Atomic Process in Plasmas

Abstract

When a high-intensity laser is irradiated onto a solid gold (Z = 79), half of the electrons is partially ionized. The multi-electron structure of such ions is not obvious. Quantum mechanics of multi-electron systems and calculations of ionization statistics are required. In this chapter, the electrons in the ion are approximated to be bound in a spherically symmetric mean field, and the isolated atom is studied.

The Hartree-Hock (HF) equation, which accurately describes atoms in many-electron systems, can be solved, but it is a daunting task. For this purpose, simple but error-prone approximations have been used, such as the HULLAC and OPAL codes, which use the para-potential method instead of a rigorous description of the HF. It is an intuitive and easy-to-understand approximation.

Once the quantum state of the bound electrons can be calculated, the statistical distribution of ionization can be obtained by solving the Saha equation for thermal equilibrium. The threshold of ionization (continuum lowering) is determined. The calculation of such an ionic structure is presented. Due to the high temperature of the plasma, interaction with thermal radiation and free electrons cause excitation, ionization, and the reverse process. Calculations of these processes will be presented.

Applications of the rate equations will be explained. In the recently introduced X-ray laser (XFEL) heating, free electrons are also non-equilibrium (non-Maxwellian). This chapter begins with a review of hydrogen and helium atoms, and then introduces the topics of atomic physics and processes from the laboratory to the universe.

5.1 Introduction

In order to solve the hydrodynamic equation of partially ionized plasma, it is necessary to determine charge distribution and average charge of plasma ions. The equation of states, transport coefficients and so on are very sensitive to the charge state. In solving a time evolution of plasma, the charge state changes in time. Under a certain condition, the local thermodynamic equilibrium (LTE) is good approximation, while the other condition requires analysis based on some non-LTE atomic process model. In general, laser produced plasma requires such non-LTE analysis to determine the charge distribution.

The charge distribution in LTE is given by solving well-known Saha relation. However, we have to know about the quantum states of bound electrons in all charge state ions in advance. Lasers irradiate not only low-Z materials, but also medium-Z and high-Z materials and the calculation of the quantum states themselves is a hard job. In the history of development of quantum mechanics, numerical methods to solve such quantum state of many electron system have been developed [1, 2].

Alongside development of such methods, a variety of simplified models have also been studied intensively. When we need only the charge state in a given condition, it is not necessary to solve Hartree-Fock equation with Slater matrix as shown in such textbooks. However, spectroscopic analysis and radiation transport demand data base obtained by such sophisticated calculation. Before going to details of the atomic model, we are required to speculate how detail atomic data are required in our specified plasma to be studied.

In non-LTE plasma we have to solve dynamics of ionization, recombination, excitation, and de-excitation. All of such processes depend on the quantum state of bound electrons. The cross section of each process σ has to be solved with quantum mechanics [3, 4]. In general, we use the perturbation method to solve Schrodinger equation. Except for special case with XFEL as we see later, it is a good approximation to assume the free electrons are in Maxwell distribution. Then, the velocity averaged rate (frequency) of the collisional atomic process is given for the corresponding cross section σ(v) in the form.

$$ \upnu =\left\langle \mathrm{n}\upsigma \left(\mathrm{v}\right)\mathrm{v}\right\rangle, $$

(5.1)

where 〈〉 represents to take average with Maxwell distribution.

Then, we can formulate so-called rate equation to the population of each quantum state of all ions. It can be expressed by a relational expression that seems simple as follows.

$$ \frac{{\mathrm{dN}}_{\mathrm{m}}^{\upvarsigma}}{\mathrm{dt}}=\sum \limits_{\mathrm{k}\ne \mathrm{m},\upeta \ne \upvarsigma}\left(-{\upnu}_{\mathrm{m}\mathrm{k}}^{\upvarsigma \upeta}{\mathrm{N}}_{\mathrm{m}}^{\upvarsigma}+{\upnu}_{\mathrm{k}\mathrm{m}}^{\upeta \upvarsigma}{\mathrm{N}}_{\mathrm{k}}^{\upeta}\right) $$

(5.2)

where ${\mathrm{N}}_{\mathrm{m}}^{\upvarsigma}$ represents the number of ions with the quantum state of bound electrons m and charge ζ. Of course, their numbers increase abruptly with the increase of atomic number Z of plasma or how detail atomic states we take account of. The first term on RHS gives the loss of ${\mathrm{N}}_{\mathrm{m}}^{\upvarsigma}$ due to the transition of (ς → η) and (m → k), while the second term gives the increase due to the transition of (η → ς) and (k → m). How many quantum states of bound electrons should be included strongly depends on how precise analysis are required. How we can reduce this task depends on modeling of atomic structure. Note that such a rate equation has been widely used in many different sciences; not only natural but also social sciences.

In plasma physics, so-called collisional radiative model (CRM) of rate equation is widely used for solving atomic process of non-LTE plasma [5]. Such codes are applied to study Astrophysical objects. The details of photo-ionized plasmas in Universe and laboratory are explained. The principle of masers and lasers are also shown with rate equations as described later. The masers and lasers are also observed in Universe.

Finally, interdisciplinary topics governed by rate equations are briefly described. Nucleosynthesis in Big-Bang and supernova explosion, Lorentz model giving chaos of weather, and virus infection are explained by showing how such rate equations are used in different natural and social sciences.

5.2 Saha Equilibrium of Charge State

In the region where the temperature is low and the density is moderately high, the collision effect by free electrons is dominant for excitation, ionization, de-excitation, and recombination even if the radiation field is very weak compared to Planckian distribution. In such a case, the ionization state of the plasma is realized while locally achieving thermal equilibrium. Such thermodynamically equilibrium (LTE) ionization distribution is called Saha equilibrium after the name of the Indian astronomer Meghnad Saha (1893–1956), who first proposed the equation for calculating the distribution of ions in different charge state in LTE.

Let us derive an equation of Saha equilibrium, which is coupled equations of the ionization distribution in the thermal equilibrium state. Here, the partition function Z is defined by Helmholtz’s free energy F and is given in the form

$$ \mathrm{F}=-\mathrm{kT}\ln \mathrm{Z} $$

(5.3)

For example, when the system is made of different types of gases (ions and electrons of different charges in the case of plasma) and their number are N₁, N₂,…, the total partition function Z of the system is divided into individual partition functions Z₁, Z₂, in the form

$$ \mathrm{Z}=\frac{{\mathrm{Z}}_1^{{\mathrm{N}}_1}}{{\mathrm{N}}_1!}\cdot \frac{{\mathrm{Z}}_2^{{\mathrm{N}}_2}}{{\mathrm{N}}_2!}\cdot \dots $$

(5.4)

Here, the partition function for N₁ particles is defined as

$$ \mathrm{Z}=\sum \limits_{\mathrm{n}}{\mathrm{g}}_{\mathrm{n}}{\mathrm{e}}^{-\frac{{\mathrm{E}}_{\mathrm{n}}}{\mathrm{kT}}} $$

(5.5)

The partition function is defined to be the sum of all possible quantum states for each particle system. Note that g_n, E_n are the degeneracy (number of states) and the intrinsic energy of all quantum states that the N₁ particle can take.

Inserting (5.5) into (5.4) and using the following Stirling formula,

$$ \mathrm{N}!={\left(\frac{\mathrm{N}}{\mathrm{e}}\right)}^{\mathrm{N}} $$

(5.6)

where “e” is the base of natural logarithm. The free energy of the formula (5.3) is obtained to be

$$ \mathrm{F}=-{\mathrm{N}}_1\mathrm{kT}\ln \left(\frac{{\mathrm{Z}}_1\mathrm{e}}{{\mathrm{N}}_1}\right)-{\mathrm{N}}_2\mathrm{kT}\ln \left(\frac{{\mathrm{Z}}_2\mathrm{e}}{{\mathrm{N}}_2}\right)-\dots $$

(5.7)

Now let’s consider ionization equilibrium. In the partially ionized state, various ionic states coexist from the neutral state to the completely ionized state of the same atom. A partial ionized plasma can be thought of as a group of ions with different charge numbers. Let Z be the atomic number of the neutral atom, we obtain

$$ \mathrm{F}=-\sum \limits_{\upzeta =0}^{\upzeta =\mathrm{Z}}{\mathrm{N}}_{\upzeta}\mathrm{kT}\ln \left(\frac{{\mathrm{Z}}_{\upzeta}\mathrm{e}}{{\mathrm{N}}_{\upzeta}}\right)-{\mathrm{N}}_{\mathrm{e}}\mathrm{kT}\ln \left(\frac{{\mathrm{Z}}_{\mathrm{e}}\mathrm{e}}{{\mathrm{N}}_{\mathrm{e}}}\right) $$

(5.8)

Here, ζ is the number of ionized electrons. The charge is Z_ζ and number is N_ζ. The second term of Eq. (5.8) is a term due to free electrons. When the volume of plasma is V, the partition function is given to be

$$ {\mathrm{Z}}_{\mathrm{e}}=2{\left(\frac{\mathrm{mkT}}{2\uppi {\mathrm{\hslash}}^2}\right)}^{3/2}\mathrm{V} $$

(5.9)

It is clear in thermal equilibrium that the Helmholtz free energy F in (5.8) should take the minimum value. That is, if the ions in the ionization state are replaced with some ions, the following relationship holds.

$$ {\updelta \mathrm{N}}_{\upzeta}=-{\updelta \mathrm{N}}_{\upzeta +1}=-{\updelta \mathrm{N}}_{\mathrm{e}} $$

(5.10)

At the same time δF = 0 should also be satisfied. From this condition and (5.8), we obtain the relation

$$ \frac{{\mathrm{N}}_{\upzeta +1}{\mathrm{N}}_{\mathrm{e}}}{{\mathrm{N}}_{\upzeta}}=\frac{{\mathrm{Z}}_{\upzeta +1}{\mathrm{Z}}_{\mathrm{e}}}{{\mathrm{Z}}_{\upzeta}} $$

(5.11)

Defining the number density, n_e = N_e/V, n_ζ = N_ζ/V, (5.11) can be written

$$ \frac{{\mathrm{n}}_{\upzeta +1}{\mathrm{n}}_{\mathrm{e}}}{{\mathrm{n}}_{\upzeta}}=2\frac{{\mathrm{Z}}_{\upzeta +1}}{{\mathrm{Z}}_{\upzeta}}{\left(\frac{\mathrm{mkT}}{2\uppi {\mathrm{\hslash}}^2}\right)}^{3/2} $$

(5.12)

This is the basic formula by Saha.

Here, we introduce a new function of the energy u_ζ and the ionization state as follows

$$ {\mathrm{Z}}_{\upzeta}={\mathrm{e}}^{-\frac{{\mathrm{E}}_0^{\upzeta}}{\mathrm{kT}}}\sum \limits_{\mathrm{n}=0}{\mathrm{g}}_{\mathrm{n}}^{\upzeta}{\mathrm{e}}^{-\frac{{\mathrm{E}}_{\mathrm{n}}^{\upzeta}-{\mathrm{E}}_0^{\upzeta}}{\mathrm{kT}}}={\mathrm{e}}^{-\frac{{\mathrm{E}}_0^{\upzeta}}{\mathrm{kT}}}{\mathrm{u}}_{\upzeta} $$

(5.13)

Then, the Eq. (5.12) is

$$ \frac{{\mathrm{n}}_{\upzeta +1}{\mathrm{n}}_{\mathrm{e}}}{{\mathrm{n}}_{\upzeta}}=2\frac{{\mathrm{u}}_{\upzeta +1}}{{\mathrm{u}}_{\upzeta}}{\left(\frac{\mathrm{mkT}}{2\uppi {\mathrm{\hslash}}^2}\right)}^{3/2}{\mathrm{e}}^{-\frac{{\mathrm{I}}_{\upzeta +1}}{\mathrm{kT}}} $$

(5.14)

$$ {\mathrm{I}}_{\upzeta +1}={\mathrm{E}}_0^{\upzeta +1}-{\mathrm{E}}_0^{\upzeta} $$

(5.15)

Here, (5.15) is the ionization energy from the ionized state ζ to ζ + 1.

Now, how is the internal excited state of each ion and its number of states determined? If it is hydrogen, the number of states is 2n² and the energy level is simple. However, it is not simple how high n should be included as bound states, because the upper n states may be free state due to the perturbation by the ions surrounding. If we think that every ion is in the ground state, we can calculate the number of states if Z is small, so the above equation seems to be solvable. However, it is necessary to consider the excited state in the partially ionized plasma in the case where the density is high and the temperature is not so high. Also, there are many states in the shell of the same main quantum number, and when a part is clogged, there is degeneracy. In that case what should one do? Although the charge distribution of thermal equilibrium state in Saha’s equation seems to be apparent at first glance, the reality needs more detailed and lengthy study. Such ionization level lowering will be discussed in Chaps. 8 and 9 in detail.

Equation (5.14) is a nonlinear algebraic equation that can be solved by considering the conservation law of the number of electrons and the number of ions. The result will be different depending on how to calculate the internal quantum state of each ion state. An example of charge state distribution as a function of temperature is shown in Fig. 5.1 for the characteristic density when aluminum foil is irradiated with intense laser, where simple energy levels are used. The ion density is 10¹⁸ cm⁻³ and temperature are from 10 eV to 10 keV. In thermodynamic equilibrium, complete ionization occurs when the temperature exceeds 1 keV. The charge state 11+ is seen in the wide temperature range is because a higher energy (temperature) of electrons is necessary for stripping off more from the bound state because ions are in a helium-like closed shell state.

Fig. 5.1

An example of charge state distribution as a function of temperature is shown for the characteristic density when aluminum foil is irradiated with intense laser, where simple energy levels are used in solving Saha equation. The ion density is 10¹⁸ cm⁻³ and temperature is from 10 eV to 10 keV

Now, let us also discuss for widely seen ionization situation in laboratory discharge experiment. When the temperature is sufficiently low, at most one electron is ionized. Try to find an approximate Saha relation exactly applied to hydrogen atoms and approximately used for any atoms. In one electron ionization state, (5.14) closes only in the case of ζ = 0. Since n_e = n₁ at the same time, if we define ionization degree as α = n_e/n, n = n₀ + n₁, we obtain the one relation

$$ \frac{\upalpha^2}{1-\upalpha}=\frac{2}{\mathrm{n}}\frac{{\mathrm{u}}_1}{{\mathrm{u}}_0}{\left(\frac{\mathrm{mkT}}{2\uppi {\mathrm{\hslash}}^2}\right)}^{3/2}{\mathrm{e}}^{-\frac{\mathrm{I}}{\mathrm{kT}}} $$

(5.16)

This is an exact relation for hydrogen, but an approximate relation for other ions. It can be applicable in the limit of I/kT ≪ 1 and α ≪ 1. Let us see the ionization degree when the ionization is triggered. With the condition α ≪ 1, the following density dependence is obtained from (5.16)

$$ \upalpha \propto \frac{1}{\sqrt{\mathrm{n}}} $$

(5.17)

Even at the same temperature, the lower the density, the higher the degree of ionization. Physical reasons can be intuitively explained in the following two ways.

• 1. Considering from the number of states, the number of bound states does not depend strongly on density, but as the space between atoms spreads wider, the number of identical energy states of free electrons increases proportionally to the volume. Therefore, many electrons gather in the free-state with a large number of states.

• 2. In the space where the density is extremely low(n = 1 cm⁻³), it is found that even if the temperature is extremely low the hydrogen is completely ionized from (5.16). This is because if free electrons collide with neutral atoms and collision ionization occurs, the probability that other free electrons are captured by hydrogen ions are extremely low. Therefore, the free electrons will freely travel around the vacuum for a long time without encountering ions.

5.3 Quantum States of Atoms

In the plasma generation process, excitation and ionization of atoms by radiation absorption and electron collisions should be considered as elementary process. Moreover, it is necessary to understand quantum-mechanical interaction of atoms, electrons and photons, such as de-excitation, recombination, which is the reverse process of the above-mentioned elementary process. In some cases, it is necessary to calculate mathematical models with complicated atomic structures. To grasp the background knowledge of such atomic process, we will need to briefly review quantum mechanics and perturbation theory. However, we only discuss the quantum mechanics which is the foundation to study the atomic process in plasmas.

In order to study atomic physics, a multi-electron wave equation is the basis. Quantum mechanics can be easily derived by the principle of correspondence with classical mechanics.

5.3.1 Hydrogen Atom

Let us show time-dependent Schrodinger equation for the wave function ϕ(r,t) of a single electron system i.e., Hydrogen atom for a given spherical potential U(r).

$$ \mathrm{H}\kern0.5em =\frac{{\mathbf{p}}^2}{2\mathrm{m}}+\mathrm{U}=\mathrm{E} $$

(5.18)

$$ \mathbf{p}=\mathrm{i}\mathrm{\hslash}\nabla, \kern2em \mathrm{E}=-\mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}} $$

(5.19)

$$ \left[-\frac{{\mathrm{\hslash}}^2}{2\mathrm{m}}{\nabla}^2+\mathrm{U}\left(\mathbf{r}\right)\right]\upphi \left(\mathrm{t},\mathbf{r}\right)=-\mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}}\upphi \left(\mathrm{t},\mathbf{r}\right) $$

(5.20)

As is well known, the steady state solution of this equation is obtained by placing the conserved energy E

$$ \upphi \left(\mathrm{t},\mathbf{r}\right)=\uppsi \left(\mathbf{r}\right){\mathrm{e}}^{-\mathrm{i}\upomega \mathrm{t}},\kern2em \upomega =\mathrm{E}/\mathrm{\hslash} $$

(5.21)

$$ \left[-\frac{{\mathrm{\hslash}}^2}{2\mathrm{m}}{\nabla}^2+\mathrm{U}\left(\mathbf{r}\right)\right]\uppsi \left(\mathbf{r}\right)=\mathrm{E}\uppsi \left(\mathbf{r}\right) $$

(5.22)

This equation can be transformed into ordinary differential equations for r by a method such as the separation of variables. At that time, it can be seen that the three quantities of intrinsic energy, orbital angular momentum, and magnetic angular momentum cannot be taken as continuous values and are quantized as a separation constant.

Let’s see why Schrodinger was able to arrive at (5.20). Naturally, without the predecessor work of analytical mechanics and electromagnetism, he never got to his idea. Analytical mechanics proves that geometric optics and wave optics can be connected by “introducing the concept of mechanics into the concept of wave” that the action function on the mass point of the Hamilton-Jacobi equation corresponds to the phase function of the wave. At the same time the boundary value problem of the Maxwell equation, for example, about the propagation of electromagnetic waves in a waveguide, it was found that the eigenvalue problem should be solved, and it was known that a specific frequency can exist only as a propagation solution. This is a concept conforming to Bohr’s quantum hypothesis. At the same time, wave hypothesis of electron by de Broglie reminds us of the dynamic representation of wave optics. In this way academics clearly shows the birth of quantum mechanics that how to successfully adopt concepts of other fields and open up by giving new interpretations when challenging new discipline.

Hydrogen is the simplest atom binding one electron by the Coulomb force of a proton. The solution is the foundation for considering complex atoms, so let us review it briefly here. In the case of electrons bound to a hydrogen nucleus, the potential is a function of the radius coordinate r only in the form

$$ \mathrm{U}\left(\mathrm{r}\right)=-\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0\mathrm{r}} $$

(5.23)

Three eigenvalues appear when substituting (5.23) into (5.22) and solving it by separating the variables, and the wave function is found to be in the form with three eigenvalues n, ℓ, m.

$$ \uppsi \left(\mathbf{r},\upsigma \right)={\mathrm{R}}_{\mathrm{n}\mathrm{\ell }}\left(\mathrm{r}\right){\mathrm{Y}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\uptheta, \upvarphi \right){\upchi}_{{\mathrm{m}}_{\mathrm{s}}}\left(\upsigma \right) $$

(5.24)

Here, R_nℓ(r) the radial wave function, ${\mathrm{Y}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\uptheta, \upvarphi \right)$ is a spherical harmonic function. Note that the spherical harmonic function is an eigen-function satisfying the boundary condition in Fourier decomposition in the two-dimensional space on the surface of a sphere. For the convenience of multi-electron atom, the spin function ${\upchi}_{{\mathrm{m}}_{\mathrm{s}}}\left(\upsigma \right)$ is also introduced.

It is well known that the eigenvalue of energy is given in the form

$$ {\mathrm{E}}_{\mathrm{n}}=-13.6\frac{1}{{\mathrm{n}}^2}\kern0.5em \left(\mathrm{eV}\right) $$

(5.25)

Here, the reason why the energy eigenvalue depends only on the principal quantum number n is that the potential of (5.23) is mathematically special and accidental degeneracy with respect to the orbital quantum number l appears. For example, when there is shielding by the electrons of the inner shell like an alkali metal, the energy of the electrons of the outermost shell does not degenerate. The energy level of the outermost shell electron of a lithium atom is shown in Fig. 5.2. Since there are two electrons in the 1s state, there are two different energy levels 2s and 2p in n = 2 state. Because of the spread of the wave function of two electrons in 1s state, the wave function of 2s of lithium shrinks more than 2s of hydrogen, and consequently the energy becomes deeper. Because of the spread of the charge distribution of the two electrons of 1s, the effective potential becomes different from (5.23) and the accidental degeneracy disappears.

Fig. 5.2

The energy level of the outermost shell electron of a lithium atom

The number of quantum states of a hydrogen atom is 2n² for each principal quantum state with its energy in (5.25). It is clear the sum of all of quantum states (n = 0~∞) diverges. In calculating Saha equilibrium for hydrogen plasma, it is necessary to avoid this divergence. Pay attention to the orbit radius of the wave function of n in the form:

$$ {\mathrm{r}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{B}}\frac{{\mathrm{n}}^2}{\mathrm{Z}} $$

(5.26)

where a_B is the Bohr radius (=0.53 A). It is clear that the orbit becomes larger with the increase of n. In plasma, an ion is surrounded by many other ions, and the free electrons also shields the nuclear charge at the center. Both effects make the large orbit electron be out of the attractive force by the nucleus and it is rather a free electron. This fact is schematically shown in Fig. 5.3, where outer orbits of bound electrons overlap with those of adjacent atoms.

Fig. 5.3

In plasma, an ion is surrounded by many other ions, and the free electrons also shields the nuclear charge at the center. Both effects make the large orbit electron be out of the attractive force by the nucleus and it is rather a free electron

Then, as shown in Fig. 5.4, the potential and bound states of an isolated atom must be modified like a plot in red as an atom embedded in plasma. The electrons at higher energy levels becomes free electrons. As the result, charge shielding by such electrons disappears to make the potential structure sallower. Such modification is especially important in high-density plasma. If it is possible to evaluate the average number of the maximum n as n*, using this as the maximum value of n in (5.5) and (5.13) makes it possible to solve the Saha Eq. (5.14) for hydrogen plasma. Such physics is called ionization potential lowering to be discussed later in the next chapter. As shown in Fig. 5.5, the effect of the lowering is clearly seen in the observed emission spectrum from hydrogen pellet injection into magnetically confined plasma. It is clear that line emissions from only three excited levels are observed, namely n* = 4 is inferred.

