Joint ICTP-IAEA Workshop on Dense Magnetized Plasma and...

2168-Presentation

Joint ICTP-IAEA Workshop on Dense Magnetized Plasma and Plasma Diagnostics

KHAN Shabbir Ahmad

15 - 26 November 2010

Department of Physics, COMSATS Institute of Information Technology CIIT

Shahzad Town, Park Road, Islamabad 44000 Federal Capital

PAKISTAN

Low Frequency Modes in Fermi degenerate Plasmas

1

Low Frequency Modes in Fermi degenerate Plasmas

Shabbir Ahmad Khan

Department of Physics, COMSATS Institute of information Technology (CIIT)Islamabad, Pakistan

2

1. Introduction

2. Fermi degenerate (quantum) plasmas parameters

3. Examples of Fermi degenerate plasmas in laboratory and universe

4. Mathematical models and motivations

5. Coupling of low frequency modes of Fermi degenerate plasmas

5.1. Significance of electron Fermi pressure

5.2 Dynamics of ions and electrons

5.3. Electron dynamics with stationary (background) ions

6. Conclusion

Outline

3

(1) Introduction

• In a traditional classical equilibrium plasma, the distribution of plasma particles is

given by the well known Boltzmann distribution function proportional to

• The mean kinetic energy of particle in a classical system is

• The average interpaticle distance is much larger than thermal de

Broglie wavelength associated with electron

• Classical plasma is a gas of hot electrons and ions with no quantum effects

• However, when plasma density is sufficiently high so that the average interparticledistance becomes of the order of de Broglie wavelength of electron, the quantum mechanical effects can’t be ignored.

• If temperature is lower than the electron Fermi temperature, the relevant statistical distribution of electrons is Fermi-Dirac. Then the energy of the system is given by

the Fermi energy.

• Dense degenerate plasmas have been introduced since long ago [Yuri Klimontovichand V. P. Silin, JETP (1952). D. Pines, Benjamin, New York (1963)] .

31−∝−= nRRR jiin

Tmkh

BB π

λ2

=

TBk23

][TBk

ε−exp

4

Introduction (cont.)

Occurrence of PlasmaContemporary Physics Education Project (CPEP)http://www.cpepweb.org/

Dense laboratory andastrophysicalPlasmas

5

(2) Quantum plasma parameters

• Average interparticle distance:

• Quantum effects appear when i.e.,

• This condition corresponds to where is the Fermi temperature:

• Plasma oscillation frequency (same): , Screening length:

• Fermi speed: ;

• The quantum coupling parameter:

13 ≥Bnλ

32

)3(22

222

nm

vmETk F

FFB πh=== 3

23

BF n

TT

λχ =⎟⎠⎞

⎜⎝⎛≅

FTT <FT

(2.1)

(2.2)

31−≅ nRin

inB R≈λ

⇒2/124

⎟⎟⎠

⎞⎜⎜⎝

⎛=

mne

pπω

31

)3( 2nm

vF πh=

p

FF

vω

λ =

31

32

2

22

3int 41 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛≅=Γ nme

EnEE

F

p

FF

Q

h

h πωλ

31−∝Γ nQ

potEE ≡int

(2.4)⇒

(2.3)

(Degeneracy parameter)

is the Fermi speed.

6

(3) Examples of quantum plasmas

From: M. Bonitz (Springer-Berlin 2010)

7

Examples (cont.)

In the past few years, the collective plasma dynamics in potentially important dense systems has been a subject of great interest [D. Melrose, Springer, New York (2008), D. Kremp, Springer, Berlin (2005), M. Bonitz, Teubner (1998), G. Manfredi, Field Inst. Comm. (2005), Springer 2009].

The collective plasma effects are found important in quantum electron gas [G. Manfredi and F. Haas, PRB (2001)], electron-hole plasmas in semiconductors [C. F. Clingshirn, Springer (2007)], quantum coulomb crystals [M. Bonitz, Springer (2009)],metallic nanostructures [G. Manfredi et. al., Springer (2009)], charged particles confined in various traps [M. Bonitz, Teubner (1998)], ultrasmall electronic devices [A. Markowich et. al., Springer (1990)], plasmonic devices [M. Marklund et al., EPL (2008)],ultra-cold plasmas [T. Killian, Science (2007)], scattering from solid-density targets [S. Glenzer et. al., PRL (2007)], laser plasmas [D. Kremp, Springer, Berlin (2005), ] and so on.

Laboratory

8

Examples (cont.)

