Physical of Warm Dense Matters

Abstract

Continuous progress of compute capability, DFT has been used to study complicated physics of warm-dense matter (WDM) to compared to a variety of experimental results in laboratories obtained by compression and heating of solids with intense lasers. The phase transition of insulator-metal of hydrogen is now hot topics in high-pressure physics (HPP). Advancement of laser technology and diagnostics have made such challenging subjects as precision science.

In twenty-first century, x-ray free electron laser (XFEL) facilities have been constructed as users’ facility. XFEL is new method to precision diagnostics of dense matters via x-ray Thomson scattering (XRTS). For bridging the experiment and theory to analyze XRTS data, Chihara formulated scattering spectra by decomposing three dynamical structure factors (DSF). Now, time-dependent DFT (TDDFT) is also solved with supercomputer to apply laser-matter interaction in quantum world.

In this chapter, whole stories and models are explained and some examples are explained regarding the application to analyze experimental data obtained with intense lasers and XFEL.

9.1 Shock Dynamic Compression and Equation of State

A static compression method to study the thermodynamic property of compressed matters was explained in Chap. 8, where a study of the metallic hydrogen with a diamond anvil cell (DAC) is described. Such static mechanical compression of matters is useful tool for study of the properties of matters at low temperature, and such study is called high-pressure physics. The extremely high pressure up to 5 Mbar is achieved at the present day. Since the property of metal is defined at the temperature of the null Kelvin, DAC would be a good device to study such high-pressure physics for the condensed matter physics.

As the readers see here, on the other hand, it is possible to compress matters to pressure higher than the static methods with DAC by use of intense lasers. The ablation pressure loaded on solid sample surface generates shocks during the time comparable to the laser pulse duration, namely order of 10⁻⁹ s (~ns). Although it is very short time, the time need to change from the initial state to the shocked state is shorter than the laser pulse and thermodynamic equilibrium relation can be used in general even in the case where phase transition happens at the shock front.

The achievable pressure depends on the laser intensity and its wavelength as shown in Chap. 3. Most of the recent experiments use the intensity less than several 10¹⁴ W/cm² and laser wavelength of 0.35 μm, and the experimental range covers the pressure up to 100 Mbar. In general, the physical quantities of equation of state are derived using Ranking-Hugoniot (RH) relation. The curve obtained from the RH relation is called the Hugoniot curve as shown in Chap. 3. This topic is well described in the textbook by Zel’dovich and Raizer [1]. The experiments to obtain the Hugoniot curve of a variety of solid materials at extremely high pressure have been done with high-explosive or nuclear underground test. The theory and experiment before the 1960s are well described in Chapter XI-2 in the book [1].

In general, the pressure of matters is divided into three components as Helmholtz free energies in (8.9).

$$ P\left(V,T\right)={P}_c(V)+{P}_i\left(V,T\right)+{P}_e\left(V,T\right) $$

(9.1)

where V = 1/ρ is the specific volume. In what follows, we use both of V and the mass density ρ for convenience. The P_c(V) is the cold pressure of matters at temperature equal to zero. This pressure is the target in the study by use of DAC. In the shock compression, the temperature also increases as the density, and the Hugoniot curve also depends on the thermal pressure by the motion of nuclei, namely ions P_i(V,T), and the contribution by thermal electrons P_e(V,T). The ion pressure includes the contribution by phonons and ion thermal motion. The Debye theory is famous at low temperature condensed matters [2].

It is better to see one example for the case of a solid at the initial state. In the book [1], the case of solid lead is shown, and its data are shown in Table 9.1. The solid lead is compressed by the pressure up to 4 Mbar and its density increases to 2.2 times the solid density. It is clear in Table 9.1 that even with heating by a shock wave up to about 2.3 eV, the cold pressure is predominant in (9.1), while the internal energy looks distributed almost equally to the three components; namely, equal energy partition. Mostly the compressed state is kept by the cold pressure in (9.1), consequently, to know P_c(V) is very important.

Table 9.1 Parameter behind a strong shock wave in lead. Pressure, energy, and temperatures are in the units of Mbar, eV, and eV, respectively. Reprinted with permission from ref. [1]. Copyright by Cambridge University Press

Progress of technology of pressure-drivers and diagnostics has led the experiments to a laboratory scale and precise data are now available experimentally. The diagnostics is essential ingredient for such research, but we don’t describe the details of the diagnostics and limit in this book only to explain its physical principle.

9.1.1 Theoretical Base for Shock Equation of State

Assume that the physical quantities in front of a shock, ρ₀, P₀ and ε₀ are known in (3.10, 3.11, and 3.12). Introducing the shock velocity, U_s, and compressed matter velocity, U_p, which is equal to the piston velocity in the laboratory frame, we can replace u₀ and u₁ with them.

$$ {U}_s={u}_0,\kern1em {U}_p={u}_0-{u}_1 $$

(9.2)

If we can obtain these velocities experimentally, we can solve (3.10, 3.11, and 3.12) to obtain the physical quantities in the shock compressed region,

$$ {\displaystyle \begin{array}{l}{\rho}_1={\rho}_0\frac{U_s}{U_s-{U}_p}\\ {}{P}_1={P}_0+{\rho}_0{U}_s{U}_p\\ {}{\varepsilon}_1={\varepsilon}_0+\frac{1}{2}\left({P}_1-{P}_0\right)\left(\frac{1}{\rho_0}-\frac{1}{\rho_1}\right)\end{array}} $$

(9.3)

With given values of U_s and U_p for single-shock dynamics, the thermodynamic state after the shock passage are given by (9.3). This curve of density vs pressure is called the Hugoniot curve.

In general, for the case of high-pressure applied to the solid or liquid samples, it is possible to neglect the initial values of P₀ and ε₀ in (9.3) and we obtain

$$ {\displaystyle \begin{array}{l}P={\rho}_0{U}_p^2\left(1-\frac{V}{V_0}\right)\\ {}\varepsilon =\frac{1}{2}{PV}_0\left(1-\frac{V}{V_0}\right)\end{array}} $$

(9.4)

where the suffix “1” is omitted for simplicity. The cold components must be given by the integral:

$$ {\varepsilon}_c=-{\int}_{V_0}^V{P}_c dV $$

(9.5)

Plotting the Hugoniot cure for P (=P_H) and the cold curve P_c as functions of the specific volume V with use of (9.4) in Fig. 9.1, it is clear graphically that

• 1. The increase of the internal energy by shock is given by the area of the triangle of O-A-C from (9.4).

• 2. The energy increase of the cold component is given by the area of O-B-C shaded by the horizontal lines as defined in (9.5).

• 3. The increase of the thermal energy by shock is the area of O-A-B shaded by the vertical lines.

Let us evaluate the energy given to the kinetic flow energy K in the shocked region. K is easily obtained from (9.2) and (9.4) that

$$ K=\frac{1}{2}{U}_p^2=\frac{1}{2}P{V}_0\left(1-\frac{V}{V_0}\right)\kern1em \Rightarrow \kern1em K=\varepsilon $$

(9.6)

It is surprising that the shock wave gives equally its energy to internal and kinetic energies regardless the strength of shock wave. The total energy given to the compressed materials is

$$ E=\varepsilon +K=P\left({V}_0-V\right) $$

(9.7)

It is informative to confirm whether E is the mechanical work done by the piston, W. The mechanical work W done by the piston per unit time is

$$ \frac{dW}{dt}=P{U}_p=P\left({V}_0-V\right){J}_0 $$

(9.8)

where J₀ is the mass increase in the shocked region per unit time,

$$ {J}_0={\rho}_0{u}_0 $$

(9.9)

Therefore, the energy given to the unit mass by the shock wave, E, is the same as the mechanical power given to the unit mass.

Fig. 9.1

Pressure-Volume (P-V) diagram with adiabatic relation P_c and Hugoniot relation PH, when the initial volume V_0c is compressed to the volume V. Since the shock wave accompanies the increase of entropy by altering flow kinetic energy to the thermal energy at the shock front, the compressed matter has extra internal energy O-A-B compared to that for the case of adiabatic compression. Reprinted with permission from ref. [1]. Copyright by Cambridge University Press

If we can obtain many Hugoniot curves starting from different initial conditions, say different initial densities or initial pressures, we can obtain the thermodynamic quantities ε(P,V) in a wide range as a table or fitting formula. Then, we can solve the hydrodynamic energy equation by altering LHS of (2.22) to the form.

$$ \frac{d\varepsilon}{d t}={\left.\frac{\partial \varepsilon }{\partial P}\right|}_V\frac{d P}{d t}+{\left.\frac{\partial \varepsilon }{\partial V}\right|}_P\frac{d V}{d t} $$

(9.10)

However, still the temperature is unknown. Recently, the temperature is also measured experimentally by measuring the radiation emission from the shocked region. It is simple that the compressed matters are optically thick and the radiation emission is assumed to be Planck radiation and the observed photons in appropriate energy is used to obtain the temperature T(V,P).

Even if the temperature data are also obtained, the thermodynamic consistency should be checked. If not, there should be some miss assumption mentioned already. In the early time of research, the temperature was not be able to be detected, but we have to derive the temperature T(P, V) in order to determine the equation of state. In such case, we can obtain the temperature from P and ε data as follows.

Start with the first law of thermodynamics.

$$ d\varepsilon = TdS- PdV $$

(9.11)

Assume that the internal energy ε is obtained as a function of density and pressure from experimental Hugoniot data, the following relation should be satisfied in LTE.

$$ d\varepsilon ={\left.\frac{\partial \varepsilon }{\partial V}\right|}_P dV+{\left.\frac{\partial \varepsilon }{\partial P}\right|}_V dP $$

(9.12)

Eliminating (9.11) with (9.12) to obtain the following relation to T,

$$ TdS=\left\{P+{\left.\frac{\partial \varepsilon }{\partial V}\right|}_P\right\} dV+{\left.\frac{\partial \varepsilon }{\partial P}\right|}_V dP $$

(9.13)

Equation (9.13) can be modified by dividing it with dS as

$$ T=\left\{P+{\left.\frac{\partial \varepsilon }{\partial V}\right|}_P\right\}{\left.\frac{\partial V}{\partial S}\right|}_P+{\left.\frac{\partial \varepsilon }{\partial P}\right|}_V{\left.\frac{\partial P}{\partial S}\right|}_V $$

(9.14)

This can be also written with use of Maxwell relation [2].

$$ T=\left\{P+{\left.\frac{\partial \varepsilon }{\partial V}\right|}_P\right\}{\left.\frac{\partial T}{\partial P}\right|}_S{\left.-\frac{\partial \varepsilon }{\partial P}\right|}_V{\left.\frac{\partial T}{\partial V}\right|}_S $$

(9.15)

In order to solve (9.14) or (9.15) to obtain T(V,P), we need to know the entropy S=S(V,P). This can be done from (9.11) with dS = 0 for a constant S_i.

$$ \varepsilon \left(V,{S}_i\right)=-\int P\left(V,{S}_i\right) dV $$

(9.16)

Solving (9.16) as implicit unknown for S_i, we obtain the so-called iso-entropy (adiabatic) curve. Once the iso-entropy relation is also obtained, the temperature is given from (9.11) in the form

$$ T={\left.\frac{\partial \varepsilon }{\partial S}\right|}_V $$

(9.17)

Then, we have obtained all thermodynamic quantities. It is better to check the thermodynamic consistency (2.37) is satisfied and how much the error bar is. It is noted that in the case where phase transitions happen, be careful to the fact that some derivatives may diverge. For example, the volume change under a constant pressure in the 1st order phase transition.

9.1.2 Shock EOS Experiments

Let us see a modern method of diagnostics widely used in laser shock experiments. The measurement of a shock wave and piston velocities are carried out with use of the Doppler shift of the irradiating laser for diagnostic purpose. In Fig. 9.2a, a schematics of a typical shock experiment with intense laser is shown, where Omega EP laser stands for laser irradiation from the left [3]. The sample of compression experiment is shown as TATB and the other target materials are for clear imaging of shock propagation in the target.

Fig. 9.2

(a) A typical target structure set-up of EOS experiment with intense lasers. The right (b) is data of VISAR showing the timing of shock passage [3]. Reprint with permission from ref. [3]. Copyright 1998 by American Institute of Physics

By measuring the Doppler shift of the reflected laser of diagnostic for VISAR, the piston velocity is observed. Shock velocity is observed by measuring the time interval of Dt between the shock arrival and exist in the TATB layer as shown in Fig. 9.2b. At the same time, measuring the self-emission spectrum from the shock region, the temperature is inferred.

It is noted that the materials are optically thin for the laser for the diagnostic coming from the left in Fig. 9.2, but optically thick to the spectrum in the self-emission of the shocked material. In case of hydrogen sample, the diagnostic laser is chosen to be visible light and the emission is chosen ultra-violet. Shock Hugoniot experiment for metal materials, we can extend the above method, too. The details of the diagnostic system called VISAR and VOP is well explained in a review paper [4].

