Tesi di Laurea di
Stefano Paleari
MAT. 074077
Università degli studi
di Milano Bicocca
Facoltà di Scienze Matematiche, Fisiche e Naturali
Corso di laurea in Fisica
Study of carbon
in high pressure regime
Relatore:
Prof. Dimitri Batani
Correlatore:
Prof. Marco Bernasconi
Ottobre 2010
Un diamante
non è per sempre
Contents
Contents
0.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
vii
1 Physical context
1.1 Carbon: Overview of Phase and Transformation Diagram
1.2 High Pressure Experimental Techniques . . . . . . . . . .
1.3 Shock Wave Physics . . . . . . . . . . . . . . . . . . . . .
1.4 Laser-Matter Interactions . . . . . . . . . . . . . . . . . .
1.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . .
1.6 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Experimental setup
2.1 Laser System . . . . . . . . . . . .
2.2 Targets . . . . . . . . . . . . . . .
2.3 Self-emission Diagnostics . . . . . .
2.4 Interferometric Diagnostic Visars
2.5 Resume . . . . . . . . . . . . . . .
2.6 Bibliography . . . . . . . . . . . .
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3 Data analysis
3.1 Timing . . . .
3.2 Velocity . . .
3.3 Reflectivity .
3.4 Temperature
3.5 Resume . . .
3.6 Bibliography
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4 Results
4.1 Porous Carbon EoS Model
4.2 Shock Dynamics . . . . .
4.3 Carbon Reflectivity . . . .
4.4 Resume . . . . . . . . . .
4.5 Bibliography . . . . . . .
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5 Conclusions
5.1 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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i
Introduction
In this thesis the properties of carbon have been investigated in the multimegabar regime, using laser-induced shock-wave compression. This work is
directed to the high-pressure liquid phases. This topic is of ongoing scientific
interest in many different fields, such as material science[Bundy et al., 1996],
planetary science and astrophysics[Guillot, 1999].
Carbon has been studied extensively for a long time, due to its uniqueness
in the diversity of its different phases. The phase diagram in the low pressure
region (under 50 GPa and for temperature below 6,000 K) is well established
and the solid graphite and diamond phases characteristics are well known. On
the contrary, the liquid high-pressure phase is almost unexplored due to the
extreme conditions required to study this state. The theoretical calculations,
also, are complicated by the sophisticated models needed to give accurate predictions and may be very demanding from a computational standpoint.
Recently there has been speculation by van Thiel and Ree [1993] and Correa
et al. [2006] about a phase transition between two liquid phases characterized
by different conductivity. A third liquid phase was predicted by Grumbach and
Martin [1996] for pressures in the range 4-10 Mbar with metallic properties.
Experimentally, the phase diagram can be explored by static or dynamic
methods. With the first approach pressure and temperature can by varied
independently in the range 0 - 1 Mbar, 0 - 103 K, while the second allows
much higher pressures and temperatures. In this last case, unfortunately, the
two quantities are correlated through Hugoniot relations and the compressed
state is accessible to measurements only for very short time. Different strategies have been developed to overcome these limitations. In particular, the
target design can be tailored to explore a definite area of the phase diagram
by means of special density material arranged in multi-layer structures. The
scheme adopted should allow to image directly carbon in high pressure state,
preventing the release into vacuum of the shock wave, thanks to the double
layer carbon-on-transparent substrate. Fabrication process was oriented to the
production of low density carbon to customize the shocked state properties.
Target production required the efforts of three different groups in Italy: Prof.
Paolo Piseri of University of Milan (Supersonic Cluster Beam Deposition technique, using Pulsed Microplasma Cluster Source), Prof. Stefano Bellucci of
Frascati National Laboratory, Rome (spray coating technique) and Prof. Alfonso Mangione University of Palermo (magnetron sputtering depositions).
As stated in the beginning, the method employed in the present work
to achieve high energy densities is laser driven shock compression. The exii
INTRODUCTION
iii
periments were performed in October, 2008 and in November, 2009 at the
GEKKO/HIPER (High Intensity Plasma Experimental Research)[Myanaga
et al., 2001], [Yamanaka, 1999] system at the Institute of Laser Engineering
(ILE), Osaka University. Several diagnostics with sub-nanosecond time resolution were used. Emissivity diagnostics recorded both space and wavelength
resolved data and interferometric velocimeters Visar measured shock velocity
and reflectivity.
Data analysis was performed mainly under MATLAB environment and several routines have been developed to extract from the images supplied by the
diagnostics the relevant quantities such as emissivity and the correspondent
emission temperature, traversing time of the shock in the carbon layer, reflectivity and velocity from fringe shift. In addition, robust numerical fitting
method of the emission spectra has been set up to calculate temperature, as
well as FFT analysis of interferometric images to calculate the fringe shift
instant by instant.
Extensive use of numerical simulations, as a very powerful tool, was made
throughout all the steps of the experimental campaign. Firstly, simulations
have been used to predict the experimental conditions, in order to optimize
the target design. Besides, during the experiment simulations suggested the
suitable laser energy shot by shot, according to the actual target characteristics.
In the end, the most important contribution is that to the shock parameters
estimation and comparison with experimental findings. The simulations were
performed with Multi [Ramis et al., 1988], a one-dimensional hydrodynamic
radiative code.
To ensure high quality simulations, adherent to the real experimental conditions, accurate modeling of the target is required. In particular, the thermodynamical properties of the materials are described by the Equation of State
(EoS), in the form of tables. Sesame[LANL] is the standard source of numerical tables, however this vast archive does not cover low density porous carbon.
Thus, the Qeos[More et al., 1988] model was adopted, owing to its high accuracy and the limited number of parameters required to build the EoS. Among
∂P
− is used by the
these parameters, the bulk modulus − defined as B = −V0 ∂V
model to account for the binding energy. To estimate this quantity, a simple
scaling-law relating density and B was devised and compared to other models
and experimental data found in literature.
The interpretation of experimental results required care in distinguishing
between reflection on carbon and reflection on substrate, as the two thresholds
for the reflectivity rise are very close. The following method was adopted
do discriminate the two cases: when an increase in reflectivity is observed
after the shock breakout, if the simulations predict pressure P below substrate
metalization threshold, carbon temperature T close to the observed spectral T
and Visars yield velocity close to the interface velocity U , carbon was imaged.
Otherwise, if simulations predict P above substrate threshold, the observed
temperature is close to the temperature predicted for the substrate and Visars
yield velocities close to shock velocity D, a substrate metalization is likely to
have occurred.
One shot fell in the very narrow window available for the observation of
the reflectivity rise in carbon. The uprising of the reflectivity above the typical
value of the insulating carbon was found at pressure of 2.4 ± 0.3 Mbar and
temperature of 14, 000 ± 2, 000 K. The last experimental finding is in line with
INTRODUCTION
iv
another experimental result, concerning shock wave release from carbon free
surface. In that case, an increase in reflectivity was found for pressure of about
1 Mbar at 12, 000 K and for six more points on the Hugoniot curve at pressure
in the range 3-8 Mbar and temperature from 20, 000 to 70, 000 K.
(3|x|2/3 + |y|5/2 − 1)(1.5|x|2/3 + 0.5|y|5/2 − 1)(x2 + (y − 0.65)2 − 0.005) = 0
INTRODUCTION
v
Riassunto
In questo lavoro di tesi la tecnica di compressione della materia attraverso
onde d’urto prodotte da laser è stata applicata allo studio delle proprietà delle
fasi liquide del carbonio alle pressioni dei megabar. Questo ramo della ricerca
sul carbonio coinvolge diversi campi, dalla scienza dei materiali[Bundy et al.,
1996], alla planetologia, all’astrofisica[Guillot, 1999].
Il carbonio è oggetto di studi da lungo tempo, in quanto presenta una
varietà di fasi differenti con proprietà interessanti. Il diagramma delle fasi a
bassa pressione (al di sotto dei 50 GPa e per temperature inferiori ai 6 000 K) è
ormai noto con precisione, così come le caratteristiche delle fasi solide, grafite e
diamante, che sono state indagate a fondo. Al contrario, le fasi liquide ad alta
pressione sono pressoché inesplorate a causa delle condizioni estreme richieste
per il loro studio. Anche le simulazioni e i calcoli teorici sono complicati, in
quanto per produrre previsioni accurate sono richiesti modelli sofisticati, i quali
necessitano di grandi potenze di calcolo.
Alcuni recenti lavori teorici[van Thiel and Ree, 1993],[Correa et al., 2006]
hanno previsto l’esistenza di due fasi liquide, caratterizzate da diversa conducibilità. Una terza fase liquida metallica è stata predetta da [Grumbach and
Martin, 1996] per pressioni nel range 4 - 10 Mbar.
Lo studio del diagramma delle fasi può essere realizzato sperimentalmente
attraverso due approcci distinti. Il primo approccio prevede la compressione
statica del campione per mezzo di celle ad incudine di diamante e permette
di variare la temperatura da pochi gradi kelvin, fino a circa 1000 K, mentre
la pressione massima raggiungibile è di circa 1 Mbar. Il secondo approccio,
la compressione dinamica per mezzo di onde d’urto, permette di raggiungere
temperature e pressioni molto più elevate ma, in questo caso, le due grandezze
sono legate dalle relazioni di Hugoniot che governano gli shock e non possono
essere variate indipendentemente. Inoltre la compressione è mantenuta solo
per tempi molto brevi (∼ ns). Per superare la prima limitazione sono state
sviluppate diverse strategie. Ad esempio, il bersaglio può avere una struttura
multistrato, composta da materiali di diversa densità, in modo che l’onda d’urto
porti il mezzo di interesse nella regione del diagramma delle fasi desiderata. Lo
schema che è stato adottato prevede un bersaglio composto da uno strato di
carbonio a bassa densità (poroso) depositato su un supporto trasparente. La
produzione di questi bersagli ha richiesto un notevole sforzo da parte di tre
diversi gruppi italiani: quello del Prof. Paolo Piseri, Università degli Studi di
Milano (deposizioni con tecnica SCBD: Supersonic Cluster Beam Deposition
utilizzando sorgenti PMCS − Pulsed Microplasma Cluster Source), quello del
Prof. Stefano Bellucci, Laboratori Nazionali di Frascati (deposizioni per spray
coating) e quello del Prof. Alfonso Mangione, Università degli studi di Palermo
(deposizioni per magnetron sputtering).
Come già anticipato, in questo lavoro di tesi la tecnica impiegata per la
compressione è la generazione di onde d’urto attraverso impulsi laser intensi.
Gli esperimenti sono stati realizzati nei mesi di ottobre 2008 e novembre 2009
presso l’Istituto di Ingegneria dei Laser (ILE) dell’Università di Osaka, con
l’utilizzo del laser GEKKO XII[Myanaga et al., 2001],[Yamanaka, 1999]. Diverse diagnostiche, basate su streak camera con risoluzione temporale inferiore
al nanosecondo hanno permesso di determinare i parametri sperimentali. Le diagnostiche passive hanno permesso di registrare l’emissività risolta nello spazio
INTRODUCTION
vi
e nella frequenza, mentre le diagnostiche attive, dei dispositivi interferometrici
basati sullo spostamento doppler della luce riflessa da una superficie riflettente
detti Visar, hanno fornito misure della velocità nello shock e della riflettività.
L’analisi dei dati è stata eseguita principalmente in ambiente MATLAB.
Sono state realizzate diverse routine per estrarre dalle immagini fornite dalle
diagnostiche le grandezze di interesse, come l’emissività e la corrispondente
temperatura di emissione, il tempo di attraversamento dello spessore di carbonio da parte dell’onda d’urto, la riflettività e, partendo dallo spostamento
delle frange, la velocità. Inoltre, sono stati sviluppati metodi di robust fitting
applicati agli spettri di emissione, per ottenere la temperatura, e metodi di
analisi, basati su trasformata di Fourier, delle immagini degli interferometri
per ottenere il valore dello spostamento delle frange istante per istante.
Durante tutte le fasi degli esperimenti, si è fatto ricorso allo strumento
molto potente delle simulazioni numeriche. In un primo momento, le simulazioni sono state utilizzate per avere informazioni sulle caratteristiche dei
bersagli, utili agli scopi dell’esperimento. Successivamente, grazie alle simulazioni basate sui parametri di densità e spessore effettivi dei bersagli, si sono
ottenute indicazioni sul valore dell’energia del laser più appropriato per ogni
tiro. Infine, il contributo più importante è stato quello fornito dalla chiarificazione della dinamica degli shock, con un alto grado di dettaglio. Tutte
le simulazioni sono state eseguite con il codice Multi[Ramis et al., 1988], in
geometria piana monodimensionale, tenendo conto del trasporto radiativo di
energia e dell’evoluzione idrodinamica.
É importante sottolineare che le simulazioni forniscono indicazioni utili ed
affidabili solamente se la modellizzazione del bersaglio e dell’impulso laser è
accurata. In particolare, le proprietà termodinamiche dei materiali sono descritte dalle equazioni di stato (EoS), sotto forma di tavole numeriche. Nella
maggior parte dei casi le tavole sono contenute nell’archivio di Sesame[LANL]
ma quella del carbonio a bassa densità non è compresa. Di conseguenza è
stato necessario impiegare il modello Qeos[More et al., 1988], scelto per la
grande qualità delle tavole prodotte e per l’esiguo numero di parametri richiesti in input. Tra questi parametri il modulo di compressibilità − definito da
∂P
− è usato all’interno del modello per stimare l’energia dei legami
B = −V0 ∂V
tra le molecole. Per valutare il modulo di compressibilità del carbonio poroso
è stata formulata una legge di scala che lega B alla densità, la cui validità è
stata confermata tramite un confronto altri modelli e dati sperimentali trovati
in letteratura.
L’interpretazione dei risultati sperimentali ha richiesto una particolare attenzione nel distinguere tra la luce riflessa dal carbonio e quella riflessa dal
substrato, dato che la soglia di pressione oltre la quale si osserva un aumento
della riflettività nel primo è appena inferiore a quella del secondo. Per discriminare i due casi è stato adottato il metodo descritto di seguito: se le simulazioni
hanno previsto pressioni P al di sotto della soglia di metallizzazione del substrato, temperature del carbonio T vicine a quelle ricavate dalle diagnostiche
di emissività, velocità dell’interfaccia U prossima a quella ottenuta attraverso
le Visar, allora è possibile concludere che il segnale osservato proviene dallo
strato di carbonio. Al contrario, se le simulazioni hanno previsto P al di sopra
della soglia critica, T del substrato vicine a quelle ricavate dalle diagnostiche di
emissività, velocità dell’onda d’urto nel substrato D prossima a quella ottenuta
attraverso le Visar, allora è possibile affermare che quanto osservato si deve
INTRODUCTION
vii
al cambiamento di riflettività del substrato.
L’aumento della riflettività del carbonio compresso è stato osservato sperimentalmente alla pressione di 2, 4 ± 0, 3 Mbar e alla temperatura di 14, 000 ±
2, 000 K. Questo risultato è in linea con un’altra serie di dati, ottenuta con una
diversa geometria dei bersagli. In quel caso lo shock, dopo aver attraversato lo
spessore di carbonio, ne provocava il rilassamento nel vuoto a causa della mancanza del substrato. L’aumento di riflettività era stato ottenuto alla pressione
di 1 Mbar a 12, 000 K e per altri sei punti sulla curva di Hugoniot nel range di
pressioni 3-8 Mbar e temperature da 20, 000 a 70, 000 K.
0.1
Bibliography
F. P. Bundy et al. The pressure-temperature phase and transformation diagram
for carbon. Carbon, 34:141–153, 1996.
A. A. Correa et al. Carbon under extreme conditions: Phase boundaries and
electronic properties from first-principles theory. PNAS, 103(5):1204–1208,
2006.
M. P. Grumbach and R. M. Martin. Phase diagram of carbon at high pressures
and temperatures. Phys. Rev. B, 54(22), December 1996.
T. Guillot. Science, 286:72, 1999.
LANL. Sesame: The lanl equation of state database, los alamos national laboratory, la-ur-92-3407, 1992.
R. M. More et al. A new quotidian equation of state (qeos) for hot dense
matter. Phys. Fluids, 31(10):3059, 1988.
N. Myanaga et al. In Proceedings of the 18th International Conference on
Fusion Energy, IAEA-CN-77, Sorrento, Italy, 2001. IAEA.
R. Ramis et al. Mulit - a computer code for one-dimensional multigroup radiation hydrodynamics. Comp. phys. comm, 49:475–505, 1988.
M. van Thiel and F. H. Ree. Phys. Rev. B, 48:3591, 1993.
C. Yamanaka. In C. Labaune et al., editors, Proceedings of the Inertial Fusion
Sciences and Applications 99, page 19, Bordeaux, 1999. Elsevier, Paris, 2000.
INTRODUCTION
viii
Acknowledgements – Ringraziamenti
Questo lavoro di tesi è stato possibile grazie al contributo di molte persone, a partire
dal Prof. Dimitri Batani che mi ha dato l’opportunità di prendere parte a una vera
attività di ricerca e ha seguito e diretto il mio lavoro, fino alle ultime fasi. Ringrazio
anche Roberto, per avermi fornito le prime basi in un campo che non conoscevo e per
aver condiviso con me i giorni di attività intensa a Osaka. Non posso inoltre dimenticare Renato, per il tempo che ha dedicato alla ricerca dei target, senza trascurare i
numerosi suggerimenti in molteplici campi.
I miei ringraziamenti vanno inoltre al Prof. Paolo Piseri, per la sua grandissima
disponibilità, per le settimane di lavoro che ha dedicato alla realizzazione dei bersagli
e per aver affidato i macchinari del suo laboratorio alle mie mani (mostrando una
notevole fiducia!). Ringrazio anche i membri del laboratorio di getti molecolari di via
Celoria, tra i quali cito Luca, Marco, Tommaso e Ajay.
My special thanks are addressed to Prof. Keisuke Shigemori and all the ILE staff,
who supported my work. In particular, I am also grateful to Yoichiro Hironaka, for
his help with Visar system, to Akiyuki Shiroshita for the profile measurements of
the targets, the image database and datasheets as well, and to Toshihiko Kadono for
the laser pulse data.
I am also thankful to Abutrab Aliverdiev, who introduced me into Multi simulations. A questo proposito i maggiori ringraziamenti li rivolgo a Tommaso Vinci,
per tutti i chiarimenti che mi ha fornito e per il servizio di help-desk continuo che ha
svolto sia via mail che dal vivo.
Ringrazio Luca Volpe per avermi aiutato contro quelle ”maledette” Hugoniot,
fornendomi programmi e spiegazioni. Un grazie anche a tutti quelli che si sono sorbiti
le mie presentazioni di prova, in vista della discussione di fronte alla commissione:
Alessio, Andrea, Claudio, Dimitri, Luca3 , Giulia, Roberto, Yas . . . Infine, non posso
dimenticare di ringraziare Daniela, per l’aiuto con le urgenze burocratiche che si sono
presentate di volta in volta.
Un ringraziamento particolare è diretto al Prof. Marco Bernasconi, correlatore di
questo lavoro di tesi, per la disponibilità dimostrata nei miei confronti. Sono inoltre
grato a Davide Donadio per i dati del carbonio poroso che mi ha fornito.
Dopo i ringraziamenti ”istituzionali” arrivano quelli meno formali, rivolti innanzitutto ai miei genitori Arduino e Mariella che mi hanno supportato (e sopportato!)
in tutti questi anni di studi.