Astrophysical environmentThe cores of Jovian planets (e.g.,Jupiter, Saturn) are composed of about 70% to 97% of compressed Hydrogen and Helium plasmas with sufficiently high plasma densities [G. Chabrier et. al. IOP Publishing (2006)]. Similarly, firm evidences of degenerate plasmas exist for two classes of compact astrophysical objects namely white dwarfs and neutron stars. [S. Shapiro and S. Teukolsky, WILEY-VCH (2004)]. The interior of white dwarf resemble a very dense solid, with an ion lattice surrounded by degenerate electrons. Dense degenerate electrons are also believed to be present in the crust and atmosphere of neutron stars. The density ranges are as follows [M. Bonitz, Springer 2010];

– Interior of Jovian planets (Jupiter, Saturn):

Hydrogen/Helium plasma

– White and Brown dwarf stars:

– Neutron stars:

32410 −≅ cmn

32826 10...10 −≅ cmn

(?)10...10 33628 −≅ cmn

9

(4) Mathematical models and motivations

Dense degenerate plasmas are very complicated systems (various aspects are not well

understood till todate). These complications arise due to implication of certain quantum

mechanical rules, for instance, quantum statistics, uncertainity priniciple, quantum

degeneracy, de Broglie length scales etc. The full description of dense degenerate

plasmas is a major challenge from theoretical perspective. Certain assumptions are

required to obtain simplified models starting from basic phenomena. To study the

dynamics of such plasmas, some motivations are as follows:

– How are the fundamental plasma models modified by quantum effects?

– What happens to the typical plasma effects e.g., Debye screening, wave-particle interactions etc.?

– What information plasma waves carry in such plasmas?

– The possibility of new modes

10

Mathematical models (cont.)

WT = Wigner transform

(Ackn. G. Manfredi)

N-body Schrodinger Vlasov

Quantumhydrodynanics

Classicalhydrodynamics

ħ → 0WTWigner

Overview

ħ→ 0

1ΓQ << 1k F <<λ

11

(5) Coupling of low frequency modes

• Quantum Hydrodynamic (QHD) model for macroscopic variable n (number density) and u (velocity) of jth magnetoplasma species:

• Frequency range: ; ;

• Electric and magnetic field perturbations:

; ;

(5.1)⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∇+−×+=+

∂∂

j

j

j

jFjjjjjjjj n

nmn

Pc

nqt

nm22

2)1().( ∇∇∇

hBvEvv

( ) ( )0=

∂

∂+⋅+

∂

∂⊥⊥ z

vnn

tn jzj

jjj v∇

∑−=∇j jjqnπϕ 42

JBcπ4

=×∇

|∂ t | pe,ck cjΩ<<ω cmeB jcj /0=Ωj=e,i for electron, ion

(5.2)

(5.3)

(5.4)

FeFiei TTTT <<,,

E −∇ − c−1∂Az/∂tz B ∇Az z Bext B0 z

12

Coupling of low frequency modes (cont.)

• Consider the non-zero electric and magnetic field perturbations in homogenous plasma.

• At very low temperature, the electron equation of state leads to the statistical pressure:

• Gradient of quantum pressure (Bohm quantum potential term):

• It gives relation of density and interparticle separation:

Very short scale lengths:

• The perturbation wavelength

( )( ) 3/53/222 35/ eeFe nmP πh=

cmR

cmncmk

in

e8

3240

16

101

101101

−

−

−

×≈⇒

×≈

×≈2

2 1

inRk <<

)].(exp[ ti ω−∝ rkonperturbati

⇒

1

22

1 4 ee

q nmkP ∇∇

h≅

( ) 13/2

02

2

1 33 ee

eFe nn

mP ∇∇ πh

≅

213/2

0inRen =

Dense plasmaparameters:

Feλλ >

(5.1.1)

(5.1.2)

(5.1.3)

5.1 Significance of electron Fermi pressure

Subscript “0” - equilibrium

Subscript “1” - perturbation

⇒3/1−∝ nRin (average interparticle distance)

13


• From QHD equations, the linearized ion velocity components:

and are electric and ion polariztion drifts resp. For electrons

and are the electron quantum and diamagnetic drifts.

• The Ampere’s law:

and the Poisson’s equation:

v i1 ≃ cB0

z ∇1 − 1ci

∂∇1∂t ≃ vE vPi

∂ tv iz1 ≃ − emi

∂1∂z 1

c∂Az1∂t

ve1 ≃ cB0

z ∇ 1 2

4men0e∇2ne1 −

2kBTFe3n0e ne1

≃ vE vqe vDe,

∂ tv ez1 ≃ eme∂∂z 1 2

4men0e∇2ne1 −

2kBTFe3n0e ne1 e

mec∂Az1∂t

vE vpi

vDe01, nn ei <<

0≅FiP

( )( ) 3/53/222 35/ eeFe nmP πh=

( )1112 4 ie nne −=∇ πϕ

( )102 4

izezz vvc

enA −=∇⊥

π

(5.2.2)

(5.2.1)

(5.2.3)

(5.2.4)

(5.2.5)

(5.2.6)

5.2 Dynamics of ions and electrons

qev

14


• The electron and ion equations after some algebra:

