
: (0.4114, -0.0921 )l :
nLIlllIllllllljllllllllllllllllllllllllllllllr
-2 0 2 4 6

meters

Figure 3.7: Position of Hall probe and NMR probes for magnetic ﬁeld measurements.
The thick lines are the locations of the Hall probe measurements; the dots are the location
of the NMR probe measurements. The origin of the coordinate system is at the center of
the entrance to the D1 magnet.

37

99999

'I' u—n
'A‘ A , .

l’it Residuals [G]
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~450

Field Strength [( lutiss]

~500

Fevmil82113n
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2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Radius [m]

Figure 3.8: Hall probe data for D2 at 1.6 Tesla. The line through the data is a 5th order
polynomial ﬁt. The ﬁt residuals are all < 0.1 Gauss.

38

J:- “o
[IIIIIII'IIIW

 

 

 

1% 1.2
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0.0:llllJLllllllllllllllll: _600:lllllllllllllll-

O 100 200 300 400 2,4 2.6 2_3 3.0

Current [Amperes] Radius [m]

Figure 3.9: L_eft_: Excitation ﬁmctions for D1 and D2 determined from NMR measure-
ments. BJEDL Field strength as a ﬁtnction of starting radius for D2 determined by Hall
probe measurements. The ﬁelds range from 0.28 T (top) to 1.6 T (bottom).
as to the magnet steel. Figure 3.10 shows three such magnets placed on the edges of the
D1 magnet steel and mapped. Following alignment of the maps with the steel, the
magnets were in their expected locations of 5.0:t0.5 mm from the steel. This error alone
will not degrade the energy resolution and is within speciﬁcations for maintaining 1 in
10,000 energy resolution [Zh97].

Position errors are always possible at some level. The frame was designed to have
the same are center as the magnet steel. However, small translational or rotational errors

can occur when placing the mapper in the steel. Careful alignment ensured there was no

rotational error for D1 or D2. There was a small translational error of 5.9 mm for D1.

39

 

meters

 

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_Tl I r1 I I I I I I I I III I I I.
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2 — —
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0.0 1.0 2.0 3.0
meters
x] I I Ij’ I I I I I I I I I
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1.76 - -
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8 1- -I
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- 1
1.68 - -
' . i 1 1 L. . l .L 1

 

 

 

2.76 2.80 2.84 2.88
meters

meters

meters

0.32

0.28

0.24

0.20

-0.28

-0.32

-0.36

-0.40

 

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n i l q
n -l
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.40xlo‘3

meters

40

Figure 3.10: Field measurement of small permanent magnets placed on the D1 magnet
steel. The magnets are 1cm x 1cm x 2cm long. The top left panel shows all the magnets.
The lines through the centers of the circular contours are 5mm from the magnet steel edge.

3.3.7 Smoothing

Smoothing of the data was sometimes necessary for two reasons: (1) to provide the

MF element in COSY with usable maps and (2) to provide accurate interpolation from the

polar coordinate system to the cartesian coordinate system. COSY takes midplane ﬁeld

data and performs an off-midplane expansion to determine the ﬁelds everywhere in the

40

gap of the m:
the dcfiVame
less numberS-
The (l.
polaied from
error follou'in
the order of tli
The St

at a speciﬁc (18

where S is the
ﬁeld values at
essentially act :
Three tj
Software: Savit.
cients are a par;
COefﬁcients are

method uses a

C
page 33 of th C
loc

gap of the magnet. Even small (<1 Gauss) high-frequency ﬂuctuations get magniﬁed in
the derivatives. Higher order terms in the transfer map then tend to diverge to meaning-
less numbers. Thus smooth data with smooth derivatives is desired.

The data provided to COSY is in a cartesian coordinate system, so the data is inter-
polated from polar coordinates to cartesian coordinates. Simulations prove that the ﬁeld
error following interpolation is greatly magniﬁed when there is noise in the data and are of
the order of the noise level. The smoothing process minimizes these errors.

The smoothing techniques use neighboring points to compute a weighted average

at a speciﬁc data point, as follows:

N

s = zci-Bi (3.5)

i: 1
where S is the smoothed result, Ci are the weighting coefﬁcients used and Bi is the raw
ﬁeld values at the points. There is more smoothing with higher N. All these techniques
essentiallyact as a ﬁlter to remove high frequency components of the ﬁeld ﬂuctuations.
Three types of coefﬁcients were made available to the user in the post processing
soﬂware: Savitzky-Golay, binomial, and Gaussian. The Savitzky-Golay weighting coefﬁ-
cients are a parabola with its extremum point at the center [Pr92]. The binomial weighting
coefﬁcients are just numbers taken from a binomial distribution [Ma83]. The Gaussian
method uses a Gaussian distribution of coefﬁcients and is exactly the process described on
page 33 of the COSY manual [Be95], with a twist. The width of the Gaussian depends on
local derivatives; higher derivatives mean a smaller width and vice versa. The Savitzky-
Golay smoothing tends to preserve higher order derivatives more accurately and is very

fast. The binomial and Gaussian methods smooth more effectively but may inaccurately

41

reproduCe Zn
3.4 Result

The P
0.35 to 1.6 TC
\I'hile acquirir
at the same Ct
D2. D2 “'35 1’

Raw 5‘
sweep differer
These could bk
desired.

The dat
polynomial. Tl'
as interpolation
than interpolatit

effect.

Figure 3.

magnets in the n

7
D. close to the n
In-‘o -

“elm region f0

contours in th

"‘9

reproduce 2nd and higher order derivatives.
3.4 Results

The production maps were taken at several current and ﬁeld settings ranging from
0.25 to 1.6 Tesla, from 50 to 450 Amps. Tables 3.1 and 3.2 summarize the settings used
while acquiring production ﬁeld maps. D1 was mapped separately from D2, but with D2
at the same central ﬁeld value. The mapper was then removed from D1 and installed in
D2. D2 was then mapped with D1 at the same central ﬁeld value.

Raw sweep data show smooth, continuous behavior. However, RMS sweep-to—
sweep differences exist at the level of :t0.0006 Tesla at the highest ﬁelds of 1.6 Tesla.
These could be interpreted as higher order aberrations by COSY, and their elimination is
desired.

The data were interpolated onto the cartesian grid by means of a second order
polynomial. This had a smoothing effect on the data and was determined to be as accurate
as interpolation within the errors of the maps. In fact, this is more accurate for noisy data
than interpolation. Choosing a larger number of points in the ﬁt had a larger smoothing
effect.

Figure 3.11 shows midplane data taken for the B=1.50 Tesla setting for each of the
magnets in the magnet coordinate system. Figure 3.12 shows two separate ﬁeld maps for
D2 close to the maximum strength of the ﬁeld. The ﬁeld proﬁle is very ﬂat in the central
magnet region for low ﬁeld settings, but signiﬁcantly curved at the highest ﬁeld setting.
All contours in the ﬁgure are separated by 1 Gauss. No rapid ﬂuctuations from measure-

ment errors remain in the maps, which is a result of the smoothing process.

42

Table 3.1: Mapping parameters used during D1 production maps.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D1 D2
I [Amperes] B [Tesla] I [Amperes] B [Tesla]
67 0.25463 50 0.25463
111 0.54328 100 0.54328
167 0.79567 149.9 0.79567
199.2 1.02964 199.9 1.02964
250.7 1.22031 255.0 1.22031
275.7 1.29650 282.3 1.29650
300.9 1.36513 310.3 1.36513
350.9 1.47982 368 1.47982
366.0 1.50819 385.1 1.50819
381.1 1.53601 402.7 1.53601

 

Table 3.2: Mapping parameters used during D2 production maps.

 

 

 

 

 

 

 

 

 

 

 

 

 

D1 D2

1 [Amperes] B [Tesla] I [Amperes] B [Tesla]
50.0 0.28421 50.3 0.28421
100.2 0.54741 100.7 0.54741
150.3 0.79840 150.4 0.79840
200.5 1.03484 201.3 1.03484
250.6 1.21966 254.7 1.21966
300.8 1.36506 310.2 1.36506
350.9 1.47998 368.0 1.47998
401.1 1.56973 425.3 1.56973
419.7 1.60008 446.5 1.60009

 

 

 

43

 

 

 

 

 

 

lllClCl‘S

Figure 3.1].

ough the Coi

 

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meters

 

 

 

IIIIFTTI1IIIIIIIIIIII'IIIIIIljleITIIIIlIlI

[LmllllllllIllJlIllllIlLJIIllLlIJJllIllI4Il

 

 

IllIII]llIllllILlllIllllIllllIllllIllllIllllIliLLIllll

-1 0 l 2 3 4

 

meters

Figure 3.11: Dipole magnetic ﬁeld data for both spectrograph dipoles. The current
through the coils was ~350 A for both, giving ﬁelds of 1.48 Tesla for both D1 and D2.

ntctc‘t‘s

Flgllre 3_ l 3.
contours are;

 

IIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIII

meters

 

llllLllIlIllllILlllIIIIIIILIIIIIIIIllllIllLl

LII
IIIIWIIT ITII IIII IIII VIII—IIII IIII IIII
I

 

 

3IIIIIIIIIIIIIIIIIIIJIIIIIII_IIIIIII

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

 

meters

Figure 3.12: A D2 ﬁeld map at 0.28 Tesla, plotted above 99% of the ﬁeld strength. The
contours are separated by one Gauss, indicating a very ﬂat central ﬁeld.

45

. C) :57
W 0"
g

0
H,
(I)
m
r—f

 

I Ij I l I I I r I I rj ijrT I I I I I I I I I I I l I Iji

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1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

meters

Figure 3.13: Field measurements for D2 at 1.6 Tesla. All contours are separated by 5
Gauss. In sharp contrast to Figure 3.12, the ﬁeld exhibits a bowing behavior characteristic
of saturation effects.

46

Figures 3.14, 3.15, and 3.16 show some interesting characteristics of the ﬁeld data.
Figure 3.14 shows that larger currents produce ﬁelds with opposite sign outside the mag-
nets, making the particles bend ﬁrst one way, then the other. The central ﬁeld also becomes
more bowed with higher ﬁeld settings, a prominent saturation effect. Figures 3.15 and
3.16 show the D2 magnetic ﬁeld proﬁle and its derivatives through order along the refer-
ence trajectory for the lowest and highest ﬁeld settings. The derivatives are somewhat
smooth, but start to get noisy at 3rd order. The inﬂuence of the coil is greatly pronounced
in Figure 3.15 compared to Figure 3.16. This is easily understood; the current in the coil
is nine times greater, but the central ﬁeld is only six times greater, due again to saturation
of the magnet steel.

The mapping served to verify general characteristics about the magnets. It also
provides a check on the magnet construction. The edge angles of the steel, the effective

lengths, and ﬁeld gradients can be measured with the maps.

47

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIIIIII [I I I lIITrI—l II I FTIIIIIIIIIIIIIIrrI I IIIIrIrII
1.5 - l —
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C:
g
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0.0 a

 

 

 

 

 

 

 

IIIIIIJLILIIIIIIIIIII

0 1 2 3 4

S [meters]

‘—
llIllllIlllLIIlllIllllIJl

0 1 2 3

Figure 3.14: D1 (left) and D2 (right) magnetic ﬁeld strength along the reference trajectory
for the ﬁeld settings from Tables 3.1 and 3.2. The curvature in the ﬂat ﬁeld region grows
with ﬁeld strength. The negative ﬁeld just outside the coil also grows with ﬁeld strength.

48

 

0.30

0.25

0.20

p
u:

B [Tesla]

I
.—
O

0.05

0.00

 

IIIIIIr'IIIIIIIIIIIIIIlII

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3 5 t'li o

11)
o

 

 

 

 

 

 

 

llIIIlIIlILIIIILIIIJJIII

 

IIIIIIIIITIT'IIII'ITIIIIT

ﬁIII

d3B/ds3 [73 m'3]

IIIIIIIIIIII_LIIIIII

 

 

0 1 2 3 4
m

400

200

O

EIIIIIIIIIIII¥IFII

1L:
:3
o

-400

-600

 

 

hI-TTrTIIIlIIIIlIIITIIIIITII

ng H._..I.. ﬂ---

IIIILI

 

 

 

 

l

 

IIIIIlIIIUIII

 

 

IIIIIIIIIIIIIIIIIIIIIII

O l 2 3 4
m

 

 

Figure 3.15: Magnetic ﬁeld proﬁle and derivatives along the reference trajectory for a D2

map at B=0.28 Tesla. The ﬁeld proﬁle is very ﬂat in the center.

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PIIIIIIIIIIILHLIIIIIIII'III-W : q
1.5— F’— T 1 6:- 1]

E ‘ 2 4?- -2
31.0: ] —_ E 2:— —j
g ' l ‘ 7 0: I
z ' l : if; = 1
0.59- l - 12-2:— J

l- \ .1 _ 1

- l j 4'— 3
().0i—IIIJIIIIIIllIIIIIIIlII\II'rl-I_I -6:L-IIIllllIllILIJJlIIllllIll—

0 1 2 3 4 0 1 2 3 4
m m

_ IIII IIIWIIIIIIIIIIIII'.I ITIIII IIII IIIIIIIIIIII
40711 I _: : ‘1 I 2
20'. l L 1000:- '3
9.3"" I ll\\ 1 .~t"" I :
E O_ \ “1 E r- .1
“t— : [ . : "1.. 0_ [lr‘j
(VI-20:- ] __ MB 1- ] -l
2 : : 5 I 2
Nam 40;- l : mm -1000— -
‘3 r. : “‘3 Z :
-60r '- : :

: : -2000— -
-80.;ll 'llll‘llllllllllllllil_ :lll llllIJIIlIlIIIIIIIlIIII:

0 1 2 3 4 0 1 2 3 4
m m

 

Figure 3.16: Magnetic ﬁeld proﬁle and derivatives along the reference trajectory for a D2
map at B=1.6 Tesla. The ﬁeld is much more curved in the center than are lower ﬁelds. The
effect of the coil is much more prominent than in Figure 3.13; the outermost shoulders in
the ﬁrst derivative are a larger fraction of the peak. The outermost peaks in the second
derivative also demonstrate this effect.

50

3.4.1 Field curvature and gradients

The region inside the magnet steel is susceptible to ﬁeld curvature due to the
limited extent and susceptibility of the steel. Substantial ﬁeld curvature was found to exist
in the azimuthal (beam) direction. The curvatures are substantially higher at higher ﬁelds.
Still, the ﬁeld gradients in the ﬂat ﬁeld region of the dipoles are within design
speciﬁcations. The curvature is normally deﬁned in terms of an arc length, 5, and its
tangent angle, 6, as follows: K=|d¢/ds|. It usually has units of [length‘l]; the inverse of the
curvature deﬁnes the radius of curvature at that point along the arc [Th84]. In this context,

the curvature is deﬁned as:

(128

2
K = —dS___ (3.6)

[ell-£312];

where B is the ﬁeld measured in Tesla and s is the arc length in meters.

 

 

Figure 3.17 shows the azimuthal ﬁeld curvatures for all mapping settings. These
were determined by making a second order polynomial ﬁt to the central portion of the
ﬁeld. The ﬁrst and second order coefﬁcients were used as the ﬁrst and second derivatives
in equation 3.6. As the steel starts to saturate, the azimuthal curvature starts to grow
exponentially. Figure 3.19 shows average radial ﬁeld gradients for the mapping settings.
These data exhibit strange behavior which is not understood. However, notice that the
peak occurs close to the knee of the azimuthal curvature curve, perhaps suggesting a satu-
ration effect.

The dipoles have normal (i.e., not superconducting) trim coils along their inner and

outer radii to reduce the radial ﬁeld gradients to tolerable, design (< l Gauss/cm) levels.

51

 

 

 

 

 

 

IOXIO-3 a. I I I I I I I I I I I I I I I I I I I I I I I I Ii I I I I I I I .l:
: :1
IE“ E g
E 8 .- D 1
<3 5 [3 é
a E 0 D1 :
CU - O J
.>. 6 '5 C1 D2 0 =
:3 - ..
3 E 9 T-
7» 4 a E
L: t: 5
:3 F' :1
.§ 2 :— 0 O '7-
2 E g 5
a 9 a
oaLIIImLILLIiLI I IIJLIIILILIIIIIILII11l

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

BNMR [Tesla]

Figure 3.17: Azimuthal ﬁeld curvature for some ﬁeld settings used during the mapping
campaign. Despite the high curvature at the largest ﬁeld settings, the ﬁeld gradients
remain within design speciﬁcations (1 Gauss/cm).

During the test mapping phase, ﬁeld maps were acquired with the trim coils on at their
optimized settings (1mm coil = Imam coil * 0.4375 [Ze93]) and with the trim coils at zero
current. The trim coils have a pronounced effect on the radial ﬁeld ﬂatness in the region of

usable ﬁeld. Figure 3.18 shows a test map at 1.56 Tesla with the trim coils on and off. The

radial ﬁeld gradient is too high unless the trim coils are on and at their optimum setting.

52

 

  

 

 

 

 

 

 

 

l_[IIIIIIIIITTIIITTIIIIIIIIIIIIIIIIIIIII'L

q

4

h -

TH I j
E . .
U 'Ib _
53 I .
:3 1- d
9‘: i :
i5 -2_ -.
an - . . ’ + -
“O - O Tr1mco1lson + 4
’- . . + ..

-3 l— + Trim corls off + _.

h- + -I

I ++ I

‘ T
‘4'_IIJ_I111IIIlllIlllLIlLllIllllIlLlIIllll'I"

 

2.7 2.8 2.9 3.0

radius [meters]

Figure 3.18: Measured radial ﬁeld gradients in D1 with (+) the trim coils off and (O) the
trim coils on at their optimized setting.

 

1.0-LIIIIIIIIII1IIIIIIIITIIIIIﬁ.