Fig. 5.4

The potential and bound states of an isolated atom are shown on the left. They must be modified like a plot in red as an atom embedded in plasma shown on the right. The electrons at higher energy levels becomes free electrons. As the result, the Coulomb field is shielded by such electrons disappears to make the potential structure sallower. Such modification is especially important in high-density plasma

Fig. 5.5

The effect of the ionization potential lowering is clearly seen in the observed emission spectrum from hydrogen pellet injection into magnetically confined plasma. It is clear that line emissions from only three excited levels are observed

5.3.2 Helium Atom

The property of electron spin and Paul’s exclusive principle is not considered in a hydrogen atom, while it becomes essential in the atom having more than one electron. This is because Pauli principle prohibits two electrons are in the same quantum state including the spin state.

The Hamiltonian of a helium-like atom is

$$ \mathrm{H}\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=-\frac{{\mathrm{\hslash}}^2}{2{\mathrm{m}}_{\mathrm{e}}}\left({\nabla}_{{\mathbf{r}}_1}^2+{\nabla}_{{\mathbf{r}}_2}^2\right)-\frac{{\mathrm{Ze}}^2}{4{\uppi \upvarepsilon}_0}\left(\frac{1}{{\mathrm{r}}_1}+\frac{1}{{\mathrm{r}}_2}\right)+\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\frac{1}{{\mathrm{r}}_{12}} $$

(5.27)

The first term is the kinetic energy of two electron whose coordinates are r₁ and r₂. The second one is the attractive force by the nucleus. The third one is Coulomb repulsive term between two electrons, where |r₁ − r₂| = r₁₂.

It is known that Fermi particles like electrons should have the anti-symmetry wave function. Describe two particle wave function as

$$ \Psi \left({\mathbf{r}}_1,{\upsigma}_1;{\mathbf{r}}_2,{\upsigma}_2\right)=\Psi \left({\uptau}_1,{\uptau}_2\right) $$

(5.28)

where τ is a simplified notation of r and σ. The function Ψ is called spin-orbitals. The anti-symmetry condition requires

$$ \Psi \left({\uptau}_1,{\uptau}_2\right)=-\Psi \left({\uptau}_2,{\uptau}_1\right) $$

(5.29)

This requirement indicates that just a product of two single electron wave function cannot be a total wave function, where two electrons are in the quantum states a and b, the two-electron wave function are in the form.

$$ \Psi \left({\uptau}_1,{\uptau}_2\right)=\frac{1}{\sqrt{2}}\left\{{\upvarphi}_{\mathrm{a}}\left({\uptau}_1\right){\upvarphi}_{\mathrm{b}}\left({\uptau}_2\right)-{\upvarphi}_{\mathrm{a}}\left({\uptau}_2\right){\upvarphi}_{\mathrm{b}}\left({\uptau}_1\right)\right\} $$

(5.30)

Inserting (5.30) into Schrodinger equation with Hamiltonian (5.28), the total energy of two electrons is given in the form:

$$ \mathrm{E}={\mathrm{E}}_1+{\mathrm{E}}_2+{\mathrm{J}}_{12}+{\mathrm{K}}_{12} $$

(5.31)

where E₁ and E₂ are a single electron energy given by hydrogen like binding energy.

$$ {\mathrm{E}}_1+{\mathrm{E}}_2=-{\mathrm{Z}}^2\left\{{\mathrm{E}}_{\mathrm{H}}(1)+{\mathrm{E}}_{\mathrm{H}}(2)\right\} $$

(5.32)

J₁₂ in (5.31) is Coulomb interaction energy of two electrons and defined by

$$ {\displaystyle \begin{array}{c}{\mathrm{J}}_{12}=\iint {\upvarphi_{\mathrm{a}}}^{\ast}\left({\uptau}_1\right){\upvarphi_{\mathrm{b}}}^{\ast}\left({\uptau}_2\right)\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\frac{1}{{\mathrm{r}}_{12}}{\upvarphi}_{\mathrm{a}}\left({\uptau}_1\right){\upvarphi}_{\mathrm{b}}\left({\uptau}_2\right){\mathrm{d}\uptau}_1{\mathrm{d}\uptau}_2\\ {}\kern1em =\iint {\left|{\upvarphi}_{\mathrm{a}}\left({\uptau}_1\right)\right|}^2\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\frac{1}{{\mathrm{r}}_{12}}{\left|{\upvarphi}_{\mathrm{b}}\left({\uptau}_2\right)\right|}^2{\mathrm{d}\uptau}_1{\mathrm{d}\uptau}_2\end{array}} $$

(5.33)

On the other hand, K₁₂ in (5.31) is the exchange interaction energy

$$ {\mathrm{K}}_{12}=\iint {\upvarphi_{\mathrm{a}}}^{\ast}\left({\uptau}_2\right){\upvarphi_{\mathrm{b}}}^{\ast}\left({\uptau}_1\right)\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\frac{1}{{\mathrm{r}}_{12}}{\upvarphi}_{\mathrm{a}}\left({\uptau}_1\right){\upvarphi}_{\mathrm{b}}\left({\uptau}_2\right){\mathrm{d}\uptau}_1{\mathrm{d}\uptau}_2 $$

(5.34)

Note that K₁₂ is null when the spins are anti-direction, while K₁₂ is finite only when the spins are in the same direction. It is clear that the total energy is lower when the spins are anti-direction for K₁₂ > 0 for same direction, but K₁₂ = 0 for anti-direction.

The energy diagram of helium atom is shown in Fig. 5.6. The left is for spins in anti-direction, while the right is for the spins in the same direction. The former is called para-helium (S = 0) and the latter is called ortho-helium (S = 1), where S is the sum of spins and S = 0 is called singlet and S = 1 is triplets, because S = 1 has three quantum state of S = −1, 0, and 1.

Fig. 5.6

The energy diagram of a helium atom

5.3.3 Many-Electron Atom

In the case of atoms with many bound electrons, there are several approaches to obtain the atomic structure numerically, depending on how precisely the bound states should be calculated. The Hamiltonian for an atom or ion with atomic number Z with N bound electrons is given as

$$ {\displaystyle \begin{array}{c}\mathrm{H}\kern1em =-\frac{{\mathrm{\hslash}}^2}{2{\mathrm{m}}_{\mathrm{e}}}\sum \limits_{\mathrm{i}}^{\mathrm{N}}{\nabla}_{{\mathbf{r}}_i}^2+\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\left[-\sum \limits_i^{\mathrm{N}}\frac{\mathrm{Z}}{{\mathrm{r}}_{\mathrm{i}}}+\frac{1}{2}\sum \limits_{\mathrm{i}}^{\mathrm{N}}\sum \limits_{\mathrm{j}\ne i}^{\mathrm{N}}\frac{1}{\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|}\right]\\ {}={\mathrm{T}}_{\mathrm{e}}+{\mathrm{V}}_{\mathrm{ne}}+{\mathrm{V}}_{\mathrm{e}\mathrm{e}}\end{array}} $$

(5.35)

The first term is total electron kinetic energy, the second is Coulomb attraction energy by the nucleus, the third is electron-electron Coulomb interaction energy.

The basic strategy to solve (5.35) is to assume the configuration of N electrons. This means to define the quantum state of N electrons with the principal quantum number and the orbital angular momentum quantum number (n, l). For example, when six electrons are bound in an ion, the configuration of the ground state is represented as (1s)²(2s)²(2p)², with spectroscopic notation for l-state (s, p, d, f, g, h… for l = 0, 1, 2, 3, 4, 5, …). This is an approximate expression of N-electron quantum state. Note that the closed shells of 1s and 2s are omitted from the expression in the above case, in general, and given as 2p². The potential force to each electron is also assumed to be spherically symmetric. This is called central field approximation. Then, the total wave function is assumed to be given by the combination of the function in (5.66) for assumed configuration of N-electrons [1, 2].

The standard numerical method of atomic structure calculation is Hartree-Fock (HF) method. The total wave function is assumed to be given by Slater matrix so that the total wave function is anti-symmetric. In this method, an iterative calculation is required until the numerical solution finally converges. Sometimes, just the production of N single wave functions is also used as the total wave function and this method is called Hartree method. This method is simpler as numeric method than the Hartree-Fock method, but no exchange interaction is included. Numerical data base obtained with Hartree-Fock calculation may be used to solve Saha equation for multi-electron ions of medium- and high-Z atom plasma. However, it is too much just for obtaining the effective charge distribution. As we see later, more simplified atomic structure model is used for plasma hydrodynamic simulations. As seen in the case of helium atom, the exchange interaction gives different energy spectrum depending on the spins of N electrons.

It is troublesome numerically to obtain the final radial wave functions of many electron system because of the iteration conversion process. Historically, more convenient way has been developed. It is parametric potential (para-potential) method [6]. So-called opacity codes have been developed to study the radiation transport in high-temperature plasma such as inside stellar objects and laboratory plasmas. As seen below, the opacity calculation demands reasonable atomic structure data. The number of atomic configurations is very huge in relatively mid and high Z atoms, it is convenient if a good approximate model is available to obtain the data of radial distribution functions in many configurations.

The well-known opacity codes, OPAL [7] and HULLAC [8] have been developed with the parametric potential method. When discussing the effective potentials in OPAL, for example, the electron configurations are assumed with two components. The first component is a “parent” configuration consisting of all the electrons in a given configuration except one. The excluded electron defines the second component or “running” electron. The parent configuration defines the effective potential for all the subshells available to the running electron. In order to incorporate the shell structure of the parent while retaining an analytic Fourier transform, OPAL introduced a potential with one Yukawa term for each occupied shell in the parent configuration.

$$ \mathrm{V}\left(\mathrm{r}\right)=\frac{\mathrm{e}}{4{\uppi \upvarepsilon}_0\mathrm{r}}\left[\left(\mathrm{Z}-\upnu \right)+\sum \limits_{\mathrm{n}=1}^{\mathrm{n}\ast }{\mathrm{N}}_{\mathrm{n}}{\mathrm{e}}^{-{\upalpha}_{\mathrm{n}}\mathrm{r}}\right] $$

(5.36)

where

$$ \upnu =\sum \limits_{\mathrm{n}=1}^{\mathrm{n}\ast }{\mathrm{N}}_{\mathrm{n}} $$

(5.37)

is the number of electrons for the parent ion, N_n the number of electrons in the shell with principal quantum number n, n* the maximum value of n for the parent configuration, and α_n the screening parameter for electrons in the shell n having principal quantum number n.

$$ {\upalpha}_{\mathrm{n}}\left(\upzeta, \upnu \right)=\left(\upzeta +1\right)\left[{\mathrm{a}}_0\left(\upnu \right)+\frac{{\mathrm{a}}_1\left(\upnu \right)}{\upzeta}+\frac{{\mathrm{a}}_2\left(\upnu \right)}{\upzeta^2}\right] $$

(5.38)

where

$$ \upzeta =\mathrm{Z}-\upnu $$

(5.39)

The table of {a_i(ν), i = 0, 1, 2} are given in Ref. [6] and i = 3 is also given. These constants are obtained so that the results are optimized by comparing to the data of Dirac-Fock calculation. Relativistic version of Hartree-Fock. One electron wave function is calculated with Dirac equation in the potential (5.36). Note that some constants are also optimized to reproduce corresponding experimental data.

It is largely due to the improvement of the capability of the computer that such detailed calculation has become possible. However, simple model of atomic structure is demanded for direct coupling of radiation transport in the integrated hydrodynamic code.

5.3.4 Term Splitting

As shown in Fig. 5.7, the ion with one excited electron with a configuration p³s¹ has the average energy for the configuration, splitting to three via exchange interaction and to six levels by L-S coupling. The energy levels given by the total wave function depend on the total orbital angular momentum L and total spin S. Each LS term is (2L + 1)(2S + 1)-fold degenerate, where

$$ \mathbf{L}=\sum \limits_i{\boldsymbol{l}}_i,\kern1em \mathbf{S}=\sum \limits_i{\mathbf{s}}_i $$

(5.40)

It may be shown that the splitting of the configuration depends on L and S. The energy levels which are characterized by certain values of L and S are called terms and the splitting is called term splitting. The coupling of angular momenta of individual electrons to a resulting orbital angular momentum and spin is referred to as LS-coupling or Russell-Saunders coupling.

Fig. 5.7

An ion with one excited electron with a configuration p³s¹ has the average energy for the configuration, splitting to three via exchange interaction and to six levels by L-S coupling. The energy levels given by the total wave function depend on the total orbital angular momentum L and total spin S

Since the term splitting depends on L and S, the total electron state referred to as configuration state functions (CSFs) are shown as the combination of orbitals and LS coupling term in the form.

$$ \Psi \left({\gamma}^{\left(2\mathrm{S}+1\right)}\mathrm{L}\right) $$

where γ represents the orbital and S and L are the total spin and angular momentum. For example, the ground state of six electrons has the following two CSFs for anti-spin or parallel spin in 2p state.

$$ 1{\mathrm{s}}^22{\mathrm{s}}^22{\mathrm{p}}^{21}\mathrm{P}\kern1em \mathrm{or}\kern1em 1{\mathrm{s}}^22{\mathrm{s}}^22{\mathrm{p}}^{23}\mathrm{P} $$

For the case of helium atom in Fig. 5.6, the parahelium and ortho-helium in S = 0 and S = 3 states, respectively.

It is almost enough to determine the quantum state of multi-electron atoms or ions by Hartree-Fock methods. However, the details of line group structure become important for analyzing experimental spectroscopic data and/or computation of radiation hydrodynamics when the line radiation transport heats non-locally cold plasma region. It is required to improve by adding the spin-orbit interaction in Hamiltonian as follows.

$$ \mathrm{H}={\mathrm{T}}_{\mathrm{e}}+{\mathrm{V}}_{\mathrm{ne}}+{\mathrm{V}}_{\mathrm{e}\mathrm{e}}+{\mathrm{V}}_{\mathrm{so}} $$

(5.41)

where the spin orbit interaction is given in the form [1, 2].

$$ {\mathrm{V}}_{\mathrm{so}}={\mathrm{a}}_{\mathrm{so}}\sum \limits_{\mathrm{i}}^{\mathrm{N}}\frac{1}{r_i}\frac{\mathrm{\partial V}\left({\mathrm{r}}_{\mathrm{i}}\right)}{\partial {\mathrm{r}}_{\mathrm{i}}}{\boldsymbol{l}}_i\cdot {\mathbf{s}}_i,\kern2em {\mathrm{a}}_{\mathrm{so}}=\frac{\upmu_{\mathrm{B}}}{\mathrm{\hslash}{\mathrm{m}}_{\mathrm{e}}{\mathrm{c}}^2} $$

(5.42)

The energy levels of atomic electrons are affected by the sum of the interaction between the electron spin magnetic moment and the current due to orbital angular momentum of each electron. It can be visualized as a magnetic field caused by the electron’s orbital motion interacting with the spin magnetic moment of the electron. This effective magnetic field can be expressed in terms of the electron orbital angular momentum. Therefore, the energy levels have more fine structure and the number of energy levels becomes very huge when high-Z ions are considered.

Finally, schematic diagram of the lowest configuration of a neutral neon atom is shown in Fig. 5.8. The levels of each configuration lie within the limited energy range shown by the corresponding colored blocks (bands). This suggests that if the line width is broader than the separation of fine structure, it is possible to model all as a band, while if the width is narrower, it is required to identify the fine structure depending on what physics should remain in our model.

Fig. 5.8

Schematic diagram of the lowest configuration of a neutral neon atom. The levels of each configuration lie within the limited energy range shown by the corresponding shaded block

5.4 Quantum Theory of Electron Transitions

It is a fundamental of quantum mechanics to study the physics of atom-photon interaction based on the perturbation method. It is also a good example of mathematical physics on the perturbation theory. Try to derive the mathematics of transition of an electron in N-electron atom or ion, when a weak external perturbation is impacted on the bound electrons. Assume that the external one is due to a photon or an electron with its energy E₀ and its wave function is given in the form.

$$ {\mathrm{V}}_{\mathrm{e}\mathrm{x}}\left(\mathrm{t},\mathbf{r}\right)={\mathrm{a}}_0{\mathrm{e}}^{-{\mathrm{i}\upomega}_0\mathrm{t}+\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}} $$

(5.43)

where ω₀ = E₀/ℏ. Note that the wavenumber k₀ has the following relation with the energy, respectively,

$$ {\displaystyle \begin{array}{ll}\upomega =\mathrm{ck}& \left(\mathrm{photon}\right)\\ {}\upomega =\frac{\mathrm{\hslash}}{2{\mathrm{m}}_{\mathrm{e}}}{\mathrm{k}}^2& \left(\mathrm{electron}\right)\end{array}} $$

(5.44)

The total wave function of the initial bound electrons is Ψ_i(τ), where “i” means the initial state and τ indicate all coordinates of electrons. In addition, the total wave function of final state after transition is Ψ_f(τ), where “f” means the final state.

Assume that the perturbation is weak and the transition is slow enough as adiabatic transition, namely it is possible to assume that the wave function during the transition is approximated as a liner combination of the two states with time-dependent coefficients.

$$ \Psi \left(\mathrm{t},\boldsymbol{\uptau} \right)={\mathrm{C}}_{\mathrm{i}}\left(\mathrm{t}\right){\Psi}_{\mathrm{i}}+{\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right){\Psi}_{\mathrm{f}} $$

(5.45)

Of course, the conservation relation should be satisfied.

$$ {\left|{\mathrm{C}}_{\mathrm{i}}\right|}^2+{\left|{\mathrm{C}}_{\mathrm{f}}\right|}^2=1 $$

(5.46)

The initial and final wave functions are stationary ones and satisfy the Schrodinger equation.

$$ -\mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}}\Psi \left(\mathrm{t},\boldsymbol{\uptau} \right)={\mathrm{H}}_0\left(\boldsymbol{\uptau} \right)\Psi \left(\mathrm{t},\boldsymbol{\uptau} \right) $$

(5.47)

where H₀ is Hamiltonian given in (5.37) for N-electron atom. The initial and final states Ψ_i(τ) and Ψ_f(τ) are eigen sates of (5.22) for N-electron Schrodinger equation. In addition to H₀, the perturbation potential energy (5.43) is included in time dependent Schrodinger equation. Assume that the wave function in the transition phase is given by the form (5.45). Then the Schrodinger equation for the perturbation terms is given as

$$ -\mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}}\left[{\mathrm{C}}_{\mathrm{i}}\left(\mathrm{t}\right){\Psi}_{\mathrm{i}}+{\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right){\Psi}_{\mathrm{f}}\right]={\mathrm{V}}_{\mathrm{ex}}\left(\mathrm{t},\boldsymbol{\uptau} \right)\left[{\mathrm{C}}_{\mathrm{i}}\left(\mathrm{t}\right){\Psi}_{\mathrm{i}}+{\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right){\Psi}_{\mathrm{f}}\right] $$

(5.48)

Integrating (5.48) by all coordinate τ after the product by the complex conjugate of the final state Ψ_f^*, the following relation is obtained.

$$ -\mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}}\left[{\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right)\right]={\mathrm{C}}_{\mathrm{i}}\left(\mathrm{t}\right)\int {\Psi_{\mathrm{f}}}^{\ast }{\mathrm{V}}_{\mathrm{ex}}\left(\mathrm{t},\boldsymbol{\uptau} \right){\Psi}_{\mathrm{i}}\mathrm{d}\boldsymbol{\uptau } +{\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right)\int {\Psi_{\mathrm{f}}}^{\ast }{\mathrm{V}}_{\mathrm{ex}}\left(\mathrm{t},\boldsymbol{\uptau} \right){\Psi}_{\mathrm{f}}\mathrm{d}\boldsymbol{\uptau } $$

(5.49)

Note that the transition is given by the change of electron configuration. For example, consider the initial state is the oxygen ground state and the final state is one of the following two.

$$ 1{\mathrm{s}}^22{\mathrm{s}}^22{\mathrm{p}}^4\to 1{\mathrm{s}}^22{\mathrm{s}}^22{\mathrm{p}}^3\mathrm{nd} $$

(5.50)

$$ 1{\mathrm{s}}^22{\mathrm{s}}^22{\mathrm{p}}^4\to 1\mathrm{s}2{\mathrm{s}}^22{\mathrm{p}}^3\mathrm{np} $$

(5.51)

(5.50) is the transition of an outer shell electron, while (5.50) is the inner-shell electron transition. It is possible to integrate (5.49) by τ except for the transiting electron coordinate r. (5.49) becomes the integral to one electron wave function (φ_i, φ_f).

$$ \frac{{\mathrm{dC}}_{\mathrm{f}}}{\mathrm{dt}}=\mathrm{i}\frac{{\mathrm{a}}_0}{\mathrm{\hslash}}{\mathrm{e}}^{\mathrm{i}\Delta \upomega \mathrm{t}}\int {\upvarphi_{\mathrm{i}}}^{\ast }{\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}{\upvarphi}_{\mathrm{f}}\mathrm{d}\mathbf{r}={\upalpha}_{\mathrm{i}\mathrm{f}}{\mathrm{e}}^{\mathrm{i}\Delta \upomega \mathrm{t}} $$

(5.52)

where

$$ \Delta \upomega =\left({\upomega}_{\mathrm{i}}-{\upomega}_{\mathrm{f}}\right)-{\upomega}_0 $$

(5.53)

$$ {\upalpha}_{\mathrm{i}\mathrm{f}}=\mathrm{i}\frac{{\mathrm{a}}_0}{\mathrm{\hslash}}\int {\upvarphi_{\mathrm{i}}}^{\ast }{\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}{\upvarphi}_{\mathrm{f}}\mathrm{d}\mathbf{r} $$

(5.54)

In deriving (5.52), it is assumed that C_i = 1. The second term in (5.49) is neglected because of the rapidly oscillating term with small C_f at the beginning, and only the first term remains because of resonance.

$$ {\mathrm{C}}_{\mathrm{f}}\left(\mathrm{t}\right)={\upalpha}_{\mathrm{i}\mathrm{f}}\frac{{\mathrm{e}}^{\mathrm{i}\Delta \upomega \mathrm{t}}-1}{\mathrm{i}\Delta \upomega} $$

(5.55)

Therefore, the probability of the j state is near the time origin

$$ {\left|{\mathrm{C}}_{\mathrm{f}}\right|}^2={\mathrm{C}}_{\mathrm{f}}^{\ast }{\mathrm{C}}_{\mathrm{f}}={\left|{\upalpha}_{\mathrm{if}}\right|}^2\frac{2\left(1-\cos \Delta \upomega \mathrm{t}\right)}{{\left(\Delta \upomega \right)}^2} $$

(5.56)

As can be seen from the uncertainty principle, the energy level has a finite width and it is necessary to integrate over the frequency ω for obtaining the transition probability of electrons. Then, changing variables like

$$ \Delta \upomega \mathrm{t}=\upxi =2\mathrm{x} $$

(5.57)

and using the relation

$$ {\int}_{-\infty}^{\infty}\frac{1-\cos \upxi}{\upxi^2}\mathrm{d}\upxi ={\int}_{-\infty}^{\infty}\frac{\sin^2\mathrm{x}}{{\mathrm{x}}^2}\mathrm{d}\mathrm{x}=\uppi $$

(5.58)

The following solution is obtained.

$$ {\left|{\mathrm{C}}_{\mathrm{f}}\right|}^2={\upnu}_{\mathrm{if}}\mathrm{t},\kern2em {\upnu}_{\mathrm{if}}=2\uppi {\left|{\upalpha}_{\mathrm{if}}\right|}^2 $$

(5.59)

where ν_if is the transition probability (in unit of s⁻¹).

