Download The Index of Refraction of Lithium Fluoride at Pressures in Excess of

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 distinction from Clarkson University with a Bachelor of Science in mechanical engineering
and physics (dual major). Interested in the Inertial Confined Fusion (ICF) campagain 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.” Submitted 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 measurements 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 November 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, ”Measurements of Strain-Induced Refractive-Index Changes in Shocked LiF Using Laser-Driven Flyer Plates.“ Contributed talk, 16th APS Topical Conference 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 StrainInduced Refractive Index Changes in Shocked LiF Using Laser-Driven
Flyer Plates.“ Contributed talk, OMEGA Laser Facility Users Group Workshop, 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 Conditions 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 throughout 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 supervision 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 appreciated. 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 Engineering was greatly appreciated.
I would like to acknowledge the close professional contacts at Lawrence Livermore 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 Development 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 window 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 experiments 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
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .
93
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
4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 133
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
5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 157
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
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
. . . . . . 109
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 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Alamogordo 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, respectively,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 nonnuclear 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 U235
92 . For
each U235
92 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 nuclear 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,
236
141
92
n + U235
92 → U92 → Ba56 + Kr36 + 3n(170 MeV),
(1.1)
92
∗
where fission fragments (Ba141
56 and Kr36 ) 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 fragments
found in equation 1.1 form is less than 10 % (other fragments may results). The probability of
fragment 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 approximately a nuclear diameter of one another overcoming the electro-static repulsive
force. Significant energy is required to fuse the elements together. The benefits 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 intensities 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 inertial 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 1011 J/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 simulations 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 understanding 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 compression, isentropic compression, isenthalpic compression, and shock wave compression). HEDP experiments have primarily used shock wave compression and
isentropic compression. Shock waves are generated in materials through laser ablation, flyer plates, high energy explosives and nuclear explosions.26, 27 Each technique 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 equations 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 experiment.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 experiments.
1.2
Relevance of This Study
EOS experiments are important to a variety of fields as they enable one to correctly determine the phase space of materials, close the fluid dynamic system of
equations, and provide insight into other high pressure phenomena. A typical experiment 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 because 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 optical window change. If these changes are not accounted for, systematic errors in
the velocity interferometer measurements will be introduced. Therefore, knowledge 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 experiments because its transparency at high pressure allows in situ particle-velocity
measurements at the sample/window interface and for spectroscopic optical measurements of samples.33 Recent experimental developments using ramp compression 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 optical 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 pressures 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 transparent 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
1.3
13
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 various 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 experimentalists 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. Diagnostics necessary for this study are then discussed. The LiF ramp compressed
refractive index analysis and results are discussed in Chapter 4. Twenty-four experiments 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 required 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 (conservation of mass, momentum and energy), a thermodynamic state relation, and the
boundary conditions specific to that system.38 Using these relations, the state
2. Fundamentals of Fluid Dynamics
16
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 sections, 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 compression) 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 discrete 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=
Z
V
ρdV.
(2.1)
2. Fundamentals of Fluid Dynamics
17
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
~ · ~n) flowing through the surface element. U
~ is the velocity permeating
velocity (U
the surface and ~n represents the normal vector to that surface (positive direction
~ The discrete mass (dm) flowing through dS in time
defined as outward on S).
interval (dt) is
~ · ~n)dSdt.
dm = ρ(U
(2.2)
Total mass (dM) leaving the volume becomes
dM = −
I
S
~ · ~n)dSdt.
ρ(U
(2.3)
2. Fundamentals of Fluid Dynamics
18
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
Z
V
∂ρ
dV = −
∂t
I
S
~ · ~n)dS,
ρ(U
(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
~ = 0.
+ ∇ · ρU
∂t
(2.5)
The mass conservation equation (2.5) relates the element density and the momentum. The single equation with two unknowns (ρ, U) is not a closed system. The
~ ) illustrates the need for a higher order moment conservation
momentum term (ρU
equation.
2. Fundamentals of Fluid Dynamics
2.1.2
19
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)
P
Applying Newton’s second law of motion ( F = ma) to the fluid element
(dM), the force (F ) is expressed as
X
F =
dU
(x(t), t)ρ(t)dx(t)A
dt
The acceleration (a) is written in terms of the element velocity (a =
(2.7)
dU
(x(t), t)).
dt
Expanding equation 2.7 the total derivative gives
X
F =
∂U
∂t
+U
∂U ρ(t)dx(t)A.
∂x
(2.8)
If a pressure (P ) is applied to the two surfaces shown in figure 2.1.2, equation
2. Fundamentals of Fluid Dynamics
20
(2.8) maybe rewritten substituting the relation for the applied pressure,
∂U
dx ∂U dx
ρ(t)dx(t)A.
+U
P (x − ) − P (x + ) A =
2
2
∂t
∂x
(2.9)
The fundamental theorem of calculus reduces equation (2.9) to the conservation
of momentum equation for one dimension,
∂U
~
∂P
∂U ρ
=−
+U
.
∂t
∂x
∂x
(2.10)
This derivation assumes that the applied pressure is the only external force acting on the element. In three dimensions, equation (2.10) transforms to a three
equation set with five unknowns,
∂U
~
~ · ∇)U
~ = −∇P + ρ~g ,
ρ
+ (U
∂t
(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
~ and P ). The gradient of
gives a four equation set with five unknowns (ρ, U
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 .
2. Fundamentals of Fluid Dynamics
21
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( 12 ρU 2 ). The rate of
change of energy stored per unit volume is simply
∂
(ρe + 12 ρU 2 ).
∂t
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
2. Fundamentals of Fluid Dynamics
22
which mass flows in and out of a fluid element. As the mass enters and leaves the
element, energy of the form (e + 12 U 2 ) 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
))y=∆y/2 ∆x∆z.
2
The fundamental
theorem of calculus reduces the rate of flow in the y-direction per unit volume
to
∂
(ρUy (e
∂y
+
U2
)).
2
A similar derivation follows for the other directions leading
~ (e +
to a rate of energy flow per unit volume: −∇ · (ρU
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
~ . Work can be performed by the surface forces
unit volume by gravity is ρ~g · U
~ . Heat generation occurs within an element if a
acting on the body or −P ∇ · U
heat source is present. This is defined as the power released per unit volume per
unit time or Q̇.
2. Fundamentals of Fluid Dynamics
23
Combing all terms, the conservation of energy equation for a fluid element is
2
1
∂
~ (e + U )) − P ∇ · U
~ − ∇ · ~q + ρ~g · U
~ + Q̇. (2.12)
(ρe + ρU 2 ) = −∇ · (ρU
∂t
2
2
To determine a relationship for the pressure, required by the conservation of momentum equation, two unknowns have been introduced (internal energy and heat
flux). This assumes the internal heat generation is known. The pressure and internal 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 thermal 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
2. Fundamentals of Fluid Dynamics
24
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 (P V ≈ T ), which defines the relationship 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 variables 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 thermodynamics 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)
2. Fundamentals of Fluid Dynamics
25
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
∆V.
(2.15)
S
Further state relations are determined using similar thermodynamic principles
and arguments. The defining relations for the enthalpy (H), free energy (F ) and
2. Fundamentals of Fluid Dynamics
26
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 equations (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 pressure 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 interdependence 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 thermodynamic state relation and the appropriate boundary conditions. Using Fourier’s
approximation to close the system of equations leaves five fluid equations and
2. Fundamentals of Fluid Dynamics
27
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 specific to a known system. For instance, many classifications of fluid flow exist,
with each classification accompanied with an assumption that may simplify the
= 0), inviscid flow (as was
equation set. Such assumptions are steady flow ( dU
dt
~ = 0). Steady
assumed in the derivation of 2.11), and incompressible flow (∇ · U
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, entropy 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
2. Fundamentals of Fluid Dynamics
28
(P ) are small and expressed as
ρ = ρ0 + ρ̃,
(2.19)
U = U0 + Ũ,
(2.20)
P = P0 + P̃ ,
(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),
∂ ρ̃
∂ Ũ
+ ρ0
∂t
∂x
∂ Ũ
∂ P̃
ρ0
+
∂t
∂x
= 0,
(2.22)
= 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
2. Fundamentals of Fluid Dynamics
29
entropy (P (ρ, S)):
∆P =
∂P
∂ρ
∆ρ +
S
∂P
∂S
∆S.
