A High Resolution Study of Proton Resonances in 51Mn

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BEAL, WILLIAM C., JR. A High Resolution Study of Proton Resonances in Mn. (Under the direction of Dr. Gary E. Mitchell and Dr. John F. Shriner, Jr.)

High-resolution measurements of the differential cross sections of the50Cr(p,p0) and 50Cr(p,p

1) reactions were performed over the energy rangeEp= 1.8045 – 3.5011 MeV at five

different scattering angles. This experiment was performed at the High Resolution Labo-ratory (HRL) at Triangle Universities Nuclear LaboLabo-ratory (TUNL); the system is capable of measuring proton-scattering data with an energy resolution of250 eV. The goal of this work is to improve the purity and completeness of the 51Mn level sequences in order to in-vestigate average level properties. The observed cross sections were fit using the multi-level, multi-channel, R-matrix code MULTI, and resonance parameters were extracted from the data. Many resonances were observed for the first time, and several others were reassigned different quantum numbers compared to previous work. A total of 185 resonances were observed, 64 of which had not been observed in previous studies.

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by

WILLIAM CHANDLER BEAL, JR.

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Department of Physics

Raleigh, North Carolina 2004

APPROVED BY:

D. Ronald Tilley

Mohamed Bourham

Gary E. Mitchell, John F. Shriner, Jr.,

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— Friedrich D¨urrenmatt, “21 Punkte zu den Physikern”

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William Chandler Beal, Jr.

Personal

Born April 22, 1971 Newton, MA

Education

B.S. in Physics, Cornell University, 1993

M.S. in Physics, North Carolina State University, 1998

Positions

Teaching Assistant, North Carolina State University, 1995–1997 Research Assistant, North Carolina State University, 1997–2004

Membership

American Physical Society

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This work would not have been possible without the support and assistance of my colleagues, friends, and family. Their contributions deserve to be recognized here.

It has been an honor and a privilege to work under the tutelage of Dr. Gary Mitchell. His wisdom in matters both academic and pragmatic will not be forgotten, and I will surely miss his stories. Dr. John Shriner has also been an invaluable source of knowledge and advice throughout my research. I cannot imagine working with a more straightforward, honest, and equitable advisor. I would also like to thank Dr. Mohamed Bourham and Dr. D. Ronald Tilley for agreeing to serve on my advisory committee, and Dr. John Kelley for attending my thesis defense and lending an additional critical eye to this work.

Incalculable thanks are due to Chris Westerfeldt, the glue that held the HRL to-gether. The depth and breadth of his knowledge in so many different areas is astounding, and his approachability, patience, and friendliness are dearly appreciated. Without Chris, there would be no HRL. My sincere gratitude goes out also to the TUNL technical staff, not only for their skill and willingness to quickly fix problems in the laboratory, but also for their continued friendship. It has been a pleasure to work just down the hall from Sidney Edwards, Bret Carlin, Patrick Mulkey, John Dunham, Richard O’Quinn, and Paul Carter during my time at TUNL.

It is thanks to the friendship of Dr. Chris Grossmann and Lance McLean that I was fortunate enough to join the research group at the HRL. Lance and Dr. Stephen Lokitz have become my close friends; they are even more important to me now when we play music or argue about sports than when we were troubleshooting dead times or repairing the accelerator. I owe a debt of gratitude to Dr. Daniel McDevitt for his dedication to collecting the data presented in this work. My thanks also go out to Dashdorj Dugersuren for his assistance in the laboratory and to Dr. Undraa Agvaanluvsan for her encouragement

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during my studies.

Finally, I wish to thank Mom, Dad, and my brothers Mike and Steve for their continued love, support, and encouragement. I am extremely lucky to have such a close family. My success is their success.

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List of Tables viii

List of Figures ix

Chapter 1 Introduction 1

Chapter 2 Theoretical Background 4

2.1 R-matrix Formalism . . . 4

2.1.1 Assumptions and Terminology . . . 5

2.1.2 External Wavefunctions . . . 6

2.1.3 Internal Wavefunctions . . . 8

2.1.4 Boundary Conditions and the R-matrix . . . 9

2.1.5 The Collision Matrix and Differential Cross Sections . . . 11

2.2 Isobaric Analog Resonances . . . 15

Chapter 3 Experimental Setup and Procedure 18 3.1 Accelerator and Control Systems . . . 18

3.2 Data Acquisition . . . 23

3.2.1 Target Chamber . . . 23

3.2.2 Electronics . . . 24

3.2.3 Generation of Yield Curves . . . 27

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3.4.1 Correction of Discontinuities in Measured Cross Sections . . . 32

3.4.2 Absolute Energy Calibration . . . 34

Chapter 4 Data-Fitting Procedure 37 4.1 MULTI . . . 37

4.2 Fitting Examples . . . 43

4.2.1 Improvement in p-wave Assignments . . . 43

4.2.2 Ambiguities in d-wave Resonances . . . 45

4.2.3 Identification of New Resonances . . . 47

4.3 Fitting Results . . . 49

Chapter 5 Results and Analysis 52 5.1 Statistical Tests . . . 52

5.2 Level Spacing Analysis . . . 56

5.3 Isobaric Analog Resonances . . . 68

5.4 Reduced Width Distributions and Strength Functions . . . 76

5.5 Level Densities . . . 83

Chapter 6 Summary and Conclusions 88

Appendix A 51Mn Resonance Parameters 91

Appendix B Data and Fits 129

Appendix C Energy Unfolding of Sequences 142

Bibliography 146

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3.1 Detector solid angles. . . 24

4.1 Allowed channel spins and relative angular momenta for 50Cr exit channels 40 4.2 Comparison of the number of resonances by sequence . . . 51

5.1 Identification of isobaric analog states in 51Mn . . . 75

5.2 Proton strength functions for 50Cr(p,p0) . . . 83

5.3 Level densities in 51Mn for pure sequences . . . 86

5.4 Level densities in 51Mn for mixed sequences . . . 86

A.1 51Mn resonance parameters . . . . 93

A.2 51Mn resonance parameters,Jπ = 1 2 + . . . 114

A.3 51Mn resonance parameters,Jπ = 1 2 −. . . . 116

A.4 51Mn resonance parameters,Jπ = 3 2 −. . . . 118

A.5 51Mn d-wave resonance parameters . . . 121

A.6 51Mn f-wave resonance parameters . . . 127

A.7 51Mn g-wave resonance parameters . . . 128

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2.1 Analog-state energy relations in the 51Cr and51Mn parent-daughter system 16

3.1 Floorplan of the High Resolution Laboratory at TUNL . . . 19

3.2 Schematic of accelerator and ion source . . . 20

3.3 HRL feedback and control loops . . . 21

3.4 Proton scattering chamber . . . 23

3.5 Data acquisition electronics for the charged particle detectors. . . 25

3.6 Sample charged particle spectrum . . . 28

3.7 Sample yield curves for 50Cr(p,p0) and 50Cr(p,p1) . . . 29

3.8 Schematic of the high-current evaporator . . . 31

3.9 Correction of discontinuities in the data . . . 34

3.10 High-Ep absolute energy calibration using a 56Fe(p,p0) resonance . . . 36

3.11 Low-Ep absolute energy calibration using a44Ca(p,p0) resonance . . . 36

4.1 Angular momentum coupling for proton elastic scattering . . . 38

4.2 Angular momentum coupling for proton inelastic scattering . . . 38

4.3 Sample shapes for s-, p-, and d-wave resonances in elastic-scattering . . . . 42

4.4 Sample shapes for f- and g-wave resonances in elastic scattering . . . 42

4.5 Sample inelastic scattering shapes for compound states Jπ = 12− and 32− . . 44

4.6 Jπ assignment based on the angular distribution of proton inelastic scattering 45 4.7 Sample inelastic scattering shapes for compound states Jπ = 32+ and 52+ . . 46

