Acoustic Waves Part 3 potx

3.1.1 Single mode Figure 2 shows the transmission rates versus frequency in the case of equilateral-triangular holes α = π/3 and isosceles-triangular holes α = 2π/9 for two opposite inc

Rectifying Acoustic Waves 49

Here we consider acoustic waves propagating through an array of triangular holes drilled in

an isotropic material In this case the equations of motion governing the displacement

vectors u(r, t) are given by

( ) ( )u i ,t j ij( ) (,t i 1,2,3)

ρ x  r = ∂σ r = (1) ( ), ( ) ( ),

ij t c ijmn n m u t

where r = (x, z) = (x,y, z) and the summation convention over repeated indices is assumed in

Eqs (1) and (2) ρ (x) and c ijmn (x) are the position-dependent mass density and elastic

stiffness tensor of the system, and σij (r, t) is the stress tensor Furthermore, we need to

impose proper boundary conditions for SAWs SAWs should satisfy the stress-free

boundary condition at the surface z = 0, or

3 0 3 0 0 1,2,3

i z c i mn n m u z i

Solving the equations with finite-difference time-domain (FDTD) method numerically, we

can obtain the time evolutions of displacement vectors u(r, t) and stress tensors σij (r, t) at

each point in the system To calculate the transmission rate through the periodic array of

triangle holes, we define the acoustic Poynting vector J i (r, t) = −  (r, t) u j σji(r, t) from the

continuity of energy flow In terms of the Fourier components of the displacement ˆu (r,ω)

and the stress tensor ˆσij (r,ω), the energy flow at frequency ω in the x direction at the

position x is expressed by

ˆ ( , ) 4 Im ˆ ( , )ˆ ( , ) .

J xω = − π∫ ⎡⎣ωu rω σ∗ rω ⎤⎦dydz (4) Hence we can determine the transmission rate T (ω) by the ratio of the element of the

acoustic Poynting vector in the x direction ˆ ( J x x D, )ω to that in the absence of scatterers

J x

ωω

ω

where x D is the detecting position which is on the right side of the scatterers for case (I) and

on the left side of the scatterers for case (II) We introduce an efficiency η(ω) to quantify

rectification by

( ) ( )( )

3.1 Bulk acoustic waves

In the section we illustrate the rectifying effects of bulk acoustic waves propagating in the

x-direction Because of the homogeneity in the z direction, the governing equations (1) and (2)

are decoupled into two independent sets; one is expressed by

( ) ( ) ( ) ( )

2 2

44 44

,, ,

,,

, ,

, , ,

zy

z zx

z zy

ρσσ

where the acoustic wave is z-polarized transverse wave, referred to as single mode Another

is the acoustic waves termed mixed modes with polarization in the x – y plane, which obey

44

,, ,

,

,,,

,,

,,

xx

y x

yy

y x

ρ

ρσσσ

x x

x x

The mixed modes consist of longitudinal and transverse waves due to the scattering by the

triangular voids since the mode conversion between the longitudinal and transverse waves

takes place for scattering

3.1.1 Single mode

Figure 2 shows the transmission rates versus frequency in the case of equilateral-triangular

holes (α = π/3) and isosceles-triangular holes (α = 2π/9) for two opposite incident directions

(I) and (II) For both types of triangular holes, there is not noticeable difference in the

transmission rates between the two incident directions at low frequency; ωa/v t < π On the

other hand, we find remarkable dependence in the transmission rates on the incident

directions at ωa/v t > π Above the threshold frequency ωtha/v t = π, the transmission rate for

(I) is approximately T = 0.5 that is the same as the magnitude predicted from the

ray-acoustics, showing small dips in magnitude at the multiples of the threshold frequency The

transmission rate for (II) is larger than that for (I), and also shows periodic dips in

magnitude with the same period as (I) The obvious difference in the transmission rates

above the threshold frequency between (I) and (II) indicates that the rectification occurs at

the wavelength comparable to the dimension of scatterers, i e a/λ >1/2 due to the linear

dispersion relation ω = kv t =2πv t/λ Although the periodic dips, which appear when ωa/v t =

nπ (n = 1, 2, 3, ), are common to both the equilateral- and isosceles-triangular holes, the

latter system has advantageous properties for rectification of acoustic waves; the

transmission rates for (II) of α = 2π/9 are larger than those for α = π/3 This indicates that

the rectification is enhanced with decreasing α

Rectifying Acoustic Waves 51

10.80.60.40.20

14121086420

ωa/vt

(II)

