This monograph is intended for master and PhD students majoring in the field of plasma physics. Specific topics of vario

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- Давлетов А.

AL-FARABI KAZAKH NATIONAL UNIVERSITY

Askar Davletov

EﬀECTIVE POTENTIALS IN NON-IDEAL PLASMA PHYSICS Monograph

Almaty «Kazakh university» 2019

UDC 523 LBC 22.3 D26 Recommended for publication by the Academic Council (Protocol №7, 25.03.2019) and the Editorial and Publishing Council of al-Farabi Kazakh National Universty (Protocol №4,16.04.2019) Reviewers: Candidate of Physical and Mathematical Sciences, PhD, Professor M.T. Gabdullin Doctor of Physical and Mathematical Sciences, Professor Yu.V. Arkhipov

Davletov A. D 26 Eﬀective Potentials in Non-Ideal Plasma Physics: monograph / Askar Davletov. – Almaty: Kazakh University, 2019. – 149 p. ISBN 978-601-04-4043-2 This monograph is intended for master and PhD students majoring in the field of plasma physics. Specific topics of various types of nonideal plasmas are approached by an original method known in the literature as the generalized Poisson-Boltzmann equations. The whole consideration is entirely based on the renormalization of a pure Coulomb interaction to take into account collective events in the medium, which allows one to both evaluate the thermodynamic characteristics of a plasma and estimate its transport coeﬃcients. UDC 523 LBC 22.3 ISBN 978-601-04-4043-2

c ⃝Askar Davletov, 2019 c ⃝al-Farabi KazNU, 2019

Contents Introduction

5

1 Chapter General approaches 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Simple renormalization theory . Bogolyubov’s chain of equations Pair correlation approximation . Solution to the master equation Density-response formalism . . . Integral equation method . . . . Conclusions . . . . . . . . . . . .

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2 Chapter Classical plasmas 2.1 2.2 2.3 2.4 2.5

Plasma parameters . . Pseudopotential model Correlation functions . Thermodynamics . . Conclusions . . . . . .

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3 Chapter Semiclassical plasmas 3.1 Plasma parameters . . . . . . 3.2 Pseudopotential model . . . . 3.3 Correlation functions . . . . . 3.4 Thermodynamics . . . . . . . 3.5 Electrical conductivity . . . . 3

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CONTENTS 3.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

4 Chapter Partially ionized plasmas 4.1 Plasma parameters . . . . . . 4.2 Pseudopotential model . . . . 4.3 Self-consistent chemical model 4.4 Correlation functions . . . . . 4.5 Thermodynamics . . . . . . . 4.6 Electrical conductivity . . . . 4.7 Conclusions . . . . . . . . . . . 5 Chapter Dusty plasmas 5.1 Plasma parameters . . . 5.2 Pseudopotential model . 5.3 Charging of dust particles 5.4 Correlation functions . . 5.5 Thermodynamics . . . . 5.6 Conclusions . . . . . . . . Bibliography

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74 . 76 . 78 . 86 . 92 . 95 . 98 . 104

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106 108 110 115 127 132 136

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Introduction This monograph is completely devoted to the application of pseudopotentials, or so-called eﬀective potentials, in the physics of non-ideal plasmas. A natural question arises what a nonideal, or as is usually said, a strongly coupled system is. In the 20th century, the theory was most fully developed of a socalled ideal plasma, i.e. the plasma, in which the interaction of particles with one another can be entirely ignored, treating their mutual influence through the created collective electromagnetic field. This means that the plasma medium was regarded as a charged ideal gas, so that corrections for two-body interactions were simply discarded. At present a nonideal plasma, in which interparticle interactions play an important role, finds its increasing number of practical applications both in experimental studies and technological designs. The concept of nonideality normally implies that the average interaction energy of plasma particles becomes of the order and, in some cases, significantly exceeds the average thermal energy of their chaotic motion. It is immediately inferred from this very definition of a nonideal plasma that the potential energy of interparticle interactions actually governs all of its properties. It seems at first sight that charged particles just interact according to the Coulomb law and nothing else can be added here. The elaborate answer to this claim constitute the pivot of the present scrutiny. Of particular interest in the following are the properties of various types of plasmas at thermodynamic equilibrium. On the one hand, 5

Introduction thermodynamic relations are the most universal and are rigorously valid for systems of an arbitrary physical nature. On the other hand, only knowledge of the specific thermodynamic characteristics of the medium enables one to predict the direction of physical and chemical processes taking place under certain conditions. From this point of view, the establishment of the thermal and caloric equations of state of matter is one of the important problems of statistical physics, to which thermodynamics itself cannot provide a positive solution because it completely ignores all specific models of matter disregarding the concept of its internal structure. This problem is also comprehensively addressed hereinafter. It should immediately be noted and directly stressed that this monograph is not aimed at reviewing the current state of the field of non-ideal plasmas. The main goal pursued is the detailed presentation of those important results that have been achieved at the Kazakhstani School of Plasma Physics, founded in 1975 by Professor Fazylkhan Baimbetov. This monograph is based on the course of lectures ”Selected topics of non-ideal plasma physics” given by the author to master and PhD students of Al-Farabi Kazakh National University majoring in the speciality ”6D060400-Physics” and the rest is organized as follows. Chapter 1 is designed to present the general approaches that are utilized throughout this monograph to describe the properties of various types of plasmas. They include the Bogolyubov chain of equations for equilibrium distribution functions, the linear density-response formalism, and the method of integral equations for evaluating correlation functions. A classical fully ionized plasma is discussed in Chapter 2, which bears more methodical character and aims to show at work all the methods, described in Chapter 1. All numerical results are presented for the simplest model of a one-component plasma to demonstrate the influence of strong interparticle interaction on various physical properties. It has been well understood for many decades that the construction of a consistent quantum-mechanical theory of plasmas with the 6

Eﬀective Potentials in Non-Ideal Plasma Physics Coulomb interaction potential encounters sophisticated mathematical diﬃculties that are uneasy to overcome. This resulted in the appearance of the so-called pseudopotential models that essentially relied on the replacement of the Coulomb interaction by some eﬀective potential, which included such characteristics of the medium as its temperature and number density. In particular, pseudopotentials have been worked out to take into account the quantum eﬀects of diﬀraction and symmetry, necessary for the description of equilibrium properties of a semiclassical plasma, which is in focus of Chapter 3. Another type of plasmas, which requires an engagement of the whole apparatus of quantum mechanics in a logically consistent manner, is a partially ionized medium. It is obvious that only the power of quantum mechanics is capable to correctly incorporate the formation of various bound states, such as atoms and molecules. However, the presence of the neutral plasma component can still be handled within the framework of the quasi-classical method, which is referred to in the literature as a chemical model of plasmas. Chapter 4 dwells on a self-consistent chemical model of partially ionized plasmas, which allows not only to study the ionization equilibrium and thermodynamic quantities, but to predict the behavior of the correlation functions and transport coeﬃcients of the system as well. The burden of Chapter 5 is to entirely concentrate on the investigation of dusty plasmas, in which micron-size particles, called grains, are present. Being immersed into an ordinary plasma, dust particles acquire such a high electric charge that the intergrain interaction energy begins to dominate. In a particular experimental situation, this usually leads to the formation of a plasma crystal, in which the mutual arrangement of dust particles bears a strong resemblance of the crystal lattice. Although dusty plasmas are not so widely used in practical applications, they are especially interesting from the fundamental point of view because the behavior of dust particles can be easily visualized and observed experimentally with a high-resolution videography, which gives a wonderful opportunity to directly verify the theoretical concepts, developed for strongly coupled Coulomb systems. 7

Introduction At the end of this introductory section I would like to express my deep and sincere gratitude to my colleagues and collaborators, Professors Fazylkhan Baimbetov, Tlekkabul Ramazanov, Yuriy Arkhipov and Igor Tkachenko, for those valuable discussions of the jointly obtained results, which have made a tremendous impact on the present monograph.

8

Chapter 1

General approaches In this chapter a generalized Poisson-Boltzmann equation for eﬀective interactions, accounting for the collective events in the medium, is obtained. The derivation is made in two diﬀerent ways. First of all, starting from the elementary arguments, based on the Boltzmann ideas, a renormalization procedure is introduced in Section 1.1. The physical idea behind the proposed approach is that the interaction of two chosen particles is naturally aﬀected by the presence of a third. Then, the same generalized Poisson-Boltzmann equation is strictly deduced from the Bogolyubov hierarchy in the pair correlation approximation. To do so, the Bogolyuboc chain of equations for equilibrium distribution functions is derived in Section 1.2 from the Gibbs distribution of the canonic ensemble. Section 1.3 consistently adopts the pair correlation approximation for the Bogolyubov chain of equations to ultimately prove for the generalized Poisson-Boltzmann equation. It is later demonstrated in Section 1.4 that the generalized PoissonBoltzmann equation, as applied to a multi-component system, turns out to be a set of linear algebraic equations in the Fourier space that can be solved analytically and presented in a concise manner. Another two approaches to benefit from are the linear density9

CHAPTER 1. GENERAL APPROACHES response formalism, outlined in Section 1.5, and the method of integral equations, elucidated in Section 1.6. The former in the random-phase approximation is shown to be equivalent to the Bogolyubov chain of equations in the pair correlation approximation, whereas the latter is frequently engaged to obtain correlation function of various non-ideal systems in weakly and moderately coupled regimes. Final conclusions and general remarks are briefly summarized in Section 1.7.

1.1

Simple renormalization theory

For the purpose of taking into account the collective events in pairwise interaction potentials, the interaction of two arbitrary chosen particles is considered in the presence of a third, see Figure 1.1. The total force, called then the macroscopic force Fmac acting on the reference, ij say ith, particle from the whole system, is written as ∑∫ mic Fmac = F + Fmic (1.1) ij ij ik P (rik , rjk )drk . k k̸=i,j

Here Fmic ij is the true microscopic force acting between the ith and the jth particles, rij = rj − ri denotes the position vector drawn from the ith particle position vector ri to the jth particle position vector rj , P (rik , rjk ) stands for the probability density of finding the kth particle at certain distances from the ith and jth particles, respectively. The summation in (1.1) is implied over all particles except for the ith and the jth. The average over the position of the kth particle is taken by means of the integration over rk in (1.1). Hence, according to (1.1), the total macroscopic force acting on the ith particle from the jth particle emerges from the true microscopic interaction between these two particles and the positionally averaged microscopic interaction with the third kth particle. By definition, the microscopic interaction force is directly related to the microscopic interaction potential φij via the nabla operator ∇i , 10

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 1.1: Interaction diagram of the ith and the jth particles in the presence of the kth particle. Integration over the position of the kth particle is implied. acting on the coordinates of the ith particle ri Fmic ij = −∇i φij .

(1.2)

Likewise, the macroscopic force Fmac is assumed to be associated ij with the eﬀective potential Φij as Fmac = −∇i Φij . ij

(1.3)

To unchain the set of equations (1.1)-(1.3) it is required to make an appropriate guess on the nature of the probability density P (rik , rjk ). Its exact form remains, strictly speaking, unknown and should be found on the basis of a more consistent approach, what is actually realized in the following Sections 1.2 and 1.3. Nevertheless, the simplest way at hand is to heuristically adopt the Boltzmann distribution ( ) Φik + Φkj 1 , (1.4) P (rik , rjk ) = exp − V kB T 11

CHAPTER 1. GENERAL APPROACHES where V stands for the whole volume of the system. The factor 1/V in (1.4) has been formally multiplied because of the normalization condition since the integration in (1.1) is taken over all possible positions of the kth particle inside the system volume. Substitution of equations (1.2)-(1.4) into equation (1.1) yields ) ( ∫ Φik + Φkj 1 ∑ ∇i Φij = ∇i φij + ∇i φik exp − drk , V k kB T

(1.5)

k̸=i,j

and linearizing the exponent in (1.5), one finally obtains ∇i Φij = ∇i φij +

) ( ∫ Φkj 1 ∑ Φik − drk . ∇i φik 1 − V k kB T kB T

(1.6)

k̸=i,j

It should be noticed that the interaction potentials depend on the modulus of the vector rij = ri − rj only. It is then a consequence of the spherical symmetry that the first two terms in integrand (1.6) are simply canceled, and, thus, the following relation holds ∫ 1 ∑ ∇i φik Φkj drk . (1.7) ∇i Φij = ∇i φij − V kB T k k̸=i,j

Assuming the particles indistinguishability, the summation in (1.7) is changed from the particle numbers to the particle species. Writing out explicitly the coordinates of all particles in the form of rai with a being the particle species, one ultimately gets ∇i Φab (rai , rbj ) = ∇i φab (rai , rbj ) ∑ Nc ∫ − ∇i φac (rai , rck )Φcb (rbj , rck )drck , V k T B c

(1.8)

where Nc denotes the total number of particles of species c in the system. 12

Eﬀective Potentials in Non-Ideal Plasma Physics Introducing the number density nc = Nc /V and acting with the nabla operator ∇i on both sides of (1.8), the generalized PoissonBoltzmann equation is suitably derived as ∆i Φab (rai , rbj ) = ∆i φab (rai , rbj ) ∑ nc ∫ − ∆i φac (rai , rck )Φcb (rbj , rck )drck , k T B c

(1.9)

which adheres the eﬀective macroscopic potential Φab (rai , rbj ) to the true microscopic interaction potential φab (rai , rbj ). Note that a set of equations (1.9) must be solved for particle species. The elementary concept, presented above, was first introduced in [1] to clarify the derivation procedure of the generalized Poisson-Boltzmann equation from the Bogolyubov chain of equations for equilibrium distribution functions, described in the forthcoming Sections 1.2 and 1.3.

1.2

Bogolyubov’s chain of equations (0)

It is known that the complete distribution function fN in the phase space of coordinates ri and momenta pi for the canonical ensemble is given by the Gibbs formula ( ) H 1 (0) exp − , (1.10) fN = 3N h N !Z θ where N stands for the number of particles in the system, θ = kB T denotes the temperature T in energetic units, H refers to the Hamiltonian of the system, and the partition function Z is written as ) ( ∫ 1 H dr1 dp1 . . . drN dpN . (1.11) Z = 3N exp − h N! θ The Hamiltonian of the system can be represented by the sum of the kinetic and interaction parts as N ∑ p2i H= + φ(r1 , . . . , rN ), 2mi i=1

13

(1.12)

CHAPTER 1. GENERAL APPROACHES in which the potential energy of interaction φ(r1 , . . . , rN ) is composed of the pairwise interactions in the following form φ(r1 , . . . , rN ) =

1∑ φij (ri − rj ). 2 i,j

(1.13)

i̸=j

Here mi stands for the mass of the ith particle, and φij (|ri − rj |) designates the potential energy of interaction between the ith and the jth particles. The summation in (1.13) is understood to be taken over all possible indexes except for i = j. It strictly follows from equations (1.10) and (1.12) that [ )] ( ) ( ∏ φ(r1 , . . . , rN ) p2i (0) exp − , (1.14) fN ∼ exp − 2mi θ θ i

i.e. at thermal equilibrium the correlations between the momentum and the configuration spaces are completely absent. It is prescribed to the fact that the potential energy of the system is independent of the momenta of the particles. Thus, the distribution in the coordinate and the momentum spaces can be treated separately from each other. To concentrate on correlations in the configuration space only, it is convenient to introduce, in accordance with (1.13) and (1.14), the normalized total distribution function as ∑ φ (|r − rj |) exp − 1 2θ i,j ij i i̸=j PN (r1 , . . . , rN ) = . (1.15) ∫ ∑ exp − 1 φ (|r − rj |) dr1 . . . drN 2θ i,j ij i i̸=j

In the momentum space, though, the usual Maxwell distribution of particle velocities is maintained in full compliance with (1.14). It is well realized that the description of the system by using the total distribution function (1.15) contains an excess of detailed information 14

Eﬀective Potentials in Non-Ideal Plasma Physics that is of no practical use. It is, therefore, conventional to define a set of partial distribution functions of a lower order as ∫ Ps (r1 , . . . , rs ) =

PN (r1 , . . . , rN )drs+1 . . . drN .

(1.16)

Let s-configuration be an arbitrary chosen set of s particles in the coordinate space. Then, according to definition (1.16), the partial distribution function of the order s comprises the probability density for s distinguishable particles to have a certain set of coordinates r1 , . . . , rs . Denote the particle species in Latin letters a, b, c, . . ., and the particle numbers in i, j, k, . . .. The general distribution function P s of the order s is introduced to measure the probability density that s particles are characterized by a set of coordinates a

r1 , . . . , a rνa , . . . , d r1 , . . . , d rνd .

Here νf designates the number of particles of species f contained in the given s-configuration, the diﬀerence between particles within each group νf is completely omitted. Obviously, the following relation holds for the conservation of the number of particles ∑ νf = s. (1.17) f

Let Nf be the total number of particles of species f in the system, and f ri be the position vector of the ith particle of the species f in the given s-configuration. It then follows from the known formulas of the combinatorics that the general P s and the partial Ps distribution functions are rigidly related to each other by P s (a r1 , . . . , a rνa , . . . , d r1 , . . . , d rνd ) = ∏ Nf ! Ps (r1 , . . . , rs ). = (Nf − νf )! f

15

(1.18)

CHAPTER 1. GENERAL APPROACHES To obtain the Bogolyubov chain of equations, the nabla operator ∇i is applied to act on the ith particle coordinates of the partial distribution function Ps in (1.16), and, with the help of (1.15), one gets ∇i Ps = −

1 3N θh N !Z

∫

N ( φ) ∑ exp − ∇i φij drs+1 . . . drN . θ j=1

(1.19)

j̸=i

The sum in equation (1.19) is evidently split into two parts N ∑

=

j=1 j̸=i

s ∑

+

j=1 j̸=i

N ∑

.

(1.20)

k=s+1

The first part implies the summation over all particles of the given s-configuration, whereas the second one incorporates all other particles in the system. As a result, the following equation holds 1∑ (∇i φij )Ps (r1 , . . . , rs ) θ j=1 s

∇i Ps (r1 , . . . , rs ) = −

j̸=i

−

1 θ

N ∑

∫ (∇i φik )Ps+1 (r1 , . . . , rs , rk )drk ,

(1.21)

k=s+1

which, after dividing by Ps and multiplying by θ, takes the form θ∇i ln Ps (r1 , . . . , rs ) = −

s ∑

(∇i φij )

j=1 j̸=i

−

N ∫ ∑

(∇i φik )

k=s+1

Ps+1 (r1 , . . . , rs , rk ) drk . Ps (r1 , . . . , rs )

(1.22)

Equation (1.22) is the sought chain of Bogolyubov equations for the partial distribution functions. It has to be emphasized that equation (1.22) is called the chain because it is not closed from the viewpoint of mathematics because it is necessary to know Ps+1 in order to find Ps . 16

Eﬀective Potentials in Non-Ideal Plasma Physics With the aid of (1.18), the Bogolyubov chain of equations is rewritten for the general distribution function as [2] θ∇i ln P s (r1 , . . . , rs ) = −

−

∑ Nc − νc ∫ c

V

s ∑

(∇i φij )

j=1 j̸=i

(∇i φik )

P s+1 (r1 , . . . , rs , rck ) c drk , P s (r1 , . . . , rs )

(1.23)

in which the summation is now performed over the particle species c.

1.3

Pair correlation approximation

The Bogolyubov chain of equations (1.23) has been actually obtained from the Gibbs distribution (1.10), which is only valid for systems at thermodynamic equilibrium. That is why it is called the chain of Bogolyubov equations for equilibrium distribution functions. For the following needed is the physical interpretation of the Bogolyubov chain (1.22), whose right hand side embodies two standard contributions. Namely, the first term is just the force acting on the ith particle from the other particles of the given s-configuration, whereas the second term stands for the average force, exerted by all other particles in the system. Thus, the right hand side of (1.22) is simply the total force acting on the ith particle from the whole system. In this respect the average energy of the ith particle in the given s-configuration is introduced as

=

s ∑ j=1 j̸=i

φij +

∑ N c − νc ∫ c

V

φik

P s+1 (rai , . . . , rbj , rck ) P s (rai , . . . , rbj )

drck . (1.24)

Acting with the nabla operator ∇i on both sides of (1.24) and taking 17

CHAPTER 1. GENERAL APPROACHES summation with (1.23) yield θ∇i ln P s (rai , . . . , rbj ) + ∇i < φsi >= ∑ N c − νc ∫ P s+1 (rai , . . . , rbj , rck ) c = φik ∇i drk . V P s (rai , . . . , rbj ) c

(1.25)

Further derivation entirely resides upon the pair correlation approximation, which is formulated as follows. An expression for the general distribution function P s of the order s is written in the most general case as s ∏ ∏ (1 + s hij (rai , rbj )) P s (rai , . . . , rbj ) = P1 (ri ) i=1

×

∏

k(2)

(1 + s hijk (rai , rbj , rck ))

k(3)

∏

. . ..

(1.26)

k(4)

From the mathematical point of view, relation (1.26) is just a functional substitution, in which new variables s hij...k represent the correlation functions of the system of s particles, whose order is strictly determined by the number of indexes that coincide with the indexes of the correlating particles. The symbol k(q) means that the product should be taken over all possible combination of q indexes from the given s-configuration. It is known that for homogenous systems of interest herein the functions P1 (rai ) = 1/V are simply constants independent neither of the sort of the particle nor its coordinate. It is the pair correlation approximation that vigorously dismiss all the correlation functions of higher than the second order, as well as their products, such that the ratio of the general distribution function of the order s + 1 to the general distribution function of the order s is found from the ansatz (1.26) as P s+1 (rai , . . . , rbj , rck ) P s (rai , . . . , rbj )

s ∏ = P1 (rck ) (1 + s hik (rai i , rck )) i=

≈

P1 (rck )

( 1+

s ∑ i=1

18

) s

hik (rai , rck )

.