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4.10 50Cr(p,p

0) and 50Cr(p,p1) data and fits at 90◦ and 165◦ . . . 50

5.1 Poisson and GOE nearest-neighbor spacing distributions . . . 55

5.2 Relative probability of small spacings in the Wigner distribution . . . 58

5.3 Elimination of s-wave level based on spacing anomaly . . . 59

5.4 Level spacing parameters x and 1x for theJπ = 12− sequence . . . 60

5.5 Level spacing parameters x and 1x for theJπ = 32− sequence . . . 61

5.6 Nearest-neighbor spacing distribution forJπ = 12+ levels . . . 63

5.7 Nearest-neighbor spacing distribution forJπ = 12− levels . . . 64

5.8 Nearest-neighbor spacing distribution forJπ = 32− levels . . . 65

5.9 Nearest-neighbor spacing distribution for d-wave levels . . . 66

5.10 Nearest-neighbor spacing distribution for f-wave levels . . . 67

5.11 Plots ofγ2 and Σγ2 for the Jπ = 1 2 + sequence . . . 69

5.12 Plots ofγ2 and Σγ2 for the Jπ = 1 2 − sequence . . . . 70

5.13 Plots ofγ2 and Σγ2 for the Jπ = 32− sequence . . . 71

5.14 Plots ofγ2 and Σγ2 for the mixed d-wave sequence . . . 72

5.15 Plots ofγ2 and Σγ2 for the mixed f-wave sequence . . . 73

5.16 Reduced width distribution for theJπ = 12+ sequence . . . 77

5.17 Reduced width distribution for theJπ = 12− sequence . . . 78

5.18 Reduced width distribution for theJπ = 32− sequence . . . 79

5.19 Reduced width distribution for the mixed d-wave sequence . . . 80

5.20 Reduced width distribution for the mixed f-wave sequence . . . 81

B.1 Differential cross section of the 50Cr(p,p 0) reaction from Ep = 1.80 MeV to Ep = 2.09 MeV . . . 130

B.2 Differential cross section of the 50Cr(p,p 1) reaction from Ep = 1.80 MeV to Ep = 2.09 MeV . . . 131

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Ep = 2.38 MeV . . . 132

B.4 Differential cross section of the 50Cr(p,p1) reaction from Ep = 2.08 MeV to

Ep = 2.38 MeV . . . 133

B.5 Differential cross section of the 50Cr(p,p0) reaction from Ep = 2.37 MeV to

Ep = 2.67 MeV . . . 134

B.6 Differential cross section of the 50Cr(p,p1) reaction from Ep = 2.37 MeV to

Ep = 2.67 MeV . . . 135

B.7 Differential cross section of the 50Cr(p,p0) reaction from Ep = 2.66 MeV to

Ep = 2.96 MeV . . . 136

B.8 Differential cross section of the 50Cr(p,p

1) reaction from Ep = 2.66 MeV to

Ep = 2.96 MeV . . . 137

B.9 Differential cross section of the 50Cr(p,p

0) reaction from Ep = 2.95 MeV to

Ep = 3.25 MeV . . . 138

B.10 Differential cross section of the 50Cr(p,p1) reaction from Ep = 2.95 MeV to

Ep = 3.25 MeV . . . 139

B.11 Differential cross section of the 50Cr(p,p0) reaction from Ep = 3.24 MeV to

Ep = 3.50 MeV . . . 140

B.12 Differential cross section of the 50Cr(p,p1) reaction from Ep = 3.24 MeV to

Ep = 3.50 MeV . . . 141

C.1 Linearization of level density forJπ = 12+ . . . 143 C.2 Linearization of level density forJπ = 1

2

. . . . 143 C.3 Linearization of level density forJπ = 3

2

. . . . 144 C.4 Linearization of level density forJπ = (3

2 +

,52+) . . . 144 C.5 Linearization of level density forJπ = (5

2 −,7

2

) . . . . 145

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Introduction

Statistical studies of the nucleus are of interest for many reasons in different areas of physics. Average properties of nuclear energy levels, such as level densities or strength functions, play an important role in predicting nuclear reaction rates. These predictions are applicable to diverse subjects, among which are stellar evolution and radiation-shielding calculations. Measuring statistical nuclear properties is also useful to test the predictions of nuclear models, including those that attempt to describe nuclei far from stability.

Often, the results of these statistical studies of nuclear levels are very sensitive to the data quality; therefore, a worthwhile goal is to establish sequences of levels that are both pure and complete. A sequence is defined as a set of levels that share particular symmetries. In the context of this work, the relevant symmetries are the spin and parity of the level,Jπ. A pure sequence contains only levels with a particular value ofJπ, and a complete sequence is missing no levels with that Jπ value.

Recent work [Wal96, Vav97, Gro00, Lok04] performed by our research group at the High Resolution Laboratory (HRL) of the Triangle Universities Nuclear Laboratory (TUNL) has sought, with much success, to establish pure and complete level sequences for nuclei in the mass range A'30–45. New methods have also been developed recently [Agv03a] to estimate the fraction of missing levels from incomplete sequences, thus improving the accuracy of nuclear level densities. In this context, a primary aim of the present work is to

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establish pure and complete level sequences for the compound nucleus 51Mn by measuring differential cross sections for the 50Cr(p,p0) and 50Cr(p,p1) reactions and determining the nuclear level parameters from resonance scattering.

These reactions have been studied previously by Moses [Mos71] and Salzmann [Sal77], and a related experiment was performed by Whatley [Wha82] in order to determine level parameters for 51Mn. However, for several reasons the current experimental system and methods of data-fitting and analysis used in the HRL are superior to those available in previous experiments. Improvements in beam-energy resolution and detector resolution permit weaker resonances to be observed, while automated cross-section measurements, faster computers, and improved data-display capabilities result in faster data collection and the ability to devote more time to careful fitting and analysis. As mentioned above, recently developed methods of statistical analysis yield improved assessments of data quality and better estimates for level densities.

The experiment presented in this work measured differential cross sections for the 50Cr(p,p

0) and 50Cr(p,p1) reactions at five different angles over the proton energy range Ep=1.80–3.50 MeV. The measurements were performed with the modified KN-3000 Van de

Graaff proton accelerator of the HRL at TUNL, using thin ( 1.5µg/cm2) 50Cr targets. The measured cross sections were fit using a computer code which utilizes R-matrix theory to calculate cross sections, and level parameters were obtained for 185 resonances in the compound nucleus 51Mn. A total of 64 resonances were observed which had not been identified in previous work. The nuclear levels were divided into sequences and statistical tests were performed to judge the data quality. Strength functions and level densities were determined from the level sequences. Where possible, comparisons were made to previous work to evaluate the relative success of the current methods.

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introduced as non-statistical effects in the nucleus that exist due to the isospin symmetry of the nuclear force. Parent and daughter states are defined, and the energy relations between the two are outlined. Chapter 3 describes the experimental equipment used and the procedure followed to measure the differential cross sections, with special attention paid to the feedback and control systems that provide excellent beam-energy resolution and automate the data collection. The data acquisition electronics and target fabrication process are described, and the necessary data corrections and calibrations are summarized. Chapter 4 illustrates the method of fitting the data and extracting resonance parameters using the R-matrix computer code MULTI. Comparisons of the current results to the results of previous work are made. Chapter 5 introduces statistical tests of data quality from random matrix theory and applies them to the current data. Nearest-neighbor spacing distributions and reduced width distributions are compared to theoretical predictions. Strength functions are compared to the results of previous work, and level densities are determined. Finally, Chapter 6 summarizes this experiment and its results.

Three appendices are included in this work. Appendix A tabulates the resonance parameters in two ways: in order ofEp, and sorted by sequence. Appendix B contains plots

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Theoretical Background

An appropriate theory of nuclear scattering is desired to provide a framework for the present study of the 50Cr(p,p

0) and 50Cr(p,p1) reactions. First developed by Wigner and Eisenbud [Wig47], the R-matrix theory of compound nuclear reactions has proven useful to describe such reactions. Elements of the theory relating to the current work will be summarized in this chapter without proof; a much more general and comprehensive treatment is developed by Lane and Thomas [Lan58]. This summary will attempt to follow the notational conventions of Lane and Thomas.

The specific topic of isobaric analog resonances (IAR) will also be briefly described here; it is necessary to consider the non-statistical effects of the IAR before performing certain statistical analyses on the compound nuclear level strengths (see Chapter 5).

2.1

R-matrix Formalism

In the context of the present study, the goal is to formulate an expression for the differential cross sections for proton elastic and inelastic scattering. In practice, however, a broader approach is taken to determine a general formula for the cross section that can be applied to many different nuclear reactions. The FORTRAN R-matrix code MULTI is an application based on the results of this approach and will be described in Chapter 4.

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2.1.1 Assumptions and Terminology

Four main assumptions are made in the formulation of the R-matrix theory. First, the use of the nonrelativistic Schr¨odinger equation HΨ = EΨ in the derivation presumes that relativistic effects can be ignored. This is reasonable since the kinetic energies of nucleons inside nuclei are less than a few percent of the rest energy. A second assumption demands that processes which create more than two reaction products do not exist or may be ignored; certainly this is the case for proton elastic and inelastic scattering. In this formulation of the theory, it is also assumed that no particles may be created or destroyed; an implication of this is that photons are excluded from consideration in the reaction process. The theory can be extended to include photons, but it will not be necessary in this case. Finally, for each pair of nuclei there is assumed to be a finite radial separation ac, the

“channel radius”, beyond which the nuclear interaction between the two vanishes. Nuclear forces are known to be short-range, so this is also a valid assumption.

