Quantum Effects on Low Frequency Waves in Dense Plasmas
By
SHABBIR AHMAD KHAN CIIT/SP04PPH004/ISB
Ph. D. Thesis
COMSATS Institute of Information Technology Islamabad Pakistan
August 2009
This work is submitted as a dissertation in
partial fulfillment of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
IN
PHYSICS
in the
Department of Physics COMSATS Institute of Information Technology
Is lamabad Pakistan
August 2009
This work is supported by Higher Education Commission (HEC) Pakistan
under
Indigenous PhD Fellowship Program (5000): No. 1751(P149)/HEC/Sch/2004
Final Approval
This thesis titled
Quantum Effects on Low Frequency Waves
in Dense Plasmas
By
SHABBIR AHMAD KHAN
Has been approved
For COMSATS Institute of Information Technology, Islamabad
External Examiner: _____________________________________________________________ Prof. Dr. Arshad Majeed Mirza Head Theoretical Physics Group, Department of Physics, Quaid‐i‐Azam University Islamabad
Supervisor: _____________________________________________________________ Dr. Hamid Saleem Adjunct Professor, Department of Physics, COMSAT Institute of Information Technology Islamabad
Head of department: ______________________________________________________________ Dr. Ishaq Ahmed Head, Department of Physics COMSATS Institute of Information Technology, Islamabad
Dean: _____________________________________________________________ Prof. Dr. Raheel Qamar, (TI)
Dean, Faculty of Sciences COMSATS Institute of Information Technology, Islamabad
DECLARATION I hereby declare that the work presented in this thesis is produced by me during the scheduled
course of time. My name in all the publications is written as S. A. Khan. It is further declared
that this thesis neither as a whole nor as a part there of has been copied out from any source
except referred by me whenever due. No portion of the work presented in this thesis has been
submitted in support of any other degree or qualification of this or any other university or
institute of learning. If any volition of HEC rules on research has occurred in this thesis, I shall
be liable to punishable action under the plagiarism rules of HEC.
Date: _________________ Signature of the student:
__________________________ (Shabbir Ahmad Khan)
(CIIT/SP04‐PPH‐004/ISB)
CERTIFICATE It is certified that the work contained in this thesis entitled “Quantum Effects on Low
Frequency Waves in Dense Plasmas” has been carried out by Mr. Shabbir Ahmad Khan
under my supervision. In all the publications his name is written as S. A. Khan.
Supervisor:
Dr. Hamid Saleem Adjunct Professor Department of Physics, COMSATS Inst itute of Information Technology (CIIT), Islamabad 44000 Pakistan, and Director General National Centre for Physics (NCP), Quaid‐iAzam University Campus, Shahdra Valley Road, Islamabad 45320, Pakistan
Submitted through:
Dr. Ishaq Ahmad Head, Department of Physics COMSATS Institute of Information Technology (CIIT), Islamabad 44000, Pakistan Prof. Dr. Arshad Saleem Bhatti Chairman, Department of Physics , COMSATS Institute of Information Technology (CIIT), Islamabad 44000, Pakistan
Dedicated to
My loving parents, caring wife and sons (Aqdis, Osama and Shehryar)
ACKNOWLEDGEMENTS All Praises to Almighty Allah, the most benevolent and merciful and the Creator of the whole
universe, who enabled me to complete this research work successfully.
I would like to express my sincere and hearty appreciation to my supervisor Dr. Hamid Saleem for his constant encouragement, constructive guidance and helpful suggestions during the course of this research work. I deeply appreciate his energy, enthusiasm, and ability to see and formulate new and exciting physical problems. I also would like to express gratitude for his moral support during crucial periods of my stay at CIIT. I acknowledge the co‐operation and encouragement extended to me by Prof. Dr. Arshad Saleem Bhatti (Chairman, Department of Physics, CIIT). Appreciations also go to Rector CIIT, Dr. S.M. Junaid Zaidi, the Head of Physics Department, Dr. Mahnaz Haseeb and all faculty members for their strenuous efforts aiming to create a vibrant scientific research environment at CIIT.
I am grateful to Prof(s). Dr(s). Kamaludin Ahmed (CIIT) and Arshad Majeed Mirza (QAU) for many useful discussions as well as Dr. Imtinan Elahi Qureshi (COMSATS) and Dr. Ehsan Ullah Khan (IIU) for initial guidance. I also thank the organizers of AS‐ICTP Trieste, Italy for inviting me to participate in plasma physics workshops and summer colleges which helped me to better understand the quantum plasmas through the discussions with experts there for which I appreciate all of them. I also acknowledge the role of Ministry of Education, Government of Azad Jammu and Kashmir for providing me study leave to carry out PhD research work. I am too grateful to PINSTECH and PAEC authorities who allowed me to visit and carry out the research work in collaboration with Theoretical Plasma Physics Group (TPPG) at PINSTECH. My heartiest thanks to all members of TPPG, especially Dr.(s) Mushtaq, Shahzad, Qamar ul Haq, Waqas, Mohsin, Sajid, and Mr(s) Sajjad and Ali Ahmad for their all time cooperation. I also strongly appreciate the cooperation of my NCP fellows Dr. Shahid, Dr. Sadiq, M. Asif and Nazia Batool.
Cordial thanks to my PhD colleagues at CIIT particularly Zafar, Rafaqat, Rizwan, Niaz, Azeem, Saeed, Saifullah, Rab Nawaz, Qazi Ahkam, Ghulam Asghar, Nasir, Fayyaz, Irfanullah, Afzal, Anwaar, Kamal, Farooq, Farah Deeba, and Nauman Muteeb (MS) for their cooperation and good company. Thanks to CIIT staff for providing support and facilities when needed, especially Khan Shaukat, Sajid Rasool, Mohsin, Sarfraz, Sajid (IT), Athar and Shams (GSR), Tanveer Baig, Azhar, Tahir, Javed and Nadir Jami (late). Finally, my deepest gratitude goes to my loving mother, my caring wife and my children who suffered but always prayed for my success and supported me. Heartiest tribute (and prayers) to my late father (who died in a road accident in 2006) who never tired caring his family in his life but could not witness this moment, Alas! I would also like to pay compliments to my uncle Mr. M. Bashir for his unceasing support during my entire education.
(Shabbir Ahmad Khan)
ABSTRACT
The low frequency electrostatic and electromagnetic waves in dense plasmas are studied using the quantum hydrodynamic formulation. Several linear and nonlinear waves in uniform as well as nonuniform plasmas are investigated taking into account the quantum diffraction and quantum statistical effects. In an inhomogenous plasma, the drift type wave can appear which doesn’t require electron temperature to be non‐zero for its existence and the electron quantum effects contribute to the wave dispersion at very short length scales. The effect of stationary dust is also discussed. It is also found that the drift wave of ultracold dense plasma can couple with Alfven wave and the linear dispersion relation is analogous to the classical plasma case. But physically, both the dispersion relations are very different. The dispersion relations are analyzed numerically for particular cases of ultracold dense plasma. In a homogenous quantum plasmas, the linear waves are studied for electron‐ion as well as stationary dust case. It is found that the quantum ion‐acoustic wave frequency in the presence of background dust increases with electron quantum effects and dust concentration. In a magnetized electron‐ion plasma, the wave frequency increases with electron number density and magnetic field. The linearly coupled electrostatic and Alfven waves are also investigated and the role of electron fermionic pressure in the wave dynamics of dense quantum plasmas is pointed out. A comparison of fermionic pressure with the quantum pressure due to Bohm potential term is presented. The limit of ultracold dense plasma is discussed in the light of this comparison. The wave dispersion properties for static as well as dynamic ions are elaborated. In the nonlinear regime, it is found that the dust concentration in unmagnetized plasma increases the amplitude and width of dust ion‐acoustic soliton whereas the increase in quantum diffraction parameter reduces the width of the soliton, but doesn’t affect its amplitude. For, magnetized electron‐ion quantum plasma, the quantum diffraction effects are found to increase the amplitude as well as width of the solitons. The increase in magnetic field shrinks the soliton keeping the amplitude constant. The results presented in this thesis are supported by numerical analysis and illustrations. The relevance of the study with the dense astrophysical and laboratory plasmas is also pointed out. Keywords: Dense quantum plasmas, low frequency waves, quantum effects, solitons, etc.
LIST OF PUBLICATIONS OF THE AUTHOR IN
INTERNATIONAL JOURNALS
[1] S. A. Khan and H. Saleem, Linear coupling of Alfven waves with acoustic type modes in
dense quantum plasmas, Phys. Plasmas 16, 052109 (2009)
[2] S. A. Khan, S. Mahmood and S. Ali, Quantum ionacoustic double layers in unmagnetized
dense electronpositronion plasmas, Phys. Plasmas 16, 044505 (2009)
[3] S. A. Khan, S. Mahmood and Arshad M. Mirza, Nonplanar ionacoustic solitons in
electron–positron–ion quantum plasmas, Chin. Phys. Lett. 26, 045203 (2009)
[4] S. A. Khan, W. Masood and M. Siddiq, Obliquely propagating dustacoustic waves in
dense quantum magnetoplasmas, Phys. Plasmas 16, 013701 (2009)
[5] H. Saleem, Ali Ahmad and S. A. Khan, Low frequency electrostatic and electromagnetic
modes of ultracold magnetized nonuniform dense plasmas, Phys. Plasmas. 15, 094501
(2008)
[6] H. Saleem, Ali Ahmad and S. A. Khan, Low frequency electrostatic and electromagnetic
modes in nonuniform cold quantum plasmas, Phys. Plasmas 15, 014503 (2008)
[7] S. A. Khan, S. Mahmood and H. Saleem, Linear and nonlinear ionacoustic waves in very
dense magnetized plasmas, Phys. Plasmas 15, 082303 (2008)
[8] S. A. Khan, S. Mahmood and Arshad M. Mirza, Cylindrical and spherical dust ionacoustic
solitary waves in quantum plasmas, Phys. Lett. A 372, 148‐153 (2008)
[9] S. A. Khan and Q. Haque, Electrostatic Nonlinear Structures in Dissipative Electron
PositronIon Quantum Plasmas, Chin. Phys. Lett. 25, 4329 (2008)
[10] S. A. Khan, A. Mushtaq and W. Masood, Dust ionacoustic waves in magnetized
quantum dusty plasmas with polarity effect, Phys. Plasmas 15, 013701 (2008)
[11] S. A. Khan and W. Masood, Linear and nonlinear quantum ionacoustic waves in dense
magnetized electronpositronion plasmas, Phys. Plasmas 15, 062301 (2008)
[12] H. Ur‐Rehman, S. A. Khan, W. Masood and M. Siddiq, Solitary waves with weak
transverse perturbations in quantum dusty plasmas, Phys. Plasmas 15, 124501 (2008)
[13] S. A. Khan and A. Mushtaq, Linear and nonlinear dust ionacoustic waves in ultracold
quantum dusty plasmas, Phys. Plasmas 14, 083703 (2007)
[14] A Mushtaq and S. A. Khan, Ionacoustic solitary wave with weakly transverse
perturbations in quantum electronpositronion plasma, Phys. Plasmas 14, 052307 (2007)
THE THESIS IS BASED UPON FOLLOWING PAPERS FROM
THE FOREGOING LIST
• S. A. Khan and H. Saleem, Linear coupling of Alfven waves with acoustic type modes in dense quantum plasmas, Phys. Plasmas 16, 052109 (2009)
• H. Saleem, Ali Ahmad and S. A. Khan, Low frequency electrostatic and electromagnetic modes in nonuniform cold quantum plasmas, Phys. Plasmas 15, 014503 (2008)
• H. Saleem, Ali Ahmad and S. A. Khan, Low frequency electrostatic and electromagnetic modes of ultracold magnetized nonuniform dense plasmas, Phys. Plasmas. 15, 094501 (2008)
• S. A. Khan, S. Mahmood and H. Saleem, Linear and nonlinear ionacoustic waves in very dense magnetized plasmas, Phys. Plasmas 15, 082303 (2008)
• S. A. Khan and A. Mushtaq, Linear and nonlinear dust ionacoustic waves in ultracold quantum dusty plasmas, Phys. Plasmas 14, 083703 (2007)
Contents
1 Introduction 5
1.1 Dense Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Characteristics of Quantum Plasmas . . . . . . . . . . . . . . . . . . . . 6
1.3 Developments in Quantum Plasmas . . . . . . . . . . . . . . . . . . . . . 13
1.4 Waves and Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Layout of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Mathematical Models for Quantum Plasmas 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Schrodinger-Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Wigner-Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Quantum Hydrodynamic Model . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Schodinger-Poisson Approach . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Wigner-Poisson Approach . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Applications of Quantum Hydrodynamic Theory . . . . . . . . . . . . . . 35
3 Linear Modes in Nonuniform Ultracold Quantum Plasmas 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Quantum Drift and Inertial Alfven Waves . . . . . . . . . . . . . . . . . 41
3.2.1 Set of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Linear Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . 44
1
3.2.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Drift, Acoustic and Inertial Alfven Waves. . . . . . . . . . . . . . . . . . 49
3.3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Linear Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 53
4 Low Frequency Linear Modes in a Homogenous Quantum Plasma 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Dust Ion-Acoustic Wave in Unmagnetized Quantum Plasmas . . . . . . . 56
4.2.1 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Ion Waves in a Quantum Magnetoplasma . . . . . . . . . . . . . . . . . . 59
4.3.1 Applications, Results and Discussions . . . . . . . . . . . . . . . . 61
4.4 Fermionic pressure and quantum pressure . . . . . . . . . . . . . . . . . . 63
4.5 Linear coupling of Alfven waves and acoustic type modes . . . . . . . . . 65
4.5.1 Dynamic ions and electrons . . . . . . . . . . . . . . . . . . . . . 66
4.5.2 Immobile ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Nonlinear Electrostatic Waves in Homogenous Quantum Plasmas 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Korteweg-de Vries equation . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Quantum Dust Ion-Acoustic Solitary Waves . . . . . . . . . . . . . . . . 79
5.3.1 Small Amplitude Waves . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Nonlinear Ion Waves in Quantum Magnetoplasmas . . . . . . . . . . . . 86
5.4.1 Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . . 86
5.4.2 Ion Solitary Wave Solution of KdV Equation . . . . . . . . . . . . 90
6 Summary 94
2
List of Figures
1-1 Temperature dependance of Fermi-Dirac distribution for electrons. . . . . 10
3-1 The SV-mode dispersion relation !(rad=sec) vs ky � 106(cm�1) is plotted
for the case of cold dense hydrogen plasma having ne s 0:5 � 1026cm�3,
nd s 0:05ni and Te = 1eV: Solid curve corresponds to equation (3.15) in-
cluding quantum corrections (!�q 6= 0) and dashed curve without quantum
corrections.(!�q = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3-2 The Inertial Alfven wave dispersion relation !(rad=sec) vs ky(cm�1) is plot-
ted in an electron-ion dense hydrogen plasma having ne s 1026cm�3and
Te = 1eV: Solid and dotted curve represents two branches of Alfven wave
whereas dashed curve corresponds to ! = qe which couples with lower
branch of Alfven wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3-3 Four modes of inhomogeneous ultracold dense plasma are shown for ky =
4:3� 105 cm�1, n0 �= 1026cm�3 and B0 = 9� 105G: The two outer curves
correspond to shear Alfven wave and inner ones to electrostatic wave. The
modes without dispersion are represented by the dashed curves. . . . . . 53
3-4 Plot of ! vs kz of four modes for relatively higher density and magnetic
�eld i.e., n0 �= 1028cm�3 and B0 �= 108G with ky = 3:5 � 106 cm�1:The
two outer (inner) curves correspond to shear Alfven (electrostatic) wave
whereas the modes without dispersion are represented by the dashed curves. 54
3
4-1 E¤ect of quantum di¤raction on a linear wave in dust contaminated quan-
tum plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4-2 In�uence of dust concentration on linear waves . . . . . . . . . . . . . . . 60
4-3 Linear dispersion relation of quantum ion wave in the dense Hydrogen
plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4-4 The Alfven wave frequency ! is plotted versus kz and k? for mobile ions
(upper panel) without and (lower panel) with inclusion of electron fermi-
onic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4-5 The quantum ion-acoustic wave frequency ! is plotted against kz and k?
(upper panel) without and (lower panel) with inclusion of the electron
fermionic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4-6 The linear dispersion relation for immobile ions is plotted with ne0 �=
1� 1024cm�3 and B0 �= 1� 108G. . . . . . . . . . . . . . . . . . . . . . . 74
5-1 The variation of soliton pro�le with quantum di¤raction e¤ects . . . . . 83
5-2 Plot of soliton�s widthW versus constant speed u0 with variation of quan-
tum parameter H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5-3 Electrostatic potential � as a function of � for di¤erent values of � . . . . 85
5-4 Variation of electrostatic potential � with � for di¤erent values of dust
concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5-5 Solitary waves solution of the ion wave in dense strongly magnetized quan-
tum plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5-6 In�uence of strong magnetic �eld on ion solitary wave . . . . . . . . . . . 93
4
Chapter 1
Introduction
The scope and fundamental properties of dense quantum plasmas are discussed in this
chapter. The light is shed on the recent developments in quantum plasmas emphasizing
the collective e¤ects. Then, the outline of this dissertation is presented.
1.1 Dense Plasmas
The plasma is an essential stage in the process of formation of matter from elementary
particles up to condensed matter. Generally, the word plasma is referred to as a statistical
system of charged particles, for instance, electrons and di¤erent ions, exhibiting collective
behaviour due to the long range coulomb forces. Plasmas are characterized by regimes
of high temperature and low density commonly found in space (e.g., interplanetary and
interstellar media) as well as in laboratory (e.g., gas discharges and thermonuclear fusion
experiments). The dynamics of a plasma is governed by internal �elds produced by the
plasma particles and the externally applied �elds [1].
The charged particle systems with su¢ ciently high density and low temperature also
exhibit plasma e¤ects, the most obvious example being the gas of free electrons in an
ordinary metal or semi-metal. Plasmas produced in laboratories by various compression
techniques, e.g., diamond anvils, launch of shock waves into matter, high current pinch
5
e¤ect, laser or ion beams etc. are some other examples of high density plasmas in which
particle number density may be upto 1024 cm�3. Dense plasmas are also found in nature.
The plasmas in the interior of Jovian planets (Jupiter, Saturn), brown and white dwarfs,
and neutron stars crust are believed to be ultradense. For instance, the density in the
neutron star interior can be upto 1036 cm�3: But the study of collective e¤ects at such
densities is very complicated. Di¤erent types of nonidealities and correlations give rise
to additional complexities in dense plasmas. The quantum mechanical e¤ects can�t be
ignored and many unusual phenomena like tunneling of electrons, pressure ionization,
condensation, and crystallization etc. can also be important. Although the temperature
of dense astrophysical plasmas is very high, yet quantum e¤ects cannot be ignored due
to restrictions of Pauli�s principle. The dense quantum plasmas provide promises of
important scienti�c applications in future [2; 3].
The full description of dense quantum plasmas is a major challenge from a theoretical
perspective. As soon as we attempt to model such a plasma, a systematic approach is
necessary using certain number of assumptions to obtain a tractable mathematical model
starting from basic phenomena. However, the collective behaviour can also be described
by using a self consistent �uid approach [2]. This provides us a relatively simpler way to
study the dynamics of dense plasmas in comparison with the complex quantum statistical
methods.
1.2 Characteristics of Quantum Plasmas
In a traditional classical equilibrium plasma, the distribution of plasma particles of species
� is given by the well known Boltzmann distribution function [4]
f� (�) =1
e�(����); (1.1)
where � = 1=kBT; kB the Boltzmann constant, T is the equilibrium temperature and
� = p2=2m�with �� and m� being the chemical potential and particle mass of � species.
