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Exchange effects in Coulomb quantum plasmas: Dispersion of waves in 2D and 3D quantum plasmas
Annals of Physics 350 (2014) 198–210
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Exchange effects in Coulomb quantum plasmas: Dispersion of waves in 2D and 3D quantum plasmas Pavel A. Andreev M. V. Lomonosov Moscow State University, Moscow, Russian Federation
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Article history: Received 9 May 2014 Accepted 10 July 2014 Available online 17 July 2014 Keywords: Quantum plasmas Exchange interaction Ion-acoustic waves Quantum hydrodynamics Non-linear Schrödinger equation
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aop.2014.07.019 0003-4916/© 2014 Elsevier Inc. All rights reserved.
abstract We describe quantum hydrodynamic equations with the Coulomb exchange interaction for three and two dimensional plasmas. Explicit form of the force densities are derived. We present non-linear Schrödinger equations (NLSEs) for the Coulomb quantum plasmas with the exchange interaction. We show contribution of the exchange interaction in the dispersion of the Langmuir, and ionacoustic waves. We consider influence of the spin polarization ratio on strength of the Coulomb exchange interaction. This is important since exchange interaction between particles with same spin direction and particles with opposite spin directions are different. At small particle concentrations n0 ≪ 1025 cm−3 and small polarization the exchange interaction gives small decrease of the Fermi pressure. With increase of polarization role the exchange interaction becomes more important, so that it can overcome the Fermi pressure. The exchange interaction also decreases contribution of the Langmuir frequency. Ion-acoustic waves do not exist in limit of large polarization since the exchange interaction changes the sign of pressure. At large particle concentrations n0 ≫ 1025 cm−3 the Fermi pressure prevails over the exchange interaction for all polarizations. We obtain a similar picture for two dimensional quantum plasmas. © 2014 Elsevier Inc. All rights reserved.
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1. Introduction Exchange effects in quantum plasmas have been considered for a long time. A lot of different approaches, methods and results were obtained in this field. Before we present a new approach to this subject based on the many-particle quantum hydrodynamic (MPQHD) method [1,2] we present a brief review of earlier results. Earlier results are focused on three dimensional electron gas. They do not include dependence on the spin polarization of electron gas. We consider three and two dimensional electron gas and present dependence on its spin polarization. Exchange interaction is a remarkable example of quantum physical effects. The exchange interaction is related to overlapping of the wave functions of interacting particles. Hence it reveals itself as a short range interaction, even when we consider the exchange part of a long range interaction. The Coulomb exchange interaction gives contribution in the spectrum of Langmuir waves in three dimensional degenerate electron gas. This spectrum without exchange contribution was obtained in Refs. [3,4]. In 1960 Kanazawa et al. [5] presented contribution of the Coulomb exchange interaction in the Langmuir wave spectrum in terms of the Green function method [6]. Their result coincides with results of Ref. [7] obtained in 1958 by Nozieres and Pines. However it differs from the results that had been obtained earlier by other authors (see Wolff [8], Pines [9], Ferrell [10], DuBois [11]). Results of Refs. [5,7] for the shift of the frequency square are 2 1ωGFM =−
3
k2
20 (3π 2 n0 )
1
2 3
2 ωLe ∼ n03 .