(1.27)

Eﬀective Potentials in Non-Ideal Plasma Physics Combining (1.27) and (1.25) gives rise to θ∇i ln P s (rai , . . . , rbj ) + ∇i < φsi > ∑ Nc − νc ∫ = φik (rai , rck )∇i s hik (rai , rck )drck . V c

(1.28)

It is generally recognized that in the thermodynamic limit the correlation functions of the homogeneous system should solely depend on the relative distances between the correlating particles, i.e. s hik (rai , rck ) = s h (|ra − rc |). Consequently, the integral on the right hand side of ik i k equation (1.28) is simply equal to zero due to the notorious spherical symmetry and, it simplifies to θ∇i ln P s (rai , . . . , rbj ) + ∇i < φsi >= 0.

(1.29)

It is enough to consider the eﬀective interaction of two particles in the configuration with s = 2 by assuming < φ2i >= Φab (rai , rbj ).

(1.30)

Then, equations (1.29), (1.30) and (1.26) in the pair correlation approximation inevitably produce ( ) P 2 (rai , rbj ) = P1 (rai )P1 (rbj ) 1 + 2 hij (rai , rbj ) ( ) Φab (rai , rbj ) = A exp − , (1.31) kB T where A = 1/V 2 denotes an integration constant, defined by the normalization condition ∫ ∫ P 2 (rai , rbj )drai drbj = 1. (1.32) It follows from (1.31) that (

Φab (rai , rbj ) 2 a b hij (ri , rj ) = exp − θ 19

) − 1.

(1.33)

CHAPTER 1. GENERAL APPROACHES Substituting expressions (1.27) and (1.29) into the Bogolyubov chain (1.23) with s = 2 and making use of equations (1.30) and (1.33), the following relation is obtained ∇i Φab (rai , rbj ) = ∇i φab (rai , rbj ) +

∑ (Nc − νc ) c

(

∫ ∇i φac (rai , rck ) exp

V

) Φcb (rck , rbj ) − drck . θ

(1.34)

At this stage it is necessary to take the thermodynamic limit Nc → ∞, V → ∞ with nc = Nc /V = const being the particle number density of species c. It is well understood that the thermodynamic limit is vital to neglect the global fluctuations of thermodynamic quantities and to avoid studying near-wall eﬀects. Linearizing the exponential term in (1.34), one finally gets [1] ∇i Φab (rai , rbj ) = ∇i φab (rai , rbj ) ∑ nc ∫ − ∇i φac (rai , rck )Φcb (rck , rbj )drck . k T B c

(1.35)

A similar linearization of the exponential term has, of course, to be carried out in (1.33) 2

hij (rai , rbj )

Φab (rai , rbj ) =− . kB T

(1.36)

After applying the operator ∇i to equation (1.35), the same generalized Poisson-Boltzmann equation (1.9) is again derived for the eﬀective interaction potential, thus, proving its validity in the pair correlation approximation for the Bogolyubov chain of equations. Expression (1.9) is actually a set of integral and diﬀerential equations for determining the eﬀective interaction potentials Φab (rai , rbj ) (macroscopic potentials) in the medium via the true microscopic potentials φab (rai , rbj ). Note that the indexes in all potentials has been changed from particle numbers to particle species. 20

Eﬀective Potentials in Non-Ideal Plasma Physics Equation (1.9) has been called the generalized Poisson-Boltzmann equation since the ordinary Poisson-Boltzmann equation, which specifically includes the Debye-H¨ uckel theory, is its consequence when substituting the Coulomb potential as the microscopic potential φab (rai , rbj ). In this sense, the proposed approach is a fundamental extension of the mean field theory and the mean force potential, originally proposed by Kirkwood [3, 4]. Note that equation (1.29) actually persists in that the mean potential explicitly coincides with the mean force potential in the pair correlation approximation for the Bogolyubov chain of equations. A few comments has to be added on the above used pair correlation approximation. First of all, according to (1.36) the pair correlation functions have been found in the first order of the interaction potential and, therefore, neglect of their product corresponds to the second order of accuracy in the interaction potential. It is strictly proved that three-body correlations appear only in that second order of the interaction potential, which fully justifies the pair correlation approximation. Secondly, while dealing with various chains of equations a natural question arises of how the truncation is implemented in a particular case. For the pair correlation approximation the rejection of the higher order correlation functions in (1.26) leads to that the partial and general distribution functions of an arbitrary order are systematically expressed through the same pair correlation function and this immediately decomposes the Bogolyubov chain of equations for the equilibrium distribution functions. A set of equations (1.9) has turned out to be very successful in describing properties of various types of plasmas and its further applications constitute the core of the present monograph.

1.4

Solution to the master equation

An application of the generalized Poisson-Boltzmann equation (1.9) to the system of interest starts from an appropriate choice of microscopic potentials φab . It is not very hard to show that the master equation (1.9) in the Fourier space turns into a set of linear algebraic 21

CHAPTER 1. GENERAL APPROACHES equations that must be solved for all particle species. For example, in the case of the three-component plasma solution to (1.9) in the Fourier space reads [1] ˜ ab (k) = 1 (φ˜ab (k) + Ac [φ˜cc (k)φ˜ab (k) − φ˜ac (k)φ˜bc (k)]+ Φ ∆[ φ˜aa (k)φ˜cc (k)φ˜dd (k) − δab Ac Ad φ˜ac (k)φ˜ad (k)φ˜cd (k) + 2 ]) φ˜aa (k)φ˜cd (k)2 + φ˜cc (k)φ˜ad (k)2 + φ˜dd (k)φ˜ac (k)2 , 2

(1.37)

with [ ] ∆ = 1 + Aa φ˜aa (k) + Aa Ab φ˜aa (k)φ˜bb (k) − φ˜ab (k)2 [ φ˜ab (k)φ˜bc (k)φ˜ac (k) φ˜aa (k)φ˜bb (k)φ˜cc (k) +Aa Ab Ac + 3 6 ] 2 2 φ˜aa (k)φ˜bc (k) + φ˜bb (k)φ˜ac (k) + φ˜cc (k)φ˜ab (k)2 − . 6

(1.38)

Here Ac = nc /kB T , φ˜ab (k) signifies the Fourier transform the microscopic potential and δab denotes the Kronecker delta, which is equal to 1 if a = b, and equals to 0, otherwise. As it is conventionally implied in tensor algebra, the summation is assumed over the repeated indexes of particle species in the above expressions (1.37) and (1.38). Final expressions for the macroscopic potentials (pseudopotentials) in the configuration space are obtained from (1.37) by the inverse Fourier transform ∫ ˜ ab (k) exp (ikr)dk. Φab (r) = Φ (1.39) Please, note, that in order to be correctly applied equations (1.37) and (1.38) require the Fourier transform φ˜ab of the microscopic potential φab to exist, which is not always the case for widely-spread theoretical models. For instance, the well-known Lennard-Jones interaction potential has no Fourier transform at all since the corresponding integral simply diverges ar r → 0. 22

Eﬀective Potentials in Non-Ideal Plasma Physics

1.5

Density-response formalism

To describe the correlations of particles in a multicomponent plasma, a formalism of the dielectric response is widely used. In its framework a real physical system is assumed to be subject to an external fictitious field, characterized by the potential Va (r, t) acting on the particles of species a. Under its influence an additional term enters into the Hamiltonian of the system, which can be written as ∑∫ Hext (t) = drρa (r)Va (r, t), (1.40) a

where ρa (r) stands for the particle charge density of species a. The Fourier transform of the potential Va (r, t) is found as ∫ ∫ ˜ Va (k, ω) = dr dtVa (r, t) exp[−i(kr − ωt)]. (1.41) The plasma reacts to the external field V˜a (k, ω) by inducing the charge density deviation δρa (k, ω) from its unperturbed value. In the framework of the linear density-response formalism the following relation holds ∑ χab (k, ω)Va (k, ω), (1.42) δρa (k, ω) = a

where χab (k, ω) denotes the response function of particles of species a and b. For example, in case of the two-component plasma the response functions χab (k, ω) are well found in the random-phase approximation to read [5] [ ] (0) χee (k, ω) = χ(0) (k, ω) 1 − χ (k, ω) φ ˜ (k) /D, (1.43) ii e i [ ] (0) χii (k, ω) = χi (k, ω) 1 − χ(0) (k, ω) φ ˜ (k) /D, ee e (0)

χei (k, ω) = χie (k, ω) = χ(0) ˜ei (k)/D, e (k, ω)χi (k, ω)φ 23

(1.44) (1.45)

CHAPTER 1. GENERAL APPROACHES and [ ][ ] (0) D = 1 − χ(0) (k, ω) φ ˜ (k) 1 − χ (k, ω) φ ˜ (k) ee ii e i (0)

−χ(0) ˜2ei (k). e (k, ω)χi (k, ω)φ

(1.46)

Here φ˜ab (k) designates the Fourier transform of the interaction poten(0) tial of particle of species a and b, and the screening function χa (k, ω) is derived as ( ) na ω (0) χa (k, ω) = − W , (1.47) kB T kvT a where vT a = (kB T /ma )1/2 stands for the thermal velocity of particle of sort a, and the Dawson integral W (z) is written in the following form [6] ∫z W (z) = 1 − z exp(−z /2) 2

exp(y 2 /2)dy 0

√ +i

π z exp(−z 2 /2). 2

(1.48)

A set of partial static dielectric functions εab (k, 0) is easily expressed (0) through the static response function χa (k, 0) (ω = 0) as ε−1 ˜ac (k)χcb (k, 0), ab (k, 0) = δab + φ

(1.49)

where δab still symbolizes the Kronecker delta. Substituting expressions (1.43)-(1.46) into equation (1.49), one finally obtains [7] na εab (k, 0) = δab + φ˜ab (k). (1.50) kB T The static pseudopotentials, taking into account the collective events in plasmas, are easily obtained in terms of the partial static dielectric functions as ˜ ab (k) = φ˜ac (k)ε−1 (k, 0). Φ (1.51) cb 24

Eﬀective Potentials in Non-Ideal Plasma Physics As usual, expression for the pseudopotential Φab (r) in the ordinary ˜ ab (k) by the inverse Fourier configuration space is recovered from Φ transform (1.39). It has to stressed that for the two-component system, the resulting Φab (r) is the same as that one obtained from the solution to the master equation (1.37), see Section 1.4. Thus, it is now proved that in the static case there is a direct correspondence between the linear density-response formalism in the random-phase approximation and the Bogolyubov chain of equations for equilibrium distribution functions in the pair correlation approximation. It is common knowledge that with a growth of the particle number density an increasingly important role is played by the non-ideality eﬀects, whose account is usually taken within the dynamic local-field correction to the random-phase approximation, considered above. Indeed, while increasing the particle number density, the average distance between plasma particles in the medium decreases and, therefore, the motion of any selected charge turns stronger aﬀected by the local field, created by all neighboring particles. The relation for the linear response functions appearing in (1.42) is written with the local-field corrections Gab (k, ω) as [8] [ ] (0) χee (k, ω) = χ(0) (k, ω) 1 − χ (k, ω) φ ˜ (k)(1 − G (k, ω)) /D, (1.52) ii ii e i [ ] (0) χii (k, ω) = χi (k, ω) 1 − χ(0) (k, ω) φ ˜ (k)(1 − G (k, ω) /D, (1.53) ee ee e (0)

χei (k, ω) = χie (k, ω) = χ(0) e (k, ω)χi (k, ω) φ˜ei (k)(1 − Gei (k, ω)/D, (1.54) where

[ ] D = 1 − χ(0) (k, ω) φ ˜ (k)(1 − G (k, ω) ee ee e [ ] (0) 1 − χi (k, ω)φ˜ii (k)(1 − Gii (k, ω) (0)

−χ(0) ˜2ei (k)(1 − Gei (k, ω)2 . e (k, ω)χi (k, ω)φ 25

(1.55)

CHAPTER 1. GENERAL APPROACHES In case of the two-component plasma, an expression for the longitudinal dielectric function is given by ∑ 1 =1+ φ˜ab (k)χab (k, ω). ε(k, ω)

(1.56)

a,b=e,i

Of particular interest is another case of the one-component plasma consisting of electrons or ions moving on a neutralizing background of opposite charges. This furnishes the neat expression for the longitudinal dielectric function (0)

ε(k, ω) = 1 −

φ˜aa (k)χa (k, ω) (0)

.

(1.57)

1 + φ˜aa (k)Gaa (k, ω)χa (k, ω)

It seems rather reasonable that as soon as the local-field correction Gaa (k, ω) takes into account the non-ideality of the plasma medium, it should somehow be determined by correlation functions of the system. The simplest approximation gives rise to the following analytical expression for the local-field correction [9] Gaa (k, ω) = 1 +

kB T C˜aa (k) , φ˜aa (k)

(1.58)

where C˜aa (k) denotes the Fourier transform of the so-called direct correlation function Caa (r), which can be obtained, for example, within the integral equation method, sketched in the coming Section 1.6.

1.6

Integral equation method

In the physics of nonideal systems various approaches are used to determine the radial distribution function and the method of integral equations really stands out of them for its reliability and fruitfulness. From the viewpoint of statistical physics of many body systems, the complete distribution function is exactly known and expressed by the Gibbs distribution, as already highlighted in Section 1.2. However, 26

Eﬀective Potentials in Non-Ideal Plasma Physics it still carries a great deal of unnecessary information, which is truly unneeded in practice. In order to calculate the thermodynamic characteristics of the medium, it is suﬃcient to know only the pair correlation function or the corresponding radial distribution function. The statistical physics of many body systems at equilibrium asserts that the direct correlation function Cab (r), mentioned in the previous Section 1.5, and the pair correlation function hab (r) are expressed through each other in terms of the Ornstein-Zernike relation ∑ ∫ hab (r) = Cab (r) + nc Cac (r − r′ )hcb (r′ )dr′ , (1.59) c

where hab (r) = gab (r) − 1

(1.60)

with nc being the particle number density of species c. As a matter of fact, equations (1.59) and (1.60) are just definitions, and, therefore, exact both in mathematical and physical senses, but they are still insuﬃcient to determine the three unknown functions. There is another general relation that is also explicit and expresses the radial distribution function via the so-called bridge function Bab (r) in the form ( ) φab (r) gab (r) = exp − + hab (r) − Cab (r) + Bab (r) . (1.61) kB T Equation (1.61) adds nothing new in terms of the possibility of evaluating the radial distribution functions, because it simultaneously introduces another one unknown function Bab (r). It is, therefore, rather instructive to consider several practical closures, proposed in the literature. One of the productive steps is to posit the hypernetted-chain approximation (HNC), which presumes the bridge correction to be zero, i.e. Bab (r) = 0. (1.62) 27

CHAPTER 1. GENERAL APPROACHES In this case, the set of equations (1.59)-(1.61) can be solved numerically for interaction potentials of special kind by using an iterative algorithm. A straightforward comparison with the results of extensive numerical simulation proved, though, that the hypernetted-chain approximation works exceptionally well for weak and moderately coupled systems only. Namely, it was discovered for one-component systems that with decreasing distances between particles, the discrepancy between the HNC approximation on the one hand and the Monte Carlo and the molecular dynamics simulation data on the other reach significant values, which stimulated further research in this direction. It was readily noted that at very large couplings, the radial distribution function becomes similar to that observed for a hard sphere system (HS), which is usually analyzed within the Percus-Yevick approximation [ ( )] φab (r) Cab (r) = gab (r) 1 − exp . (1.63) kB T Equation (1.63) demands for the radial distribution function to vanish in the regions, where the potential of hard spheres turns to infinity, which allows one to numerically solve the resulting set of equations and completely determine all the unknown functions for a HS system. Zeroing the radial distribution function at distances, smaller than the diameter of the interacting particles, is a natural physical requirement of the HS model expressing the inability of particles to penetrate into one another. To understand it, one has to recall the physical meaning of the radial distribution function, which is the probability density of finding two particles at some distance from each other. It should be specially emphasized that the results of computer simulations for HS systems by the Monte Carlo and molecular dynamics methods have completely confirmed the validity of the Percus-Yevick closure (1.63). This success in the description of HS systems [10] provoked further inquiries, putting forward a number of possible approximate solutions for an arbitrary interaction potential. The primary idea was to use the 28

Eﬀective Potentials in Non-Ideal Plasma Physics HS model as a reference by representing the interaction in the system as a sum of two parts, one of which is the potential of hard spheres, and the other is rigidly defined by the real interaction potential. All these methods somehow exploited the fact that the bridge function was weakly dependent on the type of the interaction potential and could, therefore, be taken from the solution of the Percus-Yevick equation for a reference HS system. This was how the simplest reference hypernettedchain (RHNC) approximation was brought to life in [11, 12]. It has to be advocated that all eﬀorts of that period were concentrated on systems with point-like particles and the diameter in a HS reference system was treated a free adjustable parameter, like it was done by Lado [13, 14]. The original insight was to minimize the system free energy at the variation of the radial distribution function. Such an approach was called the modified hypernetted-chain (MHNC) approximation. Finally, Rosenfeld and Ashcroft [15–17] went further and suggested minimizing the free energy by varying all the parameters of the system, which provided simultaneous control over the fulfillment of some additional constraints that appeared in the modified hypernetted-chain approximation. Another practically interesting approach is provided by the RogersYoung closure, which is written in the form [18] ] ( ) [ φab (r) exp[γab (r)f (r)] − 1 exp − , (1.64) gab (r) = 1 + f (r) kB T where γab = hab (r) − Cab (r) and the auxiliary function f (r) is defined as f (r) = 1 − exp(−αr). (1.65) If f (r) = 0 or α = 0, than equation (1.64) is reduced to (1.63), i.e. the Percus-Yevick closure is recovered. In case od f (r) = 1 or α → ∞, equation (1.64) turns into equation (1.61) with Bab (r) = 0, i.e. the HNC closure is recovered as well. The free parameter α in (1.65) is traditionally picked out to assure the thermodynamic consistency between the pressure and the com29

CHAPTER 1. GENERAL APPROACHES pressibility equations of state, which are briefly discussed in Section 2.4.

1.7

Conclusions

This chapter has dealt with the general methods that are to be used in the sequel to study properties of various types of plasmas. The master equation that has been approached from various sides is the generalized Poisson-Boltzmann equation. In particular, simple renormalization theory has been presented, stemming from the illustrating hypothesis that the interaction between two particles is aﬀected by the presence of a third. Moreover, it has been strictly proven that the Bogolyubov chain of equations for equilibrium distribution functions in the pair correlation approximation is fully equivalent to the static limit of the linear density-response formalism in the random-phase approximation. The resulting solution to the master equation has been explicitly written out for the case of a three-component system. Finally, a brief description of the integral equation method has been invoked since it is of extensive use in the due course of this monograph.

30

Chapter 2

Classical plasmas This Chapter 2 is dedicated a classical plasma, which is confined in some fixed volume in space and has a suﬃciently high temperature, so that quantum eﬀects can be completely ignored. Strictly speaking, noncontradictory theoretical description of the plasma medium within the framework of classical statistical mechanics is impossible. Indeed, the real plasma contains at least two sorts of particles, which causes the so-called classical collapse when the probability density of finding two oppositely charged particles grows infinitely at zero separation. It is understood that to fundamentally overcome this problem the quantum eﬀects and the bound states formation must be taken into account at rather short distances. Nevertheless, this Chapter 2 deals with the classical approach to static properties of the plasma under the assumption that the plasma medium retains its local quasi-neutrality, such that the following relation holds ∑ na ea = 0. (2.1) a

It is shown below how the imposition of equality (2.1) guaranties that the expressions, obtained in the classical statistical physics, can be made convergent for a multi-component plasma to avoid the classical collapse of unlikely charged particle species. 31

CHAPTER 2. CLASSICAL PLASMAS However, there exists a theoretical model of plasmas to which the classical statistical physics is surely applicable. It is referred to as a onecomponent plasma (OCP) that incorporates a single sort of particles. It is obvious from what has been said above that the classical collapse is prevented in the OCP by treating only likely charged species. Real systems, that are encountered in various practical situations, can only approach an OCP model. For example, if the electron subsystem of the plasma is strongly degenerate, one can consider a system of ions moving on the neutralizing background of electrons that have completely lost their mobility. Another fruitful pattern is delivered by an electron gas model when attempting to study high-frequency processes such that the ions can be considered immobile for rather short period of time because of the large ion-to-electron mass ratio.

2.1

Plasma parameters

In this Chapter 2 all numerical calculations are performed for the OCP, which consists of the single particle species with the number density n and the electric charge e. The whole system is assumed to be at thermodynamic equilibrium, which is characterized by the temperature T . It is quite easy to apprehend that the medium parameters n and T completely determine all the plasma extensive characteristics, but it turns out that everything depends on their specific combination, which is defined as e2 , (2.2) Γ= akB T where a = (3/4πn)1/3 is the mean interparticle spacing. The dimensionless parameter (2.2) is then called the coupling parameter since it represents the ratio of the average Coulomb interaction energy to the thermal kinetic energy. It can be said that Γ measures the non-ideality of the plasma medium, i.e. an extent to which the interpartcle interactions aﬀect the properties of the OCP. In particular, for Γ ≪ 1 the OCP is considered to 32

Eﬀective Potentials in Non-Ideal Plasma Physics be in a weakly coupled regime, whereas at Γ ≥ 1 the OCP should be treated as a moderately and at Γ ≫ 1 as a strongly coupled system. It has to be immediately mentioned that at Γ = Γm ≈ 175 the first order phase transition has been observed and interpreted as the Wigner crystallization.

