UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Measurement of shock-induced luminescence in silicon Xu, Guang 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A6_7 X82.pdf [ 5.02MB ]
Metadata
JSON: 831-1.0084959.json
JSON-LD: 831-1.0084959-ld.json
RDF/XML (Pretty): 831-1.0084959-rdf.xml
RDF/JSON: 831-1.0084959-rdf.json
Turtle: 831-1.0084959-turtle.txt
N-Triples: 831-1.0084959-rdf-ntriples.txt
Original Record: 831-1.0084959-source.json
Full Text
831-1.0084959-fulltext.txt
Citation
831-1.0084959.ris

Full Text

M E A S U R E M E N T OF S H O C K - I N D U C E D L U M I N E S C E N C E IN SILICON By Guang X u B. Sc. University of Science and Technology of China M . Sc. Institute of Physics, Academia Sinica A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April,1991 © Guang Xu , 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract Shock induced luminescence in silicon was measured with a time resolution of 20 ps. Pol-ished silicon wafers were irradiated with 0.53/xm laser light in a pulse of 2.7 ns F W H M at intensities ranging from 2 x 1017W/m2 to 5 x 1017W/m2. This produced a strong shock in the solid. The shock speeds were determined by measuring the shock tran-sit times through targets of different thickness (68/im and 82 fim). The corresponding shock pressure was found to range from 3 Mbar to 6 Mbar. Shock induced luminescence at wavelengths of 430nm and 570nm were recorded using a streak camera which was calibrated for ablsolute response. Substantial disagreement was observed between the lu-minosity data and the theoretical prediction which assumes thermal equilibrium between electrons and ions in the shock wave. An electron-phonon thermal relaxation model was proposed which treated the processes of both equilibration and thermal diffusion in an electron temperature gradient behind the shock front. Calculations including such a gra-dient yielded good agreements with data. The electron-phonon coupling constant g in shock-compressed Si was estimated to be WW-ru^K'1 in the pressure range of interest. Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement x 1 Introduction 1 1.1 Research in shock wave and matter at high pressure 1 1.2 Effort in determining the temperature of a shocked solid 2 1.3 Present investigation 4 1.4 Thesis outline 5 2 Shock wave theory and numerical simulation 6 2.1 Laser-driven ablation 6 2.2 Rankine - Hugoniot relation 8 2.3 QEOS 12 2.3.1 QEOS structure 12 2.3.2 Ion EOS 14 2.3.3 Electron EOS 16 2.3.4 Correction for chemical bonding 18 2.3.5 Equation of state of silicon 20 2.4 Numerical simulation 26 in Table of Contents Abstract ii List of Tables iv List of Figures v Acknowledgement vi 1 Introduction 1 1.1 Research in shock wave and matter at high pressure 1 1.2 Effort in determining the temperature of a shocked solid 2 1.3 Present investigation 4 1.4 Thesis outline 5 2 Shock wave theory and numerical simulation 6 2.1 Laser-driven ablation 6 2.2 Rankine - Hugoniot relation 8 2.3 QEOS 12 2.3.1 QEOS structure 12 2.3.2 Ion EOS 14 2.3.3 Electron EOS 16 2.3.4 Correction for chemical bonding 18 2.3.5 Equation of state of silicon 20 2.4 Numerical simulation 26 iii 2.4.1 Physics contents in L T C 26 2.4.2 Numerical method in LTC 31 2.4.3 Simulations for shock compressed silicon 33 3 The experiment 38 3.1 Experimental facility 38 3.1.1 Laser facility 38 3.1.2 Irradiation conditions 38 3.2 Calibration of the streak camera 42 3.3 Measurement of the shock speed 48 3.3.1 The experimental set up 48 3.3.2 Experimental results 49 3.4 Shock-induced luminescence measurement 57 4 Comparison of data to calculations 66 4.1 Discrepancy between data and calculation 66 4.1.1 Simulation of the unloading process 66 4.1.2 Luminescence calculation 69 4.1.3 Comparison of the temporal profiles 76 4.2 Nonequilibrium electron temperature model 77 4.2.1 Nonequilibrium electron temperature in shock-compressed material 77 4.2.2 A relaxation-diffusion model 82 4.2.3 Electron-phonon coupling constants estimated for the shocked silicon 89 4.3 Comparison of the calculation to the data at different pressures 91 5 Summary and Conclusions 101 5.1 Summary 101 iv 5.2 New contributions 102 5.3 Future work 102 Bibliography 104 v List of Tables 3.1 Table of average irradiances 42 3.2 Values of r 0 at different gain setting and wavelengths 45 3.3 Table of shock speeds and effective irradiances 56 4.4 Table of shock parameters used in the simulation 69 vi List of Figures 2.1 Schematic diagram of a solid target irradiated by laser radiation at high intensity 7 2.2 Schematic diagram of an ideal shock transition 10 2.3 Schematic diagram of Hugoniot curve (solid) for an initial state (Po, Vo, EQ) projected on the P — V plane 11 2.4 Cold curve of Si 21 2.5 Hugoniot curve of Si 22 2.6 Three dimensional EOS (pressure-temperature-density) diagram for Si. . 23 2.7 Three dimensional EOS (entropy-temperature-density) diagram for Si. . 24 2.8 Schematic diagram of the two zones 34 2.9 Calculated ablation pressure (solid line) and shock pressure (dash-dotted line) as a function of time 35 2.10 Calculated profiles before the shock break out (a) and after the shock break out (b) 36 3.11 Schematic diagram of the laser system 39 3.12 Contour plot of a typical laser focal spot intensity distribution 40 3.13 Experimental set up for absolute intensity calibration of the streak camera. 43 3.14 Exposure time r versus signal count C for A=430 nm, gain=5 46 3.15 Exposure time r versus signal count C for A = 570 nm, gain=5 47 3.16 Experimental set up for the luminescence measurement 50 3.17 Streak record of the shock emission and the fiducial signal 51 vii 3.18 Profiles of the shock emission and the laser fiducial 52 3.19 Shock trajectory 53 3.20 Shock trajectory 54 3.21 Shock trajectory 55 3.22 A streak image of the shock emission at A = 430nm and for Va = (2.0 ± 0.1) x 10 4m/5 58 3.23 A streak image of the shock emission at A = 570nra and for Vs = (2.0 ± 0.1) x 10 4 m/ 5 59 3.24 Temporal profile of the shock emission 60 3.25 Inverse of rise time versus shock speed for the 570 nm measurement. . . . 61 3.26 Snap-shots of the spatial profiles of the shock emission 63 3.27 Photon fluxes at three shock speeds for measurements at A = 430nm. . . 64 3.28 Photon fluxes at three shock speeds for measurements at A = 570nm. . . 65 4.29 Peak value in photon flux at A = 430ram 67 4.30 Peak value in photon flux at A = 570ram 68 4.31 Profiles of (a) electron temperature and (b) electron density during the unloading process 70 4.32 Calculated shock emission at (a) A = 430rara and (b) A = 570nra of Si shocked to 6.1 Mbar 75 4.33 Temporal profile of the data and the calculations for A = 430nm 78 4.34 Temporal profile of the data and the calculations for A = 570nra 79 4.35 A moving frame y is attached to the shock front at y=0 83 4.36 Saddle node and its stable and unstable manifolds 87 4.37 Qualitative analysis of Eqs. (4.139) 88 4.38 Numerical solution of (a) fi(y) and (b) 0 X - fx{y) 90 V l l l 4.39 Calculated photon flux for various g 92 4.40 Comparison of results of calculation with data at A = 430nra 93 4.41 Comparison of results of calculation with data at A = 570nm 94 4.42 Comparison of results of calculation with data at A = 430nm 95 4.43 Comparison of results of calculation with data at A = 570nm 96 4.44 Comparison of results of calculation with data at A = 430ram 97 4.45 Comparison of results of calculation with data at A = 570nm 98 4.46 Peak values in photon flux for A = 430nm 99 4.47 Peak values in photon flux for A = 570nm 100 ix Acknowledgement I would like to express my sincere gratitude to my research supervisor, Dr. Andrew Ng, for his support, his effort to obtain the resources required for our experiment, and his meticulous guidance throughout the courses of this work. I am especially grateful to Dr. Peter Celliers for his continuous help in performing the experiment and the computation, as well as for happily sharing his experimental and theoretical expertise throughout our cooperation. I would also like to thank Andrew Forsman for his time in cooperating the experiment, and Alan Cheuck for his technical support and great effort to obtain all the necessary supplies. The financial assistance of U B C Physics Department is also appreciated. Finally I would like to thank my wife Shuren for her understanding and patience which made my work all possible. x Chapter 1 Introduction 1.1 Research in shock wave and matter at high pressure Shock wave research has developed rapidly since 1950. It provided new opportunities to understand the properties of matter under extreme conditions in terms of high tem-perature and high pressure which are not accessible by other laboratory means. Today it has become a multi-displinary field which involves condensed matter physics, plasma physics, atomic and molecular physics, geophysics, chemistry and material sciences. The equation of state is the most important information on the thermodynamic prop-erties of matter at high pressure. The literature on high pressure equation of state studies is very extensive. General reviews of physics of matter at high energy density have been given by Caldirola and Knoepfel [1], and Zel'dovich and Raizer [2]. Recent research on thermodynamic properties in the pressure range below 10 Mbar is reviewed by Ross [3] with special emphasis on phenomenological models for the interpretation of shock wave data. Tabular equation of state data such as S E S A M E [4] is collected at several lab-oratories. A new quotidian equation of state (QEOS) is developed by More [6] et al. Extensive shock wave experiments have been carried out to verify theoretical predictions [5]. However the equation of state of many materials at high pressure is still uncertain due to the lack of data. Therefore more experimental measurements are needed to test theoretical estimates. Techniques to achieve high pressure can be separated into two categories, namely, 1 Chapter 1. Introduction 2 static pressure loading and dynamic pressure loading. Using a piston cylinder [7] and a multianvil apparatus [8], static pressure loading can generate a pressure form hundreds of Kbar to 5 Mbar [7, 9, 10]. Similar pressures can be obtained from dynamic pressure loading using explosive systems [11, 12], or two stage gas guns [13, 14], while higher pressure can be achieved with high power lasers [15, 16, 17, 18, 19, 20] and nuclear explosions [21]. Among them high power lasers provide a practicable way to achieve ultra high pressures (up to 10 tbar) in the laboratory with large control flexibility. The mechanism of pressure loading by a high power laser is discussed in Sec. 2.1. 1.2 Effort in determining the temperature of a shocked solid Traditional studies of the behavior of shock-compressed materials assess the mechanical response of these material to shock compression, for example, the change of density with pressure. Since this approach can not directly constrain the temperature of the high pressure state, other means are needed to provide experimental constraints on a complete equilibrium thermodynamic description (i.e. pressure-density-temperature) of these materials. Therefore direct temperature measurements of shocked materials are an important route to understand physical properties of matter under shocked conditions. The temperature behind the front of a shock wave in transparent material was first measured from optical pyrometry by Zel'dovich, Kormer, Sinitsyn and Kuryapin [22]. In these experiments the surface brightness of a shock front propagating through a transpar-ent material was observed. The surface brightness was converted to temperature under the assumptions that the heated region bounded by the shock front radiates as a black body. The measurements were made both in the red and blue regions of the spectrum so that not only the brightness temperature but also the color temperature could be deter-mined. The comparison of experimental results to temperatures estimated from theory Chapter 1. Introduction 3 showed reasonable agreement. However, the overwhelming majority of solids are opaque. It is extremely difficult to determine the temperature in opaque solids with the same method. The problem arises from the release of the free surface when the shock emerges and the extremely short duration in which the luminosity of the shock front is observed before the release. Take metal as an example, it is opaque even in very thin layers (less than 10~7 m). A shock with a velocity of 104 m/sec would travel through such a layer in less than 1 0 - 1 1 sec. Even if one had a detector with a temporal resolution of 10~ 1 2 sec to capture the luminescence signal before the free surface releases, it would still be almost impossible to ensure simultaneous emergences of the shock at different points of the free surface. To avoid the release of the free surface, Urtiew and Grover [26], Lysenga [27] and Bass et al. [28] used window materials to constrain the surface of the opaque material. In their experiments, radiation from the opaque solid was viewed through the transparent or semi-transparent window. However, due to the mismatch of impedance between the opaque solid and the window, this method is still subject to release problem, i.e., the brightness of the interface between the two different materials does not give the temperature of the shock front but the temperature of the released material. The shock temperature can then be calculated from the interface temperature provided the equations of state for both the opaque solid and the transparent window are known. This makes the temperature measurement dependent on the properties of the window material. An alternative method is to determine the shock temperature by measuring the lumi-nosity of the material under going release. The shock temperature can be inferred from the temperature of the vaporized material during unloading if the proper thermodynamic and atomic models of the gas are given. Such an experiment was performed by Ng et al. [29]. Besides equation of states, this method requires additional information such as ion-ization and thermal conductivity. The interpretation of the shock temperature strongly Chapter 1. Introduction 4 depends on the validity of all these models. Although today's technology still does not allow us to apply the direct optical mea-surement of temperature on highly opaque materials such as metals, the fast streak camera with resolution of a few picoseconds has yielded the ability to perform such mea-surement on large variety of solids with intermediate opacity such as semiconductors. 