5.5 Photo-excitation and Ionization

Calculate the transition probability by photon interaction. As mentioned in Vol. 1, Hamiltonian of an electron including interaction with photon in the vacuum is given in the form [9]

$$ {\displaystyle \begin{array}{l}\mathrm{H}=\frac{{\mathbf{P}}_{\mathrm{c}}^2}{2\mathrm{m}}=\frac{{\left(\mathbf{p}+\mathrm{e}\mathbf{A}\right)}^2}{2\mathrm{m}}={\mathrm{H}}_0+\frac{\mathrm{e}}{\mathrm{m}}\mathbf{p}\cdot \mathbf{A}+\frac{{\mathrm{e}}^2}{\mathrm{m}}{\mathbf{A}}^2\\ {}{\mathrm{H}}_0=\frac{{\mathbf{p}}^2}{2\mathrm{m}}\end{array}}, $$

(5.60)

where A is the vector potential of photon field. Assume that only one electron interacts with the photon for the transition and the other electrons are not necessary to consider in the following analysis. Use the following corresponding relation of the operators.

$$ \mathrm{E}\to \mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}},\kern2em \mathbf{p}\to -\mathrm{i}\mathrm{\hslash}\nabla $$

(5.61)

Then, Schrodinger equation is obtained as

$$ \mathrm{i}\mathrm{\hslash}\frac{\partial }{\mathrm{\partial t}}\Psi =\left[{\mathrm{H}}_0+\mathrm{i}\mathrm{\hslash}\frac{\mathrm{e}}{\mathrm{m}}\mathbf{A}\cdot \nabla +\frac{{\mathrm{e}}^2}{2\mathrm{m}}{\mathbf{A}}^2\right]\Psi $$

(5.62)

It is clear that the second term on RHS is linear perturbation and the third term is non-linear perturbation. Here, the analysis is limited to the case of the linear theory and the third term is neglected. Assume that the vector potential is due to plane electromagnetic wave polarized in the x-direction.

$$ \mathbf{A}\left(\mathrm{t},\mathbf{r}\right)={\mathrm{A}}_0{\mathbf{i}}_{\mathrm{x}}{\mathrm{e}}^{-\mathrm{i}\left({\upomega}_0\mathrm{t}-{\mathbf{k}}_0\mathbf{r}\right)} $$

(5.63)

Comparing (5.43) and (5.62), the assumed coefficient a₀ is in the present case found to be an operator.

$$ {\mathrm{a}}_0{\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}=-i\mathrm{\hslash}\frac{\mathrm{e}}{\mathrm{m}}{\mathrm{A}}_0{\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}{\mathbf{i}}_{\mathrm{x}}\cdot \nabla $$

(5.64)

In order to calculate α_if in (5.54), we have to calculate

$$ {\upalpha}_{\mathrm{i}\mathrm{f}}=\frac{\mathrm{e}}{\mathrm{m}}{\mathrm{A}}_0\left\langle \mathrm{i}\right|{\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}{\mathbf{i}}_{\mathrm{x}}\cdot \nabla \left|\mathrm{f}\right\rangle $$

(5.65)

It is not easy to directly integrate (5.65). Let’s examine the relationship between the photon energy and the energy level of electrons. In the case of hydrogen, the wave function of electrons is at most the extent of the Bohr radius. However, its energy level is about 10 eV, and the wave number of the photon with energy of 10 eV is k = 8 × 10⁴ cm⁻¹. This is because of the difference of the dispersion relation shown in (5.44). Therefore, the exponent of (5.65) is very small such as

$$ \left\langle {\mathrm{k}}_0\mathrm{r}\right\rangle \sim {10}^{-3} $$

(5.66)

This means it is possible to use Taylor expansion.

$$ {\mathrm{e}}^{\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}}\approx 1+\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}+\frac{1}{2}{\left({\mathbf{k}}_0\cdot \mathbf{r}\right)}^2+.\dots $$

(5.67)

When only the first term unity is taken, the wavelength of the radiation field corresponds to an infinite. Such assumption is called dipole approximation. The second term in (5.67) gives electric quadrupole transition and magnetic dipolar transition, while they are neglected in the following analysis.

5.5.1 Dipole Transition Matrix Element

In integrating (5.65), the corresponding relation is used.

$$ -\mathrm{i}\mathrm{\hslash}\nabla =\mathbf{p}=\mathrm{m}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\mathrm{t}}=-\mathrm{i}\mathrm{m}\upomega \mathbf{r} $$

(5.68)

(5.65) is rewritten by setting ω in (5.68) as ω_if.

$$ {\upalpha}_{\mathrm{if}}=\frac{\mathrm{e}}{\mathrm{\hslash}}{\mathrm{A}}_0{\upomega}_{\mathrm{if}}\left\langle \mathrm{i}\right|\mathrm{x}\left|\mathrm{f}\right\rangle $$

(5.69)

In general, the transition from i to j defines the dipole matrix element as.

$$ {\mathbf{r}}_{\mathrm{ij}}=\left\langle \mathrm{i}\right|\mathbf{r}\left|\mathrm{j}\right\rangle $$

(5.70)

The matrix of (5.65) is its x component of (5.70). The value of the matrix element is large as the overlapping of the wave functions of i and j becomes stronger. The one electron wave function is possible to be given in the form (5.66). The radial distribution of the electron, rR(r) can be imaged from that of an electron in hydrogen atom in Fig. 5.9. It is seen that the closer the principal quantum number is, the stronger the overlapping in (5.70). However, note that the integral on θ and φ in (2.97) gives the selection rule to be explained later.

Fig. 5.9

The radial distributions of an electron, rR(r), of a hydrogen atom

Since the electric field E₀ = ω₀A₀, (5.65) is shown as

$$ {\left|{\upalpha}_{\mathrm{if}}\right|}^2=\frac{{\mathrm{e}}^2}{{\mathrm{\hslash}}^2}{\mathrm{E}}_0^2{\left|{\mathrm{x}}_{\mathrm{if}}\right|}^2 $$

(5.71)

Now, since the polarization direction is along x, the transition probability is symmetric around the x axis and θ is the angle formed by r and the x axis, x = r cos θ

$$ {\displaystyle \begin{array}{l}\frac{1}{\uptau_{\mathrm{if}}}={\upnu}_{\mathrm{if}}=\frac{4}{3}\frac{\uppi^2}{\mathrm{c}{\mathrm{\hslash}}^2}\left(\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\right)I\left(\upomega \right){\left|{\mathbf{r}}_{\mathrm{if}}\right|}^2,\\ {}\mathrm{I}\left(\upomega \right)=\frac{1}{2}{\upvarepsilon}_0{\mathrm{c}\upomega}^2{\mathrm{A}}_0^2\end{array}} $$

(5.72)

Here, I(ω) is the photon energy flux overlap with the transition energy spectrum and the relationship 〈cos²θ〉 = 1/2 is used. Consequently, τ_if represents the transition time. When the number density of atoms or ions with the same configuration of the initial state is n_i, the transition cross section σ_if is given by the relation.

$$ {\upnu}_{\mathrm{i}\mathrm{f}}={\mathrm{n}}_{\mathrm{i}}{\upsigma}_{\mathrm{i}\mathrm{f}} $$

(5.73)

5.5.2 Einstein’s A, B Coefficients

(5.72) gives coefficients of absorption and emission due to induced process, but in fact there is spontaneous emission process that cannot be solved by the above perturbation theory. Consider two energy levels as a simple quantum system. The upper level is indicated by 2, and the lower level is indicated by 1. Then, considering the three processes shown in Fig. 5.10, the number of the state 1 (N₁), evolves according to the following coupled equation.

$$ \frac{{\mathrm{dN}}_1}{\mathrm{dt}}={\mathrm{A}}_{21}{\mathrm{N}}_2+{\mathrm{B}}_{21}\mathrm{I}\left(\upomega \right){\mathrm{N}}_2-{\mathrm{B}}_{12}\mathrm{I}\left(\upomega \right){\mathrm{N}}_1 $$

(5.74)

At the same time, a similar equation governs the evolution of N₂. Here, A is the spontaneous emission coefficient, which is derived from the fact that the state other than the ground level is unstable and has a finite lifetime. Then, B₂₁ and B₁₂ indicate induced emission coefficient and absorption coefficient, respectively.

Fig. 5.10

Energy diagram of two-level system showing absorption, spontaneous emission, and stimulated emission

When both atoms and radiation fields are completely in thermal equilibrium, the right side of Eq. (5.74) must be balanced. In other words, when the photon field is Planck distribution (see Appendix) and the electron population is Boltzmann distribution, the detail balance relation should be satisfied.

$$ {\mathrm{A}}_{21}{\mathrm{N}}_2+{\mathrm{B}}_{21}\mathrm{I}\left(\upomega \right){\mathrm{N}}_2-{\mathrm{B}}_{12}\mathrm{I}\left(\upomega \right){\mathrm{N}}_1=0 $$

(5.75)

where I(ω) = (Planck distribution).

$$ {B}_T\left(\nu \right)=\frac{2h{\nu}^3/{c}^2}{\exp \left( h\nu /T\right)-1} $$

(5.76)

The electron equilibrium population should satisfy the relation

$$ \frac{{\mathrm{N}}_2}{{\mathrm{N}}_1}=\frac{{\mathrm{g}}_2}{{\mathrm{g}}_1}\exp \left(-\frac{{\mathrm{E}}_2-{\mathrm{E}}_1}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right) $$

(5.77)

It is noted that B₁₂ has already been given by (5.72), so it is possible to explicitly obtain the remaining B₂₁ and A₂₁ from the three simultaneous equations. Equation (5.72) gives B₁₂ as follows

$$ {\mathrm{B}}_{12}=\frac{2}{3}\frac{\uppi}{{\mathrm{\hslash}}^2}\left(\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0}\right){\left|{\mathbf{r}}_{12}\right|}^2 $$

(5.78)

Using (5.78), B₂₁ and A₂₁ are obtained as follows.

$$ {\mathrm{B}}_{21}=\frac{{\mathrm{g}}_1}{{\mathrm{g}}_2}{\mathrm{B}}_{12}, $$

(5.79)

B₁₂ and A₂₁ are written as

$$ {\mathrm{A}}_{21}=\frac{2{\mathrm{h}\upnu}^3}{{\mathrm{c}}^2}{\mathrm{B}}_{21} $$

(5.80)

Since the three coefficients are proportional to the dipole moment, we introduce a dimensionless quantity of order of unity called oscillator strength

$$ {\mathrm{f}}_{\mathrm{ij}}=\frac{2\mathrm{m}\upomega}{3\mathrm{\hslash}}{\left|{\mathbf{r}}_{\mathrm{ij}}\right|}^2 $$

(5.81)

Then, the spontaneous emission coefficient is given in [10].

$$ {\mathrm{A}}_{21}=8.0\times {10}^9{\left(\frac{\mathrm{\hslash \upomega }}{13.6\mathrm{eV}}\right)}^2{\mathrm{f}}_{21}\kern1em {\mathrm{s}}^{-1} $$

(5.82)

In order to estimate the size of f_ij intuitively, rewrite (5.81) as

$$ {\mathrm{f}}_{12}=\frac{2}{3}\frac{\mathrm{\hslash \upomega }}{{\mathrm{E}}_0}\frac{{\left|{\mathbf{r}}_{12}\right|}^2}{{\mathrm{a}}_{\mathrm{B}}^2}, $$

(5.83)

where E₀ is the energy of hydrogen ground state.

We found that Einstein’s A and B coefficients are proportional to the oscillator strength. From the integral in the spherical coordinates in the calculation of the oscillator strength, it is found that the selection rule is derived. What remains is the integral of radial direction r. Radial wave function is a solution oscillating to positive and negative except for the ground state (Fig. 5.9). It is clear that when the principal quantum numbers of |1〉 and |2〉 states are too far each other, the integral becomes that of positive and negative oscillating functions and becomes smaller. Oscillator strength takes a large value when both principal quantum numbers are close to each other. In addition, it can be expected that the larger ℏω/E₀ in (5.83) the value of ${\left|{\mathbf{r}}_{12}\right|}^2/{\mathrm{a}}_0^2$ the smaller. In fact, the magnitude of the oscillator strength is a constant value, and in the case of hydrogen atoms, it is known to be as shown in Table 5.1 [10].

Table 5.1 Oscillator strength of hydrogen atom. The values are for the principal quantum number n = 1–5 [10]. The positive values are the absorption oscillator strength and the negative ones the emission oscillator strength. The larger values mean easier transition from the initial state to the final state

In addition, the following sum rule called Fermi’s golden rule is satisfied.

$$ \sum \limits_{\mathrm{j}}{\mathrm{f}}_{\mathrm{ij}}=1 $$

(5.84)

Note that the summation of (5.84) represents all states for the complete system, including free electron states.

5.5.3 Selection Rule

Carry out the integration of (5.70). As is well known each Cartesian coordinate is transferred to the spherical coordinate as follows,

$$ {\displaystyle \begin{array}{l}\mathrm{x}=\mathrm{r}\sin \uptheta \cos \upvarphi \\ {}\mathrm{y}=\mathrm{r}\sin \uptheta \sin \upvarphi \\ {}\mathrm{y}=\mathrm{r}\cos \uptheta \end{array}} $$

(5.85)

The spherical harmonics in (5.24) is defined as

$$ {\mathrm{Y}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\uptheta, \upvarphi \right)={\mathrm{P}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\uptheta \right){\mathrm{e}}^{\mathrm{im}\upvarphi} $$

(5.86)

where P_l^m is Legendre fold function. The integral (5.70) by θ coordinate has the following form,

$$ \int_{-1}^1{\upxi \mathrm{P}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\upxi \right){\mathrm{P}}_{{\mathrm{\ell}}^{\hbox{'}}}^{{\mathrm{m}}^{\hbox{'}}}\left(\upxi \right)\mathrm{d}\xi $$

(5.87)

Legendre fold function has a formula.

$$ {\upxi \mathrm{P}}_{\mathrm{\ell}}^{\mathrm{m}}\left(\upxi \right)=\frac{\mathrm{\ell}+\mathrm{m}}{2\mathrm{\ell}+1}{\mathrm{P}}_{\mathrm{\ell}-1}^{\mathrm{m}}\left(\upxi \right)+\frac{\mathrm{\ell}-\mathrm{m}+1}{2\mathrm{\ell}+1}{\mathrm{P}}_{\mathrm{\ell}+1}^{\mathrm{m}}\left(\upxi \right) $$

(5.88)

Inserting (5.88) into (5.87), it is easy to show that the integral has finite value only for the case,

$$ \Delta \mathrm{\ell}=\pm 1 $$

(5.89)

In addition, the integral to φ coordinate requires the following selection rule, too.

$$ \Delta m=0,\pm 1 $$

(5.90)

5.6 Photo Excitation and De-excitation

In case where ions are partially ionized, spontaneous transition from the upper-level q to the lower-level p has been derived in (5.78). Rewriting it with the oscillator strength of radiation f_p,q, we obtain

$$ \mathrm{A}\left(\mathrm{q},\mathrm{p}\right)=\frac{{\mathrm{e}}^2{\upomega}^2}{2{\uppi \mathrm{mc}}^3{\upvarepsilon}_0}\frac{\mathrm{g}\left(\mathrm{p}\right)}{\mathrm{g}\left(\mathrm{q}\right)}{\mathrm{f}}_{\mathrm{p},\mathrm{q}}\kern1em \left[{\mathrm{s}}^{-1}\right] $$

(5.91)

Here, g(p) and g(q) are the number of states at the levels p and q, respectively. The oscillator strength f_q,p of radiation emission is related to the absorption oscillator strength f_p,q as follows.

$$ {\mathrm{f}}_{\mathrm{q},\mathrm{p}}=\frac{\mathrm{g}\left(\mathrm{p}\right)}{\mathrm{g}\left(\mathrm{q}\right)}{\mathrm{f}}_{\mathrm{p},\mathrm{q}} $$

(5.92)

Regarding the oscillator strength of absorption in the case of hydrogen, there is a classical expression by Kramers (p. 269 in [10]),

$$ {\mathrm{f}}_{\mathrm{p},\mathrm{q}}=\frac{2^6}{3\sqrt{3\uppi}}\frac{1}{2{\mathrm{p}}^5}\frac{1}{{\mathrm{q}}^3}\frac{1}{{\left({\mathrm{p}}^{-2}-{\mathrm{q}}^{-2}\right)}^3}{\mathrm{g}}_{\mathrm{bb}} $$

(5.93)

Here, g_bb is a gaunt factor, which is a correction factor for matching with quantum mechanically accurate calculation. The subscript “bb” means a transition from a bound state to a bound state.

It is valuable to note a simple property of f_p,q. For a given p, f_p,q becomes smaller in proportion to q⁻³ for larger q. In addition, the oscillator strength between higher levels such as f_p,p + 1 tends to the following form.

$$ {\mathrm{f}}_{\mathrm{p},\mathrm{p}+1}\simeq \left(\mathrm{p}+1\right)/5 $$

(5.94)

Let’s find the photo excitation cross section. From the relation (5.72), we see that the reciprocal of the transition time from p to q is B_p,qI(ω). This should be equal to ρ(ω)σ where ρ(ω) is the number density of photons at thermal equilibrium (Planck distribution) and we obtain the cross section of spontaneous emission.

$$ \upsigma \left(\mathrm{p},\mathrm{q}\right)=\frac{1}{4}\frac{{\mathrm{c}}^3{\mathrm{h}}^2}{{\left(\mathrm{\hslash \upomega}\right)}^2}\mathrm{A}\left(\mathrm{q},\mathrm{p}\right) $$

(5.95)

Figure 5.11 shows the p and q dependence of A (q, p).

Fig. 5.11

The p and q dependence of Einstein’s A coefficient A (q, p) [4]. Reprinted with kind permission by T. Fujimoto

The cross section of the photo de-excitation is automatically obtained from the relation of detailed balance by using the cross section of the photo excitation. In thermodynamic equilibrium condition, RHS of (5.74) should balance, namely (5.75) should be satisfied. This is called the principle of detailed balance. That is, if the cross-section of photo-excitation is given, the cross-section of photo de-excitation in the reverse process should be automatically determined with use of (5.77) and (5.78). We insert Planckian intensity distribution into (5.74), and we use the relation (5.76). If we put these relationships into (5.74), we obtain the photo de-excitation cross section.

$$ \upsigma \left(\mathrm{q},\mathrm{p}\right)={\mathrm{B}}_{\mathrm{q},\mathrm{p}}{\mathrm{B}}_{\mathrm{T}}\left(\upomega \right) $$

(5.96)

Here, B_T(ω) is energy flux of the Planck distribution define by ℏωcρ(ω) and (5.76). B_q,p in (5.96) is obtained by using (5.77).

5.7 Photoionization and Photo-recombination

Let’s calculate the cross section of photoionization where a photon interacts an atom and one electron becomes free electron by absorption of photon energy. The way of thinking is the same as in the above photo-excitation case, and this time we think about the transition including the electron free state from the bound state. We have to start with finding the wave function of free electron whose energy derived from Schrodinger equation is positive and the wave function is infinitely spread. This results in a wave function including the spherical Bessel function derived by the scattering problem. However, as the calculation becomes complicated, let’s calculate the cross-section within Born approximation here. In Born approximation, free electron is assumed plane wave and is written in the following normalized form

$$ {\uppsi}_{\mathrm{f}}={\left(\frac{1}{2\uppi}\right)}^{3/2}{\mathrm{e}}^{\mathrm{i}\mathbf{kr}} $$

(5.97)

That energy is

$$ {\mathrm{E}}_{\mathrm{f}}=\frac{{\mathrm{\hslash}}^2{\mathrm{k}}^2}{2\mathrm{m}} $$

(5.98)

Consider the photo-ionization cross section for the case where the initially bound electron |i〉 is ionized and becomes a final state of free electron |f〉. Then, as has been derived in (5.65), we obtain the following form as the transition probability

$$ {\upalpha}_{\mathrm{if}}=\frac{\mathrm{e}}{\mathrm{m}}{\mathrm{A}}_0\left\langle \mathrm{i}\right|{\mathrm{e}}^{-\mathrm{i}\mathbf{k}\cdot \mathbf{r}}{\mathbf{e}}_{\mathrm{p}}\cdot \nabla \left|\mathrm{f}\right\rangle $$

(5.99)

Try to integrate (5.99) explicitly by assuming the case of |i〉 being the 1s of a hydrogen atom. The radial wave function of the hydrogen 1s is given as

$$ {\uppsi}_{\mathrm{i}}={\left(\frac{1}{{\uppi \mathrm{a}}_0^3}\right)}^{1/2}{\mathrm{e}}^{-\mathrm{r}/{\mathrm{a}}_0} $$

(5.100)

Inserting (5.100) and (5.97) into (5.99), the integration is easily carried out with assumption of the dipole approximation (5.67) in the form.

$$ \int {\mathrm{e}}^{-\mathrm{r}/{\mathrm{a}}_0}{\mathrm{e}}^{-\mathrm{i}\mathbf{k}\cdot \mathbf{r}}\mathrm{d}\mathbf{r}=\frac{8\uppi}{{\mathrm{a}}_0{\left({\mathrm{a}}_0^{-2}+{\mathrm{k}}_{\mathrm{f}}^2\right)}^2} $$

(5.101)

With use of (5.101), the cross section (5.73) for photo-ionization is derived to be the following form.

$$ \upsigma \left(\upomega \right)=32\upalpha \frac{\mathrm{\hslash}}{{\mathrm{m}\upomega \mathrm{a}}_0^5}\frac{{\mathrm{k}}_{\mathrm{f}}^2}{{\left({\mathrm{a}}_0^{-2}+{\mathrm{k}}_{\mathrm{f}}^2\right)}^4}\int {\cos}^2\uptheta \mathrm{d}\Omega $$

(5.102)

where θ is the angle formed by the polarization direction of light and the direction of the wave number of free electrons. In (5.100) α is the fine-structure constant.

$$ \upalpha =\frac{{\mathrm{e}}^2}{2{\upvarepsilon}_0\mathrm{hc}}=1/137 $$

(5.103)

Note that in deriving (5.102) we have assumed the wave number of free electrons is short, that is, the energy of light is sufficiently larger than the binding energy of the ground state;

$$ {\mathrm{k}}_{\mathrm{f}}>>\frac{1}{{\mathrm{a}}_0} $$

(5.104)

In addition, (5.102) can be written for any hydrogen-like ions of atomic number Z in the form,

$$ \upsigma \left(\upomega \right)=\frac{128\uppi}{3}\upalpha \frac{\mathrm{\hslash}}{\mathrm{m}\upomega}{\left(\frac{\mathrm{Z}}{{\mathrm{k}}_{\mathrm{f}}{\mathrm{a}}_0}\right)}^5 $$

(5.105)

(5.105) can be written finally as follows.

$$ \upsigma \left(\upomega \right)=\frac{16\sqrt{2}\uppi}{3}\upalpha {\left(\frac{\mathrm{Z}}{{\mathrm{a}}_0}\right)}^5{\left(\frac{\mathrm{\hslash}}{\mathrm{m}\upomega}\right)}^{7/2} $$

(5.106)

As you can see in (5.106), the photo-ionization cross section strongly depends on the photon energy.