0.5

  

EIIIII

 

Radial ﬁeld gradient [Gauss/cm]

 

 

0.0 _ \\g 7.
- —B— D2 -
.5

‘1.0JIIIIIIIIIIIIIIIIIIIIIIIII

 

0.4 0.6 0.8 1.0 1.2 1.4 1.6
Field Strength [Tesla]

Figure 3.19: Average radial gradients measured for D1 and D2 during the mapping cam-
paign. Despite the apparently erratic pattern, the general pattern is reproduced for both
dipoles. The error bars are i15%. The design limit is 1 Gauss/cm.

3.4.2 Effective length

To determine the effective length of the dipoles, ﬁeld data from the maps was
interpolated onto the central, reference trajectory. The fringe ﬁelds of the dipoles
effectively make them longer. The effective length was determined by performing a
Simpson integration along this trajectory, then dividing the ﬁeld integral by the NMR
reading at that ﬁeld setting. Figure 3.20 shows the measured lengths for D1 and D2

respectively.

 

r’r I I I I I I I I I I I I I I I I I I I IIT’I I I I

3.81

3.80

1’

 

 

3.79

3.78

3.77

Effective Length [m]

3.76

3.75

IIIIIIIIIIIIIIIIIIIIIIIIIIITIIIIIII
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIh

 

 

I 1.1 I I I I III,IIJ I I I I I I,l I I I I I I I l I L f

0.4 0.6 0.8 1.0 1.2 1.4 1.6
BNMR[T]

 

Figure 3.20: Effective length for both dipoles from ﬁeld data, calculated by dividing the
integrated ﬁeld along the reference trajectory by the appropriate NMR reading (from
Tables 3.1 and 3.2).

The strong ﬁeld dependence of the effective length is typical of these types of
magnets. Outside the magnet gap, the ﬁeld contribution from the coil is opposite in direc-

tion. At higher ﬁelds, a higher current/central ﬁeld ratio is necessary to account for satura-

tion effects. This adds strongly to the negative ﬁeld outside the magnet gap and is largely

54

responsible for this phenomenon.
3.4.3 Edge angles

The inﬂuence of saturation, ﬁnite susceptibility and other magnetic sources affect
the edge angle measurements with ﬁeld maps. Nevertheless, edge angles were computed
for all of the midplane ﬁeld maps. A straight line ﬁt to a ﬁeld contour (B=0.4*Bmax) at the

entrance and exit edges determine the angles.

Table 3.3: Entrance and exit angles for the D1 magnet.

 

 

 

 

 

 

 

 

 

 

 

 

BNMR [Tl Entrance[°] Exit [°] Difference [°]
0.25463 0005 29.73 29.735
0.54328 -0.07 29.70 29.87
0.79567 -0.04 29.79 29.83
1.02964 -0.07 29.86 29.93
1.22031 -0.09 29.75 29.84
1.29650 -0.08 29.76 29.84
1.36513 -0.05 29.72 29.77
1.47982 0.07 29.71 29.78
1.50819 -0.05 29.69 29.74
1.53601 -0. 14 29.64 29.78

Average = -0.07i0.04 29.74i0.06 29.81i0.06

 

 

 

 

 

 

The entrance and exit angles as well as the angle between the two faces are listed
in Tables 3.3 and 3.4. The averages and standard deviations are given in the last row of
the table.

The slight lowering of the thirty-degree edge angle is possibly due to the increased
inﬂuence of the other magnet and saturation effects or the limited extent of the steel on the
faces. Using a different contour has a small effect on the measured angle unless very high

or very low contours are used. This doesn’t mean that the actual steel angle is different

55

from its designed 30° (in fact, this was measured to be 29.96° for each dipole). This can

only be interpreted as a true magnetic effect due to the sum of magnetic sources.

Table 3.4: Entrance and exit angles for the D2 magnet.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BNMR [Tl Entrance[°] Exit [°] Difference [°]
0.28421 -0.005 29.83 29.835
0.54741 0.005 29.83 29.825
0.79840 0.04 29.71 29.67
1.03484 -0.04 29.85 29.81
1.21966 -0.09 29.80 29.89
1.36506 0.01 29.74 29.73
1.47998 0.003 29.775 29.77
1.56973 0.007 29.77 29.78
1.60008 0.005 29.63 29.635

Average = -0.01i0.04 29.77i0.06 29.77i0.08

 

3.4.4 Alignment of both dipoles

As a ﬁnal check of the maps, Figure 3.21 shows two maps at the same ﬁeld settings
plotted in the overlap region between the two magnets. The contours are separated by 20
Gauss. The contours line up to better than 2 Gauss, indicating that the two sets of maps
are aligned.
3.5 Conclusions

In the future, if one were to map these dipoles again, the best way to do it would be
to use a zero dead time system, such as has been developed for the K500 cyclotron. The
dead time correction used here, while accurate to :6 Gauss, is insufﬁcient without sub-
stantial post processing effort. The most likely source of this error is the position mea-

surement along the path. The optosensor is very sensitive to collimation of its light source

56

 

3.5 _.

 

3.0 - ~-

 

meters
I

2.5-’

 

 

 

15 41 LII] I1 1 1IIJ 1 1 I1 1 1

 

meters

Figure 3.21: Contour plot of the overlapping ﬁeld region of two maps at the same current
and ﬁeld settings. The contours are separated by 20 Gauss, and the region contoured is
that $0.0 Gauss. The overlap of the contours is on the few Gauss level, indicating correct
position and ﬁeld calibration for the two dipoles.

57

and distance to the position strip from its light collector. Any variation in this distance (on
the level of 0.01”) can cause a change in triggering time, creating an error in position.
Eliminating the dead time would reduce the dependence of the integrated ﬁeld value on
this position and may be enough to reduce the errors to the Gauss level as desired.

The data presented here were intended to be used in the MF element in COSY.
Unfortunately, this method doesn’t provide meaningful transfer maps beyond the 2nd
order. Simulations show that the Gaussian interpolation (image charge ﬁtting) used in
COSY’s MF element is adequate to describe the data at the 1 Gauss level at the 2.3 mm
resolution used during mapping. However, this doesn’t consider higher order derivatives,
which may be unreliably described. Using larger image charge widths compensates for
this somewhat, but then also fails to reproduce the ﬁelds in the regions with the largest
gradients. Another possibility is that errors in the ﬁelds are just too large for reliable
offplane expansion of the ﬁelds. This doesn’t seem likely as the ﬁeld ﬂuctuation can be
reduced to the ~l .0 Gauss level.

The good news is that the mapping illuminated various characteristics of the mag-
nets that can be used in a ﬁeld model representation of the magnets. The advantage to ﬁeld
models is that the higher order derivatives are continuous and, usually by deﬁnition,
adhere to Maxwell’s equations. The azimuthal curvatures, the radial ﬁeld gradients, and
the decrease in effective edge angles, as well as the fringe ﬁeld data is very useful in deter-
mining a realistic model of the ﬁeld. Thus, accurate transfer maps can be generated by

simply using an accurate ﬁeld model.

58

Chapter 4

Spectroscopy of 10Li

4.1 Motivation
4.1.1 Halo nuclei

A nuclear halo is a simple quantum mechanical effect arising from low separation
energy. This allows the valence nucleons to penetrate quantum-mechanically into the
potential barrier, increasing the nuclear RMS radius appreciably. While stable nuclei
have nearly identical neutron and proton radii, the distributions for halo nuclei can differ
substantially. Neutron halo nuclei are those in which only the neutrons have a larger
radius and similarly for proton halos. The differences in RMS radii can be as large as 6 ﬂu
or more. Nuclear halo states occur in nuclei having orbital angular momentum quantum
numbers of 1=0,1. For higher angular momentum, the states are more deeply bound due to
the angular momentum term in the nuclear potential and hence the wavefunctions are
more conﬁned. Proton halos are also possible, but are less striking due to the added poten-
tial barrier provided by the Coulomb force. Several review articles describe the character-
istics of nuclear halos [Ri92, Ha95, Or98].

Halos are of interest because of their inhomogenous structure and low nuclear den-

59

sity compared to stable nuclei. Most stable nuclei have a nuclear radius well described by
r=r0A1/3, where A is their nucleon number and r0=1.2 frn. Implicit in this scaling law is
the fact that nuclear density is constant. Halo nuclei, by contrast, not only have substan-
tially lower nuclear density but the nucleus is inhomogenous. A core of normal nuclear
density is surrounded by a halo of sparse nuclear matter. One can ask whether nuclear
models that work well for stable nuclei, such as the shell model does, apply to such nuclei
or whether the nuclear structure laws are different for these halo nuclei. By studying halo
nuclei one h0pes to answer these questions.

Neutron halos which have neutrons in s-wave states lead to even larger matter dis-
tributions than those in p-wave states owing to the lack of an angular momentum barrier.
Consider an s-wave neutron bound by Sn=100 keV in a Wood-Saxon potential. Its asymp-
totic wave function is m(rke‘k’, with the characteristic length k2=2uSn/h, leading to an
RMS neutron radius of ~11 frn, compared with 3.4 frn for Sn=l MeV.

Another novel feature recently discovered in nuclei is the neutron skin [Ta92].
These are also extended nuclear matter distributions, like halos, but have distinctly differ-
ent features. Unlike halos, they are characterized by proton and neutron RMS radii that
differ by 5] fm and the matter in this skin is dense compared to halo matter. The skin is
thought to arise from differences in proton and neutron Fermi energies; a larger difference
meaning a thicker skin. Examples of skin nuclei are 6He and 8He.

4.1.2 Lithium-11

Lithium-ll has been the subject of intense study in the last 15 years. It has been
the most studied of the halo nuclei. The interest in halo nuclei and “Li was started by

experimental results in the 1980’s. Tanihata et al. used radioactive nuclear beams of 790

60

MeV per nucleon to measure total interaction cross sections [Ta85]. The interaction cross
sections were surprisingly large for the neutron rich species 6He, 8He, and 11Li. The devi-
ation from the expected rerA“3 value was much larger for 11Li than either of the two
helium isotopes. The nuclei were suspected of having extraordinarily large matter distri-
butions and therefore low nuclear densities. This is in contrast to stable nuclei where
nuclear densities are essentially constant.

Magnetic dipole and electric quadrupole measurements performed at ISOLDE at
CERN in 1987 gave essentially the same results for 9Li and 11Li. This demonstrated that
the proton distributions for 9Li and 11Li were very similar [Ar87]. This was interpreted as
evidence that some or all of the neutrons, but not the protons, are responsible for 1lLi’s
increased radius.

This was conﬁrmed by Blank et al. [B192], who measured the interaction cross
sections together with the charge changing cross sections. (A charge-changing reaction is
any that changes the proton number of the projectile.) The 8’9’11Li charge changing cross
sections are shown to be constant while the total interaction cross section increased mono-
tonically, again interpreted as evidence that the neutron distribution was responsible for
the increased radii.

Kobayashi et al. found surprisingly narrow momentum distributions for 9Li frag-
ments following 11Li breakup at 790 MeV/u [K088]. The sudden approximation is valid
in high energy breakup and the fragment momentum distributions were interpreted as a
direct measure of the intrinsic momentum distributions. In other words, the width of the
fragment momentum distributions are inversely proportional to the extent of the valence

wavefunctions; a narrow fragment momentum distribution corresponds to a large spatial

61

wavefunction.

In a case like 11Li the neutrons in the halo have a much larger radius (~10 frn) than
the 9Li core (3 fm), and greater than 90% of the halo wave function is outside the core
[A197]. It is therefore appropriate but not necessary to use a three body model to describe
llLi properties, where its properties are determined by the n-n and n-9Li interactions.

4.1.3 Lithium-10

Using a three body model to interpret the above experiments requires knowledge
of its binary subsystems. The n-n interaction is well understood, but the n-9Li interaction
is not. The 10Li nucleus is an unbound system with a half-life of ~10'21 seconds, making
it difﬁcult to study experimentally. A simple shell model suggests that the ground state
character of the 10Li system would be a neutron in the 0p1/2 orbit coupled to the 3/2'
ground state conﬁguration of 9Li leading to two states with 1+ and 2+ character. How-
ever, it was suggested as early as 1977 that the ground state character of the 10Li system
might mimic that of 11Be [Ba77], in which an s-wave intruder state (the 151/2) drops
below the 0p1/2 orbit giving rise to 1' or 2' states.

The llLi breakup data also suggest at least some component of s-wave in the 10Li
ground state. The unexpectedly narrow momentum distributions and large breakup cross
sections can’t be explained using a (0p1/2)2 conﬁguration for the 11Li ground state
[Th94]. Adding an sl/2 component to the 11Li ground state ﬁxes these discrepancies.
Consequently, this means that the 10Li system should have both s-wave and p-wave reso-
nances close to threshold.

The experimental difficulties in studying 10Li are well documented. Currently,

there are 10 published results regarding the low-lying structure of 10Li, and they are sum-

62

marized in Table 4.1.

Table 4.1: Previous measurements of 10Li states below -Sn=l .0 MeV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Year Ref. Reaction ‘Sn [MCVI Width[MeV] assignment
1975 W175 9Be(9Be,813)10Li 080(25) 1.2(3) g.s.
1990 Am90 11301131310141 0.15(15) <0.4 sl/2 g.s.
I994 YO94 113(7Li,8B)l0Li <0.10 <0.23 81/2 g.s.
054(6) 036(2) pl/2
1993 Kr93 180 fragmentation <0.15 OR sl/2, gs
~2.5 sl/2, excited
state
1995 Zi95 ”Li breakup a = 0-(-50) sl/2, g.s.
frn
1997 8097 986(13C,12N)10Li 053(6) 030(8) pl/2
1036(12C,12N)10L1 024(6) pl/Z
1997 2197 ”Li breakup 021(5) 0.12
0.62(10) 0.6(1)
1998 Be98 “Li breakup a=-1.75(75) sl/2 g.s.
030(5) p1/2
1999 T1199 130 fragmentation a < -20 fm <0.030 81/2, g.s.
(SNDS) OR s1/2, excited
~25 state

 

The ﬁrst experimental observation was in 1975 by Wilcox et al. who used the

9Be(9Be,8B)10Li reaction at 120 MeV to produce 10Li. They used two silicon telescopes

at 14° and 30° in the lab to detect the 8B and recoiling IOLi, respectively. There is one

prominent peak in their spectrum at Sn = 08010.25 MeV, which they attribute to the

ground state. The energy resolution of this spectrum is not very good, and the peak may

be a composite of two or more peaks. In addition, there are only ~30 counts in the

observed peak (Figure 4.1).

63

 

 

I I I I I I r

20~IOI 4

 

 

 

121 MeV ’Be "CI ’13:. '8) "e
'5 F. Blob. I4° I760 PC .4
‘ (348-370
IO 1.. I -l
”B g.s.
l
; 11 111111 1
3o . (hi 4
’Be ( ’Be .'81 '°Li
singles
20 '- 6950 ,u.C "
S to r
O
U W
0 1 1 1 t I
(c) .s . ’ael’ae. '8) '°Li
'0 " L' g.s. coincidence
6950 ac

 

 

 

 

 

elk l
”1
° 60 u

Figure 4.1: The ﬁrst experimental observation of 10Li [Wi75] using the 9Be(9Be,8B)lOLi
reaction at 121 MeV. The resolution and statistics are poor in the 10Li energy spectrum
(bottom panel).

 

80 90
Particle energy (MeV)

 

 

 

 

1 L

1
92 3’! 95 50' MeV

 

 

10 ‘ ‘ protons/MeV per rr“ meson

Figure 4.2: The llB(rt',p)10Li reaction [Am90]. It is difﬁcult to tell whether the peak is a
single state or a composite.

Amelin et al. published the next experimental observation of 10Li in 1990. They
used a (71',p) reaction on a 11B target to produce 10Li. Their spectrum shows a rise near
zero separation energy, and has a peak at -Sn=150 :1: 150 keV. They speculate this is an 5-
state and that it is the ground state. Again, the resolution is too poor to determine whether
the peak is a composite structure or one peak (Figure 4.2). The observation of a 240 keV
p-wave state that will be discussed shortly makes it possible this state contributes to the
observed peak.

In 1993, Kryger et al. performed an experiment using Sequential Decay Neutron
Spectroscopy (SNDS)[Kr93]. The nucleus of interest is generated in a fragmentation
reaction and then decays in ﬂight into a charged fragment and a neutron. The energy of

both are measured, and thus the velocity of the neutron relative to the charged fragment is

65

deduced. The center of mass energy difference is then extracted. Kryger et al. used frag-
mentation of 18O to produce 10Li, and 7He. The observed relative velocity of the 6He-n
system agrees well with the known ground state in 7He at Er = 440 :l: 30 keV. They see a
peak at zero relative velocity in the 9Li-n system which they attribute to an s-wave ground
state OR a state coupled to the ﬁrst excited state of 9Li at 2.7 MeV (the absolute uncer-
tainty is due to the relative nature of the measurement). There are shoulders on the spec-
trum which can be'ﬁt with a p-wave state near 500 keV.

Young et al. [Yo94] performed another binary reaction spectroscopic measurement
of the 10Li nucleus. The reaction they chose was 11B(7Li,gB)10Li at E(7Li)=l31 MeV.
They observe a sharp rise near zero separation energy in the 10Li energy spectrum, and see
one prominent peak at ~500 keV. Their ﬁtting procedure can ﬁt either one p-wave state at
500 keV or an s-wave state at <100 keV, and a p-wave state at 540 keV. The statistics for

this experiment are low and they describe the evidence for the s-wave state as “weak”.

 

 

 

 

 

 

 

 

 

 

 

 

 

20 I I I— I I T I I I I I I II I I
r g l L I
r I I I
16 - 3 2
13 b H
E, r “a(’u.‘3)‘°u data
_ +
Q 10 l-w p I van fit.
3 ------ p wave 111
S
o
o
5 ll
0 I J _I I I I J—
200 220 240
Channel

Figure 4.3: Results from the Young, et al. experiment [Yo94]. A prominent peak is
observed at -Sn=540 keV.