1.3 Present investigation The objective of this work was to assess the possibility of measuring the shock tempera-ture of a semiconductor before the shock arrives at the free surface and begins to release. Using laser-driven shock waves in Si targets and a fast streak camera (5 ps response time), we measured the shock-induced luminescence signal. As a semiconductor, Si has a band-to-band absorption edge at 1.1 eV which means that its opacity decreases rapidly from violet to red. The absorption coefficient at 430 nm is about 3.5 x 106 m~l while at 570 nm is 6 x 105 m"1. The surface of the Si wafer used in the experiment was polished to be optically smooth. With a time resolution of 20 ps, we were able to resolve the temporal history of the shock luminescence at 430 nm and 570 nm before shock release. The absolute response of the streak camera was calibrated at 430 nm and 570 nm using a tungsten filament. Numerical calculations were performed using the QEOS [6] equation of state, Lee and More's conductivity model [32], and a self-consistent model for emis-sion and absorption calculations. Large discrepancies were observed between theoretical prediction and data in terms of the temporal shape and the absolute intensity of the lumi-nescence signal. An electron-phonon thermal relaxation model was introduced to explain the disagreement. By assuming the value of the electron-phonon coupling constant g to be (1 ~ 2) x 1017W/m~3K~1, the model yielded satisfactory results compared with data. Similar values of g have been reported in many metals. A first-principle calculation of Chapter 1. Introduction 5 g for shock compressed Si would be needed to elucidate electron-phonon relaxation pro-cesses in the shocked materials. On the other hand, this thermal relaxation would render direct measurement of Hugoniot temperatures unattainable in metals or other materials which are metallic under the shock pressure . 1.4 Thesis outline A brief introduction to high pressure physics and to shock induced luminescence studies has been given here. Chapter 2 introduces the basics of shock wave theory, the equation of state model QEOS, and the numerical simulation code LTC. Chapter 3 presents details of the experiment including the introduction to the laser facility, calibration of the streak camera, and the luminescence measurement. The luminescence measurement yields both shock speed and the luminescence intensity. In Chapter 4 we compare the data with the simulations. Detail of the luminescence calculation is given. A review of electron-phonon relaxation studies is given, and finally a relaxation-diffusion model is proposed. Chapter 5 gives the summary of this report and the discussion of the future work. Chapter 2 Shock wave theory and numerical simulation 2.1 Laser-driven ablation The process of laser-driven ablation is shown schematically in Fig. 2.1. The laser is focused onto the target surface with irradiances of typically 10 1 6-10 1 9 W/m2. Heated by the laser radiation through inverse bremsstrahlung absorption, A thin layer of the target is vaporized and highly ionized, resulting in the formation of a coronal plasma. Most of the laser energy is absorbed in the plasma where the electron density is lower than the critical density which is defined (in M K S units) by the following relation [30] nCT = — (2.1) where u>£ is the (angular) frequency of the laser light, e0 is the permittivity of free space, and me and e are, respectively, the electron mass and charge. The laser light can not penetrate the critical layer but is either reflected or absorbed. The bremsstralung absorption coefficient KV is given by (in M.K.S . units) [31] - = 5 < 5 ^ p | ^ * * <"> where h, k are Plank and Boltzmann constants respectively; T is the electron tempera-ture; N+,Ne are the particle density of ions and electrons respectively. The absorbed laser energy heats the coronal plasma to temperatures of the order of KeV. Consequently, large temperature and density gradients exist between TI c t layer and the cold solid. This results in a large thermal flux in this region known as the ablation 6 Chapter 2. Shock wave theory and numerical simulation 7 CORONAL PLASMA FOIL ^ lO^ 'W/cm LASER ^> XL<1^ jm SHOCK COMPRESSED SOLID • SHOCK FRONT ABLATION FRONT ABLATION ZONE Figure 2.1: Schematic diagram of a solid target irradiated by laser radiation at high intensity. Chapter 2. Shock wave theory and numerical simulation 8 zone. The cold material in this region is ablated and expelled out into the surrounding vacuum due to thermal expansion. The point where the ablation zone joins the cold material is called the ablation front. As a consequence of momentum conservation, the expanding material drives a strong shock into the target. The amplitude of the shock wave is determined by the ablation pressure which is in turn determined by the laser irradiance. The shock wave propagates much faster than the ablation front, therefore the amount of compressed mass is much more than that of the ablated mass. 2.2 Rankine - Hugoniot relation The thermodynamic states of a material under shock compression are uniquely deter-mined by the equation of state. The equation of state (or EOS) is usually represented by a two dimensional surface in the three dimensional phase space of pressure, density and temperature (or internal energy). A reversible thermodynamic process can be described by a continuous path on the EOS surface connecting the initial and final states. On the other hand, the irreversible shock process is generally described as a jump from the initial state to the shocked state on the EOS surface. The changes occurring at the shock front results from a nonequilibrium process which can not be described by an equilibrium EOS. Given the initial state, the final state is governed by the Rankine-Hugoniot rela-tion, or simply the Hugoniot, which results from the conservation of mass, momentum and energy across the shock front. These constraints are: V_ Vo D-u D (2.3) Du (2.4) P-Po = V0 E-E0 = ±(P + P0)(V0-V) (2.5) Chapter 2. Shock wave theory and numerical simulation 9 Here P0, V0, E0 (P,V,E) are the initial (shocked) pressure, specific volume and internal energy per unit mass respectively. D is the shock speed and u is the particle speed behind the shock front measured in the laboratory frame in which the particle speed ahead of shock front is zero. Such an ideal shock transition is illustrated in Fig. 2.2 There are five parameters associated with the shocked state: D,u,P,V and E. We have three conservation equations (2.3) - (2.5), and the equation of state P = P{V,E). (2.6) Any one of these five parameters can then be expressed as a function of another param-eter. This yields a curve on the EOS surface which gives the locus of points reached by single-shock compression known as the Hugoniot curve. A schematic diagram of the Hugoniot curve projected onto the P — V plane is shown in Fig. 2.3. The isothermal and the isentrope passing through the initial state (Po, Vo, E0) are also shown. Note that the three different routes of compressing the material from Vo to V fall into three categories: dS < 0 through the isothermal path; dS = 0 through the isentropic path; and dS > 0 through the shock process where dS is the change in entropy. Entropy increases during shock compression simply follow from the fact that shock is an adiabatic and irreversible process. The Hugoniot is always tangential to the isentrope at the initial point (Po, V0, Eo). This can be easily seen by considering an infinitesimal shock with P = P0 + dP. It follows from Eq. (2.5) that as dP -> 0, dE = P0dV which is exactly the isentropic process, where dS — 0. Projected on the P — V plane, a straight fine drawn between (P0, V0, E0) and (P, V, E) has a slope which is proportional to D2. This can be shown by combining Eq. (2.3) and Eq. (2.4) which give P-P0 = ^2(Vo-V). (2.7) Chapter 2. Shock wave theory and numerical simulation P.V.E u D X Figure 2.2: Schematic diagram of an ideal shock transition. Chapter 2. Shock wave theory and numerical simulation 11 VOLUME Figure 2.3: Schematic diagram of Hugoniot curve (solid) for an initial state (P0,V0,E0) projected on the P — V plane. Also included in the diagram are the isentrope (dashed) and isotherm (dash-dot) passing through the initial state. (P, V, E) is an arbitrary shocked state. The Rayleigh line (dotted) is also shown. Chapter 2. Shock wave theory and numerical simulation 12 This is called the Rayleigh line which provides a graphic interpretation of the relationship between shock pressure, compression and shock speed. In the case of a solid, the response of the material is usually more complicated than that by simple hydrostatic compression especially when the shock pressure is low. The deformation of the solid is described by a stress-strain tensor. In the limit of a strong shock the shear stress is negligible and the deformation is nearly isotopic. When the solid undergoes a phase transformation, the Hugoniot curve will show a cusp at the transition pressure. Detailed reviews of the shock wave process and shock-induced phase transformations can be found in the works of Zel'dovich and Raizer [31], and Duvall and Graham [33]. 2.3 QEOS A new quotidian equation of state (QEOS) was developed by More et al. [6]. It is a gen-eral purpose equation of state model for use in hydrodynamic simulations of high-pressure phenomena. QEOS is a self contained theoretical model that requires no external data base. The inputs are simply the material compositions and properties of the cold solid (its density, bulk modulus or sound speed). Therefore, in principle, it can be appUed to any material given this information. The outputs of the code include pressure, en-ergy, entropy, Helmholtz free energy and thermodynamic derivatives (e.g. dp/dp, dp/dT , etc.). In this work we used QEOS to calculate the Hugoniot of Si and to simulate the hydrodynamic motion driven by a shock wave. 2.3.1 QEOS structure QEOS adopts the usual assumption [31, 34] that, to a sufficient degree of accuracy, electron and ion contributions can be considered additive. Accordingly, the Helmholtz Chapter 2. Shock wave theory and numerical simulation 13 free energy per unit mass is F(p, T e , J-) = typ, Ti) + Fe(P, Te) + Fb(p). (2.8) Here Fi is the ion free energy calculated from formulas in Sec. 2.3.2, from which an ideal gas law at high temperature (T 3> T m e / t ) and a Debye-Griineisen law in the solid phase (T < Tmeit) can be derived. Fe is the semiclassical electron free energy obtained from the spherical-cell Thomas-Fermi theory, including ionization and core-repulsion effects (Sec. 2.3.3). Fb is a semi-empirical correction for chemical bonding effects in the solid phase which is independent of temperature. Fb can also represent identical-particle-exchange or other quantum effects. The pressure p, entropy S and energy E are obtained from the free energy (2.8) according to the following equations: BF BF P e = p 2 ^ , Se=-%±, Ee=Fe+TeSe, (2.9) Op Ole BF- BF-Pi=P2^, S< = -W> E{ = F{ + TiSi, (2.10) P b = p 2 ^ , Se = - ^ = 0, Eb = Fb. (2.11) F and E are energies per gram and p is the mass density (g/cm2). The total pressure, entropy and energy are given by the sums of the various contributions. The complete EOS is thermodynamically consistent because the individual components are thermody-namically consistent. For example, for the electron component, (2.12) Chapter 2. Shock wave theory and numerical simulation 14 Fe is given by a tabular representation of the Thomas-Fermi free energy which exactly satisfies this equation. Fi and Fb are derived from analytic formulas which automatically obey the consistency condition. In order to deal with the cold solid phase with reasonable accuracy, the following conditions are required: (1) zero total pressure at the correct normal-density pa, (2) approximately correct density jump for the shock wave starting from the cold solid (i.e. a correct shock speed), and (3) smooth variation of pressure and energy between the cold solid and the high-pressure plasma phase. 