6
For such plasmas, the classical coupling parameter may be de�ned as
�C =jhUijhKi _
3pn�T
; (1.2)
where hUi = 12
Pi6=j
eiej�ris the two-particle Coulomb interaction (potential) energy, hKi =
32kBT is the average kinetic energy and n� is the particle number density. The average
interparticle distance �r is given by
�r = hri � rji _1
3pn�� (1.3)
The parameter �C may also be written in the form�
1n��
3D
�2=3where �D =
�2kBT4�n�e2
�1=2is the Debye screening length. The ordering �C << 1 corresponds to collisionless and
�C ' 1 to collisional regime in classical plasmas. So, a classical plasma can be said
collisionless (ideal) when long-range self-consistent interactions (described by the Poisson
equation) dominate over short-range two-particle interactions (collisions).
When the density is very high , the average interparticle distance �r become comparable
to thermal de Broglie wavelength of a charged particle de�ned as
�B� =hp
2�m�kBT; (1.4)
where h is the Planck�s constant. Then the degeneracy e¤ects cannot be neglected i.e.,
1 . n��3B� and the quantum mechanical e¤ects alongwith collective (plasma) e¤ects be-
come important at the same time. Such plasmas are also referred to as quantum plasmas.
Some common examples are electron gas in an ordinary metal, high-density degenerate
plasmas in white dwarfs and neutron stars, and so on. From quantum mechanical point
of view, the state of a particle is characterized by the wave function associated with the
particle instead of the trajectory in phase space, and the Heisenberg uncertainty prin-
ciple leads to the fundamental modi�cations of classical statistical mechanics. The de
Broglie wavelength has no role in classical plasmas because it is too small compared to
7
the average interparticle distances. There is no overlapping of the wave functions and
consequently no quantum e¤ects. So the plasma particles are considered to be point like
and treated classically.
However, in quantum plasmas, the overlapping of the wave functions associated with
the particles takes place which introduces novel quantum e¤ects [2]. It is clear from (1.4)
that the de Broglie wavelength depends upon mass of the particle and thermal energy.
That is why, the quantum e¤ects associated with electrons are more important than the
ions due to smaller mass of electron which quali�es electron as a true quantum particle.
The behaviour of many particle system is now essentially determined by statistical laws.
The plasma particles with symmetric wave functions are termed as Bose particles and
those with antisymmetric wave function are called Fermi particles. We can subdivide
plasmas into (i) quantum (degenerate) plasmas if 1 < n��3B� and (ii) classical (non-
degenerate) plasmas if n��3B� < 1. The border between the degenerate and the non-
degenerate plasmas is roughly given by
n��3B� = n�
�hp
2�m�kBT
�3= 1: (1.5)
For quantum plasmas, the Boltzmann distribution function (1.1) is strongly modi�ed to
Fermi-Dirac or Bose-Einstein distribution functions in a well known manner. For Fermi
particles, i.e., for plasma particles with spin 1=2; 3=2; 5=2; � � �; the distribution function
takes the form
f� (�) =1
[e�(����) + 1]; (1.6)
which is called the Fermi-Dirac distribution function. Similarly, the plasma particles with
spin 0; 1; 2; 3; � � � (or bosons), obey the Bose-Einstein distribution function given by
f� (�) =1
[e�(����) � 1] � (1.7)
The di¤erent signs in the denominators of (1.6) and (1.7) are of particular importance
8
at low temperatures. For fermions, this leads to the existence of the Fermi energy (Pauli
principle), and for bosons, to the possibility of the macroscopic occupation of the same
quantum state which is the well known phenomenon of Bose-Einstein condensation.
Let us consider a degenerate Fermi gas of electrons at absolute zero temperature. The
electrons will be distributed among the various quantum states so that the total energy
of the gas has its least possible value. Since each state can be occupied by not more
than one electron, the electrons occupy all the available quantum states with energies
from zero (least value) to some largest possible value which depends upon the number of
electrons present in the gas. The corresponding momenta also starts from zero to some
limiting value. This limiting momentum is called the Fermi momentum pF given by [5]
pF = }�3�2n
�1=3; (1.8)
where } is the Planck�s constant divided by 2� and n is the number density of electrons.
Similarly, the limiting energy is called the electron Fermi energy �F which is
�F =p2F2m
=}2
2m
�3�2n
�2=3; (1.9)
with m being the electronic mass. The Fermi-Dirac distribution function (1.6) becomes
a unit step function in the limit T ! 0. It is zero for � < � and unity for � < �. Thus
the chemical potential of the Fermi gas at T = 0 is same as the limiting energy of the
fermions i.e.,
� = �F ; (1.10)
as shown in Fig. (1-1). It is worth mentioning here that the statistical distribution of
electrons changes from the Maxwell-Boltzmann to Fermi-Dirac whenever T approaches
the so-called Fermi temperature TF ; given by
kBTF � �F =}2
2m
�3�2n
�2=3 � (1.11)
9
Figure 1-1: Temperature dependance of Fermi-Dirac distribution for electrons.
It means that the quantum e¤ects are important when 1 . TF=Te. In dense plasmas,
the plasma frequency !p = (4�n0e2=m)
1=2 becomes su¢ ciently high due to very large
equilibrium particle number density. Consequently, the typical time scale for collective
phenomena (!p)�1 becomes very short. The thermal speed vT = (kBT=m)
1=2 is su¢ -
ciently smaller than the Fermi speed given by
vF =
�2�Fm
�1=2=}2
m
�3�2n
�1=3 � (1.12)
which is the speed of an electron at the Fermi surface. With the help of plasma frequency
and Fermi speed, we can de�ne a length scale for electrostatic screening in quantum
plasma i.e., the Fermi screening length �F = vF=!p which is also known as the quantum-
mechanical analogue of the Debye length in dense electron gas.
The adiabatic equation of state for a quantum electron gas at very low temperatures
can be written as [2]
P = P0
�n
n0
� ; (1.13)
10
where P0 (n0) is the equilibrium pressure (density). The exponent = (d+ 2) =d
where d = 1; 2; 3 denotes the dimensionality of the system. Any thermodynamic process
which is typically characterized by a relatively faster change of state so that the system
undergoing change doesn�t have time to exchange signi�cant amount of heat with its
surroundings is called an adiabatic process. The adiabatic compressions are faster than
the heat conduction.
If the electrons behave as an almost completely degenerate gas, the equation of state
becomes relatively simpler. In this case, the electrons obey Fermi-Dirac statistics and
are restricted to occupy energy states according to Pauli�s exclusion principle. The Fermi
pressure increases with increase in number density of electrons. It is evident from equation
(1.4) that the de Broglie wavelength associated with an electron is much larger than that
of an ion. Also the Fermi temperature of ions is much less than the Fermi temperature
of electrons. That is why in many cases, the ion dynamics is considered classical in �uid
approximation in dense systems. The temperature de�ned by the relation kBTF �= �F is
called the degeneracy temperature of electrons and the condition T < TF always holds
in dense degenerate plasmas.
The pressure law for degenerate electron gas can be written by using equation (1.13).
In three dimensions, = 5=3 and P0 = (2=5)n0�F which re�ects fully 3D equilibrium
leading to P = (}2=5m) (3�2)2=3n5=3 for kBT << �F corresponding to the pressure of
dense Fermi gas of electrons at absolute zero or nearly absolute zero temperature [5]. For
one dimensional case, = 3 and P = (mv2F=3n20)n
3:
Like classical plasmas, a coupling parameter can be de�ned in a quantum plasma.
For strongly degenerate electron gas, the interaction energy may still be given by hUi,
but the kinetic energy is now replaced by the Fermi energy. This leads to the quantum
coupling parameter
�Q =hUi�F
_3pn
TF� (1.14)
Since TF _ n2=3 which shows that �Q _ n�1=3, so the peculiar property of quantum
plasma is that it increasingly approaches the more collective (ideal) behaviour as its
11
density increases. Another useful form of �Q is as follows
�Q =
�1
n�F
�2=3=
�}!p�F
�2=3; (1.15)
which shows the resemblance with classical coupling parameter. It is important to men-
tion here that �D is the range of screened Coulomb potential (Debye length) in a classical
Maxwellian plasmas (non-degenerate) which is associated with mean kinetic (thermal)
energy hKi = d2kBT of plasma species. On the other hand, in a dense degenerate Fermi
plasma, one can�t de�ne average kinetic energy in the same way as it is done in classical
case. Instead of average kinetic energy, the Fermi energy is used to de�ne the screening
length called the Fermi screening length �F which is a function of density only [2; 3]:
The screening length �F is signi�cant in the temperature regime T < TF . The para-
meter �=T in Fermi-Dirac distribution (1.6) is large and negative in non-degenerate
plasmas, and is larger and positive in completely degenerate plasmas. This shows that
exp(�=kBT ) << 1 in non-degenerate limit and exp(�=kBT ) >> 1 in the completely
degenerate limit corresponding to � �! �F and �F=kBT >> 1:
Since the fermions are spin half particles, the magnetic dipole moment of the electron
associated with spin introduces a magnetic dipole force and spinor e¤ects, as well as new
current sources in the Ampere law [6; 7]. The magnetic moment of the electron may be
written as
m = ��B h j�j ih j i ; (1.16)
which gives a contribution �B �m to the energy, where is the electron wave function,
B is the magnetic �eld, �B (= e}=2mc) is the Bohr magneton and � = (�1; �2; �3) gives
the Pauli spin matrices i.e.,
�1 =
0@0 1
1 0
1A ; �2 =
0@0 �ii 0
1A and �3 =
0@1 0
0 �1
1A : (1.17)
The spin e¤ects are signi�cant when energy di¤erence between two spin states is com-
12
parable to thermal energy. The spin e¤ects associated with ions are usually smaller due
to their larger mass. A variety of new dynamic e¤ects are associated with microscopic
and macroscopic spins e.g., new modes and wave particle resonances may appear in spin
1=2 plasmas. The transverse motion of the charged particles in magnetized plasmas can
also be quantized into Landau levels. Apart from the spin degeneracy, the Landau level
is degenerate by itself which tends to smear out at high temperatures [8]. The e¤ects of
the Landau quantization are important in variety of situations in dense plasmas at high
magnetic �elds:
1.3 Developments in Quantum Plasmas
The �eld of quantum plasmas has been introduced since long ago. Klimontovich and Silin
[9] derived a general kinetic equation for quantum plasmas and studied the dispersion
properties of electromagnetic waves. Some other developments of that time include the
equilibrium theory of quantum plasmas using a procedure similar to Feynmann�s meth-
ods in �eld theory [10], dielectric formulation of quantum statistics in random phase
approximation [11]; and the self consistent �eld approach to many-electron problem [12].
For nonequilibrium homogenous systems, kinetic equations have been derived by Balescu
[13]. Guernsey [14] used an approach originally developed by Bogoliubov to present a
uni�ed theory of equilibrium and nonequilibrium quantum plasmas.
Pines [15] studied the dynamics of quantum plasmas with particular attention to the
relationship between individual particle and collective behaviour. Emphasizing the exci-
tation spectrum of quantum plasmas, theoretical investigations of the author describes
the dispersion properties of electron plasma oscillations involving the electron tunnel-
ing. A general theory of electromagnetic properties of the electron gas in a quantizing
magnetic �eld was also developed treating the electrons quantum mechanically [16; 17].
Since the pioneering work of these authors which laid foundations of quantum plasmas,
many theoretical studies have been done in the subsequent years. Bezzerides and Du-
13
Bios [18] have studied the quantum electrodynamic properties of nonthermal plasmas by
developing a many-particle kinetic model. Hakim and Heyvaerts [19] investigated the
relativistic quantum plasmas using the covariant Wigner function formalism.
Quantum plasmas have received much attention during the last decade due to variety
of reasons. The main reason for this interest is the manifold applications of quantum plas-
mas from nanoscience [20] to astrophysics [21]. There are two main kinds of many-particle
e¤ects in quantum plasmas: (a) collective (mean �eld) e¤ects leading to synchronized
excitations like plasma oscillations, waves, instabilities, etc., (b) correlation e¤ects such
as polarization, screening, pressure ionization, etc. Having known that the quantum
mechanical e¤ects play a crucial role in the dynamics of quantum plasmas; various theo-
retical and computational approaches of quantum statistical theory have been employed
to describe such systems in di¤erent limits [2; 4; 22; 23]:
Plasmas found in the astrophysical domain, e.g., in the interior of Jovian planets,
white dwarfs, neutron stars etc. have enormously high densities and magnetic �elds
[8; 21]. Despite 105K . T; these ultradense plasmas possess strong quantum e¤ects
and exhibit �uid and crystal properties in a quantum sea of electrons [3; 24]. Quantum
e¤ects are also found important in dense laboratory plasmas in intense laser �elds [25]
and laser-based inertial fusion experiments [26]:
A spectrum of phenomena which recently became more important reveals the exis-
tence of several quantum aspects of the physics of accelerated particles with the frontiers
of several other disciplines, such as plasma physics, radiation beam physics, mesoscopic
and condensate physics, and so on [27]. Most of these phenomena introduce a sort of
quantum correction to the leading classical behaviour of the system. For example, quan-
tum excitation plays a role for stability of the longitudinal electron beam dynamics in
the high-energy accelerating machines [28]: Similarly, numerical phase space investiga-
tions have shown that quantum corrections can substantially a¤ect the particle beam
trajectories [29].
The investigations of quantum plasma dynamics span from non-relativistic regime
14
with its description in Schrodinger or Heisenberg picture of non-relativistic quantum
theory to strongly relativistic domain where quantum electrodynamics (QED) and quan-
tum �eld theory become applicable [30]. The search for the relevance of classical and
quantum physics has been an area of interest since the early days of quantum mechanics.
In 1926, Madelung proposed the hydrodynamic description of quantum mechanics and
demonstrated that the Schrodinger equation can be transformed in hydrodynamic form
[31]. The Madelung interpretation is closely related to the work of de Broglie and later
on popularized by Bohm in 1952 [32]. In Madelung �uid description, the wave function,
say ; being a complex quantity, is represented in terms of modulus and phase. When
substituted in Schrodinger equation, it leads to the pair of nonlinear �uid equations for
the density and the current velocity. One is the continuity equation giving the conser-
vation of probability and other is the Navier-Stoke type momentum balance equation
which introduces the quantum potential term. The hydrodynamic or �uid approach is
relatively simpler in comparison to the complex description used in the statistical mod-
els. This description has been popular in many important areas of physics ([33; 34]
and refs. therein) and used by Manfredi and collaborators in the recent years to de-
rive the quantum hydrodynamic model [2; 35; 36]. The electron dynamics is described
by a set of hydrodynamic equations (typically, continuity and momentum conservation)
that include quantum e¤ects via a Bohm-like potential. Quantum statistics and the new
force associated with quantum Bohm potential introduces the pressure e¤ects of pure
quantum origin. Several studies have appeared in literature in the recent years showing
the important contribution of Bohm potential (quantum pressure) and quantum statis-
tical e¤ects in dense plasmas [37; 38; 39; 40]. Also the magnetohydrodynamic (MHD)
model and multistream model for quantum plasmas have been developed [41; 42]. Some
further developments include quantum electrodynamic e¤ects [43], solitons and vortices
in quantum plasmas [44], quantum Hall-MHD equations [45], waves and instabilities
[46; 47; 48; 49], trapping in quantum plasmas [50], Landau damping e¤ects [51] and
turbulence in quantum plasmas [52].
15
Marklund and Brodin [6; 7] have derived the multi�uid equations for spin 1/2 quan-
tum electron plasma using the Pauli equation. Similarly, Brodin and Marklund [53] have
studied the spin and QED e¤ects in quantum plasmas in the framework of Maxwell-
Fluid equations. The authors point out di¤erent limits where the Bohm potential, Fermi
pressure, spin and QED e¤ects become important. Brodin et. al. [54] have shown the
signi�cance of quantum e¤ects in a relatively higher temperature regime by studying the
Alfven waves in the presence of electron spin. The spin-up and spin-down electrons are
treated as di¤erent �uids.
Solid-state plasmas are known since many years [55]. The miniaturization in device
technology has entered in nanometer scale. The dynamics of charges on such ultrasmall
scales shows the important role of collective plasma e¤ects [34]: Such e¤ects are also
recognized in dense metallic systems [35], nanostructures [56], quantum dots and quantum
wires [57], quantum wells, nanotubes and quantum diodes [58; 59], nonlinear quantum
optics [60]; microplasmas [61] and exotic ultracold plasmas [62].
1.4 Waves and Instabilities
Plasmas may con�ne enormous number of particles (e.g., electrons, ions, neutrals, dust
etc.) within a small volume. To describe the motion of these particles requires a cor-
responding enormous number of modes. In quantum plasmas, the situation becomes
more complicated due to implications of certain quantum mechanical rules, for instance,
quantum statistics, uncertainty principle, de Broglie length scales etc. That is why, the
modeling of quantum plasmas is not an easy task. To obtain a simpler and realistic math-
ematical model to study the dynamics of dense plasmas, a certain number of assumptions
are required. The resulting model throws some light on peculiarity of quantum e¤ects in
dense plasmas. It has been known for a long time that a set of quasi-�uid equations can
be derived from Schrodinger equation [31; 63]: The �uid models are simpler to handle
and numerically e¢ cient. Moreover, basic physical phenomena in a dense plasma can
16
be described by using �uid models which incorporate the lower order quantum e¤ects.
The simpli�cation of N-body Schrodinger equation using Madelung �uid approach leads
to Schrodinger Poisson (SP) model which is a useful approaches to study the hydrody-
namic behaviour of quantum plasmas [2]. Using the Wigner phase space interpretation
of quantum mechanics [64], Manfredi and Haas have derived an e¤ective SP system for
completely degenerate electron gas [35]. In this e¤ective SP model, the Schrodinger equa-
tion is nonlinear as it includes an e¤ective potential depending on the modulus of the
wave function. The pressure of electrons contains two parts i.e., the quantum pressure
contributed by Bohm potential and the classical like pressure which is function of density.
In linear limit, it was found that the dispersion relation resembles the results of classical
Vlasov-Poisson model. But the system at very low temperature obey Fermi-Dirac equi-
librium. The model has also been used to study the nonlinear stationary solutions and
the two-stream instability.
The well established phenomena of classical plasma physics like electrostatic and elec-
tromagnetic wave dynamics, wave-wave and wave-particle interactions, dielectric prop-
erties, etc. show signi�cant modi�cations and sometime behaves in anomalous ways
at quantum scales. This has motivated the studies of numerous types of collective ef-
fects in degenerate quantum plasmas in the past few years using some form of quantum
transport models. This include linear waves and nonlinear structures in homogenous
and inhomogenous quantum plasmas [37; 39; 44; 46; 48; 65; 66; 67], quantum plasma
instabilities [40; 42; 49; 68; 69; 70], modi�ed plasma modes [71; 72; 73] and quantum
and electrodynamic corrections to spin plasmas [43; 53; 74].