2.2

Pseudopotential model

This Section 2.2 presents the very first example of the application of the generalized Poisson-Boltzmann equation to a classical Coulomb system, whose particles interact via the pure Coulomb microscopic potential ea eb φab (r) = , (2.3) r where ea , eb are the charges of the interacting particles, separated by the distance r. To proceed with the classical plasma, the Fourier transform of the Coulomb potential φab (r) is needed hereinafter, which is found as follows ∫∞ 4π φ˜ab (k) = rφab (r) sin(kr)dr. (2.4) k 0

The integral in (2.4) for φab (r) from formula (2.3) actually diverges, but formally multiplying it by exp(−ar), integrating over r and, then, taking the limit a → 0, the Fourier transform of the Coulomb potential acquire the following form φ˜ab (k) =

4πea eb . k2

(2.5)

Substituting the Coulomb potential (2.3) into (1.35), which is simply the generalized Poisson-Boltzmann equation, gives rise to the following Fourier transform of the macroscopic interaction potential ˜ ab (k) = 4πea eb , Φ k 2 + r12 D

33

(2.6)

CHAPTER 2. CLASSICAL PLASMAS where

∑ 4πna e2 1 a = 2 k T rD B a

(2.7)

with rD being called the Debye radius. Taking the inverse Fourier transform, one finally gets the macroscopic potential, which is referred to in the literature as the DebyeH¨ uckel potential ( ) ea eb r Φab (r) = exp − . (2.8) r rD Straightforward comparison of expressions (2.3) and (2.8) clearly demonstrates that the Debye-H¨ uckel potential vanishes exponentially at distances r ≥ rD and this phenomenon is called the screening. At small interparticle separations, though, the screening in the DebyeH¨ uckel potential has absolutely no eﬀect on the interparticle interactions because Φab (r) ≈ φab (r) at r ≪ rD .

2.3

Correlation functions

One of the important correlation functions, introduced in the previous Chapter 1, is the radial distribution function, representing the probability density of finding two particles at a certain distance from each other. Using expression (1.36) with the aid of formula (2.8), it is easy to obtain gab (r) = 1 + hab (r) = 1 −

Φab (r) kB T

( ) r ea eb exp − . =1− rkB T rD

(2.9)

A few remarks has to be made concerning expression (2.9). First of all, the radial distribution function (2.9) is a monotonic function of the distance r, which is the distinguishing trait of a gas-like phase. Secondly, it is a strong physical requirement that the radial distribution function should be zero at very small distances, which is untrue for 34

Eﬀective Potentials in Non-Ideal Plasma Physics formula (2.9) since gab (r) → ±∞ at r → 0. Thus, expression (2.9) incorrectly describes the behavior of the radial distribution function at all interparticle separations, which is, thus, the main deficiency of the Debye-H¨ uckel approximation. To practically observe the picture of the radial distribution function, the HNC approximation, described in Section 1.6, has been solved numerically for the OCP model [19] at diﬀerent values of the coupling parameter Γ and the results are summarized in Figure 2.1. It is seen that when the coupling parameter grows the strong first peak appears in the curve of the radial distribution function to resemble the shortrange order formation in ordinary liquids. In the same Figure 2.1 the radial distribution function (2.9) for Γ = 0.5 is drawn in dot-dashed line to show how the validity of the Debye-H¨ uckel approximation is undermined at small interparticle spacing. It has to be admitted, though, that expression (2.9) is still in very good agreement with the HNC results at rather large distances for an appropriate value of the Coulomb coupling parameter Γ. It is now possible to make a few general comments concerning the behavior of the radial distribution function g(r). It remains smooth at relatively small couplings Γ and rapidly vanishes, which is perfectly reminiscent of a dilute gas structure. As the coupling parameter Γ grows, positions of all neighboring particles turn more correlated due to strengthening of the reciprocal interaction, which results in a modulation of the radial distribution function extending over a few interparticle distances. Although, those oscillations largely fade away with distance, which is characteristic of the short-range order of an ordinary liquid phase. It is also known that when the coupling reaches some critical value Γm , for the OCP it is, for instance, Γm ≈ 175, the system undergoes a first order phase transition, such that the short-range order is spontaneously amplified to turn into the long-range order, typical for a periodic crystalline structure. It is worth mentioning that in all cases the non-ideality eﬀects have enormous impact both an static and dynamic properties of the system under investigation. Another important correlation function, which is of particular inter35

CHAPTER 2. CLASSICAL PLASMAS

Figure 2.1: Radial distribution function g(R) of the OCP as a function of the reduced distance R = r/a, calculated within the HNC approximation for various values of the coupling parameter Γ. Dashed line: Γ = 0.5; dotted line: Γ = 5; solid black line: Γ = 50; dot-dashed line: formula (2.9) at Γ = 0.5. est in the physics of strongly coupled systems, is the static structure factor Sab (k), defined via the radial distribution function gab (r) as follows ∫ √ (2.10) Sab (k) = δab + na nb (gab (r) − 1) exp(ikr)dr, where δab denotes the Kronecker delta. It is remarkable that the static structure factor together with its dynamical counterpart is a quantity that can be directly measured in experiments on x-ray scattering. On substituting equation (2.9) into (2.10) one arrives at the following expression for the static structure factor in the Debye-H¨ uckel approximation Sab (k) = δab + (−1)δab 36

1 2 ), rDa rDb (k 2 + kD

(2.11)

Eﬀective Potentials in Non-Ideal Plasma Physics −1 in which kD = rD stands for the Debye wavenumber inverse to the Debye screening radius rD and the partial Debye radius is defined as √ kB T rDa = . (2.12) 4πna e2a

It is inferred from (2.11) that the static structure factor in the Debye-H¨ uckel approximation is a monotonic function of the wavenumber, which is quite analogous to the behavior of the radial distribution function (2.9). To illustrate the dependence of the static structure factor on the wavenumber in a broad range of couplings, Figure 2.2 is plotted to reproduce the results of the HNC calculations for the OCP [19]. It is again concluded that the first high maximum is observed in the static structure factor at large values of the coupling parameter Γ, and a very satisfactory coincidence is discovered between expression (2.11) and the HNC data in the weakly coupled regime.

2.4

Thermodynamics

Thermodynamic properties, such as the equation of state and the correlation energy, play a significant role in describing the processes taking place in various installations as well as in evaluating their optimal parameters. One of the most important thermodynamic characteristics that determine the properties of the plasma is its internal energy, which is widely used to evaluate the specific heat, the free energy and other thermodynamic functions. The statistical theory of equilibrium systems, consisting of a large number of particles, precisely claims that the internal energy can be evaluated using the relation 3 E = N k B T + UN 2

(2.13)

where N stands for the total number of particles in the system, and the correlation energy UN is obtained via the radial distribution function 37

CHAPTER 2. CLASSICAL PLASMAS

Figure 2.2: Static structure factor S(ka) of the OCP as a function of the reduced wavenumber ka, calculated within the HNC aproximation for various values of the coupling parameter Γ. Dashed line: Γ = 0.5; dotted line: Γ = 5; solid black line: Γ = 50; dot-dashed line: formula (2.11) at Γ = 0.5. as UN = 2πV

∑

∫∞ φab (r)gab (r)r2 dr

na nb

a,b

(2.14)

0

with V being the volume of the system. Direct evaluation of the correlation energy in the Debye-H¨ uckel approximation renders impossible because the corresponding integrals, entering formula (2.14), just diverge. In order to overcome this obstacle the quasi-neutrality condition (2.1) is used to rewrite (2.14) in the following way

UN = 2πV

∑ a,b

∫∞ φab (r) (gab (r) − 1) r2 dr.

na nb

(2.15)

0

On substituting formula (2.9) into (2.15), the correlation energy in 38

Eﬀective Potentials in Non-Ideal Plasma Physics the Debye-H¨ uckel approximation is found as UN = −

kB T V 3 . 8πrD

(2.16)

Similarly to the internal energy (2.14), the plasma pressure is expressed by means of the radial distribution function as ∫ 2π ∑ dφab (r) P = Pid − gab (r)r3 dr na nb 3 dr ∞

a,b

(2.17)

0

with the ideal gas contribution Pid = kB T

∑

na .

(2.18)

a

To derive the equation of state in the Debye-H¨ uckel approximation it is again necessary to stipulate for the quasi-neutrality condition (2.1), which helps to rewrite formula (2.17) as follows ∫ dφab (r) 2π ∑ na nb (gab (r) − 1) r3 dr. P = Pid − 3 dr ∞

a,b

(2.19)

0

With the aid of expression (2.9), formula (2.19) gives rise to P = Pid −

kB T 3 . 24πrD

(2.20)

Finally, the free energy in the Debye-H¨ uckel approximation is derived using the very well known thermodynamic relation ( ) F UN ∂ =− 2 (2.21) ∂T T T and on putting in equation (2.16) yields F = Fid − 39

kB T V 3 , 12πrD

(2.22)

CHAPTER 2. CLASSICAL PLASMAS where Fid stands for the integration constant that denotes the free energy of an ideal system, which cannot be conceived within the framework of classical statistical physics. It is a power of quantum statistical mechanics to derive an exact expression for Fid , which will be used in Section 4.3. For the OCP, formulas (2.16), (2.20) and (2.22) depend on the coupling parameter Γ as follows √ UN 3 3/2 =− Γ , (2.23) N kB T 2 √ P 3 3/2 =1− Γ , (2.24) nkB T 6 √ 3 3/2 F =− Γ , (2.25) N kB T 3 where N denotes the total number of particles in the system. It is seen from the above expressions that the validity of the above expressions is radically restricted in the coupling parameter Γ, since, for example, the pressure of a real system should always remain positive. The above obtained expressions for thermodynamic quantities in the Debye-H¨ uckel approximation are, thus, only valid in the weakly coupled regime. There is no need here to repeat the results of the HNC calculations for the OCP model, since the following fitting formulas were developed [20] in a wide range of the coupling parameter 1 < Γ < 160 UN = aΓ + bΓ1/4 + cΓ−1/4 + d, (2.26) N kB T P UN =1+ , (2.27) nkB T 3N kB T ( ) F = aΓ + 4 bΓ1/4 − cΓ−1/4 N kB T +(d + 3) ln Γ − (a + 4b − 4c + 1.135), (2.28) where a = −0.89752, b = 0.94544, c = 0.17954 and d = −0.80049, respectively. 40

Eﬀective Potentials in Non-Ideal Plasma Physics It is worthwhile mentioning in closing of this Section 2.4 that there is an alternative way to evaluate the equation of state, which for the OCP reads ( ) ∫ ∂n kB T = 1 + n (g(r) − 1) dr. (2.29) ∂P T Formula (2.29) is called the compressibility equation of state and for a genuine radial distribution function it must give, in principle, the same result as formula (2.17). For an approximate radial distribution function, though, formulas (2.29) and (2.17) may produce peculiarly diﬀerent outputs, which is called the thermodynamic inconsistency. For example, for the Debye-H¨ uckel approximation of the OCP the right hand side of expression (2.29) completely vanish, which is an absolutely absurd result corresponding to the greatest extent of the thermodynamic inconsistency. This again totally demonstrates the restraint of the Debye-H¨ uckel approximation, already mentioned above.

2.5

Conclusions

This Chapter 2 has exclusively enlarged on the static properties of classical plasmas. In particular, the Debye-H¨ uckel approximation has been neatly derived from the generalized Poisson-Boltzmann equation, which has resulted in the psedopotential, known as the Debye-H¨ uckel potential. In contrast to the Coulomb potential, the Debye-H¨ uckel potential consistently treats the collective events in the plasma to describe the screening phenomena of the electric field at large interparticle distances. The radial distribution functions in the Debye-H¨ uckel approximation have been shown valid only for relatively small values of the coupling and at rather large interparticle separations. This conclusion has been drawn from the straightforward comparison with the results of the HNC calculations for the one-component plasma. Quite a similar situation has been discovered for the static structure. Thermodynamic characteristics of the classical plasma have been obtained in the Debye-H¨ uckel approximation under the assumption of 41

CHAPTER 2. CLASSICAL PLASMAS the total plasma quasi-neutrality, which has been proved essential for converging of integrals at short distances. Accurate expressions for thermodynamic quantities of the one-component plasma has been presented in the strong coupling regime.

42

Chapter 3

Semiclassical plasmas It is well known that the microscopic potential of the Coulomb interaction φab (r) of two isolated charged particles takes the form of (2.3). In the sequel, an agreement, accepted in the physics of nonideal plasmas, is used to call the interaction potential energy of particles simply a potential. The interaction force due to the Coulomb potential (2.3) has a longrange character, therefore, the interaction of particles in plasmas is greatly influenced by their surrounding, i.e. a significant role is played by the collective events to cause the screening phenomena. In this regard, the pseudopotentials were introduced into the plasma physics, and the Debye-H¨ uckel eﬀective potential (2.8) of the self-consistent field serves as an outstanding example. Another important class of thermodynamic pseudopotentials refers to eﬀective potentials that take into account the quantum eﬀects of diﬀraction and symmetry at short distances. Their derivation is naturally based on a direct match between the Slater quantum-mechanical sum and the classical Boltzmann factor since both represent the probability densities of discovering two particles at some distance from each other. In doing so the true particle interaction potential φ(r) is replaced in the partition function (1.11) by some pseudopotential φ(r, T ) of the particle interaction. In view of the importance of this method for 43

CHAPTER 3. SEMICLASSICAL PLASMAS the subsequent presentation, it is briefly described herein following [21]. Consider a basic system of two particles with an interaction potential φ(r). The probability density of finding two particles at a distance r is determined in the classical Boltzmann statistics by the quantity exp[−φ(r)/kB T ], whereas in the quantum mechanics it is written via the Slater sum S2 (r, T ) of two particles as follows ( ) ∑ Eα 2 6 S2 (r, T ) = 2λ |Ψα (r)| exp − , (3.1) kB T α where Ψα and Eα are the wave functions in the center-of-mass system and the corresponding eigenvalues of the energy of two particles, λ denotes the thermal de Broglie wavelength, defined below. In equation (3.1), the summation extends over all states of the discrete and continuous spectra. Having formally presented the Slater sum in the classical form, it is not diﬃcult to originate the following expression for the eﬀective pairwise potential (pseudopotential) φ(r, T ) = −kB T ln S2 (r, T ).

(3.2)

Thus, the pseudopotential φ(r, T ) is such a potential, which in the classical statistics gives the same particle distribution that is produced by the potential φ(r) in the quantum case at the same system temperature. This is how the quantum statistics is reduced to the classical statistics with a fictitious pseudopotential and it is this approach, which is called semiclassical, and the corresponding medium is referred to as a semiclassical plasma. Note that a somewhat distinct version of the calculation of the pseudopotential was proposed in [22, 23] to treat many-body eﬀects. For a fully ionized plasma the true microscopic potential φ(r) is always the Coulomb potential (2.3). In spite of the notorious fact that in this case the eigenfunctions and the energy eigenvalues of the Schr¨odinger equation are all written in an analytical form, the evaluation of the Slater sum (3.1) is a cumbersome task that can only be 44

Eﬀective Potentials in Non-Ideal Plasma Physics performed numerically, and from formula (3.2) one can obtain tables of the pseudopotential at diﬀerent values of the system temperature. An analysis of those data showed [24, 25] that the eﬀective pairwise potential, which takes into account the quantum eﬀects of diﬀraction and symmetry, can be approximated by a simple analytical expression [ ( )] r ea eb 1 − exp − φ˜ab (r, T ) = r λab ) ( r2 , (3.3) + δae δbe kB T ln 2 exp − 2 λee π ln 2 where λab = ~/(2πµab kB T )1/2 designates the thermal de Broglie wavelength, µab = ma mb /(ma + mb ) stands for the reduced mass of interacting particles of masses ma and mb , respectively. Akin procedure for the calculation of the pseudopotential, stemming from the perturbation theory, was proposed by Kelbg [26]. In particular, the following expression was deduced { ( ) r2 ea eb 1 − exp − 2 φ˜ab (r, T ) = r λab [ ( )]} √ r r 1 − erf , (3.4) + π λab λab where erf(x) is the error function. Later, this pseudopotential was refined by the direct comparison of the Slater sum and its derivative at zero separation with the exact value corresponding to the wave packet of two particles [27–29]. Pseudopotentials of the form (3.3) and (3.4) are widely used to study various physical properties of a semiclassical plasma. All of the above mentioned pseudopotentials actually depend on the medium temperature and density, and, therefore, are not authentic potentials in the traditional mechanistic concept, since in classical mechanics the interaction energy cannot contain the thermodynamic characteristics. To partly justify the introduction of such pseudopotential models in plasma physics it is necessary to restate that in view 45

CHAPTER 3. SEMICLASSICAL PLASMAS of the theoretical description the Coulomb potential (2.3) is responsible for divergences that arise due to its infiniteness at r → 0, and its long-range character at r → ∞. This Chapter 3 focuses on the application of the general methods, described in Chapter 1. to a fully ionized two-component plasma that is treated semiclassically by exploiting (3.3). In particular, Section 3.1 introduces the dimensionless parameters appropriate for the description of the plasma state. In case of a fully ionized two-component hydrogen plasma they simply include the coupling and the density parameters that characterize a need to account for the non-ideality and quantum eﬀects, respectively. The other dependent parameter is the degeneracy parameter that imposes the use of the corresponding statistics, classical or quantum [30]. Section 2.2 is intended for a pseudopotential model that simultaneously take into account the quantum eﬀects of diﬀraction and symmetry at short distances and the screening phenomena at long interparticle separations. The characteristic feature of such a macroscopic potential is that it is finite at the origin and vanishes quickly with distance. A very interesting scenario is observed when the scales of action of the quantum eﬀects and the screening phenomena turn comparable. Section 3.3 concentrates on the study of the radial distribution functions and the corresponding static structure factors. The approaches, employed here at full strength, are the generalized Poisson-Boltzmann equation, the linear density-response formalism and the integral equation method. Thermodynamic properties are investigated in Section 3.4 with the purpose of comparison of various methods with the obtained simple formulas for the equation of state and the correlation energy of the semiclassical two-component plasma. In the weak coupling regime the analytical formulas are developed, mimicking the Debye-H¨ uckel approximation of classical plasmas. The same is done in Section 3.5 but for the electrical conductivity, which is strictly related to the structural characteristics, studied in Section 3.3. The consideration severely relies on the Green-Kubo formula, 46

Eﬀective Potentials in Non-Ideal Plasma Physics in which the Coulomb logarithm is obtained within the ChapmanEnskog method in the second order of the Lagerre polynomials. A comparison with other results and data are also provided in order to verify the validity of the expressions obtained. Main inferences are shortly discussed in Section 3.6.