(2.24)
ρ
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
∆ρ.
(2.25)
S
As previously stated, state relations are path independent and equation (2.25)
can be simplified to
P̃ =
∂P
∂ρ
ρ̃.
(2.26)
S
Combining this result, with the conservation of mass and momentum equations
(2.22, 2.23) for small perturbations yields the one dimensional wave equation,
2
∂ 2 Ũ
2 ∂ Ũ
=
C
,
E
∂t2
∂x2
2
∂ 2 ρ̃
2 ∂ ρ̃
=
C
,
E
∂t2
∂x2
(2.27)
(2.28)
where
CE2
=
∂P
∂ρ
,
S
(2.29)
2. Fundamentals of Fluid Dynamics
30
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
Ũ = f1 (x − CE t) − f2 (x + CE t),
ρ̃ =
ρo
ρo
f1 (x − CE t) +
f2 (x + CE t),
CE
CE
(2.30)
(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 perturbation has split into two separate waves traveling in opposite directions. The
dt
trajectory ( dx
) 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.
2. Fundamentals of Fluid Dynamics
31
Time (t)
Initial Perturbation
Wave Splitting
Characteristics
Position (x)
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 perturbations 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
1
∂P
∂U
∂P
±
= 0,
+ (U ± CE )
+ (U ± CE )
∂t
∂x
ρCE ∂t
∂x
where the sound speed (CE ) need not be constant.
(2.33)
2. Fundamentals of Fluid Dynamics
32
As was shown for acoustic waves, disturbances propagate along specific pathlines or characteristics. For one dimensional isentropic flow, there are two families
of characteristics that propagate in opposite directions and the solutions to equations (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
=
±
∂
∂
+ (U ± CE ) .
∂t
∂x
(2.34)
where (±) indicates the forward and backwards traveling waves. Substituting this
assumption into equation (2.33) gives
1 dP
dU
±
= 0 and
dt
ρCE dt
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 CE2 =
(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 ±
Z
dP
= U ± σ,
ρCE
(2.36)
2. Fundamentals of Fluid Dynamics
33
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 = ρ0 ργ
and CE2 = γρ0 ργ−1 ,
(2.37)
where γ is the ratio of specific heats or the isentropic exponent. The Reimann
invariants are defined as
J± = U ±
2
CE .
γ−1
(2.38)
The velocity and sound speed may then be expressed in terms of these invariants,
U=
J+ + J−
2
and CE =
γ−1
(J+ − J− )
4
(2.39)
as can the characteristic equations
C± :
dx
dt
±
=
γ+1
3−γ
J± +
J∓
4
4
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,
dx
dt +
depends solely on
the J− invariant, and vise versa. Therefore, if J− is constant everywhere,
dx
dt +
too is
2. Fundamentals of Fluid Dynamics
34
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) =
Z
t
0
t
− τt
w dt = −U0 τ
− (1 − e ) .
τ
(2.41)
Analytic Solution
Characteristics propagating in the negative direction originate from the undisturbed region. Negative characteristics cannot emanate from the piston front
because they are bounded by that surface. For this reason, the invariant J− is
2. Fundamentals of Fluid Dynamics
35
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− invariants. 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,
2
2
CE = constant = −
CE0 ,
γ−1
γ −1
J− = U −
−2
(CE0 − CE ),
γ −1
γ−1
= CE0 +
U.
2
U =
CE
(2.42)
(2.43)
(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
U,
2
(2.45)
2. Fundamentals of Fluid Dynamics
36
and upon integration
x = (CE0 +
γ+1
U)t + f (U).
2
(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 determined from the negative Reimann invariant (J− ) or equation (2.44) which,
expressed in terms of w, becomes
CE (w) = CE0 +
γ−1
w.
2
(2.48)
Substituting this relation into equation (2.47) gives
γ −1
f (w) = X(t) − w + CE0 +
w t.
2
(2.49)
The time dependence is removed using the expressions for the piston velocity
2. Fundamentals of Fluid Dynamics
(t = −τ ln(1 +
w
))
U0
37
t
and position (X(t) = −U0 τ [ τt − (1 − e− τ )]),
w
γ+1
.
w + U0 ln 1 +
f (w) = −wτ + τ CE0 +
2
U0
(2.50)
The transcendental equation for velocity as a function of space (x) and time (t)
becomes
γ+1
γ+1
U
x = CE0 +
U0 +
U t + −Uτ + τ ln 1 +
U + CE0 , (2.51)
2
U0
2
which is only valid in the interval of X(t) < x < CE0 t. CE0t is the lead characteristic 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
2. Fundamentals of Fluid Dynamics
38
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.
2. Fundamentals of Fluid Dynamics
39
piston velocity,
2
CE0 ,
γ−1
2
=
CE0 + 2w.
γ −1
J− = −
(2.52)
J+
(2.53)
Using equation (2.39), the velocity and sound speed at the intersection of every 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 trajectory 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 intersection. 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
2. Fundamentals of Fluid Dynamics
40
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 characteristics 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 properties 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-
2. Fundamentals of Fluid Dynamics
41
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 perspectives: 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
2. Fundamentals of Fluid Dynamics
42
that the fluid element mass is conserved in both systems or
ρ(t)dx = ρ0 dh,
(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 describe the derivative taken along a fluid path. In Eulerian and Lagrangian systems
the material derivative is defined as27
D
∂
=
+ U · ∇(Eulerian),
Dt
∂t
D
∂
=
(Lagrangian).
Dt
∂t
(2.56)
(2.57)
Using these conditions, the one-dimensional mass and momentum conservation
equations are transformed from the Eulerian into the Lagrangian coordinate system,
∂ρ
ρ2 ∂U
=−
,
∂t
ρ0 ∂h
1 ∂P
∂U
=−
.
∂t
ρ0 ∂h
(2.58)
(2.59)
Conservation of mass (2.58) and momentum (2.59) equations are used to determine the characteristics in Lagrangian coordinate system. Through substitution
2. Fundamentals of Fluid Dynamics
43
one finds,
1
∂P
∂U
∂P
∂U
±
= 0,
± CL
± CL
∂t
∂h
CL ρ0 ∂t
∂h
(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
CE .
ρ0
(2.61)
Following the techniques outlined in Section 2.4, the characteristic equations become
1
dU ±
dP = 0 and
ρ0 CL
dh
dt
Z
dP
.
ρ0 CL
= CL along C± .
(2.62)
±
The Riemann invariants become
J± = U ±
2.6
(2.63)
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 propagating to the right in an ideal gas with known thermodynamic properties. The
2. Fundamentals of Fluid Dynamics
44
initial velocity disturbance of the wave (U(x, t0 )) is known. The positive characteristics 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 equations 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 following 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 ρ1 ν1 = ρ0 ν0 in Lagrangian space. In the Eulerian
2. Fundamentals of Fluid Dynamics
45
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.
2. Fundamentals of Fluid Dynamics
46
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 ,
E1 +
P1 (D − U1 )2
P0 (D − U0 )2
+
= E0 +
+
.
ρ1
2
ρ0
2
(2.65)
(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 standard 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
2. Fundamentals of Fluid Dynamics
47
(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. Hugoniots 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 nonprinciple Hugoniots can be determined.27 Through a simple manipulation of the
Rankine-Hugoniot equations (2.64, 2.65, 2.66), one can show that
1
E1 (P1 , ρ1 ) − E0 (P0 , ρ0 ) = (P1 + P0 )
2
∗
ρ1 − ρ0
ρ1 ρ0
.
