· ii curriculum vitae dayne eric fratanduono was ﬁrst interest in engineering when he attended...

The Index of Refraction of Lithium Fluoride at Pressures

in Excess of 100 GPa

by

Dayne Eric Fratanduono

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Doctor Thomas R. Boehly

Professor David D. Meyerhofer

Department of Mechanical Engineering

Arts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of Rochester

Rochester, New York

2010

ii

CURRICULUM VITAE

Dayne Eric Fratanduono was first interest in engineering when he attended

Clarkson University from 2002 to 2006. In 2006 he graduated with great distinc-

tion from Clarkson University with a Bachelor of Science in mechanical engineering

and physics (dual major). Interested in the Inertial Confined Fusion (ICF) cam-

pagain at the Laboratory for Laser Energetics at the University of Rochester, he

pursued further research studies. At the University of Rochester he was awarded

the prestigious Sproull Fellowship from the fall of 2006 until the fall of 2008. He

received the Horton fellowship over the remainder of this time at the University.

In 2008, he received a Master of Science degree in Mechanical Engineering and

remained at the University of Rochester to fullfill the requires of the Doctoral

degree in Mechanical Engineering. His thesis was performed at the Laboratory

for Laser Energetics under the direction of Dr. D.D. Meyerhofer and Dr. T.R.

Boehly which focused on material properties in the high energy density regime.

Publications and selected professional presentations include:

• D.E. Fratanduono, T.R. Boehly, M.A. Barrios, D.D. Meyerhofer, J.H. Eg-

iii

gert, R.F. Smith, D.G. Hicks, P.M. Celliers, D.G. Braun and G.W. Collins.

“Refractive Index of Lithium Fluoride at Pressures up to 800 GPa.” Sub-

mitted to Phys. Rev. Letter in September, 2010.

• M.A. Barrios, D.G. Hicks, T.R. Boehly, D.E. Fratanduono, J.H. Eggert,

P.M. Celliers, G.W. Collins, and D.D. Meyerhofer. “High-precision mea-

surements of the equation of state of hydrocarbons at 1-10 Mbar using

laser-driven shock waves,” Physics of Plasmas, 17, 056307 (2010).

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, J. H.

Eggert, R. Smith, D. G. Hicks, P. M. Celliers, and G. W. Collins, “Measures

of Strain-Induced Refractive-Index Changes in Ramp-Compressed Lithium

Fluoride.” Contributed poster, OMEGA Laser Facility Users Workshop,

Rochester, NY, 28-30 April 2010.

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, R.

Smith, J. H. Eggert, D. G. Hicks, P. M. Celliers, G. W. Collins, and R.

Rygg, “Measurements of Strain-Induced Refractive Index Changes in LiF

Using Direct-Drive Ramp Compression.“ Contributed talk, 51st Annual

Meeting of the APS Division of Plasma Physics, Atlanta, GA, 2-6 Novem-

ber 2009.

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, R.

iv

Smith, J. H. Eggert, D. G. Hicks, P. M. Celliers, and G. W. Collins, ”Mea-

surements of Strain-Induced Refractive-Index Changes in Shocked LiF Us-

ing Laser-Driven Flyer Plates.“ Contributed talk, 16th APS Topical Con-

ference in Shock Compression of Condensed Matter, Nashville, TN, 28

June-3 July 2009.

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, J.

Eggert, R. Smith, D. G. Hicks, and G. Collins, ”Measurements of Strain-

Induced Refractive Index Changes in Shocked LiF Using Laser-Driven

Flyer Plates.“ Contributed talk, OMEGA Laser Facility Users Group Work-

shop, Rochester, NY, 29 April-1 May 2009.

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, D.

G. Hicks, P. M. Celliers, S. Wilks, and R. Smith, ”Optical Properties of

Materials at High Pressure Using ’Sandwich’ Targets.“ Contributed talk,

50th Annual Meeting of the APS Division of Plasma Physics, Dallas, TX,

17-21 November 2008.

• D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, D.

G. Hicks, P. M. Celliers, S. Wilks, and J. E. Miller, ”Nonequilibrium Con-

ditions in a Shock Front.“ Contributed talk, 49th Annual Meeting of the

APS Division of Plasma Physics, Orlando, FL, 12-16 November 2007.

v

ACKNOWLEDGEMENTS

I am deeply indebted to all the individuals who provided assistance through-

out my academic career. Most importantly, I would like to thank mother, father

and sister for their constant encouragement and support, without them this work

would not have been possible. I am very appreciative for the guidance and super-

vision provided to me by Dr. Tom Boehly and Professor David Meyerhofer of the

Laboratory for Laser Energetics (LLE) at the University of Rochester. The fruitful

discussions, constant encouragement and difficult questions were greatly appreci-

ated. LLE has been a wonderful institution to advance my scientific knowledge

and I have greatly enjoyed my time there. The financial support that I received

from the Laboratory of Laser Energetics and the Department of Mechanical En-

gineering was greatly appreciated.

I would like to acknowledge the close professional contacts at Lawrence Liv-

ermore National Laboratory. The ability to collaborate with leading scientists in

the field of high pressure science was inspiring. Most specifically, I would like to

vi

note the guidance I received from Dr. J. H. Eggert, Dr. D. G. Hicks, Dr. R. F.

Smith, Dr. P. M. Celliers and Dr. R. E. Collins.

Specially thanks to my colleagues at the Laboratory for Laser Energetics. Dr.

M.A. Barrios, my office mate and group member, for the long discussions. The

experimental technicians and systems scientist who were extremely helpful in the

development and execution of this experiment. Most specifically, I would like to

thank Andrew Sorce for his due diligence in ensuring that the ASBO diagnostics

was in prime operating condition for my campaigns. Steve Stagnitto for time

spent explaining the OMEGA laser system and ensuring that experiments were

performed without error. Mark Bonino and target fabrication for their willingness

to build targets with strict requirements. Lastly, Bob Boni for making time to

discuss streak cameras and various diagnostics. I am very greatly for that time,

and my understanding of those diagnostics is a direct result of his teachings.

This work was supported by the U.S. Department of Energy Office of Inertial

Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302,

the University of Rochester, and the New York State Energy Research and Devel-

opment Authority. The support of the DOE does not constitute an endorsement

by DOE of the views expressed in this work.

vii

ABSTRACT

The compression of materials to high pressure can alter their optical properties

in ways that provide insight into the resulting structural changes. Under strong

shock compression, transparent insulators transform into conducting fluids as a

result of pressure-induced reduction of the band gap and thermal promotion of

electrons across that gap. A new ramp compression technique; direct-drive shaped

ablation, is used to compress LiF to 800 GPa without generating shocks thereby

producing high pressures at significantly lower temperatures than would be created

by shock waves. In this study, ramp compressed lithium fluoride (LiF) is observed

to remain transparent to 800 GPa, pressures seven times higher than previous

shock compression experiments. The ramp compressed refractive index of LiF is

measured at pressures up to 800 GPa and depends linearly on density over this

range. This is the highest pressure refractive index measurement made to date.

The linear dependence of the refractive index and density is examined using a

single-oscillator model. This model indicates that the linear behavior is a result of

monoatomic closure of the band gap. Extrapolation of these results indicates that

viii

the band gap closure (metallization) will be greater than 5,000 GPa, well above

the Goldhammer-Herzfeld criterion for metallization (∼ 2,860 GPa). The high

metallization pressure of LiF is attributed to its large band gap and isoelectronic

counterparts that exhibit high metallization pressures.

The high pressure transparency of LiF has technical utility as an optical win-

dow for materials studies since the transparency at high pressure allows in situ

measurements of samples confined by that window. The observed transparency

and measurement of LiF refractive index enables advancement of those experi-

ments to higher pressure regimes.

Contents ix

CONTENTS

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 High Energy Density Physics . . . . . . . . . . . . . . . . . . . . . 7

1.2 Relevance of This Study . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2. Fundamentals of Fluid Dynamics . . . . . . . . . . . . . . . . . . . 15

2.1 Governing Equations of Fluid Dynamics . . . . . . . . . . . . . . 16

2.2 Remarks on the Conservation Equations . . . . . . . . . . . . . . 26

2.3 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Planar Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Eulerian and Lagrangian Coordinates . . . . . . . . . . . . . . . . 40

2.6 Shock Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.7 Hugoniot Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Contents x

2.8 Isentropic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 51

3. Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1 Interferometric Measurements Through Optical Windows . . . . . 53

3.2 Shock Refractive Index Experiments . . . . . . . . . . . . . . . . 60

3.3 Isentropic Refractive Index Experiments . . . . . . . . . . . . . . 65

3.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


4. Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Weighted Mean and Orthogonal Regression . . . . . . . . . . . . . 112

4.3 LiF Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4 LASNEX Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.5 Temperature Measurements . . . . . . . . . . . . . . . . . . . . . 129


5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.1 VISAR Window Corrections . . . . . . . . . . . . . . . . . . . . . 135

5.2 Classical Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3 Single-Oscillator Model . . . . . . . . . . . . . . . . . . . . . . . . 143

Contents xi

5.4 Metallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150


6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Appendix 182

A. Direct Drive Laser Ablation Scaling . . . . . . . . . . . . . . . . . 183

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . 185

A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

A.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

B. Weighted Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . 193

C. LiF Shock Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

C.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . 200

C.2 Unsteady Shock Breakout of an Optical Window . . . . . . . . . . 203

C.3 Steady Shock Breakout of an Optical Window . . . . . . . . . . . 209

C.4 Analysis of Shot 58815 . . . . . . . . . . . . . . . . . . . . . . . . 211

C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

List of Tables xii

LIST OF TABLES

4.1.1 Shot Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.2 Fused Silica Etalon Parameters . . . . . . . . . . . . . . . . . . 98

4.2.1 Orthogonal Fitting Parameters . . . . . . . . . . . . . . . . . . 119

5.3.1 Dispersion Parameters for the Alkali-Halides with NaCl-Type

Lattice Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4.1 Metallization Pressure for Various Materials . . . . . . . . . . . 154

5.4.2 Band Gap Energy for Various Materials . . . . . . . . . . . . . 155

A.2.1 Shot Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 186

A.3.1 Laser Ablation Scaling . . . . . . . . . . . . . . . . . . . . . . . 190

B.1.1 Apparent and True Weighted Mean Values . . . . . . . . . . . . 193

C.4.1 Analysis of Shot 58815 . . . . . . . . . . . . . . . . . . . . . . . 213

List of Figures xiii

LIST OF FIGURES

1.0.1 Fission Product Yields . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Sample EOS Target . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Mass Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Momentum Element . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3 Energy Element . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Acoustic Perturbation for a System Initially at Rest . . . . . . 31

2.4.1 Receding Piston . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.2 Sample x-t Diagram for Characteristics Analysis. . . . . . . . . 38

2.4.3 Comparison of Analytic and Numeric Techniques . . . . . . . . 40

2.6.1 Wave Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.2 Shock Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7.1 Hugoniot and Isentrope of LiF . . . . . . . . . . . . . . . . . . 48

2.8.1 Hugoniot and Isentrope Temperature Dependence of LiF . . . . 51

3.1.1 Velocity Window Correction of an Optical Window . . . . . . . 54

List of Figures xiv

3.1.2 Velocity Window Correction of a Shocked Optical Window . . . 57

3.2.1 Gas-Gun Experimental Configuration . . . . . . . . . . . . . . . 60

3.2.2 Collision Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.3 LiF Shocked Refractive Index Data . . . . . . . . . . . . . . . . 64

3.2.4 LiF Melt Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 Target Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.2 Characteristics Numbering Scheme . . . . . . . . . . . . . . . . 69

3.3.3 Pulse Shape Design . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4.1 Mach-Zhender Interferometer . . . . . . . . . . . . . . . . . . . 83

3.4.2 Sample VISAR Data. . . . . . . . . . . . . . . . . . . . . . . . 86

3.4.3 VISAR Configuration . . . . . . . . . . . . . . . . . . . . . . . 87

3.4.4 Processing VISAR Data . . . . . . . . . . . . . . . . . . . . . . 88

4.1.1 Shot 57575: VISAR Data . . . . . . . . . . . . . . . . . . . . . 99

4.1.2 Shot 57575: Backwards Characteristics Diagram . . . . . . . . . 101

4.1.3 Shot 57575: Forwards Characteristics Diagram . . . . . . . . . 103

4.1.4 Monte-Carlo Error Analysis . . . . . . . . . . . . . . . . . . . . 107

4.1.5 Shot 57575: Apparent versus True Particle Velocity . . . . . . 109

4.1.6 Shot 57575: Refractive Index . . . . . . . . . . . . . . . . . . . 110

4.1.7 Apparent versus True Velocity of All Experiments . . . . . . . . 111

4.2.1 Weighted Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

List of Figures xv

4.2.2 Ratio of the Apparent to True Velocity . . . . . . . . . . . . . . 118

4.2.3 Comparison of the Weighted Mean and Data Using Pulse Shape

RM3702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.2.4 Target Design with an Embedded Gold Layer . . . . . . . . . . 122

4.2.5 Comparison of Weighted Mean and Embedded Gold Layer Target

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.1 Refractive Index of LiF to 800 GPa . . . . . . . . . . . . . . . . 125

4.4.1 Characteristics Analysis of Shot 56113 with Shock Formation in

the LiF Window . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4.2 Comparison of Hydrocode Simulations and the Method of Char-

acteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5.1 Shot 57575: Temperature Measurement . . . . . . . . . . . . . 130

4.5.2 Shot 57577: Temperature Measurement . . . . . . . . . . . . . 132

5.2.1 Lorentz Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2.2 LiF Refractive Index at Various Frequencies . . . . . . . . . . . 142

5.3.1 LiF Refractive Index in the Optical Region . . . . . . . . . . . 146

5.3.2 LiF Single Oscillator Model . . . . . . . . . . . . . . . . . . . . 149

5.4.1 Xenon Band Broadening . . . . . . . . . . . . . . . . . . . . . . 152

A.3.1 Ablation Pressure Versus Laser Intensity . . . . . . . . . . . . . 189

List of Figures xvi

A.3.2 Shot 54944: Ablation Pressure . . . . . . . . . . . . . . . . . . 190

C.1.1 Shock Release Target Design . . . . . . . . . . . . . . . . . . . 201

C.1.2 Shot 58815: VISAR Data . . . . . . . . . . . . . . . . . . . . . 202

C.2.1 Shock Front in an Optical Window . . . . . . . . . . . . . . . . 204

C.2.2 Shock Breakout of an Optical Window. . . . . . . . . . . . . . . 205

C.2.3 Density and Refractive Index Profiles at Various Stages of Shock

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

C.4.1 Shot 58815: Velocity Profile . . . . . . . . . . . . . . . . . . . . 213

C.4.2 Shock Refractive Index Measurements . . . . . . . . . . . . . . 214

C.4.3 Shot 58815: Velocity Profile Difference . . . . . . . . . . . . . . 215

Foreword 1

FOREWORD

The author was the principal investigator (PI) for all experiments analyzed and

discussed in the body of this thesis. Chapter 4 and 5 (Experimental Results and

Discussion) is based on the publication submitted to Physical Review Letters:

D.E. Fratanduono, T.R. Boehly, M.A. Barrios, D.D. Meyerhofer, J.H. Eggert,

R.F. Smith, D.G. Hicks, P.M. Celliers, D.G. Braun and G.W. Collins, “Refractive

Index of Lithium Fluoride at Pressures up to 800 GPa,” submitted on September

24th, 2010. The author of this thesis performed all the analysis described herein

and in that publication. LASNEX simulations (Section 4.4) were performed by

David Braun at Lawrence Livermore National Laboratory (LLNL) to verify the

consistency of the analysis technique outlined here. The remaining analysis and

text in this thesis was developed by the author and the guidance of colleagues at

LLNL.

The experiment discussed in appendix C was performed by Hye-Sook Park of

LLNL. Analysis, derivation and discussion using that data was performed solely

by the author.

1. Introduction 2

1. INTRODUCTION

The first test of a nuclear weapon occurred on July 16th, 1945 at the Alam-

ogordo Bombing range in a remote part of New Mexico,1 beginning the scientific

exploration of the field of high energy density physics.2 This detonation increased

scientific interest in atomic physics, fission, and fusion. Further testing ensued,

and after the Second World War, the United States deemed the development

of nuclear weapons essential to national security. For the next fifty years, the

Unites States and the former Soviet Union carried out aggressive campaigns to

increase their understanding of nuclear weapons. During this period, the United

States and the former Soviet Union performed 1,030 and 715 nuclear tests, re-

spectively,1,3 accounting for over 80% of the world’s nuclear detonations. Tests

were conducted to gather information on nuclear devices, nuclear phenomena, and

material properties at extreme conditions.

Eight countries have since developed and detonated nuclear weapons (United

States-1945, Soviet Union-1949, Great Britain-1952, France-1960, China-1964,

India-1974, Pakistan-1998 and North Korea-2006).4 During this period, numer-

1. Introduction 3

ous nuclear agreements were proposed among these countries and others with the

goal of halting or minimizing detonations. In 1954, the first agreement to stop

nuclear testing was proposed by the prime minister of India. Three years later

(1957), President Eisenhower announced a moratorium on nuclear testing. At

this time, the United States, Great Britain and the former Soviet Union (the only

countries in possession of nuclear weapons) agreed to halt testing.1 However in

1960, France detonated its first nuclear device causing the Soviet Union to resume

testing, with the United States and Britain following. In the ensuing years, the

United States, Soviet Union and Great Britain along with other non-nuclear states

agreed to the Limited Test Ban treaty outlawing the testing of nuclear weapons

in the atmosphere, outer space and underwater.5 It was not until 1996 that an

agreement banning all nuclear testing was adopted by a majority of nations and

nuclear detonations nearly ended.

The Unites States signed the Comprehensive Nuclear Test Ban Treaty in

1996,6,7 with their last nuclear explosion occurring four years prior.4 The United

States has not yet ratified the treaty, but is currently abiding by its provisions.

Great Britain, China, France and Russia followed along with seventy other non-

nuclear states. These nations agreed to cease nuclear explosions and refrain from

causing or encouraging other nations to carry out nuclear testing. This ban on

nuclear testing caused the United States to pursue other forms of scientific re-

1. Introduction 4

Fig. 1.0.1: Fission product yields for thermal nuclear fission of U23592 . For

each U23592 atom that undergoes fission, the probability of by-products

(in mass number) is shown.10

search to ensure the viability of their nuclear stockpile. The most recent is the

Stockpile Stewardship and Management Program (SSMP)8 whose purpose is to

maintain and enhance the safety, security and reliability of the United States nu-

clear weapons stockpile without conducting underground testing.9 The specific

aims of the SSMP are to establish a critical understanding of the material science,

hydrodynamic and hydro-nuclear phenomenon in the high energy density regime

(defined as material pressures greater than 100 GPa).

Coincident with these events, research in large-scale fission and fusion energy

production began. In 1934 Oliphant et al.11 discovered the fusion reaction while

findings by Hahn and Strassmann in 1939 demonstrated the first fission reaction.12

1. Introduction 5

These processes illustrate that large scale energy production is possible through

nuclear means. Nuclear fission is a process in which nuclear decay is initiated by

a neutron.13 A neutron colliding with U235 results in the nuclear reaction,

n + U23592 → U236

92 → Ba14156 + Kr92

36 + 3n(170 MeV), (1.1)

where fission fragments (Ba14156 and Kr92

36) are created.∗ Equation 1.1 shows that a

single neutron can be used to initiate decay in an U235. The advantage of fission is

that the neutron (neutral particle) can easily penetrate the electron cloud of the

Uranium atom. The reaction generates three neutrons that can then be used to

initiate a chain reaction leading to a self sustaining energy source. These reactions

are easily initiated and sustained explaining their usefulness for power generation.

Nuclear fission has a number of difficulties and environmental concerns. As

figure 1.0.1 shows, various fission fragments are created during Uranium fission.

A majority of these fission fragments are highly unstable (radioactive) and their

environmental impact is of concern.14 Proper treatment, storage and segregation

from the environment is difficult. Significant engineering technologies are required

to control the chain reactions that occur during the fission process. Over 400

nuclear reactors have been developed with two reactor meltdowns occurring.15 It

∗ For this reaction a variety of nuclear fragments form. The probability that nuclear fragmentsfound in equation 1.1 form is less than 10 % (other fragments may results). The probability offragment formation is shown in figure 1.0.1.

1. Introduction 6

is because of these concerns that the safety and viability of nuclear fission reactors

are questioned.

Nuclear fusion represents a much safer form of nuclear energy production. A

typical sustainable fusion reaction uses deuterium and tritium,

D + T → He4(3.5 MeV) + n(14.1 MeV). (1.2)

This reaction is significantly more difficult to produce than fission. For deuterium

and tritium to undergo a nuclear reaction, their nuclei must come within approx-

imately a nuclear diameter of one another overcoming the electro-static repulsive

force. Significant energy is required to fuse the elements together. The bene-

fits of fusion reactions, when compared with their fission counterpart, is that the

nuclear by-products are an inert gas and a neutron. The neutron carries away

the majority of the energy generated during the process and could be used for a

power plant. Most importantly, fusion energy production exceeds that of fission.

The fission reaction of equation 1.1 releases 207 MeV per reaction or 0.9 MeV per

nucleon. In the fusion reaction, 17.2 MeV is generated or 3.5 MeV per nucleon.

The energy density of D and T is higher than that of U235. The economical and

environmental benefits of nuclear fusion motivates research in this field.16

In the 1950’s various confinement techniques were developed to initiate fusion:

1. Introduction 7

Princeton University invented the magnetic confinement stellerator,17 Los Alamos

National Laboratory created the magnetic pinch18 and the Soviet Union developed

a toroidal model of a magnetic thermonuclear reactor that later evolved into the

tokamak.19 These magnetic confinement devices aimed to create the high density

and temperatures required in fusion reactions by compressing and heating plasmas

confined by magnetic fields. Initial attempts were unsuccessful due to plasma

instabilities.20

With the development of the laser in 1960,21 the ability to generate high inten-

sities created an additional pathway to controlled fusion. The concept of inertial

confinement fusion was developed in which a fuel capsule is compressed to high

temperature and pressure conditions using laser energy. Programs, such as the

Laboratory for Laser Energetics were developed to advance this field. Due to

the high pressures and material densities required to initiate fusion, the iner-

tial confinement fusion community began scientific research into high pressure,

temperature and density matter.22 Consequently, inertial confinement fusion and

SSMP become intertwined due to their common interests in high energy density

physics.

1.1 High Energy Density Physics

High energy density2 has been defined as energy densities greater than 1011J/m3

which typically corresponds to material pressure exceeding 100 GPa. Recent ad-

1. Introduction 8

vances in driver techniques (lasers, particle beams, Z-pinch generators, magnetic

flyer plates) have produced greater heating and compression enabling scientists

to explore and develop a fundamental understanding of matter at ever increasing

energy density. These advances have initiated new discoveries that have benefited

nuclear initiatives (Stockpile Stewardship and Management Program) and inertial

confinement fusion while initiating new innovations and ideas.2

High energy density physics (HEDP) encompasses many scientific fields (not

limited to Astrophysics, Laser Plasma Interactions, Fluid Dynamics, Condensed

Matter, and Equation of State Physics). This work is most relevant to the field

of Equation of State (EOS) Physics that uses compression experiments to study

material properties at high pressure. EOS data are needed for hydrodynamic sim-

ulations of inertial confinement fusion, to confirm theoretically predicted states

of matter, and to aid in the understanding of solid-state dynamics at high strain

rates.23 In astrophysics, understanding the evolution of a giant planet requires

analysis of the thermodynamic and transport properties of compressed hydrogen

and other materials at pressures greater than 100 GPa. Geophysicists understand-

ing of the earths interior requires knowledge of the phase diagram of iron at high

pressure and determination of its melt-line (in the pressure/temperature plane).24

Hydrodynamic codes solve the three conservation equations (mass, momentum

and energy) which requires one additional equation: EOS or constitutive relation

1. Introduction 9

for closure.25 Understanding material behavior in the high energy density regime

assists these efforts.

Many techniques are employed to compress materials (i.e. isothermal com-

pression, isentropic compression, isenthalpic compression, and shock wave com-

pression). HEDP experiments have primarily used shock wave compression and

isentropic compression. Shock waves are generated in materials through laser ab-

lation, flyer plates, high energy explosives and nuclear explosions.26,27 Each tech-

nique transfers momentum to the surface of the target, inducing stress waves that

coalesce into a shock wave. A shock wave is a discontinuity in density and pressure

that carries energy as it propagates through a medium. The wave, propagating

faster than the local sound speed, causes a “step-like” change in the properties

of the material and the flow variables on each side of the discontinuity. These

changes are related by the Rankine-Hugoniot conditions: the conservation equa-

tions for mass, momentum and energy across the shock. These relations include

terms for the particle and shock velocities. In many experiments, measurements of

these velocities is essential for material studies. Entropy increases across the shock

front, indicating an irreversible, dissipative process that will cause an increase in

temperature.27

Isentropic compression is a reversible process that can be achieved through the

use of diamond anvil cells,28 magnetic confinement,29 and laser ablation.30 This

1. Introduction 10

technique requires that material be gradually compressed to maintain constant

entropy, creating a continuous pressure and density profile within the material.

