 Open access peer-reviewed chapter

# Plasma Kinetic Theory

Written By

Kashif Chaudhary, Auwal Mustapha Imam, Syed Zuhaib Haider Rizvi and Jalil Ali

Submitted: April 25th, 2017Reviewed: September 5th, 2017Published: December 20th, 2017

DOI: 10.5772/intechopen.70843

From the Edited Volume

## Kinetic Theory

Edited by George Z. Kyzas and Athanasios C. Mitropoulos

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## Abstract

The description of plasma using fluid model is mostly insufficient and requires the consideration of velocity distribution which leads to kinetic theory. Kinetic theory of plasma describes and predicts the condition of plasma from microscopic interactions and motions of its constituents. It provides an essential basis for an introductory course on plasma physics as well as for advanced kinetic theory. Plasma kinetics deals with the relationship between velocity and forces and the study of continua in velocity space. Plasma kinetics mathematical equations provide aid to the readers in understanding simple tools to determine the plasma dynamics and kinetics as described in this chapter. Kinetic theory provides the basics and essential introduction to plasma physics and subsequently advanced kinetic theory. Plasma waves, oscillations, frequencies, and applications are the subjects of kinetic theory. In this chapter, mathematical formulations essential for exploring plasma kinetics are compiled and described simplistically along with a precise discussion on basic plasma parameters in simple language with illustrations in some cases.

### Keywords

• plasma parameters
• kinetic theory
• particle distribution

## 1. Introduction

Plasma is the fourth state of matter, and it is defined as “a quasineutral gas of charged and neutral particles which exhibits collective behavior.” As plasma contains charged particles, these charged particles move around and generate local concentrations of positive or negative charges (collective behavior) which give rise to electric fields. Motion of these charges also generates currents and hence magnetic fields . Therefore, the macroscopic forces acting in plasma are totally different from ordinary gasses and hence remarkable differences in their physical properties are observed. The salient features of the plasma can be understood by investigating the behavior of the electrons, by far the most mobile-charged particle in plasma . Plasma physics deals with the equilibrium and non-equilibrium properties of a statistical system of charged particles. Microscopic degrees of freedom arising from the motion of individual particles describe the system. These statistics therefore theoretically treat the macroscopic behavior of such a system .

The knowledge of plasma parameters helps to understand the dynamics of plasma. Electrons being dominant mobile species play an important role in the behavior of the plasma. The most important of these parameters include plasma temperature, electron density, Debye shielding, and Debye length. Plasma is transient in nature. Therefore, the plasma is generally characterized on the basis of instantaneous observations. Charged particles, neutrals, and molecules coexist in plasma under various circumstances. Conditions in the plasma strongly depend on the distribution of charged particles, where electrons being lighter and highly mobile play a dominant role. Therefore, plasma is generally represented through parameters which are derived from the behavior of electrons in the plasma. When an external point charge is introduced or a localized unbalanced charge is formed in the plasma, readjustment of charge density occurs to neutralize the effect by shielding its electric field. The electrons being more mobile than heavier ions move toward or away from the unbalanced charge faster than ions. It gives rise to oscillations which are referred to as electron oscillationsor plasma oscillations. The frequency at which these oscillations take place is called as plasma frequency. This phenomenon of shielding or screening a foreign charge or an unbalanced charge inside the plasma is known as Debye shieldingor sometimes referred to as Debye screening. It is specific for plasma. Charges keep accumulating around the foreign or unbalanced charge until the static electric field of the unbalanced charge is screened and the balance is restored. A sphere of charges that is created around the unbalanced charge is called as Debye sphere and its radius is known as Debye length. A detailed discussion on two of the most important plasma parameters electron densityand plasma temperatureis provided in Sections 2.7 and 2.8.

Comprehensive mathematical details on certain aspects of plasma are found in published literature. The authors aim to provide readers with an overall view of fundamental mathematical relations explaining the kinetics of plasma that are compiled simplistically in one chapter. This can be an easy reference for the researchers interested in plasma kinetics.