Before proceeding, it is useful to make some procedural and notational comments. In this formulation the channel-spin representation of R-matrix theory will be utilized. This representation is chosen for compound nuclear reactions with unpolarized beams and targets because the cross sections are incoherent in channel spin; the total cross section is then a weighted average of the cross sections for each individual channel spin. In this scheme, the projectile and target spins I~1 and I~2 are coupled to form a channel spin~s=I~1+~I2 which can have magnitudes between |I1+I2 | and |I1−I2 |. The projection component of the channel spin is labelledν.

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the reaction channel completely. Alternatively, ~`can be coupled to the channel spin ~s to form the total spinJ~with projectionM; in this case, the reaction channel is described with

{α(I1I2)sν JM}.

The following quantities are defined for convenience:

Mc≡Mα=

Mα1Mα2 Mα1 +Mα2

the reduced mass of the system,

kc ≡kα=

·

2MαEα

~2 ¸1

2

the wave number,

vc ≡vα=

~kc Mc

the relative speed,

ηc ≡ηα =

Zα1Zα2e 2

~vα the Coulomb field parameter, and ρc ≡ρα=kcrc a dimensionless radial coordinate.

Mα1,Zα1 andMα2,Zα2 are the masses and charge numbers of the two particles in the pair α, and Ec ≡ Eα is the energy of the pair’s relative motion. Only channels with Eα > 0

are considered; these are called “open” channels. Channels for which Eα is negative are

“closed” channels.

We now wish to solve the Schr¨odinger equationHΨ =EΨ and write the solution Ψ in a form that will yield an expression for the reaction cross section. To do this, the problem is separated into three parts: solution in the region external to the channel radiusac, solution

in the complementary internal region, and the specification of boundary conditions on the wavefunctions at each channel surfaceSc defined byrc=ac.

2.1.2 External Wavefunctions

The coordinate transformation (xi, yi, zi) → (R, q~ t) from a Cartesian system into

a “center-of-mass motion plus internal motion” system is combined with the separation of all space into regions internal (rc < ac) and external (rc > ac) to the range of the nuclear

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particles.) In the external region, the potential in the Hamiltonian is due only to the Coulomb interaction. This permits the separation of variables and allows the solution to the Schr¨odinger equation in the external region to be written as the product

Ψext = Φ(R)~ χ(~rα)ψα1(qα1)ψα2(qα2) (2.1)

in which Φ describes the center-of-mass motion,χ describes the relative motion of α1 and α2, and ψα1 and ψα2 describe the internal states of α1 and α2. The total Hamiltonian is written asH =H0+Hc+Hα1+Hα2, with

H0 = − ~2 2M∇

2

R,

Hc = −

~2 2Mα∇

2

rα+Vc(rα), (2.2)

Hα1 = Tαint1 +Vαint1,

Hα2 = Tαint2 +Vαint2;

Tint and Vint are the kinetic and potential energies of the internal motion.

The center-of-mass motion can henceforth be neglected by assuming the center of mass is at rest. The symmetry of the total Hamiltonian with respect to spin implies that the channel-spin wavefunctions can be written as a linear combination of products of the individual spin wavefunctions,

ψαsν =

X

i1+i2=ν

(I1I2i1i2 |sν)ψα1(I1,i1)ψα2(I2,i2), (2.3)

where (I1I2i1i2 |sν) are the Clebsch-Gordon coefficients associated with the spin coupling (described, for example, in Sakurai [Sak94]).

Since the Coulomb-interaction term Vc(rα) in the relative-motion Hamiltonian Hc

is a central potential, χcan be further separated into the form

χ ·

i`Ym

` (θα, φα)

¸

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(where theY`m are spherical harmonics), so that the functionsuαs` are solutions to a radial Schr¨odinger equation, · d2 dr2 α −

`(`+ 1) r2

α −

2Mα

~2 (Vαs`−Eα) ¸

uαs`(rα) = 0, (2.5)

with Vαs` = Zα1Zα2e2/rα. The solutions to this differential equation for open

chan-nels (Eα > 0) are the regular and irregular Coulomb functions Fc ≡Fα`(ρα, ηα`) and

Gc ≡Gα`(ρα, ηα`), respectively [Jah60], where ρ and η have been defined in Section 2.1.1.

The spatial solutions for open channels are written in the form of incoming and outgoing waves,

Ic+= (Gc−iFc) eiωc and Oc+= (Gc+ iFc) e−iωc, (2.6)

with

ωc ≡ωα`= `

X

n=1

tan−1³ηα n

´

. (2.7)

The full, normalized, orthogonal channel wavefunctions representing incoming and outgoing waves of unit flux can then be written for the external region (rc > ac) as

Iαs`νm+ =

³

i`Y`m´ I +

α`

rα√vα

ψαsν and O+αs`νm =

³

i`Y`m´ O +

α`

rα√vα

ψαsν. (2.8)

These will be useful later when we write the total external wavefunction as a linear combi-nation of the incoming and outgoing channel waves,

Ψext=X

c

(xcIc++ycOc+), (2.9)

to investigate the nature of the outgoing-wave amplitudesyc for a single incoming channel.

2.1.3 Internal Wavefunctions

In the internal region (rc < ac) no attempt is made to write a specific form of the

potentialV(~rα); otherwise the solutions to the internal-region Schr¨odinger equation would

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to write the internal solutions ΨJM to the Schr¨odinger equationHΨJM =EΨJM as linear combinations of mutually orthogonal, normalized eigenfunctions XλJM:

ΨintJM =X

λ

AλJXλJM. (2.10)

TheXλJM are solutions to the internal-region Schr¨odinger equation with energy eigenvalues

EλJ; they are determined using boundary conditions on the eigenvalues at each channel

surface Sc. The expansion coefficients AλJ are determined from boundary conditions on

the wavefunctions at each channel surface.

2.1.4 Boundary Conditions and the R-matrix

The quantities

Vc=

s ~2 2Mcac

uc(ac) and Dc =

s ac~2

2Mc

· duc

drc

¸

rc=ac

(2.11)

are defined in terms of the values and derivatives of the radial wavefunctionsuc evaluated

on the channel surfaces Sc. The ratio

Dc

Vc

=ac

· u0

c

uc

¸

rc=ac

(2.12)

is also useful because it incorporates the logarithmic derivative at the channel surface to which a boundary condition will be applied. We also define the “surface functions,”

φαs`νm ≡

·

i`Y`m(θα, φα)

¸

ψαsν, (2.13)

which are simply the total channel wavefunctions for the external region with the radial factorsuαs`removed. The surface functions are mutually orthogonal and normalized on the

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thatVc andDc may be written as surface integrals:

Vc =

s ~2 2Mcac

Z

Sc

φ∗cΨ dS, (2.14)

Dc =

s ~2 2Mcac

Z

Sc

φ∗cn(rcΨ) dS

= Vc+

s ac~2

2Mc

Z

Sc

φ∗cnΨ dS, (2.15)

where Ψ is the complete wavefunction of the system. The orthonormality of the φ allows the inversion of these expressions to yield

Ψ = X

c

r 2Mcac

~2 Vcφc, (2.16)

∇nΨ =

X

c

r 2Mc

ac~2

(Dc−Vc) φc. (2.17)

Similar expressions exist for the internal-region wavefunctions Xλ when the overlap

inter-grals are performed with theXλ rather than the Ψ; the definitions are:

γλc =

s ~2 2Mcac

Z

Sc

φ∗cXλdS, (2.18)

δλc = γλc+

s ac~2

2Mc

Z

Sc

φ∗cnXλdS. (2.19)

Based on the completeness of the XλJ, the energy-dependent coefficients AλJ in

Equation (2.10) may be written

AλJ =

Z

τ

X∗

λJMΨintdτ. (2.20)

whereτrepresents the internal volume bounded bySc. We now choose two possible solutions

to the internal-region Schr¨odinger equation, Ψ andXλ, so that

HΨ =EΨ and HXλ=EλXλ. (2.21)

Multiplying the first expression in Equation (2.21) byX∗

λ and the complex conjugate of the

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Theorem is applied, reducing the volume integral over τ to an integral over the surface

S. This procedure is combined with the imposition of the boundary condition on the logarithmic derivative at the channel surface,

δλc

γλc

=Bc, (2.22)

to yield an expression for the expansion coefficients Aλ:

Aλ=

1 Eλ−E

X

c

γλc(Dc−VcBc). (2.23)

Substituting this into Equation (2.10), one can multiply both sides byφ∗

c0 and integrate over

Sc to obtain the boundary condition on the value of the total wavefunction at the channel

surface,

Vc0 =

X

c

Rc0c(Dc−VcBc), (2.24)

where the elements of the R-matrix are defined by Rc0c

X

λ

γλc0γλc

Eλ−E

. (2.25)

2.1.5 The Collision Matrix and Differential Cross Sections

Recalling Equation (2.9), the general expression for the external-region wavefunc-tion, the elements of the collision matrix U (matrix quantities are denoted in bold) are defined implicitly as

xc0 =−

X

c

Uc0cyc. (2.26)

Evidently, U describes the effect of the nuclear scattering by relating the amplitudes of outgoing channel waves to the amplitudes of particular incoming channel waves. The goal is now to express the Uc0c in terms of the R-matrix elements Rc0c.