Quasi-isentropic compression is an approximation to an isentrope that minimizes

the entropy increase. These methods differ from shock wave compression in that

many states can be reached at a nearly constant temperature during a single ex-

periment.27 Shock and ramp compression are two techniques that are commonly

used to explore the phase space of materials in the high energy density regime.

Ramp compression to greater than 100 GPa is a relatively new technique that

provides access to states of matter previously inaccessible to laboratory experi-

ments.

1.2 Relevance of This Study

EOS experiments are important to a variety of fields as they enable one to cor-

rectly determine the phase space of materials, close the fluid dynamic system of

equations, and provide insight into other high pressure phenomena. A typical ex-

periment involves compression of a material and the measurement of the resulting

material or compression wave velocity. A sample target geometry for laser driven

HEDP experiments is shown in figure 1.2.1a. Laser ablation drives compression

waves that traverse the sample and reach the rear surface. Diagnostics measure

the velocity in the target and or at the rear surface to determine the compressed

material properties.

1. Introduction 11

Fig. 1.2.1: Sample EOS target geometry for high pressure experiments

In some cases, optical windows are used to observe compressed materials that

are confined by a “window” through which that material is observed. An example

of this style of target is shown in figure 1.2.1b. Optical windows are used be-

cause they suppress complicated wave interactions when studying materials that

undergo phase transformation,31 and confine fluids/melted materials enabling in

situ measurements.32 The simplest example of the need for high pressure windows

can be understood by imagining a target design in which the EOS of an ideal gas

is to be studied. Due to experimental constraints, the gas must be confined in a

compression cell. Velocity interferometry is used to determined the characteristics

of the compressed gas. To view the gas, a transparent optical window is required.

As the cell compresses, the material properties and the refractive index of that op-

tical window change. If these changes are not accounted for, systematic errors in

the velocity interferometer measurements will be introduced. Therefore, knowl-

edge of the high pressure behavior of an optical window is required for precise

interferometry measurements.

1. Introduction 12

Lithium Fluoride (LiF) is of interest for shock and ramp compression exper-

iments because its transparency at high pressure allows in situ particle-velocity

measurements at the sample/window interface and for spectroscopic optical mea-

surements of samples.33 Recent experimental developments using ramp compres-

sion have created a need for a transparent window, whose refractive index is known

at high pressures (≫ 100 GPa). Before this work, LiF had been characterized only

to 115 GPa.34

Velocities are measured in dynamic compression experiments through opti-

cal windows using interferometry.33,35 Knowledge of the optical properties of the

compressed window is required to properly correct velocity measurements.36,37 A

new technique to determine the refractive index of materials at high pressures is

presented. This study fully characterizes ramp compressed LiF windows to pres-

sures seven times higher than previous studies (800 GPa). This will enable the

scientific community to extend measurements of various materials to pressures

previously inaccessible. The results suggest that this window will remain trans-

parent at significantly higher pressures by linking the dependence of the refractive

index on the density to an effective single-oscillator model. This model implies

that the band gap closes with increasing density and predicts a high metallization

pressure for LiF (∼ 4200 GPa), suggesting that LiF will be a vital optical window

for extreme pressure experiments.

1. Introduction 13

1.3 Thesis Outline

Chapter 1 described the history and development of high energy density physics

as an emerging scientific field. It discussed the needs of this research area to vari-

ous scientific communities. Most specifically, the area of EOS physics, the subject

of this work. The relevance of this body of knowledge will prove useful to experi-

mentalists and theorists throughout this field. Fluid dynamic and thermodynamic

equations central to the body of this work are derived in Chapter 2. To familiarize

the reader with these equations and their applications, solutions to various systems

are presented. The fundamental differences between shock and ramp compression

are discussed. Chapter 3 discusses previous techniques used to measure the high

pressure refractive index of shocked LiF. Equations relating measured observables

to isentropically compressed refractive index are discussed. This relationship is

used to design an experiment to measure the ramp compressed refractive index of

LiF. Pulse shapes, drive specifications and target constraints are identified. Di-

agnostics necessary for this study are then discussed. The LiF ramp compressed

refractive index analysis and results are discussed in Chapter 4. Twenty-four ex-

periments are examined in detail. Chapter 5 discusses the implications of these

measurements on optical interferometry experiments and the dependence of the

refractive index and density is examined using a single-oscillator model. The

1. Introduction 14

metallization pressure for LiF is predicted using this model and compared to the

metallization pressure of other materials. Chapter 6 summarizes and concludes

the findings of this study.

Three appendices are provided. Appendix A discusses the laser intensity to

ablation pressure scaling law for ramp compression of diamond targets using a

351 µm laser. Appendix B is a tabular appendix that contains the values of the

apparent and true particle velocity determined from the weighted mean of all

experiments. Appendix C discusses a new experimental technique to measure the

shocked refractive index of an optical window.

2. Fundamentals of Fluid Dynamics 15

2. FUNDAMENTALS OF FLUID

DYNAMICS

Fluid mechanics is the study of fluids and the forces imparted upon them.

It is often divided into two parts; the study of stationary fluids (fluid statics)

and fluids in motion (fluid dynamics). As with all systems, there are conservation

laws governing the dynamic and static processes. In principle, there are an infinite

number of fluid dynamic conservation equations, where each equation depends on

a higher order moment (i.e. the conservation of mass equation depends upon a

momentum term).

Due to the infinite equations governing fluid dynamics, assumptions are re-

quired to close the system of equations. These assumptions may deal directly

with the characteristics of the fluid flow or with the thermodynamic properties.

Fluid mechanics is most commonly defined by three laws of mechanics (conserva-

tion of mass, momentum and energy), a thermodynamic state relation, and the

boundary conditions specific to that system.38 Using these relations, the state


of a moving fluid with known thermodynamic properties (velocity, density, and

pressure) can be defined as a function of position and time.27 In the following sec-

tions, the conservation equations for fluid dynamics are derived. Fluid derivations

identify the need for a thermodynamic relation to complete the set of equations.

Specific fluid systems (acoustic perturbations, isentropic flow, and shock com-

pression) are discussed and analyzed using the derived equations. The method of

characteristics, a solution technique central to the body of this work, is introduced

and employed to solve the case of isentropic flow.

2.1 Governing Equations of Fluid Dynamics

Derivation of the conservation equations for fluid dynamics are taken from the

Hydrodynamic Instability lectures given by Professor Riccardo Betti.39 While dis-

crete elements are referenced, they contain a large number of atoms or molecules.

These derivations assume that discrete elements within the flow are macroscopic

and the fluid can be regarded as a continuous medium.40

2.1.1 Conservation of Mass

Consider a volume (V ) enclosed by a surface (S) as shown in figure 2.1.1. The

mass (M) of the fluid element is defined as the integral of the density (ρ) over the

volume,

M =

∫

V

ρdV. (2.1)


Fig. 2.1.1: Mass Element

The mass of the fluid element may also be defined as a function of the fluid

traveling through the surface. Assume a surface element dS, as shown, with

velocity (~U ·~n) flowing through the surface element. ~U is the velocity permeating

the surface and ~n represents the normal vector to that surface (positive direction

defined as outward on ~S). The discrete mass (dm) flowing through dS in time

interval (dt) is

dm = ρ(~U · ~n)dSdt. (2.2)

Total mass (dM) leaving the volume becomes

dM = −∮

S

ρ(~U · ~n)dSdt. (2.3)


Fig. 2.1.2: Momentum Element

This defines the mass flux traveling through the surface. Combining equation

(2.3) and the time derivative of equation (2.1) gives

∫

V

∂ρ

∂tdV = −

∮

S

ρ(~U · ~n)dS, (2.4)

for a fixed volume. This represents the integral form of mass conservation. Using

the divergence theorem, equation (2.4) is reduced to the differential form,

∂ρ

∂t+ ∇ · ρ~U = 0. (2.5)

The mass conservation equation (2.5) relates the element density and the momen-

tum. The single equation with two unknowns (ρ, U) is not a closed system. The

momentum term (ρ~U) illustrates the need for a higher order moment conservation

equation.


2.1.2 Conservation of Momentum

Consider a discrete fluid element in one dimension as shown in figure 2.1.2.

The element is defined such that the mass within the volume remains constant

(i.e. a Lagrangian fluid element). The element, with fixed cross section (A) and

variable length (dx), is free to move along the x-axis in time. The discrete mass

(dM) of the element is defined as the density (ρ) times the element volume,

dM = ρ(t)dx(t)A. (2.6)

Applying Newton’s second law of motion (∑

F = ma) to the fluid element

(dM), the force (F ) is expressed as

∑

F = dUdt

(x(t), t)ρ(t)dx(t)A (2.7)

The acceleration (a) is written in terms of the element velocity (a = dUdt

(x(t), t)).

Expanding equation 2.7 the total derivative gives

∑

F =(∂U

∂t+ U

∂U

∂x

)

ρ(t)dx(t)A. (2.8)

If a pressure (P ) is applied to the two surfaces shown in figure 2.1.2, equation


(2.8) maybe rewritten substituting the relation for the applied pressure,

(

P (x − dx

2) − P (x +

dx

2))

A =(∂U

∂t+ U

∂U

∂x

)

ρ(t)dx(t)A. (2.9)

The fundamental theorem of calculus reduces equation (2.9) to the conservation

of momentum equation for one dimension,

ρ(∂~U

∂t+ U

∂U

∂x

)

= −∂P

∂x. (2.10)

This derivation assumes that the applied pressure is the only external force act-

ing on the element. In three dimensions, equation (2.10) transforms to a three

equation set with five unknowns,

ρ(∂~U

∂t+ (~U · ∇)~U

)

= −∇P + ρ~g, (2.11)

where the force of gravity (g) is included for clarity and in most applications can

be neglected.41 Combining the conservation of mass and momentum equations

gives a four equation set with five unknowns (ρ, ~U and P ). The gradient of

pressure term (∇P ) describes the work done on the element. As observed in the

derivation of mass conservation equation, we see that conservation of momentum

depends on a higher order energy term, P .


Fig. 2.1.3: Energy Element

2.1.3 Conservation of Energy

The first law of thermodynamics states that in an isolated system the total

energy remains constant. The energy is free to change forms (e.g. transformation

of kinetic to potential energy), while obeying this principle. Various mechanisms

exist by which energy is transformed within a closed system and the conservation

of the energy equation relates these mechanisms to the total energy of the system.

The transfer of energy is a dynamic process requiring rate equations to describe

energy conservation. The total energy of a system is described by the energy

stored in the form of internal energy (e) and kinetic energy(12ρU2). The rate of

change of energy stored per unit volume is simply ∂∂t

(ρe + 12ρU2). This represents

the total energy within the discrete volume. The energy of the element can be

changed by mass flow into and out of the volume, heat transfer across the surface,

work performed on and energy generated within the volume.

The net rate of energy flow across a unit volume is affected by the rate at


which mass flows in and out of a fluid element. As the mass enters and leaves the

element, energy of the form (e+ 12U2) is deposited or removed. Thermal energy is

transfered by the movement of particles from one region to another and is termed

convection. For the fluid element shown in figure 2.1.3, the convection across

the y + ∆y/2 face per unit time is (ρUy(e + U2

2))y=∆y/2∆x∆z. The fundamental

theorem of calculus reduces the rate of flow in the y-direction per unit volume

to ∂∂y

(ρUy(e + U2

2)). A similar derivation follows for the other directions leading

to a rate of energy flow per unit volume: −∇ · (ρ~U(e + U2

2)). The rate of heat

transfer follows a similar derivation. The heat transfer is defined as the transfer

of thermal energy through conduction between neighboring elements. The heat

flux (q) describes the transfer of energy between element faces of figure 2.1.3 and

the rate of heat transfer per unit volume is −(∇ · ~q).

The two remaining energy terms to be derived are work performed on and

energy generated within the volume. Work is performed on a element volume

through gravitational effects or external forces. The rate of work performed per

unit volume by gravity is ρ~g · ~U . Work can be performed by the surface forces

acting on the body or −P∇ · ~U . Heat generation occurs within an element if a

heat source is present. This is defined as the power released per unit volume per

unit time or Q.


Combing all terms, the conservation of energy equation for a fluid element is

∂

∂t(ρe +

1

2ρU2) = −∇ · (ρ~U(e +

U2

2)) − P∇ · ~U −∇ · ~q + ρ~g · ~U + Q. (2.12)

To determine a relationship for the pressure, required by the conservation of mo-

mentum equation, two unknowns have been introduced (internal energy and heat

flux). This assumes the internal heat generation is known. The pressure and inter-

nal energy can be related through thermodynamic state relations. However, the

heat flux has created the need for an additional conservation equation. It is easy

to see how a recursion relation exists for the equations governing fluid mechanics

(“n” unknowns with “n-1” equations) where each equation depends on a higher

order momentum. To close the system of equations, thermodynamic properties

of the fluid and assumptions regarding the heat flux are required. In the simplest

case, the heat flux may be neglected or approximated using Fourier’s Law of ther-

mal conduction (q ≈ −k∇T ). The temperature (T ) and thermal conductivity (k)

are related to the thermodynamic properties of the fluid that are required for a

complete set of equations.

2.1.4 Thermodynamic State Relations

Thermodynamics characterizes the relation among measurable independent

quantities that describe the current state of the system. These parameters are


path independent and do not characterize dynamic changes to the system. The

basic laws of thermodynamics are used to determine the mathematical relations

among these independent quantities. The thermodynamic state describes the set

of values that must be specified to exactly reproduce the current state of the

system.42

Thermodynamic properties are defined by state variables. The interrelation

among the variables is termed the state relation or equation of state (EOS). The

most commonly known EOS is the ideal gas law (PV ≈ T ), which defines the rela-

tionship among pressure (P ), volume (V ) and temperature (T ). In many systems,

an analytic representation of the state variable interdependence is unknown as is

the number of quantities required to specify the state. The number of state vari-

ables required to delineate a system is determined by experimental observations

and fundamental thermodynamic laws governing the system.42

The relation between state variables is determined from the laws of thermody-

namics that describe the transport of heat and work in a closed system. The first

law states that energy is conserved within a system. A change in internal energy

(U) is related to the work (w) done on the system (denoting the negative sign)

and the addition of heat (q),

dU = δq − δw. (2.13)


Fundamentally, work and heat addition are path dependent processes and are not

state variables. This is noted in the notation as δ where as changes state relations

are expressed as ∆U = Uf − Ui due to their path independence. Equation (2.13)

can be described in terms of state relations by assuming that the work done on

a system is reversible and is equivalent to the pressure times a discrete change in

volume (δw = P∆V ). The second law of thermodynamics (entropy increases in

time) states that entropy increase is related to the discrete heat transfer divided

by the temperature (∆S = δq/T ).43 The relation between the state variables

defined by the first and second law of thermodynamics is

∆U = T∆S − P∆V. (2.14)

This equation may be rewritten such that state relation depends on two state

variables U(S, V ),

∆U =

(

dU

dS

)

V

∆S +

(

dU

dV

)

S

∆V. (2.15)

Further state relations are determined using similar thermodynamic principles

and arguments. The defining relations for the enthalpy (H), free energy (F ) and


Gibbs free energy (G) are

∆H = T∆S + V ∆P, (2.16)

∆F = −S∆T − P∆V, (2.17)

∆G = −S∆T + V ∆P. (2.18)

For a thermodynamic system, there are eight defining parameters with four equa-

tions (2.15-2.18). To define a thermodynamic system, four parameters must be

specified, allowing the calculation of all other thermodynamic quantities using

the equations above. For instance, common numerical equations of state, such as

Sesame44 or Quotidian Equation of State,45 define the internal energy and pres-

sure as functions of density and temperature (U(ρ, T ), E(ρ, T )). Such a definition

is sufficient in describing the thermodynamic properties of a system and the in-

terdependence of the formulas derived can be used to determine all of the other

parameters.

2.2 Remarks on the Conservation Equations

Fluid mechanics is described by the three laws of fluid mechanics, the thermo-

dynamic state relation and the appropriate boundary conditions. Using Fourier’s

approximation to close the system of equations leaves five fluid equations and


seven unknowns. The thermodynamic equation of state provides two additional

equations closing the set and determining the fluid flow.

The hydrodynamic equations can further be simplified using assumptions spe-

cific to a known system. For instance, many classifications of fluid flow exist,

with each classification accompanied with an assumption that may simplify the

equation set. Such assumptions are steady flow (dUdt

= 0), inviscid flow (as was

assumed in the derivation of 2.11), and incompressible flow (∇ · ~U = 0). Steady

flow assumes that conditions may vary in spatial location but are constant in time,

inviscid flow assumes that the viscosity is zero, and incompressible flow assumes

that changes in pressure do not effect the volume of the fluid. Each assumption

reduces the complexity of the hydrodynamic equation set. If further is known

about the fluid flow, assumptions in terms of the state variables maybe made

(e.g. isentropic, isobaric, adiabatic etc...). For instance, isentropic flow occurs

when changes in the fluid system are small and gradual. For such a flow, en-

tropy is assumed constant and the state relation, 2.14, is reduced to a simple form

∆U = −P∆V .

2.3 Acoustic Waves

To gain a general understanding of the governing equations of fluid dynamics,

consider a small sound disturbance centered on the origin in a one-dimensional

spatially uniform system. Changes in the density (ρ), velocity (U) and pressure


(P ) are small and expressed as

ρ = ρ0 + ρ, (2.19)

U = U0 + U , (2.20)

P = P0 + P , (2.21)

where the subscript, 0, and tilde denote the initial state of the system and the

infinitesimal perturbation, respectively. The mass and momentum conservation

equations for the acoustic waves are determined by linearizing equations (2.5) and

(2.11),

∂ρ

∂t+ ρ0

∂U

∂x= 0, (2.22)

ρ0∂U

∂t+

∂P

∂x= 0, (2.23)

assuming the system is initially at rest (U0 = 0). The derived equations relate

the density, velocity and pressure of the system and this two equation set is not

sufficient to close the system of equations; requiring a thermodynamic relation.

It has been shown that a thermodynamic state variable can be described by

two other parameters. The pressure may then be described by the density and


entropy (P (ρ, S)):

∆P =

(

∂P

∂ρ

)

S

∆ρ +

(

∂P

∂S

)

ρ

∆S. (2.24)

For acoustic disturbances, the changes in fluid properties are small and develop

gradually over time. The entropy is assumed to be constant and the dependence

on entropy in equation (2.24) can be removed,

∆P =

(

∂P

∂ρ

)

S

∆ρ. (2.25)

As previously stated, state relations are path independent and equation (2.25)

can be simplified to

P =

(

∂P

∂ρ

)

S

ρ. (2.26)

Combining this result, with the conservation of mass and momentum equations

(2.22, 2.23) for small perturbations yields the one dimensional wave equation,

∂2U

∂t2= C2

E

∂2U

∂x2, (2.27)

∂2ρ

∂t2= C2

E

∂2ρ

∂x2, (2.28)

where

C2E =

(

∂P

∂ρ

)

S

, (2.29)


and CE is assumed to be constant.

General solutions to the wave equations are written as the superposition of

two particular solutions with the waves propagating in opposite directions.46 The

solutions are

U = f1(x − CEt) − f2(x + CEt), (2.30)

ρ =ρo

CE

f1(x − CEt) +ρo

CE

f2(x + CEt), (2.31)

(2.32)

where f1 and f2 are arbitrary functions determined from the initial conditions.27

The solutions illustrate that the disturbances travel at velocity of CE which is

formally referred to as the sound speed.

Consider a small perturbation for a system that is initial at rest (figure 2.3.1).

The perturbation is shown in blue and at some time later (red), where the per-

turbation has split into two separate waves traveling in opposite directions. The

trajectory ( dtdx

) that the waves trace in x-t space is referred to as the pathline or

characteristic (shown in black) and is inversely related to the sound speed (CE)

of the material. In a more rigorous approach, the next section considers a system

in which the disturbances are not small, but entropy remains constant.


Position (x)

Tim

e (t

)

Initial PerturbationWave SplittingCharacteristics

Fig. 2.3.1: Acoustic perturbation for a system initially at rest.

2.4 Planar Isentropic Flow

The equations of conservation of mass (2.5) and momentum (2.11) are used

to determine the wave equation for one-dimensional isentropic flow. The acoustic

wave solution (Section 2.3) assumes that entropy is constant and that the pertur-

bations are small. However, in the case of planar isentropic flow, no constraint

is placed upon the size of the perturbation. The conservation equations cannot

be linearized (as was performed in the acoustic wave analysis). The conservation

equations are reduced to the simple form,

[

∂U

∂t+ (U ± CE)

∂U

∂x

]

± 1

ρCE

[

∂P

∂t+ (U ± CE)

∂P

∂x

]

= 0, (2.33)

where the sound speed (CE) need not be constant.


As was shown for acoustic waves, disturbances propagate along specific path-

lines or characteristics. For one dimensional isentropic flow, there are two families

of characteristics that propagate in opposite directions and the solutions to equa-

tions (2.33) can be found along these characteristics. The derivatives can be

defined along the forward and backwards propagating characteristics

(

dx

dt

)

±

= U ± CE or

(

d

dt

)

±

=∂

∂t+ (U ± CE)

∂

∂x. (2.34)

where (±) indicates the forward and backwards traveling waves. Substituting this

assumption into equation (2.33) gives

dU

dt± 1

ρCE

dP

dt= 0 and

(

dx

dt

)

±

= U ± CE along C±, (2.35)

where C± are the positive and negative characteristics. The pressure, density

and sound speed are uniquely related through the thermodynamic relation C2E =

(dP/dρ)S. A complete solution to equation (2.35) can described by two functions

in the x,t plane; the velocity (U) and one of the three thermodynamic relations.27

Integrating equation (2.35) gives

J± = U ±∫

dP

ρCE

= U ± σ, (2.36)


where J± are arbitrary constants commonly referred to as the Reimann invariants

for isentropic flow. If we assume a perfect gas with constant specific heats, the

velocity and sound speed of the gas can be expressed in terms of the Reimann

invariants. Assume that

P = ρ0ργ and C2

E = γρ0ργ−1, (2.37)

where γ is the ratio of specific heats or the isentropic exponent. The Reimann

invariants are defined as

J± = U ± 2

γ − 1CE. (2.38)

The velocity and sound speed may then be expressed in terms of these invariants,

U =J+ + J−

2and CE =

γ − 1

4(J+ − J−) (2.39)

as can the characteristic equations

C± :

(

dx

dt

)

±

=γ + 1

4J± +

3 − γ

4J∓ for J± = constant. (2.40)

It is important to note that the Reimann invariants possess a very important

property: since J+ is constant along the C+ characteristics, dxdt +

depends solely on

the J− invariant, and vise versa. Therefore, if J− is constant everywhere, dxdt +

too is


constant. This important property is fundamental to understand the propagation

of characteristics. From this approach, we see that if the Reimann invariants are

known, all other parameters can be determined. This general approach is applied

to the simple case of a receding piston to demonstrate the power of this technique

(taken from the work of Zel’dovich27). Two separate solutions (analytical and

numeric) of this example are discussed in the following section.

2.4.1 Receding Piston

Imagine a one-dimensional semi-infinite system, in which a perfect gas with

constant specific heats occupying infinite half space (x > 0) is constrained by a

piston at (x = 0). Initially (t < 0), the system is at rest with constant density,

pressure and sound speed. At t = 0, the piston is withdrawn with velocity (w =

−U0(1 − e−t/τ )) where U0 and τ are positive, real constants (figure 2.4.1). The

piston path (X(t)) is described by the integral of the piston velocity,

X(t) =

∫ t

0

w dt = −U0τ

[

t

τ− (1 − e−

tτ )

]

. (2.41)

Analytic Solution

Characteristics propagating in the negative direction originate from the undis-

turbed region. Negative characteristics cannot emanate from the piston front

because they are bounded by that surface. For this reason, the invariant J− is


Fig. 2.4.1: Semi-infinite ideal gas, occupying half space (x > 0), is

bound by a piston at x = 0. At t = 0 the piston is withdrawn from the

gas and the corresponding fluid flow is determined.

constant over all of the (x,t) space, greatly simplifying the problem. As shown

previously, the slopes of the C+ characteristics are affected only by the J− invari-

ants. In this case, the C+ characteristics will be straight lines due to the constant

values of the J− invariants. The fluid velocity and sound speed are related using

this invariant,

J− = U − 2

γ − 1CE = constant = − 2

γ − 1CE0, (2.42)

U =−2

γ − 1(CE0 − CE), (2.43)

CE = CE0 +γ − 1

2U. (2.44)

Substituting equation (2.44) into the relation for the slope of the positive

characteristic (2.35) gives

(

dx

dt

)

+

= U + CE = CE0 +γ + 1

2U, (2.45)


and upon integration

x = (CE0 +γ + 1

2U)t + f(U). (2.46)

f(U), a constant of integration, is determined by solving equation (2.46) at the

piston interface. The arbitrary function defined at that location is

f(w) = X(t) − [w + CE(w)]t, (2.47)

where w denotes the velocity at the piston interface. The sound speed is de-

termined from the negative Reimann invariant (J−) or equation (2.44) which,

expressed in terms of w, becomes

CE(w) = CE0 +γ − 1

2w. (2.48)

Substituting this relation into equation (2.47) gives

f(w) = X(t) −[

w + CE0 +γ − 1

2w

]

t. (2.49)

The time dependence is removed using the expressions for the piston velocity


(t = −τ ln(1 + wU0

)) and position (X(t) = −U0τ [ tτ− (1 − e−

tτ )]),

f(w) = −wτ + τ

[

CE0 +γ + 1

2w + U0

]

ln

(

1 +w

U0

)

. (2.50)

The transcendental equation for velocity as a function of space (x) and time (t)

becomes

x =

[

CE0 +γ + 1

2U

]

t + −Uτ + τ ln

(

1 +U

U0

)[

U0 +γ + 1

2U + CE0

]

, (2.51)

which is only valid in the interval of X(t) < x < CE0t. CE0t is the lead charac-

teristic that represents the interface between the stationary gas and the gas set

in motion by the receding piston. Since the velocity has been determined as a

function of space and time, the corresponding relation for the sound speed may

be calculated. All other parameters are determined from the assumed EOS.