## 2. Plasma kinetic equations

Plasma physics involves phenomena that are related to dynamical processes in statistical mechanics. It is thus very significant to study the properties and structure of the basic kinetic equations governing the dynamical behavior of plasma . The dynamical behavior of a system of N-interacting particles is generally investigated using the Liouville equation. A microscopic distribution function could be used to describe the behavior of such a system. A six-dimensional phase-space distribution function called “Klimontovich distribution function,” which obeys a continuity equation in the phase space, is defined. The system of charged particles can then be described by Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy equations.

### 2.1. Klimontovich equation

To further formalize the kinetic theory, we introduce Klimontovich’s microscopic description and derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon equations . To introduce the Klimontovich equation, we consider a classical system containing Nidentical particles in a box of volume V; n ≡ N/Vdenoting the average number density. Each particle in the box is characterized by an electric charge qand mass m.

In the six-dimensional phase space consisting of the position rand velocity v, each of the particle has its own trajectory; for ith particle,

Xi(t)[ri(t),vi(t)].E1

The microscopic density of the particles in the phase space may be expressed by the summation of the six-dimensional delta functions as

NXt1ni=1nδXXitE2

where X ≡ (r, v). N(X; t) is known as the Klimontovich distribution functionwhich satisfies the continuity equation in the phase space,

dNdt=Nt+ẋ.NX=0.E3

In phase-space coordinates, Eq. (3) can be written as

dNdt+v.Nr+v̇.Nv=0E4

where v̇is the acceleration at point (r, v).

The electromagnetic acceleration is very important in plasma physics,

v̇=qmErt+vcBrt.E5

The electric and magnetic fields E(r, t) and B(r, t) consist of two separate contributions: those applied from the external sources and those produced from the microscopic fine-grained distribution of the charged particles.

E(r,t)=Eext(r,t)+e(r,t),B(r,t)=Bext(r,t)+b(r,t).E6

The microscopic fine-grained fields e(r, t) and b(r, t) are determined from a solution of Maxwell equations,

e+1cbt=0,b1cet=4πcqnvNXtdv,.e=4πqnNXtdv1,.b=0.E7

For a given N(X; t), the solution to these set of equations can generally be written; the solution, when substituted in Eq. (5), would amount to taking account of both electromagnetic and electrostatic interactions between the particles. The electromagnetic interactions are usually negligible as compared with the electrostatic interactions for a nonrelativistic plasma; hence, the microscopic fields become.

ert=qnrNXtrrdX,brt=0.E8

Substituting Eq. (8) into Eq. (6), we get an expression for the acceleration in terms of N(X; t). Eq. (3) can be written with the aid of such an expression as

t+LXVXXNXtdXNXt=0E9

L(X), as a single particle operator is defined by

LXv.r+qmEextrt+vcBextrt.vE10

And V(X, X) is a two-particle operator arising from the Coulomb interaction defined by

VXXq2nmr1rr.vE11

Eq. (9) is known as Klimontovich equation. The equation describes the space–time evolution of the microscopic distribution function.

### 2.2. Liouville distribution

The fine-grained distribution function, which is precise in describing the microscopic conditions of many particles, would not by itself correspond to the coarse-grained quantities in the macroscopic view. There is a need to introduce an averaging process based on the Liouville distribution over the 6 N–dimensional phase space to establish a connection between them.

In T-space, the microscopic state of the system is expressed by a point.

{Xi} ≡ (X1, X2, ……., Xn), called a system point.

By following a normal procedure of the ensemble theory in statistical mechanics, it can assume Nreplicas which are microscopically identical to the system under consideration. Ncan be chosen to be very large so that it can approach infinity when it requires. These Nreplicas are characterized by different microscopic configurations; the system points are scattered over the T-space. Liouville distribution function  can then be defined as D({Xi}; t) in the T-space as

DXitdXilimnNo.of system points in the infinitesimal volumedXiinTspace withingXiNE12

which by definition satisfies the normalization condition.

D({Xi};t)d{Xi}=1.E13

The Nsystem points distributed in the T-space do not interact with each other, behaving like an ideal gas. The distribution function D({Xi}; t) therefore satisfies a Liouville-type continuity equation

Dt+Ẋi.DXi=0E14

The distribution is conserved along a trajectory in the phase-space distribution.