Spatial nonuniformities are common feature of plasmas. These nonuniformities sup-
port various modes, for example, macroscopic MHD type modes or small scale drift
type modes. Many types of nonlinear phenomena like instabilities, turbulence and the
formation of nonlinear structures like solitons, shocks, vortices, double layers etc. may
occur as well. The electron and ion mass di¤erence and correspondingly their slow and
rapid response to the electric �eld in the presence of a density gradient is a typical fea-
17
ture of the drift wave. Shokri and Rukhadze [75] have studied the quantum drift waves
in a two-component inhomogenous quantized plasma in the presence of strong mag-
netic �eld. They have taken the cases of quantum electrons (Te << }!ce) and classical
ions (}!ci << Ti) as well as ultra quantum case when both electrons and ions manifest
quantum behaviour (Te << }!ce; Ti << }!ci). The authors have shown that the volume
quantum drift waves may exist in such plasma which under certain circumstances, can
become unstable. Haas [41] has developed the quantum magnetohydrodynamic (QMHD)
model and established the conditions for ideal magnetohydrodynamic equilibrium. The
QMHD model has been employed to investigate new drift-like linear modes in a quan-
tum magnetoplasma containing an equilibrium density inhomogeneity [71]. The authors
have pointed out the excitation of such modes due to sheared plasma �ows, and their
contribution to the cross-�eld charged particle transport in quantum plasmas, e.g., in
microelectromechanical systems and dense ionized media.
Shukla and Ali [76] have investigated the existence of electromagnetic drift modes in
nonuniform quantum magnetoplasmas with �2ek2? << 1 where �e = c=!pe. Their results
show that the electron quantum nature signi�cantly alters the wave frequencies. The free
energy source is the density gradient, which couples to di¤erent modes and may support
instability on account of the pressure force at quantum scales associated with the elec-
trons. We have studied the low frequency (! << !ci) modes in inhomogenous magnetized
quantum plasmas and obtained the dispersion relation of linearly coupled drift wave and
the inertial Alfven wave in an electron-ion quantum plasma [77] for the case �2ek2? 6= 0.
In the presence of stationary dust background, the system leads to the electrostatic Shukla-
Varma mode with quantum corrections. The numerical studies corresponding to dense
laboratory and astrophysical quantum plasmas have been presented in detail to elaborate
the relevant length scales for the signi�cance of quantum e¤ects.
The quantum pressure (due to Bohm Potential) is purely a quantum mechanical
phenomena having no classical analogue. The existence of drift-type wave is possible in
an inhomogenous quantum plasma which ceases to exist in classical plasmas. It can make
18
the �uctuations of plasma parameters possible in ultracold temperature limits leading to
the low frequency modes. Keeping this in mind, we have studied the basic electrostatic
and electromagnetic modes in the ultracold nonuniform dense magnetoplasma [78]. We
have obtained a coupled linear dispersion relation which contains quantum drift wave,
electrostatic wave and Alfven wave. Interestingly, it is analogous in form to the classical
case [79] but very di¤erent physically. It shows that the drift like mode may exist even
if the gas temperature is negligibly small. We have de�ned an e¤ective temperature in
energy units which is based on the quantum pressure term in electron momentum equation.
The results are analyzed numerically pointing out the relevance of this work to the dense
plasmas.
Haas et. al. [37] have derived the one-dimensional QHD model for unmagnetized
quantum plasmas. The authors have studied the quantum ion-acoustic waves in linear
and nonlinear regime and found several new e¤ects of purely quantum origin. The quan-
tum e¤ects of di¤raction are represented by the parameter H = }!p=2kBTF which is
proportional to the ratio between the electron plasmon energy and the electron Fermi
energy. The presence of charged impurities (dust) in quantum plasma leads to quantum
dust contaminated (dusty) plasma. Collective modes in such plasmas have been studied
by Shukla and collaborators [46; 72; 80]: The authors have shown that the quantum
nature of electrons in a dense dust contaminated plasma leads to various new e¤ects
and discussed the importance of such modes in laboratory systems e.g., metallic and
semiconductor nanostructures and microelectromechanical systems. The dense plasmas
of astrophysical domain are also believed to be contaminated with heavier species (dust)
[46]. The force associated with quantum electrons acts like a pressure force leading to
novel e¤ects. Keeping this in view, we investigate the low frequency linear waves in ho-
mogenous quantum plasmas which may or may not be magnetized. We have shown that
the phase speed of dust ion-acoustic wave in unmagnetized quantum plasma increases with
dust concentration and quantum e¤ects of di¤raction [81]. In the presence of uniform
magnetic �eld, we study the obliquely propagating ion waves in electron ion quantum
19
plasma. We have shown that the dispersion caused by quantum e¤ects is possible only
in a very short wavelength regime [82]. In the absence of quantum e¤ects, the linear
dispersion relation looks like the case of classical plasmas. It is seen that the di¤raction
parameter increases with magnetic �eld and decreases with electron number density.
For very dense plasmas, the Fermi temperature of electrons is much larger due to
very high density. So, even in the simpli�ed hydrodynamic model, it is reasonable to
compare the statistical pressure arising due to fermionic character of electrons and the
quantum pressure due to quantum Bohm potential in the ultracold magnetoplasma. In
this context, we study the shear Alfven waves and electrostatic waves in a homogenous
dense quantum magnetoplasma using QMHD model [83]. The dispersive contribution of
electron quantum e¤ects on coupled electrostatic and electromagnetic modes is discussed
for dynamic as well as static ions. The dominant role of electron fermionic pressure is
highlighted and its comparison with the quantum pressure arising due to quantum Bohm
potential is presented indicating its limits in ultracold dense plasmas. For illustrative
purpose, the results are analyzed numerically. The relevance of the study with the dense
astrophysical and laboratory plasmas is pointed out with possible consequences.
One can �nd coherent structures (solitons) by using the generic Korteweg-de Vries
(KdV) model [84] of the �uid approximation of classical plasma. It is recognized that the
presence of charged dust impurities in quantum plasma give rise to new modes [46; 72].
The behaviour of massive dust particles in quantum plasmas is essentially classical since
the de Broglie wavelength associated with the dust particle is much smaller than the
average interparticle distance. However, quantum corrections appearing in dense low
temperature dust contaminated plasmas due to electron quantum behaviour have an
e¤ect on the ion-acoustic waves. We have studied the nonlinear ion-acoustic waves in
unmagnetized dense quantum plasma in the presence of stationary dust [81]. For this
purpose, a KdV equation is derived and its localized solution is presented. The results
are analyzed numerically and the e¤ects of dust density and quantum pressure on solitons
are discussed.
20
When a dense quantum plasma is immersed in a uniform external magnetic �eld,
the strength of magnetic �eld and the angle of propagation of wave with the magnetic
�eld contribute to the dispersion of nonlinear waves. We have investigated the obliquely
propagating ion solitary waves in a dense quantum magnetoplasma using quantum hy-
drodynamic formulation [82]. The in�uence of the quantum e¤ects on ion acoustic type
soliton has been noticed. The dependence of solitary pulse on propagation angle and the
plasma number density is investigated and results are discussed in various limits. Possible
applications of our results in dense plasmas of laboratory and astrophysical environment
are also discussed in some detail.
1.5 Layout of the Thesis
In chapter one, we have given the brief introduction and properties of dense quantum
plasmas outlining the recent developments in this area. In the next chapter, the well
known quantum plasma models namely the Schrodinger-Poisson and the Wigner-Poisson
model have been described. We have also presented the basic derivations of quantum
hydrodynamic model in the same chapter. In chapter three, we have studied the low fre-
quency electrostatic and electromagnetic linear modes in nonuniform ultracold quantum
plasmas. Further, we have investigated the coupled electrostatic and electromagnetic
modes i.e., the quantum drift waves, quantum ion acoustic waves and Alfven waves in
the same chapter. In chapter four, we have presented the study of linear waves in ho-
mogenous quantum plasmas emphasizing the e¤ects of fermionic pressure. A quantitative
comparison of fermionic pressure and quantum pressure due to Bohm potential is made.
The dispersion relations of dust ion-acoustic waves in unmagnetized and ion-acoustic type
waves in a magnetized quantum plasma are also presented. The coupled electrostatic and
electromagnetic modes of electron ion quantum magnetoplasma is also discussed in this
chapter for dynamic as well as static ions. In chapter �ve, we have given the nonlinear
studies of quantum dust ion-acoustic waves in unmagnetized quantum plasma as well as
21
ion solitary waves in an electron ion quantum magnetoplasma. The summary of the work
is presented in chapter six.
22
Chapter 2
Mathematical Models for Quantum
Plasmas
The mathematical models which are widely used as a tool for description of the dense
quantum plasmas are discussed. Schrodinger-Poisson and Wigner-Poisson models are
brie�y described and quantum hydrodynamic model is presented in some detail. Further-
more, a few important applications of quantum hydrodynamic theory are pointed out.
2.1 Introduction
Most physical systems-atoms, molecules, solids, �uids, gases, plasmas etc. involve many
particles and hence are called many-particle or many-body systems. The number of
particles in atoms or molecules (electrons, nucleons etc.) may be upto few hundreds
only. But in solids, �uids, gases and plasmas, one deals with truly very large number
of particles. The dynamics of plasmas is typically peculiar due to many reasons. A
plasma consists of freely moving charged particles, which produce charge and current
densities and, therefore, an electromagnetic �eld through which the plasma particles
interact. In general, the dynamics of the plasma particles and that of the �eld have to
be dealt self consistently. For many problems, it is su¢ cient to account only for the
23
Coulomb interaction between the plasma particles. The long range Coulomb interaction
leads to the collective behaviour of the plasma particles such as dynamic screening and
plasma oscillations. In theoretical description, the long range character leads to special
di¢ culties in the determination of thermodynamic or transport properties. Binary or
few-particle approximations are not appropriate to describe the collective interaction of
plasma particles. Additional di¢ culties arise when one deals with dense low temperature
quantum plasmas. Various new e¤ects at very short time scales make the picture very
complicated. Theoretical description of such plasmas demands the knowledge of various
types of many-body e¤ects which is very challenging task. However, a reductive principle
of research is usually fruitful, for which one successively builds more complex models
based on previous results. Drastic simpli�cations are needed in some cases which provides
only a partial description of quantum plasmas.
2.2 Schrodinger-Poisson Model
Quantum mechanically, N identical particles are truly indistinguishable. The underlying
basis for this is twofold. First, to describe a particle, we cannot specify more than a
complete set of commuting observables. In particular, there exists no mechanism to
tag the particles as in classical mechanics. Second, due to uncertainty principle, the
concept of the path of a particle becomes meaningless. Even if the position of a particle
is exactly determined at a time, it is not possible to specify its coordinates at the next
instant. Thus identical particles lose their identity (individuality) in quantum mechanics
and we can only use the probabilistic approach for a certain particle to be located at a
certain position. The motion in the centre-of-mass coordinates of the system is not of
interest, since in the absence of external �elds, the centre of mass is either at rest or
moves with a constant velocity. Then, how does one describe the dynamics of the system
of N particles?
This description become possible only by generalization of the single particle dynam-
24
ics. In dense quantum plasmas, there are enormously large number of particles within
a small volume. An accurate description of a quantum plasma requires the solution of
N -particle Schrodinger equation. The Schrodinger equation is the fundamental model
and one of the corner stones of quantum theory but can never be solved for N particle
wave functions [85]. To deal with the N -body problem in Schrodinger�s picture, a dras-
tic but useful, and somewhat justi�able simpli�cation can be achieved by assuming the
neglect of two-body and higher order correlations. The small values of quantum coupling
parameter given by (1.14) validate this assumption of weak correlations. Then, the N -
particle wave function (x1; x2; � � ��; xN ; t) can be written as the product of N individual
(one-particle) wave functions 1(x1; t) 2 (x2; t) � � � � N(xN ; t). For fermions, none of the
product wave functions should be identical. This leads to the solution of the N -particle
Schrodinger eigenvalue equation to be the product of uncorrelated systems such that the
measurement of the properties of one particle can be made independent of the others.
Necessarily, the entanglement of states is disregarded in this assumption. Then the N
wave functions i (x; t) lead to the N independent Schrodinger equations
i}@ i (x; t)
@t= � }
2
2m� i (x; t) + e' (x; t) i (x; t) ; i = 1; 2; � � � �N; (2.1)
where ' (x; t) is the self-consistent electrostatic potential given by the Poisson equation
�' (x; t) = 4�e
NXi=1
pi j i (x; t)j2 � n0
!: (2.2)
For simplicity of notations, only one spatial dimension is considered where � = @2=@x2.
Electrons having absolute charge e and mass m are globally neutralized by a �xed ion
background with density n0. The ion density may also be a continuous function of the
position coordinates ni (x). All the electrons are supposed to be in well de�ned quantum
states i (x; t). In other words, we are dealing with a statistical mixture of N pure states
i (x; t). The probabilities of occupation pi of various quantum states for Fermi particles
are de�ned through the Fermi-Dirac distribution i.e., pi =�e(���F )=kBT + 1
��1where the
25
conditionNPi=1
pi = 1 always holds: Such type of model was originally developed by Hartree
to study the self consistent e¤ects of atomic electrons on the Coulomb potential of the
nucleus.
The Schodinger-Poisson model (2.1)-(2.2) is simple and numerically e¢ cient that
contains two main ingredients for a quantum plasma, namely, long-range self-consistent
interactions and a quantum mechanical equation of motion. But neglects the dissipation,
spin and relativistic corrections. These e¤ects may be of importance in more realistic
models. Nevertheless, it is useful to start with simpli�ed models that capture the main
features of quantum plasmas and are convenient to study the macroscopic properties of
such systems. In some sense, this model is the quantum mechanical analogue of Vlasov-
Poisson model as most of the assumptions of both models are the same. For instance, the
collisions are neglected, only electrostatic interactions are taken into account and single
particle wave function is used.
2.3 Wigner-Poisson Model
In the standard formulation of quantum mechanics, the probability density �(x) in posi-
tion space x is given by square of the magnitude of the wave function, �(x) = j (x)j2 :
When (x) is known, it is easy to visualize the distribution �(x): Similarly, the dis-
tribution in the momentum space is also straight forward. It would be desirable to
have a function that displays the probability distribution simultaneously in coordinates
and momenta. In 1932, Wigner [64] suggested a phase space representation of quantum
mechanics by means of joint distributions of probabilities (more precisely, the quasiprob-
abilities). Wigner�s original goal was to �nd quantum correction to classical statistical
mechanics where the Boltzmann factors are expressed as functions of both coordinates
and momenta. The Wigner�s formalism has attracted considerable attention in various
disciplines of physics. The Wigner function has also been the objective of a detailed the-
oretical analysis ([34; 35] and refs. therein). Moreover, the Schrodinger-Poisson system
26
can be obtained in a completely equivalent form by making use of Wigner function.
The Wigner distribution for quantum mixture of states i(x; t); each characterized
by occupation probability pi is given by
W (x; v; t) =m
2�}
NXi=1
pi
Z 1
�1d� �i
�x+
�
2; t
�� i
�x� �
2; t
�eimv�=}; (2.3)
where one dimensional case is assumed with 0 6 pi satisfying the normalization conditionNPi=1
pi = 1; and sum extends over all states: It can describe the evolution of W (x; v; t)
under the action of electrostatic potential ' (x; t) : The underlying idea is that the quan-
tum transport can be seeded into generalized transport equations that are in the spirit
of Boltzmann transport equation, appropriately extended with terms that represents
quantum corrections. But the Wigner function doesn�t necessarily stay nonnegative in
its evolution process for some regions of phase space. Unlike the classical case, it can
therefore not be interpreted as a true probability density. However, it is real, normaliz-
able to unity and give averages just like the classical statistical case. For instance, the
expectation value (spatial average) of some quantity C (x; v) may be de�ned like
hCi =R R
W (x; v)C (x; v) dxdvR RW (x; v) dxdv
� (2.4)
It should be noted that some terms in C may not commute. Then it is necessary to estab-
lish a well de�ned ordering rule (or Weyl quantization rule) between classical variables
and quantum mechanical operators which de�nes methods of correspondence between
functions of operators and functions of commuting variables. Similarly, the Wigner func-
tion gives the correct spatial density n(x; t) in the form
n(x; t) =
Z 1
�1W (x; v; t)dv =
NXi=1
j i (x; t)j2 � (2.5)
In the absence of collisions and scattering, the time evolution of the Wigner function is
27
governed by following equation
@W
@t+ v
@W
@x+iem
2�}2
Z Zd�d�veim(v��v)�=}
�'
�x+
�
2
�� '
�x� �
2
��W (x; �v; t) = 0;
(2.6)
with ' (x; t) being the self consistent electrostatic potential. In order to account for
' (x; t), (2.6) has to be coupled with the Poisson equation
�' (x; t) = 4�e[n(x; t)� n0]; (2.7)
where n0 is the uniform background ion density. The assumptions and limitations of
Schrodinger-Poisson system are also valid for Wigner-Poisson model. However it also
allows one to work with pure as well as mixed states. Although theoretically equivalent
to Scrodinger-Poisson model, the computational treatment of quantum phase space in
Wigner-Poisson system is numerically expensive.
2.4 Quantum Hydrodynamic Model
The quantum hydrodynamic (�uid) model is a generalization of classical �uid model of
plasmas where the transport equations are expressed in terms of conservation laws for
particles, momentum and energy. The quantum hydrodynamic (QHD) model is a reduced
model that allows straightforward investigation of the collective dynamics rather than to
deal with complexities of Schrodinger-Poisson (2N equations) or Wigner-Poisson (phase
space dynamics) models. The Schodinger-Poisson as well as the Wigner-Poisson systems
lead to the set of QHD equations making use of the standard de�nitions of macroscopic or
averaged quantities like density, velocity, pressure etc. [2; 35]. Both of these approaches
produce equivalent results and the outcome is a simpli�ed model.
28
2.4.1 Schodinger-Poisson Approach
The Schrodinger-Poisson system (2.1)-(2.2) is particularly convenient to derive the quan-
tum hydrodynamic model, since it makes direct use of macroscopic quantities, such as
density and average velocity. The Schrodinger-Poisson equations may be taken as the
quantum-mechanical analog of Dawson multistream model of classical plasmas [86] which
results in continuity and momentum conservation equations of electrons in the framework
of Vlasov model. The N streams considered in Dawson model represent in�nitely thin
�laments of plasma each with some number density, velocity and probability. The same
line of reasoning has been used to the system (2.1)-(2.2) in quantum case by applying
the Madelung representation of the wave function to each stream [2; 42]. The pure state
i(x; t) is de�ned according to the relation
i = Ai(x; t) exp [iSi(x; t)=}] � (2.8)
where Ai (x; t) is the real amplitude and Si(x; t) is the real phase. The density ni and
the velocity vi of each stream are thus given by
ni = j ij2 = A2i ; vi = @xSi=m; (2.9)
Using (2.8)-(2.9) into (2.1)-(2.2) and separating the real and imaginary parts, the Schrodinger-
Poisson system reduces to the following equations
@ni@t
+@
@x(nivi) = 0; (2.10)�
@
@t+ vi
@
@x
�vi =
e
m
@'
@x+}2
2m2
@
@x
�@2xpnipni
�; (2.11)
where the Poisson equation is given by
@'
@x= 4�e(
NXi=1
ni � n0)� (2.12)
29
The set of equations (2.10)-(2.12) constitute the quantum multistream model which re-
duces to classical multistream model [86] in the limit }! 0. The last term on the r.h.s
of (2.11) refers to a purely quantum mechanical e¤ect which is alternatively interpreted
as the quantum pressure or the gradient of quantum Bohm potential.