2.3.2 Ion EOS In QEOS the physical properties of ions are treated completely independent of the elec-trons except for the constraint that the charge densities be equal. Using Cowan's ion equation of state model [35], Fi is written as kT Fi{p>T) = AMpf{u>w)- ( 2 - 1 3 ) AMP is the mass per atom, u and w are scaling variables defined by ®D{P) u = T ' Tm w = ~^r-T where &D is Debye temperature and T m (p) is the melting temperature. The solid phase corresponds to w > 1 and the region where w < 1 is called the fluid phase. The following functional forms of / are assumed for different phases: (1) fluid: w < 1 Chapter 2. Shock wave theory and numerical simulation 15 / = - y + V / 3 + ' l o g ^ , (2.14) 1 1 I w (2) high-temperature solid: w > l,u < 3 / = - l + 31og« + ^ - ^ l (2.15) (3) low-temperature solid: w > l,u > 3 + (2 ,6 , Eqs. (2.14) - (2.15) are determined to satisfy the following well known physical laws or relations: (1) ideal gas law, (2) fluid scaling law [36, 37, 38], (3) the Lindemann's melting law [39], (4) the Dulong-Petit's law [39], (5) the Griineisen's pressure law [39], (6) the Debye lattice theory, and (7) the third law of thermodynamics. Detailed discussions of these can be found in More's work [6]. In order to determine 0 ^ and T m , a "reference density" is defined as * - / = 9^s (5/<™3)> (2-17) QD and T m are given by 1 68 £ 6 + 2 kQD = • * >x, (eV), (2.18) D Z + 22 (1 + 0 2 £ 2 6 + 1 0 / 3 kTm = 0 .325__^- (eV). (2.19) Here, Z is the atomic number, b — 0.6Z1/9 and £ = p/pref. Chapter 2. Shock wave theory and numerical simulation 16 2.3.3 Electron EOS Feynman, Metropolis, and Teller [2] formulated a Thomas-Fermi (TF) theory of hot dense plasmas in terms of the spherical cell model for compressed atoms, a physical description which has proven extremely fruitful. In QEOS this original T F cell model is used with improved numerical methods but without exchange or other corrections. In the T F theory the electrons are treated as a charged fluid surrounding the nucleus. The properties of this electron gas are obtained from finite temperature Fermi-Dirac statistics. Effect of the plasma environment are introduced by the ion-sphere model. Each nucleus is located at the center of a spherical cavity of radius R0 = (3/47rni)1/'3 and the cavity contains enough electrons to be electrically neutral while other ions are assumed to remain outside the sphere. Within the ion sphere an electrostatic potential V(r) is calculated by solving the Poisson equation A V = 4xen(r) - ±irZe8(r). (2.20) Here 8(r) is the delta function defined by /oo 8{r)dr = 1, (2.21) -oo 6(r) = 0 for r ± 0, (2.22) and n(r) is the total electron number density for a finite-temperature semiclassical elec-tron gas which is given by ^ ^ / ^ / M ' M ^ M - ^ ) , (2 23) where Chapter 2. Shock wave theory and numerical simulation 17 tl \ M L <V2l2rn- eV(r)-p ! . . f(r,p) = [1 + exp( — )J , (2.24) C l = (l/27r 2)(2m/^ 2) 2 / 3 , (2.25) and r°° xudx FM = / — r — ^ • (2.26) Jo 1 + exp(a; + y) Here f(r,p) is Fermi-Dirac distribution function. The electron chemical potential \i is determined by the requirement that the cell be neutral, i.e., J n(r)dsr = Z. (2.27) The potential V(r) is self consistent in the sense that it must simultaneously satisfy both Eq. (2.20) and (2.23). In order to construct the thermodynamic functions, the kinetic and electrostatic (including electron-nucleus and electron-electron) energies are respectively given by K = Cl{kT)'" j M - ^ y ^ A (2-28) Um = -f ^n{r)d\ (2.29V U ^ I ^ ^ M r ' . (2.30) 2 J \r — r'\ The internal energy, Helmholtz free energy, and entropy per gram are respectively Ee = (K + Uen + Uee)/AMP, (2.31) Chapter 2. Shock wave theory and numerical simulation 18 Fe = {ZIL ~\K- Uee)/AMP, (2.32) Se = {^K - -Zu + Um + 2Uee)/(AMpT), (2.33) and the pressure is pe = ^Cl(kTf2F3/2(-u./kT). (2.34) The ionization state Z {p,T) is taken to be ~ ATT Z= - j R X R o ) - (2-35) 2.3.4 Correction for chemical bonding The TF model gives a positive pressure produced by the electron gas at T ^ 0. To insure zero pressure for cold matter at normal density, a semi-empirical bonding correction is given by Eb = E0{1- exp 6[1 - (pJPf/3}}. (2.36) where pa is the normal solid density and E0 and b are positive numbers which characterize the chemical bond strength and range. The bonding correction to the pressure is Pb idEb •\E0bp. 3 \ps 2 / 3 exp b 1_ i'l P 1/31 (2.37) (2.38) Chapter 2. Shock wave theory and numerical simulation 19 The parameters Eo and b are determined by requiring that the total pressure and bulk modulus be correct for the initial cold solid. The experimental bulk modulus is required to be B = p{^bfL) =p{^ + Pj)-{b + 2)ET1- (2-39) This equation, together with the condition that Vtot«i(Ps,0) = 0 (2.40) will determine E0 and b. The above treatment is sufficient as far as the condensed phase is concerned. In order to apply the model to the gas phase, the bonding energy has to be modified such that it yields the correct vaporization energy. In this work the bonding energy was treated separately for the compressed (p > p3) and the expanded (p < p0 ) states. The energy given by Eq. (2.36) was only used for compressed states. For the expanded states, another form of bonding correction was used which yields a bonding pressure of pb = B0(l + yf2{a + y- 2{3y2 + 4 7 y 3 ) , (2.41) where 1 (2.42) and only —1/2 < y < 0 is considered (corresponding to 0 < p < po )• B0, a, /3, and 7 are modeling parameters constrained by both Eq. (2.39) and Eq. (2.40) together with PtotaldV = QV (2.43) Chapter 2. Shock wave theory and numerical simulation 20 where V = 1/p and Qv is the vaporization energy. The details of these are discussed in the work of Celliers [40]. 2.3.5 Equation of state of silicon QEOS is constructed to describe a solid at high temperature and high pressure, which behaves like a hot dense plasma. By imposing zero pressure at normal condition one can extend QEOS to describe cold material approximately. However, when the cold material undergoes a phase-transition especially with a volume collapse, the single group of input parameters (density and bulk modulus at room temperature) is not sufficient to characterize the cold solid. Therefore piecewise construction of the isotherm at room temperature (cold curve) is needed to obtain reasonable prediction in the low temperature range. The calculation of equation of state is then based on the cold curve which is governed by the density, bulk modulus, and the pressure at the phase-transition region for each phase. A rich variety of phase-transitions in Si have been reported since the work of M i -nomura and Drickamer [41]. In the range of 0 ~ 0.5 Mbar, Si shows the following sequence of phase changes: cubic diamond to body centered tetragonal (B-Sn) at 0.11 Mbar [42, 43, 44], 3 - Sn to simple hexagonal (ah) at 0.13 ~ 0.16 Mbar [45, 46], sh to an intermediate phase at 0.34 Mbar [45], then to hexagonal-close-packed (hep) above 0.4 Mbar, and finally hep to face centered cubic (fee) at 0.78 Mbar [47]. As far as QEOS modeling is concerned, the transition from cubic diamond to 3-Sn is the most prominent transition in terms of density change. Therefore a two phase construction of the cold curve was performed by fitting the curve to the data from Olijnyk et al. [45] and the data from Duclos et al. [47]. Fig. 2.4 shows the experimental data and the fitted curve. The phase transition pressure was taken from the work of Olijnyk; the density and the bulk modulus of the Chapter 2. Shock wave theory and numerical simulation 21 Figure 2.4: Cold curve of Si. Shown in the figure are the experimental data from Olijnyk et al. (open circle) and Duclos et al. (solid circle), The cold curve fitted on these data is indicated by solid line. The calculated cold curve form the Hugoniot data (see next page) is indicated by dashed line. Chapter 2. Shock wave theory and numerical simulation 22 Figure 2.5: Hugoniot curve of.Si. Shown in the figure are the experimental data from Gust and Royces (circle), and Pavlovskii (triangle). The Hugoniot curve fitted to these data is indicated by the dashed line. The calculated Hugoniot based on the cold data (see last page) is indicated by the solid line. Chapter 2. Shock wave theory anH „ • , r y a n d ™™encal simulation 23 Figure 2.6: Three dimensional EOS ( p r e s s u r e t e ^ ipressure-temperature-density) di agram for Si Chapter 2. Shock wave theory and numerical simulation 24 Figure 2.7: Three dimensional EOS (entropy-temperature-density) diagram for Si. Chapter 2. Shock wave theory and numerical simulation 25 higher pressure phase was chosen to yield the optimal fit. At room temperature To, the volume is generally a function of pressure V(p, To) which can be Taylor expanded at p=0: V on p, the higher order terms should be included in a polynomial fit. Fig. 2.6 shows the pressure as a function of temperature and density based on the cold curve constructed above. The entropy as a function of temperature and density is shown in Fig. 2.7. There are two ways to construct the Hugoniot curve. In the first method, the Hugo-niot curve of Si is calculated using the equation of state based on the cold curve data. Fig. 2.5 shows the calculated Hugoniot curve (solid line) compared with the experimental Hugoniot data form Gust and Royces [48] and from Pavlovskii [49]. The data suggest a higher phase transition pressure. In addition the curve in the high density phase deviates slightly from the data. In the second method, one can obtain directly the Hugoniot curve by fitting to the Hugoniot data. The cold curve can thus be calculated from the fitted Hugoniot curve. Such a calculated cold curve has been included in Fig 2.4 (dashed line). The comparison of these two cold curves indicates that the difference between them is not very large. The Hugoniot curves yielded from these two methods also show little difference. We used both methods in the hydrodynamic simulation to test the sensitivity of the hydrodynamic variables to the different Hugoniots. The results showed very little differences, which indicates that the cold curve and the Hugoniot constructed through either method is acceptable. The bulk modulus is — ( V(P,To) = V(0,To) 1 + v(o,r0) dp 1 dV(p,T0) V(0,To) dp i Chapter 2. Shock wave theory and numerical simulation 26 2.4 Numerical simulation Numerical simulations were performed on shock compression of silicon with the hydrody-namic code LTC (Laser Target Code) which was developed at UBC by Celliers [50]. LTC was adapted from the laser fusion code M E D U S A [51] with substantial modifications. A complete description of the code was given in [50]. This section presents a brief outline of the physics contents and the numerical techniques, and the treatment of the equation of state using QEOS. 2.4.1 Physics contents in L T C LTC solves the one-dimensional fluid equations in a Lagrangian formalism. The target material is regarded as a compressible fluid composed of electrons and ions with temper-atures which may in general be different. In addition a radiation field with a separate radiation temperature interacts with the material through the Rosseland and Planck opacities. In the simpler "standard" description the electron and ion temperature are treated as equal and no radiation transport is considered. Lagrangian formalism means that the calculation follows the time evolution of indi-vidual fluid elements. A Lagrangian coordinate m at laboratory coordinate r and time t is defined by . m(x,t) = [" p(x',t)dx' (2.45) JXi (t) where X{(t) is the position of the left end of the target and p(x,t) is the density profile. The target is characterized by the following variables which are functions of the lagrangian coordinate m and t. Variables Definitions Units Chapter 2. Shock wave theory and numerical simulation V specific volume m2/kg u fluid velocity m/s Pe electron pressure J/m3 = Pi ion pressure J/m3 = Ee electron internal energy J/kg Ei ion internal energy J/kg U . radiation energy J/kg Te electron temperature K Ti ion temperature K Tu radiation temperature K F heat flux W/m2 tip Planck opacity m2/kg X Rosseland opacity m2/kg These variables are governed by a set of differential equations representing tion of mass, momentum and energy: Chapter 2. Shock wave theory and numerical simulation 28 J > M + ^ f = - f £ W 4 < < - n (2.50) e-,BF BfU_!