Numerical Solution

In a similar approach, the characteristics are calculated and propagated in

the (x,t) plane to determine the fluid flow. The Reimann invariants (J±) are

determined from the boundary conditions of the problem. As previously stated,

the J− invariant is constant everywhere and J+ invariant is determined from the


Fig. 2.4.2: Sample x-t diagram for characteristics analysis. Reimann

invariants J+ and J− are show in blue and red, respectively. The piston

trajectory is shown in black. The intersection of the solid blue and red

lines is determined using the boundary conditions and the slope of the

characteristics.


piston velocity,

J− = − 2

γ − 1CE0, (2.52)

J+ =2

γ − 1CE0 + 2w. (2.53)

Using equation (2.39), the velocity and sound speed at the intersection of ev-

ery Reimann invariant is determined. The difficulty in this technique arises in

determining the location and time of each intersection. The trajectory of each

characteristic is determined from

(

dx

dt

)

±

= U ± CE0. (2.54)

The values at each intersection must be determined in sequence since the trajec-

tory of later characteristics depend on the interaction of previous ones. A sample

(x-t) for characteristic analysis is shown in figure (2.4.2).

Mapping all of the characteristics provides the location and time of each inter-

section. From this geometrical interpretation, similar results are produced when

compared to those obtained in the analytic approach (to within round-off errors).

Solutions using both methods are shown in figure 2.4.3. The numerical solution

does not provide values at every location in (x-t) space and interpolation of the


Fig. 2.4.3: Velocity (U), Sound Speed (CE) and density (ρ) profiles

arising from the motion of a receding piston in a semi-infinite system.

Analytic (solid line) and numeric (points) solutions are shown. Both

solutions produce identical results (to within round-off errors).

results is required. This can be circumvented by increasing the density of charac-

teristics to better approximate the analytic solution.

2.5 Eulerian and Lagrangian Coordinates

Often it is useful to transform the conservation equations from the Eulerian

coordinate system (of which they were derived) into the Lagrangian system. For

one-dimensional systems, Lagrangian coordinate systems often enable simplified

exact solutions to fluid flow equations.27 Lagrangian flow describes fluid proper-

ties along the pathline of the individual particles whereas Eulerian flow describes

the state of a fluid at a specific location through which various particle pass.47

Examples in this chapter have focused on the use of the Eulerian coordinate sys-


tems since readers are more familiar with such a coordinate system. However, as

is shown in the following chapter, the Lagrangian coordinate system is used to

describe the fluid flow specific to the body of this work.

Eulerian and Lagrangian systems can be visualized by imagining two perspec-

tives: a passenger in a boat versus observing the passage of that boat. Sitting in a

boat as it travels downstream describes Lagrangian flow. A specific particle (boat)

is tracked through the fluid along specific pathlines. Standing on the bank and

observing the passage of the boat represented the Eulerian system. The passage of

particles at a specific location in time is observed. Each of the coordinate systems

possesses qualities that make them optimal in certain situations. For instance,

if one desired to measure the flow rate through a channel it would be easier to

measure the current at a specific location (Eulerian) than to determine the fluid

path of each particle to calculate the flow rate (Lagrangian).47 In one-dimensional

systems, the Lagrangian coordinates describes the location and trajectory of each

particle allowing one to observe particle movement through the domain of the

system.

Transformations of one-dimensional fluid equations from Eulerian to Lagrangian

coordinates is straight forward. In the Lagrangian coordinate system, the fluid

trajectory of a particle is described by the fluid trajectory (h) and time (t). In

Eulerian coordinates, this is represented as x(h, t). Conservation of mass requires


that the fluid element mass is conserved in both systems or

ρ(t)dx = ρ0dh, (2.55)

where ρ0 describes the initial density of the fluid element.40 Transforming between

coordinate systems requires use of the material (convective) derivative that de-

scribe the derivative taken along a fluid path. In Eulerian and Lagrangian systems

the material derivative is defined as27

D

Dt=

∂

∂t+ U · ∇(Eulerian), (2.56)

D

Dt=

∂

∂t(Lagrangian). (2.57)

Using these conditions, the one-dimensional mass and momentum conservation

equations are transformed from the Eulerian into the Lagrangian coordinate sys-

tem,

∂ρ

∂t= −ρ2

ρ0

∂U

∂h, (2.58)

∂U

∂t= − 1

ρ0

∂P

∂h. (2.59)

Conservation of mass (2.58) and momentum (2.59) equations are used to deter-

mine the characteristics in Lagrangian coordinate system. Through substitution


one finds,[

∂U

∂t± CL

∂U

∂h

]

± 1

CLρ0

[

∂P

∂t± CL

∂P

∂h

]

= 0, (2.60)

where h is the Lagrangian coordinate and CL is the Lagrangian sound speed. The

Lagrangian sound speed is related to the Eulerian sound speed (CE) by

CL =ρ

ρ0CE. (2.61)

Following the techniques outlined in Section 2.4, the characteristic equations be-

come

dU ± 1

ρ0CLdP = 0 and

(

dh

dt

)

±

= CL along C±. (2.62)

The Riemann invariants become

J± = U ±∫

dP

ρ0CL. (2.63)

2.6 Shock Formation

If characteristics with increasing sound speed are allowed to propagate for an

infinite time they will intersect. After this point, these characteristics become

multivalued which is physically unrealistic. Consider the case of a wave propa-

gating to the right in an ideal gas with known thermodynamic properties. The


initial velocity disturbance of the wave (U(x, t0)) is known. The positive charac-

teristics are defined as straight lines with slope dx/dt = (γ +1)U/2+CE0. Figure

2.6.1a shows an initial disturbance and its corresponding characteristics in Figure

2.6.1b and at later times (t1 and t2). As time progresses, the wave profiles steepen

(figure 2.6.1c) and eventually “overshoots” becoming multivalued (figure 2.6.1d).

When the characteristics cross, the method of characteristics is multivalued and a

continuous solution does not exist. The wave profile develops into a discontinuity

in velocity, pressure and density. The solution for systems in which wave profiles

steepen into a discontinuity is the starting point for the development of shock

wave theory.

Realistically, “overshooting” does not occur. As the wave profiles deform they

become extremely steep and discontinues are formed, preventing the system from

becoming multi-valued. To understand this discontinuity the conservation equa-

tions are applied to the fluid flow. Consider a shock discontinuity propagating

through an ideal gas with initial values (ρ0, P0, CE0, U0) and unknown state fol-

lowing the discontinuity (ρ1, P1, CE1, U1) as shown in Figure 2.6.2. Assume that

the shock propagates with velocity D and ν1 is the post-shock fluid velocity with

respect to the shock front (ν0 = D − U0 and ν1 = D − U1). Using the equation

of mass conservation (2.5) and noting that ∂ρ/∂t = 0, we find that ∂(ρν)/∂t = 0.

Integrating this relation gives ρ1ν1 = ρ0ν0 in Lagrangian space. In the Eulerian


Fig. 2.6.1: Wave deformation. Diagram of a finite amplitude wave

propagating to the right. Figure (a) depicts the initial disturbance

as a sinusoidal function. As time progresses, the wave profile steepen

(figure (c)) and eventually “overshoots” becoming multivalued as shown

in figure (d). Characteristics are illustrate at the times relative figure

(a), (c) and (d). Figure (b) shows that as time advances characteristics

intersect for finite amplitude waves.

Fig. 2.6.2: Thermodynamic properties before before and after a shock

front in the shock reference frame. Density (ρ), pressure (P ), and

particle velocity (U) before and after the shock front. Shock propagates

with velocity (D) and the mass velocities (ν) through the shock front

are shown.


frame of reference, this is the conservation of mass flux equation,

ρ1(D − U1) = ρ0(D − U0). (2.64)

Using the same approach, the conservation equations for momentum and energy

can be determined in Eulerian space,

P1 + ρ1(D − U1)2 = P0 + ρ0(D − U0)

2, (2.65)

E1 +P1

ρ1+

(D − U1)2

2= E0 +

P0

ρ0+

(D − U0)2

2. (2.66)

These equations relate the flow variables at each side of the discontinuity and are

called the Rankine-Hugoniot relations. No assumptions have been made regarding

the properties of the fluid and equations 2.64, 2.65, and 2.66 represent the general

conservation equations across a discontinuity or shock front.27

2.7 Hugoniot Curves

Shock Hugoniot curves represent possible states that are achievable due to a

shock discontinuity. The principle Hugoniot is determined from the initial stan-

dard density and pressure conditions. For any material there is only one principle

Hugoniot and an infinite number of reshock Hugoniots that may originate at any

point along the principle Hugoniot.48 The first two Rankine-Hugoniot equations


(equations 2.64 and 2.65) have seven unknowns. If the initial conditions are known

(P0, ρ0, and U0) are known, that leaves two equations with four unknowns. Hugo-

niots are experimentally determined by measuring two parameters, typically the

shock (D) and fluid velocity (U). In many materials, a linear relation is observed

between the shock and fluid velocity,49

D = C0 + sU, (2.67)

where C0 is often, but not always, the sound speed under standard conditions

and s is typically the derivative of the bulk modulus at zero pressure.∗ The shock

conservation equations (2.64, 2.65, and 2.66) can be used to determine all other

parameters. Due to the arbitrary change in the entropy across the shock front,

a fully-defined EOS requires a measurement of the thermal state of the shocked

material.

If the EOS for a material is known, the corresponding principle and non-

principle Hugoniots can be determined.27 Through a simple manipulation of the

Rankine-Hugoniot equations (2.64, 2.65, 2.66), one can show that

E1(P1, ρ1) − E0(P0, ρ0) =1

2(P1 + P0)

(

ρ1 − ρ0

ρ1ρ0

)

. (2.68)

∗ It is important to note, that this linear dependence is not observed in all materials and oftena higher order fit is required to accurately represent the relation.


Fig. 2.7.1: Hugoniot and Isentrope for LiF calculated using Sesame

table 727150 at standard conditions.

By specifying the initial conditions within a known EOS (P (T, ρ), E(T, ρ)), the

Hugoniot can be found. For instance, using the Sesame table50 7271 for LiF the

principal Hugoniot curve is shown in figure 2.7.1. The Hugoniot curve (red line)

describes the possible end states for shocked LiF with standard initial conditions

(T = 298 k, ρ = 2.64 g/cc) shown as the black point. For a single shock ex-

periment, the LiF would compressed to a known end state (red point). The line

connecting the initial and final state is referred to as the Rayleigh line. It is im-

portant to note that a shocked material only experiences the beginning and end

state unlike the curve for isentropic compression. The blue isentrope in figure

2.7.1 is described in the following section.


2.8 Isentropic Curves

Similar to the Hugoniot curves, the isentrope can be determined using the

known thermodynamic relations of a material. The thermodynamic relation (Sec-

tion 2.14) for internal energy (E) can be described in terms of the temperature

(T), entropy (S), pressure (P) and density (ρ),

∆E = T∆S +P

ρ2∆ρ, (2.69)

where the specific volume (V) has been removed using the relation V = 1/ρ. For

isentropic systems, the entropy remains constant, reducing the state relation to

∆E =P

ρ2∆ρ, (2.70)

for ∆ρ/ρ ≪ 1. Equation 2.70 determines the isentrope for a material with known

thermodynamic properties. The isentrope for LiF is determined from standard

conditions using Sesame table 727150 and is shown in figure 2.7.1 as the blue

line. To reach the final state (blue point), the material follows the isentrope with

each intermediate state experienced as long as the pressure rise is gradual enough.

This is fundamentally different from the Hugoniot case in which the material only

experiences the initial and final states. Isentropic compression experiments have


a special desirability because in a single experiment a continuum of states can be

achieved. Moreover, since entropy addition increases the temperature, isentropic

compression is typically lower temperature than shock compression.

In juxtaposing the Hugoniot and isentrope for LiF, figure 2.7.1 shows that

for the same final pressure, the isentrope achieves a higher density. In isentropic

compression, more energy is used in compression compared to shock compression

because the entropy remains constant. For the shock case, higher temperatures

are reached, as shown in figure 2.8.1, because the entropy increases. The rapid

increase in temperature along the Hugoniot is due to the large shock-induced

increase in entropy. For an isentropic compression wave propagating in a solid,

the relationship among the temperature, density and entropy is

T

T0

=

(

ρ

ρ0

)Γ0

exp

(

∆S

CV

)

, (2.71)

where T is the temperature, ρ is the density, Γ0 is the Gruneisen parameter,27

S is the entropy of the system, Cv is the specific heat at constant volume, and

the subscript 0 indicates the initial value. For the isentropic case (∆S = 0), the

temperature increases by

T

T0

=

(

ρ

ρ0

)Γ0

, (2.72)


Fig. 2.8.1: Hugoniot and isentrope temperature dependence of LiF

calculated using Sesame table 727150 at standard conditions.

while for the shock case, the increase in entropy at the shock front will cause an

exponential increase in temperature.

2.9 Concluding Remarks

In the previous sections, the conservation of mass, momentum and energy

equations were derived. From these derivations, it was evident that there are

an infinite set of conservation equations. To close the system of equations, a

thermodynamic relation and assumptions regarding the heat flux are required.

In many systems, further understanding of the thermodynamic processes enables

additional assumptions to be made, reducing the complexity of the system of


equations. Fluid dynamic equations were applied to three separate cases (acoustic

waves, planar isentropic flow, and shock formation). The solutions to each of

these cases were provided. The evolution of isentropic compression waves into

shock waves was discussed. Determination of isentropic and Hugoniot curves from

EOS state data was shown. In terms of this work, understanding of the planar

isentropic flow and its solutions are most important. Fundamental understanding

of the method of characteristics and the limitations are pivotal to this research.

3. Experimental Technique 53

3. EXPERIMENTAL TECHNIQUE

Interferometric measurements of flow velocities made through optical windows

are discussed in this chapter. Equations relating the observed (apparent) particle

velocity to the (true) particle velocity when measured through an optical window

are derived for both shock and ramp compression. Previous experimental tech-

niques that measured the shock-induced refractive index of an optical window are

explored. These techniques are extended to an experimental design to measure

the isentropically compressed refractive index of LiF. The target design and ex-

perimental constraints are identified. Diagnostics used to measure the required

velocities and temperatures are discussed.

3.1 Interferometric Measurements Through

Optical Windows

Materials that remain transparent at extremely high pressures are useful in

high energy density experiments. Shock and ramp compression experiments often

employ optical windows that confine the samples but allow in situ particle-velocity


measurements at the sample/window interface. Velocities are measured using

interferometry whose sensitivity depends on the refractive index of the material

and the window. The optical window can be compressed during the experiment

and changes in its refractive index must be accounted for in velocity measurements.

If unaccounted for, systematic errors are introduced. The dependence of the

refractive index on the compression of the window is required to provide accurate

velocity measurements.

The effects on interferometry measurements are considered in the following ex-

amples. Consider a system in which the velocity of a reflecting surface is observed

through an optical window as shown in figure 3.1.1. The optical path length in

the window (Zw) is expressed as the integral of the refractive index (n(x, t)) over

Fig. 3.1.1: Velocity of a reflecting surface observed through an optical

window.


the window length (xfs(t) − x(t)),

Zw(t) =

∫ x(t)

xfs(t)

n(x, t)dx, (3.1)

where x(t) is the position of the reflecting surface and xfs(t) is the free surface

position. For velocity interferometry measurements that occur at a reference

plane in vacuum, the total optical path length (ZT ) from the observer (xV ) to

the reflecting surface (x(t)) is expressed as

ZT (t) =

∫ x(t)

xfs(t)

n(x, t)dx +

∫ xfs(t)

xV

dx. (3.2)

Now consider that the reflecting surface (x(t)) is a piston that moves with the

true particle velocity U(t), compressing the window material. Since the window

material is constrained to move with the piston, its velocity is also the true parti-

cle velocity (U(t)). The optical thickness represents the apparent position of the

reflective surface. This apparent position depends on the motion of the interface

(U(t)) and the refractive index of the window. The time derivative of the total

optical path length determines the apparent particle velocity. The difference be-

tween the apparent particle velocity and the true particle velocity is attributed

solely to changes in the refractive index of the window. Taking the time derivative


of 3.2 determines the apparent particle velocity (Uapp),

Uapp(t) =d

dt

[

∫ x(t)

xfs(t)

n(x, t)dx +

∫ xfs(t)

xV

dx

]

,

Uapp(t) =d

dt

[

∫ x(t)

xfs(t)

n(x, t)dx

]

+d

dt[xfs(t) − xV ] ,

Uapp(t) =d

dt

[

∫ x(t)

xfs(t)

n(x, t)dx

]

+ Ufs(t), (3.3)

where Ufs is the window free surface velocity. To determine the true velocity

(dx/dt = Utrue) of the reflecting interface (i.e. the velocity that would be measured

if the window were not present) knowledge of the refractive index as a function

of time and space is required. Equation 3.3 is simplified for the case of shock and

ramp compressed windows in the next section.

3.1.1 Shock Compressed Window

Figure 3.1.2 shows an optical window that is compressed by a single steady

transparent shock whose position is defined as xD(t). Prior to shock arrival at xfs,

the window free surface is at rest (Ufs = 0). Let n0 and n represent the spatially

uniform refractive indices ahead of and behind the shock, respectively. Equation


Fig. 3.1.2: Velocity of a reflecting surfaces observed through an optical

window.

3.3 becomes

Uapp(t) =d

dt

[

∫ x(t)

xD(t)

ndx +

∫ xD(t)

xfs(t)

n0dx

]

,

Uapp(t) =d

dt[n(x(t) − xD(t)) + n0(xD(t) − xfs)] ,

Uapp(t) = n(Utrue − D) + n0D. (3.4)

The true velocity of the reflecting surface as viewed through the shock compressed

window is

Utrue =Uapp + (n − n0)D

n. (3.5)

Thus, the shock compressed refractive index (n) can be determined if the true and

apparent interface velocities are measured and the Hugoniot (D(UTrue)) is known.


3.1.2 Ramp Compressed Window

Hayes36 derived the relation for the dependence of the true and apparent ve-

locities for isentropically or ramp compressed windows. The derivation assumes

that waves are simple and that shocks do not occur in the window. Hayes showed

that for simple waves the apparent and true particle velocities are directly related

to refractive index and density gradients within the window

dUapp

dUtrue= n − ρ

dn

dρ. (3.6)

Equation 3.6 can be expressed in integral form

n(ρ) = ρ(n0

ρ0−

∫ ρ

ρ0

f(ρ′)

ρ′2dρ′

)

, (3.7)

where f(ρ) represents the derivative of the apparent to true particle velocity and

n0, ρ0 are the required boundary conditions. Equation 3.7 illustrates that the

influence of the refractive index in ramp compressed optical windows with density

gradients is much more complicated than in single shock experiments. However,

by measuring Uapp and Utrue over a range of ramp compressed pressures, the

refractive index can be determined.


3.1.3 Optical Windows With Refractive Index That

Varies Linearly With Density

Velocity corrections for optical windows that possess a linear behavior of re-

fractive index and density (n = a + bρ) are straightforward. Recall equation 3.3

for shocked windows and assuming the free surface is stationary (Ufs = 0),

Uapp(t) =d

dt

[

∫ x(t)

xfs(t)

n(x, t)dx

]

. (3.8)

Substituting the linear relations for refractive index and density

Uapp(t) =d

dt

[

a(x(t) − xfs(t)) + b

∫ x(t)

xfs(t)

ρ(x, t)dx

]

. (3.9)

The second term of 3.9 represents the derivative of mass conservation of the win-

dow and is zero. This gives

Utrue(t) =Uapp(t)

a. (3.10)

Substituting this linear relation into the equation 3.6, for ramp compressed win-

dows, gives identical results indicating that corrections to the velocity measure-


Fig. 3.2.1: Experimental configuration shock-compressed gas gun ex-

periment.34

ments have a constant value. Various materials have demonstrated this behavior

such as quartz, sapphire and lithium fluoride (LiF).37

3.2 Shock Refractive Index Experiments

Numerous studies have been performed to measure the refractive index of

shocked windows34,51–54 with the earliest performed by Kormer.51 These exper-

iments used explosives and gas guns as drivers to compress the samples. The

experimental approach was to collide a flyer plate into a buffer with a window

attached to the rear surface (shown in figure 3.2.1). The flyer plate collision gen-

erates a shock that propagates across the buffer and into the optical window.

Velocity interferometry measures the flyer plate velocity prior to impact and the

apparent window interface velocity.

Collisional analysis48 is used to determine the true particle velocity at the


window interface. This requires that the EOS of each material be known. A

graphical representation of the collisional analysis is shown in figure 3.2.2. At the

time of the collision, continuity requires that the pressure and particle velocity

at the interface must be continuous (termed impedance matching). If Hflyer and

Hbuffer represents the flyer plate and buffer Hugoniots, respectively, impedance

matching requires that

Hflyer(Uflyer − Up) = Hbuffer(Up). (3.11)

The value of Up that satisfies this condition determines the particle velocity at

the contact interface (Ucollision). The collision between the flyer plate and buffer,

generates a shock that propagates across the buffer and into the window. The

analysis assumes that the shock is steady and does not decay as it travels through

the buffer. If Hwindow describes the window Hugoniot, then continuity requires

Hbuffer(2Ucollision − Up) = Hwindow(Up). (3.12)

The particle velocity (Up) that satisfies this equation determines the true particle

velocity at the buffer/window interface. The collisional analysis is simplified when

the flyer-plate and buffer are of the same material or when the flyer plate collides

directly with the LiF window. In this case, the analysis reduces to the single


equation

Hflyer(Uflyer − Up) = Hwindow(Up). (3.13)

Using equation 3.5, the shocked refractive index is expressed as

nS =UApp − n0D

UTrue − D. (3.14)

By measuring the flyer plate velocity prior to collision, the true velocity is de-

termined using the collisional analysis previously outlined. The Hugoniot of the

optical window determines the shock velocity. Lastly, by measuring the apparent

interface velocity post-collision, the refractive index is determined.

Experiments conducted by Wise and Chhabildas34 to measure the refractive in-

dex of Lithium Fluoride (LiF) windows used various impactor and buffer materials

(Al, Cu, Be, Ta, LiF, Al2O3) with known Hugoniots. The authors performed both

symmetric (identical flyer-plate and buffer material) and non-symmetric impacts.

The flyer plate and apparent particle velocities were measured using velocity in-

terferometry. The collision analysis outlined above determined the true particle

velocity. Using equation 3.14, the shocked refractive index was determined. The

results indicated that up to 115 GPa, the refractive index of shocked compressed

LiF demonstrates a linear behavior with density as shown in figure 3.2.3. These

are the highest published shocked refractive index measurements of LiF made to


Fig. 3.2.2: Geographical representation of collisional analysis to deter-

mine the true particle velocity. The flyer plate Hugoniot (blue line) is

reflected through the measured flyer plate velocity (blue point). The

intersection of this Hugoniot with the buffer Hugoniot (red line) deter-

mines the conditions of the flyer plate and buffer collision (red point).

The buffer Hugoniot (dashed red line) is reflected through this point

and the intersection with the window Hugoniot (black line) determines

the true particle velocity.


Fig. 3.2.3: Shock refractive index measurements made by Wise and

Chhabildas.34 The data shows a linear dependence over the measured

density, with pressure up to 115 GPa.

date. Observations and predictions suggest that shock-driven LiF becomes opaque

between 130 and ∼ 280 GPa when shock compressed.51,55 This is due, in part, to

melting by the shock.