3.1

Plasma parameters

This Chapter 3 is devoted to a two-component, high temperature plasma consisting of the ions that have the electric charge Ze, the mass mi , and the number density nt as well as of the electrons that have the electric charge −e, the mass me , and the number density ne . The ionic subsystem of such a plasma can be characterized by the average interionic distance )1/3 ( 3 (3.5) a= 4πni and by the dimensionless coupling parameter Γ=

(Ze)2 . akB T

(3.6)

As usual, the coupling parameter measures the ratio of the average potential energy of the Coulomb interaction between ions in the plasma to their average kinetic energy of thermal motion. Again, the case of Γ ≤ 1 corresponds to weakly coupled plasmas, in which the Coulomb interaction can be accounted for by the usual methods of perturbation theory, and Γ > 1 stands for strongly coupled plasmas. The electronic subsystem can be described by the dimensionless density parameter ( )1/3 3 m e e2 rs = (3.7) 4πne ~2 and by the degeneracy parameter ( )2/3 kB T 4 rs ϑ= =2 Z 5/3 , EF 9π Γ 47

(3.8)

CHAPTER 3. SEMICLASSICAL PLASMAS where EF = ~2 (3π 2 ne )2/3 /2me denotes the Fermi energy of electrons and ~ designates the reduced Planck constant. The case of complete electron degeneracy corresponds to the inequality ϑ ≪ 1, whereas at ϑ ≫ 1 the electron gas can be treated within the framework of classical statistical mechanics. In what follows all numerical calculations are carried out for a hightemperature, hydrogen (Z = 1) plasma of the very high density typical for main sequence stars. Such a plasma has a pressure of about 1011 atm and a temperature of about 107 K. In this case both the coupling and the density parameters have numerical values close to unity, for example, for the Sun rs = 0.4 and Γ = 0.1. The degeneracy parameter (3.8) is of the same order of magnitude and, consequently, a significant contribution to the properties of such a plasma is made by both the collective events and the quantum-mechanical eﬀects of diﬀraction and symmetry. It is sometimes useful to have the formulas that accomplish a transition from the dimensionless parameters (3.6) and (3.7) to the ordinary temperature and the number density. In case of a hydrogen plasma, they acquire the following form ne = ni =

T =

3.2

1.6107 × 1024 −3 cm , rs3 3.1571 × 105 K. rs Γ

(3.9)

(3.10)

Pseudopotential model

This Chapter 3 actively exploits the pseudopotential model (3.3), whose Fourier transform is found as φ˜ab (k) =

4πea eb + δae δbe kB T ln5/2 2π 3 2 2 + k λab ) ( 2 2 ) k λee π ln 2 × exp − . 4

k 2 (1

48

(3.11)

Eﬀective Potentials in Non-Ideal Plasma Physics Of particular significance is the derivation of an analytical expression for the pseudopotential, taking into account the quantum eﬀects of diﬀraction and symmetry together with the screening phenomena. For this purpose the solution to the master equation, obtained in Section 1.4, is utilized for the two-component semiclassical plasma to yield [7] { [ 2 1 1 1 4πe ˜ ee (k) = + 4 2 Φ 2 2 2 2 2 ∆ k (1 + k λee ) k rDi (1 + k λee )(1 + k 2 λ2ii ) ] ( ) ( 2 )} 1 k 1 +A 1+ 2 2 exp − − , (3.12) 2 2 2 2 2 4b (1 + k λei ) k rDi (1 + k λii ) [ 1 1 1 + 4 2 2 2 2 2 2 k (1 + k λii ) k rDe (1 + k λee )(1 + k 2 λ2ii ) ] ( 2 )} A 1 k + 2 2 , (3.13) − exp − 2 2 2 2 2 4b (1 + k λei ) k rDi (1 + k λii )

2 2 ˜ ii (k) = 4πZ e Φ ∆

{

2 1 ˜ ei (k) = − 4πZe Φ , ∆ k 2 (1 + k 2 λ2ei )

1 1 + 2 2 2 2 + k λee ) k rDi (1 + k 2 λ2ii ) [ ] 1 1 1 + 2 2 2 2 − k rDe k rDi (1 + k 2 λ2ee )(1 + k 2 λ2ii ) (1 + k 2 λ2ei )2 ( ) ( 2) A 1 k + 2 . 1+ 2 2 exp − 4b rDe k rDi (1 + k 2 λ2ii )

∆=1+

(3.14)

2 (1 k 2 rDe

(3.15)

Here the following notation is used for b = (λ2ee π ln 2)−1 and A = √ kB T ln 2 πb−3/2 /4e2 , and rDa = (kB T /4πna e2a )1/2 denotes the partial Debye screening radius. The pseudopotentials are restored in the configuration space from expressions (3.12)-(3.14) by the inverse Fourier transform ∫ 1 ˜ ab (k) exp (−ikr) . Φab (r) = dk Φ (3.16) (2π)3 49

CHAPTER 3. SEMICLASSICAL PLASMAS It is rather rational to consider some limiting cases of the expressions, obtained right above. Case A. rDa → ∞, then Φab (r) = φab (r),

(3.17)

i.e in the absence of the screening eﬀects, the pseudopotential coincides with the microscopic potential (3.3) as it should be. Case B. λab → 0, then ) ( ea eb r Φab (r) = exp − , (3.18) r rD where

∑ 4πnc e2 1 c = , 2 kB T rD c=e,i

(3.19)

i.e. in the absence of the quantum eﬀects, the pseudopotential coincides with the the Debye-H¨ uckel potential (2.8). Case C. λab → 0, rDa → ∞, then Φab (r) =

ea eb , r

(3.20)

i.e. in the absence of both the collective and the quantum eﬀects, the pseudopotential coincides with the Coulomb potential (2.3). Case D. λab ≪ rDa , then [ ( ) ( )] ea eb r r Φab (r) = exp − − exp − + r rD λab ) ( r2 . (3.21) +δae δbe kB T ln 2 exp − 2 λee π ln 2 The last novel expression diﬀers from the microscopic potential (3.3) by the presence of the term exp(−r/rD ) in the square brackets instead of 1. Figures 3.1-3.3 depict electron-electron, electron-proton and protonproton pseudopotentials (3.16) of a semiclassical hydrogen plasma at 50

Eﬀective Potentials in Non-Ideal Plasma Physics the values of the dimensionless parameters rs = 1.0 and Γ = 0.1. For clarity, in the same figures the microscopic potential (3.3) and the Debye-H¨ uckel potential (2.8) are also plotted. It is easy to see that the macroscopic potentials coincide with the Debye-H¨ uckel potential at large distances and are finite at the origin as the microscopic potentials. It should be emphasized that Figures 3.1-3.3 correspond to the limiting Case D, for which the macroscopic potential curves remain strictly monotonic. In this particular case the proton-proton pseudopotential is hardly diﬀerent from the Debye-H¨ uckel potential due to the very minute value of the thermal de Broglie wavelength of the proton-proton pair. When increasing the plasma number density, local maxima and minima in the curves of the pseudopotentials occur, see Figures 3.4-3.6, which indicates the short-range order formation in the system. This non-monotonic behavior of the macroscopic potential is a result of the competition between the quantum eﬀects and the screening phenomena when the scales of their action, i.e. the de Broglie wavelength and the Debye radius, become comparable. This possibility is realized because the quantum eﬀects, in contrast to the electrostatic forces, cannot be screened. As it is seen from Figure 3.6, the oscillation amplitudes in the proton-proton potential are much smaller than the oscillation amplitudes in the electron-electron potential in view of the condition λpp ≪ λee .

3.3

Correlation functions

In the previous Section 3.2 a procedure has been described to obtain the pseudopotential Φab (r), taking into account both the quantummechanical eﬀects of diﬀraction and symmetry, and the screening phenomena as well. In the pair correlation approximation, see Section 1.3, the radial distribution function gab (r) of particles of sorts a and b is expressed by Φab (r) . (3.22) gab (r) = 1 − kB T At rather small values of the coupling parameter Γ and suﬃciently 51

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.1: Electron-electron pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 1 and Γ = 0.1. Dashed line: the Debye-H¨ uckel potential (2.8); dotted line: the microscopic potential (3.3); solid line: the pseudopotential (3.16).

Figure 3.2: Electron-proton pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 1 and Γ = 0.1. Dashed line: the Debye-H¨ uckel potential (2.8); dotted line: the microscopic potential (3.3); solid line: the pseudopotential (3.16).

52

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.3: Proton-proton pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 1 and Γ = 0.1. Dashed line: the Debye-H¨ uckel potential (2.8); dotted line: the microscopic potential (3.3); solid line: the pseudopotential (3.16).

Figure 3.4: Electron-electron pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel potential (2.8); solid line: the pseudopotential (3.16).

53

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.5: Electron-proton pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel potential (2.8); solid line: the pseudopotential (3.16).

Figure 3.6: Proton-proton pseudopotential of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel potential (2.8); solid line: the pseudopotential (3.16).

54

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.7: Electron-electron radial distribution function of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 1 and Γ = 0.1. Dashed line: the Debye-H¨ uckel theory (2.9); solid line: the present result (3.22). large values of the density parameter rs , the radial distribution function (3.22) retains its monotonicity at all separations, as displayed in Figure 3.7 for the electron-electron radial distribution function at rs = 1.0 and Γ = 0.1. In case of small Γ when λab ≪ rDa , with the aid of (3.21), the following analytical expression for the radial distribution function is recovered in the weakly coupled regime [31, 32] [ ( ) ( )] r r ea eb exp − − exp − − gab (r) = 1 − rkB T rD λab ( ) r2 −δae δbe ln 2 exp − 2 . (3.23) λee π ln 2 With the growth of Γ, local maxima and minima occur in the curve of the electron-electron radial distribution function, which implies the short-range order formation in the system. This inference follows directly from the physical meaning of the radial distribution function as the probability density of finding two particles at some distance from 55

CHAPTER 3. SEMICLASSICAL PLASMAS each other. Such a nonmonotonic behavior takes place only if the following inequality holds √ 1 πrs Γ≥ . (3.24) 2 6 It is assured by (3.24) that if the dimensionless density parameter rs is suﬃciently small, the short-range order formation in the plasma becomes possible even if the coupling parameter Γ is not too large. This is demonstrated in Figures 3.8-3.10, where the electron-electron, electron-proton and proton-proton distribution functions are drawn at rs = 0.5 and Γ = 1.0. Quite similar to the pseudopotential behavior, the oscillating amplitude in the proton-proton radial distribution function is significantly smaller than that in the corresponding electron-electron radial distribution function, because the proton-proton thermal de Broglie wavelength is too small in magnitude as compared to all other lengths. It is salient from Figure 3.9 that the electron-proton radial distribution function preserves its monotonicity even in this case. It should be noted that the short-range order formation is a result of the interplay between the quantum-mechanical eﬀects and the screening phenomena, when the scales of their action becomes comparable, as reflected by (3.24). As the coupling parameter Γ grows the non-ideality eﬀects come to play a significant role, which can be handled within the integral equation method, outlined in Section 1.6. In particular, the HNC approximation has been solved with the microscopic potential (3.3) in order to draw Figure 3.11, which represents the radial distribution functions of the semiclassical hydrogen plasma at rs = 2 and Γ = 50. The electron-electron and the electron proton radial distribution functions reveal highly pronounced non-monotonic behavior, whereas the protonproton radial distribution function remains almost smooth as it is the case for the weakly coupled regime, catched by the analytical formula (3.22). The classical theory of linear density-response formalism, underlying Section 1.5, provides the following relation between the dynamic 56

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.8: Electron-electron radial distribution function of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel theory (2.9); solid line: the present result (3.22).

Figure 3.9: Electron-proton radial distribution function of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel theory (2.9); solid line: the present result (3.22).

57

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.10: Proton-proton radial distribution function of a semiclassical hydrogen plasma as a function of the reduced distance at rs = 0.5 and Γ = 1.0. Dashed line: the Debye-H¨ uckel theory (2.9); solid line: the present result (3.22).

Figure 3.11: HNC results for radial distribution functions of a hydrogen semiclassical plasma as a function of the reduced distance at rs = 2 and Γ = 50. Solid line: electron-electron; dashed line: electron-proton; dotted line: proton-proton.

58

Eﬀective Potentials in Non-Ideal Plasma Physics structure factors Sab (k, ω) and the response functions χab (k, ω) Sab (k, ω) = −

kB T Imχab (k, ω). πω

(3.25)

In turn, the static structure factor Sab (k) is related to the dynamic structure factor as ∫ +∞ 1 Sab (k) = √ dω χab (k, ω). (3.26) na nb −∞ With the help of (3.25) and (3.26) the fluctuation-dissipative theorem proves that kB T Sab (k) = − √ χab (k, 0). (3.27) na nb Collecting expressions (1.43)-(1.45) for the static response functions χab (k, 0) from Section 1.5, one finally obtains [7] √ na nb ˜ Sab (k) = δab − Φab (k), (3.28) kB T ˜ ab (k) stands for the Fourier transforms of the pseudopotentials where Φ (3.12)-(3.14) and δab is again the Kronecker delta. When the inequality λab ≪ rD is fulfilled, formula (3.28) is considerably simplified and, in the case of a two-component plasma, gives rise to Sab (k) = δab + (−1)δab

1 rDa rDb (1 +

k 2 λ2ab )(k 2

2) + kD

.

(3.29)

Note that formula (3.29) is distinct from formula (2.11) by the factor (1 + k 2 λ2ab ) in the denominator of the second term on the right hand side that accounts for the quantum eﬀects of diﬀraction. Figures 3.12-3.15 compare the values of the electron-electron static structure factors (3.29) with the results of [20], in which the linear density-response formalism was used with the local-field corrections and the degeneracy of the electrons was taken into account by means 59

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.12: Electron-electron static structure factors of a semiclassical hydrogen plasma as a function of the reduced wavenumber at ϑ = 10 and Γ = 0.1. Solid line: formula (3.28); dashed line: the results of [20]. of the Fermi-Dirac distribution. In the present approach, the exchange eﬀects are only taken into account in the interaction model, namely, in the second term of the microscopic potential (3.3). A thorough analysis allows one to conclude that the electron-electron static structure factors are in good agreement for small values of Γ, and the electron-proton static structure factors practically coincide in a broad range of the plasma parameters. This means that as the coupling parameter Γ increases the corrections due to the local-field become significant for the semiclassical plasma. The same is applicable to an increase in ϑ when the many-body quantum eﬀects play a crucial role. It is worthwhile mentioning that an increase in the degeneracy parameter ϑ can also be handled within the semiclassical approach by incorporating the quantum statistical eﬀects, like it was done, for example, in [30]. In this case neat analytical approximation can be developed as well but a detailed consideration of [30] lies out of the scope of the present monograph. Nevertheless, it should be stressed that a much better agreement is achieved with the results of [20], especially for the equation of state and the correlation energy. 60

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.13: Electron-electron static structure factors of a semiclassical hydrogen plasma as a function of the reduced wavenumber at ϑ = 1 and Γ = 0.1. Solid line: formula (3.28); dashed line: the results of [20].

Figure 3.14: Electron-electron static structure factors of a semiclassical hydrogen plasma as a function of the reduced wavenumber at ϑ = 1 and Γ = 0.5. Solid line: formula (3.28); dashed line: the results of [20].

61

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.15: Electron-electron static structure factors of a semiclassical hydrogen plasma as a function of the reduced wavenumber at ϑ = 0.1 and Γ = 0.1. Solid line: formula (3.28); dashed line: the results of [20]. The above discussed again supports the necessity to simultaneously harness the non-ideality eﬀects and strong plasma degeneracy when the plasma density considerably rises.

3.4

Thermodynamics

One of the most important thermodynamic characteristics that governs the properties of the plasma is its internal energy, which is widely used to evaluate the specific heat, the free energy and other thermodynamic quantities, as it has been already shown in Section 2.4. The statistical theory of equilibrium systems proves that the internal energy can be calculated using the relation 3 E = N k B T + UN , 2

(3.30)

in which the correlation energy UN is expressed through the radial 62

Eﬀective Potentials in Non-Ideal Plasma Physics distribution function as UN = 2πV

∫∞ ∑ 0

a,b

[ na nb

] ∂φab (r) φab (r) − T gab (r)r2 dr. ∂T

(3.31)

It is necessary to make a comment concerning the appearance of formula (3.31), which is basically diﬀerent from formula (2.14). In complete accord with the physical sense of the semiclassical approach one has to substitute a real interaction potential in the partition function (1.11) by a pseudopotential. On the one hand, the Helmholtz free energy is expressed via the partition function Z as F = −θ ln Z and, on the other hand, the correlation energy is related to the free energy with the help of (2.21). This actually results in the appearance of the second term in the square brackets of (3.31), because in order to obtain the correlation energy one has actually to take a derivative of the partition function over the temperature T and in this case the microscopic potential is temperature dependent. In case of small values of Γ the integration in formula (3.31) can be accomplished analytically using the radial distribution functions (3.23) 2 , including, in particular, and dropping out terms of the order λ2ab /rD the quantum eﬀects of symmetry, the correlation energy is obtained as follows 15πV ∑ na nb e2a e2b kB T V + λab . (3.32) UN = − 3 4 kB T 8πrD a,b

The first term in (3.32) corresponds to the Debye-H¨ uckel contribution, whereas the second one appears due to the ring diagrams in the quantum group expansion [33]. As expected, the weakening of the interaction between charged particles due to quantum eﬀects of diﬀraction is responsible for an increase in the correlation energy of the plasma. Figures 3.16 and 3.17 indicate the results of calculations of the correlation energy of a semiclassical hydrogen plasma as a function of the coupling parameter Γ at the fixed values of ϑ = 10 and ϑ = 5. In those figures, the solid line shows the results of calculations via formula (3.31) using the radial distribution functions (3.22), dashed line represents the 63

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.16: Correlation energy of a semiclassical hydrogen plasma as a function of the coupling parameter Γ at ϑ = 10. Solid line: present results [31]; dashed line: the Debye-H¨ uckel theory (2.16); solid triangles: [34]; open circles: [35]; solid squares: [36]. Debye-H¨ uckel theory, solid triangles correspond to the linear densityresponse theory with the local field corrections, open circles are based on the known results of quantum statistics using the Pad´e approximation, solid squares show the HNC results for the same microscopic potential (3.3). It is rather remarkable that a very good agreement is achieved between the sophisticated approaches, mentioned above, and the simplest approximation, proposed here. With decrease of the degeneracy parameter ϑ an inflection point appears in the curve of the correlation energy. This is due to the increasing role of the quantum symmetry eﬀects, which make an opposite contribution to that of the diﬀraction eﬀects. The equation of state is still evaluated via formula (2.17) because, in contrast to (3.31), one has to virtually take the derivative of the partition function over the system volume, of which the microscopic potential is surely independent. For rather small values of Γ using the radial distribution functions 2 that include the (3.23) and dropping out terms of the order λ2ab /rD 64

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.17: Correlation energy of a semiclassical hydrogen plasma as a function of the coupling parameter Γ at ϑ = 5. Solid line: present results [31]; dashed line: the Debye-H¨ uckel theory (2.16); solid triangles: [34]; open circles: [35]; solid squares: [36]. quantum eﬀects of symmetry, the following expression for the pressure is restored in the weakly coupled regime P = Pid −

3π ∑ na nb e2a e2b kB T + λab . 3 2 kB T 24πrD

(3.33)

a,b

In this expression the first term corresponds to an ideal gas pressure, and the second is the classical Debye-H¨ uckel approximation (2.20). It is natural that the quantum eﬀects of diﬀraction lead to a positive contribution to the pressure, see the third term in equation (3.33), because they virtually weaken the electrostatic interaction. Figures 3.18 and 3.19 graphically exhibit the equation of state of a semiclassical hydrogen plasma at the fixed values of the density parameter rs = 1.0 and rs = 2.0. As it can be seen from these figures, the results, obtained on the basis of (2.17), agree quite well with the DebyeH¨ uckel approximation (2.20) at small values of the coupling parameter Γ, which follows directly from equation (3.33). Up until Γ ∼ 0.5 65

CHAPTER 3. SEMICLASSICAL PLASMAS

Figure 3.18: Equation of state of a semiclassical hydrogen plasma as a function of the coupling parameter Γ at rs = 1.0. Solid line: present results [32]; dashed line: the Debye-H¨ uckel theory (2.20); open triangles: [37]; solid squares: [36]; open circles: [38]; solid triangles: [39]; solid squares: [33, 40]; open squares: [41]. there is reasonable agreement with the virial expansion (solid squares) and semiclassical molecular dynamics (open circles). At Γ ∼ 1 there are small discrepancies with the results of computer simulations (solid squares, solid triangles, open triangles and open squares). It should be noted that the results, depicted as solid squares, are considered the most reliable and are also in reasonable agreement with the data, presented in the graphs.

3.5

Electrical conductivity

Electrical conductivity is one of the key quantities that determines the transport properties of plasmas. Knowledge of the transport coefficients, and the conductivity in particular, is absolutely necessary for the construction of a hydrodynamic model of plasmas, within which one can explore various electrodynamic, including nonlinear wave, processes in isotropic and magnetized media. 66

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.19: Equation of state of a semiclassical hydrogen plasma as a function of the coupling parameter Γ at rs = 2.0. Solid line: present results [32]; dashed line: the Debye-H¨ uckel theory (2.20); open triangles: [37]; solid squares: [36]; open circles: [38]; solid triangles: [39]; solid squares: [33, 40]; open squares: [41].

For a matter to be conductive it must contain free charge carriers. Thus, an electrical conductivity is defined as an intrinsic property of the medium that quantifies how a particular material hampers the flow of the electric current. The electrical conductivity of plasmas is mostly due to electrons that scatter on ions, neutral atoms and molecules while subject to an external electric field. It is known that in plasmas the electrical conductivity varies tremendously with the number density of charged particles. It is well known that the transport coeﬃcients are calculated in the kinetic theory, based on the Boltzmann equation, which, with respect to the plasma, is written with either the Landau or the Balescu-Lennard collision integrals. Two perturbative approaches exist in theory to solve the kinetic equation. The first was proposed by Chapman and Enskog and actually resides on the expansion of the solution in the orthonormal set of the Laguerre polynomials. The second is the Grad method of moments, in which the solution is expanded in series of the orthogonal 67

CHAPTER 3. SEMICLASSICAL PLASMAS Hermite-Chebyshev polynomials. Herein the Chapman-Enskog method is used in the second order approximation of the Laguerre polynomials. In this case, the normalized electrical conductivity σ ∗ = σ/ωpe takes the form (

∗

σ = 1.93

3π 2

)1/2

1 4πΓ3/2 LC

,

(3.34)

where ωpe = (4πne e2 /me )1/2 is the plasma frequency, and LC denotes the so-called generalized Coulomb logarithm. Using the Green-Kubo formula, the expression for the Coulomb logarithm LC can be written in terms of the static structure factors Sab (k) (3.28) of Section 3.3 as [42] ∫∞ LC = 0

[ ] 2 (k) dk See (k)Sii (k) − Sei . k (1 + k 2 λ2ei )2

(3.35)

Thus, formulas (3.34) and (3.35), along with expression (3.28), constitute a very eﬀective scheme for calculating the electrical conductivity of a two-component semiclassical plasma. Under the condition λab ≪ rD , the substitution of (3.29) into (3.35) leads to the simplest formula for the Coulomb logarithm ( ) rD 1 LC = ln − . (3.36) λei 2 Formula (3.36) is a semiclassical analog of the classical formula of Spitzer [43], in which the Landau parameter e2 /kB T should be replaced by the thermal de Broglie wavelength λei . Figures 3.20 and 3.21 illustrate the normalized electrical conductivity (3.34) of a semiclassical hydrogen plasma with the Coulomb logarithm (3.35) as a function of the coupling parameter Γ at the fixed values of the density parameter rs and make a comparison with results of other approaches. 68

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 3.20: Normalized electrical conductivity of a hydrogen semiclassical plasma σ ∗ as a function of the coupling parameter Γ at rs = 0.4. Solid line: present results [44]; dashed line: analytical formulas (3.34) and (3.36); open squares: [42]; solid circles: [45]; open triangles: [46]; solid diamonds: [47].