* dt dm c and the equation of state Ee = Ee{V,Te), Pe = Pe(V,Te), (2.52) Ei^EMTi), Pi = Pi(V,Ti). (2.53) Eq. (2.46) and (2.47) represent fluid continuity and momentum conservation respec-tively; Eq. (2.48) and (2.49) represent conservation of energy for the electron and ion fluids. The additional terms Qe and Qi in Eq. (2.48) and (2.49) are heat sources due to the various transport and deposition processes which are described by QE = HE + Xei + ALAE - CKP{AOT* - U), (2.54) Qi = HI-XEI, • (2.55) Here, c is the speed of light, cr is the Stefan-Boltzmann constant, He and Hi are the electron and ion heat flow respectively; A\AE is the laser energy absorbed, and Xei is the energy exchange between electrons and ions. The electron and ion heat flow terms are given by p (2.56) Chapter 2. Shock wave theory and numerical simulation 29 Hi = -S7-KiVTi. (2.57) P where Ke and K ; are the thermal conductivities for electrons and ions respectively. The electron thermal conductivity can be specified in a tabular form such as that provided by the SESAME library. Spitzer [52] conductivity is used if such a table is not available. The Spitzer form of conductivity is given by 715/ 2 Ke = 1.955 x 10-9eg t ' t , (2.58) Z log A where Z 4- 0 24 eSt = 0.095— :7-^7 7r, (2.59) 1 + 0.24Z' v ' log A = 16.34 - l o g ( T e 3 / V 1 / 2 ^ _ 1 ) . (2-60) Zp ne = —. Here AM is the atomic mass, Z is the charge state of the ions. The thermal conductivity of the ions is given by a similar expression [51] (2.61) The electron ion energy exchange Xei is given by [51] Xei = 0.59 x lQ-8ne(Ti - Te)T~3/2Aj}Z2 log A. (2.63) Chapter 2. Shock wave theory and numerical simulation 30 The mechanism of laser energy deposition is assumed to be inverse bremsstrahlung absorption [51] up to the critical density given by Eq. (2.1). The absorption coefficient, a is 13.51 3* Z l o g A  a ~ A ' ( l - / 3 2 ) i / 2 T 3 / 2 V-M) where 3 = ne/nCT < 1. The laser flux at the position r, $; a a(r, £), is related to the laser flux at the boundary Ro by $ia,{r,t) = $la,(R0,t) exp(— / a(r',t)dr') (2.65) The laser power absorbed by a cell of size A r and mass A m at r is Alas(r,t) = $ t a .(r ,0 V K A m (2.66) The last term CKv(\(TT^ — U) in Eq. (2.54) represents the coupling of the electrons to the radiation field. One can define a radiation temperature TT from U by T,= ( - ) . (2.67) which is not necessarily the same as the kinetic temperatures of the electrons or ions. In the single temperature description the electrons and ions are characterized by the same temperature T. Accordingly the fluid is described by a single internal energy E which yields the pressure p and the temperature T. P = P(T, V), E = E(T, V), (2.68) and Eqs. (2.48), (2.49) are replaced by a single equation: ajp A T / Chapter 2. Shock wave theory and numerical simulation 31 In calculations neglecting radiation transport, Eq. (2.50) and (2.51) are ignored and the radiation coupling term CKp(AaTe1^4 — U) is set to be zero. 2.4.2 Numer ica l method in L T C In LTC the Lagrangian mesh consists of a set of cells of mass Arrij identified by j. At the beginning of the nth time step, the average values of U,V,Ee, and U are known for each cell, and F and r are known at each cell boundary. A l l other quantities are expressed as functions of these variables and are therefore known. The solution to Eq. (2.46) - (2.53) is obtained in two stages for each time step. In the first stage the hydrodynamic motion of the cell is calculated by advancing the coordinate r and velocity u in an explicit hydrodynamic scheme. This is done by solving a simplified version of the system of equations given in Eq. (2.46) - (2.49): dV du n « " 5^ = 0 <2 7 0 ) du dp , § + ^ = 0 (2.72) dt dm Here e = Ee + Ei + u2/2 and p = Pe + P{. Note that the heat source terms Qe and Qi have been dropped, hence only adiabatic motion is described here. The method of solving Eqs. (2.70) - (2.72) follows the piecewise paraboHc method (PPM), a scheme developed by Colella and Woodward [53]. The solution yields the variables r (hence V and p) and u for the next time step. As Eqs. (2.70) - (2.72) do not include the heat source terms, another stage of calculation is needed. Chapter 2. Shock wave theory and numerical simulation 32 In this stage, the goal is to solve the energy equations (2.49) - (2.53) self-consistently by including all of the temperature and density dependent source terms. Both the electrons and ions are treated identically in the numerical method, therefore we will not distinguish them in the following discussion. The energy equations are written in the form dT dp dV Here Cv and BT are defined by the equation of state, dE dE C v ~ d f (2.74) Replacing the term pdV/dt by —dE j dt\hya-TO and rewriting H as p~ldf dr(ndT / dt), a dp rearrangement of two sides of Eq (2.73) gives: dt p dt dr w here dp dE hydro The term (Q — H) stands for the heat flow in addition to thermal conduction (laser heating for instance). To solve the heat equation the Crank-Nicholson time centered differencing scheme is used as in M E D U S A [51], which yields a tridiagonal system in the unknown T which is solved using Gauss elimination. Note that the left hand side of Eq. (2.75) contains the derivatives of T while the right hand side contains only functions of T. An iteration of T can thus be performed to solve the equation. The heat conduction H and the internal energy E is evaluated in the iteration until a convergence of E has reached. Chapter 2. Shock wave theory and numerical simulation 33 The radiation diffusion equations are solved using the method of Mihalas [54]. Eq. (2.51) is rewritten as an iteration eqation: Fn+l = -pc\T^- + ^Fn (2.77) om At Here Fn and Fn+i are the values of F at the iteration steps n and n+1 respectively, and A r is given by A ; 1 = A " 1 + P X , Xt = cAt. Eq. (2.77) is substituted into Eq. (2.50) to obtain an expression for Z7 n + 1 . Differen-tiation of the resulting expression gives a tridiagonal system which is solved by Gauss elimination. A boundary condition of F(X0,t) — cU(X0,t) has to be satisfied by the solution. The self-consistent solution is obtained when the convergence of U is reached. 2.4.3 Simulations for shock compressed silicon As an example, the numerical code was used to examine the process of shock compression in a Si target. The laser pulse was assumed to have a Gaussian shape with 2.7 ns F W H M and peak irradiance of 4.7 X 1013W/cm2. The Si target was taken to be 90um thick and a mesh of 200 cells was used. The mesh was more finely zoned near the front surface where the laser light was absorbed and more coarsely zoned deeper into the target where compression took place. This was done by dividing the mesh into two regions as shown in Fig. 2.8. The finer zone contained 100 cells which covered a depth of 5.095776/tm, and the ratio of adjacent cell masses was chosen to be 0.950054. The coarser zone contained 100 uniform cells which covered the rest of the target. The geometric ratio of the cell masses and the thicknesses of the two zones were chosen such that the cell masses across the interface of the two zones were equal and the smallest cell in the finer zone was Chapter 2. Shock wave theory and numerical simulation 34 REGION 2 REGION 1 100 CELLS 100 CELLS LASER Figure 2.8: Schematic diagram of the two zones. Chapter 2. Shock wave theory and numerical simulation 35 800.0 Figure 2.9: Calculated ablation pressure (solid line) and shock pressure (dash-dotted line) as a function of time. Also shown is the laser intensity (dotted line). Time zero is taken to be the time of the peak laser intensity. cr P p i 3 cn o_ ST 3 CD * to p 3 CD 3 "> S. P ET: cn 03 cn £ | D-P P rt- cr 3* n to _ CD P CD o ^ 3 O . O i-rj 3 _^  0 cr CD to £ ° " O P p CD cn *0 I-I O 22 3 CD cr CD 3" CD 3. § era CD CT CD cn cr o n 2 ^ CD fD a >-> ^ P> - • r—t-> O v 3 cr CD cn cr o o P < to CD p o p 3 3 P < to O T) P OP P - i P o s> id p OP p H T ) CD cr CD CD P cn era CD l!L cT cr cr CD cn cr o 0! P DENSITY Kg/M**3 DENSITY Kg/M**3 "103 q . q , q . q » qo q , q , TEMPERATURE K TEMPERATURE K Chapter 2. Shock wave theory and numerical simulation 37 0.003/im. The reason for using very fine zones at the laser absorbing region is to reduce the numerical noise on the shock profile. Fig 2.9 shows the ablation pressure, pressure at the shock front and the laser power as a function of time. The shock front is not well defined until the spatial profile of pressure shows a sharp jump. The ablation pressure is higher than the shock pressure when the laser intensity increases, and it drops faster than the shock pressure when the laser intensity decreases. Two snap shots of the profiles of pressure, density and temperature are shown in Fig. 2.10. Fig. 2.10(a) shows the profiles before the shock front reaches the rear surface of the target and Fig. 2.10(b) shows the profiles when the release has taken place. Three regions can be distinguished in the figure. In Fig. 2.10(a), they are (i) the cold solid which is to the left of the shock front; (ii) compressed material which is between the shock front and the ablation front; (iii) ablation region and coronal plasma which is to the right of the ablation front. In Fig. 2.10(b) the cold solid region no longer exists. Instead there is the rarefaction region which is to the left of the rarefaction front. The rarefaction wave propagates in the opposite direction to the shock wave but with a faster speed. Once the release starts, the temperature at the rear surface drops so rapidly that it reduces to one half of the temperature of the shock front within a picosecond. Chapter 3 The experiment 3.1 Experimental facility 3.1.1 Laser facility Schematic diagram of the laser facility is shown in figure 3.11. It consists of a Quantel solid-state laser chain, a second harmonic conversion crystal, beam steering optics, and associated beam monitoring diagnostics for recording the incident laser pulse shape and energy. The laser chain begins with a Nd-YAG (neodymium-yttrium aluminum garnet) oscillator which is passively Q-switched by a dye cell to produce a Gaussian seed pulse with a duration of 2.7ns (FWHM) at a wavelength of 1.064 am in a TEM0o mode. The horizontally polarized seed pulse is amplified by a Nd-YAG preamplifier and four Nd-glass amplifiers. Six vacuum spatial filters are present in the chain to reduce nonuniformities in the beam. In addition two pairs of polarizer are used to remove the depolarized radiation produced by thermal birefringence in the first two Nd-glass amplifiers. The IUJ (1.064 fim) output beam with pulse energies up to 30 J is converted to 2u> (0.532/im) radiation using a potassium dihydrogen phosphate (KDP) type II crystal. The maximum energy of the 2a; pulse is typically 15 J . It is steered into the target chamber by a series of 5 dichroic mirrors. In this experiment, a low energy beam was split from the third dichroic mirror and steered into the streak camera as a timing mark (fiducial). 3.1.2 Irradiation conditions 38 Chapter 3. The experiment 39 OSC /f-LXr—H VSF v PA DM y A \ DM VSF VSF VSF - T ^ ^ - ^ A T J ^ V C ^ | A 2 | / M X H s PI P2 P3 P4 SHG-A3 VSF -LX3- A 4 VSF \ >- FIDUCIAL TYPICAL LASER PULSE PHOTO DIODE ENPRGY METER Figure 3.11: Schematic diagram of the laser system. OSC: oscillator; PA: preamplifier; A1-A4: amplifiers; VSF: vacuum spatial filter; P1-P4: polarizer; DM: dichroic mirror; ND: neutral density filter. Also shown is a typical laser pulse shape. Chapter 3. The experiment 40 Figure 3.12: Contour plot of a typical laser focal spot intensity distribution. Chapter 3. The experiment 41 The laser was focussed onto the target with a f/2.2 lens (focal length 110mm). The focal spot size and the focused intensity distribution at the target plane were measured for a range of lens positions. This was done by imaging the target plane onto either a Polaroid camera (for spot size measurements) or an optical streak camera (for intensity distribution measurement). The streak camera was operated in focus mode and hence was functioning essentially as a video camera. The overall magnification of the imaging optics and streak camera system was found by placing a grid with known spacing at the target plane and comparing it to the camera image. The magnification of the imaging optics for the Polaroid camera was approximately 80. Each pixel on the streak camera corresponded to 3.4 micron on the target plane. Fig. 3.12 shows the time integrated intensity distribution of a typical laser focal spot. The spatial resolution of this measurement was approximately 10pm. The focal spot appeared to be confined in a region of approximately 150pm in diameter. The significant spatial intensity modulation indicated that local irradiances at some "hot spots" were much higher than the average irradiance over the entire laser spot. In simulating the experiment with a one-dimensional hydro code it is necessary to chose an effective laser irradiance as an input parameter. This presents some difficulty. The laser intensity distribution at the target plane is not uniform. The usual procedure is to characterize the irradiance by an average irradiance that represents the average irradiance over an area containing 90 percent of the laser energy. This may or may not yield a satisfactory effective irradiance particularly if plasma effects, such as lateral thermal transport or laser beam self-focusing, become important. We define an average irradiance $ a as following: *• = Tlh <3'78> Here EL is the incident laser pulse energy, TL is F W H W of the incident laser pulse, Chapter 3. The experiment 42 ND 0.0 0.3 0.6 Average irradiance(1017W/m2) 3.3 ± 0.1 1.7 ± 0.2 0.8 ± 0 . 0 1 Table 3.1: Table of average irradiances A is the area inside the contour corresponding to 10% of the maximum laser intensity. In this experiment the irradiance was varied by placing different neutral density (ND) filters in the path of the main laser beam. Table 3.1 shows the ND values of the filters and the corresponding average irradiances. 