(2.68)
It is important to note, that this linear dependence is not observed in all materials and often
a higher order fit is required to accurately represent the relation.
2. Fundamentals of Fluid Dynamics
48
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 experiment, 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 important 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. Fundamentals of Fluid Dynamics
2.8
49
Isentropic Curves
Similar to the Hugoniot curves, the isentrope can be determined using the
known thermodynamic relations of a material. The thermodynamic relation (Section 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
∆ρ,
ρ2
(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
∆ρ,
ρ2
(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
2. Fundamentals of Fluid Dynamics
50
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 Grüneisen 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)
2. Fundamentals of Fluid Dynamics
51
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
2. Fundamentals of Fluid Dynamics
52
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 techniques 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 experimental 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
3. Experimental Technique
54
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 examples. 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.
3. Experimental Technique
55
the window length (xf s (t) − x(t)),
Zw (t) =
Z
x(t)
n(x, t)dx,
(3.1)
xf s (t)
where x(t) is the position of the reflecting surface and xf s (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) =
Z
x(t)
xf s (t)
n(x, t)dx +
Z
xf s (t)
dx.
(3.2)
xV
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 particle 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 between 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
3. Experimental Technique
56
of 3.2 determines the apparent particle velocity (Uapp ),
d
Uapp (t) =
dt
Uapp (t) =
Uapp (t) =
d
dt
d
dt
"Z
"Z
"Z
x(t)
n(x, t)dx +
xf s (t)
x(t)
Z
#
x(t)
#
d
[xf s (t) − xV ] ,
dt
n(x, t)dx + Uf s (t),
xf s (t)
#
dx ,
xV
n(x, t)dx +
xf s (t)
xf s (t)
(3.3)
where Uf s 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 xf s ,
the window free surface is at rest (Uf s = 0). Let n0 and n represent the spatially
uniform refractive indices ahead of and behind the shock, respectively. Equation
3. Experimental Technique
57
Fig. 3.1.2: Velocity of a reflecting surfaces observed through an optical
window.
3.3 becomes
d
Uapp (t) =
dt
Uapp (t) =
"Z
x(t)
ndx +
xD (t)
Z
xD (t)
#
n0 dx ,
xf s (t)
d
[n(x(t) − xD (t)) + n0 (xD (t) − xf s )] ,
dt
Uapp (t) = n(Utrue − D) + n0 D.
(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(UT rue )) is known.
3. Experimental Technique
3.1.2
58
Ramp Compressed Window
Hayes36 derived the relation for the dependence of the true and apparent velocities 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
dn
=n−ρ .
dUtrue
dρ
(3.6)
Equation 3.6 can be expressed in integral form
Z ρ
n
f (ρ′ ) ′ 0
,
n(ρ) = ρ
−
′ 2 dρ
ρ0
ρ0 ρ
(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. Experimental Technique
3.1.3
59
Optical Windows With Refractive Index That
Varies Linearly With Density
Velocity corrections for optical windows that possess a linear behavior of refractive index and density (n = a + bρ) are straightforward. Recall equation 3.3
for shocked windows and assuming the free surface is stationary (Uf s = 0),
Uapp (t) =
d
dt
"Z
x(t)
#
n(x, t)dx .
xf s (t)
(3.8)
Substituting the linear relations for refractive index and density
"
#
Z x(t)
d
a(x(t) − xf s (t)) + b
Uapp (t) =
ρ(x, t)dx .
dt
xf s (t)
(3.9)
The second term of 3.9 represents the derivative of mass conservation of the window and is zero. This gives
Utrue (t) =
Uapp (t)
.
a
(3.10)
Substituting this linear relation into the equation 3.6, for ramp compressed windows, gives identical results indicating that corrections to the velocity measure-
3. Experimental Technique
60
Fig. 3.2.1: Experimental configuration shock-compressed gas gun experiment.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 experiments 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 generates 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
3. Experimental Technique
61
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 Hf lyer and
Hbuf f er represents the flyer plate and buffer Hugoniots, respectively, impedance
matching requires that
Hf lyer (Uf lyer − Up ) = Hbuf f er (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
Hbuf f er (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
3. Experimental Technique
62
equation
Hf lyer (Uf lyer − Up ) = Hwindow (Up ).
(3.13)
Using equation 3.5, the shocked refractive index is expressed as
nS =
UApp − n0 D
.
UT rue − D
(3.14)
By measuring the flyer plate velocity prior to collision, the true velocity is determined 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 index of Lithium Fluoride (LiF) windows used various impactor and buffer materials
(Al, Cu, Be, Ta, LiF, Al2 O3 ) 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 interferometry. 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
3. Experimental Technique
63
Fig. 3.2.2: Geographical representation of collisional analysis to determine 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) determines 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.
3. Experimental Technique
64
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 experimental 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
3. Experimental Technique
65
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 estimated 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
3. Experimental Technique
66
of the corresponding EOS,
dn
dUapp
=n−ρ .
dUtrue
dρ
In this work, the refractive index of ramp compressed LiF is measured using a twosection 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 (Uf s )
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 onedimensional analysis is justified. The technique, outlined by Rothman56 and
Maw,57 corrects for the wave interactions at the free surface to provide particle velocities 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 apparent velocity (Uapp ). The technique is to measure the free surface velocity (Uf s )
of the piston material and using a backwards characteristics scheme determine the
3. Experimental Technique
67
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
dh
dt
= CL ,
(3.15)
B = Up − σ along
dh
dt
= −CL ,
(3.16)
The notation for Reimann invariants J+ and J− are replaced with F and B, respectively.
3. Experimental Technique
68
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
dP
,
ρ0 CL
1/2
ρ ∂P
.
=
ρ0 ∂ρ
σ =
CL
Z
(3.17)
(3.18)
Up and σ are determined from the intersection of the ith positive and j th negative characteristics where i and j represent the indexing of the characteristics as
illustrated in figure 3.3.2. That value is given by
1
Upi,j = (F i + B j ),
2
1
σ i,j = (F i − B j ).
2
(3.19)
(3.20)
The negative characteristics that emerge from the undisturbed material (t ≤ 0)
must have B = 0. Using these equations, two characteristics schemes are developed to propagate the characteristics that determine the fluid flow.
3. Experimental Technique
69
Fig. 3.3.2: Index of characteristics is shown. Blue and red lines represent the forward and backwards characteristics. The indexing of characteristics is defined and the intersection of the F3 and B2 characteristics
is illustrated for clarity.
3. Experimental Technique
3.3.1
70
Backwards Characteristics Scheme
The backwards characteristics scheme uses the free surface velocity measurement (Uf s ) to determine the ablation pressure (P (0, t)). When the compression
wave reaches the free surface, it releases along the reflected isentrope to zero pressure. The free surface velocity is directly related to the particle velocity by the
velocity doubling rule or
Uf s = 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 = B i = Uf s (tif s ),
(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. Experimental Technique
71
3.20). The location and time of these intersections is
i,j
h
i,j
t
CLi,j−1CLi−1,j (ti−1,j − ti,j−1 ) + CLi,j−1hi−1,j + CLi−1,j hi,j−1
=
,
CLi,j−1 + CLi−1,j
(3.24)
CLi,j−1ti−1,j + CLi,j−1 ti,j−1 + hi−1,j − hi,j−1
,
=
CLi,j−1 + CLi−1,j
(3.25)
where CLi,j = CL (σ i,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 characteristics 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)),
i
i
F = 2U(0, t ) = 2σ(0, t ) = 2
Z
0
P (0,ti )
dP
.
ρ0 CL
(3.26)
At the intersection of the forward traveling characteristics with the window interface, the impedance matching boundary conditions are applied,
P1 (σ1 ) = P2 (σ2 ),
(3.27)
Up1 = Up2 ,
(3.28)
3. Experimental Technique
72
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 compression waves do not reach the window free surface and reflect back into the interaction region
prior to the termination of the experiment.
‡
Experiment termination is defined as the time at which peak compression is observed at the
rear of the piston sample.
