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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