Modern shock experiments are done with gas gun, Z-pinch or laser facilities [5]. In the case of Z-pinch and some case of lasers, soft x-ray source generated by such drivers has been used, since the radiation is smooth flux to the ablator of targets to generate planer shocks. By use of such shock Hugoniot experiments, the pressure and density curves are obtained experimentally as already shown in Fig. 8.13.

9.1.3 Shock Experimental Results

Even if the band gap of compressed matters is wide enough in the shock experiments as indicating the gap of insulator in Fig. 8.8, they can show the properties of metal, namely insulator-metal transition (IMT) is observed in shock experiments. It is easy to understand the physics, because the band gap disappears by the shock compression as shown in Fig. 8.5. This is purely insulator-metal transition to be discussed later. At the same time, however, the Fermi-Dirac distribution of the electrons expands the electron distribution to higher energy levels and some fraction of electrons can freely run in the conduction band, when the matter is heated by the shock wave to the temperature where the thermal energy is comparable or larger than the band gap energy.

Experimental result of the insulator-to-metal transition of fluid molecular hydrogen was initially reported by the experiment with gas gun [6]. High shock pressures are generated by impact of a hypervelocity impactor accelerated by a gas gun onto the front surface of aluminum sample holder. The experiment is designed so that a shock wave reverberating hold between Al₂O₃ anvils to compress liquid H₂ or D₂ to pressures of 0.93–1.8 Mbar. Thanks to the multi-shock compression, the temperature is kept at a few 0.1 eV, roughly 10 times lower than a single shock case at the same pressure. In this experiment, electric conductivities are measured. It is measured that the resistivity decreases almost 4 orders of magnitude from 0.9 to 1.4 Mbar and then plateaus to 1.8 Mbar.

This pressure 1.4 Mbar for insulator-to-metal transition is very low compared to the static experiment result 5 Mbar by another static method. Shock compression is also used to clarify the critical pressure to change the insulator to metal in fluid deuterium at relatively low temperature. The experimental data are plotted in Fig. 9.3 in Ref. [7] to find the phase transition boundary curve, where the black solid line inferred by experimental data is the plasma phase transition. It is easily understood such phase transition from the images combined both of Figs. 8.5 and 8.8. The boundary with the black open circles means the points where the band gap becomes about 2 eV, almost semiconductor.

Fig. 9.3

Many experimental data of solid deuterium Hugoniot curves compared to a variety of theoretical curves. Reprint with permission from ref. [8]. Copyright 1998 by American Physical Society

There have been published many experimental data for many materials. In Fig. 9.3, many of experimental Hugoniot data of deuterium are compared to calculated curves [8]. This is the present status of shock compression experiments, where still error bars are not so small and several different theoretical models of equation of state (EOS) look relatively well explain the experimental data.

It was already mentioned that to obtain the dense matter EOS in a wide range, we must get experimental data by changing the initial condition of density and/or pressure. Using the shock wave transmitting from the quartz reference plate, the Hugoniot curves of the different initial density and pressure can be obtained [9]. Since we know the details of the shock Hugoniot of quartz, the shock compressed state of the fluid hydrogen is evaluated with the initial parameters of shock experiment. In Fig. 9.4, the experimental data are plotted for hydrogen and deuterium for four different initial pressures (density), 1 kbar (0.029 mol/cm³), 3 kbar (0.044 mol/cm³), 7 kbar (0.061 mol/cm³), and 15 kbar (0.079 mol/cm³) [9]. The corresponding solid lines are theoretical ones from density-functional-theory with molecular dynamics (DFT-MD) methods. It is seen that small change of the initial pressure and density alters the Hugoniot curve substantially.

Fig. 9.4

The Hugoniot data of hydrogen/deuterium with different initial condition, where the initial density is altered by use of DAC. Reprint with permission from ref. [9]. Copyright 1998 by American Institute of Physics

9.2 Equation of State of Hydrogen at High Pressure

The compressible fluid dynamics becomes important to study and apply the dynamics of plasmas produced by intense lasers, because the energy flux (W/cm²) is extremely high on the surface of the matters irradiated by such lasers and the pressure over 1 Mbar (100 GPa) is easily generated as shown in Chap. 3. Any solid matters are highly compressed and heated under one million atmosphere pressure. To know the dynamics of such plasmas, realistic equation of state (EOS) should be prepared before any hydrodynamic analysis.

Here, a brief review of the equation of state (EOS) for the liquid or solid hydrogen is given since most of the important physics are included in the case of hydrogen. Such high-pressure properties of the matters have been studied in the community of high-pressure physics for a long time and the standard technique to produce high-pressure has been done with Diamond Anvil Cell (DAC) method.

The DAC uses two diamonds as shown in Fig. 9.5 to apply high pressure to a small sample sandwiched by two diamonds with help of mechanical pressure to them [10]. With advancement of technology, the maximum pressure approached to about 5 Mbar (500 Gpa). It is noted that DAC can compress matters while keeping the sample temperature low enough. So, it cannot be used to obtain experimental data for the wide range of matter temperature at high-pressure. Note that the laser shock wave method has appeared as an alternative way to study the high-pressure physics in higher pressure region.

Fig. 9.5

The image of structure of the diamond anvil cell (DAC). A small sample is compressed by the force on the both diamonds

The physics of molecular bonding of the hydrogen gas was explained already. It is well known that the hydrogen becomes molecular solid at extremely low temperature. The phase diagram of hydrogen at high pressure is shown in Fig. 9.6 [11]. Compared to Figs. 9.4 and 9.6 is more precisely in high-pressure and low-temperature region, based on the recent theoretical results reported in [12]. Hydrogen has four different solid states at very low temperature. At the atmospheric pressure it is normal molecular solid (Solid I), but it changes the lattice structure at high pressure to Solid I, II, and III [13]. The physics in such low temperature region is out of the present textbook and the readers can obtain the image of the lattice structures drawn in [13].

Fig. 9.6

Phase diagram of hydrogen at high pressure and relatively low temperature. The melting boundary is shown with red curve, and the boundary of plasma phase transition is shown with blue curve. The critical point is the triple-phase point. It is predicted that the solid metallic hydrogen appears at high-pressure under very low temperature as shown with red dot line. Reprinted with permission from ref. [11]. Copyright by PNAS

Insulator to metal transition of hydrogen at high pressure is very fundamental in the condense matter physics. Over 80 years ago Wigner and Huntington predicted that if solid molecule hydrogen was sufficiently compressed in the T = 0 K limit, the molecules would dissociate to form atomic metal hydrogen (MH) [14]. In this famous old paper, the authors calculated the electron wave functions with Slater matrix in so-called Wigner-Sitz cell. It is concluded that the total electron state become in free state to form metallic hydrogen at pressure of 0.25 Mbar (25 Gpa).

9.2.1 Insulator Metal Transition

Before talking about the insulator-metal transition (IMT) at high-pressure, let us remember the definition of materials. The solid materials are classified as metals, semiconductors, and insulators at T = 0 K based on the band gap theory as schematically shown in Fig. 9.4. The bands in quantum energy levels are classified as valence and conduction bands. Electrons in the valence band cannot move and all electrons in the insulators are in the valence band. On the other hand, electrons in the conduction band can easily move in the materials and the metals have enough electrons in the conduction band. The semiconductors are in the middle. If the insulators have energy gap over which electrons easily jump up to the conduction band with absorption of photon energy of the ultra-violet component of wavelength 400 nm (~3.2 eV), the materials are defined as semiconductors.

In the case of insulators, electrons are filled in the density of state below Fermi-energy as explained in Fig. 8.5, and the band gap is wide not allowing electrons in the conduction band. It is, however, noted that the band gap is a function of the average atomic distance of the matters, and it changes as the matters are compressed as shown in Fig. 8.11. In the metals, enough electrons are in the energy level higher than the Fermi energy level, which is the zero energy in the case of an atomic energy. When a condensed matter is compressed and the energy levels of the bands change so that the electrons located in the valence band becomes free, the compressed matters show the properties of metals thanks to such free electrons. This is a rough explanation of the IM transition of hydrogen at high pressure for the temperature T = 0 case.

As increase of the pressure of hydrogen sample, the semiconductor state appears where external ultra-violet photons are absorbed at the sample surface. Then, after the metal transitions the photons are reflected almost completely by the free electron plasma current near the surface. These two phenomena are regarded the evidence of IM transition at high-pressure. This transition is shown above 4 Mbar in Fig. 9.6.

It is also suggested from Fig. 9.6 that at high pressure lower than the critical value of the MI transition, substantial free electrons in the conduction band can be supplied if the temperature of the matters are increased by k_BT > ΔE, where ΔE is the energy of the gap. Such transition from insulator to metal at relatively high temperature in the compressed matter is studied by Zel’dovich and Landau [15]. Such phase transition is plotted with blue line as plasma phase transition in Fig. 9.6. In this case, the hydrogen phase changes from the solid to liquid insulator and then to metallic liquid as shown in Fig. 9.6.

9.2.2 Computational Studies

There are many computational methods to obtain the band structures and electron density structures in high-pressure regime based on quantum many-body problem. Simply saying, it is numerical modeling to solve many electrons Hartree-Fock equation in many nuclei, say more than 10²⁰ electrons should be solved consistently as shown in Chap. 8.5. It is impossible even with supercomputers in near future.

One trial for the compressed matters in an early time has been done with so-called discrete-variational Xα method (DX- Xα) [16]. They considered clusters with a crystal structure of neon. Solving many wave functions for one set of nucleus and electrons with appropriate boundary conditions and they are used for the base functions for variational methods for total wave functions of all electrons in the cluster. Numerical atomic basis orbits are solutions of the one-electron wave equation containing the spherically symmetric potential

$$ V(r)={V}_C+{V}_{ex}+{V}_W $$

(9.18)

Here V_C is the Coulomb potential due to the nucleus and direct part of the inter-electronic interaction. V_ex is the Slater exchange potential, which is of local assumption and proportional to the (density)^1/3 as shown in (15.4.33). V_W is a model potential of the wall corresponding to the ion sphere.

In Fig. 9.7, the energy levels of all electrons in 13 neon nuclei are plotted as functions of interatomic distance normalized by Bohr radius. All electrons energy levels tend to those of a single atom for the case where the atomic distance is large. With decrease of the distance the ground state energy decreases thanks to the potential of the neighbor nucleon, but the energy levels of the states whose wave functions are wider and affected by the other electrons are spread as already pointed out schematically in Fig. 8.11. It is noted that most of the electrons at the upper energy levels are free electrons and they are jumping around the cluster. They can be regarded as free electrons and the energy levels of the free state are speculated going down in Fig. 9.7 as the density increases. There are finite number of eigen states in Fig. 9.7, because of finite number of the base functions in the computation.

Fig. 9.7

Computational result of eigen-energies of totally 13 neon nuclei are compressed with 10 × 13 (=130) electrons. Total wave functions have 130 energy levels. At far interatomic distance, the energy levels of each isolated atom is obtained, while as decrease of the distance, they are separated and makes band structure [16]. Reprint with permission from ref. [16]. Copyright 1998 by American Physical Society

Fig. 9.8

The phase transition is clearly observed computationally. The radial distribution function is the radial distribution of probability of the surrounding nuclei. From orange to blue lines, the increase of the density changes the probability dramatically. Reprint with permission from ref. [18]. Copyright 1998 by American Physical Society

More precise calculations have been done with ab initio molecular dynamics (MD), quantum Monte Carlo (QMC) and density functional theory (DFT). Computations have been done for hydrogen at pressure up to several Mbar and temperature above the melting line up to 1500–2000 K. The first-order phase transition in liquid hydrogen, between a low conductivity molecular state and a high conductivity atomic state has been observed. The phase transition is characterized by the abrupt change of electric conductivity and the radial distribution function, which are derived consistently from the computational results [42, 43].

In Fig. 9.8, it is very clear that the phase transition dramatically happens at the density around 1.03 g/cm³ as the proton-proton radial distribution function (pair-correlation function) g(r) in blue changes to that in red [18]. In the computation, the density is changed with a constant temperature 700 K, isotherm. It is seen in Fig. 9.8 that the pair function before the phase transition behaves like a lattice, while once the phase transition happens the ordered structure disappears, and protons are located randomly in such metallic state.

In Fig. 9.9, the resultant electric conductivities are shown for the case of four different temperatures [19] and three temperatures [17]. The abrupt increase of the conductivity is seen at high pressure and such phase transitions happen at lower pressure at higher temperature. These transitions correspond to the plasma phase transition in Fig. 9.6. Such insulator-metal transition can be experimentally observed by detecting abrupt change of reflectivity of probing laser light from the pressured sample.

Fig. 9.9

Pressure dependence of electric conductivity at given temperatures. The phase transition is clearly observed. Reprint with permission from ref. [18]. Copyright 1998 by American Physical Society

9.2.3 Experimental Evidence of Insulator Metal Transition

As mentioned before, the hydrogen metal state appears at more pressure than that at which the band structure becomes of semiconductor. With optical measurement of solid hydrogen at T = 100 K under the pressure of 300, 306, 311, and 316 GPa with DAC, the transmission spectra show photon absorption edge and its photon energy decreases as the pressure increases [20]. Since the photons of visible light are completely absorbed by the hydrogen, it is called black hydrogen. In this experiment, the technical maximum pressure was 320 Gpa, but the data extrapolate to the higher pressure, consequently indicating the prediction that the metal hydrogen should be observed at about 450 GPa when the gap closes.