The above calculation can be also expressed using the concept of the absorption oscillator strength in the free state f_p,ε. Assuming that the energy of incident light is hν, the cross-section of photoionization is analytically obtained with respect to hydrogen-like case. Ionization cross section can be written with the oscillator strength f_p,ε in the form;

$$ {\upsigma}_{\mathrm{p},\upvarepsilon}\left(\upnu \right)=\frac{e^2}{4{\mathrm{mc}\upvarepsilon}_0}\frac{{\mathrm{d}\mathrm{f}}_{\mathrm{p},\upvarepsilon}}{\mathrm{d}\upnu} $$

(5.107)

Here, f_p,ε is the continuous absorption oscillator strength for the transition from the bound state p to a free state with kinetic energy ε. In the case of hydrogen atoms, the expression of f_p,θ in (5.93) can be used and extended to the bound-free transition. Replacing the final state to free state as q → iκ, a pure imaginary number, we obtain the following form;

$$ {\mathrm{f}}_{\mathrm{p},\upkappa}=\frac{2^6}{3\sqrt{3\uppi}}\frac{1}{2{\mathrm{p}}^5}\frac{1}{\upkappa^3}\frac{1}{{\left({\mathrm{p}}^{-2}+{\upkappa}^{-2}\right)}^3}{\mathrm{g}}_{\mathrm{bf}} $$

(5.108)

where κ should satisfy the energy conservation relation

$$ \mathrm{h}\upnu ={\mathrm{z}}^2\mathrm{R}\left(\frac{1}{{\mathrm{p}}^2}+\frac{1}{\upkappa^2}\right) $$

(5.109)

where R is the Rydberg constant (=13.6 eV), and z is equal to 1 in the case of hydrogen and the charge number of the hydrogen-like ion. About the hydrogen-like atom whose charge number is z and who has one-electron in the s-state, we have the following relation.

$$ {\mathrm{df}}_{\mathrm{p},\upvarepsilon}={\mathrm{f}}_{\mathrm{p},\upkappa}\mathrm{d}\upkappa ={\mathrm{f}}_{\mathrm{p},\upkappa}\frac{{\mathrm{h}\upkappa}^3}{2{\mathrm{z}}^2\mathrm{R}}\mathrm{d}\upnu $$

(5.110)

Using (5.110) the photoionization collision cross section is given

$$ {\upsigma}_{\mathrm{p},\upvarepsilon}\left(\upnu \right)=\upalpha \uppi \frac{2^6}{3\sqrt{3}}{\left(\frac{{\mathrm{p}}^2{\mathrm{a}}_{\mathrm{B}}}{\mathrm{z}}\right)}^2\frac{1}{{\mathrm{p}}^3}{\left(\frac{{\mathrm{z}}^2\mathrm{R}/{\mathrm{p}}^2}{\mathrm{h}\upnu}\right)}^3{\mathrm{g}}_{\mathrm{bf}} $$

(5.111)

Here, α is the fine structure constant (=1/137). This equation is equivalent to (5.107). When considering the case of a hydrogen atom and looking for the value at the point of hν = R, the cross section of the ground state (p = 1) is calculated

$$ \upsigma =7\times {10}^{-18}\kern1em \left[{\mathrm{cm}}^2\right] $$

For the case of hydrogen, the photoionization cross section is shown in Fig. 5.12. In the log-log figure, the straight line is a power law as given in (5.111). The following features are known for the cross section.

• 1. There is a threshold for hν for photoionization, requiring the photon energy more than the ionization energy

• 2. Photo-ionization cross section from the s-state is proportional to 1/(hν)³

• 3. For higher energy photon, the photo-ionization cross section is larger for the lower energy level transition as seen in Fig. 5.12.

Regarding the iron atom (Fe), the photoionization coefficient from the K shell of iron is shown in Fig. 5.13 as a function of photon energy. As can be seen from Fig. 5.12, the ionization cross section is maximum at the threshold of photon energy, and abruptly decreases by the power of energy as the energy increases. Another characteristic feature is that as shown in Fig. 5.13. the ionization cross section from the grand state 1s hardly depends on ionization state. In other words, when high energy photons are incident on atoms with high-Z value, it is necessary to always consider inner shell ionization.

Fig. 5.12

The photoionization cross section of a hydrogen atom [4]. Reprinted with kind permission by T. Fujimoto

Fig. 5.13

The photoionization coefficient from the K shell of iron is shown as a function of photon energy [3]. Reprinted with kind permission by D. Salzmann

As evidenced by the discussion above, when a photon energy high enough to ionize 1s-electron, the 1s electron is ionized predominantly even if there are 2s electrons. Since the photo-ionization cross section is larger for the inner shell, it can be seen that the inner shell ionization is dominant. In this way, photoionization plasma needs to be modeled by taking account of the ionization progresses after the inner shell vacancy is produced.

Now, photo recombination, which is the reverse process of the photoionization, is an elementary process in which a free electron is captured to a bound state (quantum state m) to release the excess energy as a photon. The problem is a two-body problem in terms of quantum mechanics; a plane wave (5.97) of the free electron is defined as an initial state of (5.99). However, the cross section of the photo-recombination is easily obtained by considering the detail balance relation in the thermodynamic equilibrium state.

5.8 Quantum Theory of Electron Impact on Atom

In plasma high energy electrons play important role in ionizing and recombining the ions via electron impact. They are called collisional ionization and recombination, respectively. Most of plasma temperature is low compared to mc² and non-relativistic analysis is enough to obtain the cross sections of such processes. The situation is the same in the cases of excitation and de-excitation as well. Therefore, the following analysis is limited to the non-relativistic case. It is, however, noted that the highly relativistic electron beam produced by accelerators have been used as diagnostic tool to study the structure of nuclei and quarks inside nuclei [9]. This is because the de-Broglie wavelength of impacting electron can resonate with the size of nucleus and elementary particles. So, the following analysis to be done by stating with Schrodinger equation can be easily extended to the relativistic case, if Dirac equation is instead used as the basic equation.

Consider the case where free electrons collide to ionize or excite atoms or ions. This is called electron impact excitation or ionization. Precise calculation requires computation using wave function of free electrons distorted by the atomic field, and analytical handling is complicated. It is now possible to solve almost exact equations in detail with computers.

The cross section of the collisional excitation is shown in Fig. 5.14 for helium atoms in the para-helium ground state already explained in Fig. 5.6. As can be seen from the Fig. 5.14, the probability of transition to the p-state is the largest, and it is also possible for transition to 2s which is the optically forbidden. It is also noted that the cross section decreases as the transition to the higher energy levels. Unlike photoionization, the cross section gradually increases from the threshold energy, and when the energy is about three times the threshold energy, the cross section becomes maximum, and thereafter decreases with the power of energy as determined by the Born approximation described later.

Fig. 5.14

The cross sections of the collisional excitation of helium atoms from the para-helium ground state as a function of an impacting electron kinetic energy

Different from the photoionization, in the electron collision, the ionization cross section of the outermost shell electron is largest, and bound electrons are peeling from the outside. It is also important to note that the photo-ionization cross section is of the order of barn (=10⁻²⁴ cm²), whereas the electron impact ionization cross section is on the order of 10⁻¹⁶ cm². Although it is not very meaningful to compare the cross section in Fig. 5.13 and one in Fig. 5.14, it is insufficient by comparison only with the cross sections. It is required to compare the ionization rates

$$ \upnu =\left\langle \mathrm{n}\upsigma \mathrm{v}\right\rangle $$

(5.112)

should be compared, taking average of the number of photons and photo energy distribution for the photo-ionization or the number of electrons and electron energy distribution in the electron impact ionization. In comparing the both ionization rates, it is necessary to evaluate the plasma temperature, density and optical depth etc. The optical depth is needed to evaluate if the photon density and energy distribution is close to the Planck distribution.

With sizes like normal laboratory plasma the density of photons is much lower than in Planck distribution and such plasma is called “optically thin plasma”. However, in huge plasmas like in the sun, radiation can be considered Planck distribution. Even in a laboratory plasma for example, when heating material with a high Z such as gold (Au: Z = 79) or uranium (U: Z = 92) with a high intensity laser, the radiation field is close to Planck distribution with the temperature of hundreds of eV.

The collisional ionization cross section of argon when peeling off electrons one by one is shown in Fig. 5.15. It can be seen that as the bound electrons are ionized and the ionization energy increases, the ionization cross section decreases. This can be interpreted intuitively that the interaction with free electrons becomes less likely as the wave function of bound electrons becomes smaller.

Fig. 5.15

The collisional ionization cross sections of argon when peeling off electrons one by one from neutral atom

5.8.1 Electron Impact to Atom

Consider the case where an electron collides on a neutral atom. There are three cases in the phenomena; elastic collision, electron collisional excitation, and electron impact ionization. These three processes are important elementary processes in plasma generation, plasma heating, and plasma transport. In plasma generation in a discharge tube, neutral atom excitation and plasma formation are the most basic atomic processes. This physics needs to be quantum mechanically analyzed, and the quantum scattering is the base for analysis.

Now let r_a be the coordinates of electrons subjected to excitation and ionization within the target atom, and denote with r the coordinates of the impact electron. The Schrodinger equation for a steady state where such phenomena continue is

$$ \left\{-\frac{{\mathrm{\hslash}}^2}{2\mathrm{m}}{\nabla}^2+{\mathrm{H}}_{\mathrm{a}}\left({\mathbf{r}}_{\mathrm{a}}\right)+\mathrm{V}\Big({\mathbf{r}}_{\mathrm{a}},\mathbf{r}\Big)-\mathrm{E}\right\}\Psi \left({\mathbf{r}}_{\mathrm{a}},\mathbf{r}\right)=0 $$

(5.113)

Here, E is the total energy of the atom and the free electron. The general solution of (5.113) is given by expanding the r_a dependence of the wave function by eigen-functions, so that we obtain the following multi electron wave function solution.

$$ \Psi \left({\mathbf{r}}_{\mathrm{a}},\mathbf{r}\right)=\sum \limits_{\upalpha}{\mathrm{F}}_{\upalpha}\left(\mathbf{r}\right){\upvarphi}_{\upalpha}\left({\mathbf{r}}_{\mathrm{a}}\right) $$

(5.114)

$$ {\mathrm{H}}_{\mathrm{a}}\left({\mathbf{r}}_{\mathrm{a}}\right){\upvarphi}_{\upalpha}\left({\mathbf{r}}_{\mathrm{a}}\right)={\mathrm{E}}_{\upalpha}{\upvarphi}_{\upalpha}\left({\mathbf{r}}_{\mathrm{a}}\right) $$

(5.115)

Here, φ_α(r_a) is the complete orthogonal functions to the Hamiltonian H_a(r_a) including all free electron eigen states. Substituting (5.114) into (5.113) and using the relation of (5.115), (5.113) becomes a simple form. Then, multiplying φ_α^∗(r_a) to the resultant form of (5.113) the following relation is obtained.

$$ \left({\nabla}^2+{\mathrm{k}}_{\upalpha}^2\right){\mathrm{F}}_{\upalpha}\left(\mathbf{r}\right)=\sum \limits_{\upbeta}{\mathrm{U}}_{\upalpha \upbeta}\left(\mathbf{r}\right){\mathrm{F}}_{\upbeta}\left(\mathbf{r}\right) $$

(5.116)

where

$$ {\mathrm{k}}_{\upalpha}^2=\frac{2\mathrm{m}}{{\mathrm{\hslash}}^2}\left(\mathrm{E}-{\mathrm{E}}_{\upalpha}\right) $$

(5.117)

$$ {\mathrm{U}}_{\upalpha \upbeta}\left(\mathbf{r}\right)=\frac{2\mathrm{m}}{{\mathrm{\hslash}}^2}\int {\upvarphi_{\upalpha}}^{\ast}\left({\mathbf{r}}_{\mathrm{a}}\right)\mathrm{V}\left({\mathbf{r}}_{\mathrm{a}},\mathbf{r}\right){\upvarphi}_{\upbeta}\left({\mathbf{r}}_{\mathrm{a}}\right)\mathrm{d}{\mathbf{r}}_{\mathrm{a}} $$

(5.118)

We think that the initial state of the electron in the atom is constantly excited to the eigenstate α (α can also be in the continuous state). Then, when there is a free electron sufficiently far from the atom before the impact, it is possible to assume the wave function of the two electrons before the impact is given by

$$ {\Psi}^0\left({\mathbf{r}}_{\mathrm{a}},\mathbf{r}\right)=\exp \left(\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}\right){\upvarphi}_0\left({\mathbf{r}}_{\mathrm{a}}\right) $$

(5.119)

As the two electrons interact in the atomic potential, the bound-bound electron transit from the initial state to the eigen-state α of the bounded electron. The free electron becomes a spherical scattering state with the center of the atom as the origin. Therefore, the wave function of the free electron is described as

$$ {\mathrm{F}}_{\upalpha}\left(\mathbf{r}\right)\to {\updelta}_{\upalpha 0}\exp \left(\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}\right)+\frac{1}{\mathrm{r}}\mathrm{f}\left(0\to \upalpha; \uptheta, \upvarphi \right)\exp \left(\mathrm{i}{\mathbf{k}}_{\upalpha}\cdot \mathbf{r}\right) $$

(5.120)

Multiply (5.120) by the ratio of the electron flow velocity, that is, the wave number ratio, and integrate (5.120) over all angles. Then, the following equation can be obtained as the collisional cross section where the bound electron is excited as |0〉 → |α〉 and at the same time the free electron is scattered.

$$ \upsigma \left(0\to \upalpha \right)=\frac{{\mathrm{k}}_{\upalpha}}{{\mathrm{k}}_0}\int \int {\left|\mathrm{f}\left(0\to \upalpha; \uptheta, \upvarphi \right)\right|}^2\sin \uptheta \mathrm{d}\uptheta \mathrm{d}\upvarphi $$

(5.121)

(5.121) is the cross-section of elastic scattering.

Based on the perturbation theory to find the solution of (5.121), so-called Born approximation is used. That is, (5.120) is assumed to be expanded as

$$ {\mathrm{F}}_{\upalpha}={{\mathrm{F}}_{\upalpha}}^{(0)}+{{\mathrm{F}}_{\upalpha}}^{(1)}+{{\mathrm{F}}_{\upalpha}}^{(2)}+\dots $$

(5.122)

In this approximation, the term V(r_a, r) in (5.113) is regarded a perturbation term. Then the first order equation is given

$$ \left({\nabla}^2+{\mathrm{k}}_{\upalpha}^2\right){\mathrm{F}}_{\upalpha}^{(1)}=\sum \limits_{\upbeta}{\mathrm{U}}_{\upalpha \upbeta}\left(\mathbf{r}\right){\mathrm{F}}_{\upbeta}^{(0)}\left(\mathbf{r}\right) $$

(5.123)

In addition, the following form is also assumed.

$$ {\mathrm{F}}_{\upalpha}^{(0)}\left(\mathbf{r}\right)={\updelta}_{\upalpha 0}\exp \left(\mathrm{i}{\mathbf{k}}_0\cdot \mathbf{r}\right) $$

(5.124)

Then, RHS of (5.123) is given. It is noted that Green function method is usually used to solve (5.116) exactly [9].

The collision cross section in the case of elastic collision is shown in Fig. 5.16. Its value is, of course, a function of the energy of the colliding electrons. It is useful to note that it is larger than the cross-section of “inelastic collision” such as collisional excitation and collision ionization.

Fig. 5.16

Elastic collision cross sections for Ar, Kr, and Xe neutral atoms as a function of impacting electron velocity. Note that such elastic collision cross section is larger than the cross-section of “inelastic collision” such as collisional excitation and collision ionization

As shown in Fig. 5.15, the ionization energy has the threshold value for the impact ionization cross section and the cross section abruptly increases. It takes the maximum value at about 2–3 times the threshold energy, and thereafter suddenly decreases. The detailed calculations of (5.121) are given in more specialized books, but a comparison between the results of Born approximation and experimental values is shown for the case of hydrogen in Fig. 5.17 [9]. Even when Born approximation is used, impact ionization of hydrogen can be obtained with such degree of accuracy. The Born approximation is appropriate at the high energy limit of the colliding electron, but the height of the peak of the cross section in the vicinity of the threshold is only about 1.5 times different. It is surprising to note that the properties of the curve are well reproduced.

Fig. 5.17

Elastic collision cross section obtained theoretically by Born approximation compared to the experimental value for the case of hydrogen atom. The horizontal and vertical axis are impacting electron momentum and collision cross sections in arbitrary units, respectively. It is seen that the theory can predict well the experiment. Reproduced from ref. [9] by permission of Person education

5.8.2 Elastic Scattering

Let us calculate the cross-section of the elastic collision. Since the coordinate dependence of (5.113) can be fixed for r_a, the wave function can be obtained by solving an equation for one electron wave function F₀(r) in the form of (5.120). Partial wave expansion, which is Fourier transformation of any function axially symmetric, is applied to the scattering component as

$$ {\mathrm{F}}_0\left(\mathbf{r}\right)=\frac{1}{\mathrm{r}}\sum \limits_{\mathrm{\ell}}{\mathrm{A}}_{\mathrm{\ell}}{\mathrm{u}}_{\mathrm{\ell}}\left(\mathrm{r}\right){\mathrm{P}}_{\mathrm{\ell}}\left(\cos \uptheta \right) $$

(5.125)

where P_ℓ is a normalized Legendre function.

By multiplying P_ℓ to (5.113) and integrating it over the angle, the following equation is obtained after the separation of variables.

$$ \frac{{\mathrm{d}}^2{\mathrm{u}}_{\mathrm{\ell}}}{{\mathrm{d}\mathrm{r}}^2}+\left\{{\mathrm{k}}_0^2-\mathrm{V}\left(\mathrm{r}\right)-\frac{\mathrm{\ell}\left(\mathrm{\ell}+1\right)}{{\mathrm{r}}^2}\right\}{\mathrm{u}}_{\mathrm{\ell}}=0 $$

(5.126)

Consider the properties of the solution of (5.126) intuitively without solving. Since the effective range of V (r) is of the order of Angstrom (10⁻⁸ cm), the inside of the parentheses in (5.126) becomes negative for large values of ℓ. In other words, it can be considered that there is only a small component of ℓ in scattering of low energy electrons. It is also clear that the increase of the free electron energy, large angle scattering described with large numbers of ℓ appears. It is noted that the partial waves with ℓ = 1, 2, 3 are called s-wave, p-wave, d-wave. Since the s-wave has no ℓ-dependency, it shows isotropic scattering. The asymptotic solution of Eq. (5.126) at large radius is known to be

$$ {\mathrm{u}}_{\mathrm{\ell}}{\left(\mathrm{r}\right)}_{\mathrm{r}\to \infty}\to \frac{1}{{\mathrm{k}}_0}\sin \left({\mathrm{k}}_0\mathrm{r}-\frac{1}{2}\mathrm{\ell \uppi }+{\upeta}_{\mathrm{\ell}}\right) $$

(5.127)

Here, η_ℓ is the “phase shift“due to scattering for the partial wave with angular quantum number ℓ. By placing (5.127) in (5.121), the scattering cross section can be found as follows.

$$ \upsigma =\frac{4\uppi}{{\mathrm{k}}_0^2}\sum \limits_{\mathrm{\ell}=0}^{\infty}\left(2\mathrm{\ell}+1\right){\sin}^2{\upeta}_{\mathrm{\ell}} $$

(5.128)

In the scattering of low energy electrons only a small number of partial waves appear with lower ℓ numbers. This fact suggests that the phase shift of larger l number is near nπ, where n is an integer. Especially when it is less than 1 eV, only s-wave appears. It is also known that the phase shift becomes nπ at certain electron energy. Then, the scattering cross section of (5.128) becomes extremely small. This phenomenon is called the Ramsauer effect. In Fig. 5.16, elastic collision cross section by the noble gas is shown. It is seen that Ramsauer effect clearly appears at the low-speed part.

In Vol. 1, the scattering of an electron in Coulomb field by classical mechanics was solved to find the formula of Rutherford scattering. Let’s compare the same electron scattering due to bare hydrogen nucleus (proton) to that obtained by the quantum mechanical analysis. The same problem can be solved exactly as the scattering problem described above. The scattering cross section thereof agrees with the classical Rutherford scattering one in the form [9].

$$ \upsigma ={\left(\frac{{\mathrm{e}}^2}{4{\uppi \upvarepsilon}_0{\mathrm{mv}}^2}\right)}^2\frac{1}{4\sin^2\left(\uptheta /2\right)} $$

(5.129)

It is useful to note that such quantum scattering of electron beam is applied to study a nuclear structure of many nuclei. In this case the potential force is due to not only Coulomb force but also nuclear force is inserted in (5.113). Historically, such electron beam measurement has been used to identify the particle distribution of nucleus and quarks in nucleon [11]. Note that in the case of relativistic electron scattering measurement by nuclei, the spin effect of electron and nucleus becomes important and the cross-section (5.129) is modified to Mott scattering cross-section as shown in this chapter [11]. With the increase of Lorentz factor β to unity, the scattering angle becomes narrower than Rutherford scattering (5.129). The Motto formula is the relativistic quantum scattering derived by stating with Dirac equation.

5.8.3 Electron Collision De-excitation and Recombination

Solving (5.113), the cross sections of the collisional excitation and de-excitation are obtained. As clear in the above formulation, the bound state wave function transits from the initial state 〈i| = 〈0| to the final state 〈f| = 〈α| in (5.118). Such an inelastic collision, the wavenumber of the scattered electron k₀ in (5.120) should be replaced with k_α in (5.117).

However, recombination due to electron collision cannot be described by the formulation above with only one electron scattering wave. It is necessary to formulate for two electron scattered wave after the ionization. This is a three-body problem, it is hard to deal with quantum mechanically. The detail mathematics is out of the scope of the book.

If the cross sections of the electron collision excitation and electron impact ionization are calculated, the electron collision de-excitation and electron collisional recombination can be obtained. This principle is called the detail balance requiring that the latter two reversal process should balance in each when the thermodynamic equilibrium is established in plasma. Then, the distribution function of free electrons is Maxwellian.

The measured cross sections of the elastic, collisional-excitation, and collisional ionization are shown in Fig. 5.18 for the case of neutral argon gas as a function of impacting electron energy. When increasing the energy of electrons, firstly elastic scattering is dominant, excitation occurs from the point where electron energy exceeds the threshold, and then electron impact ionization becomes dominant in further high energy region. By injection of high-energy electron beams to any neutral gas, it is possible to generate plasma.

Fig. 5.18

The cross sections of elastic, collisional excitation, and collisional ionization as function of impacting electron kinetic energy for an argon atom

5.9 Atomic Process in Maxwellian Free Electrons

Even if we have some detail database via computation about photon and electron atomic processes, it is hard job to chase the time evolution of all atomic states. This is because the atomic process cross sections discussed are functions of photon energy and electron kinetic energy. If the photon field is not Planck distribution or the free electron energy distribution is not Maxwellian, the atomic process demands a huge computation. This resembles to the case why the hydrodynamic approximation is used to study the macroscopic plasma dynamics, instead of solving kinetic equation to the velocity distribution.

Most of the plasma analysis, the free electron distribution is assumed to be Maxwellian, even when the electron distribution in the bound state is not necessarily Boltzmann distribution. Regarding the photon energy distribution, it is usual that the photon field is neglected in laboratory plasmas, while it is assumed Planckian in stellar objects. This is valid by evaluating the optical thickness of the plasmas. Most of laboratory plasmas are optically thin except for the plasma of high-Z atom like gold.