66

MeV

 

      

 

 

 
  

  

      
     

 

‘°Ba(‘2c,"N)‘°Li, ELab=357

 

 

 

 

 

 

 

 

 

 

 

  
 
 

 

  

 

 

 

200 _ﬁ ' TV ' 4
E- EnlMeV] 35927:? 83’; :3
m 160 :- o.4° W” "0:31 I i €
-+-' —1
C T .
3 f ' '1
o : i ' 1
O 120 7 ”1H (L! N ‘.
80 ﬁ “1 ' €
‘mh. .. ~:
10 5 10 . 0
ER( L1) [MeV]
‘Be(‘3c,‘=N)‘°Li, Bub-336.4Mev
‘00 :5 5.353.536 (’°Li)[MeVl
F V NW
80 6 30° 1
£3 3
g 60 1
o .
0 4o -3
20 . 17N ' .3
'l
O _ 'Mx

 

 

Figure 4.4: Two measurements by the Bohlen group [B097]. The measurements show

evidence for states at -Sn = 240 and 530 keV.

67

 

 

A group at the Hahn-Meitner Institute in Berlin have made many spectroscopic
measurements using binary transfer reactions. They have performed several measure-
ments of the 10Li nucleus, ﬁrst published in 1993 [3093], then again in 1997 [13097]. In
the 1997 paper they re—analyzed the data from the 1993 paper with an asymmetric, Breit-
Wigner line shape. The 9Be(13C,12N)10Li and the 10Be(12C,12N)10Li reactions were used
to populate states in 10Li. Two prominent peaks at -Sn = 240 :t 60 and -Sn=530 :t 60 keV
are identiﬁed as two p-wave states, the 1+ and 2+, but with no speciﬁc assignment. In all
their experiments the spectra have signiﬁcant background contamination, and the statistics
are generally low. Extracting the location of the states then becomes very dependent on
the background subtraction (Figure 4.4,).

In 1995 a group from CS] in Germany published results from a 11Li and 11Be
breakup experiments [Zi95]. They measured momentum distributions of the neutrons fol-
lowing breakup to ascertain the population of states in 10Li. They see evidence for a 5-
wave ground state close to threshold with a scattering length in the range of -5 to ~50 fm.

In 1997 Zinser et al. measured energy distributions of the neutrons and 9Li frag-
ments following breakup of 11Li [Zi97]. The spectrum was ﬁt with an s-wave shape at
200 keV and a p-wave shape at 600 keV, not accounting for experimental resolution
effects. Bertsch et al. re-analyzed this experiment with a three-body approach to the 11Li
system and included experimental resolution in the ﬁtting procedure [Be98]. They see an
s-wave state with a relatively high scattering length (a~-2 fm) peaking at 80 keV, and a p-
wave at 300250 keV.

Recently, Thoennessen et al. [Th99] published a new SNDS experiment which is

essentially a repeated Kryger et al. experiment but with better resolution. Their ﬁndings

68

are consistent with a s-wave state at <50 keV (again either the ground state or coupled to
the ﬁrst excited state of 9Li) and a shoulder which may be attributed to a p-wave state at
540 keV. A p-wave state at ~250 keV can not be ruled out from their ﬁt [Th99].

These experimental results taken together start to paint a picture of 10Li which has
a very low lying s-wave ground state (1' or 2') and two p-wave states separated by 300
keV at ~250 and ~550 keV. However, it is very desirable to ﬁnally perform one clean,
high-statistics, absolute mass measurement of the states involved. This is the goal of the
experimental work described in this chapter.

4.2 Experimental setup
4.2.1 Reaction and targets

The 9Be(9Be,8B)10Li reaction was chosen because it has been reported [Wi75] to
have an relatively large cross-section (ocm=30 nb/sr) for this type of process. In addition,
the 8B ejectile has a much different rigidity than other competing reactions which provides
a relatively contaminant-free focal plane. A 40 MeV/u 9Be beam was used to bombard
three targets: 0.94 mg/cm2 beryllium, 1.1 mg/cm2 carbon, and a 0.45 mg/cm2 beryllium
foil on a formvar backing.

The binary reaction 9Be(9Be,8B)10Li was used to populate states in 10Li. The
exact mechanism for particle transfer (sequential single particle transfers or as groups) is
uncertain. Because it is a binary reaction (only two products), the kinematics are com-
pletely determined, meaning a 8B with a certain energy and scattering angle deﬁnes
exactly the parameters of the residual 10Li nucleus.

4.2.2 8800

The 8800 was operated in the dispersion-matched, energy-loss mode (see chapter

69

2). The primary beam was focussed at the object position of the spectrograph, then trans-
ported to the target position of the spectrograph. There it reacted with the targets and the
reaction products were swept to the focal plane. The spectrograph was set with the center
of the aperture at 5° corresponding to an angular coverage of 3.5-8 degrees in the labora-
tory.
4.2.3 Detectors

The focal plane detectors consisted of two position detectors and a plastic scintilla-
tor. The position detectors measured the two transverse positions and angles of the parti-
cles as well as their energy loss. A Scrn thick plastic scintillator was located behind these
detectors for total energy measurement.

All ions from reactions in the target stopped in the scintillator. The scintillator is a
5 cm thick piece of plastic designed to scintillate when particles strike it. Light guides are
mounted on each end to enhance the collection of the light in the photomultiplier tubes.
The light travels through the plastic and light guides and is collected in the photomultiplier
tubes on either end of the scintillator. The light collection time is of the order of nanosec-
onds from the scintillator. The amount of light collected in the scintillator is a function of
the ion’s atomic number, mass, and total energy. It can be parameterized as given in a
Nuclear Instruments and Methods article published by Bechetti [Be76].

The position detectors are cathode readout drift chambers (CRDCs). The CRDCs
were ﬁlled to a pressure of 140 Torr with 80% CF4 and 20% C4H10. Figure 4.5 shows a
schematic of their operation. Ions traveling through the gas create an ionization. A con-
stant vertical electric ﬁeld in the detector move the electrons toward an anode wire, where

charge ampliﬁcation takes place in the high electric ﬁeld close to the wire. The anode wire

70

is placed below a grounded Frisch grid and held at a constant voltage, typically 1400
Volts. The electrons are collected on the anode wire. Cathode pads are located in front
of and behind the anode wire. The charge collected on the anode wire induces a positive
charge on the cathode pads, and each individual pad is read. A Gaussian ﬁt to the charge
distribution is performed, and the centroid is used as the x (dispersive) position in the
detector. The signals from the cathode pads are ampliﬁed by small circuit boards placed
directly on the detectors.

The vertical (y) position is determined by measuring the drift time of the electrons
to the anode wire. The typical driﬂ time of the electrons to the anode wire are 0-20 us,
depending on their vertical position. The fast response of the scintillator is advantageous
for timing the electron drift to the anode wire in the CRDCs. Measuring the time between
the scintillator signal and the anode wire signal provides a direct vertical position mea-
surement of the particle track with a 0.3 mm resolution.

The total charge collected on the cathode pads depends on the nuclear species and
its energy loss in the detector gas, and can be used as a AE~z2 measurement for particle
identiﬁcation. Charge resolution was barely suﬂicient to separate particle groups and
would have beneﬁted from energy loss measurements in an ion chamber.

The particle's ﬂight time is also measured relative to the cyclotron radiofrequency
pulses. Reactions in the target make fragments with different speeds, and hence different
ﬂight times to the focal plane. This ﬂight time measurement can be used in conjunction
with the total energy measurement (given by the scintillators) and energy loss (given by
the CRDCs) to identify nuclear species that arrive at the focal plane.

4.2.4 Electronics

71

Figure 4.6 is a schematic of the electronics used in this experiment. The plastic
scintillator behind the focal plane generates the master gate for the electronics. The signal
from each PMT on the scintillator is ampliﬁed and send to a constant fraction discrimina-
tor module (CFD) to provide a logic pulse. A master gate is generated by requiring coinci-
dence of the two PMT logic pulses on the scintillator. The master gate is used to open a 20
us gate to wait for an anode signal (the wait gate). The master gate is logically AND-ed
with the (NOT Busy) signal from the data acquisition front end CPU to provide the “mas-
ter gate live” signal. This is also the start for the drift time in the CRDCs and the stop for
the ﬂight time information, taken relative to the RF pulses for the cyclotron. The anode
wire signal is used as the stop for the drift time in the CRDCs. The TAC output (0-10V for
0-20 microseconds) was input to a 12-bit ADC. Each pad (there are 224 in each CRDC)
is read by its own fast encoding and reading ADC (FERA) (charge integrators), following
ampliﬁcation by the front end electronics (FEE) cards. A 500 us gate for the FERA read
is opened by the AND of the wait gate and the anode pulse. The pedestals on the FERAs
were set so that a channel with no data has a digital output of zero. When the FERA mod-
ule is operated in zero suppression mode, all channels that equal zero are ignored, ensur-
ing that only those channels with signiﬁcant signal heights are read. This reduces the
deadtime substantially. Reading each FERA channel every event takes Sits/channel *
224channel/detector * 2detectors = 2240 microseconds, limiting the data rate to $450

events/second; this limit is increased to ~20,000 events/second in zero suppressed mode.

72

 

Particle track/ 'V

I
/. .

 

 

Ionization

 

_ V
e Tdrift~15ps Frisch grid

 

 

 

 

1

 

'4 Anode wire

 

 

 

 

 

TOQDC'SI I I IIIVIIIIIIIII'I IIIII'IIIIT\CIathodepads

E

1200

E

o
Oﬁlrllll

800
600

Charge collected [an]
a
8

200

 

 

 

A _— __-_ A.

A‘

‘ﬁ
50 100 150 200 250

Pad #

 

 

 

Figure 4.5: Schematic of the CRDC operation. A particle track ionizes the gas as it passes
through the detector. The electrons drift to the anode wire where they are collected. An
induced image charge on the cathode pads provides horizontal position information, while
driﬂ time of the electrons to the anode wire provide vertical position information. (Note
the cathode pad locations are really on the front and back of the detector; they are shown

schematically on the bottom here for clarity.)

73

 

El Scintillator:

 

Up CFD

 

AND W Master Gate

 

 

 

 

 

Down

 

 

 

lU

 

 

AND . MG Live

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Busy
CRDCAnode a} CFD (Slave) L——stan
TAC — ADC S80l_TAO
O stop
AND —l gate
FERA

l Cathode pad Shaping
Pre-amp ampliﬁer

 

 

 

front-end electronics board

 

 

 

Figure 4.6: Schematic of 8800 focal plane detector electronics. Only one CRDC is shown
and only one cathode pad is shown from the overall electronics to simplify the diagram.

74

4.2.5 Calibrations

4.2.5.1 Tracking detector position calibrations

The ray reconstruction depends on the measured positions and angles of the frag-
ments in the focal plane. Masks with well-deﬁned hole and slit patterns were placed in
front of (upstream) the CRDCs and illuminated with beam particles to calibrate the detec-
tor positions. The dispersive direction is already calibrated in the sense that the cathode
pads are known to be separated by 2.54mm and ﬁxed in position. A beam-axis hole (pre-
viously optically aligned with the beam axis) provides the optical center of the detectors.
Other holes at several known locations from the beam axis hole provide the vertical posi-
tion calibration points. Figure 4.7 is a picture of the calibrated position spectra of the

$801 mask.

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

010 JIIIIIIIIIIII—[IIIIlIIIIIITWrIIIrIFITIITijj—IIIIIIIIIIIII
r— u- ' -1
m _:
33 .._ __ 1
33' 0.05:; .. -_ - 1
E :t‘ 1 ' : t: f ’ g :
goon-tin. new».
U) .. ..
8. -0.05 r’ \ slits/ 1
>- I J
_010 :l—lllllulLlllllllllllllllllllllJllJI’lllllllllllllllllllu
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
X position [meters]
0_10_r1rr]rrrrIrrrrrrrrTfﬁirlrrrr llIII—TII—IIFTIW[ITWIIIIﬂrTIIIC‘
r—1 [- _.
t r
H —' 5 —‘
Q, 0.05_ < o ,
E i 8 ‘l g i
c: P c 3 er cccc c
.9 0.00, _
3: : (D :
(I) .. .4
8.. -0.05:— 1
>- I I
_0IOPIIIIIIIIIIIIlllllllllllllllll ItllllllllllllllLlLlllllllIlI‘
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
X position [meters]

 

Figure 4.7: Data taken with the position mask placed in front of the 8801 CRDC. The top
panel is the raw data taken with a small cut in the dispersive angle, theta, and the bottom
panel is a picture of the prominently illuminated holes.

Throughout the experiment the measured drift times of the electrons to the anode
wire changed in a systematic way, causing the y position calibration to drift. Eventually
they reached an asymptotic value at the end of the experiment. This phenomenon was
caused by the gas mixture changing in the CRDCs. Drift times decreased, and the col-
lected charge increased over time, suggesting that gas contaminants were being purged.
Thus, the drift time distributions (8801 and 8802 TAC parameters) shrunk throughout the
experiment (Figure 4.8).

The drift behavior can be measured. The masks can be repeatedly inserted for

position calibrations throughout the experiment. However, this was not done because the

76

drift was not discovered until after the experiment had ended.

The width of the TAC distributions only depends on the gain (mm/channel) and
the vertical width of the beam if the spectrograph magnets don’t change. Thus, the verti-
cal position calibration can be tracked throughout the experiment by measuring the width
of the TAC distributions. The center of the distributions will also change and can be
adjusted accordingly. The best measure of this, lacking repeated mask measurements, is
to use a constant like the spectrograph acceptance. The gains and offsets were modiﬁed
for each run to keep the spectrograph acceptance constant throughout the experiment.
Since the y position in the focal plane strongly depends on ¢ angle, the known reaction

kinematics were used to validate this procedure.

77

 

 

 

 

 

 

 

 

 

3 @ - ssoi TAC f I
O } .._. ‘ I-
th r _ F.
3 o ‘ OOOOOQDQOOQOQOOOO @0000 089:
.r: ht .
U + ’ +*+ + F
++ + + ++1T++++++++++ +++H 444.:
IIIIIIIIIIIIIIIIIIIIIIITTT—[IIIIIIITIIW'IIIIIIIII
12:00 AM 12:00 AM 12:00 AM 12:00 AM
10/8/96 10/9/96 10/10/96 10/11/96
Run time
l[11111111lllLlllLllllllllllllll!lllllllllJl!lll
.89 (D, 00 8802 _TAC 1 C
-l 6) ’
2000-..“- -- ~- 02031300000 ----
“g I CC; 0069000000000 (moo co):
- r-
'5 woo-diam- t: + + __
1200 _: _ t;— . . +,+ ++++++~H~++ _.{_.-l;1,_.+..[,~ ‘H- ‘—
‘ Z , + Q i , t
TTTIIIII'IIIII]ITIII'IIIII'IIIIIIIIIIIrTIIIIII
12:00 AM 12:00 AM 12:00 AM 12:00 AM
10/8/96 10/9/96 10/10/96 10/11/96
Run time

 

 

 

Figure 4.8: Detector y-position (TAC) driﬁing over time during this experiment. The cir-
cles are the centers of the distribution and the (+) are the widths. Error bars are not plotted
and ﬂuctuations are due to low statistics.

78

4.2.5.2 Elastic and inelastic scattering

Scattering the beam off the different targets used in the experiment serves many
purposes. Sweeping the elastic peak across the focal plane allows the spectrograph bend-
ing radius as a function of focal plane position to be calibrated. (In most cases the scatter-
ing populates excited states in the target nucleus through inelastic scattering, which are
also useful in the bending radius calibration.) The resolution and operation of the detec-
tors were checked in this fashion. Since ray-reconstruction is used to achieve the ultimate
resolution, these strongly populated reactions provide veriﬁcation of the coefficients used
in ray-reconstruction. Target contaminants may be present and knowledge of these is
important in deciphering the energy spectra of the reactions. Target contaminants were
found by scattering the 9Be beam off the different targets and measuring the energy as a
function of angle. Different target constituents have different reaction kinematics and
therefore are very apparent in an energy-angle plot. The different elastic scattering kine-
matics provided a very accurate method to determine absolute scattering angle.
4.2.5.3 Spectrograph bending radius calibration

Figure 4.9 shows the spectrograph bending radius calibration taken from the elas-
tic scattering of the beam off the carbon target. The data points were taken from runs at

four different ﬁeld settings which swept the peaks across the focal plane.

79

 

 

 

 

,__. 2.88 .LIJ.'...'..!.].11!..'..[..'...'..'.._'..l.'.-.'..!.[I.!-.'...'..!.I..'..'...'.-.'..[..'...'.I..'.I.'...'..!..'..Ill.I..I.lI.I..'...'..].I.U.!.l. '
:2 I. . ‘ , . , -
cu - .
3 1
m - -
.a - - -
15 2.84 - .
(‘6 C -
m .
on t -
5 2.82 — ----- -
"U I I
c:

cu - . , ‘ ; : .
m 2-80 . it"; It"‘.‘1".‘"."l"."'."."."i' ."."I'.'l‘ ".1"; t t"; 11111". t11'1111'1’1"."t"Li’L13“.‘L””.’L."i”.‘T

400 500 600 700 800 900
Channel

Figure 4.9: Spectrograph bending radius calibration using elastically scattered 9Be for the
10Li experiment.

4.2.5.4 Scattering angle calibration

The kinematics of the 9Be beam scattering off 12C and 1H are dramatically differ-
ent. Aligning the kinematics with the spectrum measures any angular offset present in the
data (beam angle, spectrograph angle, etc.). Figure 4.10 shows such a plot for the beam
scattering off the carbon target. The procedure strongly populated states in carbon (0.0,
4.4, 7.6, 9.6) as well as the ground state in hydrogen, and states at 0.0, ~6 and ~7 MeV in
16O. A scattering angle offset of 0.252’c0.03° was necessary to align the calculations with

the data.