3.2 Calibration of the streak camera A Hamamatsu streak camera model C1370 is used to record the shock-induced lumines-cence. In order to evaluate the brightness temperature of the target, the streak camera was first calibrated for absolute response. The experimental set up is given in Fig. 3.13. A tungsten filament of 0.5mm diameter was placed at the target plane. A portion of the tungsten surface was imaged through a 50mm f/1.4 lens and a 600mm f/11 achromat onto the streak camera entrance slit with a magnification of 12. The tungsten filament was heated by an electric current to a temperature of 2600K which was measured using an optical pyrometer. The response of the camera at wavelengths of 430 nm and 570 nm was obtained by recording the tungsten emission through lOnm-band-pass filters centered at the appropriate wavelengths. The streak camera was operated in a streak mode and measurements were made for various combinations of sweep speed, entrance slit width and gain settings. The streak camera signal in photoelectron counts is determined by the following factors: (A) source of emission: brightness at a certain wavelength; Chapter 3. The experiment 43 VACUUM CHAMBER PYROMETER WINDOW STREAK CHAMBER TUNGSTEN FILAMENT f/1.4 LENS INTERFERENCE v FILTER WINDOW > 600mm F.L. ACHROMAT CAMERA SLIT MECHANICAL SHUTTER Figure 3.13: Experimental set up for absolute intensity calibration of the streak camera. Chapter 3. The experiment 44 (B) optics: overall magnification, effective aperture (f-number), and transmission; (C) streak camera settings: streak speed, entrance slit width and gain. In this experiment the magnification and f-number were fixed and only two different filters were used (centered at 430 nm and 570 nm). The objective of the calibration was to measure the camera output signal for different camera settings at these two wavelengths for a known source brightness, so as to obtain a formula relating the photon flux collected by the optics to the camera signal count for various camera settings and wavelengths. In the calibration, the brightness of the tungsten filament in terms of photon flux was given by Planck's gray body radiation function, nXc = —[expl — ) - - coM), (3-79) where n\c is the number of photons emitted per unit area, unit time and unit wavelength interval in a cone angle 9 which corresponds to the collection angle of the 50 mm, f/1.4 lens which is related to the f-number by tanO = 7~~r ~ ; — v , (3.80) 2 X ( / — number) where A is the wavelength, e is the emissivity of tungsten obtained from pubbshed data [55], T is the temperature of the tungsten, c, h, and k are the speed of light,the Planck constant, and the Boltzmann constant respectively. The camera exposure time r is determined by the streak speed vc and the entrance slit width Dc: (3.81) First we measured the signal counts versus different exposure times at the highest gain setting for both wavelengths. The count numbers were averaged over the central image area on the screen of the streak camera monitor, where the shock luminescence Chapter 3. The experiment 45 Gain setting 5 4 3 ro at 430 nm (ps) 1.03 2.59 11.22. r 0 at 570 nm (ps) 0.42 1.06 4.63 Table 3.2: Values of r 0 at different gain setting and wavelengths signal would be recorded. Fig. 3.14 and Fig. 3.15 show that the signal count C is related to r by r = r 0 C 1 / a . (3.82) where a is found to be approximately 1.2 by a linear regression. Each data point was statistically weighted by the square of its standard deviation. r 0 is a function of gain and wavelength and can be obtained from the r-intercept on the log-log plot. For gain=5 we found the value of r 0 to be 1.03ps at A = 430mn, and 0.42ps at A = 570ram. The value of r 0 for both wavelengths drifted down by a factor of 2 between two calibration measurements which were four months apart. The later values were used because most of the shock luminescence measurements were taken just prior to the time of the second calibration. At a lower gain setting, the values of r 0 for the two wavelengths differed by a similar factor. This was obtained by comparing the exposure-count curves for two different gains when measured without the filter. Table 3.2 shows the values of r 0 for various gains and wavelengths. The photon flux collected in the shock luminescence measurements is given by the following expression, n\ = — nxcfoptica- (3.83) T n\c is given by Eq. (3.79) and foptica is the adjustment factor for taking into account the changes made in the optics since the calibration measurement. Chapter 3. The experiment 46 1 0 0 0 z r Figure 3.14: Exposure time r versus signal count C for A =430 nm, gain=5. Chapter 3. The experiment 47 Figure 3.15: Exposure time r versus signal count C for A = 570 nm, gain=5. Chapter 3. The experiment 48 It should be noted that the count C presented here was obtained by subtracting the background count from the original count. One problem was that the background level drifted during the experiment, which caused the large error bars in C. This problem also affected the accuracy of the luminescence measurement. A sensitivity test was performed on the variation of the values of a and r 0 . A arbitrary choice of a = 1 yielded a comparable confidence level in fitting the calibration curve but a shifted r 0 . The factor of the shift was estimated to be about 1.5 for both wavelengths. We used the best-fit values of a and r 0 and regarded the factor of the shift of 1.5 as the uncertainty. 3.3 Measurement of the shock speed Shock speed is an important parameter in characterizing the shock wave and can be measured directly. For opaque solids it can be determined by observing the luminous radiation emitted from the rear surface of the solids as the shock emerges at this surface. The emission starts near the time of the shock wave break out and continues as the released material expands into the vacuum. By recording the time when the luminous signal starts for various thicknesses of the target, one can map out the shock trajectory (space-time history) and thus determining the shock speed. 3.3.1 The experimental set up The experimental set up is illustrated in Fig. 3.16. Polished single crystal silicon wafers with thicknesses of 68.3/im and 82Apm were used as targets. The target front surface was irradiated by the focused laser beam at normal incidence with irradiation conditions given in Sec. 3.1.2. The irradiance was controlled by varying the ND filters placed in the path of the laser beam. The target rear surface was imaged onto the entrance slit of the streak camera with a f/1.4, 50mm f.l. lens and a 600mm f.l. achromatic (same as Chapter 3. The experiment 49 in the calibration measurement). The imaging lens was covered by a shield to block the stray light from the laser-produced plasma. To determine the timing of the shock arrival at the target rear surface relative to the incident laser pulse, a reference beam (fiducial beam) split from the laser beam was recorded on the streak camera through an optical fiber. Interference filters with center wavelengths of 570 nm or 430 nm were placed at the entrance of the streak camera to stop the 0.53fim laser light from entering the camera. The camera resolution used for the shock speed measurement was 22 ps/pixel which yielded an actual time resolution of about 90ps governed by the width of the entrance slit. The dynamic range of the streak camera was approximately 100. The test results from the manufacturer indicate that the streak speed was uniform to within 1.3 percent. 3.3.2 Experimental results After introducing a proper time delay between the arrival of the main laser pulse at the target plane and the arrival of the fiducial signal at the streak camera, the luminescence signal of the shock break out and the fiducial signal were recorded on the same streak picture. Fig. 3.17 shows the streak records for the 68.3/xm and 82.4/zm Si targets both irradiated at an average irradiance of 3 X 10 1 7W/m 2. Temporal profiles of the shock emission were obtained by spatially integrating over the central ~ 40/zm of the break out region. These are presented in Fig. 3.18. The timing of the fiducial was characterized by the center of the fiducial signal. The shock break out time was characterized by the half intensity point of the shock signal. A shock transit time is then defined to be the time interval between fiducial time and the shock break out time. The shock trajectory was determined by repeating the transit time measurement as a function of target thickness at same irradiance. Fig. 3.19 shows the measured shock trajectory in Si for an average irradiance of 3 x 10 1 7W/m 2. Each data point is an average of at least 4 measurements with the error bars given by the standard error. Also shown Chapter 3. The experiment 50 DICHROIC MIRROR MAIN LASER BEAM FOCUSING LENS OPTICAL FLAT VACUUM CHAMBER LIGHT SHIELD STREAK CAMERA NEUTRAL DENSITY FILTER f /1 .4 LENS INTERFERENCE FILTER > 600mm F. L. ACHROMAT FIDUCIAL Figure 3.16: Experimental set up for the luminescence measurement. Chapter 3. The experiment 51 X (micron) Figure 3.17: Streak record of the shock emission and the fiducial signal. The target thickness was 68.3 am in (a) and was 82.4 am in (b). D„ is the width of shock break out region. Chapter 3. The experiment 52 Figure 3.18: Profiles of the shock emission and the laser fiducial. The target thickness was 68.3 fim in (a) and was 82.4 am in (b). Chapter 3. The experiment 53 100 50 -_ 40 — i i i i — | i i i — i — | i i — i — i — | — i i i i — 1 1.5 2 2.5 3 TIME (ns) Figure 3.19: Shock trajectory. The data is obtained from measurements at an average irradiance of 4.3 x l017\V/m2. Curve (a), (b), and (c) are calculated shock trajectories for irradiances of 5.4 x 1017 W/m2, 4.7 x 1017W/m2, and 4.2 x lQi7W/m2 respectively. The corresponding shock speeds are 2.08 x 104m/.s, 2.02 x 104m/.s, and 1.96 x 10 4m/s. Chapter 3. The experiment 54 Figure 3.20: Shock trajectory. The data is obtained from measurements at an average irradiance of 3.5 x 10 1 7 W/m 2 . Curve (a), (b), and (c) are calculated shock trajectories for irradiances of 3.9 x IQ17W/m2, 3.5 x 10 1 7 V^/m 2 , and 3.2 x \0l7W/m2 respectively. The corresponding shock speeds are 1.88 x 10 4m/s, 1.81 x 104m/.s, and 1.76 x 104m/.s. Chapter 3. The experiment 55 Figure 3.21: Shock trajectory. The data is obtained from measurements at an average irradiance of 2.5 x 1017 W/m2. Curve (a), (b), and (c) are calculated shock trajectories for irradiances of 2.2 x 1Q17 W/m2, 2.1 x lO17 W/m2, and 2.0 x 1017 W/m2 respectively. The corresponding shock speeds are 1.53 x 104m/.s, 1.50 x 10 4m/s, and 1.45 x 10 4m/s. Chapter 3. The experiment 56 ND niters 0.0 0.3 0.6 Shock speed Vs (10 4m/s) 2.0 ± 0 . 1 1.8 ± 0 . 2 1.5 ± 0.1 Effective irradiance(10 l7W/m2) 4.7 ± 0 . 7 3.5 ± 0.4 2.1 ± 0 . 1 $A ( l017W/m2) 3.3 ± 0 . 1 1.7 ± 0.2 0.8 ± 0.01 (l017W/m2) 4.3 ± 1.4 3.5 ± 1.4 2.5 ± 1.2 D3 (fim) 130 ± 20 105 ± 20 85 ± 20 Table 3.3: Table of shock speeds and effective irradiances in the figure is the shock trajectory calculated from 1-dimensional hydrodynamic (LTC) simulations. The irradiances in the simulations were chosen to give the best fit to the data. The shock speed was determined from the calculated trajectories which yielded the values of (2.0 ± 0 . 1 ) x 104m/.s . The uncertainty corresponds to the two extremes of the calculation as shown in Fig. 3.19. Similarly, the effective irradiance was found to be (4.7 ± 0 . 7 ) x 1017W/m2 which differs from the estimated average irradiance of (3.3 ± 0 . 1 ) x 1017W/m2. The shock trajectories at the lower irradiances (with attenuations using ND0.3 or ND0.6 filters) were obtained in a similar way. Fig. 3.20 and Fig. 3.21 show the data and the simulation results. Table 3.3 summarizes the shock speed, effective irradiances and the average irradiances defined in Sec. 3.1.2. Also shown in Table 3.3 is an irradiance $* defined by EL (3.84) Here EL is the incident laser pulse energy, T £ is F W H M of the incident laser pulse, D„ is the width of the shock break out region shown in Fig. 3.17. One can see that $* is in reasonable agreement with the effective irradiance. Chapter 3. The experiment 57 3.4 Shock-induced luminescence measurement The experimental set up for the shock luminescence measurement is the as same as that for the shock speed measurement except that the streak camera was set at a higher sweep speed. The ultimate time resolution was 4.8ps per pixel. However, because of the width of the entrance slit, the actual time resolution was about 20ps. Fig. 3.22 and Fig. 3.23 present two typical streak images taken at two wavelengths (430 nm and 570 nm). The spectral selection of the recorded radiation was realized by installing interference filters in front of the camera entrance slit as in the calibration measurement. The spatial resolution was 20^m. The signal counts were averaged along the X axis over a width of 4 to 10 pixels. The different laser irradiances were obtained by inserting neutral density filters in the path of laser beam as in the shock speed measurement. The signal counts were converted into collected photon flux by applying Eq. (3.83) obtained from the calibration measurement. Fig 3.24 shows the shock emission at a shock speed of 2 x 10 4m/s at two wavelengths. The signal profiles at the two wavelengths differed from each other significantly: the 430 nm signal shows a sharp jump to the peak while the 570 nm signal has a rising foot before the peak. It can be seen that the total rise time (from the noise level to the peak) of the 430 nm signal was about 80 ps while the foot of the 570 nm signal shows a much longer rise time. This is due to the fact that the absorption coefficient of Si is 3.5 x 1 0 6 m - 1 at 430 nm and 5.9 x 1 0 5 m _ 1 at 570 nm. To analyze the rise time of the shock emission signals at 570 nm, we have plotted the inverse of the rise time versus the shock speed at two irradiances in Fig 3.25. The dotted line represents the predicted results given by trise = 77— (3.85) Vact where tT{se is the 1-e folding rise time, V„ is the shock speed, and a is the absorption coefficient. The data shows reasonable agreement with the prediction which suggests Chapter 3. The experiment 58 X (micron) Figure 3.22: A streak image of the shock emission at A == 430n,m and for V, = (2.0 ± 0.1) x 10 4 m/ 5 . Chapter 3. The experiment 59 SHOCK 1500 X (micron) Figure 3.23: A streak image of the shock emission at A = V, = (2.0 ± 0.1) x 10 4m/s. 570nm and for Chapter 3. The experiment 60 X -200 0 200 400 600 -200 0 200 400 600 TIME (ps) Figure 3.24: Temporal profile of the shock emission: (a) at 430 nm, (b) at 570 nm. The shock speed is (2.0 ± 0.