(1)
Plasma wave dispersion with the exchange interaction was considered in 1962 in terms of the trace of two-particle density matrix (see for instance formula (1) in Ref. [12]). An explicit form of the exchange potential is given by formula (16) of Ref. [12] (see Ref. [13] as well). This potential has the following form:
φex =
1 2
4π e f
1F (k′ , q) ′ dk − 1F |k − k′ |
f (k′ + q, k′ ; ω)
|k − k′ |2
dk′ ,
(2)
where
1F = F (|k + q|) − F (|k|),
(3)
here one has assumed that the one-particle density matrix [14] of electrons in wave-vector space has the following form: R(1) (k, k′ ; s′ , s; t ) = F (|k|) + f (k, k′ ; s′ , s; t )
(4)
f (k + q, k; ω) = Tr(s) [f (k + q, k; s′ , s; ω)],
(5)
and
with s, s are spin coordinates, F (|k|) is the equilibrium Fermi–Dirac distribution, and f is a small perturbation. We see that the exchange potential is presented in a linear approach obtained in terms of distribution function. Contribution of the exchange interaction in the spectrum of longitudinal waves is presented in Ref. [13]. This contribution coincides with results of Refs. [5,7]. In Ref. [12] it was shown that the Coulomb exchange interaction gives contribution in the electromagnetic wave spectrum. It was found despite the fact that the exchange interaction appears as an extra potential in addition to the scalar potential of electromagnetic field (see Eq. (4) of Ref. [12]), one can expect that it gives no trace in the spectrum of transverse waves. Generalization of spectrum (1) was presented in Refs. [15,16]. Hedin and Lundqvist [17] discussed contribution of the Coulomb exchange interaction in density functional theory. Results of Ref. [17] were applied in paper [18] for the study of wide parabolic ′
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quantum wells (see formula (3) for exchange potential). The exchange potential presented in Refs. [17,18] is negative. That reveals an attraction between electrons. Its explicit form is Vex = −0.985
e2
ϵ
n
1 3
1+
0.034 1
aB n 3
ln 1 + 18.376aB n
1 3
,
(6)
where ϵ is the (uniform) static dielectric constant, and aB = ϵ h¯ 2 /(me2 ) is the Bohr radius with contribution of medium by means of the dielectric constant. This formula consists of two parts. The first term is the exchange interaction. The second term is the correlation contribution [18]. Formula (6) shows that the correlations increase contribution of the exchange interaction. This extra contribution is of the order of 50% percent of the exchange interaction. This potential was included in the QHD model at consideration of finite width thin metal films [19]. The exchange part of potential (6) gives the following shift of the 3D Langmuir wave spectrum 2 1ωLDT =−
0.985
k2
12π (n0 ) 23
2 ωLe .
(7)
This result in 5.24 times more than the earlier result presented by formula (1). A review of local field corrections in electron gas was presented in 1975 by Kugler [20]. Hohenberg–Kohn–Sham local density theory [17,21–23] contains a NLSE for an effective one-body wave function of quasielectrons and quasiholes (see for instance formula (2.2) in Ref. [17]). Sometimes result (6) has been used for consideration of 2D plasmas [24]. It does not seem correct. For instance, if we consider the Fermi pressure of 2D gas it appears in terms of 2D particle concentration and it has a structure, which differs from the structure of 3D Fermi pressure. Similar situation happens with the potential and force field of the Coulomb exchange interaction. It has to be derived for 2D case independently. It should be done in terms of the 2D particle concentration as we do in this paper for 2D plane electron gas. We have not obtained contribution of the correlations in terms of the MPQHD. This paper is dedicated to the Coulomb exchange interaction in the MPQHD. The exchange–correlation potential (6) shows that there is a lot of work for future development of the MPQHD method. It is important to develop a method of derivation of the correlation potential. New approaches to hydrodynamic description of quantum plasmas were considered in 1999 by Kuz’menkov and Maksimov [1] and Kuzelev and Rukhadze [25], where were considered spinless Coulomb quantum plasmas. The self-consistent field approach was the center of attention of these papers. Nevertheless a general form of the exchange interaction for bosons and fermions, in terms of the MPQHD, was derived in Ref. [1]. In 2000–2001 attention shifted towards spin-1/2 quantum plasmas [2,26–28]. The explicit form of the exchange interaction for the Coulomb and the spin–spin interaction was derived in Ref. [28]. Contribution of these interactions in properties of many-electron atoms was described there. During period 2001–2007 most of researches considered spinless quantum plasmas [29,30] (for review see Refs. [31,32]). A great interest to quantum plasmas of spinning particles arose since 2007. Marklund and Brodin drew attention to the field by their papers [33,34], some examples are presented below. Dispersion properties of spin-1/2 quantum plasmas have been under consideration for many years (see Refs. [27,33,35–54]). It was shown that