The quantities Vc andDc of Equations (2.14) and (2.15) can be written in terms of

the external surface functions I+

c andO+c using Equation (2.9),

Vc =

~ √ 2 · ρ− 1 2

c Ocxc+ρ−

1 2

c Icyc

¸

(2.27)

and Dc =

~ √ 2 · ρ− 1 2

c Oc0xc+ρ−

1 2

c Ic0yc

¸

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where the primes denote differentiation with respect to r. Substituting back into the R-matrix relation of Equation (2.24), collecting terms in xc and yc, and writing in

ma-trix notation, the collision mama-trix then becomes

U=hρ−12O−R ³

ρ12O0−Bρ− 1 2O

´i−1h

ρ−12I−R ³

ρ12I0−Bρ− 1 2I

´i

. (2.29)

The matrixU is now written in terms of known functions and their derivatives, evaluated at the channel surfaces, and the internal-region quantitiesEλ and γλc. Its elements can be

expressed further as

Ucc0 =ei(ωc+ωc0−ϕc−ϕc0)

·

δcc0+ 2iP

1 2

c 1

1Rcc0Lc0Rcc 0P 1 2 c0 ¸ (2.30) by defining

Lc ≡

· ρc O0 c Oc ¸

rc=ac

−Bc = [Sc−Bc] +iPc. (2.31)

Pc, the penetrability, is related to the penetration of the channel wavefunction through the

channel surface and can be written in terms of the Coulomb functions as

Pc =

· ρc

F2

c +G2c

¸

rc=ac

. (2.32)

The shift function Sc has the form

Sc =

· ρc

FcFc0+GcG0c

F2

c +G2c

¸

rc=ac

(2.33)

and is so named because it appears as a factor in the level shift ∆λ which will be defined

shortly. The ϕc and ωc are the phase shifts due to hard-sphere and Coulomb scattering,

respectively. The hard-sphere phase shift is defined as

ϕc≡tan−1

µ Fc

Gc

, (2.34)

and the Coulomb phase shift is given by Equation (2.7).

It can be shown that, when Equation (2.30) is evaluated more fully, the quantity

∆λ ≡ −

X

c0

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is subtracted from the energy Eλ in the denominator of the second term. This suggests

that ∆λ represents a shift in the resonance energy from the internal-region eigenenergy.

The presence of Bλ in the expression further implies that this shift is partly due to the

boundary conditions imposed on the wavefunctions at the channel surfaces. The quantity

Γλc= 2Pcγλc2 (2.36)

also appears in the evaluation of the U-matrix elements and is called the partial laboratory width for the channel c. Accordingly, the quantities γ2

λc are called the reduced channel

widths, and the unsquared quantities γλc are the reduced width amplitudes. The total

laboratory width for a level is then

Γλ=

X

c

Γλc. (2.37)

It is interesting to note that the partial width has been separated into two factors, one parameterizing the effects of the reaction in the internal region (the reduced widthγλc2 ), and one describing the external kinematics (the penetrability Pc). The penetrability decreases

with increasing `, implying that higher-` states will tend to contribute less to the total width than lower-`states. Since the penetrability effects have been removed in the reduced widths, theγ2

λcprovide an effective measure of the nuclear coupling to the entrance channel.

Lane and Thomas derive an expression for the differential cross section from the U-matrix elements; the result will be quoted here without proof. The derivation involves writing the external wavefunctions as the sum of an incident plane wave and outgoing waves only,

Ψ(general) = Ψ(plane wave) +X

cc0

¡ e2iωcδ

cc0−Uc0c¢ Oc0yc, (2.38)

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them to obtain the differential cross section. The result is dσαs,α0s0

dΩα0

= π

k2

α|

Cα0(θα0)|2δα0s0,αs

+ 1

k2

α(2s+ 1)

X

L

BL(α0s0, αs)PL(cosθα0) (2.39)

+

π k2

α(2s+ 1)

X

J`

(2J+ 1)δα0s0`0,αs`Re£iTJ

α0s0`0,αs`

¤

Cα0(θα0)P`(cosθ),

where

BL(α0s0, αs) =

1 4(−1)

(s−s0) X

J1J2`1`2`01`02

Z(`1J1`2J2, sL)

× Z(`01J1`02J2, s0L) ³

TJ1

α0s0`0

1,αs`1 ´ ³

TJ2

α0s0`0

2,αs`2 ´∗

, (2.40)

TαJ0s0`0,αs` = e2iωα0`0δα0s0`0,αs`−UαJ0s0`0,αs`, (2.41)

and

Cα=

1

4π ηαcsc 2

µ θα

2 ¶

e−2iηαlog[sin(θα2 )]. (2.42) The Z coefficients are the Z coefficients of Biedenharn et al. [Bie52] with the phase con-vention of Huby [Hub54]. The P`(cosθ) are the Legendre polynomials with the phase

convention of Condon and Shortley [Con51]. The first term in Equation (2.39) represents Coulomb scattering, the second term represents resonance scattering, and the third term mixes the two. Thus, for example, it is possible to observe interference effects in an isolated resonance if several channels contribute. The first and third terms in the differential cross section vanish for all but elastic scattering.

In this experiment, an unpolarized proton beam was used to bombard unpolarized targets. The appropriate differential cross section in this case is then obtained by averaging over the projectile polarization statessand summing over the ejectile states s0, yielding

α,α0

dΩα0 =

1

(2I1+ 1)(2I2+ 1) X

ss0

αs,α0s0

dΩα0 . (2.43)

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2.2

Isobaric Analog Resonances

The symmetry of the nuclear force under the interchange of neutrons and protons leads to the definition of the isospin quantum number T. The isospin of a single nucleon is 12; by convention, itsz-component is tn= 21 for neutrons and tp =−12 for protons. The

z-component of total nuclear isospin is then given byT = 12(NZ). Isobaric analog states are defined as the nuclear states that form a multiplet of a given isospin. Both nuclei in a parent-daughter analog pair contain Anucleons, and the daughter is formed by converting a proton from the parent into a neutron. In other words, if the parent is denoted ZAN+1, then the daughter is defined byZ+1AN. AlthoughT to a very good approximation is a good

quantum number under nuclear forces alone, the Coulomb interaction significantly mixes isospin in the total nuclear system due to the fact that the proton carries charge while the neutron does not. The energy difference between parent and daughter can then be written ∆Ec−δ, where ∆Ec is the difference in energy due to the Coulomb force andδis the energy

equivalent of the proton-neutron mass difference. An analog state in the daughter has the sameJπ value as its corresponding state in the parent.

The energy relations between the parent and daughter configurations are illustrated in Figure 2.1. From the figure it is evident that the Coulomb energy difference can be expressed as

∆Ec =Bn+Ecm−Ex, (2.44)

whereBn is the binding energy of the last neutron in the parent nucleus (sometimes called

the neutron separation energy Sn of the parent), Ecm is the proton energy in the

center-of-mass frame, and Ex is the excitation energy of the parent state. The quantity Bp in

Figure 2.1 is the binding energy of the last proton in the daughter nucleus, equivalent to the proton separation energy Sp of the daughter. If ∆Ec can be estimated, then Equation

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semi-50Cr + p

Daughter (Analog) State

51Mn

51Cr Bp = 5.2715 MeV

Bn = 9.26062 MeV

δ = -0.822 MeV ∆Ec ≈8.439 MeV

Parent State Bp

Ecm

∆Ec - δ

Ex

Bn

50Cr + n

δ

Figure 2.1: Analog-state energy relations in the 51Cr and 51Mn parent-daughter system. Bp, the binding energy of the last proton in the daughter nucleus, and Bn, the binding

energy of the last neutron in the parent nucleus, are taken from the 2003 A = 51 mass evaluation of the Atomic Mass Data Center [Nnd04]. ∆Ec, the Coulomb energy difference

between the parent and daughter nucleus, is estimated using the semi-empirical formula of J¨anecke [J¨an69].

empirical approximation to ∆Ec was given by J¨anecke [J¨an69],

∆Ec =

C1Z<+C2 A13

, (2.45)

whereAis the parent mass,Z<is the proton number of the target, andC1andC2are empiri-cally determined constants. For theA= 51 system studied in this work,Bn= 9.26062 MeV

[Nnd04], and using J¨anecke’s values C1= 1389 keV and C2 =−2041 keV yields a Coulomb energy difference estimate ∆Ec ≈ 8.439 MeV. Thus the comparison between the

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to be Ecm≈Ex−0.822 MeV. In general ∆Ec can vary greatly, thereby effecting a large

uncertainty in this calculation.