(I)

α=2π/9 α=π/3

Fig 2 Transmission rate versus frequency The dashed and solid lines indicate the transmission

rate for α = π/3 and 2π/9, respectively Each case of (I) and (II) is bundled with an ellipse

Within the ray acoustics approximation, the transmission rate of (II) is expected to be 1 for

α < π/3 and 0.5 for α > π/2, and varies as T = (1/2) + cosα for π/3 < α < π/2 On the other

hand, the transmission rate of (I) becomes 0.5, independent of α For finite wavelength, the

transmission rate changes with α as shown in Fig 2, showing a larger transmission rate at

α = 2π/9 than that at α = π/3 From the results, we expect that the rectification effects decay

with increasing α To investigate the angle dependence, we examine the change in the

transmission rates for variation of α Generating a wave packet having a Gaussian spectral

distribution of central frequency ωC = (5π/2)(v t/a) with Δω = (π/2)(v t/a), we evaluate the

transmission rate for the wave packet, defining

−Δ

= ∫∫

Figure 3 plots the transmission rates defined by Eq (7) versus α The difference in the

transmission rates decreases with increasing α However, the rectification effects survive for

α > π/2 We also find that the transmission for (I) is slightly larger than 0.5 We can regard

these deviations from the predictions based on the ray acoustics as diffraction effects

The threshold frequency for the rectification and the periodic change in the transmission

rate originate from the interference effects Because of the periodic structure in the

y-direction, the wavenumber in the y direction is discretized in unit of n

aπ , so that the dispersion relation of the acoustic waves becomes subband structure given by

angle α = π/3 together with the corresponding transmission rate When the incident waves

with k y = 0 are elastically scattered, the waves are transited to the waves with finite k y

However, below the threshold frequency ωth a/v t, there is no waves with finite k y, so that the

incident waves in the x-direction are scattered only forward or backward, even if the waves

are scattered from the legs of triangles, resulting in the transmission rates independent of

the incident-wave directions

1 0.8 0.6 0.4 0.2 0

2.5 2

1.5 1

0.5 0

α

2π/9 π /3 π/2

(I) (II)

Fig 3 Transmission rate defined by Eq (7) versus the summit angle α The thick dashed and

solid lines indicate the transmission rate for cases (I) and (II), respectively The thin solid

lines indicate the transmission rates for (I) and (II) based on the ray acoustics

Fig 4 (a) Transmission rate versus frequency for single modes through single-array of

triangular holes with α = π/3 The labels (I) and (II), designated by red and blue solid line,

respectively, indicate the incident direction of acoustic waves The vertical dashed lines

indicate the positions of nπv t/a, where n = 1, 2, where v t are the velocity of transverse

waves (b) Dispersion relation of single modes within the empty-lattice approximation

Redirection of the incident waves for scattering occurs only in the frequency region above

the threshold frequency Since each dispersion relation of the waves with finite k y becomes

minimum at k x = 0, the density of states diverges, resulting in remarkable scattering into the

waves with finite k y and k x = 0 when the frequency matches the subband bottoms The

Rectifying Acoustic Waves 53

geometry of the scatterers enhances or suppresses the redirection depending on the incident

directions of the wave Hence, the rectification occurs only above the threshold frequency

and the dips in the transmission rates take place

3.1.2 Mixed mode

The transmission rates versus frequency for mixed modes are shown in Fig 5, when the

longitudinal waves are transmitted We assumed the matrix made of tungsten, whose the

mass density ρ and the elastic stiffness tensors C11, C44 are 19.317g cm–3 and 5.326×1012dyn

cm–2, 1.631 × 1012dyn cm–2, respectively The velocities of bulk longitudinal and transverse

waves are v l = 5.25 × 105cm s–1 and v t = 2.906 × 105cm s–1 (Kittel, 2004)

The red and blue solid lines indicate two different incident directions (I) and (II),

respectively The two transmission rates agree for ωa/v t < π, and we can see the difference

between the transmissions for ωa/v t > π, although it is not as large as that for the single

modes, manifesting rectification of the mixed modes Unlike the single modes, we can see

two kinds of periodic changes in transmission rates above ωa/v t = π; one is periodic

modulation with period Δωa/v t = π, indicated by the black dashed vertical lines, and another

is periodic variations with period Δωa/v t = π × v l/v t ~ 1.807π, indicated by the green ones In

addition, some aperiodic dips in the transmission rate indicated by the arrows appear above