80

 

8 IITIITII'IIIIFIIIIIIIIIIIIITI.III'I|IIII]I‘I‘ I!ITIIFT\IIIIIII
u

Scattering Angle [deg]

K]!
III I—IIj IIII IIII IIII
l I ,’l l
lllLLJLlLlllllllllllllll

 

 

 

8

Figure 4.10: Elastic scattering of 9Be beam at 40 MeV/u from a carbon target consisting
of carbon with H20 contamination. The solid lines are known kinematics for scattering
off carbon nuclei populating states, from right to left, g.s., 4.44, 7.65,and 9.64 MeV. The
dashed lines are scattering from 16O and populating g.s., 6.1, and 7.1 MeV. The dot-
dashed line is elastic scattering off lH nuclei. Aligning the known kinematics with the
spectrum provides a very accurate scattering angle calibration.

81

4.2.5.5 Ray reconstruction

As described in Chapter 2, the spectrograph has signiﬁcant aberrations which must
be known and removed. This was the purpose of the mapping effort described in Chapter
3. The ﬁelds of Q1, Q2, D1, and D2 were modeled as fringe ﬁelds with ﬂat central
regions. This ﬂat-ﬁeld model is valid for the low magnetic ﬁelds used in this experiment,
but may not be valid for higher ﬁeld settings at which radial gradients and azimuthal cur-
vature of the spectrograph dipole ﬁelds become substantial.

The fringe ﬁelds were determined by interpolating the ﬁelds from the ﬁeld maps
onto the central trajectory, then ﬁtting the ﬁeld proﬁle with a 5th order (6 term) Enge func-

tion. The effective ﬁeld boundaries were chosen by matching the integral on either side of

the ﬁeld boundary:
s0 00
des = f(l——B)ds (4.1)
_oo 50

where so is the ﬁeld boundary location. Appendix B lists the coefﬁcients used in the
reconstruction process.
The useful reconstruction variables are 6 (the fractional kinetic energy) and 0 and

4», the two angles. The reaction scattering angle was constructed as follows:

 

O = atan(J(mn0)2+(tan(¢+¢,))2)+80 (4. 2)
where G) is the reaction scattering angle, 0 and 4) are the reconstructed angles emerging
from the target, $5 is the angle position of the spectrograph, and 80 is the measured
angular offset described in section 4.2.7.2.

The energy of the ion observed in the spectrograph is a ﬁtnction of the Q-value of

the reaction and the kinematic energy shifts. The ejectiles lose more energy as they are

82

scattered to larger angles. This energy dependence can be calculated exactly. These
known kinematic affects can be used to combine all events with the same reaction Q-
value by adding the amount of energy necessary to shift that event to a single angle (usu-

ally 0 degrees), as follows:

15(6),) = 5(9) +az(@—@o)2+a4(@—®o)4 (4. 3)
where a2 and a4 are determined from relativistic kinematics, E(®) is the measured energy
at the measured angle 8, and B(OO) is the energy at 80. This parameterization is used
because of the ease and speed of the calculation. The kinematic energy shifts were
calculated for (~30 = 0.0 degrees.

4.3 Results

Nuclear species were identiﬁed using the energy loss, total energy, and time-of-
ﬂight measurements described earlier. Particle identiﬁcation in the spectrograph focal
plane starts by assuming all ions reach the detectors with the same average rigidity. The
particle’s total energy and speed is then calculated and used to make- simulated particle
identiﬁcation plots. The measured and calculated patterns are compared to identify each
group of ions.

Figure 4.11 is a spectrum of energy loss in the CRDCs vs. measured time-of-ﬂight
to the ﬁrst scintillator. The data was ﬁltered to select only the 11C and 8B groups. Subse-
quent ﬁltering was performed using total energy measurements from the stopping scintil-
lator. The resulting contamination was <1% for the 13B spectrum, and 2.5% for the 10Li
spectrum. The 11C group is very clearly separated, and the contamination for that group

is well below 0.1%. Table 4.2 lists the measured cross sections for all of the reactions.

83

Table 4.2: Reaction cross sections from the 10Li experiment. The uncertainties on
the cross sections are purely statistical. A 10-50% systematic error from the beam
current measurement is not included.

 

 

 

 

 

 

 

 

 

Reaction Target Area ‘ 0 [nb/sr]
10Li(lowest peak) 0.94 mg cm'2 Be 305 2601-15
l3B(g.s.) 1.1 mg cm'2 C 44 420i60
7He(g.s.) 0.94 mg cm'2 Be 14534 12300i100
10Be(g.s.) 1.1 mg em-2 (3 1379 13 120:1:350

 

84

 

 

 

Summed AB in CRDCs [a.u.]

 

 

 

Time of Flight [a.u.]

Figure 4.11: Measured energy loss vs. time-of-ﬂight spectra. Contamination in the 8B
gate arises mainly from scattered beam.

85

4.3.1 10Be and 7He energy spectra

The (9Be,“C) reaction on the carbon and beryllium targets provided additional
calibration and veriﬁcation points by measuring the location of states in the nuclei 10Be
and 7He. The ground state mass of 10Be [Au93] and its ﬁrst excited state at
3.36803(11)MeV [Aj88] are known to better than 1 keV. These states were included in the
energy calibration. Figure 4.12 shows an energy vs. scattering angle plot of the
12C(9Be,llC)IOBe reaction. It clearly shows that the ground and ﬁrst excited states of
10Be are populated, as well as states in the ejectile 11C nucleus. Figure 4.13 shows just an
energy spectrum of the 11C ejectiles from this reaction, as well as Gaussian ﬁts used to
extract the peak locations. There is no problem ﬁtting the isolated ground state peak with
a Gaussian shape. The location of the ﬁrst excited state is more difﬁcult to ascertain due to
the complex, non-Gaussian background surrounding it. Nevertheless, the spectrum was ﬁt
with a sum of 5 Gaussians, and the position error in ﬁrst excited peak was increased by the
covariance of the peak parameters surrounding it. i

The ground state of 7He is known at Sn=-4403:30 keV with a width of
F=160i30keV [Au93,Aj88]. Figures 4.14 and 4.15 are the data taken populating the 7He
nucleus. The peak ﬁtting procedure for the 7He ground state is not straightforward
because it is an unbound nucleus and has an asymmetric shape with an energy-dependent
width. The line shape is folded with the experimental resolution function and ﬁt to the
data. The location and width of the peak is within errors of their expected values. (See
section 4.4.3.1 for details about the ﬁtting procedure.) This procedure conﬁrms the beam
energy and relative target thicknesses between the carbon and beryllium targets.

The experimental resolution function was determined to be a mixture of Gaussian

86

and Lorentzian shapes. Figure 4.16 shows the effect of using a pure Gaussian (top) con-
trasted with using the mixture. The function is a sum with 25% Lorentzian and 75%
Gaussian. This mixture improves the ﬁt substantially in the low (<0) energy tail. How-

ever, it doesn’t appreciably affect the width or location of the peak.

87

 

o
Illjllllll-

llilll'lqnlrilll-lq

Scattering Angle [deg]
O\

 

 

LrJ
rIIIllllllllIITIrIT‘lIIﬂ‘I—IFII

 

 

5
4 .
. -_ ""'_ .1
1111111'11I1111l1111l1r\\rtiril‘rr-l 1114
.60x10‘3 —40 —20 0
6.

Figure 4.12: Spectrum of 11C ejectiles from 12C(9Be,“C)lOBe reaction populating the
ground and excited states in 10Be. The solid lines overlaid on the spectrum are the calcu-
lated kinematics for known states in 10Be. The dashed lines are the 11C 2.0 MeV excited
state corresponding to the reactions (from right to left) 12C(9Be,l]C(2.0))10Be(g.s.) and
12C(9Be,1'(:(2.0))1013e(3.368).

88

 

IrIIlrIjIrIIIIII

 

300

10Be (0.0)

200

Counts

 

°Be (3.368)

100

 

 

 

III]IIIIIIIII'IIII'IIIIIIIIIIIIII'I

 

 

   

 

lllllllllllllllllLlll

0 100 200 400

300
Channel

II'IIIIIIIII'IIIIlrrIIlITIIIIIIII

10

Allllllllllllllllllljllllllllllllll

 

500

 

1 .gh-‘-I I' ,. ‘~.. ! _ ‘-'— i '
- '- “m‘ \

...... .r . . . w» - : e .
94 W .1.

.lst'o

ﬁt residuals

400

300

200

.IIIIITIIIIII

Counts

100

N]

IIII]

x :-
ILIIIIII ..

280

 

  

O rlr111l1111l111111111

220 240 260
Channel

 

300

3|IIIWITIVIIIIIIIIIWIFIIIIIIIFIIITIIIIFTIIIIIIIIIII
“to":

llllllllllllllllllll

 

Figure 4.13: Top: Spectrum of 11C ions in the S800 focal plane populating states in 10Be.
The top panel shows the entire focal plane extent. Bottom: Five Gaussians were ﬁt to the
lower portion of the spectrum to determine the location of the g.s. and 3.368 MeV states in

10Be.

89

 

9 IllllllI'IIlllllllllIIIIIIIIIITIIIIIIIIIILIIIIIIIIIIIIIII

llll

   

   

Scattering Angle [deg]
as

I_IllIIIIIIIIIIIIIIIIIIIIIII

.
- I -t. __ ._.__ _'_ _. — - . .
- _ - .

 

 

3 llllllIIIIIIIIl—lLl-llzrlluugll—I'llllﬂlTL;-L1lI-IIIIIHI:IIII
—60x10'3 -50 -40 -30 -20 —10 0
6.

l

 

Figure 4.14: Same as ﬁgure 4.12, but using the 9Be(9Be,“C) reaction to make 7He.

90

é

 

 

 

 

 

 

 

 

 

h I I I I l I I I I I I I I l I I I I I I I I I l I I I I I I I I I ..
a -l
1500 :- 3; T -
._ ,3 r I} -
d.) - .I ..
= l- l] x -1
I:
e t / g .
im’ 1000 - ,t‘ \ —
g i f I\ I
U : f I III:
\ I
500 — r, I —.
F If a It ‘
C 7/ K u‘ i
i- 3' KM“
omffi/I/I l l I L1 1 l - - I I n l l I l l l l
-l.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
15,,m lMeV]
2000 - I i I I I I I I I I I I I l r I I I l I I I I l I I I I I I I I I .1
.. 7 .1
: He :
1500 — E -. a
E 3 ﬁ {it I
t: 1‘
5 : f I ‘
J: i .1
2 1000 - \ -
.22 , .
C
e - r -
U I , it Iri‘
500 r— ! .1
J I
I- ‘41
0 I l I l l I l I I l I n 1 I L 1 I 1 1

 

 

 

.N
u:

-0.5 0.0 0.5 1.0 1.5 2.0
Ecm [MeV]

I
b

Figure 4.15: Fits t0 the 7He ground state using two different resolution functions. The top
panel shows a purely Gaussian resolution function, whereas the bottom shows an
improved ﬁt (<0 MeV) by including some Lorentzian shape in the resolution ﬁinction.
The mixture doesn’t affect the peak location (430i30 keV) 0f the ﬁt, but has a small affect
on the extracted width (120280 versus 150:t30 keV).

91

4.3.2 13B energy spectrum

The 12C(9Be,88)13B reaction provided a quality 8B energy calibration. The reac-
tion populates numerous states in the 13B nucleus ranging from the 0-10 MeV. The higher
lying states provide the most useful energy calibration because they are closest to the 10Li
Q-value. Figure 4.12 shows a '3 B energy spectrum. The top portion of the ﬁgure shows
all the peaks. The bottom panel shows Gaussian ﬁts to the spectrum of the known states in
13 B. The ground state is the only clearly separated peak. Because of the substantial back-
ground above 7 MeV in the spectrum, peaks above 7 MeV were excluded from the energy
calibration.

The resolution for this peak is E/AE=1300, limited by the intrinsic angular spread
in the beam. The target thickness broadening has an effect, but is E/AE~3000. The much
larger effect is the kinematic broadening due to the beam emittance. The primary beam
typically has an emittance 5-10 It mm-mrad in both the dispersive and non-dispersive
directions, usually a little better in the dispersive direction [St98]. These angles smear out
the kinematics of the reaction substantially, especially at larger angles.

Directly measuring these divergences by transmitting the beam to the focal plane
is the preferred method of determining the intrinsic angular divergence. This option was
unavailable in the present experiment. Instead, the beam divergence was estimated
through simulation. The prescription is to ﬁrst assume that the resolution is entirely dom-
inated by kinematic and target thickness broadening, a valid assumption for energy resolu-
tions of ~1 in 103. The angular distribution for an isolated state is measured and entered
into the simulation. The 0 and 4) acceptances are measured and the target thickness contri-

butions are known. These are entered into the simulation, leaving only the incoming

92

angular spread, (136,134)), as free parameters. Finally, these angular widths are adjusted
until the resolution matches that of the measured peak. The resulting full angular widths

were 4.4 milliradians in each direction.

93

 

 

 

 

 

 

 

 

 

 

50 r-I-I-I-rr-I-r-I-rI-rr-r-rnw-r-rr-rm
" 1
.— \o" 133 -l
I t I
40 — t5 -
I- m n
. oo 3“ —: - .
: 2% vi 2 :
_ r —I (\ _
*2 30 _ :3 I ; m; .
g I °° 8 1
U 20*- ] u- _-
I- I’! fig! l \d ([5 ‘
.. 1 . o0 -
10 '— ' _§ ""
: I 1
0 ,_ < _ - ,,lIJ.,,..'_I‘,
O 100 200 300 400 500
Channel
—"’- lO-
3
g -5
:10
t...
13B '2‘
£9 1
I: -
= _
o I _
U h : 1
till '2‘ 3
II I“. .II II _
. 2!. r {I =.- ,, j
’r' '=
1' 5.. g“. Y, i : 1

 

 

 

Np-
0‘.
Cl
DJ
8

Channel

Figure 4.16: Top: Energy spectrum of 8B ions in the focal plane populating states in the
13B nucleus. Bottom: Multiple Gaussian ﬁts used for energy calibration of the focal
plane.

94

 

 

 

 

- I I j I I I I I I I I I I I I I I I I -
1.0 — . _
F lorentznan .
: ----- gaussian f :
0.8 _— —— mixture 5‘ 1
- r, = 270 keV f .
.. LII] ‘-
0.6 — I _
: / :
0.4 — ”j \ __
Z l \‘a -
I. ll: \‘\ 1
0.2 : If? I‘\ .\i 1
0.0 ..... ~“11‘Léfu/l/jill l 1 l 1 l ‘\l‘-_l_. ‘ xx" ..... d
- 1.0 -O.5 0.0 0,5 1.0

Energy [MeV]

Figure 4.17: Resolution function used for the (gBe,SB) reaction (solid line). All three
peaks have the same FWHM.

Adjusting the resolution width for different target thickness and kinematic broad-
ening effects one arrives at the total resolution width of 300 keV for the 8B energy spec-

trum from the 10Li reaction, which is 270 keV in the 10Li c.m. system.

4.3.3 10Li energy spectrum

Figure 4.19 shows an energy spectrum for the 8B ejectiles following the
12C(9Be,gB)l38 and 9Be(9Be,8B)10Li reactions. The 8B ions were selected, and kine-
matic energy shifts were applied event by event to shiﬁ the energy to what it would be at
0°. This moves all events with the same Q-value into one peak. The loLi spectrum exhib-
its the sharp rise at threshold characteristic of unbound nuclei and similar to that observed
for the 7He spectrum shown earlier.

A pre-existing NSCL computer program called RELMASS was used to calibrate

95

the energy within the 10Li system. The program is described in detail Brian Young’s thesis
[Yo93]. The program accepts spectrograph magnetic ﬁeld settings, calibration reactions,
target thickness, beam energy, and scattering angles as input. For energy calibration, it
takes a known state in a nucleus and deduces a bending radius. RELMASS then ﬁts a
polynomial to the bending radius as a function of focal plane position for a series of mea-
sured ﬁeld vs. position measurements with a particle of known momentum.

For mass measurements, the process is inverted; the program uses a focal plane
position to deduce bending radius and then the energy of the state in the nucleus. The cal-

ibration data and subsequent ﬁt are shown in Figure 4.19.

96

 

—n
O

 

 

 

 

 

 

 

'1 —IITIIIIIlIlITrrrIWFIIIIIITIrTTTI[III-

.55; 0.5:— I -E

I.._.I l. I :

'a 00— T I E T T T ‘

:3 E I l' t 13

3 -05? l —E

L: -1.0-‘— —‘

-— A

— —J

300: _

L" 'l

T) 280—— ‘j

g . .

5 260— -:

240; L

220—; lALlllllJlllllllldlllllillllllllll—
2.805 2.810 2.815 2.820 2.825 2.830 2.835 2.840

BplT-m]

Figure 4.18: Rigidity calibration for this experiment. The line through the data is a linear
ﬁt. The points surrounded by sciIuares are calibration points from the 12C(SlBe,llC)10Be
reaction; all others are from the 2C(9Be,SB)13’B reaction.