Figure 3.21: Normalized electrical conductivity of a hydrogen semiclassical plasma σ ∗ as a function of the coupling parameter Γ at rs = 1.0. Solid line: present results [44]; dashed line: analytical formulas (3.34) and (3.36); open squares: [42]; solid circles: [45]; open triangles: [46]; solid diamonds: [47]; open circles: [48]. 69

CHAPTER 3. SEMICLASSICAL PLASMAS An approach, analogous to the one proposed here, was used in [42], in which the static structure factors were calculated on the basis of the HNC approximation with the microscopic potential (3.3) and, then, substituted into expressions (3.35) and (3.34), respectively. The results of [42] lie systematically below the values of the electrical conductivity, obtained within the present scheme, and decrease monotonically with the growth of the coupling parameter Γ. It should, first, be noted that there is a minimum in the electrical conductivity curve, also found in [45–47], and, secondly, the analytical formula, based on the expressions (3.35) and (3.34), well describes the results of [42, 45, 46] in the region of small values of the coupling parameter Γ. The appearance of a minimum in the electrical conductivity has a clear physical meaning. Indeed, in case of a fully ionized plasma the electrical conductivity is mainly governed by the electron-ion collisions. The electron-ion attractive electrostatic forces have a characteristic range of action rD in the plasma, whereas the repulsive forces due to the quantum eﬀects have a characteristic range of action λei . Therefore, the minimum position, corresponding to the weakest scattering of the electrons by the protons in the hydrogen plasma, is precisely determined by the equality of those characteristic ranges, rD ∼ λep , which provides the minima location at √ πrs . (3.37) Γmin ≈ 3 To comprehensively test estimate (3.37) the normalized electrical conductivity (3.34) is compiled in Table 3.1 at diﬀerent values of the coupling Γ and the density rs parameters. Its careful analysis confirms that formula (3.37) describes quite well the minimum position of the electrical conductivity for suﬃciently small values of the density parameter rs . Note that the approach, presented above, corresponds to the classical consideration of the electrical conductivity in a constant external electric field when all dynamical eﬀects are completely discounted. 70

Eﬀective Potentials in Non-Ideal Plasma Physics

Table 3.1: Normalized electrical conductivity σ ∗ of a hydrogen semiclassical plasma for various values of the density rs and the coupling Γ parameters. rs Γ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

= 1.3 σ∗ 11.2297 5.3204 3.5796 2.7668 2.3021 2.0042 1.7986 1.6493 1.5369 1.4499 1.3812 1.3261 1.2813 1.2446 1.2143 1.1893 1.1685 1.1513 1.1372 1.1255

rs Γ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

= 1.0 σ∗ 11.1835 5.6885 3.8791 3.0340 2.5517 2.2435 2.0319 1.8794 1.7656 1.6785 1.6106 1.5571 1.5145 1.4805 1.4534 1.4318 1.4147 1.4015 1.3915 1.3842

rs Γ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

71

= 0.7 σ∗ 12.7167 6.2806 4.3748 3.4867 2.9829 2.6640 2.4481 2.2953 2.1840 2.1013 2.0394 1.9930 1.9584 1.9331 1.9153 1.9035 1.8968 1.8942 1.8951 1.8991

rs Γ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

= 0.4 σ∗ 14.4776 7.5201 5.4673 4.5256 4.0061 3.6909 3.4898 3.3591 3.2747 3.2226 3.1938 3.1824 3.1843 3.1967 3.2175 3.2451 3.2782 3.3159 3.3576 3.4024

CHAPTER 3. SEMICLASSICAL PLASMAS

3.6

Conclusions

The pseudopotential model of interparticle interactions in semiclassical plasmas has been put forward in this Chapter 3 to take into account both the collective events and the quantum-mechanical eﬀects of diﬀraction and symmetry. The original idea has been to start from the microscopic potential that takes into account the quantum eﬀects and to apply the generalized Poisson-Boltzmann equation to incorporate the screening phenomena as well. It turns out that the macroscopic potential, derived in such a way, remains finite at the origin and vanishes with distance with the characteristic length, defined by the Debye radius. As it has been demonstrated in Chapter 1, the same result is achieved from the linear density-response formalism in the pair correlation approximation. Based on the proposed pseudopotential model, the correlation functions have been studied of a fully ionized hydrogen plasma and the possibility of the short-range order formation has been demonstrated. The domain of the plasma parameters has been specified, in which such a formation turns possible and it has been understood that it is a result of the competition between the quantum-mechanical eﬀects and the screening phenomena when their scales of action become comparable. The oscillations in the proton-proton radial distribution function have been found less pronounced since the corresponding wavelength always remains much smaller than the screening Debye radius. In the strongly coupled regime the radial distribution functions of a fully ionized hydrogen plasma have revealed salient non-monotonic behavior as a function of distance. The analytic expression has been obtained for the static structure factors of semiclassical two-component plasmas and detailed comparison has been provided with the results of other approaches. The correlation energy and the equation of state of a fully ionized hydrogen plasma have been evaluated and a good agreement has been found with other theoretical approaches and computer simulation data. In the weakly coupled regime the analytic expressions for the corre72

Eﬀective Potentials in Non-Ideal Plasma Physics lation energy and the pressure of semiclassical two-component plasmas have been derived, which fundamentally correspond to the DebyeH¨ uckel approximation of classical plasmas, discussed in the previous Chapter 2. The electric conductivity of a fully ionized hydrogen plasma has been calculated and compared with the results of other authors. The minimum in the curve of the electrical conductivity has been discovered and its physical justification has provided the estaimate of the minimum position that has been numerically verified. The important analytical expression for the generalized Coulomb logarithm has been obtained for the weakly coupled regime.

73

Chapter 4

Partially ionized plasmas Investigations of various properties of partially ionized plasmas have been under way for almost the century and are still of huge interest. The reason is that partially ionized plasmas are encountered in astrophysical objects, such as giant planets and star atmospheres, and also appear in the context of various technological problems of industrial plasma engineering. This strongly necessitates our exact knowledge of various properties of partially ionized plasmas in a wide range of their parameters. The most consistent approach to studying equilibrium properties of partially ionized plasmas consists in utilizing the physical picture, which essentially relies on application of the quantum statistical mechanics to a Coulombic system of many particles [49–54]. In this framework bound states of electrons and nuclei, i.e. atoms and molecules, arise quite naturally and are not considered as independent species. Such an approach, however, faces very well known mathematical difficulties, thereby making the so-called chemical picture of wider use [55–60]. Strictly speaking, a chemical model calculation of thermodynamic quantities is only justified for an ideal partially ionized plasma, in which interparticle interactions are totally discarded. In this particular case a partially ionized plasma is treated a mixture of ideal gases of electrons, ions and neutrals. Therefore, the Helmholtz free energy of 74

Eﬀective Potentials in Non-Ideal Plasma Physics the whole system is represented by an additive sum of the corresponding free energies of separate plasma components, which is then called the linear mixing rule. All of the aforesaid presumes that at high temperatures and low pressures the system obeys ordinary classical laws. For instance, the specific thermal capacity at constant volume does not depend on temperature at all and, in case of electrons, i.e. fermions with the half-integer spin, the spin susceptibility remains inversely proportional to the temperature according to the Curie law. With pressure increase and/or temperature decrease, the plasma undergoes a transition to a weakly non-ideal state, in which the role of interparticle interactions turns more pronounced. Nevertheless, the linear mixing rule is again applied within standard chemical models, i.e. the free energy of the system is still considered additive, thus treating the charged and neutral constituents of the plasma as statistically independent subsystems. Unfortunately, such a procedure implies that a standard expression for the free energy of partially ionized plasmas is not thoroughly deduced but actually postulated by formally inserting corresponding contributions. It is, thus, inferred that chemical models, developed earlier, have at least two serious disadvantages. First of all, an expression for the free energy is not precisely derived but constructed by means of the linear mixing rule. It was shown that this might result in some thermodynamic inconsistencies [61–63] and success is known to strongly depend on an adequate choice of the corresponding contributions. Secondly, only thermodynamical properties stay in focus, since postulation of the free energy expression hinders any further investigations of other plasma characteristics such as, for instance, transport coeﬃcients. Developed in this Chapter 4 is a self-consistent chemical model of partially ionized plasmas, which is radically deprived of the above mention drawbacks. Section 4.1 defines the dimensionless parameters of a partially ionized plasma of hydrogen that contains electrons, protons and neutral atoms. In Section 4.2 the macroscopic potentials are studied under the as75

CHAPTER 4. PARTIALLY IONIZED PLASMAS sumption that the ionization equilibrium is determined by the Saha equation. An expression for the system free energy is obtained in Section 4.3 by virtually engaging the generalized Poisson-Boltzmann equation and its minimization provides the ionization equilibrium in the system. Occasionally, a systematic approach is developed to determine a characteristic length of the short-range order in partially ionized plasma subsystems. In contrast to ordinary chemical models, its self-consistent version allows one to observe the correlation functions in a broad range of plasma parameters, which is realized in Section 4.4. The gist of the proposed chemical model is an eﬀective potential taking into account collective events in the medium, which enables one to calculate both thermodynamic and transport properties of partially ionized plasmas, which is respectively done in Section 4.5 and 4.6. Section 4.7 concludes this Chapter 4 with a brief discussion of the main results.

4.1

Plasma parameters

Of interest in the sequel is a partially ionized hydrogen plasma consisting of three particle species: free electrons with the number density ne , free protons with the number density np and hydrogen atoms (neutrals) with the number density nn . A typical partially ionized plasma may also contain hydrogen molecules and other charged clusters [64, 65] but their presence is omitted everywhere below, thus, restricting the present consideration to the high-temperature domain, where the molecule formation is tackled by the thermal dissociation process as proved by the extensive numerical simulations [66]. To characterize the state of the plasma medium, the Wigner-Seitz radius is defined as ) ( 3 1/3 (4.1) a= 4πn 76

Eﬀective Potentials in Non-Ideal Plasma Physics and, then, the density parameter is introduced as ( rs =

3 4πn

)1/3

me e2 a , = 2 ~ aB

(4.2)

where n = np +nn is the total number density of protons in the system, aB = ~2 /me e2 designates the first Bohr radius, ~ signifies the reduced Planck constant, me and −e stand for the electron mass and electric charge, respectively. To describe the strength of interparticle interactions, the Coulomb coupling parameter is evaluated as Γ=

e2 , akB T

(4.3)

where kB denotes the Boltzmann constant and T is the plasma temperature. It is noteworthy that the real coupling between particles in the system is always less in magnitude than that defined in (4.3) since the definition of the Coulomb coupling parameter contains the Wigner-Seitz radius which is, without fail, less than the mean interparticle spacing. It is obvious from the pure physical point of view that to describe the state of partially ionized plasmas it is suﬃcient to know three parameters, i.e. the total number density, the system temperature and the ionization degree. It should be noted, though, that knowledge of the dimensionless density and coupling parameters is enough since the ionization equilibrium is determined by physical conditions in the medium. Generally speaking, the plasma composition is mainly governed by two competing processes, i.e. thermal ionization and recombination. However, at thermal equilibrium the ionization degree is independent of the details of those processes and can principally be evaluated from the thermodynamical point of view by minimizing the system free energy. Such an approach is consistently implemented below beyond the linear mixing rule. 77

CHAPTER 4. PARTIALLY IONIZED PLASMAS Since the system is assumed abandoned by hydrogen molecules, the temperature of the medium should exceed the dissociation energy of hydrogen, which is equal to 4.477eV. In view of formula (3.10), this requires the fulfillment of the inequality rs Γ < 6.077.

4.2

Pseudopotential model

As usual, in order to initiate the application of the generalized Poisson-Boltzmann equation, it is necessary to make an appropriate choice of the microscopic potentials that describe the interaction between all plasma components. The microscopic potentials for the neutral component are chosen for a hydrogen plasma in the form [67] ) ( ) ( 1 2r 2 1 + exp − , (4.4) φpn (r) = −φen (r) = e r aB aB ( √ ) e2 2r φnn (r) = exp − , r aB

(4.5)

whereas the charged component is assumed to interact via the Coulomb potential φee (r) = φpp (r) = −φep (r) =

e2 . r

(4.6)

A few remarks has to be made concerning the choice of the microscopic potentials. First of all, it was thought upon derivation of expressions (4.4) and (4.5) that all the atoms are in their ground states and, thus, all excited states are completely omitted. Furthermore, polarization phenomena, associated with the deformation of electronic clouds in atoms, such as those in the Lennard-Jones potential, are dropped out as well. This comes from the fact that formulas (4.4) and (4.5) represent the average atomic fields treating the atomic electron as an 78

Eﬀective Potentials in Non-Ideal Plasma Physics electron cloud with the probability density, determined by the unperturbed ground state wave function and further integration is meant over the atomic electron position. While a more realistic atom-atom microscopic potential displays a potential well, the same microscopic potential (4.5) is monotonic but still capable of catching the repulsion between atoms at very short distances, which may validate its usage for relatively high densities of interest herein. To prove for a successful accomplishment of the solution, obtained in Section 1.4, it is essential for microscopic potentials to have Fourier transforms. Unfortunately, this is not the case for the van der Waals interactions, induced by quantum fluctuations of atomic dipole moments, due to divergence at small interatomic distances, as evidenced by the Lennard-Jones potential. It has to be advocated that a physically meaningful result for a microscopic potential definitely implies the existence of its Fourier transform. Thus, a stricter derivation of the microscopic potentials is in need to correctly describe interatomic interactions at all separations, which is, thus, one of the obvious provisions for future improvements. The Fourier transforms of the microscopic potentials (4.4)-(4.6) are simply found as φ˜pn (k) = −φ˜en (k) =

φ˜nn (k) =

4πe2 (k 2 + 8/a2B ) , (k 2 + 4/a2B )2

4πe2 , (k 2 + 2/a2B )

φ˜ee (k) = φ˜pp (k) = −φ˜ep (k) =

(4.7)

(4.8) 4πe2 . k2

(4.9)

The above expressions are insuﬃcient to obtain numerical results for the macroscopic potentials since what remains unknown is the exact proportion between the number densities of free electrons and protons on the one hand and the number density of atoms on the other. 79

CHAPTER 4. PARTIALLY IONIZED PLASMAS For the sake of simplicity it is reasonable to use the ionization equilibrium, provided by the Saha equation [49, 68] ( )3/2 ( ) ne np m e kB T I =2 exp , (4.10) nn 2π(1 + me /mp )~2 kB T where I = −me mp e4 /2(me + mp )~2 stands for the ground state energy of the hydrogen atom. It has to be stressed that the Saha equation is only valid for a system of non-interacting particles, which is not the case in the present scrutiny. The non-ideality of the plasma is known to be responsible for lowering of the ionization energy of hydrogen atoms right up to the socalled Mott transition when it actually turns zero. A noncontradictory description of the lowering of the ionization potential is the central goal of the self-consistent chemical model, developed below. Now the pseudopotential can be restored from the solution to the master equation (1.37)-(1.39), supplied by Section 1.4. Figures 4.1-4.3 display the macroscopic potentials at Γ = 0.8 and rs = 5 as compared to the microscopic potentials (4.4)-(4.6) and the Debye-H¨ uckel potential (2.8). Accurate analysis drives us to conclude the following. (i) The macroscopic interaction potential of charged particles remains monotonic in a wide range of plasma parameters and systematically lies in between the Coulomb and the Debye-H¨ uckel potentials. (ii) The atom-atom interaction is practically not aﬀected by the presence of the charged component. (iii) The macroscopic interaction potential between the charged particle and the atom reveals a nonmonotonic behavior, which turns especially evident while increasing the coupling parameter Γ. Such a non-monotonic behavior with the appearance of a local extremum can be explained as follows. According to the microscopic potential (4.4), an atom attracts electrons, which, thus, form an electron cloud around it. And, although, the atom itself repels a proton, the presence of that electron cloud can simply cause an eﬀective attraction of protons at certain distances from the atom. This becomes physically obvious from the renormalization theory for the eﬀective potential, presented in Section 1.1. Note that the attrac80

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.1: Electron-electron macroscopic potential of a partially ionized hydrogen plasma as a function of the reduced distance at rs = 5 and Γ = 0.8. The ionization equilibrium is provided by the Saha equation (4.10). Solid line: macroscopic potential [69]; dashed line: the Debye-H¨ uckel potential (2.8); dotted line: the Coulomb potential (4.6). tion of an electron by an atom may ultimately result in the formation of an H− ion. Finally, some limiting cases of formulas (1.37)-(1.39) for the eﬀective potentials are of utmost significance. In case of highly ionized plasmas nn ≪ ne , np , expressions (1.37)(1.39) simplify to give [70] ( ) r e2 exp − , Φee (r) = Φpp (r) = −Φep (r) = r rD

(4.11)

( ) ( ) e2 1 1 2r Φpn (r) = −Φen (r) = + exp − + 1 − β aB r aB ( ) ( ) βe2 2r β(2 − β)e2 r exp − − exp − , (4.12) (1 − β)2 r aB (1 − β)2 r rD 81

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Figure 4.2: Proton-atom macroscopic potential of a partially ionized hydrogen plasma as a function of the reduced distance at rs = 5 and Γ = 0.8. The ionization equilibrium is provided by the Saha equation (4.10). Solid line: macroscopic potential [69]; dotted line: microscopic potential (4.4).

Figure 4.3: Atom-atom macropotential of a partially ionized hydrogen plasma as a function of the reduced distance at rs = 5 and Γ = 0.8. The ionization equilibrium is provided by the Saha equation (4.10). Solid line: macroscopic potential [69]; dotted line: microscopic potential (4.5). 82

Eﬀective Potentials in Non-Ideal Plasma Physics ( √ ) ( ) e2 2r β 2 (2 − β)2 e2 r Φnn (r) = exp − + exp − − r aB (1 − β)4 r rD ( ) βe2 2r [ 3 exp − 4r (1 − β)3 + 24a3B β(2 − β)3 + aB 24(1 − β)4 a3B r ] +3a2B r(1 − β)(13 − 5β 2 ) + 6aB r2 (1 − β)2 (5 − 3β) , (4.13) √ where rD = kB T /8πne2 and β = (aB /2rD )2 . As expected, the classical Debye screening has appeared in the interaction of charged particles. The interatomic interaction has also undergone a slight deviation due to the presence of charged particles. These expressions truly indicate that the Debye-H¨ uckel theory is included in the constructed pseudopotential model as its limiting case of full ionization. In case of weakly ionized plasmas nn ≫ ne , np expressions (1.37)(1.39) again simplify to yield [70] e2 β 2 (3 − β)2 e2 Φee (r) = Φpp (r) = −Φep (r) = + r (1 − β)4 r ( √ ) ( ) 2(1 + β)r βe2 2r − − exp × exp − aB aB 48(1 − β)4 a3B r [ 2 × 3aB r(13 + 29β − 65β 2 + 23β 3 ) +4r3 (1 − β)3 + 48a3B β(3 − β)2 ] +6aB r2 (5 − β)(1 − β)2 , ( ) ( ) e2 1 1 2r Φpn (r) = −Φen (r) = + exp − 1 − β aB r aB ( ) 2βe2 2r β(3 − β)e2 + exp − − (1 − β)2 r aB (1 − β)2 r ) ( √ 2(1 + β)r × exp − , aB 83

(4.14)

(4.15)

CHAPTER 4. PARTIALLY IONIZED PLASMAS ( √ ) 2(1 + β)r e2 Φnn (r) = exp − , (4.16) r aB √ √ where rn = kB T /4πne2 and β = (aB / 2rn )2 . In this case the screening in the interaction of charged particles is out of power, although, the neutral component slightly modifies the pure Coulomb interaction. In full congruence with the pair correlation approximation (1.36) the non-monotonic behavior of the macroscopic potential between a charged particle and an atom is to be interpreted as the short-range order formation in the subsystem of neutral gas - plasma. From the theoretical point of view it is uneasy to predict the characteristic length of the structures that are formed under appropriate external conditions. In spite of this, the present approach provides a systematic tool to evaluate the characteristic length of the structures, which can be employed in the following way. The Fourier transforms of the microscopic potentials are simply rational functions of k 2 . In their turn the macroscopic potentials are expressed in terms of the microscopic potentials in a rational way, therefore, the former are found as rational functions of k 2 as well. It is elementary to prove that the power of the denominator does not exceed k 12 and a partial fraction expansion formally yields ˜ ab (k) = Φ

6 ∑ 4πci , 2 k − ki2 i=1

(4.17)

where ki are the roots, in general complex, of the equation ∆(k) = 0, see (1.38), and ci denotes some numerical coeﬃcients. The nature of the polynomial is such that the roots of the equation ∆(k) = 0 simply alternate as follows ki = ±Reki ± iImki

(4.18)

and, then, taking into account the roots that lie in the upper half of the complex plane, the residue theory immediately gives rise to ∑ ci cos(Reki r) exp(−Imki r). (4.19) Φab (r) = r 84

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.4: Characteristic length λ of the short-range order in a partially ionized hydrogen plasmas as a function of the coupling parameter Γ. Solid line: rs = 10; dashed line: rs = 5; dotted line: rs = 1. Formula (4.19) unambiguously demonstrates that the short-range order formation in the system becomes possible if the solution to equation ∆(k) = 0 produces roots located in the upper half of the complex plane with a nonzero real part. Then, the characteristic length of the structures is determined, in accordance with (4.19), as λi =

2π . Reki

(4.20)

Figure 4.4 presents the dependence of the characteristic lengths of structures in the hydrogen plasma as a function of the coupling parameter Γ at fixed values of the density parameter rs . With the growth of the coupling parameter the characteristic length decreases, tending to a constant value, and, at the same tine, it goes to infinity with Γ → 0. This is quite consistent with the physical picture, since the decrease of the coupling parameter is equivalent to an increase in temperature and, therefore, the thermal motion should eﬀectively disrupt the structures formed. A natural question arises whether there is only one characteristic 85

CHAPTER 4. PARTIALLY IONIZED PLASMAS length or there may be a few. Numerical inquiry of this issue shows that at the density parameter rs = 1, unlike the cases with rs = 5 and rs = 10, a second characteristic length appears with another typical size, which is much greater in magnitude than the first one, but reveals quite an analogous behavior. It has to be admitted that this is quite a surprising finding that demands further elaborate investigation.