We can now perform a statistical averaging of a fine-grained quantity A(X, X ’ , ……; {Xi(t)} defined at a set of points (X, X, …) in the six-dimensional phase space. With the aid of Liouville distribution, we follow this way:

<AXXt>=dXiDXi;t)AXXXiE15

With respect to the conservation property, this average can be transformed equivalently into an average over the initial distribution, such that

<AXXt>=dXi0DXi0;t)AXXXi0E16

where {Xi(0)}; t)} represents the coordinates of the system points in T-space at tunder the condition that it is located at {Xi(0)} when t= 0.

### 2.3. BBGKY hierarchy

The distribution functions can be obtained through a statistical average of products of Klimontovich functions. A shorthand notation and numerals 1, 2, 3……, etc., in place of X, X, X′′……., etc., can be used to simplify the presentations.

The Klimontovich Eq. (9) can therefore be written as

t+L1N1t=V12N1tN2td2.E17

The Liouville average of this equation can be obtained by using the methods in Eq. (16). The averaging process commutes with differential operators involved in Eq. (17). Now, with the aid of a single-particle distribution function

<NXt>=f1XtE18

The average of the second term defines the two-particle distribution function:

f2XXt;<NXtNXt>=1nδXXf1Xt+f2XXt.E19
t+L1f11t=V12{1nδ12f11t+f212td2E20

For an arbitrary function y(1, 2, …; t), it can be proved from symmetry considerations that

V12δ12y12..td2=0E21

Consequently,

t+L1f11t=V12f212td2E22

It can start from an equation as well

t+L1+L2N1tN2t=V13+V23N1tN2tN3td3E23

Eq. (23) can be derived from a combination of Klimontovich equations. After averaging this equation with respect to the Liouville distribution and Eqs. (16) and (17), an equation involving f1, f2, and f2 can be obtained. This equation can then be simplified with the aid of Eqs. (21) and (22), the result yields

t+L1+L21nV12+V21f212t=V13+V23f3123td3.E24

Similarly, it is considered that a Klimontovich equation for a product of an arbitrary number of the Klimontovich functions performs a statistical average of the equation. We therefore obtain the BBGKY hierarchy equations expressed as

i=1sLi1nijsVijfs1.st]=i=1sVis+1)fs+1(1.s+1tds+1.E25

The set of equations in Eq. (25) is the basis for the kinetic theory of plasmas.

### 2.4. Vlasov’s equation

For identical non-interacting particles, Liouville’s equation can be written in T-space. Introducing two properties of identical non-interacting particles such as the distribution function fand the Hamiltonian function qsimplifies the problem. The distribution function written as a function of 6Nvariables and time factorizes to a product of Nfunctions, each involving only the coordinates and momenta of one particle, and time.

The Hamiltonian function of Nnon-interacting particles is the sum of Nterms, each involving only the coordinates and momenta of one particle. For identical particles, the terms of the Hamiltonian are also identical. For weakly or occasionally interacting particles, the decomposition of finto a product of factors and of qinto a sum of terms is identical.

For a collection of Nidentical interacting particles, Liouville’s theorem can be written in the μ-space as

ft+13fxjxjt+fpjpjt=ft=ftintE26

The right-hand side of Eq. (26) can be evaluated using the exact form of the interaction terms in the 6N + 1 Hamiltonian function variable. It is assumed that the Hamiltonian expression involves no magnetic terms. Under such conditions, p = Mv. If the coordinate system in the μ-space is changed from x, y, z, px, py, pz to x, y, z, vx, vy, vz, then the corresponding volume elements will be in the ratio M3. Hence, if the figurative points density in the (r, p) space is constant according to Eq. (26),

ft+13fxjxjt+fvjvjt=ft=0E27

If magnetic terms are included in the Hamiltonian function, then p = Mv + qA. When the coordinates are changed, the ratio of the corresponding volume elements can be calculated by using Jacobian. The Jacobian is a determinant calculated by taking the partial derivative of any coordinate of one system with respect to all the coordinates of the second system. For instance, in the physical space, the Jacobian is

Jacobian=x1x2x1y2x1z2y1x2y1y2y1z2z1x2z1y2z1z2

The value of the determinant is equal to the ratio of the corresponding volume elements. The ratio of the volume elements, M3, is constant even in the presence of magnetic forces.