It is useful to summarize the atomic processes to be taken into account for studying the physics of laboratory and astrophysical plasmas. It is also necessary to include the free-free radiation (called Bremsstrahlung radiation) and free-free absorption (inverse-Bremsstrahlung) already discussed in Vol. 1.

• 1. Photo excitation

• 2. Photo de-excitation (spontaneous and stimulated emission)

• 3. Photo-ionization

• 4. Photo-recombination

• 5. Electron collisional excitation

• 6. Electron collisional de-excitation

• 7. Electron impact ionization

• 8. Electron collisional recombination (two-electron recombination):

• 9. Bremsstrahlung radiation

• 10. Inverse-Bremsstrahlung (absorption):

In the photo excitation, we need to know the detailed distribution of atomic energy levels and line-profile of radiation emission and absorption. The line profile in frequency space is determined by synthesized effects such as natural width, Doppler broadening, Stark broadening. Especially the theory of Stark broadening at high density plasma is still the theme of the forefront of research. The discussion is relatively simple if the photon distribution is Planckian. However, in the laboratory plasma and the interstellar plasma whose density is relatively low, the radiation spectrum in plasma is generally very different from Planck distribution. Therefore, usually the photo-ionization and excitation process are neglected in determining the atomic states in plasma.

Assume that the plasma is close to the thermodynamic equilibrium state and the electron velocity distribution is Maxwellian. Then, the number of atomic process for an ion per unit time and unit volume, ν is given as an averaged value over the free electron velocity distribution.

$$ \upnu =\left\langle \mathrm{n}\upsigma \mathrm{v}\right\rangle $$

(5.130)

Here, n is the number density of atoms or ions, v is the velocity of electrons, σ is the velocity-dependent cross section, and 〈〉 means to take average value with the electron velocity distribution. Considering that n can be put out from the averaging and the free electrons have a Maxwell distribution of the temperature T, then

$$ \upnu =\mathrm{n}\int {\upsigma \mathrm{vf}}_{\mathrm{M}}\left(\mathrm{v}\right)\mathrm{dv}=4\uppi {\left(\frac{\mathrm{m}}{2\uppi \mathrm{T}}\right)}^{3/2}\int {\mathrm{v}}^3\upsigma \left(\mathrm{v}\right)\exp \left(-\frac{{\mathrm{m}\mathrm{v}}^2}{2\mathrm{T}}\right)\mathrm{dv} $$

(5.131)

Converting (5.131) to the integral to the kinetic energy $\upvarepsilon =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathrm{mv}}^2$, the followings is obtained

$$ \upchi =\frac{\upnu}{\mathrm{n}}=\frac{2^{3/2}\sqrt{\mathrm{m}}}{\sqrt{\uppi}{\mathrm{T}}^{3/2}}{\int}_0^{\infty }{\upvarepsilon}^{3/2}\upsigma \left(\upvarepsilon \right)\exp \left(-\frac{\upvarepsilon}{\mathrm{T}}\right)\mathrm{d}\upvarepsilon $$

(5.132)

where χ is called rate coefficient. It is found that the rate coefficient for excitation and ionization due to electron collision is a function only of temperature.

The coefficients of the atomic process involving the collision by free electrons are obtained by substituting the cross section as a function of electron energy into (5.132) and performing integration. However, it should be noted that the velocity distribution function of free electrons is limited to Maxwell distribution. For example, in the case of impact ionization, as shown in Fig. 5.15, the cross-section of the integrand of (5.132) is a function that rapidly rises from the ionization energy. However, Maxwell distribution has the shape as shown in Fig. 5.19, which has the maximum value. When the temperature is low, the tail component of the Maxwell distribution contributes greatly to those products as shown in Fig. 5.19. It is noted that depending on generation and heating process, plasma is not necessarily Maxwell distribution. Particularly, since the tail component of the velocity distribution is often elongated, it is better approximated by the Maxwellian distribution with two temperatures. It should keep in mind in using to the average values. It is useful to note that nuclear reaction cross sections such as DT fusion reaction also have strong dependence on the particle energy and the corresponding integrand in (5.132) has a peak called Gamow peak.

Fig. 5.19

Maxwellian averaged cross section in the case of impact ionization (see Fig. 5.15). The cross-section rapidly rises from the ionization energy like σ_Aor σ_T, while Maxwell distribution has the shape like v²f_e(v) as shown in the figure. When the temperature is relatively low, only the high-energy component of the Maxwell distribution contributes greatly to the averaged cross sections

The following point should also be noted. The electron velocity distribution has been assumed to be isotropic (spherically symmetric in speed space). However, when energy transport is strong in a certain direction, or when electrons confined by magnetic field are heated, the distribution function may become anisotropic. When ions in such plasma are excited by free electrons in an anisotropic velocity distribution, it is known that the excited quantum state can be biased, and line emission via transition to the ground state by spontaneous emission is polarized. The science field for studying physics of polarized emission is called “polarization spectroscopy“, and studies are progressing with space physics observations and laboratory plasmas. Polarization spectroscopy is used as a method to investigate the structure of the magnetic field, especially in the universe.

5.9.1 Rate Coefficient of Electron Collision Excitation

The rate coefficient of velocity-averaged electron collisional excitation is

$$ {\displaystyle \begin{array}{c}{\upchi}_{\mathrm{m},\mathrm{n}}=\left\langle {\upsigma}_{\mathrm{m},\mathrm{n}}\mathrm{v}\right\rangle \\ {}\approx 16\uppi {\left(\frac{2{\uppi \mathrm{E}}_{\mathrm{H}}}{3\mathrm{m}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{a}}_0^2{\mathrm{f}}_{\mathrm{m},\mathrm{n}}\frac{{\mathrm{E}}_{\mathrm{H}}}{{\mathrm{E}}_{\mathrm{m},\mathrm{n}}}{\left(\frac{{\mathrm{E}}_{\mathrm{H}}}{\mathrm{T}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{g}}_{\mathrm{m},\mathrm{n}}\exp \left(-\frac{{\mathrm{E}}_{\mathrm{m},\mathrm{n}}}{\mathrm{T}}\right)\end{array}} $$

(5.133)

Here, m < n, E_H = 13.6 eV = R (Rydberg constant), a₀ is the Bohr radius, g_m,n is the gaunt factor. f_{m, n} is the oscillator strength of the absorption defined in (5.83). (5.133) reduces to

$$ {\upnu}_{\mathrm{m},\mathrm{n}}^{\mathrm{bb}}={\mathrm{n}}_{\mathrm{e}}\left\langle {\upsigma}_{\mathrm{m},\mathrm{n}}\mathrm{v}\right\rangle =4.3\times {10}^{-6}{\mathrm{f}}_{\mathrm{m},\mathrm{n}}\frac{{\mathrm{n}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{e}\mathrm{V}}^{3/2}}\frac{\mathrm{T}}{{\mathrm{E}}_{\mathrm{m},\mathrm{n}}}\exp \left(-\frac{{\mathrm{E}}_{\mathrm{m},\mathrm{n}}}{\mathrm{T}}\right)\kern1em \left[{\mathrm{s}}^{-1}\right] $$

(5.134)

Here, Gaunt factor g_mn = 0.275 was used as approximate value.

5.9.2 Rate Coefficient of Electron Impact Ionization

The impact ionization coefficient (5.67) can be calculated in principle by setting the initial state to be bound electrons in atoms and the wave function after collision to be a plane wave (free electron) in a certain direction. However, it is easier to start with the formula by Thomson (1912) that has been used historically as the simplest expression given in (6.83) in [12].

$$ {\upsigma}_{\mathrm{m},\mathrm{c}}=4{\uppi \mathrm{a}}_0^2{\left(\frac{{\mathrm{E}}_{\mathrm{H}}}{\mathrm{E}}\right)}^2\left(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{m},\mathrm{c}}}-1\right) $$

(5.135)

where E is the electron kinetic energy and E_mc is the ionization energy from the bound state 〈m|. When this ionization cross section is multiplied by the Maxwell, the following rate is obtained after the integration.

$$ {\displaystyle \begin{array}{c}{\upchi}_{\mathrm{m},\mathrm{c}}=\left\langle {\upsigma}_{\mathrm{m},\mathrm{c}}\mathrm{v}\right\rangle \\ {}=8{\uppi \mathrm{a}}_0^2{\left(\frac{2{\mathrm{E}}_{\mathrm{H}}}{\uppi \mathrm{m}}\right)}^{1/2}{\left(\frac{{\mathrm{E}}_{\mathrm{H}}}{{\mathrm{E}}_{\mathrm{m},\mathrm{c}}}\right)}^{3/2}{\upbeta}^{-1/2}\left[{\upbeta \mathrm{e}}^{-\upbeta}-{\upbeta \mathrm{E}}_1\left(\upbeta \right)\right]\end{array}} $$

(5.136)

where

$$ \upbeta =\frac{{\mathrm{E}}_{\mathrm{m},\mathrm{c}}}{\mathrm{T}} $$

(5.137)

The definition of E₁ and the approximate expression for the large value of β are

$$ {\mathrm{E}}_1\left(\upbeta \right)={\int}_{\upbeta}^{\infty}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}}\mathrm{dx}={\mathrm{e}}^{-\upbeta}\left(\frac{1}{\upbeta}-\frac{1}{\upbeta^2}+\cdots \right) $$

(5.138)

Using (5.138), (5.136) becomes

$$ {\upchi}_{\mathrm{m},\mathrm{c}}=8{\uppi \mathrm{a}}_0^2{\left(\frac{2{\mathrm{E}}_{\mathrm{H}}}{\uppi \mathrm{m}}\right)}^{1/2}{\left(\frac{{\mathrm{E}}_{\mathrm{H}}}{{\mathrm{E}}_{\mathrm{m},\mathrm{c}}}\right)}^{3/2}{\upbeta}^{-1/2}{\mathrm{e}}^{-\upbeta} $$

(5.139)

As a more precise expression, Lotz has obtained the following formula fitted well with one-electron data of high Z [3].

$$ {\upchi}_{\mathrm{m},\mathrm{c}}=8{\uppi \mathrm{a}}_0^2{\left(\frac{2{\mathrm{E}}_{\mathrm{H}}}{\uppi \mathrm{m}}\right)}^{1/2}{\left(\frac{{\mathrm{E}}_{\mathrm{H}}}{{\mathrm{E}}_{\mathrm{m},\mathrm{c}}}\right)}^{3/2}{\upbeta}^{-1/2}\left[0.69{\mathrm{e}}^{-\upbeta}\mathrm{f}\left(\upbeta \right)\right] $$

(5.140)

where

$$ \mathrm{f}\left(\upbeta \right)={\upbeta \mathrm{e}}^{\upbeta}{\mathrm{E}}_1\left(\upbeta \right) $$

(5.141)

This function varies from 0.34 to 0.90 for β = 1/4 to 8. Here, if you obtain the rate coefficient of impact ionization after inserting the values into (5.139), the impact ionization frequency is

$$ {\upnu}_{\mathrm{m},\mathrm{c}}^{\mathrm{bf}}=2.15\times {10}^{-6}{\mathrm{T}}_{\mathrm{e}\mathrm{V}}^{-3/2}{\mathrm{n}}_{\mathrm{e}}{\upbeta}^{-2}{\mathrm{e}}^{-\beta}\kern1em \left[{\mathrm{s}}^{-1}\right] $$

(5.142)

5.9.3 Detailed Balance and Collisional Rates

The rate coefficient of collisional de-excitation is obtained from the rate coefficient of collisional excitation based on the detailed balance. The governing equation for transition of bound electrons due to electron collision with respect to the quantum states m and n can be written as follows.

$$ \frac{{\mathrm{dN}}_{\mathrm{m}}}{\mathrm{dt}}=-\frac{{\mathrm{dN}}_{\mathrm{n}}}{\mathrm{dt}}={\upnu}_{\mathrm{n},\mathrm{m}}^{\mathrm{bb}}{\mathrm{N}}_{\mathrm{n}}-{\upnu}_{\mathrm{m},\mathrm{n}}^{\mathrm{bb}}{\mathrm{N}}_{\mathrm{m}} $$

(5.143)

This relation should be balanced in the thermal equilibrium state. Using the rate coefficient of (5.134), the following is obtained.

$$ {\displaystyle \begin{array}{c}{\upnu}_{\mathrm{n},\mathrm{m}}^{\mathrm{bb}}=\frac{{\mathrm{N}}_{\mathrm{m}}}{{\mathrm{N}}_{\mathrm{n}}}{\upnu}_{\mathrm{m},\mathrm{n}}^{\mathrm{bb}}\\ {}=\frac{{\mathrm{g}}_{\mathrm{m}}}{{\mathrm{g}}_{\mathrm{n}}}{\mathrm{e}}^{{\mathrm{E}}_{\mathrm{n}}-{\mathrm{E}}_{\mathrm{m}}}{\upnu}_{\mathrm{m},\mathrm{n}}^{\mathrm{bb}}\end{array}} $$

(5.144)

Now, in collisional recombination coefficient is also derived from the detailed balance, using the impact ionization coefficient. In case of the high-density state, it becomes necessary to use the Fermi-Dirac distribution for electrons, but here the rate coefficient at the limit of low density satisfying the condition,−μ/T >  > 1, is obtained as follows from the detailed balance with (5.139).

$$ {\upnu}_{\mathrm{cm}}^{\mathrm{fb}}=3.55\times {10}^{-28}{\mathrm{g}}_{\mathrm{m}}\frac{{\mathrm{n}}_{\mathrm{e}}^2}{{\mathrm{T}}_{\mathrm{e}\mathrm{V}}^3}{\upbeta}^{-2}\kern0.5em \left[{\mathrm{s}}^{-1}\right] $$

(5.145)

5.9.4 Rate Coefficient of Photo-recombination

The rate coefficient of photo recombination is obtained from the rate coefficient of photoionization in the local thermal equilibrium (LTE) state where photo distribution is Planckian. In LTE, the ionization cross section of (5.107) needs to be averaged by the radiation field of the Planck distribution. Planck distribution is a function of the temperature. The coefficient of optical recombination averaged by the energy distribution of electrons and photons is a function of only the quantum state of bound electrons in the ion.

When the cross section of photo-recombination is calculated from the detailed balance, the general theory is very complicated. Limit the study to the case only when a captured electron forms a hydrogen-like ion of charge state z from the charge state z-1. Write the photo recombination cross section as σ_ε,p, where the suffix indicates that ε is the kinetic energy of a free electron and that the free electron emits a photon of energy hν and is captured in the state of the principal quantum number p. The following energy conservation should be satisfied.

$$ \mathrm{h}\upnu =\frac{1}{2}{\mathrm{mv}}^2+\frac{{\mathrm{z}}^2{\mathrm{I}}_{\mathrm{H}}}{{\mathrm{p}}^2},\kern1em \upvarepsilon =\frac{1}{2}{\mathrm{mv}}^2 $$

(5.146)

In LTE, the rate coefficient of photoionization given by (5.107) needs to be equal to the rate coefficient of photo recombination given by σ_ε,p., namely the relation becomes as follows using the Planck distribution.

$$ {\mathrm{n}}_0^{\upzeta +1}{\mathrm{n}}_{\mathrm{e}}{\mathrm{f}}_{\mathrm{M}}\left(\mathrm{v}\right){\mathrm{d}\mathrm{vv}\upsigma}_{\upvarepsilon, 0}=\sum \limits_{\mathrm{p}}{\mathrm{n}}_{\mathrm{p}}^{\upzeta}{\uprho}_{\upnu}{\mathrm{d}\upnu \mathrm{c}\upsigma}_{\mathrm{p},\upvarepsilon}\left(1-{\mathrm{e}}^{-\mathrm{h}\upnu /\mathrm{T}}\right) $$

(5.147)

Here, f_M is the normalized Maxwell distribution, and ρ_ν is the photon density of the Planck distribution. The last parenthesis on R is a term for subtracting the number of photons by stimulated emission. In addition, the relationship dν/(vdv) = h/m is satisfied by (5.146).

The cross section of photo recombination is obtained from (5.147) as follows

$$ {\upsigma}_{\upvarepsilon, 0}=\sum \limits_{\mathrm{p}}\frac{{\mathrm{n}}_{\mathrm{p}}^{\upzeta}}{{\mathrm{n}}_0^{\upzeta +1}{\mathrm{n}}_{\mathrm{e}}}\frac{\mathrm{m}}{\mathrm{h}}\frac{\uprho_{\upnu}\mathrm{c}}{{\mathrm{f}}_{\mathrm{M}}\left(\mathrm{v}\right)}\left(1-{\mathrm{e}}^{-\mathrm{h}\upnu /\mathrm{T}}\right){\upsigma}_{\mathrm{p},\upvarepsilon} $$

(5.148)

where ρ_ν is given by (B2.3), and f_M is (B1.19). The first term of (5.148) is given by Saha’s solution found in (5.14). Inserting the constants to (5.148), it is shown as

$$ {\upsigma}_{\upvarepsilon, 0}=2.8\times {10}^{-21}\frac{{\mathrm{z}}^2}{\upvarepsilon_{\mathrm{eV}}}\upvarphi \left(\frac{{\mathrm{z}}^2{\mathrm{I}}_{\mathrm{H}}}{\upvarepsilon}\right)\kern2em \left[{\mathrm{cm}}^2\right] $$

(5.149)

The dimensionless function φ (x) is a constant of order unity only depending on the principal quantum number. For the case where p = 1 contributes the most, its value is about 1, and (5.149) is approximated as follows

$$ {\upsigma}_{\upvarepsilon, 0}\approx 3\times {10}^{-21}\frac{{\mathrm{z}}^2}{\upvarepsilon_{\mathrm{eV}}}\kern2em \left[{\mathrm{cm}}^2\right] $$

(5.150)

5.10 Bremsstrahlung Emission and Absorption

Via collisions of free electrons with ions, photons are emitted. Since the above atomic processes via photons are important to be installed in the radiation hydrodynamic simulation to be discussed later, it is better to study the radiation process via free-free electron interaction. This photo-emission is called Bremsstrahlung. The reverse process contributes photo-absorption by free electrons. Note that this process is very important as classical absorption of laser photons as studied in Vol. 1. Calculate the cross-sections of Bremsstrahlung and its inverse process. Do not calculate strictly, try intuitive derivation, and finally match numerical coefficients to the exact solutions. Keep in mind that an electron is accelerated by an ion.

The charged particles under acceleration radiates electromagnetic waves. The electric dipole p has the following relationship for charged particles 1 and 2:

$$ \mathbf{p}={\mathrm{e}}_1{\mathbf{r}}_1+{\mathrm{e}}_2{\mathbf{r}}_2=\upmu \left(\frac{{\mathrm{e}}_1}{{\mathrm{m}}_1}-\frac{{\mathrm{e}}_2}{{\mathrm{m}}_2}\right)\mathbf{r} $$

(5.151)

Here, μ is the reduced mass, and r is the inter-particle distance. As can be seen, dipole radiation does not occur in collisions between electrons. Radiation emission by two electrons can be made by quadrupole effect. Consider radiation emitted when an electron collides at high speed with an iron of electric charge Z via Coulomb force. Then, electron is decelerated to run away from the ion.

In the case of the collision parameter b introduced in Chap. 2 and the typical acceleration is

$$ \left|\ddot{\mathbf{r}}\right|=\upalpha =\frac{1}{\mathrm{m}}\frac{{\mathrm{e}}^2\mathrm{Z}}{4{\uppi \upvarepsilon}_0{\mathrm{b}}^2} $$

(5.152)

Also, the time interval to feel acceleration force is

$$ \Delta \mathrm{t}=\frac{2\uppi \mathrm{b}}{\mathrm{v}} $$

(5.153)

Therefore, the energy radiated by one collision is

$$ \Delta \mathrm{E}=\mathrm{P}\Delta \mathrm{t}=\frac{\uppi}{4{\mathrm{c}}^3{\upvarepsilon}_0}{\left(\frac{{\mathrm{e}}^2\mathrm{Z}}{4{\uppi \upvarepsilon}_0\mathrm{m}}\right)}^2\frac{1}{{\mathrm{vb}}^3} $$

(5.154)

where P is the power of Larmor radiation emission.

Introduce a new physical quantity q_ν defined as

$$ {\mathrm{dq}}_{\upnu}=\Delta \mathrm{E}\times 2\uppi \mathrm{bdb} $$

(5.155)

Note that this has dimensions of (energy) × (area). Assuming that the reciprocal of the frequency ν of the light emitted is roughly equal to Δt in (5.153), the following relation is obtained after inserting (5.154) to (5.155).

$$ {\mathrm{dq}}_{\upnu}=\frac{{\mathrm{e}}^6{\mathrm{Z}}^2}{32{\mathrm{m}}^2{\mathrm{c}}^3{\upvarepsilon_0}^3}\frac{1}{{\mathrm{v}}^2}\mathrm{d}\upnu $$

(5.156)

This is a rough calculation. It is known that the exact coefficient is larger by $4/\sqrt{3}=2.3$ after solving the trajectory mathematically,

$$ {\mathrm{dq}}_{\upnu}=\frac{{\mathrm{e}}^6{\mathrm{Z}}^2}{8\sqrt{3}{\mathrm{m}}^2{\mathrm{c}}^3{\upvarepsilon_0}^3}\frac{1}{{\mathrm{v}}^2}\mathrm{d}\upnu $$

(5.157)

For a given free electron distribution f (v), the energy radiated per unit time, volume, and frequency is

$$ {\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{e}}\mathrm{f}\left(\mathrm{v}\right){\mathrm{dvvdq}}_{\upnu}\left(\upnu \right) $$

(5.158)

The minimum velocity v_m at which electrons are not captured by ion is

$$ \frac{1}{2}{{\mathrm{m}\mathrm{v}}_{\mathrm{m}}}^2=\mathrm{h}\upnu $$

(5.159)

Maxwell distribution of temperature T is

$$ \mathrm{f}\left(\mathrm{v}\right)\mathrm{dv}=\sqrt{\frac{2}{\uppi}}{\left(\frac{\mathrm{m}}{\mathrm{T}}\right)}^{3/2}{\mathrm{v}}^2{\mathrm{e}}^{-\frac{{\mathrm{m}\mathrm{v}}^2}{2\mathrm{T}}}\mathrm{dv} $$

(5.160)

Integrate Eq. (5.159) from the minimum velocity v_m in (5.159), the spectral emission is obtained.

$$ {\mathrm{J}}_{\upnu}\mathrm{d}\upnu ={\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{e}}{\int}_{{\mathrm{v}}_{\mathrm{m}}}^{\infty}\mathrm{f}\left({\mathrm{v}}^{\hbox{'}}\right){\mathrm{dv}}^{\hbox{'}}{\mathrm{v}}^{\hbox{'}}{\mathrm{dq}}_{\upnu}\left({\mathrm{v}}^{\hbox{'}}\right) $$

(5.161)

Inserting (5.160) and (5.157) to (5.161), the spectral emission power is obtained.

$$ {\mathrm{J}}_{\upnu}\mathrm{d}\upnu =\frac{1}{4}{\left(\frac{2}{3\uppi \mathrm{mT}}\right)}^{1/2}\frac{{\mathrm{e}}^6{\mathrm{Z}}^2}{{\mathrm{mc}}^3{\upvarepsilon_0}^2}{e}^{-\frac{\mathrm{h}\upnu}{\mathrm{T}}}\mathrm{d}\upnu $$

(5.162)

This is the energy spectrum of Bremsstrahlung emission of plasma from unit volume and unit time.