It is reported that the Wigner-Huntington insulator-metal transition predicted in 1935 was experimentally observed with the diamond anvil cell (DAC) [21]. Production of the metallic hydrogen has been a great challenge to condense matter physics and applications. At a pressure of 495 GPa hydrogen is found to become metallic with reflectivity as high as 91%. This critical pressure is about 10 times higher than the value predicted by Wigner-Huntington about 80 years ago, while it was near the value predicted by the extrapolation of the photon absorption edge mentioned above at15 years ago. Comparison with a theoretical model on reflectance inferred the plasma frequency of about 33 eV at T = 5.5 K, which corresponds to the free electron density of 7 ~ 9 × 10²³ cm⁻³. This number is consistent with the value of estimated atom number density. This accomplishment is regarded to the production of Wigner-Huntington transition to the atomic metallic hydrogen in the laboratory [21].

9.3 Radial Distribution Function and Strongly Coupled Plasma

9.3.1 Interaction Energy in Ideal Plasma

As explained in Chap. 2 in Volume 1, the Debye shielding is given for the ideal plasma in the form.

$$ \rho (r)=-\frac{1}{4\pi }{k}_D^2\frac{q}{r}\mathit{\exp}\left(-{k}_Dr\right) $$

(9.19)

where q = Ze or = −e for the central charge of ion or electron, respectively. Then the Coulomb interaction energy of such Debye shielded charged particle can be obtained as

$$ \int \rho (r)\frac{q}{4\pi {\varepsilon}_0r}d\boldsymbol{r}=-\frac{q^2{k}_D}{4\pi {\varepsilon}_0} $$

(9.20)

Summing up two cases of an ion at the center and the electron at the center, and dividing by two for avoiding double counting, the following total energy density of plasma particles is obtained

$$ {E}_{tot}={E}_{kin}+{E}_{int}=\frac{3}{2}\left({n}_i+{n}_e\right)T-\frac{1}{8\pi }{k}_D^3T $$

(9.21)

The Coulomb interaction due to Debye potential is attractive force for both of ion and electron because the interaction energy of the second term of RHS in (9.21) is negative.

The ratio of the interaction energy to thermal energy is a dimensionless parameter proportional to the plasma Λ defined as

$$ \Lambda =4\pi n{\lambda}_D^3 $$

(9.22)

where n = n_i = n_e is assumed for simplicity and λ_D is the Debye length. The Λ is roughly equal to the number of particles in the Debye sphere, which is required to be much larger than unity for the ideal plasma. For hydrogen plasma the energy ration is

$$ \frac{E_{kin}}{E_{int}}=24\pi n{\lambda}_D^3=3\varLambda $$

(9.23)

The plasma Λ is nothing without the ration of both energies and it approaches the unity as the increase of the interaction energy. It is noted that as Coulomb logarithm approaches unity, the collisional coefficients based on Debye shielding model becomes not applicable and some new theory is required. The plasma Λ is also related to the ion-ion coupling parameter Γ_ii defined in the form:

$$ {\varGamma}_{ii}=\sqrt[3]{3}{\varLambda}^{-2/3} $$

(9.24)

It is important to study the physical properties of such strongly coupled plasma to study the physics near the centers of stars and giant planets. Even in the industrial application of plasma or planet formation with accretion disk in Universe, for example, so-called dust plasma plays important role in the evolution of system. In case of such dusty plasma, the coupling parameter Γ_ii approaches to unity even though the plasma number density is very low compared to the solid density. The dusty plasma is made of many giant particles whose charge is thousands of the elementary charge, namely an effective Z of each cluster of particles is very large. In such case, plasma becomes strongly coupled even if the density is not so high.

In addition, it is useful to enumerate the reason why Debye shielding model is violated near the Coulomb logarithm near unity:

• 1. The interaction energy of the order of kinetic energy means T ≈ eϕ in Deby theory and Taylor expansion cannot be used in this case.

• 2. In such condition, the number of charged particles in Debye sphere with radius λ_D is only a few particles and the statistical assumption like Boltzmann distribution cannot be used in deriving (9.19).

In such strongly coupled plasma, Rutherford scattering, and Debye shielding cannot be used to obtain the transport coefficients due to Coulomb scattering in plasma.

9.3.2 Strongly Coupled Plasmas

The radial probability function of neighboring ions has been given statistically in (9.19). As we see above, however, it is not a good approximation with Debye shielding, and we must ask more precise study. In case of high-density and relatively low temperature plasma, the electron Fermi energy is high and spatially uniform density distribution of electron can be a good assumption. Such plasma is called one component plasma (OCP) and its ion radial distribution function and related physics have been studied, for example, by S. Ichimaru [22].

In Fig. 9.10, the radial distribution function g(r) calculated by Monte Carlo method is shown for different ion coupling parameter Γ_ii from 0.1 to 140 [23]. For the case of a small Γ_ii showing week coupling, the distribution is like that suggested by (9.19). With the increase of the ion coupling parameter, the probability inside the ion sphere is going to vanish and large coupling makes the function overshoot outside the ion sphere radius. In this case the neighbor ions can frequently approached around the sphere the radius of which is the value of the peak in Fig. 9.11. It is known that at the coupling parameter around 178, the liquid-solid phase transition is observed in such OCP. This phase transition is called glass phase transition because the ions make lattice structure in a short distance, but it like an amorphous and no crystal structure is observed over a long distance.

Fig. 9.10

Radial distribution function of one-component plasmas with different ion-ion coupling parameters. The radius is normalized by the average ion distance. Reprint with permission from ref. [23]. Copyright 1998 by American Physical Society

Fig. 9.11

Image of a snap shot of the most probable ion configurations suggested by an example of radial distribution function. Reprint from ref. [23] with kind permission from Springer Science + Business Media

Such strongly coupled plasma has been studied to know the physics inside of white dwarfs, giant planets, and so on. To know the ion distribution, for example, inside of the white dwarfs, where electron strongly degenerates and OCP is good approximation. The precise radial distribution function is very interesting to know the fusion reaction due to the overlapping of two ions wave functions at an extremely high density such as the center of white dwarf characterized by the density 10⁹ g/cm³. In addition, the surface of neutron stars is also in such strongly coupled plasma state.

The classical statistical Monte Carlo is applicable to ion-ion coupling, but it is doubtful to apply to the centered ion and electrons interaction. This is because electrons must be treated quantum mechanically. One possibility is to use the density-functional method explained in Chap. 8.4. To know the physics simpler, however, the electron density profile obtained after solving Thomas-Fermi model is widely used for the electron radial distribution function around the centered ion. Therefore, using the electron density profile or electrostatic potential around the central ion, the transport coefficients can be calculated by use of the effective Coulomb logarithm explained later in this chapter.

9.4 Quantum Scattering of Electrons in Coulomb Field

In high density near the solid density, the scattering analysis should be done quantum mechanically. For example, the de Broglie wavelength of electron with kinetic energy equal to that of the 1s state of Hydrogen atom is about Bohr radius 0.5Å. This is the same order of the average ion distance near solid density. If the charge density distribution statistically averaged around each ion can be obtained by some method, the differential cross section of Coulomb scattering can be calculated based on Schrodinger equation for one particle as shown in Chap. 5. It is noted that the following non-relativistic derivation can extend to the relativistic case in replacing the wavenumber of electron wave function with the momentum vector for relativistic collision.

As was discussed in Chap. 5, Born approximation is the basic method on such scattering problem. Born approximation is widely used, not only atomic physics, but also nuclear physics to study charge distribution inside nucleus, including quarks. In nuclear physics, relativistic scattering should be considered because injected electron energy is in the rage of 100 MeV, but here the impinging electron energy is in the rage of 1–100 eV and non-relativistic treatment is enough.

When the density distribution screening the central charge q is assumed to be given as a function of the radial coordinate r, the potential V(r) can be obtained by solving the Poisson equation.

$$ {\varepsilon}_0{\nabla}^2V(r)=- q\delta (r)-{\rho}_{sc}(r) $$

(9.25)

For a given potential and no change of the charge around the point charge, one electron Schrodinger equation reduces to a simple equation for scattering problem:

$$ \left\{-\frac{{\mathrm{\hslash}}^2}{2m}{\nabla}^2+V(r)-E\right\}\psi \left(\boldsymbol{r}\right)=0 $$

(9.26)

This can be written in the form

$$ \left({\nabla}^2+{k}_0^2\right)\psi =\frac{2m}{{\mathrm{\hslash}}^2} V\psi $$

(9.27)

Where k₀ is the wavenumber of the incident electron wave defined to be

$$ {k}_0^2=\frac{2 mE}{{\mathrm{\hslash}}^2} $$

(9.28)

9.4.1 Born Approximation

Here the scattering is elastic one and no energy change happens, namely the energy E in (9.26) should be kept constant. The homogeneous solution of (9.27) for V = 0 is the plane wave being incident to the target and its solution is

$$ \psi ={\psi}_0=\mathit{\exp}\left(i{\boldsymbol{k}}_0\cdotp \boldsymbol{r}\right) $$

(9.29)

In solving (9.27) mathematically to obtain a formal solution, it is very useful to know that LHS of (9.27) is Helmholtz equation. Using Green function to the Helmholtz homogeneous term, (9.27) can be given in the following integral equation form.

$$ \psi \left(\boldsymbol{r}\right)={\psi}_0\left(\boldsymbol{r}\right)-\frac{2m}{{\mathrm{\hslash}}^2}\int \frac{e^{i{k}_0\left|\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right|}}{4\pi \left|\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right|}V\left({\boldsymbol{r}}^{\prime}\right)\psi \left({\boldsymbol{r}}^{\prime}\right)d{\boldsymbol{r}}^{\prime } $$

(9.30)

where ψ₀(r) is the general solution of the homogeneous equation of (9.27) in neglecting RHS and homogeneous solution is the incident wave (9.29). Of course, (9.30) is not the final solution of (9.27), but it should be noted that (9.30) gives us the exact solution when the interaction potential V(r) is given. It is possible to solve (9.30) with use of computer, then the well-known iteration method will be used starting from an appropriate initial solution. It is reasonable to start with solution obtained by Born approximation which will be described below. The solution of Born approximation is good enough in many cases as will be explained later.

In calculation of the scattering by the central charge, we would like to know the wave function after the scattering. This means the coordinate r in (9.30) can be assumed to be very far from the central region where the potential V(r) is large enough and cannot be neglected in the integral (9.30). Namely, if r > > r’ then the following approximation can be used.

$$ \left|\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right|\approx \sqrt{{\left(\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right)}^2}=\sqrt{r^2-2\boldsymbol{r}\cdotp {\boldsymbol{r}}^{\prime }+r{\prime}^2}\approx r-\boldsymbol{r}\cdotp {\boldsymbol{r}}^{\prime }/r $$

$$ \frac{1}{\left|\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right|}\approx \frac{1}{r\left|1-\boldsymbol{r}\cdotp {\boldsymbol{r}}^{\prime }/{r}^2\right|}\approx \frac{1}{r}\left(1+\frac{\boldsymbol{r}\cdotp {\boldsymbol{r}}^{\prime }}{r^2}\right)\approx \frac{1}{r} $$

(9.31)

This is the first order expansion to the small value r’/r. In the solution the scattered direction is r/r and its wave number is equal to k₀ because of elastic scattering, we can introduce the scattered wavenumber k₁ as follows.

$$ {\boldsymbol{k}}_1={k}_0\frac{\boldsymbol{r}}{r} $$

(9.32)

It is noted that the first order term of (9.31) is important in the exponential term but can be neglected in the denominator of the integral. Inserting (9.31) in the exponential term in (9.30) and using the scattered wave momentum (9.32), (9.31) is written to be

$$ \psi \left(\boldsymbol{r}\right)\to {e}^{i{\boldsymbol{k}}_0\cdotp \boldsymbol{r}}+\frac{e^{i{k}_0r}}{r}f\left(\theta \right)\kern1em \left(r\to \infty \right) $$

(9.33)

$$ f\left(\theta \right)=-\frac{m}{2\pi {\mathrm{\hslash}}^2}\int {e}^{-i{\boldsymbol{k}}_1\cdotp \boldsymbol{r}\prime }V\left({\boldsymbol{r}}^{\prime}\right)\psi \left({\boldsymbol{r}}^{\prime}\right)d{\boldsymbol{r}}^{\prime } $$

(9.34)

$$ \theta =\frac{1}{k_0^2}{\boldsymbol{k}}_0\cdotp {\boldsymbol{k}}_1 $$

(9.35)

In many cases, we assume that the potential in (9.34) is spherically symmetric [V(r) = V(r)] and the scattering is axially symmetric and scattering angle depends only the angle θ made by the incident direction and scattered direction as defined in (9.35). It is informative to rewrite (9.34) in the following form for the amplitude of scattered wave,

$$ {\displaystyle \begin{array}{l}f\left(\theta \right)=-\frac{m}{2\pi {\mathrm{\hslash}}^2}\left.\phi \right|V\left|\psi \right.\\ {}\phi \left(\boldsymbol{r}\right)={e}^{i{\boldsymbol{k}}_1\cdotp \boldsymbol{r}}\end{array}} $$

(9.36)

where ϕ(r) is the plane wave of the incident electron wave function.