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Experimental Setup and Procedure

Proton elastic and inelastic scattering cross sections on 50Cr were measured at five different angles within the incident proton energy range 1.80 – 3.50 MeV. This experiment was performed at the Triangle Universities Nuclear Laboratory (TUNL) in the High Reso-lution Laboratory (HRL), which houses a single-ended Van de Graaff accelerator capable of producing stable proton beams in the energy range 0.9 – 4.0 MeV for use in scattering and capture experiments. Several control and feedback systems provide excellent beam energy resolution as well as automate the data collection process [Wes95]; large quantities of high quality data can be accumulated in a relatively short time. Figure 3.1 illustrates the layout of the various beamline components and target chambers in the laboratory. These will be described in this chapter. The data acquisition system and target-making process will also be summarized, followed by a description of the procedures used to create one continuous data set from the fragmented set of measured yield curves.

3.1

Accelerator and Control Systems

A model KN-3000 positive ion accelerator (Figure 3.2) provides the proton beam for the experiment. Inside a large gas-storage tank is mounted a horizontally oriented

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KN 3000 Van de Graaff

Electrostatic Analyzer Analyzing

Magnet

(p,p) Chamber

(p,γ) Chamber

Figure 3.1: Floorplan of the High Resolution Laboratory at TUNL.

insulating column with a drive motor at the base end and an alternator at the terminal end. The motor drives a wide rubberized cotton charging belt. Charge supplied by an external current-regulated power supply is fed through the tank wall and onto the belt, which carries the charge to the terminal to be deposited onto a hollow stainless steel dome. A large positive voltage of up to 4 MV can thus be built up on the dome to serve as the accelerating potential. Moderate voltage stabilization is achieved with a capacitive pickoff circuit in the tank wall and a corona circuit which utilizes moveable stainless steel needles to take small amounts of charge from the dome via corona discharge. The charge on the dome is bled back to the base though a series of 69 column-mounted 600-MΩ resistors across adjacent equipotential planes; this process results in a uniform potential gradient down the length of the column. Voltage isolation is achieved by pressurizing the surrounding tank with a dry (typically 30 ppm water) insulating mixture of nitrogen and carbon dioxide.

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Figure 3.2: Schematic of the KN-3000 Van de Graaff accelerator and ion source.

containing both protons and singly ionized molecular hydrogen, H+2, which are extracted from the bottle using a small probe voltage. Once the positive ions leave the bottle, they are subject to the large accelerating voltage outside the dome and travel down the tube. They are first focused by a negatively biased focus electrode at the terminal end of the tube. Upon exiting the accelerator, the ion beam passes through the main analyzing mag-net. The strength of its magnetic field is set via a control computer and a programmable high-current power supply. Due to their different charge-to-mass ratios, the constituent protons and H+2 ions are separated into two distinct beams as they traverse the magnet, exiting 25◦and 17to the left of the initial beam direction, respectively. The protons travel down the 25◦ leg toward the experimental chamber, which will be described in the next sec-tion of this chapter. The 17◦ leg carries the H+

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KN Van de Graaff Accelerator

Corona Controller

Digital Tesla

Meter Power SupplyMagnet

Analyzing Magnet Digital Voltmeter Personal Computer Electrostatic Analyzer

- High Voltage Supply &

Divider

+ High Voltage Supply & Divider Optical Receiver &Amp High Voltage Target Rod Amplifier Fiber Optic Cable Preamp & Optical Transmitter/ Receiver Elastic Scattering Chamber Chamber

CPO Control Slits

Figure 3.3: HRL feedback and control loops.

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signal is also monitored by the control computer that governs the field strength of the main analyzing magnet; slow drifts in the slit signal are corrected for by changing the magnetic field slightly to re-center the beam on the image slits. Finally, a “homogenizer” system amplifies the slit-difference fluctuations and applies them as a bias voltage to the target. The homogenizing voltage floats at a 3-kV DC level so that ripples as large as ±3 kV can be corrected for with only a positive-polarity amplifier. Because variations in the proton beam energy track identically with those in the H+2 beam energy (since they have the same source), the target bias effectively moderates the fluctuations and improves the overall beam energy resolution.

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90 108

135

165

150

Beam Collimator

Target

Vacuum Faraday cup Port

Assembly

Viewing Ports

Figure 3.4: Top view of proton scattering chamber with beam collimator and detectors.

3.2

Data Acquisition

3.2.1 Target Chamber

The proton beam is tuned down the 25◦ leg using two sets of left-right and up-down magnetic steerers and a set of quadrupole focusing magnets mounted along the beamline. The beam travels through a water-cooled collimator assembly and into the proton elastic scattering chamber illustrated in Figure 3.4. An aluminum ladder that holds up to four thin 50Cr targets (described in Section 3.3) plus a tuning ring is lowered through the top of the chamber into the beamline. The targets are sufficiently thin such that most of the proton beam passes through the target and reaches the Faraday cup, where the total beam on target is measured. A typical tune for this experiment, measured with the target ladder in its tuning-ring position, consists of roughly 7µA of beam on target with less than 50 nA on the tuning ring and less than 5 nA measured on the last collimator.

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Table 3.1: Detector solid angles. θlab Ω (msr)

90◦ 0.53 108◦ 0.90 135◦ 1.54 150◦ 1.83 165◦ 2.04

electrons that contribute background counts to the spectra. Two different types of charged-particle detector were used for these measurements: Silicon surface barrier (SSB) detectors were used to collect data in the proton energy range 1.80 MeV – 3.24 MeV, and passivated implanted planar silicon (PIPS) detectors were used to measure the remainder of the data up to Ep= 3.50 MeV. In each case, the detectors had an active area of 50 mm2 and were

300µm thick. Table 3.1 lists the effective solid angles subtended by each detector after collimation. The detector geometries were chosen to yield approximately equal counting rates in all five detectors.

3.2.2 Electronics

Figure 3.5 shows the electronics setup for data acquisition, including the beam current integration and various data inhibits. An event seen by a SSB/PIPS detector is converted into a voltage pulse by a preamplifier located outside the scattering chamber; the pulse is then sent to the control room for further processing. A linear amplifier integrates and shapes the input pulse and provides two output pulses, one unipolar and one bipolar. The unipolar signal encodes the energy of the event observed by the detector; it is delayed by 900 ms and sent to a multiplexer module operating in a gated mode. Multiplexing is necessary as a single ADC is used to digitize the event pulses.

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3. EXPERIMENT AL SETUP AND PR OCEDURE 25 4 Faraday Cup Manual & Beam Limit Data On/Off Digital Current Meter PC Data On/Off Crate Inhibit &

Alarm CAMAC Crate

CAMAC Crate Terminator

Preset Scaler

Buffer

Inhibit Out

Hex Scaler (2) Router Decoder ADC/Buffer Interface Scaler Interface (2) Router Encoder Multiplexer Multiplexer Gate Box ADC system busy Delay Line Linear Amplifier Timing SCA Gated SCA Delay

Gate & Delay Generator Same for each detector Charged Particle Detector Preamp Clock Branch Driver (MBD) VAXStation 3200 LAM Enable Panel Crate Controller 1 25 unipolar out bipolar out ADC Gate Target Rod Voltage

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for the multiplexer. The carbon events comprise a large fraction of the protons detected. If they are not vetoed before digitizing, they swamp the ADC with counts and cause in-creased pileup and large dead times in the electronics. The bipolar signal is sent to a timing single channel analyzer (TSCA) which allows only pulses within a narrow range of voltage amplitude to produce a unipolar output pulse. The window on the allowed pulses is set to correspond to protons scattered elastically from 12C, and the TSCA output is then sent as a gate signal to a gated single channel analyzer (SCA). Coincident as an input at the SCA with the carbon-event gate signals from the TSCA is the 1-µs-delayed bipolar output from the linear amplifier; those bipolar signals which correspond to carbon events are vetoed inside the SCA. In addition, the SCA further suppresses unwanted events by providing ad-justable overall upper and lower discrimination thresholds for the signals. Thus the output from the SCA corresponds only to those proton events within the energy range of interest for the reaction, minus the protons scattered from the carbon foil target backing. The SCA output is then shaped by the multiplexer gate module to serve as the gate input to the multiplexer.

Delayed unipolar signals from the linear amplifier are sorted by the multiplexer and encoded to include information indicating from which detector each energy signal originates. The routing information is converted into a gate signal for the ADC by a buffer interface, and the digitized ADC output and its routing information are then stored in a CAMAC buffer module. When a memory block in the buffer module becomes full, the buffer signals the microprogrammable branch driver (MBD) to transfer the data into memory on the VAX microcomputer running the XSYS data acquisition software [Sod87].