(5.162) indicated that Bremsstrahlung shows a straight line in the semi-logarithmic graph as shown in the experimental data of Fig. 5.20 [13]. Since this inclination represents temperature, it is used for temperature measurement of optically thin plasma. By integrating over the spectrum, the radiant energy per unit time and unit volume can be obtained as follows.

$$ \mathrm{J}={\int}_0^{\infty }{\mathrm{J}}_{\upnu}\mathrm{d}\upnu =1.7\times {10}^{-26}{\mathrm{Z}}^2{\mathrm{n}}_{\mathrm{e}}{\mathrm{n}}_{\mathrm{i}}\sqrt{{\mathrm{T}}_{\mathrm{e}\mathrm{V}}}\kern1em \left[\mathrm{W}/{\mathrm{cm}}^3\right] $$

(5.163)

where n_e and n_i is in unit of cm⁻³ and T is in eV unit.

Fig. 5.20

Bremsstrahlung emission spectrum shows a straight line in the semi-logarithmic graph in the experimental data. Reproduced from Ref. [13] by permission of John Wiley & Sons Ltd

Now, consider the absorption coefficient by the inverse-Bremsstrahlung process. From the detail balance, the photo-absorption by free electrons in the Planck distribution should balance with the emission of (5.158). That is

$$ {\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{e}}{\mathrm{U}}_{\upnu, \mathrm{P}}\mathrm{d}\upnu \mathrm{f}\left(\mathrm{v}\right){\mathrm{cdva}}_{\upnu}\left(1-{\mathrm{e}}^{-\mathrm{hv}/\mathrm{T}}\right) $$

(5.164)

Here, a_ν is the spectral absorption coefficient, U_ν,P is Planck distribution, which is defined as

$$ {\mathrm{U}}_{\upnu, \mathrm{P}}={\mathrm{h}\upnu \uprho}_{\upnu} $$

(5.165)

The last term in Eq. (5.164) is required due to stimulated emission effect.

The energy conservation relation is given as

$$ \frac{1}{2}{{\mathrm{mv}}^{\hbox{'}}}^2=\frac{1}{2}{\mathrm{mv}}^2+\mathrm{h}\upnu $$

(5.166)

where v and v′ are the electron velocity after and before the photon absorption, respectively. Define the emission cross section σ_ν by

$$ {\mathrm{dq}}_{\upnu}={\mathrm{h}\upnu \mathrm{d}\upsigma}_{\upnu} $$

(5.167)

The spectral absorption coefficient is shown as

$$ {\mathrm{a}}_{\upnu}=\frac{{\mathrm{c}}^2\mathrm{v}{\hbox{'}}^2}{8{\uppi \upnu}^2\mathrm{v}}\frac{{\mathrm{d}\upsigma}_{\upnu}}{\mathrm{d}\upnu} $$

(5.168)

This is the absorption coefficient of the inverse-bremsstrahlung. With use of (5.167) and (5.157), the spectral absorption coefficient is obtained.

$$ {\mathrm{a}}_{\upnu}=\frac{{\mathrm{e}}^6{\mathrm{Z}}^2}{64\sqrt{3}{\uppi}^2{\mathrm{hcm}}^2{\upvarepsilon}_0^3{\upnu}^3\mathrm{v}} $$

(5.169)

This was derived by Kramers in 1923. When this is integrated with respect to the Maxwell distribution, the absorption coefficient is obtained.

$$ {\mathrm{K}}_{\upnu}=4.1\times {10}^{-37}{\mathrm{Z}}^2\frac{{\mathrm{n}}_{\mathrm{e}}{\mathrm{n}}_{\mathrm{i}}}{{{\mathrm{T}}_{\mathrm{e}\mathrm{V}}}^{7/2}{\mathrm{x}}^3}\kern1em \left[{\mathrm{cm}}^{-1}\right],\kern1em \mathrm{x}=\frac{\mathrm{h}\upnu}{\mathrm{T}} $$

(5.170)

Here, the densities are in units of [cm⁻³].

5.11 Rate Equations

In general, the ionization and excitation of each ion at a local point should be solved as a function of time, because such processes have typical times to be some steady sate. As we have seen in this chapter, the reaction rates are calculated when the physical values of electrons and ions are gives with the information of photon fields in the case of local thermodynamic equilibrium being satisfied. It is useful to note that rate equations to be explained here are widely solved in many different kinds of problems. Of course, the time evolution of atomic process in plasma is a good example, while we can enumerate the following physical phenomena, where different rate equations with corresponding reaction cross sections control the phenomena, while they are the same or similar form mathematically.

Different rate equation manages the phenomena in chemical reactions, nuclear reactions, spread of infectious diseases, and so on. It is known that the big bang produces the light elements up to He and Li during the first several minutes as shown in Fig. 5.21. This is called the big bang nuclear-synthesis. Since the Universe starts from a point by phase transition of vacuum state and extremely high-energy Universe starts to expand and cooled rapidly. Then, the density also decreased rapidly, the rate equation to the nuclear fusion processes of all elements provides the time evolution given in Fig. 5.21. The heavier elements than He and Li are produced mainly inside stars using long time. Relating to the hydrodynamic instabilities to be studied in Vol. 3, the supernova explosions are known to be the place where heavy elements such as irons are produced in extremely high-temperature and density conditions.

Fig. 5.21

Time evolution of the big-bang nucleosynthesis

It is useful to obtain the feeling about the difference of chemical, atomic, and nuclear reactions. It is clear that each of three becomes important at the difference temperature. Chemical reaction is via bonding of different molecules whose binding energy is roughly 0.1–1 eV. Considering the effect of the contribution of tail component shown in Fig. 5.19, most of chemical reactions become dominant around the temperature of T = 0.01–0.1 eV, namely room temperature to thousands of degrees. The atomic reactions studied in details above are characterized by the electron binding energy in ion. Of course, it depends on ionization stage and Z-number of an ion, while the binding energy is in the range of 10 eV–1 keV. Therefore, when the plasma temperature is approaching to the binding energy, many atomic reactions take place.

Nuclear reaction is evaluated by the binding energy of nucleus. It is roughly about 1 MeV. Therefore, the temperature approaches to 10–100 eV, the nuclear reactions should be considered. Of course, whether such reactions may be dominant or not is strongly depends on the density. Since the reaction cross section is the orders of the size of molecule for chemical, atom for atomic, and nucleus for nuclear reactions. It is useful to have idea about how large their sizes. Molecule is about 10 Å (10⁻⁷ cm), atom 1 Å (10⁻⁸ cm), and nucleus 1 fm (10⁻¹³ cm). It is clear that the reaction cross section proportional to the square of radius is very small for the nuclear reaction and extremely-high density or extremely long time is required for substantial reaction. Such nuclear reaction is usually possible only in stars with high-density-high-temperature.

Three coupled nonlinear rate equations known as the Lorentz equation is famous to give chaotic variation of three quantities as a function of time with selection of coefficients of the rates in the equation. Rate equation is rather simple compared to, for example, hydrodynamic equations described in the previous chapter. The chaos and stochasticity in the Lorentz equation is interesting subject, while let us study the rate equations of atomic process by assuming that there are no chaotic phenomena.

In order to analyze the process of plasma formation from gas, it is necessary to solve the temporal evolution of excitation and ionization of gas atoms by free electrons and radiation. Then, the de-excitation and recombination are also treated self-consistently. The equation governing the temporal evolution is called rate equation for atomic processes. In general, we have to consider many atomic processes as schematically shown in Fig. 5.22. Consider here only the atomic process due to particles, while the atomic processes induced by external photons are neglected. This assumption is acceptable for most of plasmas in laboratory, namely optically thin plasma. It is also good approximation of plasmas in space whose density is low enough.

Fig. 5.22

Basic atomic processes to be solved in rate equation in laser plasma, where only single electron excitation is included for a simple atomic model with each configuration depending only on the principal quantum number, n. Reprint from Ref. [5] with kind permission from Springer Science + Business Media

Consider an ion. Let the ionization degree be ζ and the quantum state of the bound electrons be m. The rate equation for the number of the ions in that state (${\mathrm{N}}_{\mathrm{m}}^{\upzeta}$) is given as the sum of seven elementary processes in the form including only the electron collision process as follows

$$ \frac{{\mathrm{dN}}_{\mathrm{m}}^{\upzeta}}{\mathrm{dt}}=-{\mathrm{A}}_1+{\mathrm{A}}_2 $$

(5.171)

$$ {\mathrm{A}}_1=\left(\sum \limits_{\mathrm{n}>\mathrm{m}}{\upnu}_{\mathrm{m},\mathrm{n}}^{\mathrm{bb}}+\sum \limits_{\mathrm{k}<\mathrm{m}}{\upnu}_{\mathrm{m},\mathrm{k}}^{\mathrm{bb}}+\sum \limits_{\mathrm{j}}{\upnu}_{\mathrm{m},\mathrm{j}}^{\mathrm{bf}}\right){\mathrm{N}}_{\mathrm{m}}^{\upzeta} $$

(5.172)

$$ {\mathrm{A}}_2=\sum \limits_{\mathrm{n}>\mathrm{m}}{\upnu}_{\mathrm{n},\mathrm{m}}^{\mathrm{bb}}{\mathrm{N}}_{\mathrm{n}}^{\upzeta}+\sum \limits_{\mathrm{k}<\mathrm{m}}{\upnu}_{\mathrm{k},\mathrm{m}}^{\mathrm{bb}}{\mathrm{N}}_{\mathrm{k}}^{\upzeta}+\sum \limits_{\mathrm{j}}{\upnu}_{\mathrm{j},\mathrm{m}}^{\mathrm{fb}}{\mathrm{N}}_{\mathrm{j}}^{\upzeta +1}+\sum \limits_{\mathrm{j}}{\upnu}_{\mathrm{j},\mathrm{m}}^{\mathrm{bf}}{\mathrm{N}}_{\mathrm{j}}^{\upzeta -1} $$

(5.173)

Here, A₁ is a homogeneous term, showing the transitions indicated as 1, 2, and 3 schematically shown in Fig. 5.23. In other words, it is a term by which an ion transits from the state of ${\mathrm{N}}_{\mathrm{m}}^{\upzeta}$ to another state by collision excitation, de-excitation, and impact ionization. In contrast, A₂ in (5.173) shows the elementary processes 4, 5, 6, and 7 of Chap. 5.9 and are shown in Fig. 5.23. This shows a term in which the state of ions changes to ${\mathrm{N}}_{\mathrm{m}}^{\upzeta}$ due to de-excitation, excitation, recombination, and ionization.

Fig. 5.23

Atomic processes to m; going-out (5.3) and coming-in (4, 5, 6, 7) to the quantum sate m of an ion with charge state ζ

Since the term due to radiation is not considered here, RHS of (5.171) is easy to understand by comparing with Fig. 5.23. While it will be explained later about the case with the effect by photons, it is useful to study the property of the solutions of (5.171). In such a case, the ionization state should be in the detail balance at the stationary state in (5.171) and the solution becomes the same as Saha solution (5.14).

How to solve the time evolution of ionization with (5.171). Given all eigen-state data of the bound states, (5.171) looks like a homogeneous coupled equation to the variables ${\mathrm{N}}_{\mathrm{m}}^{\upzeta}$. If so, it is easy to solve (5.171) by obtaining the eigen-values of the determinant for the matrix. However, (5.171) is not a linear equation to ${\mathrm{N}}_{\mathrm{m}}^{\upzeta}$. The rate coefficients are functions of the free electron density n_e, and it is given so that the charge neutrality is satisfied. Therefore, it is necessary to solve (5.171) numerically with the iteration method.

5.11.1 Corona Equilibrium (CE)

The Saha equilibrium is relatively easier to calculate the charge distribution of ions by using some atomic structure model for partially ionized atoms. The LTE assumption is, however, applicable to the limited cases such as the inside of the sun or low temperature plasmas. In general, plasmas in laboratory and observed plasma in the universe are rather of thermodynamically non-equilibrium (non-LTE) in many cases. However, solving the rate equation including all atomic processes requires a special computation technique and super-computing. Historically, non-LTE steady state models have been proposed. Typical examples of corona equilibrium (CE) model and collisional radiative equilibrium (CRE) model being used widely are explained below.

The corona equilibrium (CE) model stems from the corona plasma of the sun, far extended plasma from the surface of the Sun as shown in white in Fig. 5.24 [14]. The CE model can be applied to the plasma characterized by extremely low density, optically thin, and high temperature. It is assumed that all bound states are in the ground states. This is because the lifetimes of the excited states are relatively short compared to the time scale of the collisional excitation. Therefore, the bound states of partially ionized atoms have only one quantum state for each. The photo-effects are all neglected.

Fig. 5.24

Large scale corona over the surface of the Sun [14]. (NASA)

In the CE model, it is enough to solve a steady state by leaving only the three terms in the rate Eq. (5.171). Then, (5.171, 5.172, and 5.173) are approximated as

$$ \frac{{\mathrm{dN}}_0^{\upzeta}}{\mathrm{dt}}=-{\mathrm{C}}_1+{\mathrm{C}}_2 $$

(5.174)

$$ {\mathrm{C}}_1={\upnu}_{\mathrm{C}}^{\mathrm{bf}}{\mathrm{N}}_0^{\upzeta} $$

(5.175)

$$ {\mathrm{C}}_2={\nu}_{\mathrm{R}}^{\mathrm{fb}}{\mathrm{N}}_0^{\upzeta +1}+{\upnu}_{\mathrm{C}}^{\mathrm{bf}}{\mathrm{N}}_0^{\upzeta -1} $$

(5.176)

Here, C₁ is the electron impact ionization, the first term of C₂ is the radiative recombination, and the second term is the electron impact ionization. The impact ionization coefficient is proportional to the electron density, and the radiative recombination coefficient is also proportional to the electron density. Therefore, the equilibrium solution of Eq. (5.174) is a function only of temperature and does not depend on the density. If the plasma is low density and high temperature, the ionization state becomes corona equilibrium (CE).

For example, in the universe, the interstellar plasma can be described by CE plasma, and the magnetically confined plasmas can be also modeled with CR plasma. Even in laser plasma, the corona equilibrium will be seen when the expansion plasma becomes sufficiently low density. Figure 5.25 shows the temperature dependence of the charge distribution of aluminum in the corona equilibrium (CE). The density is 10¹⁸ cm⁻³, four order of less than LTE case in Fig. 5.1. It can be seen less ionization in CE compared to LTE.

Fig. 5.25

Temperature dependence of the charge distribution of aluminum plasma in the corona equilibrium (CE). The density is 10¹⁸ cm⁻³

It is important to know that observing the line emission from CE plasma can be used to identify the plasma temperature as follows. Consider the line of hydrogen 2p-1s transition. Since the ground state of hydrogen atom is N₀ = N_1s in (5.174). Using the electron collision excitation coefficient (5.134), the intensity of the Lyman α line of 2p-1s transition is found to be proportional to the excitation rate to 2p as

$$ \frac{{\mathrm{dN}}_{2\mathrm{p}}}{\mathrm{dt}}={\left({\upnu}_{1\mathrm{s},2\mathrm{p}}^{\mathrm{bb}}{\mathrm{N}}_{1\mathrm{s}}\right)}_{\mathrm{coll}} $$

(5.177)

The amount of Lyman α radiation emitted from the unit area in unit time can be known by identifying the product of the density and depth by another method. Then, (5.177) is the relation to measure the plasma temperature.

Figure 5.26 shows the corona equilibrium of iron (Fe), the most abundant metal element in the universe [15]. The temperature of the photosphere of the sun is 6000 K (about 0.6 eV), but the temperature of the corona plasma outside is as high as 1 keV evaluated with CE model. It corresponds to the temperature at the right end of the horizontal axis in Fig. 5.26. There is no conclusion on why the corona plasma, which is downstream of the plasma flow from the sun, has a higher temperature than the upstream by three orders of magnitude. For example, some theory suggests many micro-magnetic reconnections are taken place in the corona and the energy of the magnetic field is converted to the thermal energy of the plasma.

Fig. 5.26

Charge distribution of the corona equilibrium of iron (Fe), the most abundant metal element in the universe. Reprint from Ref. [15] with kind permission from Cambridge University Press

5.11.2 Collisional Radiative Model (CRM)

Time development of ion charge and atomic structure distribution in plasma has been studied by solving the rate equations without the radiation field. Adding the radiative decay terms in (5.171, 5.172, and 5.173), the time evolution of dynamic plasma can be solved, for example, by coupling with hydrodynamic code. Such atomic process is called collisional radiative model (CRM). This code can be extended to study the radiation effect on the atomic process by coupling with radiation transport with detail photons spectrum. Most of the study can be done by including radiative recombination and photo-ionized plasma has been studied [16].

In the case of long-time evolution of plasma like those in Universe and magnetic confinement, stationary state assumption is valid. Such model of CRM is called collisional radiative equilibrium (CRE). The CRE connects continuously between LTE and CE plasma atomic state. As shown in Fig. 5.27 [16], CRE model tends to Saha equilibrium at the high-density limit and CE at the low density and high temperature limit. Since the CRE model without radiation pumping is a function of temperature and density, it is widely used for laser plasmas and radiation-hydrodynamic computations for optically-thin condition. In the CRE model without radiation, the following assumptions are adopted.

• 1. Since the radiation intensity is weak enough compared to the Planck radiation, the excitation and ionization due to radiative process are neglected.

• 2. Including all except for the above two, the stationary solutions of (5.171) are solved as functions of given temperature and density.

In the plasma where the time scale of the rate coefficients of all atomic processes is sufficiently faster than the time scale of the change of plasma, for example, the time scale of fluid plasma change, quasi-steady state is satisfied. Then, CRE becomes the solution of the rate equation. So, the comparison of the time scale is important. The temporal evolution of the ionization or recombination becomes important as follows.

Fig. 5.27

Comparison of three steady state equilibrium atomic process models. The CRE connects continuously to the LTE and CE in the low density and high-density limits. Reprint from Ref. [5] with kind permission from Springer Science + Business Media

Ionizing Plasma

When the interstellar gas is abruptly heated by a shock wave such as a blast wave of a supernova explosion, it takes time for the heated plasma to reach the equilibrium. Meanwhile, ionization progresses gradually. It is referred to as ionizing plasma. It is considered that ionization progresses due to the collisions of free electrons, so unless the product of density and time (nt) is not less than a certain value, it does not settle to the equilibrium state. In the interstellar space, the density is extremely low, so that the equilibrium time will be tens of thousands of years.

Recombining Plasma

When high temperature plasma is suddenly cooled, such as by expanding into a vacuum like solar wind from the corona plasma and laser produced plasma, recombination progresses slowly far from thermal equilibrium or corona equilibrium. Since the probability of transition to the quantum state of the outer shell having a high energy level is high in collisional recombination, the distribution of the bound electrons of the ion may be larger in the upper level. This is called the negative temperature state, and in such a plasma, as explained in the following section, maser or laser amplification by induced emission becomes possible.

It is important to note that for the purpose to couple with radiation hydrodynamic code, it is better to solve the rate equations with less atomic states. If the average charge state is well predicted with such a simple model, it is better to install such a model in the integrated code to solve time dependent ionization. It is, however, important to compare radiation emission spectra with many experiments, it is required to know the detail atomic data as seen in Fig. 5.7. It is clever way if it is possible to reproduce the detail energy levels and the other radiation transport related data from the simple atomic model. Such research has been done, for example, based on simple screened hydrogen model [17]. It has been developed to be able to study spectroscopic data with laser-plasma experiments.

5.12 Masers and Lasers

5.12.1 Principle of Laser and Maser

The principle of laser and maser using different energy transitions of atoms in solids and gases is based on the rate equation of the atomic and molecular states of matters. For simplicity, consider a system of atomic bound states consisting of four levels as shown in Fig. 5.28. The Level 1 is the ground state and Level 4 is assumed to have many levels like conduction band in solid. Such a system is called a four-level system, and electrons move between levels. Assume that the Level 4 to Level 3 is a non-radiative transition, that is, thermal relaxation occurs due to the influence of, for example the phonons or surrounding electrons. Then the rate equation for such four levels is

$$ \frac{{\mathrm{dN}}_1}{\mathrm{dt}}=-{\upnu}_{14}{\mathrm{N}}_1+\left({\upnu}_{21}+{\mathrm{A}}_{21}+{\mathrm{S}}_{21}\right){\mathrm{N}}_2 $$

(5.178)

$$ \frac{{\mathrm{dN}}_3}{\mathrm{dt}}={\upnu}_{23}{\mathrm{N}}_2-\left({\upnu}_{32}+{\mathrm{A}}_{32}\right){\mathrm{N}}_3+{\mathrm{S}}_{43}{\mathrm{N}}_4 $$

(5.179)

$$ \frac{{\mathrm{dN}}_4}{\mathrm{dt}}={\upnu}_{14}{\mathrm{N}}_1-\left({\upnu}_{41}+{\mathrm{A}}_{41}+{\mathrm{S}}_{43}\right){\mathrm{N}}_4 $$

(5.180)

$$ {\mathrm{N}}_2=\mathrm{N}-\left({\mathrm{N}}_1+{\mathrm{N}}_3+{\mathrm{N}}_4\right) $$

(5.181)

Here, ν_mn is the transition probability from state m to state n. In the solid laser, a strong light source from the outside excites from the Levels 1 to 4 in Fig. 5.28. In the above equations, the coefficient of photo-excitation by external light is set in the rate coefficient ν₁₄, A_mn is the spontaneous decay coefficient, and S_mn represents the relaxation rate by interaction with the degree of freedom in the solid such as lattice vibration. (5.181) is the conservation equation of the number of electrons.

Fig. 5.28

Schematics of excitation by external photon for laser emission in a typical four-level atom

Steady state (d/dt = 0) should be realized if the strong light source for laser excitation is continuously irradiated to this atomic system. When the third and fourth terms on RHS of (5.179) are large, and if the first and fourth terms are balanced in (5.180), the following equation is obtained.

$$ \frac{{\mathrm{dN}}_3}{\mathrm{dt}}={\upnu}_{14}{\mathrm{N}}_1-{\mathrm{A}}_{32}{\mathrm{N}}_3 $$

(5.182)

In the steady state,

$$ \frac{{\mathrm{N}}_3}{{\mathrm{N}}_1}=\frac{\upnu_{14}}{{\mathrm{A}}_{32}} $$

(5.183)

If there is no pumping light source and the material is in thermal equilibrium, the following relation should be satisfied.

$$ \frac{{\mathrm{N}}_3}{{\mathrm{N}}_1}=\frac{{\mathrm{g}}_3}{{\mathrm{g}}_1}\exp \left(-\frac{{\Delta \mathrm{E}}_{31}}{\mathrm{T}}\right) $$

(5.184)

However, if the photo-excitation by the external source is strong and the spontaneous emission coefficient is relatively small, the following relation may be satisfied.

$$ \frac{{\mathrm{N}}_3}{{\mathrm{N}}_1}\frac{{\mathrm{g}}_1}{{\mathrm{g}}_3}>1 $$

(5.185)

In other words, a “negative temperature” state of temperature T < 0 is realized by the definition of (5.184). This is called “population inversion“. This also indicates that the population inversion is also realized between the Levels 3 and 2 in Fig. 5.28. When passing light with the energy of hν = ΔE₃₂, the light is amplified due to the stimulated emission. In addition, the light is amplified to keep coherency (same phase) in the amplification process. In gas media, electron beam may be also used to generate a population inversion via the electron collisional excitation. For example, this method is adopted for a carbon dioxide (CO₂) gas laser with a wavelength of 10.6 μm which is widely used for industrial purpose. The energy levels of the vibration of the carbon dioxide molecule are used for laser emission.