Let us de�ne the global density n (x; t) and global average velocity u (x; t) as follows
n(x; t) =
NXi=1
pini ; u(x; t) =
NXi=1
pininvi � hvii � (2.13)
Multiplying (2.10) and (2.11) by pi, the probability of occupation of state i and summing
over i = 1; 2; ���; N , we obtain the continuity and momentum equations for global averaged
quantities n and u of electron �uid obeying Fermi-Dirac statistics as given below
@n
@t+@ (nu)
@x= 0; (2.14)�
@
@t+ u
@
@x
�u =
e
m
@'
@x+}2
2m
@
@x
�@2xpnpn
�� 1
mn
@P
@x; (2.15)
where
P (x; t) = mn
24 NXi=1
piniv2i =n�
NXi=1
pinivi=n
!235 � (2.16)
In the model (2.14)-(2.15), it is assumed that the pressure P = P (n) which leads to
the appropriate classical equation of state to obtain the closed system of equations,
andNPi=1
pi�@2xpni=pni�) (@2x
pn=pn) : The second assumption is generally valid for
wavelength larger than �F : On the r.h.s. of (2.15), the last term corresponds to the
quantum statistical pressure e¤ects which results from fermionic nature of the electrons
at low temperatures. The equations (2.14)-(2.15) are commonly known as the quantum
hydrodynamic equations which takes into accounts the e¤ects of quantum di¤raction and
quantum statistics. Another useful form of this model is achieved by using an e¤ective
wave function (x; t) based on the global quantities i.e., the density n(x; t) and the
30
velocity u(x; t) as,
=pn(x; t) exp [iS(x; t)=}] ; (2.17)
with n = jj2 and u = @xS=m. This leads to the nonlinear Schrodinger equation of the
form
i}@(x; t)
@t= � }
2
2m
@2
@x2(x; t)� e' (x; t) (x; t) + Veff (n)(x; t); (2.18)
where Veff (n) = Veff�jj2
�=R n d�n
�ndP (�n)d�n
is the e¤ective potential. For one-dimensional
case, = 3 and P = (mv2F=3n20)n
3 (where vF is the electron Fermi velocity) which leads
to the e¤ective potential Veff = (mv2F=2n20) jj
4 :
2.4.2 Wigner-Poisson Approach
The classical distribution function tells us how the particles are distributed in phase
space at equilibrium. The classical �uid (or hydrodynamic) models are usually derived
by taking moments of suitable kinetic equation e.g., the Vlasov equation. The analo-
gous procedure can be applied to Wigner phase-space distribution W (x; v; t) to get the
quantum hydrodynamic equations [2; 34; 35] by taking the moments of Wigner equa-
tion (2.6). The lower-order moments are related to physically relevant quantities such
as the particle density, average velocity, and pressure etc. Since, for all hydrodynamic
approaches, the moments obey an in�nite hierarchy of equations, whereby the equation
for the ith order moment requires the knowledge of the i + 1th moment. Therefore, a
closure assumption has to be made, which allows us to establish a relationship between
the electron pressure and density, thus closing the QHD system of equations. De�ning
31
the averages i.e., density, velocity and pressure in a standard way
n =
ZWdv =
NXi=1
pi j ij2 ; (2.19)
u =1
n
ZWvdv =
i}2mn
NXi=1
pi
� i@ �i@x� �i
@ i@x
�; (2.20)
P = m
�ZWv2dv � nu2
�; (2.21)
and using the Taylor expansion of the the wave function form
f (x� a) = f (x)� af (x)@x
+a2
2!
@2f (x)
@x2� ��; (2.22)
for a << x, equation (2.6) leads to the continuity and momentum equations in the
familiar form of classical �uid equations
@n
@t+@ (nu)
@x= 0; (2.23)�
@
@t+ u
@
@x
�u =
e
m
@'
@x+
1
mn
@P
@x� (2.24)
However, an exotic pressure term (second term on the r.h.s. of (2.24)) appears in mo-
mentum equation. We will notice in a short while that the quantum e¤ects are actually
con�ned in this term.
For each state i(x; t) = Ai(x; t) exp(iSi(x; t)=}), its complex conjugate is �i (x; t) =
Ai(x; t) exp(�iSi(x; t)=}). For real amplitude Ai and real phase Si, the density and
32
velocity are de�ned by (2.9). The derivatives of i and �i lead to
@ i@x
=@Ai@x
eiSi=} +i
}Ai@Si@x
eiSi=}; (2.25)
@ �i@x
=@Ai@x
e�iSi=} � i
}Ai@Si@x
e�iSi=}; (2.26)
@2 i@x2
=@2Ai@x2
eiSi=} + 2i
}@Ai@x
@Si@x
eiSi=} +i
}Ai@2Si@x2
eiSi=} � Ai}2
�@Si@x
�2eiSi=}; (2.27)
@2 �i@x2
=@2Ai@x2
e�iSi=} � 2 i}@Ai@x
@Si@x
e�iSi=} � i
}Ai@2Si@x2
e�iSi=} � Ai}2
�@Si@x
�2e�iSi=}� (2.28)
Using (2.25)-(2.28) in (2.21), some obvious calculations lead to the pressure relation
containing the terms proportional to }2: The terms which don�t contain } constitute the
classical pressure Pc given by
Pc =1
2mn
Xi;j
pipjA2iA
2j
"�@Si@x
�2� 2@Si
@x
@Sj@x
+
�@Sj@x
�2#;
= mn
24 NXi=1
piniv2i =n�
NXi=1
pinivi=n
!235 � mn�v2i�� hvii2
�; (2.29)
which follows from the replacement of a dummy index i ! j. Equation (2.29) is the
function of density i.e Pc(n) which is the standard relation for pressure arising through
the dispersion of velocities. This analogy is the reason of calling it as a classical pressure
although the averages take into account the quantum nature of system. Similarly, the
pressure term proportional to }2 (also called quantum pressure) is given by
Pq =}2
2m
Xi
pi
"�@Ai@x
�2� Ai
�@2Ai@x2
�#;
=}2
2m
Xi
pi
"�@pni
@x
�2�pni
�@2pni
@x2
�#� (2.30)
So, the pressure term in (2.24) can be written as P = Pc + Pq: For statistical mixture of
pure states, all amplitudes may be equal such that Ai(x) = A(x) which leads to n = A2.
33
So (2.24) leads to
�@
@t+ u
@
@x
�u =
e
m
@'
@x� 1
mn
@Pc@x
+}2
2m2
@
@x
�@2xpnpn
�(2.31)
The set of (2.23) and (2.31) represents the reduced model in quantum hydrodynamic
approximation using Wigner formalism and is analogous to (2.14)-(2.15). Using the de�-
nition of e¤ective wave function given by (2.17) based on the global quantities n(x; t) and
u(x; t), one obtains the nonlinear Schrodinger�s equation as already given by (2.18). When
coupled with Poisson equation (2.7), it constitute the complete e¤ective Scrodinger-
Poisson system.
In the presence of magnetic �eld, the �uid equations take the form [53]
@n
@t+r � (nu) = 0; (2.32)�
@
@t+ u �r
�u =
q
m(E+ u�B)� 1
mnrP + }2
2mr�1pnr2pn
�: (2.33)
It is worth mentioning that P = Pt + PF where Pt is thermal pressure and PF is the
fermionic pressure. For dense plasmas at low temperature, PF is su¢ ciently large as
compared to Pt.
Generally, the hydrodynamic approach gives better results over the distances larger
than the Fermi screening length �F : Further, it enables one to perform perturbation cal-
culations in the same fashion as in the classical case. Analytical results can be obtained
using linearization procedure like the classical plasmas and all the linear phenomena can
be described by the quantum dispersion relation which gives insight of the main role of
quantum e¤ects. The properties of quantum electron plasma (with neutralizing ionic
background) can be measured with good accuracy in hydrodynamic approximation. The
system is also e¢ cient in reduction to some well known mathematical models for ana-
lytical studies of certain nonlinear e¤ects. However, some limitations are also associated
with QHD approach. For example, to treat the system computationally is not straight
forward as some technical di¢ culties may arise due to the third order derivative of den-
34
sity in the Bohm potential term. The wave-particle interactions can�t be studied with
�uid models in both classical and quantum plasmas. Similarly, it takes into account only
higher order quantum e¤ects and wavelengths shorter than the Fermi screening length of
electron can�t be treated properly. The later restriction is also analogous to the classical
�uid models, which are valid for the wavelengths longer than the Debye length. Beside
this, QHD model has obvious advantages due to its simplicity to deal with higher or-
der quantum e¤ects. The kinetic model is rigorous but too complicated to understand
fundamental dynamics of quantum plasmas.
2.5 Applications of QuantumHydrodynamic Theory
Hydrodynamic models were developed several times in the past in condensed-matter
physics, particularly for applications to semiconductors [87] and metal clusters [88] where
they are frequently referred to as time-dependent Thomas-Fermi models. The results
are generally validated by using simulation techniques. There has been a recent surge
of interest in QHD models for technologically important quantum systems, especially
resonant tunneling diodes [33; 89], self consistent quantum electron gas [35]; the metallic
and semiconductors nanostructures [38; 56]; carbon nanotubes [90], charged quantum
�uids [91], plasmonics [92]; numerical simulation of ultra-integrated devices [93]; as well
as mathematical modeling [34]: Quantum transport models similar to QHD model have
been employed in the study of superconductivity [94], super�uidity [95] and Bose-Einstein
condensation [96]:
Fluctuations in quantum plasmas excite the oscillatory patterns. The spatiotemporal
patterns that appear (either stationary or propagating) are called modes which is actually
a relation of wave propagation vector k and wave frequency !. In highly compressed
quantum plasmas, the tunneling of electrons takes place even near the absolute zero
temperature which provides the manifestation of the Bohm potential. The gradient of
quantum Bohm potential shows the e¤ect of wave function spreading which gives rise to
35
dispersive-like term. The QHD model for semiconductors reveals that it is responsible
for di¤erential resistance e¤ects [33]: In low temperature dense plasmas, the fermionic
pressure is of most signi�cance. The dispersive e¤ects of fermionic pressure and quantum
Bohm potential have been veri�ed experimentally in X-ray scattering experiments made
in Laser-produced plasmas [97]. Dense laboratory and astrophysical quantum plasmas
can be con�ned by external magnetic �elds and may also have density gradients. In
this context, quantum e¤ects become important in a magnetohydrodynamics limit [41]:
The spin 1/2 nature of electrons lead to search for new e¤ects in quantum MHD model
[74]: Recently, an ultracold plasma instability is observed experimentally [98]. This high-
frequency electron drift instability arises due to the coupling between the electron drift
wave and electron cyclotron harmonic, which has large wave numbers corresponding to
wavelengths close to the electron gyroradius.
The coincidence of di¤erent areas of physics at ultra-small scale and their useful
applications motivate us to study the dense quantum plasmas.
36
Chapter 3
Linear Modes in Nonuniform
Ultracold Quantum Plasmas
The basic low frequency electrostatic and electromagnetic modes in nonuniform dense
quantum plasmas are investigated. The dispersive drift wave in electron ion plasma and
quantum corrections to low frequency waves in the presence of stationary dust are dis-
cussed . The drift mode is also discussed using the e¤ective temperature de�ned via the
quantum Bohm potential. The drift wave studied in this chapter doesn�t require electron
temperature for its existence. However, it couples with electrostatic and Alfven waves
analogous to classical plasma case. The aim is to point out the weak quantum e¤ects on
the linear dispersion relation of low frequency electrostatic and electromagnetic waves in
cold dense plasmas.
3.1 Introduction
Drift waves in classical plasmas are known since long ago by theoretical predictions
[99] and experimental veri�cations [100], and have been studied intensively in the past
decades [101; 102]. Drift waves are low frequency waves in comparison with the ion
gyrofrequency !ci. The perpendicular (with respect to the magnetic �eld) wave number
37
k? is much larger than the parallel wave number kk. Due to the presence of the pressure
gradient in the system, the plasma equilibrium condition generates a current, which
�ows in such a way that it reduces the ambient magnetic �eld. This current is called
diamagnetic current. The diamagnetic current is not a result of guiding center drift, but
it is due to an e¤ective mass �ow created by the density imbalance in neighboring Larmor
circles. Thus the drift modes propagate in the direction of electron diamagnetic drift.
The �nite ion-temperature does not e¤ect the picture of the basic mode but it can make
some corrections to it.
Drift wave resembles with ion-acoustic wave, in which electrons provide elasticity
through the pressure and ions provide the inertia. The main di¤erence is that in the case
of ion acoustic wave, energy equipartition holds between the ion kinetic energy and the
potential energy, while in the case of long wavelength drift mode, it does not hold. In
drift wave, the ion kinetic energy is subdominant and the energy density is dominated by
the potential energy. For large parallel wave vector, drift wave can couple with the ion
acoustic wave. The electron and ion mass di¤erence, and correspondingly their rapid and
slow response to the electric �eld in the presence of a density gradient is a typical feature
of the drift wave. The drift wave is also considered to play a crucial role in particle and
energy transport in plasma. Sometimes it is also known as universal mode.
Alfven waves are fundamental normal modes of plasma which travel along magnetic
�eld lines and can be excited in any electrically conducting �uid permeated by a magnetic
�eld. Afven wave was �rst deduced by Hannes Alfven from the equations of electromag-
netism and hydrodynamics [103]. This wave is now called the shear or torsional Alfven
wave. Experimental veri�cation of Alfven waves was found by Lundquist seven years
later while studying the waves in conducting liquid mercury [104]. Physical intuition
of Alfven waves can be obtained if we imagine a uniform magnetic �eld permeating a
plasma �uid with a uniform �ow initially normal to the magnetic �eld lines. The �uid
�ow will distort the magnetic �eld lines so that they become curved. Then the curvature
of the magnetic �eld lines produces Lorentz force on the �uid which opposes further
38
curvature as predicted by Lenz�s law. The Lorentz force changes the momentum of the
�uid pushing it to minimize �eld line distortion and reverse the �uid velocity so that
the equilibrium is restored. The process of �eld line distortion is then reversed until the
cycle is completed restoring the equilibrium con�guration of the system. The restoring
force provides the basis for transverse oscillations of magnetic �eld and therefore the
Alfven wave. These waves are known to be an important mechanism for transporting
energy and momentum in di¤erent systems. That is why the Alfven waves have been the
subject of intense study in the succeeding decades in laboratory, space and astrophysi-
cal plasmas [105; 106]. Recently, quantum and spin magnetohydrodynamic models have
been developed and properties of the Alfven waves in dense quantum plasmas are studied
[41; 54; 73; 74]. This relatively new area is getting much attention of the researchers.
Theoretical analysis of oscillation spectrum of an inhomogenous plasma placed in a
very strong quantizing magnetic �eld has been presented by Shokri and Rukhadze [75]
using Hamiltonian description. It is shown that the quantum drift waves may exist in such
plasmas which become unstable under certain circumstances. The quantum behaviour
of the electrons is manifested in a strong magnetic �eld viz., Te << }!ce where Te and
!ce are thermal energy and Larmor frequency of electrons. For electrons in metals, this
condition is satis�ed when 106 G < B0 and for electrons and holes in semiconductors it
holds when 104�105G < B0 at room temperature. The presence of very strong magnetic
�elds i.e., B0 > 1010G, leads to the ultra quantum case where both electrons and ions
show quantum behaviour with Te << }!ce and Ti << }!ci.
The inhomogeneities in dense plasmas lead to drift waves at quantum scales ([40; 71]
and refs. therein). Shukla and Sten�o [71] have shown that there exists new low fre-
quency (in comparison with the ion gyrofrequency) electrostatic modes in inhomogenous
quantum plasmas which may not exist in the absence of the quantum e¤ects. They
have shown that the local dispersion relation for electrostatic waves in electron-ion quan-
tum plasma with spatial density inhomogeneity strongly depends on quantum behaviour
of electrons and exists only in quantum plasmas. For collisional quantum plasma, the
39
drift waves become unstable [40] and may cause cross-�eld charged particle transport in
inhomogeneous, magnetized quantum plasmas.
The high-frequency drift waves in ultracold plasmas in the presence of crossed mag-
netic and electric �elds have been identi�ed in a recent experiment [98]. The drifting of
electrons relative to the ions across the magnetic �eld causes the electron drift waves. The
electron drift instability arises due to the coupling between the electron drift wave and
electron cyclotron harmonic, which has large wave numbers corresponding to wavelengths
close to the electron gyroradius.
Dusty plasmas are found in interstellar media, interplanetary spaces, dense molecular
clouds, planetary rings, cometary tails and comae, as well as low-temperature laboratory
plasmas e.g., plasma processing, radiofrequency discharges, plasma coating and tokamak
edges. The dusty plasmas contain the dust particles which can be macromolecules,
clusters of molecules, small sub-micron sized grains, micron-sized grains, larger grains
etc. whose charge and mass varies according to their size. Due to variety of applications
in space and industry, dusty plasmas have received growing interest of the researchers.
The collective modes in dusty plasma have been extensively studied both experimentally
and theoretically in the past three decades [107; 108; 109].
Once a dust grain is charged, it responds to electric and magnetic forces in addition
to gravitational forces which are negligible for electrons and ions. For a typical dust
grain size a and average intergrain distance d, the situation a << �D << d corresponds
to isolated screened dust particles in a plasma and a << d << �D to a dusty plasma
exhibiting collective behaviour [109]. The quantum e¤ects on ordinary dust �uids are
negligible because the de Broglie wavelength associated with a dust particle is negligibly
small. The ratio of de Broglie wavelength to average intergrain distance is also negligible
for heavy dust.
However, ultrasmall sized materials like metallic and semiconductor micro and nanos-
tructures e.g., nanoelectromechanical systems (NEMS) in dense plasmas can be treated
like dust particles. Similarly, the metallic nanoparticles/nanotubes can be regarded as
40
charged dust particles surrounded by degenerate electrons and holes and non-degenerate
ions [47]. The density of electrons in metals is very high e.g., we have typically ne0 '
1023cm�3: The dynamics of electrons and holes is quantum mechanical in nature because
the de Broglie wavelength associated with them is signi�cant. Quantum e¤ects associ-
ated with dense Fermi gas of electrons in the presence of nanometer sized structures in
a laboratory system or charged heavy particles (dust) in the astrophysical environments
(e.g., supernova remnants) greatly in�uence the wave dynamics [46; 48].
3.2 Quantum Drift and Inertial Alfven Waves
The propagation of quantum drift waves may be described on the basis of quantum mag-
netohydrodynamics and Poisson equation [71] in addition to the Hamiltonian description
of Ref. [75]: The ion dynamics provides a possibility of low frequency modes which are
in�uenced by quantum e¤ects. When plasma density is spatially nonuniform, the elec-
tron quantum e¤ects lead to alter the electromagnetic drift wave frequency due to mode
coupling [77]:
3.2.1 Set of Equations
Let us consider an ultracold dense magnetized plasma embedded in a constant external
magnetic �eld B0 = B0z along z-axis with density gradientrnj0 = x�dnj0dx
�along x-axis
where j = e; i; d denotes electron, ion and dust species respectively. The steady state in
the presence of stationary dust demands
ne0 + Zdnd0 = ni0; (3.1)
where dust is assumed to be negatively charged and the subscript naught "0" denotes
the background quantities. The continuity and motion equations for jth species from
41
(2.32)-(2.33) can be written as
@nj@t
+r � (njvj) = 0; (3.2)
mjnj(@t + vj �r)vj = qjnj
�E+
1
c(vj �B)
��rPj +
�}2nj2mj
�r�1pnjr2pnj
�;
(3.3)
where qj; nj, vj; mj and } are electrostatic charge, density, �uid velocity, particle mass,
and the Planck�s constant divided by 2�, respectively. We assume that the pressure term
rPj in (3.3) is negligible for ions and hence the ions behave classically. In the limit
j@tj � !pe; ck; the Ampere law may be written as
r�B = (4�=c)J� (3.4)
Equation (3.3) leads to the electron equation
me (@t + ve �r)ve = �e�E+
1
c(ve �B)
��rPe +
}2
2me
r�r2pnep
ne
�; (3.5)
which leads to the parallel and perpendicular components of the velocity vectors, respec-
tively, as
me@tvez1 = �eEz1 +}2
4mene0@zr2ne1; (3.6)
and
ve?1 ' �c
B0
�r?'1 � z+
}2
4eme
r?