\)m/s. Chapter 3. The experiment 61 0 . 0 2 0 0 . 0 1 5 H c/T y 0 . 0 1 0 o >s 0 . 0 0 5 0 . 0 0 0 H 0 5 10 15 20 25 30 SHOCK SPEED (KM/s) Figure 3.25: Inverse of rise time versus shock speed for the 570 nm measurement. Eq. (3.85) is shown in dotted line. Chapter 3. The experiment 62 that the shock speed obtained'by fitting the calculated trajectory to data be reasonable. Several spatial profiles of the shock emission were analyzed. These shots were taken at the higher magnification (3.4/im/pixel). Fig. 3.26 shows the several snap-shots of the spatial profile of the shock emission. The shock emerged with a width of about 20p,m and then broadened into about 60/xm within 40 ps. Recalling that the laser focal spot extended over a region of 150/im in diameter and had hot spots localized at a smaller area, it appears that the hot spots played an important role in driving the shock and yielded a smaller breakout region. Fig. 3.27 and Fig. 3.28 summarized the results of shock luminescence measurement at three shock speeds and two wavelengths. Comparison of these data to theoretical prediction is presented in the next chapter. Chapter 3. The experiment 63 Figure 3.26: Snap-shots of the spatial profiles of the shock emission. The solid curve is the average emission over a 20 ps duration starting from the moment when the lumines-cence emerged. The dashed and dash-dotted curves correspond to the spatial profiles in subsequent 20 ps intervals. Chapter 3. The experiment 10 X ZD. 10 A O lO-ci-10 14000 20000 SHOCK SPEED (M/S) 25000 Figure 3.27: Photon fluxes at three shock speeds for measurements at A = 430nm. Chapter 3. The experiment 65 10' X ZD. o 10' o 10 Q_ 5 H 10 14000 20000 SHOCK SPEED (M/S) 25000 Figure 3.28: Photon fluxes at three shock speeds for measurements at A = o70nm. Chapter 4 Comparison of data to calculations 4.1 Discrepancy between data and calculation Numerical simulations using the LTC code were performed with the laser parameters given Table 4.4. The shock induced luminescence was calculated using the electrical conductivity given by Lee and More [32]. Fig. 4.29 and 4.30 show the comparison of the results of calculation to the data. Discrepancies of one to two orders of magnitude are observed. The calculations using Spitzer conductivity are also presented and yield worse agreement. The detail of the calculation is given below. 4.1.1 Simulation of the unloading process In order to calculate the whole process of the shock emission, it is necessary to explore the hydrodynamics taking place in the released target and to understand especially how the electron temperature and density profiles evolve with time. Numerical simulations using LTC code were performed. At time zero, the target was assumed to be shock compressed with uniform temperature and density. It is then free to release into the vacuum. The simulation considered a 2 fim target divided into 200 cells with geometric ratio of 0.95. The hydrodynamics of the release was calculated for a duration of 75 ps. Thicker targets were used to simulate a longer release history. Heat transfer was described by either Lee and More's thermal conductivity or Spitzer conductivity. Fig. 4.31 presents the 66 Chapter 4. Comparison of data to calculations 67 Figure 4.29: Peak value in photon flux at A = 430nra. The curves are obtained from the calculations. The solid circles with error bars are the data. Chapter 4. Comparison of data to calculations 68 1 0 H • • • • — i — • — • — - — . — i 14000 . 20000 25000 SHOCK SPEED (M/S) Figure 4.30: Peak value in photon flux at A = 570nm. The curves are obtained from the calculations. The solid circles with error bars are the data. Chapter 4. Comparison of data to calculations 69 Shock speed V3 (104m/.s) 2.0 1.8 1.5 Effective Irradiance(1017W/m2) 4.7 3.5 2.1 Hugoniot Temperature (10 4K) 5.50 4.43 2.77 Table 4.4: Table of shock parameters used in the simulation Lagrangian spatial profiles of the electron temperature and the electron density (using Lee and More's conductivity). In the diagram the material is released to the left and the rarefaction wave propagates to the right. The oscillations in the profiles are believed to be caused by the liquid-vapor phase transition. Note that the electron temperature decreases rapidly once the release begins. On the other hand the critical density (about 10 2 7ra~ 3 for visible light) layer moves inward into the target to expose the hotter material. This competition determines the feature of the luminescence signal during the release. 4.1.2 Luminescence calculation Previous work in luminescence calculation on shocked metal by Parfeniuk [56] used bremsstrahlung emissivity to evaluate the gray body radiation from a given temperature and density gradient. That scheme ignored the reflection and refraction of the electro-magnetic wave inside the nonuniform emitting medium. The validity of the approach became questionable when large gradients of optical properties exist. For instance, one has to put a cut-off function into the calculation to deal with emissions from regions with electron densities higher than the critical density. A more rigorous treatment based on the propagation of E M waves in a stratified medium was used by Celliers [57]. The basic idea is to regard the unloading material as a conducting medium with a complex dielectric function which varies with space and time. The spectral emissivity is related by Kirchoff's law to the absorption coefficient which is given by the dielectric function at that layer. By solving the electromagnetic wave equation in the stratified medium, the Chapter 4. Comparison of data to calculations 70 lOOOOOq L l J 10 I i—i i 11111| i—i i 111111 1—i i 11111| i i i 11111 0.001 0.01 0.1 1 10 LAGRANGEAN COORDINATES (um) Figure 4.31: Profiles of (a) electron temperature and (b) electron density during the unloading process. The shock pressure is 6.1 Mbar ( corresponding to V, = 2 x 10 4m/s). Chapter 4. Comparison of data to calculations 71 total radiation intensity emitted can be calculated. The shocked material was treated as a dense plasma. The dielectric function is de-termined by its conductivity: ATCCT e = e T + i — (4.86) where e is the complex dielectric function, er is the real part of e, cr is the conductivity, and UJ is the angular frequency of the electromagnetic wave. Numerical conductivity data for Si from Lee and More's model [58] was employed for cr. Let us start from Maxwell's equations: V D=p (4.87) V x i = - ^ (4.88) V B= 0 (4.89) y i = J + ^ f (4-90) where E denotes the electric field, B denotes the magnetic field, D and H are defined by D = eE and B = pH, p is the magnetic permeability, J is the current density which is related to E by J = uE. (4.91) Let us consider a plane electromagnetic wave propagating through a stratified medium, with the plane of incidence taken to be the T/z-plane. For a transverse electromagnetic wave, Ey = Ez = 0 and Maxwell's equations reduce to the following six scalar equations: (4.92) Chapter 4. Comparison of data to calculations 72 dHr dHr dz dx = 0, (4.93) 8HY 8HX —Hx = 0, (4.95) dEx iu>p — - —Hv = 0, (4.96) 8EX iufi dy It follows that Hz = 0. (4.97) where n 2 = e/z, k0 = - = ^ . (4.99) c A 0 To solve Eq. (4.98), one assumes that Ex(y,z) = Y(y)U(z). (4.100) Equation (4.98) then becomes 1 d2Y Y dy2 1 d2U Ullz2 n2kl <f(log p) 1 dU dz U dz (4.101) This yields [73] Chapter 4. Comparison of data to calculations 73 Y = const.eikaay (4.102) so that Ex = U(z)ei{koay-wt). (4.103) Similarly, the magnetic field will be given by Hy = V(z)ei{-koay-ut\ (4.104) Hz = W(z)ei{koay-ut). (4.105) U , V, W are related by the following equations: V = ik0[aW + eU], (4.106) U' = ik0pV, (4.107) aU + pW = 0. (4.108) A pair of simultaneous first order differential equations of U and V is obtained U' = ik0pV, (4.109) a 2 V' = ik0(e )U. (4.110) Using Eq. (4.101) to (4.110), We have the following second-order differential equations + kl(n2 - a2)U = 0, (4.111) dz2 dz dz Chapter 4. Comparison of data to calculations 74 Eq. (4.111) and ( 4.112) can be solved analytically in a homogeneous dielectric medium, but has to be solved numerically in our problem because the dielectric function varies with z and the dependence on z is in general not an analytical function. The purpose of solving the above equations is to calculate the emissivity at each layer and the transmitted emission from that layer to the boundary. The transmission is calculated by solving Eq. (4.111) and (4.112) numerically. Once the values of U and V at the boundary of the medium is given, the values of U and V at any z are uniquely determined. Thus, the ratio of Ex(z) to Ex at the boundary can be calculated. An absorption coefficient « a (z ) is denned by <•(.) = (4,13) The emissivity ne(z) is defined by p» = %^W{7j-^^ <4'114> where Pu is the power of the emitted radiation per unit IM per unit solid angle, T(z) is the temperature at z, h and k are the Planck constant and the Boltzmann constant respectively. At local equilibrium, the emissivity Ke at the layer between z and z + dz is equal to the absorption coefficient « a , Ke{Z) = na{z). (4.115) Here E(z) is given by the solution of U and V from Eq. (4.111) and (4.112). Thus, a radiating layer at z can be characterized by its local temperature T(z) and emissiv-ity ne(z), a n d its contribution to the total radiation observed can be calculated using equations Eq. (4.111) and (4.112). Figure 4.32: Calculated shock emission at (a) A = 430nm and (b) A = olOnm of Si shocked to 6.1 Mbar. In each plot the upper curve corresponds to calculation using Spitzer Conductivity and the lower curve corresponds to that using Lee and More's conductivity. Chapter 4. Comparison of data to calculations 76 The radiation emitted by the shocked material before shock break out at the free surface was also calculated with this method using the dielectric function of cold Si (under normal condition) which is known [59, 60]. Fig. 4.32 shows the calculated photon flux collected by the optics for each wavelength. The silicon is shocked to 6.1 Mbar which corresponds to a shock speed of 2.0 x 104m/.s. The results using both Lee and More's conductivity model and the Spitzer conductivity are presented. Time zero is defined as the time of shock break out. The oscillations before time zero are caused by the interference generated in the layer between the shock front and the Si free surface. As the shock front approaches the free surface this layer becomes thinner and thinner, thus giving rise to the temporal interference pattern. The slope of the rising edge of the signal, averaged over the interference oscillations, is determined by the shock speed and the absorption coefficient of cold Si. The dielectric function e = 6! + ze2 for cold Si at 430 nm and 570 nm was calculated from published data of refractive index n and absorption coefficient a [61] through 2 „ 2 (4.116) nac e2 = . (4.117) 4.1.3 Comparison of the temporal profiles In order to compare the result of calculations with the measured signal, one has to take into account the finite temporal resolution of the detector (streak camera). A Gaussian window with a full width of half maximum of 20ps was assumed for the streak camera resolution. The convoluted photon flux at any time t is obtained by integrating of the calculated temporal history of the luminescence emission with the Gaussian time window Chapter 4. Comparison of data to calculations 77 (of the streak camera) centered at that time t. Fig. 4.33 and Fig. 4.34 present the results of the convoluted calculations as well as the data for both wavelengths at a shock speed of 2.0 x 10 4m/5. The lower curve was obtained using by Lee and More's conductivity and the upper curve was obtained using by Spitzer model. The thin curves with symbols are the data. There are three data curves in Fig. 4.33 and Fig. 4.34. Fig. 4.33 and Fig. 4.34 show tremendous discrepancy between data and calculation in terms of both the peak intensity and the temporal shape. The disagreement is evident for both wavelengths before time zero. Although the curve corresponding to Lee and More's model touches the data after lOOps at 570 nm, it is obvious that the calculation shows a faster decay than the data (same for the 