5.12.2 Masers and Lasers in Universe

C. Towns, an awardee of the Nobel Prize in Physics in 1964 for the invention of the principle of laser and maser, has moved to the field of radio astronomy in 1967, and in 1968 his group discovered the maser emission of water molecules coming from Orion constellation [18]. Strong infrared emission from stars creates the population inversion in the rotational levels of the water molecules around young-age stars and high intensity radio waves are generated via induced emission. The maser is coherent emission of microwave due to excited molecule. H₂O (water) maser and SiO masers are typical examples. The former mainly stems from the star-birth region, while the latter is from star-death region.

In 1992 water molecule maser from the Active Galaxy M 106 (NGC 4258) was observed [19]. This observation concluded from the spatial distribution of maser source that there is a black hole with a mass of 36 million times the solar mass at the center. The maser source molecules are excited by the light emitted by plasma falling into the black hole, and the population inversion is maintained. The maser’s energy is likely to reach 210 times the total energy emitting from the sun’s surface. The spatial variation of the Doppler shift of the maser light was identified to be Kepler motion with which the mass of super-massive Black Hole was inferred.

It is also reported that strong ultra-violet laser is observed by Hubble Space Telescope (HST), and it can be interpreted theoretically due to the population inversion of a four-level atom of Fe II [20]. The observed line was 250 nm, in the UV spectrum of gas closed to the η-Carina, the most active and luminous star in the Galaxy. It is inferred that accidental wavelength coincidence between a strong line of the most abundant elements (H, He) and Fe II absorption line makes the photo-excitation possible. The image of laser emission from the gas pumped by the η-Carina is shown in Fig. 5.29.

Fig. 5.29

The image of laser emission by amplified spontaneous emission from the gas pumped by extreme photons in the η-Carina. (NASA)

Note that the induced emission in Universe is not the same as lasers and masers in laboratory. The induced emission in Universe is so-called amplified spontaneous emission (ASE) of radiation. In lasers in laboratory is designed so that a seed light pass through the media with population inversion and the emission is controlled.

It is impressive to copy the words by Towns in [18]. “Both masers and lasers have been in the universe for billions of years. I didn’t have to invent them. As is clear from this example, there must be many more and more secrets hidden in the universe.”

5.13 Photo-ionized Plasma

Photo-ionized plasmas in Universe emits non-thermal spectra being observed by telescopes for wide range of photon-energy. Especially line emissions from relatively cold objects in space are mainly due to photo-ionization by hot compact objects in the vicinity of line-emitting large objects. Therefore, study of photo-ionized plasma and spectrum from such a plasma is important to study the compact object such as black-holes, neutron-stars, white dwarfs, etc. In this section, the observation of photo-ionized plasmas in Universe is briefly introduced and x-ray photo-ionized plasma experiments in laboratory as model experiment for space is also reviewed.

5.13.1 Planetary Nebula

In the space, the planetary nebulae (PN) are observed, many of which newly found by Hubble Space Telescope (HST). The image of NGC7009 is shown in Fig. 5.30. The colorful images of the planetary nebulae are not powered by the ultraviolet lights from the central white dwarf. It excites and ionized the surrounding gas to keep the emission. It emits intrinsic emission lines in the process of recombination and de-excitation to the ground state. They are emission nebula consisting of an expanding, glowing shell of ionized gas ejected from red giant stars late in their lives. The term “planetary nebula” is a misnomer because they are unrelated to planets or exoplanets. The term originates from the planet-like round shape of these nebulae observed by astronomers through early telescopes. Though the modern interpretation is different, the old term is still used.

Fig. 5.30

Emission image and spectra of a planetary nebula NGC7009, a photo-excited plasma in Universe [21]. (NASA)

All planetary nebulae form at the end of the life of a star of intermediate mass, about 1–8 solar masses. It is expected that the Sun will form a planetary nebula at the end of its life cycle [20]. They are a relatively short-lived phenomenon, lasting perhaps a few tens of thousands of years, compared to considerably longer phases of stellar evolution. Once all of the red giant’s atmosphere has been dissipated, energetic ultraviolet radiation from the exposed hot luminous core, called a planetary nebula nucleus ionizes the ejected material. Absorbed ultraviolet light then energizes the shell of nebulous gas around the central star, causing it to appear as a brightly colorful planetary nebula.

It is noted that many forbidden line emissions are observed, because the density is very low and almost no collisional de-excitations are taken place. As the result, the long life-time metastable states decay via higher order effect in (5.67). The lines indicated with the brackets like [Ne III] in Fig. 5.30 are the forbidden lines within the dipole transitions. X-ray lines from planetary nebulae are observed [21].

5.13.2 XFEL and Inner-Shell Ionization

X-ray free-electron lasers (XFELs) have been used to develop new science with coherent hard x-rays. After accelerating electron beams relativistic, coherent x-rays are generated and amplified in passing through a magnetic undulator device. For example, European-XFEL [22] has output of 3–25 keV with focused intensity of 10^17–18 W/cm² for 2–100 fs pulse. It is 3 mJ at 5 keV with 10¹⁰ photons/pulse. Such x-ray is strong enough to study the inner-shell ionization process of high-Z atoms. When the ionization potential is lower than the XFEL energy, the inner-shell ionization is dominant in the atomic process.

As we see in Fig. 5.12, 1s electron in K-shell is dominantly ionized, while 2s and 2p electrons are dominantly ionized if the ionization potential of K-shell is larger than the x-ray energy. When a vacancy is produced in such an inner shell, it is known that the subsequent process is Auger decay or fluorescence process. The atomic process induced by such inner-shell ionization is schematically shown in Fig. 5.31 [23]. In order to analyze experimental results, all rate coefficients are calculated by developing XATOM code, the details of the code are given in [23]. In obtaining the orbital wave functions, Hartree-Fock-Slater method is used in the code.

Fig. 5.31

Diagrams of X-ray-induced physical processes treated by XATOM. P photoionization, A Auger (Coster–Kronig) decay, F fluorescence, SO shake-off, S Rayleigh and Compton X-ray scattering, RS resonant elastic X-ray scattering. Reprint from Ref. [23] with kind permission from Springer Science + Business Media

In particular, XATOM is designed for describing complex interactions between atoms and intense XFEL pulses. During the XFEL–atom interaction, if single-photon absorption is saturated, multiphoton absorption occurs via a sequence of single-photon ionization and accompanying relaxation processes like fluorescence and Auger decay. The X-ray multiphoton absorption usually yields highly charged states, involving a variety of different multiple-hole states. XATOM calculates atomic data – orbitals and orbital energies, and cross sections and rates for X-ray-induced processes – for all individual electronic configurations, including multiple-hole states, of arbitrary atomic species. More detail description of the theory of atomic photo-effect is given in [24].

Since the x-ray intensity is high, deep inner-shell multiphoton ionization has been observed experimentally [25]. In the experiment XFEL-SACLA was used to irradiate neutral xenon (Xe) gas to measure the multiply ionized Xe ions. At highest charged ions of Xe²⁶⁺ have been observed. The ionization process was theoretically studied to obtain a model shown in Fig. 5.32. It is one typical pathway yielding Xe²⁴⁺ [25]. The plot illustrates that the total energy of the system varies in the course of the ionization steps. After L-shell photoionization (blue arrows), the energetically excited core-hole state relaxes via a series of Auger and Coster-Kronig decays (green arrows), and/or fluorescence (yellow arrows). Note that another photoionization occurs before the atom fully relaxes to the ground configuration. We find it useful to view the multiphoton multiple ionization dynamics occurring in a single atom in terms of quantum evaporation of electrons: x rays heat up the atomic system to highly excited states, and then the system relaxes primarily by emitting electrons with characteristic energies. The excess energy is shared among the electrons via electron-electron collisions, resulting in the ejection of 24 electrons in total. For each photoionization. Note that this multi-photon process is not the nonlinear process discussed in Vol 1 regarding multi-photon ionization by intense lasers, but liner process sequence in one pulse.

Fig. 5.32

An exemplary pathway of multiphoton multiple ionization of Xe at 5.5 keV. The black solid line with dots indicates the ground-configuration energy for given charge states, and the energy of neutral Xe is set to zero. Reprint with permission from Ref. [25]. Copyright 1998 by American Physical Society

The population inversion and subsequent x-ray laser phenomenon have been demonstrating with the inner-shell photo-ionization by XFEL-LCLS at SLAC [26]. There have been a lot of studies on x-ray lasers in laboratory with use of plasma as briefly shown below. XFEL made it possible to pump new atomic X-ray lasers with ultrashort pulse duration, extreme spectral brightness and full temporal coherence. X-ray laser in keV energy regime based on atomic population inversion and driven by rapid K-shell photo-ionization are demonstrated using pulses from an XFEL through the physical process shown in Fig. 5.33 [26]. A population inversion of the Kα transition is experimentally demonstrated in singly ionized neon at x-ray energy of 849 eV. Strong amplified spontaneous emission is observed from the end of the excited plasma. This resulted in femtosecond-duration, high-intensity X-ray pulses of much shorter wavelength and greater brilliance than achieved with previous atomic X-ray lasers.

Fig. 5.33

Level scheme. Population inversion of the 1s 1 2s 2 2p6 -to-1s 2 2s 2 2p5 transition is created by K-shell photo-ionization of neutral neon. The Auger decay time of the inverted state (2.4 fs) dominates the kinetics of the system in the small-signal-gain regime. The lower lasing state is depleted by K-shell photo-ionization. Reprint from Ref. [26] with kind permission from Springer Nature Publ.

It is useful to briefly describe about x-ray laser research before the XFEL era. X-ray spectroscopy is one of the most powerful diagnostics in laser plasma from the beginning of research. H. Griem gave a review on diagnostics and modeling of dense plasma, emphasizing density and temperature measurements [27]. Most of the x-ray laser pumping has been studied for the case of collisional excitation scheme in laser produced plasma, while the energy conversion efficiency was very low in UV range, say ~60 eV.

Since the XFEL x-ray source is mono-energetics and photo-ionization plays essential role to generate free electrons from target atoms, it is not appropriate to assume the free electrons have Maxwell distribution during the short time of x-ray pulse. Time-dependent calculations of electron energy distribution functions (EEDF) in the presence of intense XFEL radiation have been studied computationally by solving Boltzmann equation of free electrons. The code is coupled with atomic codes for photo-ionization and related atomic processes. The condition of simulation is that argon gas of atomic number density 1.6 × 10¹⁹ cm⁻³ is irradiated by a pulse duration of 40 fs, intensity 2 × 10¹⁷ W/cm² XFEL with x-ray photon energy of 1.07 keV. In Fig. 5.34, the time evolution of the free electron distribution functions is shown from 1 fs to 40 fs [28].

Fig. 5.34

Time evolution of free electron distribution in argon gas of atomic number density 1.6 × 10¹⁹ cm⁻³ irradiated by a pulse duration of 40 fs, intensity 2 × 10¹⁷ W/cm² XFEL with x-ray photon energy of 1.07 keV. Time evolution is shown from 1 fs to 40 fs. The distribution function is far from Maxwellian. Reprint from Ref. [28] with permission from Institute of Physics Publ

At 1 ps, the peak of EEDF is located at the excess energy of K-shell ionization of neutral atom around 200 eV. Then, the peak around 700–900 eV is due to electrons ejected by Auger process. As time increases and the ionization potential increases in ionized argon, the photo-ionized electrons appear in the energy spectrum lower than 200 eV. In Fig. 5.35 long time evolution after x-ray irradiation is shown [28]. It is clear that it takes about 2 ps so that the free electron is thermalized to be Maxwellian. Simulation was also carried out for the case of irradiation of black-body radiation of the radiation temperature T_r = 100 ~ 300 eV [29]. FEDF is relatively smooth compared to the XFEL irradiation case.

Fig. 5.35

Long time evolution of electron distribution function after Fig. 5.34 by XFEL irradiation. It is clear that it takes about 2 ps so that the free electron is thermalized to be Maxwellian. Reprint from Ref. [28] with permission from Institute of Physics Publ

5.13.3 Photo-ionization in X-Ray Binary

It is very challenging to study photo-ionized plasma, because it is hard to produce black-body radiation with radiation temperature more than 100 eV in laboratory to ionize very abundant atom such as Si, Fe, etc. with atomic numbers more than 10. However, such bright radiation sources in x-ray regime have been observed in Universe with x-ray satellites such as ASCA, Suzaku (Japan), Chandra (US), Newton (EU). In studying compact objects such as black-holes, neutron stars, and white dwarfs, binary system made of such a compact star and a normal star emits strong x-rays with witch the property of compact objects can be studied with use of x-ray spectra etc. I order to analyze the observed x-ray spectrum, modeling of photo-ionized plasma is essential. Such simulation codes should be verified and validated via comparison with appropriate experiment in laboratory. Here, the topics of photo-ionized plasma in Universe and related laboratory experiments are reviewed. Always, the computational modeling is discussed by comparing to both of the observational and experimental data.

The photo excitation and ionization processes become important in a variety of astrophysical phenomena. A familiar photo-ionized plasma is clearly observed by telescope as the edge of neutral hydrogen in clouds with H_a line emissions. For example, Eagle Nebula boundary observed is the surface where the photo-ionization by UV radiation generated by massive stars nearby. The UV light ionizes molecular cloud to ionize neutral hydrogen atoms to be proton, HII. This region is called HII region [30]. HII region is related to the birth of many stars in the molecular clouds.

On the other hand, photo-ionization of so-called metal in universe, where metal means higher Z atoms than most abundant H and He. Since high-Z ions emits x-ray line emissions, they are good targets to be observed to study the physics of very energetic radiation source. Cygnus X-1 and Cygnus X3 are well known x-ray binary [31]. They are located around 7.4 kpc (20 k light years), however emitting very luminous x-rays. The schematic of such a binary system is shown in Fig. 5.36. The companion star is a massive normal star with very high-mass stellar wind to donate the matter to the compact object. Since the matter has angular momentum to form the accretion disc around the compact object. Since the compact object like black-hole absorb the mass via strong gravitation, the matter of accretion disc is continuously falls into the black hole. During the process losing the angular momentum, the excess energy heats the matter and the matter near the compact object becomes extremely high temperature plasma emitting almost Planckian radiation. The radiation temperature becomes almost 1 keV and the radiation photo-ionized the accreting plasma and the surface of the companion star whose radius is much larger than the compact one. In case of Cygnus X3, the binary system is rotating around the center of mass with 4.8 h period.

Fig. 5.36

A schematic of x-ray binary system consisting of a compact object and a main sequence star. (Image by NASA)

How the plasma is strongly affected by photo-ionization in atomic process is measured with the photo-ionization parameter ξ defined by

$$ \upxi =\frac{\mathrm{L}}{{\mathrm{n}}_{\mathrm{e}}{\mathrm{R}}^2}\kern1em \left[\mathrm{erg}/\mathrm{cm}/\mathrm{s}\right] $$

(5.186)

where L is the total luminosity of the compact object and R is the radius of the most x-ray emitting plasma region of the accretion disk by photo-ionization. The n_e is the electron density to measure the photo-recombination. This (5.186) is a rough estimation of the ratio of the photo-ionization rate to the photo-recombination rate, namely.

$$ \mathrm{R}=\frac{\left\langle {\upsigma}_{\mathrm{p}\mathrm{i}}{\mathrm{n}}_{\mathrm{p}}\mathrm{c}\right\rangle }{\left\langle {\upsigma}_{\mathrm{p}\mathrm{r}}{\mathrm{n}}_{\mathrm{e}}\mathrm{v}\right\rangle}\propto \frac{{\mathrm{I}}_{\mathrm{p}}}{{\mathrm{n}}_{\mathrm{e}}} $$

(5.187)

In (5.187), the photo-ionization and photo-recombination cross sections are of course dependent on each transition, while neglecting this difference. In addition, the photo-ionization also depends on the photon spectrum n_p(ν). After neglecting such dependencies, roughly speaking the radiation intensity divided by the electron density would be a good measure of the effect of photo-ionization in plasma. Note that our photo-ionization plasma can be realized when the condition ξ >> 1 is satisfied in the unit of (5.186).

In a detail analysis of the spectra from Cygnus X-3 has been done [32]. In order to analyze the data, the ionization parameter should be identified. The XSTAR code for photoionization equilibrium model with a 1.72 keV Planck radiation temperature illumination spectrum is used. In the XSTAR code, atomic process is coupled with energy balance relation to determine electron temperature consistently. Note that XSTAR code is well developed but is zero-dimensional model, no spatial structure, and radiation transport is not included in the code. This becomes important in laboratory experiments as mentioned below. In Fig. 5.37, the charge state distribution as a function of the ionization parameter ξ is plotted. It is seen that around ξ = 10² the abundance of He-like silicon has the peak. The bright emission of He-like silicon Kα line suggests the ionization parameter ξ ~ 10² for Cygnus 3 binary system. It is also concluded that the density must be higher than 10¹² cm⁻³ in the region responsible for most of the emission.

Fig. 5.37

Charge distribution of silicon atom as a function of the ionization parameter ξ. Reprinted with permission from Ref. [32]. Copyright by American Astronomical Society

The electron temperature obtained from the energy balance relation is also calculated. It is about 20 eV at ξ ~10². This is lowest estimation of the temperature of photo-ionized plasma, because the plasma is assumed to be optically thin and the emitted radiation due to photo-recombination and photo de-excitation are assumed all escape from the system. This may be due to the fact that the specification of the spatial profile of the binary plasma system is very tuff.

In Fig. 5.38, the observed line emissions from H-like and He-like silicon ions (Si XIV and Si XIII) are shown. The brightest line near 2 keV is from Kα line of H-like silicon. The line-like peaks from He-like silicon are also observed near 1.85 keV. In photo-ionized plasma, it is usual to observe three lines around 1.83–1.88 keV [33]. They are evaluated to be by the resonance (r), inter-combination (i), and forbidden (f) lines as indicated in Fig. 5.38. In Fig. 5.39, such three different line transitions are shown for helium-like ions, where w, (x,y), and z correspond to the resonance, inter-combination, and forbidden transitions [34]. The forbidden lines are possibly observed from plasma in low density as we saw in the planetary nebulae in Fig. 5.30.

Fig. 5.38

The observed line emissions from H-like and He-like silicon ions (Si XIV and Si XIII) . The brightest line near 2 keV is from Kα line of H-like silicon. The line-like peaks from He-like silicon are also observed near 1.85 keV. The resonance (r), inter-combination (i), and forbidden (f) lines are speculated. Reprinted with permission from Ref. [32]. Copyright by American Astronomical Society

Fig. 5.39

The three different line transitions in Fig. 5.38 are shown for helium-like ions, where w, (x,y), and z correspond to the resonance (r), inter-combination (i), and forbidden (f) transitions [34]. Credit: D. Porquet, et al., A&A,376, 1113, 2001, reproduced with permission © ESO

In Fig. 5.38, the model calculation with only the photo-ionized plasma at rest gives the blue spectrum, being very different intensity profile near all lines. The red spectrum, on the other hand, is given by taking account additionally the wind plasma component from the surface of the companion normal star [32]. It is complicated to explain the physical reason and recommended to refer the paper by Kallman et al. Even in such a case, we have to consider many atomic processes as schematically shown in Fig. 5.22.

Active Galactic Nuclei (AGN) [31] extremely far from our Galaxy are strong X-ray source in the deep sky, and about 70% of x-ray observed from far from our Galaxy are inferred from many AGNs. The physical process of x-ray generation is the same as x-ray binary in our Galaxy, namely from accretion disk. However, the accreting matter is absorbed by a huge black-hole with mass of 10⁶–10¹⁰ solar mass. Some of them also observed as maser source as described previously. The x-ray spectra from AGNs have been observed and studied for example in [35]. Theoretical models are fitted to the observed optical/UV spectra and the maximum accretion disk temperature was always about 2 eV. However, the observed x-ray spectrum has a power law in high-energy region because of the non-thermal process in complicated geometry near the center of the massive black-hole and accretion disk. So, effective ionization parameter to silicon is relatively high.

A possibility of x-ray laser emission from such Planckian radiation pumping was studied theoretically by us [36]. We have asked ourselves whether x-ray lasers can exist in Astrophysical objects. As a model, we used the 1s2p ¹P₁ populations in He-like ions, which are generated by K-shell photo-ionization of Li-like ions as typically observed in X-ray binaries as mentioned above. Although we have fixed to Planckian radiation source, it is concluded that it is difficult to produce population inversion for this transition. If the low energy part of Planckian radiation is absorbed in the atmosphere of a companion star, it may be possible to produce such population inversion inside the star near the surface. We are required to study more about the condition of x-ray binary or other candidates where such a condition is realized.

5.13.4 Photo-ionized Plasmas in Laboratory

The photo-ionized plasmas are model experiment of x-ray binary system, while there are several differences due to limit of laboratory plasma. One is time dependence of atomic state, second is expanding flow of plasma, the third is optical depth especially of line radiation transport. Of course, the radiation source such as more than 1 keV radiation temperature is not possible in laboratory so far. However, photo-ionized plasma itself is an interesting problem from view point of atomic process, opacity, line profile modeling, radiation transfer, and so on.

The photo-ionized plasma has been studied for looking for better coupling with the absorbing plasma in hohlraum targets, where laser energy is converted to radiation energy to be absorbed by fusion pellet in the hohlraum. In the National Ignition Facility (NIF) ignition campaign, almost Planckian radiation with radiation temperature T_r = 250–300 eV has been used for implosions under the absorption of laser energy of 0.8 ~ 1.1 M Joule [37]. The radiation continues for about 5 ns and it is expected that the photo-ionized plasma is almost in steady-state. Z-pinch machine is demonstrated to be an intense x-ray sources and proposed to be also applicable to study the physics of photo-ionized plasma in Universe [38]

Preliminary design and experiments of photo-ionized plasma were reported for the cases with Z-pinch [39] and laser induced gold cavity [40] radiations. The radiation temperature is in the rage of 80 ~ 200 eV. The absorption and emission spectroscopy of such photo-ionized plasma have been studied. In addition, scalability to very low-density plasma in Universe was also discussed. Application of such photo-ionized experiments to astrophysics has been discussed internationally [41] and The Z Astrophysical Plasma Properties (ZAPP) collaboration has been initiated for applying the Z-pinch for a variety of astrophysical model experiments [42].