The Born approximation can be used if the scattered wave component of the second term of (9.33) is much smaller than the incident component of the first term of (9.33). Then, the perturbation theory can be used to assume that the wave function in the second term of (9.33) can be replaced with that shown in (9.29). The following relation is obtained

$$ f\left(\theta \right)=-\frac{m}{2\pi {\mathrm{\hslash}}^2}\int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}\prime }V\left({r}^{\prime}\right)d{\boldsymbol{r}}^{\prime } $$

(9.37)

Where the following the wave vector made of the incident and scattered wave vectors is introduced.

$$ \boldsymbol{q}={\boldsymbol{k}}_0-{\boldsymbol{k}}_1 $$

(9.38)

Its absolute value is easily calculated with use of (9.32) as follows.

$$ {\displaystyle \begin{array}{c}q=\left|{\boldsymbol{k}}_0-{\boldsymbol{k}}_1\right|=\sqrt{k_0^2+{k}_1^2-2{k}_0{k}_1\cos \theta}\\ {}={k}_0\sqrt{2\left(1-{\cos}^2\theta \right)}=2{k}_0\sin \left(\theta /2\right)\end{array}} $$

(9.39)

It should be noted that (9.37) is proportional to the Fourier transformation of the scattering potential V(r) with respect to the wave vector defined in (9.38). For the spherically symmetric potential, (9.37) can be written

$$ f\left(\theta \right)=-\frac{m}{\hslash^2}{\int}_0^{\infty }{r}^{\prime 2}\ {d r}^{\prime }{\int}_0^{\pi}\mathit{\sin}{\theta}^{\prime }{d\theta}^{\prime}\mathit{\exp}\left({iqr}^{\prime}\mathit{\cos}{\theta}^{\prime}\right)V\left({r}^{\prime}\right) $$

(9.40)

After integration to the angle, (9.40) become the following simple form

$$ f\left(\theta \right)=-\frac{2m}{{\mathrm{\hslash}}^2q}{\int}_0^{\infty }{r}^{\prime }V\left({r}^{\prime}\right)\mathit{\sin}\left({qr}^{\prime}\right){dr}^{\prime } $$

(9.41)

where q is given in (9.39).

9.4.2 Differential Cross Section

As is defined in (9.33) the function f(θ) is clear to provide the differential scattering cross section in the form.

$$ \frac{d\sigma}{d\varOmega}={\left|f\left(\theta \right)\right|}^2 $$

(9.42)

Let us calculate the scattering cross section for the case of the Debye potential given in (5.7). Inserting (5.7) to V in (9.41) and using the formula

$$ {\int}_0^{\infty}\mathit{\exp}\left(- ax\right)\mathit{\sin}(bx) dx=\frac{b}{a^2+{b}^2} $$

(9.43)

(9.41) reduces to the following form.

$$ f\left(\theta \right)=-\frac{meQ}{2\pi {\varepsilon}_0{\mathrm{\hslash}}^2}\frac{1}{q^2+{k_D}^2} $$

(9.44)

Since q is given in (9.39), the following differential scattering cross section is obtained for the Debye potential

$$ \frac{d\sigma}{d\varOmega}={\left|f\left(\theta \right)\right|}^2={\left(\frac{Z{e}^2}{4\pi {\varepsilon}_0{E}_0}\right)}^2\frac{1}{{\left[4{\mathit{\sin}}^2\left(\theta /2\right)+{\left({k}_D/{k}_0\right)}^2\right]}^2} $$

(9.45)

In deriving (9.45), the charge of the central scatter is assumed Q = Ze and the following relation is used

$$ {E}_0=\frac{{\mathrm{\hslash}}^2{k_0}^2}{2m} $$

(9.46)

where E₀ is the kinetic energy of the incident electron. In the limit of Rutherford scattering, Debye screening is not considered, namely k_D ➔ 0. In this limit, the well-known differential cross section is obtained.

$$ \frac{d\sigma}{d\varOmega}={\left(\frac{Z{e}^2}{8\pi {\varepsilon}_0{E}_0}\right)}^2\frac{1}{{\mathit{\sin}}^4\left(\theta /2\right)}=\frac{{b_0}^2}{{\left(1-{\mathit{\cos}}^2\theta \right)}^2} $$

(9.47)

Here the impact parameter b₀ is used. The Rutherford scattering differential cross section is exactly reproduced using Quantum mechanical Born approximation derived in (9.47).

Then, the total cross section of the scattering by Rutherford formula is calculated by changing as x = 1 −  cos θ and the total cross section by Coulomb scattering is given to be

$$ {\sigma}_C=\int \frac{d\sigma}{d\varOmega}\left(1-\mathit{\cos}\theta \right) d\varOmega =2\pi {b_0}^2{\int}_{x_{min}}^1\frac{1}{x} dx $$

(9.48)

In (9.48), the term in the second eq. 1 −  cos θ is a weight factor to the contribution of the cross section since the scattering with the angle of θ can change the initial momentum to the fraction of 1 −  cos θ. Thanks to this contribution, a simple integration shown in the last term in (9.48) is obtained.

$$ {\sigma}_C=2\pi {b_0}^2\mathit{\ln}\left(1/{x}_{min}\right) $$

(9.49)

To avoid infinity by accumulation of small angle scattering in Rutherford model, the minimum angle scattering is introduced, namely

$$ {x}_{min}={\theta_{min}}^2/2 $$

(9.50)

How to evaluate the minimum angle in the present model is same as the minimum impact factor introduced in previous section. The Debye shielding is very important for converging the integral. A simple evaluation of the minimum angle should be given by the following evaluation

$$ {\theta}_{min}\approx \frac{\left(\frac{Z{e}^2}{4\pi {\varepsilon}_0{\lambda}_D^2}\frac{\lambda_D}{v}\right)}{mv} $$

(9.51)

Where the scattering to the electron by the central charge is calculated up to the radius of Debye length. Taking an average energy of the plasma electron as mv² ≈ T, (9.51) can be approximated as

$$ {\theta}_{min}\approx \frac{1}{4\pi n{\lambda}_D^3}=\frac{1}{\varLambda } $$

(9.52)

Where Coulomb log defined in (9.22) is used. Finally, the total cross section of electrons scattering by Coulomb field in plasma is obtained as

$$ {\sigma}_C=4\pi {b_0}^2\mathit{\ln}\left(\varLambda \right) $$

(9.53)

The simply evaluated cross section in Volume 1 is obtained after more precise calculation starting from Rutherford differential cross section given in (9.47).

Finally, note that with use of computer, it is possible to solve the exact solution of (9.25) and (9.27) for the case of thermodynamic equilibrium condition, for example, assuming Fermi-Dirac distribution. This is the next step after the above Born approximation and the incident wave is also modified by the Coulomb potential of the ventral charge. This is called distorted wave method.

9.4.3 Density Distribution and Form Factor

From (9.41) the differential scattering cross section can be obtained provided that the potential structure by the screening charge distribution is given. Mathematically more convenient if (9.41) is an integration of the density profile instead of the potential obtained solving Poisson equation. By use of Green function theorem, this purpose has been accomplished. The Green theorem to volume integral with Laplacian operator is given

$$ {\int}_V dV\left(A{\nabla}^2B-B{\nabla}^2A\right)={\oint}_Sd\boldsymbol{S}\cdotp \left(A\nabla B-B\nabla A\right) $$

(9.54)

Where A and B are arbitrary functions defined in space. If the volume is taken so that the integrand of RHS sufficiently disappears on the integral surface, RHS vanishes in the surface integration. When any combination of two scalar functions A and B satisfies this condition, LHS of (9.54) always vanishes for the infinite volume integral, namely

$$ {\int}_V dVA{\nabla}^2B={\int}_V dVB{\nabla}^2A $$

(9.55)

By use of (9.55) to the integral (9.37), the following relation is obtained.

$$ \int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}V\left(\boldsymbol{r}\right)d\boldsymbol{r}=-\frac{1}{q^2}\int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}{\nabla}^2V\left(\boldsymbol{r}\right)d\boldsymbol{r} $$

(9.56)

Then, (9.56) and Poisson equation

$$ {\nabla}^2\left[\frac{V\left(\boldsymbol{r}\right)}{e}\right]=-\frac{1}{\varepsilon_0}\rho \left(\boldsymbol{r}\right) $$

(9.57)

gives the following surprising relation to the integrand of (9.37)

$$ \int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}V\left(\boldsymbol{r}\right)d\boldsymbol{r}=\frac{e}{\varepsilon_0{q}^2}\int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}\rho \left(\boldsymbol{r}\right)d\boldsymbol{r} $$

(9.58)

$$ f\left(\theta \right)=-\frac{m}{2\pi {\mathrm{\hslash}}^2}\frac{e}{\varepsilon_0{q}^2}\int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}\rho \left(\boldsymbol{r}\right)d\boldsymbol{r} $$

(9.59)

It is easily found that the integration of (9.58) is Fourier transformation of the charge density distribution around the scattering center charge.

Let us examine how the differential scattering cross section defined by (9.42) and (9.37) is conveniently reduces to Rutherford scattering case with use of (9.53). Inserting the charge density as Zeδ(r) in (9.59) the following relation is obtained.

$$ f{\left(\theta \right)}^2={\left(\frac{Z{e}^2m}{2\pi {\mathrm{\hslash}}^2{\varepsilon}_0}\right)}^2\frac{1}{q^4} $$

(9.60)

It is easily seen that using (9.39), (9.60) reproduced Rutherford scattering given in (9.47).

9.4.4 Form Factor and Nucleus Charge Experiment

Introducing the concept of form factor of electron scattering by the screened Coulomb field around the nucleus with charge Ze, (9.59) gives the following definition to the differential scattering cross section

$$ \frac{d\sigma}{d\varOmega}={\left(\frac{d\sigma}{d\varOmega}\right)}_{Rutherford}{\left|F(q)\right|}^2 $$

(9.61)

Where F(q) is called form factor defined as

$$ F(q)=\int {e}^{i\boldsymbol{q}\cdotp \boldsymbol{r}}{\rho}_N\left(\boldsymbol{r}\right)d\boldsymbol{r} $$

(9.62)

F(q) has been widely used to study the charge distribution of nucleus in nuclear physics experiment. The density distribution in (9.62) is normalized one so that its volume integral becomes equal to unity.

$$ {\rho}_N\left(\boldsymbol{r}\right)=\frac{1}{Ze}\rho \left(\boldsymbol{r}\right) $$

(9.63)

It should be noted that the form factor.

If mono-energetic electron beam is available and strongly coupled plasma can be kept for a long time, experimental data of the scattering can be used to obtain the density distribution around the central charge. Through such experiment the angular distribution of the differential scattering cross section is obtained and the experimental data of angular distribution of scattering is compared with that given in (9.61). Then, the form factor obtained experimentally can be used to determined statistically averaged radial density distribution through the inverse-Fourier transform

$$ {\rho}_N\left(\boldsymbol{r}\right)=\frac{1}{{\left(2\pi \right)}^3}\int {e}^{-i\boldsymbol{q}\cdotp \boldsymbol{r}}F\left(\boldsymbol{q}\right)d\boldsymbol{q} $$

(9.64)

Once the density distribution is obtained, the effective Coulomb collision cross section will be used to calculate the revised transport coefficients in wide range of plasma parameter. However, it has not been done in plasma, because it is difficult to maintain such plasma for a long time so that enough scattering data are available.

In the beginning of nuclear physics, Rutherford used alpha particles to study inside the atom and he found vary rare scattering of large angle from a foil target. He derived the Rutherford scattering differential cross section shown in (9.47) and concluded that the radial spread of the nucleus is smaller than 10⁻¹² cm. However, proton or such alpha particles are not convenient to study the charge distribution of nucleus, because they interact with nuclear force, and they also have inner structures. Electron is more convenient because it has no internal structure and Coulomb interaction is most dominant compared to another forces.

The study of charge distribution of nucleus with form factor has been done with 500 MeV electron beam accelerated by LINAC in Stanford University in the early 1950s. In the case of nuclear physics scattering, beam electrons are highly relativistic, and we must follow this case.

Example of form factor F(q) obtained in early experiments are plotted in Fig. 9.12 where electron beam of energy from 150 to 300 MeV was impinged and solid circles are experimental data for three different nuclei. Note that the axis of q is given in the unit of momentum with relation p = ℏq and the momentum is given in the unit of (energy)/c. In this case, instead of Rutherford scattering, Mott scattering formula is used. In Mott formula additional physics due to spin and magnetic interaction is added to Rutherford formula.

Fig. 9.12

The form factor F(q) obtained in early experiments. The electron beam of energy from 50 to 300 MeV was impinged and solid circles are experimental data for three different nuclei

For convenience to the readers, the density distribution and its form factor is shown in Fig. 9.13 for several different density distributions.