�r2ne1
ne0
�� z�; (3.7)
where subscripts "1" and "?" denote the perturbed quantities and direction perpen-
dicular to the ambient magnetic �eld, and r? = x@=@x + y@=@y with x and y being
the unit vectors along x and y axes respectively. We have assumed (kBTFe)ne1 <<
(}2=4me)r2ne1 where kB is the Boltzmann constant and TFe is the electron Fermi tem-
42
perature. We de�ne the electric �eld perturbation as
E1 = �r?'1 � (@z'1 + (1=c) @tAz1)z; (3.8)
where ' is the electrostatic potential and Az is the z-component of magnetic vector
potential. The magnetic �eld perturbation is de�ned as
B1 =r?�Az1: (3.9)
The electron continuity equation along with (3.7) leads to
�@t �
!2pe!ce
�en�4qer2@y
�ne1 �
c
B0ne0�en@y'1 + @z(ne0vez1) = 0; (3.10)
where �en = j 1ne0dne0dxj and �qe =
�}2
4m2e!
2pe
�1=4is the electron quantum wavelength. As-
suming that the ion parallel velocity viz is negligibly small, the Ampere law (3.4) yields
r2?Az1 =4�e
cne0vez1; (3.11)
From (3.6) and (3.11), we obtain
�1 + �2ek
2?�@tAz1 = �c@z'1 �
c}2
4emene0@zr2ne1; (3.12)
where �e = c=!pe is the electron skin depth. Then using (3.10)-(3.12) we have @2t �
!2pe!ce
�en�4qer2@y@t �
c2�4qe�1 + �2ek
2?�@2zr2?r2
!ne1 =
c2
4�e�1 + �2ek
2?�@2zr2?'1
+c
B0ne0�en@y@t'1; (3.13)
with !ce = eB0=mec being the electron cyclotron frequency. Note that (3.13) is the same
as equation (26) of Ref. [76] if �2ek2? << 1. In what follows, we present the linear wave
43
analysis using Fourier transform method.
3.2.2 Linear Dispersion Relation
From ion continuity equation, we have
@tni1 � ni0c
B0!ci@tr2?'1 + �in@y'1 = 0 (3.14)
where �in = j 1ni0dni0dxj. Assuming that the linear perturbation is proportional to exp[i(kyy+
kzz � !t)]; we take two cases of interest (a) : The electron-ion plasma, and (b) : The
electron-ion plasma with some concentration of heavy species (dust) in the background.
In the limit 1 << �2ek2?, (3.13) and (3.14) yield
! =
�zdnd0ni0
���dnkyk2?
�!ci � !�q� (3.15)
Equation (3.15) represents the Shukla-Varma (SV) mode in the presence of stationary
dust where �dn = j 1nd0dnd0dxj and the quantum drift frequency !�q = [
!2pe!cekyk
2�en�4qe]. The
quantum drift wave ! ' !�q is di¤erent from classical drift wave in a sense that it exists
in a cold electron plasma. For !�q = 0, (3.15) reduces to classical SV mode which is (7) of
Ref. [110] if the dissipative damping is ignored. Note that in a pure electron -ion plasma
the SV mode ceases to exist.
In case of electron-ion quantum plasma, the electromagnetic dispersion relation can
be obtained by using (3.13) and (3.14) along with quasi-neutrality ne1 ' ni1 which leads
to
�!(! � !�q)� 2qe
���inkyk2?
+!
!ci
�=
!2A�1 + �2ek
2?� !!ci
+�enkyk2?
!2 =
c2k2z!ci�1 + �2ek
2?�!2pi
! +!2pe!2pi
�enkyk2?
!ci!ce
!2; (3.16)
where qe =qk2zk
2?k
2c2�4qe=�1 + �2ek
2?�. It is to be noted that the terms !�q and qe
44
depends on �qe which shows the in�uence of quantum nature of electrons, an indication
that the mode (3.16) exist only in quantum plasmas. The dispersion relation (3.16)
reduces to (28) of Ref. [76] for ae = 1. But we think that the factor �2ek2? 6= 0 is
important to include. In the classical limit (}! 0), (3.16) gives the well-known linear
dispersion relation of slow shear Alfven waves in a homogeneous plasma as
!2 =v2Ak
2z
1 + �2ek2?� (3.17)
Note that the term qe in (3.16) is modi�ed due to the factor �2ek2? which depends on
the electron number density.
3.2.3 Results and Discussions
Using the parameters as mentioned in Ref. [76] for dense molecular clouds i.e. ne0 = ni0 �
2�10�3cm�3 and k � 0:01cm�1, we have !pe � 2:5�103rad=sec, and �2ek2? � 1010 >> 1.
Therefore, it is important to take into account the electron inertial term in equation of
motion. It is also necessary to note that corresponding to the choice k � 0:01cm�1, we
assume �en = �in = 10�4cm�1so that the local approximation remains valid. The value
of the quantum wavelength turns out to be of the order of 10�2cm while the perturbation
wavelengths are of the order of � ' 6 � 102cm: For Te ' 0:1eV; �B ' 9 � 10�8 and
ne0�3B <<< 1: Hence there is no contribution of quantum e¤ects in this system. If
we consider the denser molecular clouds having stationary dust with ne0 = 0:95cm�3,
ni0 = 1cm�3, nd0 = 0:05cm�3, Zd = 1, B0 = 2 � 10�6G, and mi = 12mp (where mp
is proton mass), then quantum wavelength becomes �qe = 3 � 10�3cm. The interstellar
medium (ISM) has Te ' (0:1 � 1)eV in general which gives the electron thermal de
Broglie wavelength �B ' 9 � 10�8 � 2:9 � 10�8cm. It is negligibly small compared to
inter-particle distance �r � n�1=3i � 1cm (which is almost the same for electrons as well).
The electron Fermi temperature TF corresponding to ne � 0:95cm�3 turns out to be
4� 10�11K. Thus in ISM we have TF <<< T and hence this plasma is described by the
45
Figure 3-1: The SV-mode dispersion relation !(rad=sec) vs ky � 106(cm�1) is plottedfor the case of cold dense hydrogen plasma having ne s 0:5 � 1026cm�3, nd s 0:05niand Te = 1eV: Solid curve corresponds to equation (3.15) including quantum corrections(!�q 6= 0) and dashed curve without quantum corrections.(!�q = 0).
classical distribution functions and Fermi-Dirac statistics is not applicable. Therefore
the parameters related to ISM used by Ref. [76] are relevant to classical plasmas.
Our aim is to investigate the weak quantum e¤ects on the linear dispersion relations
of electrostatic and electromagnetic waves in cold dense plasmas. As an illustration we
choose the hydrogen plasma [111] with ni � 1026cm�3. Let us suppose that ne � 0:5niand the dust (or impurity) concentration is nd � 0:05ni while Zd � 10. Corresponding to
ne � 0:95� 1026cm�3 one obtains TFe � 9� 106K. If this plasma is produced at electron
temperature Te � 1eV, then Te << TF and hence the classical plasma equations can not
describe the dynamics of the system. The electron de Broglie wavelength in this case turns
out to be ' 2� 10�8cm while �r ' n�1=3i ' n
�1=3e ' 2� 10�9cm. Therefore, the quantum
e¤ects are important to be investigated. Let us assume that the strong magnetic �eld
B0 � 104Tesla is applied externally to such a laboratory plasma. We have used quantum
hydrodynamic equations to obtain the linear dispersion relation for low frequency waves
46
Figure 3-2: The Inertial Alfven wave dispersion relation !(rad=sec) vs ky(cm�1) is plottedin an electron-ion dense hydrogen plasma having ne s 1026cm�3and Te = 1eV: Solid anddotted curve represents two branches of Alfven wave whereas dashed curve correspondsto ! = qe which couples with lower branch of Alfven wave.
in a dense ultracold plasma. So the temperature e¤ects are not under consideration,
therefore the wave particle interactions are not important. For the above parameters,
we obtain !ci � 9:6 � 1011rad=sec, !pe � 3:9 � 1017rad=sec, �e ' 7:5 � 10�8cm and the
interparticle distance n�1=3i � 1 � 10�9cm. Taking kz � 10�3ky and varying ky from
1� 105-7� 106cm�1, we have plotted the dispersion relation (3.15) for SV-mode in Fig.
(3-1). The cold quantum plasma drift wave frequency !�q and frequency corresponding to
SV-mode lies very close to each other for smaller values of ky. In the shorter wavelength
the two modes become separate. It may be mentioned here that in the shorter wavelength
region, the interatomic quantum interactions should be taken into account. Therefore
we do not go beyond 7� 106cm�1. For the case of electron-ion plasma we again consider
the example of a magnetized Hydrogen plasma with ne0 ' ni0 ' 1026cm�3, Te � 1eV
and B0 � 8� 106G. Then we �nd �B � 2:6� 10�8cm � 10�r and vA ' 1:7� 105rad=sec.
Corresponding to ky � 5�106cm�1 and kz � 500cm�1 we obtain � ' 1:2�10�6cm, !A �
47
8:7�107rad=sec, !�q � 5:9�109rad=sec and !qe � 3:7�108rad=sec. The electrostatic and
electromagnetic waves can be in�uenced by the quantum interactions because � � 102�B.
In the dispersion relation the three roots, one corresponding to qe and two corresponding
to Alfven waves are coupled as shown in Fig. (3-2). The positive frequency branch of
Alfven wave couples with the drift frequency !�q which itself is independent of kz. In
classical plasmas the drift wave exist due to hot electrons while in the cold dense plasmas
the quantum drift frequency !�q is independent of Te. However, the frequency qe also
appears in the dispersion relation and it depends on kz as well. The third root in the
dispersion relation represents a wave corresponding to qe. The negative branch of Alfven
wave couples with it. The coupling of the low frequency drift and Alfven modes may
provide useful information on ambient plasma conditions at quantum scales.
48
3.3 Drift, Acoustic and Inertial Alfven Waves.
The drift wave studied in this section depends on the e¤ective temperature de�ned via the
electron quantum di¤raction term and doesn�t need electron temperature for its existence
which is quite di¤erent phenomenon compared to the classical drift waves [78]. Such type
of drift mode can exist only in dense quantum plasmas even if the electron temperature
is ignored (Te = 0). The dispersion relation has an interesting analogy with the classical
plasma case of drift waves [79]:
3.3.1 Basic Equations
Let us de�ne �jn = j 1nj0�dnj0dx
�j =constant, where j = e; i denotes electron, ion species
and the density inhomogeneity in the x-direction i.e.,rnj0(x) is given in Sec. (3.2.1). To
study the dynamics of low frequency electrostatic and electromagnetic modes, we use the
equations (3.2)-(3.4) which leads to the perpendicular velocity component of jth species
as
vj?1 'c
B0E?1 � z+
�}2
4m2j!cj
�r?
�r2nj1
nj0
�� z� 1
!cj@t(vj?1 � z);
' vE + vqj + vPj; (3.18)
where !cj = eB0=mj0 and we have assumed j@tj << !cj. The parallel component of
velocity may be written as
@tvjz1 'qjmj
Ez1 +}2
4m2j
@z
�r2nj1
nj0
�� (3.19)
For ion dynamics, we obtain the same equations as in the case of classical plasmas since
the ions are assumed to behave classically in the limit TFi << TFe. Then (3.18)-(3.19)
for ions yield,
vi?1 'c
B0E?1 � z+
c
B0!ci@tE?1 = vE + vpi; (3.20)
49
and
viz1 ' �e
mi
@z'1 �e
mic@tAz1; (3.21)
where vE and vpi are the electric and ion polarizations drifts, respectively. The electric
and magnetic �eld perturbation is given by the equations (3.8) and (3.9) respectively.
Let us assume an e¤ective temperature Tq arising due to Bohm potential term-a contri-
bution of quantum electrons which will be discussed later. Then we may de�ne a diamag-
netic drift velocity similar to the case of classical drift wave as vqD = (cTq=eB0)rlnn0�z
where ne0 = ni0 = n0. The ion continuity equation in linear limit can be written as,
@tni1 +rn0:vE1 + n0r � vpi + n0@zviz1 = 0� (3.22)
In the limit j@tj � !pe; ck; the Ampere law (3.4) yields
r2?Az1 =4�n0e
c(vez1 � viz1) ; (3.23)
and the electron continuity equation can be expressed as
@tne1 +rn0 � vE1 + n0@zvez1 = 0� (3.24)
Since Jz1 ' en0(viz1� vez1), therefore (3.22)-(3.24) along with quasi-neutrality condition
ne1 ' ni1 give a relation between '1 and Az1 as,
Az1 '�
c
kzv2A
�!'1 (3.25)
where vA = B0=4�min0 is the speed of Alfven wave and perturbed quantities are assumed
to be proportional to exp[i(kyy+kzz�!t)]. In the ion continuity equation, we use (3.21)
for viz1 and obtain
!
�ni1n0
�' !�q�1 � !(�2qk2?)�1 +
c2qk2z
!
��1 �
!
ckzA
�; (3.26)
50
where !�q = vqD � k is the quantum drift wave frequency depending on e¤ective tempera-
ture Tq = }2k2=4me which is purely a quantum contribution of electrons. It is analogous
in shape to the classical drift wave frequency !�e = (cTe0=eB0)rlnn0�z when the electron
pressure is de�ned by ideal gas law Pe0 = n0Te0. In (3.26) we have de�ned �1 = e'1=Tq
and A = eAz1=Tq. The electron equation of motion (3.5) leads to
ve?1 'c
B0E?1 � z�
}2
4m2e!ce
r?
�r2ne1
n0
�� z = vE + vq; (3.27)
and
(@t + vqD �r)vez1 = �e
me
Ez1 +}2
4m2e
@z
�r2ne1
n0
�� e
mec(vqD �B1)z � (3.28)
Approximating Jz1 ' �en0vez1 in electron dynamics, because me << mi, the Ampere
law (3.23) yields
vez1 ' ��
c
4�n0e
�k2?Az1; (3.29)
and equation (3.28) becomes
(! � !�q)vez1 'ekzme
��'1 +
!
ckzAz1
�� e
mec!�qAz1 +
Tqkzmen0
ne1; (3.30)
From electron continuity equation using (3.29) and (3.30), we obtain
ne1n0' �1 �
1
ckz(1 + �2ek
2?)�! � !�q
�A (3.31)
3.3.2 Linear Dispersion Relation
In order to obtain the dispersion relation, we use the equations (3.25), (3.26) and (3.31)
along with quasi-neutrality condition ne1 � ni1. This gives the linear dispersion relation
51
of low frequency waves in a magnetized dense quantum plasma of the form
�(! � !�q)! �
c2qk2z
1 + �2ek2?
�!2 � !2A
1 + �2ek2?
�!2 � !�q! � c2qk2z
�=
��2qk
2?
1 + �2ek2?
�!2A!
2 (3.32)
where !A = kzvA. In the limit �2ek2? << 1, above equation reduces to
�(! � !�q)! � c2qkzz
�(!2 � !2A) = (�2qk2?!2A)!2 (3.33)
It is interesting to point out the analogy of dispersion relation (3.32) with the classical
plasma dispersion relation. The dispersion relation of classical electron ion plasma in the
presence of ion thermal e¤ects as given by (4.47) of Ref. [79] is
�!(! � !�e)� c2sk2z
� �!(! � !�i )� v2Ak2z
�= �2k2?v
2Ak
2z(! � !�i )!; (3.34)
where the ion sound speed is de�ned as cs = (Te=mi)1=2; the ion Larmor radius at elec-
tron temperature is de�ned as � = cs=!ci, !�i = � TiTe!�e and Ti is the ion temperature.
The opposite signs of electron and ion drift wave frequencies are due to di¤erent charge
states of electron and ion. Mathematical expression of classical drift wave frequency
!�e is analogous to the corresponding quantum drift wave frequency !�q but physics of
both the waves is very di¤erent. Classically, the drift wave exists due to the electron
temperature. However, the dispersion relation (3.32) for cold dense plasma shows the
existence of drift wave due to Tq which appears through quantum correction term in elec-
tron momentum equation: Note that the pressure term has been neglected assuming the
ultracold electrons. This is a very special situation but interesting because the ultracold
dense quantum plasma has the electromagnetic linear dispersion relation analogous to
the classical case [79]. The result (3.32) ceases to exist if the electron quantum di¤raction
e¤ects are ignored. It is evident from (3.32) that the replacement of cq, �q and !�q by
cs; � and !�e, respectively, gives the result analogous in form to (3.34) if Ti = 0 there.
52
Figure 3-3: Four modes of inhomogeneous ultracold dense plasma are shown for ky =4:3� 105 cm�1, n0 �= 1026cm�3 and B0 = 9� 105G: The two outer curves correspond toshear Alfven wave and inner ones to electrostatic wave. The modes without dispersionare represented by the dashed curves.
3.3.3 Results and Discussions
As an illustration, we apply our results to the dense plasmas found in the astrophysical
environment (white dwarfs) with n0 � 1026cm�3 and B0 ' 106G. In laboratory plasmas,
the choice of magnetic �eld of the order of 106G is also relevant. It is noted that the
Fermi temperature for electron gas becomes TFe � 9 � 106K for n0 � 1026cm�3 and we
assume that the real gas temperature Te is very small, such that Te << TFe and n�3B
>> 1. We obtain �r � n�1=30 ' 1�10�9 cm and de Broglie wavelength �B ' 9�10�8 cm if
real gas temperature is assumed to be Te ' 0:1eV. Using quantum hydrodynamic model
treating the cold ions as a classical gas, we consider the wavelength of perturbations of
the order of (10�5 � 10�6) cm and hence the weak quantum corrections to such waves
become relevant.
In Fig. (3-3), the frequencies of the four modes of ultracold dense plasma are plotted
against kz. One branch of the shear Alfven wave and one branch of the electrostatic wave
53
Figure 3-4: Plot of ! vs kz of four modes for relatively higher density and magnetic �eldi.e., n0 �= 1028cm�3 and B0 �= 108G with ky = 3:5�106 cm�1:The two outer (inner) curvescorrespond to shear Alfven (electrostatic) wave whereas the modes without dispersionare represented by the dashed curves.
are in�uenced by the quantum drift wave !�q near kz � 4 � 103cm�1. Since Ti = 0 has
been assumed, therefore the second branch of the Alfven wave remains a straight line in
this �gure. The second branch of the electrostatic wave has also the e¤ects of dispersion
similar to the classical case [79]. Similarly, the coupled dispersion relation (3.33) has
been plotted for relatively higher densities and magnetic �elds as shown in Fig. (3-4).
The behaviour of the coupled modes in this case is similar to the previous one but the
wave frequency range is increased.
54
Chapter 4
Low Frequency Linear Modes in a
Homogenous Quantum Plasma
The propagation of long wavelength waves in homogenous quantum plasmas are studied.
The low frequency electrostatic and shear Alfven waves are investigated. The e¤ects of
stationary dust on the waves is also pointed out. The role of electron fermionic pressure
on wave dispersion for dynamic as well as static ions is discussed. A comparison of
statistical pressure term with quantum Bohm potential term is made and the results are
analyzed numerically.