4.3

Self-consistent chemical model

Until very recently an ordinary chemical model of the plasma was constructed as follows. An expression for the free energy was actually postulated in the form of independent contributions from neutral and charged components of the system, thus, neglecting their interrelation. This is called the linear mixing rule. It is obvious that such a method is only valid in case of very small values of the coupling parameter when the correlations in the system remain really weak. Increasing the coupling parameter requires a more accurate microscopic account of the mutual interinfluence of the charged and neutral components. To go beyond the Saha equation and the linear mixing rule, a general expression for the system free energy is exactly deduced below from the generalized Poisson-Boltzmann equation. The expression for the radial distribution functions gab (r) is taken from Section 1.3 in the form Φab (r) gab (r) = 1 + hab (r) = 1 − , (4.21) kB T and the correlation energy is straightforwardly expressed through the radial distribution functions as ∫∞ ∑ UN = 2πV na nb φab (r)gab (r)r2 dr. (4.22) 0

a,b

In contrast to (3.31) the correlation energy is again found via expression (4.22) since the microscopic potentials (4.4)-(4.6) are independent of the temperature. 86

Eﬀective Potentials in Non-Ideal Plasma Physics In its turn the free energy is written via the correlation energy as a sum of ideal and excess parts as follows F = Fid + Fexc =

(

) eV −Ne kB T ln − Np kB T ln Np λ3p ( ) ∫ UN eV Σn −Nn kB T ln − T dT , 3 Nn λn T2 2eV Ne λ3e

)

(

(4.23)

where λa = (2π~2 /ma kB T )1/2 stands for the thermal de Broglie wavelength and Σn is the atomic partition function. One can envisage formula (4.23) in order to come to a conclusion that the ideal part of the free energy, appearing as a constant in equation (2.22), cannot be derived within the classical statistical mechanics because it comprises essentially quantum parameters as the de Broglie wavelength and atomic partition function. Substituting equations (4.21) and (4.22) into formula (4.23) gives the following expression for Fexc V ∑ VT Fexc = na nb φ˜ab (0) + 2 16π 3 kB a,b ∫ ∫ ∑ ˜ ab (k) Φ na nb dkφ˜ab (k) dT × . (4.24) T3 a,b

Note that expression (4.24) for the excess free energy correctly takes into account mutual interaction between all particles in the system. It is well established that the approximation of an absolutely ideal system is of no physical sense, since in this case the atomic partition function simply diverges. Thus, a more thorough account of interactions between the particles is needed to provide a finite result. One of the possible options for the choice of the atomic partition function is the following expression, proposed by Planck and Larkin [49] ) ] [ ( ∞ ∑ In In 2 −1− , (4.25) Σn = Σ P L = 2n exp kB T kB T n=1

87

CHAPTER 4. PARTIALLY IONIZED PLASMAS where In = −I/n2 refers to the energy eigenvalues of an isolated hydrogen atom. A second, in fact rougher, option is the cut-oﬀ partition function of the form [71] ) ( ∞ ∑ In 2 Σn = Σcut = ωn (4.26) 2n exp kB T n=1

with the formfactor in the spirit of the Saha equation { 1, n = 1 ωn = . 0, n ̸= 1

(4.27)

The choice of the atomic partition function in the form of (4.26) and (4.27) is only justified at low temperatures when major contribution to the atomic partition function is made by the ground state, while all other contributions remain exponentially small. Indeed, the energy levels of the hydrogen atom are arranged in such a way that the energy of the first excited state is comparable to the ionization potential, and, thus, the atoms are all either ionized or in the ground state. Expression (4.24) is a function depending on the number of free electrons Ne , the number of free protons Np and the number of atoms Nn . It is well known that for a fixed volume and temperature the free energy of the system has to be minimal, which allows one to determine the ionization degree of the plasma medium by minimizing expression (4.24). To demonstrate the possibility of minimizing the free energy, Figure 4.5 represents a characteristic shape of its dependence on the ionization degree α = np /(np + nn ) at Γ = 0.5 and rs = 5. Thus, by minimizing the free energy, the ionization degree α = Np /(Np + Nn ) is eﬀectively obtained as a function of the dimensionless coupling Γ and density rs parameters, what is graphically depicted in Figures 4.6-4.9. Thus, the ionization equilibrium can be determined in a wide range of plasma parameters and careful analysis of the figures drives one to the following conclusions. It seems rather natural that with increasing temperature and, hence, decreasing the coupling parameter, the ionization degree grows tending 88

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.5: Free energy of a partilly ionized hydrogen plasma as a function of the ionization degree α at Γ = 0.5 and rs = 5. Dashed line: the ideal part of the free energy; solid line: the total free energy with the cut-oﬀ atomic partition function (4.26) and (4.27).

to a limiting case of full ionization. With increasing the number density or decreasing the density parameter, the ionization degree, calculated using the Planck-Larkin atomic partition function, goes to unity, whereas in all other cases the ionization degree first reaches a maximum value and, then, vanishes. Thus, application of the Planck-Larkin atomic partition function discloses more physically meaningful picture since growth of the number density should somehow initiate the pressure ionization leading to a full ionization of the plasma. Of course, this does not mean that the Planck-Larkin atomic partition function includes the density eﬀects arising from the spectrum modification, but still implies that from the energetic point of view it is advantageous for a system to undergo a transition to a fully ionized state. In Figure 4.10 an additional comparison is made of the ionization degree of a hydrogen plasma with the path integral Monte Carlo simulations [66] and quite a good agreement is found for the coupling parameters Γ < 0.5. Figures 4.6-4.10 clearly reveal that the account of particle inter89

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Figure 4.6: Ionization degree of a hydrogen plasma as a function of the density parameter rs at Γ = 0.2. Dotted line: the Saha equation (4.10); solid line: the Planck-Larkin atomic partition function (4.25); dashed line: the cut-oﬀ atomic partition function (4.26) and (4.27).

Figure 4.7: Ionization degree of a hydrogen plasma as a function of the density parameter rs at Γ = 0.8. Dotted line: the Saha equation (4.10); solid line: the Planck-Larkin atomic partition function (4.25); dashed line: the cut-oﬀ atomic partition function (4.26) and (4.27). 90

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.8: Ionization degree of a hydrogen plasma as a function of the coupling parameter Γ at rs = 2.0. Dotted line: the Saha equation (4.10); solid line: the Planck-Larkin atomic partition function (4.25); dashed line: the cut-oﬀ atomic partition function (4.26) and (4.27).

Figure 4.9: Ionization degree of a hydrogen plasma as a function of the coupling parameter Γ at rs = 12.0. Dotted line: the Saha equation (4.10); solid line: the Planck-Larkin atomic partition function (4.25); dashed line: the cut-oﬀ atomic partition function (4.26) and (4.27). 91

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Figure 4.10: Ionization degree of a hydrogen plasma as a function of the coupling parameter Γ at rs = 10. Dotted line: the Saha equation (4.10); solid line: the Planck-Larkin atomic partition function (4.25); dashed line: the cut-oﬀ atomic partition function (4.26) and (4.27); black circles: data of [66]. actions in a partially ionized plasma always leads to higher values of the ionization degree as compared to the Saha equation, which can be interpreted as lowering of the ionization potential [72, 73].

4.4

Correlation functions

This Section is devoted to the correlation functions with a purpose to underscore the power of the self-consistent chemical model since an ordinary chemical picture cannot provide any clue to determination of, for example, the radial distribution functions of partially ionized plasmas. In particular, formula (4.21) allows one to study the radial distribution function as soon as the ionization equilibrium has been found in the preceding Section 4.3. Owing to formula (4.21), the behavior of the radial distribution 92

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.11: Electron-atom radial distribution function of a partially ionized hydrogen plasma as a function of the reduced distance at rs = 10. Dotted line: Γ = 0.5; dashed line: Γ = 0.75; solid line: Γ = 1.0. functions is quite similar to that one of the macroscopic potentials and the following conclusions may be drawn from a detailed numerical calculations. The distribution of the charged component of partially ionized plasmas has a monotonic character in quite a wide range of plasma parameters, whereas a maximum appears in the plasma-gas distribution, see Figure 4.11 that shows the electron-atom radial distribution function for various values of the coupling parameter Γ and the fixed value of the density parameter rs = 10. It is seen that the depth of the minimum of the electron-atom radial distribution function grows while the coupling parameter Γ increases. In accord with the physical meaning of the radial distribution function, mentioned several times above, this indicates the appearance of the short-range order in the corresponding subsystem. It should be noted that until very recently the appearance of the short-range order was believed possible only for the coupling parameter much greater than unity. As the number density grows or the density parameter decreases, 93

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Figure 4.12: Electron-atom static structure factor of a partially ionized hydrogen plasma as a function of the reduced wavenumber ka and the coupling parameter Γ at rs = 5.

the distribution of the charged component changes insignificantly, remaining monotonous, whereas in the distribution of the plasma-neutral gas the first extremum shifts toward larger distances due to stronger interparticle interactions. It can be also claimed that the distribution of the plasma component slightly depends on the coupling parameter and remains smooth at all interparticle separations. In the distribution of the plasma-gas subsystem the situation is exactly opposite. It has a pronounced non-monotonic character, which becomes particularly evident when the coupling parameter rises. This is surely due to the fact that an atom attracts free electrons that form an electron cloud around it, so that the eﬀective attraction between that atom and a proton is substantially reinforced. Such an interpretation is a straightforward consequence of the renormalization procedure of particle interactions, proposed in Section 1.1. Formula (3.28) can be used to observe the whole picture of the dependencies of the static structure factors of the system on the plasma parameters, but note that, in contrast to the radial distribution func94

Eﬀective Potentials in Non-Ideal Plasma Physics tions, the static structure factors are also determined by the number densities of the correlating particles, which directly depend on the plasma composition. A detailed numerical analysis evidences that the electron-electron, electron-proton and atom-atom structure factors retain their smoothness in a broad range of plasma parameters. Conversely, the electronatom structure factor has a maximum, see Figure 4.12, whose height increases with the growth of the coupling parameter to manifest strengthening of correlations in the system. Such a maximum clearly proves in favor of the short-range order formation in the plasma-neutral gas subsystem.

4.5

Thermodynamics

Since the microscopic potentials (4.4)-(4.6) are both temperature and volume independent, the equation of state can be evaluated with the help of formula (2.19). Figures 4.13 and 4.14 display the pressure of the partially ionized hydrogen plasma as a function of the density parameter rs at fixed values of the coupling parameter. The pressure is normalized to its maximum possible value of Pid = 2nkB T , and, thus, varies from 1, corresponding to a fully ionized plasma, to 0.5, standing for a completely neutral gas of atoms. Figures 4.13 and 4.14 reveal that for suﬃciently large values of the coupling parameter Γ the choice of the atomic partition function does not seriously aﬀect the result. This is due to the fact that at high values of Γ the system temperature turns lower and, hence, the eﬀect of excited levels of atoms vanishes. It has to be clearly stressed that observed is the eﬀect of finite size of atoms, i.e. a sharp growth of the plasma pressure with increase in the number density or, equivalently, decrease in the density parameter. Although, all the particles in the present considerations are treated point-like, the interaction between atoms strongly increases when the average interatomic distance approaches the first Bohr radius, see for95

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Figure 4.13: Equation of state of a partially ionized hydrogen plasma as a function of the density parameter rs at Γ = 0.2. Dashed line: for the cut-oﬀ partition function (4.26) and (4.27); dotted line: for the Saha equation (4.10); solid line: for the Planck-Larkin partition function (4.25). mula (4.6), which results in a rapid growth of the pressure at rs ∼ 1. The purpose of the following is to compare the equation of state, derived within the self-consistent chemical model, with the exact quantummechanical result of the activity expansions [49, 74], which reads [π ∑ ∑ κ3 ζa ζb + (βea eb )3 ln(κλab ) βP = ζa + 12π 3 a a,b ] +2πλ3ab K0 (ξab ; sa ) , (4.28) na = ζa

∂ (βP ) , ∂ζa

(4.29)

√ where β = 1/kB T , ξab = −βea eb /λab , λab = ~/ 2µab kB T , µab = ma mb /(ma + mb ). Here the following virial functions of Ebeling are introduced K0 (ξab ; sa ) = Q(ξab ) + δab

(−1)2sa E(ξab ), 2sa + 1 96

(4.30)

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.14: Equation of state of a partially ionized hydrogen plasma as a function of the density parameter rs at Γ = 1.0. Dashed line: for the cut-oﬀ partition function (4.26) and (4.27); dotted line: for the Saha equation (4.10); solid line: for the Planck-Larkin partition function (4.25). ( ) √ ξ ξ2 π ξ3 C 1 Q(ξ) = − − − + ln 3 − 6 8 6 2 2 √ ∞ ∑ πζ(n − 2) ( ξ )n ( ) + , 2 Γ n2 + 1

(4.31)

n=4

E(ξ) =

√ √ π ξ π 2 π2 3 + + ln 2 ξ + ξ 4 2 4 72 ( ) ∞ √ ∑ π(1 − 22−n )ζ(n − 2) ξ n ( ) + , 2 Γ n2 + 1 n=4

(4.32)

with the Euler constant C and the Riemann function ζ (n − 2) of argument n − 2. It has to be remarked that the set of equations (4.28) and (4.29) has been inverted in a non-perturbative way to catch up the strong departure from an ideal fully ionized plasma since there is a term in (4.28) 97

CHAPTER 4. PARTIALLY IONIZED PLASMAS that naturally reproduces the recombination process. Thus, Figures 4.15 and 4.16 make a comparison of both methods, mentioned above, at the fixed values of the density parameter rs = 5 and rs = 10. For quite large values of the number density the choice of the atomic partition function has minor influence on the result, which agrees very well with the quantum mechanical activity expansion. The same holds for suﬃciently large values of the coupling since this eﬃciently reduces the contribution of excited states to the atomic partition function. It should also be noted that the results, calculated by the Saha equation, systematically underestimate the pressure because the Saha equation permanently predicts lower values of the ionization degree of the medium.

4.6

Electrical conductivity

Developed above in Section 4.2 is an eﬀective macroscopic potential of particle interactions in a partially ionized plasma. Although it is only valid for a plasma at thermal equilibrium it is used in the remainder to assess the electrical conductivity assuming that the deviation from the equilibrium state is only slight. Accurate knowledge of the eﬀective macroscopic potentials gives a wonderful opportunity to calculate phase shifts, which determine the scattering cross sections, which, in its turn, are needed to evaluate the transport coeﬃcients. However, it is clear from the derivation of the macroscopic potentials that such a strategy for computing transport quantities excludes purely dynamical collective eﬀects that are seen in time-displaced correlations and in the corresponding structure factors. It is, however, rather instructive that an ordinary chemical model is only suitable for studying the thermodynamic characteristics of the system, whereas the self-consistent chemical model, proposed above, is capable of predicting transport coeﬃcients as well. Since the ionization degree has been calculated above in Section 4.3, it, thus, becomes possible to obtain the behavior of macroscopic potentials as functions of the distances between particles. The phase 98

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.15: Equation of state of a partially ionized hydrogen plasma as a function of the coupling parameter Γ at rs = 5. Dashed line: for the cut-oﬀ partition function (4.26) and (4.27); dotted line: for the Saha equation (4.10); solid line: for the Planck-Larkin partition function (4.25); solid circles: formulas (4.28) and (4.29) with A0 = π/2; open squares: formulas (4.28) and (4.29) with A0 = 0.

Figure 4.16: The same as in Figure 4.15 but for rs = 10.

99

CHAPTER 4. PARTIALLY IONIZED PLASMAS shifts at the scattering process can be found from the macroscopic potentials by solving the Calogero equation [75] [ ]2 d ab 2µab δl (r) = − 2 Φab (r) cos δlab (r)jl (kr) − sin δlab (r)nl (kr) (4.33) dr ~ k with the initial condition δlab (0) = 0. Here δlab (r) is a phase shift in the scattering of particles of species a and b, jl (kr) and nl (kr) denotes the Ricatti-Bessel functions of first and second kinds, respectively, E = ~2 k 2 /2µab designates the relative kinetic energy of scattering particles with the reduced mass µab = ma mb /(ma + mb ). The Calogero equation (4.33) has been solved numerically with the macroscopic potentials and the phase shift is exemplified in Figure 4.17 for the electron-proton scattering. A thorough assessment of the numerical results shows that at a fixed value of the density parameter rs , increasing the coupling parameter Γ makes the phase shift decrease at infinite distances, which is prescribed to the growing role of the collective phenomena resulting, in particular, in the screening of charged particles interaction. Similarly, at a fixed value of the coupling parameter Γ, decreasing the number density (or increasing the density parameter rs ) makes the phase shift rise due to the weakening of the influence of the collective events on the interaction between the particles in the medium. In all cases the phase shifts diminish while increasing the orbital quantum number l, since at fixed energy of the scattering particle an increase in l corresponds to the growth of the impact parameter, and, hence, to some depletion in the scattering intensity. The transport diﬀerential cross section stems from the phase shifts at infinite separation as Qab T (k) =

∞ [ ]2 4π ∑ ab ab (l + 1) sin δ (∞) − δ (∞) . l+1 l k2

(4.34)

l=0

In Figure 4.18 the typical behavior of the electron-atom transport diﬀerential cross section is shown as a function of the reduced wavenumber that marks the relative kinetic energy of scattering particles. 100

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.17: Electron-proton phase shifts as a function of the dimensionless interparticle distance at rs = 5, Γ = 0.1 and ka = 1. Solid line: l = 0; dashed line: l = 1; dotted line: l = 2.

Figure 4.18: Electron-atom transport diﬀerential cross section of a partially ionized hydrogen plasma as a function of the reduced wavenumber ka. Solid line: rs = 5 and Γ = 0.1; dotted line: rs = 5 and Γ = 1; dashed line: rs = 10 and Γ = 0.1; dot-dashed line: rs = 10 and Γ = 1. 101

CHAPTER 4. PARTIALLY IONIZED PLASMAS On the basis of the above consideration, one can draw a general conclusion that both an increase in the particle number density and a decrease in the temperature of the plasma medium must lead to vanishing of the transport cross sections for the particle scattering. To numerically evaluate the electrical conductivity of a partially ionized plasma, the following formula, proposed by Frost [76], is of particular significance 4 e2 σ= √ 3 2πme (kB T )5/2

∫∞ 0

ne E exp(−E/kB T ) dE. en np Qep T (E)/γE + nn QT (E)

(4.35)

It is required to explain some contributions to expression (4.35), closely related to the study of the electrical conductivity of a partially ionized hydrogen plasma. The specific feature of the electromagnetic interaction manifests itself in that the interelectron correlations have a significant impact on the electrical conductivity even at low values of the coupling parameter Γ. To treat them rigorously, an electronelectron collision integral should be added to the electron-proton collision integral, which finally results in the correction containing the term γE = 0.582 [77]. Thus, the first term in the integrand denominator of (4.35) takes into account both electron-proton and electron-electron scattering processes, whereas the second term in the integrand denominator of (4.35) accounts for the scattering of electrons by hydrogen atoms. Both contributions are strictly proportional to the number densities of free protons and atoms, respectively. Figure 4.19 displays the graphical dependence of the electrical conductivity of a partially ionized hydrogen plasma on the coupling parameter Γ at the fixed value of the density parameter rs = 10. It definitely testifies that the electrical conductivity of a partially ionized plasma vanishes while increasing the coupling parameter Γ. The physical reason for this is the lowering of the ionization degree, i.e. a decrease in the number density of free electrons, since a rise in the coupling parameter is responsible for a decline in the system temperature. Besides, the growth of the coupling parameter also leads to the 102

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 4.19: Dimensionless electrical conductivity σ/ωp with ωp = (4πne2 /me )1/2 of a partially ionized hydrogen plasma as a function of the coupling parameter Γ at rs = 10. Solid line with circles: for partially ionized plasmas; dashed line: for fully ionized plasmas. increase of the scattering cross section. Comparison with the case of a fully ionized plasma shows that the contribution from the scattering on atoms is numerically not very significant and reaches about 10% when the coupling parameter grows. It should be borne in mind, however, that for the case of fully ionized plasmas the Debye radius has been recalculated taking into account the partial ionization of the medium. Elaborate numerical examination clearly demonstrates that the decrease in the density parameter, i.e. the increase of the number density, gives rise to an increase of the electrical conductivity since the number density of charged particles also grows due to the lowering of the ionization potential. Table 4.1 shows the comparison of the calculated electrical conductivity of a partially ionized plasma with the available theoretical calculations and experimental data. The chief in Table 4.1 is the comparison with experimental data σexp . It is seen that the results of the present approach σp for a partially ionized plasma and σf for a fully ionized plasma are in a reason103

CHAPTER 4. PARTIALLY IONIZED PLASMAS

Table 4.1: Comparison of the dimensionless values of the electrical conductivity of a partially ionized hydrogen plasma [81]. Γ 0.175 0.165 0.170

rs 117.22 102.40 86.37

σ, [78, 79] 3.781 4.954 6.183

σ, [80] 3.083 4.122 5.271

σexp , [82–85] 2.457 3.618 4.530

σp 1.516 2.379 3.951

σf 2.467 2.578 2.389

able agreement with the experimental data, and the discrepancy may be prescribed to the presence of hydrogen molecules, especially pronounced for the first two lines in Table 4.1. In general, the deviations of the theoretical works [78,79] and [80] from the experimental data are within the experimental error which, regrettably, remains quite large.