magnetic moment (spin) evolution in quantum plasmas leads to new branches of wave dispersion [36–38,43,44,48,51]. The last of these Ref. [51] shows dispersion of spin wave in the spin-1/2 two dimensional electron gas. Contribution of the exchange interaction in dispersion properties of spin-1/2 quantum plasmas was considered in 2008 (see Ref. [41]). It was demonstrated that the exchange interactions give potential force field depending on the particle concentration and the spin density. In linear approximation the exchange interactions can be combined with the Fermi pressure, so we have an effective shifted Fermi velocity [41]. A description of the spin–plasma wave propagating perpendicular to an external magnetic field was given in 2006 by Vagin et al. (see Ref. [36]), Andreev and Kuz’menkov in 2007 (see Ref. [37]), and Brodin et al. in 2008 (see Ref. [38]). In Refs. [36,38] the wave is considered in terms of the kinetic
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equation, the quantum hydrodynamics was applied for consideration of the wave in Ref. [37]. Spin wave propagating parallel to an external magnetic field was obtained by Misra et al. in 2010 (see Ref. [43]) and Andreev and Kuz’menkov in 2011 (see Ref. [48]). Influence of the spin–orbit interaction on these waves was considered in 2011–2012 (see Refs. [44,48]). Charge of particles creating the electric field is essential for the plasma wave existence. However the spin waves can exist in systems of neutral particles, since the magnetic field plays a main role in spin wave propagation. It was suggested in Ref. [37] that the spin-1/2 quantum plasmas can support spin waves propagating by magnetic field with no electric field contributing in their propagation. Dispersion of such waves was considered in Refs. [37,44]. Three branches of these waves were obtained. Dispersion of one of them depends on the Fermi velocity. Hence the exchange interaction gives contribution in this wave dispersion via the shifted Fermi velocity [41]. Spin waves in systems of neutral particles existing due to the long range spin–spin interaction are considered in Ref. [55] in terms of quantum kinetics. This quantum kinetics was derived in Ref. [55] as direct generalization of the method of many-particle quantum hydrodynamics [1,2,28,44,50,56,57]. This kinetics differs from recent generalization of the Wigner kinetics for spinning particles [58,72–75,59]. All mentioned effects are based on scalar g-factor theories. Effect of tensor g-factor on the spectrum of eigenmodes in spin-1/2 quantum plasmas was considered by Vagin et al. (see Ref. [42]). Let us mention excellent applications of exchange interaction in terms of many-particle quantum hydrodynamics. They were presented in Ref. [60] for the system of neutral quantum particles with the short range interaction. The famous Gross–Pitaevskii equation, for the inhomogeneous non-ideal Bose–Einstein condensates, together with its analog for ultracold fermions, and their generalizations was derived there. Recent achievements in field of quantum plasmas with the exchange interactions can be found in Refs. [61,62], where authors applied the Wigner kinetics [63]. In this paper we consider the contribution of the Coulomb exchange interaction in the spectrum of quantum plasma waves including ion-acoustic waves. Recently a model for classic and quantum plasmas including the finite size of ions was developed (see Ref. [64]). Contribution of the finite size of ions in dispersion of ion-acoustic waves was obtained in Ref. [64]. This paper is organized as follows. Two fluid quantum hydrodynamics for Coulomb plasmas with the exchange interaction is presented in Section 2. In Section 3 we give applications of the developed model to the Langmuir and ion-acoustic waves in two- and three dimensional quantum plasmas. In Section 4 a brief summary of obtained results is presented. 2. Model In this paper we present a set of quantum hydrodynamic equations for spinless Coulomb quantum plasmas without derivation. Method of direct derivation of the quantum hydrodynamic equations from many-particle Schrödinger equation can be found in Ref. [1] (Coulomb interaction with the exchange part for Bose and Fermi particles), [2,28,56] (spin-1/2 charged particles), [57] (charged particles baring electric dipole moment), [65] (spinless semi-relativistic quantum plasmas). Some details of derivation can be also found in Ref. [60] dedicated to quantum neutral particles with exchange interactions. The QHD equations for the electron subsystem in electron–ion quantum plasmas are
∂t ne + ∇(ne ve ) = 0,
(8)
and me ne (∂t + ve ∇)ve + ∇ pe −
h¯ 2 4me
ne ∇
△ne ne
−
(∇ ne )2
2n2e
1 = qe ne Eext + [ve , Bext ] − q2e ne ∇ G(r, r′ )ne (r′ , t )dr′ c
− qe qi ne ∇
G(r, r′ )ni (r′ , t )dr′ + FC ,e ,
(9)
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and the QHD equations for ions are
∂t ni + ∇(ni vi ) = 0,
(10)
and mi ni (∂t + vi ∇)vi + ∇ pi −
h¯ 2 4mi
ni ∇
△ ni ni
1
= qi ni Eext + [vi , Bext ] − q2i ni ∇
− qe qi ni ∇
−
2n2i
G(r, r′ )ni (r′ , t )dr′
c
(∇ ni )2
G(r, r′ )ne (r′ , t )dr′ + FC ,i .