4.1 Introduction
The wave associated with ion oscillations in plasma is the ion wave or ion-acoustic wave
which is well understood classically. Analogous to sound wave in air, ion-acoustic (IA)
wave can propagate in plasmas, having thermal e¤ects. Ordinary sound waves propagate
from one layer of medium to next by collisions among the molecules, while ion acoustic
wave can propagate through the intermediary of the electric �eld without collisions,
where ions transmit vibrations to each other because of their charge. This is basically
an electrostatic longitudinal wave and can exist in unmagnetized as well as magnetized
55
plasmas. In quantum plasmas at extremely low temperature, Haas et. al. have proposed
for the �rst time that the quantum ion acoustic wave can exist in quantum plasmas
and wave dispersion appear due to quantum e¤ects associated with electrons [37]. The
normalized dispersion relation obtained by the authors is of the form
!2 =k2 (1 +H2k2=4)
1 + k2 (1 +H2k2=4); (4.1)
where H = }!pe=2kBTFe is the measure of quantum di¤raction e¤ects. The classical
result of ion acoustic wave can be recovered in the limit }! 0. The frequency of the ion
waves is lower than the ion plasma frequency.
4.2 Dust Ion-Acoustic Wave in Unmagnetized Quan-
tum Plasmas
In classical plasmas, it is well known that the two normal modes of unmagnetized, weakly
coupled dusty plasmas are the dust acoustic and dust ion-acoustic waves, which were
theoretically predicted by Rao, Shukla and Yu [112] and Shukla and Silin [113]. In the
dust acoustic wave, the pressure exerted by the inertialess electrons and ions provide the
restoring force, while the dust mass gives the inertia. The frequency range of the wave is
much lower and the phase velocity much smaller than the ion acoustic velocity. On the
other hand, the phase velocity of the dust ion-acoustic wave is larger than both dust and
ion thermal velocities. The inertia is provided by dust and ions while the restoring force
comes from the electron pressure. Another way to view this mode is to treat dust as
immobile. Both these waves have been experimentally observed in several dusty plasma
experiments.
When charged dust impurities are present in dense quantum plasmas, they modify the
long wavelength waves and quantum corrected modes may appear [71, 72]. In extremely
low-temperatures dense plasmas with dust background, the electron quantum behaviour
56
play a vital role in the dynamics of charged particles. A number of e¤orts have been made
theoretically to investigate the collective modes in a dust contaminated quantum plasmas
([46] and refs. therein). The authors have pointed out the importance of their results in
electronic micro and nano electromechanical devices. These ultrasmall systems may be
contaminated by the presence of highly charged dust impurities in their environment. In
an unmagnetized dense quantum plasma, the immobile charged dust impurities interact
via self consistent electrostatic potential and the low phase velocity (viz., vFi << !=k <<
vFe) dust ion-acoustic wave is in�uenced by the quantum e¤ects of electrons [81].
Let us consider a three component unmagnetized, ultracold quantum plasma com-
prised of inertialess electrons, inertial ions and negatively charged immobile dust parti-
cles. The dynamics of this system in one spatial dimension may be described by the set of
quantum hydrodynamic equations comprising the continuity and momentum equations,
@nj@t
+@ (njuj)
@x= 0; (4.2)�
@
@t+ uj
@
@x
�uj = �
qjmj
@'
@x+}2
2m2j
@
@x
�@2xpnj
pnj
�� 1
mjnj
@Pj@x
; (4.3)
coupled through Poisson equation,
@2'
@x2= 4�e(ne + Zdnd � Zini); (4.4)
where j = e; i for electrons and ions, respectively. Here nj, uj, mj and Pj are the
perturbed number density, �uid velocity, mass and �uid pressure, respectively, ' is the
electrostatic wave potential, nd is the dust number density, Zi (Zd) is the ion (dust)
charge number and } is the Planck�s constant divided by 2�. We assume that Zi = 1 and
qj = �e(e) for electron (ion). When plasma density �uctuates, the dust particles remains
static in the background of the perturbed plasma. Therefore, under the quasineutrality
condition, (4.4) leads to ne ' ni � Zdnd0: At equilibrium, we have ni0 = ne0 + Zdnd0
where nj0 is unperturbed number density of jth species. For pressure, we assume the
57
equation of state for zero-temperature fermion gas (1.13) for = 3 which leads to
Pj =1
3
mjv2Fj
n2jon3j ; (4.5)
where vFj = (2kBTFj=mj)1=2 is the Fermi speed and TFj is the Fermi temperature of jth
species. In (4.3), the second term corresponds to quantum statistical e¤ects of plasma
particles and third term represents the quantum di¤raction e¤ects. Linearizing the ion
continuity and momentum equation, electron (inertialess) momentum equation and Pois-
son equation, and assuming that all quantities are varying as eikx�i!t, we obtain the linear
dispersion relation of dust ion-acoustic wave in quantum dusty plasma as
!2 =k2c2q
��1 +H2k2�2Fe=4
�+ �(1�Nd)
�1 + k2�2Fe
�1 +H2k2�2Fe=4
��(1�Nd)
�1 + k2�2Fe
�1 +H2k2�2Fe=4
�� ; (4.6)
whereH = }!pe=2kBTFe is the measure of quantum di¤raction e¤ects, cq = (2kBTFe=mi)1=2
is the quantum ion-acoustic speed, �Fe = (2kBTFe=4�ne0e2)1=2 is the Fermi screen-
ing length, !pe = (4�ne0e2=me)
1=2 is the electron plasma frequency, � = TFi=TFe and
Nd =Zdnd0nio
. For TFi << TFe, � = 0 and (4.6) leads to
!2 =k2c2q
�1 +H2k2�2Fe=4
�(1�Nd)
�1 + k2�2Fe
�1 +H2k2�2Fe=4
�� � (4.7)
In the limit �2Fek2 << 1;the dispersion relation (4.7) can be written as
!2 =k2c2q
�1 +H2k2�2Fe=4
�(1�Nd)
� (4.8)
Introducing the normalization
u�i ! ui=cq , n�j ! nj=nj0 , t�! t!pi , x�! !pix=cq , '�! e'=2kBTFe; (4.9)
58
yields the linear dispersion relation of the form
!2 =k2 [(1 +H2k2=4) + �(1�Nd) f1 + k2 (1 +H2k2=4)g]
(1�Nd) [1 + k2 (1 +H2k2=4)]� (4.10)
The dispersion relation (4.10) shows that the phase velocity is a¤ected by the quantum
corrections (via quantum di¤raction and statistics e¤ects) and concentration of dust
particles. In the absence of dust particles (Nd = 0) and ion Fermi temperature (� = 0);
we retrieve the dispersion relation of Ref. [37] for electron-ion quantum plasma given by
equation (4.1).
4.2.1 Results and Discussions
It is obvious from relation (4.10) that the wave frequency is directly related to the quan-
tum parameter } and the dust density ratio Nd. For illustration, we plot the dispersion
relation (4.10) for di¤erent values of normalized quantum di¤raction parameter H and
concentration of stationary dust Nd in Figs. (4-1) and (4-2) respectively. Due to large
mass di¤erence of electrons and ions, TFi << TFe, the e¤ect of parameter � is negligibly
small. In Fig. (4-1), it is shown that the wave frequency increases with H fastly for
smaller values of k. When k becomes larger, the increase in frequency becomes smaller
and smaller and the curve separation also reduces. The linear variation of normalized
wave frequency ! as a function of normalized wave number k for di¤erent values of dust
particle concentration Nd is shown in Fig. (4-2). It is seen that the phase velocity in-
creases with Nd.The classical result of ion acoustic wave can be recovered in the limit
}! 0.
4.3 Ion Waves in a Quantum Magnetoplasma
We study the obliquely propagating, linear electrostatic waves in a dense homogenous
quantum magnetoplasma consisting of electrons and ions. For this, let us consider the
59
Figure 4-1: Normalized dispersion relation for the quantum dust ion-acoustic wave withH = 0 (solid line), H = 1:5 (dashed line) and H = 3 (dotted line). Other parametersare � = 0:3 and Nd = 0:2.
Figure 4-2: Normalized wave frequency as a function of normalized wave number fordi¤erent values of dust concentration i.e. Nd = 0:1 (solid line), Nd = 0:2 (dashed line)and Nd = 0:5 (dotted line). Other parameters are � = 0:03 and H = 0:5.
60
hydrodynamic equations describing the dynamics of low frequency (! << !ci where
!ci = eB0=mic is the ion cyclotron frequency) ion waves given by (3.2)-(3.3) and Poisson
equation
r � E = 4�e(ni � ne); (4.11)
where r = x@=@x + y @=@y + z @=@z. The electrostatic electric �eld perturbation is
de�ned as E = �r' and external magnetic �eld B = B0z; where z is the unit vector
along the z-axis and B0 is the strength of uniform �eld. We assume that the ions behave
classically in the limit TFi << TFe: The pressure e¤ects of only quantum electrons are
relevant obeying the pressure equation (1.13).
We assume that the perturbed quantities are varying as eik:�r�i!t with k being the
wavenumber and ! being the wave frequency. Linearizing (3.2),(3.3) and (4.11), and
disregarding the terms proportional tome=mi, we obtain the following dispersion relation
of quantum ion wave in magnetized quantum plasma comprising of electrons and singly
charged ions,
!2 =c2qk
2z(1 + �
2qk2=4)
1 + �2qk2?(1 + �
2qk2=4)
; (4.12)
where �q = cq=!ci is the ion Larmor radius at electron Fermi temperature and �q =
}= (2mekBTFe)1=2 is the quantum di¤raction length scale. If �q = 0, (4.12) is similar
to the classical plasma case. But the system still exhibits the quantum nature due to
Fermi-Dirac distribution of electrons in equilibrium.
4.3.1 Applications, Results and Discussions
To examine the dispersive properties of ion waves in this system, we choose some typical
plasma parameters which are representative of laboratory and superdense astrophysical
plasmas [46]. If dense laboratory Hydrogen plasma with n0 w 1 � 1026cm�3 (TFe w9� 106K) is produced at electron temperature Te = 1eV; then TFe >> Te holds and the
electron de Broglie wavelength becomes �B w 2:9 � 10�8cm. The interparticle distanceturns out to be �r w 2 � 10�9cm and the degeneracy parameter becomes ne0�
3B >> 1.
61
Figure 4-3: The frequency ! vs the wavenumber k is plotted for linear dispersion relationof quantum ion wave in the dense laboratory Hydrogen plasmas for neo = nio = 1 �1025cm�3 and Bo = 1� 106G with kz s 10�2k? (cm�1)
The non-dimensional di¤raction parameter in dense quantum magnetoplasma may be
de�ned as He = }!L=2kBTFe, where !L = (!ce!ci)1=2 is the lower-hybrid resonance
frequency. In the presence of uniform external magnetic �eld B0 = 104G (1T) which can
be applied in laboratory plasma experiments, He w 1:7� 10�9 since He / B0=n2=3e0 . The
quantum di¤raction length becomes �q w 7 � 10�10 and the second term in parenthesis
in the linear dispersion relation (4.12) is found to be very small i.e., �2qk2=4 <<< 1:
This shows that the di¤raction e¤ects in this system and the dispersion caused by the
quantum di¤raction is vanishingly small. However, this system doesn�t follow the classical
equations and hence we need a quantum description. In this case, the quantum e¤ects
appear through quantum statistical pressure term only.
We consider a relatively low density magnetized Hydrogen plasma with n0 w 1 �
1025cm�3 and B0 = 106G. For this plasma we have �q w 1:5� 10�9 and He w 8� 10�7
which are again very small values. The linear dispersion relation for this case is plotted
in Fig. (4-3).
Now we present an example of astrophysical plasmas. The studies of plasmas in dense
astrophysical bodies have been motivated by the ultrahigh plasma densities and enormous
62
magnetic �elds (� 1012 G) and the tentative evidence for �elds as strong as � 1015 G
[8; 21]. We apply our results to highly magnetized superdense plasmas corresponding to
two di¤erent sets of parameters.
First, taking number density n0 = 1�1026cm�3 in the presence of very strong uniform
magnetic �eld B0 = 1�109G; we get !ci w 9:5�1012rad=sec; cq w 3:8�107cm=sec , �q w4�10�6cm; � << 1; the quantum di¤raction length �q w 6:9�10�10cm and the parameterHe = 0:0002: Assuming kz s 10�2k? and varying k? from 5� 102 � 2:4� 105cm�1 leads
to �qk? w 0:002 � 0:9. The perturbation wavelength � w 0:01 � 2:6 � 10�5cm and the
term �2qk2=4 <<< 1 which shows that the dispersion due to quantum di¤raction is still
negligibly small.
Second, for the case of very dense plasma in a very strong magnetic �eld with n0 =
1 � 1028cm�3 and B0 = 5 � 1012G , the quantum di¤raction length becomes �q w1:5� 10�10cm, quantum ion-acoustic speed cq w 1:7� 108cm=sec, the ion Larmor radius�q w 3�10�9cm, !ci w 5:7�1016 rad=sec, � w 0:13 and He = 0:05. For kz s 4�104cm�1
and k? = 2:7�108cm�1 where kz w 10�4k?, we obtain � w 2:2�10�8cm and �qk? w 0:8.This shows that the quantum di¤raction e¤ects are important for dispersion in this case
but the relevant length scale is very short.
4.4 Fermionic pressure and quantum pressure
For dense electron gas in metals with equilibrium density ne0 ' 1023cm�3, the typical
value of Fermi screening length is of the order of Ångstrom while the plasma oscillation
time period (!�1pe ) is of the order of the femtosecond. The electron-electron collisions can
been ignored for relatively short time scales [35]. The Fermi temperature is very large
in such situations i.e., TFe ' 9 � 104K. So, even in the simpli�ed QMHD model, it is
reasonable to compare the statistical pressure term arising due to the fermionic charac-
ter of electrons and the quantum Bohm potential term in the ultracold magnetoplasma
[83]. Here, by taking into account the quantum statistical e¤ects (Fermi pressure), it
63
is pointed out that the contribution of fermionic pressure to the dispersive properties
of low frequency electrostatic and electromagnetic waves in comparison with the quan-
tum di¤raction e¤ects (quantum pressure) is signi�cant for dynamic, as well as static
ions. The shear Alfven waves can be coupled with the electrostatic quantum ion-acoustic
type waves. However, the inhomogeneity e¤ects, collisions and the spin e¤ects are not
considered in the model.
Let us consider two-component dense homogenous magnetoplasma consisting of elec-
trons and ions. The plasma is embedded in a uniform magnetic �eld B0z; where B0 is
the strength of magnetic �eld and z is the unit vector in the z-axis direction. The low
frequency (in comparison with the ion cyclotron frequency) electric, and magnetic �eld
perturbations are de�ned as E = �r'� c�1 (@Az=@t) z and B? =r?Az � z; where Azis the component of vector potential along z-axis. We start from the set of �uid equa-
tions for quantum plasmas i.e., the continuity equation (3.2) and the equation of motion
(3.3) along with Ampere law (3.4) and Poisson equation (4.11). The pressure term in
(3.3) contains both the fermion pressure PF and thermal pressure Pt. For very low tem-
perature plasma by assuming that the ions behave classically in the limit TFi << TFe;
the pressure e¤ects of only quantum electrons are relevant. In this situation the Fermi
pressure which is contribution of the electrons obeying the Fermi Dirac equilibrium is of
most signi�cance. Then the �uid pressure (third term in (3.3)) is given by the equation
of state for spin 1/2 electrons (1.13) as follows
PFe =}2
5me
�3�2�2=3
n5=3e : (4.13)
In the linearized form, the gradient of Fermi pressure leads to
rPFe1 '}2
3me
�3�2ne0
�2=3rne1; (4.14)
where the perturbation is assumed to be proportional to exp[i (k � r� !t)]. The index 0
and 1 is used to denote the equilibrium, and perturbed quantities, respectively. The last
64
term in expression (3.5)-the gradient of the so called Bohm potential in the linear limit
may be written as
rPq1 '}2k2
4me
rne1 (4.15)
where Pq has the dimensions of pressure. Notice that (3�2)2=3 ' 9:6 and n2=3e0 = 1
�r2where
�r is the average interparticle distance. If k s 106cm�1 is assumed, then at metallic
electron densities i.e., ne0 s 1024cm�3, we have, �r ' 10�8cm; which shows that
k2 <<1
�r2: (4.16)
The inequality (4.16) shows that the variation of the quantities should be on length scales
that are larger than �Fe. The �uid model is a useful approach on such scales. We study
the low frequency linear waves for a very simple physical picture, retaining quantum
e¤ects due to both the terms.
4.5 Linear coupling of Alfven waves and acoustic type
modes
From (3.3), the linearized velocity components of ions in the perpendicular and parallel
directions may be written as,
vi?1 'c
B0
�z�r?'1 �
1
!ci
@r?'1@t
�' vE + vPi; (4.17)
@tviz1 ' � e
mi
�@'1@z
+1
c
@Az1@t
�; (4.18)
with vE and vpi being the electric and polarization drifts, respectively. The components
of electron velocities in perpendicular and parallel directions can be written, respectively,
as,
65
ve?1 'c
B0z�r?
�'1 +
~2
4men0er2ne1 �
2kBTFe3n0e
ne1
�;
' vE + vqe + vDe; (4.19)
@tvez1 'e
me
@
@z
�'1 +
~2
4men0er2ne1 �
2kBTFe3n0e
ne1
�+
e
mec
@Az1@t
; (4.20)
where vqe and vDe are de�ned as the electron quantum and diamagnetic type drifts,
respectively, j@tj � !pe; ck; and ne0 = ni0 = n0. The continuity equation (3.2) can be
expressed as
@tnj1 + n0 (r:vE1 +r:vpj + @zvjz1) = 0: (4.21)
and the Ampere law (3.4) is given by
r2?Az1 =4�n0e
c(vez1 � viz1) � (4.22)
4.5.1 Dynamic ions and electrons
First, we consider the dynamics of both the species i.e., ions and electrons. The subtrac-
tion of electron and ion continuity equations leads to
@
@t(ne1 � ni1)� n0r:vpi + n0
@
@z(vez1 � viz1) = 0: (4.23)
Using expressions (4.11), (4.17) and (4.22) in the above equation, we obtain
@
@t
�r2 + c2
v2Ar2?�'1 +
c@r2?Az1@z
= 0; (4.24)
66
where vA = B0=p4�n0mi is the speed of Alfven wave, and we have de�ned the current
as Jz1 ' en0(viz1 � vez1). Ion continuity equation along with (4.17) and (4.18) yields,
@2ni1@t2
� n0c
B0!ci
@2r2?'1@t2
� n0e
mi
�@2'1@z2
+1
c
@
@z
@Az1@t
�= 0: (4.25)
Eliminating Az1 from (4.24) and (4.25) and Fourier analyzing, we obtain,
ni1n0' 1
!2
���2qk2?!2 +
c2qk2z
!2A
�!2A � !2
�1 +
vAk2
c2k2?
����1; (4.26)
where we have de�ned the quantum ion-acoustic speed as cq =pTq=mi, the ion Larmor
radius at e¤ective electron temperature as �q = cq=!ci, the Alfven wave frequency as
!A = kzvA and �1 = e'1=Tq: The e¤ective temperature of electrons (in energy units) is
de�ned as Tq = (~2k2=4me + 2kBTFe=3) ; which is a pure quantum mechanical e¤ect. The
�rst term in Tq corresponds to quantum Bohm potential, and the second term represents
the electron Fermi energy. So the parameters cq and �q contain the contribution of both
the terms. The Poisson equation (4.11) gives
ne1n0
=ni1n0�c2qk
2
!2pi�1; (4.27)
with !pi�=p4�n0e2=mi
�being the ion plasma frequency. Using (4.26) and (4.27), we
obtainne1n0
=1
!2
���2qk2?!2 +
c2qk2z
!2A
�!2A � !2
��c2qk
2
!2pi!2��1: (4.28)
The electron parallel equation of motion leads to,
@vez1@t' e
me
�@'1@z
+1
c
@Az1@t
�� Tqmen0
@ne1@z
: (4.29)
67
From Ampere law we �nd, vez1 = c4�n0er2?Az1+ viz1; which on using in expression (4.29),
along with (4.24) leads to
ne1n0' 1
!2A
�!2A �
�1 + �2ek
2?��1 +
v2Ak2
c2k2?