3. Experimental Technique
73
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 subsequent 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
3. Experimental Technique
74
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
3. Experimental Technique
75
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 provides 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.
3. Experimental Technique
76
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 Highpower lasers typically produce drive pulses of a few nanoseconds that readily produce high-pressure (> 100 GPa) shock waves. These have been used in reservoircoupled experiments where a shocked reservoir material releases across a vacuum
gap and the hydrodynamics of the releasing shocked material determines the target 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 ablation. The laser directly irradiates the sample; there is no intermediate energy
conversion nor coupling and the ablation pressure is controlled by temporal shap-
3. Experimental Technique
77
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 significant 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 ablation scaling law is required. An initial scaling law was provided by Hicks for
direct-drive shock compression of low-Z materials67
0.6
TW
P [GPa] = 55.3 I
,
cm2
(3.32)
3. Experimental Technique
78
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
0.75
TW
.
Pablation [GPa] = 42 I
cm2
(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 mentioned, the pulse duration is the primary factor in determining the target thickness. 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 thicknesses 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-
3. Experimental Technique
Fig. 3.3.3:
79
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.
3. Experimental Technique
80
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 Å 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
3. Experimental Technique
81
(∼ 80%). A 1000 Å 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 temperature. A velocity interferometer system for any reflector (VISAR)33 measures the
free surface velocity (Uf s ) 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.
3. Experimental Technique
82
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 1
n−
,
c
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
1
d=h 1−
.
n
(3.35)
3. Experimental Technique
83
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.
3. Experimental Technique
84
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
U(t),
c
(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
∆λ(t),
λ2
(3.38)
3. Experimental Technique
85
and the velocity is defined as
U(t) =
λ
∆N(t).
2τ
(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
3. Experimental Technique
86
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) + 2πf0 x + δ0 ],
(3.40)
where A(x, t) and B(x, t) are functions representing the varying background in-
3. Experimental Technique
87
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(2πif0 x) + C ∗ (x, t) exp(−2πif0 x),
(3.41)
where
C(x, t) =
B(x, t)
exp(iφ(x, t)).
2
(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.
3. Experimental Technique
88
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
3. Experimental Technique
89
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) =
Z
∞
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((2πf0 x + δ0 )i),
(3.47)
where D(x, t) is a complex function. The phase is defined as
φ(x, t) = −2πf0 x − δ0 + arctan
Im[D] Re[D]
,
(3.48)
3. Experimental Technique
90
and the amplitude variation (B(x, t)) is given by
B(x, t) =
Re[D]
.
cos(φ(x, t) + 2πf0 x + 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 )
± n,
2π
(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,
dn −n0
,
λ0 δ= 2
n0 − 1 dλ λ=λ0
(3.52)
3. Experimental Technique
91
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),
1
U(t − τ1 ) = VPF1 (∆φ(t)1 ± n),
2
1
U(t − τ2 ) = VPF2 (∆φ(t)2 ± m),
2
(3.53)
(3.54)
and if the VPF’s of these interferometers are not integer multiples of one another, 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 subnanosecond temporal resolution. The camera output intensity, I, can be related
to the source irradiance through the following equations,71
B∆xWs Ωlens
I=
ηM 2
Z
dλTx (λ)SR(λ)Ls (λ, T ),
(3.55)
3. Experimental Technique
92
where the source radiance (Ls (λ, T )) is given by Planck’s law,
L(λ, T ) =
1
2hc2
.
5
hc/λT
λ e
−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)
hc
λ0
(3.58)
T0 is photon energy detected,
T0 =
3. Experimental Technique
93
and A is defined as
A=
2B∆xΩlens < Tx SR > hc2 G
,
ηM 2 λ50
(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 accurately.71 The reflectivity (R) of the target is determined from equation 3.49. The
gray-body temperature is defined as
T =
3.5
T0
ln(1 +
A(1−R)
)
I
.
(3.60)
Concluding Remarks
The effect of shocked and ramp compressed optical windows on interferometry 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-
3. Experimental Technique
94
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
4. Analysis and Results
96
the impedance of the glue did not affect the velocity measurements. A 1000 Å
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
Å 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.
4. Analysis and Results
97
Table 4.1.1: Shot Specifications
Shot
54939†
54940†
54941†
54944†
54945†
54946†
54948†
55857‡
Pulse
Shape
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
J
J
J
J
J
J
J
J
Sweep
Speed
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
55859
55860‡
RM3503 370 J
RM3503 378 J
9ns/15ns
9ns/15ns
15a/7a
15a/7a
56109‡
RM3503 508 J
9ns/5ns
18a/7a
56112
56113
57569
57570
57571
57572
57574
57575
57576
57577⋆
57579⋆
57581⋆
57583
RM3503
RM3503
RM3504
RM3504
RM3504
RM3504
RM3504
RM3702
RM3702
RM3702
RM3702
RM3702
RM3702
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/15ns
9ns/5ns
9ns/5ns
9ns/5ns
9ns/5ns
9ns/5ns
9ns/5ns
18a/7a
18a/7a
18a/7a
18a/7a
18a/7a
18a/7a
18a/7a
7a/18a
7a/18a
7a/18a
7a/18a
7a/18a
7a/18a
∗
†
‡
⋆
Energy
362
480
601
422
558
421
532
442
384 J
387 J
711 J
767 J
774 J
767 J
623 J
270 J
345 J
272 J
362 J
362 J
N/A
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.
Etalon∗ Thickness
Coating
18a/7a
18a/7a
15a/7a
15a/7a
15a/7a
15a/7a
15a/7a
15a/7a
Ti
Ti
Ti
Ti
Ti
Ti
Ti
Al
43 µm
46 µm
46 µm
46 µm
46 µm
46 µm
43 µm
15-[3 Au]40µm
46 µm
10-[3 Au]25 µm
10-[1 Au]35 µm
45 µm
45 µm
97 µm
99 µm
95 µm
97 µm
103 µm
46 µm
45 µm
44 µm
45.5 µm
45.5 µm
46 µm
Al
N/A
N/A
Au
Ti
Ti
Ti
Ti
Ti
Ti
Ti
Ti
N/A
Ti
Ti
Ti
4. Analysis and Results
98
Table 4.1.2: Etalon Parameters
Etalon Thickness (µm)
7a
15a
18a
7.2095
15.1318
18.2268
Delay (ps)
37.32
78.35
94.37
VPF (
µm
ns fringe
6.0965
3.2906
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
4. Analysis and Results
99
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 profiles determined from the VISAR measurements. The blue and red lines
corresponds to the diamond free surface velocity and the diamond/LiF
interface velocity, respectively.
4. Analysis and Results
100
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 corresponds 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 characteristic 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 characteristics 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 characteristics scheme discussed in Section 3.3.2. The graphical representation of the
4. Analysis and Results
101
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.
4. Analysis and Results
102
forward characteristics is shown in figure 4.1.3a. To account for the finite gap between 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 characteristics 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 at
the diamond/LiF interface. For shot 57575 this corresponds to 5.4 ns.
4. Analysis and Results
103
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.
4. Analysis and Results
104
agreement with experimental termination. The effect of shock formation on interface 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 measured 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 measurement (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 decompression 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.
4. Analysis and Results
105
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 variables from the density distributions.76 Random numbers (z) are chosen from a
normal distribution with mean 0 and standard deviation of 1,
h −z 2 i
1
P (z)dz = √ exp
dz.
2
2π
(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 experiment). Four Monte-Carlo variables were defined for the simulations. These correspond 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-
4. Analysis and Results
106
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 estimate 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
dn
=n−ρ .
dUtrue
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
4. Analysis and Results
107
Fig. 4.1.4: Monte-Carlo simulation of shot 57575. The black points represent 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.