(11)
The set of Eqs. (8), (9) and (10), (11) are coupled to each other by means of the last terms in the Euler equations (9) and (11). In the equation set (8)–(11) we assumed that thermal pressure is isotropic: αβ pa = pa δ αβ , where a stands for species of particles. Eqs. (8) and (10) are continuity equations for electrons and ions correspondingly. These equations show conservation of particle number of electrons and ions. Eqs. (9) and (11) are the momentum balance (Euler) equations for electrons and ions. The first terms on the left-hand side of Euler equations ma na (∂t + va ∇)va are the kinematic part. The second terms are the gradient of the thermal pressure or the Fermi pressure for degenerate electrons and ions. It appears as the thermal part of the momentum flux related to distribution of particles on states with different momenta. The third terms are the quantum Bohm potential appearing as the quantum part of the momentum flux [66]. Form of the quantum potential was analyzed in terms of the density matrix in Refs. [67,68]. On the right-hand sides of the Euler equations we present interparticle interaction and interaction of particles with external electromagnetic fields. The first group of terms on the right-hand side of the Euler equations describe interaction with the external electromagnetic fields. The second term in the Euler equation for electrons (9) describes the electron–electron Coulomb interaction. The third term is the Coulomb action of ions on electron motion. The second term in the Euler equation for ions (11) gives the ion–ion Coulomb interaction. The Coulomb field of electrons acting on ions is presented by the third term in Eq. (11). The last terms on the right-hand sides of Eqs. (9) and (11) describe the Coulomb electron–electron and ion–ion exchange interactions FC ,a correspondingly. These two terms are the main subject of this paper. Below we consider their contribution in the plasma wave dispersion. We can rewrite the Euler equations (9) and (11) in terms of the self-consistent electric field
Ea(int) = −qa ∇
G(r, r′ )na (r′ , t )dr′ ,
(12)
where Ea(int) is the electric field created by particles of species a = e, i. Hence Eqs. (9), (11) attain a more familiar form me ne (∂t + ve ∇)ve + ∇ pe −
h¯ 2 4me
ne ∇
△ne ne
−
(∇ ne )2
2n2e
1 = qe ne Eext + Eint + [ve , Bext ] + FC ,e ,
(13)
c
and the Euler equation for ions is mi ni (∂t + vi ∇)vi + ∇ pi −
1
h¯ 2 4mi
ni ∇
△ ni ni
−
2n2i
= qi ni Eext + Eint + [vi , Bext ] + FC ,i . c
(∇ ni )2
(14)
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Physical meaning of terms in the non-integral Euler equations (13) and (14) is similar to that described above for Eqs. (9) and (11). Interparticle interactions are presented in (13) and (14) in terms of internal electric field satisfying the Maxwell equations. The × Eint = 0 and ∇ Eint = 4π ρ , where ρ3D = a qa na , and ρ2D = Maxwell equations ∇ δ(z ) a qa na,(2D) , with Eint = a Ea(int) . The two dimensional charge density presented here ρ2D = ρ2D (x, y, z ) is explicitly presented as two dimensional layer in the three dimensional physical space. Two dimensional particle concentration is a function of two space coordinates na,(2D) = na,(2D) (x, y) in plane z = 0. The equation of state for degenerate 2D Fermi gas is pa,2D = π h¯ 2 n2a,2D /(2ma ). In 3D case one 5/3