�!2 +
me
mi
�!2A �
�1 +
v2Ak2
c2k2?
�!2���1:
(4.30)
where �e = c=!pe is the electron collisionless skin depth and the small term in the curly
bracket appears from ion parallel velocity component. From (4.28) and (4.30), we obtain
the linear dispersion relation of low frequency coupled electrostatic and electromagnetic
modes in the dense cold magnetoplasma as,"�1 +
v2Ak2
c2k2?
�!2 +
me
mi
�1 + �2ek
2?� ��1 + v2Ak
2
c2k2?
�!2 � !2A
��
c2qk2z�
1 + �2ek2?� �1 + v2Ak
2
c2k2?
�#!2�
!2A�1 + �2ek
2?� ��1 + c2qk
2
!2pi
�!2 � c2qk2z
�=
�2qk2?!
2A�
1 + �2ek2?�!2:
(4.31)
Since me=mi << 1; therefore the coupled dispersion relation (4.31) reduces to"�1 +
v2Ak2
c2k2?
�!2 �
c2qk2z�
1 + �2ek2?� �1 + v2Ak
2
c2k2?
�#!2�
!2A�1 + �2ek
2?� ��1 + c2qk
2
!2pi
�!2 � c2qk2z
�=
�2qk2?!
2A�
1 + �2ek2?�!2; (4.32)
In the limit �2ek2? << 1; (4.32) can be written as��1 +
v2Ak2
c2k2?
�!2 � c2qk2z
�1 +
v2Ak2
c2k2?
��!2�
!2A
��1 +
c2qk2
!2pi
�!2 � c2qk2z
�= �2qk
2?!
2A!
2: (4.33)
If we assume vA << ck?=k in a quasi-neutral limit ne1 t ne1; (4.33) leads to
�!2 � c2qk2z
� �!2 � !2A
�= �2qk
2?!
2A!
2; (4.34)
68
which is the same as equation (18) of Ref. [78] if the density inhomogeneity is neglected
there in the limit kBTFene1 << (}2=4me)r2ne1:
In the case of negligibly small parallel ion current, equation (4.32) yields,
!2 =k2zv
2A�
1 + �2ek2?�0@ 1�
1 +v2Ak
2
c2k2?
� + �2qk2?
1A : (4.35)
Expression (4.35) shows the e¤ects of electron inertia on shear Alfven wave frequency, at
quantum scale lengths of electrons in a dense ultracold magnetoplasma. If the electron
inertia is neglected (4.35) may be written as
!2 = k2zv2A
0@ 1�1 +
v2Ak2
c2k2?
� + �2qk2?
1A : (4.36)
For vA <<ck?k; we have
!2 = k2zv2A
�1 + �2qk
2?�; (4.37)
which shows the dispersion of shear Alfven wave frequency due to quantum e¤ects asso-
ciated with the electrons in a dense quantum magnetplasma.
4.5.2 Immobile ions
Now we assume that the ions are immobile in the background of dynamic electrons i.e.,
vi = ni1 = 0. Then the Ampere law leads to
vez1 =c2
4�n0er2?Az1: (4.38)
69
The perpendicular component of electron �uid velocity from (3.3) becomes
ve?1 'c
B0z�r?
�'1 +
~2
4men0er2ne1 �
2kBTFe3n0e
ne1
�+
c@r?
B0!ce@t
�'1 +
~2
4men0er2ne1 �
2kBTFe3n0e
ne1
�: (4.39)
On using (4.11), (4.22) and (4.39) in the electron continuity, we obtain
@
@t
�r2'1 +
!2pe!cer2?
�'1 +
~2
4men0er2ne1 �
2kBTFe3n0e
ne1
��+ c
@
@zr2?Az1 = 0; (4.40)
where !ce = eB0=mec is the electron cyclotron frequency. Using (4.20), (4.24) and (4.38)
along with (4.40), we have
�1� �2er2?
� @2@t2
�r2'1 +
!2pe!cer2?
�'1 +
1
n0e
�}2r2
4me
� 2TFe3
�ne1
���
c2r2?@2
@z2
�'1 +
1
n0e
�}2r2
4me
� 2TFe3
�ne1
�= 0: (4.41)
Combining (4.11) and (4.41), and Fourier transforming the resulting equation, we obtain
the linear dispersion relation as follows
!2 =v2Aek
2z
�1 + �2qk
2��
1 + �2ek2?� h1 + �2qk
2 +v2Aek
2
c2k2?
i ; (4.42)
where vAe = B0=p4�n0me; �q =
vq!pe
and vq =pTq=me: The above equation shows that
the wave frequency strongly depends on the quantum nature of electrons which gives
rise to the dispersion due to the fermionic pressure and di¤raction e¤ects. The last term
in square brackets in denominator is negligibly small in general. If Fermi pressure is
neglected, expression (4.42) reduces to the result (9) of Ref. [73]. However, it is seen
that the contribution of Fermi pressure is dominant in the wave dispersion as compared
with the quantum pressure arising due to Bohm potential term.
70
Figure 4-4: The Alfven wave frequency ! is plotted versus kz and k? for mobile ions:Case (a) corresponds to the frequency without the e¤ect of fermionic pressure, and case(b) corresponds to ! with the e¤ect of fermionic pressure where ne0 �= 1� 1026cm�3 andB0 �= 1� 108G.
71
4.5.3 Applications
We numerically analyze the quantum e¤ects arising due to Fermi pressure and Bohm
potential on the waves using typical parameters relevant to the dense plasmas of compact
astrophysical objects e.g., white dwarf stars [3, 8], as well as dense electron Fermi gas near
metallic densities [35]. The density in the interior of white dwarf stars is of the order of
1026cm�3: For such densities, we have typically vFe �= 108cm=sec; �Fe �= 4� 10�9cm and
�r �= 2� 10�9cm: For vte < vFe; we have �Be > �Fe and ne0�3Be > 1 where vte is electron
thermal speed. We choose k? s 106cm�1 so that the wavelengths of the perturbation
are much larger than �Be and Rin. Since we have assumed kz << k? in deriving the
dispersion relation, therefore we take kz=k? s 0:002: The dispersion relation (4.32) is
analyzed numerically in Figs. (4-4) and (4-5) for Alfven wave and electrostatic waves,
respectively, to demonstrate the wave dispersion due to quantum e¤ects. We use the high
magnetic �elds of the order 108G which are within the limits of dense astrophysics plasmas
[8, 53]. For above mentioned parameters we �nd �2ek2?�= 0:003, �q �= 0:3 � 10�5cm,
vA �= 2�106cm=sec and ck?=k >> vA: The overall contribution of the quantum e¤ects in
wave dispersion is weak but the e¤ect of the electron Fermi pressure is more important as
compared with the corresponding quantum di¤raction term. It may be mentioned here
that TFe is a function of density and assumes very large values. The dispersion relation
(4.42) is plotted in Fig. (4-6) for ne0 ' 1024cm�3. Such densities are relevant to the
dense Fermi gas of electrons with the background of stationary positive ions [35] as well
as the plasma in the interior of Jovian planets (Jupiter, Saturn) [3]. For ! << !ce; it is
found that �2qk2?�= 0:03 which is due to the dominant contribution of electron fermionic
pressure since the dispersive e¤ects due to quantum di¤raction term are negligibly small.
The approximations and the assumptions made in deriving the dispersion relation (4.42)
are satis�ed in the parameter range used for numerical work. The �uid model may be
used for physical understanding of the waves in dense plasmas even if �2qk2? > 1 since we
have T << TFe.
72
Figure 4-5: The quantum ion-acoustic wave frequency ! from (4.32) is plotted against kzand k?: Case (a) shows the wave frequency when the e¤ect of fermionic pressure is notincluded, and case (b) when included. Other parameters are the same as in Fig. (4-4)
73
Figure 4-6: The linear dispersion relation (4.42) for immobile ions is plotted with ne0 �=1� 1024cm�3 and B0 �= 1� 108G.
74
Chapter 5
Nonlinear Electrostatic Waves in
Homogenous Quantum Plasmas
In this chapter, we study the propagation of low frequency nonlinear electrostatic waves
in homogenous dense quantum plasmas with and without the background of charged dust
particles. The in�uence of quantum e¤ects in unmagnetized as well as magnetized cases
is discussed with illustrations.
5.1 Introduction
Plasmas are rich in waves. The dynamics of a plasma is governed by the self consistent
electric and magnetic �elds. The linear wave propagation is generally studied by assuming
a harmonic wave solution proportional to ei(k�r�!t) in the small amplitude limit. When
the amplitudes of the waves are su¢ ciently large, the nonlinearities cannot be ignored.
The nonlinearity makes the system highly complicated and its analysis di¢ cult. For a
nonlinear system, a small disturbance such as a slight change of the initial conditions,
can result in a big di¤erence in the behaviour of the system at a later time. Then
the wave propagation cannot be described by the linear theory. The nonlinearities in
plasmas enter through the harmonic generation (involving �uid advection), trapping
75
of particles in the wave potential, the nonlinear Lorentz force, ponderomotive force etc.
Sometimes, the nonlinearities in plasmas contribute to the localization of waves which give
rise to di¤erent types of interesting coherent structures, for instance solitary waves, shock
waves, double layers, vortices and so on. These structures have profound consequences
in the system which are important from both theoretical and experimental point of view.
Such structures commonly arise due to competition between nonlinearity, dispersion and
dissipation and important to understand the particle or energy transport mechanisms.
Among the nonlinear structures, solitary waves or solitons have received great atten-
tion of researchers because of their wide ranging applications in physics. The solitons
were �rst observed in water by J. S. Russel more than one and half century ago. In
1895, Dutch mathematicians Korteweg and de Vries derived a model equation which de-
scribes the unidirectional propagation of certain types of waves (solitary waves) of long
wavelengths in water of relatively shallow depth [84]. This equation is now known as
Korteweg-de Vries equation or KdV equation in short. Korteweg and de Vries showed
that periodic solutions of that equation, which they named cnoidal waves, can be found
in a closed form without further approximation. Moreover they found a localized solution
which represents a single hump or positive elevation. This hump was the solitary wave
which was discovered incidentally by J. S. Russel.
The KdV equation remained in obscurity until 1965 when Zabusky and Kruskal dis-
covered that two such solitary waves or solitons emerge unchanged from a collision [114].
The discovery of this remarkable stability property of soliton caused a tide of research
in the subsequent years which is still going on. The KdV equation is a generic model
which has been widely used to investigate the stable solitary waves known as KdV soli-
tons in media with weak dispersion. Initially the main interest in the soliton theory was
related to the hydrodynamics. Then it attracted the plasma physics. Its connection with
nonlinear Schrodinger equation made it popular in condensed matter physics, biophysics,
Bose-Einstein condensates and quantum plasmas. With KdV as a prototype, the soliton
theory is an interdisciplinary topic, where many ideas from mathematical physics, sta-
76
tistical mechanics, solid state physics and quantum theory mutually bene�t each others.
The solitons appear as solutions of particular nonlinear wave equations which often have
a certain universal signi�cance. Many applications of the soliton theory in di¤erent areas
are based on similar model equations and thus allow uni�ed theoretical approaches.
An arbitrary pulse propagating in a dispersive medium spreads out as it moves ahead.
If the nonlinear interactions are signi�cant, the points of large amplitude of the wave
over take the points of small amplitude, the wave steepens and ultimately breaks down.
However, if the dispersion and wave breaking e¤ects delicately compromise with each
other in such media, a stable stationary structures-the solitons can be formed. Solitons
are localized structures which travel with permanent shape and constant velocity. They
are robust against perturbations, and can cross over without change of their shapes and
velocities. They are considered as independent dynamic entities.
5.2 Korteweg-de Vries equation
The Korteweg-de Vries equation in simpli�ed form may be given by,
@u
@t+ u
@u
@x+ �
@3u
@x3= 0; (5.1)
where � is a constant having dimensions of L3
T. The second term on the l.h.s of (5.1) is
the nonlinear term and third one is the dispersive term. The nonlinear term steepen the
wave as it moves. When the wave steepening progresses, the contribution to the higher
derivative term becomes more and more e¤ective which prevent the unlimited steepening.
Ultimately it becomes comparable to the nonlinear term and the wave steepening is
stopped [115].
Assume that the balance between nonlinearity and dispersion produces a stable wave
structure in the �uid which is moving with some constant speed say u0 and de�ne a
stationary coordinate � such that � = x� u0t and t = � measured from the center of the
77
localized solution. Then equation (5.1) can be written in (�; �) frame as,
@u
@�+ (u� u0)
@u
@�+ �
@3u
@�3= 0: (5.2)
For co-moving solution of (5.2), the time derivative should vanish. The transformed
equation then becomes a third order ordinary di¤erential equation in a single variable �
given by
(u� u0)@u
@�+ �
@3u
@�3= 0: (5.3)
Integration of the above equation leads to,
u2
2� u0u+ �
@2u
@�2+ C1 = 0; (5.4)
where C1 is arbitrary constant. Multiplication of both sides of (5.4) with @u@�and integra-
tion gives1
6u3 � u0
2u2 + C1u+
1
2�
�@u
@�
�2+ C2 = 0; (5.5)
with C2 being another constant.
Using the boundary conditions for a localized solution such that u = @u@�= 0 as
� ! �1; the constants vanishes i.e., C1 = C2 = 0. Then (5.5) can be written as
du
up(3u0 � u)
=d�p3�: (5.6)
Substituting u = y2 in the above equation and integrating, (5.6) leads to
Z y
0
2d�y
�yp(3u0 � �y2)
=
Z �
0
1p3�d� (5.7)
or2p3u0
sech�1�
yp3u0
�=
�p3�
(5.8)
The lower limit starts from 0 which does not bring any loss of generality since the starting
78
point can be transformed linearly. Then (5.8) can be written as
y =p3u0 sech
�ru0�
�
2
�(5.9)
or
u(x� u0t) = 3u0 sech2�r
u0�
(x� u0t)2
�(5.10)
Equation (5.10) describes the propagation of a stationary bell-shaped wave with velocity
u0 along x direction without any change in shape. The parameter 3u0 gives the amplitude
and 2q
�u0gives the width of the pulse. The high speed solutions have a narrower width
than low speed solutions and greater amplitude corresponds to larger speed. The solution
(5.10) does not change or slow down during the course of propagation and remains stable
against disturbances and collisions.
5.3 Quantum Dust Ion-Acoustic Solitary Waves
Haas et. al. have studied the quantum ion-acoustic solitary waves in the framework
of KdV equation by establishing a quantum hydrodynamic model [37]: Such studies are
also carried out in dense dusty plasmas and their relevance is shown in astrophysical
environment as well as ultra-small scale electronic devices which may be contaminated
with the presence of charged dust impurities [46, 71, 72]. The presence of heavy charged
impurities (dust) give rise to quantum corrected dust modes over the length scales larger
than �Fe. The low frequency (viz., kvFi << ! << kvFe) electrostatic waves propagating
in dense plasma are a¤ected by quantum e¤ects of electrons. We study the weakly
nonlinear dust ion-acoustic waves in this section [81]. The KdV equation (5.10) is used
to obtain the co-moving analytical soliton solution.
Let us consider a three component unmagnetized, ultracold quantum plasma com-
prised of inertialess electrons, inertial ions and negatively charged immobile dust parti-
cles. The dynamics of dust ion-acoustic wave in such plasma is governed by normalized
79
quantum hydrodynamic equations (4.2)-(4.4) of the form
@'
@x� ne
@ne@x
+H2
2
@
@x
�@2xpnepne
�= 0; (5.11)
@ui@t+ ui
@ui@x
+@'
@x+ �ni
@ni@x
= 0; (5.12)
@ni@t
+@
@t(niui) = 0; (5.13)
@2'
@x2+ ni � "ine �Nd = 0; (5.14)
where the normalization scheme is the same as given by (4.9) andH is the non-dimensional
quantum di¤raction parameter in the unmagnetized quantum plasma. We assume that
the fermionic pressure e¤ects are given by (4.5).
Normalized equations (5.11)-(5.14) constitute the Quantum Hydrodynamic (QHD)
model for a three-component dust contaminated quantum plasma. The quantum correc-
tions appear in (5.11) through the 2nd and 3rd terms on the right hand sides, and in
(5.12) through the fourth term. The 3rd term in (5.11) appears due to quantum cor-
relation of density �uctuations and this type of quantum e¤ect is known as quantum
di¤raction and it is taken into account by the term proportional to }2. This contribution
is some time called quantum pressure or quantum Bohm potential. It is noted that the
quantum Bohm potential for the dynamics of ion has been ignored because of its large
inertia. The other quantum contribution (2nd term in (5.11) and 4th term in (5.12)) is
due to quantum statistics and is included in this model through one dimensional equa-
tion of state for fermions. Integrating once the electron momentum equation (5.11) with
boundary conditions ne = 1 and ' = 0 at jxj ! �1, we obtain
' = �12+n2e2� H2
2
�@2xpnepne
�(5.15)
80
5.3.1 Small Amplitude Waves
In order to study the behaviour of nonlinear waves in this system, we apply the reductive
perturbative method to (5.12)-(5.15) to obtain the Korteweg-de Vries (KdV) equation
for nonlinear small but �nite amplitude quantum dust ion-acoustic wave. The plasma
parameters can be expanded as:
nj = 1 + �nj1 + �2nj2 + � � �;
uj = �uj1 + �2uj2 + � � �; (5.16)
' = �'1 + �2'2 + � � �;
where � is a small (0 < � � 1) expansion parameter characterizing the weakness of the
nonlinearity. Introducing the stretched variables in standard form
� = �1=2(x� �t);
� = �3=2t; (5.17)
with � being the normalized linear constant. Substituting (5.16) and (5.17) into (5.12)-
(5.15) and combining terms of lowest order in � i.e.(� � and �3=2) we obtain
�ni1 � �2ui1 = '1�; � =
�1 + �(1�Nd)(1�Nd)
�1=2� (5.18)
Collecting the next higher order of � , we get
��@ni2@�
+@ui2@�
+@ni1@�
+@
@�(ni1ui1) = 0;
��@ui2@�
+@'2@�
+ �@ni2@�
+@ui1@�
+ u1@ui1@�� �ni1
@ni1@�
= 0;
ne2 � '2 +'212� H2
4
@2'1@�2
= 0; (5.19)
@2'1@�2
+ ni2 � (1�Nd)ne2 = 0;
81
Using �rst order solutions in the above set of equations, some obvious calculations lead
to the following KdV equation in terms of '1
@'1@�
+ A'1@'1@�
+B@3'1@�3
= 0; (5.20)
with coe¢ cients
A =1 + (3�2 + �)(1�Nd)2
2�(1�Nd); (5.21)
B =1
2�(1�Nd)
�1
(1�Nd)� H2
4
�� (5.22)
The steady state solution of the KdV equation (5.20) is obtained by transforming the
independent variables � and � into new coordinate � = � � u0� and � = � , where u0 is
the normalized constant speed of wave frame. The possible stationary solution of (5.20)
is then given as
� = 'm sech2
�� � u0�W
�; (5.23)
where '1 � � and W and 'm are the normalized width and amplitude of soliton, which
are given as
'm =3u0A; (5.24)
and
W =
r4B
u0� (5.25)
5.3.2 Results and Discussions
It is seen that the coe¢ cients of the nonlinear and dispersive terms i.e., A and B are
modi�ed in the presence of stationary dust. The non-dimensional quantum parameter
H appears only in dispersive coe¢ cient B: This is due to the fact that we are using the
small amplitude approximation where ne1=ne0 � � and � << 1 in which the nonlinear
part of 3rd term (higher order correction term) on r.h.s. of (5.15) is ignored and we
82
Figure 5-1: Electrostatic potential � as a function of �(= � � u0�) with H = 0 (solidline), H = 1:5 (dashed line), H = 3 (dotted line). Other parameters are arbitrarily takenas u0 = 2, Nd = 0:2, � = 0:01.
are left with purely dispersive contribution of H in KdV equation (5.20). If one uses
arbitrary amplitude approximation, the nonlinear contribution of H dependent term can
be incorporated. Due to small amplitude limit in our treatment, only dispersive coe¢ cient
B depends upon H not the nonlinearity coe¢ cient A.