4.7

Conclusions

In this Chapter 4 a novel approach to the chemical model of partially ionized hydrogen plasmas has been presented, which resides on the generalized Poisson-Boltzmann equation and allows one to determine both the ionization equilibrium and the correlation functions. The analysis of the numerical results, implemented above, has concluded that the charged and neutral components of the plasma medium are closely interrelated and their reciprocal influence is responsible for the short-range order formation in the system of interest. Consequently, the contributions of charged and neutral components to the free energy can no longer be considered independent, especially for moderately and strongly coupled regimes. Interparticle interactions have been found to increase the ionization degree in comparison with the ideal system case and the choice of the atomic partition function in the Planck-Larkin form has shown more realistic behavior of the ionization degree at high densities. The pair distribution functions have supplied us with an opportunity 104

Eﬀective Potentials in Non-Ideal Plasma Physics to study the equation of state of partially ionized hydrogen plasmas in a wide range of plasma parameters. The sophisticated comparison with the exact quantum-mechanical expansion and quantum Monte Carlo simulations has been made to observe a fairly good agreement at relatively low densities and high temperatures. The generalized Poisson-Boltzmann equation has provided the effective macroscopic interaction potentials, which have made it possible to evaluate the transport properties and a reasonable agreement with the available theoretical and experimental results has been found for the electrical conductivity of a partially ionized hydrogen. Despite of these important advances there are many ways to improve the approach, developed above. First of all it is desirable to amend the pseudopotential model (4.4) in order to include the polarization phenomena and finite size of atoms (the so-called hard core eﬀect). This goal demands far more accurate determination of the microscopic potential model, in which the perturbation of the ground state electron orbit is to be taken into account. Another point of interest is the inclusion of excited states of atoms that could be done by considering each excited state as a new entity. This will only increase a number of equations in the generalized Poisson-Boltzmann equation, but will provide the populations of excited levels of atoms. For hydrogen plasmas it is also advisable to thoroughly engage in the analysis the formation of hydrogen molecules and other charged clusters [64, 65]. Of course, the approach developed may easily be extended to other physical situations of interest such as, for example, to partially ionized alkaline, or even dusty plasmas.

105

Chapter 5

Dusty plasmas In recent decades, considerable attention has been paid to dusty plasmas [86, 87] that arise as ordinary plasmas, in which particles of micron sizes, called grains, are introdyced. It is known that dust particles in a plasma acquire such a high, mostly negative, electric charge [88, 89] that their mutual interaction energy substantially prevails over their kinetic energy of chaotic motion [90, 91]. At the same time, dust particles, in their arrangement with respect to one another, form a structure resembling a crystal lattice, which is known as a plasma crystal [92, 93]. Dusty plasmas are often found in both nature and laboratory [94,95]. For example, they are under scrutiny in astrophysical applications and space experiments [96,97], produced by the contact of plasmas with the chamber walls of installations, designed for controlled thermonuclear fusion [98, 99], employed in medicine to treat cancer [100, 101], and the like. A dusty plasma also retains its fundamental importance, since the behavior of dust particles is easily taped in experiment using highresolution video cameras [102, 103], which provides a firm ground for verification of the theoretical concepts, developed for open and nonideal many-body systems [104, 105]. In modern literature on the physics of dusty plasmas, heuristic ap106

Eﬀective Potentials in Non-Ideal Plasma Physics proaches are mainly used to study the thermodynamic properties of the dust component, which are more or less based on engaging the Yukawa (Debye-Hckel) potential [106,107] with a constant particle charge. Ultimately, such an approximation cannot be considered satisfactory, since the derivation of the Yukawa potential essentially exploits the fact that the dust particles should be point-like, but in this case they could not acquire an electric charge at all by absorbing plasma particles. Such a contradiction necessitates a logically consistent investigation, which capitalizes on the finite dimensions of dust particles. It has to be remarked, though, that the size of dust particles has no direct influence on the properties of the dusty plasmas since under ordinary experimental conditions the packing fraction parameter, introduced below in Section 5.1, remains quite low. Nevertheless, the dust grain charge strongly depends on its size, which means that the coupling of the dust component is no longer a free variable parameter. Moreover, it is one of the goals of this Chapter 5 to establish the relation between the dust grain charge on the one hand and the parameters of the buﬀer plasma on the other. In the following Section 5.1 the dimensionless plasma parameters are defined to describe the state of the dusty plasma and its surrounding medium. Section 5.2 strictly focuses on the derivation of the pseudopotential of intergrain interaction that simultaneously takes into account finite dimensions of dust particles, the polarization phenomena and the screening eﬀects due to the buﬀer plasma. Charging of dust particles is scrutinized in Section 5.3 within the orbital motion limited approximation in order to establish the influence of polarization phenomena on the dust grain charge. Section 5.4 is primarily dedicated to investigation of the correlation functions behavior in a broad range of dusty plasma parameters. On the basis of Section 5.4 the dust component contribution to thermodynamic quantities is evaluated and compared with the results of computer simulations in Section 5.5. Major outcomes of the whole Chapter 5 are stated in the final Sec107

CHAPTER 5. DUSTY PLASMAS tion 5.6.

5.1

Plasma parameters

For the sake of simplicity it is assumed in the rest that the dust particles are immersed in a two-component hydrogen plasma consisting of free electrons with the electric charge −e and the number density ne , and free protons with the electric charge e and the number density np . The dust particles are supposed to be solid hard balls of radius R and to possess the electric charge −Zd e with Zd being the charge number, such that the local plasma quasi-neutrality is assured by imposing the condition np = ne + Z d nd . (5.1) To describe the state of the buﬀer hydrogen plasma the density parameter is introduced as ap rs = , (5.2) aB where aB = ~2 /me e2 stands for the first Bohr radius, ~ symbolizes the reduced Planck constant, me designates the electron mass and ap = (3/4πnp )1/3 is the mean interproton spacing. Another dimensionless parameter, relevant to the description of the buﬀer plasma, is the coupling parameter, defined as Γ=

e2 , ap kB T

(5.3)

where kB denotes the Boltzmann constant, T designates the medium temperature. Note that formula (5.3) estimates the ratio of the mean Coulomb interaction energy of two protons to their average kinetic energy of thermal motion. To relate the dusty component to the buﬀer plasma, it suﬃces to define the Havnes parameter [108] P =

Zd nd ne

108

(5.4)

Eﬀective Potentials in Non-Ideal Plasma Physics that characterizes the ratio of the charge densities of the dust and electron components, as well as the screening parameter ad , (5.5) κ= rD where rD = (kB T /4π(ne + np )e2 )1/2 refers to the Debye radius and ad = (3/4πnd )1/3 signifies the mean intergrain distance. The dust coupling parameter has the same physical meaning as (5.3), but is intended to measure the nonideality of the dust component in the form Z 2 e2 . (5.6) Γd = d ad kB T Herein the finite-size eﬀects of dust particles are in close focus, which necessitates a standard definition of the packing fraction as follows 4 η = πnd R3 . (5.7) 3 It is claimed in the√literature that the packing fraction cannot exceed the critical value of 2π/6 ≈ 0.74 for the most dense packing of hard balls of the same radius. Knowledge of the dimensionless parameters (5.2)-(5.7), introduced above, is quite enough to restore numerical values of all plasma parameters. From its practical point of view only parameters (5.3)-(5.7) are needed to conduct numerical calculation since all lengths are easily expressed in terms of the first Bohr radius for a classical system of interest herein. It is rather timely to make a few comments concerning the choice of the dimensionless plasma parameters. First of all it is quite clear from the physical point of view that the charge number Zd depends on the state of the surrounding buﬀer plasma and can independently be found using, for instance, the orbital motion limited approach [109]. It was shown [110] that such a technique proved fruitful to be confirmed experimentally. Secondly, stochastic heating of the dust particles, caused by fluctuating electric fields, can result in that the dust temperature may turn distinct from the temperature of the buﬀer plasma [111] or the charge number of the grains can even evolve in time [112, 113]. 109

CHAPTER 5. DUSTY PLASMAS

5.2

Pseudopotential model

It is inevitable in fundamental researches that the true microscopic interaction between all of the plasma constituents (i.e. the electrons, the protons and the dust grains, denoted in the following by the subscripts e, p and d, respectively) is described by the Coulomb potential as e2 φee (r) = φpp (r) = −φep (r) = , (5.8) r φed (r) = −φpd (r) =

Z d e2 , r

φdd (r) =

Zd2 e2 . r

(5.9)

It should be stressed here that the Coulomb potentials (5.8) and (5.9) are infinite at r → 0 and very slowly vanishes with the distance, which result in the well-known diﬃculties in the theoretical description of plasma properties. Of use in the following is the renormalization theory of plasma particle interactions, see Section 1.1, in which an appropriate averaging procedure over particles positions is carried out to account for the collective events in plasmas. No doubt, such a renormalization has to exclude the particles positions, located inside the dust grains. Thus, to take into account the finite size eﬀects in the interaction potentials, the following substitution [114], φ(p,e)d (r) → φ(p,e)d (r + R), φdd (r) → φdd (r + 2R) is made in the microscopic potentials (5.9) to yield φed (r) = −φpd (r) =

Z d e2 , r+R

φdd (r) =

Zd2 e2 . r + 2R

(5.10)

It is clear that formulas (5.10) imply that there is no mutual penetration of the dust particles into one another and there is no intrusion of the electrons and protons of the buﬀer plasma into the dust particles, which can only result in changing of their electric charge already taken into account by introducing the parameter Zd . Note that unlike the Coulomb potentials (5.8) and (5.9), expressions (5.10) remain finite at the origin. 110

Eﬀective Potentials in Non-Ideal Plasma Physics The Fourier transforms of the Coulomb potentials (5.8) are found as follows φ˜ee (k) = φ˜pp (k) = −φ˜ep (k) =

4πe2 , k2

(5.11)

whereas the Fourier transforms of the microscopic potentials (5.10) take the form 4πZd e2 4πZd e2 R φ˜pd (k) = −φ˜ed (k) = − + k2 k [ ] 1 × Ci(kR) sin(kR) + cos(kR)(π − 2Si(kR)) , 2 4πZd2 e2 8 πZd2 e2 R φ˜dd (k) = − k2 k [ ] 1 × Ci(2kR) sin(2kR) + cos(2kR)(π − 2Si(2kR)) . 2 Here Ci(x) = −

∫∞ cos t x

t

dt and Si(x) =

∫x sin t 0

t

(5.12)

(5.13)

dt represent the integral

cosine and sine functions, respectively. The purpose of the following is to obtain an interaction potential of two isolated grains, immersed into the buﬀer plasma of electrons and protons. To do so the generalized Poisson-Boltzmann equation is used in the following form [115] ∆i Φab (rai , rbj ) = ∆i φab (rai , rbj ) ∑ nc ∫ − ∆i φac (rai , rck )Φcb (rbj , rck )drck . k T B c=e,p

(5.14)

In contrast to (1.9), the summation on the right hand side of (5.14) is only performed over the electrons and protons of the buﬀer plasma, c = e, p, whereas the number density of dust particles is assumed to be zero since of interest is just the interaction of two isolated dust particles. 111

CHAPTER 5. DUSTY PLASMAS Equation (5.14) is actually a relation to determine the eﬀective interaction potential, or the pseudopotential, Φab , through the true microscopic potentials φab . It can be clearly seen that the pseudopotentials definitely take into account the screening due to the buﬀer plasma since they inevitably incorporate the number densities of electrons and protons. In the Fourier space the set of equations (5.14) turns into a set of linear algebraic equations, whose solution is readily found for the interaction of two isolated dust grains as [ ] φ˜dd (k) + (Ae + Ap ) φ˜ee (k)φ˜dd (k) − φ˜2ed (k) ˜ Φdd (k) = , (5.15) 1 + (Ae + Ap )φ˜ee (k) with Ae,p = ne,p /kB T . The interaction pseudopotential of two isolated dust particles in the configuration space is then restored from (5.15) by the inverse Fourier transform ∫ ˜ dd (k) exp(−ikr)dk. Φdd (r) = Φ (5.16) In Figure 5.1 the pseudopotential of dust particles interaction is drawn as a function of the intergrain spacing at diﬀerent values of the packing fraction. It is seen that growth in the packing fraction results in a decrease in the intergrain potential since the corresponding dust coupling parameter diminishes. It is sensible to consider one limiting case of (5.15) and (5.16). Namely, if R → 0, i.e. the dust particles are treated as point-like charges to completely drop out the finite size eﬀects, then, combining all previous equations yields the well used Yukawa potential ( ) Z 2 e2 r Φdd (r) = d exp − . (5.17) r rD It is inferred from the above consideration that formulas (5.15) and (5.16) treat simultaneously the finite size of the dust particles and the shielding of the electric field by the buﬀer plasma. Quite an identical result can be achieved using the linear density-response formalism [116]. 112

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.1: Interaction pseudopotential of dust particles as a function of dimensionless intergrain distance at Γ = 0.2, P = 5 and κ = 4. Dotted line: η = 0.125; dashed line: η = 0.008; solid line: η = 0.002.

The only disadvantage of the present model is that the plasma as a whole is assumed to be at equilibrium, which cannot be true for real dusty plasma experiments due to the absorption and shadowing eﬀects. Deviations of the particle distribution functions from the Maxwellian are easily incorporated to show that the Yukawa potential (5.17) is violated at rather large distances between grains [117–119]. In a certain domain of plasma parameters when the free flight path turns out to be less than the Debye shielding radius, an important role is played by interparticle collisions that lead to trapping ions around a dust particle [120] and, sometimes, necessitate the hydrodynamical description of the plasma shielding eﬀect [121]. Another challenging problem is to take into account the dipole moments of dust particles, which are responsible for anisotropy of interparticle interactions [122]. It has to be mentioned that it is occasionally possible to determine the interaction potential experimentally, which was done, for example, for rf-discharge plasmas by solving the inverse problem in the frame113

CHAPTER 5. DUSTY PLASMAS work of the Langevin dynamics [123]. If the grains are assumed to be metallic charged balls, than the polarization phenomena turn out to play an essential role in interparticle interactions. This situation can be handled within the method of charge images, such that the interaction microscopic potentials between the dusty plasma components are found as follows [124]: φee (r) = φpp (r) = −φep (r) = φed (r) =

e2 , r

e2 R 3 Zd e2 − 2 2 , r 2r (r − R2 ) Zd e2 e2 R 3 − 2 2 , r 2r (r − R2 )

φpd (r) = −

Z 2 e2 φdd (r) = d R

1 − 1 ∞ , ∑ (−1)n+1 sinh β sinh nβ

(5.18) (5.19) (5.20)

(5.21)

n=1

where cosh β = r/2R. It is seen from (5.19) and (5.20) that the electrostatic induction results in that the interaction of a charged particle with a metallic charged ball comprises, together with the pure Coulomb interaction, an additional term corresponding to the attraction of the charged particle with an induced image charge of opposite sign. It can be learnt from formula (5.21) that an infinite number of image charges must be considered in describing the interaction between two charged metallic balls. Applying again the procedure to count distances between the surfaces of the interacting particles, as described above, the Fourier transforms of microscopic potentials (5.18)-(5.21) take the form: φ˜ee (k) = φ˜pp (k) = −φ˜ep (k) =

4πe2 , k2

114

(5.22)

Eﬀective Potentials in Non-Ideal Plasma Physics φ˜pd (k) = −

4πZd e2 4πZd e2 R + [Ci(kR) sin(kR) k2 k ] 1 πe2 R + cos(kR)(π − 2Si(kR)) − 2 k ×[2Ci(kR) sin(kR) − 2Ci(2kR) sin(2kR)

+ cos(kR)(π − 2Si(kR)) − cos(2kR)(π − 2Si(2kR)],

φ˜ed (k) =

(5.23)

4πZd e2 4πZd e2 R − [Ci(kR) sin(kR) k2 k ] 1 πe2 R + cos(kR)(π − 2Si(kR)) − 2 k ×[2Ci(kR) sin(kR) − 2Ci(2kR) sin(2kR)

+ cos(kR)(π − 2Si(kR)) − cos(2kR)(π − 2Si(2kR)], 4πZd2 e2 Zd2 e2 Rf (k) 8πZd e2 R + − φ˜dd (k) = k2 k k ] [ 1 × Ci(2kR) sin(2kR) + cos(2kR)(π − 2Si(2kR)) , 2

(5.24)

(5.25)

where f (k) is a known interpolating function. Using formulas (5.15) and (5.16) it is straightforward to study the intergrain interaction potential that accounts for the electrostatic induction, which is realized in Figure 5.2. A more versatile behavior on the packing fraction is discovered in this particular case.

5.3

Charging of dust particles

The problem of theoretical calculation of the dust particle charge in a plasma is closely related to the theory of a Langmuir probe, which is frequently encountered in the field of plasma diagnostics. The standard approach here is to apply the orbital motion limited approximation [125], which assumes that the buﬀer plasma remains quasi-neutral and 115

CHAPTER 5. DUSTY PLASMAS

Figure 5.2: Interaction pseudopotential of dust particles as a function of dimensionless intergrain distance at Γ = 0.2, P = 5 and κ = 4. Dotted line: η = 0.125; dashed line: η = 0.008; solid line: η = 0.002.

Maxwellian far away from the dust grain, and the mean free paths of plasma particles are much greater than the characteristic size of the sheath. This allows one to consider only ballistic trajectories of electrons and ions, and further use of the conservation laws of energy and angular momentum makes it possible to derive the corresponding absorption cross sections, and, hence, to evaluate the charge of the dust particle, or the current-voltage characteristics of the Langmuir probe. The classical version of the orbital motion limited approximation implies that eﬀective interaction energies between the plasma particles and the dust grain, that include a centrifugal component, are monotonic functions of the distance between them, which is not always accurate. Taking into account the shielding results in the appearance of the so-called absorption radius eﬀect, which is caused by the onset of local maxima in the curve of the eﬀective interaction energy [126, 127]. At the same time there is a need to treat the anisotropy of the ion flow near the dust grain, which is due to the accelerating field of the plasma sheath [128]. 116

Eﬀective Potentials in Non-Ideal Plasma Physics It is easy to imagine that the orbital motion limited approximation presumes that the electron and ion fluxes on the dust surface depend on their spatial distribution and the charge of the dust particle itself. In this sense, the equilibrium charge of the dust grain, originally derived from the equality of electron and ion fluxes, is completely determined by the parameters of the buﬀer plasma and independent of both the material the dust particle is made of and of elementary processes taking place on its surface. That is why the orbital motion limited approximation works rather well only for the dust particles, whose dimensions are small compared to the Debye radius [129]. It is obvious that the above presented interpretation, in spite of its attractiveness, is unsatisfactory from the physical point of view, since it essentially relies on the idea that the surface of the dust particle is a perfect absorber of incoming electrons and ions [130]. To avoid such an unjustified assumption, an attempt was made in [131,132] to develop a true microscopic theory that takes into account the near-surface states of electrons and ions appearing as a result of the polarization of dust particles. Moreover, the electron emission from the surface of dust particles [133], which is determined by the work function of electrons, and the secondary electron emission [134] should be thoroughly involved into the analysis. Further consideration of the charging process is carried out within the orbital motion limited approximation, in which all trajectories of plasma particles, i.e. electrons and protons, are considered ballistic such that the interparticle collisions are completely ignored. To justify such an approach the mean free paths of plasma particles ℓe(p) should be much greater than the Debye screening radius rD , which, in its turn, should significantly exceed the dust grain radius R, i.e. R ≪ rD ≪ ℓe(p) , It is believed in the classical treatment of the dust particle charging that the material of the dust is a perfect absorber, i.e. all the plasma particles, that reach the grain surface, are inevitably taken up. This typically leads to that the grain charge is solely determined by the buﬀer plasma parameters and, thus, independent of the dust material. 117

CHAPTER 5. DUSTY PLASMAS On the other hand it is known that a charged double-layer exists near the surface of solids, and whenever an attempt is undertaken to pull an electron out of a solid, the polarization phenomena come to play an essential role to cause an additional attraction. This results in that to extract an electron from the bulk of a solid it is necessary to perform some work, which is referred to as a work function. The main idea of this Section 5.3 is to account for those polarization eﬀects, which should ultimately lead to a true microscopic theory for the charge of the dust particle in a plasma. Assume that a dust particle absorb a proton with the fixed energy E and the impact parameter ρ. It is known [135] that this process is governed by the eﬀective potential energy, defined as ρ2 , (5.26) r2 where φpd (r) is determined by expression (5.20). At the impact parameter ρ = 0, the eﬀective potential energy of interaction between the proton and the dust particle is a monotonically increasing function of the distance, it is negative everywhere and tends to −∞ when the proton approaches the surface of the dust particle. Thus, the proton with the impact parameter ρ = 0 is surely absorbed by the grain. At the fixed energy, an increase in the impact parameter results in that a maximum appears in the curve of the eﬀective potential energy, whose height grows while increasing ρ. It is then evident that protons with small values of the impact parameter are absorbed by the dust particle, but at a certain value, ρ = ρcr , the height of the maximum of the eﬀective potential energy turns equal to the total energy of the proton, thereby causing its rebound. Obviously, this value ρcr fully determines the absorption cross section as σpd = πρ2cr . All above said is summarized in Figure 5.3, which shows that the protons with the energy E/kB T = 1 and the impact parameters ρ = 0 and ρ = 1 are absorbed by the dust particle, and those with the impact parameter ρ = 2 are scattered. The black line in Figure 5.3 corresponds to the value ρ = ρcr ≈ 1.6, which separates the whole region of the proton impact parameters into the absorption and rebound domains. eﬀ Upd (r, ρ, E) = φpd (r) + E

118

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.3: Eﬀective interaction energy between the proton and the dust particle as a function of the reduced distance r/R at E/kB T = 1 and ΓR = e2 /RkB T = 0.1. Dotted line: ρ = 0; dot-dashed line: ρ = 1; solid line: ρ = ρcr ≈ 1.6; dashed line: ρ = 2; gray line: the total energy of the proton E/kB T = 1.