similarly finds pa,3D = (3π 2 )2/3 h¯ 2 na,3D /(5ma ). These equations of state take place for unpolarized fermions at zero temperature pa,ND = pa,ND↑↓ , N = 2 or 3, where subindex ↑↓ means that in each occupied quantum state we have two particles with opposite spins. Hence we have two fermions in each state with energy lower than the Fermi energy εF , fermions of each pair have opposite spins. When system of spin-1/2 fermions is polarized then distribution of fermions looks like one electron in each state with energy lower than 22/3 εF ,3D for 3D mediums, and 2εF ,2D for 2D mediums. For polarized 5/3
systems equations of state appear as pa,3D↑↑ = 22/3 (3π 2 )2/3 h¯ 2 na,3D /(5ma ) for 3D mediums, and
pa,2D↑↑ = 2π h¯ 2 n2a,2D /(2ma ) for 2D mediums, where subindex ↑↑ means that all particles have same spin direction. We may consider partly polarized particles, then we need to introduce the ratio of polarizability η =
|n↑ −n↓ | , n↑ +n↓
with indexes ↑ and ↓ meaning particles with spin up and spin down.
Here we have that instead p0 = p↑ + p↓ , with p↑ = p↓ = p0 /2 for η = 0, we find p = p˜ ↑ + p˜ ↓ , with p˜ ↑ = 25/3 p↑ = 22/3 p0 and p˜ ↓ = 0 for η = 1 in 3D case. Similarly we have p0 = p↑ + p↓ , with p↑ = p↓ = p0 /2 at η = 0, we obtain for η = 1 p = p˜ ↑ + p˜ ↓ , with p˜ ↑ = 4p↑ = 2p0 and p˜ ↓ = 0 for 2D case. In the general case of partially polarized system of particles we can write 5/3
pa,3D⇕ = ϑ3D (3π 2 )2/3 h¯ 2 na,3D /(5ma ) for 3D mediums, and pa,2D⇕ = ϑ2D π h¯ 2 n2a,2D /(2ma ) for 2D mediums, with
ϑ3D =
1 2
[(1 + η)5/3 + (1 − η)5/3 ],
(15)
and
ϑ2D = 1 + η2 ,
(16)
where ⇕ stands for partially polarized systems, that means that part of states contains two particles with opposite spins and other occupied states contain one particle with same spin direction. Considering two electrons one finds that full wave function is anti-symmetric. If one has two electrons with parallel spins one has that wave function is symmetric on spin variables, so it should be anti-symmetric on space variables. In the opposite case of anti-parallel spins one has anti-symmetry of wave function on spin variables and symmetry of wave function on space variables. Considering energy of two electron Coulomb interaction one finds it has two parts: the classic like part C and the exchange part A. For the parallel (anti-parallel) spins one obtains E↑↑ = C − A (E↑↓ = C + A). Parallel (anti-parallel) configuration of spins decreases (increases) energy of the Coulomb interaction. Systems of unpolarized electrons then the average numbers of electrons with different directions of spins equal to each other, we find that the average number of particles with parallel and anti-parallel spins is the same. Consequently we have that average exchange interaction equals to zero. In partly polarized systems the number of particles with different spins are not the same. In this case a contribution of the average exchange interaction appears. At full measure it reveals in fully polarized system then all electrons have same direction of spins. In accordance with the previous discussion we find that exchange interaction, for this configuration, gives attractive contribution in the force field.