Un grazie speciale va a Chiara, che mi è stata sempre vicina, anche quando ero a
10 000 Km di distanza.
Grazie a Robi e ai suoi libri, grazie a mio zio Piero, a mia nonna Ebe e a mia zia
Maria, che mi hanno sempre incoraggiato.
E ora i ringraziamenti verso tutti i miei amici, non quelli di facebook − dal quale
riesco ancora a tenermi lontano − ma quelli che mi sono stati accanto, quelli per cui
sono ”Pale”, ”il Pale”, ”il professore”, ”o’ professore”, ”Monza” oppure ”Stefano”.
L’ultimo grazie è per coloro che in questo anno di tesi sono stati per me dei
compagni, non solo di lavoro: Alessio, Alessandro, Claudio, Daniela, Dimitri, Giulia,
Kadhim, Luca3 , Roberto, Saad, Yas . . .
INTRODUCTION
ix
This thesis is divided into five chapters. The first illustrates the physical background, both theoretical and experimental, which constitutes the basis of the
present work. The second describes the experimental setup and the third explains the methods developed to analyse the data. The fourth chapter shows
the results achieved. In the last chapter, conclusions are drawn.
Physical context
1
This chapter is focused on the physical background that support the present
work. An introduction on the carbon-related research topics in the high energy
density physics will be given and recent theoretical results will be reported.
Next, the main experimental techniques employed to achieve high pressures
will be illustrated, with a special attention to shock compression methods and
shock wave dynamics. Then, laser-matter interaction will be studied in the
case of shock generation by intense beams of nanosecond duration. In the end
a very powerful tool supporting the experimental activity will be analyzed: the
numerical hydrodynamic-radiative Multi simulation code.
1.1
Carbon: Overview of Phase and Transformation
Diagram
Elemental carbon has been known since prehistory, and diamond is thought
to have been first mined in India more than 2,000 years ago, although recent
archaeological discoveries point at the possible existence of utensils made of
diamond in China as early as 4,000 before Christ. Therefore, the properties of
carbon and its practical and technological applications have been extensively
investigated for many centuries. In the last few decades, after the seminal work
of Bundy and coworkers [Bundy et al., 1955] in the 1950s and ’60s, widespread
attention has been devoted to studying carbon under pressure [Bundy et al.,
1996].
Carbon is unique among the elements in the diversity of its different phases.
The diamond phase is a three-dimensional tetrahedral network of fourfold coordinated atoms and is the hardest known material. The graphite phase is
made up of planes of threefold coordinated atoms and is the strongest twodimensional material known, but behaves as a lubricant because of the weak
bonding between planes. The vapor consists of chains of twofold coordinated
atoms.
The binding energy between atoms of carbon is very large; for example, the
cohesive energy of diamond is 717 kJ/mol. This property is also demonstrated
by the extremely high melting temperatures of its solid forms (∼ 5,000 K).
In addition, once carbon atoms are locked into a given phase configuration,
typically a large amount of activation energy is required to produce a different
stable phase; in other words, very high temperatures are often required to ini1
CHAPTER 1. PHYSICAL CONTEXT
2
tiate spontaneous transformations from one solid phase to another.
In addition to the well known crystalline forms of carbon, i.e. graphite and diamond, there are ’amorphous’ forms, such as glassy carbon and carbon black,
and possibly metastable solid forms referred to as carbines. The latter are believed to be solid condensations of linear molecules of carbon but remain controversial. Also, there are the more recently discovered crystalline forms of pure
carbon molecules, the fullerenes such as C60 buckyballs and C70 buckyfootballs.
Because of the high cohesive and activation energy, carbon polymorphs typically exist metastably well into a T ,P region where a different solid phase is
thermodynamically stable. For example, diamond survives indefinitely at room
conditions where graphite is the thermodynamically stable form. Conversely,
except at very high temperatures, graphite stubbornly persists at pressures
far into the diamond stability field. The same is true of fullerene and amorphous carbons. There are two preferred forms of electronic bonding of carbon
atoms in the solid state: the sp2 type in which a given atom is bonded to
three equidistant nearest neighbors 120◦ apart in a plane, as in graphite; the
sp3 type in which a given carbon atom is bonded to four equidistant nearest
neighbors arranged in tetrahedral symmetry, as in diamond. The amorphous
or glassy carbon forms are thought to be nano-zonal mixtures exhibiting these
two types of bonding.
The properties of other phases, such as the liquid or the high-pressure metallic
solid, are almost unknown because of the extreme conditions of temperature
and pressure that are needed to study them experimentally. Experimental data
are scarce because of difficulties in reaching megabar (1 Mbar = 100 GPa)
pressures and thousands of Kelvin regimes in the laboratory. Theoretically,
sophisticated and accurate models of chemical bonding transformations under
pressure are needed to describe phase boundaries. In most cases, such models
cannot be simply derived from fits to existing experimental data, and one needs
to resort to first-principles calculations, which may be very demanding from a
computational standpoint.
Nevertheless, the phase diagram of C at high pressures and temperatures is
of ongoing scientific interest in materials science, planetary science, and astrophysics. Major issues include a predicted first-order phase transition in the
liquid [Glosli and Ree, 1999b], the melting curve of graphite [Togaya, 1997],
the slope and magnitude of the diamond melting curve, the nature of the liquid
state [Grumbach and Martin, 1996], and the nature of C in the deep interiors
of Uranus and Neptune. The description of high-pressure phases is essential
for developing realistic models of planets and stars [Guillot, 1999]. Carbon is
a major constituent (through methane and carbon dioxide) of giant planets.
High pressures are thought to produce methane pyrolysis with a separation of
the carbon phase and the possible formation of a diamond or metallic layer
[Ross, 1981],[Ancillotto et al., 1997] and [Nellis et al., 1997]. Metalization of
the carbon layer in the mantle of these planets (the ”ice layers”) could give a
high electrical conductivity and, by the dynamo effect, be the source of the
observed large magnetic fields [Ness et al., 1989], but in this case also a contribution from water metalization could be present.
CHAPTER 1. PHYSICAL CONTEXT
3
The phase diagram
The T ,P phase and ’reaction’ diagram in the low pressure regime for pure
carbon is presented in Fig. 1.1 [Bundy et al., 1996]. The topology of stability
Figure 1.1: P ,T phase and transition diagram for carbon in low pressure regime
(P < 50 GPa). Solid lines represent equilibrium phase boundaries. A: commercial synthesis of diamond from graphite by catalysis; B: P/T threshold of
fast (less than 1 ms) solid-solid transformation of graphite to diamond; C: P/T
threshold of very fast transformation of diamond to graphite; D: single crystal
hexagonal graphite transforms to retrievable hexagonal-type diamond; E: upper ends of shock compression/quench cycles that convert hex-type graphite
particles to hex-type diamond; F: upper ends of shock compression/quench cycles that convert hex-type graphite to cubic-type diamond; B,F,G: threshold
of fast P/T cycles, however generated, that convert either type of graphite or
hexagonal diamond into cubic-type diamond.
fields of the thermodynamically stable phases is quite simple:
i the boundary between the graphite and diamond stable regions which runs
from 1.7 GPa/O K, to the graphite/diamond/liquid triple point at about
12 GPa/5,000 K;
ii the melting line of graphite extending from the graphite/liquid/vapor triple
point at 0.011 GPa/5,000 K to the graphite/diamond/liquid triple point at
12 GPa/5,000 K;
iii the diamond melting line that runs to higher P and T above the triple
point.
Although there has often been theoretical speculation about the phase diagram in the multi-Mbar regime, predictions have been hampered by the lack of
CHAPTER 1. PHYSICAL CONTEXT
4
a comprehensive model for covalent liquids. Density-functional based molecular dynamics [Car and Parrinello, 1987] provides just such a comprehensive
model. The density-functional treatment of the electronic degrees of freedom
allows an accurate description of all the types of bonding that might occur in
carbon, while molecular dynamics offers a means to simulate various thermodynamic states. One of the most impressive applications of this method has been
to help establish the nature of the melting of the diamond phase of carbon, as
discussed below. Fig. 1.2 shows the phase diagram of carbon in high pressure
Figure 1.2: Phase diagram of carbon in high pressure regime (0.01 ÷ 10 Mbar)
from Wang et al. [2005]. The solid line and up triangles indicates the graphitediamond boundary from Ref. [Bundy et al., 1996]; the graphite-liquid boundary
(solid line and down triangles) is from an experimental work by Togaya [1997];
the rectangle gives the uncertainty on the experimental triple point [Bundy
et al., 1996]; the solid line with open circles is the diamond melting line from
Ref. [Wang et al., 2005] and the dashed line is the extrapolation to the triple
point; the dotted line is the empirical melting curve from Ref. [Glosli and Ree,
1999a] and the dashed-dotted line is the empirical melting curve from Ref.
[Ghiringhelli et al., 2005].
regime proposed by Wang et al. [2005] and based on ab initio density functional theory. They combined first-principles molecular dynamics with modern
thermodynamic integration techniques to compute the free energy of solid and
liquid carbon in an extended range of pressures and temperatures. In Fig.
1.2, the solid line and up triangles indicates the graphite-diamond boundary
from Ref. [Bundy et al., 1996]; the graphite-liquid boundary (solid line and
down triangles) is from an experimental work by Togaya [1997]; the rectangle
gives the uncertainty on the experimental triple point [Bundy et al., 1996]; the
solid line with open circles is the diamond melting line from Ref. [Wang et al.,
2005] and the dashed line is the extrapolation to the triple point; the dotted
line is the empirical melting curve from Ref. [Glosli and Ree, 1999a] and the
dashed-dotted line is the empirical melting curve from Ref. [Ghiringhelli et al.,
2005].
CHAPTER 1. PHYSICAL CONTEXT
5
The diamond melting line
Prior to the mid-1980s it was thought that the behavior of cubic-type diamond
would be analogous to that of the diamond-structure forms of the heavier Group
IV elements Si and Ge. Experiments demonstrated that diamond-structure Si
and Ge melt to form metallic liquids of higher density than the solid, which
requires dTm /dP to be negative. It was also known experimentally that at
higher pressures, diamond-structure Si and Ge undergo transitions at room
temperature to denser, metallic phases [Young, 1991]. By analogy, the earlier
T,P phase diagrams for carbon showed the melting line of diamond with a negative slope, extending to a diamond/metallic solid/liquid triple point at higher
pressure and lower temperature. The melting line of the (supposed) metallic
solid was believed to have a positive slope, as found in Si and Ge [Bundy,
1980].
This picture was brought into question by the experiment of Shaner et al.
[1984]. In this experiment a graphite sample was shock compressed to a series of high-temperature, high-pressure states. The sound velocity within the
shocked material was monitored for possible discontinuous changes that would
signal a first-order phase transition. From previous work it was known that
graphite collapses to diamond when sample pressures reach the range 300–600
kbar. However, beyond this no further phase changes were detected up to 1.4
Mbar and 5,600 K. This was interpreted to mean that diamond is in a solid
state at this phase point. Since this temperature is higher than the triple point
temperature, the slope of the phase boundary must be positive. A positive
slope for the melting curve was also found by Galli et al. [1990] who simulated
melting and freezing using first-principles molecular dynamics methods. Despite the difficulties associated with the persistence of metastable superheated
or supercooled states, they were able to estimate that the melting temperature
at pressures of approximately 1 Mbar is between 6,500 and 8,000 K. They also
showed conclusively that upon melting the pressure of the system increases.
Thus, from the Clausius-Clapeyron relation, they concluded that the slope of
the melting curve is positive. Grumbach and Martin [1996] predicted a change
in slope of the diamond melting curve at pressures above ∼ 6 Mbar, due to a
change in the characteristic coordination of the liquid. This is based on the
fact that, as discussed later, the sixfold liquid is more dense than the fourfold
liquid and so the slope of the phase boundary between diamond and the sixfold
liquid is expected to be of opposite sign than the slope of the diamond-fourfold
liquid boundary.
Regarding the metallic character of diamond, a result of electronic structure
calculations and first-principles molecular dynamics using density-functional
theory by Correa et al. [2006] is that the energy gap remains finite even at
temperatures where the solid is likely to be superheated (i.e., metastable).
More precisely, the gap is predicted to increase with pressure (from 4.4 eV at
P = 0 to 6.9 eV at P = 1,100 GPa, at zero temperature) and to decrease
with temperature (from 6.8 eV at T = 0 to 3.6 eV at melting temperature
Tm = 6750 K, for a pressure of 1,000 GPa. The liquid, instead, showed a finite
density of states at the Fermi level and a finite conductivity at zero frequency.
Thus, they concluded that diamond remains an insulator in the solid phase,
and an insulator-to-metal transition occurs only upon melting.
CHAPTER 1. PHYSICAL CONTEXT
6
Liquid carbon
The only well-known phases are the diamond and graphite solids and the vapor. Very little is known experimentally about the liquid. Motivated by a
theoretical prediction that two phases of liquid carbon, one insulating and one
metallic may exist[Gustafson, 1986], experiments have focused on establishing
the electronic nature of the liquid at low pressures for a long time. Using a
high intensity laser pulse it is possible to heat a small area of a carbon sample,
while a second probe beam measures the reflectivity of the heated area. Reitze
et al. [1992] employed a 90-fs laser pulses followed by optical measurements
of 10 fs time resolution. They find that the reflectivity increases during the
first 1-2 ps after the pulse but then after about 10 ps decreases to below initial reflectivity (prior to the pulse). They argue that the initial increase in
reflectivity is due to a phase change to the liquid, while the decrease is due to
hydrodynamic expansion of the surface. They estimate a moderate resistivity
for the liquid phase from the reflectivity data, for which the title ”metal” is
barely applicable. A difficulty of the laser-pulse approach is that it is difficult
to estimate the temperature and pressure of the excited state of the sample.
The experiments are done on such a short time scale that there is very little
expansion of the volume of the heated material. The final conclusion is that the
nature of the liquid state is still not fully established by experiments. Recently
there has been speculation by van Thiel and Ree [1993] that there is a phase
transition between two liquid phases. A model was developed which includes
two liquid phases similarly to graphite and diamond solid phases. Liquid-1 has
mostly graphitic character (threefold coordination) while liquid-2 has mostly
tetrahedral character (fourfold coordination). Grumbach and Martin [1996]
have also found that the character of the liquid changes from about fourfold to
about sixfold coordination in the pressure range 4–10 Mbar. This could occur
either as a rapid but continuous change of a single phase or discontinuously at
a first-order phase transition. Due to the difficulties of finding a discontinuity
in the equation of state and to the lack of knowledge of the melting temperature of diamond at these pressures, it is not possible to directly identify a
first-order phase transition. Determining this would be of great interest, since
structural changes between amorphous phases are quite uncommon [Aasland
and McMillan, 1994], [Poole et al., 1994]. However, they do find two distinct
liquid structures among the metastable simulations which they performed. It is
possible to conclude from this that sixfold structures are unstable at pressures
below ∼ 6 Mbar. The proposed phase diagram is shown in Fig. 1.3. It is
worth noting that the phase boundaries between solid (graphite or diamond)
and liquid is in accordance with the diagram shown in Fig. 1.2.
1.2
High Pressure Experimental Techniques
There are many method to achieve high pressures, based on two different approaches: static or dynamic. The first compresses matter at rest, while the
second makes use of shock waves. A brief resume of the main techniques is
presented hereafter.
CHAPTER 1. PHYSICAL CONTEXT
7
Figure 1.3: Proposed phase diagram for liquid carbon after Grumbach and
Martin [1996]
Static methods
Diamond anvil cell
A diamond anvil cell (also known with the acronym DAC) consist in two diamond anvils separated with a gasket compressing the specimen statically, as
shown in Fig. 1.4. Typical diameter of the diamonds is about 100 µm on
Figure 1.4: Diamond anvil cell
the tip, so that applying 10 kN pressure on the specimen can be as high as
1 Mbar. The compressed target is accessible through the diamond windows
to various diagnostics, such as optical probes or X-rays. Also, the sample can
be heated, with a continuous laser or using electrical current, for example, or
cooled with cryogenic liquids. The main advantage of this technique is that the
high pressure state is maintained for a long time, on a wide range of temperatures, enabling studies of crystalline structure, optical properties or transport
properties (refractive index or thermal and electrical conductivity).
CHAPTER 1. PHYSICAL CONTEXT
8
Dynamic methods
Typically a shock wave is produced by means of explosions, impact or laser
irradiation and the sample is in high pressure state only for a very short time,
so measures are difficult and mostly indirect.
Chemical explosions
Chemical explosions can generate shock waves of the order of kbars if the
target is in contact with the explosive or of the order of Mbars if the explosive
accelerates a metal plate that impacts on the target at high speed, as shown
in Fig. 1.5. The time of observation of the compressed sample are of the order
of µs.
Figure 1.5: Shock wave generation by means of chemical explosions: the slow
and fast explosives are arranged to produce a plane shock wave in the metal
plate. In case (a) the plate is in contact with the sample, while in case (b) an
empty space let the metal to be accelerated during the explosion, in order to
impact on the sample at very high speed
Gas gun
The principle is to accelerate a bullet at high speeds towards the sample. The
detonation pushes a piston to compress the hydrogen gas contained in a cylinder. The bullet is kept separated from the gas by a valve until the explosion
breaks the valve and the bullet is set into motion. Typical speeds are of the
order of 10 Km/s and pressure can rise up to 1 Mbar.
Z-pinch
An intense current (of the order of 107 A) flows for about 10 ns in a cylinder
made of a hundred of small tungsten wires. The wires get ionized very quickly
and the plasma so generated is compressed along the axis (usually indicated
as z-axis) by the magnetic field produced by the intense current. The plasma
can reach temperatures as high as the keV and emits intense X-rays that heat
the sample up to hundreds of eV.
CHAPTER 1. PHYSICAL CONTEXT
9
Nuclear explosions
The principles is the same as in chemical explosions, where the explosive is
replaced by a nuclear weapon located inside a thickness of rocks. This technique
allows to reach the highest pressures but it is also the most problematic. After
the signature on September 24th, 1996 and the ratification on February 1st,
1999 by the Italian government of the Comprehensive Nuclear Test Ban Treaty1
it is not possible to realize experiments of compression by nuclear explosion.
United Kingdom and France ratified the treaty on April 6th, 1998, United
States, Israel and China signed but did not ratify the Ban while India, North
Korea, Pakistan and Iran did not sign yet.
Intense lasers
As shown in Fig. 1.6, a high intensity laser pulse (I > 1011 W/cm2 ) is focused
on the surface of the sample. Immediately a hot plasma is created and its
expansion generates a shock wave that starts propagating inside the medium,
in the opposite direction. This approach will be described more carefully in
section 1.4.
Figure 1.6: Laser-induced shock wave: the laser beam is focused on an absorbing layer that is ablated at high rates, producing a plasma corona expanding
in the vacuum and a shock wave propagating inside the medium
Exploration of the phase diagram
The investigation of the phase diagram requires to compress and heat (or cool)
matter in a wide range of temperatures and pressures. Static methods have
a big advantage in this, as temperature and pressure can be varied independently, while dynamic methods are constrained by the conservation laws of
shocks that correlate P and T (as will be shown in section 1.3). However, the
1 http://www.ctbto.org
CHAPTER 1. PHYSICAL CONTEXT
10
highest pressures and temperatures cannot be achieved with DACs and dynamic approaches are necessary. Moreover, the use of pre-compressed, porous
or multi-layer target overcome the constraints and enable the exploration of
regimes different from the states that standard-density and bulk materials can
reach. For example, in shock wave experiment with a one layer target, the
accessible state are those lying on the Hugoniot curve, as will be explained in
the next section. Fig. 1.7 shows three Hugoniot curves of carbon, drawn for
different initial densities. For lower density, at the same pressure, the temperature is higher. In a different way, multi-layer target can be used to achieve
pressures above or below Hugoniot exploiting the reflection of the shocks at
the interfaces between the layers. In the experiment presented in this work,
both of these strategies are applied.