4. Analysis and Results
108
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
otal
δUTTrue
=
h
2 dU
2
T rue
δUT rue +
δtT rue
dt
dU
2 dU
2 i1/2
T rue
T rue
+
τetalon +
τsweep
,
dt
dt
(4.4)
(4.5)
where δUT rue and δtT rue are the uncertainties in the timing and velocity determined from the Monte-Carlo simulations, τetalon is the timing uncertainty corresponding 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
4. Analysis and Results
109
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 isentrope50 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
4. Analysis and Results
110
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. Extrapolation 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.
4. Analysis and Results
111
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.
4. Analysis and Results
112
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 profile 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 between the apparent and true particle velocities.76 The weighted mean (ȳ) is defined
as
and the variance of the mean as
P
(yi/σi2 )
P
ȳ =
,
1/σi2
σµ = P
1
,
(1/σi2)
(4.6)
(4.7)
4. Analysis and Results
113
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 twentyfour 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) =
where
dUtrue
dUapp
dUtrue
σapp ,
dUapp
(4.8)
represents 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) =
σtrue (total) =
q
q
2
2
σtrue
+ σtrue
(equiv),
(4.9)
2
2 /1.28)2 .
σtrue
+ (σapp
(4.10)
4. Analysis and Results
114
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 combined 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
N
X
yi − ȳ 2
),
χ =
(
σi
i=1
2
(4.13)
where N is the number of measurements, yi are the experimental values, ȳ is the
4. Analysis and Results
115
Fig. 4.2.1: The weighted mean performed using all twenty-four experiments is shown in black with uncertainties.
The orthogonal fit
performed using existing data is shown as the dashed line. Shock measurements 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
4. Analysis and Results
116
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 orthogonal 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
4. Analysis and Results
117
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
4. Analysis and Results
118
Fig. 4.2.2: Ratio of the apparent to true velocity is shown. Black
points correspond to the weighted mean off all twenty-four experiments. 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 respectively. The pressure scaling along the top axis corresponds to the LiF
isentrope.
4. Analysis and Results
119
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
±0.004
Ramp
8.79
1.28
0.000
6.91
3.10
9.96
& Glue
± 0.01
±0.009
34
Wise
1.39
1.289
0.000
1.09
0.87
2.89
± 0.003 ± 0.002
Lalone54
0.48
1.276
0.01
0.38
0.25
0.61
± 0.005
±0.02
Jensen53, 78 0.56
1.26
-0.2
0.46
0.33
0.56
± 0.04
±0.6
Ramp
1.49
1.273
0.001
1.17
0.61
6.23
& Shock
±0.008
±0.003
Ramp, 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.
4. Analysis and Results
120
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.
4. Analysis and Results
121
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 preheat 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 energies. 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
4. Analysis and Results
122
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 corresponds to the LiF isentrope.
4. Analysis and Results
123
velocities for ramp compression experiments
dUapp
dn
= f (ρ),
=
dρ
dUtrue
Z ρ
n
f (ρ′ ) ′ 0
n(ρ) = ρ
.
−
′ 2 dρ
ρ0
ρ0 ρ
n−ρ
(4.15)
(4.16)
This integral equation requires a boundary condition; the highest pressure refractive index measurements made by Wise and Chhabildas is used.34 Uncertainties
are estimated from the deviation of measured values from linearity. The boundary 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. Results 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 encompass 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)
4. Analysis and Results
124
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 nonlinear 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 window 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 hydrodynamic 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.
4. Analysis and Results
125
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 orthogonal 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. Analysis and Results
4.4
126
LASNEX Simulations
One-dimensional hydrodynamic LASNEX82 simulations are performed to validate 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 termination of the experiment, generating a release fan that may impede the interface
velocity. Recall, that the characteristics analysis is valid only for isentropic compression. 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
4. Analysis and Results
127
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.
4. Analysis and Results
128
Fig. 4.4.2: Comparison of hydrocode simulations and method of characteristics 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 characteristics 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
4. Analysis and Results
129
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 results.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)
4. Analysis and Results
130
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 Å 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 consisted of a 46 µm CVD diamond piston with a titanium coated LiF window attached to half of the rear surface. Peak compression (∼ 280 GPa) occurs at 5.5
4. Analysis and Results
131
ns. The large increase in temperature that occurs at ∼ 0.1 ns is due to the formation 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 formation of the opaque elastic limit in diamond. Emission is not observed at the LiF
interface due to the opaque 1000 Å 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 selfemission 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
4. Analysis and Results
132
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 corresponds 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.
4. Analysis and Results
4.6
133
Concluding Remarks
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 measurements 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 interferometry 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 index 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 refractive 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 compared 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 refractive 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
~ = ρf ,
∇·E
ǫo
~
~ = − ∂B ,
∇×E
∂t
~ = 0,
∇·B
~
~ = µo J~ + ǫo µo ∂ E ,
∇×B
∂t
(5.3)
(5.4)
(5.5)
(5.6)
~ is the electric field, ρf is the free charge density, B
~ is the magnetic field,
where E
~ = ǫE,
~
and c is the speed of light in vacuum. Assuming a linear medium (D
~ = B/µ)
~
H
that is free of sources (ρf = 0) with zero conductivity (σ = 0), the
solutions to Maxwell’s equations are
~ r , t) = Eo ei(~k·~r−ωt) ê,
E(~
(5.7)
~ r , t) = Ho ei(~k·~r−ωt) ĥ,
H(~
(5.8)
~ is the electric displacement field, H
~ is the magnetizing field, E0 and H0
where D
are constants, ê and ĥ 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=
r
ǫµ
.
ǫo µo
(5.10)
For materials in which the dielectric response is frequency dependent, the refractive 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,
ñ =
s
ǫµ iσ 1+
.
ǫo µo
ω
(5.11)
The propagation of light through a medium can be described in terms of the
complex refractive index,88
ñ = n + iκ,
(5.12)
where the real part of ñ 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. Electronic 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 crystal 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 + ωp2
X
j
ωj2
fj
,
− ω 2 − iγj ω
(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 =
s
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 ≈ ñ2 .
(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 experimentally 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 measurements, 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 describe 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 Wemple 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 measurements as a function of density indicates that the linear behavior is due to a
reduction in the band gap energy.91
5. Discussion
5.3
143
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 + ωp2
X
j
ωj2
fj
,
− 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 /(ω12 − ω 2 )) 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,
X fj ω2 1
+
,
2
2
ω
ω
j
j
j6=1
(5.17)
and equation 5.16 takes the form
ǫ(ω) = 1 + ωp2
X fj ω2 f1
2
1
+
.
+
ω
p
2
2
ω12 − ω 2
ω
ω
j
j
j6=1
(5.18)
Equation 5.18 reduces to the single-oscillator model by retaining terms to order
ω2,
ǫ(ω) − 1 ≈
E02
F
,
− (~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 = E0 Ed ,
(5.20)
5. Discussion
145
leading to the relation
n2 − 1 =
Ed E0
.
− ~2 ω 2
E02
(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
x
−
.
Ed E0 Ed
(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 regression. 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 NaCltype crystal lattice.91 The oscillator energy (E0 ), the strength of interband transition (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 NaCltype lattice structure.
Crystal
LiF
NaF
KF
NaCl
KCl
RbCl
CsCl
KBr
RbBr
KI
RbI
E0 (eV)91
16.7
15
14.8
10.3
10.5
10.4
10.6
9.2
9.1
7.7
7.7
Ed (eV)91
15.4
11.3
12.3
13.6
12.3
12.2
14
12.4
12.1
12.8
12.1
Ex (eV)98
12.9
10.66
9.88
7.96
7.79
7.51
7.85
6.71
6.64
5.88
5.73
E0 /Ex
1.29
1.41
1.5
1.29
1.35
1.38
1.35
1.37
1.37
1.31
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 H2 O 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 H2 O, 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 metallization.
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 measurements.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 examining 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 compression 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
4π
=
Na ρα
2
n +2
3
(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 2
n+1
.