Fig. 9.13

The charge distributions of nuclei obtained with the experimental data of form factors

Is it possible to do the same kind of experiment for the ions in plasma? Can we, however, obtain the form factor to infer the electron density distribution around an ion? It is possible of atoms at a fixed position, namely solid-state matters, molecules, etc. It is difficult to apply it for experiment in the case of plasmas in which the ions are moving with thermal velocity. In the future, however, very short pulse electron beam and its diagnostic technique are developed, the form factor of the screened ions can be experimentally studied.

It is useful to note that the density distribution of very small scale of plasmas is now experimentally studied with x-ray pulse from XFEL, which is called Small-Angle X-Ray Scattering (SAXS) [24]. The principle and mathematics are the same as described above. The difference is that X-ray beam is used instead of the electron wave packets and the charge distribution is replaced by the x-ray refraction index determined by the electron density distribution.

SAXS has been developed as a particularly suitable technique to characterize structure and form factors of colloidal systems in solution and therefore to probe nanometer-scale structure. The combination of microfluidics and SAXS provides a powerful tool to investigate phase transitions at different molecular levels and relevant timescales. This method can be a new one to study the density structure of dense plasmas produced by ultra-intense lasers.

9.5 Coulomb Log Λ in Dense Quantum Plasmas

We should think about the minimum angle of scattering to avoid divergence in (9.42). It is also important to discuss about why we need to exclude the contribution from large angle scattering more than θ = 90 degree to total scattering cross section in (9.48). In a large angle scattering, it is not appropriate to use classical mechanics, because the kinetic energy of impact electron is less than the Coulomb interaction energy. Born approximation of quantum scattering is not applicable either. Therefore, the integral equation of (9.30) should be solved directly. Of course, such calculation has been carried out using computer, however, it is easily understood that this contribution is negligible as far as the plasma is ideal Λ >> 1. This is the reason way there is also limitation of analysis based on Rutherford formula.

Such quantum deflection contribution has been studied, and the following modified formula is proposed [25]

$$ \mathit{\ln}\left(\varLambda \right)=\frac{1}{2}\mathit{\ln}\left[1+{\left(\frac{b_{max}}{b_{min}}\right)}^2\right]=\frac{1}{2}\mathit{\ln}\left(1+\frac{\lambda_D^2}{b_0^2+{\lambda}_{dB}^2}\right) $$

(9.65)

where electron de-Broglie wavelength

$$ {\lambda}_{dB}=\frac{\mathrm{\hslash}}{\sqrt{mT}} $$

(9.66)

has been introduced to take account of quantum deflection near head-on collision. The modified form in (9.65) is reasonable because it guarantees a positive value of the Coulomb log for a strongly coupled plasmas. In addition, the quantum diffraction effect is modeled in the last term in (9.65).

The better fitting formula for the Debye length including the quantum diffraction effect is also obtained by solving Schrodinger equation directly. The modified Debye length is given in (9.65) with the form

$$ {\lambda}_D\to {\lambda}_D\mathit{\exp}\left\{\frac{1.65-0.4 ln\varLambda}{{\left( ln\varLambda \right)}^{0.65}+1}\right\} $$

(9.67)

where lnΛ evaluated in (9.65) is installed in (9.67). The Debye length re-calculated in (9.67) is used to obtain the fitting formula to the Coulomb logarithm.

This modified Coulomb logarithm has been compared to another theoretical and computational results in Fig. 9.14 [25]. In Fig. 9.14, the fitting formula of (9.65) and (9.67) is plotted with the line (T-M) for the classical case, where de Broglie correction is neglected to compare with the more precise computation of molecular dynamics (MD). The diamond symbols are MD results and the red line is the fitted curve to the MD result. It is seen that the good agreement is obtained with MD results. The blue and orange dashed curves are the theoretical results based on perturbation methods 50].

Fig. 9.14

The effective Coulomb log derived by taking account of strongly coupled quantum effect at high-density plasma. Reprint from ref. [25]. Copyright 2012, with permission from Elsevier

In the MD simulation, quantum effect is not included. The green lines are two theory cases with the de Broglie cut included. Then, the denominator in (9.65) will be increased and the value of Coulomb log decreased as shown in Fig. 9.14. It is noted that the horizontal axis is the coupling parameter defined to be

$$ g={b}_0/{\uplambda}_D. $$

(9.68)

In Fig. 9.14, MD simulation has been carried out with the parameters in the gray region in the upper figure, from weekly coupled plasma g = 0.003 to strongly coupled plasma with g = 20, where the other formulae are calculated with fixed temperature of T = 500 eV. More details are given in the reference.

It is important to note that in the region of large g, Coulomb log approaches zero and it means that running electrons are almost not scattered by the ion charges to provide high conductivity. This is a general property of the dense Quantum plasmas. In such a case, dominant conduction is attributed to the electrons with Fermi energy. The conductivity, for example, the inside of white dwarfs is very high due to the above reason.

9.6 Density Fluctuation and Dynamic Structure Factor

So far, most of the plasmas have been assumed to have constant density in space. If the electron density is completely uniform as well as the charge is also smeared out uniformly, any photons are not scattered from such mathematically idealized plasmas. The ion individuality and electron density fluctuation are subject to Thomson scattering and incident electromagnetic waves are scattered, when the cut-off frequency of plasma is lower than the incident electromagnetic wave frequency ω_cr < ω, where ω is the incident wave frequency and ω_cr is the cut-off (critical) frequency of the plasma with free electron density n_e. The cut-off electron density n_cr is given as

$$ {n}_{cr}=\frac{\varepsilon_0m}{e^2}{\omega}^2,\kern1.5em {n}_{cr}={10}^{27}{\left({\mathrm{\hslash}\omega}_{keV}\right)}^2\left[1/{cm}^3\right] $$

(9.69)

Note that the cut-off density of x-rays is higher than most of electron density of solids.

In measuring the plasma density and temperature, optical diagnostics have been used with use of scattered spectra of laser probe from a local point of plasma. This is common methods used for a variety of plasmas. Depending on the density of plasma, infrared to X-ray coherent light sources are used in laser plasmas.

In this book, it is not mentioned in detail about the optical probe of plasma for relatively low-density plasma such as ablating plasma generated by intense lasers. Since the optical probe measurement, the plasma density should be lower than the cut-off density of plasmas. In addition, the photon number of the optical probe should be large enough so that clear signal of the scattered photons is detected.

For the case of measuring the high-energy density plasmas and warm-dense matters, the optical method demands higher-cut-off density probe. If the matter is at rest such as solid, molecule and so on, then x-rays from synchrotron radiation source (SRS) has been widely used to study the electron density distribution due to the Thomson scattering, the physics of which has been shown in Volume 1. The principle is simple. The electrons irradiated by the optical probe oscillate by the electric field of x-rays and emit the electromagnetic waves with the same frequency. Since the brightness of the synchrotron radiation is relatively low as shown in Fig. 9.15 [26] a long-time exposure is required.

Fig. 9.15

Brightness of photon sources as a function of photon energy (eV). The synchrotron radiation source has been used widely as x-ray source like shown as Spring-8 etc., while XFEL source is monoenergetic and brighter x-ray source such as LCLS etc. Reprint with permission from ref. [26]. Copyright 1998 by American Physical Society

Since the electron density distribution fluctuates in HEDP or WDM with thermal velocity and phase velocity of collective motion, on the other hand, brighter x-ray source with short pulse is required for diagnostics of them. As shown in Fig. 9.15, the relatively large facilities of x-ray free-electron lasers (XFEL) have been constructed in US (LCLS: https://lcls.slac.stanford.edu/), Japan (SACLA: http://xfel.riken.jp/), Europe (European XFEL: xfel.eu). XFELs are also characterized by high-repetition rate pulses, typically 10 HZ. In addition, each pulse consists of many ultra-short pulses. It is 2–100 fs in European XFEL with about 10¹⁰ photons in 25 keV x-ray operation and 10¹² photons for 5 keV.

The ultra-short pulse allows to take snap shots of the scattered x-ray signal in high-density plasma with finite temperature. Not only the snap shots, but the frequency of the plasma waves is also high enough to obtain the oscillation frequencies of such collective modes in dense plasmas. As shown in Chap. 2, the electron plasma wave and ion acoustic wave are typical collective modes in plasmas without external magnetic field. The scattered x-ray spectra can be related to the plasma parameters of density and temperature via comparing to the wave theory of plasmas. Note that the damping process and non-Maxwellian effects also affect the profile of the scattered x-ray spectra, and the kinetic theory of plasma waves and Landau damping to be described in Volume 4 should be applied to the theoretical study.

9.6.1 X-Ray Scattering Diagnostics

In Fig. 9.16, a typical configuration of WDM experiment is shown [27]. The drive laser is irradiated on the target to compress and heat the target material, where the laser intensity profile is shown in the inset figure as two step intensity for pre-compression and main compression. Then, a LCLS XFEL is injected to the compressed WDM. The x-ray diffraction (XRD) and forward x-ray Thomson scattering (FXRTS) and backward one (BXRTS) were measured.

Fig. 9.16

A typical configuration of WDM experiment. The drive laser is irradiated on the thin foil target to compress and heat the target material. The laser intensity profile is shown in the inset figure as two step intensity for pre-compression and main compression. Then, a LCLS XFEL is injected to the compressed WDM. Reprinted by permission from Macmillan Publisher Ltd: ref. [27], copyright 1993

In any plasmas, Thomson scattering is induced by two different processes. They are called non-collective scattering and collective scattering. Thomson scattering in linear and nonlinear cases discussed in Volume 1 corresponds to the non-collective scattering. The scattering of light due to a single electron has been considered. When XFEL is irradiated to a test sample matter, the non-collective scattering is observed as total sum of the scattered x-ray from all electrons in XFEL interacting region.

Since strongly coupled plasmas such as WDM, ion-ion pair corelates via Coulomb force and the pair correlation function as shown in Fig. 9.11, has a characteristic distance as the case for the coupling parameter larger than unity. This means that each scattered x-ray with phase information from each electron cloud overlaps on the imaging plate. This interference makes the image to reflect the information of ion-ion pair correlation. This is the case of non-collective scattering used for XRD imaging.

This is the case of conventional XRD method been used historically to study the atomic structure of crystal and so on. With use of XFEL, coherent and bright x-ray beam can be used to apply the conventional XRD with higher resolution. In addition, the pulse duration of XFEL is very short enough to obtain the pair correlation function even for plasmas such as WDM where ion thermal velocity is small enough to obtain a snap shot with XFEL pulse.

It has been studied in Volume 1 that an incident laser interacts with waves in plasmas to make the plasma waves unstable and the frequency-shifted scattered light is produced. This is called parametric instability due to non-liner ponderomotive force of intense lasers in plasmas. For example, laser is scattered by electron plasma wave, it is called as stimulated Raman scattering. Note that Raman has found the scattering for the case without instability, and it is called Raman scattering.

The same physical phenomenon as Raman scattering is now clearly observed in high-density plasma thanks to bright-coherent XFEL source. The scattering of incident x-ray with the plasma wave due to plasma collective physics corresponds to the collective scattering. In plasma without external magnetic field, electron plasma wave sand ion acoustic waves are spontaneously induced by thermal fluctuation and decay due to Coulomb collision and Landau damping. The dispersion relations of the waves and the damping rates are functions of plasma temperature and density if assuming Maxwell distribution functions. With such scattering spectra, plasma density and temperature can be measured for WDM.

As will be clear with plasma kinetic theory in Volume 3, the plasma waves have relatively long-life time when their wavelength is longer than the Debye length.

$$ k{\lambda}_{De}<\kern0.5em 1 $$

(9.70)

For separating which the non-collective or collective Thomson scattering is dominant from plasma scattering, the nondimensional parameter α is introduced [28, 29].

$$ \alpha =\frac{1}{{k\lambda}_{De}} $$

(9.71)

The collective scattering can be used for high-energy x-ray beam α > 1. Note that XFEL wavenumber k in (9.71) should be replaced by the wavenumber of the plasma waves scattering x-rays. It is approximately acceptable because of the following consideration.

As has been shown in Fig. 9.17 (also see Fig. 4.6 in Volume 1), three waves have to satisfy the matching condition.

$$ {\omega}_0={\omega}_s+\omega \kern1em {\boldsymbol{k}}_0={\boldsymbol{k}}_s+\boldsymbol{k} $$

(9.72)

where (ω₀, k₀), (ω_s, k_s), and (ω, k) represent the incident, scattered and plasma waves, respectively.

Fig. 9.17

Schematics of the incident (X Rays, k₀), and the scattering (Detector, k_s) and the plasma mode (k) satisfying (9.72). The scattered lights are blue shift and red sifts by the electron thermal motions. Reprint with permission from ref. [29]. Copyright 1998 by American Physical Society

In the case without extra explanation, it is assumed that the vectors k₀ and k_s have finite angle and the wavenumber of the plasma waves also has roughly the same absolute values as k₀ and k_s.