Figure 1.7: Comparison between main Hugoniot curves of carbon, for different
initial densities and phases. Graphite has ρ0 = 2.2 g/cm2 , while porous carbon
has ρ0 = 0.55 g/cm2 and diamond has ρ0 = 3.51 g/cm2
1.3
Shock Wave Physics
In order to achieve high energy density states, the shock compression is the
simplest and most effective approach. Hence, in this section the physics of
shock waves will be examined, starting from the mechanism responsible for
their generation and the relationships governing their dynamics.
From a mathematical point of view, a wave is a field. The surface of a lake is
a natural example: the height of the water varies both in 2-dimensional space
and time. The surface is described by a function h of the quantities x and t
where h((x), t) is the height of the water at point x at time t. A shock wave
is defined as a discontinuity in the thermodynamical quantities fields. In the
case of the surface of the lake, a shock wave would be a kind of propagating
”wall” of water.
CHAPTER 1. PHYSICAL CONTEXT
11
Fluid dynamics
For the sake of simplicity, it is possible to limit ourselves to non-viscous fluids,
initially at rest with respect to the laboratory frame. When a small perturbation dependent only on time and on coordinate x is applied, the mechanical
state of the fluid can be described through pressure P , density ρ and velocity
u as follows:
P (x, t) = P0 + ∆P (x, t)
ρ(x, t) = ρ0 + ∆ρ(x, t)
u(x, t) = u(x, t)êx
(1.1)
(1.2)
(1.3)
where ∆ρ ρ0 and ∆P P0 .
To describe a moving fluid, two coordinates systems are possible, both presenting advantages and disadvantages depending on the specific context in
which they are applied. The Lagrangian description follows the properties of
the fluid varying throughout the motion, while the Eulerian description observes the properties of the fluid flowing in a fixed point in space, at a certain
time[Paleari, 2008]. The two descriptions are related through the relationship:
∂
D
=
+ (u · ∇)
Dt
∂t
(1.3)
The starting point in the description of continuous media are the conservation
laws, stated in the two possible coordinate systems as follows.
mass conservation
∂ρ
+ ∇ · (ρu) = 0 Euler
∂t
Dρ
+ ρ∇ · u = 0 Lagrange
Dt
(1.3)
momentum conservation
1
∂u
+ u · ∇u = − ∇P Euler
∂t
ρ
Du
ρ
= −∇P
Lagrange
Dt
(1.3)
also called Euler’s equation and valid for non-viscous fluids.
energy conservation
∂
ρu2
u2
ρ +
= −∇ ρu +
+ P u + ρQ Euler
∂t
2
2
D
DV
+P
=Q
Lagrange
Dt
Dt
(1.3)
where is the specific energy of the fluid and Q is the specific energy
introduced by external sources into the fluid. In absence of external
sources of energy, Q = 0 and
D
DV
+P
=0
Dt
Dt
S indicates the specific entropy.
yielding
DS
=0
Dt
(1.3)
CHAPTER 1. PHYSICAL CONTEXT
12
Equation of state
The equation of state (EoS) of a system constituted of a single phase is completely determined by the thermodynamical potentials. This definition depends
on the particular quantities chosen to describe the state: for example, it is possible to associate to the density ρ and to the temperature T the free energy F
that satisfies the relation
F = − TS
P
dF = dρ − SdT
ρ
(1.4)
(1.5)
From the differential form 1.5 it is possible to calculate the pressure and the
entropy as
∂F
P = ρ2
(1.6)
∂ρ T
∂F
(1.7)
S =−
∂T ρ
then, substituting P from 1.7 into eq. 1.4 the internal energy is obtained again
∂
F
= F + T S = −T 2
(1.7)
∂T T ρ
More generally, from the knowledge of the free energy F (ρ, T ), a relationship
between three of the four quantities P , ρ, T and can be written, such as
f (P, ρ, ) = 0
(1.7)
This relation is called equation of state. A model of equation of state is thermodynamically coherent if it satisfies
∂2F
∂2F
=
∂ρ∂T
∂T ∂ρ
(1.7)
or, in other words, if dF is an exact differential form. Expanding, it is possible
to obtain
∂
∂P
2
P −ρ
=T
(1.7)
∂ρ T
∂T ρ
Linear acoustics
The wave equation can be obtained expanding the mass conservation law 1.3
and the Euler’s equation 1.3 at the first order in u:
∂∆ρ
∂u
∂ρ
+ ∇ · ρu = 0
+ ρ0
'0
∂t
∂t
∂x
−→
∂u
∂u ∂∆P
+ u · ∇u = − ρ1 ∇P
ρ0
+
'0
∂t
∂t
∂x
(1.7)
Next, the variation of pressure can be related to the variation of density considering that the transformation is isentropic (it is reversible and so rapid that
CHAPTER 1. PHYSICAL CONTEXT
13
no heat exchange occurs): the pressure is a function of the density and the
entropy, so that developing in the neighborhood of (P0 , ρ0 ) yields
∂∆P
∂∆ρ ∂P
'
(1.7)
∂x
∂x
∂ρ S(P0 ,ρ0 )
and it is possible to write the last of the eq. 1.3 in the form
∂u
∂P
∂∆ρ
ρ0
+
=0
∂t
∂ρ S ∂x
(1.7)
Combining this with mass-conservation law (the first of the eq. 1.3), it is
possible to write
1 ∂2u
∂2u
1 ∂ 2 ∆ρ
=
(1.7)
=
−
ρ0 ∂x∂t
∂x2
c2 ∂t2
that is the wave equation. More clearly:
2
∂2u
2∂ u
=
c
2
∂t2
∂x ∂P
c2 =
∂ρ S
(1.8)
(1.9)
and c is regarded as the speed of sound in the medium. The solutions are of
the form
u(x, t) = U1 f1 (x − ct) − U2 f2 (x + ct)
U1
U2
∆ρ(x, t) = ρ0
f1 (x − ct) +
f2 (x + ct)
c
c
U
U2
1
2
∆P (x, t) = ρ0 c
f1 (x − ct) +
f2 (x + ct)
c
c
(1.10)
(1.11)
(1.12)
(1.13)
These equation were derived under the hypothesis that the perturbation is
small, so that u · ∇u is negligible with respect to ∂u/∂t. As
∂u
∼ cU f 0
∂t
u · ∇u ∼ U 2 f 0
(1.14)
(1.15)
the conditions ∆ρ ρ0 and ∆P P0 are satisfied provided that U c.
Shock waves
When the condition U c is not satisfied, a shock wave can appear. From the
mass conservation law in the form
∂ρ
∂ρ
∂u
+u
+ρ
=0
∂t
∂x
∂x
(1.15)
recalling that c2 = (∂P/∂ρ)S it is possible to write
1 ∂P
u ∂P
∂u
+
+c
=0
ρc ∂t
ρc ∂x
∂x
(1.15)
CHAPTER 1. PHYSICAL CONTEXT
14
Adding and subtracting eq. 1.3 and the second of the 1.3, the two equations
below follows:
∂u
∂u
1 ∂P
∂P
+ (u + c)
+
+ (u + c)
= 0
(1.16)
∂t
∂x
ρc ∂t
∂x
∂u
∂u
1 ∂P
∂P
+ (u − c)
−
+ (u − c)
= 0
(1.17)
∂t
∂x
ρc ∂t
∂x
In this form, it is easy to define two families of curves C+ and C− on the
plane (x, t), called characteristics, along which the integration of the differential
equations is immediate:
dx
= u(x, t) + c(x, t)
dt
dx
:
= u(x, t) − c(x, t)
dt
C+ :
(1.18)
C−
(1.19)
The existence and uniqueness theorem states that
1. each of the two families of curves covers the whole plane (x, t) – existence
of the solution;
2. two or more curves of the same family never cross; if two curves C 1
and C 2 pass through the same point (x1 , t1 ), they must have the same
tangent and so they would be the same curve everywhere – uniqueness
of the solution.
From the definitions 1.18 and 1.19 it is possible to see that the equations 1.16
and 1.17 are two differential forms along the characteristics. The integration
yields
Z Z
1
dρ
du + dP = u +
c
≡ J+
(1.20)
ρc
ρ
C+
C+
Z Z
1
dρ
du − dP = u −
c
≡ J−
(1.21)
ρc
ρ
C−
C−
and the integration constants J± are called Riemann invariants. As far as the
compression can be regarded as isentropic, it is possible to rewrite, for example,
P and c with respect to ρ. Resolving the equations 1.20 and 1.21, the function
ρ(u) can be obtained, as well as P (u) or c(u)2 . Eq. 1.18 can be integrated as
follows
x = [u + c(u)]t + G(u)
(1.21)
and inverting:
u = G −1 (x − [u + c(u)]t)
(1.21)
The solution is a wave traveling in the x-direction with a speed of u + c(u). In
the same way, another solution of a wave traveling in the opposite direction at
speed c(u) − u can be obtained. The simple waves are deformed as time passes
because the propagation velocity c(u) ± u is not constant but it depends on the
amplitude of the wave.
2 It
is interesting to note that c depends only on u
CHAPTER 1. PHYSICAL CONTEXT
15
Ideal gas
In the case of the ideal gas, the calculation can be carried out explicitly. The
equation of state can be expressed as P = ρkT /m, the isentropic curves are
given by P = Kργ and the sound speed is
c=
∂P
∂ρ
1/2
=
p
γ−1
γKρ 2
(1.21)
S
So, the thermodynamical quantities P and ρ can be written as functions of c
only:
2γ
2
ρ ∝ c γ−1 and P ∝ c γ−1
(1.21)
Thus, pressure and density profiles are similar to the profile of c. The Riemann
invariants can be written explicitly as
Z p
γ−3
2
γKρ 2 dρ = u ±
J± = u ±
c
(1.21)
γ
−
1
C±
Considering a wave propagating forward in the x-direction and choosing the
integration constant appropriately to have u = 0 and c = c0 in the undisturbed
fluid, the invariant J− is equal to −2c0 /(γ − 1) and follows
u=
2
(c − c0 ),
γ−1
or c = c0 +
γ−1
u
2
(1.21)
This last equation shows an important physical phenomenon: the propagation
velocity is bigger than c0 if u > 0 and smaller than c0 if u < 0. As the velocity
profile is not propagated rigidly, a discontinuity emerges from the velocity
profile as illustrated in Fig. 1.8. The physical reason for the distortion lies in
the fact that the wave crests travel relatively faster, due to the higher velocity
with which they are propagated through the fluid (higher speed of sound), as
well as due to the fact that they are carried forward faster together with the
fluid (higher gas velocity). On the other hand, the valleys travel relatively
slower, since in this region both velocities are slower. Moreover, if the analytic
solution is extended over a sufficient long period the characteristics get close
and close until they intersect. This mean that the solution in the form of a
simple wave is, in this case, valid only for a limited period, up to the time when
discontinuities are formed. Besides, the hypothesis of isentropic compression
is not valid, as the propagation of a discontinuity is a non-reversible process
and so the entropy is increased due to the shock. In the case of ideal gas, the
entropy is given by
P0 V0
S − S0 = cv log
(1.21)
T0 (γ − 1)
PV
where γ indicates, as usual, the ratio between the specific heats, cv = T (γ−1)
and V0 = 1/ρ0 is the specific volume. In a shock, the change in entropy is
[Zel’dovich and Raizer, 2002]
γ P0 V0
P (γ − 1)P + (γ + 1)P0
∆S =
log
(1.21)
T0 (γ − 1)
P0 (γ + 1)P + (γ − 1)P0
and increases with pressures and tends to 0 when P/P0 → 1.
CHAPTER 1. PHYSICAL CONTEXT
16
Figure 1.8: Diagram showing the steepening and ”overshooting” of a finite
amplitude wave in nonlinear theory. The figure shows velocity profiles at successive instant of time. To compare these waves, the combination x − ct has
been plotted along the abscissa. The wave form (d) corresponds to a physically
unrealistic condition. Actually, the wave has the form (e) with discontinuities
CHAPTER 1. PHYSICAL CONTEXT
17
The Hugoniot curves
The previous section described how a shock wave is generated, here the laws
governing the propagation of the discontinuity are presented. Generally speaking, the laws of conservation of mass, momentum and energy that form the
basis for the equations of inviscid flow of a nonconducting fluid do not necessarily assume continuity of the flow variables. These laws can also be applied to
those flow regions where the variables undergo a discontinuous change. From
a mathematical point of view, a discontinuity can be regarded as the limiting
case of very large but finite gradients in the flow variables across a layer whose
thickness tends to zero. Applying the general laws of conservation of mass, momentum and energy it is possible to find the unknown quantities such as the
density ρ and the pressure P in the compressed region, and the propagation velocity of the discontinuity through the undisturbed fluid D. The parameters of
the undisturbed gas ρ0 , P0 and the fluid velocity3 U are assumed to be known.
A mass of fluid equal to ρ0 Dt contained in a column of unit cross sectional
area is set in motion at time t. This mass occupies a volume (D − U )t, that
is, the density of the compressed fluid ρ satisfies the condition [Zel’dovich and
Raizer, 2002]
ρ(D − U )t = ρ0 Dt
(1.21)
The mass ρ0 Dt acquires a momentum ρ0 Dt · U which, according to Newton’s
law, is equal to the impulse due to the pressure forces. The resultant force
acting on the compressed fluid is equal to the difference between the pressure
on the piston side and on the side of the undisturbed fluid, that is,
ρ0 DU t = (P − P0 )t
(1.21)
Finally, the increase in the sum of the internal and kinetic energies of the
compressed gas is equal to the work done by the external force acting on the
piston, P U t,
U2
= PUt
(1.21)
ρ0 Dt − 0 +
2
Dividing these equations by t, we obtain a system of three algebraic equations
which van be used to express the three unknown quantities P , ρ and D in terms
of the known quantities U , ρ0 and P0 as the thermodynamic relationship (P, ρ)
is assumed to be known. Rearranging the previous equations, it is possible to
note that if D is the propagation velocity of the discontinuity through the
stationary fluid, then u0 = −D is the velocity at which the undisturbed gas
flows into the discontinuity. Likewise, D − U is the propagation velocity of the
discontinuity with respect to the gas moving behind it and u = −(D − U ) is
the velocity of the fluid flowing out of the discontinuity. With this notation
the law of conservation of mass reads
ρu = ρ0 u0
(1.21)
Using 1.3, the law of conservation of momentum takes the form
P + ρu2 = P0 + ρ0 u20
(1.21)
3 In the case of shocks produced by means of a piston compressing a gas, the fluid velocity
coincides with the speed of the piston.
CHAPTER 1. PHYSICAL CONTEXT
18
Using 1.3 and 1.3, the law of conservation of energy becomes
+
P
u2
P0
u2
+
= 0 +
+ 0
ρ
2
ρ0
2
(1.21)
These equations relate the flow variable at the surface of the discontinuity into
which the fluid is flowing normal to the surface. It is important to note that
these equations do not require any assumptions regarding the mechanical properties of the fluid and express only the general laws of conservation of mass,
momentum and energy, forming a system of three algebraic equations with
six variables u0 , ρ0 , P0 , u, ρ and P . It is assumed that the thermodynamic
properties of the fluid (the function (P, ρ)) are known. From the knowledge
of the state of the undisturbed fluid ρ0 and P0 , and the value of a parameter
describing the strength of the shock wave (for example, the pressure behind
the wave front P or the velocity of the ”piston” creating the wave U ), all the
remaining variables can be calculated.
Now, some general relationships which follow from the conservation laws expressed by the eq. 1.3 – 1.3 will be derived. In place of the density, the specific
volumes V0 = 1/ρ0 and V = 1/ρ are introduced. From eq. 1.3 follows
u0
V0
=
V
u
(1.21)
Eliminating the velocities u0 and u from eq. 1.3, 1.3
P − P0
V0 − V
P − P0
u2 = V 2
V0 − V
u20 = V02
(1.22)
(1.23)
are obtained. If the shock wave is created in the undisturbed fluid by the motion
of a piston, the following equation for the flow velocity of the compressed fluid
(equal to the ”piston” velocity) with respect to the undisturbed fluid can be
found:
p
|U | = u0 − u = (P − P0 )(V0 − V )
(1.23)
A useful formula for the difference between the kinetic energy of the fluid on
each side of the discontinuity in a coordinate system in which the shock is at
rest is the following
1 2
1
(u − u2 ) = (P − P0 )(V0 + V )
2 0
2
(1.23)
Other practical relationships are the following:
D
D−U
P − P0 = ρ0 DU
ρ = ρ0
(1.24)
(1.25)
Substituting eq. 1.22 and 1.23 into the energy equation 1.3 yields the relationship between the pressure and the specific volume on each side of the
discontinuity
1
(P, V ) − 0 (P0 , V0 ) = (P + P0 )(V0 − V )
(1.25)
2
CHAPTER 1. PHYSICAL CONTEXT
19
By analogy with the equation relating the initial and final pressures and volumes during adiabatic compression of a fluid, the relation 1.3 is termed shock
adiabatic or the Hugoniot relation. The Hugoniot curve is represented by the
function
P = H(V, P0 , V0 )
(1.25)
which in many practical cases, when the thermodynamic function = (P, V )
has a simple form, can be found explicitly.
Hugoniot curves differ appreciably from ordinary isentropes or isentropic adiabatics. While the ordinary isentrope belongs to a one-parameter family of
curves P = P(V, S), where the only parameter is the entropy S, the Hugoniot
curve is a function of two parameters, the initial pressure P0 and volume V0 .
In order to cover all the curves P = P(V, S) it is sufficient to traverse a onedimensional series of values of the entropy S. In order to cover all the curves
P = H(V, P0 , V0 ) it is necessary to construct an ”infinity squared” of curves
corresponding to all possible values of P0 and V0 .
Ideal gas with constant specific heats
The shock wave equations for a perfect gas with constant specific heats are
particularly simple. It is convenient to use this case to explain all the qualitative
relationships governing the changes in the variables across a shock wave. In
the case of the ideal gas, the energy for unit volume can be written as
= cv T =
1
PV
γ−1
(1.25)
and the Hugoniot curve in the explicit form is
P
(γ + 1)V0 − (γ − 1)V
=
P0
(γ + 1)V − (γ − 1)V0
(1.25)
from which the specific volume ratio is given by
V
(γ − 1)P + (γ + 1)P0
=
V0
(γ + 1)P − (γ − 1)P0
(1.25)
The temperature ratio follows from
T
PV
=
T0
P0 V0
(1.25)
It is interesting to note that the density ratio across a very strong shock wave,
where the pressure behind the wave front is much higher than the initial pressure, does not increase with increasing strength indefinitely, but it approaches
a certain finite value. This limiting density or volume ratio across the shock
wave is a function of the specific heat ratio only, and is equal to
V0
γ+1
ρ
=
=
ρ0
V
γ−1
(1.25)
The limiting density ratio for a mono-atomic gas with γ = 5/3 is equal to 4.