vjt=aj=Fj/M, where a= acceleration and F= external force. Introducing an operator ∇v,

v=ivx+jvy+kvzE28

where i, j, and kare the unit vectors in the vx, vy, and vzdirections, respectively. In a vectorial form, Eq. (27) now becomes

ft+v.f+FM.vf=0E29

Eq. (29) is known as Vlasov’s equation. Fis the sum of the electric, magnetic, and gravitational forces resulting from external fields and the macroscopic forces resulting from the plasma itself. If we consider the viscous-like forces, Vlasov’s equation becomes

ft+v.f+v.FMf=0E30

### 2.5. Maxwell’s equations

Maxwell’s equations express the relations between electric and magnetic fields in a medium. Consider a current Iflowing in an element of length dl. The magnetic flux density dBproduced by this current at a point P, a distance rfrom the element, and making an angle θ with its axis is known as Ampere’s law

dB=KIdlsinθ/r2E31

where Kis a constant of proportionality defined as

K=μ/4πE32

where μ= permeability of the medium. For vacuum, μ = μ0 = 4π ⨯ 10−7 H/m. The total magnetic flux density Bproduced at point Pby the current flowing in a long conductor is

B=μI4πsinθr2dlE33

For infinite and linear conductor,B=μI2πr; where r = radial distance from Pto the linear conductor as shown in Figure 1. Figure 1.Flux density near a straight wire in which currentIflows where sinθdl = rdθ andr0 = r sin θ.

Figure 1. Flux density near a straight wire in which current Iflows where sinθdl = rdθ and r0 = r sin θ.

If Bis integrated around the path that encloses the wire, then

B.dl=μIE34

A magnetic field vector His introduced to make the equation independent of the medium. It is such that

B=μHE35

Introducing Eq. (35) into Eq. (34) and the current Iby the surface integral of the conduction current density Jover the area described by the path of integration of H,

H.dl=sJ.dSE36

The total current density = conduction current density σE; σ= conductivity of the wire and E = electric field. D/t = displacement current density; D = electric flux density. Applying Stokes’ theorem, Eq. (36) can now be written in a general form as

H=J+DtE37

According to Faraday’s law, the total e.m.f (V) induced in a closed loop as a function of the total flux Φm/dtproducing the e.m.f is given as.

V=dΦm/dtE38

If the total flux linkage Φt is nΦm, then V = −dΦt/dt. The total flux through the circuit is equal to the integral of Bover the area bounded by the circuit. Therefore,

V=ddtsB.dSE39

The change in magnetic field produces an electric field E, thus

V=E.dlorE=VE40

Combining Eqs. (39) and (40), the induced e.m.f is

E.dl=Bt.dSE41

Applying Stokes’ theorem on Eq. (41),

E=BtE42

Gauss’ law states that the surface integral of the normal component of the electric flux density Dover any closed surface equals the total enclosed charge q. Dis proportional to the electric field with permittivity ɛ of the medium as the constant of proportionality (D∝ ɛE). ɛ = ɛ0 in free space.

Replacing qwith the integral of the charge density ɐe over the volume enclosed by the surface S, the vectorial form

.D=ɐeE43

For magnetic field, the integral of Bover a closed surface is always equal to zero, thus

.B=0E44

Eqs. (37) and (42)(44) are known as Maxwell’s equations. The equations are satisfied in all plasma physics phenomena.

### 2.6. Liouville’s theorem

In relation with the Boltzmann approach, most of the problems of statistical mechanics are best studied in multidimensional spaces called “phase spaces.” Consider a μ-space, which is a six-dimensional and makes use of coordinates and the three components of momentum x, y, z, px, py, pz or any set of Lagrangian coordinates for a point together with the associated generalized momenta. In this space, each plasma particle is represented by a point. If only one degree of freedom exists, the μ-space can be represented on a plane of Figure 2.

Figure 2. Trajectory of an oscillating point in the μ-space.

In the μ-space, the distribution function fis a function of seven variables. Thus, the probability of finding a particle in a given volume element depends only on the coordinates and momenta of this particle, not on those of the other particles. This is a simplified version of Liouville’s theorem for a large number of non-interacting particles.