97

4.3.3.1 Fitting procedure

The expected line shapes for an unbound nucleus are asymmetric resonance shapes
with an energy dependent width because the separation energy changes dramatically over
the width of the state. The line shape for a complicated transfer reaction is nearly impossi-
ble to deduce exactly, so resonances are usually calculated using one-nucleon scattering.
For example, Young [Yo94] made calculations of neutrons scattering off 9Li to deduce the
approximate line shapes for s-wave and p-wave scattering. Several parameterizations
have been established [Yo94, Be98], but there is no clear favorite. In particular, there may
be reason to expect that the s-wave line shapes depend strongly on the initial states
[Th99]. An R-matrix parameterization is a good approximation [Mc94] for the p-waves

and ﬁts the calculated p-wave line-shapes of Young very well. The form is as follows:

NE)

 

0(E) = A (4. 4)

2

2 F
(E-E,) +2—

where I“(E)~E'+1/2, so the cross section exhibits the well-known E3/2 behavior at
threshold. The s-wave shapes were determined by ﬁtting a modiﬁed Breit-Wigner
distribution to the calculations of Young [Yo94]. The width is much more energy
dependent than the p—waves. The parameterization was adjusted to reproduce the energy

dependence of the width that Young calculated [Yo94].

98

 

ITIIIIIIIIrIﬁIIjIIIIIITTIIIII

40 13B

l

30 ,

Counts

l .
20 '11 1 1M1

if :1 W11 l?
11%; f 1 HO

1
l
i
‘1

10 If :11 I 1   ' l

40

liJllllllllllllllllll

 

 

 

Li

 

30

Counts

20

 

r IIII'IIj—rlITjIlrII

10

 

 

0 114 Llllllliljlbli ‘thll’llll .E_ll.-.._.l- "
50 100 150 200 250 300 350

Channel

Figure 4.19: Energy spectrum of 8B ions from both the reactions 12C(9Be,i‘B)13’B (top)
and 9Be(9Be,BB)loLi (bottom), shown for comparison.

The statistics of the spectrum are low and hence a maximum likelihood ﬁtting pro-
cedure is appropriate. The ﬁtting procedure is used to maximize the likelihood that the

data set describes a Poisson distribution. The likelihood function is deﬁned as follows:
N
L = HP(YI»31) (4- 5)
i = l
where P(yi,ai) is the probability of observing yi, and ai is a vector ﬁlled with the ﬁt

99

parameters. Minimizing the negative logarithm of L is the same as maximizing L but
more useful for ﬁtting routines which try to minimize a ﬁgure of merit. The distribution is

assumed to be Poisson in nature, and hence this equates to minimizing the function

N
‘1"(L) ‘3 EYiln(f(xis ai))_f(xivai)" My!) (4. 6)
i=1

where Yi is the measurement at point Xi and f(xi,ai) is the ﬁt evaluated at point Xi with
parameters ai.

The ﬁtting model consisted of a background and line-shape contributions. The
background shape was a constant plus a three-body contribution which arises from mak-
ing 9Li+n+BB in the ﬁnal state [8156]. The form is a EU2 shape at threshold leveling off to
a constant at roughly 3 MeV. The data were ﬁt with this background plus one p-wave,
two p-waves, or one s-wave and one p-wave shapes.

The background shape and size have a very small affect on the locations of the
peaks, but a dramatic affect on the widths. Therefore, the background size was adjusted
by ﬁtting one p-wave peak to the data such that the size of the ﬁt region didn’t affect the
peak locations or widths. The resulting background shape was then held constant for all of
the ﬁts.

Figure 4.20 shows all the ﬁts to the 10Li energy spectrum. A single p-wave ﬁts the
data with a peak at 500(60) keV, and a width of 420(60) keV. When using a combined ﬁt,
the spectrum can be ﬁt with one p-wave at 525(60) keV and either an s-wave right at

threshold (<30 keV) or an additional p-wave at 100 keV. Table 4.3 summarizes the ﬁt

100

parameters.

Table 4.3: Fitted parameters for 10Li energy spectrum.

 

 

 

 

 

 

 

 

States included in the ﬁt Epeak [keV] F cm [keV]
one p-wave 500i60 . 400i60
two p-waves 100i60, 525160 50i60, 345i60

one s-wave, one p-wave <50, 515i60 <100, 350i60

 

Uncertainties of the parameters were calculated by summing all contributions in
quadrature. There are systematic contributions to the absolute energy scale from target
thickness (30 keV), beam energy (10 keV) and scattering angle (10 keV). The uncertainty
of the width and shape of the resolution function each contributes 10 keV. These uncer-
tainties contribute 40 keV systematic error to the peak positions. The relative energy scale
(keV/channel) is much less sensitive to the target thickness and beam energy contribu-
tions. This reduces the systematic uncertainty of the width substantially, from 40 to 10
keV. Statistical uncertainties on the ﬁt parameters contain contributions from the individ-
ual parameter variance and the covariance between other parameters. These depend on
each parameter and range from 10-60 keV. The statistical uncertainty of one parameter
was determined by adjusting it until the ﬁt test parameter changed by unity. These were
added in quadrature with the systematic uncertainties to arrive at the uncertainties in Table
4.3.

All ﬁts are of very similar quality, making interpretation difﬁcult. There is clearly
a peak at ~500 keV, no question. The region near threshold needs more serious consider-
ation. The structure peaking at ~100 keV could be either an s-wave right at threshold or a
p-wave at ~100 keV. Past experiments and three-body calculations taken together [K088,

Ta85, Kr93, Yo94, Th94, Th99] would favor an s-wave interpretation for this low lying

101

structure. However, this claim can’t be made from the present data alone. In fact, when
considering the degrees of freedom, the single p-wave ﬁt is the best with a LV=O.95. Nev-

ertheless, the other ﬁts are equally valid given the low statistics in the energy spectrum.

 

 

 

 

 

   
 

 

 

10

 

.- I I I I I I T I I I I I Ij I I I_II_I I Iﬁ I I I I I I I I I I I I l I I I j q
' 0 9Be(gBe,88)lOLi data ‘
30 l- _
- b ----- background -
I c ' — l p-wave ﬁt " I
' + --------- 2 p-wave fit a
25 — ._ _
_ __1 ----- s-wave and p—wave ﬁt -
I- O _L ‘ u-
t l .
20 I- t, -'
g l. ‘ F ' .- .
a : 3
I 1
C

  
   

 

 

 

 

 

[111'1L1111 llllllllllllllIlLllIllllIJlll

-l O l 2 3
Ecm [MeV]

 

Figure 4.20: The 10Li energy spectrum gathered in this experiment. The three different
ﬁts are of similar quality, but the single p-wave ﬁts the data best because it has the fewest
degrees of freedom. The background values were held constant for all three ﬁts.

102

 

 

4.4 Discussion

The ﬁrst measurement of this reaction reported one state at 800(250) keV [Wi75].
It is conceivable that the ﬁtting procedure used in that experiment was a symmetric ﬁanc-
tion and hence would move the peak toward higher energy. Nevertheless, 800 keV value
reported there was not corroborated by this measurement.

This particular transfer reaction did not populate 240 keV state reported by Bohlen
et al. [8097] and Bertsch et al. [Be98]. The Bohlen group used a charge exchange reac-
tion to populate the states (see Table 4.1), and it is possible that the reactions have differ-
ent selectivity. It is also possible that the state was misidentiﬁed due to the plethora of
other peaks surrounding the supposed peak. Another measurement of this reaction with
higher resolution and fewer target contaminants would be desirable.

The analysis of Bertsch et al. [Be98] favors an s-wave state peaking at 0-100 keV
and a p-wave state at 250-350 keV. The resolution in that experiment is insufﬁcient to
resolve the two states but the best ﬁt is obtained by including both.

Comparison of data from the present experiment with that of Young’s is quite
remarkable. Both spectra exhibit the same relative size and shape between 0 and 3 MeV,
seen in Figure 4.21. The low-lying rise at threshold is present as well as the large, broad
peak at ~500 keV in both spectra.

This measurement also agrees with the 530:}:60 keV measurement by Bohlen et al.
[B097]. This measurement has the same net nucleon transfer as the present experiment (-
p,2n) and perhaps suggests the two reactions should have the same selectivity.

The work of Kryger et al. [Kr93] and Thoennessen et al. [Th99] indicate a low-

lying s-wave state and a p-wave state at 540 keV. While their resolution was insufﬁcient,

103

it still appears that the 240 keV state wasn’t populated in those experiments either.

All these facts lead to a question: Is there really a 240 keV state? Certainly there
has to be another p-wave state. It could be very low-lying, or it could be in with broad
~540 keV state seen in other reactions. The only way to'clear up this situation is to do
more experiments. A coincident inverse kinematics measurement such as 9Li(d,p)10Li
where both the protons and 9Li fragments are detected might populate all states and should
determine spin-parity assignments of the states involved because the (d,p) reaction is well
understood. This experiment has been performed with the 8800 spectrograph and results

should be available shortly [Ko].

104

”B(7Li,8B)‘°Li, E = 131 MeV

Counts

9Be(9Be,83)mLi, E = 361 Me

iii I;

?

 

 

 

 

 

 

-1 0 1 2 3
Ecm [MeV]

Figure 4.21: Two different reactions measuring the 10Li energy spectrum. The similari-
ties between the two are striking.

4.5 Conclusions

In summary, another binary reaction measurement of the 10Li system was made.
One p-wave state was observed at 500i60 keV, with a width of 400i60 keV corroborating
several previously reported results. A small excess of counts right at threshold could be

interpreted as a p-wave at 100i60 keV or an s-wave state at <50 keV, but the evidence for

105

such states is weak at best. Since there are no deﬁnitive ﬁnal measurements available for
the 10Li system, each additional measurement contributes valuable knowledge to the over-
all picture. Future experiments will be performed with the goal of continuing this clariﬁ-

cation process.

106

 

 

Chapter 5

Spectroscopy of 27P and 23A1

5.1 Motivation
5.1.1 Explosive hydrogen burning and the rp process

Explosive hydrogen burning is a phenomenon that occurs in extremely hot and
dense stellar environments such as occur during X-ray and novae bursts, and Type II
supernovae. The extreme temperatures (108 - 109 K) and densities (104-108 g cm’3) in
such environments are necessary for the protons to overcome the Coulomb barrier and
fuse with other light nuclei. X-ray and nova bursts occur in binary systems where a small,
massive object like a white dwarf accretes hydrogen from a companion star, like a red
giant. A thermonuclear ﬂash occurs once enough material has been accreted onto the
white dwarf and some of the surface material is ejected and contributes to galactic element
production. The massive companion in an x-ray burster system is usually thought to be a
neutron star or black hole so no material escapes the star’s gravity, and hence it contributes
nothing to galactic elemental production. While the hydrogen burning in a supernovae
does not contribute to the energy production or the explosion, observation of the proton

rich nuclei produced in the event can shed light into and put constraints on the explosion

107

processes.

The hot CNO cycle and the rp-process are the two main processes which comprise
explosive hydrogen burning. The HCNO cycle is fairly well understood, but the rp-pro-
cess is not. The rp—process is a series of rapid proton capture reactions and [3+ decays that
lead to the production of proton rich nuclei. As the environment gets hotter and denser the
rp-process moves toward the proton drip line and advances to higher mass nuclei. The
capture rates are often based on limited nuclear data or calculations and therefore are very
uncertain. Reactions near the valley of stability often have high Q-values (~5 MeV) so
statistical (Hauser—Feshbach) calculations can be used to determine energy levels because
the level density is high. Reactions on nuclei far away from stability typically have low
Q-values, so the reaction rate is dominated either by resonant capture or direct capture to
the ground state. Not only is it experimentally difficult to study these nuclei, but Hauser—
Feshbach calculations fail due to the low level density. Recently, shell model calculations
have been used to predict excited state locations in proton rich nuclei, but experimental
veriﬁcation of the levels is still desired [Ch92].

Gamma ray observations of explosive events can provide insight into the workings
of the event and how the ejected material is mixed into the interstellar medium. Under-
standing gamma-ray observations of these events requires knowledge of the production
rate and half-life of these isotopes. Typically the light emitted from the explosive event at
any given time after the explosion is dominated by the decay of one or a few nuclear spe-
cies, such as 22Na, 44Ti or 26Al. Understanding their production and half-life can provide
information about the complex workings such as energy generation and nucleosynthesis

for the explosive event.

108

  
  
  

T9= 0.4
p = lOiig/cm3

rp process

24 26 28

    

22

 

Figure 5.1: The rp-process network for speciﬁc temperature and density conditions,
reprinted from [Ch92].

This chapter will describe spectroscopic measurements of two nuclei which impact
the production of 22Na, 26Al, and other heavier mass elements under certain conditions
during an explosive event.

5.1.2 Stellar Reaction Rates

Many (p,y) reactions in the rp-process proceed through resonant capture and it nec-
essary to know which parameters are important for calculating the capture rates. What
follows is a derivation of the resonant capture rate, taken from [R088].

The gas in a star is normally nondegenerate, the motion of the nuclei can be
described non-relativistically, and the gas is in thermal equilibrium. Hence, the velocity

distribution of any nuclear species in a stellar environment is well described by a Max-

109

well-Boltzmann form:

¢(v) = 4nvz(2$(T)3/2exp(—%n£;) (5. l)

where m is the mass of the species and T is the temperature of the gas. The total reaction

 

rate for a reaction whose cross section is 0(0) is given by the integral:
r = NxNy(ov) = NxNyfo ¢(v)vo(v)dv (5. 2)

where the product NXNy is the number density of pairs of the two species x and y, and v is
the relative velocity of the two species. If the two species are identical, then r is divided by
two to eliminate double-counting. Calculation of the total reaction rate involves the
integral over all velocities, but because there are two species with different masses, the
reaction rate involves a double integral over both distributions:

0000

(00) = ff¢(vx)¢(vy)vo(v)dvxdvy (5.3)
00

Transforming the two velocities UK and vy into center-of-mass coordinates v and V, the
double integral changes to two independent integrals. Performing the integration over the

variable V (the velocity of the center-of-mass) the reaction rate per particle pair is

(00) =f:¢(v)vo(v)dv. (5.4)

Inserting the velocity distribution with the reduced mass u, and using the center-of-mass
l 2 .
energy E = itw we amve at:

l/2 l °°

Wfo o(E)Eexp(-I%,)dE (5. 5)

“who

up

If the reaction proceeds through a narrow resonance then the reaction cross section

110

takes on a Breit-Wigner form:

2 21+1 rarb

08MB) = 3‘" (21 +1)(2J + 1)(1 +512) 2
1 2 (E—Er)2+(E—)

 

 

(5. 6)
4

The resonance is assumed narrow if F « E r. Over this small energy range the Maxwell-

Boltzmann energy distribution is ﬂat, takes on the value at the resonance energy and

comes out of the integral. Performing just the cross section integral:

so I‘ r
_ 2 2 a b
L) oBW(E)dE _ 211: x u) r

 

(5. 7)

where (1) is the statistical factor in Equation 5.5. The resonance strength is deﬁned as

(1) = (Drarb
Y I" '

 

For radiative proton capture, the proton widths for the reaction are determined by
the Coulomb and centrifugal barriers. These depend on the radial wave functions and the
form of the potential taken for the nuclear part of the interaction, and are given by:

F,(E) = 21,65) 1/2

n

P1(E, Run),2 (5. 8)

where Rn is the radius in the nucleus where the capture occurs PI(E,Rn) is the probability
of penetrating the barrier for a given angular momentum l, and 012 is generally called the
reduced width of the state. This is essentially the probability that the capture leaves the
resulting nucleus in that state. The penetrability is deﬁned as the absolute square of the
ratio of the radial wave functions at inﬁnity and inside the barrier, and depends on the
energy and angular momentum of the resonance.

Finally, the resonant capture reaction rate can be expressed as:

111

 

Adult».

if». ...P.

 

3/

271sz exp (7%) (5. 9)

<ov> = 131)

ukT
Because of the exponential dependence, uncertainties of 100 keV in the resonance energy
can mean orders of magnitude uncertainty in the reaction rate. The uncertainty also enters
through the resonance strength but is usually less dramatic. The proton widths are
typically linear functions of the resonance energy over the uncertainty range. The gamma

E2)“, where 2» is the multipolarity of the transition, and depend on the initial

widths go as
and ﬁnal quantum numbers, so the spin-parity assignments for the levels involved are also
needed to determine the resonance strengths. Experimental determination of spin-parity
assignments is preferred but not always available. Reaching very proton rich species with
such reactions requires special targets which maybe difﬁcult to manufacture. In the
absence of experimental information shell model calculations provide accurate spin-parity
assignments of the levels involved.

Factoring in constants, a usable form of the reaction rates derived above is as fol-

lows [He95]:

—11.605E,[MeV]
T9

 

N,(ov) -_- 1.54x10“(AT,)‘3/2wy[MeV]exp( )cm3s“mole" (5.10)

where the reduced mass A is given by ApAT/(Ap+AT) and AI) is the proton mass and AT is

the target mass. The resonance strength is given by

 

my _ (21+1) r,,rY

‘ 2(21T+1) r (5'11)

where J is the spin of the compound nucleus and IT is the spin of the target. The proton

width is normally determined by I‘, = CZS - I‘ where C28 is the single particle

spa

spectroscopic factor and I“Sp is the single particle width. The I‘Sp is determined by

112

calculating phase shifts of the protons scattering off a Woods-Saxon potential whose depth
is determined by matching the resonance energies.

The gamma decay widths I“I are determined from the reduced electromagnetic
transition probabilities and have an energy dependence speciﬁc to the type of transition.
The reduced transition probabilities are usually determined using shell model calculations.
The transitions of importance here are either E2 or Ml transitions and the gamma widths

are given by:

I‘Ez[eV] = 8.13x10'7EY5[MeV]B(E2)[e2fm4] (5. 12)

and

I‘M,[ev1 = 1.16x10'2E,3[MeV]B(Ml)[u~2] (5. 13)

Direct reaction cross section measurements are the preferred approach for deter-
mining these capture reaction rates. However, the cross sections become prohibitively
small at stellar energies because of the repulsion due to the Coulomb barrier so other
experimental methods must be employed to gain information about the levels. Indirect
spectroscopic measurements have been successfully used to determine energy levels in
proton rich nuclei using multinucleon transfer reactions. The (7Li,8He), (3He,8Li), and
(4He,8He) reactions are examples of reactions that have been successfully used in the past

for spectroscopic measurements of proton rich nuclei [Wi88,Wi98,Be75,Be77,Sc97].