For a diatomic gas γ = 7/5 (assuming that the vibrational modes gave not
been excited) and the limiting density ratio is 6; if, on the other hand, full
CHAPTER 1. PHYSICAL CONTEXT
20
vibrational excitation is assumed, then γ = 9/7 and the density ratio limit is
8. In reality, at high temperatures and pressures, the specific heats and the
specific heat ratio are no longer constant because of molecular dissociation and
of ionization. Even in this case, however, the density ratio remains finite and
does not increase without limit; generally it does not exceed 11–13.
Shock propagation at the interface between two different
materials
As anticipated before in section 1.2, at the interface between two different materials interesting phenomena occur, such as shock wave reflection, that can be
exploited to achieve off-Hugoniot states.
The interface is an equilibrium surface where the fluid velocities and the pressures of the two materials are equal. On the other hand, the density and the
temperature are determined by the equation of state, thus they can be different
in the two media. The final state locus in the (P, U ) plane is called shock polar ;
Fig. 1.3 shows the shock polar for graphite, according to mis [1992]. If M is
Figure 1.9: Shock polar of carbon according to SESAME[mis, 1992]. The slope
of the OM line is called shock impedance
the final state of the shock, the slope of the OM line is called shock impedance.
After relation 1.25, P = ρ0 DU in strong shock limit, the shock impedance Z
is
Z = ρ0 D
(1.25)
When a shock is transmitted from a medium (A) to a medium (B), two possibilities have to be considered:
ZA < ZB : a shock at higher pressure is transmitted to material (B) and a
shock is reflected back to (A), with the same pressure of the shock in (B).
ZA > ZB : a shock wave at lower pressure is transmitted to (B) and a rarefaction wave propagates backward into (A). This case is quite similar to
the shock breakout at a free surface, with the difference that the release
is only partial
CHAPTER 1. PHYSICAL CONTEXT
21
ZA < ZB
The shock wave compress material (A) to the state named A in the (P,U) plane
(Fig. 1.10) along the Hugoniot HA . From this point a reflected shock wave
Figure 1.10: Pressure profile produced after the reflection of a shock wave at the
surface between a low impedance medium (A) and a high impedance medium
(B) and schematic representation in the plane (P,U)
0
. This curve is the shock polar HAc ,
moves back into (A) along the curve HAc
called hot polar, symmetrized by a vertical axis passing through point A. The
0
with
pressure of the shock transmitting in (B) can be found intersecting HAc
the (B) shock polar, HB . From the figure it is also possible to note that the
reflection of the wave brings (A) away from its original Hugoniot state, to a
higher pressure state. This phenomenon allows the exploration of the phase
diagram by the use of multi-layer targets.
ZA > ZB
A releasing wave is reflected in material (A). In the (P, U ) plane (see Fig.
1.11), the pressure in the releasing medium (A) is given by the intersection of
the isentrope, calculated from the point A on the Hugoniot HA , with the HB
polar. The shock release decreases progressively the pressure in medium (A).
Figure 1.11: Pressure profile produced after the reflection of a shock wave at the
surface between a high impedance medium (A) and a low impedance medium
(B) and schematic representation in the plane (P,U)
CHAPTER 1. PHYSICAL CONTEXT
1.4
22
Laser-Matter Interactions
The interaction between laser and matter, for ”long” pulses (>100 ps), can be
described according to three regimes, depending on the intensity on the target:
low flux : laser radiation can only melt the specimen;
medium flux : target can be heated up to the vapor phase;
high flux : for Ilaser > 1012 W/cm2 the radiation can create a completely
ionized plasma.
The generation of shock waves is achieved in the last case, when the hot plasma,
created at the surface of the target, expands backwards into vacuum. For reaction to this expansion, the medium is compressed.
In this section, the mechanisms responsible for the absorption of laser energy
will be analyzed, as well as the transport of energy inside matter and the expansion of the plasma, in order to estimate the ablation pressure that produces
the shock wave.
Absorption
The dispersion relation of an electromagnetic wave with frequency ω and wave
vector k, is the following:
k 2 c2 = ω 2 − ωp2
(1.25)
where ωp is the plasma frequency, related to the electronic density ne , electronic
charge e, electronic mass me and dielectric constant in vacuum ε0 through
s
ne e2
(1.25)
ωp =
ε0 me
As the surface of the target is ablated, the laser pulse propagates into a plasma
with an increasing electronic density profile. Thus, a critical density is defined
equating the laser frequency and plasma frequency:
nc [cm−3 ] =
ε0 me 2
1.1 × 1021
ω
=
e2
λ[µm])2
(1.25)
At this density, the wave vector vanishes and there is no propagation. For
example, if λ = 1.054 µm, nc = 2 × 1021 cm−3 , and for λ = 0.527 µm,
nc = 4 × 1021 cm−3 . The region in which the electronic density is below
critical density is called interaction zone, or corona. This region where the laser
propagates is generally isotherm (with T ∼ keV, for a nanosecond duration and
1013 W/cm2 intensity pulse). Here, many nonlinear phenomena (parametric
instabilities) account for the absorption of the laser in the plasma. The two
main mechanisms are the collisional and resonant absorption.
Resonant absorption
This mechanism arises when the the laser beam impinges at oblique angle and
the polarization is in the plane of incidence. In this case a component of the
electric field interacts with the electrons resonantly and generates a longitudinal
CHAPTER 1. PHYSICAL CONTEXT
23
plasma wave of high amplitude, close to the critical density.
The electric field associated to this wave can be much bigger than the field of
the laser wave and so it can accelerate hot-electrons that preheat the target
in front of the shock wave. Generally, this effect is unwelcome, as alters the
thermodynamical state of the target in an uncontrollable way. The theoretical
and experimental studies shows that the resonant absorption is negligible with
respect to the collisional absorption[Vinci, 2006] if
Ilaser [W/cm2 ] ≤
1014
= 3.5 × 1014 [W/cm2 ]
(λ[µm])2
(1.25)
a condition satisfied in the shots analyzed in the present work.
Collisional absorption
The collisional absorption, also called inverse Bremsstrahlung, is a three-body
process: a photon is absorbed after the collision of an electron and a ion. The
absorption of the laser intensity is given by
dIlaser
= −αIlaser
dx
(1.25)
where α is the absorption coefficient, in m−1 . Referring to the classical DrudeLorentz model, the effect of the laser pulse on the plasma is described by
α=
νei ωp2
2
ω c<(n̂)[1 + (ν
ei /ω)
2]
(1.25)
p
where <(n̂) = 1 − ne /nc is the real part of the optical index and νei is the
electrons-ions collision frequency. This is obtained from the Fokker-Planck
equation[Shkarovsky et al., 1966]:
νei = 3.6Z 2 ni
log Λ
3/2
(1.25)
Te
where log Λ is the Coulomb logarithm given by
log Λ = log λD b0
(1.25)
where λD is the Debye length and b0 is the impact parameter corresponding
to a 90o deflection. The Coulomb logarithm ranges from 5 to 10 in the case of
laser produced plasmas. In the typical solid target compressions, νei ω and
the absorption coefficient can be written as follows:
α=
3.6Z 2 ni ωp2 log Λ
p
3/2
Te ω 2 c 1 − ne /nc
(1.25)
The absorption coefficient is a decreasing function of T but it increases as
ne increases. α(ne ) shows a sharp peak for ne = nc , thus the absorption is
concentrated at the critical density.
CHAPTER 1. PHYSICAL CONTEXT
24
Figure 1.12: Energy absorption and transport in a low atomic number target
(Z < 10)
Energy transport
Figure 1.12 shows mass density ρ and temperature T profiles in the different zones that characterize the interaction of laser with a low atomic number
medium (Z < 10). It is possible to identify three regions: the corona, the
conduction zone and the shock zone. In the shock zone, the density is higher
than the solid density. In the conduction zone, the energy transport is essentially due to thermal electronic conduction. The temperature and the velocity
decrease from the critical surface to the ablation front.
According to Spitzer and Härm [1953], the heat flux is given by the formula
5/2 dTe
QSH = −k0 T0
dx
(1.25)
where k0 is the thermal electronic conductivity. This formula is valid as long as
the characteristic size of the temperature gradient is bigger than the electron
mean free path. In the conduction zone the gradients can be very steep and
the eq. 1.4 is no longer applicable. Experiments realized by Gray and Kilkenny
[1980] shows that the heat flux is reduced by a factor between 10 and 100 with
respect to that predicted by the classical formula. Thus, the following heuristic
relationships are used:
dTe
Q = sgn
min(|QSH |, f Ne kTe )
(1.26)
dx
dTe
Q−1 = (QSH )−1 + (sgn
f Ne kTe )−1
(1.27)
dx
CHAPTER 1. PHYSICAL CONTEXT
25
where f ∼ 0.06 is called flux limiting factor. The comparison between the
previous equations and more elaborated theoretical models[Vinci, 2006] shows
that eq. 1.27 describes heat transport in presence of steep thermal gradients
with a high degree of accuracy.
Ablation
The ablation process, consequent to the plasma expansion into vacuum, is
responsible for the shock wave generation. Studying the plasma expansion it is
possible to obtain some scale laws in order to estimate the ablation pressure,
equal to the shock pressure, and the ablation rate.
For a spatial homogeneous focal spot, it is possible to restrict the description
to a one-dimensional geometry. Thus, in the plasma, the conservation laws
take the form
∂ρ ∂(ρv)
+
=0
(1.28)
∂t
∂x
∂(ρv)
∂
= − P + ρv 2
(1.29)
∂t ∂x ∂ 3P
ρv 2
∂
P
v2
+
=−
ρv + +
+ q + Ilaser δ(x − xc )(1.30)
∂t
2
2
∂x
ρ
2
The pressure is given by
P = ne kB Te + ni kB Ti = ρ
Z +1
kB T
Am0
given by h ≡ + P/ρ =
where cs =
Z +1
kB T = ρc2s
Am0
(1.30)
1/2
is the isothermal sound velocity. The enthalpy is
γ P
γ+1 ρ
=
γ
2
γ+1 cs .
The heat flux in a plasma is
q = −K0 T 5/2
∂T
∂x
(1.30)
where K0 ' 10−11 [S.I.]
Mora [1982] showed that if the laser intensity is above the threshold
Ic [W/cm2 ] ' 2 × 1013 λ−5 [µm]
Z∗
3.5
3/2 A
2Z
5/4
τ 3/2 [ns]
(1.30)
where Z ∗ is the mean ionization degree, then the ablation pressure Pablation
and the ablation rate ṁa are given by the following expressions
14
2
2
3
− 23
Pablation [Mbar] = 12.3(I[10 W/cm ]) (λ[µm])
1
4
ṁa [kg/s · cm2 ] = 150(I[1014 W/cm2 ]) 3 (λ[µm])− 3
1
A 3
2Z
23
A
2Z
(1.31)
(1.32)
If, on the converse, the intensity is below the critical intensity stated in eq. 1.4,
as showed by Fabbro et al. [1985], then the pressure and the ablation rate can
CHAPTER 1. PHYSICAL CONTEXT
26
be obtained through the relations:
14
2
3
4
− 14
Pablation [Mbar] = 11.6(I[10 W/cm ]) (λ[µm])
3
4
ṁa [kg/s · cm2 ] = 143(I[1014 W/cm2 ]) 4 (λ[µm])− 3
7 − 81
A 16 Z ∗ τ [ns]
(1.33)
2Z
3.5
28 ∗
− 41
A
Z τ [ns]
(1.34)
2Z
3.5
In the present work, laser light irradiated carbon targets (A = 12.01 and
Z ∗ ∼ Z = 6) with a pulse duration of about 2.4 ns and wavelength of 1.054
µm and 0.527 µm in 2008 and 2009 respectively. Thus, the critical intensity is
Ic ' 1.3 × 1014 W/cm2
Ic ' 4.1 × 1015 W/cm2
year
year
2008
2009
(1.35)
(1.36)
above Imax = 2.3 × 1013 W/cm2 , the maximum intensity reached during the
two campaigns. The ablation pressure can be estimated according to 1.33:
Pablation [Mbar] = 0.30(I[1012 W/cm2 ])3/4
Pablation [Mbar] = 0.35(I[1012 W/cm2 ])3/4
1.5
year
year
2008
2009
(1.37)
(1.38)
Numerical Simulations
Numerical simulations are a very powerful tool throughout all the steps of
an experimental campaign. Firstly, simulations can be used to predict the
experimental conditions, in order to optimize the target design. Besides, during
the experiment, they are important, for example, to tailor the laser energy
shot by shot, accounting for the actual target characteristics. In the end, the
most important contribution is that to the shock parameters estimation and
comparison with experimental findings, as clarified in chapter 4.2.
Multi
Multi stands for MULTIgroup readiation transport in MULTIlayer foil and
it is a lagrangian, hydrodynamic, implicit, 1-dimensional code accounting for
radiation, developed at Max Planck Institut für Quantenoptik by Ramis et al.
[1988]. The code has been further developed at LULI.
Multi is an implicit numerical code, in the sense that the physical quantities
at time n are expressed as functions of the values assumed at time n − 1 and
n+1, while in an explicit code only time n−1 affects the values of quantities at
time n. The implicit numerical codes are numerically more stable than explicit
ones, at the expense of higher complexity.
Multi is written in Lagrangian coordinates, thus the cell motion follows the
fluid. In this coordinate system, there is no flux of matter between adiacent
cells and the hydrodynamic equations assume a simpler form.
The properties of the matter are described through external thermodynamic
tables, such as Sesame or MPQeos (see section 1.5). Moreover, the equation
of states for electronic and ionic population can be treated separately, but this
is not necessary in the range of time and energies employed in this work. In
fact, ionic and electronic temperature are the same almost everywhere, apart
from two regions:
CHAPTER 1. PHYSICAL CONTEXT
27
in the plasma corona the laser releases its energy to the electrons. The electrons themselves heat the ions by collisions until equilibrium is reached
and then matter is set into motion, as almost only the ions contribute to
the mass.
in the shock front , on the converse, matter is set into motion by the ion
wave and electrons are heated by collisions.
However, due to the high collision frequency, relaxation time is small (∼ ps)
compared to the hydrodynamic time scale (∼ ns) and the two temperatures
can be regarded as equal.
Heat flux and laser-matter interaction
Multi neglects the ion ionic thermal flux. The electronic heat transfert is
calculated from eq. 1.27 as the harmonic mean between the classic heat flux
following Spitzer and the limited flux. Inside the corona, the energy deposition
is defined as the difference between the energy of the incident laser beam and
the that of the reflected beam, at each cell.
Radiation transport
Multi divides the radiation space of frequencies and direction in N different
groups. Each group is defined by a frequency domain and a direction cosine.
The local balance of absorption and emission in the plasma is described by the
radiation transfert equation, just stated as
1 ∂Iν
+ Ω · ∇ν = kν (1 − e −hν/kB T )(Iνp − Iν )
c ∂t
(1.38)
where k is a convently defined opacity. The integration of eq. 1.5 reduces
the variables from four (the position x, the direction Ω, the frequency ν and
the time t), to two (position and time only) yielding N differential equations
that are resolved with finite differences method. The opacity k is contained
in tables, as well as the equation of state. It is also possible to exclude the
radiation transport in the simulation in order to study the hydrodynamics
alone.
Mesh
Multi is designed do handle multilayer targets, made of different materials.
Each layer is described by the thickness, the atomic and mass number of the
medium and the mesh geometry. To ensure numerical stability, it is necessary
that the layers merge smoothly, limiting the mass difference between adjacent
cells below 10%. Besides, In order to accurately describe laser energy deposition, the first cells should be ten times smaller than laser wavelenght (. 50
nm).
Incident laser
The incident laser beam is parametrized by the wavelength and the time profile,
given either as a combination of up to four simple forms (gaussian, linear, flat
top, . . . ) or as an arbitrary profile.
CHAPTER 1. PHYSICAL CONTEXT
28
Variables
The code calculates, at each time step set by the user and in each cell, many
quantities of interest. Among them, the more useful were
1. position in the lab frame
2. density
3. temperature
4. pressure
5. velocity
Numerical tables
As the hydrodynamic equations are not a closed system, it is necessary to find
an independent relation between the variables like the equation of state (see
section 1.3. The EoS describes the thermodynamical properties of the fluid
through a functional relation between density, pressure, energy, temperature
and entropy. If the equation of state is given by an analytical formula, often
some hypothesis of the fluid model have been done and so the EoS domain
will be limited. In particular, the simple analytical formulas do not describe
regions where phase transitions occur.
An equation of state can be given in the form of tables, determined from the
best physical models avaliable and, in principle, they should be consistent with
the experimental findings. Tables offer the possibility to be more precise than
a purely theoretical model.
Sesame tables
The Sesame equation of state Library is a standardized, computer-based library of tables for the thermodynamic properties of materials. The thermodynamic data stored in the Library include tables of pressure P and energy E,
each as a function of density ρ and temperature T . The typical density and
temperature ranges are from 10−6 to 104 g/cm3 and 0 to 105 eV respectively.
The complete description of a medium is split into different tables, coded by
three figure numbers. The standard thinking for Sesame is that an EoS is composed of three parts; the zero temperature isotherm (cold curve), the thermal
ionic, and thermal electronic. Each is modeled separately and the total EOS,
the 301 table, is constructed by summing the three. Other tables, 304, 305,
and 306, come from the individual parts. By convention the zero point motion
of the ions is contained in the thermal ionic contribution. The thermal electronic contribution is calculated using finite temperature Thomas-Fermi-Dirac
(TFD) theory. For the cold curve we look at three different density regions.
We usually have shock wave data from the ambient density of the solid to some
high density. Over this range we use the shock data and the above fit for the
Debye temperature to calculate the cold curve from Mie-Grüneisen theory. For
higher densities an interpolation is made to TFD theory with a smooth connection to the Mie-Grüneisen region. For low densities a simple analytic form is
again smoothly merged into the Mie-Grüneisen region. The other parameters
CHAPTER 1. PHYSICAL CONTEXT
29
for expansion are selected to obtain the correct cohesive energy and gas-liquid
critical point. In a few special cases the Mie-Grüneisen procedure has been
replaced by ab initio band-structure calculations.
There also exist Sesame libraries for opacity and conductivity data. The lower
limit on the opacity data is about 1 eV. The opacity library includes Rosseland and Plank mean opacities (502 and 505 respectively), electron conductive
opacity (503) and mean ion charge (504). The accuracy of the table is limeted
by the interpolation mesh in the thermodynamical plane, as well as by the
precision of the parameters employed in the models.
Qeos model
The quotidian equation of state (Qeos)[More et al., 1988] is a general-purpose
equation of state model for use in hydrodynamic simulation of high-pressure
phenomena. Electronic properties are obtained from a modified Thomas-Fermi
statistical model, while ion thermal motion is described by a multiphase equation of state combining Debye, Grüneisen, Lindemann and fluid-scaling laws.