As was shown in Section 2.8, ramp compression experiments achieve much

lower temperatures than shock compression. Figure 3.2.4 illustrates the Hugoniots

(red line) and isentrope (blue line) for LiF.50 Included in the figure are experi-

mental and theoretical molecular dynamic predictions for the melt line of LiF.51,55

The figure shows that at ∼160 GPa, the shock Hugoniot crosses the melt line in


Fig. 3.2.4: LiF Hugoniot (red) and isentrope (blue) calculated using

sesame table 7271.50 Shock, diamond anvil cell (DAC) and molecular

dynamic (simulations) of the LiF melt are included. Boehler’s esti-

mated melt line (black) is included. A large discrepancy between the

results of Kormer and Boehler are observed in determining the shock

melting of LiF.51, 55

the region where LiF becomes opaque. Isentropic compression is able to achieve

significantly higher pressures without crossing the melt line suggesting that LiF

will remain transparent at pressures well above 160 GPa enabling refractive index

measurements at substantially higher pressure than previously demonstrated.

3.3 Isentropic Refractive Index Experiments

The derivation by Hayes (3.6) shows that the refractive index of a window is

determined by measuring the true and apparent particle velocities with knowledge


of the corresponding EOS,

dUapp

dUtrue= n − ρ

dn

dρ.

In this work, the refractive index of ramp compressed LiF is measured using a two-

section target consisting of a piston with an optical window attached to half of its

rear (undriven) surface. Planar compression waves are driven into the front surface

of the target, traverse the piston, and reach the rear surface. These compression

waves extend across the two sections (bare and window) on the rear of the target,

producing distinct compression features in those two sections due to dissimilar

boundary conditions (free surface vs. impedance matching). The free surface (Ufs)

and the apparent interface (Uapp) velocities are measured simultaneously. The true

interface velocity (Utrue) is determined using the method of characteristics.

Velocity measurements at the free surface and piston/window interface are

made and the characteristics relations in Lagrangian form (section 2.4) are used.

The transverse dimension of the target and drive are sufficient such that a one-

dimensional analysis is justified. The technique, outlined by Rothman56 and

Maw,57 corrects for the wave interactions at the free surface to provide particle ve-

locities within the sample. Adaptation of this method allows for a time-dependent

calculation of the true interface velocity (Utrue) that can be compared to the ap-

parent velocity (Uapp). The technique is to measure the free surface velocity (Ufs)

of the piston material and using a backwards characteristics scheme determine the


Fig. 3.3.1: Target design used to determine the refractive index of an

optical window. The drive produces uniform compression regions across

both sections of the target. The piston free surface and piston/window

interface experience the same loading. The boundary conditions at

corresponding interfaces are shown.

applied ablation pressure (P(h=0,t)), where h=0 defines the location of ablation.

Once the applied ablation pressure is known, a forward characteristics scheme is

used to determine the true particle velocity at the piston/window interface (Utrue).

Each step requires different boundary conditions.

For simplicity the forward (F) and backwards (B) characteristics∗ are defined

as

F = Up + σ along dhdt

= CL, (3.15)

B = Up − σ along dhdt

= −CL, (3.16)

∗ The notation for Reimann invariants J+ and J− are replaced with F and B, respectively.


where Up is the particle velocity, σ is the Lagrangian strain, h is the Lagrangian

depth and CL is the Lagrangian sound speed. The Lagrangian strain and sound

speed are related to the thermodynamic properties (density (ρ) and pressure (P ))

by

σ =

∫

dP

ρ0CL, (3.17)

CL =ρ

ρ0

(

∂P

∂ρ

)1/2

. (3.18)

Up and σ are determined from the intersection of the ith positive and jth nega-

tive characteristics where i and j represent the indexing of the characteristics as

illustrated in figure 3.3.2. That value is given by

U i,jp =

1

2(F i + Bj), (3.19)

σi,j =1

2(F i − Bj). (3.20)

The negative characteristics that emerge from the undisturbed material (t ≤ 0)

must have B = 0. Using these equations, two characteristics schemes are devel-

oped to propagate the characteristics that determine the fluid flow.


Fig. 3.3.2: Index of characteristics is shown. Blue and red lines repre-

sent the forward and backwards characteristics. The indexing of charac-

teristics is defined and the intersection of the F3 and B2 characteristics

is illustrated for clarity.


3.3.1 Backwards Characteristics Scheme

The backwards characteristics scheme uses the free surface velocity measure-

ment (Ufs) to determine the ablation pressure (P (0, t)). When the compression

wave reaches the free surface, it releases along the reflected isentrope to zero pres-

sure. The free surface velocity is directly related to the particle velocity by the

velocity doubling rule or

Ufs = 2Up. (3.21)

The free surface boundary condition requires that

P (L, t) = σ(L, t) = 0, (3.22)

where L is the Lagrangian thickness of the piston. These conditions together

require that

F i = Bi = Ufs(tifs), (3.23)

which define the F and B characteristics. The velocity and Lagrangian stress are

found at the intersection of every positive and negative characteristic (equation


3.20). The location and time of these intersections is

hi,j =Ci,j−1

L Ci−1,jL (ti−1,j − ti,j−1) + Ci,j−1

L hi−1,j + Ci−1,jL hi,j−1

Ci,j−1L + Ci−1,j

L

, (3.24)

ti,j =Ci,j−1

L ti−1,j + Ci,j−1L ti,j−1 + hi−1,j − hi,j−1

Ci,j−1L + Ci−1,j

L

, (3.25)

where Ci,jL = CL(σi,j). The initial conditions (U(0, ti), σ(0, ti)) are determined

when the ith characteristic reaches the loading surface (h = 0). The ablation

pressure is found using the piston’s equation of state: P (0, ti) = P (σ(0, ti)). This

scheme is valid until the backwards characteristics reach the loading surface (plane

at which the ablation pressure is applied) because interaction of these character-

istics with the loading surface are uncertain.

3.3.2 Forward Characteristics Scheme

The forward characteristics (F ) are determined from the ablation pressure

(P (0, ti)),

F = 2U(0, ti) = 2σ(0, ti) = 2

∫ P (0,ti)

0

dP

ρ0CL

. (3.26)

At the intersection of the forward traveling characteristics with the window inter-

face, the impedance matching boundary conditions are applied,

P1(σ1) = P2(σ2), (3.27)

Up1 = Up2, (3.28)


where subscripts 1 and 2 indicate the piston and window, respectively. Using

equations 3.19 and 3.20, one may write

Up1 = F1 − σ1 = Up2 = B2 + σ2. (3.29)

Assuming no negative characteristics in the window (B2 = 0)† equation 3.27 is

rewritten

P1(σ1) = P2(F1 − σ1). (3.30)

The value of σ1 that satisfies this equation is used to determined the particle

velocity at the interface (Utrue = Up1 = Up2).

3.3.3 Target Design

The technique described requires a two section target that consists of a piston

with a window attached to half of the rear surface (figure 3.3.1). Characteristic

schemes require that that reflected characteristics do not reach the loading surface

prior to experimental termination.‡ Thus, the window must be sufficiently thick

such that reflected characteristics from the window free surface do not perturb

the flow at the piston/window interface.

† The assumption that B2 = 0 requires that the window be sufficiently thick such that com-pression waves do not reach the window free surface and reflect back into the interaction regionprior to the termination of the experiment.

‡ Experiment termination is defined as the time at which peak compression is observed at therear of the piston sample.


The proper piston thickness requires balancing the applied strain rate, the

material properties, and the limits of the driver to obtain the desired pressure

profile. As a material is compressed, its sound speed increases such that subse-

quent compression waves can overtake predecessors. If the applied pressure rises

too quickly, the compression waves will coalesce and form a shock (terminating

the analysis) as shown in section 2.6. Stiffer materials can be ramped more rapidly

while maintaining shockless compression. Materials with high Lagrangian sound

speed and a large Bulk Modulus are ideal for piston materials.58 Diamond is used

as a piston because its low compressibility allows it to be rapidly compressed to

high pressures without shocking. Recently, the ramp wave response of chemical

vapor deposited (CVD) diamond was experimentally determined to 800 GPa.59

The use of an experimentally determined quasi-isentrope increases the utility of

diamond as a piston material such that errors in the diamond isentrope are greatly

reduced when compared to theoretical predictions.

The method of characteristics was used to specify the ablation pressure and

determine the optimal diamond piston thickness. Characteristics were tracked

in the diamond piston and LiF window to ensure that shock formation does not

occur prior to the conclusion of the experiment. The duration of the applied

ablation pressure is determined by the OMEGA laser system capabilities. Single

laser pulse durations can be from 1 to 3.7 ns. The arrival time of each beam can


be adjusted using the path length adjustment system allowing a laser irradiation

of significantly longer durations. In addition to these constraints, the diamond

piston introduces further considerations with respect to the ablation pressure.

When stressed to the elastic limit (EL), diamond generates a two-wave structure60

that consists of an elastic precursor propagating at the elastic sound speed and

an inelastic wave traveling at a reduced velocity. If the two-wave structure turns

on during the bulk of the compression, large uncertainties in the refractive index

determination are introduced in the vicinity of the EL. Therefore, the ablation

profiles are designed such that the initial pressure pulse stresses the diamond

above its EL, initiating the two wave structure and this effect is easily accounted

for in the analysis. When the elastic wave of diamond (∼ 80 GPa)59 is impedance

matched to LiF, the minimum achievable pressure of LiF is ∼ 30 GPa.

An initial estimate of the target thickness can be determined from the sound

speed of the elastic wave (∼ 20µm/ns)59 and the pulse duration. The minimum

required diamond thickness (T) to optimize the pulse duration is

T =(Elastic Wave Sound Speed)(Pulse Duration)

2. (3.31)

From this we see that the pulse duration is the limiting factor in determining

the optimum diamond piston thickness. For a 3.7 ns pulse, the minimum target


thickness is ∼ 40 µm. Characteristic analysis shows that the arrival time of

the reflected elastic wave at the ablation surface is slowed due to interaction of

forward traveling waves, slightly reducing the constraint on the diamond thickness.

Furthermore, the determination of the refractive index requires the derivative of

the apparent to true particle velocities (equation 3.6). Steep compression profiles

introduce significant errors in this determination while gradual compression rates

introduce fewer errors. The data quality and the peak compression achieved must

therefore be balanced. For these experiments pulse shapes were designed to have

durations of 3.7 and 7 ns.

The applied ablation profiles were determined such that shock formation did

not occur in the diamond piston or LiF window. Three ablation profiles were

defined such that the LiF window achieved pressures ranging from 30 to 800 GPa.

Once the ablation profile is known, the drive technique and ablation pressure to

laser power scaling are required to determine the laser pulse profile.

3.3.4 Driver

Experiments were performed on the OMEGA laser61 at the University of

Rochester’s Laboratory for Laser Energetics. OMEGA is a 60 beam Nd:Glass

laser where the laser light is frequency converted from 1054 to 351 nm which pro-

vides up to 30 kJ of energy on target in the UV (the maximum energy per beam is

∼500 J). The beams are arranged to achieve uniform ablation on spherical targets.


To produce planar compression up to 12 beams are used to irradiate the target.

Six of these beams arrive at an angle of 23.2 to the normal with the other six

beams arriving at angles of 47.8. The lower angle beams (23.2) are used when

all 12 beams are not required. Laser light is smoothed using distributed phase

plates62 to produce spots that having a planar region of about 876 µm diameter.

In laser-driven experiments, high pressures are produced by the ablation of

target material. The ablation can be driven by either x-rays from laser-driven

hohlraums (indirect drive)59 or by direct laser irradiation (direct drive).63 High-

power lasers typically produce drive pulses of a few nanoseconds that readily pro-

duce high-pressure (> 100 GPa) shock waves. These have been used in reservoir-

coupled experiments where a shocked reservoir material releases across a vacuum

gap and the hydrodynamics of the releasing shocked material determines the tar-

get loading.64,65

Recent advances in the ability to control the temporal shape of laser pulses

have enabled shaped-ablation ramp compression. Laser-driven halfraums have

been used to produce ramp compression in diamond targets by directly ablating

the sample material59 (indirect-drive shaped-ablation ramp compression). The

present experiments demonstrate ramp compression by direct-drive shaped ab-

lation. The laser directly irradiates the sample; there is no intermediate energy

conversion nor coupling and the ablation pressure is controlled by temporal shap-


ing of the laser pulse. The benefits of this technique are that relatively lower

energies are required compared to the other techniques and many gigapascals are

achieved in a few nanoseconds.66 The compression profile is directly controlled by

adjusting the laser pulse profile using the appropriate ablation pressure to laser

intensity scaling law. Long pulse shapes produce the highest data quality because

gradual compression rate introduces fewer errors in determining the derivative

of the apparent to true particle velocities. It was determined that achieving a

smooth transition when stacking two beams was difficult and introduced signifi-

cant deviations in the measured profiles. Thus, single 3.7 ns ramp profiles were

predominantly used.

3.3.5 Laser Pulse Shape

To develop isentropic compression in diamond samples, the direct-drive ab-

lation scaling law is required. An initial scaling law was provided by Hicks for

direct-drive shock compression of low-Z materials67

P [GPa] = 55.3

(

I

[

TW

cm2

])0.6

, (3.32)


where P is the ablation pressure and I is the laser intensity. The scaling law for

diamond was determined from the present experimental results to be

Pablation[GPa] = 42

(

I

[

TW

cm2

])0.75

. (3.33)

The ablation analysis technique is discussed in the Appendix A. The laser power

is determined from the spot size of the laser pulse and total energy. Due to

the requirement that the transverse dimension of the two compression regions be

sufficiently large such that each undergo identical compression, distributed phase

plates62 were used to produce uniform planar region of ∼876 µm in diameter.68

Using this cross sectional area, the laser power is determined.

Three pulse shapes were designed for these experiments. As previously men-

tioned, the pulse duration is the primary factor in determining the target thick-

ness. The pulse shapes are shown in figure 3.3.3a. Figure 3.3.3 contains plots of

the laser pulse power, the corresponding ablation pressure, the predicted diamond

free surface velocity and the predicted diamond/LiF interface velocity.

3.3.6 Target Specification

Laser pulse durations were 3.7 and 7 ns correspond to required diamond thick-

nesses of ∼ 46 µm and 100 µm respectively. The thicknesses were chosen to be

slightly larger to ensure that the backward characteristics do not reach the ab-


Fig. 3.3.3: Three pulse shapes designs for these experiments. Fig-

ures (a) and (b) show the laser power and ablation pressure for the

three designed pulse shapes. Using the method of characteristics, the

free surface velocity and true particle velocity are calculated, figures

(c) and (d) respectively. The diamond target thickness used in these

simulations were 46 µm for pulse shapes RM3503 and RM1134 and 100

µm for RM3504. Figures (c) and (d) are normalized to the arrival time

of the elastic wave.


lation surface prior to laser termination. The cross section of diamond samples

ranged from 1.1 mm to 2.5 mm square. The larger cross section of the diamond

samples was an engineering control to ensure that the laser spot did not miss

the target. The cross section of the diamond is sufficient such that the wave

interactions at the edges of the target do not influence the fluid flow.

Chemical vapor deposition (CVD) and ultra-pure nano-crystalline diamond

were used. A 500 µm thick LiF window was mounted over half of the rear surface of

these diamond targets. The thickness ensures that reflected waves do not interfere

with diamond/LiF interface. High-purity LiF was orientated with the [100] axis

along the pressure loading direction. A 1000 A coating of aluminum was applied

to the ablation side of the diamond to prevent the low-intensity leading edge of

the laser from penetrating the target before the ablation plasma is formed. The

rear surface of the LiF had an anti-reflection coating to reduce ghost reflections

in the optical measurements of the apparent interface velocity.

A velocity interferometer system for any reflector (VISAR)33,35, 69, 70 was used

to measure the diamond free surface velocity and LiF interface velocity. The

reflectivity of these layers must be taken into account to achieve sufficient signal to

noise ratio in the VISAR measurements. The interface reflectivity of uncoated LiF

mounted on diamond was inadequate to achieve significant signal to noise ratio.

After arrival of the elastic wave, a substantial decrease in reflectivity was observed


(∼ 80%). A 1000 A metallic coating was applied to the diamond/LiF interface

to increase the reflectivity of that surface for interferometric measurements of the

interface velocity. The material layer was sufficiently thin so that it did not affect

the characteristics analysis. Three coating materials were used for this purpose:

titanium, aluminum and gold. Titanium was predominantly used because the

change in reflectivity over the pressure range was closest to that of the diamond

free surface creating comparable reflectivity.

During target fabrication, samples were glued only at the edges such that

the compressed diamond released directly into the LiF where there was no glue.

A finite gap between the diamond and LiF window was inevitable in the target

fabrication process. The gap was measured to be less than > 3 µm in all samples

and was verified by the delay in elastic wave arrival times between the free surface

and interface measurements.§

3.4 Diagnostics

Two diagnostics were used to measure the velocity and experimental temper-

ature. A velocity interferometer system for any reflector (VISAR)33 measures the

free surface velocity (Ufs) and the apparent particle velocity (Uapp). A streaked

optical pyrometer (SOP) provides optical emission measurements to determine the

§ Initial targets were faced glued (thin layer of epoxy between the diamond and LiF window).The impedance of the glue reduced the interface particle velocity, introducing significant errors.


gray body temperature of the target.71 These diagnostics and their data reduction

techniques are discussed below.

3.4.1 Velocity Interferometer System for Any Reflector

Theory

VISAR measures the velocity history of a reflecting surface. A schematic of the

Mach-Zhender interferometer used in these experiments is shown in figure 3.4.1.

Coherent laser light (probe beam) with wavelength λ is reflected off a moving

target and imaged through a Mach-Zhender interferometer. At the first beam

splitter, the light is separated into two legs. The light traveling through leg two

is delayed by a time τ using an etalon. The etalon, typically made of fused silica,

increases the optical path length of leg 2, imposing a delay. The etalon delay is

τ =2h

c

(

n − 1

n

)

, (3.34)

where h and n are the thickness and refractive index of the etalon, respectively,

and c is the speed of light. The etalon adjusts the focal plane of the corresponding

leg and this is accounted for by applying a translation distance (d) to the location

the etalon mirror defined as

d = h(

1 − 1

n

)

. (3.35)


Fig. 3.4.1: Schematic of Mach-Zhender interferometer

This ensures that the recombined images at the output beam splitter are both in

focus.

A second beam splitter recombines the two beams such that the detector,

or streak camera, observes a brightness that depends upon the phases of the

recombined beams (interference). The recombined image contains phase data

(brightness record) regarding the difference in velocities at times separated by τ .

For constant velocities, the change in phase is zero and a constant light amplitude

is observed.

The relation between the target velocity and changes in the observed light

amplitude (interference pattern) are understood through the following example.72

Assume that the light from leg 2 is delayed by Nλ where N is an integer number.


The length of this delay is written as

Nλ = cτ. (3.36)

Since N is an integer, both beams arrive in phase at the output beam splitter

(constructive interference). If the length of leg 2 is held constant, then changes

in the wavelength of the probe beam will directly affect the amplitude of light

observed at the detector. If the wavelength decreases by 1/2, the combined beams

will arrive out of phase and destructively interfere. The Doppler shifts in the

probe beam are observed as changes in light amplitude. These changes are directly

related to the velocity of the target since movement of the target induces a Doppler

shift in the probe beam wavelength or

∆λ(t) =−2λ

cU(t), (3.37)

where U(t) is the time varying velocity of the reflecting surface. Note that the

change in wavelength is related to the total change in the amplitude (∆N) up to

time t by taking the derivative of equation 3.36,

∆N(t) =−cτ

λ2∆λ(t), (3.38)


and the velocity is defined as

U(t) =λ

2τ∆N(t). (3.39)

In this configuration, the light amplitude depends on the integrated field of

view at the output beam splitter. A spatial dimension is imposed at the image

plane by tilting the output beam splitter. This changes the relative optical path

of light across the output field, creating a series of fringes that establish “zero”

phase. Tilting the output beamsplitter produces a shear in the phases of the beams

creating a linear fringe pattern across the field of view. The spatial dimension

of the image enables simultaneous measurements at different target regions. A

sample pattern is shown in figure 3.4.2a, where the x-axis is time and the y axis

is space. At t1 a shift in fringes is observed that corresponds to a change in target

velocity. The velocity profile determined from those fringes is shown in figure

3.4.2b.

Application

A two dimensional image is relayed to a streak camera slit producing a one

dimensional spatial image that is “streaked” in time. The image is recorded on

with time on x-axis and a one-dimensional image of the target on the other.

VISAR consists of a probe beam, an interferometer, and an imaging system. The


Fig. 3.4.2: Sample VISAR data. (a) The “fringe comb” observed at

the image plane. (b) The corresponding velocity profile.

configuration of this system on OMEGA is shown in figure 3.4.3. The probe beam,

Nd-YAG laser light (532 nm), is reflected off the target. The reflected light is

collimated, imaged at the output beam splitter of a Mach-Zehnder interferometer,

and this image is relayed to the slit of the streak camera.70 Analysis of this image

to determine the target velocity is discussed below.

The target velocity is related to the fringe phase (φ(x, t)) of figure 3.4.4a. The

fringe phase is determined by fitting the fringe intensity, S(x, t), to the sinusoidal

function

S(x, t) = A(x, t) + B(x, t)cos[φ(x, t) + 2πf0x + δ0], (3.40)

where A(x, t) and B(x, t) are functions representing the varying background in-


Fig. 3.4.3: Configuration of the VISAR and SOP on OMEGA.73

tensity and the amplitude variation respectively, and f0 is the spatial frequency.70

The desired phase can be obtained using a Fourier-transform method.35,74

In this technique, equation 3.40 is rewritten in terms of complex conjugates

S(x, t) = A(x, t) + C(x, t) exp(2πif0x) + C∗(x, t) exp(−2πif0x), (3.41)

where

C(x, t) =B(x, t)

2exp(iφ(x, t)). (3.42)

Taking the complex logarithm of equation 3.42 yields

log(C(x, t)) = log(B(x, t)/2) + iφ(x, t), (3.43)

where the desired phase information is separated from the amplitude information.


Fig. 3.4.4: Phase extraction procedure using the Fourier method to

determine the velocity. (a) VISAR Raw data (b) Fourier Spectrum (c)

unwrapped phase (d) velocity field (e) final velocity (line out over the

velocity field). Figure taken from Celliers et al.35


C(x, t) is determined by taking the spatial Fourier transform of equation 3.41,

s(f, t) = b(f, t) + c(f − f0, t) + c∗(f + f0, t), (3.44)

where the Fourier transformation of functions are represented in lowercase,

s(f, t) =

∫ ∞

−∞

S(x, t) exp(−2πixf)dx. (3.45)

The background information (b(f, t)) is separated from the phase information by

the constant fringe frequency (f0). s(f, t) is filtered such that c(f−f0, t) is selected

(figure 3.4.4b),

d(f, t) = c(f − f0, t). (3.46)

Taking the inverse Fourier transformation of d(f, t) produces

D(x, t) = C(x, t) exp((2πf0x + δ0)i), (3.47)

where D(x, t) is a complex function. The phase is defined as

φ(x, t) = −2πf0x − δ0 + arctan(Im[D]

Re[D]

)

, (3.48)


and the amplitude variation (B(x, t)) is given by

B(x, t) =Re[D]

cos(φ(x, t) + 2πf0x + δ0). (3.49)

In equation 3.48, φ(x, t) is not uniquely determined; an integer number of 2π

can be added or subtracted. The relation between the fringe phase and the total

change in fringe amplitude up to time t is

∆N(t) =φ(t) − φ(t0)

2π± n, (3.50)

where the n is an integer that represents the 2π ambiguities. The velocity is

determined from equation 3.39 as shown in figure 3.4.4e. Difficulties arise in this

technique when Doppler shifts occur at rates faster than c/τ due to constructive

interface occurring at integer multiples (referred to as 2π ambiguities). This is

resolved by using two interferometers with different VPF’s. The velocity per fringe

(VPF) is defined as

VPF =λ

2τ(1 + δ), (3.51)

where δ depends on the dispersion of the etalon,

δ =−n0

n20 − 1

λ0dn

dλ

∣

∣

∣

λ=λ0

, (3.52)


and n0 is the refractive index of the etalon at the probe wavelength. Etalons

are chosen such that the sensitivities of the two interferometers are not integer

multiples. The Doppler shift is found by comparing the predicted velocities of

different VPF integer fringe shifts. Since both interferometers observed the same

velocity (where subscripts 1 and 2 represent the different interferometers),

U(t − 1

2τ1) = VPF1(∆φ(t)1 ± n), (3.53)

U(t − 1

2τ2) = VPF2(∆φ(t)2 ± m), (3.54)

and if the VPF’s of these interferometers are not integer multiples of one an-

other, the integers n and m can be determined uniquely and the velocity profile

is determined.