5.1.3 Why study ”P?

The level structure of 27F governs the proton capture reaction rate on 26Si and dra-
matically affects the procession of the rp-process through the waiting point nucleus 26Si
(t1/2=2.2 seconds). (A waiting point nucleus is one with a half-life comparable to the

duration of the explosive event.) If the proton capture rate is small enough, the rp—process

113

must wait for 2(’Si to beta decay before proceeding to higher masses. Also, if 26Si remains
at the end of an explosive event, it beta decays through an excited state of 26Al directly to
the ground state of 26Mg, bypassing the ground state of 26Al. It is the [3+ decay of the
26Al ground state to an excited state of 26Mg that produces the characteristic 1.8 MeV
gamma ray. The important point to note here is that the characteristic 1.8 MeV gamma ray
from 26Al beta decay is n_o_t produced. Therefore, it becomes important to predict the
amount of 26Si produced during the event in order to understand the gamma ray observa-
tory observations of 26Al gamma rays from explosive events. Several reactions contrib-
ute to the population of this nucleus, shown schematically in Figure 5 .2. Proton capture
on 25A1 and beta decay of 26P increase the population, while proton capture on 26Si and its
beta decay decrease the population. This is just a subset of the much more complex net-

work of reactions in the rp process, shown in Figure 5.1.

 

 

 

 

 

 

A ‘iilihb
\
258i 268i
V [3+

 

 

 

 

 

Figure 5.2: Subset of the chart of the nuclides schematically showing the rp-process reac-
tions affecting the abundance of 26Si. This is just a small subset of the total reaction net-
work (see Figure 5.1).

The proton threshold in 27F is at 0.897(35) MeV. Experiments have reported only
one excited state at l.66(4) MeV [Be77] using the 32S(3He,8Li)27P reaction at 70 MeV

and 7° (see Figure 5.3). Recently a shell model calculation suggested there is a state at

l.l8:t0.10 MeV, but there is no experimental report of this state. Examination of Benen-

114

 

son et al. spectrum indicates the possibility of a state at ~l .2 MeV, but it wasn’t reported.
A level at ~1.2 MeV would change the proton capture rate by orders of magnitude at cer-
tain temperatures. Network calculations at these temperatures then show a substantial
depletion of the 26Si nucleus, and an enhancement of 27P and 288. Thus, experimental

veriﬁcation of this level is highly desired and another measurement would prove useful.

 

v I I I 1 r 1
323(3He, 8L1127P
12- 70 MeV -
EX 3 i.66 7o

 

 

 

+

 

COUNTS PER CHANNEL

J
1‘

 

 

 

 

 

g.
I Illlll

v
I I

 

[State at ~l .2 MeVﬂ
L o 4
s.

l
CHANNEL NUMBER

Figure 5.3: Benenson’s spectroscopic measurement of 27F [Be77]. There appears to be a
state at ~1.2 MeV, but wasn’t reported.

5.1.4 Why study 23A]?

The level structure of 23A1 determines the proton capture reaction rate on 22Mg. In
the 23A] case, the Q-value for the (p,y) reaction is very low at 0.13 MeV and thus com-
petes with photodisintegration. Processing more 22Mg into 23Al and beyond decreases the
amount of 22Na produced in the explosion. There has been one observation of an excited

state at 0.462t0.06 MeV by Wiescher et al. [Wi88] using the 24Mg(7Li,8He)23Al reaction at

115

190 MeV but was not resolved from the ground state (see Figure 5.2).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30'- 24' 7 I. I '23 4
Mg( LI, He) Al
High energy
shoulder is
ﬁrst exerted state 20 _ z
m
p.
2 H
D _ _
O
o
10 - ‘
O “l l I l l
500 550 600 650
CHANNEL

Figure 5.4: Previous spectroscopic measurement of 23Al. The ﬁrst excited state is not
resolved from the ground state [Wi88].

Using the same reaction and the 8800 spectrograph, it should be possible to resolve the
two states and measure its location to 5 keV accuracy. Since this level is so close to the
proton threshold, an uncertainty of 60 keV can mean orders of magnitude uncertainty in
the reaction rate. Thus, resolving the ﬁrst excited state from the ground state is highly
desirable. An uncertainty of 5 keV would translate into an acceptable 10% uncertainty in
the reaction rate.

5.2 Experiment

The experiment employed the 8800 spectrograph in the dispersion-matched,

116

 

energy-loss mode. A primary beam of 50.1(3) MeV/u 7Li entered the 8800 at the object
position, the beam was transmitted to the target position in the main scattering chamber of
the 8800, and targets of 2538i and 24Mg were used to populate states in 27P and 23A1 with
the binary transfer reaction (7Li,8He). The beam interacted with the targets, and the 8He2+
ions were swept to the focal plane detectors by the spectrograph magnets. The angular
coverage of the spectrograph was 0-12° in the laboratory.

The targets used were 0.94 mg cm‘2 93c, 0.34 mg cm‘2 10B mounted on a 30 ug/
cm2 carbon backing, 0.63 mg cm'2 24Mg, and 1.4 mg cm'2 “atSi. The transverse size of the
targets is modestly important for background reduction, especially given how thin the tar-
gets are. Since the image in the main scattering chamber is dispersive (12 cm/%), and
there is roughly a 0.08% momentum spread in the primary beam, the beam spot size will
be about 1 cm on target. Beam tails can react with the target frame and produce an unde-
sirable background in the focal plane. Targets of at least 12 mm in diameter were chosen
to minimize this kind of background.

The natural silicon target was a 2” diameter circular silicon wafer obtained from a
silicon chip manufacturer, Virginia Semiconductor, placed in a 45 mm diameter circular
frame. The target was extremely uniform (at the 0.1% level) and was oxygen free but con-
tained the natural abundances of other silicon isotopes. The target was 1.4(1) mg/cm2
thick as speciﬁed by the manufacturer and was veriﬁed with alpha-particle energy loss.
Prior to the experiment, it was not known whether the 4.3% 29Si and 3.7% 3OSi contami-
nants in the target would pose contamination problems.

The 24Mg target was 99.9% enriched and was 063(2) mg/cm:2 thick mounted in a

20 mm square frame. The thickness was measured with alpha-particle energy loss.

117

 

The target thicknesses were veriﬁed by measuring the energy loss of the beam
through them. This method is only good to 10% absolute target thickness because of the
uncertainty in stopping power of the target materials for a 7Li beam of 50 MeV/u. Alpha-
particle energy-loss measurements (typically done with “alphas from 241Am and mixed
thorium sources) are much more accurate for the silicon and magnesium targets because
of the quantity of existing stopping power data (see [Zi9l]). Nevertheless, beam measure-
ments were within error of the alpha-particle measurements.

The 7Li primary beam was ﬁrst transported to the focal plane of the spectrograph
to check the dispersion matching. Ideally, the width of the peak at the focal plane should
be 5 0.01% of the total beam energy. Figure 5.5 shows a spectrum of the position in the
dispersive direction at the ﬁrst tracking detector. The peak has a non-Gaussian shape with
a full width of AE/E=0.03%, but this is acceptable because the resolution was limited by
kinematic broadening due to the primary beam divergence (A0~5mrad, A¢~5 mrad

FWHM) to 0.05-0.1%.

118

 

l
t L-
""1
w

 

l
2000 *- I -1

 

 

 

 

1500 — 1
1 ‘1 FWHM = 9.074 channels
.9 1 I‘-
c: i i
5 1000 — “ 7 "
O 1'! "1
U ,1 1
,I I.
/ \
500 —- #5

 

 

ﬂ: \3:
1860 1870 1880 1890 1900 1910 1920 1930
Horrzontal posmon [channels]

 

 

Figure 5.5: Primary beam spot at the focal plane, used to check the dispersion matching of
the spectrograph for the experiment. The line through the data points is a single Gaussian
ﬁt; the error bars are statistical. The beam was transmitted to the focal plane from the
object position with the spectrograph in dispersion matched mode. The FWHM of the
peak corresponds to an energy resolution of E/AE~4000.

The detector positions were calibrated by inserting masks just upstream of the
tracking detectors and illuminating the holes with primary beam, as described in Chapter
4. The x-position (dispersive direction) is “hard-wired” in the detectors, so only the center
of the detector has to be determined. The y-position is determined by drift time of the
electrons from the ionization to the anode wire, and depends on several factors including
gas mixture and drift voltage. Gradual changes in the y-position through time was
observed in this experiment as in the loLi experiment (see Chapter 4). The magnitude of
the effect (~10%) was much smaller than in the 10Li experiment, and a much better moni-

toring method was available. The 7Li2+ charge state of the beam provided a very nice

monitor of the drifting y position as well as magnet fluctuations.

119

 

Particle identiﬁcation in this experiment was straightforward. Not only were the
particle groups clearly separated with no overlap, but a 7Li2+ charge state of the beam
comes strongly and provided a constant reference. Figure 5 .6 shows a particle identiﬁca-
tion spectrum for this experiment. Singly charged alpha particles from the reaction arrive
at the focal plane at the same time. The 8He2+ and 4He” lose similar amounts of energy
in the ion chamber and CRDCs, but the 8He2+ ions have twice the total energy. Hence, the

total E-TOF spectrum was used to identify the 8He2+ ions.

 

 

250—
=3
53 200-
>»
no
I-
a) . _> .
a 150- '.':.I . '
o " " V
H w

l 00 - .

 

 

 

l l l 1 l l l
50 100 150 200 250 300 350
Time of ﬂight [a.u.]

Figure 5.6: Particle identiﬁcation spectrum for this experiment.
5.3 Results

Energy calibrations were performed by inserting the 98c and 10B targets and
observing the (7Li,8He) reaction without changing the spectrograph settings. Calibrations
using the 9Be(7Li,8He)8B reaction were performed regularly throughout the experiment;
the 10B(7Li,8He)9C measurements were done near the end of the experiment.

The ground state Q-values of the 10B(7Li,8He)9C and 9Be(7Li,8He)gB reactions

120

 

 

are -33.5534(74) and -28.2636(7l) MeV, respectively [Au95]. The errors on these values
are dominated by the 7 keV 8He mass error. Three peaks were populated in the 8B
nucleus: the ground state and excited states at 0.774(6) and 232(3) MeV. The ﬁrst excited
state in 8B (0.774 MeV) was used in the calibration, but the 8B (2.32 MeV) state was not
used. Only the ground state of 9C was populated. A magnet excursion roughly halfway
through the experiment was accounted for by observing the peak shift in the
9Be(7Li,8He)8B reaction, and applying the same shift to all spectra following the excur-
sion.

The energy and scattering angle of each event were reconstructed using the tech-
niques described in Chapter 4 and Appendix B lists the reconstruction coeﬁicients used.
Energy spectra were created by applying kinematic shifts to all events to shift the energy
to what it would be at 0°. In other words, all events having the same Q-value appear in the
same peak after the shifts are applied. Figure 5.7 shows kinematic spectra for the two cal-
ibration reactions before shifting the energies.

The expected line shape from a kinematic simulation including ﬁnite beam emit-
tance is a Lorentzian. All data points in the ﬁts were weighted according to Poisson statis-
tics. The two lowest states in 8B were ﬁt with a double Lorentzian with identical widths
plus constant and linear backgrounds. Holding the widths of the two states equal is appro-
priate because the intrinsic state width does not contribute to the width of the peak; the
width of the peak is dominated by kinematic broadening effects described in Chapter 4.
The ground state of 9C was ﬁt with a pure Lorentzian plus a constant background. The
two states of 23A1 were ﬁt with a double Lorentzian with identical widths plus a constant

background. Allowing the width of the second state to vary gives an identical result with

121

 

no signiﬁcant change in )8. The ground state peak of 27F was ﬁt by a single Gaussian plus
a constant background. The next two higher-lying peaks were ﬁt with a double Gaussian
with equal widths plus a constant background. The widths of all of these 27P peaks were
equal within errors. The large, broad structure around channel 140 was ﬁt with a skewed
Gaussian (an additional x3 term in its argument) and a constant background. Fitting all
peaks with Gaussian or Lorentzian line shapes had a negligible effect on the results. The
ﬁts to the 23Al and 27F energy spectra are shown in Figures 5.10 and 5.11 respectively.

The 2S’Si and 3OSi contaminants in the silicon target don’t appear to cause signiﬁ-
cant spectrum contamination, illustrated in the lower left panel in Figure 5.7 and the bot—
tom panel in Figure 5.8.

Figure 5.9 is a plot of the energy calibration for this experiment.

122

 

 

 

 

12 I-IIIIIIIIIIIIII 12IIIIIIIIIIIIIII

.;

1.;
11‘

>33 '5' ‘
, ii," .ﬂ

Scattering Angle [deg]

  

Scattering Angle [deg]

rilirilrilliLJirrrlrr

llllllllll'lllljlllll

 

  

 

 

 

 

0 I..| i . .1.. .

—60xlo'3 —2o 0 20

 

 

0 3
-60x10” -20 0

Fractional Energy Fractional Energy

 

Ix)
O

 

 

 

 

N

III'III'IIIIII

12 Illllll'llllll

I’ll-.IIIII

   

   

Scattering Angle [deg]

lllI-lllllllllllllllll

 

llllllllllll‘llllllll
Scattering Angle [deg]

 

 

 

 

OiiirlrriLirrlrrr

 

0
-60x103 -2o 0 20 —60xlo'3 -20 0 20

Fractional Energy Fractional Energy

 

 

 

 

 

Figure 5.7: All kinematic spectra for this experiment. The fractional energy is relative to
the central energy and deﬁned as (E-E0)/E0. The resolution is better at lower angles
because of the reduced kinematic broadening effects at lower scattering angles (see Chap-
ter 4).

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30 5 __
25 E- 10B(7Li,8He)9C| j
20 E- _:
15 '. _:
10 1 é
5 E- _:
200 3 % T : 1 J J I . a. a n . -‘
5 9Be(7Li,8He)8BI E
150 '- ~ 1
100 ' '5
50 :— _:
3 1; :
g 30 h. 4. = I : + ‘r : § 1 t : : 4 : t : .L } ‘
25 — 24Mg(7Li,8He)23A1J _
20 — , _
—1
A
288i(7Li,8He)Z7P| E
uni
30 __
20 1 _:
10 i . _:
o . . ‘ ‘ . . ' MW

150 200 250 300

Channel

Figure 5.8: 8He energy spectra for the listed reactions at E(7Li)=50.l MeV/u and
Glab=O-12°.

124

 

 

 

 

 

 

      

 

 

 

,__. I I I I I T I I I I I I I I I I F] I I I I I I l I I I I I I I I I I I I I I
31>) 20 _ I _
£5. _ I 3
7:; T
.6 O- I 1 _
g -20 — 1,
L 8
B 0.774
323 — ( ) _
—. 322 [_ Slope = 76.22 i 0.23 keV/channel] _
>
0 321 P _
Z
320 - _
E6 9 8B(g.s.)
“c’. 319 - C( _
LL]
mg 318 — _
317 _ _
316 1 1 l m 1 1 1 l 1 1 1 1 l 1 4 1 1 1 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 14 1 1 l 1 1
240 260 280 300
Channel

Figure 5.9: 8He energy calibration for this experiment. The line through the data is a lin-
ear ﬁt. Errors on the points are statistical uncertainties (translated into energy) and mass
errors of the calibration nuclei summed in quadrature. The mass errOrs are taken from the
1995 Mass Evaluation Update [Au95].

125

 

 

 

 

 

 

 

 

 

 

25 .1. I I I I I I I I I I I I I I I I I I I I I I I I I I I I l I I I l l "I'
' A ’7 ‘
- Wr- h! (A. u
n M on u
. x :1 -
A
20 - A T (f: In "ft -1
' ”7 1| "c1 V? ‘V '
' "3 g; 53: '
- < If" .4 "ft .
15 _ R __ o I»: :7 N _
V) - "’ . < .1
"" N M

g - 0 :1 N -
o h J— < «-
U 1 ‘5 a .
10 ﬁ —
.J— -
5 i E i 1
- 11 111 I I H

1- ’01 ﬂ? L0

0 + 1 1 1 1 I 1 1 1 1 I r 1 1 1 § 9 9' L -1 -1 1 1 1 1 I 1 3;:

 

 

 

 

 

 

 

 

130 140 150 160 170 180 190 200
Channel

Figure 5.10: The energy spectrum of the 24Mg(7Li,8He)23Al reaction.

 

ITIIrrTTIIIIIII TTIIIIIIIIIIIIIIIIITTrleWTIIIIIIIII

 

     

 

E f\
40 :' m.
: Cl
: 1 .E‘
I I N
: 1| A
30 :— v’? 8“ f:
33 : {3 —: "a
5 : "I: .4 C 39
o : o v CL. 1:.
U ' D— t\ l‘
20- CiSG N N
.. Q“ i
[\
" N
10 I—
E . .22).“: b‘JU- 5:I' .- ..' . SE
0 f _' iii'iiil‘ 521.1135». AEL'CIC. :-
120 140 160 200

 

Channel

Figure 5.11: The energy spectrum of the 28Si(7Li,8He)27P reaction.

126

Table 5.1: Experimental observations of states in 23Al.