The theory gives smooth and usable predictions for ionization state, pressure,
energy, entropy and Helmholtz free energy. Qeos model is particularly well
suited for the use in hydrodynamical simulation codes, as the calculated pressure and energy exactly satisfy the condition of thermodynamic consistency,
results are smooth functions of ρ and T and are remarkably free of numerical
noise. Additionally Qeos is a self-contained theoretical model that requires
no external database. The inputs are simply the material composition and
properties of the cold solid (density and bulk modulus). In particular, bulk
modulus is responsible for the correction for chemical bonding. Actually the
TF-theory in its basic form gives large positive pressure values for densities
near solid density. In real matter though, the repulsive Coulomb forces from
the electrons are compensated by attractive forces due to chemical bonding, so
that the total pressure at standard conditions vanishes. In order to account
for the attractive exchange forces, a semi-empirical bonding correction term,
inspired by the well-known Morse potential for a diatomic molecule, is added
to the energy and a corresponding term is added to pressure. There is no
bonding contribution to the entropy. The Qeos model uses two parameters
in the bonding correction. They are determined by the requirement that the
total pressure at solid density vanishes and the solid bulk modulus be correct.
1.6
Resume
Carbon has been studied extensively due to its uniqueness in the diversity of its
different phases. The phase diagram in the low pressure region (under 50 GPa
and for temperature below 6,000 K) is well established and the solid graphite
and diamond phases characteristics are acknowledged. On the contrary, the
liquid high-pressure phase is almost unknown due to the extreme conditions
required to study this state. The theoretical calculations, also, are complicated
by the sophisticated models needed to give accurate predictions and may be
very demanding from a computational standpoint.
Recently there has been speculation by van Thiel and Ree [1993] that there
is a phase transition between two liquid phases characterized by different conductivity. A third liquid phase was predicted by Grumbach and Martin [1996]
CHAPTER 1. PHYSICAL CONTEXT
30
for pressures in the range 4-10 Mbar with metallic properties.
The phase diagram of carbon at high pressures and temperatures is of ongoing
scientific interest in many different fields, such as material science, planetary
science and astrophysics.
Experimentally, the phase diagram can be explored by static or dynamic methods. With the first approach pressure and temperature can by varied independently in the range 0 - 1 Mbar, 0 - 103 K, while the second allows much more
higher pressures and temperatures. In this last case, unfortunately, the two
quantities are correlated through Hugoniot relations and the compressed state
is accessible to measurements only for very short time. Different strategies
have been developed to overcome these limitations. In particular, the target
design can be tailored to explore a definite area of the phase diagram by means
of special density material arranged in multi-layer structures.
In particular, the method employed in the present work to achieve high energy densities is the shock compression produced by intense laser irradiation.
A deeper insight into this method is given in sections 1.3 and 1.4, where the
physics of shock waves and the interaction between laser and matter are presented. The relation governing the dynamics of shocks are presented, in the
form of the Hugoniot relationships, as well as the methods to exploit shock
wave reflection at interfaces to explore the phase diagram. Besides, the mechanisms responsible for the generation of multi-megabar pressure by an intense
laser pulse are examined, with focus on the absorption and transport of energy
in the three characteristic regions of the corona, the conduction zone and the
shock zone.
At chapter close, numerical simulation methods have been presented. A brief
review of Multi code is given, with stress on the most important features such
as the capability of accurately simulate the hydrodynamics and the radiation
transport in laser-shocked matter. However, in addition to the sophisticated
calculation techniques, high precision results can be achieved only with accurate modelling of the experimental conditions. For this purpose, great attention
has been paid to characterize the actual material employed in the experiments.
1.7
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la-ur-92-3407, 1992.
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F. P. Bundy, W. A. Bassett, M. S. Weathers, R.J Hemley, H. K. Mao, and A. F.
Goncharov. The pressure-temperature phase and transformation diagram for
carbon. Carbon, 34:141–153, 1996.
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A. A. Correa, S. A. Bonev, and G. Galli. Carbon under extreme conditions: Phase boundaries and electronic properties from first-principles theory.
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D. R. Gray and J. D. Kilkenny. The measurement of ion acoustic turbolence
and reduced thermal conductivity caused by a large temperature gradient in
a laser heated plasma. Plasma Phys., 22:81–111, 1980.
M. P. Grumbach and R. M. Martin. Phase diagram of carbon at high pressures
and temperatures. Phys. Rev. B, 54(22), December 1996.
T. Guillot. Science, 286:72, 1999.
P. Gustafson. An evaluation of the thermodynamical properties and the p, t
phase diagram of carbon. Carbon, 24(2):169–176, 1986.
P. Mora. Theroetical model of absorption of laser light by a plasma. Phys.
Fluids, 25(6):1051, 1982.
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quotidian equation of state (qeos) for hot dense matter. Phys. Fluids, 31
(10):3059, 1988.
W. J. Nellis et al. J. Chem. Phys., 107:9096, 1997.
N. F. Ness et al. Science, 246:1473, 1989.
S. Paleari. Alcuni aspetti matematici della teoria dei fluidi debolmente viscosi.
Bachelor’s thesis, Università degli Studi di Milano - Bicocca, 2008.
P. H. Poole et al. Phys. Rev. Lett., 73:1632, 1994.
R. Ramis, R. Schmalz, and J. Meyer-Ter-Vehn. Mulit - a computer code for
one-dimensional multigroup radiation hydrodynamics. Comp. phys. comm,
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T. Vinci. Les chocs raditifs générés par les lasers à haute énergie: une opportunitè pour l’astrophysique de laboratoire. PhD thesis, Ecole Polytechnique,
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Ya. B. Zel’dovich and Yu. P. Raizer. Physics of shock waves and hightemperature hydrodynamic phenomena. Dover, Mineola, New York, 2002.
Experimental setup
2
The experimental results discussed in the present thesis were obtained in two
series of experiments conducted in October, 2008 and in November, 2009 at
the GEKKO/HIPER (High Intensity Plasma Experimental Research) system
at the Institute of Laser Engineering (ILE), Osaka University.
In this chapter the laser system will be presented briefly, as well as the different
diagnostic and the target design. The main aim of the experiments is to study
the reflectivity of carbon varying pressure and temperature in the Mbar - few eV
regime. For this purpose several diagnostics were used to probe temperature,
shock velocity, shock uniformity and reflectivity.
2.1
Laser System
The HIPER laser facility [Myanaga et al., 2001] is an irradiation system on
the GEKKO XII (GXII): Nd glass laser system at ILE [Yamanaka, 1999]. The
facility provides one-dimensional compression by smoothed laser beams with
short wavelength and high intensity. In the system, twelve beams of the GXII
are bundled in an F/3 cone angle. The focusing system is represented in Fig.
2.1. Only three beams were used in the experiments with a wavelength of
1053 nm (fundamental, ω) in 2008 and with a wavelength of 527 nm (second
harmonic, 2ω) in 2009. The conversion to second harmonic is operated when
the coherent electromagnetic wave propagates in a non-linear dielectric medium
(in this case a KDP crystal). The utilization of shorter wavelength is justified
by a number of advantages: the absorption of the laser light is more efficient;
the reduction of hot electrons suppresses the preheating; the hydrodynamic is
more efficient. Kinoform wave plates (KPPs) [Dixit et al., 1996] were installed
for all the beams to obtain a uniform irradiation pattern. The temporal shape
of the laser pulse was approximately square with a full width half maximum
(FWHM) of 2.4 ns and a rise and fall time of 100 ps, as shown in Fig. 2.2. The
focal-spot diameter was typically 1 mm. In 2008 experiments the target was
usually kept slightly defocused, yielding a spot diameter on target of about
1.5 mm. The total energy of the laser pulse is estimated with a calibrated
pyrometer. The light intensity on the target can be calculated as follows: the
energy of a pulse is given by the integral of the intensity over the target face
33
CHAPTER 2. EXPERIMENTAL SETUP
(a) Schematic optical arrangement
(b) Arrayed KDP cells
(c) Final mirror bundle
Figure 2.1: HIPER focusing system
Figure 2.2: GXII laser pulse – time profile
34
CHAPTER 2. EXPERIMENTAL SETUP
and time
Z
∞
E=
Z
dt
0
35
dxdyI(x, y; t)
R2
Approximating the laser spot as circular with flat top and abrupt edge
I(x, y; t) = I0 ξ(t)I(x2 + y 2 < r)
the maximum intensity in the spot is given by
I0 =
πr2
E
R∞
0
dtξ(t)
(2.0)
Inserting in the expression above the actual values one obtains
• I0 [Wcm−2 ] = 2.77 × 1010 [cm−2 s−1 ] · E[J] for 2008 experiments.
• I0 [W cm−2 ] = 4.36×1010 [cm−2 s−1 ]·E[J] for 2009 experiments, accounting
for the transmittance of the KPP.
A schematic view of the experimental configuration is shown in Fig. 2.3.
Four diagnostics systems were used to measure a target rear side event at
Figure 2.3: Laser beams configuration
the same time. Two of them collected the self-emission giving space-resolved
and frequency-resolved emission intensity (SOP and SSOP respectively) and
the other two recorded the reflection of a probe laser from the target rear side.
An injection-seeded Q-switched Nd:YAG (Yttrium Aluminum Garnet) laser
was used as the probe light. A notch filter for the 532 nm YAG wavelength
was installed to prevent the self-emission measurement from the probe light
signal.
2.2
Targets
The design of the targets is quite simple: a layer of carbon is deposited on
a transparent substrate, the main laser beam hits the carbon and the rear
CHAPTER 2. EXPERIMENTAL SETUP
36
surface is imaged by the self-emission (passive) diagnostics and the active diagnostics, as illustrated in Fig. 2.4. This design has two strong points: firstly,
Figure 2.4: Target design.
the substrate prevent the carbon to release into vacuum after shock breakout
and reflects the shock wave back due to its higher density, raising the pressure;
secondly the interface between carbon and substrate is accessible to the diagnostics.
Two distinct series of targets have been produced at the Molecular Beam Laboratory (Università degli Studi di Milano, Italy) and at Enea Laboratory, Frascati, Italy. In the first series, the substrate was fused silica (SiO2 ) of circular
shape (4 mm of diameter and 100 µm thickness) with anti-reflective coatings for
the wavelength of the probe laser on the bottom side and the carbon layer was
deposited with supersonic cluster beam deposition (SCBD) technique [Milani
et al., 2001]. The apparatus for the deposition of cluster beams is represented
in Fig. 2.5. A supersonic beam is schematically described as a gas stream
expanding very rapidly from a high pressure region through a nozzle to a low
pressure region. The characteristics of the beam are mainly determined by
the size and shape of the nozzle and by the pressure difference between the
two regions. Supersonic beams provide high intensity and directionality allowing deposition with high growth rates. This is essential to produce a layer of
several (from 10 to 50) microns with steady deposition conditions, as the process can be considered stationary only if the deposition time is less than two
hours. Carbon clusters are produced by a pulsed micro plasma cluster source
(PMCS) [Barborini et al., 1999] (see Fig. 2.6). A confined plasma discharge
ablates a graphite rod in the cluster source. The vaporized carbon atoms are
quenched by a pulse of helium and condense in clusters. The source consist
in a ceramic body with a channel drilled to intersect perpendicularly a larger
cylindrical cavity. The other side of the cavity is closed by a removable nozzle.
The solenoid valve delivers He pulses with an opening time of about 220 µs.
An intense helium jet is formed toward the cathode surface facing the valve. In
this region the gas density is substantially higher than in the rest of the cavity.
During standard operation the mean He pressure in the source cavity is roughly
30 Torr. Just before the valve closing, a voltage of about 7 kV is applied to the
CHAPTER 2. EXPERIMENTAL SETUP
37
Figure 2.5: Schematic representation of the supersonic beam apparatus for the
deposition of cluster beams
Figure 2.6: Expanded view of the pulsed cluster source and of the region near
the skimmer where a shock wave is formed. The trajectories of the heavy and
light particles are schematically shown. Due to separation effects, films with
different nanostructures can be deposited by placing a substrate to intercept
different regions of the beam
CHAPTER 2. EXPERIMENTAL SETUP
38
electrodes, causing the firing of the discharge and the production of the plasma
localized in the region of the higher gas density. The ablation occurs when
helium plasma strikes the cathode surface removing atoms via sputtering. The
electrode surface where ablation occurs is very small and the cathode is constantly rotated along its axis by a stepping motor external to vacuum to allow
constant ablation conditions. The kinetic energy of the clusters is lower than
0.2 eV/atom, well below the binding energy of carbon atoms in the clusters.
At cluster impact there is thus no substantial fragmentation of the aggregates.
In order to control the cluster mass distribution it is possible to modify the
PMCS working condition, such as the time delay between the introduction of
the gas pulse and the firing of the discharge, changing the cluster aggregation
conditions preventing or favoring the growth of large aggregates; the design of
the nozzle, exploiting aerodynamic gas effects to focus light clusters; the background pressure in the source chamber, producing a shock wave in front of the
skimmer, causing mass separation effects. By placing substrates in different
portions of the beam, films from different precursor clusters can be grown.
The second series of targets was made with spray coating technique. This is a
liquid phase chemical synthesis involving the natural assembling of a precursor (graphite dissolved in isopropyl alcohol) to from a solution supersaturated
with the product (carbon cluster) [Kelsall et al., 2005]. Thermodynamically
this is an unstable situation and ultimately results in nucleation of the product.
The nature, size and morphological shape of the precipitated structures can be
controlled by parameters such as temperature, pH, reactant concentration and
time. The deposition was followed by an annealing at 350 ◦ C for 30 minutes to
increase the density of the carbon layer. Lithium fluoride substrates of square
shape (3x3 mm) and 300 µm thickness were used.
Target characteristics are resumed in table 2.1.
#
31709
31718
31725
33062
33063
33065
33073
33085
carbon
thickness
32.3 µm
37.5 µm
48.1 µm
15 µm
12 µm
5 µm
5 µm
17 µm
carbon
density
0.55 g/cm3
0.55 g/cm3
0.55 g/cm3
0.55 g/cm3
0.55 g/cm3
0.8 g/cm3
1 g/cm3
0.55 g/cm3
substrate
type
SiO2
SiO2
SiO2
LiF
LiF
SiO2
SiO2
LiF
Table 2.1: Target characteristics
2.3
Self-emission Diagnostics
This kind of diagnostic allow to record the intensity of the self-emission of the
rear face of the target, with a time resolution better than 500 ps. The sensitivity is in the visible wavelength domain and ranges from 380 to 700 nm.
At shock wave breakout matter is heated to temperatures of several thousands
degrees and accelerated to speeds in the order of ten Km/h. To measure all
CHAPTER 2. EXPERIMENTAL SETUP
39
the parameters of interest, with sufficient space and time resolution, the most
useful instrument is the streak camera (SOP). The streak camera is a device
to measure ultra-fast light phenomena and delivers intensity vs. time vs. position (or wavelength) information. No other instruments which directly detect
ultra-fast light phenomena have better temporal resolution than the streak
camera. Since the SOP is a two dimensional device, it can be used to detect
several tens of different light channels simultaneously. Used in combination
with proper optics, allows to measure time variation of the incident light with
respect to position and, used in combination with a spectroscope, time variation of the incident light intensity with respect to wavelength is measured
(SSOP). Fig. 2.7 shows the operating principle of the streak camera. The light
Figure 2.7: Operating principle of the streak tube
being measured passes through a slit and is formed by the optics into a slit
image on the photo-cathode of the streak tube. For illustration purpose, in the
figure are represented four optical pulses which vary slightly in terms of both
time and space and which have different optical intensities, passing through
the slit and arriving at the photo-cathode. The incident light on the photocathode is converted into a number of electrons proportional to the intensity of
the light, so that these four optical pulses are converted sequentially into electrons. They then pass through a pair of electrodes, where they are accelerated
against a phosphor screen. As the electrons produced from the four optical
pulses pass between a pair of sweep electrodes, high voltage is applied at a
timing synchronized to the incident light (see Fig. 2.8). This initiates a high-
Figure 2.8: Operation timing
speed sweep, the electrons are swept from top to bottom and those arriving at
slightly different times are deflected in slightly different angles in the vertical
direction. After the deflection, the electrons enter the MCP (micro-channel
CHAPTER 2. EXPERIMENTAL SETUP
40
plate), an electron multiplier consisting of many thin glass capillaries with internal diameters ranging from 10 µm to 20 µm, bundled together to from a
disk-shaped plate with a thickness of 0.5 mm to 1 mm. The internal walls of
each individual channel are coated with a secondary electron emitting material,
so that as the electrons come flying through the channels, they bump against
the walls and repeated impact causes them to multiply in number. In this way
the electrons are multiplied several thousands of times and impact against the
phosphor screen, converting the information back into light. The phosphor image corresponding to the optical pulse which was the earliest to arrive is placed
in the uppermost position, with the other images being arranged in sequential
order from top to bottom. Also, the brightness of the various phosphor images
is proportional to the intensity of the respective incident optical pulses. The
image is recorded by a CCD camera mounted in contact with the screen. The
cameras have 12 bits of bit depth and resolutions of 640 × 512, 672 × 512,
1242 × 1152 and 1344 × 1024 pixels.
SOP
Each shot in the two series of experiments discussed in the present work, was
imaged by two self-emission diagnostics. A SOP measured space and timeresolved emissivity, giving information on shock planarity, time of shock breakout, preheating and time profile of emission. The main characteristics of this
device are summarized in table 2.2. The field of view (portion of the target
year
2008
2009
resolution
672 × 512 px
1344 × 1024 px
field of view (on target)
1.5 mm
0.55 mm
time window
6.0 ns
6.9 ns
Table 2.2: SOP main features
imaged by the SOP) has been calculated from the image of a test pattern (see
Fig. 2.9a). Also, it was found that the vertical position on the screen vs. time
follows approximately a linear relation (see Fig. 2.9b). The deviation from linearity is small (hundreds of picoseconds) and when higher accuracy is needed,
time is calculated with a quadratic relation.
Spectrometer SOP
A spectrometer coupled with the streak camera resolves the spectrum of the
emission from the target. In this case the calibration is critical to obtain reliable
measures. For this reason the light of a Hg low pressure lamp was employed as
a reference for the wavelength axis. The Hg lamp was chosen for the clearness
and broad distribution of its spectral lines, as can be seen in Fig. 2.10a. To
correct the distortions of the SSOP images due to small flaws in the imaging
system, a spatial transformation was applied to every image. A fourth order
polynomial mapping was employed to compensate for the distortions. Fig.
2.11 shows the 2-dimensional mapping applied to a test pattern. To obtain
accurate spectral information, it is crucial to determine carefully the detection
sensitivity of the imaging system. The light emitted from the target before it is
CHAPTER 2. EXPERIMENTAL SETUP
(a) Field of view estimation.
41
(b) Pixels vs. time relation.
Figure 2.9: Calibrations of SOP diagnostic in 2009 experiment
(a) Spectrum of a Hg lamp, used for calibrat-(b) Peak wavelength vs. image coordinate, in
ing the wavelength scale. The x axis is thepixels. The relation, to a good approximation,
position in the image, in pixels. Correspond-is linear.
ing wavelength of the peaks is also indicated.
Figure 2.10: 2009 SSOP calibration
converted to a digital signal by the CCD control unit, has to go through many
passages. At first it travels through different optical elements, characterized by
an overall transmittance Θ(λ). When the light reaches the photo-cathode, it
is converted to electrons inside the streak tube and converted again into light
at the phosphor screen, with an efficiency Ξ(λ). At the end, the screen light is
detected by the CCD in proportion C[counts/intensity] to its intensity. Thus,
it is possible to retrieve the spectrum emitted by the target dividing the SSOP
spectrum by η(λ) ≡ Θ · Ξ · C.
The function η(λ) was estimated using two reference lamps, with known spectrum, in order to cover both the short and long wavelength region. Fig. 2.12
refers to 2009 SSOP. The main characteristics of this device are summarized
in table 2.3.