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This result allows us to find the force field of the Coulomb exchange interaction
FC ,a(3D) =
3 24/3 q2a
= 6q2a
6
3
π
3√ 3 na ∇ na
π
na ∇ n1a /3 .
(17)
We obtain the force field of exchange interaction for 2D quantum plasmas in the following form: FC ,a(2D) = 23/2
√
2π
24 arsh1
π
2
q2a
√
na ∇ na
√ β √ = 8 π 2 q2a na ∇ na π where we introduce β ≡ 24 arsh1 = 21.153.
(18)
The force fields (17) and (18) are obtained for fully polarized systems of identicalparticles. For partially polarized particles the force fields reappear as FC ,a(3D) = ζ3D q2a FC ,a(2D) = ζ2D
√
2π
√ 2
β q π2 a
√
3 3 3
π
na ∇ na and
na ∇ na , with
ζ3D = (1 + η)4/3 − (1 − η)4/3
(19)
ζ2D = (1 + η)3/2 − (1 − η)3/2 .
(20)
and We should mention that coefficients ζ3D ∼ η and ζ2D ∼ η are proportional to spin polarization. Limit cases of ζ3D and ζ2D are ζ3D (0) = 0, ζ3D (1) = 24/3 , ζ2D (0) = 0 and ζ2D (1) = 23/2 . If we do not apply a magnetic field to systems under consideration we need to have systems, where self-organization of equilibrium state leads to formation of population levels with uncompensated spin, such that it is in ferromagnetic domains. Otherwise we have no contribution of the Coulomb exchange interaction. Force fields of exchange interaction (17) and (18) are potential fields. Thus they do not give contribution in dispersion of transverse waves. They affect longitudinal waves and waves with complex polarization: longitudinal–transverse waves. Consequently electromagnetic waves are not affected by the Coulomb exchange interaction in 3D and 2D mediums in the absence of external fields. Force fields of the Coulomb exchange interaction can be presented as a product of the particle concentration on the gradient of a function when it gives no contribution in equations of the vorticity evolution (see Refs. [51,69–71]). Many-particle quantum hydrodynamic equations can be represented in form of the non-linear Schrödinger equation (NLSE). Let us consider evolution of electrons at motionless ions. We introduce the wave function in the medium defined in terms of hydrodynamic variables
Φ=
√
n exp ı
mS
h¯
,
(21)
where S is the potential of velocity field. Let us mention that the NLSE can be derived for eddy-free motion of electrons. Definition (21) can be √ √applied for three dimensional and low dimensional systems of particles |Φ3D | = n3D , and |Φ2D | = n2D , with [n3D ] = cm−3 , [n2D ] = cm−2 . To derive the NLSE we need to differentiate function (21) with respect to time. After some calculations we find the NLSE for electrons in three dimensional quantum plasmas ıh¯ ∂t Φ3D =
−
h¯ 2 △ 2me
2/3
+ ϑ3D
(3π 2 )2/3 h¯ 2 ne
ıh¯ ∂t Φ2D =
−
h¯ 2 △ 2me
+ ϑ2D
−3
2me
For 2DEGs we obtain
π h¯ 2 me
n−
2β
√
2π
π2
ζ
3
3
π
2 2D qe
ζ3D q2 n1/3 + qe ϕ Φ3D .
√
(22)
n + qe ϕ Φ2D .