It is informative to calculate Debye length of plasma λ_De and the wavelength of x-ray from XFEL, λ_X.

$$ {\displaystyle \begin{array}{l}{\lambda}_{De}=0.74\ {\left(\frac{T_{eV}}{n_{20}}\right)}^{\frac{1}{2}}\kern0.75em \left[ nm\right]\\ {}{\lambda}_X=1.2{\left[{\left(\mathrm{\hslash}\omega \right)}_{keV}\ \right]}^{-1}\kern0.75em \left[ nm\right]\end{array}} $$

(9.73)

where n₂₀ is the electron density in unit of 10²⁰ cm⁻³.

9.6.2 Dynamical Structure Factor (DSF)

So far, the physical reason for Thomson scattering has been explained. It is better to explain the theory of the Thomson scattering from plasma with use of dynamical structure factor (DSF) S(k, ω). The definition of DSF is given as

$$ S\left(\boldsymbol{k},\omega \right)=\frac{1}{2\pi V}\int_{-\infty}^{\infty } dt\left\langle {\rho}_{\boldsymbol{k}}\left({t}^{\prime }+t\right){\rho}_{-\boldsymbol{k}}\left({t}^{\prime}\right)\right\rangle \mathit{\exp}\left( i\omega t\right) $$

(9.74)

Where Fourier decomposed density fluctuation ρ_k(t) is defined to be,

$$ {\rho}_{\boldsymbol{k}}(t)=\int_{-\infty}^{\infty }d\boldsymbol{r}\left[n\left(\boldsymbol{r},t\right)-{n}_0\right]\mathit{\exp}\left(-i\boldsymbol{k}\bullet \boldsymbol{r}\right) $$

(9.75)

The electron density perturbation at a given time n(r, t) − n₀ consists of thermal noise and the enhanced fluctuation by the resonance in plasma, namely electron and ion waves. The DSF sheds light on the oscillation frequency of the waves in plasma. Namely, DSF S(k, ω) has all information of the density fluctuation in the plasma.

In addition, the structure factor (SF) S(k) is defined with DSF as follows.

$$ S\left(\boldsymbol{k}\right)=\frac{1}{N}\left\langle {\left|{\rho}_{\boldsymbol{k}}(t)\right|}^2\right\rangle =\int_{-\infty}^{\infty }S\left(\boldsymbol{k},\omega \right) d\omega $$

(9.76)

This SF is related to the pair (radial) distribution function g(r) given in Fig. 9.10.

$$ g\left(\boldsymbol{r}\right)=1+\frac{1}{n}\int_{-\infty}^{\infty}\frac{d\boldsymbol{k}}{{\left(2\pi \right)}^3}\left[S\left(\boldsymbol{k}\right)-1\right]\mathit{\exp}\left(i\boldsymbol{k}\bullet \boldsymbol{r}\right) $$

(9.77)

Note that the position r is a vector in g(r), while the electron cloud is spherically symmetric around the nucleolus, consequently g(r) is only a function of radius.

The differential scattering cross section defined as fraction of the scattered x-ray to the angle θ between the incident and scattered X-rays per solid angle Ω and the frequency interval Δω of ω = ω_i − ω_s is given in the form [29, 30].

$$ \frac{\partial^2\sigma }{\partial \omega \partial \boldsymbol{\varOmega}}={\sigma}_T\frac{\omega_s}{\omega_0}\frac{1}{2}\left(1+{\mathit{\cos}}^2\theta \right)S\left(k,\omega \right) $$

(9.78)

$$ k=\left|{\boldsymbol{k}}_0-{\boldsymbol{k}}_s\right|=2{\omega}_0\mathit{\sin}\left(\theta /2\right)/c $$

This indicates that we can evaluate the plasma parameters from Thomson scattering spectrum, if we have the theory relating to the DSF S(k, ω) as a function of the plasma parameters such as temperature and density. Note that if the data quality is very fine, electron distribution function can be determined from the scattered spectrum within some assumption.

In general, DSF consist of three terms, namely

• 1. Peaked density profile by bound electros and free electrons forming Debye shielding around each nucleus. They move with ion motion, but the ion thermal velocity is very small so that Doppler shift by the ion motion can be neglected. The form factor of the bound and Debye shielding electron distribution and the ion-ion pair correlation function determines the structure factor and it can be regarded as elastic scattering (ω = 0). Let us write this component as the elastic component, S_elas(k).

• 2. The longitudinal collective motions in plasma have the density structure according to the dispersion relation of the plasma waves. They are the electron plasma wave and ion acoustic wave as discussed in Chap. 2. Including the case of WDM, the dispersion relation with strongly coupling and quantum effects are used for the electron plasma wave [20, Chap. 2 in Ref. 18]. Since they have the frequencies of electron and ion plasma frequencies, the frequency shifts are observed. They are collective mode and in-elastic scattering. Let us write this as the collective component, S_collec(k, ω).

• 3. Another inelastic component in DSF is related to the photoionization of x-ray by a bound electron, the physics of which has been discussed in Chap. 5. In the present case, the process is not absorption of x-ray photon by a bound electron, but the process is inelastic photo absorption by the bound electron. Let us write it as S_be(k, ω).

It is shown by J. Chihara that the total DSF can be written in the form [31, 32].

$$ S\left(k,\omega \right)={S}_{elas}(k)+{S}_{collec}\left(k,\omega \right)+{S}_{be}\left(k,\omega \right) $$

(9.79)

$$ {S}_{elas}\left(k,\omega \right)={\left[f(k)+q(k)\right]}^2{S}_{ii}(k) $$

(9.80)

$$ {S}_{collec}\left(k,\omega \right)={Z}_f{S}_{ee}^0\left(k,\omega \right)\kern0.5em $$

(9.81)

$$ {S}_{be}\left(k,\omega \right)={Z}_b\int {d\omega}^{\prime }{S}_{ce}\left(k,\omega -{\omega}^{\prime}\right){S}_s\left(k,{\omega}^{\prime}\right) $$

(9.82)

where f(k) and q(k) are the form factor of bound electrons and Debye shielding free electrons, and S_ii(k) is the ion-ion pair correlation function. Z_f and Z_b are the numbers of free and bound electrons in the statistically averaged ion. The real spectra of ${S}_{ee}^0\left(k,\omega \right)$ by one free electron is discussed soon.

9.6.3 Elastic Scattering (X-Ray Diffraction: XRD)

In 1914, von Laue observed these diffraction patterns by irradiating metal with X-rays. Braggs and his son also developed his X-ray crystallography in 1915. Each person’s achievements have come to fruition in each year as the Nobel Prize in Physics.

With the advent of X-ray scattering technology in recent years, it has become possible to measure the physical characteristics of high-density plasma and use it for research on high-energy density physics.

When the parameter α in (9.71) is small enough by inserting the parameters in (9.73), Coherent-XRD image can be obtained to provide the ion-ion pair correlation. In Fig. 9.16, the image on XRD is the scattered x-ray intensity image in two-dimensional plate (say, x and y). Since the electrons are located randomly and the scattered x-ray is a snapshot for a few femto-second, the scattering image is like the scattered electron de-Broglie wave image discussed previously. This is the non-collective Thomson scattering method.

The x-ray is an electromagnetic wave and its propagation and scatter in electron density are given by the following equation from Maxwell equations.

$$ {c}^2{\nabla}^2E+\left({\omega}^2-{\omega}_{pe}^2\right)E=0 $$

(9.83)

where the electron plasma frequency is only a function of the electron density:

$$ {\omega}_{pe}^2\propto {n}_e\left(\boldsymbol{r},t\right) $$

(9.84)

Comparing (9.83) with Schrodinger equation for quantum electron scattering (9.26), the mathematics of electron scattering in each potential field can be directly applied to the analysis of coherent XRD image. In the case of x-ray scattering by the electron cloud around a central ion, the potential in (9.26) is replaced as

$$ V(r)\kern0.75em \Longleftrightarrow \kern0.5em -{n}_e(r) $$

(9.85)

In the case of x-ray scattering, Coulomb potential by the central ion should be neglected, namely, it is enough to neglect the term due to Rutherford scattering in (9.61). It is noted that in WDM many electrons are still in bound state in each ion, but the bound electrons also contribute to Thomson scattering with the incident frequency as far as the absolute value of binding energy is smaller than the x-ray photon energy.

Therefore, the forma factor F(q) is used to obtain the electron density profile. The foam factor is also used in identifying a single gold nano-scale particles of size from 10 nm to 80 nm in water solution. The conventional small-angle x-ray scattering (SAXS) has been used to reproduce the electron density profile in the gold by reproducing with the resultant form factor [33]. SAXS is now widely used to measure a time evolution of clumpy density structure in WDM, for example, the growth of ripple amplitude of a surface unstable to Rayleigh-Taylor instability [24].

As a typical experiment, liquid iron scattering measurement with XRD is explained. In this study, the experiment measured the density of liquid iron at pressures up to 116 GPa and 4350 K via static compression using a laser-heated diamond-anvil cell (LH-DAC) [34].

In studying the physics of planets’ inside, it is required to know the physical property of main component of dense metallic cores of planets. This is not only true for Earth, but also for Mercury and Mars, which are expected to have partially molten cores. Density (ρ) and longitudinal sound velocity (VP) are the primary observables of Earth’s liquid outer core.

Therefore, laboratory measurements of these properties at high pressure are of great importance to understand Earth’s and other planets’ core composition and behavior. While determination of density for crystalline materials under high pressure and temperature (P-T) is relatively straight-forward by in situ x-ray diffraction.

The density of liquid iron has been determined via static compression experiments following an innovative analysis of diffuse scattering from liquid [34]. The longitudinal sound velocity was also obtained to 45 GPa and 2700 K based on inelastic x-ray scattering measurements. Combining these results with previous shock-wave data, we determine a thermal equation of state for liquid iron.

In Fig. 9.18, ion-ion pair correlation function is obtained as the radial distribution function [34]. The scattered intensity in the x-y plane at the rear of the liquid iron is shown in Fig. 9.18a, where the scattered angle is used to obtain the scattered wavenumber Q (nm⁻¹). By assuming the spherically symmetric electron cloud distribution around each ion core, the date (a) can be transformed to the radial distribution function shown in Chap. 9.3.

Fig. 9.18

(a); Scattered wavenumber dependence of form factors by elastic scattering of XRD for iron at high pressures produced by DAC. (b); The form factor is inverse-transformed with Fourier method to convert the radial distribution functions. The reduce of the first peak of the radial distribution give the density compressed with high pressure. Reprint with permission from ref. [34]. Copyright 1998 by American Physical Society]

In Fig. 9.18b, the radial distribution function (ion-ion correlation function) g(r) is shown. It is clearly seen the over-shooting oscillation profile is seen indicating that the liquid iron is strongly coupled plasma as shown in Fig. 9.10. Note that low frequency density perturbation due to acoustic wave in the ion liquid has also observed in the spectrum of the scattered x-ray with angle Q = 3 [nm⁻¹].

9.6.4 Collective Thomson Scattering

It is complicated calculation is required to understand the physics behind Thomson scattered x-ray spectra from WDM where plasma waves are excited and damping. The detail derivation of general relativistic case has been given in the book [28] or a review paper [29]. Here, we try to explain the basic physics affecting the scattered spectra from WDM with a finite temperature.

Recall the simple formula of Thomson scattering by an electron oscillating without initial velocity. Thomson scattering cross section σ_T defined as

$$ {\sigma}_T=\frac{8\pi }{3}{r}_e^2,\kern1em {r}_e=\frac{e^2}{4\pi {\varepsilon}_0m{c}^2} $$

(9.86)

where r_e is the electron classical radius and they are given as

$$ {\sigma}_T=67\ \left[{fm}^2\right],\kern2em {r}_e=2.8\ \left[ fm\right]. $$

(9.87)

The scattered power by a single electron to the angle θ made by the incident and scattered directions is given as

$$ \frac{d{P}_s}{I_i d\omega d\boldsymbol{\Omega}}=\frac{3}{16\pi }{\sigma}_T{\mathit{\cos}}^2\theta \delta \left(\omega -{\omega}_i\right) $$

(9.88)

Note that the scattering cross section is independent of incident electromagnetic wave frequency. In (9.88), I_i, ω_i, and P_s are incident intensity and frequency and scattered power. To observe bright scattering photons, total large number of electrons and bright light source are key technology.

From an electron located at r = r_j(t) is oscillated by the incident XFEL with its acceleration α_j in the form, where we assume the XFEL is linearly polarized in the vector E_i direction.

$$ {\boldsymbol{\alpha}}_j=-\frac{e}{m}{\boldsymbol{E}}_{\boldsymbol{i}}\sin \left\{\omega t-\boldsymbol{k}\bullet {\boldsymbol{r}}_j\left(\mathrm{t}\right)\right\} $$

(9.89)

So, the angle θ in (9.89) is the angle of the vector E_i and the vector R − r_j(t), where R is a fixed point of the diagnostic window at a far from the plasma. Since the angle θ is also the electron position dependent and write as θ_j below.

In the case of incoherent light source, (9.89) can be applicable by taking the summation to all electrons. For the coherent photon source, the interferometry of the scattered photons from different electrons become important and it has an essential information for Thomson scattering diagnostic. For this purpose, it is necessary to obtain the retarded electric and magnetic fields at the observation point R as sum of all electrons at r = r_j(t).