It is also found that in the absence of dust particles (Nd = 0), (5.20) reduces to
equation (36) of Ref. [37] which is e� i quantum plasma case. However, (36) of Ref. [37]
doesn�t possess the soliton solution at a critical value of H = 2 because the dispersion
term disappears at this value. In our model, the presence of stationary dust particles in
quantum plasma modi�es B and the disappearance of B shifts from H = 2 to a greater
value because Nd < 1. This critical value of H (= Hcr) can be found by taking some
arbitrary value of parameter Nd. For example, when Nd = 0:1; � = 0:01, B = 0 when
Hcr�= 2:108. In such situation, KdV leads to the �rst order nonlinear partial di¤erential
equation which yields the formation of a shock due to free streaming. For all other possible
values of H, the soliton solution (5.23) holds with di¤erent character for H greater or
83
Figure 5-2: Plot of soliton�s widthW versus constant speed u0 with variation of quantumparameter H, i.e. H = 0 (solid line), H = 1:5 (dashed line) and H = 3 (dotted line).Other parameters are � = 0:1 and Nd = 0:1.
smaller than Hcr. We have noticed that for 0 6 H 6 Hcr; i.e., H2 < 4= (1�Nd) with
0 < Nd < 1; B > 0 and the soliton solution demands u0 to be positive. In this case,
we have a compressive solitons. On the other hand, for H > Hcr or H2 > 4= (1�Nd) ;
u0 should be negative for the soliton solution (5.23) to exist and we have a rarefactive
soliton. The dependence of the sign of dispersive coe¢ cient on quantum parameter H in
electron ion quantum plasma has been discussed by Ref. [37] in detail.
Now we use some typical normalized parameters to elaborate the quantum e¤ects on
dust ion-acoustic solitary wave. The quantum di¤raction correction does not a¤ect the
amplitude of wave potential, but shrinks its width as shown in Fig. (5-1) which is further
veri�ed by Fig. (5-2), in which the e¤ect of quantum di¤raction parameter on width
of dust ion-acoustic soliton is shown It is also shown that quantum statistical e¤ects
through ion Fermi temperature, decreases the soliton amplitude as shown in Fig. (5-3).
With increasing dust concentration both the width and amplitude of ion acoustic solitary
wave increases as evident from Fig. (5-4).
84
Figure 5-3: Electrostatic potential � as a function of � for di¤erent values of � Otherparameters are.H = 1, Nd = 0:2 and u0 = 2.
Figure 5-4: Variation of electrostatic potential � with � for di¤erent values of dustconcentration
85
5.4 Nonlinear Ion Waves in Quantum Magnetoplas-
mas
The basic set of equations describing the dynamics of ion waves in the presence of a
constant external magnetic �eld and in the absence of dissipation is given by (Sec. 2.5)
mj
�@uj@t
+ (uj �r)uj�= qj
�E+
1
c(uj �B)
�� 1
njrPj +
}2
2mj
r r2pnjpnj
!; (5.26)
@nj@t
+r: (njuj) = 0; (5.27)
r � E = 4�e(ni � ne); (5.28)
where subscript j = e; i denotes electrons and ions, respectively and r = x@=@x + y
@=@y+ z @=@z. In the above equations uj; mj, nj, Pj and qj are the �uid velocity, mass,
number density with equilibrium value nj0, pressure and charge of jth species, c is the
speed of light in vacuum, } is the reduced Plank�s constant and E and B are electric
and magnetic �elds. The external magnetic �eld is de�ned as B = B0z; where z is the
unit vector along the z-axis and B0 is the strength of uniform �eld, and the electrostatic
electric �eld perturbation is de�ned as E = �r'. The pressure of electrons is obtained
by assuming the equation of state for a one dimensional zero-temperature Fermi gas as
given by (4.5) i.e.,
Pe =1
3
mev2Fe
n2eon3e; (5.29)
where vFe = (2kBTFe=me)1=2 is the Fermi speed of an electron. We assume that the
condition TFi << TFe is satis�ed and only the pressure e¤ects of quantum electrons are
relevant.
5.4.1 Korteweg-de Vries Equation
For small but �nite amplitude weakly nonlinear ion waves in quantum magnetoplasma,
we derive the Korteweg-de Vries (KdV) equation by employing reductive perturbative
86
method (RPM). Equations (5.26)-(5.28) are normalized using the parameters
u�i;ez ! ui;ez=cq , n�j ! nj=nj0 , t�! t!ci , r�!rcq=!ci , �z = z!ci=cq , '�! e'=2kBTFe�
The resulting ion momentum and continuity equations in normalized form can be written
as
@uix@t
+ (uix@
@x+ uiy
@
@y+ uiz
@
@z)uix = �
@'
@x+ uiy;
@uiy@t
+ (uix@
@x+ uiy
@
@y+ uiz
@
@z)uiy = �
@'
@y� uix;
@uiz@t
+ (uix@
@x+ uiy
@
@y+ uiz
@
@z)uiz = �
@'
@x; (5.30)
@ni@t
+@ (niuix)
@x+@(niuiy)
@y+@(niuiz)
@z= 0:
The electron momentum equation leads to
@�
@z� ne
@ne@z
+H2e
2
@
@z
�r2pnepne
�= 0; (5.31)
and the Poisson�s equation becomes
�r2' = ne � ni (5.32)
where � = !2ci=!2pi. The primes from normalized quantities has been omitted for simplic-
ity. Integration of electron equation with boundary conditions ne = 1, ' = 0 at in�nity
gives the relation
' = �12+n2e2� H2
e
2
�r2pnepne
�(5.33)
where the non-dimensional parameter He = }!L=2kBTFe represents the quantum di¤rac-
tion e¤ects in this system, !L = (!ce!ci)1=2 is the lower-hybrid resonance frequency and
!ce = eB0=mec is the electron cyclotron frequency. The perturbed quantities nj, uj and
87
' are expanded as a power series in � as
nj = 1 + �nj1 + �2nj2 + � � �;
uix;y = 0 + �3=2uix;y1 + �2uix;y2 � ��;
uiz = 0 + �uiz1 + �2uiz2 + � � �; (5.34)
' = 0 + �'1 + �2'2 + � � �;
and independent variables are rescaled in a standard fashion given by
� = �1=2(lxx+ lyy + lzz � �t);
� = �3=2t; (5.35)
where � (0 < � � 1) is the parameter proportional to the amplitude of perturbation and �
is linear constant speed of the wave normalzed by cq. The directional cosines of the wave
vector k along the x-,y- and z-axis are given by lx, ly and lz such that l2x+l2y+l
2z = 1. Using
(5.35)-(5.34) in (5.30) and (5.32)-(5.33), and collecting the lowest orders of ��� �; �3=2
�,
we get
uix1 = �ly�@'1@�
�; uiy1 = lx
�@'1@�
�; (5.36)
� = lz and uiz1 = ni1 = ne1 = '1� (5.37)
88
The next higher order of ��� �2; �5=2
�leads to
uix2 = �lx
�@2'1@�2
�, uiy2 = �ly
�@2'1@�2
�; (5.38)
�lz@'2@�
+ �@uiz2@�
=@uiz1@�
+ lzu(1)iz1
@uiz1@�
; (5.39)
��@ni2@�
+ lz@uiz2@�
+ lx@uix2@�
+ ly@uiy2@�
= �@ni1@�� lz
@
@�(ni1uiz1) ; (5.40)
'2 � ne2 ='212� H2
e
4
�@2'1@�2
�; (5.41)
ne2 � ni2 = �@2'1@�2� (5.42)
From (5.39) and (5.40), we obtain
l2z@'2@�
+ �2@ni2@�
= f1; (5.43)
where f1 = �2lz @'1@� �3l2z'1
@'1@�� l2z(l2x+ l2y)
@3'1@�3
: Using (5.41) in (5.42) and di¤erentiating
once, the resulting equation along with (5.43) gives
f1 � l2zf2 = 0 (5.44)
with f2 = '1@'1@�+(�� H2
e
4)@
3'1@�3
: Equation (5.44) leads to the KdV equation in normalized
parameters as,@'1@�+2lz'1
@'1@�+lz2
�1� l2z + � � H2
e
4
�@3'1@�3
=0; (5.45)
and in un-normalized parameters as,
@'1@�+
�2elzmicq
�'1@'1@�+
�lzcq�
2q
2
��1� l2z + � � H2
e
4
�@3'1@�3
=0� (5.46)
89
where l2x + l2y = 1� l2z and � = lz from (5.37) has been used. If we introduce lx = ly = 0
in (5.45), we obtain
@'1@�+
�2e
micq
�'1@'1@�+
c3q2!2pi
�1� H2
4
�@3'1@�3
=0: (5.47)
where the non-dimensional quantum parameter H = }!pe=2kBTFe in this case: It may be
mentioned here that the condition � << 1 holds in general. For such situations, quasi-
neutrality is a good approximation. However, for the case of dense plasmas in presence
of very high magnetic �elds, � can�t be ignored. We notice that the dispersion term
is strongly a¤ected by the parameter He, the representative of the quantum di¤raction
e¤ects present in the system, the angle of propagation and the background magnetic
�eld.
We have used lz = k=kz = cos � where � is the angle between the directions of the
wave propagation vector k and the external magnetic �eld B0 k z: We are studying the
electrostatic wave, and hence lz 6= 0: Also the perturbation scheme (5.34) is valid for
small but �nite amplitude limit only, and not for large �:
5.4.2 Ion Solitary Wave Solution of KdV Equation
To obtain the steady state solution of (5.46), we transform the independent variables �
and � to a new coordinate � = � � u0� where u0 is constant velocity. This leads to the
possible solution of the form
� = 'm sech2� �W
�(5.48)
with '1 � � and the peak soliton amplitude 'm and soliton width W is given by
'm =3cqmiu02elz
, W =
242 �lzcq�2q��1 + � � l2z �
H2e
4
�u0
351=2 (5.49)
Now, we investigate the soliton properties with the variation of number density, quan-
90
tum di¤raction parameter and magnetic �eld. In Fig. (5-5), the e¤ect of variation of
number density is shown on the amplitude and width of the soliton corresponding to
di¤erent values of B0. It is seen that the amplitude as well as the width of soliton
increases with increase in plasma number density. But the range of spatial scale and
potential is di¤erent for all the cases. Since He increases with increase in is magnetic
�elds, it is seen that the di¤raction e¤ects are too small even at very high magnetic �elds.
The increase in magnetic �eld decreases the soliton width while the amplitude of soliton
remains una¤ected as shown in Fig. (5-6).
It is interesting to note that the results in Sections (5.3) and (5.4) i.e, equation
(5.20) and (5.45) lead to the same equation. The relation (1.11) shows that the Fermi
temperature contains mass of the particle in the denominator. As mi << me; we can
ignore the Fermi pressure of ions (� = 0) in (5.20) in the limit TFi << TFe: Then, setting
Nd = 0 in (5.20), we obtain the same equation as (5.47).
91
Figure 5-5: Plot of soliton solution � vs � for neo = 1 � 1026cm�3(solid line), neo =1:5 � 1026cm�3(dotted line) and neo = 2 � 1026cm�3(dashed line) in dense magnetizedHydrogen plasma with Bo = 1 � 106G (upper panel), in dense strongly magnetizedastrophysical plasma with same density and Bo = 1 � 109G (middle panel), and neo =1�1028cm�3(solid line), neo = 1:5�1028cm�3(dotted line) and neo = 2�1028cm�3(dashedline) with Bo = 5� 1012G (lower panel) where lz = 0:8
92
Figure 5-6: Soliton solution is plotted for dense Hydrogen plasmas with neo = 1 �1026cm�3, Bo = 1�106G (solid line) andBo = 2�106G (dotted line) (upper panel) and fordense strongly magnetized astrophysical plasma with neo = 1�1026cm�3, Bo = 1�109G(solid line) and Bo = 2 � 109G (dashed line) (middle panel) and Bo = 1 � 1012G (solidline), Bo = 2� 1012G (dashed line).(lower panel).
93
Chapter 6
Summary
The results of the studies on low-frequency linear and nonlinear waves are summarized
here.
The propagation of low frequency electrostatic and electromagnetic modes in homoge-
nous as well as inhomogenous dense plasmas is studied using �uid equations including
quantum e¤ects. In chapters 1 and 2, brief description of quantum plasmas is given and
the mathematical models are presented in some detail. In chapter 3, linear dispersion
relations of coupled quantum drift and inertial Alfven waves in an electron-ion quantum
plasma are investigated. The quantum e¤ects on low frequency waves in the presence
of stationary dust are also discussed. The ultracold dense inhomogenous plasmas can
support a drift like mode which is independent of electron temperature having ! �= !�q:
The linear dispersion relation shows that if the quantum drift wave frequency is zero i.e.,
!�q = 0, the dispersion relation (3.15) reduces to classical Shukla-Varma mode which is
equation (7) of Ref. [76] if the dissipative damping is ignored. Using typical parameters,
the length scales for propagation of such waves are discussed and the relevance of the
results with dense laboratory and astrophysical plasmas is described. The three roots
of the dispersion equation, one corresponding to qe and two corresponding to Alfven
waves are shown in Fig. 4-2. The positive frequency branch of Alfven wave couples with
the drift frequency !�q which itself is independent of kz. The third root in the dispersion
94
relation represents the wave corresponding to qe:
The drift wave investigated in Sec. (3.3) exists due to the e¤ective temperature
de�ned through the quantum Bohm potential of electrons given by Tq = }2k2=4me (in
energy units). The dispersion relation shows the coupling of three basic low frequency
modes in quantum plasmas which contains the quantum drift wave, the ion acoustic wave
and Alfven wave. The interparticle distance and the de Broglie wavelength in cold dense
plasmas can become of the order of an Angstrom (10�8cm). The collective phenomena like
wave propagation in such systems should be de�ned very carefully which contains only the
weak quantum e¤ects in the limit �B < � within the �uid framework. It is interesting to
point out the analogy of dispersion relation (3.32) with the classical plasma case [79]. If cq,
�q and !�q are replaced by the corresponding classical parameters i.e., the ion sound speed
cs = (Te=mi)1=2, the ion Larmor radius at electron temperature �s = cs=!ci and classical
drift wave frequency !�e, respectively, then the dispersion relation (3.32) becomes similar
to (4.47) of Ref. [79] for Ti = 0. This is very special situation but interesting because
the ultracold dense quantum plasma has the electromagnetic linear dispersion relation
analogous to the well known classical plasma result. In the dense ultracold plasma case,
the drift wave can exist even if the temperature of electrons is ignored. For illustration,
the frequencies of the four modes of ultracold dense plasma are plotted against kz. One
branch of the shear Alfven wave and one branch of the electrostatic wave are in�uenced
by the quantum drift wave !�q near kz � 4 � 103cm�1. Since Ti = 0 has been assumed,
therefore the second branch of the Alfven wave remains a straight line in this �gure.
The second branch of the electrostatic wave has also the e¤ects of dispersion similar to
the classical case. Similarly, the coupled dispersion relation (3.32) for the plasma with
relatively high density and magnetic �elds is investigated numerically. In both the cases,
the behaviour of the coupled modes is found similar but the wave frequencies are di¤erent
due to di¤erent number densities and magnetic �elds.
In chapter 4, the linear waves in homogenous dense quantum plasmas are inves-
tigated. For magnetized dense electron-ion plasma, the linear dispersion relation of
95
ion waves propagating obliquely with respect to the external magnetic �eld B0 in a
dense magnetized electron-ion quantum plasma shows that the quantum e¤ects are im-
portant only at very short length scales. The quantum di¤raction length de�ned as
�q = ~= (2mekBTFe)1=2 is much smaller than the perturbation wavelength. If �q = 0
is assumed, then relation (4.12) seems to be similar to the well known classical plasma
dispersion relation. But the system still exhibits the quantum nature due to Fermi-Dirac
distribution of electrons in equilibrium. The quantum di¤raction e¤ects are proportional
to B0=n2=3e0 where ne0 is equilibrium number density. The parameter representing the
quantum di¤raction e¤ects is He s 1 for Bo s 1013G. This shows that the wave disper-
sion caused by the parameter He is vanishingly small in laboratory systems as the choice
of ambient magnetic �eld is limited in this case. However, in the case of dense plasmas
found in the astrophysical objects like neutron stars and white dwarfs, the magnetic �eld
is extremely high and plasma densities are enormously large [3; 8; 21; 24]. Therefore, it
is important to consider the quantum di¤raction e¤ects in such systems. The ion-acoustic
wave pro�le in unmagnetized dense quantum plasma is a¤ected by the presence of charged
background dust: The presence of stationary dust particles in quantum plasma controls
the dispersive properties of the wave. The quantum di¤raction e¤ects associated with
electrons in case of unmagnetized quantum plasma are proportional to n1=6eo . It is found
that the concentration of the dust particles and the quantum di¤raction e¤ects increase
the phase speed of the wave.
The self-consistent dynamics of low frequency linear modes in dense homogenous
electron-ion quantum magnetoplasmas are also studied and a generalized dispersion re-
lation is obtained which shows the coupling of Alfven waves with the electrostatic wave.
The dispersion relation reveals the dispersive e¤ects associated with fermionic pressure of
electrons and quantum Bohm potential. By numerical calculations, it is found that the
electron fermionic pressure is dominant for static as well as dynamic ions. A comparison
of the fermionic pressure term with the Bohm potential term is also presented. The lin-
ear dispersion relation has also been obtained for dense plasma with the background of
96
stationary ions in the presence of high magnetic �eld. In the case of immobile ions, only
electromagnetic part survives and the dispersion enters through the electron quantum
e¤ects.
The nonlinear electrostatic waves are studied for unmagnetized dense homogenous
plasma with the background of stationary dust as well as for magnetized electron ion
plasma. The reductive perturbative technique is employed to obtain the KdV equation
whose co-moving solution gives the quantum ion-acoustic solitary structures. The KdV
solitons are stable one and show the in�uence of quantum e¤ects of electrons. In dust con-
taminated plasma case, the amplitude of the KdV solitons doesn�t change with quantum
pressure while the width of the soliton decreases. The increase in the dust concentration
increases both the amplitude and width of the soliton. In the absence of dust, the re-
sults of Ref. [37] are recovered for electron-ion quantum plasma case. For electron-ion
magnetoplasma, the soliton pro�le shows strong dependence on number density, quan-
tum pressure and magnetic �eld. It is seen that the amplitude as well as width of the
soliton increases with increase in plasma number density since TFe � constant � n2=3eo :
The increase in magnetic �eld decreases the soliton width while the amplitude of soliton
remains una¤ected.
The studies carried out in this thesis on the basis of nonrelativistic quantum hydro-
dynamic model can be useful to understand the dynamics of low frequency modes in
dense plasma systems. For future studies, we think that the low frequency modes in the
presence of spin e¤ects will be interesting which may lead to new modes and interesting
modi�cations.
97
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