430 nm). This discrepancy suggests the presence of certain unknown mechanism inside the shocked material before the unloading which yields abnormally low luminescence. What could such a mechanism be? 4.2 Nonequilibrium electron temperature model 4.2.1 Nonequilibrium electron temperature in shock-compressed material Let us review the microscopic process which takes place in shock-compressed Si. The energy gap between the conduction band and the valence band for normal Si is about l . l e V and this band gap is reduced when the pressure increased. A pressure of more than 200 kbar will causes the closure of the band gap thus transforming Si into a metallic state [62]. The valence band electrons thus become free electrons as soon as the lat-tice is compressed by a shock wave strong enough to close the band gap. On the other hand, shock compression heats the lattice by dissipating the mechanic energy into ther-mal energy. The lattice can reach thermal equilibrium in 10 _ 1 2sec to 10~13sec through phonon-phonon interactions [23, 24]. The free electrons acquire thermal energy from the lattice through electron-phonon interactions. Although the momentum relaxation time Chapter 4. Comparison of data to calculations 78 TIME (ps) Figure 4.33: Temporal profile of the data and the calculations for A = 430nm. The corresponding shock speed is 2 x 10 4m/s. Chapter 4. Comparison of data to calculations TIME (ps) Figure 4.34: Temporal profile of the data and the calculations for A = 570nm. Th corresponding shock speed is 2 x 104m/.s. Chapter 4. Comparison of data to calculations 80 between electrons and phonons in solids is very short (10 1 3sec approximately, depend-ing on temperature [25]), the energy transfer from the phonons to the electrons takes a much longer time since the electrons acquire only a small amount of energy from each collision with the phonons. If this energy relaxation time is longer than the phonon-phonon relaxation time, there could be a gradient in electron temperature behind the shock front with the electron temperature lagging the lattice temperature. Further more if this relaxation time is long compared to the time for the shock to propagate through a distance of one optical depth, this gradient in electron temperature could affect the observed emission. Since the cold electrons at the shock front absorbs the emission from the hot material deep inside, one can never observe the true Hugoniot temperature. A quantitative understanding of the energy transfer process is therefore crucial for predict-ing the shock induced luminescence. Moreover, thermal relaxation between electrons and the lattice also reveals fundamental information about electron-phonon coupling which has wide impUcations. Back in 1956, Kaganov, Lifshits and Tantarov discussed the electron-phonon relax-ation process and calculated the heat transfer coefficient [63]. In principle, the relaxation time for electron temperature T can be obtained from electron-phonon energy exchange rate (dE/dt)ep and the electron heat capacity Ce: where the energy exchange rate can be calculated from the electron and phonon dis-tribution functions and the electron-phonon matrix elements. In the high temperature hmit, i.e., when both the electron and lattice temperatures are higher than the Debye temperature, the energy exchange rate has the form (4.118) Chapter 4. Comparison of data to calculations 81 (-^)ep=g(T-Q). (4.119) Here 0 is the lattice temperature, g is called the electron-phonon heat transfer coef-ficient (or electron-phonon coupling constant) which depends on the material properties but is independent of the electron temperature. Recent work by Allen [64] related g to some interesting parameters in the theory of superconductivity. For the degenerate electrons as in metals and in shocked silicon, Ce is given by Ce = CfT, (4.120) where Cf= ^ • (4-121) Here ne is electron density, kg is the Boltzmann constant, and TV is the Fermi tempera-ture. So Eq. (4.118) and Eq. (4.119) yield CfT^=g(T-Q). (4.122) The relaxation time T. is defined as TT = (4.123) 9 In 1960s, Kormer and Zel'dovich investigated the kinetics of electron heating in shock-compressed alkali-halides. To explain the observed dependence of emission intensities on pressure in their experiment, they proposed a phenomenological model for thermal relaxation between the electrons and the lattice [23, 24]. In their model, electron heating is affected by two processes: thermal excitation from the valance band to conduction band and heat transfer from the phonons to the free electrons. A relaxation time of Chapter 4. Comparison of data to calculations 82 1 0 - 9 to l O - 1 0 sec was estimated to give reasonable agreement between the data and the calculations for the shock-induced luminescence. Recent developments in ultra-short-pulse lasers made it possible to directly investigate electron-phonon relaxation in metals [65, 66, 67]. Several experiments have yielded values of g or rr in metals [67, 68, 69, 70, 71]. Fujimoto et al. estimated a value for tungsten of gw = (5 ~ 10) x 1017Wm~3K'1. Elsayed-Ali et al. reported a value for Copper of gcu = 1017Wm~3K~1. Lately Corkum et al. reported a smaller value of gcu = l016Wm~3K~1 for copper and gMo = 2 x l016Wm~3K~1 for molybdenum. Schoenlein et al. obtained a rT = 2 ~ Zps in gold which yields gAu = (2 ~ 3) x 1016Wm~3K~1. Groeneveld et al. reported a value for silver of gAg = 3.5 x lO^Wm^K-1. We propose that thermal relaxation between the electrons and the lattice is also taking place in the shock-compressed Si. The excitation of electrons from valance band to conduction band is assumed to be accomplished immediately after the silicon is shock-compressed. Thermal diffusion of electrons is taken into account in our model. The shock emissions are calculated with various values of g to fit our data. There are no published data of g for Si so far; we estimate that g$i — (1 ~ 2) X 1017W M—ZK'1 for shock-compressed Si at pressures of 3 to 6 Mbar. 4.2.2 A relaxation-diffusion model By taking into account of both thermal relaxation and heat diffusion, the electron tem-perature as a function of time and space can be described by dT(x t) B2 Ce-^=g(Q-T(x,t))+K—T(x,t). (4.124) Here K is the thermal conductivity. In the case of steady shock propagation, a self similar solution of the form Chapter 4. Comparison of data to calculations 83 \ \ \ —+us -> X Figure 4.35: A moving frame y is attached to the shock front at y=0. T{x,t) = f{vt - x) (4.125) is expected, where v is the shock speed. We denote the variables in the shocked region with subscript 1 and those in the cold region with subscript 2, and define y = vt — x. From Eq. (4.124) CeivMv) = 5(0! - /i(y)) + «i/i(y), (4.126) Ce2vf2{y) = 5 ( © 2 - fiiy)) + ^hiy)-fi(y) and f2(y) satisfy the boundary conditions: (4.127) lim fi = Qi, y —*oo (4.128) Um f2 = 0 2 , y — - o o (4.129) Chapter 4. Comparison of data to calculations 84 MO) = / 2(0) = T0, (4.130) « i / i (0) - « 2 / 2 ( 0 ) = ( C e l - Ce2)T0(v - vv). (4.131) Eq. (4.131) follows from energy conservation for the electrons in an arbitrarily thin layer at the shock front. vp is the particle velocity behind the shock front. As region 2 (y < 0) contains cold Si, it has a low free electron density n e 2 at normal conditions. The free electrons are nondegenerate at room temperature therefore C e 2 = | fcsn e 2 is constant. Eq. (4.127) can be easily solved to yield /2(y) = 02 + (r0 - 0 2 )e A 2 a , • (4.132) where A ; = c . ; + y y r ^ i ( 4 i 3 3 ) According to QEOS ionization calculations, the shocked Si is 3 ~ 4 times ionized. The corresponding Fermi energy is above 20 eV. Therefore the electrons are degenerate even at a temperature of 5 eV. The electron heat capacity is given by Eq. (4.120). Eq. (4.126) then becomes CfvfMMy) = </(©! - My)) + KJM. (4.134) In general « x is also a function of temperature. As an approximation we treat KX as a constant because the Lee and More's conductivity model shows that «i is not sensitive to electron temperature for temperatures of a few eV. Eq. (4.134) is a nonlinear differential equation of the second order. It can be written as two coupled, first-order equation by defining Chapter 4. Comparison of data to calculations V = h{y), g = A(y). Then We define dimensionless variables Eq. (4.137) gives Cfv t = V Eq. (4.139) has only one parameter A which is given by A = ( C > 0 i )2 K\9 We seek a solution of Eq. (4.139) such that Chapter 4. Comparison of data to calculations 86 lim £ = 1, (4.141) T—>00 lim 77 = 0. (4.142) T —•OO One can check that indeed £ = 1,7/ = 0 is a fixed point of Eq. (4.139) and is the only one. The Jacobi matrix of Eq. (4.139) is ( ° 1  v A(rj + 1) An The eigenvalues of the Jacobi matrix at £ = 1,77 = 0 are (4.143) A + y/A2 + AA K = \ > 0, (4.144) A - VA2 + 4A A_ = 2 < (4.145) It is obvious from Eq. (4.144) and Eq. (4.145) that (1,0) is a saddle node illustrated in Fig. 4.36. It implies that in order to reach (1,0) when r —> 0 0 the initial point (£(0), 7/(0)) must be on a curve which is a branch of the stable manifold of the saddle point. Recalling that the boundary condition Eq. (4.131) forms another restraint on the initial point (£(0),T?(0)) which is given by Here 7,(0) = A(e2(0) + ^ ( 0 ) - f e 2 ) - (4.146) Chapter 4. Comparison of data to calculations 87 stable Figure 4.36: Saddle node and its stable and unstable manifolds. ^ = ^ f (4-148) K2X2Q2 vC}Ql Fig. 4.37 shows the qualitative analysis of Eq. (4.139). The trajectory starting at (£(0),77(0)) which is the crossing point of Eq. (4.146) and the upper branch of the stable manifold moves into (1,0) as r —> 00. The electron temperature at the shock front, To, is determined by To = £(0)0!. (4.149) Numerically (£(0),T;(0)) was obtained by computing Eq. (4.139) in the reversed di-rection starting at e(l,A_) where e is a very small number (e ~ 10 - 1 0 ) . A Runge-Kutta solver [74] was employed to solve the equations until it reaches the right branch of the parabola given by Eq. (4.146). Fig. 4.38(a) shows the numerical solution f\{y) as well as ./^(y) given by Eq.( 4.132) for 0 ! = 5.5 x 10 4/f, 0 2 = 300/f, v = 2x 10 4m/s, and g is assumed to be 10 1 7 Wm'3K~l. was obtained from Lee and More's thermal conductivity data, K2 was obtained from the published data [72]. f\(y) approaches 0 ! approximately exponentially with a rise time determined by both g and K^. TO show this, a logarithm plot of 0 a — f\(y) is presented in Chapter 4. Comparison of data to calculations 88 Figure 4.37: Qualitative analysis of Eqs. (4.139). Chapter 4. Comparison of data to calculations 89 Fig. 4.38(b) for two different values of g for the same shock conditions. Besides the effect on the rise time, To also plays an important role in determining the emission intensity. From Fig. 4.37 one can see that as A oc g~l a lower value of g will yield a steeper rise of the parabola (4.146) and hence move (£(0),T?(0)) towards £ = 0. Although the upper branch of the stable manifold moves up when g is decreased, it is restrained in the area under the curve n = | — 1. Therefore T0 decreases when g is reduced. When g —> oo, A —> 0; it is obvious from Fig. 4.37 that (£(0),T?(0)) —> (1,0) which yields the step profile fi = T0. In summary, the finite electron-phonon coupling constant g has two effects on the electron temperature profile: a lower T 0 than the lattice temperature at the shock front , and a gradient with an approximately exponential approach to the lattice temperature. When g is reduced, T 0 decreases and the temperature gradient becomes less steep. In the next section one will see how this gradient affects the shock emission. 