Such complicated calculation is shown in [28]. Here we limit the description qualitatively and show the formula of the scattered spectrum. It is easy to know that the delta-function of (9.89) becomes Gaussian spectrum because the thermal motion of electrons leads to Doppler shift of the scattered spectrum. This means the spread of the scattered spectrum has the information of electron temperature. In addition, propagating waves inside the plasma gives the shift of the scatted spectrum as shown in (9.72).

In WDM, plasma waves are spontaneously indued and decay due to damping. A simple damping oscillation model is

$$ \frac{d^2}{d{t}^2}X+\gamma \frac{d}{dt}X+{\omega}_0^2X=0 $$

(9.90)

Then, an approximate solution is

$$ X(t)\kern0.75em \propto \kern0.75em \mathit{\exp}\left[i\left(\omega -{\omega}_0\right)t-\gamma t\right] $$

(9.91)

Laplace transformation gives the Lorentzian spectrum.

$$ X\left(\omega \right)\kern0.75em \propto \kern0.75em \frac{\gamma }{{\left(\omega -{\omega}_0\right)}^2+{\gamma}^2} $$

(9.92)

It is natural to assume that there will be two peaks with shifts from the incident frequency at ω = ω_i ± ω_P. The wave number k_P is uniquely determined by the relation (9.72). These two peaks are easily understood from the following mathematics of the trigonometric formula. As already derived in (4.6.5) in Volume 1, the source term to generate scattering electromagnetic waves in plasma is given as

$$ \left({\omega}_s^2-{c}^2{k}_s^2-{\omega}_{pe}^2\right){E}_s=-{\omega}_{pe}^2 cos\theta \delta {n}_P{E}_i $$

(9.93)

where the suffix “i, s, p” represents the physical quantities shown in (9.72).

Use (9.72) so that incident and scattered electromagnetic field satisfies the dispersion relation, LHS in (9.93) vanishing, the following relation is obtained to RHS with the sinusoidal plasma wave and incident wave.

$$ \delta {n}_P{E}_i\propto \mathit{\sin}\left({\omega}_it-{\boldsymbol{k}}_i\bullet \boldsymbol{r}\right)\mathit{\sin}\left({\omega}_Pt-{\boldsymbol{k}}_{\boldsymbol{p}}\bullet \boldsymbol{r}\right)\propto \mathit{\sin}\left\{\left({\omega}_i-{\omega}_P\right)t-\left({\boldsymbol{k}}_i-{\boldsymbol{k}}_{\boldsymbol{p}}\right)\bullet \boldsymbol{r}\right\}+\mathit{\sin}\left\{\left({\omega}_i+{\omega}_P\right)t-\left({\boldsymbol{k}}_i+{\boldsymbol{k}}_{\boldsymbol{p}}\right)\bullet \boldsymbol{r}\right\} $$

(9.94)

Under the matching condition of (9.72), the first term in RHS of (9.94) becomes the source to generate the scattering electromagnetic wave resonantly, because the dispersion relation in LHS in (9.93) is satisfied. This low frequency component is called Stokes line in the cases of Raman scattering.

On the other hand, the second term in (9.94) generates higher frequency component of the scattered electromagnetic wave. As will see below, this component can be observed for the case with high temperature plasma. It is called anti-Stokes line in the Raman scattering.

For the case where the plasma wave is the electron plasma wave (Bohm-Gross wave) given in (11.7.17), it is possible to measure the electron density. In addition, the spread of the scattered light is related to the damping process shown in (9.92); consequently, it is also possible to determine the thermal spread of the electron distribution function or electron temperature, because the damping by the electron Landau damping or Coulomb collision damping is a function of electron temperature.

To predict the scattered wave intensity, it is necessary to know the electron density fluctuation symbolically written as δn_P in (9.93). The density fluctuation in plasma is determined by the balance between the thermal excitation of the fluctuation and the dissipation process. It is well known that the fluctuation-dissipation theory provides the fluctuation spectrum.

Dynamic structure factor (DSF) of the collective modes in plasma S_collec(k, ω) is related to the plasma dielectric function ε(k, ω) by use of the fluctuation-dissipation theory [28].

$$ {S}_{collec}\left(k,\omega \right)=\frac{\epsilon_0\mathrm{\hslash}{k}^2}{\pi {e}^2}\frac{1}{1-\mathit{\exp}\left(\mathrm{\hslash}\omega /{T}_e\right)}\mathcal{\operatorname{Im}}\left[\frac{1}{\epsilon \left(k,\omega \right)}\right] $$

(9.95)

$$ \epsilon \left(k,\omega \right):\mathrm{longitudinal}\ \mathrm{plasma}\ \mathrm{dielectric}\ \mathrm{function} $$

For example, when a simple fluid model with a constant dissipation rate γ is used to derive the dielectric function of plasma wave as derived in Chap. 2, the dielectric function of a plasma ϵ(k, ω) is easily obtained by regarding X in (9.90) as electrostatic field.

$$ \epsilon \left(k,\omega \right)=1-\frac{\omega_{pe}^2}{\omega^2}-i\frac{\gamma }{\omega } $$

The imaginary part of ϵ(k, ω) has the Lorentzian form (9.92). Weaker the damping, stronger the peak intensity of scattering, and the width of the DSF of such collective scattering becomes wide as the damping increases.

In addition, the detail balance relation for ℏω ≫ T_e yields the following relation

$$ {S}_{collec}\left(-k,-\omega \right)=\mathit{\exp}\left(-\mathrm{\hslash}\omega /{T}_e\right){S}_{collec}\left(k,\omega \right) $$

(9.96)

This relation says that the up-shift scattering is much weaker than the lower-shift scattering. On the other hand, in the ideal plasma α ≫ 1, the plasma temperature is much higher than the photon energy ℏω ≪ T_e, the up-shift component in (9.94) is almost the same as the down-shift component.

$$ {S}_{collec}\left(-k,-\omega \right)={S}_{collec}\left(k,\omega \right) $$

(9.97)

This is the case of the optical diagnostics of laser produced plasmas and magnetically confined plasmas, and the in-elastic scattering by the collective plasma wave modes in (9.94) appears in both sides of the original photon energy as we see later.

9.6.5 Plasma Diagnostics with Optical Thomson Scattering

Optical probe for measuring the plasma parameters of laser produced plasma in ablating region, relatively lower density than the solid, has been used for diagnostics. For example, the higher-harmonic laser is used for such diagnostic to study the plasma generate by more powerful fundamental frequency lasers [Chap. 5 in Ref. 53].

In this case, the laser produced plasma is lower density and higher temperature compared to WDM and the condition α ≫ 1 is satisfied. Therefore, the atomic and Debye electron cloud is too small to scatter the optical probe and the most of the scattering is due to the collective-inelastic scattering, the second term in RHS in (9.94).

In Fig. 9.19, typical spectra of the collective DSF are shown [35]. In Fig. 9.19, the three density cases are plotted with different temperature and different incident photon energy.

Case (a):

The incident probe is laser with its wavelength λ = 532nm (ℏω = 2.3eV), the plasma density is 10¹⁹ cm⁻³. This is a typical case of diagnosing laser-produced ablation plasma with the second harmonic light probe. The frequency is normalized by the plasma frequency ω_pl. In this case, ℏω_pl = 0.117eV. The colors of lines correspond to different plasma temperatures. T_e = 200, 600, and 3000 eV for black (a1), red (a2), and blue (a3), respectively. Both side intensities are the same, because ℏω/T_e≪1. The width of the both spectra become wider as increase of temperature, because the damping of the plasma wave becomes higher due to Landau damping.

Case (b):

This figure is the case with TS light source is λ=4.13 nm (ℏω = 30eV), so-called EUV light source. In this case, ℏω_pl = 1.17eV. The colors of lines correspond to different plasma temperatures. T_e = 0.8, 2.0, and 8.0eV for black (c1), red (c2), and blue (c3), respectively. The asymmetry of Stokes and anti-Stokes lines appears in the case of (c1) with ℏω/T_e=2. Although the anti-Stokes line increases as the temperature increases, the scattered spectra become broader due to the increase of Landau damping effect. In addition, the coupling parameter Γ is almost unity in these three cases, therefore the fluctuation amplitude per one free electron ${S}_{ee}^0\left(k,\omega \right)$ in (9.81) is relatively small compared to the case a). Note that the spread of a3 and c3 looks same in the normalized frequency means the damping in c3 is about ten time larger than a3.

Case (c):

For the case of x-ray probe with λ=0.26 nm (ℏω=4.77 keV), the even the high-density plasma is transparent to the x-ray, since the cut-off density of 10²³ cm⁻³ is ℏω_pl = 11.7eV, much lower than the x-ray photon energy. The colors of lines correspond to different plasma temperatures. T_e = 0.5, 3.0, and 13.0eV for black (e1), red (e2), and blue (e3), respectively. In most of the case, ℏω/T_e > 1 and the anti-Stokes lines disappears. In such WDM situation, the Fermi energy contributes to the plasma wave dispersion relation and the peak of the resonance is shifted near ω ≈ 2ω_pl. In addition, the fluctuation amplitude per one free electron ${S}_{ee}^0\left(k,\omega \right)$ in (9.81) becomes smaller compared to the above two cases. This means extremely bright x-ray source such as XFEL is dispensable for such weak signal measurement.

Fig. 9.19

Theoretical spectra of DSF, ${S}_{ee}^0,$for the collective scattering by the electron plasma wave. Figure (a) is the case of ablating plasma with relatively low density and optical Thomson scattering. With the increase of the electron temperature increases from a1 to a3, the damping effect is enhanced to make the structure broad. Figure (b) is the scattering of EUV light from higher density plasma. The asymmetry of the structure appears reflecting the relation (9.96). Figure (c) is x-ray scattering near solid density. Note that the amplitude decreases from (a) to (c) drastically. Reprint with permission from ref. [35]. Copyright 1998 by American Physical Society

With increase of coupling parameters from the top to bottom in Fig. 9.19, the intensity of DSF per one free electron becomes weaker since the collective motion of free electrons are prohibited by Coulomb interactions in strongly coupled plasma.

9.6.6 WDM Experiment with XRTS Diagnostics

As the integrated experiments, all three contributions in (9.79) are identified to be used to measure the temperature, density, and ionization degree. Let us see two examples of laser shocked WDM experiments for the case of strongly coupled plasma in the region of α < 1. In such case with x-ray source, the wavelength of x-ray is shorter than the Debye length and the x-ray is refracted by the Debye shielding electron cloud. So, the elastic scattering by the atomic and Debye cloud foam factors in (9.80) becomes important.

Before the completion of XFEL facilities, laser-generated hard x-ray line emissions have been applied to the x-ray Thomson scattering diagnostics of dense plasmas. Here, one example using NIF laser is reported with 1.1 MJ hohlraum implosion of a spherical CH solid with 1.15 mm radius [36]. The line x-rays are produced by another laser beams irradiating on zinc solid and the helium-like ions of Zn emitting 9.0 keV photon energy is used for TS x-ray scattering source.

The x-ray spectrum from XRTS data have been fitted with theory of DSF explained above. The best fit of the WDM scattering zone averaged temperature and charge state of carbon ions are concluded that [36].

$$ \left\langle T\right\rangle =86\pm 20 eV,\kern0.5em \left\langle {Z}_C\right\rangle =4.92\pm 0.15, $$

while an integrated radiation-hydrodynamic code HYDRA results

$$ \left\langle T\right\rangle =109 eV,\kern0.5em \left\langle {Z}_C\right\rangle =4.40, $$

In the analysis of XRTS spectrum, it is shown that the fraction of the elastic scattering to the total one is 0.24, while this ratio from HYDRA simulation is 0.35. Such a difference suggested some improvement of the ionization model in HYDRA compared to XRTS data, because the collective scattering is proportional to the number of free electrons.

The HYDRA simulations have been carried out with two different ionization lowering models. One is Thomas-Fermi model and the other is Stewart-Pyatt mode. How to theoretically model the ionization potential lowering will be discussed next section and it is concluded that both models cannot well predict the pressure ionization of WDM [36]. It is insisted that XRTS can be a precise diagnostics of modeling high-pressure effects in WDM.

Along with such experiments, XFELs are now widely used to study the plasma properties in WDM compressed by intense lasers. In Fig. 9.20, x-ray spectrum of incident XFEL with photon energy of 8 keV scattered from WDM is shown [37]. The target is a solid aluminum foil and the scattered spectrum before the compression, aluminum solid state, is shown with bule line. The elastic scatter component around 8 keV and a peak at lower energy is observed.

Fig. 9.20

XRTS spectra from aluminum foil before compression (blue) and after compression (red). The profile of the low energy hump by the collective scattering by plasma wave changes as green and pink by changing the electron temperature slightly. The shift of this peak position gives the compressed density. Reprinted by permission from Macmillan Publisher Ltd: ref. [37], copyright 1993

Aluminum at room temperature (blue line) is metal and three electrons from each atom are not in the bound state. It is a kind of plasma with free electrons with the Fermi energy. The plasma wave in such quantum plasma provides the peak of the collective mode on the left as seen in Fig. 9.20. This is due to the collective contribution to DSF shown in (9.81).