CHAPTER 2. EXPERIMENTAL SETUP
42
Figure 2.11: Test pattern before (on the right) and after (on the left) the
fourth order polynomial mapping applied to SSOP images. The most evident
correction is a shearing of the horizontal lines, while another minor rescaling
involve the spacings between vertical lines
(a) Xe lamp.
(b) Tungsten-halogen lamp
Figure 2.12: Effective spectral response of the 2009 SSOP imaging system
year
2008
2009
resolution
1024 × 1024 px
1344 × 1024 px
spectral range
740 – 356 nm
726 – 357 nm
time window
6.0 ns
9.8 ns
Table 2.3: SSOP main features
CHAPTER 2. EXPERIMENTAL SETUP
2.4
43
Interferometric Diagnostic Visars
The velocity of short duration high-amplitude shock waves and high-speed
motion are difficult to measure due to their fast reaction times. One measurement tool frequently used is Visar (Velocity Interferometer System for
Any Reflector). Visar is an optical-based system that utilizes Doppler interferometry techniques to measure the complete time-history of the motion of
a surface. Laser light is focused to a point onto the target and the reflected
light is collected, routed through an unequal leg interferometer, and the resulting interference pattern is recorded with a streak camera. The image is
then analyzed for the amount of Doppler shift during a given time. The predecessors of the modern Visar are the Wide Angle Michelson Interferometer
(WAMI) and the Lockheed Laser Velocimeter [Barker and Hollenbach, 1972].
The WAMI, first described in 1941, was a modified Michelson interferometer
that allowed a diffuse target to be used as a reflecting source, instead of the
traditional mirrored surface required with displacement interferometers. This
new interferometer also used an etalon to temporally lengthen the path of one
leg of the interferometer, while keeping the apparent image paths equal. This
path difference makes Doppler shifted light interference possible. The longer
the path length, the larger the Doppler shift for a given velocity, which provides greater sensitivity when measuring slower events. The Lockheed Laser
Velocimeter used the same techniques as the WAMI and is acknowledged as
the first linear velocity interferometer to be used for shock physics. Barker and
Hollenbach used the previous developments of the Lockheed Laser Velocimeter
to build a laser interferometer for shock physics at Sandia National Laboratories. That system was improved upon, and uses the acronym ”Visar” (Velocity
Interferometer System for Any Reflector). Subsequent developments and improvements to Visar were the advent of the Push/Pull Visar, developed by
Hemsing, where the previously unused optical information of the interferometer was added to the primary signal, causing a twofold increase in signal and
a reduction in optical noise. A later development that had significant impacts
on the ease and portability of Visar was the Fixed Cavity Visar. This innovation greatly simplified the use of the system while providing an extremely
stable interferometer.
Description of the interferometry system
The Visar system used in 2009 is represented in Fig. 2.13. The main characteristics of these devices are summarized in table 2.4. The light reflected from
the target rear face follows the green path on the figure: at first it is distributed
to the two Visars by a beam splitter and then it enters the two double-etalons
where, as will be described immediately after, the interference take place. The
resulting signal is recorded by the two streak cameras. A double-etalon is an
optical component made of two different transparent materials with different
optical index n, joined at one side. The main task of the double-etalon is to
split the light beam into two paths, introducing a delay τ in one path with
respect to the other preserving the spatial coherence of the beam and finally
recombine the two beams. In Fig. 2.14 are shown, in a simplified picture, the
principles of a double-etalon system. Light enters from the left into the etalon
and it is partially reflected at the interface. The light-green path (in the na
CHAPTER 2. EXPERIMENTAL SETUP
44
Figure 2.13: Diagram of 2009 Visar system. Visar Left (or taka) is colored
in blue, while Visar Right (or waka) is colored in orange
year and name
2008, VIS 2
2008, VIS 3
2009, VIS L
2009, VIS R
resolution
672 × 512 px
640 × 512 px
672 × 512 px
672 × 512 px
field of view
1.5 mm
0.86 mm
0.48 mm
0.78 mm
VPF 1
4.5 Km/s
12.65 Km/s
7 Km/s
18 Km/s
Table 2.4: SOP main features
time window
6.0 ns
10 ns
5.6 ns
12.8 ns
CHAPTER 2. EXPERIMENTAL SETUP
45
Figure 2.14: Double-etalon system
medium) is shorter than the dark-green path (in the nb medium) by a time τ
given by the difference in the optical paths, but the apparent lengths are made
equal adjusting the position of the movable mirror (on the right). This ensures
that interference can be produced even with light from a diffusive surface, such
as the target rear face. The tilt of the movable mirror by a small angle α
produces a path difference in the plane of Visar and fringes are produced on
the streak camera entrance slit, where the interference of two beams yield an
intensity profile given by
Ientrance
slit (x)
= 2E02 (1 + cos(ωτ + kx sin α))
(2.0)
This equation can be obtained considering the interference of the electric fields
of the two beams on the slit:
E1 (t) =
=
E2 (t) =
=
E0 exp{i(ωt − k · r1 )}
E0 exp{i(ωt − kz)}
E0 exp{i(ω · (t − τ ) − k · r2 )
E0 exp{i(ω · (t − τ ) − kz − kx sin α)}
(2.1)
(2.2)
(2.3)
(2.4)
and
Ientrance
slit
= (E1 + E2 )(E1 + E2 )∗
(2.4)
When the reflecting surface of the target is moving at a speed u, the light is
doppler-shifted according to
1 − u(t)
2u(t)
c
λ(t) = λ0
' λ0 1 −
(2.4)
c
1 + u(t)
c
where non-relativistic approximation was used, as u/c ∼ 10−4 . As λ changes,
a fringe shift
2πc
φ = ∆ωτ = ∆
τ
(2.4)
λ
is induced.
CHAPTER 2. EXPERIMENTAL SETUP
46
Visar formulas
As the interest of Visar is its use as a ultra-fast velocimeter, it is important
to relate the fringe variation to the reflecting surface velocity with the best
accuracy. The most general equation relating velocity u to fringe count F is
the following [Barker and Schuler, 1974]:
λ0 F (t)
1
(2.4)
u t− τ =
2
(1 + δ)
2τ 1 + ∆ν
ν0
The leading term is (λ0 /2τ )F (t)/(1 + δ) ≡ VPF (Velocity per Fringe) and
it is calculated from the Visar parameters and, if necessary, corrected with
data from calibration shots. Equation 2.4 considers the effect of the etalon
dispersion, included in the factor 1 + δ. Etalon dispersion cause the Dopplershifted light to see a slightly different index of refraction from the unshifted
light. Typical corrections are of the order of 3%.
The factor 1 +∆ν/ν0 corrects for the effect of index of refraction variation with
shock stress in transparent window material. Fig. 2.15 shows the motion of the
shock and release waves in a distance-time plane. Light back-scattered from
the interface between carbon and substrate is used to generate the fringe signal
F related to the interface velocity by eq. 2.4. ∆ν = νm − ν0 is a frequency
correction defined as the difference between the measured Doppler frequency
shift νm and the frequency shift ν0 that would be observed in the absence of
window refractive-index effects (ν0 = 2u/λ0 ). Now, it is useful to write eq.
2.4 in terms of a velocity correction instead of a frequency correction [Setchell,
1979]:
λ0 F (t)
1
− ∆u
(2.4)
u t− τ =
2
2τ 1 + δ
where the velocity correction is
∆u = (∆ν/ν0 )u(t − τ /2)
(2.4)
The τ /2 time-centering can be neglected, as τ /2 ' 25 ps and the typical time
resolution is 200 ps, so the previous equation can be written as ua = um − ∆u,
where ua is the actual velocity and um is the VPF without window material.
∆u can be related to shock velocity and to the refractive index as follows.
Consider a pair of laser light rays incident upon the reflecting interface at this
time, separated in time by the initial period τ , where ca is the speed of the
laser light in air. As a ray propagates through the substrate, its local slope is
given by
dt
±n(x, t)
=
(2.4)
dx
ca
where n(x, t) is the local refractive index. Following reflection from the VISAR
interface, the pair of rays emerge from the window with period τf . The Doppler
frequency shift νm is simply
νm = τf−1 − τi−1
(2.4)
In the present case the shock wave at t1 (see Fig. 2.15) is an abrupt discontinuity followed by a uniform state having refractive index n1 . A direct geometrical
CHAPTER 2. EXPERIMENTAL SETUP
47
Figure 2.15: Wave motion in the distance-time plane for the laser-induced
shock-wave compression experiment. The dashed curves represent the trajectories of a pair of light rays whose frequency is altered by passing through the
wave motion and reflecting from the interface
calculation of the trajectories of the two rays using eq. 2.4 results in [Setchell,
1979]
νm = (2/λ)[ua n1 − D(n1 − n0 )]
(2.4)
where n0 is the refractive index within the unshocked window and D is the
shock velocity. The interferometer fringe signal is [Barker and Schuler, 1974]
F (t) = τ (1 + δ)νm
(2.4)
Combining eq. 2.4, 2.4 and 2.4:
∆u(t1 ) =ua (n1 − 1) − D(n1 − n0)
1 λ0 F (t)
ua =
+ (n1 − n0 )D
n1 2τ 1 + δ
(2.4)
Yet, to obtain the corrected (actual) speed ua it is necessary to know n1 and D.
These values can be estimated from empirical formulas and shock laws, such as
Hugoniot relations, or numerical simulations. Another useful approach is the
CHAPTER 2. EXPERIMENTAL SETUP
48
following. Using mass conservation law (ρ1 = ρ0 D/(D − ua )) it is possible to
write
λ0 F (t)
ρ1 (n0 − 1) − ρ0 (n1 − 1)
= ua 1 +
(2.4)
2τ 1 + δ
ρ1 − ρ0
At first order, the refractive index is considered to vary with density as follows
n = n0 +
dn
(ρ − ρ0 )
dρ
(2.4)
−1
dn
n 0 − ρ0
dρ
(2.4)
yielding
ua =
λ0 F (t)
2τ 1 + δ
The refractive index of a material can be related to density through different
empirical models. As a starting point, it is possible to consider the GladstoneDale relation [Gladstone and Dale, 1863] dρ/ρ = dn/(n − 1), which can be
written in terms of conditions across a steady shock wave as:
n = n0 +
n0 − 1
(ρ − ρ0 )
ρ0
(2.4)
If this relation is combined with the shock jump conditions, the velocity correction ∆u given in eq. 2.4 is identically zero and ua = um = VPF. Thus,
velocity corrections due to refractive index effects in a window correspond to
the window material departing from Gladstone-Dale behavior. To go beyond
Gladstone-Dale model, it is possible to consider the relation between refractive index and polarizability described by a dispersion formula such as the
Lorentz-Lorentz or Drude equations. For the present purpose it is useful to
write [Setchell, 2002]
Lorentz − Lorentz
Drude
(n2 − 1)/(n2 + 2) = KρP
n2 − 1 = KρP
(2.4)
where K is a constant, ρ is density and P is the polarizability. Differentiating
these expressions with respect to ρ results in:
!
6n
dn
∂P
∂P
dT
n2 − 1 1
=K P +ρ
+ρ
,
K= 2
2
2
(n + 2) dρ
∂ρ T
∂T ρ dρ
n + 2 ρP
1 ∂P
dn
(n2 − 1)(n2 + 2)
ρ ∂P
dT
+ρ
=
1+
dρ
6nρ
P ∂ρ T
P ∂T ρ dρ
|
{z
}
| {z }
τ0
−Λ0
(2.4)
for the Lorentz-Lorentz model and
dn
n2 − 1
=
dρ
2nρ
dT
1 − Λ0 + ρτ0
dρ
(2.4)
for Drude model. Λ0 is a strain-polarizability coefficient and τ0 is a temperaturepolarizability coefficient [Waxler and Cleek, 1973]. The term in square brackets
in eq. 2.4 and 2.4 represent the change in refractive index due to change both
CHAPTER 2. EXPERIMENTAL SETUP
Material
Fused silica
Lithium fluoride
49
τ0 × 105
2.4
0.0
Λ0
0.17
0.62
Table 2.5: Strain and temperature coefficients of polarizability, derived from
[Davis and Vedam, 1967]and [Zel’dovich and Raizer, 1966]
with density and temperature. For the target substrates used this thesis work,
the coefficients are reported in table 2.5.
As indicated in figure 2.15, additional wave motions will occur in the window material later in time. At time marked t2 the substrate experiences both
the initial shock wave and the subsequent rarefaction wave. Assuming that
rarefaction waves can be represented by a simple, steady discontinuity propagation with a single velocity, we find for the trajectories of a pair of rays
calculated at time t2 :
νm = (2/λ0 )[ua n1 − D(n1 − n0 ) − R1 (n2 − n1 )]
(2.4)
where R1 is the velocity of the rarefaction wave and n2 is the refractive index
within the uniform region between the rarefaction and the reflecting interface.
The previous equation is actually more general and holds if R1 represents a
second shock discontinuity. For present purpose eq. 2.4 leads to:
∆u(t2 ) = ua (n2 − 1) − D(n1 − n0 ) − R1 (n2 − n1 )
Combining this with eq. 2.4 yields
1 λ0 F (t)
+ (n1 − n0 )D + R1 (n2 − n1 )
ua =
n2 2τ 1 + δ
A comparison with eq. 2.4 finally gives
−1
λ0 F (t)
n1
n1
dn
ua =
n 0 − ρ0
+ 1−
R1
2τ 1 + δ
dρ
n2
n2
|
{z
}|
{z
}
single−shocked window
(2.4)
(2.4)
(2.4)
rarefaction correction
The values of n1 and n2 can be obtained from eq. 2.4, 2.4 and 2.4, estimating
ρ1 , ρ2 and R1 through shock laws or numerical simulations.
In the case that the shock is strong enough to metalize the window material,
the probe laser reflects on the shock front. Noting that no change in refractive
index happen before the shock arrival, eq. 2.4 can be rearranged as follows
D=
2.5
λ0 F (t) 1
2τ 1 + δ n0
(2.4)
Resume
The compression of carbon to Mbar regime requires complex and sophisticated
technological devices:
• the laser system provides a large focal spot and a flat-top intensity profile,
to assure strong and steady shocks with a high degree of front planarity;
CHAPTER 2. EXPERIMENTAL SETUP
50
• target design allow to image directly carbon in high pressure state, preventing the release into vacuum of the shock wave, thanks to the double
carbon-transparent substrate layer. Fabrication process is oriented to the
production of low density carbon to tailor the shocked state properties;
• diagnostics with sub-nanosecond time resolution give important information on many characteristic quantities. Self-emission diagnostics give
information on shock planarity, time of shock breakout, preheating and
emission resolved both in time and wavelength;
• Visar diagnostic is a ultra-fast interferometry velocimeter. In the chapter were derived the most important formulas that apply to the case
of experimental relevance. If the window material remains transparent
under shock compression, the interface speed u is related to the fringe
movement F (t), the Visar parameters λ0 , τ , δ and the medium property
ρ0 , n0 , dn/dρ through eq. 2.4
1 λ0 F (t)
+ (n1 − n0 )D
u=
n1 2τ 1 + δ
Considering the rarefaction wave, the velocity reads (eq. 2.4)
ua =
λ0 F (t)
2τ 1 + δ
−1
dn
n1
n1
n 0 − ρ0
+ 1−
R1
dρ
n2
n2
The formula contains more parameters to account for the intermediate
shock state. At last, if the shock is strong enough to metalize the substrate, the Visar formula reads (eq. 2.4)
D=
2.6
λ0 F (t) 1
2τ 1 + δ n0
Bibliography
E. Barborini, P. Piseri, and P. Milani. Rev. Sci. Instrum., 72:2261, 1999.
L. M. Barker and R. E. Hollenbach. Laser interferometer for measuring high
velocities of any reflecting surface. Journal of applied physics, 43(11), 1972.
L. M. Barker and K. W. Schuler. Correction to the velocity-per-fringe relationship for the visar interferometer. Journal of applied physics, 45(8), August
1974.
T. A. Davis and K. Vedam. J. Appl. Phys., 38:4555, 1967.
S. N. Dixit et al. Opt. Lett., 21:1715, 1996.
J. H. Gladstone and T. P. Dale. Researches on the refraction, dispersion and
sensitiveness of liquids. Phil. Trans. Roy. Soc., 153:317–343, 1863.
R. W. Kelsall, I. W. Hamley, and M Geoghegan, editors. Nanoscale Science
and Technology. John Wiley and Sons, Ltd, 2005.
CHAPTER 2. EXPERIMENTAL SETUP
51
P. Milani, P. Piseri, E. Barborini, A. Podesta,́ and C. Lenardi. Cluster beam
synthesis of nanostructured thin films. J. Vac. Sci. Technol. A, 19(4):2025,
2001.
N. Myanaga et al. In Proceedings of the 18th International Conference on
Fusion Energy, IAEA-CN-77, Sorrento, Italy, 2001. IAEA.
R. E. Setchell. Refractive index of sapphire at 532 nm under shock compression
and release. J. Appl. Phys., 91(5):2833, March 2002.
R. E. Setchell. J. Appl. Phys., 50:8186, 1979.
R. M. Waxler and G. W. Cleek. J. Res. Natl. Bur. Stand., 77A:755, 1973.
C. Yamanaka. In C. Labaune, W.J. Hogan, and K. A. Tanaka, editors, Proceedings of the Inertial Fusion Sciences and Applications 99, page 19, Bordeaux,
1999. Elsevier, Paris, 2000.
Ya. B. Zel’dovich and Yu. P Raizer. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena, page 700. Academic, New York,
1966.
Data analysis
3
The aim of this chapter is to present a detailed discussion of the data analysis.
This process consists in various steps to allow an as accurate as possible determination of the shock and thermodynamic parameters that characterizes each
shot.
3.1
Timing
The first step is the synchronization of the four diagnostics employed. Simplifying, a triggering signal is generated about 200 ns before the arrival of the
laser on the target. This signal travels through different cables towards the
delay boxes. These devices allow a coarse shift of the active (recording) window associated with each diagnostic in order to catch the event of interest.
The trigger finally reaches the streak cameras and the recording starts. Due to
the different cable and electronic circuit delays, the trigger signal reaches the
diagnostics at slightly different times.
A reference shot is performed before the main experiments in order to calculate
with good accuracy these time shifts. This is a low energy shot, using only the
rod amplifiers in the laser chain resulting in about 1 J total energy, on a glass
stoke. The emission of the heated up glass is recorded and the timing is calculated from the images, as shown in Fig. 3.1. In the main shots, the shock-wave
traveling time – the time between the generation of the shock-wave on the front
face of the target and the shock breakout at the carbon-substrate interface –
is calculated from the reference time found before, accounting for variations in
the delay boxes setup. The shock traveling time is the most critical parameter
in the comparison between experiment and simulations, as it depends strongly
on laser intensity, carbon layer thickness and density.
For the self-emission diagnostics, the shock breakout time is characterized by
a sharp increase of the signal, while in the Visars it produce a fringe shift and
a keen change in reflectivity.
3.2
Velocity
In the previous chapter it was shown that Visars can record the carbonsubstrate interface velocity or the shock velocity in the substrate. In this
52
CHAPTER 3. DATA ANALYSIS
53
Figure 3.1: Timing shot 2009. Aligning the onset of the glass self-emission
gives the actual time delay
section the calculation of the fringe shift from the image will be illustrated.