In the six-dimensional μ-space, the particles are conservative just as they are in the ordinary space. Hence, the conservation theorem may be applied in the phase space,

Bt+6.fv6=0E45

where ∇6 = six-dimensional divergence and v6 is a six-dimensional velocity vector whose components are exact time derivatives of the six coordinates of the μ-space. We now write ∇ temporarily x1, x2, x3 instead of x, y, z,

Bt+13fẋjxj+fṗjpj=0E46

or

ft+13[fxjẋj+fpjṗj+fẋjxj+ṗjpj]=0E47

Hamilton’s canonical equations yield

qjqj+ṗjpj=qjqpjpjqqj=2qpjqj2qqjpj=0E48

where q = Hamiltonian. Eq. (47) now becomes

ft+13fxjxjt+fpjpjt=ft=0E49

Eq. (49) is known as Liouville’s equation.

### 2.7. Boltzmann’s equation

The Boltzmann equation provides the statistical analysis of all the individual positions and momenta of each particle in the fluid (macro-system) at an instant, that is, number of particles in a particular level and their distribution among different levels . It gives relative number of atoms in different excitation states as a function of temperature and refers to certain number of atoms or ions in a particular excitation state with respect to the ground state. Boltzmann equation gives a mathematical description of the state of a system and how it changes. It describes a quantity called the distribution function, f, which depends on a position, velocity, and the time. The function fdetermines the average number of particles having velocities within a small range from νto ν + Δνand coordinates within a small range from r to r + Δrin time Δt.

In a hot dense gas, the atoms constantly experience collisions with each other, which lead to excitation to the different possible energy levels. The collisional excitation follows radiative de-excitation in timescales of the order of nanoseconds. For a constant temperature and pressure, a dynamic equilibrium is established between collisional excitations and radiative de-excitations, which lead to particular distribution of the atoms among different energy levels. Most of the atoms are at low-lying levels. The number of atoms at higher levels decreases exponentially with energy level. At low temperature, the faster the population drops at the higher levels. Only at very high temperatures, high-lying energy levels are occupied by an appreciable number of atoms. Boltzmann’s equation gives the distribution of the atoms among the various energy levels as a function of energy and temperature.

Let us consider a system at local thermal equilibrium (LTE) with a constant volume consisting of “N” atoms and each of atoms has “m”possible energy levels. Suppose there are “Nj” atoms in energy level “Ej.” The total number Nof atoms is given as

N=i=1mNiE50

The total energy “E” of the system can be written as

E=i=1mNiEiE51

A number of ways in which “N1” atoms from total atoms “N”can occupy the first level areNN1. In the same manner, the total number of ways to arrange “N2” atoms from the remaining “N-1”atoms isN1N2and so on. Thus, the total number of microstates “X”in the system, that is, the number of ways to arrange “N”atoms of the system, is given as

X=NN1N1N2N2N3..Nm1NmE52

By solving the above binomial

X=N!N1!N2!.Nj!..Nm!E53
X=N!i=1mNi!E54

Taking log on both sides of Eq. (53),

lnX=lnN!lnN1!lnN2!.lnNj!..lnNm!E55

By applying Stirling’s approximations to the factorials of all variables,

lnXlnN!N1lnN1N1N2lnN2N2.E56
lnXlnN!i=1mNilnNi+NE57

Let us maximize the lnXwith respect to one microstate “Nj,” in a manner that is consistent with constrains of Eqs. (50) and (51). Lagrangian multiplier for the most probable occupation of the jth level is given as

lnXNj+λNNj+μENj=0E58

where μ and λ are Lagrangian’s multipliers.

By adding values from Eqs. (50), (51), and (56) in Eq. (58),

lnN!N1lnN1N1N2lnN2N2.Nj+λN1++NjNj+μE1N1++EjNjNj=0E59

On operating differential,

lnNj+λ+μEj=0E60
λ+μEj=lnNjE61
Nj=eλ+μEjE62
Nj=eλeμEjE63
Nj=CeμEjE64

In general form by multiplying both sides by “Nj”, Eq. (60) can be written as

i=1mNilnNi++μE=0E65

From Eqs. (57) and (65),

lnXlnN!N++μE=0E66
lnX=lnN!λ+1NμEE67

The change in the internal energy of the system in terms of thermodynamics equation can be written as

dE=TdSPdVE68

where “T” is the temperature, “P” is the pressure, and “S” is the entropy of the system.