ωa/v t = π These dips shift when the shape of the triangular hole changes Very interestingly

there is no rectification in high frequency regions (ωa/v t > 13) because, for the waves

impinging on the summit, the mode conversion from longitudinal waves to transverse ones

is strongly caused and the scattered transverse waves return to the incident direction

3.2 Surface acoustic waves

Figure 6(a) shows the frequency dependences of the transmission rates for SAWs with the

incident-wave directions (I) and (II) which are denoted by red and blue solid lines,

respectively For numerical evaluation, the matrix is assumed to be polycrystalline silicon

regarded as an isotropic medium, where the mass density ρ and the stiffness tensors C11, C44

are 2.33g cm–3 and 1.884 × 1012dyn cm–2, 0.680 × 1012dyn cm–2, respectively (Tamura, 1985)

Then the velocities of bulk longitudinal and transverse waves are v l = 8.99 × 105cm s–1 and v t

= 5.40 × 105cm s–1, respectively The equation for the velocity of a Rayleigh wave in an

isotropic medium with a surface is given by

where ξ = v R/v t (v R is the velocity of Rayleigh wave) (Graff, 1991) Solving Eq (9), we obtain

ξ = 0.914 A wave packet with z-polarized vector is used as an incident wave in order to

excite SAWs in the system Below the threshold frequency corresponding to the wavelength

of SAWs equivalent to the periodicity of the array, both the transmission rates are coincident

because the waves with long wavelength cannot recognize the geometrical difference

However, above the threshold frequency, the transmission rate shows obvious rectification of

SAWs as well as periodic dips with respect to frequency, resulting from the strong interference

effects of scattered SAWs We also find the periodic structure of the transmission rate of case

(II) is more pronounced than that of case (I) because the former makes the mode conversion

more accessible than the latter due to the geometry of the scatterers

(II) (I)

10.80.60.40.20

181614121086420

ωa/vt

Fig 5 Transmission rate versus frequency for mixed modes through single-array of

triangular holes with α = π/3 (I) and (II), designated by red and blue solid line, respectively,

indicate the incident direction of acoustic waves The vertical black dashed lines (as shown

in Fig 4) and green ones indicate the positions of nπv t/a and nπv l/a, where v l and v t are the

velocity of longitudinal and transverse waves, respectively, and n is a positive integer (n = 1,

2, 3, ) The arrows indicate the dips whose positions depend on the geometry of

triangular holes such as the summit angle α and the length of base a

20

10

0 -10 -20

Fig 6 (a) Transmission rate versus frequency for SAWs through single-array of triangular

holes with α = π/3 The labels (I) and (II), designated by red and blue solid line,

respectively, indicate the incident directions of SAWs The vertical dashed lines indicate the

positions of 2nξπv t/D, where n = 1, 2, where ξ = v R/v t (v R and v t are the velocity of

Rayleigh and transverse waves, respectively) (b) Dispersion relation of SAWs within the

framework of empty-lattice approximation

Figure 6(b) shows the dispersion relation of SAWs within the framework of empty-lattice

approximation to reveal the origin of the periodic dips in Fig 6(a) Within the empty-lattice

approximation the subband structures due to the periodicity of the y direction appear in the

dispersion relation, which are given by

Rectifying Acoustic Waves 55

2 , 0, 1, 2,

x t

relation of Rayleigh waves 2 ( )2

by 2 /

ω= + + π where 2πn/D is the reciprocal

lattice vector in the y direction The dips in Fig 6(a) correspond to the band edges of the

subband structure, manifesting that the periodic dips in the transmission are due to the

Bragg reflection of SAWs in the y direction It should be noted that the shift of the band

edges for the SAWs is modified by a factor of ξ as much as that for bulk transverse waves

Figure 7 shows the efficiency for the rectification of SAWs which is denoted by black solid

line The thin blue lines indicate the efficiency for bulk transverse waves as reference The

efficiency for SAWs is lower than that for bulk waves because of the mode conversion from

SAWs to bulk waves due to the triangular scatterers

Figure 8 (a) shows the transmission rate versus frequency for shear horizontal (SH) modes

through the single-array of triangular holes For excitation of SH waves in the system, we

use a wave packet with y-polarization vector as an incident wave The threshold frequency

above which the rectifying effect occurs becomes exactly 2πv t/D where v t is the velocity of