3.4.2 Streaked Optical Pyrometer

The temperature of the target is determined from the optical self emission

brightness temperature. The streaked optical pyrometer (SOP)71 uses the VISAR

telescope to relay the target self emission to a streak camera having a sub-

nanosecond temporal resolution. The camera output intensity, I, can be related

to the source irradiance through the following equations,71

I =B∆xWsΩlens

ηM2

∫

dλTx(λ)SR(λ)Ls(λ, T ), (3.55)


where the source radiance (Ls(λ, T )) is given by Planck’s law,

L(λ, T ) =2hc2

λ5

1

ehc/λT − 1. (3.56)

B is the binning of the CCD, ∆x is the length of one pixel, Ws is the slit width, Ω

lens is the solid angle viewed by the lens, η is the sweep rate, M is the magnification

of the system, λ is the wavelength, Tx is the product of the transmission/reflection

spectrum of the optical elements in the system, SR is the wavelength-dependent

sensitivity of the streak camera system, Ls is source radiance given by Planck’s

law, h is Planck’s constant, c is the speed of light, and T is the temperature.71

The SOP has two narrow wavelength bands: a red-channel is created by using

a long-pass filter with a cutoff wavelength of 590-nm and the streak cameras

insensitivity to wavelengths greater than ∼ 850 nm. Similarly, a blue-channel is

defined by a short pass filter with a cutoff of 500 nm and a long pass filter with

a cutoff of 390 nm.71 When used with a relatively narrow wavelength band, the

wavelength is approximated as a delta function and equation 3.55 becomes

T =T0

ln(1 + AI). (3.57)

T0 is photon energy detected,

T0 =hc

λ0(3.58)


and A is defined as

A =2B∆xΩlens < TxSR > hc2G

ηM2λ50

, (3.59)

where G is the gain and λ0 is the central wavelength of the band.

In these experiments, the reflectivity of the target is determined from the

VISAR measurements. Consequently, the optical emission measurements must

be corrected for the emissivity of the target to determine the temperature accu-

rately.71 The reflectivity (R) of the target is determined from equation 3.49. The

gray-body temperature is defined as

T =T0

ln(1 + A(1−R)I

). (3.60)


The effect of shocked and ramp compressed optical windows on interferom-

etry measurements was discussed. The single shock experimental technique to

determine the compressed refractive index of an optical window was examined in

detail. It was shown that a similar approach can be used to determine the ramp

compressed refractive index of an optical window. Experimental constraints re-


garding this technique were illustrated and diagnostics required to determine the

compressed refractive index were described

4. Analysis and Results 95

4. ANALYSIS AND RESULTS

The LiF ramp compressed refractive index analysis and results are discussed in

this Chapter. Twenty-four experiments are analyzed in detail. Seven experiments

contained a thin layer of glue and the technique used to correct for the impedance

of the glue is discussed. In all experiments, a strong linear dependence is observed

between the apparent and true velocity resulting in a linear refractive index as a

function of density. Concerns regarding x-ray preheat and the large oscillations

observed on specific shots when determining the apparent and true velocity are

discussed in detail. Hydrocode simulations are performed to verify the accuracy

of the method of characteristics. Temperature measurements of LiF are provided.

4.1 Data Analysis

Twenty-four experiments were conducted to determine the ramp compressed

refractive index of LiF. The analysis method is discussed in detail and illustrated

using data acquired from shot 57575. The target for this shot consisted of a

46 µm diamond piston with a 500 µm LiF window attached to half of its rear

surface. The LiF window was edge glued to the diamond piston to ensure that


the impedance of the glue did not affect the velocity measurements. A 1000 A

aluminum coating was applied to the ablation side of the diamond to prevent

the low-intensity leading edge of the laser from penetrating the target before the

ablation plasma was formed. It had an aluminum shine through barrier and a 1000

A titanium coating on the LiF. Four beams, using pulse shape RM3502 (shown in

figure 3.3.3a), irradiated the target with a total energy of 270 joules. The sweep

speeds for camera 1 and 2 were 9 ns and 5 ns, respectively. This corresponds to

a temporal resolution of 11 ps and 6 ps.∗ Target specifications for all shots used

in this study are found in table 4.1.1. Table 4.1.2 contains the relative etalon

parameters determined from equations 3.34 and 3.51.

The VISAR data for shot 57575 camera 1 is shown in figure 4.1.1a. Before

t = 0 the fringes are horizontal because neither the free nor embedded surfaces

are moving. The fringes in the top half result from probe light reflected off the

embedded diamond/LiF interface. On the bottom, they are from the reflection off

the diamond free surface. At laser initiation (t = 0) a sudden change in the free

surface reflectivity is observed. This is because the diamond is transparent and

the signal from that section is a combination of reflection from the free surface

and the aluminum coating on the front surface. The drop in signal is attributed

to the ablation of the aluminum coating on the front surface and formation of

∗ Cameras that used a 15 ns sweep speed had a temporal resolution of 16 ps.


Table 4.1.1: Shot Specifications

Shot Pulse Energy Sweep Etalon∗ Thickness CoatingShape Speed

54939† RM3503 362 J 9ns/15ns 18a/7a 43 µm Ti54940† RM3503 480 J 9ns/15ns 18a/7a 46 µm Ti54941† RM3503 601 J 9ns/15ns 15a/7a 46 µm Ti54944† RM3503 422 J 9ns/15ns 15a/7a 46 µm Ti54945† RM3503 558 J 9ns/15ns 15a/7a 46 µm Ti54946† RM3503 421 J 9ns/15ns 15a/7a 46 µm Ti54948† RM3503 532 J 9ns/15ns 15a/7a 43 µm Ti55857‡ RM3503 442 J 9ns/15ns 15a/7a 15-[3 Au]- Al

40µm55859 RM3503 370 J 9ns/15ns 15a/7a 46 µm Al55860‡ RM3503 378 J 9ns/15ns 15a/7a 10-[3 Au]- N/A

25 µm56109‡ RM3503 508 J 9ns/5ns 18a/7a 10-[1 Au]- N/A

35 µm56112 RM3503 384 J 9ns/15ns 18a/7a 45 µm Au56113 RM3503 387 J 9ns/15ns 18a/7a 45 µm Ti57569 RM3504 711 J 9ns/15ns 18a/7a 97 µm Ti57570 RM3504 767 J 9ns/15ns 18a/7a 99 µm Ti57571 RM3504 774 J 9ns/15ns 18a/7a 95 µm Ti57572 RM3504 767 J 9ns/15ns 18a/7a 97 µm Ti57574 RM3504 623 J 9ns/15ns 18a/7a 103 µm Ti57575 RM3702 270 J 9ns/5ns 7a/18a 46 µm Ti57576 RM3702 345 J 9ns/5ns 7a/18a 45 µm Ti57577⋆ RM3702 272 J 9ns/5ns 7a/18a 44 µm N/A57579⋆ RM3702 362 J 9ns/5ns 7a/18a 45.5 µm Ti57581⋆ RM3702 362 J 9ns/5ns 7a/18a 45.5 µm Ti57583 RM3702 N/A 9ns/5ns 7a/18a 46 µm Ti∗ Etalon parameters are given in table 4.1.2.† Targets contained a thin glue layer.‡ Targets contained a gold preheat shield.⋆ Nano-crystalline diamond was used.


Table 4.1.2: Etalon Parameters

Etalon Thickness (µm) Delay (ps) VPF ( µm

ns fringe)

7a 7.2095 37.32 6.096515a 15.1318 78.35 3.290618a 18.2268 94.37 2.7318

the opaque plastic wave in the diamond. This abrupt drop in reflectivity triggers

an immediate reduction of current within the streak tube. The space charge of

the electron beam is affected, causing a rapid change in the magnification of the

streak tube and an artificial shift in the positions of the fringes. Note that this

shift is symmetric about the center of the data record with fringes shifted in the

vertical direction away from center. Since the current is constant after that abrupt

change, the streak camera equilibrates and the fringe position stabilizes.

At 2.2 ns, the elastic precursor reaches the rear diamond surface and the fringe

position abruptly changes in response to the velocity of that surface. At 3.5 ns,

the LiF window undergoes compression. The delay in arrival of the elastic wave is

attributed to a gap between the diamond piston and LiF window. The diamond

free surface velocity propagates at a velocity of ∼ 2.2 µm/ns at breakout. This

velocity and the delay time of the LiF window compression is indicative of the

gap thickness (a gap 3 µm or less is inevitable in the target fabrication process).

After this time, fringes move continuously to higher displacement (velocity) as the

pressure increases. The resulting diamond free-surface velocity profile (blue) and


Fig. 4.1.1: (a) Shot 57575 VISAR data corresponding to camera 1.

Time is shown on the x-axis with the spatial dimension on the y-axis.

VISAR measurements at the diamond free surface and diamond/LiF

interface are made simultaneously. (b) The corresponding velocity pro-

files determined from the VISAR measurements. The blue and red lines

corresponds to the diamond free surface velocity and the diamond/LiF

interface velocity, respectively.


the apparent interface velocity (red) are shown in figure 4.1.1b . Zero velocity for

both portions of the target is chosen as the fringe position after the space-charge

induced shift at t=0.

The diamond free surface velocity is backwards propagated to determine the

applied ablation pressure using the backwards characteristics scheme discussed in

Section 3.3.1. Figure 4.1.2a shows a graphical representation of the characteristic

calculations. The 0 µm depth corresponds to the ablation surface and 46 µm cor-

responds to the free surface (diamond thickness for shot s57575). Time increases

vertically, and depth (in Lagrangian coordinates) increases to the right. The slope

of the characteristics is the inverse Lagrangian sound speed (CL). Each character-

istic line is color coded by the pressure colorbar shown at the right (the temporal

profile of the inferred ablation pressure shown in figure 4.1.2b). Characteristics

propagate to the right, at the pressures they were initiated, until they reach the

rear surface. A zero-pressure boundary condition is imposed for the free surface,

producing a reflected wave. The analysis is invalid once the free surface character-

istics reach the loading surface at t ∼ 4.8 ns because interaction of characteristics

with the ablation region is unknown.

The applied ablation pressure (shown in figure 4.1.2b) is used to determine

the true particle velocity at the diamond-LiF interface using the forward charac-

teristics scheme discussed in Section 3.3.2. The graphical representation of the


Fig. 4.1.2: (a) Shot 57575 backwards characteristics. The diamond free

surface velocity measurement is used as the boundary condition at the

Lagrangian depth of 46 µm. The characteristics are color coded by the

pressure colorbar shown on the right. (b) The ablation pressure profile

in time calculated for the Lagrangian depth of 0 µm.


forward characteristics is shown in figure 4.1.3a. To account for the finite gap be-

tween the diamond and LiF window, a time constraint is place on the boundary

condition at the diamond/LiF window interface. The apparent interface velocity

is used to determine when LiF compression occurs (t ∼ 3.5 ns). Prior to this time,

the free surface boundary condition is imposed on the diamond piston. At later

times (t > 3.5 ns) the impedance matching condition is used. This adequately

accounts for the finite vacuum gap between the diamond and LiF. The thickness

of this gap is determined by integrating the diamond free surface velocity profile

until gap closure at 3.5 ns. The estimated gap thickness for this shot is ∼ 2.6 µm.

Shock formation in the LiF window is predicted at ∼5.4 ns when the character-

istics cross. This occurs at just over 59 µm downstream of the interface. At this

time a release fan would be generated that propagates backwards and impedes

the interface.75 The characteristics analysis does not account for shock formation

or the generation of the release fan. It is estimated that a release fan reaches

the interface at ∼5.9 ns as shown in figure 4.1.3. In this case, shock formation

is predicted after the conclusion of the experiment (∼5.5 ns),† and the estimated

arrival of the release fan is ∼0.4 ns after peak compression. This indicates that

shock formation in the LiF window does not influence the experimental results.

The characteristic corresponding to the laser termination (black line) is in close

† The experiment conclusion is defined as the time at which peak compression is observed atthe diamond/LiF interface. For shot 57575 this corresponds to 5.4 ns.


Fig. 4.1.3: (a) Shot 57575 forward characteristics. Prior to ∼ 3.5 ns the

free surface boundary condition is imposed to account for the finite gap

between the diamond and LiF window. The points corresponding to

the predicted shock formation, laser termination and peak compression

are shown. The characteristics are color coded by the pressure colorbar

shown on the right. (b) The true velocity (black line) is determined

from the forward characteristics at the Lagrangian depth of 46 µm. The

free surface velocity (blue line) and apparent velocity (red line) are also

shown.


agreement with experimental termination. The effect of shock formation on in-

terface measurements and validation of the method of characteristics is discussed

in Section 4.4.

Figure 4.1.3b shows the measured free-surface velocity (blue curve) and mea-

sured interface velocity (red curve). The calculated true interface velocity is shown

in black. A noticeable feature is that the apparent interface velocity (red curve)

exhibits a deceleration (at ∼5.5 ns) that is not evident in the free surface mea-

surement (blue curve). Deceleration or pull-back results when two decompression

waves, traveling in opposite directions, intersect in the bulk material to produce a

region of tension. The decompression waves in this study correspond a relaxation

wave that originates at the loading surface after laser termination and decom-

pression waves originating at the free surface or diamond/LiF interface. Since

the diamond/LiF interface undergoes impedance matching (i.e. higher pressure),

an elevated state of stress exists in that section. This supports a decompression

wave (tension) and deceleration is observed. The deceleration is not observed at

the diamond free surface because the diamond free surface has released to zero

pressure (by definition of a free surface). One reason that deceleration is not

observed at the diamond free surface is that the compressed diamond fractured

and the resulting structure has no tensile strength, a requirement for deceleration.


Refractive index measurements cannot be made at the onset of pull-back because

information about the drive pressure is lost.

A Monte-Carlo procedure was performed to determine the errors associated

with the calculated true particle velocity. This procedure randomly samples vari-

ables from the density distributions.76 Random numbers (z) are chosen from a

normal distribution with mean 0 and standard deviation of 1,

P (z)dz =1√2π

exp[−z2

2

]

dz. (4.1)

The Monte-Carlo variable (y) is expressed as the sum of the known value (x)

and the uncertainty (σ) times the random number (z) chosen from the normal

distribution,

y = σz + x. (4.2)

1,000 simulations were performed for each camera (2,000 simulations per experi-

ment). Four Monte-Carlo variables were defined for the simulations. These cor-

respond to the precision of fringe shift measurements (2.5 % of a fringe), the

uncertainties in the diamond and the LiF isentropes, and in the gap closure time.

The uncertainty in the diamond isentrope provided by Bradley et al. is used.59 A

conservative 10 % error in pressure is assumed for the LiF isentrope. Experiments

conducted by Ao et al.77 to measured the LiF isentrope show that the discrep-


ancies between the experimentally determined isentrope and Sesame 727150 is ∼

3 % of the Lagrangian sound speed over a pressure range of 0 to 114 GPa. This

corresponds to a 6 % error in the pressure, which justifies the conservative esti-

mate of 10 % in pressure for the LiF isentrope over the pressure range of 30 to

800 GPa. The uncertainties in the diamond and LiF isentropes are chosen to best

preserve the shape of the isentrope. The uncertainty in the gap closure time was

estimated to be ±0.1 ns. It was later determined that errors associated with the

gap closure time are significantly less than all other sources. The results of the

Monte-Carlo simulations for shot 57575 (camera 1) are shown in figure 4.1.4. The

individual black points represent the Monte-Carlo simulation and the red line is

the mean of these points with 1-sigma error bars in velocity and time. The inset

shows the estimation of the 1-sigma error bars from the Monte-Carlo calculation.

Recall that the apparent particle velocity as a function of true particle velocity

determines the refractive index,

dUapp

dUtrue

= n − ρdn

dρ. (4.3)

A plot of the apparent versus true particle velocity is shown in figure 4.1.5. Errors

associated with the apparent particle velocity correspond to the precision of fringe

shift measurements (2.5% of a fringe). Errors in the calculated true particle


Fig. 4.1.4: Monte-Carlo simulation of shot 57575. The black points rep-

resent true velocity values determined using the Monte-Carlo routine.

The red line is the mean of the Monte-Carlo simulations with 1-sigma

error bars in time and velocity shown. The inset is an enlargement of

the Monte-Carlo simulation illustrating the standard deviations of the

data.


velocity correspond to the 1-sigma uncertainties in velocity and time added in

quadrature. This includes the errors associated with the camera resolution and

etalon delay. The total error in the true particle velocity is determined from

δUTotalTrue =

[(

δUTrue

)2

+(dUTrue

dtδtTrue

)2

(4.4)

+(dUTrue

dtτetalon

)2

+(dUTrue

dtτsweep

)2]1/2

, (4.5)

where δUTrue and δtTrue are the uncertainties in the timing and velocity deter-

mined from the Monte-Carlo simulations, τetalon is the timing uncertainty corre-

sponding to the etalon delay and τsweep is the timing uncertainty corresponding

to the streak camera temporal resolution.

Solving equation 4.3, the refractive index as a function of density for shot

57575 is determined (figure 4.1.6). This requires the derivative of the apparent

to true particle velocity. Small deviations in their ratio produces large deviations

in the refractive index. The boundary condition required to solve equation 3.6 is

discussed in Section 4.3. The propagation of uncertainties through equation 3.6 are

not straightforward. To remedy these issues, multiple experiments are conducted

in which a weighted mean and orthogonal fit76 are performed to the determine the

relation (Uapp(Utrue)) and uncertainties associated with the orthogonal fit. The


Fig. 4.1.5: The apparent versus true particle velocity for shot 57575

camera 1.

uncertainties in the orthogonal fitting parameters are easily propagated though

equation 4.3.

A total of seventeen target shots were analyzed in the manner described above.

These are shown as an ensemble of blue-green like-colored points in figure 4.1.7a.

The pressures corresponding to these velocities are determined from the LiF isen-

trope50 as shown on the top axis. Seven additional that targets employed ∼2 µm

of glue to fill the gap between the diamond and the LiF window are shown as

the ensemble of red-yellow points. At low pressures, the compressibility of the

glue and reverberations within it cause the data to deviate from the general trend

of the vacuum-gap data. Once the glue “rang up” to higher pressure, the glue


Fig. 4.1.6: LiF refractive index determined using shot 57575 (black

line). Shock measurements made by Wise,34 Lalone,54 and Jensen53, 78

are included as yellow, brown and red squares respectively. Extrapo-

lation of the linear fit proposed by Lalone et al.54 is shown as the red

line. The pressure scaling along the top axis corresponds to the LiF

isentrope.


Fig. 4.1.7: Apparent versus true particle velocity of all experiments

conducted in this study. The like colored points (blue-green) result

from the seventeen targets that did not contain a glue layer. The peak

compression of those targets is 500 GPa. The remaining seven points

(red-yellow) are the results of those shots that contained glue which

reach 800 GPa. The pressure scaling along the top axis corresponds to

the LiF isentrope.


data follows the trend in the vacuum-gap data. To account for delay in arrival

times of the compression waves at the LiF interface, the diamond thickness in

the characteristics analysis is increased to account for the relative thickness of the

glue. Although this modification is not without error, it is a better approximation

than to neglect that thickness. The glue thickness is estimated from the delay in

the arrival time of the elastic wave and the estimated Lagrangian sound speed of

5 µm/ns of the glue. Errors associated with the glue thickness are incorporated

in the Monte-Carlo routine with an estimated error of ± 1 µm. The large error

bars shown in figure 4.1.7 are due to the steep rate of rise of the pressure pro-

file and uncertainties associated with the glue thickness. The weighted mean and

orthogonal fit of all twenty-four experiments are discussed in the following section.

4.2 Weighted Mean and Orthogonal Regression

A weighted mean of the twenty-four experiments determines the relation be-

tween the apparent and true particle velocities.76 The weighted mean (y) is defined

as

y =

∑

(yi/σ2i )

∑

1/σ2i

, (4.6)

and the variance of the mean as

σµ =1

∑

(1/σ2i )

, (4.7)


where yi and σi are the measured value and uncertainties of the points to be

averaged and the sums are over the twenty-four (i) experiments. The apparent

velocity is considered the dependent variable and the values are equally spaced

by ∆Uapp = 0.01 µm/ns. Prior to performing the weighted mean on all twenty-

four measurements, the values obtained from camera one and camera two for a

single measurement are combined using the same technique. To determine the

weighted mean, error measurements in both the apparent and true velocity are

combined. The equivalent error in the true velocity due to uncertainties in the

apparent velocity is described as

σtrue(equiv) =dUtrue

dUapp

σapp, (4.8)

where dUtrue

dUapprepresents the inverse slope of the apparent to true velocity and is

estimated to be 1/1.28 for these measurements. The total error in the true particle

velocity is

σtrue(total) =√

σ2true + σ2

true(equiv), (4.9)

σtrue(total) =√

σ2true + (σ2

app/1.28)2. (4.10)


Once the errors in the weighted mean are determined (equation 4.7), they are

converted to errors in the apparent velocity through the same technique,

σapp(total) = 1.28σtrue(total). (4.11)

Therefore, errors associated with the apparent and true velocity have been com-

bined into a single term. Since the weighted mean is performed on measurements

using cameras 1 and 2, the errors bars in figure 4.1.7 are shown only for the

apparent velocity measurements.

The twenty-four measurements are combined using the weighted mean. A

reduced chi squared test is performed to ensure that errors are not underestimated.

χ2ν is defined as

χ2ν =

χ2

ν, (4.12)

where ν corresponds to the number of degrees of freedom and χ2 represents the

measure of agreement between the observed and expected values. χ2 is defined as

χ2 =

N∑

i=1

(yi − y

σi)2, (4.13)

where N is the number of measurements, yi are the experimental values, y is the


Fig. 4.2.1: The weighted mean performed using all twenty-four ex-

periments is shown in black with uncertainties. The orthogonal fit

performed using existing data is shown as the dashed line. Shock mea-

surements made by Wise,34 Lalone,54 and Jensen53, 78 are included as

yellow, brown and red squares respectively. The pressure scaling along

the top axis corresponds to the LiF isentrope.

weighted mean and σi is the uncertainties in yi. For χ2ν values less than one, the

errors are underestimated and increased such that χ2ν becomes unity.

Figure 4.2.1 shows the weighted mean (black points) of the data from figure

4.1.7 using the associated errors discussed above. As a record, the values of the

weighted mean are included in Appendix B. The large errors between 700-800

GPa occur because only a single experiment reached those pressures. Included


in the figure are previous shock measurement data from Wise,34 Lalone,54 and

Jensen.53,78

A second-order orthogonal polynomial regression is performed to determine the

relation between the true and apparent particle velocities.76,79 In the orthogonal

fit, the value of each coefficient is independent of higher-order terms, diagonalizing

the covariance. The form of the fit is

Uapp(Utrue) = a0 + a1(Utrue − β) + a2(Utrue − γ1)(Utrue − γ2). (4.14)

The coefficients a0, a1 and a2 (the centroid, average slope, and average curvature,

respectively) are found by minimizing χ2, the goodness-of-fit parameter. The fit

requires additional parameters, β, γ1 and γ2. Errors are not assigned to these

parameters because they only depend on the independent variable. The orthogo-

nal fit is performed using various combinations of the data sets to best determine

the relation between the true and apparent particle velocity. The orthogonal fit

is determined for the seventeen experiments that did not include the glue layer

as well as for all twenty-four experiments. Those results are shown in table 4.2.1.

Furthermore, orthogonal fitting was performed on the shock measurement data

from Wise,34 Lalone,54 and Jensen.53,78 Due to the observed commonality between


the shock and ramp compression measurements, orthogonal fitting is performed

in which results of all experiments are combined.

The errors determined in the orthogonal fit are not a true representation of

deviations within the data. This is observed when shots are removed at random

and orthogonal regression performed. To remedy this issue, uncertainties in the

orthogonal coefficients are determined by randomly removing one to four shots

and performing orthogonal fitting. The coefficients determined from 100 such

groupings are averaged and the standard deviations provided (table 4.2.1). The

orthogonal fit (grey dashed line) corresponding to the shock and ramp compression

experiments is shown in figure 4.2.1. A plot of the ratio of the apparent to true

velocity is shown in figure 4.2.2 illustrating that the values of the weighted mean

deviate about the orthogonal fit. It was found that 66 % of the weighted mean

error bars encompass the orthogonal fit using the ramp, glue and shock data. This

percentage is in good agreement with one standard deviation (68.2 %), indicating

a strong correlation between the data and orthogonal fit.

Two concerns, one regarding the large deviations of single shots from linearity

observed in figure 4.1.7 and secondly the effect of x-ray preheat on LiF refractive

index, are addressed. Steep gradients of the velocity profiles produce large errors

in the apparent versus true particle velocity plot for individual shots, since the

correlation of events is limited by the etalon delay and temporal resolution of


Fig. 4.2.2: Ratio of the apparent to true velocity is shown. Black

points correspond to the weighted mean off all twenty-four experi-

ments. The first order orthogonal fit using all existing data is shown as

the grey dashed line. Shock measurements made by Wise,34 Lalone,54

and Jensen53, 78 are included as yellow, brown and red squares respec-

tively. The pressure scaling along the top axis corresponds to the LiF

isentrope.