The fitting method is based on the Fourier transform applied to each row of the
Visar image to individuate the fringe pattern. Then the phase information
is extracted and, going down the rows, the fringe shift vs. time relation is
obtained. Fig. 3.2 shows diagrammatically the steps involved. The wavenumber correspondent to the fringes is found calculating the Fourier transform
using the FFT algorithm of each row in a selected region of the image. The area
of selection is chosen in the early-time region of the image, to have clear fringes.
The spectrum is the square modulus of the Fourier coefficients corresponding
to components at different wave-number. In the process a high-pass filter is
applied to suppresses the low part of the spectrum, corresponding to patterns at
frequencies lower than that of the fringes. Finally, the frequency of the fringes
is found as the most intense spectral component, with highest recurrence in the
selected region (see Fig. 3.3). Once the wave-number of interest is individuated,
the FFT is applied row by row (i.e. at constant time) on the whole image and
the phase of the relevant Fourier complex term is recorded. Due to imperfect
alignment of the CCD or aberrations in the imaging system, the fringes can have
a small angle respect the vertical, introducing a positive or negative slope in the
phase vs. time plot. This systematic error can be easily corrected subtracting
the background line. The velocity is obtained from the phase according to the
formulas of the previous chapter. This calculation, as well as most of the other
presented in this work, has been developed under MATLAB environment.
3.3
Reflectivity
The reflectivity of the carbon surface is one of the most interesting information
for this thesis work, as gives information on the conductivity of the compressed
CHAPTER 3. DATA ANALYSIS
54
Figure 3.2: Logic-block diagram of the fringe-to-velocity algorithm
Figure 3.3: Resume of the FFT results. On bottom right, it is shown the spectrum of a single row; the Fourier component of the fringes is the clearly visible
peak at the center. On the bottom left, the histogram shows the recurrence of
the peak frequencies; the most recurrent frequency is the fringe one. On the
top, the fringe signal (in red) is plotted over the noisy image (in blue)
CHAPTER 3. DATA ANALYSIS
55
state reached in the shock. Visar images yield also the reflectivity of the
surface where the probe laser reflects. As this laser intensity varies on a time
scale of ns, it is necessary a reference image taken before the firing of the main
laser. The relative reflectivity is calculated as follows
R=
main shot signal − background
reference signal − background
(3.0)
and is normalized to unity before shock breakout. The result is shown in Fig.
3.4.
Figure 3.4: Example of time-resolved reflectivity calculated from Visar images.
Probe intensity taken from reference image is also shown
3.4
Temperature
Temperature is the last important parameter in the knowledge of the thermodynamical state of the shocked matter. Generally, temperature is determined
observing the self-emission of the target. There are three main cases differing
in the characteristics of the rear face of the target.
a Releasing plasma: if the material is opaque, the emission is visible only after
the shock breakout on the rear face. This is the case of shocks in Al, Fe, etc.
b Propagation through a material initially transparent: the emission is visible
along the propagation of the shock through the unshocked thickness. This
is the case of shocks in water, hydrogen, etc.
c Propagation through a window material, transparent even in the shocked
state. This is the case of shocks below a critical threshold in high gap
material, like glass or lithium fluoride.
The cases of interest for this work are the third, when the window material
remains transparent, and the second, when the window material metalizes or
become opaque after the shock.
CHAPTER 3. DATA ANALYSIS
56
Black body and gray body approximations
The most general expression for the light emitted by a body at a temperature
T is the Planck law [Planck, 1901], in this form
B(λ, T ) = e(λ)
1
2hc2
hc
5
λ e λkT − 1
(3.0)
where e(λ) is the emissivity at the wavelength λ. In the black body approximation e = 1 for all wavelength. The less restrictive gray body approximation
considers e in the range 0 - 1, independent of λ. The planck curves can be
seen as a family of one parameter (the temperature) curves and have special
mathematical properties:
• for fixed λ, are monotonically increasing functions of T ;
• for T fixed have an absolute maximum at
λmax =
257.6eVnm
2.989 × 10−3 mK
=
T
T [eV]
(3.0)
(Wien’s displacement law);
• the integral on all the wavelengths for fixed T yields
Φ = σT 4
,
σ=
2π 5 k 4
= 5.67 × 10−8 W/m2 K4
15h3 c2
(3.0)
(Stefan-Boltzmann law);
• the ratio between the radiance at two different wavelengths is a monotonic
function of T .
Experimental methods
There are many different strategies in the experimental determination of the
temperature starting from the self-emission, illustrated in Fig. 3.5. Accordingly, the temperature obtained is named after the particular strategy adopted:
radiation temperature : it is the measure of the total power emitted for
unit area, regardless of the wavelength. The temperature is related to
the energy by the Stefan-Boltzmann law
Z
P = dλI(λ) = σT 4
brilliance temperature : the temperature is determined measuring the radiance at a fixed wavelength.
spectral temperature : the temperature is determined fitting a region of
the spectrum to find the planck curve that realizes the best accordance.
color temperature : T is calculated from the ratio of the radiance at two
different wavelengths, which is a monotonic function of the temperature.
CHAPTER 3. DATA ANALYSIS
57
Figure 3.5: Black body radiation at three different temperatures and different
temperature determination strategies
pyrometric temperature : the temperature is inferred from the power emitted in a limited range of the spectrum. The relation can depart strongly
from Stefan-Boltzman law.
In this work the temperature will be the spectral temperature, derived from
the images of the Spectrometer SOP. An effective pyrometric temperature can
be also obtained from the SOP diagnostic, as explained later on.
Spectral temperature
In order to obtain accurate values for the temperature, a multi-step process
was devised, as can be seen in Fig. 3.6. The main problem is that to achieve a
Figure 3.6: Diagram of the main steps of the spectral temperature calculation
high time resolution (better than 1 ns), light must be focused on a narrow slit
and high sweep rates are used. The signal arriving at the MCPs is weak and
CHAPTER 3. DATA ANALYSIS
58
the high gain necessary for the detection increase the overall noise. So, before
fitting, the image quality is improved removing the aberrations of the imaging
system and compensating the noise averaging each pixel with its close neighbors
(smoothing). Also, the transmittance and responsivity of the whole system is
taken into account, as explained in the previous chapter. The variation of the
CCD sensitivity on the image is considered, too.
The core part of the algorithm is the fitting routine. The test function is a
planckian written in this form
A·
1
T5
exp
λ−5
1.439
−1
λT
(3.0)
As the height of the planck curve increases as T 5 the normalization parameter
A is rescaled by 1/T 5 to improve numerical convergence. The constant 1.439
is given by hc/k in [cmK].
Two different fitting methods are applied and the results are averaged with
the respective weight (if the two values are consistent). Robust methods are
employed to increase accuracy and reduce the noise, as the weight of outliers
(i. e. data points lying far from the bulk) is reduced. For this purpose in
one fit absolute residuals are minimized within the parameters (Least Absolute
Residuals – LAR), instead of the squares (Least Squares Method – LSM); in
the other fit, a recursive procedure is used to assign an effective weight to each
data point [DuMouchel and O’Brien, 1989]. Errors on the fitted parameters
are evaluated with bootstrap method: a set of parameters is obtained by fitting
a large number m of simulated data sets. These simulated data sets (bootstrap
replicates) are simply obtained by random sampling, with replacement, of the
experimental data set (in our case a single raw of the Spectrometer SOP image).
The bootstrap estimate of the standard error on parameters p = (T, A) is given
by
"
#1/2
m
1 X
2
σB (p) =
(pi − pa )
(3.0)
1 − m i=1
where
m
pa =
1 X
pi
m i=1
(3.0)
or other robust averages.
Pyrometry
Temperature can be measured also from SOP images, as the self-emission is
recorded. Assuming that the target emits like a gray body, with emissivity e,
the CCD counts N (t) are given by the convolution of the light spectrum with
the frequency dependent sensitivity of the imaging system η(λ) multiplied by
the conversion efficiency C:
Z
(3.0)
N (t) ' ẽ(t)SC dλB(λ, T (t))η(λ)
|
{z
}
F(T )
CHAPTER 3. DATA ANALYSIS
59
where and ẽ is the emissivity averaged on the spectral window of the SOP, S is a
system parameter depending on the geometry and B is the planck distribution
as written in eq. 3.4.
The function F(T ) can be evaluated for a finite numbers of T values calculating
the integral numerically. The results F(Ti ) can be interpolated with a spline 1
to obtain the function F for continuous T . Figure 3.7 shows the calculated
F(T ) in the case of SOP diagnostic employed in the experiments. It is possible
to note that for values bigger than 104 K the function is linear in T , while for
low temperatures it follows the Stefan-Boltzman law F ∝ T 4 .
For normal incidence, the emissivity e(λ) of a plane parallel specimen is given
Figure 3.7: Function F(T ) is the convolution of a planck spectrum at temperature T with the global efficiency η of the imaging system. In the low
temperature limit F follows the Stefan-Boltzman’s law, as expected while in
the high temperature regime it increases linearly with T . F is normalized to 1
for a temperature of 5,000 K
by [Ravindra et al., 1998]
e(λ) = (1 − R(λ)
1 − T (λ)
1 − R(λ)T (λ)
(3.0)
where R is the true reflectivity and T is the true transmissivity. R and T are
related to the fundamental optical parameters – n, the refractive index and k,
the extinction coefficient – by the following relations:
(n − 1)2 + k 2
(n + 1)2 + k 2
T = exp(−αδ) = exp(−4πkδ/λ)
R =
(3.1)
(3.2)
where α is the absorption coefficient and δ is the thickness of the material.
Thus, from eq. 3.4, for a perfect opaque body like the carbon layer in the
target, since T = 0, Kirchoff’s law follows as
e(λ) = 1 − R(λ)
(3.2)
1 Splines are smooth piecewise polynomials that can be used to represent functions over
large intervals, where it would be impractical to use a single approximating polynomial
CHAPTER 3. DATA ANALYSIS
60
and the average on the relevant wavelengths yields ẽ(t) = 1 − R̃(t). R(t, λ =
532nm) ' R̃(t) is obtained from Visar diagnostics.
This procedure allow the estimation of the temperature from the intensity of
the signal recorded by the SOP up to a multiplicative constant, lacking an
absolute calibration. Thus, a reference point is always necessary. Usually the
temperature from Spectrometer SOP is used for this purpose.
A pyrometric estimation of the temperature, hence, can be obtained from the
intensity of the light ISOP (t) according to
Temission (t) =
F −1 (ISOP (t)/Iref )
1 − R̃(t)
(3.2)
where F −1 : I 7→ T indicates the inverse of the function F : T 7→ I and Iref is
the reference light intensity level corresponding to a temperature of 5,000 K.
Figure 3.8 shows spectral and pyrometric (emission) temperature measured in
two shots, reported here as examples. As noted before, pyrometric temperature
is defined up to a multiplicative constant and in each plot the high and low
bounds are shown.
(a) Temperature is consistent with emission from carbon
(b) Temperature is consistent with emission from the silica substrate
Figure 3.8: Representative plots of the temperature obtained in the experiments. Temperature obtained in Multi simulations is also shown for comparison
3.5
Resume
This chapter presents the main steps of the data analysis. After accounting for
the time delays between the various diagnostics, the shock wave transit time
in carbon is established. Subsequently, the other information of interest are
extracted from the images: Visars yield the velocity and the reflectivity of the
interface or the shock front in the metalized substrate; spectrometer SOP allow
the determination of the spectral temperature of the emitting surface, while
CHAPTER 3. DATA ANALYSIS
61
from SOP it is possible to calculate a ’pyrometric’ temperature. Numerical
fitting and data analysis tool were developed under Matlab environment.
3.6
Bibliography
W. H. DuMouchel and F. L. O’Brien. Integrating a robust option into a multiple regression computing environment. In Computer Science and Statistics:
Proceedings of the 21st Symposium on the Interface, Alexandria, VA, 1989.
American Statistical Association.
M. Planck. On the law of distribution of energy in the normal spectrum.
Annalen der Physik, 4:553, 1901.
N. M. Ravindra, W. Chemn, F. M. Tong, A. K. Nanda, and A. C. Speranza. Temperature-dependent emissivity of silicon related materials and
structures. IEEE Transactions on semiconductor manufacturing, 11(1):30,
February 1998.
Results
4
This chapter reports the main results attained in this thesis work. At first,
a satisfying equation of state was devised, in order to describe the particular
porous carbon employed in the targets. Such a description is of great importance to obtain accurate numerical simulations of the shock dynamics. The
numerical approach is a very powerful and accurate tool in clarifying the hydrodynamics at the energy and time scales typical of laser induced shock compression experiments. In this way, the various processes occurring in the few
nanoseconds accessible to the diagnostics are displayed. In conclusion, taking
advantage of all the previous theoretical and numerical results, together with
the latest data analysis methods developed, the results are presented.
4.1
Porous Carbon EoS Model
In the experiments performed at ILE during the two campaigns 2008 and 2009,
porous carbon has been employed in the targets, thus in order to perform
realistic and as accurate as possible simulations, an EoS model for the material
is required. Unfortunately, the Sesame database includes only diamond and
bulk graphite libraries and the porous carbon EoS tables have been generated
with the computer program MPQeos developed by Kemp and ter Vehn [1998]
implementing the Qeos model.
Testing
A first test for the model has been carried out comparing the table generated
by MPQeos with the Sesame [mis, 1992] table of graphite (ρ = 2.205 g/cm3
and B = 34 GPa) and with experimental data from Nellis et al. [2001].
Hugoniot curves has been calculated from Sesame and Qeos tables, in the
(D,P ), (U ,D), (ρ,P ) and (T ,P ) planes (figure 4.1). Qeos and Sesame are in
good agreement for shock pressure below approximately 1 Mbar but for stronger
shocks the Hugoniot curves calculated from the two models separate. Qeos
curves are smoother and show a slightly better agreement with the experimental
data from Nellis. In the (T ,P ) plane, where no experimental data are available,
both models are in accordance for shock pressure up to 12 Mbar. Thus, Qeos
is expected to be a reliable and quite accurate model, well suited for the use
with hydrodynamical simulation code.
62
CHAPTER 4. RESULTS
63
Figure 4.1: Comparison between Qeos, Sesame and experimental Hugoniot
data for graphite
Porous carbon bulk modulus
The parameters needed to generate a EoS table with MPQeos are the atomic
number Z, the atomic weight A, the cold density ρ0 and the bulk modulus
B. To estimate this quantity, a simple scaling-law relating density and B was
devised and compared to other models and experimental data found in literature.
Bulk modulus is strongly affected by both porosity and chemical bonding.
Graphite (sp2 -bonded carbon) has a lower bulk modulus than diamond (sp3 bonded carbon) and porous graphite-like carbon has a lower bulk modulus than
bulk graphite as well. Unfortunately there is not a simple relation linking density and chemical bonding with bulk modulus. Some models can be devised to
obtain quantitative estimation of the bulk modulus for the material of interest,
however such models give reliable predictions only in a limited density range.
The simplest calculation is to estimate bulk modulus B from the relation[Kittel,
1995]
1 (V − V0 )2
B
2
V0
∂2E
B = −V0
∂V 2
E =
(4.1)
(4.2)
where E is the bonding energy. Assuming that E scales as the number of
bonds, it is possible to write
X
E=N
i (d)ρ
i
CHAPTER 4. RESULTS
64
where the summation is extended over each molecular bond with energy i , d
is the bond length, and N is the number of molecules. From this assumption
it follows that
!
d2 X ∂ 2 i (d)
ρ
(4.2)
B=−
9 i
∂d2
The term in parenthesis depends on the bond type and geometry, so for fixed
carbon structure, B is expected to scale approximately as density.
A different approach was developed by Donadio et al. [2004] using molecular
dynamic (MD) simulations to reproduce the SCBD deposition [Milani et al.,
2000] conditions. They were able to find a scale relation between bulk modulus
B and average size of carbon clusters hN i
−α
B ∝ hN i
where α ' 0.87 assuming for the clusters an impact energy of about 0.1eV /atom.
−0.23
From their data it is possible to infer the scaling relation ρ ∝ hN i
, thus
giving
B ∝ ρ3.8
(4.2)
An earlier empirical model was that developed by Hashin and Hasselman
[Hashin, 1962],[Hasselman, 1962] for porous graphite. In this case B is related
to the porosity φ of the material through
B
1−φ
=
B0
1 + cφ
(4.2)
where B0 is the bulk modulus of standard non-porous material and c is a
parameter. Knowing that
φ=
Vpores
ρporous
=1−
Vtotal
ρbulk
it is possible to rewrite 4.1 as
ρ
B
=
B0
ρ0 + c(ρ0 − ρ)
From an analysis by Mackenzie [1950], for an isotropic solid containing spherical
holes all of the same size, it is possible to approximate
c≈
3B0
4G0
where G0 is the shear modulus. In the case of graphite the formula gives c ' 2,
while Boey and Bacon [1986] found c ' 5 − 10.
In order to check the validity of the theoretical models, it is necessary to make
a comparison with experimental data. Table 4.1 shows the bulk modulus of
different carbon structures, along with the corresponding density.
In figure 4.2 are summarized the theoretical and experimental findings discussed above, in the range of density 0 − 2.2 g/cm3 . For density lower than
1 g/cm3 both Hashin & Hasselman model and linear approximation give results in good accordance, thus for density of 0.55 g/cm3 a bulk modulus of
approximately 1.4 ± 0.6 GPa is expected.
CHAPTER 4. RESULTS
ρ[g/cm3 ]
3.515
3.5
3.26
2.8
2.7
2.35
2.2
2.05-2.28
1.8
1.76
1.7
1.2-1.3
1-1.3
∼1
∼1
B[GPa]
444.8
435
334
289
> 67
248
34
135
144.5
2.5-7.3
11.3
2.1
2-3.7
3.7
38.4
notes
diamond, 100% sp3 , 0%H
diamond, micro-crystalline, CVD, 100% sp3 , 0%H
ta-C (tetrahedrally bonded amorphous carbon)
ta-C
ta-C, surface layer
ta-C:H (hydrogenated ta-C), 70% sp3 , 30%H
graphite single crystal, 100% sp2
a-C:H, 70-50% sp3 , 28-15%H
a-C:N, magnetron sputtering
porous (20%) graphite
fullerene C60
Supersonic Cluster Beam Deposition
nanostructured a-C, 10-20% sp3
SCBD
MD simulation of SCBD carbon
Table 4.1: Bulk modulus of different carbon structures
Figure 4.2: Resume of theoretical and experimental findings
65
ref.
[boo, 2003]
[boo, 2003]
[boo, 2003]
[boo, 2003]
[boo, 2003]
[boo, 2003]
[boo, 2001]
[boo, 2003]
[boo, 2003]
[boo, 2001]
[boo, 2003]
[Casari et al., 2001]
[boo, 2003]
[Bottani et al., 1998]
[Donadio et al., 2004]
CHAPTER 4. RESULTS
66
Low density carbon EoS
To be sure that the table are reliable, it is necessary to check that in the
whole range the EoS tables calculated with different B are consistent and that
the error on B yields a small error in the EoS. At first, Hugoniot curves were
calculated from different low density carbon tables and the results are shown in
figure 4.3. From this comparison, it is possible to argue that in the range 0.5 - 10
Figure 4.3: Comparison between Hugoniot curves from Qeos tables for low
density carbon
GPa the actual value of B affect only slightly the EoS table. Secondly, the effect
of different bulk modulus values on the whole shock generation and propagation
dynamics has been investigated with the hydrodynamical-radiative simulation
code Multi. In order to study shock velocity D, particle velocity U , pressure
at shock breakout Pbo and temperature at shock breakout Tbo dependency on
B, a few simulations have been performed reproducing conditions as close as
possible to real experimental ones. Two target types are considered, 10 and 30
µm respectively of carbon on a thick quartz substrate under two different laser
intensities irradiation (1 × 1012 and 3 × 1013 W/cm2 ). As general trend, all the
examined quantities (D, U , Pbo and Tbo ) decrease slightly as B increases and
the uncertainty in the bulk modulus value of 0.6 GPa yields the accuracies of
the quantities taken from simulations reported in table 4.1.