SH waves Above the threshold frequency, the transmission rates exhibit dips periodically at

multiples of the threshold frequency due to the same mechanism as the SAWs and bulk

waves However, the SH waves are inefficient compared to the SAWs as shown in Fig 8(b)

The inversions between the transmissions of cases (I) and (II) occur around ωD/v t ≈18

4 Summary and future prospects

We proposed an acoustic-wave rectifier and numerically demonstrated the rectification

effects on bulk waves as well as SAWs above the threshold frequencies The rectification

mechanism is due to the geometric effects of the asymmetric scatterers on acoustic wave

scattering, which is enhanced by interference among the scattered waves The threshold

frequency for the rectification results from the periodic arrangement of scatterers Hence, it

is possible to tune the rectifier by adjusting the position of the scatterers The findings of this

work can be applied not only to sound waves in solids or liquids but also to optical waves,

leading to new devices in wave engineering

1 0.5

0

30 25

20 15

10 5

0

T

SAW

Fig 7 Efficiency for the rectification of SAWs (α = π/3) The solid black and thin blue lines

indicate the efficiencies for SAWs and bulk single modes (T), respectively The efficiency of

the SAW rectifiers is slightly lower than that of the bulk single mode

(a) (b)

(II)

(I)

1 0.5

0

30 25

20 15

10 5

0

1 0.5

0

Fig 8 (a) Transmission rate versus frequency for SH modes through single-array of triangular

holes with α = π/3 The labels, (I) and (II), designated by red and blue solid line, respectively,

indicate the incident direction of acoustic waves (b) Efficiency for the rectification of SH

Graff, K F (1991) Wave Motion in Elastic Solids, Dover Publications

Kittel, C (2004) Introduction to Solid State Physics, 8 edition, Wiley

Krishnan, R., Shirota, S., Tanaka, Y & Nishiguchi, N (2007) High-efficient acoustic wave

rectifier, Solid State Communications 144(5-6): 194–197

Liang, B., Yuan, B & Cheng, J.-C (2009) Acoustic diode: Rectification of acoustic energy

flux in one-dimensional systems, Phys Rev Lett 103(10): 104301

Linke, H., Sheng, W., Löfgren, A., Xu, H.-G., Omling, P & Lindelof, P E (1998) A quantum

dot ratchet: Experiment and theory, Europhys Lett 44(3): 341

Shirota, S., Krishnan, R., Tanaka, Y & Nishiguchi, N (2007) Rectifying acoustic waves,

Japanese Journal of Applied Physics 46(42): L1025–L1027

Song, A M., Lorke, A., Kriele, A., Kotthaus, J P., Wegscheider, W & Bichler, M (1998)

Nonlinear electron transport in an asymmetric microjunction: A ballistic rectifier,

Phys Rev Lett 80(17): 3831–3834

Tamura, S (1985) Spontaneous decay rates of la phonons in quasi-isotropic solids, Phys

Rev B 31(4): 2574–2577

Tanaka, Y., Murai, T & Nishiguchi, N (n.d.) in preparation

2National Centre for Physics, Quaid-i-Azam University Campus, Islamabad 43520

3Department of Physics, Government College Bagh 12500, Azad Jammu & Kashmir,

Pakistan

1 Introduction

The underlying physics of nonconventional quantum plasmas has been introduced long ago Analytical investigations of collective interactions between an ensemble of degenerate electrons in a dense quantum plasma dates back to early fifties The general kinetic equations for quantum plasmas were derived and the dispersion properties of plasma waves were studied (Klimotovich & Silin, 1952) It was thought that the quantum mechanical behaviour of electrons, in the presence of heavier species modifies the well known properties of plasma The dynamics of quantum plasmas got particular attention in the framework of relationship between individual particle and collective behavior Emphasizing the excitation spectrum of quantum plasmas, theoretical investigations describe the dispersion properties of electron plasma oscillations involving the electron tunneling (Bohm & Pines, 1953; Pines, 1961) A general theory of electromagnetic properties

of electron gas in a quantizing magnetic field and many particle kinetic model of thermal plasmas was also developed treating the electrons quantum mechanically (Zyrianov

non-et al., 1969; Bezzerides & DuBois, 1972) Since the pioneering work of these authors which laid foundations of quantum plasmas, many theoretical studies have been done in the subsequent years The rapidly growing interest in quantum plasmas in the recent years has several different origins but is mainly motivated by its potential applications in modern science and technology (e.g metallic and semiconductor micro and nanostructures, nanoscale plasmonic devices, nanotubes and nanoclusters, spintronics, nano-optics, etc.) Furthermore, quantum plasmas are ubiquitous in planetary interiors and in compact astrophysical objects (e.g., the interior of white dwarfs, neutron stars, magnetars, etc.) as well as in the next generation intense laser-solid density plasma interaction experiments Such plasmas also provide promises of important futuristic developments in ultrashort pulsed lasers and ultrafast nonequilibrium phenomena (Bonitz, 1898; Lai, 2001; Shukla & Eliasson, 2009)