 

 

 

 

 

 

 

 

Mass Excess Excited State Loc. Prev. Exp. M.E.
[MeV] [MeV] [MeV]

6.773(28) 0.00 6.767(25)[Be75]
7.322(30) 0.550(20) 7.23(6) [Wi88]
8.545(44) 1.773(35) --
9.348(42) 2.575(34) __
9.95(33) 3.204(21) --
10.47(36) 3.699(24) --

 

 

 

Table 5.2: Experimental observations of states in 27R

 

 

 

 

 

 

Mass Excess Excited State Loc. Prev. Exp. M.E.
[MeVI [MeV] [MeVl lBe77]
-0.670(4l) 0.00 -0.753(35)
0.529(42) 1.199( 19) —-
0.945(44) 1.615(21) 0.91(4)
2.783(46) 3.453(22) --

 

 

 

127

 

 

 

 

‘31)? :i I. ‘

 

 

 

 

 

 

 

 

— 3.699
3.453
_ 3.204
_ 2.575
_ 1.773 1 66
12615
— 1.199
0.859
26 .
0.550 3”"
—— 0.46
0.123
22 0 0
M + 27
g P 23 A] p

 

 

 

 

 

Figure 5.12: New level schemes for 23Al and 27P following this experiment. The thick
lines are measurements reported here; the thin lines are previous measurements. The
excited state locations are given relative to the ground state of the nucleus from their
respective measurements. The proton separation energies are a weighted average of the
present ground state mass measurements and those of Benenson et al. in the seventies
[8e75, Be77].

128

 

E4- ._

The errors on the energies are a quadrature sum of the calibration mass errors. As
described in Chapter 4, RELMASS was used to determine the masses of the states in the
nuclei 23Al and 27R The beam energy, target thicknesses, scattering angle (0°), and
channel locations were input as calibrations. The mass eXcess of all peaks in 23Al and 27P
are given as output and are summarized in Tables 5.1 and 5.2.

Absolute mass errors were a combination of statistics, calibration mass errors,
beam energy, target thickness, and scattering angle, which dominated (21 keV for 23A1
and 36 keV for 27P). The errors in excited state location quoted in Tables 5.1 and 5.2 are
purely statistical. The astrophysical signiﬁcance of these measurements requires that res-
onant energies (energies above the proton threshold) be quoted. The Tables give a mass

excess for each state, excited state location within the nucleus, and location with respect to

the proton separation energy of the nucleus.
5.4 Discussion

5.4.1 The Isobaric Multiplet Mass Equation (IMME)

The IMME is an empirical equation which relates the masses of isobaric analog
states to each other based on their z-axis projection of isospin [Be79]. The equation is a
good predictor of masses among isobaric multiplets and is as accurate as the input data
allow. Its form is a quadratic polynomial relating the mass excess of multiplet states as
ME = a + bTZ + cTzz. The IMME coefﬁcients for many isobaric multiplets have been
determined [Be79].

The 23Al and 27P ground states belong to isobaric quartets of T=3/2. Figure 5.13
shows the A=23, J“=5/2*, T=3/2 isobaric quartet mass excesses plotted as a ﬁinction of T2.

The ground state 23Al is the TZ=-3/2 member of the quartet. The A=23 T=3/2 IMME coef-

129

 

ﬁcients calculated using the current measurement along with the other three measurements

 

[A289] agree with those published in the 1979 Benenson and Kashy review article. Com-
bining the previous value of the 23Al mass excess with this one gives a mass excess of

6.770(19) keV.

 

N
O

_TIII[IIVTIIIIIIIIII IIIIIIIITIIIIIIIIIIq -g
I
1 I

 

Residuals [keV]
0

$115
0O
[‘1'

1111

 

 

a = 0.2864 i 0.0063

 

 

 

 

 

: 1

._. 63' b=-3.9731i0.0088 ‘;

a, ; c=0.2317d:0.0077 .-
4

2 2 a

a 2.— '1

Q) )- ...

a .. —I

w “E ‘2

8 -2; L

2 2 2

-4:' ‘:

:11111111111111111111141111111111111111:

-2 -1 0 l 2

T

Z

Figure 5.13: Mass excesses for the A=23, J“=5/2+, T=3/2 isobaric quartet. The line
through the data is a quadratic ﬁt; the box lists the coeﬂ‘icients from the ﬁt with errors in
MeV.

Figure 5.14 shows the A=27, J”=1/2+, T=3/2 isobaric quartet mass excesses plot-
ted as a function of T2. The ground state of 27P is the Tz=-3/2 member. There is a small
disagreement with the present mass excess of -0.670(4l) MeV and the previous measure-
ment of -0.753(3 5) MeV, but both measurements agree, within errors, with the IMME pre-

diction of -7l6(16) MeV in the review article. A weighted average of the two

measurements yields a mass excess for the ground state of 27P of -715(27) keV. This

130

value is used for determining the IMME coefﬁcients in Figure 5.15 and Table 5.3.

 

 

 

 

 

 

 

 

 

 

 

 

 

F“ -IIITIIIIIIIIIIIIIFf IlllllIlllllIllllIl—I
E 40_— . 1.
T: 20: t
"g 0_ “:5 .. = .
“E 40.— 1
m '40;— . j
E a = -8.1199 1 0.0023 3

-21.- b=-4.6224:t0.0046 :

E 4:— c=0.2082i0.0029 _:
I. .

:2; -6:— ‘3
8 : 3
:2 -8r ':
Lu : I
a -10r 1
«3 ._ q
2 -12:— 1
»- -l

-14E— —1‘

" llllllllillllllLJl IIIJIILLlillllllLtl-i

-2 -1 0 1 2

T

Z

Figure 5.14: Mass excesses of the A=27.l“=1/2+ T=3/2 isobaric quartet. The line through
the data is a quadratic ﬁt; the box shows the ﬁt coefficients and their errors in MeV. The
two solid squares are this measurement (lower) and Benenson’s measurement.

131

Table 5.3: IMME coeﬂicients for the A=23 and A=27, T=3/2 isobaric multiplets.

 

 

 

 

 

 

 

 

 

 

A=23, J“=5/2+, T=3/2 A=27, J“: “2*, T=3/2
Coef. Present Past [Be79] Present Past [Be79]
a 0.286(6) 0.287(4) -8. 120(2) -8.120(2)
b -3.97 3(9) -3.97 3(8) -4.622(5) 4.629(5)
0 0.232(8) 0.230(4) -0.208(3) 0.207(3)

 

 

In the review article, the possibility of a cubic term or “d” coefﬁcient is discussed.
The cause of such a term would be the signiﬁcant expansion of the wave function in the
lowest isospin member of the multiplet due to Coulomb repulsion. If the wave function
was 51/2 in nature, the lack of a centrifugal barrier may cause additional expansion. The
27F nucleus is a good test of this because it is proton rich and its ground state is 31/2- When
the multiplet is ﬁt with a cubic, with d=l(5) keV, consistent with zero. The A=23 case was
also ﬁt with a cubic, yielding d=8(13) keV, again consistent with zero.
5.4.2 Astrophysical implications

The measurement of the excited states have a direct effect on the proton capture
rates described earlier in this chapter. Speciﬁcally the excited state at 0.550(19) MeV in
23Al and the newly reported 1.199(19) MeV in 27F. In these speciﬁc cases (23Al and 27P),
the proton and gamma widths have been previously determined for the levels that domi-
nate the resonant capture rates [Wi88, He95]. Equations 5.12 and 5.13 describe how the
gamma widths scale with energy. The proton width is insigniﬁcant for the 23Al case
because I‘Y<<1"p. In the case of 27F, the proton and gamma widths are of the same order,
so both must be scaled slightly. Alex Brown has made several calculations illustrating

how the 26Si(p,y) proton width scales with resonance energy. The dependence is linear

132

 

 

over the small range of use. The widths and resonance strengths are listed in Tables 5.4
and 5.5.

Figures 5.15 and 5.16 show the difference in reaction rates based on the old and
new experimental information. At temperatures between 0.1 and 3 GK the 26Si(p,y)27P
reaction rate is greatly enhanced. This leads to a signiﬁcant depletion of 26Si and 26A1 and
a substantial enhancement of 27F and, at higher temperatures, 28S. See [He95] for details.
The 22Mg(p,y)23Al reaction rates are very similar to those published in [Wi88] but slightly
lower with improved error bars. This will cause a slight enhancement of the 22Na in the

temperature range 0.1-1.0 GK compared to previous predictions.

Table 5.4: Resonance parameters the levels that dominate resonant proton capture in
the 22Mg(p,y)23Al reaction.

13* Ep rY [eV 1 1“,, [eV 1 yr my [eV]
0.550(20) 0.430 7.63E-07 32 53+ 7.63E-07

 

 

 

 

 

 

 

 

 

 

Table 5.5: Resonance parameters the levels that dominate resonant proton capture in
the 26Si(p,y)27P reaction.

12* Ep rY [eV 1 1“,, leV] ﬁt my [eV]
1.199(19) 0.385 1.43503 23313—03 30+ 17713-03

 

 

 

 

 

 

 

 

 

 

133

 

o!
U.

I
O

3 -1 -1
N A <0v> [cm mole s ]
3

Figure 5.15:

llllll‘lALAAl l lllllllLJ—LAL IJLLJLJJJH

 

 

  

 

 

 

 

 

 

 

 

 

' -22 23 '
: Mg(p,y) Al ”’/ :
"’
- ' I _
I
- x _
I

.1 h
‘ II] resonant capture (previous result) '
resonant capture (this experiment) '

i , - - - direct capture _
u I, I-

I
- I -
I -
I
I I‘lIIII]""1 I I llllll 'I T I IIIIII '
234567 234567 234567 2
0.1 l 10
T9
Astrophysical reaction rates using data from past and present experiments.

134

llllllllllll

llllllllnnnn

 

3 -1 -1
NA <ov> [cm mole s ]

-20

 

 

 

 

 

10

26 - 27
81(1),?) P
3
3.23
:5?
2:3
33 ,
."7 - I
3"
3.43:"
.33
2‘3;
33
’ 3.:
I 3.2?
I f
I31:
I if}:
12‘”
l I I l
4 5 6 7 2

 

 

 

 

 

llllllll....l
’l/ d—
4 I I ’ -
ram-ﬂ I’ r \-
., 2 2 I’ ’,,.-.1.=.._\__
'3? I "
.53; / I-
55:" r
1:"; I .-
I
I _
l
I -
I
I I—
h—
. l-
[j resonant capture (past experiment) _
resonant capture (this experiment) _.
- - - direct capture -
3 4 5 6 7 2 3 4 .5 6 7 2
1 10
T9

Figure 5.16: Astrophysical reaction rates due to past and present experiments. Note that
the calculations of Herndl et al. [He95] agree with the present experiment.

5.5 Conclusions

High precision mass measurements (6m/msl.6x10'6) were made of several

nuclear states including the ground states of two proton rich nuclei. The spectroscopic

measurements in the two nuclei represent the ﬁrst experimental report of some of the

excited states in the nuclei. Astrophysical proton capture reaction rates for the rp process

were calculated and found to deviate slightly from previous calculations but are now on a

much ﬁrmer basis due to the smaller uncertainties. These will impact the mass flow dur-

135

ing explosive hydrogen burning events in speciﬁc temperature and density conditions, and
it will be left up to the astrophysicist to run network calculations using the new, improved

nuclear data.

136

 

 

 

 

 

Chapter 6

Summary

The S800 spectrograph was built at the NSCL and commissioned in September
1996. The magnetic ﬁelds of the spectrograph dipoles were measured to a precision of 1
Gauss after smoothing, and the data were used for transfer map calculations necessary to
provide the high energy resolution of the S800. Features of the ﬁelds such as azimuthal
curvatures, radial ﬁeld gradients, and effective lengths were measured. Measurements of
the central ﬁeld region indicate that the maximum ﬁeld gradient is 51 Gauss/cm, which
meets the design criteria.

The S800 spectrograph was used for spectroscopic measurements of the exotic
nuclei IOLi, 23’Al, and 27R The 9Be(9Be,8B)10Li reaction was used to populate states in
the particle unbound nucleus 10Li. Deﬁnitive evidence for a p-wave state at -Sn=500(60)
keV, corresponding to a 9Be(9Be,8B)IOLi reaction Q-value of -33.59(6) MeV, was pre-
sented and agrees with previous spectroscopic measurements of 10Li. A small excess of
counts right at threshold is tantalizing but, given the experimental resolution, provides lit-
tle information about low-lying s-wave or p-wave states. The experiment represents one

more measurement with which to draw conclusions about the 10Li nucleus. Future mea-

137

 

"s

 

 

surements are necessary to clarify the 10Li situation. Possible future studies include re-
measurement of the 10Be(12C,12N)mLi reaction, either with an improved 10Be target or
with a 10Be beam, as a conﬁrmation of the existence of the state at ~250 keV is desired.

The rp-process nuclei 23A1 and 27F were studied spectroscopically through the
reactions 24Mg(7Li,8He)23Al and 28Si(7Li,8He)27P. The ground state mass excesses and
low lying excited states were determined by measuring the Q-value of the
24Mg(7Li,8He)23Al and 288i(7Li,8He)27P reactions which were Q=-37.396(28) and
Q=-37.347(4l), respectively. This measurement of the ﬁrst excited state of 23A1 at
550(20) keV substantially reduces the error of the 22Mg(p,y)23Al reaction in the rp process
network and will reduce the uncertainty on the amount of the gamma ray emitter 22Na pro-
duced in explosive hydrogen burning situations where the rp process is triggered.

The ﬁrst experimental report of the ﬁrst excited state in 27F at 1.199(19) keV sig-
niﬁcantly increases the 26Si(p,y)27P reaction rate in certain temperature and density condi-
tions during explosive hydrogen burning. The effect can be as large as 3 orders of
magnitude at T=0.4 GK and will have a signiﬁcant effect on mass and reaction ﬂow. This
new rate also affects the amount of the gamma ray emitter 26Al produced in explosive stel-
lar events. The uncertainties on the excited states of 23Al and 27F are still large when con-
sidering their effect on reaction rates and even higher resolution measurements are still
desired though not critical.

A recent review article points out that the dominant uncertainty in galactic 26Al
production is the rate of the 25Al(p,y)26Si reaction in these explosive hydrogen burning
scenarios in novae, which will bypass the production of the 26Al ground state via

25Al([3+v)25Mg(p,y)26Al [Ka98]. The 25Al(p,y)268i reaction proceeds through several

138

 

states just above threshold states in 26Si. The special characteristics of the S800 are ide-
ally suited for spectroscopic study of 26Si with transfer reactions. For example, the
27Al(7I-i,8He)26Si or the 28Si(p,t)‘7'6Si reactions could be used to populate states near the
proton threshold. The high resolution and larger cross section possible with the (p,t) reac-
tion makes this a particularly attractive option.

High resolution studies of exotic, unstable nuclei and those with astrophysical sig-
niﬁcance are now possible and will continue to be performed with the S800. Until direct
measurements of (p,y) reactions are possible, the 8800 will continue to provide invaluable,
indirect astrophysical information. Higher precision mass measurements using very low
cross section transfer reactions are now possible and will continue. Experiments using the
high intensity radioactive beams to be delivered by the coupled cyclotron project coupled

with the capabilities of the S800 will undoubtedly be fruitful.

139

 

 

 

Appendix A

Mapper Post-Processing Software

A.1 Introduction

The program D1_D2 was written to read and process raw map ﬁles from the map-
ping campaign described in Chapter 3. The software was written in VAX Fortran for the
Digital Equipment Corporation’s Alpha servers at the NSCL. This appendix will brieﬂy
describe how to use the software, and input and output ﬁle formats.

A.2 How to use the software

To run the program, type:

RUN DISK$S800:[CAGGIANO.MAPPER.D1D2]D1_D2.EXE

Depending on the options selected, it may take up to 2 hours of CPU time to pro-
cess a map, so the program must be run in batch mode. The program also uses some large
arrays, so the user will need access to 80 MB of working space.

All that is needed to run the program is the name of an input ﬁle which contains all
the important information necessary to process the map. Map ﬁle names, calibration ﬁle
names, current settings, how to process the map, etc. is all passed to the program through

this input ﬁle. The input ﬁle is a simple text ﬁle containing keywords followed by appro-

140

 

 

 

priate parameters. The parameters vary depending on the keyword detected. The user
speciﬁes the name of the input ﬁle and the program reads commands from the ﬁle. The
commands don't have to occur in any particular order. If a command is not encountered,
default values are used.

The following are allowed commands and parameters:
ADC <switch> <resolution>

This command controls the resolution used for the raw voltage from the ADC's. If
switch=1, then the resolution is set to the number of bits speciﬁed by <resolution>. If
switch=0, then the full resolution is used (20 bits).
SMOOTH <switch> <Pl> <P2> <P3> <P4> <P5>

The switch instructs the program to smooth the data. The data will be smoothed
<switch> if set to an integer greater than zero. If <switch> > 0, the number selects the
smoothing method used. P1, P2, and P3 have different meanings depending on the
method of smoothing selected. Table A.1 lists all the choices and their meanings in con-
text.

In all cases except the gaussian weighted average, P3 is an integer which tells the
user how to treat the ends of the data array if the weighted sum option is used:

1 = REPEAT (repeat the last point)

2 = BOUNCE (use the last points in reverse order)

3 = WRAP (as if the data was one period of a sinusoidal

wave)
4 ZERO (additional points are = 0.0)

5 = NOTHING (no smoothing on the ends)
6 = FIT (extrapolate the data by linear fit)

In the gaussian smoothing case, the width of the gaussian is a ﬁlnction of the local

derivatives, given by:

141

 

 

 

 

*l—‘l / 10

0(X) = (0m - amine (A-l)
where 0min, omax, and x0 are P1, P2, and P3 speciﬁed by the user, and x is the value of the
local derivatives. Typical values are omin=l, omax=10, x0=2. The maximum gradients
encountered in the ﬁeld maps are ~7 T/m which is a useful number when deciding how to

set x0. P4 is the number of times to apply the smoothing algorithm. P5 is a switch for

smoothing in the polar angle coordinate; 0=off, l=on.