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CAMAC crate when operating parameters fall outside acceptable limits. The gated SCA output is also sent to a gate and delay generator (G&D) module which produces TTL pulses that are counted by another scaler in the CAMAC crate; these are sent to the VAX and compared to the total number of counts in the corresponding spectrum to calculate a dead time for that detector channel.

3.2.3 Generation of Yield Curves

Charged particle spectra are stored in five 512-channel data areas allocated by XSYS; a sample spectrum is shown in Figure 3.6. During a single “run,” data are collected at a specific proton beam energy for a predetermined amount of BCI. The preset BCI is chosen to yield approximately 1% error in counting statistics for Coulomb scattering events. At lower beam energies, a preset value of 400µC of BCI achieves this; a value of 500µC becomes necessary at higher energies as the Rutherford scattering cross section decreases steadily with increasing Ep.

The XSYS data acquisition software works in conjunction with associated VMS command files and programs to generate and display yield curves for proton elastic and inelastic scattering over a range of incident proton energies. When the preset BCI is reached at the end of a run, a dormant VAX subprocess becomes active and executes several tasks. First, the VAX transmits to the accelerator control computer the new Ep setting based

on a “step size” parameter set by the experimenter. The area under each 50Cr(p,p 0) and 50Cr(p,p

1) peak in each spectrum, defined by software “windows” set in XSYS, is then plotted as a single point in a separate yield curve data area. A linear background taken from the region surrounding the peak is subtracted from the area. Next, the spectra are cleared from memory, the data and preset scalers are reset to zero, and the yield curve index represented on the x-axis of each yield curve data area is incremented by one. The summing windows around the peaks of interest are shifted accordingly in XSYS to track the reactions kinematically.

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0 1000 2000 3000 4000

0 100 200 300 400 500

Counts

Channel

50Cr(p,p 1)

16O(p,p 0)

50Cr(p,p 0)

Ep = 3.2200 MeV

Θlab = 135°

Figure 3.6: Sample charged particle spectrum. Protons scattered from the carbon target backing that would appear in the flat region just below channel 300 have been removed electronically. The three peaks labelled are used for energy calibration.

examples in Figure 3.7 show elastic and inelastic scattering yields measured in 100-eV steps by the detector at 135◦. At lower beam energies where 50Cr proton resonances are isolated, data were taken in 400-eV steps until possible resonant structure was observed in the yields. The beam energy was then stepped back down, and the region where structure was observed was covered in more detail with 100-eV energy steps. At higher energies, only 100-eV steps were taken because of the emergence of many small resonances in the inelastic scattering and much more detailed structure in the elastic channel.

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0 200 400 600 800

0 50 100 150 200 250 300 350

Yield curve index

50Cr(p,p 1)

Θlab = 135°

0 4000 8000 12000 16000

Counts

50Cr(p,p 0)

Θlab = 135°

Figure 3.7: Sample yield curves for 50Cr(p,p0) and 50Cr(p,p1) measured in 100-eV steps from 3.220 MeV to 3.251 MeV. The first data point in each plot represents the summed area of its corresponding reaction peak in Figure 3.6.

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3.3

Targets

The 50Cr(p,p

0) and 50Cr(p,p1) reactions were measured for this experiment using

∼1.5-µg/cm2-thick50Cr targets. Because a goal of this work is to observe small and narrow resonances in the scattered proton yields, targets must be thin enough such that energy loss due to Coulomb straggling of protons through the target is smaller than the energy spread of the beam. If this condition is not met, the smaller structures in the yield curves tend to get washed out. For a proton beam energy of 1.8 MeV, a thickness of 1.5-µg/cm2 50Cr results in an average energy loss of 160 eV through the target. As beam energy increases, the proton energy loss in the target decreases. Compared to the typical beam energy resolution of & 230 eV, these targets are thin enough to maintain high yield-curve resolution while still keeping the data count rates high.

A 50Cr target is fabricated by evaporating metallic chromium powder onto a thin (5µg/cm2) carbon foil that spans the inner area of a stainless steel annulus. First the carbon foils are floated from slides onto the target rings using distilled water to avoid contamination. The rings are then suspended by a frame which is affixed above a high-current evaporator, illustrated in Figure 3.8. Clean glass slides are used to cover the top of the frame to prevent evaporated material from being deposited on the wrong side of the carbon foils. A few milligrams of 50Cr powder are added to a 0.010”-thick open tantalum boat, and the boat is fixed between a pair of electrodes. The chromium powder used for this experiment was 96.8% isotopically pure 50Cr, with a 3% impurity of52Cr and much smaller percentages of 53Cr and 54Cr. An aperture and a moveable shutter are mounted just above the boat to regulate the evaporation process, and a piezoelectric quartz-crystal thickness monitor is positioned adjacent to the frame containing the target rings to monitor the evaporation rate. The chamber is covered with a bell jar and evacuated to< 1µTorr.

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P P P P P P Shutter

Aperture

X X X X X X Target

rings »»»»»

Thickness monitor

((((((((

Ta boat & sample

Figure 3.8: Schematic of the high-current evaporation chamber used to make targets.

the thickness monitor was observed to yield a 50Cr target layer 1.5µg thick. Target thicknesses are verified by measuring the yields from proton Coulomb scattering.

44Ca and 56Fe targets were also employed in this experiment to measure proton elastic scattering resonances that were used for absolute energy calibration of the 50Cr data (Section 3.4). The fabrication process for both of these types of targets involves the chemical reduction of an oxide sample and is described in detail by Lokitz [Lok04].

3.4

Data Correction and Energy Calibration

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procedures are summarized here.

3.4.1 Correction of Discontinuities in Measured Cross Sections

Two types of discontinuities between adjacent segments were observed in the stripped data: shifts in Ep, and anomalous shifts in measured yield. The energy shifts appeared

fre-quently between segments, and were generally due to either a voltage breakdown in the tank or a drift in the tune that changed the beam optics through the ESA. Though oc-curring less frequently, the anomalous changes in yield were more troublesome and fell into two categories: discrete step-like jumps in Coulomb scattering yields in the middle of a data segment, and a gradual degradation in off-resonance yields beyond the natural drop in Coulomb yields that occurs as Ep increases. Both were likely caused by the deterioration

or partial breaking of the target, conditions that could sometimes be identified during an experiment by the development of broad low-energy tails on peaks in the charged particle spectra. In later experiments it became easier to identify deteriorating target conditions as the symptoms became more familiar, therefore corrections for degradations in yield were needed only among a few of the earliest data segments measured.

To correct for the energy shifts between adjacent segments, a single segment was first chosen as a reference. A fixed ∆Ep was calculated between it and the neighboring segment,

equal to the difference in the observed resonance energy between the two segments for an “overlap” resonance that was common to both. This process was then iterated outward from the reference segment to calculate a cumulative ∆Ep for each segment corresponding to the

amount each Ep value in the segment would need to be shifted. It should be noted that

this procedure only takes care of local shifts in beam energy; a global energy calibration of the data set requires the further steps described in Section 3.4.2. Local shifts in Ep

between adjacent segments were usually less than 500 eV, but over the entire range of data the cumulative effect could result in a ∆Ep of several keV.

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dis-continuity before running them through the MULTI6 R-matrix code and then reassembling worked well to remove the anomalies. In addition to calculating the reaction cross sections, MULTI6 normalizes the yield curve data to Coulomb scattering (Chapter 4), so the jumps in off-resonance yield from one sub-segment to the next vanished upon normalization.

In the cases where target deterioration caused a gradual degradation in the off-resonance yields beyond the normal decrease in Coulomb cross section, the following formula was used to scale the yields linearly with increasingE:

σ0(Ep) =σ(Ep)S(Ep) (3.1)

whereσ0(E

p) is the corrected cross section at energyEp, calculated from the measured cross

section σ(Ep) and the energy-dependent scaling factor

S(E) = S(Emin) + S(Emax)−S(Emin)

Emax−Emin

³

EEmin

´

. (3.2)

The quantity Emin/max is the proton energy at the beginning/end of the segment to be scaled. S¡Emin/max¢ is defined as the ratio of σ¡Emin/max¢ in an adjacent segment with overlapping data atEmin/max toσ¡Emin/max¢ in the segment to be scaled. In other words, the S are just normalization factors for the yields at the endpoints, where the normaliza-tion is taken with respect to overlapping data from an adjacent segment which shows no degradation effects. In one special case, two adjacent segments both needed to be scaled as they both exhibited the degradation effect; the data in this case were normalized at the common endpoint with respect to the calculated Coulomb cross section. In all cases the low-end normalization factorS(Emin) was never found to be less than 0.95, and the largest

S(Emax) calculated was 1.07.