Table 4.2.1: Parameters Resulting from the Orthogonal Fit (Eq. 4.14)

a0[km/s] a1 a2[s/km] β[km/s] γ1[km/s] γ2[km/s]Ramp 5.22 1.28 0.001 4.12 2.84 7.46

± 0.01 ±0.004Ramp 8.79 1.28 0.000 6.91 3.10 9.96& Glue ± 0.01 ±0.009Wise34 1.39 1.289 0.000 1.09 0.87 2.89

± 0.003 ± 0.002Lalone54 0.48 1.276 0.01 0.38 0.25 0.61

± 0.005 ±0.02Jensen53,78 0.56 1.26 -0.2 0.46 0.33 0.56

± 0.04 ±0.6Ramp 1.49 1.273 0.001 1.17 0.61 6.23

& Shock ±0.008 ±0.003Ramp, Glue 3.06 1.275 0.001 2.41 0.71 9.53

& Shock ± 0.008 ± 0.002

the camera. Recall that this uncertainty contributes to the total uncertainty

through the time derivative of the velocity profile and errors are proportional

to the gradient of that profile. This accounts for the large error bars observed in

many shots and observed deviations of shots from linear behavior. To reduce these

deviations pulse shape RM3702 was designed to produce a more gradual velocity

profile. Shot 57575 is a demonstration of this technique, the shot described in the

previous section. Comparison of this shot with the weighted mean, shows that

there is considerably less deviation from linearity (figure 4.2.3) in these points as

compared to the experiments shown in figure 4.1.7. The data all fall within the

error bars of the weighted mean fit indicating that the observed deviations are a

results of the experimental technique and not changes in the refractive index.


Fig. 4.2.3: Comparison of the weighted mean and data acquired using

a gradual applied pressure (RM3702). The pressure scaling along the

top axis corresponds to the LiF isentrope.


Concerns regarding the effects of x-ray preheat on the refractive index of the

LiF window were addressed using specially designed targets to prevent x-ray pre-

heat from affecting the window. These targets consisted of a 10 to 15 µm diamond

ablator, a 1 or 3 µm gold layer to prevent x-ray preheat backed with 25 to 40 µm

diamond sample. The target design is shown in figure 4.2.4. When the ablation

plasma is formed, it acts as a thermal source that typically produces photons of

1 to 2 keV in energy. A 3 µm gold layer transmits less than 0.15 % of those ener-

gies. The tail of that thermal distribution is of the order of 5 keV photon energy

that corresponds to a transmission of 1.9 % through the gold barrier layer.80 It

is reasonable to expect that this thermal shield prevents x-ray penetration of the

LiF window. The results of shot 55857 which contained a 3 µm gold shield is

shown in figure 4.2.5 (blue data). Excellent agreement between this measurement

and the weighted mean suggests that the effects are x-ray preheat on the LiF

window are inconsequential. This is expected since the direct ablation method is

quite efficient and requires far less laser intensity than hohlraum drive or reservoir

coupled experiments.

4.3 LiF Refractive Index

The refractive index of LiF is inferred from the orthogonal fitting performed in

the previous section. Recall from Section 3.1.2 that the relation of the refractive

index (n), density (ρ) and the measured apparent (Uapp) and true (Utrue) particle


Fig. 4.2.4: Target design with embedded gold layer preheat shield

Fig. 4.2.5: Comparison of the weighted mean with an embedded gold

layer target (shot 55857). The pressure scaling along the top axis cor-

responds to the LiF isentrope.


velocities for ramp compression experiments

n − ρdn

dρ=

dUapp

dUtrue

= f(ρ), (4.15)

n(ρ) = ρ(n0

ρ0−

∫ ρ

ρ0

f(ρ′)

ρ′2dρ′

)

. (4.16)

This integral equation requires a boundary condition; the highest pressure refrac-

tive index measurements made by Wise and Chhabildas is used.34 Uncertainties

are estimated from the deviation of measured values from linearity. The bound-

ary condition is ρ = 4.23 ± 0.06 g/cc, n = 1.461 ± 0.003 and P = 108 ± 8 GPa.

Both first and second order fits corresponding to the ramp, glue and shock data

are used to determine the refractive index. The weighted mean (black line), first

order (blue line) and second (red line) order results are shown in figure 4.3.1. Re-

sults from Wise,34 Lalone,54 and Jensen53,78 are shown as the yellow, orange and

red squares respectively. The corresponding pressure is shown in the top axis.

The error bars corresponding to the first and second order orthogonal fit en-

compass the weighted mean and previous shock measurements. This indicates

that there is no discernable difference in the refractive index determined using

shock or ramp compression. Using the first order orthogonal fit, the solution to

equation 4.15 is

n = 1.275[±0.008] + 0.044[±0.002]ρ. (4.17)


Recall from Section 3.1.3 that the correction factor to VISAR measurements for

materials that possess a linear refractive index is simply the zero density refractive

index or a = 1.275[±0.008]. The second order orthogonal fit possesses a slight non-

linear behavior that is attributed to fitting the second order terms to the residuals

of the first order.

It should be noted that one previous shock-release refractive index study found

a strong non-linear behavior that was not observed in this nor other studies.81

That experiment consisted of direct laser ablation of an iron target with a LiF win-

dow attached to the rear surface. Interface velocity measurements made through

a LiF window were compared to hydrodynamic simulations performed using the

applied laser intensity. The authors claimed that the only explanation for the

discrepancy in the observed interface velocity when compared to the hydrody-

namic simulations was due to changes in the refractive index. They found the

refractive index to be non-linear from 100 to 250 GPa. In that study, a thin glue

layer, < 1µm, was applied to the Fe/LiF interface. As observed in this study, the

impedance of the glue layer causes a significant reduction in the apparent velocity.

If unaccounted for, this significantly alters the refractive index calculation. This

issue was not addressed in that study and may explain the perceived non-linear

behavior.


Fig. 4.3.1: The refractive index of LiF determined using the weighted

mean (black line), first order (blue line) and second (red line) order or-

thogonal fit to the ramp and shock data. Results from Wise,34 Lalone,54

and Jensen53, 78 are shown yellow, orange and red squares, respectively.

Notice that there is little difference between the first and second order

plots due to the small second order parameter (a2) and that the error

bars of both fits encompass the weighted mean and shock data. The

pressure scaling along the top axis corresponds to the LiF isentrope.


4.4 LASNEX Simulations

One-dimensional hydrodynamic LASNEX82 simulations are performed to val-

idate the method of characteristics. Hydrocodes model the complex behavior

of continuous media and are not limited to special cases of fluid flow, as is the

method of characteristics. LASNEX simulations were performed to understand

the effects of shock formation in LiF windows downstream from the diamond/LiF

interface. In several early experiments, shock formation occurred prior to termi-

nation of the experiment, generating a release fan that may impede the interface

velocity. Recall, that the characteristics analysis is valid only for isentropic com-

pression. Radiation hydrodynamic simulations were performed for shot 56113 to

determine the effects of LiF shock formation on the true interface velocity. The

target consisted of a 45 µm CVD diamond piston with a titanium coated LiF

window attached to half of the rear surface. The diamond/LiF interface reached

∼ 570 GPa. Figure 4.4.1 shows that the arrival of the release fan due at the

diamond/LiF interface due to shock formation in the LiF window occurs prior to

peak compression.

The LASNEX simulations use a diamond EOS with a Steinberg-Guinan strength

model83 to recover the diamond elastic limit and the LiF sesame table 7271.50 The

pressure drive is applied 10 µm inside the target-front surface to account for the


Fig. 4.4.1: Characteristics Analysis of shot 56113 with early shock

formation in LiF window. Prior to ∼ 2.6 ns the free surface boundary

condition is imposed to account for the finite gap between the diamond

and LiF window. Points corresponding to predicted shock formation,

laser termination and peak compression are shown. Characteristics are

color coded by the pressure colorbar shown on the right.


Fig. 4.4.2: Comparison of hydrocode simulations and method of charac-

teristics for shot s56113. (a) LASNEX ablation pressure is determined

iteratively by matching the LASNEX free surface velocity (black) with

the VISAR measurement (blue). (b) The ablation pressure is then used

to determine the true interface velocity. Comparison of the LASNEX

results (black) and method of characteristics (red) is shown.

material that is ablated by the laser. This applied pressure is iterated until the

simulated free surface velocity best agrees with the measured free surface velocity

(figure 4.4.2a). The characteristic analysis for shot 56113 is shown in figure 4.4.1.

Shock formation is predicted at 4.8 ns. The arrival of the release fan is predict at

4.9 ns, 0.3 ns before peak compression is achieved.

The calculated interface velocity determined using the method of character-

istics and LASNEX simulations is shown in figure 4.4.2b. Figure 4.4.2b shows

that the hydro-code simulations and characteristics analysis infer nearly identical


true interface velocities. The black line is the LASNEX predicted true interface

velocity and the red line is the true interface velocity calculated using the method

of characteristics. The excellent agreement between the two techniques confirms

the accuracy of the characteristic model and indicates that shock formation in the

LiF window does not significantly perturb the interface velocity. The excellent

agreement between the two techniques validates the accuracy of the experimental

analysis. It should be noted that the step like discontinues that are observed in

LASNEX calculated free surface and true interface velocity are attributed to the

limitations of the Steinberg-Guinan strength model and knowledge of the diamond

elastic limit.

4.5 Temperature Measurements

Grey-body temperature measurements were made for both the diamond free

surface and diamond/LiF interface. The temperature of the diamond free surface

remains below 6000 K for all experiments and are in agreement with published re-

sults.59 Measurements of the diamond/LiF section were dominated by emission of

the diamond piston. This precluded direct measurement of the LiF self-emission.

The absorption (α), transmission (τ), and reflectivity (ρ) are interrelated by

α + τ + ρ = 1. (4.18)


Fig. 4.5.1: Temperature measurements corresponding to shot 57575.

Blue and red lines correspond to the CVD diamond free surface and the

titanium coated LiF interface, respectively. The black line corresponds

to the LiF sesame 727150 predicted temperature. The large increase in

temperature at the diamond free surface corresponds to the formation

of the ablation plasma when the laser is initiated. The sudden reduction

is due to the formation of the opaque elastic limit. This is not observed

at the LiF interface due to the 1000 A coating of titanium. Compression

of the LiF window does not occur until 3.5 ns.

If the transmission dominates, and the absorption is minimal, then Kirchhoff’s

law indicates that the self-emission will be negligible.84 Such a low self-emission

cannot be observed in the presence of strong emission from the diamond anvil

SOP measurements for shot 57575 are shown in figure 4.5.1. The target con-

sisted of a 46 µm CVD diamond piston with a titanium coated LiF window at-

tached to half of the rear surface. Peak compression (∼ 280 GPa) occurs at 5.5


ns. The large increase in temperature that occurs at ∼ 0.1 ns is due to the forma-

tion of the ablation plasma at laser initiation and the partial transparency of the

diamond piston. The sudden reduction in that emission is caused by the forma-

tion of the opaque elastic limit in diamond. Emission is not observed at the LiF

interface due to the opaque 1000 A titanium coating. The diamond/LiF interface

temperature is lower than that of the diamond free surface. This is due to the

thermal conductivity of the titanium layer between the diamond and LiF window.

The LiF interface observed temperatures higher than predicted by LiF equation

of state50 (black line of figure 4.5.1) which is attributed to the self-emission of the

diamond sample.

Experiments were conducted using nano-crystalline diamond to reduce the self-

emission since the higher opacity of the nano-crystalline diamond compared to the

CVD diamond shields the SOP from the laser plasma emission. Results of one such

shot (57577) is shown in figure 4.5.2. The target consisted of a nano-crystalline

diamond piston and uncoated LiF window mounted on half of the rear surface.

The higher reflectivity of the nano-crystalline diamond compared to the CVD

diamond enabled the use of uncoated LiF windows. Peak compression occurs at

5.4 ns corresponding to a pressure of ∼260 GPa. The diamond free surface (blue)

and LiF interface (red) were observed to have nearly identical temperatures. The

agreement between these measurements indicates that the self-emission of the LiF


Fig. 4.5.2: Temperature measurements corresponding to shot 57577.

Blue and red lines correspond to the nano-crystalline diamond free

surface and uncoated LiF interface, respectively. The black line corre-

sponds to the LiF sesame 727150 predicted temperature. Compression

of the LiF window does not occur until 2.3 ns. The agreement between

these measurements indicates that the self-emission of the LiF is small

and of the order of the error bars.

is small and of the order of the error bars. Figure 4.5.2 shows that the SOP can

resolve temperatures of ∼ 1000 K. This indicates that the LiF window remains

below ∼1000 K for these experiments. This is in agreement with Sesame table

7271 for LiF50 that predicts that the temperature at 800 GPa to be 800 K.



Analysis and data reduction to determine the LiF ramp compressed refractive

index was discussed in detail. A linear relation between the apparent and true

particle velocity was observed up to 800 GPa. This was used to determine the

refractive index as function of density for LiF. It was found that the refractive

index depends linearly on the density up to 800 GPa. Hydrocode simulations were

performed which verified the accuracy of the method of characteristics. Concerns

regarding x-ray preheat and the large oscillations the apparent to true particle

velocity measurements on single shot basis were addressed. Temperature mea-

surements of the diamond free surface are consistent with published results and

the LiF self-emission was compromised by the diamond piston but its temperature

appears to remain below 1000 K for these experiments.

5. Discussion 134

5. DISCUSSION

Materials whose refractive index depends linearly on density are commonly

used as optical windows for high pressure experiments that use velocity inter-

ferometry system for any reflector (VISAR).33 The observed transparency and

measurement of LiF refractive index to 800 GPa is important for advancing those

experiments to higher pressure regimes. The linear behavior of the refractive in-

dex and density of LiF and the implications on VISAR experiments is examined

in Section 5.1.

The theory of classical propagation of light through a medium and the Lorentz

oscillator model is introduced in Section 5.2 to explain the dependence of the re-

fractive index on the dielectric function. A single-oscillator model is derived and

applied to the LiF refractive index as a function of density (Section 5.3). The

model indicates that the linear behavior of the refractive index and density is

related to the band gap energy. In Section 5.4, the LiF metallization pressure is

predicted by extrapolation of those results. The metallization pressure is com-

pared to the Goldhammer-Herzfeld metallization85,86 of LiF and other large band

5. Discussion 135

gap insulators. It is postulated that the high metallization pressure of LiF is due

to the large band gap and that it is isoelectronic with Helium and Neon.

5.1 VISAR Window Corrections

Optical windows that are used in VISAR experiments require corrections due

to changes in their refractive index.37 For windows that exhibit a linear refractive

index (n) as a function of density (ρ),

n = a + bρ, (5.1)

the true particle velocity (Utrue) is related to the apparent particle velocity (Uapp)

by

Utrue(t) =Uapp(t)

a, (5.2)

for both ramp and shock compression (the corrections were discussed in detail

in Section 3.1).36,37 Shocked quartz, sapphire and LiF exhibit a linear refractive

index as a function of density.37 This study demonstrated that ramp compressed

LiF is transparent and its refractive index depends linearly on density up to 800

GPa and is in agreement with studies along the Hugoniot.34,53, 54 This extends the

achievable pressure range of ramp compression experiments that require optical

windows to 800 GPa. The linear behavior of shock and ramp compressed refractive

5. Discussion 136

index suggests that changes in the refractive index are dominated by pressure

effects. Therefore, the linear behavior of the refractive index and density will also

be observed for multi-shock experiments in which thermal excitation is insufficient

to produce appreciable conduction electrons.

5.2 Classical Propagation

The response of large band gap insulators (dielectrics) to electromagnetic fields

is of particular concern to many areas of study (i.e. optics, physics, engineering).

Transparent dielectric materials are used as lenses, prisms, films and windows.

For these applications an understanding of the index of refraction (n) is critical

in determining the propagation of light. The index of refraction is a measure of

the dielectric function (ǫ), which describes the response and behavior of materials

to electric fields that vary with space and time. The relation between the refrac-

tive index (n) and the dielectric response (ǫ) of a material can be derived using

Maxwell’s equations in vacuum by replacing the free space variable (ǫo, µo) with

the materials reduced dielectric equivalents (ǫ, µ).87 Maxwell’s equations88 in free

5. Discussion 137

space become

∇ · ~E =ρf

ǫo

, (5.3)

∇× ~E = −∂ ~B

∂t, (5.4)

∇ · ~B = 0, (5.5)

∇× ~B = µo~J + ǫoµo

∂ ~E

∂t, (5.6)

where ~E is the electric field, ρf is the free charge density, ~B is the magnetic field,

and c is the speed of light in vacuum. Assuming a linear medium ( ~D = ǫ ~E,

~H = ~B/µ) that is free of sources (ρf = 0) with zero conductivity (σ = 0), the

solutions to Maxwell’s equations are

~E(~r, t) = Eoei(~k·~r−ωt)e, (5.7)

~H(~r, t) = Hoei(~k·~r−ωt)h, (5.8)

where ~D is the electric displacement field, ~H is the magnetizing field, E0 and H0

are constants, e and h are polarization unit vectors, ω is the angular frequency

and k is the wavenumber. The angular frequency and wavenumber are related by

k = ω√

ǫµ, (5.9)

5. Discussion 138

and the refractive index (n) is defined as

n =

√

ǫµ

ǫoµo

. (5.10)

For materials in which the dielectric response is frequency dependent, the refrac-

tive index varies with frequency and this is termed wavelength dispersion.

Equation (5.10) is a simple model that ignores the material’s conductivity

(i.e. the conduction electrons) and accounts only for the core electrons.88 With

the inclusion of conductivity, a more rigorous equation for the complex index of

refraction can be derived that includes the response of all electrons within the

material,

n =

√

ǫµ

ǫoµo

(

1 +iσ

ω

)

. (5.11)

The propagation of light through a medium can be described in terms of the

complex refractive index,88

n = n + iκ, (5.12)

where the real part of n is the normal refractive index (n) and the imaginary

part (κ) is the extinction coefficient. The real part of the refractive index refers

to the reduction of light velocity within a medium due to phase lag caused by

atomic oscillations.88 The extinction coefficient (κ) is related to the absorption

5. Discussion 139

in the medium. These two quantities are directly related to the electronic band

structure.

The dielectric function of a crystal lattice can be described using a classical

model that includes multiple oscillators each with their own resonant frequency.

These oscillators influence the propagation of light through that medium. Elec-

tronic resonances describe the interband absorption (band gap) required to excite

an electron from the valence to conduction band. These oscillations are in the

range from 1014 to 1015 Hz. In ionic materials (compounds in which the crys-

tal lattice is held together through ionic bonds) dipole oscillations of charged

atoms from their equilibrium positions gives rise to vibrational oscillations. These

molecular vibrations are associated with strong absorption in the infrared or the

frequency range of 1012 to 1013 Hz. Various other resonance modes exist within

a crystal lattice (such as free electron oscillators), but are not important in this

work.

The Lorentz oscillator89 model suggests that the dielectric function (ǫ) of a

crystal is related to the multiple resonances that occur,

ǫ(ω) = 1 + ω2p

∑

j

fj

ω2j − ω2 − iγjω

, (5.13)

where ωp is the plasma frequency, ω is the light frequency, ωj is the frequency of a

5. Discussion 140

particular resonance, fj is the strength of that resonance, and γj is the damping

of that resonance. The plasma frequency is defined as

ωp =

√

4πNe2

me, (5.14)

where N is the number of atoms per unit volume, e is the charge of an electron,

and me is the mass of an electron. In this model, the nucleus is assumed to be

immobile due to its large mass compared to that of the electron. The dielectric

function is related to the complex refractive index by equation 5.10. For materials

in the optical region the relative permeability (µ) is assumed to be unity and the

dielectric function is approximated by

ǫ ≈ n2. (5.15)

Using this model, the refractive index and absorption are estimated for LiF (shown

in figure 5.2.1). Two resonances with frequencies equal to 3.2 × 1015 Hz and

9.3× 1012 Hz are chosen to correspond with the electronic and vibrational modes

of LiF,90 respectively. The damping has been set to 5% of the central frequency.

The strength of each resonance is chosen such that the features are discernible

at that frequency. As the frequency approaches resonance, a discontinuity in

the refractive index is observed. The extinction coefficient (κ) is zero everywhere

5. Discussion 141

Fig. 5.2.1: The real part of the refractive index (n) and extinction

coefficient (κ) for a hypothetical solid using the electronic and vibration

modes of LiF determined using the Lorentz oscillator model.90 The

damping has been set to 5% of the central frequency and the strength

of transitions chosen to make the resonance features visible.

except in regions near the resonance. Passing through a resonance with increasing

frequency causes a reduction in the real part of the refractive index.

The LiF90 refractive index and extinction coefficient that have been experi-

mentally determined are shown in figure 5.2.2. Electronic and vibrational modes

are observed at 3.2×1015 Hz and 9.3×1012 Hz, respectively. The refractive index

measurements of this study were made at a wavelength of 532 nm or 5.64 × 1014

Hz. This is illustrated as the circle shown in figure 5.2.2. For these measure-

ments, the wavelength was bounded by the electronic and vibrational resonance

frequencies. Comparison of this data with the simple calculation performed using

5. Discussion 142

Fig. 5.2.2: Experimentally determined LiF index of refraction data.90

The extinction coefficient (κ) is of the order 10−8 in the transparent

region. The probe frequency used in this thesis (5.64×1014 Hz) is shown

as the white circle.

the Lorentz oscillator model shows that a model of this form can adequately de-

scribe the measured data. This requires knowledge of the resonance modes, the

strength of those resonances, and the damping associated with those modes. At

wavelengths near that of the probe beam, a linear relation between the refractive

index and frequency is observed. The single-oscillator model, proposed by Wem-

ple and DiDomenico,91 suggests that for large band-gap insulators, the optical

properties in the transparent region are dominated by the electronic resonance

at higher frequency. Application of this model to the LiF refractive index mea-

surements as a function of density indicates that the linear behavior is due to a

reduction in the band gap energy.91

5. Discussion 143

5.3 Single-Oscillator Model

The single-oscillator model91 has been successfully used to explain the optical

properties of various composite and amorphous materials at photon energies below

the electronic resonance (interband absorption edge).92–95 This model addresses

the frequency dependence of the dielectric function in the transparency region

(i.e. the region between the electronic and vibrational modes). It assumes that

the effects of vibrational modes on the optical properties are small compared to

the electronic modes and the former’s effect can be neglected. Due to the high

transparency in this region, the damping is assumed to be zero and the Lorentz

oscillator model (equation 5.13) reduces to the Kramers-Heisenberg,96

ǫ(ω) = 1 + ω2p

∑

j

fj

ω2j − ω2

, (5.16)

where ǫ is the dielectric response, ω is the frequency, ωp is the plasma frequency, ωj

is the frequency of a particular resonance and fj is the strength of that resonance.

The Kramers-Heisenberg dispersion relation represents the interband transitions

as individual oscillators where each electron contributes one mode to the dielectric

function.

Wemple and DiDomenico91 showed that the summation of all resonances near

the absorption edge can be approximated assuming ω < ωj . The first oscillator

5. Discussion 144

(f1/(ω21 − ω2)) is retained and the higher frequency oscillators are assumed to

occur at frequencies significantly greater than the probe frequency (ω ≪ ωj and

j > 1). The remaining oscillators in equation 5.16 are Taylor expanded,

∑

j 6=1

fj

ω2j

(

1 +ω2

ω2j

)

, (5.17)

and equation 5.16 takes the form

ǫ(ω) = 1 + ω2p

f1

ω21 − ω2

+ ω2p

∑

j 6=1

fj

ω2j

(

1 +ω2

ω2j

)

. (5.18)

Equation 5.18 reduces to the single-oscillator model by retaining terms to order

ω2,

ǫ(ω) − 1 ≈ F

E20 − (~ω)2

, (5.19)

where F and E0 are related to the combination of all oscillator strengths (fn)

and frequencies (ωn). Wemple and DiDomenico91 showed, using experimental

data, that F is related to the single-oscillator energy (E0) and a measure of the

strength of interband optical transitions (Ed) by

F = E0Ed, (5.20)

5. Discussion 145

leading to the relation

n2 − 1 =EdE0

E20 − ~2ω2

. (5.21)

Ed and E0 are determined from dispersion data in the transparent regime for ionic

and covalent materials.

If y = 1/(n2 − 1) and x = 1/(~ω)2, equation 5.21 can be written in the simple

form

y =E0

Ed− x

E0Ed. (5.22)

By plotting the dispersion in the form of 1/(n2−1) vs. (~ω)2, the single-oscillator

energy and the strength of optical transitions are determined using linear regres-

sion. The dispersion data for LiF in the optical range (black points) is shown

in figure 5.3.1 with the linear regression (red dashed line) used to determine the

single-oscillator strength and energy.97 The ambient values of Ed and E0 were

determined by fitting the refractive index to measured values97 in the range 332

nm < λ < 732 nm (the region near the probe laser). These values corresponded

to 16.66 eV and 15.38 eV for E0 and Ed respectively.

In a survey of over 100 solid and liquid insulators, the single-oscillator model

has been shown to fit the energy-dependent refractive index well. Wemple and

DiDomenico91 empirically found that the oscillator energy (E0) was approximately

5. Discussion 146

Fig. 5.3.1: LiF refractive index in the optical region used to determine

the single-oscillator energy (E0) and the strength of interband optical

transitions (Ed). The black points indicate the dispersion data97 and

the red dashed line is the linear regression to that data.

5. Discussion 147

related to the lowest direct band gap energy (Et) by

E0 ≈ 1.5Et, (5.23)

suggesting that the refractive index is directly related to the optical band gap

energy.