Contrary to classical plasmas, the number density of degenerate electrons, positrons/holes

in quantum plasmas is extremely high and they obey Fermi-Dirac statistics whereas the

temperature is very low Plasma and quantum mechanical effects co-exist in such systems

and many unusual effects like tunneling of electrons, quantum destabilization, pressure

ionization, Bose-Einstein condensation and crystallization etc may be equally important

(Bonitz et al., 2009) Their properties in equilibrium and nonequilibrium are governed by

many-body effects (collective and correlation effects) which require quantum statistical

theories and versatile computational techniques The average inter-particle distance n–1/3

(where n is the particle density) is comparable with electron thermal de Broglie wavelength

λBe (= =/mv te , where = is Planck’s constant divided by 2π, m is the electronic mass and v te is

thermal speed of electron) The overlapping of wave functions associated with electrons or

positrons take place which leads to novel quantum effects

It was recognized long ago that the governing quantum-like equations describing collective

behavior of dense plasmas can be transformed in the form of hydrodynamic (or fluid)

equations which deals with macroscopic variables only (Madelung, 1926) Here, the main

line of reasoning starts from Schrodinger description of electron The N-body wave function

of the system can be factored out in N one-body wave functions neglecting two-body and

higher order correlations This is justified by weak coupling of fermions at high densities

The coupling parameter of quantum plasmas decreases with increase in particle number

density For hydrodynamic representation, the electron wave function is written as ψ = n

exp(iS/=) where n is amplitude and S is phase of the wave function Such a decomposition

of ψ was first presented by Bohm and de Broglie in order to understand the dynamics of

electron wave packet in terms of classical variables It introduces the Bohm-de Broglie

potential in equation of motion giving rise to dispersion-like terms In the recent years, a

vibrant interest is seen in investigating new aspects of quantum plasmas by developing

non-relativistic quantum fluid equations for plasmas starting either from real space Schrodinger

equation or phase space Wigner (quasi-) distribution function (Haas et al., 2003, Manfredi &

Haas, 2001; Manfredi, 2005) Such approaches take into account the quantum statistical

pressure of fermions and quantum diffraction effects involving tunneling of degenerate

electrons through Bohm-de Broglie potential The hydrodynamic theory is also extended to

spin plasmas starting from non-relativistic Pauli equation for spin-1

2 particles (Brodin &

Marklund, 2007; Marklund & Brodin, 2007) Generally, the hydrodynamic approach is

applicable to unmagnetized or magnetized plasmas over the distances larger than electron

Fermi screening length λFe (= v Fe/ωpe, where v Fe is the electron Fermi velocity and ωpe is the

electron plasma frequency) It shows that the plasma effects at high densities are very short

scaled

The present chapter takes into account the dispersive properties of low frequency

electrostatic and electromagnetic waves in dense electron-ion quantum plasma for the cases

of dynamic as well as static ions Electrons are fermions (spins=1/2) obeying Pauli’s

exclusion principle Each quantum state is occupied only by single electron When electrons

are added, the Fermi energy of electrons εFe increases even when interactions are neglected

(εFe ∝ n2/3) This is because each electron sits on different step of the ladder according to

Pauli’s principle which in turn increases the statistical (Fermi) pressure of electrons The de

Broglie wavelength associated with ion as well as its Fermi energy is much smaller as

compared to electron due to its large mass Hence the ion dynamics is classical Quantum

Dispersion Properties of Co-Existing Low Frequency Modes in Quantum Plasmas 59

diffraction effects (quantum pressure) of electrons are significant only at very short length

scales because the average interparticle distance is very small This modifies the collective

modes significantly and new features of purely quantum origin appear The quantum

ion-acoustic type waves in such system couples with shear Alfven waves The wave dispersion

due to gradient of Bohm-de Broglie potential is weaker in comparison with the electrons

statistical/Fermi pressure The statistical pressure is negligible only for wavelengths smaller

than the electron Fermi length For plasmas with density greater than the metallic densities,

the statistical pressure plays a dominant role in dispersion

2 Basic description

The coupling parameter for a traditional classical plasma is defined as

3,

is the average kinetic energy, k B is the Boltzmann constant and T is the system’s temperature