In approximately fifty segments of yield curve comprising the data set, there were only five that had to be corrected for some type of anomalous shift in the yield, and all but one of them occurred below Ep= 2.5 MeV. Figure 3.9 illustrates one such case. A

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0 100 200 300

2.15 2.20 2.25 2.30 2.35

Ep (MeV)

50Cr(p,p

0) After corrections

Θlab = 165°

0 100 200 300

d

σ

/d

(mb/sr)

50Cr(p,p

0) Before corrections

Θlab = 165°

Figure 3.9: Plots of the 50Cr(p,p0) cross section at 165◦ before and after correcting for discontinuities in the data. The dashed line represents the R-matrix calculation of the reaction cross section. Note the energy shift as well as the elimination of the degradation effect.

3.4.2 Absolute Energy Calibration

Once the data had been corrected for local shifts in energy from segment to seg-ment, a global energy calibration was performed on the entire data set. Two well-studied resonances in other proton elastic scattering reactions were used to establish absolute refer-ence energies for the calibration at low and high proton energies. The s-wave resonance at Ep= 1.8840 MeV in44Ca(p,p0) was measured as a secondary standard against the 7Li(p,n) threshold at 1.8806 MeV by Wimpey [Wim74]; the calcium resonance was observed in the current work at 1.8795 MeV. The d-wave resonance at Ep= 3.2369 MeV in56Fe(p,p0) was

observed in this experiment at 3.2361 MeV; similarly, it is a secondary standard based on the13C(p,n) threshold at 3.2357 MeV as summarized by Nelson [Nel83].

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used as a fixed point for the energy calibration of the entire data set. It is crucial that no systematic problems arise to cause an energy shift between the collection of data on related resonance pairs. In this work the56Fe resonance was used to align a primary fixed point, the nearby 1700-eV-wide s-wave resonance in 50Cr(p,p0) at 3.2356 MeV (Figure 3.10). Using this calibration point alone, the corresponding global energy shift results in a discrepancy in energy calibration of 2.85 keV at the other fixed point, a 145-eV p-wave that should be located at Ep= 1.8973 based on the 44Ca calibration (Figure 3.11). The solution was

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40 80 120 160

3.230 3.235 3.240

d

σ

/d

(mb/sr)

Lab Ep (MeV)

Θlab = 165° 50

Cr(p,p0)

56Fe(p,p 0)

Figure 3.10: Absolute energy calibration of the50Cr(p,p) data at high Epusing a well-known

d-wave resonance in56Fe(p,p 0).

120 160 200 240

1.880 1.885 1.890 1.895

d

σ

/d

(mb/sr)

Lab Ep (MeV)

Θlab = 165°

50Cr(p,p 0)

44Ca(p,p 0)

Figure 3.11: Absolute energy calibration of the50Cr(p,p) data at low Ep using a well-known

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Data-Fitting Procedure

Cross sections for the50Cr(p,p0) and50Cr(p,p1) reactions were measured for proton energies between 1.8045 MeV and 3.5011 MeV in energy steps of 100 – 400 eV. The data were taken simultaneously at five different scattering angles: 90◦, 108, 135, 150, and 165◦. From these measurements it is possible to extract resonance parameters that describe the energy levels populated in the 51Mn compound nucleus; statistical properties of the nuclear levels can then be analyzed (Chapter 5). This chapter summarizes the method used to obtain the parameters with the R-matrix computer code MULTI. Examples of the fitting procedure are presented for illustrative cases, and current results are compared to those from previous work to highlight improvements in the experimental setup and method.

4.1

MULTI

MULTI is a FORTRAN computer code initially written by Sellin [Sel69] to calculate differential and total cross sections for nuclear reactions based on the R-matrix formalism of Lane and Thomas [Lan58] described in Chapter 2. This code has been extensively revised by a number of members of our group; the current version is MULTI6. The code allows for multiple interfering channels and levels to contribute to the cross sections; this

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50Cr 51Mn Jπ 50Cr ¡¡ ¡¡ ¡¡ ¡¡µ@ @ @ @ @ @

@@R

0+

p p0

~

J = ³~0 ~12´ ~`0 ~` = ~`0 for elastic scattering

s0 = 12 only exit channel

Figure 4.1: Angular momentum coupling for proton elastic scattering in the channel-spin scheme. The incident proton (intrinsic spin 12) carries angular momentum` relative to the target, and the scattered proton exits with relative angular momentum `0. Since the 50Cr ground state is 0+, the parity of the compound state is determined by `, and the only possible exit-channel spin is 12.

50Cr

51Mn

50Cr

­­ ­­ ­­ ­­ ­­ ­ Á@ @ @ @ @ @

@@R

?

γ (783 keV) 2+

0+ p

p1

~

J = ³~2 ~12´ ~`0 exit channelss0 = ¡3

2, 52 ¢

various`0 for each

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is necessary for the Cr(p,p) reactions in which many overlapping resonances are observed above Ep= 2.5 MeV. The calculation includes contributions from Coulomb, hard-sphere,

and resonant scattering.

MULTI accepts two files as input, a data file and a resonance-parameter file. The data file is created in the “stripping” (data-processing) procedure described in Section 3.2.3; it consists of tabulated peak areas for each reaction and scattering angle, indexed byEp and

run number. The parameter file contains the parameters used to calculate the cross section. In addition to the resonance parameters, the file incorporates other quantities necessary to perform the cross section calculation. These quantities include kinematic parameters such as target and projectile mass, model-dependent parameters such as the hard-sphere scattering radius, and code-specific instructions that denote the type of calculation to be performed (differential versus total cross section, number of scattering angles, number of reactions, etc.).

Calculation of the50Cr(p,p) cross section near a resonance requires the specification of the following parameters: the resonance energy Ep, the total angular momentum and

parity of the compound stateJπ, the channel spins0 for the elastic-scattering exit channel (which is equal to s for the entrance channel in the case of elastic scattering on spin-zero targets), the relative angular momentum `0 (equal to `) between the reaction products, and the elastic-scattering laboratory resonance width Γp. Additionally, where inelastic

scattering is observed,s0,`0, and an inelastic-channel partial width Γ

p0 must be specified for

each inelastic channel.1 Figures 4.1 and 4.2 illustrate the angular momentum coupling for each type of reaction, and Table 4.1 lists the allowed combinations for the most commonly observed resonances.

An important input to MULTI which is not directly related to the R-matrix cal-culation is the resolution function. To simulate proton energy loss in the target and the beam energy spread, a subroutine in the code calculates a combination Gaussian-Lorentzian

1

The prime superscript may be omitted ons and`when the channel being referred to is evident from

(52)

Table 4.1: Allowed channel spins and relative angular momenta for 50Cr exit channels, for compound-state Jπ up to 52+ (elastic-channel d-wave resonances). Though values of `0 up to 4 are listed for the inelastic channels, values of `0 2 were usually sufficient to fit the data.

Jπ channel channel spins0 `0

p0 12 0

1 2

+ 3

2 2

p1 5

2 2

p0 12 1

1 2

− 3

2 1

p1 5

2 3

p0 12 1

3 2

− 3

2 1,3

p1 5

2 1,3

p0 12 2

3 2

+ 3

2 0,2

p1 5

2 2,4

p0 12 2

5 2

+ 3

2 2,4

p1 5

(53)

function based on a pair of numbers in the parameter file. The resolution function is de-fined by a Gaussian full-width-at-half-maximum (FWHM) and a ratio of Lorentzian width to Gaussian width; the Lorentzian part comprises the leading edge of the function, and the Gaussian part makes up the trailing edge. This resolution function is then convoluted with the cross-section calculation to model the effects of overall resolution. The proper resolu-tion funcresolu-tion parameters are determined empirically for each segment of data by fitting a narrow but well-defined resonance, preferably one with an observed inelastic-channel width. If the total width is appreciably smaller than the Gaussian width for that segment, then the spread of the resonance shape is primarily determined by the resolution of the system. In the current work a Lorentzian-to-Gaussian ratio of 1.2 was used for all segments, while the Gaussian FWHM ranged between 230 eV and 350 eV due to varying operating conditions for the accelerator over the course of the experiment.

As configured for the current work, MULTI performs the cross-section calculations and produces two files of output. The first file lists the global cross-section calculation parameters, parameters specific to each resonance, normalization factors for the data, and the reduced widths for each resonance-scattering exit channel. The raw data are normalized to the calculated fit for elastic scattering, and the reduced widths γ2 are calculated from Γ = 2P γ2, whereP is the Coulomb penetrability for that channel. The second output file contains the normalized data and calculated cross sections for each reaction at each angle, indexed by Ep. The contents of this file can then be plotted to observe the agreement

between the normalized data and the calculated cross-section.