Equation 5.23 was re-examined specifically for the alkali-halides with the NaCl-

type crystal lattice.91 The oscillator energy (E0), the strength of interband transi-

tion (Ed), and the exciton energy (Ex) are shown in table 5.3.1. Et is compared to

the exciton energy. In ionic materials, the energy required to promote an electron

from the valence to conduction band is less than the band gap energy due to the

Coulomb attraction between the electron-hole pair. This attraction reduces the

required energy to promote an electron to the valence band and is termed the

exciton energy. It was found that the oscillator energy (E0) is best related to the

exciton energy (Ex) or the direct band gap (Et) for alkali-halides with NaCl-type

lattice structure by

E0 ≈ 1.36Ex. (5.24)

The single-oscillator model was applied to the pressure induced closing of the

H2 band gap over a density range exceeding an order of magnitude.92–94 Ten years

later, it was found that the simple model successfully predicted the emergence of

5. Discussion 148

Table 5.3.1: Dispersion parameters for the Alkali-Halides with NaCl-

type lattice structure.

Crystal E0 (eV)91 Ed (eV)91 Ex (eV)98 E0/Ex

LiF 16.7 15.4 12.9 1.29NaF 15 11.3 10.66 1.41KF 14.8 12.3 9.88 1.5

NaCl 10.3 13.6 7.96 1.29KCl 10.5 12.3 7.79 1.35RbCl 10.4 12.2 7.51 1.38CsCl 10.6 14 7.85 1.35KBr 9.2 12.4 6.71 1.37RbBr 9.1 12.1 6.64 1.37KI 7.7 12.8 5.88 1.31RbI 7.7 12.1 5.73 1.34

excitonic absorption into the visible.99 Taken together, these studies show that the

H2 exciton energy shifts from 14.5 eV to 2 eV with a slightly sublinear dependence

on density over nearly 15-fold compression. The single-oscillator model has also

been applied to compressed H2O ice, demonstrating a linear reduction of the band

gap over 2.3-fold compression.95 In that experiment, the refractive index was

determined over the wavelength range of 560 nm to 740 nm at various pressures.

Using this technique, Ed and E0 were determined over the pressure range of that

study. It was found that Ed was insensitive to changes in pressure.100 For both H2

and H2O, the data supports the assumptions that Ed is independent of density,

E0 is proportional to the minimum optical band gap energy, and the gap closes

nearly linearly with density.

5. Discussion 149

Fig. 5.3.2: Density dependence of the single-oscillator model (Et).

Weighted mean (black) and orthogonal fit (blue) with estimated error

bars. The exciton energy98 (green point) and Goldhammer-Herzfeld

metallization85, 86 (red point) are shown. Extrapolation suggests that

LiF remains transparent well above the Goldhammer-Herzfeld metal-

lization.

5. Discussion 150

In this work, the single-oscillator model is applied to the LiF refractive index

data at 532 nm ((~ω)2 = 5.43 eV) shown in figure 5.3.1. Fixing Ed to its ambient

value, E0 is calculated as a function of density using equation 5.21. The lowest

direct optical transition, Et, (proportional to the band gap) is determined using

the proportionality proposed in equation 5.24. The results are shown in figure

5.3.2. The refractive index data determined from the weighted mean is shown in

black, and the linear orthogonal fit in blue. At ambient pressure, Et corresponds

to the intense exciton (green circle) observed in ultra-violet absorption measure-

ments.101 The model suggests that the linear behavior of the refractive index as a

function of density is the result of a monoatomic decrease in the band gap energy.

The metallization pressure predicted by the Goldhammer-Herzfeld criterion85,86 is

shown as a red circle in the figure 5.3.2. Metallization is discussed in the following

section.

5.4 Metallization

Metallization is defined as zero band gap at absolute zero (0 k). This describes

the point at which the valence and conduction bands overlap and electrons are

free to travel throughout the crystal lattice. At sufficiently high pressures, all

insulators become metallic due to band gap closure. This is understood by ex-

amining the electronic band structure of a crystal lattice. As atoms are packed

closely together to form the crystal lattice, the orbitals of those atoms overlap and

5. Discussion 151

the discrete eigenstates broaden into degenerate energy bands. Electronic bands

of different energy arise due to the interaction among the electronic states of the

nearest neighbors. An example of the band broadening of Xenon under compres-

sion is shown in figure 5.4.1.102 The conduction (5s and 5p) and valence bands

(6s, 5d and 6p) are shown. Electrons become delocalized and mobile within the

conduction band. As the molar volume decreases (density increases) the 5s and

5p bands of Xenon broaden and the band gap decreases. At high compression,

the valence and conduction bands cross (intersection of 5p and 5d bands) and

compressed Xenon becomes metallic.

The earliest predictions of metallization were made almost simultaneously by

Goldhammer85 and Herzfeld.86 Both began with the Lorentz-Lorenz equation,88

n2 − 1

n2 + 2=

4π

3Naρα (5.25)

that relates the refractive index (n) to the density (ρ), the polarizability (α) and

Avogadro’s number (Na). The authors noted that a unity reflectivity (R) requires

an infinite refractive index. This is shown from the definition of reflectivity,

R =(n − 1

n + 1

)2

. (5.26)

5. Discussion 152

Fig. 5.4.1: Band structure energy calculation for Xenon with atomic la-

bels.102 Electron degeneracies are shown as the shaded region. Valence

bands (5s and 5p) broaden under compression. The band gap closes

between the valence and conduction band (5d) at high compression (∼11 cm3/mol).

5. Discussion 153

For a infinite refractive index, equation 5.25 reduces to

1 =4π

3Naρα, (5.27)

defining the condition for metallization. For constant polarizablities, this simple

model has been shown to be in good agreement with experimentally determined

metallization pressures for a variety of materials.103–105 However, discrepancies

have been observed when comparing Goldhammer-Herzfeld metallization pres-

sure with experimental results and band structure calculations.97,106, 107 These

discrepancies are attributed to the simplicity of the Goldhammer-Herzfeld model

as well as the assumption of constant polarizability.

Table 5.4.1 compares the predicted Goldhammer-Herzfeld metallization85,86

pressure with experimental results and band-structure calculations for various

materials. Experimental results indicate that change in the refractive index are

dominated by compression and insensitive to temperature changes. This is demon-

strated by similar results for shock and ramp compression where the changes in

temperature were significantly greater for shock compression, while refractive in-

dex results are identical. Therefore linear extrapolation of the data in figure 5.3.2

to zero band gap energy is a good estimate of the metallization pressure of LiF.

The first five compounds in the table correspond to alkali-halides and the

5. Discussion 154

remaining are noble gases. For potassium iodide (KI) the Goldhammer-Herzfeld

criterion is a good approximation to the metallization pressure determined using

optical absorption. However, for rubidium iodide (RbI) the results indicate that

this model can differ by up to 50% from the experimentally determined optical

absorption value. Therefore, the discrepancy between the Goldhammer-Herzfeld

criterion for LiF and the calculated metallization pressure of this study is not

unreasonable.

Table 5.4.1: Metallization Pressure for Various Materials

Goldhammer- Optical Absorption Band StructureHerzfeld85,86 Measurements PredictionsPredictions

ρ (g/cc) P (GPa) ρ (g/cc) P (GPa) ρ (g/cc) P (GPa)LiF97 11.2 2,860 – – >14.2† >5,000†

NaF97,104 8.8 300 – – 9.9 455RbI97,106, 107 10.4 127 9.62 85 10.4 122KI97,106, 107 8.85 125 8.7 115 9.51 155CsI97,108, 109 10.7 87 – 111 9.02 100He110 8 5,000 – – 21.3 25,700Ne111 20 6,000 – – 78.8 158,000Ar112,113 9.6 – – – < 7.1 <230Kr114,115 13.3 – – – 12.8 316Xe114,116 12.9 150 – 150 12.9 132† Extrapolation of this study.

The anion and cation that comprise the alkali-halides are isoelectronic with

the noble gases. Isoelectronicity is defined as two elements or ions which posses

the same number of electrons or the same electronic configuration. The simi-

5. Discussion 155

Table 5.4.2: Band Gap Energy for Various Materials

Alkali-Halides EG (eV) Noble Gas EG (eV)LiF98 14.2 He110 19.8NaF98 11.5 Ne117 21.5KF98 10.8 Ar118 14.3RbF98 10.3 Kr118 11.7CsI117 6.1 Xe117 9.32

lar metallization pressures of CsI and Xe can be attributed to their comparable

electronic configuration and band gap energies (table 5.4.2). Of the all alkali-

halides, LiF has the highest predicted metallization pressure (table 5.4.1). Li+

and F− are isoelectronic with helium (He) and neon (Ne), respectively. He and

Ne have the largest band gap and the highest metallization pressures predicted

of all monatomic materials due to the predicted intershell band overlap.119 Band

structure calculations performed by Boettger119 found that metallization of He

and Ne is due to (n+1,l+1) conduction band overlapping with the (n,l) valence

band. The metallization of the heavy noble gases (Argon, Krypton, Xenon) occurs

at significantly lower pressures due to intrashell band overlap of the (n,l) valance

band with the (n,l+1) conduction band.119

Augmented plane-wave band structure calculations were performed on LiF.120

The calculations determine the electronic band structure by approximating the

electron energy states within the crystal lattice using spherical potentials centered

at each atom with constant potentials in the interstitial region.88 Calculations in-

5. Discussion 156

dicate that the valence band consists of the 2p state of F− and the conduction

band is comprised of the 2s state of Li+, 2p state of Li+ and the 3d state of

F−.120 An extension of Boettger’s119 results suggests that metallization of LiF

occurs when the 2p state of F− crosses the 3d state of F−. Band structure cal-

culations of LiF would provide insight into the band overlapping processes that

cause metallization.

All alkali-halides, except LiF and sodium fluoride (NaF), contain at least one

species that is isoelectronic with a heavy noble gases. The heavy noble gases

exhibit significantly lower metallization pressures than the light noble gases, as

shown in table 5.4.1. The cation and anion of NaF are isoelectronic with Ne;

the monoatomic element with the highest metallization pressure. However, band

structure calculations indicate that the metallization pressure is 455 GPa,104 well

below that of Ne and LiF. Comparing the band gap energies for those materials

(Ne, LiF, and NaF), NaF has the smallest band gap (table 5.4.2), suggesting that

the electronic band structure plays a crucial role in the metallization pressure.

The low metallization pressure of NaF may also be attributed to the additional

energy bands contributed by the Na− cation (2p) when compared to the Li− anion

in LiF. Materials being isoelectronic with the light noble gases is not sufficient in

producing high metallization pressure. The high metallization pressure of LiF is

5. Discussion 157

attributed to the large band gap energy (largest of all the alkali-halides) and its

isoelectronicity with He and Ne.


The linear dependence of the refractive index and density of LiF was examined

using the single-oscillator model. This model predicted that the linear dependence

of the refractive index on density is the result of monoatomic closure of the band

gap. Extrapolation of these results suggests that metallization will occur at 4200

GPa, well above the Goldhammer-Herzfeld criterion.85,86 Comparison of LiF with

its isoelectronic counterparts (He and Ne) suggests that the high metallization

pressure is attributed to the intershell band overlap. The high metallization pres-

sure indicates that LiF will remain transparent to pressures at least six times

higher than observed in this study. If true, LiF will prove to be a valuable win-

dow material for extremely high pressure ramp compression experiments.

6. Conclusion 158

6. CONCLUSION

The optical properties of LiF under extreme pressure were examined in this

study. Using direct-drive shaped laser ablation, LiF was ramp compressed to

800 GPa at the OMEGA laser facility. A specially designed two section target

was used to determine the refractive index of ramp compressed LiF. The target

consisted of a diamond pusher with a LiF window mounted on half of the rear

surface. Laser pulse profiles were designed to prevent shock formation in both

sections of the target. VISAR measurements were made simultaneously in both

sections of that target determining the free surface and apparent particle velocity.

The method of characteristics and the free surface velocity was used to determine

the true particle velocity. The accuracy of that technique was verified through

LASNEX simulations.82 The relation between the apparent and true particle

velocity determined the refractive index as a function of density.

LiF remained transparent up to 800 GPa, pressure seven times higher than

previous shock experiments. Under strong shock compression, transparent insu-

lators transform into conducting fluids as a result of pressure-induced reduction

6. Conclusion 159

of the band gap and thermal promotion of electrons across that gap. The reduced

temperature of ramp compression enable significantly higher pressure to achieved

in LiF while remaining transparent. The refractive index was measured from 30

to 800 GPa; pressure seven times higher than previous shock experiments. As was

found with low pressure shock experiments, the refractive index depends linearly

with density up to 800 GPa. These are the highest pressure refractive index mea-

surements made to date. Measurements indicate that LiF temperature remained

below 1,000 K in these experiments which is consistent with equation of state

predictions.

A single-oscillator model was used to infer the pressure-induced band gap clo-

sure of ramp compressed LiF. Results indicate that the linear behavior of the

refractive index on the density is a direct result of pressure-induced closure of the

band gap. Extrapolation of these results indicates that LiF will remain trans-

parent to at least 5,000 GPa, well above the Goldhammer-Herzfeld criterion of

∼ 2,860 GPa. The high metallization pressure of LiF is attributed to its large

band gap and a structure that is isoelectronic with helium and neon. Helium and

neon have the highest metallization pressure of all monatomic materials due to

the predicted intershell band overlap. From those predictions, it is hypothesized

that metallization of LiF will occur when the valance 2p state of F- crosses with

the 3d conduction band of F-.

6. Conclusion 160

The high pressure transparency of LiF has technical utility at high pressure as

an optical window for material studies. The transparency at high pressure allows

in situ measurements of samples confined by that window. The observed trans-

parency and measurement of LiF refractive index to 800 GPa enables advancing of

those experiments to higher pressure regimes. Extrapolation the single-oscillator

model indicates that LiF will remain transparent to pressures at least six times

higher. If true, LiF will prove to be a valuable window material for high pressure

science.

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Appendix 182

APPENDIX

A. Direct Drive Laser Ablation Scaling 183

A. DIRECT DRIVE LASER

ABLATION SCALING

The scaling of ablation pressure to laser intensity is determine for ramp com-

pression of diamond targets irradiated by 351 nm light. The pressures are gener-

ated in diamond samples by direct laser ablation in the range of 100 to 970 GPa.

The diamond free surface velocity measurements are backwards integrated to the

ablation front to determine the ablation pressure. The laser intensity on target is

determined by optical streak camera power measurements and fully characterized

smooth laser spots. The ablation pressure (P ) is found to scale with the laser

intensity (I) as P [GPa] = 42[±3](I[TW/cm2])0.75[±0.01].

A.1 Introduction

High intensity lasers are increasingly used in the study of matter under ex-

treme conditions by creating loading through laser ablation. Accurate knowledge

of the scaling of ablation pressure with laser intensity is important for the design


of those experiments. A variety of techniques have been used to determine the

ablation parameters of materials such as x-ray spectroscopy,121 time-resolved x-

ray radiography,122 layered-target burn-through measurements,123 time-resolved

streak record of visible emission124 and velocity measurements.125 Those studies

were often plagued with problems resulting from the presence of laser hot spots

and edge effects associated with small laser spots required to reach high inten-

sities. With the development of laser smoothing techniques and more powerful

lasers,61,62, 126 the presence of hot spots and edge effects was diminished allowing

a more accurate determination of the scaling laws.127

Dynamic-loading experiments driven by lasers contain two classes: shock com-

pression and ramp compression. Until recently, shock compression experiments

were more predominant than ramp compression due to the pulse-shaping require-

ments for the former.58 Recent advances in the ability to control the temporal

shape of laser pulses have enabled shaped-ablation ramp compression with high

precision. The experiments described in this thesis demonstrate ramp compres-

sion by direct-drive shaped ablation. Here the laser directly irradiates the sample;

there is no intermediate energy conversion nor coupling. The ablation pressure is

controlled by temporal shaping of the laser intensity in time. Ramp compression

allows for the pressure scaling to be observed over a range of pressures and not

for a single point that is customary for shock-wave studies.


Recently, direct-drive shaped ablation was developed to ramp compress tar-

gets.66 Free surface velocity measurements and knowledge of the target equation

of state enable the determination of the pressure profile within the sample.56,57

Design of such experiments requires accurate knowledge not only of the targets

thermodynamic properties, but also the laser-intensity to ablation pressure scal-

ing. Accurate laser-intensity ablation pressure scaling is required to achieve a

desired pressure profile over a prescribed distance without generating shocks.58

The laser-intensity ablation scaling is determined for diamond in the next section.

The diamond free surface velocity is backwards integrated to determine the ab-

lation pressure. That calculated drive pressure is compared to the intensity on

target, determined by the optical streak camera power measurements. An appro-

priate scaling law, using the functional form proposed by Lindl,128 is determined

and compared with previous results.

A.2 Experimental Technique

The experiments were performed on the OMEGA laser61 at the University of

Rochester’s Laboratory for Laser Energetics. Pressure profiles that compressed

diamond from 100 to 970 GPa in 3.7 ns were produced using laser pulses that

ramped from to 3.0 × 1012 to 7.1 × 1013 W/cm2 with a ∼ t3 shape. The target

specifications for the twelve experiments conducted were shown in table A.2.1.


The 351 nm wavelength laser light is smoothed using distributed phase plates62

to produce spots that had a planar region of about 876 µm in diameter.68

Table A.2.1: Shot Specifications

Shot Pulse Shape Peak Intensity Peak Ablation† Thickness(TW/cm2) Pressure (GPa) (µm)

54939 RM3503 44 660 4354940 RM3503 57 850 4654944 RM3503 57 810 4654945 RM3503 71 970 4654946 RM3503 56 710 4654948 RM3503 57 770 4356112 RM3503 53 710 4556113 RM3503 52 720 4557575 RM3702 19 370 4657576 RM3702 28 440 4557577 RM3702 24 420 4457579 RM3702 31 540 45.5† Determined using the method of characteristics.56,57

The diamond thicknesses for 3.7 ns ramp-compression experiments are in-

cluded in table A.2.1. The cross section of diamond samples ranged from 1.1

mm to 2.5 mm square. Chemical vapor deposition (CVD) and ultra-pure nano-

crystalline diamond were compressed with no discernable difference observed in

their ablation pressure scaling. A 1000 A coating of aluminum was applied to the

ablation side of the diamond to prevent the low-intensity leading edge of the laser

from penetrating the target before the ablation plasma is formed.

The diamond free surface velocities were measured using a 532-nm probe beam


and a line-imaging velocity interferometer for any reflector (VISAR) discussed in

detail in Section 3.4.1.33,35 This device detects the Doppler shifts of the probe

light reflected off of the moving portions of the target. That light is imaged

through a Mach-Zehnder interferometer onto the slit of an optical streak camera

producing a series of fringes streaked in time. The position (phase) of those fringes

is proportional to the velocity of the reflecting surface. The fringe position is

measured within an accuracy of 2.5% of a fringe using Fourier analysis of the streak

record. This is confirmed by the excellent agreement between the velocities derived

from the two (redundant) VISAR channels. The free surface measurements are

backwards integrated using the method of characteristics56,57 to determine the

ablation pressure as discussed in Section 3.3.1. The diamond isentrope measured

by Bradley59 up to 800 GPa and extrapolated to higher pressures was used. The

minimum achievable pressure in these experiments is 100 GPa due to the elastic

limit of diamond.59 At pressures below that, the intermediate two-wave structure

of the elastic limit at lower pressures introduces large uncertainties in the method

of characteristics. The pulse shapes are designed such that the elastic limit is

reached, followed by ramp compression. To determine the ablation scaling law,

the free surface measurements are backwards integrated to the ablation front.129

LASNEX82 simulations estimate the ablation front to be 3 µm (± 1µm) from the

diamond loading surface


The laser intensity is determined by the optical streak camera power measure-

ments and the fully characterized smooth laser spots.68 Laser power waveforms

are obtained using P510 streak cameras,130 where the P510 indicates the Phillips

streak tube used within the camera. The camera directly measures the power of

each beam on target, and is calibrated to within 50 ps of the laser arrival time

at target. The intensity on target is calculated using focal spot produced by the

distributed phase plates and accounting for the beam angle relative to the normal

of the diamond sample. Up to twelve beams irradiate the target. The beams

are grouped into two clusters with angles of 23.2(±0.1) and 47.8(±0.1). The

beam intensity on target is modeled using a Super-Gaussian functional depen-

dence (e−r2/r2

0) with r0 = 438µm.

A.3 Results

The calculated ablation pressure versus laser intensity is shown in figure A.3.1

with estimated error bars. Linear regression is performed to determine the weighted

mean of all twelve shots (as performed in Section 4.2) and is shown in black. Ac-

cording to Lindl,128 the ablation pressure (P ), scales with laser intensity (I) as

P = P0IC0 , (A.1)


Fig. A.3.1: Ablation Pressure vs. Laser Intensity. Results of all twelve

shots are shown as the shaded blue lines with error bars. The regression

of that data using the function form proposed by Lindl128 is shown in

black.

where P0 is the ablation scaling constant and C0 is less than 1. These constants

are determined by “linearizing” equation A.1,

ln P = lnP0 + C0 ln I, (A.2)

and performing linear regression of the weighted mean. The functional form for

direct drive laser ablation of diamond at the wavelength of 351 nm is

P [GPa] = 42[±3](

I[TW/cm2])0.75[±0.01]

, (A.3)

and is shown as the black line of figure A.3.1.


Fig. A.3.2: Comparison of the ablation pressure determined from the

method of characteristics (red line) and the ablation pressure deter-

mined using the scaling law (equation A.3) from the calculated laser

intensity (blue line) for shot 54944.

Table A.3.1: Laser Ablation Scaling

Ref. λ Pulse Intensity Type Mat. P (GPa)†‡

(nm) (ns) (W/cm2)Fabbro131 260 0.5 1013 − 1015 Planar Al 56.9 × I0.75

ShockGoldsack121 530 1.0 7 × 1012− Spherical Al 18.3 × I0.95

6 ×1013 Shock1050 1.0 7 × 1012− Spherical Al 13.1 × I1.14

6 ×1013 Shock

Dahmani132 1060 0.7 1013 − 1015 Planar C & Si 30 × I2/3

Shock ×Z3/16

This Work 351 3.8 5 × 1012− Planar Dia. 42 × I0.75

7 ×1013 Ramp† I is in units TW/cm2.‡ Z is the atomic number.


Figure A.3.2 compares the ablation pressure calculated from the method of

characteristics (red) and the ablation pressure determine using equation A.3 (blue)

for shot 54944. Excellent agreement is observed between these two measurements.

Various studies have been performed to determine the laser ablation scaling law

for planar and spherical shock experiments. The results of those experiments are

summarized in table A.3.1.

Aluminum planar shock results of Fabbro et al.131 are in close agreement with

the diamond ramp compression scaling results of this thesis. The difference in the

wavelength of these studies is accounted for using the functional form proposed

by Lindl.128 Lindl finds that for direct drive the ablation pressure scales as

P ∼(I

λ

)2/3

, (A.4)

where P is the ablation pressure, I is the laser intensity and λ is laser wavelength.

Using equation A.4 and replacing the exponential with the experimental result,

the functional form proposed by Fabbro for aluminum, adjusted for the wavelength

dependence, becomes

P [GPa] = 20.7(I[TW/cm2]

λ[µm]

)0.75

, (A.5)


and ablation pressure scaling for ramp compressed diamond is

P [GPa] = 19[±1](I[TW/cm2]

λ[µm]

)0.75[±0.01]

. (A.6)

Excellent agreement is observed for the ablation scaling of aluminum and diamond.

A.4 Conclusion

An ablation pressure scaling law has been determined for direct-drive ramp

compression using diamond targets over the range of 100 to 970 GPa. Ablation

pressure is determined by backwards integrating free surface diamond velocity

measurements using VISAR to the ablation front determined from LASNEX sim-

ulations. The laser intensity is calculated from the power-on-target measurements

and the characterized laser spot. The ablation pressure (P ) and laser intensity (I)

scales as P [GPa] = 42[±3](I[TW/cm2])0.75[±0.01] for the laser wavelength of 351

nm. The scaling law is in agreement with the functional form proposed by Lindl

and corresponds well with previous direct drive planar experiments of aluminum

at a similar laser wavelength and intensity.

B. Weighted Mean Values 193

B. WEIGHTED MEAN VALUES

As a matter of record, the values determined from the weighted mean of the

apparent and true particle velocity are tabulated in this Appendix (table B.1.1).

The first fifty values correspond to shock refractive index measurements made by

Wise,34 Lalone54 and Jensen.53 The remaining values correspond to the weighted

mean determined in Chapter 4. The number of data points in the weighted mean

has been reduced by quoting every tenth value.