(5.26)
5. Discussion
152
Fig. 5.4.1: Band structure energy calculation for Xenon with atomic labels.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
Na ρα,
3
(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 pressure 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 demonstrated by similar results for shock and ramp compression where the changes in
temperature were significantly greater for shock compression, while refractive index 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
97
LiF
NaF97, 104
RbI97, 106, 107
KI97, 106, 107
CsI97, 108, 109
He110
Ne111
Ar112, 113
Kr114, 115
Xe114, 116
†
GoldhammerHerzfeld85, 86
Predictions
ρ (g/cc) P (GPa)
11.2
2,860
8.8
300
10.4
127
8.85
125
10.7
87
8
5,000
20
6,000
9.6
–
13.3
–
12.9
150
Optical Absorption
Measurements
ρ (g/cc)
–
–
9.62
8.7
–
–
–
–
–
–
P (GPa)
–
–
85
115
111
–
–
–
–
150
Band Structure
Predictions
ρ (g/cc)
>14.2†
9.9
10.4
9.51
9.02
21.3
78.8
< 7.1
12.8
12.9
P (GPa)
>5,000†
455
122
155
100
25,700
158,000
<230
316
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
LiF98
NaF98
KF98
RbF98
CsI117
EG (eV)
14.2
11.5
10.8
10.3
6.1
Noble Gas EG
He110
Ne117
Ar118
Kr118
Xe117
(eV)
19.8
21.5
14.3
11.7
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 alkalihalides, 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 calculations 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.
5.5
Concluding Remarks
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 pressure 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 window 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 insulators 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 measurements 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 closure 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 transparent 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 transparency 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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161
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Appendix
APPENDIX
182
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 compression of diamond targets irradiated by 351 nm light. The pressures are generated 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 extreme conditions by creating loading through laser ablation. Accurate knowledge
of the scaling of ablation pressure with laser intensity is important for the design
A. Direct Drive Laser Ablation Scaling
184
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 xray 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 intensities. 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 compression and ramp compression. Until recently, shock compression experiments
were more predominant than ramp compression due to the pulse-shaping requirements 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 compression 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.
A. Direct Drive Laser Ablation Scaling
185
Recently, direct-drive shaped ablation was developed to ramp compress targets.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 scaling. 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 ablation pressure. That calculated drive pressure is compared to the intensity on
target, determined by the optical streak camera power measurements. An appropriate 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.
A. Direct Drive Laser Ablation Scaling
186
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
54939
54940
54944
54945
54946
54948
56112
56113
57575
57576
57577
57579
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
RM3503
RM3702
RM3702
RM3702
RM3702
†
Peak Intensity
(TW/cm2 )
44
57
57
71
56
57
53
52
19
28
24
31
Peak Ablation† Thickness
Pressure (GPa)
(µm)
660
43
850
46
810
46
970
46
710
46
770
43
710
45
720
45
370
46
440
45
420
44
540
45.5
Determined using the method of characteristics.56, 57
The diamond thicknesses for 3.7 ns ramp-compression experiments are included 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 nanocrystalline diamond were compressed with no discernable difference observed in
their ablation pressure scaling. A 1000 Å 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
A. Direct Drive Laser Ablation Scaling
187
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
A. Direct Drive Laser Ablation Scaling
188
The laser intensity is determined by the optical streak camera power measurements 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 dependence (e−r
A.3
2 /r 2
0
) with r0 = 438µm.
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. According to Lindl,128 the ablation pressure (P ), scales with laser intensity (I) as
P = P0 I C0 ,
(A.1)
A. Direct Drive Laser Ablation Scaling
189
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 = ln P0 + 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
0.75[±0.01]
P [GPa] = 42[±3] I[TW/cm2 ]
,
and is shown as the black line of figure A.3.1.
(A.3)
A. Direct Drive Laser Ablation Scaling
190
Fig. A.3.2: Comparison of the ablation pressure determined from the
method of characteristics (red line) and the ablation pressure determined 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.
Fabbro131
λ
(nm)
260
Pulse
(ns)
0.5
Intensity
(W/cm2 )
1013 − 1015
Goldsack121
530
1.0
1050
1.0
Dahmani132
1060
0.7
7 × 1012 −
6 ×1013
7 × 1012 −
6 ×1013
1013 − 1015
This Work
351
3.8
†
‡
I is in units TW/cm2 .
Z is the atomic number.
5 × 1012 −
7 ×1013
Type
Mat.
P (GPa)†‡
Planar
Shock
Spherical
Shock
Spherical
Shock
Planar
Shock
Planar
Ramp
Al
56.9 × I0.75
Al
18.3 × I0.95
Al
13.1 × I1.14
C & Si 30 × I2/3
×Z3/16
Dia.
42 × I0.75
A. Direct Drive Laser Ablation Scaling
191
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 2/3
λ
,
(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 ] 0.75
λ[µm]
,
(A.5)
A. Direct Drive Laser Ablation Scaling
192
and ablation pressure scaling for ramp compressed diamond is
I[TW/cm2 ] 0.75[±0.01]
.
P [GPa] = 19[±1]
λ[µm]
(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 simulations. 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
(µm/ns)
0.48
0.31
0.58
0.38
0.19
0.10
0.25
0.74
0.98
0.34
0.51
0.45
Apparent Velocity
(µm/ns)
0.59 ± 0.008
0.38 ± 0.008
0.73 ± 0.008
0.48 ± 0.003
0.25 ± 0.003
0.13 ± 0.003
0.32 ± 0.003
0.95 ± 0.018
1.25 ± 0.008
0.43 ± 0.002
0.64 ± 0.004
0.58 ± 0.002
True Velocity
(µm/ns)
5.57
5.59
5.62
5.65
5.68
5.71
5.74
5.78
5.82
5.86
5.90
5.96
Apparent Velocity
(µm/ns)
7.09 ± 0.075
7.14 ± 0.081
7.19 ± 0.082
7.24 ± 0.086
7.29 ± 0.094
7.34 ± 0.095
7.39 ± 0.102
7.44 ± 0.108
7.49 ± 0.107
7.54 ± 0.111
7.59 ± 0.127
7.64 ± 0.134
B. Weighted Mean Values
194
Table B.1.1: Apparent and True Weighted Mean Values (cont.)
True Velocity
0.67
0.69
1.01
0.11
0.30
0.65
0.66
0.66
0.70
0.70
0.70
0.81
0.84
0.84
0.89
0.98
0.98
0.99
1.00
1.01
1.01
1.12
1.13
1.13
1.14
1.36
1.36
1.37
1.37
1.38
Apparent Velocity
0.85 ± 0.003
0.88 ± 0.007
1.28 ± 0.014
0.14 ± 0.010
0.38 ± 0.010
0.82 ± 0.010
0.85 ± 0.010
0.84 ± 0.010
0.90 ± 0.010
0.89 ± 0.010
0.89 ± 0.010
1.03 ± 0.010
1.07 ± 0.010
1.06 ± 0.010
1.14 ± 0.010
1.25 ± 0.010
1.25 ± 0.010
1.26 ± 0.010
1.30 ± 0.016
1.29 ± 0.010
1.31 ± 0.013
1.43 ± 0.012
1.45 ± 0.010
1.44 ± 0.010
1.47 ± 0.010
1.74 ± 0.010
1.74 ± 0.010
1.76 ± 0.010
1.76 ± 0.010
1.77 ± 0.010
True Velocity
6.01
6.07
6.05
6.10
6.15
6.22
6.28
6.33
6.37
6.42
6.47
6.51
6.55
6.60
6.65
6.69
6.71
6.76
6.82
6.87
6.91
6.94
6.95
7.03
7.06
7.08
7.09
7.11
7.11
7.12
Apparent Velocity
7.69 ± 0.134
7.74 ± 0.152
7.79 ± 0.138
7.84 ± 0.13
7.89 ± 0.137
7.94 ± 0.131
7.99 ± 0.125
8.04 ± 0.135
8.09 ± 0.136
8.14 ± 0.129
8.19 ± 0.135
8.24 ± 0.124
8.29 ± 0.116
8.34 ± 0.122
8.39 ± 0.111
8.44 ± 0.101
8.49 ± 0.11
8.54 ± 0.102
8.59 ± 0.094
8.64 ± 0.102
8.69 ± 0.096
8.74 ± 0.097
8.79 ± 0.110
8.84 ± 0.119
8.89 ± 0.128
8.94 ± 0.150
8.99 ± 0.135
9.04 ± 0.132
9.09 ± 0.146
9.14 ± 0.131
B. Weighted Mean Values
195
Table B.1.1: Apparent and True Weighted Mean Values (cont.)