142

 

 

 

Table A. l: Smoothing parameters and their meanings

 

 

 

 

 

 

 

 

 

 

 

 

 

Smoothing
Switch method P 1 P2 P3
50 None -- -- --
1 Savitzky- Number of polynomial end effect
Golay points order
weighted
average
2 Binomial Number of -- end effect
weighted points
average
3 average Number of -- end effect
points
4 not used -- -- --
5 Gaussian minimum maximum decay con-
weighted sigma sigma stant
average
6 Polynomial polynomial -- --
ﬁt over order
entire radial
range

 

BADPASS <switch> <nbad> <nbs_max> <nsig>

 

 

If switch = I then bad passes are smoothed by local radial ﬁts. <nbad> controls the
azimuthal (phi) sample number where the radial passes are checked for bad passes. The
program ﬁts the radial data at the <nbad>th polar sample. The program chooses the
<nbs_max> worst samples that have a deviation greater than <nsig>*(the standard devia-
tion from the ﬁt) and replaces them with the ﬁtted value.

INPUT <map ﬁle name> <integer current>

The map ﬁle name is the name of the ﬁle which contains the binary raw data from

a S800 dipole map. This ﬁle is generated by reading the raw integer data from a text ﬁle

143

and writing it back out in “unfonnatted” mode from a FORTRAN program. This is done
to conserve disk space. The integer current is the rough current that the map was taken at.
Allowed values are 50,100,150,...,400 for D1 and the same for D2 with 420 added. The
valid numbers are given in Tables 3.1 and 3.2 and are‘always rounded to the nearest 25
Amps. So, 111 Amps is labeled “100”, and 366 Amps is labeled “375”. The label is used
to lookup the appropriate calibration values from the calibration ﬁles and to construct the
output ﬁle names.
INTERP <switch> <spacing in mm> <# of points to use> <interp. method>

If switch = 1 then interpolation from polar to cartesian coordinates is performed.
The spacing of the xy grid has to be speciﬁed in mm. The number of points to use is the
number of points to use in one dimension. For example, if NPTS = 4, then 16 surrounding
points will be used to determine the value of the ﬁeld at the speciﬁed point. The interpola-
tion method is an integer which selects the type of interpolation to be done. The following
is a list of allowed types:

11 one-dimensional cubic spline interpolation (Akima), from IMSL

12 one-dimensional b-spline interpolation (Akima), from IMSL

l3 one-dimensional Newton's divided difference method, from CERN library

14 one-dimensional Lagrangian interpolation (3rd order), from Numerical Recipes

15 one-dimensional Singular Value Decomposition ﬁt, from Num. Rec.

21 two-dimensional area weighted average of surrounding 4 points

22 two-dimensional average of surrounding 4 points

23 two-dimensional distance weighted average of surrounding 4 points

FILES <nmr probe data ﬁle name> <hall probe data ﬁle name>

144

 

 

 

The two ﬁle names where the calibration data is kept.
F F T <switch> <P1> <P2>

If switch = 1 then a FFT is performed on each single pass of the data, with cuts at
N=Pl and N=P2 in the fourier amplitudes. Default is no FFT.
ALIGN <switch>

lf switch = 1 then the odd and even passes at the same radial coordinate are aligned
with each other by interpolating the even pass some distance and minimizing the sum-
squared difference between each pass.
CALIBRATE <switch>

If switch = 1 then the map is calibrated according to the two calibration ﬁles speci-
ﬁed in the FILES command, using the integer current speciﬁed in the INPUT command to
determine which data from the calibration ﬁles to use.
ROTATE <switch> <X0> <Y0> <angle>

If switch = l, the the map is rotated about the point (X0,Y0)(in meters) by the
angle (in degrees).
TRANSLATE <switch> <dx> <dy>

If switch = 1 then the map is translated by the amount (dx,dy) (in meters).
PASSES <switch>

If switch = 0, then the average of even and odd passes is used.

If switch = 1, then only odd passes are used (default).

If switch = 2, then only even passes are used.
POSCAL <switch>

This option exists for maps of the small, permanent magnets used to calibrate the

145

 

 

 

absolute position of the mapper with respect to the magnet steel. If the switch=1, then the
program reads in the residual ﬁeld map and subtracts the appropriate portion of the map
from the permanent magnet map. This removes the contribution to the ﬁeld from the
residual magnetization and gives a more accurate position determination. Calibration of
the map is not performed either. If switch=0, the map is assumed to be an ordinary ﬁeld
map.

A.3 Sample input ﬁle

The following is a sample input ﬁle:

adc 1 18

smooth 1 25. 2. 6. 10. 1.
BADPASS 1 1100 S 2.0

INPUT [.DlRAW]D14OOTCO_UNF.MAP 400
INTERP 1 5.0 50 15

FILES D1_NMR.DAT [.D1HALL]D1_HALL_PROBE.DAT
FFT 0 3 5

ALIGN 1

CALIBRATE 1

ROTATE 0 3.0412 0.0 -.115

TRANSL 1 -0.0059 0.0

PASSES 2

POSCAL 0

Notice that the program is not case sensitive; the keywords can have any combina-
tion of capital letters in them. The program only reads the ﬁrst 3 letters of the keyword ;
the rest of the letters are just for the reader’s ediﬁcation. Spacing and format don’t matter
as long as at least one space or tab exists between all parameters.

Proceeding from the ﬁrst line down, the ADC resolution was set to 18 bits. The
data was smoothed according to the Savitzky-Golay algorithm using 2nd order polyno-
mial weighting. The ends were extrapolated with a linear ﬁt, the smoothing algorithm will

be applied 10 times, and polar angle smoothing will be done once. Bad passes were ﬁxed

146

using the 1100th angular sample to check for deviations. Deviations greater than 20 will
be replaced by the ﬁtted value. The ﬁve worst passes were ﬁxed. The data were taken
from D1 and the raw map data is contained in a ﬁle called
[ . D1RAW]D1400TCO_UNF .MAP and the current setting for the map was ~400 Amps
for each of the dipoles. The data was interpolated onto a cartesian grid from the polar grid
using a 50x50 grid with spacing of 5 mm. The 15 indicates 1D interpolations using a
polynomial ﬁt. The calibration data was stored in the ﬁles D1_NMR.DAT
[ .DlHALL]D1_I-IALL_PROBE. DAT (NMR and ball probe respectively) and the data
will be calibrated. No FFT ﬁltering was performed. The odd and even passes were
aligned with each other. The map was not rotated, but translated -5.9 mm in the x direc-
tion. The output ﬁles contained only even pass data and the map was an ordinary ﬁeld

map, not a position calibration map.
A.4 Output ﬁles

There are two sets of output ﬁles. One set is called <map ﬁle preﬁx>_rphi_c*.out.
The rphi tells the user that the data contained in that ﬁle is in polar coordinates. The 0"
determines which coil the map corresponds to - C1 is the coil #1, etc. The ﬁle is in for-
tran unforrnatted (binary) format. Each record of the ﬁle contains all of the data for
PHI=constant, R=(rmin,rmax). Thus, the number of records in the ﬁle is equal to the
number of phi samples in the map. There is a header on the ﬁle which is two integers: NR
and NP, the number of samples in the r and phi directions, respectively. NP will be the
number of lines in the ﬁle following the header line, while the NR will specify how many
numbers are in each line (record).

The other set of output ﬁles are called <map ﬁle preﬁx>_xy_c*.out. The ﬁles con-

147

 

 

 

tain the map data on a cartesian grid, whose pitch is speciﬁed by the user in the command
INTERP. The header on this ﬁle contains an exact replica of the input ﬁle preceeded by
the date and time stamp and the input ﬁle name. The input ﬁle is followed by the 5 charac-
ter string <EOH>, followed by the following parameters:

XMIN,XMAX,YMIN,YMAX,SPACING

NX,NY
These parameters specify the extent of the cartesian grid that holds the data. The data is
on a square whose lower left corner is (XMIN,YMIN) and whose upper right corner is
(XMAX,YMAX). There are NX x NY points on the grid, separated by (SPACING). All
coordinates are in meters. Here is a sample of the header of an output ﬁle, which was used
to process a D1 ﬁeld map taken at B=1.56 Tesla:

FILE WAS CREATED ON 13-NOV-l998 02:40:29 . 84

PARAMETER FILE USED: D1_400 . INP

adc O 20

smooth 0 25. 2. 6. 1. 1.
BADPASS 1 1100 5 2.0

INPUT [.DlRAW]D1400TCQ_UNF.MAP 400
INTERP 1 5.0 50 15

FILES D1_NMR.DAT [.DlHALL]Dl_I-IALL_PROBE.DAT
FFT 0 .3 5

ALIGN 1

CALIBRATE 1

ROTATE 0 3.0412 0.0 -.115

TRANSL 1 -0.0059 0.0

PASSES 2

POSCAL o

<EOH>

The body of the data is zero suppressed to conserve disk space. The ﬁrst four num-
bers of the record are integers that contain the extent of the non-zero portion of the maps.

(Usually, only the ﬁrst two integers are non zero.) The rest of the record is written for a

148

 

EV

 

 

given Y value.

The sample output ﬁle header above describes a typical set of map processing
parameters used to provide Daniel Bazin with ﬁeld data with which he created inverse
transfer maps for 8800 experiments. For D1 maps, the even passes (the ones in which
gravity helped the motion) were used. For D2 maps, the odd passes (again, the ones in

which gravity helped the motion) were used.

149

 

Appendix B

Inverse transfer maps

This appendix lists all of the inverse map coefﬁcients used to reconstruct energy
and scattering angle in both experiments. Table B.l contains all coefficients up through
3rd order for the 10Li experiment (Chapter 4). Table 82 contains all coeﬂ‘icients through
3rd order for the 23Al and 27F experiment (Chapter 5).

In COSY language, the dispersive angle is called a and the vertical angle is called
b, i.e. a=0 and b=¢. The reconstructed coefficients are a, y, b, and d in the scattering
chamber as a function of x, a, y, b in the focal plane. Positions and angles are expressed in
meters and radians. The d variable is the reconstructed fractional energy, d=(E-E0)/E0.
All coeﬂicients of the form (al*) are grouped together, as are the (y|*) coefficients, etc.
The coefﬁcients for each reconstructed variable are grouped according to order and sorted
from lowest order (at the top) to highest order (at the bottom). For example, the third

order a coefﬁcient (alxbz) has the value -1.311 rad m'l rad‘2 in Table B. 1.

150

 

 

Table 8.1: Inverse map coeﬂ‘icients for the 10Li experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coef. # Value Order x a y b
a coefficients
1 1.859E-02 l l 0 0 0
2 -l .015E+00 l 0 l 0 0
3 -l .323E—02 2 2 0 0 0
4 1 .991E-Ol 2 l l 0 0
5 6.234E-02 2 0 2 O 0
6 1.635E-02 2 0 0 2 O
7 -1.283E-01 2 0 0 1 l
8 1.059E-01 2 0 0 O 2
9 3.537E-03 3 3 O 0 0
10 -5.004E-02 3 2 1 O 0
ll 1.412E-01 3 1 2 0 0
12 -3.467E+00 3 0 3 0 0
l3 4.078E-02 3 l 0 2 0
14 -4.230E+00 3 0 l 2 0
15 3.864E-01 3 l 0 l 1
l6 3.508E+00 3 0 l l 1
17 -l .331E+00 3 l O 0 2
18 -1.840E+00 3 0 l 0 2
y coefﬁcients

1 -5.527E-01 1 0 0 l 0
2 9.537E-01 l O 0 O l
3 -3.267E-02 2 l 0 l 0
4 3.705E-01 2 0 l l 0
5 -1.501E-01 2 l 0 0 1
6 -4.029E+00 2 0 1 0 l
7 -3.218E-02 3 2 0 l 0
8 -l .297E-01 3 l l l 0
9 7.813E+00 3 0 2 l 0
10 ~2.645E-01 3 0 0 3 0
l 1 2.089E—Ol 3 2 O 0 1
12 3.403E-01 3 l l 0 l
13 -4.810E+00 3 0 2 0 l
14 3.178E+01 3 O 0 2 l
15 -2. 199E+02 3 0 0 l 2

 

151

 

 

 

 

Table B. l: Inverse map coefficients for the 10Li experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coef. # Value Order x y b
16 5.034E+02 3 O 0 3
b coefficients
1 -7.183E-01 l p 0 0 1 0
2 -4.317E-01 1 0 O O l
3 9.235E-02 2 l 0 l 0
4 2.173E+00 2 0 1 l 0
5 —3.506E-01 2 l 0 0 l
6 -9.858E+00 2 O 1 0 l
7 -9.809E-02 3 2 O 1 0
8 1.533E-01 3 l 1 l 0
9 -5.302E+00 3 0 2 1 0
10 -5.845E+00 3 0 0 3 0
l l 5.418E-01 3 2 0 0 l
12 1.939E+00 3 l 1 0 1
l3 5.329E+00 3 0 2 O l
14 7.659E+01 3 0 O 2 l
15 -4.741E+02 3 0 0 l 2
l6 1.057E+03 3 0 0 0 3
d coeﬂ‘icients

l 2.057E-Ol l l 0 0 0
2 1.201E-02 1 0 l 0 0
3 1.632E-02 2 2 0 0 0
4 1.575E-Ol 2 l l 0 0
5 5.7941-3-01 2 0 2 0 0
6 9.261E-02 2 0 O 2 0
7 -4.582E-Ol 2 0 0 1 l
8 1.143E+00 2 0 0 0 2
9 1.27lE-03 3 3 0 0 0
10 -l .783E-02 3 2 1 0 O
11 1.216E-01 3 l 2 0 0
12 -1.922E+00 3 0 3 0 0
l3 3.922E-02 3 l 0 2 O
14 -l .859E+00 3 0 l 2 0
15 —4.23 l E-02 3 l 0 l l
16 1.468E+00 3 0 1 l l
17 -5.444E—02 3 l 0 0 2
18 3.97lE-02 3 0 l 0 2

 

152

 

Table B.2: Inverse map coefficients for the rp-process nuclei experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coef. # Value Order x a y b
a coefﬁcients
1 1.57lE-02 l l 0 0 0
2 -9.232E-01 l 0 l 0 0
3 -l.l9lE-02 2 2 0 0 0
4 1.854E-Ol 2 I l O 0
5 5.844E—02 2 0 2 0 0
6 1.457E-02 2 0 0 2 0
7 -l .l69E-01 2 0 0 l l
8 1.084E-Ol 2 0 0 0 2
9 3.257E-03 3 3 0 0 0
10 -4.881E-02 3 2 l 0 0
ll 1.833E-01 3 1 2 0 O
12 -4.525E+00 3 0 3 0 0
l3 7.037E-02 3 1 0 2 0
l4 -6.106E+00 3 0 l 2 0
15 3.017E-01 3 l 0 l 1
l6 6.683E+00 3 O l l l
17 -l.219E+OO 3 1 O 0 2
18 -3.008E+00 3 0 l O 2
y coefﬁcients
1 -2.315E-01 l 0 0 l 0
2 7.835E-01 l 0 0 0 l
3 -9.625E-02 2 l 0 l 0
4 -l .844E-01 2 0 l l 0
5 -l .720E-02 2 l 0 0 l
6 -8.269E-01 2 0 l 0 l
7 6.873E-03 3 2 O 1 0
8 -l.656E-Ol 3 l l l 0
9 9.953E+00 3 0 2 l 0
10 2.533E+00 3 O 0 3 0
ll 1.633E-02 3 2 O 0 l
12 -6.366E-01 3 1 l 0 1
l3 -6.798E+00 3 0 2 0 1
l4 4.384E+00 3 0 0 2 l
15 -5.764E+Ol 3 0 0 l 2
16 1.418E+02 3 0 0 0 3
b coefficients

 

 

 

153

Table B.2: Inverse map coeﬂicients for the rp-process nuclei experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coef. # Value Order x a y b
l -l .266E+00 l 0 0 l 0
2 -3.319E-02 1 0 0 O l
3 1.319E-01 2 1 0 l O
4 2.901E+00 2 0 l l 0
5 -5.266E-01 2 l O 0 1
6 -l.432E+01 2 0 l 0 l
7 -l .459E-Ol 3 2 0 l 0
8 2. 1 7913-01 3 l l l O
9 -7.316E+00 3 O 2 l 0
10 -8.94lE+00 3 0 0 3 0
l l 8.108E-01 3 2 0 O l
12 2.863E+00 3 l l 0 l
13 7.910E+00 3 0 2 0 l
14 1.126E+02 3 0 0 2 1
15 -6.941E+02 3 0 O l 2
l6 1.561E+03 3 0 O 0 3

d coefficients
1 2.054E-01 l 1 0 0 0
2 -3.053E-03 l 0 l O O
3 1.622E-02 2 2 0 0 0
4 1.661 E-Ol 2 1 l O 0
5 5.71 lE-Ol 2 0 2 0 0
6 9.321E-02 2 0 0 2 0
7 -4.66lE-01 2 0 0 l l
8 1.156E+00 2 0 0 0 2
9 1.412E-03 3 3 0 0 0
10 -2.253E-02 3 2 l 0 0
l l 1.644E-Ol 3 l 2 0 0
12 -2.642E+00 3 0 3 0 0
l3 5.480E-02 3 l 0 2 0
14 -2.949E+00 3 0 l 2 0
15 -5.982E-02 3 l 0 l 1
l6 2.996E+00 3 0 1 1 1
l7 -6.261E-02 3 l 0 O 2
18 -5.324E-01 3 0 l 0 2

 

 

 

 

 

 

 

154

 

[A188]
[A196]

[Am90]

[Ar87]
[Au93]
[Au95]
[Ba77]
[Be75]
[Be76]
[Be77]
[Be79]
[Be93]
[Be95]

[Be98]
[B156]
[B192]
[B093]
[B097]
[Ch92]

[Da85]
[De92]
[En79]
[Ha95]
[He95]
[Ka98]

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