The average interparticle distance r is given by

ordering ΓC << 1 corresponds to collisionless and ΓC  1 to collisional regime 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 , r become comparable to thermal de Broglie wavelength of

charged particles defined by

,2

where h is the Planck’s constant Here, the degeneracy effects cannot be neglected i.e.,

3

11nλB and the quantum mechanical effects along with collective (plasma) effects become

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 quantum particle is characterized by the wave function associated with the particle

instead of its trajectory in phase space The Heisenberg uncertainty principle leads to

fundamental modifications of classical statistical mechanics in this case The de Broglie

wavelength has no role in classical plasmas because it is too small compared to the average

interparticle distances There is no overlapping of the wave functions and consequently no

quantum effects So the plasma particles are considered to be point like and treated

classically

However, in quantum plasmas, the overlapping of wave functions takes place which

introduces novel quantum effects It is clear from (3) that the de Broglie wavelength

depends upon mass of the particle and its thermal energy That is why, the quantum effects

associated with electrons are more important than the ions due to smaller mass of electron

which qualifies electron as a true quantum particle The behavior of such 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<nλB3 and (ii) classical (nondegenerate) plasmas if nλB3< The border between 1

the degenerate and the non-degenerate plasmas is roughly given by

For quantum plasmas, the Boltzmann distribution function is strongly modified to Fermi-

Dirac or Bose-Einstein distribution functions in a well known manner, i.e.,

where β = 1/k B T; ε is the particle energy and µ is the chemical potential The ‘+’ sign

corresponds to Fermi-Dirac distribution function (for fermions with spin 1/2, 3/2, 5/2, )

and ‘_’ sign to Bose-Einstein distribution function (for bosons with spin 0, 1, 2, 3, ) The

different signs in the denominators of (5) are of particular importance at low temperatures

For fermions, this leads to the existence of Fermi energy (Pauli principle), and for bosons, to

the possibility of 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 (Landau

& Lifshitz, 1980) This limiting momentum is called the Fermi momentum p F given by

Dispersion Properties of Co-Existing Low Frequency Modes in Quantum Plasmas 61

The Fermi-Dirac distribution function becomes a unit step function in the limit T → 0 It is

zero for µ < ε and unity for ε < µ Thus the chemical potential of the Fermi gas at T = 0 is

same as the limiting energy of the fermions (µ = ε F) The statistical distribution of plasma

particles changes from Maxwell-Boltzmann ∝ exp(–ε/k B T) to Fermi-Dirac statistics ∝ exp

[(β(ε – ε F ) + 1)]–1 whenever T approaches the so-called Fermi temperature T F , given by

( )

23

It means that the quantum effects are important when 1 1 T F /T In dense plasmas, the

plasma frequency ωp = (4πne2/m)1/2 becomes sufficiently high due to very large equilibrium

particle number density Consequently, the typical time scale for collective phenomena

(ωp)–1 becomes very short The thermal speed v T = (k B T/m)1/2 is sufficiently smaller than the

Fermi speed given by

(10)

With the help of plasma frequency and Fermi speed, we can define a length scale for

electrostatic screening in quantum plasma i.e., the Fermi screening length λF (= v Fe/ωpe)

which is also known as the quantum-mechanical analogue of the electron Debye length λDe

The useful choice for equation of state for such dense ultracold plasmas is of the form

(Manfredi, 2005)

0 0,

n

P P n

γ

⎛ ⎞

= ⎜⎜ ⎟⎟

where the exponent γ = (d + 2) /d with d = 1, 2, 3 denoting the dimensionality of the system,

and P0 is the equilibrium pressure In three dimensions, γ = 5/3 and P0 = (2/5) n0εF which

pressure, which is significant in dense low temperature plasmas The Fermi pressure

increases with increase in number density and is different from thermal pressure

Like classical plasmas, a coupling parameter can be defined in a quantum plasma For

strongly degenerate plasmas, the interaction energy may still be given by 〈U〉, but the kinetic

energy is now replaced by the Fermi energy This leads to the quantum coupling parameter

which shows that ΓQ ∝ n–1/3 So the peculiar property of quantum plasma is that it

increasingly approaches the more collective (ideal) behavior as its density increases