The change in the internal energy at a constant volume is given as

ESV=TE69
SEV=1TE70

Boltzmann’s equation for entropy is

S=klnXE71
S=klnXE72

From Eqs. (67) and (72),

S=klnN!λ+1NμEE73

Differentiate with respect to the total energy at a constant volume

SEV=kμE74

By comparing Eqs. (70) and (74)

1T=kμE75
μ=1kTE76

By adding value of “μ”in Eq. (64)

Nj=CeEjkTE77

To calculate the value of “C”, let us change the subscript “j”to “i”for Eq. (77) and take summation from “1”to “m”.

N=Ci=1meEikTE78

and

C=Ni=1meEikTE79

Thus, Eq. (64) can be written as

Nj=NeEjkTi=1meEikTE80
NjN=eEjkTi=1meEikTE81

In a system, most of the energy levels in an atom are degenerated, that is, atoms have several states with the same energy. To find out the population of an atom at a particular level, the population of each constituent state is required to be added together. Thus, each term in Eq. (81) must be multiplied by the statistical weight (degeneracy) “ϖ” of the level

NjN=ϖjeEjkTi=1mϖieEikTE82

The term “i=1mϖieEikT” is called the partition function. The Eq. (82) gives the relative number of atoms in state “j”with respect to the total number of atoms in the system. The number of atoms in level “j”can also be compared with the number of atoms at the ground level

NjNo=ϖjeEjkTϖoeEokTE83
Nj=NoϖjeEjkTUE84

and the number of atoms in level 2 relative to level 1, where level 2 is higher than level 1

N2N1=ϖ2eE2kTϖ1eE1kTE85
N2N1=ϖ2ϖ1eE2E1kTE86

The Einstein A coefficient gives the probability of spontaneous emission. A quantum of radiation is emitted by an atom when it de-excites from an excited level to a lower level, which is given as

=ΔEE87

where “ν”is the frequency of emitted radiation and “ΔE = E2-E1is the energy difference between two atomic states (or level) “E2” higher level and “E1” lower level.

Suppose the number of downward transitions per unit time is merely proportional to the number of atoms “N2” at a higher state, then, the number of transition per unit time is given as

N2̇=A21N2E88

where “A21” is proportionality constant known as Einstein coefficient for spontaneous emission for transition from level “E2” to level “E1.

As “A21N2” is the downward transition per unit time from “E2” to “E1, thus the rate of emission of energy from these “N2”, that is, radiant power or flux, is given as

Φ=A21N2hυ2E89

Here, “ν21” represents the frequency of radiation due to transition from level “E2” to level “E1”. As the radiation is emitted isotropically, thus the intensity is

I21=A21N2hυ214πE90

From Eqs. (84) and (90)

I21=NoA21ϖ2hυ214πUeE2kTE91
I21=NoA21ϖ2hc4πλ21UeE2kTE92

#### 2.7.1. Boltzmann plot

Taking log and solving Eq. (92),

lnI21λ21A21ϖ2=E2kT+lnNohc4πUE93

#### 2.7.2. Intensity ratio method

Consider two different emission lines from level i → jand m → n, where iand mare higher energy levels, jand nare lower energy levels. By using Eq. (92),

Iij=NoAijϖihc4πλijUeEikTE94
Imn=NoAmnϖmhc4πλmnUeEmkTE95

Taking ratio of Eqs. (94) and (95) and solving for temperature “T”,

lnIijλijAmnϖmImnλmnAijϖi=EiEmkTE96
T=EiEmklnIijλijAmnϖmImnλmnAijϖiE97

### 2.8. Saha equation

The Boltzmann equation gives only the relative number of atoms or ions in a particular excitation state with respect to the ground state and it does not provide the total number of atoms that have been ionized. In order to determine the total abundance of a given element, it is necessary to know how the atoms are distributed among their several ionization stages. To quantify the number of atoms/ions in different ionization states, Saha’s equation is used which gives an expression for the total number of ions in an ionization state relative to lower ionization state.