4.2.3 Electron-phonon coupling constants estimated for the shocked silicon A gradient of electron temperature with T0 < 0 results in lower emission when the shock approaches the free surface and also when the release takes place. Unloading simulations were performed with an initial electron temperature gradient. Fig. 4.39 shows the photon flux calculated for various values of g without convolution for temporal resolution of the detector. Only Lee and More's conductivity model was used here. The calculation corresponds to the shock condition of Va = 2 x 104m/.s. To compare the new calculation with the data, the calculated photon flux in Fig. 4.39 was convoluted with a 20ps F W H M Gaussian window and presented in Fig. 4.40 and Fig. 4.41 together with the data. Comparing these to Fig. 4.33 and Fig. 4.34, it is evident that the relaxation model yields a much better agreement with the data. The value of g can be estimated as (1 ~ 2) x 1017Wm~3K~1. Note that at 570 nm the Chapter 4. Comparison of data to calculations 60000 90 40000 20000 0 H 1 0 0 0 0 4 1 0 0 0 d © 1 0 0 1.0 2.0 3.0 4.0 5.0 6.0 \ S = 10"WM-*K-' 9 = 2 x iO"WM-*K-' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Figure 4.38: Numerical solution of (a) fi(y) and (b) 0 ! — fi(y) Chapter 4. Comparison of data to calculations 91 calculated curve shows a jump around t=0 as the data. 4.3 Comparison of the calculation to the data at different pressures To verify the value of g estimated in the previous section, we performed the same lumines-cence calculation for shock speeds of 1.8 x 10 4m/s and 1.5 x 10 4m/s (the corresponding shock pressures are 4.7 Mbar and 2.8 Mbar respectively). The results are resented in Fig. 4.42, 4.44, 4.43, and 4.45. It appears that g = (1 ~ 2) x IQ^Wm^R-1 is a reasonable estimate for shock pressures of 4.7 ~ 6.1 Mbar and it suggests a lower value of g at a shock pressure of 2.8 Mbar. At present it is not well understood how g depends on the shock condition in terms of pressure or shock speed. It will be very interesting to have a first principle calculation of g and to compare it with data. Fig. 4.46 and Fig. 4.47 summarize the data and the calculated peak photon flux as a function of shock speed. The error bars in the data take into account uncertainties in absolute intensity calibration, fluctuations of the signal and statistic errors resulting from shot-to-shot variations. The region bounded by the curves for g = 1 X 1017Wm~3K~l and g = 2 X l017Wm~3K"1 shows a reasonable overlap with the data for both wavelengths. One may argue that g = 2 x 10i17Wm~3K~x is a better choice at 6.1 Mbar, g = 1 x 1017Wm~3K~x is a better choice at 4.7 Mbar and g might be less than 1 x l017Wm~3K~1 at 2.8 Mbar. Figure 4.39: Calculated photon flux for various g: (a)A = 430nm; (b)A = 570nm. Chapter 4. Comparison of data to calculations 93 TIME (ps) Figure 4.40: Comparison of results of calculation with data at A = 430nm. The corre-sponding shock speed is 2 x I04m/'s. Chapter 4. Comparison of data to calculations 94 -200 -100 0 100 200 TIME (ps) Figure 4.41: Comparison of results of calculation with data at X = 570nm. The corre-sponding shock speed is 2 x I04m/s. Chapter 4. Comparison of data to calculations 95 TIME (ps) Figure 4.42: Comparison of results of calculation with data at A sponding shock speed is 1.8 x 104m/s. = 430ram. The corre-Chapter 4. Comparison of data to calculations 96 -200 -100 0 100 200 TIME (ps) Figure 4.43: Comparison of results of calculation with data at A = 570ram. The corre-sponding shock speed is 1.8 x 10 4m/3. Chapter 4. Comparison of data to calculations 97 g = 2 x 1 0 1 7 W M - 3 A ' - ' TIME (ps) Figure 4.44: Comparison of results of calculation with data at A = 430ram. The corre-sponding shock speed is 1.5 x 104m/.s. Chapter 4. Comparison of data to calculations 98 Figure 4.45: Comparison of results of calculation with data at A = 570ram. The corre-sponding shock speed is 1.5 x 10 4m/s. Chapter 4. Comparison of data to calculations 99 Figure 4.46: Peak values in photon flux for A = 430ram. Chapter 4. Comparison of data to calculations 100 10 1 0 N X z> § zn Q_ 10 1 0 N 1 0 N 14000 i i i g —> oo Spitzer 5 -» oo 4—I 9 = 2 x 10"WM-3A'-' g = 10 , 7 WM- 3 A'-' 20000 25000 SHOCK SPEED (M/S) Figure 4.47: Peak values in photon flux for A = 570nm. Chapter 5 Summary and Conclusions 5.1 Summary This report presented the measurement of shock-induced luminescence in silicon with temporal resolutions of up to 20ps. The streak camera was calibrated for absolute re-sponse at wavelengths of 430 nm and 570 nm. The shock-induced luminescence was measured for three different shock velocities. The QEOS equation of state developed by More et al. was used in the hydrodynamic code LTC for the simulations of laser gener-ated shock waves in silicon. With an improved radiation calculation scheme, luminosities were calculated using Lee and More's conductivity model. The comparison between the calculated and measured luminosity showed significant disagreement. The data showed anomalously low intensities with a different temporal history. An electron-phonon relaxation and electron thermal diffusion model was pro-posed which suggested that the electron temperature behind the shock wave may be not uniform. The calculation taking into account the electron temperature gradient yielded a good agreement with the data. The electron-phonon coupling constant for shock com-pressed silicon at 3 to 6 Mbar was estimated to be (1 ~ 2) x 1017Wm~sK~1. Should this model become plausible, one can use luminescence measurements to study electron-phonon relaxation process in shocked materials. 101 Chapter 5. Summary and Conclusions 102 5.2 New contributions The results presented in this thesis offer the first information about the electron-phonon thermal relaxation precess in matter under shock compression. The anomalous low in-tensity of shock-induced luminescence observed suggested the importance of the effect of non-equilibrium between electrons and lattice. The thermal relaxation-diffusion model proposed here yielded a reasonable agreement between predictions and data. The val-ues of the electron-phonon coupling constant in shocked materials were unknown. Our estimated value for silicon has a similar order of magnitude as the values found in some metals. The implementation of QEOS equation of state in the LTC code made it capable of simulating the laser generated shock waves in any solid, given the material compositions and the properties of the cold solid. The wave solution treatment for calculating emis-sion from the shock heated solid allowed us to verify different conductivity models by comparing the calculated temporal luminosity profiles with data. 5.3 Future work A high-power laser as a means of generating shock wave has some disadvantages. The fluctuations in pulse energy and pulse shape cause large uncertainties in the ablation pressure hence affecting the reproducibility in the shock speed and Hugoniot temper-ature. One solution is to employ the two-stage light-gas gun technique in which the shock pressure and hence the shock speed and Hugoniot temperature can be very well determined. Therefore the error bars in shock speeds can be much reduced. Another advantage of using the gas gun is that the spatial extent of the shock front is much larger (of the order of two centimeters in diameter) which yields a more planar shock front. A planar shock front will reduce two dimensional effects in shock propagations. Chapter 5. Summary and Conclusions 103 Due to internal gating problems of the streak camera, we were unable to exploit the fastest sweep speed which would allow a time resolution of 4ps. Once this problem is solved, we can push the resolution of the measurement to 4ps so that further details of the temporal history of the luminescence signal will be revealed. A first principle calculation of the electron-phonon coupling constant g in shocked solid is needed to model the experiment. Similar experiments with different solids can be performed to expand the investigation of the electron-phonon thermal relaxation process in solid under shock compression. A further improvement of the LTC code to enable two-temperature calculations using QEOS will improve the simulation of the unloading process with nonequilibrium electron temperatures. Bibliography [1] See Physics of High Energy, edited by. P. Caldirola and H. Knoepfel, Proceedings of the International School of Physics "Enrico Fermi", course 48 (Academic, New York, 1971). [2] R. P. Feynman, N . Metrpolis, and E. Teller, Phys. Rev. 75, 1561 (1949). [3] M . Ross, Rep. Prog. Phys. 48, 1 (1985). [4] Los Alamos National Laboratory report LA-10160-MS, by K. Holian (1984). [5] LASL Shock Hugoniot Data, edited by. S. P. Marsh ( University of California Press, Berkeley, 1980). [6] R. M . More, K . H. Warren, D. A. Young, and G. B. Gimmerman, Phys. Fluids, 31, 3059 (1988) [7] P. W. Bridgman, The Physics of High Pressures, (G. Bell & Sons, London 1958). [8] E. K . Graham, J . Geophys. Res. 91, 4630 (1986). [9] P. M . Bell, Rev. Geophys. 21, 1394 (1983). [10] W. J . Nellis, J . A. Moriarty, A. C. Mitchell, M . Ross, R. G. Dandrea, N . W. Ashcroft, N. C. Holmes, and G. R. Gathers, Phys. Rev. Lett. 60, 1414 (1988). [11] L. V . Al'tshuler, A. A. Bakarova and R. F. Trunin, Sov. Phys. J E T P 15, 65 (1962). [12] R. G. McQueen and S. P. Marsh, J . Appl. Phys. 31, 1253 (1980). 104 Bibliography 105 A. C. Mitchell and W. J . Nellis, J . Appl. Phys. 52, 3363 (1981). [13 [14 [15 [16 [17 [18 [19 [20 [21 [22 [23 [24 [25 W. J . Nellis in High pressure Measurement Techniques edited by G. N Peggs (Appl. Sciences, New York, 1983). J . Nuckolls, L. Wood, A . Theisson, and G. Zimmerman, Nature 239, 139 (1972). K . A. Brueckner and S. Jorna, Rev. Mod. Phys. 46, 325 (1974). C. G. M . van Kessel and R. Sigel, Phys. Rev. Lett. 33, 1020 (1974). L. Veeser and J . Solem, Phys Rev. Lett. 40, 1391 (1978). R. J . Trainor, J . W. Shaner, J . M . Auerbach, and N. C. Holmes, Phys. Rev. Lett. 42, 1154 (1979). R. J. Trainor, N . C. Holmes, R. A. Anderson, E. M . Campbell, W. C. Mead, R. J . Olnecs, R. E. Turner, and F. Ze, Appl. Phys. Lett. 43, 542 (1983). C. E. Ragan III, M . G. Silbert, and B. C. Diven, J . Appl. Phys. 48, 2860 (1977). Ya. B. Zel'dovich, S. B. Kormer, M . V . Sinitsyn, and A. I. Kuryapin, Sov. Phys. Doklady 3, 938 (1958). S. B. Kormer, Sov. Phys. Uspekhi, 11, 229 (1968). Ya. B. Zel'dovich, S. B. Kormer, and V . D. Urlin, Sov. Phys. J E T P 28, 855 (1969). E. M . Conwell, High Field Transport in Semiconductor, in Solid State Physics, Supplement 9, edited by Fredereik Seitz, (Academic Press, New York and London, 1967). [26] P. A. Urtiew and R. Grover, J . Appl. Phys. 48, 1122 (1977). Bibliography 106 [27] G. A. Lyzenga, Shock Temperatures of Materials, Ph.D. dissertation, (Cahfornia Institute of Technology, 1980). [28] J. D. Bass, B. Svendsen and T. J. Ahrens, The temperatures of Shock-compressed iron, in High Pressure Research in Geophysics and Geochemistry, edited by M . Manghnami and Y . Syono (Terra, Tokyo, 1981). [29] A. Ng, D. Parfeniuk, P. Celliers, L. Da Silva, R. M . More and Y . T. Lee, Phys. Rev. Lett. 57, 1594 (1986). [30] J. J. Duderstadt and G. A. Moses, Inertial Comfinement Fusion (Wiley &; Sons, Now York, 1982), Chapter 5. [31] Ya. B. Zel'dovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamics Phenomena ( Academic Press, New York 1966). [32] Y . T. Lee and R. M . More, Phys. Fluids, 27, 1273 (1984). [33] G. E. Duvall and R. A. Graham, Rev. Mod. Phys. 49, 523 (1977). [34] See AIP document No. PAPS PFLDA-31-3059-245 for 245 pages of Lawrence Liver-more National Laboratory report "Atomic Physics in Inertial confinement Fusion" [Report No. UCRL-84991(1981) by R. M . More]. [35] Los Alamos National Laboratory report LA-7313-MS, by C. W. Cranfill and R. M . More (1978). [36] R. Grover, J. Chem. Phys. 55, 3435 (1971). [37] M . Baus and J. P. Hansen, Phys. Rep. 59, 1 (1980). [38] W. G. Hoover, G. Stell, E. Goldmark, and G. Degani, J. Chem. Phys. 63, 5434 (1975). Bibliography 107 [39] N . F. Mott and H. Jones, Theory of the Properties of Metals and Alloys (Dover, New York, 1958). [40] P. M . Celliers, To be published. [41] Minomura and H.G. Drickmaker, J . Phys. Chem. Solids, 23, 451 (1962). [42] T. J . Dyuzheva, S. S. Kabalkina, and U. P. Novichvok, Sov. Phys. J E T P 47, 931 (1978). [43] J . C. Jamieson, Science 139, 762 (1963). [44] A. Werner, J . A . Sanjurjo, and M . Cardona, Solid State Commun. 44, 155 (1982). [45] H. Olijnyk, S. K . Sikka, and W. B. Holzapfel, Phys. Lett. 103A, 137 (1984). [46] J . Z. Hu and I. L. Spain, Solid State Commun. 51, 263 (1984). [47] S. J . Duclos, Y . K . Vokra, and A. L. Ruoff, Phys. Rev. Lett. 58, 775 (1987). [48] W. H. Gust and E. B. Royce, J . Appl. Phys. 42, 1897 (1971). [49] M . N . Pavlovskii, Sov. Phys. Solid State, 9, 2514 (1968). [50] P. M . Celliers, Ph.D. Thesis, University of British Columbia (1987), unpublished. [51] J . P. Christiansen, D. E. T. F. Ashby, and K. V . Roberts, Comp. Phys. Commun. 7, 271 (1974). [52] C. E. Max, C. F. Mckee, and W. C. Mead, Phys. Fluids, 23, 1620 (1920). [53] P. Colella and P. R. Woodward, J . Comp. Phys. 43, 174 (1984). [54] D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics (Oxford University Press, New York, 1984). Bibliography 108 [55] Handbook of Chemistry and Physics 36th ed. p. 2691, (Chemical Publishing Com-pany, 1954-1955). [56] D. A. Parfeniuk, Ph.D. Thesis, University of British Columbia (1987), unpublished. [57] P. M . Celliers, To be published. [58] R. M . More, private communication. [59] W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955). [60] D. E. Aspnes and A. A. Studna, Phys. Rev. B. 27, 985 (1983). [61] Properties of Silicon (INSPEC, New York, 1988). [62] F. P. Boundy, J. Chem. Phys. 41, 3809 (1964). [63] M . I. Kaganov, I. M . Lifshitz and L. V . Tanatarov, Zh. Eksp. Teor. Fiz. 31, 232 (1956) [Sov. Phys. J E T P 4, 173 (1957)]. [64] P. B. Allen, Phys. Rev. Lett. 59, 1460 (1987). [65] S. I. Anisimov, V . A . Benderskii, and G. Farkas, Usp. Fiz. Nauk 122, 185 (1977) [Sov. Phys. Usp. 20, 467 (1977)]. [66] R. Yen, J . Liu, and N. Bloembergen, Opt. Commun. 35, 277 (1980). [67] J . G. Fujimoto, J . M . Liu, and E. P. Ippen, and N . Bloembergen, Phys. Rev. Lett. 53, 1837 (1984). [68] H. E. Elsayed-Ali, T. B. Norris, M . A. Pessot, and G. A. Mourou, Phys. Rev. Lett. 58, 1680 (1987). [69] P. B. Corkum, F. Brunei, and N . K . Sherman, Phys. Rev. Lett. 61, 2886 (1988). Bibliography 109 [70] R. W. Schoenlein, W. Z. Lin, and J . G. Fujimoto, Phys. Rev. Lett. 58, 1680 (1987). [71] Rogier H. M . Groeneveld, Rudolf Sprik, and Ad Lagendijk, Phys. Rev. Lett. 64, 784 (1990). [72] Silicon edited by M . Neuberger and S. J . Welles, (EPIC, Hughes Aircraft Company, 1969). [73] M . Born and E. Wolf, Principles of Optics (Pergamon Press, 1964). [74] See U B C R K C , Computing Center, University of British Columbia. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0084959/manifest

Comment

Related Items