Table B.1.1: Apparent and True Weighted Mean Values

True Velocity Apparent Velocity True Velocity Apparent Velocity

(µm/ns) (µm/ns) (µm/ns) (µm/ns)

0.48 0.59 ± 0.008 5.57 7.09 ± 0.075

0.31 0.38 ± 0.008 5.59 7.14 ± 0.081

0.58 0.73 ± 0.008 5.62 7.19 ± 0.082

0.38 0.48 ± 0.003 5.65 7.24 ± 0.086

0.19 0.25 ± 0.003 5.68 7.29 ± 0.094

0.10 0.13 ± 0.003 5.71 7.34 ± 0.095

0.25 0.32 ± 0.003 5.74 7.39 ± 0.102

0.74 0.95 ± 0.018 5.78 7.44 ± 0.108

0.98 1.25 ± 0.008 5.82 7.49 ± 0.107

0.34 0.43 ± 0.002 5.86 7.54 ± 0.111

0.51 0.64 ± 0.004 5.90 7.59 ± 0.127

0.45 0.58 ± 0.002 5.96 7.64 ± 0.134


Table B.1.1: Apparent and True Weighted Mean Values (cont.)


0.67 0.85 ± 0.003 6.01 7.69 ± 0.134

0.69 0.88 ± 0.007 6.07 7.74 ± 0.152

1.01 1.28 ± 0.014 6.05 7.79 ± 0.138

0.11 0.14 ± 0.010 6.10 7.84 ± 0.13

0.30 0.38 ± 0.010 6.15 7.89 ± 0.137

0.65 0.82 ± 0.010 6.22 7.94 ± 0.131

0.66 0.85 ± 0.010 6.28 7.99 ± 0.125

0.66 0.84 ± 0.010 6.33 8.04 ± 0.135

0.70 0.90 ± 0.010 6.37 8.09 ± 0.136

0.70 0.89 ± 0.010 6.42 8.14 ± 0.129

0.70 0.89 ± 0.010 6.47 8.19 ± 0.135

0.81 1.03 ± 0.010 6.51 8.24 ± 0.124

0.84 1.07 ± 0.010 6.55 8.29 ± 0.116

0.84 1.06 ± 0.010 6.60 8.34 ± 0.122

0.89 1.14 ± 0.010 6.65 8.39 ± 0.111

0.98 1.25 ± 0.010 6.69 8.44 ± 0.101

0.98 1.25 ± 0.010 6.71 8.49 ± 0.11

0.99 1.26 ± 0.010 6.76 8.54 ± 0.102

1.00 1.30 ± 0.016 6.82 8.59 ± 0.094

1.01 1.29 ± 0.010 6.87 8.64 ± 0.102

1.01 1.31 ± 0.013 6.91 8.69 ± 0.096

1.12 1.43 ± 0.012 6.94 8.74 ± 0.097

1.13 1.45 ± 0.010 6.95 8.79 ± 0.110

1.13 1.44 ± 0.010 7.03 8.84 ± 0.119

1.14 1.47 ± 0.010 7.06 8.89 ± 0.128

1.36 1.74 ± 0.010 7.08 8.94 ± 0.150

1.36 1.74 ± 0.010 7.09 8.99 ± 0.135

1.37 1.76 ± 0.010 7.11 9.04 ± 0.132

1.37 1.76 ± 0.010 7.11 9.09 ± 0.146

1.38 1.77 ± 0.010 7.12 9.14 ± 0.131




1.39 1.8 ± 0.024 7.16 9.19 ± 0.130

1.39 1.79 ± 0.010 7.22 9.24 ± 0.173

1.41 1.82 ± 0.010 7.23 9.29 ± 0.147

2.64 3.4 ± 0.010 7.26 9.34 ± 0.139

3.78 4.82 ± 0.026 7.27 9.39 ± 0.157

3.78 4.81 ± 0.045 7.30 9.44 ± 0.138

4.09 5.27 ± 0.017 7.34 9.49 ± 0.137

4.09 5.28 ± 0.033 7.36 9.54 ± 0.159

1.52 1.84 ± 0.103 7.40 9.59 ± 0.140

1.58 1.89 ± 0.105 7.45 9.64 ± 0.136

1.60 1.94 ± 0.111 7.48 9.69 ± 0.155

1.62 1.99 ± 0.114 7.51 9.74 ± 0.135

1.64 2.04 ± 0.119 7.56 9.79 ± 0.129

1.65 2.09 ± 0.127 7.58 9.84 ± 0.147

1.67 2.14 ± 0.137 7.61 9.89 ± 0.128

1.70 2.19 ± 0.145 7.71 9.94 ± 0.155

1.75 2.24 ± 0.044 7.73 9.99 ± 0.183

1.84 2.29 ± 0.043 7.76 10.04 ± 0.158

1.87 2.34 ± 0.042 7.79 10.09 ± 0.148

1.90 2.39 ± 0.044 7.81 10.14 ± 0.175

1.96 2.44 ± 0.042 7.86 10.19 ± 0.149

1.99 2.49 ± 0.049 7.91 10.24 ± 0.137

2.02 2.54 ± 0.050 7.92 10.29 ± 0.169

2.05 2.59 ± 0.055 7.97 10.34 ± 0.141

2.09 2.64 ± 0.055 7.99 10.39 ± 0.133

2.10 2.69 ± 0.056 8.11 10.44 ± 0.188

2.14 2.74 ± 0.055 8.15 10.49 ± 0.160

2.18 2.79 ± 0.052 8.20 10.54 ± 0.146

2.23 2.84 ± 0.053 8.23 10.59 ± 0.207

2.24 2.89 ± 0.069 8.26 10.64 ± 0.170




2.28 2.94 ± 0.062 8.30 10.69 ± 0.154

2.31 2.99 ± 0.057 8.44 10.74 ± 0.210

2.36 3.04 ± 0.058 8.52 10.79 ± 0.142

2.42 3.09 ± 0.044 8.54 10.84 ± 0.123

2.47 3.14 ± 0.046 8.55 10.89 ± 0.19

2.51 3.19 ± 0.049 8.61 10.94 ± 0.163

2.56 3.24 ± 0.049 8.66 10.99 ± 0.118

2.64 3.29 ± 0.047 8.80 11.04 ± 0.071

2.68 3.34 ± 0.049 8.81 11.09 ± 0.059

2.71 3.39 ± 0.051 8.82 11.14 ± 0.055

2.75 3.44 ± 0.053 8.83 11.19 ± 0.058

2.78 3.49 ± 0.050 8.84 11.24 ± 0.056

2.81 3.54 ± 0.050 8.88 11.29 ± 0.057

2.84 3.59 ± 0.050 8.89 11.34 ± 0.059

2.88 3.64 ± 0.045 8.90 11.39 ± 0.058

2.91 3.69 ± 0.048 8.87 11.44 ± 0.111

2.95 3.74 ± 0.054 8.88 11.49 ± 0.137

2.98 3.79 ± 0.049 8.93 11.54 ± 0.108

3.04 3.84 ± 0.057 8.94 11.59 ± 0.128

3.07 3.89 ± 0.055 8.94 11.64 ± 0.148

3.11 3.94 ± 0.051 8.99 11.69 ± 0.111

3.16 3.99 ± 0.054 9.21 11.74 ± 0.209

3.25 4.04 ± 0.058 9.18 11.79 ± 0.872

3.28 4.09 ± 0.053 9.25 11.84 ± 0.128

3.31 4.14 ± 0.056 9.28 11.89 ± 0.215

3.34 4.19 ± 0.052 9.26 11.94 ± 0.075

3.38 4.24 ± 0.047 9.31 11.99 ± 0.133

3.42 4.29 ± 0.048 9.35 12.04 ± 0.216

3.45 4.34 ± 0.049 9.33 12.09 ± 0.089

3.48 4.39 ± 0.047 9.38 12.14 ± 0.138




3.51 4.44 ± 0.052 9.42 12.19 ± 0.211

3.54 4.49 ± 0.057 9.41 12.24 ± 0.105

3.57 4.54 ± 0.057 9.46 12.29 ± 0.137

3.60 4.59 ± 0.059 9.48 12.34 ± 0.205

3.62 4.64 ± 0.062 9.49 12.39 ± 0.126

3.65 4.69 ± 0.062 9.53 12.44 ± 0.136

3.68 4.74 ± 0.065 9.56 12.49 ± 0.214

3.70 4.79 ± 0.064 9.59 12.54 ± 0.146

3.73 4.84 ± 0.064 9.62 12.59 ± 0.131

3.77 4.89 ± 0.064 9.62 12.64 ± 0.214

3.81 4.94 ± 0.059 9.69 12.69 ± 0.160

3.86 4.99 ± 0.061 9.70 12.74 ± 0.115

3.92 5.04 ± 0.061 9.70 12.79 ± 0.22

3.97 5.09 ± 0.066 9.83 12.84 ± 0.173

4.10 5.14 ± 0.087 9.82 12.89 ± 0.101

4.15 5.19 ± 0.092 9.87 12.94 ± 0.199

4.18 5.24 ± 0.095 10.00 12.99 ± 0.142

4.22 5.29 ± 0.096 10.05 13.04 ± 0.062

4.27 5.34 ± 0.099 10.40 13.09 ± 0.124

4.30 5.39 ± 0.097 10.55 13.14 ± 0.394

4.35 5.44 ± 0.103 10.33 13.21 ± 0.486

4.39 5.49 ± 0.104 10.39 13.29 ± 0.467

4.43 5.54 ± 0.106 10.45 13.36 ± 0.449

4.45 5.59 ± 0.104 10.52 13.44 ± 0.427

4.49 5.64 ± 0.105 10.58 13.51 ± 0.406

4.53 5.69 ± 0.106 10.64 13.59 ± 0.379

4.57 5.74 ± 0.106 10.70 13.66 ± 0.352

4.62 5.79 ± 0.107 10.75 13.74 ± 0.325

4.64 5.84 ± 0.105 10.81 13.81 ± 0.296

4.69 5.89 ± 0.110 10.86 13.88 ± 0.265




4.82 5.94 ± 0.101 10.91 13.96 ± 0.250

4.86 5.99 ± 0.093 10.95 14.04 ± 0.242

4.90 6.04 ± 0.085 10.99 14.11 ± 0.239

4.91 6.09 ± 0.083 11.03 14.19 ± 0.240

4.90 6.14 ± 0.080 11.07 14.26 ± 0.240

4.94 6.19 ± 0.079 11.11 14.34 ± 0.232

4.97 6.24 ± 0.081 11.16 14.41 ± 0.216

4.98 6.29 ± 0.076 11.08 14.49 ± 0.244

4.99 6.34 ± 0.073 11.12 14.56 ± 0.215

5.01 6.39 ± 0.073 11.18 14.63 ± 0.197

5.03 6.44 ± 0.068 11.24 14.71 ± 0.176

5.07 6.49 ± 0.069 11.38 14.84 ± 0.355

5.10 6.54 ± 0.072 11.50 15.00 ± 0.295

5.15 6.59 ± 0.069 11.65 15.15 ± 0.219

5.19 6.64 ± 0.069 11.85 15.30 ± 0.197

5.27 6.69 ± 0.067 11.30 15.45 ± 3.272

5.32 6.74 ± 0.062 11.50 15.60 ± 2.780

5.37 6.79 ± 0.060 11.64 15.75 ± 2.376

5.40 6.84 ± 0.061 11.84 15.90 ± 2.212

5.43 6.89 ± 0.061 12.04 16.05 ± 1.653

5.47 6.94 ± 0.064 12.21 16.20 ± 1.188

5.50 6.99 ± 0.072 12.37 16.35 ± 0.879

5.53 7.04 ± 0.073 12.61 16.50 ± 0.473

C. LiF Shock Release 199

C. LIF SHOCK RELEASE

This appendix describes a new technique for determining the shocked index

of refraction of an insulator. Experiments were conducted by Hye-Sook Park

of Lawrence Livermore National Laboratory to determine the effects of shock-

released LiF windows on optical interferometry measurements. A planar shock

was driven through a tantalum target with two LiF windows of different thickness

attached to rear surface. Simultaneous measurements were made through a thin

and thick LiF window. Shock breakout was observed in the thin window and

compared to measurements made through the thick window, where the shock is

still within the solid LiF. The effects of that shock breakout on optical interferom-

etry measurements were examined by comparing the measured velocities in both

sections. At shock breakout, a discontinuity in the VISAR record is observed.

The derivation shown below indicates that this discontinuity is directly related

to the shocked refractive index of the window. A new measurement technique to

determine the refractive index in that released material is proposed.


C.1 Experimental Design

Indirect-drive reservoir-coupled compression, shown in figure C.1.1, was used

to compress two LiF samples of differing thickness. In this design, a gold hohlraum

is driven using 40 OMEGA61 beams. The pulse shape is a 1 ns square pulse and

the total energy on target is 20,000 J. The laser drive generates x-rays that drive

the ablation of a 25 µm beryllium anvil. A shock is generated in the beryllium

that is impedance matched into a 12.5% BrCH 200 µm thick reservoir. The

shocked reservoir material eventually rarefies as it propagates across a 400 µm

gap and stagnates at a 10 µm tantalum pusher. This stagnation produces ramp

compression in the tantalum that in turn compresses the LiF windows. Two LiF

windows, of differing thicknesses, are attached to the rear surface of the tantalum.

One window is sufficiently thick such that compression waves do not reach the

rear surface prior to the conclusion of the experiment. The second window is 10

times thinner than the thick window. The compression waves in that window

reach the rear surface and release to zero pressure prior to the conclusion of the

experiment.

VISAR measurements are made simultaneously through both windows to de-

termine the effects of the released window on interferometry measurements. The

tantalum/LiF interface motion is measured using a 532-nm probe beam and a


Fig. C.1.1: Shock Release Target Design.

line-imaging velocity interferometer for any reflector (VISAR) discussed in detail

in Section 3.4.1.33,35 This device detects the Doppler shifts of the probe light

reflected off of the moving portions of the target. That light is imaged through a

Mach-Zehnder interferometer onto the slit of an optical streak camera producing

a series of fringes streaked in time. The position (phase) of those fringes is pro-

portional to the velocity of the reflecting surface. The fringe position is measured

within an accuracy of 2.5% of a fringe using Fourier analysis of the streak record.

A single experiment∗ (shot 58815) was conducted. In that experiment, the

compression profile of the tantalum was not properly controlled and a shock was

generated in the tantalum and LiF windows. The VISAR data for shot 58815 is

shown in figure C.1.2. At shock breakout of the thin LiF window, a discontinu-

∗ This experiment was designed and performed by Hye-Sook Park of Lawrence LivermoreNational Laboratory.


Fig. C.1.2: VISAR data for shot s58815.


ity in fringe position is observed. The fringe discontinuity can be understood by

deriving the correction to the measured apparent particle velocity when observed

through an optical window at times prior to and coincident with shock break-

out. The difference between these observed apparent particle velocities defines

the discontinuity at breakout.

C.2 Unsteady Shock Breakout of an Optical

Window

The discontinuity in fringe position is explained by examining the optical path

length of the LiF window prior to and coincident with shock breakout. The change

in optical path length determines the observed discontinuity. In this derivation,

no assumption is made regarding the shock steadiness.

C.2.1 Prior to Shock Breakout

Prior to the shock breakout (t < tB.O.), the shock position in the window is

shown in figure C.2.1. Recall from Section 3.1 that the apparent particle velocity

(Uapp(t)) is determined by the time derivative of the integral of the optical path

length (equation 3.2) or

UApp(t) =d

dt

[

∫ xint(t)

xfs(t)

n(x, t)dx]

+ Ufs(t), (C.1)


Fig. C.2.1: Shock propagating through an optical window.

where xint(t) is the interface position, xfs(t) is the free surface position, nS(x, t) is

the compressed refractive index of the optical window, and Ufs(t) is window the

free surface velocity. Prior to shock breakout, the window free surface is stationary

and equation C.1 reduces to

UApp(t < tB.O.) =d

dt

[

∫ xint(t)

xfs

n(x, t)dx]

. (C.2)

The optical path length of the shocked (nS(x, t)) and unshocked (n0) regions is

expressed as

UApp(t < tB.O.) =d

dt

[

∫ xint(t)

xD(t)

nS(x, t)dx +

∫ xD(t)

xfs

n0dx]

, (C.3)

where xD(t) defines the shock front position and n0 is the initial refractive index

of the window. Leibniz integral rule relates the differentiation of an integral whose


Fig. C.2.2: Shock Breakout of an Optical Window.

limits are functions of the differential variable,133

d

dt

∫ V (t)

U(t)

f(x, t)dx = V ′(t)f(V (t), t) − U ′(t)f(U(t), t) +

∫ V (t)

U(t)

∂f(x, t)

∂tdx. (C.4)

Applying equation C.4 to equation C.3 gives the relation for the observed apparent

particle velocity or

Uapp(t < tB.O.) = Uint(t)nS(xint, t) − D(t)nS(xD, t) (C.5)

+

∫ xint(t)

xD(t)

∂nS(x, t)

∂tdx + n0D(t), (C.6)

where D(t) is the shock velocity and Uint(t) is the true interface velocity. The

integral term in equation C.6 is directly related to the shock steadiness.

C.2.2 Shock Breakout

When the shock reaches the free surface (t = tB.O.), as shown in figure C.2.2,

a change in VISAR fringe position is observed. Rederiving the apparent particle


velocity, the free surface (xfs) is no longer stationary (Ufs 6= 0). Recall equation

C.1,

UApp(t = tB.O.) =d

dt

[

∫ xint(t)

xfs(t)

nS(x, t)dx]

+ Ufs(t).

At shock breakout, the window is completely compressed by the shock and the

integral cannot be separated into parts. Applying Leibniz’s rule (equation C.4),

Uapp(t = tB.O.) = Uint(t)nS(xint, t) − Ufs(t)nS(xfs, t) (C.7)

+

∫ xint(t)

xfs(t)

∂nS [x, t]

∂tdx + Ufs(t), (C.8)

where the integral term is related to the shock steadiness. At shock breakout,

there is a sudden change in the observed apparent particle velocity. Define this

change as ∆ or

∆ = Uapp(tB.O.) − Uapp(tB.O. − ǫ), (C.9)

where 0 < ǫ ≪ tB.O.. Substituting the equations C.6 and C.8 and taking the limit

as ǫ → 0 gives

∆ = Ufs(tB.O.)(1 − n(xfs, tB.O.)) + D(tB.O.)(n(xfs, tB.O.) − n0) (C.10)

+

∫

∂n(x, t)

∂t

∣

∣

∣

tB.O.

δ(xfs)dx. (C.11)


Notice that the dependence upon the shock steadiness and interface conditions

have been removed. The integral in equation C.11 can be simplified by examining

the density and refractive index profiles of the window (figure C.2.3) at various

times.

Figure C.2.3 shows that the free surface refractive index is n0 at all times

except at shock breakout. The rear surface refractive index is described by the

function

nS(x = xfs, t) = n0 + (nS(xfs, tB.O.) − n0)δ(

xfs −∫ t

0

D(t′)dt′)

. (C.12)

Substituting this into the integral of equation C.11 gives

∫ xfs ∂n(x, t)

∂t

∣

∣

∣

tB.O.

dx = (n(xfs, tB.O.) − n0)D(tB.O.), (C.13)

and ∆ is redefined as

∆ = Ufs(tB.O.)(1 − nS) + 2D(tB.O.)(nS − n0), (C.14)

where shocked refractive index of the window at breakout nS(xfs, tB.O.) has been

replaced with the simpler notation nS. Recall that no assumptions were made

regarding the shock steadiness within the window. The discontinuity at shock


Fig. C.2.3: Density and refractive index profiles at various stages of

shock propagation. The profiles indicate that the refractive index at

the free surface is n0 at all times except at breakout. After shock break-

out, as shown in figure d, the window releases to standard conditions

assuming the shock pressure is below the melt and the window is in

the solid state.


breakout depends on parameters at the free surface. By measuring ∆, D and Ufs

at breakout, the refractive index of the compressed window is determined. If the

EOS of the window material is known, then the number of measurable parameters

reduces to two.

C.3 Steady Shock Breakout of an Optical

Window

Equation C.14 is rederived assuming a steady shock with known shocked re-

fractive index (nS) to check the accuracy of equation C.14. Since shock steadiness

does not influence the discontinuity at shock breakout, a derivation assuming a

steady shock should arrive at the same result. Recall equation C.1 or

UApp(t) =d

dt

[

∫ x(t)

xfs(t)

n(x, t)dx]

+ Ufs(t).

Prior to shock breakout, the free surface is stationary (Ufs = 0) and the refrac-

tive index following the shock (nS) is constant. For steady shocks equation C.3


transforms to

UApp(t < tB.O.) =d

dt

[

∫ xint

xD

nSdx +

∫ xD

xfs

n0dx]

,

UApp(t < tB.O.) =d

dt

[

nS(xint − xD) + n0(xD − xfs)]

,

UApp(t < tB.O.) = nS(Uint − D) + Dn0. (C.15)

At shock-breakout (t = tB.O.), the window refractive index is uniform (nS) and

Ufs 6= 0,

UApp(t = tB.O.) =d

dt

[

∫ xint

xfs

nSdx +

∫ xfs

xV

dx]

. (C.16)

For steady shocks, equation C.16 transforms to

UApp(t = tB.O.) = nS(Uint − Ufs) + Ufs +dnS

dt(x − xfs), (C.17)

where the temporal derivative (dnS/dt) of the refractive index is required. From

the steady shock assumption, the derivative of the refractive index can be ex-

pressed as

dn

dt=

nS − n0

L/D, (C.18)

leading to the general equation

UApp(t = tB.O.) = nS(Uint − Ufs) + Ufs + D(ns − n0). (C.19)


Equation C.11 defined ∆ as

∆ = UApp(tB.O.) − UApp(tB.O. − ǫ).

The change in the apparent particle velocity at shock breakout for steady-shock

is

∆ = Ufs(1 − nS) + 2D(nS − n0),

which is identical to the unsteady shock case (equation C.14).

C.4 Analysis of Shot 58815

The velocity profile for shot 58815 is shown in figure C.4.1. Measurements

made through the thick window (blue) are compared with measurements made

through the rarefied window (red). At ∼42 ns, a discontinuity is observed corre-

sponding to shock breakout in the thin LiF window. At ∼47 ns the rarefaction

wave reaches the tantalum pusher and the interface accelerates. The observed ap-

parent particle velocity prior to shock-breakout and the magnitude of the change

in fringe position are measured. Due to the limitations of this experiment, the

free surface velocity (Ufs) and shock velocity (D) are not measured. To determine

the shocked refractive index, the steady shock assumption is made such that the

measured apparent particle velocity can be related to the free surface velocity.


Proper design of future experiments can circumvent this issue. Recall the steady

shock assumption or

Ufs(t) = 2Uint(t). (C.20)

The true interface velocity is related to the measured apparent particle velocity

through equation C.15,

UApp(t < tB.O.) = nS(Uint − D) + Dn0, (C.21)

and the change in fringe position (∆) is defined as

∆ = 2Uint(1 − nS) + 2D(nS − n0). (C.22)

A two equation set with three unknowns is determined by measuring the apparent

particle velocity and change in fringe position at breakout. The system of equa-

tions is closed with the corresponding LiF equation of state.50 Values obtained

in this experiment can be found in table C.4.1. A plot of the refractive index

versus density determined in this experiment (yellow point) is shown in figure

C.4.2 with previous measurements made by Wise34 (blue square) and Lalone54

(red diamond).

Figure C.4.3 is a plot of the difference between the measurements made in the


Table C.4.1: Analysis of Shot 58815

Asbo 1 Asbo 2 Weighted MeanUapp (µm/ns) 4.382 ± 0.031 4.331 ± 0.023 4.349 ± 0.0183∆ (µm/ns) -1.911 ± 0.118 -1.926 ± 0.044 -1.925 ± 0.044D (µm/ns) 9.755 ± 0.089 9.675 ±0.042 9.700 ± 0.038Uint (µm/ns) 3.427 ± 0.066 3.368 ±0.032 3.386 ± 0.029ρS (g/cc) 4.067 ± 0.023 4.047 ± 0.011 4.053 ± 0.010nS 1.456 ± 0.016 1.451 ± 0.008 1.452 ± 0.007

Fig. C.4.1: Shot 58815 velocity profiles.


Fig. C.4.2: Shocked refractive index measurements from various stud-

ies. Value determined in this experiment is shown in yellow. Exper-

iments conducted by Wise34 and Lalone54 are shown as blue square

and red diamonds, respectively. Red dashed line corresponds to the fit

proposed by Lalone.


Fig. C.4.3: Plot of the difference between the apparent interface

velocities observed through the thick window and the released apparent

interface velocities. After shock breakout of the thin window, we see

that prior to the reverberation the difference is constant.

compressed and released material. Between shock breakout and the arrival of the

rarefaction wave at the tantalum/LiF interface, a constant ratio is observed. Thin

LiF measurements are made through a rarified material whose density varies from

2.63 to 4.28 g/cc. These measurements are consistent with those made through

the thick LiF window (taking the ∆ offset into account). The only functional form

that satisfies these conditions is for the refractive index to depend linearly with

density over this range.


C.5 Conclusion

The shock breakout of a transparent optical window generates a discontinuity

in the velocity observed through that window. Comparison of measurements made

through released and compressed LiF windows indicates that the discontinuity is

directly related to the shocked refractive index at breakout. The derivation of this

phenomenon shows that the discontinuity at breakout is independent of the shock

steadiness and from that derivation, a new shock refractive index measurement

technique is presented.

· ii curriculum vitae dayne eric fratanduono was ﬁrst interest in engineering when he attended...

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