Quantum plasma is assumed to be collisionless when ΓQ << 1 because the two body

correlation can be neglected in this case This condition is fulfilled in high density plasmas

since εF = εF (n) In the opposite limit of high temperature and low density, we have

3

1>>nλB and the system behaves as a classical ideal gas of free charge carries Another

useful form of ΓQis as follows

F F

n

ωλ

Suppose that the N-particle wave function of the system can be factorized into N

one-particle wave functions as Ψ( , ,···,x x1 2 xN)=ψ1( )x1 ψ2( )····x2 ψ N(xN) Then, the system is

described by statistical mixture of N states ψα, 1,2, ,α= N where the index α sums over

all particles independent of species We then take each ψα to satisfy single particle

Schrodinger equation where the potentials (A, φ) is due to the collective charge and current

densities For each ψα, we have corresponding probability pα such that

11

fluid description, we define ψα= nαexp(iSα/ )= where nα and Sα are real and the velocity

of αth particle is vα= ∇Sα/mα−(qα/m cα ) A Next, defining the global density

∂+ ∇ =

The last term in (16) is the statistical pressure term For high temperature plasmas, it is simply

thermal pressure However, in low temperature and high density regime, the Fermi pressure is

significant which corresponds to fermionic nature of electrons and P is given by equation (12)

In the model (15)-(16), it is assumed that pressure P = P(n) which leads to the appropriate

Dispersion Properties of Co-Existing Low Frequency Modes in Quantum Plasmas 63

equation of state to obtain the closed system of equations For typical length scales larger than

λFe, we have approximated the =-term as ( 2 ) ( 2 )

known as the quantum fluid equations which are coupled to the Poisson’s equation and

Ampere’s law to study the self -consistent dynamics of quantum plasmas (Manfredi 2005;

Brodin & Marklund, 2007) This model has obtained considerable attention of researchers in

the recent years to study the behaviour of collisionless plasmas when quantum effects are

not negligible Starting from simple cases of electrostatic linear and nonlinear modes in two

component and multicomponent plasmas, e.g., linear and nonlinear quantum ion waves in

electron-ion (Haas et al., 2003, Khan et al., 2009), electron-positron-ion (Khan et al., 2008,

2009) and dust contaminated quantum plasmas (Khan et al., 2009), the studies are extended

to electromagnetic waves and instabilities (Ali, 2008) Some particular developments have

also appeared in spin-1/2 plasmas (Marklund & Brodin, 2007; Brodin & Marklund, 2007;

Shukla, 2009), quantum electrodynamic effects (Lundin et al., 2007) and quantum

plasmadynamics (Melrose, 2006) It is to mention here that the inclusion of simple collisional

terms in such model is much harder and the exclusion of interaction terms is justified by

small value of ΓQ

4 Fermionic pressure and quantum pressure

For dense electron gas in metals with equilibrium density n e0 1023cm–3, the typical value of

Fermi screening length is of the order of Angstrom while the plasma oscillation time period

1

(ωpe− ) is of the order of femtosecond The electron-electron collisions can been ignored for

such short time scales The Fermi temperature of electrons is very large in such situations

i.e., T Fe  9 × 104K which shows that the electrons are degenerate almost always (Manfredi &

Haas, 2001) The Fermi energy, which increases with the plasma density, becomes the

kinetic energy scale The quantum criterion of ideality has the form

2 31

Q Fe

e n

Γ ≈ <<

The parameter ΓQ decreases with increasing electron density, therefore, a degenerate

electron plasma becomes even more ideal with compression So, even in the fluid

approximation, it is reasonable to compare the statistical pressure term arising due to the

fermionic character of electrons and the quantum Bohm-de Broglie potential term in the

ultracold plasma

Let us consider two-component dense homogenous plasma consisting of electrons and ions

The plasma is embedded in a very strong uniform magnetic field B0ˆz ; where B0 is the

strength of magnetic field and ˆz is the unit vector in z-axis direction However, plasma

anisotropies, collisions and the spin effects are not considered in the model for simplicity

The low frequency (in comparison with the ion cyclotron frequency Ωci = eB0/m i c, where e,

m i and c are the magnitude of electron charge, ion mass and speed of light in vacuum,

respectively) electric and magnetic field perturbations are defined as