For a system at local thermal equilibrium with a constant volume, the Boltzmann equation for the number of ions “Ni” relative to atoms/ions “Ni1” in the ground state of ionization state “i”can be written as (by using Eq. (85))

NiN1i=ϖieEikTϖ1ieE1ikTE98

Taking the sum of all the excited states “j” of ionization state “i”, Eq. (98) can be written as

NjiN1i=j=1ϖjieEjikTϖ1ieE1ikTE99
NjiN1i=j=1ϖjieEjikTZ1E100

where Z1=ϖ1ieE1ikT, which represents the number of atoms at the ground level. The same expression for the ionization state “i + 1” will include not only the excitation states of ion “i + 1” but also the free electrons. Let us consider the energy of the free electron “Ee” and momentum “Pe” with “Ee = Pe2/2me”. Then, at any state of the system of ion “i + 1” and an electron is characterized by the total energy “Eji+1Ee” and statistical weight “ϖji+1ϖePe”. As the energies of free electrons are continuous, the Boltzmann expression analogous to Eq. (100) over all possible momenta of free electron can be written as

Nji+1N1i=1ϖ1ij=1ϖji+1eEji+1E1ikT0ϖePeeEekTdPeE101

The ionization potential of the ion “i”can be defined as

χi=E1i+1E1iE102

Thus, we can write

Eji+1E1i=Eji+1E1i+1+χiE103

Eq. (101) will become

Nji+1N1i=1ϖ1ij=1ϖji+1eEji+1E1i+1+χikT0ϖePeeEekTdPeE104
Nji+1N1i=1ϖ1ieE1i+1kTj=1ϖji+1eEji+1+χikT0ϖePeeEekTdPeE105
Nji+1N1i=1Z1j=1ϖji+1eEji+1+χikT0ϖePeeEekTdPeE106

where Z1=ϖ1ieE1i+1kT, which represents the number of atoms at the ground level. Although the momenta of the free electrons have a continuous distribution, but according to Heisenberg’s uncertainty principle the electrons within a phase-space volume ΔVare indistinguishable, unless they have an opposite spin orientation, that is,

ΔVPe3=h3E107

There are two possible distinguishable electron states within phase space ofΔVPe3=h3. Thus, the statistical weight for the free electron can be written as

ϖePedPe=2ΔVPe3h3E108

If the distribution of electron momenta is isotropic, then

ΔPe3=4πPe2dPeE109

If “ne” is the number density of electron, then the differential volume per electron will be

ΔV=1neE110

Thus, Eq. (108) can be written as

ϖePedPe=8πPe2neh3dPeE111

We can write the integral in Eq. (106) as

0ϖePeeEekTdPe=8πneh30Pe2eEekTdPeE112

The integral can be transformed into an integral over variable “z” such that

z=EekT=Pe22meKTE113
Pe=2meKTZE114
dPe=122meKTzdzE115

By replacing values of Peand dPein Eq. (112)

0ϖePeeEekTdPe=4πneh32meKT320zezdzE116

The integral on the right-hand side of Eq. (116) is the Gamma function of the argument 3/2, which is Γ3/2=1/2Γ1/2=π/2. By adding this value in Eq. (116)

0ϖePeeEekTdPe=4πneh32meKT32π/2E117
0ϖePeeEekTdPe=2neh32πmeKT32E118

From Eqs. (106) and (118),

Nji+1N1i=1Z1j=1ϖji+1eEji+1+χikT2neh32πmeKT32E119

By dividing Eqs. (100) and (119),

Nji+1Nji=j=1ϖji+1eEji+1+χikTj=1ϖjieEjikT2neh32πmeKT32E120
Nji+1Nji=j=1ϖji+1j=1ϖji2neh32πmeKT32eEji+1+χiEjikTE121
Nji+1Nji=j=1ϖji+1j=1ϖji2neh32πmeKT32eEj1Eji+χikTE122

For a particular state,

Ni+1Ni=22πmeKTneh332ϖi+1ϖieEi+1Ei+χikTE123
ne=22πmeKTh332Niϖi+1Ni+1ϖieEi+1Ei+χikTE124

Eq. (90) can be written as

Ni+1=4πhIi+1λi+1Ai+1cE125
Ni=4πhIiλiAicE126

From Eqs. (124)(126),

ne=22πmeKTh332IiλiAi+1ϖi+1Ii+1λi+1AiϖieEi+1Ei+χikTE127