Hence, ρcr is obtained from the following equation eﬀ max Upd (r, ρcr , E)r≥R = E.

(5.27)

The numerical solution to equation (5.27) is found as follows. For a fixed value of the proton energy E it is necessary to find such ρ = ρcr that the maximum of eﬀective potential energy (5.26) should exactly be equal to the total energy E. It is obvious that the proton absorption cross section σpd = πρ2cr grows when the dust charge increases. The maximum in the eﬀective interaction energy can be obtained eﬀ (r, ρ, E)/dr = 0 and since its position is very from the equation dUdp close to the dust particle surface, as witnessed by Figure 5.3, it is suitable to search for an approximate solution to equation (5.27) in the form of r = R + δ with δ ≪ R, which yields δ=√

R 32Eρ2 e2 R

+ 17 − 16Zd

119

(5.28)

CHAPTER 5. DUSTY PLASMAS and for the absorption cross section of protons ( Z d e2 2 2 σpd = πρcr = πR 1 + RE [√ ]) 2 e RE + 1 + 16Zd + 32 2 − 3 . 8RE e

(5.29)

If the polarization eﬀects are neglected, the following classical result is recovered ( ) Zd e2 C σpd = πR2 1 + . (5.30) RE Figure 5.4 makes a comparison of the absorption cross section of protons, calculated from expression (5.27) with formulas (5.29) and (5.30) at ΓR = e2 /RkB T = 0.1 and Zd = 15. Since the polarization eﬀects lead to an additional attraction of the proton by the dust particle, they are responsible for an increase in the corresponding absorption cross section. It is quite natural that formula (5.29) describes more accurately the behavior of the absorption cross section than formula (5.30), which is only valid for pure Coulomb interaction. Consider the interaction of an electron with the same spherical dust particle. Due to the mutual repulsion, the electron absorption is only possible when its energy reaches the critical value Ec determined as Ec = max φed (r),

(5.31)

where φed (r) is taken from expression (5.19). Under the assumption that the rebound of the electron occurs close to the dust particle surface, series expansion allows one to roughly solve equation (5.31) as ) ( 5 1√ e2 a (5.32) Zd + − 17 + 16Zd . Ec = R 8 8 Note that in case of the pure Coulomb interaction between the electron and the dust grain the critical energy is exactly found from the 120

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.4: Proton absorption cross section by the dust particle as a function of the reduced energy of the incident proton at ΓR = e2 /RkB T = 0.1, Zd = 15. Dotted line: formula (5.30); dashed line: formula (5.29); solid line: exact result from equation (5.27).

energy conservation law as follows EcC =

Zd e2 . R

(5.33)

Figure 5.5 provides a comparison of exact expression (5.31) with approximate formulas (5.32) and (5.33). It can be seen that equation (5.32) better describes the exact data obtained from formula (5.31) than expression (5.33), which completely ignores the polarization of the dust particle. At the same time, it is rather obvious that the polarization phenomena are responsible for an additional attraction of electrons, and, thus, the value of the critical energy is reduced as compared to the case of pure Coulomb interaction (5.33). Let a dust particle absorb an electron with the fixed energy E and the impact parameter ρ. Again, this process is governed by the eﬀective potential energy defined as eﬀ Ued (r, ρ, E) = φed (r) + E

121

ρ2 . r2

(5.34)

CHAPTER 5. DUSTY PLASMAS

Figure 5.5: Critical energy of electrons as a function of the dust particle charge at ΓR = e2 /RkB T = 0.1. Solid line: formula (5.33); dashed line: formula (5.32); solid line: exact result.

The analysis, implemented above for the absorption of the proton, is simply repeated for the absorption of the electron to find ρcr from the equation eﬀ max Ued (r, ρcr , E)r≥R = E (5.35) and, then, to obtain an approximate solution in the form ( Z d e2 2 2 σed = πρcr = πR 1 − RE [√ ]) 2 e RE − 1 − 16Zd + 32 2 + 3 8RE e

(5.36)

that recovers the classical result for the Coulomb interaction potential ( ) Zd e2 C 2 σed = πR 1 − . (5.37) RE It is known that the proton flux on the surface of the dust particle is obtained from the relevant absorption cross section by integrating 122

Eﬀective Potentials in Non-Ideal Plasma Physics over the velocity distribution function as: ∫ Jp = np vσpd fp (v)dv,

(5.38)

where fp (v) = (2πvT2 p )−3/2 exp(−v 2 /2vT2 p ) stands for the Maxwell dis√ tribution with the thermal velocity vT p = kB T /mp and mp being the proton mass. Substituting expression (5.29) into (5.38) gives rise to the following analytical approximation for the proton flux on the dust grain surface √ √ ( [ ] 8πkB T e2 1 + 16Zd 3 a 2 Jp = nR 1 + Zd + − + mp RkB T 8 8 √ ( 2 ) πe2 e (1 + 16Zd ) + exp 8RkB T 32RkB T √ 2 (1 + 16Z ) e d × erfc ,(5.39) 32RkB T where the error function and its complementary counterpart are math∫z ematically defined as erf(z) = 1 − erfc(z) = √2π exp(−t2 )dt. 0

In case of the pure Coulomb interaction, expression (5.39) reduces to the classical result of the form √ ( ) 8πkB T Z d e2 2 C nR 1 + . (5.40) Jp = mp RkB T It is clear that the proton flux grows as a function of the grain charge due to their reciprocal attraction, as evidenced by Figure 5.6. In Figure 5.7 a comparison is made of the proton flux on the dust particle, calculated from expression (5.38), with formulas (5.39) and (5.40) for the fixed value of the charge number Zd = 15. The polarization eﬀects lead to an additional attraction of protons by the dust particle and are responsible for an increase in the corresponding flux. 123

CHAPTER 5. DUSTY PLASMAS

Figure 5.6: Normalized proton flux on the surface of the dust particle as a function of the dust charge number Zd . Dotted line: ΓR = 0.1; dashed line: ΓR = 0.25; solid line: ΓR = 0.5.

Figure 5.7: Normalized proton flux on the surface of the dust particle as a function of the coupling parameter ΓR at Zd = 15. Dotted line: formula (5.40); dashed line: formula (5.39); solid line: exact result from formula (5.38).

124

Eﬀective Potentials in Non-Ideal Plasma Physics Figure 5.7 reveals that analytical formula (5.39) neater follows the behavior of the proton flux than formula (5.40), which is valid for the case of the pure Coulomb interaction. It is rather surprising to observe that the corresponding relations are quite linear. Quite an analogous procedure provides the following approximate expression for the electron flux on the dust grain surface √ [( ) 8πkB T e2 ( √ a 2 Je = nR 1− 17 + 16Zd − 2 me 8RkB T √ ( ( ))) 2 e 17 + 16Zd 5 × exp − Zd + − RkB T 8 8 √ √ ( ( 2 ) πe2 e (21 + 16Zd − 4 17 + 16Zd ) + 1 + exp − 8RkB T 32RkB T √ √ e2 (21 + 16Zd − 4 17 + 16Zd ) × 8πRkB T √ √ 2 (21 + 16Z − 4 17 + 16Z ) e d d + erf − 32RkB T )] ( 2 e (16Zd − 1) (5.41) × exp − 32RkB T that, in the classical case of the Coulomb interaction, is simplified to √ ( ) Z d e2 8πkB T 2 C nR exp − . (5.42) Je = me RkB T The charge of the dust particle is ultimately found by equating the electron and proton fluxes on its surface as Je = Jp .

(5.43)

It has to be stressed that a solution to equation (5.43) corresponds to the detailed equilibrium. Of course, the protons cannot be permanently absorbed by the dust particle and they simply neutralize at the grain 125

CHAPTER 5. DUSTY PLASMAS surface. In this respect it is quite important what physical processes take place near the dust particle surface, that is why a great attention is paid herein to polarization phenomena that is at least partly responsible for the work function generation. With the aid of expressions (5.39) and (5.41), equation (5.43) can be approximated in the simple analytical form √ ( [ ] √ me e2 1 + 16Zd 3 1+ Zd + − mp RkB T 8 8 √ √ ( 2 ) e (1 + 16Zd ) πe2 e2 (1 + 16Zd ) exp erfc + 8RkB T 32RkB T 32RkB T [( ) e2 ( √ = 1− 17 + 16Zd − 2 8RkB T √ ( ( ))) √ e2 5 17 + 16Zd πe2 exp − Zd + − + (1 RkB T 8 8 8RkB T √ ( 2 ) e (21 + 16Zd − 4 17 + 16Zd ) + exp − 32RkB T √ √ e2 (21 + 16Zd − 4 17 + 16Zd ) × + 8πRkB T √ √ 2 e (21 + 16Zd − 4 17 + 16Zd ) + erf − 32RkB T ( 2 )] e (16Zd − 1) × exp − (5.44) 32RkB T that asserts the following classical result for the Coulomb interaction ( ) ( ) √ me Zd e2 Z d e2 1+ = exp − . (5.45) mp RkB T RkB T Figure 5.8 demonstrates a comparison of the dust particle charge, calculated from expression (5.43), with the approximate formulas (5.44) 126

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.8: Electric charge of the dust particle [136] as a function of the parameter ΓR = e2 /RkB T . Dotted line: solution to equation (5.45); dashed line: solution to equation (5.44); solid line: exact result from (5.43).

and (5.45) as a function of the parameter ΓR = e2 /RkB T . Since the polarization eﬀects lead to a stronger increase in the electron flux than the proton flux on the dust particle surface, this finally causes growth of the grain charge. Solutions to equations (5.44) and (5.45) better describe the behavior of the grain charge at low values of the coupling parameter since the polarization plays less significant role in this case. It is not surprising that formula (5.44) treats more accurately the dust particle charge than formula (5.45).

5.4

Correlation functions

It has already been stressed in Section 5.2 that the present consideration essentially stems from the insight that the system of metallic hard balls is to be replaced by the system of point-like charges. This has immediate eﬀect on the interaction between two particular grains since if the number density of dust particles is left unchanged, then, the 127

CHAPTER 5. DUSTY PLASMAS average interaction energy will inevitably decrease because the distance between grains is now counted between their surfaces. Thus, the grain number density should be adjusted for the average interaction energy to stay the same, which is simply achieved by the idea of van der Waals, when he introduced his famous correction for the finite size of atoms to the ideal gas equation of state. In particular, the eﬀective number density of dust particles neﬀ d is proposed to take the following form [137] neﬀ d =

nd . 1 − η/η0

(5.46)

√ Here η0 = 2π/6 stands for the packing fraction of the hexagonal packing of hard balls, which is believed to be the most compact of all possible packings in the theory of condensed matter physics. The idea is to only consider the volume available to the dust particles such that the grains should completely lose their mobility when the packing becomes the most compact and the distance between the surfaces of two adjacent dust particles turns zero. In this particular case, the eﬀective number density of dust particles turns infinite and the real average distance between them becomes equal to 2R, as it should be. Neither the number density of dust particles nor its eﬀective counterpart does enter interaction potential (5.15) and (5.16) that virtually holds for the interaction energy of two isolated grains, whose shielding is performed by the electrons and protons of the buﬀer plasma. It is, therefore, fully justified to further apply the constructed eﬀective potential model in well-tested theoretical approaches and computer simulation techniques treating the collective events for dust particles, as it is routinely done in a one-component plasma model. One of the reliable methods for studying system correlation functions is the method of integral equations, presented in Section 1.6. In particular, the Ornstein-Zernike relation in the hypernetted-chain approximation (HNC) is numerically solved with the eﬀective number density, introduced above, in order to obtain the radial distribution 128

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.9: Radial distribution function of grains as a function of the reduced distance r/ad at Γ = 0.2, P = 5, κ = 4 and η = 0.125. Solid line: HNC without polarization phenomena; dashed line: HNC with polarization phenomena; solid circles and triangles: corresponding Monte-Carlo simulation data.

Figure 5.10: Radial distribution function of grains as a function of the reduced distance r/ad at Γ = 0.2, P = 5, κ = 4 and η = 0.002. Solid line: HNC without polarization phenomena; dashed line: HNC with polarization phenomena; solid circles and triangles: corresponding Monte-Carlo simulation data.

129

CHAPTER 5. DUSTY PLASMAS function of dust particles, whose non-monotonic behavior is portrayed in Figures 5.9 and 5.10 to clearly demonstrate the short- or even longrange order formation. A comparison is also made with the MonteCarlo simulations with the same interaction potential between grains to find a fairly good agreement at rather high values of the dust coupling. To analytically treat the collective events in dust particles interactions, the generalized Poisson-Boltzmann equation (1.9) is again used in the following form ∆i Ψdd (ri , rj ) = ∆i Φdd (ri , rj ) ∫ nd ∆i Φdd (ri , rk )Ψdd (rj , rk )drk , − kB T

(5.47)

where Ψdd is a pseudopotential that takes into account the collective events in the interaction of dust particles. Solution to equation (5.47) is found in the Fourier space as [138] ˜ dd (k) = Ψ

˜ dd (k) Φ . nd ˜ 1+ Φdd (k) kB T

(5.48)

It has been shown above that the static structure factor S(k) of dust particles can be expressed in terms of the Fourier transform of the pseudopotential (5.48) as follows Sdd (k) = 1 −

1 nd ˜ Ψdd (k) = . nd ˜ kB T 1+ Φdd (k) kB T

(5.49)

The last formula together with (5.15) provides an important analytical result for the static structure factor of the dust particles, which is valid for the weakly coupled regime when the corresponding coupling parameter of the dust component Γd =

Zd2 e2 P 2 κ5 = √ a d kB T 9 3(P + 2)5/2 Γ2 130

(5.50)

Eﬀective Potentials in Non-Ideal Plasma Physics is evaluated to be less than 1. It has to be explicitly emphasized that application of the generalized Poisson-Boltzmann equation (5.47) implies that the dust particles are viewed upon as a one-component plasma, in which the electrons and ions serve as a neutralizing background that fulfils the screening phenomena. Analytical formula (5.49) for the static structure factor secures its monotonic behavior on the wavenumber at all values of plasma parameters, which is due to the linearization procedure, actually used at the derivation of the generalized Poisson-Boltzmann equation. Figures 5.11 and 5.12 show the static structure factor (5.49) as a function of the wavenumber at diﬀerent values of the plasma parameters. It can be seen that an increase in the Havnes parameter P leads to a decrease in the static structure factor values since the coupling parameter Γd virtually goes down. More complicated behavior is observed by varying the packing fraction η, which is clearly indicated in Figure 5.12. It can, therefore, be inferred that account of the finite size eﬀects strongly influences the static structure factor of the dust particles even in the weakly coupled regime. To go beyond the validity range of the proposed analytical expression, the Ornstein-Zernike relation in the hypernetted-chain approximation (HNC) is numerically solved to obtain the static structure factor of the dust grains, whose strong non-monotonic behavior is discovered in Figures 5.13 and 5.14 to demonstrate the short- or even long-range order formation in the plasma parameters domain, Γd ≫ 1. Namely, an increase in the packing fraction η leads to the shift of the first peak of the static structure factor to smaller values of the wavenumber, whereas an increase in the screening parameter κ results in the dramatically pronounced non-monotonic behavior appropriate for the long-range order formation. The maxima and minima, interpreted as an order formation, turn less pronounced, when the polarization phenomena are properly taken into account since they always weaken repulsion between the dust par131

CHAPTER 5. DUSTY PLASMAS

Figure 5.11: Analytical formula (5.49) for the static structure factor S(k) of the dust component as a function of the dimensionless wavenumber kad at Γ = 0.1, κ = 3 and η = 0.037. Solid line: P = 1; dashed line: P = 3; dotted line: P = 10.

ticles. Analogous inference can absolutely be made in case of a growth in the packing fraction η. Quite a similar behavior of the correlation function was observed in the Percus-Yewick and superposition approximations with further experimental verification for the gas-discharge plasmas [139].

5.5

Thermodynamics

In this Section 5.5 of interest is the correlation energy (2.15) of the dust component of the plasma, which is written in the form ∫∞ UN =

Φdd (r) (g(r) − 1) r2 dr

2πn2d V

(5.51)

0

with the radial distribution function, determined by the static structure factor (5.49). 132

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.12: Analytical formula (5.49) for the static structure factor S(k) of the dust component as a function of the dimensionless wavenumber kad at Γ = 0.1, P = 1, and κ = 3. Solid line: η = 0.125; dashed line: η = 0.008; dotted line: η = 0.001.

Figure 5.13: Static structure factor S(k) of grains as a function of the dimensionless wavenumber kad at Γ = 0.2, P = 5 and κ = 4 with the polarization phenomena contribution. Solid line: η = 0.125; dashed line: η = 0.008; dotted line: η = 0.002.

133

CHAPTER 5. DUSTY PLASMAS

Figure 5.14: Static structure factor S(k) of grains as a function of the dimensionless wavenumber kad at Γ = 0.2, P = 5 and κ = 4 without the polarization phenomena contribution. Solid line: η = 0.125; dashed line: η = 0.008; dotted line: η = 0.002.

Figures 5.15 and 5.16 show the dependence of the correlation energy of the dust component on the plasma parameters and their analysis drives one to the following conclusions. The correlation energy hardly depends on the screening parameter, while the dependence on the coupling parameter remains quite strong. It is due to the interaction potential, which depends linearly on the coupling parameter and only asymptotically changes with the variation of the screening parameter. Comparison of the correlation energy, calculated on the basis of the Monte Carlo method and obtained by solving the HNC approximation, shows good agreement for suﬃciently large values of the dust coupling parameter. It should be noted that for small values of the dust coupling parameter, a discrepancy between the results obtained by Monte Carlo simulations and the solution of the HNC approximation has been discovered. The point is that for small values of the dust coupling parameter, the Monte Carlo method is poorly convergent, for which either a very large number of configurations or a very large number of par134

Eﬀective Potentials in Non-Ideal Plasma Physics

Figure 5.15: Normalized correlation energy of the dust component as a function of the screening parameter at Zd = 100 and η = 0.008. Solid line: ΓR = 0.1; dashed line: ΓR = 0.3; dotted line: ΓR = 0.5; solid squares, circles and triangles: the results of the Monte-Carlo simulations.

Figure 5.16: Normalized correlation energy of the dust component as a function of the coupling parameter at Zd = 100 and η = 0.008. Solid line: κ = 2; dashed line: κ = 5; dotted line: κ = 7; solid circles and triangles: the results of the Monte-Carlo simulations.

135

CHAPTER 5. DUSTY PLASMAS ticles in the base cell are required. Both of these factors result in a significant increase in the modeling time. Consequently, it is confirmed that the Monte Carlo simulation technique and the HNC approximation are considered complementary to each other, since the Monte Carlo method is recognized to be the most reliable for very large values of the coupling parameter, whereas the HNC approximation works very well in the weak and moderately coupled regimes.

5.6

Conclusions

A pseudopotential model of intergrain interaction in dusty plasmas has been proposed in this Chapter 5 to take into account finite size eﬀects, the electrostatic induction and the screening phenomena. Dust particles have been treated as conductive charged balls, which has validated the use of the charge image method. Such a pseudopotential model can be applied in a variety of theoretical approaches and computer simulation techniques since it does exclude the number density of dust particles. The generalized Poisson-Boltzmann equation has been utilized to appropriately treat the shielding of electric fields by the buﬀer plasma. The main idea behind the whole consideration has been to substitute the hard ball system of interest by a system of point-like particles with the properly introduced eﬀective number density as it is regularly done in the van der Waals equation of state of real gases. In particular, the HNC approximation has been iteratively solved with the eﬀective number density of grains and highly pronounced peaks in the curve of the radial distribution function and the static structure factor unveils the short- and long-range order formation in the system. It is concluded on the basis of the above stated results that the polarization eﬀects weaken the intergrain interactions as compared to the case of taking into account the finite size eﬀects only, which manifests itself in that the corresponding peaks in the correlation functions de136

Eﬀective Potentials in Non-Ideal Plasma Physics crease in height with a shift to smaller values of the distance or the wavenumber, respectively. The Monte-Carlo simulations have shown a satisfactory agreement for the radial distribution function at relatively high values of the dust coupling. This Chapter 5 has studied the static structure factor of the dust particles in a weakly coupled regime taking into account the finite size of grains. In particular, the simple analytical expression for the static structure factor has been proposed and it has been shown that its behavior remains monotonic as a function of the wavenumber for the plasma parameters, relevant to its range of the validity. To go beyond that range, the hypernetted-chain approximation has been numerically solved and highly pronounced extrema in the curve of the static structure factor have been discovered. In the course of Chapter 5 the problem of the calculation of the electric charge of dust particles, immersed into a buﬀer hydrogen plasma, has been persistently addressed. Consideration has started from the orbital motion limited approximation, which implies the collisionless ballistic trajectories of plasma particles in an electric field of the charged dust grain. It has been demonstrated that the polarization eﬀects lead to a substantial modification of the calculation technique, resulting in that the proton and electron fluxes on the grain surface strongly depend on its charge and the coupling parameter of the buﬀer plasma. In particular, the proton flux grows linearly with increasing the grain charge and the coupling parameter, which is explained by their mutual attraction. The opposite pattern is observed for the the electron flux since the electrons are repelled by the negatively charged dust particle. The influence of polarization eﬀects on the grain charge has been studied to show that it increases when the coupling parameter grows, which is prescribed to the studied behavior of the electron and proton fluxes on the dust grain surface. Finally, the correlation energy of the dust component of the plasma has been evaluated to find satisfactory agrement with the results of the Monte Carlo simulations at rather large values of the dust coupling.

137

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