(54)

100 200 300

165°

Jπ 1/2+

0 1/2 − 1 3/2 − 1 3/2 + 2 5/2 + 2 100 200 300 150° 200 400 135° 400 600 108° 400 600 800 d σ /d Ω (mb/sr) 50Cr(p,p

0) Θlab = 90°

`

Figure 4.3: Sample 50Cr(p,p

0) shapes for isolated s-, p-, and d-wave resonances.

0 500 1000

165°

Jπ 5/2−

3 7/2 − 3 7/2 + 4 9/2 + 4 0 200 400 600 150° 200 400 600 135° 200 400 600 108° 400 600 800 1000 d σ /d Ω (mb/sr) 50Cr(p,p

0) Θlab = 90°

`

Figure

Figure 2.1: Analog-state energy relations in the energy of the last neutron in the parent nucleus, are taken from the 200351Cr and 51Mn parent-daughter system.Bp, the binding energy of the last proton in the daughter nucleus, and Bn, the binding A = 51 mas

Figure 2.1:

Analog-state energy relations in the energy of the last neutron in the parent nucleus, are taken from the 200351Cr and 51Mn parent-daughter system.Bp, the binding energy of the last proton in the daughter nucleus, and Bn, the binding A = 51 mas p.28
Figure 3.2: Schematic of the KN-3000 Van de Graaff accelerator and ion source.

Figure 3.2:

Schematic of the KN-3000 Van de Graaff accelerator and ion source. p.32
Figure 3.3: HRL feedback and control loops.

Figure 3.3:

HRL feedback and control loops. p.33
Figure 3.5: Data acquisition electronics for the charged particle detectors.

Figure 3.5:

Data acquisition electronics for the charged particle detectors. p.37
Table 4.1: Allowed channel spins and relative angular momenta for 50Cr exit channels, forcompound-state Jπ up to 52+ (elastic-channel d-wave resonances)

Table 4.1:

Allowed channel spins and relative angular momenta for 50Cr exit channels, forcompound-state Jπ up to 52+ (elastic-channel d-wave resonances) p.52
Figure 5.2: The relative probability of small spacings in the Wigner distribution.Thedashed lines indicate the values of x for which the probability of finding a smaller x are0.1%, 1%, 2%, and 5%

Figure 5.2:

The relative probability of small spacings in the Wigner distribution.Thedashed lines indicate the values of x for which the probability of finding a smaller x are0.1%, 1%, 2%, and 5% p.70
Figure 5.3: Elimination of a level in the> 100 is less than 0.0001. The mistaken fit wasdue to relatively poor resolution locally in the data

Figure 5.3:

Elimination of a level in the> 100 is less than 0.0001. The mistaken fit wasdue to relatively poor resolution locally in the data p.71
Figure 5.4: Values of the level-spacing parameters x1 sequence, andx for the Jπ =12−comparing previous and current work.

Figure 5.4:

Values of the level-spacing parameters x1 sequence, andx for the Jπ =12−comparing previous and current work. p.72
Figure 5.5: Values of the level-spacing parameters x1 sequence, andx for the Jπ =32−comparing previous and current work.

Figure 5.5:

Values of the level-spacing parameters x1 sequence, andx for the Jπ =32−comparing previous and current work. p.73
Figure 5.6: Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 12+lines represent the Poisson probability density.

Figure 5.6:

Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 12+lines represent the Poisson probability density. p.75
Figure 5.7: Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 12−lines represent the Poisson probability density.

Figure 5.7:

Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 12−lines represent the Poisson probability density. p.76
Figure 5.8: Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 32−lines represent the Poisson probability density.

Figure 5.8:

Nearest-neighbor spacing distribution P levels. The dashed lines represent the Wigner probability density, and the dottedJ(x) and integrated distribution F(x) forπ = 32−lines represent the Poisson probability density. p.77
Figure 5.9: Nearest-neighbor spacing distribution PP(x) and integrated distribution F(x)for Jπ = ( 32+, 52+) d-wave levels

Figure 5.9:

Nearest-neighbor spacing distribution PP(x) and integrated distribution F(x)for Jπ = ( 32+, 52+) d-wave levels p.78
Figure 5.10: Nearest-neighbor spacing distribution PPfor(x) and integrated distribution F(x) Jπ = ( 52−, 72−) f-wave levels

Figure 5.10:

Nearest-neighbor spacing distribution PPfor(x) and integrated distribution F(x) Jπ = ( 52−, 72−) f-wave levels p.79
Figure 5.11: Plots ofis identified as the isobaric analog of the2 2  Ex = 2762 sequence

Figure 5.11:

Plots ofis identified as the isobaric analog of the2 2 Ex = 2762 sequence p.81
Figure 5.12: Plots of γof the 3055.9-keV, 3134.8-keV, 3766.8-keV, and 4040.0-keV levels in8141.6 keV, and a fragmented pair at 8382.1 keV and 8401.8 keV are identified as the analogs2  sequence

Figure 5.12:

Plots of γof the 3055.9-keV, 3134.8-keV, 3766.8-keV, and 4040.0-keV levels in8141.6 keV, and a fragmented pair at 8382.1 keV and 8401.8 keV are identified as the analogs2 sequence p.82
Figure 5.13: Plots of7558.8-keV level is identified as the analog of the 3125.9-keV level instate at 3770.5 keV inand 7367.6 keV are identified as analog fragments of the 2890.2-keV level in 512widths at 8151.5 keV, 8177.9 keV, and 8203.0 keV comprise the fr

Figure 5.13:

Plots of7558.8-keV level is identified as the analog of the 3125.9-keV level instate at 3770.5 keV inand 7367.6 keV are identified as analog fragments of the 2890.2-keV level in 512widths at 8151.5 keV, 8177.9 keV, and 8203.0 keV comprise the fr p.83
Figure 5.14: Plots ofstate at 3977.4 keV in 51and Σ8339.3 keV is identified as the analog of the 3953.2-keV state in��levels at 8356.5 keV and 8362.0 keV are identified as the fragmented analog of the2 2  51Cr, and the pair ofπ  5 level at γγfor the mixed d-

Figure 5.14:

Plots ofstate at 3977.4 keV in 51and Σ8339.3 keV is identified as the analog of the 3953.2-keV state in��levels at 8356.5 keV and 8362.0 keV are identified as the fragmented analog of the2 2 51Cr, and the pair ofπ 5 level at γγfor the mixed d- p.84
Figure 5.15: Plots of51Cr state at 3351.1 keV, adopted as�−�, 5−, 7−23222at 7794.6 keV is identified as the analog of the2 2  π 5− level γand Σγfor the mixed f-wave sequence.The J=.

Figure 5.15:

Plots of51Cr state at 3351.1 keV, adopted as�−�, 5−, 7−23222at 7794.6 keV is identified as the analog of the2 2 π 5− level γand Σγfor the mixed f-wave sequence.The J=. p.85
Figure 5.16: Reduced width distribution and summed reduced width distribution for theJπ = 12+ sequence

Figure 5.16:

Reduced width distribution and summed reduced width distribution for theJπ = 12+ sequence p.89
Figure 5.17: Reduced width distribution and summed reduced width distribution for theJπ = 12− sequence

Figure 5.17:

Reduced width distribution and summed reduced width distribution for theJπ = 12− sequence p.90
Figure 5.18: Reduced width distribution and summed reduced width distribution for theJπ = 32− sequence

Figure 5.18:

Reduced width distribution and summed reduced width distribution for theJπ = 32− sequence p.91
Figure 5.19: Reduced width distribution and summed reduced width distribution for themixed d-wave sequence

Figure 5.19:

Reduced width distribution and summed reduced width distribution for themixed d-wave sequence p.92
Figure 5.20: Reduced width distribution and summed reduced width distribution for themixed f-wave sequence

Figure 5.20:

Reduced width distribution and summed reduced width distribution for themixed f-wave sequence p.93
Table 5.2: Proton strength functions in 50Cr(p,p0).

Table 5.2:

Proton strength functions in 50Cr(p,p0). p.95
Table 5.3 lists the calculated missing fractions fw and fs, the combined missing

Table 5.3

lists the calculated missing fractions fw and fs, the combined missing p.97
Table A.1: 51Mn resonance parameters

Table A.1:

51Mn resonance parameters p.105
Table A.2: 51Mn resonance parameters, Jπ = 12

Table A.2:

51Mn resonance parameters, Jπ = 12 p.126
Table A.3: 51Mn resonance parameters, Jπ = 12

Table A.3:

51Mn resonance parameters, Jπ = 12 p.128
Table A.4: 51Mn resonance parameters, Jπ = 32

Table A.4:

51Mn resonance parameters, Jπ = 32 p.130

References

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