The experimental result obtained by Thomson scattering from the laser-compressed aluminum WDM is shown with gray spiky line “single shot signal”. With use of theoretical models of DSF in (9.79), it is concluded that the shift of the peak by the plasma wave is due to the density compression of 2.3 times the solid density, and a factor 2.8 over the cold scattering peak at energy 8 keV is due to the heating to 1.75 eV. Of course. Thomson scattering spectra are a function of scattered angle from the incident X-ray direction. Fig. 9.20 is obtained at the angle of 13 degree from XFEL incident direction, corresponding to FXRTS in Fig. 9.16. In the present WDM, the ionization energy of the bound electrons is higher than the plasma temperature. The ionization is not affected by laser compression and Z_f = 3 does not change by the shock wave.

In Fig. 9.20, the collective (inelastic) scattering component spectrum is shown with three humps with green, red, and purple colors. As shown in the box in the figure, the difference among three is the difference of assumed electron temperature, T_e = 1.75–0.5 eV, 1.75 eV, and 1.75 + 0.5 eV, respectively. The increase of the electron temperature leads to the enhancement by the plasma wave damping by Landau damping, causing the decreases of the peak amplitude. Note that the peak of the spectrum hump of the collective mode is very sensitive to the electron temperature.

In Ref. [38], sensitivity to the theoretical model to the elastic scattering in (9.80) is discussed in detail. In the elastic component in (9.80), the atomic form factor f(k) does not change after the shock compression, while it is pointed out that the Debye screening form factor q(k) is very sensitive to the temperature. Taking account of the Debye shielding effect of each atom, the increase of the scattered x-ray peak at 8 keV is evaluated due to the increase of the temperature to 1.75 eV. Of course, note that the increase or decrease of the elastic scattering component is a function of the scattering angle and the above statement is not universal.

Throughout the Thomson scattering, the decomposition of DSF shown in (9.79) proposed by J. Chihara has been used as a standard model. The Chihara formula has been compared to the DSF spectra directly obtained from the time-dependent density-functional-theory (TD-DFT) [39]. It is informative to cite this conclusion and copy it here.

The authors presented a method for the direct calculation of the DSF for warm dense matter, independent of Chihara decomposition, by applying real-time TDDFT to configuration drawn from thermal Mermin DFT-MD calculations. Comparison of the result s with state-of-the-art models applied within the Chihara picture illustrates some subtle differences between the two, though it generally supports the use of the Chihara formalism as an inexpensive alternative to the very detailed and computationally intensive TDDFT calculations.

9.7 Ionization Potential Depression (Continuum Lowering)

It has been clear that the ionization degree is very sensitive in Thomson scattering spectra. In high-density plasmas, the pressure ionization and resultant ionization potential lowering is important physics to determine the ionization degree, and transport phenomena stemming from mainly the free electrons. As mentioned in Saha LTE ionization model, the ionization potential lowering is essential to determine the number of free electrons.

The evaluation of the lowering of ionization potential which is also called ionization potential depression (IPD) and continuum lowering has been studied from the beginning of modern plasma physics in application to statistical mechanics [40] and astrophysics [41]. Let us cite them as “EK” and “SP” theoretical models simply below. More than last 50 years, SP model has been widely used to study dense plasmas. In the first experiment possible to study with XFEL, however, demonstrated that EK theoretical model well explain solid density plasma experiment [42].

In this experiment, the x-ray photon absorption edge by the bound-free photo-absorption has been measured from WDM irradiated by XFEL. However, more precise atomic model with detail configuration has been studied about the energy state of K-shell electrons [43]. This pointed out the change of K-shell electron energy as the ionization proceeds and the above conclusion has not been confirmed. With more detail atomic code, the experimental data of [42] has been analyzed again to clarify the improvement of the atomic model [44].

Let us see the basic theory of IPD and discuss the experimental data with XFEL.

9.7.1 Theoretical Models

In SP model, they proposed to use Thomas-Fermi model to the spherically symmetric potential around the centered ion as explained previously to be used as statistically averaged potential. Different from the ion-sphere model, the boundary should be taken to the infinite radius and the effect of neighboring ions is also included to the Thomas Fermi model. The electrons in bound state and free states are assumed to be in Fermi-Dirac distribution and ions are classical as given in Boltzmann distribution.

Then, SP has modified the Poisson equation mathematically to the following form with changing variables as

$$ y=\frac{e\phi}{T},\kern1em x=\frac{r}{\lambda_D} $$

(9.98)

where the Debye length is contributed by both of electrons and ions, while it is assumed that both have the same temperature T. Then, the Poisson equation with replacement of electron density with Fermi-Dirac distribution can be reduced to the following form.

$$ \frac{1}{x}\frac{d^2}{d{x}^2}(xy)=\frac{1}{Z+1}\left\{\frac{F\left(y-\alpha \right)}{F\left(-\alpha \right)}-\mathit{\exp}\left(- Zy\right)\right\} $$

(9.99)

where

$$ \alpha =\frac{\mu }{T} $$

(9.100)

and F is so-called Fermi-Dirac integral defined to be

$$ F\left(\eta \right)={\int}_0^{\infty}\frac{t^{1/2}}{e^{t-\eta }+1} dt $$

(9.101)

Finally, (9.99) should be solved to satisfy the following two boundary conditions.

$$ y\left(\infty \right)=0,\kern1em xy\to \frac{Z{e}^2}{\Big(4\pi {\varepsilon}_0}{\lambda}_DT\Big)\kern1em \left(x\to 0\right) $$

(9.102)

The normalized chemical potential α is an eigen value obtained so that the solution should satisfy the boundary conditions (9.102).

In SP paper, they have solved (9.99) numerically for wide range of density and temperature. At the same time, SP also obtained approximate solutions in the inner and outer region and tried to match the both functions to obtain the depressed ionization energy. Then, SP modified the function so that it can well reproduce the numerical results. The resultant formula of IPD by SP can be given in the following form.

$$ \varDelta {U}_{SP}=\frac{3}{2}\frac{Z{e}^2}{4\pi {\varepsilon}_0a}\left\{{\left(1+{X}^3\right)}^{2/3}-{X}^2\right\} $$

(9.103)

where a is the ion sphere radius and X and a are both defined to be

$$ X=\frac{\lambda_D}{a},\kern1em \frac{4\pi }{3}{a}^3{n}_i=1 $$

(9.104)

It is useful to know the physical image of (9.103). The ionization potential depression compared to the case of isolated ion is given with (9.103) and its high temperature limit, X > > 1,

$$ \varDelta {U}_{SP}\approx \frac{Z{e}^2}{4\pi {\varepsilon}_0a}\frac{1}{X}=\frac{Z{e}^2}{4\pi {\varepsilon}_0}\frac{1}{\lambda_D} $$

(9.105)

In such ideal plasma, the potential is shielded by the Debye shielding effect. This potential can be Taylor expanded near the centered ion (r < < λ_D) as

$$ {U}_{DH}\approx \frac{Z{e}^2}{4\pi {\varepsilon}_0r}\left(1-\frac{r}{\lambda_D}\right)\kern2.00em \left(a<<r<<{\lambda}_D\right) $$

(9.106)

The second term is the depressed ionization potential and is the same as that by SP.

In the opposite limit for strongly coupled plasma with X < < 1, (9.103) is reduced to

$$ \varDelta {U}_{SP}\approx \frac{3}{2}\frac{Z{e}^2}{4\pi {\varepsilon}_0a} $$

(9.107)

This limiting case is also easily understood as follows. In strongly coupled plasma, the ion-sphere model is good image, and we assume the situation that the electron density is very high because of Fermi energy and the charge neutrality within the ion sphere (IS) is reasonable. Then, the potential inside the sphere can be expressed as

$$ {U}_{IS}=\frac{Z{e}^2}{4\pi {\varepsilon}_0r}-f(r)=0\kern1em at\ r=a $$

(9.108)

Where the function f(r) is the potential contribution by electron density. This gives us the ionization potential depression f(a), which is the same as (9.107) except the factor 3/2.

The second formula, although their work was earlier historically by Ecker and Kroll [40], is given in the following form:

$$ \varDelta {U}_{EK}=\frac{Z{e}^2}{4\pi {\varepsilon}_0}g $$

(9.109)

$$ g=\left\{\begin{array}{c}1/{\lambda}_D\kern4.50em for\kern1em {n}_i<{n}_{cr}/\left(1+Z\right)\\ {}1/a\times C{\left(1+Z\right)}^{1/3}\kern1em for\kern1em {n}_i>{n}_{cr}/\left(1+Z\right)\end{array}\right. $$

$$ {n}_{cr}=\frac{3}{4\pi }{\left(\frac{T}{Z^2{e}^2}\right)}^3 $$

In (9.109), C is originally a function of density and temperature, but C = 1 is found to give a good agreement with recent experimental data.

9.7.2 Experimental Evidence of IPD

There have been a lot of experimental study on the theory of IPD, but it was very difficult to measure IPD directly, especially in dense plasma. The first IPD experiments have carried out with XFEL laser at SLAC, Stanford Univ. [45]. In these experiments, a monochromatic x-ray pulse of well-defined photon energy is focused to spots of ∼10 μm² on thin foils of various materials. Typical intensities achieved are on the order of 10¹⁷ Wcm⁻², sufficient to heat the irradiated regions to temperatures exceeding 100 eV on femtosecond timescales, and to drive resonant and non-linear atomic processes. The intense x-ray pulse can drive x-ray photoionization of inner-shell electrons, provided the photon energy is higher than the shell’s ionization edge, and bound-bound transitions leading to excited atomic configurations, if the resonance energies are within the bandwidth of the x-ray pulse. Recombination into the core holes created by this interaction produces strong x-ray emission that has been spectrally resolved to identify ionization edges

In Fig. 9.21, in the vertical axis, the emitted photon intensity spectrum from heated aluminum plasma is plotted as a function of the photon energy between 1470 eV to 1670 eV [44]. The image is obtained after many shots of XFEL by changing the photon energy of XFEL pulses. The data are shown for the irradiation of XFEL with photon energy from 1470 eV to 1800 eV, which is shown in the horizontal axis. The strong intensities of about eight lines are clearly seen with the threshold FEL photon energy increasing from low energy to higher energy. These lines are identified due to the electron transition 2p-1s of partially ionized aluminum atom. The FEL photons are predominantly absorbed by K-shell electrons as explained in Chap. 5, if the FEL photon energy is higher than the threshold energy of bound-free transition as shown in Fig. 9.21 for the case of iron.

Fig. 9.21

Varying the photon energy of XFEL, the x-ray emission spectra have been measured to evaluate the ionization potential depression (IPD). The experimental data are compared to simulation with two different ionization potential depression theoretical models. Reprinted by permission from Macmillan Publisher Ltd: ref. [44], copyright 1993

Just after a hole is generated in the K-shell, another electron fulfills the hole to emit the line radiation. One possible process is radiative transition of bound electron from L-shell to K-shell and this line emissions are observed and shown in Fig. 9.21. The energy gap between L-shell and K-shell changes as ionization stage changes in the aluminum atom. As the ionization stage change from Al-IV stage to Al-XI, its gap increases as shown by the line positions in the horizontal axis of Fig. 9.21. Therefore, the threshold XFEL energy gives us the information of energy depth of K-shell electron measured from the real free energy state which is determined by plasma effect.

The integrated emission intensity is plotted in the above in Fig. 9.21, and compered to computational results with two different ionization potential depression models. It is found that both theoretical models well reproduce the present experimental result [44].

In Ref. [46], another analysis of the possibility is reported to obtain highly precise measurements of the ionization potential depression (IPD) in dense plasmas with spectrally resolved x-ray scattering. In this method, the advantage is that the electron temperature and the free electron density are simultaneously determined. So, more precise study is expected. A proof-of-principle experiment at the LCLS probing isochorically heated carbon samples, demonstrates the capabilities of this method and motivates future experiments at x-ray free electron laser facilities.

In Fig. 9.22, the experimental data of XRTS spectrum are shown with blue dots. The spectrum is analyzed theoretically with Chihara decomposition formula. A model fit to the scattering spectrum using the Chihara decomposition and assuming local thermal equilibrium provides a stable fit giving Te = 21.7 eV, Z = 1.71, and an IPD of 24 eV. The model fit total spectrum is plotted in red in Fig. 9.22 showing a good agreement with the XRTS data. It is mentioned that looking at the absolute value of the IPD obtained from the fit (24 eV) is in very good agreement with the Stewart–Pyatt prediction for the best fit plasma parameters (25.3 eV) and does not agree with modified Ecker–Kroell (47.7 eV). It is noted that before such XFEL facility, there was no way to check the theoretical model with precise experimental way, while XFEL has made such comparison possible to motivate better modeling of the ionization potential depression (IPD).

Fig. 9.22

The ionization potential depression is determined experimentally from XFEL Thomson scattering measurement. Reprint from ref. [46] with permission from Institute of Physics, (Courtesy of D. Kraus)