• The effective temperature of electrons (in energy units):

• Parameters and contain contribution of both the terms-the Fermi pressure and the quantum potential

ni1n0≃ 12 −q

2k 22 cq

2k z2

A2 A

2 − 2 1 vAk 2

c2k21

ne1n0≃ 1A

2 A2 − 1 e

2k 2 1 vA2 k 2

c2k 22 me

miA

2 − 1 vA2 k 2

c2k 22 1

e c/pe

A k zvA

cq Tq /mi

q cq /ci

qTe /11 ϕ=Φ

( )3/24/22FeBeq TkmkT += h

cq q

Collisionless electron skin depth:

Alfven wave frequency:

Ion Larmor radius at electron effective temperature:

Quantum ion-acoustic speed:

Normalized potential:

Purely quantum mechanical effect

(5.2.7)

(5.2.8)

(5.2.9)

A1 eAz1Tq

15


• The linear dispersion relation of the low frequency coupled electrostatic and electromagnetic waves

• For and

• For small ion parallel current:

• For and

1 vA2 k 2

c2k 22 me

mi1 e2k 2

1 vA2 k 2

c2k 22 − A

2 −cq

2k z2

1 e2k 2

1 vA2 k 2

c2k 22 −

A2

1 e2k 2

1 cq

2k 2

pi2 2 − cq

2k z2

q2k 2A

2

1 e2k 2

2

1/ <<ie mm 122 <<⊥keλ

1 vA2 k 2

c2k 22 − cq

2k z2 1 vA

2 k 2

c2k 22 −

A2 1

cq2k 2

pi2 2 − cq

2k z2 q

2k 2A22

vA ck /k

(5.2.10)

(5.2.11)

(5.2.12)

[S. A. Khan and H. Saleem, SCIYO, Rijeka 2010; PoP 2008]

( ) ( ) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

++= ⊥

⊥⊥

2222

222

22

22

11

1k

kvk

q

kckv

e

Az

Aρ

λω

122 <<⊥keλ

2 k z2vA

2 1 q2k 2

Dispersion due to quantum effects of electrons

(5.2.13)

16


• In quasi-neutrality limit with , (5.2.11) leads to

which is the same equation as obtained by H. Saleem, Ali Ahmed and S. A. Khan (PoP

2008) when density inhomogeneity and is dropped.

5.3. Electron dynamics with stationary (background) ions

• Dispersion relation

Fermi pressure and quantum pressure both contribute in dispersion

ceΩ<<ω

2 vAe

2 k z21 q

2k 2

1 e2k2 1 q

2k 2 vAe2 k2

c2k2

cmeB ece /0=Ω

eAe mnBv 00 4/ π= q vqpe eqq mTv /=

(5.2.14)

(5.3.1)

11 ee nn ≈ vA ck /k

( )( ) 222222222 ωωρωωω AqAzq kkc ⊥=−−

17


(i). Dynamic ions and electrons• Application to white dwarf stars

(ii). Degenerate Fermi gas of electrons with stationary ions

,

5.4 Numerical results

3260 10 −≅ cmn G8

0 10≅B

Dispersion term

⇒ 130 >Ben λ

e2k2 ≅ 0.003 q ≅ 0.3 10−5cmk 106cm−1 ⇒002.0/ ≅⊥kkz

vA ck /k FeQ PP <<

3240 10 −≅ cmn

q2k2 ≅ 0.03 ceΩ<<ω

rad/s15105.1 ×≅Ωce16106 −

⊥ ×≅ cmk ⊥≅ kk z 5001

⇒rad/s12103.3 ×≅ω

Fee TT <<

inR>λinR

k 1<⊥

cm9102 −×≅inR

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• Ion-acoustic branch of (5.2.10) in presence of electron Fermi pressure

• Ion-acoustic branch of (5.2.10) in the absence of Fermi pressure

3260 10 −≅ cmn

G80 10≅B

002.0/ ≅⊥kkz

White dwarf data

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• Alfven branch in presence of Fermi pressure

• Alfven branch in the absence of Fermi pressure

• Dispersion curve of Alfven wave (5.3.1) for electrons with background of stationary ions ( )324

0 10 −≅ cmn

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(6) Conclusion

• The low frequency electrostatic and electromagnetic modes in dense homogenous as well as inhomogenous plasmas have been studied using quantum hydrodynamic approximation.

• Low frequency electrostatic and electromagnetic modes co-exist in such dense plasmasand couple analogously to the classical plasma modes. However, physical conditions ofboth the cases are very different.

• The electron Fermi pressure has the main role in wave dispersion ( ).

• The applications of the results are also discussed and it is pointed out that the resultsare relevant only to highly dense plasmas either magnetized or unmagnetized.

• Very high density systems are too complicated to be studied at large with quantum hydrodynamic approach in our opinion due to their complexities.

3/20eFe nT ∝

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