True Velocity
1.39
1.39
1.41
2.64
3.78
3.78
4.09
4.09
1.52
1.58
1.60
1.62
1.64
1.65
1.67
1.70
1.75
1.84
1.87
1.90
1.96
1.99
2.02
2.05
2.09
2.10
2.14
2.18
2.23
2.24
Apparent Velocity
1.8 ± 0.024
1.79 ± 0.010
1.82 ± 0.010
3.4 ± 0.010
4.82 ± 0.026
4.81 ± 0.045
5.27 ± 0.017
5.28 ± 0.033
1.84 ± 0.103
1.89 ± 0.105
1.94 ± 0.111
1.99 ± 0.114
2.04 ± 0.119
2.09 ± 0.127
2.14 ± 0.137
2.19 ± 0.145
2.24 ± 0.044
2.29 ± 0.043
2.34 ± 0.042
2.39 ± 0.044
2.44 ± 0.042
2.49 ± 0.049
2.54 ± 0.050
2.59 ± 0.055
2.64 ± 0.055
2.69 ± 0.056
2.74 ± 0.055
2.79 ± 0.052
2.84 ± 0.053
2.89 ± 0.069
True Velocity
7.16
7.22
7.23
7.26
7.27
7.30
7.34
7.36
7.40
7.45
7.48
7.51
7.56
7.58
7.61
7.71
7.73
7.76
7.79
7.81
7.86
7.91
7.92
7.97
7.99
8.11
8.15
8.20
8.23
8.26
Apparent Velocity
9.19 ± 0.130
9.24 ± 0.173
9.29 ± 0.147
9.34 ± 0.139
9.39 ± 0.157
9.44 ± 0.138
9.49 ± 0.137
9.54 ± 0.159
9.59 ± 0.140
9.64 ± 0.136
9.69 ± 0.155
9.74 ± 0.135
9.79 ± 0.129
9.84 ± 0.147
9.89 ± 0.128
9.94 ± 0.155
9.99 ± 0.183
10.04 ± 0.158
10.09 ± 0.148
10.14 ± 0.175
10.19 ± 0.149
10.24 ± 0.137
10.29 ± 0.169
10.34 ± 0.141
10.39 ± 0.133
10.44 ± 0.188
10.49 ± 0.160
10.54 ± 0.146
10.59 ± 0.207
10.64 ± 0.170
B. Weighted Mean Values
196
Table B.1.1: Apparent and True Weighted Mean Values (cont.)
True Velocity
2.28
2.31
2.36
2.42
2.47
2.51
2.56
2.64
2.68
2.71
2.75
2.78
2.81
2.84
2.88
2.91
2.95
2.98
3.04
3.07
3.11
3.16
3.25
3.28
3.31
3.34
3.38
3.42
3.45
3.48
Apparent Velocity
2.94 ± 0.062
2.99 ± 0.057
3.04 ± 0.058
3.09 ± 0.044
3.14 ± 0.046
3.19 ± 0.049
3.24 ± 0.049
3.29 ± 0.047
3.34 ± 0.049
3.39 ± 0.051
3.44 ± 0.053
3.49 ± 0.050
3.54 ± 0.050
3.59 ± 0.050
3.64 ± 0.045
3.69 ± 0.048
3.74 ± 0.054
3.79 ± 0.049
3.84 ± 0.057
3.89 ± 0.055
3.94 ± 0.051
3.99 ± 0.054
4.04 ± 0.058
4.09 ± 0.053
4.14 ± 0.056
4.19 ± 0.052
4.24 ± 0.047
4.29 ± 0.048
4.34 ± 0.049
4.39 ± 0.047
True Velocity
8.30
8.44
8.52
8.54
8.55
8.61
8.66
8.80
8.81
8.82
8.83
8.84
8.88
8.89
8.90
8.87
8.88
8.93
8.94
8.94
8.99
9.21
9.18
9.25
9.28
9.26
9.31
9.35
9.33
9.38
Apparent Velocity
10.69 ± 0.154
10.74 ± 0.210
10.79 ± 0.142
10.84 ± 0.123
10.89 ± 0.19
10.94 ± 0.163
10.99 ± 0.118
11.04 ± 0.071
11.09 ± 0.059
11.14 ± 0.055
11.19 ± 0.058
11.24 ± 0.056
11.29 ± 0.057
11.34 ± 0.059
11.39 ± 0.058
11.44 ± 0.111
11.49 ± 0.137
11.54 ± 0.108
11.59 ± 0.128
11.64 ± 0.148
11.69 ± 0.111
11.74 ± 0.209
11.79 ± 0.872
11.84 ± 0.128
11.89 ± 0.215
11.94 ± 0.075
11.99 ± 0.133
12.04 ± 0.216
12.09 ± 0.089
12.14 ± 0.138
B. Weighted Mean Values
197
Table B.1.1: Apparent and True Weighted Mean Values (cont.)
True Velocity
3.51
3.54
3.57
3.60
3.62
3.65
3.68
3.70
3.73
3.77
3.81
3.86
3.92
3.97
4.10
4.15
4.18
4.22
4.27
4.30
4.35
4.39
4.43
4.45
4.49
4.53
4.57
4.62
4.64
4.69
Apparent Velocity
4.44 ± 0.052
4.49 ± 0.057
4.54 ± 0.057
4.59 ± 0.059
4.64 ± 0.062
4.69 ± 0.062
4.74 ± 0.065
4.79 ± 0.064
4.84 ± 0.064
4.89 ± 0.064
4.94 ± 0.059
4.99 ± 0.061
5.04 ± 0.061
5.09 ± 0.066
5.14 ± 0.087
5.19 ± 0.092
5.24 ± 0.095
5.29 ± 0.096
5.34 ± 0.099
5.39 ± 0.097
5.44 ± 0.103
5.49 ± 0.104
5.54 ± 0.106
5.59 ± 0.104
5.64 ± 0.105
5.69 ± 0.106
5.74 ± 0.106
5.79 ± 0.107
5.84 ± 0.105
5.89 ± 0.110
True Velocity
9.42
9.41
9.46
9.48
9.49
9.53
9.56
9.59
9.62
9.62
9.69
9.70
9.70
9.83
9.82
9.87
10.00
10.05
10.40
10.55
10.33
10.39
10.45
10.52
10.58
10.64
10.70
10.75
10.81
10.86
Apparent Velocity
12.19 ± 0.211
12.24 ± 0.105
12.29 ± 0.137
12.34 ± 0.205
12.39 ± 0.126
12.44 ± 0.136
12.49 ± 0.214
12.54 ± 0.146
12.59 ± 0.131
12.64 ± 0.214
12.69 ± 0.160
12.74 ± 0.115
12.79 ± 0.22
12.84 ± 0.173
12.89 ± 0.101
12.94 ± 0.199
12.99 ± 0.142
13.04 ± 0.062
13.09 ± 0.124
13.14 ± 0.394
13.21 ± 0.486
13.29 ± 0.467
13.36 ± 0.449
13.44 ± 0.427
13.51 ± 0.406
13.59 ± 0.379
13.66 ± 0.352
13.74 ± 0.325
13.81 ± 0.296
13.88 ± 0.265
B. Weighted Mean Values
198
Table B.1.1: Apparent and True Weighted Mean Values (cont.)