From the previous analysis it is possible to claim that Qeos is a reliable
model to calculate equations of state and it can be assumed as a starting point
to simulate shock dynamics in unknown material, due to the smoothness of
the thermodynamical functions and the stability under the variation of material parameters like bulk modulus. Thus, the low density table of carbon so
generated is well suited for the present work purpose.
CHAPTER 4. RESULTS
67
Quantity
shock velocity D
particle velocity U
pressure P
temperature T
Accuracy
±3%
±2%
±1%
±5%
Table 4.2:
4.2
Shock Dynamics
The dynamics of the shocks has been studied through the numerical simulations. As showed in section 1.5, Multi simulations accurately reproduce
experimental conditions. The actual laser time profile, measured by the dedicated diagnostic, is included in the simulations and the porous carbon EoS
table is employed.
This study evidenced that:
• as the laser intensity raises, weak shock waves generated in the carbon
layer by the foot of the pulse merge into a single, strong shock wave about
500 ps after t = 0 (see Fig. 4.4);
Figure 4.4: Weak shock waves, generated by the foot of the pulse, merge
with the strong wave created as laser intensity raises; the plot shows
density vs. position and time; below, the intensity of the laser pulse is
shown
• the preheating, due to high energy (∼ keV) photons produced in the
corona, is negligible;
• when the shock wave reaches the interface between carbon and substrate,
a shock wave propagates inside the substrate and a re-shock wave is
CHAPTER 4. RESULTS
68
reflected back into the carbon, as predicted by the conservation laws (see
Fig. 4.5). The transmitted wave lies on the substrate main Hugoniot;
Figure 4.5: Reflection of the shock wave. In (a) the density plot is shown,
while in (b) pressure is plotted. The interface between carbon and substrate is an equilibrium surface, where P and U are continuos
• a rarefaction wave appears when the reflected shock arrives at the ablation front; the dynamics is similar to the release in vacuum − or in a low
impedance medium − with the condition that pressure is fixed by the
laser intensity, according to eq. 1.37, here repeated
Pablation [Mbar] = 0.30(I[1012 W/cm2 ])3/4
Pablation [Mbar] = 0.35(I[1012 W/cm2 ])3/4
year 2008
year 2009
(4.3)
(4.4)
as the laser intensity varies rapidly in time, the pressure depends on the
instant at which the reflected shock reaches the ablation front;
• the density profile is piecewise flat only short after shock reflection (red
line in Fig. 4.6), at longer times it becomes smoother, as the release wave
propagates. This behavior affects the response of the Visars because the
index of refraction depends on density, as explained in section 2.4.
Two typical regimes were identified, depending on the thickness of the carbon
layer.
for a thick carbon layer (thickness & 25 µm) the reshocked (highest pressure) state is maintained for a time ranging between 1 and 2 ns, before
the arrival of the rarefaction wave. Moreover, as the reshock wave reaches
the ablation front when the laser is almost off, the rarefaction wave drops
the pressure to zero, like in the case of shock release in vacuum (see Fig.
4.7.
for a thin carbon layer (thickness . 25 µm) the reshock state is maintained
only for 500 ps or less. In this case, however, the reshock wave reaches
the ablation front when the laser pulse is close to the maximum intensity
CHAPTER 4. RESULTS
69
Figure 4.6: Density profile at two instants; the profile is piecewise flat only
in the first nanoseconds, when the release wave propagates it is smoothed
out; the plot shows the main structures produced by the various shock
and release waves
(a) Density plot
(b) Pressure plot
Figure 4.7: Shock dynamics in the case of thick carbon layer. The high
pressure state is maintained for a long time, while, at the arrival of the
rarefaction wave, pressure drops to zero.
CHAPTER 4. RESULTS
70
and the pressure does not drop to zero. As can be seen in Fig. 4.8, after a
few hundreds ps, the pressure at the interface follows the laser intensity,
according to eq. 4.3.
(a) Density plot
(b) Pressure plot; laser pulse intensity is also shown
Figure 4.8: Shock dynamics in the case of thin carbon layer. The high
pressure state is maintained only for a short time. On the other hand,
the rarefaction wave is generated when the laser is at full intensity, thus
the pressure of the release wave is equal to Pabl .
4.3
Carbon Reflectivity
The main aim of the experimental campaign is to investigate the phase transitions of carbon between metallic and insulator states. The reflectivity, measured by the Visars, is used to probe the conductivity state, while pressure
and temperature are inferred from the other diagnostics.
A limit in probing high pressures, with the two layer / two material targets, is
that the substrate is also metalizing when compressed above a certain threshold. Silica is reported to metalize for pressures above 1 Mbar[Hicks et al., 2005],
while lithium fluoride is reported to remain transparent up to 3 Mbar[Huser
et al., 2004]. For this reason, it is critical to recognize when substrate metalization occurs and distinguish between carbon and substrate reflectivity. Unfortunately, due to image noise and difficult experimental conditions, there is
not a single, conclusive evidence of the metalization of the window material,
but there are many hints that can indicate it. In the case that an increase in
reflectivity is observed, carbon metalization is likely to have occurred if
• the simulations predict substrate pressure below metalization threshold
• the temperature determined by both SOP and Spectrometer SOP is consistent with the carbon temperature found in the simulations
• Visars velocity is close to the predicted fluid velocity U
CHAPTER 4. RESULTS
71
• the pressure given by the shock polar of the substrate (see Fig. 4.9) is
below metalization threshold
(a) Silica
(b) Lithium fluoride
Figure 4.9: Shock polars of SiO2 and LiF ; the insets show the low velocity
part of the curve with more detail
Otherwise, substrate metalization is likely to have occurred if
• the simulations predict substrate pressure above metalization threshold
• the temperature determined by both SOP and Spectrometer SOP is consistent with the shock front temperature of the window, found in the
simulations
• Visars velocity is close to the predicted shock velocity D
• the pressure given by the Hugoniot curve of the substrate, in the plane
D, P (see Fig. 4.10) is above metalization threshold
In the case of silica substrates, window metalization has always been observed because pressure exceeded the threshold value of 1 Mbar in all the shots.
Pressures ranged from 1 Mbar (shot # 18, Imax = 3.5 × 1012 W/cm2 ) to 3.5
Mbar (shot # 73, Imax = 19 × 1012 W/cm2 ).
In the case of the target with lithium fluoride substrate, opacization / metalization of the window has been observed in high intensity shots, such as #
85 (P = 3 Mbar and Imax = 13 × 1012 W/cm2 ) or # 63 (P = 4 Mbar and
Imax = 23 × 1012 W/cm2 ).
The case of interest in this study is shot # 62, where the substrate
remains transparent and carbon in the high pressure state is imaged.
The data analysis was carried out as described hereafter.
At first the planarity of the shock was checked, ensuring that the emission front
recorded by the SOP was flat (see Fig. 4.11). The breakout time was taken as
the instant at which the emission reaches half maximum (see Fig. 4.12) and
reflectivity drops (see Visar L - taka in Fig. 4.13). The best estimation, from
the three diagnostic, is ∆tbreakout = 1.1 ± 0.1 ns.
Fig. 4.14 shows the reflectivity, calculated from Visar. The shock breakout
CHAPTER 4. RESULTS
(a) Silica
72
(b) Lithium fluoride
Figure 4.10: Hugoniot curve in the plane D, P calculated for SiO2 and LiF;
the insets show the low velocity part of the curve with more detail
Figure 4.11: SOP image of shot # 62; the contour lines evidence the planarity
of the shock at breakout
CHAPTER 4. RESULTS
73
Figure 4.12: Emission recorded by SOP and Spectrometer SOP; the blue line is
the wavelength integrated signal detected by the spectrometer, consistent with
the high time resolution SOP (black line); the red line is the emission from the
center of the shock front
Figure 4.13: Diagnostic resume, shot # 62
CHAPTER 4. RESULTS
74
is easily identified with the main signal drop at 1.1 ns, while a small decrease
is found short before breakout, due to preheating of the target. However, the
most important feature is the peak after breakout, corresponding to a quick increase in reflectivity of the carbon layer and the subsequent drop caused by the
emergence of the relaxation wave at the interface. As in Fig. 4.14 reflectivity
Figure 4.14: Reflectivity recorded by Visar and pressure from Multi, shot
#62; preheating, shock breakout and carbon metalization are evidenced
R and pressure taken from Multi simulation are plotted together, the sudden
decrease of carbon reflectivity can be easily related to the falloff in pressure
below the metalization threshold.
In order to validate the value of the maximum pressure given by the simulation, the interface velocity U was calculated from the fringe shift recorded by
the Visar (Fig. 4.15). Pressure is then obtained from U through the shock
polar of LiF (Fig. 4.16). To conclude the characterization of the thermodynamical state reached in shot # 62, the temperature was calculated fitting the
spectra recorded by the Spectrometer SOP, as explained in section 3.4. The
temperature was also estimated from the emission recorded by the SOP, with
the procedure described in section 3.4. The result is shown in Fig. 4.17. In
the high pressure reflective state, temperature is estimated to be in the range
12,000 - 16,000 K, corresponding to 1 - 1.4 eV.
4.4
Resume
In this work, the main efforts were directed to devise as accurate as possible
numerical simulations of the shock dynamics in the two layer / two material
targets. As a first step, an equation of state for the porous carbon was obtained,
starting from the Qeos [More et al., 1988] model and tailoring the parameters
to the actual material. In this framework, the dependency of bulk modulus
on the density, in carbon, was extensively studied with both analytical and
heuristic approaches.
CHAPTER 4. RESULTS
75
Figure 4.15: Interface velocity calculated from fringe shift recorded by Visar
L - taka; one fringe jump is considered; the black line shows velocity calculated
according to eq. 2.4, while the red line shows velocity calculated according to
eq. 2.4, corresponding to the Kormer model ( dn
dρ ' 0.064); after the reflectivity
drop, only ghost fringes are visible and the velocity is meaningless
Figure 4.16: Lithium fluoride shock polar; according to the Visar, the fluid
velocity is 6.8±0.7 Km/s, thus the Hugoniot relation yields P = 2.6±0.4 Mbar
CHAPTER 4. RESULTS
76
Figure 4.17: Spectral and pyrometric temperature of shot #62; the solid black
line is the weighted spline approximation of spectral data (gray diamonds); the
light blue line is the pyrometric temperature, defined in section 3.4
Then, relying on accurate material description, Multi code was applied to the
investigation of the shock dynamics, stressing on the shock reflection and the
generation of the rarefaction waves in order to characterize the pressure conditions at the interface between carbon and substrate. Simulation were also used
to recognize the onset of the metalization of the window material, comparing
the substrate pressure with the threshold for metalization and comparing the
Visar velocity with the predicted shock velocity.
The chapter closes presenting the detailed analysis of shot # 62, which shows
evidences of a phase transition of carbon to a reflective state. A safe estimation
of the thermodynamical parameters yields
P = 2.5 ± 0.3 Mbar
T = 14, 000 ± 2, 000 K
4.5
(4.5)
(4.6)
Bibliography
Amorphous Carbon. INSPEC, 2003.
Brillouin spectroscopy of a-C films. 2001.
Sesame: The lanl equation of state database, los alamos national laboratory,
la-ur-92-3407, 1992.
S. Y. Boey and D. J. Bacon. Deformation of polycristalline graphite under
pressure. Carbon, 24(5):557–564, 1986.
C. E. Bottani, A. C. Ferrari, A. Li Bassi, P. Milani, and P. Piseri. Mesoscopic
elastic properties of cluster-assembled carbon films. Europhys. Lett, 42(4):
431–436, 1998.
CHAPTER 4. RESULTS
77
C. S. Casari, A. Li Bassi, C. E. Bottani, E. Barborini, P. Piseri, A. Podestá,
and P. Milani. Acoustic phonon propagation and elastic properties of clusterassembled carbon by brillouin light scattering. Phys. Rev. B, 64:085417,
2001.
D. Donadio, L. Colombo, and G. Benedek. Elastic moduli of nanostructured
carbon films. Phys. Rev. B, 70:195419, 2004.
M. P. Grumbach and R. M. Martin. Phase diagram of carbon at high pressures
and temperatures. Phys. Rev. B, 54(22), December 1996.
Z. Hashin. J. Appl. Mech., page 143, 1962.
D. P. H. Hasselman. J. Amer. Ceram. Soc., 45:452, 1962.
D. G. Hicks, T. R. Boehly, P. M. Celliers, J. H. Eggert, E. Vianello, D. D.
Meyerhofer, and G. W. Collins. Shock compression of quartz in the highpressure fluid regime. Phys. Plasm., 12:082702, 2005.
G. Huser, M. Koenig, A. Benuzzi-Mounaix, E. Henry, T. Vinci, B. Faral,
M. Tomasini, B. Telaro, and D. Batani. Interface velocity of laser shocked
fe/lif targets. Phys. Plasm., 11(10):L61, October 2004.
A.J. Kemp and J. Meyer ter Vehn. An equation of state code for hot dense
matter, based on the qeos description. Nuclear Instruments and Methods in
Physics Research, 415:674–676, 1998.
C. Kittel. Introduction to solid state physics. Wiley, seventh edition, 1995.
J. K. Mackenzie. Proc. Phys. Soc. Lond., 63:2, 1950.
P. Milani, P. Piseri, Barborini, A. Podesta, and C. Lenardi. Cluster beam
synthesis of nanostructured thin films. J. Vac. Sci. Technol. A, 19(4):2025–
2033, 2000.
R. M. More, K. H. Warren, D. A. Young, and G. B. Zimmerman. A new
quotidian equation of state (qeos) for hot dense matter. Phys. Fluids, 31
(10):3059, 1988.
W.J. Nellis, A. C. Mitchell, and A. K. McMahan. Carbon at pressures in the
range 0.1-1 tpa. Journal of applied physics, 90(2):696, 2001.
M. G. Tomasini. Studio dell’equazione di stato del ferro e del carbonio a
pressioni dell’ordine dei mbar generate da shock indotti da laser. Master’s
thesis, Università degli Studi di Milano, 2001.
Conclusions
5
In this work, carbon has been investigated in multi megabar pressure regime,
using laser driven shock compression. The properties of carbon in this state
are almost unknown because of the extreme conditions needed to study it
experimentally and data are scarce. Theoretical works predicts the existence
of different liquid phases, with different conductivity and the investigation is
aimed to probe these predictions. The experiments discussed along this thesis
were performed at the HIPER facility, an irradiation system on the GEKKO
XII (GXII): Nd glass laser system at ILE, Osaka. The facility provided onedimensional compression by smoothed laser beams with short wavelength and
high intensity. Two layer targets (porous carbon on transparent window) were
used, to reach off-Hugoniot states through shock reflection at the interface.
Streak camera and Visars diagnostics imaged the surface between carbon and
the substrate.
A reliable model for the equation of state of porous carbon was adopted.
This model is based on the Qeos description tailored to reproduce the particular properties of the material, by means of the computer program MPQeos.
Extensive use of Multi simulations allowed the clarification of the shock dynamics at the nanosecond and sub-nanosecond time scale, yielding significant
contributions to the experimental analysis.
Among the number of shots performed in two experimental campaigns, the
most remarkable result is the individuation of a reflective phase of liquid carbon
at 2.5 Mbar and 14,000 K in the plane P , T . Fig. 5.1 illustrates the up-todate phase diagram of carbon as well as the porous carbon and the graphite
Hugoniots, the regimes of interest in geophysics and planetology research and
the experiment outcome.
The last experimental findings are in line with another result, obtained
without the window material[Tomasini, 2001]. The targets were made of carbon
steps (with density ρ0 = 1.45 g/cm3 ) deposited on a CH/Al substrate, as shown
in Fig. 5.2. The shock releases from the free surface of carbon into vacuum
and the high pressure state is mantained only for a time of the order of 10 - 100
ps. Despite the short time scale of the phenomenon, an increase in reflectivity
was measured for seven points lying, to a first approximation, on the Hugoniot
curve at pressures in the range 1-8 Mbar and temperatures from 12, 000 to
70, 000 K. A typical outcome of this kind of experiments is reported in Fig.
5.3. The brighter stripes are due to the light reflected from the high reflective
78
CHAPTER 5. CONCLUSIONS
79
Figure 5.1: Phase diagram of carbon in high pressure and temperature regime,
after Grümbach and Martin; the blue solid line shows the porous carbon Hugoniot, the dashed-dotted black line indicates the graphite Hugoniot and the experimental point of reflective carbon is indicated by the red square; due to
reflection of the shock at the interface, this point lies well above the Hugoniot;
the regimes of interest in geophysics and planetology are traced with black
lines.
Figure 5.2: Experimental setup to investigate the reflectivity of carbon under
shock-wave release from free surface
CHAPTER 5. CONCLUSIONS
80
Figure 5.3: Reflectivity measurement of carbon (in blue) compared with aluminium reflectivity (in red)
aluminium substrate, while the dark lines are the opaque carbon steps. When
the shock breaks out from the aluminium a signal drop is observed. On the
contrary, when the shock breaks out from the carbon steps, a reflectivity peak
is observed. Carbon reflectivity is normalized with respect to the unperturbed
aluminium value RAl = 0.92. A raw estimation of the maximum value of
reflectivity in shocked carbon yields RC ' 0.28.
Fig. 5.4 summarizes all the points on the P, T plane where high reflective
carbon was observed. However, in this former experiment, the setup did not allow to accurately determine the temperature and pressure and these quantities
have been estimated only roughly.
5.1
Perspectives
The next step in data analysis is a quantitative determination of the high pressure state of carbon. Only then conductivity could be estimated and compared
to theoretical predictions.
New target design should be devised for further experiments. The advantage of the present scheme is the possibility to determine the temperature and
pressure of the shocked state with good accuracy, thus future improvements
may be oriented to overcome the limitation on the maximum pressure attainable before substrate opacization / metalization. An alternative substrate could
be MgF2 , a material with an energy gap comparable to that of LiF but denser,
CHAPTER 5. CONCLUSIONS
81
Figure 5.4: Summary of the experimental data avaliable on reflective high
pressure phase of carbon. The red square is the point obtained after the experiment at ILE (cp. Fig. 5.1) and the gray diamonds are the points from
Tomasini [2001]. As in the previous figure, the blue solid line shows the porous
carbon Hugoniot, the dashed-dotted black line indicates the graphite Hugoniot and the solid phases indicated with G – graphite, D – diamond and BC-8
– distorted diamond structure, semimetallic, are taken from [Grumbach and
Martin, 1996].
which in principle could remain transparent at higher pressures. A different
approach can be based on shock release from carbon free surface, with no window constraints. However, as stated before, in this second scheme the high
pressure state is retained only for a few picoseconds and it is very difficult to
determine pressure and temperature in the releasing material.

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