Precise experimental result is first reported for Z-pinch photo-ionization plasma, where the Z-pinch radiation spectrum was observed and the radiation temperature was measured to be 165 eV [43]. Charge distribution of photo-ionized iron plasma observed in the experiment is compared with three different atomic process codes including photo-ionization. Two of them are standard codes to analyze the photo-ionization plasma emission spectrum compared to X-ray satellite data, while the third one is FLYCHK mentioned in [17]. It is noted that the absorption spectra have been compared to the detail line profiles predicted with HULLAC code. And the resultant charge distribution was compared to three codes. It is found that the averaged charge is <Z > =16 and three codes well fit to the experimental data [43]. In Fig. 5.40, the charge distributions are shown for the case assuming a constant temperature of iron plasma [17, 43]. The left is without external radiation source, while the right is with external radiation heating, where the temperature is the plasma temperature. It is clear that charge distribution is insensitive to the iron plasma temperature for more than 150 eV in this experimental condition.

Fig. 5.40

Charge distribution calculated for photo-ionization experiment with Z-machine. The left is without radiation and the right is with external radiation heating, where the temperature is the plasma temperature. It is clear that charge distribution is insensitive to the iron plasma temperature for more than 150 eV in this experimental condition. Reprint with permission from Ref. [17]. Copyright 1998 by American Physical Society. Reprint from Ref. [43] with kind permission from Springer Science + Business Media

It is speculated that compared to AGN x-ray flux, the x-ray flux for contributing silicon photo-ionization is almost the same as that obtained by Z-pinch x-ray with T_r = 165 eV [44]. The absorption spectra and emission spectra have been analyzed with two different codes. For the condition measured in Z-pinch experiment, both codes result over ionization of Si⁺⁹ and Si⁺⁸ the absorption spectrum show almost the L-shell electrons are already photo-ionized and only two electrons remain. In the computational model, the inner-shell ionization from M-shell and subsequent Auger process may be overestimated. It is suggested that so-called resonant Auger destruction (RAD) are over-estimated in the codes [44].

In order to relate such laboratory experiment to the binary system observation, higher radiation has been generated by use of radiation from an imploded spherical target. By use of Gekko-XII laser system, twelve beams irradiate a target to generate almost 0.5 keV radiation temperature. Total energy of 4 kJ with pulse duration of 1.2 ns at green light (0.53 μm) are impinged on a plastic target with the diameter 505 μm and thickness 6.4 μm [33]. Although the pulse duration of the radiation is 160 ps, it is enough time to generate He-like silicon with the density n_e = 10²⁰ cm⁻³ located near the imploded core. Observation data from Cygnus X3 and Vela-X1are compared to the experimental data for the line emissions from He-like silicon [33]. The line inferred as the forbidden transitions only expected in low density astrophysical plasma [32] is also observed. However, time dependent simulation for the experiment cannot reproduce this hump in spectrum as shown in Fig. 5.41 [45]. On the other hand, this hump is generated in computation in [46, 47], although strength of three humps are not reproduced. It is still an open question.

Fig. 5.41

Comparison of experimental spectrum and time-dependent numerical spectrum for photo-ionization experiment. The time dependent simulation for the experiment cannot reproduce this hump given in the experimental spectrum. Reprinted with permission from Ref. [45]. Copyright by American Astronomical Society

The inter-combination line (1s^{2 1}S-1s2p ³P and 1s^{2 1}S-1s2p ³P) is strengthened by satellite lines from the Li-like species which has similar energies. Namely, 1s²3p-1s2p3p transition has energy around 1.855 keV. So-called satellite lines of Li-like ions.

Mancini et al. [48] has carried out sophisticated computer simulation with Boltzmann code for free electrons used in [29]. The Boltzmann code is coupled with radiation hydrodynamic code. It is shown that photo-excitation is very important to control radiation cooling rate and the evaluated electron temperature is found to be lower than predicted by CLOUDY and XSTAR codes. In addition, high-energy tail component generated by photo-ionization affects the population of excited state enhancing the radiation cooling rate.

Appendix-B: Thermal Equilibrium Statistical Mechanics

Plasma consists of a large number of particles. Even with a super-computer, it is not possible to follow the motions of all particles in systems in laboratory or in space. We need to use statistical mechanics to describe any macroscopic dynamics of such plasmas. Here, the minimum knowledge of the statistical mechanics is reviewed on the topics that will be required to study plasma physics in the present book.

5.1.1 B-1 Boltzmann Distribution

Assume that many particles exist in an isolated system. Consider that the number of the particles N is sufficiently large, while exchanging energy and momentum inside the system. In the system, each particle energy is quantized and each energy state is symbolically indicated as a k-state. Assume that n₁ particles in state 1, n₂ in state 2, n_i in state i and all the particles are somewhere in a certain energy state. Then, the probability for the system takes the combination of P (n₁, n₂,,,, n_i,,,,) in each energy state is

$$ P\left({n}_1,{n}_2,\cdots, {n}_i,\cdots \right)=p\left({n}_1\right)p\left({n}_2\right)\cdots p\left({n}_i\right)\cdots $$

(B1.1)

Here, p (n_i) is the number of combinations where n_i particles exist in i state. That is, p (n_i) obviously

$$ p\left({n}_i\right)={C}_n^{n_i}=\frac{n!}{n_i!\left(n-{n}_i\right)!} $$

(B1.2)

Calculating p (n_i) from 1 to subsequent number, and using the fact that the second term of the denominator cancels with the numerator of the next number, the probability is obtained as follows.

$$ P\left({n}_1,{n}_2,\cdots, {n}_i,\cdots \right)=A\frac{n!}{\prod {n}_i!} $$

(B1.3)

Here, Π means to take the product for all i. In (B1.3), A is a normalization constant.

Let’s find the distribution when the combination of the arrangement of the number of particles in each energy state is the maximum. For that purpose, we use the following Stirling formula.

$$ \mathit{\ln}\left(n!\right)\approx {\int}_0^n\mathit{\ln}(x) dx $$

(B1.4)

Then, we write ${\bar{n}}_i$ the number of i states where P in (B1.1) is the maximum value. Carrying out the Taylor expansion around the number of particles in each state with the maximum probability, and using (B1.2) and (B1.3), we obtain

$$ \mathit{\ln}P\left({n}_1,{n}_2,\cdots, {n}_i,\cdots \right)=\mathit{\ln}P\left({\bar{n}}_1,{\bar{n}}_2,\cdots, {\bar{n}}_i,\cdots \right)-{\sum}_i\mathit{\ln}\left({\bar{n}}_i\right)\delta {n}_i $$

(B1.5)

Here, δn_i with the energy E_i is the deviation from the maximum probability and they have to satisfy the following constraint conditions.

$$ {\sum}_i\delta {n}_i=0,\kern1.5em {\sum}_i{E}_i\delta {n}_i=0 $$

(B1.6)

(B1.6) are the condition of particle number and energy conservations, respectively.

The first term on RHS of (B1.5) is a constant value.

Also, around the maximum probability the first derivative needs to disappear. With combination of the restrictive condition (B1.6), the following relation should be satisfied.

$$ {\sum}_i\left(\mathit{\ln}\left({\bar{n}}_i\right)-\mathit{\ln}C+\frac{E_i}{T}\right)\delta {n}_i=0 $$

(B1.7)

where C and T are constants to be determined for convenience later. In the vicinity of the maximum probability, (B1.7) requires that the inside of the parentheses be balanced, so the following relational expression is obtained

$$ {\bar{n}}_i= Cexp\left(-\frac{E_i}{T}\right) $$

(B1.8)

Here, C is determined so that the number of all particles is N. T is the temperature of the system, multiplied by the Boltzmann constant and is a unit of energy. (B1.8) shows just the Boltzmann distribution. Any particles that weakly interact in a system should be Boltzmann distribution in the thermal equilibrium.

It was assumed above that each energy state has one particle at most, but it is easy to extend the case where the energy state is degenerate or even continuous. If the eigen-state “i” is degenerate with g_i, the distribution function is modified as

$$ {\bar{n}}_i=C{g}_i\mathit{\exp}\left(-\frac{E_i}{T}\right) $$

(B1.9)

5.1.1.1 Maxwell Distribution

Maxwell derived the velocity distribution of gas system by assuming that there must be a steady state velocity distribution for many particles in gas (1860). Using the Boltzmann distribution given in (B1.9), it is easy to derive Maxwell distribution for freely moving particles in gas and plasma. Note that the number of states changes depending on how to define the freedom, say one, two, or three-dimension in space. Maxwell distribution is that for freely moving particles system of Boltzmann distribution, so it is also called Maxwell-Boltzmann distribution.

For the sake of simplicity, assume that plasma particles only move in the x direction in one dimension. Let’s assume that the plasma is trapped in the potential of length L which correspond the size of box containing the particles. Then, the energy state of the free particle is discretized, and the wave function of electron or ion must satisfy the following condition.

$$ \psi =\mathit{\sin}\left(\frac{p_xL}{\hslash}\right),\frac{p_xL}{\hslash }=\pi k\ \left(\mathrm{k}:\mathrm{integer}\right) $$

(B1.10)

It is clear that the number of states of one-dimensional free particles is equally spaced in the velocity (momentum) space. Therefore, the number of states is constant. From this fact, the velocity distribution function of the plasma particles with free thermal motion is given by (B1.8).

$$ f\left({v}_x\right)= Cexp\left(-\frac{m{v_x}^2}{2T}\right) $$

(B1.11)

Using the relation of integration;

$$ {\int}_{-\infty}^{\infty }{e}^{-\alpha {x}^2} dx={\left(\frac{\pi }{\alpha}\right)}^{1/2} $$

(B1.12)

The velocity distribution function of free particles in one-dimensional space is

$$ f\left({v}_x\right)={\left(\frac{m}{2\pi T}\right)}^{1/2}\mathit{\exp}\left(-\frac{m{v_x}^2}{2T}\right) $$

(B1.13)

This is a Gaussian distribution, and its dispersion shows thermal spread.

In three dimensions, the upper one-dimensional distribution is used also in the y and z directions, and it is obtained by taking three products. However, note that in that case, it is the probability of finding particles in the volume of the velocity space dv_xdv_ydv_z. Since it can be considered that there is no anisotropy of the distribution function in the thermal equilibrium state, the distribution function is spherically symmetric in the velocity space. It should be only a function of the absolute value of the velocity

$$ {v}^2={v_x}^2+{v_y}^2+{v_x}^2 $$

(B1.14)

Then, we can write a three-dimensional velocity distribution as a function of v. Note that the number of states increases with v. That is, using the spherical coordinates of the velocity space

$$ g(v)=4\pi {v}^2 dv $$

(B1.15)

Using (B1.9), the velocity distribution function is obtained as

$$ f(v)=4\pi {\left(\frac{m}{2\pi T}\right)}^{3/2}{v}^2\mathit{\exp}\left(-\frac{m{v}^2}{2T}\right) $$

(B1.16)

Here, the normalization constant is obtained with the following relation

$$ {\int}_0^{\infty }{x}^2{e}^{-\alpha {x}^2} dx=\frac{\pi^{1/2}}{4{\alpha}^{3/2}} $$

(B1.17)

Since the relation 1/2mv² = ε is satisfied, the distribution function for the kinetic energy ε is also derived from (B1.16). The change of variables must be done in the integral system of the distribution. That is

$$ mvdv= d\varepsilon \to dv=\frac{1}{\sqrt{2m}}\frac{d\varepsilon}{\sqrt{\varepsilon }} $$

(B1.18)

By rewriting (B1.16) with use of (B1.18), the energy distribution function is obtained as a function of kinetic energy.

$$ f\left(\varepsilon \right)=\frac{2}{\sqrt{\pi }{T}^{3/2}}\sqrt{\varepsilon}\mathit{\exp}\left(-\frac{\varepsilon }{T}\right) $$

(B1.19)

Note that, as is well known, the number of states increases proportionally to $\sqrt{\varepsilon }$ for free electrons in plasma or solid.

5.1.2 B-2 Bose-Einstein Distribution and Planck Distribution

Particles with spin integer or zero are allowed to stay as many as in one quantum state. The distribution function at this time can be easily obtained as follows. (B1.9) is the probability that particles of quantum state i exist. It is enough to add the probability that particles 1, 2, and 3, exits at the same energy state. By use of the probability

$$ {x}_i=\mathit{\exp}\left(\frac{\mu -{E}_i}{T}\right) $$

(B2.1)

The probability in this case is

$$ {\displaystyle \begin{array}{c}p\left({x}_i\right)=\kern-5em {x}_i+{x_i}^2+{x_i}^3+.\dots \dots \\ {}\kern5.00em =\frac{x_i}{1-{x}_i}=\frac{1}{1-\exp \left(\frac{E_i-\mu }{T}\right)}\end{array}} $$

(B2.2)

Here, μ can be regarded as a normalized constant determined from particle density.

5.1.2.1 Planck Distribution

The energy distribution of the radiation field in thermal equilibrium with the substance at the temperature T is Planck distribution. Photon is a Bose particle of spin 1. The number density of photons of energy hν can be described as follows with use of (B2.2).

$$ \rho \left(\nu \right)=\frac{8\pi }{c^3}{\nu}^2\frac{1}{e^{\frac{h\nu}{T}}-1}=\frac{8\pi }{c^3}\frac{\nu^2{e}^{\frac{- h\nu}{T}}}{1-{e}^{\frac{- h\nu}{T}}} $$

(B2.3)

In order to derive (B2.3), we first need to calculate the number of states of photons with energy hν.. Let us assume that the radiation field is in thermal equilibrium in a square cavity of its length L. The photons, which are electromagnetic waves, are quantized so that the electric field has a node at the boundary. Then the wavenumber of allowed photon is calculated simply

$$ {k}_x=\frac{\pi n}{L},\kern1.25em n=0,1,2,\cdots $$

(B2.4)

The wavenumbers should be quantized. By applying this relation also in the y and z directions, we can see that the number of state is a function of only the absolute value k of the wavenumber. Calculating the volume of photons (k, k + dk) in k space and set it as g (k)dk, we obtain

$$ g(k) dk=\frac{1}{V}{\left(\frac{L}{\pi}\right)}^3\times \frac{\pi }{2}{k}^2 dk=\frac{k^2 dk}{2{\pi}^2} $$

(B2.5)

where V is the volume of system and V = L³. Furthermore, since electromagnetic waves are transverse waves and the degree of freedom of polarization direction is 2 which should be multiplied to (B2.5). Using the frequency ν of photons instead of wavenumber,

$$ k=\frac{2\pi \nu}{c} $$

(B2.6)

Then, (B2.5) becomes

$$ g\left(\nu \right) d\nu =\frac{8\pi {\nu}^2 d\nu}{c^3} $$

(B2.7)

There was a classical theory of Rayleigh-Jeans until Planck distribution was found. They considered the photons in the cavity as having an energy of T as an average per photon from the energy equilibrium distribution and the energy distribution of photons was considered to be the value multiplied by T to (B2.7). However, it is obvious that it diverges in the area of large ν.

The probability of having energy E_i is

$$ {P}_i=\frac{e^{\frac{-{E}_i}{k_BT}}}{\sum_i{e}^{\frac{-{E}_i}{k_BT}}} $$

(B2.8)

In the classical image, the photon field has not been determined that the state of ν has the energy of hν, and the photon field is quantized in the harmonic oscillator field of the classical frequency ν. Then, the energy level is shown with integer n,

$$ {E}_i=\left(n+\frac{1}{2}\right) h\nu $$

(B2.9)

where we neglect the contribution from n = 0, since it is the energy determined by the uncertainty principle and cannot be observed. Among the small set of this frequency ν, the probability is found to be

$$ {P}_n=\frac{{\left({e}^{-\frac{h\nu}{k_BT}}\right)}^n}{\sum_{n=1}^{\infty }{\left({e}^{-\frac{h\nu}{k_BT}}\right)}^n}={\left({e}^{-\frac{h\nu}{k_BT}}\right)}^n\left(1-{e}^{-\frac{h\nu}{k_BT}}\right) $$

(B2.10)

And the average energy of photons of frequency ν is

$$ \bar{\varepsilon}={\sum}_{n=1}^{\infty }{P}_n nh\nu =\frac{h\nu}{e^{\frac{h\nu}{k_BT}}-1} $$

(B2.11)

By multiplying (B2.11) with (B2.7), the photon number density between frequencies (ν,ν + dν) is obtained to be

$$ \rho \left(\nu \right) d\nu =\frac{8\pi }{c^3}{\nu}^2\frac{1}{e^{\frac{h\nu}{T}}-1} d\nu $$

(B2.12)

(B2.12) is the Planck radiation distribution.

5.1.3 B-3 Fermi-Dirac Distribution

Only one particle can exist in the same quantum state for spin-half particles; they are for example, electrons, protons, and neutrons. It is due to Pauli’s exclusion principle. Even in plasma, the density becomes high enough, such as inside the white dwarf, the free electrons degenerate. The electrons in accelerating fuel should be controlled degenerate in laser fusion. In such degenerate plasma, free electrons have average energy of Fermi energy although the temperature is relatively low. Let’s obtain such Fermi-Dirac distribution in a simple way.

The idea is the same as in the case of the Bose-Einstein distribution, but in the state of energy E_i only one electron can exist. Since the probability that n particles exist is proportional to x_iⁿ, the probability of existence in the quantum state |i⟩ is obtained as follows using (B2.1) where the probability in the case of no particle and one particle at the state should be denominator

$$ p\left({x}_i\right)=\frac{x_i}{1+{x}_i} $$

(B3.1)

(B3.1) can be rewritten to the Fermi-Dirac distribution

$$ p\left({E}_i\right)=\frac{1}{1+\mathit{\exp}\left(\frac{E_i-\mu }{T}\right)} $$

(B3.2)

The three distribution functions are plotted for μ = 0 in Fig. B.1.

Fig. B.1

Bose-Einstein, (gold) Boltzmann, (red) and Fermi-Dirac (blue) distributions (μ = 0)

5.1.3.1 Fermi Energy

Inside metals free electrons are running around even at a relatively low temperature. Fermi particles such as electrons with T = 0 and density n are distributed in the form of a step function in Fig. B.1. The maximum energy is called Fermi energy E_F. Derive the Fermi energy E_F.

Apply the calculation of photons (B2.5) to electrons waves, assume T = 0, and the wavenumber space is filled up to k_F. Since the degree of freedom of the electron’s spin is 2, the number of electrons of the Fermi sphere is

$$ n=2{\int}_0^{k_F}g(k) dk=\frac{1}{3{\pi}^2}{k}_F^3 $$

(B3.3)

Using this k_F, Fermi energy is

$$ {\displaystyle \begin{array}{l}{E}_F=\frac{{\left(\mathrm{\hslash}{k}_F\right)}^2}{2m}=\frac{{\mathrm{\hslash}}^2}{2m}{\left(3{\pi}^2n\right)}^{2/3}\\ {}{E}_F=3.3\times {10}^{-15}{\left(n\left[{cm}^{-3}\right]\right)}^{2/3}\ \left[ eV\right]\end{array}} $$

(B3.4)

The average energy of Fermi particles is calculated as

$$ \left\langle {E}_F\right\rangle =\frac{2\int_0^{k_F}\frac{{\mathrm{\hslash}}^2}{2m}{k}^2g(k) dk}{n}=\frac{3}{5}{E}_F $$

(B3.5)

The typical values of Fermi energy are as follows.

• 1. Metal: The number density of metal free electrons is given with n = 10²² to ²³ [cm⁻³] in the solid state. (B3.4) gives

$$ {E}_F=2\sim 10\ eV $$

(B3.6)

• 2. White dwarf: A white dwarf is a star with about solar mass and sustained by balance between the degenerate pressure of electrons and self-gravity. The electrons are almost completely degenerate. It is a compact star of the size of the Earth. It has been observed in an old galaxy as shown in Fig. B.2. Several white dwarfs are shown in a stretched view on the right and they are encircled by white circles. Since a solar mass is in the radius of the Earth, a typical electron density is n = 10³⁰ [cm⁻³] (about ρ = 10⁶ [g/cm³]). Fermi energy at such a high density is

$$ {E}_{\mathrm{F}}\approx 3\times 1{0}^5\ eV=0.3\ MeV $$

(B3.7)

• 3. Nucleus: The inverse of the volume of a proton is n = 10³⁷ [cm⁻³], and the size of the nucleus is decided by balancing of the degenerate pressure of the nucleon and the strong force There. Therefore, using this density and the mass of protons, Fermi energy is

$$ {E}_F\approx 3\times 1{0}^7\ eV=30\ MeV $$

(B3.8)

• 4. Neutron star: A neutron star is a compact star left at the center, when a massive star gravitationally collapses and type II supernova explosion occurs. It is a compact object whose radius is about 10 km and mass is about the solar mass. A typical density is 2 × 10¹⁴ g/cm³, namely the size of cubic sugar is about the total weight of 7 billion people in the world. This number density is n = 10³⁸ [cm⁻³], and its Fermi energy is

$$ {E}_F\approx 1\times 1{0}^9\ eV=1 GeV $$

(B3.9)

Consider the case of a binary star where a white dwarf forms a binary star with the main sequence star. The mass of the white dwarfs increases with time due to mass accretion from the surface of the companion star. As already seen above, Fermi energy (B3.7) is about the rest mass energy of an electron; therefore, it is necessary to consider the relativistic effect as the white dwarf mass increases. The gravitational force can be supported by non-relativistic electron degeneration pressure. However, it is necessary to consider the fact that the pressure is the momentum flux density per unit area

$$ {P}_F= npv $$

(B3.10)

where p is the average momentum of electrons, v is the average velocity of electron. In non-relativistic cases

$$ {\displaystyle \begin{array}{ll}p=\sqrt{2{mE}_F},& v=p/m\\ {}& {P}_F\kern0.5em \upalpha \kern0.5em {n}^{5/3}\end{array}} $$

(B3.11)

Fig. B.2

Observation image of an old galaxy and several white dwarfs are seen in the stretched view (in all circles). Image by HST, NASA

And the density-dependency is strong. This 5/3 power law prevents the collapse due to self-gravity. However, as the mass increases in time, the density further increases and the electrons become relativistic degenerate. In this case, since the speed of the particle flux is limited by the speed of light, the pressure dependence on the density is weakened. In other words, the star becomes “soft” at the dense central region, since the relativistic Fermi pressure is

$$ {\displaystyle \begin{array}{ll}p=\sqrt{2{mE}_F},& v=c\\ {}& {P}_F\kern0.5em \upalpha \kern0.5em {n}^{4/3}\end{array}} $$

(B3.12)

It is found that the hydrostatic solution of a white dwarf is not obtained in the relativistic region where the pressure depends on the power of 4/3 of density. Namely, gravitational collapse should be taken place near the central point as mass increases.

When the gravitational collapse happens near the center of the star, the main component of the star, carbon and oxygen, starts thermonuclear reaction. The produced thermal energy explodes the white dwarf. Such phenomenon is the Type Ia supernova explosion. The explosion is believed to occur when the mass exceeds the critical mass (this is called Chandrasekhar critical mass, 1.26 times the solar mass with a simple calculation, which is estimated to be about 1.44 times recently). In Fig. B.3, the radius of a white dwarf obtained with the equation of state with no relativistic effect and relativistic effect is plotted as a function of the total mass of the white dwarf. It is clear that taking account of the relativistic effect, there is no steady state solution of such white dwarf. Therefore, it is considered that the mass at the time of explosion is constant and the luminosity from Type Ia supernovae is also equal. Due to such reason, it can be used as the standard light source of the universe.

Fig. B.3

The radius of white dwarfs with different mass. Non-relativistic case gives physical solutions, while relativistic Fermi pressure cannot sustain the gravity for the case with more than Chandrasekhar mass