True Velocity
4.82
4.86
4.90
4.91
4.90
4.94
4.97
4.98
4.99
5.01
5.03
5.07
5.10
5.15
5.19
5.27
5.32
5.37
5.40
5.43
5.47
5.50
5.53
Apparent Velocity
5.94 ± 0.101
5.99 ± 0.093
6.04 ± 0.085
6.09 ± 0.083
6.14 ± 0.080
6.19 ± 0.079
6.24 ± 0.081
6.29 ± 0.076
6.34 ± 0.073
6.39 ± 0.073
6.44 ± 0.068
6.49 ± 0.069
6.54 ± 0.072
6.59 ± 0.069
6.64 ± 0.069
6.69 ± 0.067
6.74 ± 0.062
6.79 ± 0.060
6.84 ± 0.061
6.89 ± 0.061
6.94 ± 0.064
6.99 ± 0.072
7.04 ± 0.073
True Velocity
10.91
10.95
10.99
11.03
11.07
11.11
11.16
11.08
11.12
11.18
11.24
11.38
11.50
11.65
11.85
11.30
11.50
11.64
11.84
12.04
12.21
12.37
12.61
Apparent Velocity
13.96 ± 0.250
14.04 ± 0.242
14.11 ± 0.239
14.19 ± 0.240
14.26 ± 0.240
14.34 ± 0.232
14.41 ± 0.216
14.49 ± 0.244
14.56 ± 0.215
14.63 ± 0.197
14.71 ± 0.176
14.84 ± 0.355
15.00 ± 0.295
15.15 ± 0.219
15.30 ± 0.197
15.45 ± 3.272
15.60 ± 2.780
15.75 ± 2.376
15.90 ± 2.212
16.05 ± 1.653
16.20 ± 1.188
16.35 ± 0.879
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 shockreleased 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 interferometry 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. LiF Shock Release
C.1
200
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 determine the effects of the released window on interferometry measurements. The
tantalum/LiF interface motion is measured using a 532-nm probe beam and a
C. LiF Shock Release
201
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 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.
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 Livermore
National Laboratory.
C. LiF Shock Release
Fig. C.1.2: VISAR data for shot s58815.
202
C. LiF Shock Release
203
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 breakout. 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
dh
UApp (t) =
dt
Z
xint (t)
xf s (t)
i
n(x, t)dx + Uf s (t),
(C.1)
C. LiF Shock Release
204
Fig. C.2.1: Shock propagating through an optical window.
where xint (t) is the interface position, xf s (t) is the free surface position, nS (x, t) is
the compressed refractive index of the optical window, and Uf s (t) is window the
free surface velocity. Prior to shock breakout, the window free surface is stationary
and equation C.1 reduces to
dh
UApp (t < tB.O. ) =
dt
Z
xint (t)
xf s
i
n(x, t)dx .
(C.2)
The optical path length of the shocked (nS (x, t)) and unshocked (n0 ) regions is
expressed as
dh
UApp (t < tB.O. ) =
dt
Z
xint (t)
xD (t)
nS (x, t)dx +
Z
xD (t)
xf s
i
n0 dx ,
(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
C. LiF Shock Release
205
Fig. C.2.2: Shock Breakout of an Optical Window.
limits are functions of the differential variable,133
d
dt
Z
V (t)
U (t)
′
′
f (x, t)dx = V (t)f (V (t), t) − U (t)f (U(t), t) +
Z
V (t)
U (t)
∂f (x, t)
dx. (C.4)
∂t
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)
Z xint (t)
∂nS (x, t)
+
dx + n0 D(t),
∂t
xD (t)
(C.5)
(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
C. LiF Shock Release
206
velocity, the free surface (xf s ) is no longer stationary (Uf s 6= 0). Recall equation
C.1,
dh
UApp (t = tB.O. ) =
dt
Z
xint (t)
xf s (t)
i
nS (x, t)dx + Uf s (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) − Uf s (t)nS (xf s , t)
Z xint (t)
∂nS [x, t]
+
dx + Uf s (t),
∂t
xf s (t)
(C.7)
(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
∆ = Uf s (tB.O. )(1 − n(xf s , tB.O. )) + D(tB.O. )(n(xf s , tB.O. ) − n0 ) (C.10)
Z
∂n(x, t) +
δ(xf s )dx.
(C.11)
tB.O.
∂t
C. LiF Shock Release
207
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
Z t
nS (x = xf s , t) = n0 + (nS (xf s , tB.O. ) − n0 )δ xf s −
D(t′ )dt′ .
(C.12)
0
Substituting this into the integral of equation C.11 gives
Z
xf s
∂n(x, t) dx = (n(xf s , tB.O. ) − n0 )D(tB.O. ),
∂t
tB.O.
(C.13)
and ∆ is redefined as
∆ = Uf s (tB.O. )(1 − nS ) + 2D(tB.O. )(nS − n0 ),
(C.14)
where shocked refractive index of the window at breakout nS (xf s , 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
C. LiF Shock Release
Fig. C.2.3:
208
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 breakout, 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.
C. LiF Shock Release
209
breakout depends on parameters at the free surface. By measuring ∆, D and Uf s
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 refractive 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
dh
UApp (t) =
dt
Z
x(t)
xf s (t)
i
n(x, t)dx + Uf s (t).
Prior to shock breakout, the free surface is stationary (Uf s = 0) and the refractive index following the shock (nS ) is constant. For steady shocks equation C.3
C. LiF Shock Release
210
transforms to
Z
Z xD
i
d h xint
nS dx +
UApp (t < tB.O. ) =
n0 dx ,
dt xD
xf s
i
dh
nS (xint − xD ) + n0 (xD − xf s ) ,
UApp (t < tB.O. ) =
dt
UApp (t < tB.O. ) = nS (Uint − D) + Dn0 .
(C.15)
At shock-breakout (t = tB.O. ), the window refractive index is uniform (nS ) and
Uf s 6= 0,
dh
UApp (t = tB.O. ) =
dt
Z
xint
nS dx +
xf s
Z
xf s
xV
i
dx .
(C.16)
For steady shocks, equation C.16 transforms to
UApp (t = tB.O. ) = nS (Uint − Uf s ) + Uf s +
dnS
(x − xf s ),
dt
(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 expressed as
dn
nS − n0
=
,
dt
L/D
(C.18)
UApp (t = tB.O. ) = nS (Uint − Uf s ) + Uf s + D(ns − n0 ).
(C.19)
leading to the general equation
C. LiF Shock Release
211
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
∆ = Uf s (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 corresponding to shock breakout in the thin LiF window. At ∼47 ns the rarefaction
wave reaches the tantalum pusher and the interface accelerates. The observed apparent 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 (Uf s ) 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.
C. LiF Shock Release
212
Proper design of future experiments can circumvent this issue. Recall the steady
shock assumption or
Uf s (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 equations 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
C. LiF Shock Release
213
Table C.4.1: Analysis of Shot 58815
Uapp (µm/ns)
∆ (µm/ns)
D (µm/ns)
Uint (µm/ns)
ρS (g/cc)
nS
Asbo 1
4.382 ± 0.031
-1.911 ± 0.118
9.755 ± 0.089
3.427 ± 0.066
4.067 ± 0.023
1.456 ± 0.016
Asbo 2
4.331 ± 0.023
-1.926 ± 0.044
9.675 ±0.042
3.368 ±0.032
4.047 ± 0.011
1.451 ± 0.008
Weighted Mean
4.349 ± 0.0183
-1.925 ± 0.044
9.700 ± 0.038
3.386 ± 0.029
4.053 ± 0.010
1.452 ± 0.007
Fig. C.4.1: Shot 58815 velocity profiles.
C. LiF Shock Release
214
Fig. C.4.2: Shocked refractive index measurements from various studies. Value determined in this experiment is shown in yellow. Experiments conducted by Wise34 and Lalone54 are shown as blue square
and red diamonds, respectively. Red dashed line corresponds to the fit
proposed by Lalone.
C. LiF Shock Release
Fig. C.4.3:
215
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. LiF Shock Release
C.5
216
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.