(23)
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Eqs. (22) and (23) contain potential of the electric field ϕ presenting the external and internal electric fields: E = −∇ϕ . Considering dynamics of two or more species we should present the wave functions in the medium for each species and derive NLSEs for each species either. 3. Applications In this section we consider small perturbations of equilibrium state described by nonzero particle concentration n0 , zero velocity field v0 = 0 and electric field E0 = 0. Assuming that perturbations are monochromatic
δn NA δ v = VA e−ıωt +ıkr , δE EA
(24)
we get a set of linear algebraic equations relatively to NA and VA . The condition of existence of nonzero solutions for amplitudes of perturbations gives us a dispersion equation. 3.1. Three dimensional quantum plasmas In classic plasmas the Langmuir waves have the following spectrum: 2 ω2 = ωLe +
γT me
k2 ,
(25)
with the three dimensional Langmuir frequency 2 ωLe ,3D =
4π e2 n0,3D me
,
(26)
T is the temperature, γ is the adiabatic index. We consider quantum plasmas, so we are interested in the low temperature properties, then carriers are degenerated. Hence we have contribution of the Fermi pressure instead of the temperature. Spectrum of the Langmuir waves in 3D quantum plasmas with the Coulomb exchange interaction appears as
ω =ω 2
2 Le,3D
− ζ3D
3
2/3
3 e2 √ (3π 2 )2/3 h¯ 2 n0e 2 h¯ 2 k4 3 k + . n0e k2 + ϑ3D π me 3m2e 4m2e
(27)
The first term in formula (27) is the 3D Langmuir frequency existing due to the Coulomb interaction in the self-consistent field approximation. The second term describes the Coulomb exchange interaction between electrons. The third term appears from the Fermi pressure. The last term describes contribution of the quantum Bohm potential. Let us consider a dimensionless form of the spectrum of 3D Langmuir waves. Introducing dimen√ sionless frequency Ω = ω/ωLe,3D and wave vector ξ = k/ 3 n0e,3D we obtain
Ω 2 = 1 − ζ3D
1
3
4π
3
π
ξ 2 + ϑ3D
(3π 2 )2/3 3
+
1 16π
ξ 2 Λ3D ξ 2 ,
(28)
where we also have a dimensionless parameter
Λ3D =
h¯ 2 √ 3 n0e , me e2
(29)
√
depending on fundamental constants h¯ , e, me , and the equilibrium concentration of electrons 3 n0e,3D . The contribution of the exchange interaction can be considered as a shift of the Fermi pressure since both of them are proportional to ξ 2 . On the other hand, the term describing the exchange interaction
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is proportional to the square of the Langmuir frequency. From this point of view we see that exchange interaction gives a shift of the Langmuir frequency and this shift is proportional to square of the wave vector ξ 2 . So exchange interaction is considerable in the short wave length limit. Increase of the particle concentration allows to increase maximal wave vector of wave propagation in the medium. Hence, in short wave length limit the first two terms reveal same dependence on the equilibrium particle concentration. For whole range of wave vectors we see that the first term ∼ n0e,3D 1/3
grows faster than the second term ∼ n0e,3D with the increase of the particle concentration. The third 2/3 term has an intermediate rate of growth being proportional to n0e,3D . However, in the short wave 2/3
length limit the third and fourth terms increase rather faster like n0e,3D k2 and k4 correspondingly. Finally formula (28) shows that the third and fourth terms can be rather large at high densities (particle concentrations) and large wave vectors. Let us estimate contribution of the exchange interaction in comparison with other terms in dispersion dependence of the Langmuir waves. Comparing exchange interaction with the Fermi pressure we should consider the ratio of the second term to the third term
χEF =
me e2
√
h2 3
¯
=
n0e,3D
1
Λ3D
≈
3
1025 n0e,3D
.
(30)
Exchange interaction prevails over the Fermi pressure than χEF > 1, that corresponds to n0e,3D < 1025 cm−3 . Hence the Coulomb exchange interaction is larger than the Fermi pressure in metals n0e,3D ∼ 1022 cm−3 and semiconductors n0e,3D ∼ 1018 cm−3 . Considering extreme astrophysical objects like white dwarfs n0e,3D ∼ 1028 cm−3 we find that Fermi pressure is larger than the Coulomb exchange interaction. The ratio between the Fermi pressure and the exchange interaction does not depend on the wave vector k. We see that at large concentrations the role of the Coulomb exchange interaction is negligible in comparison with the Fermi Pressure even for fully polarized fermions. At small concentrations the role of the exchange interaction increases, but it never overcomes the self-consistent part of the Coulomb interaction presented by the square of Langmuir frequency. Hence the spectrum of the Langmuir waves is stable. Let us consider this problem closer. It is useful to introduce the average 1 interparticle distance a = √ 3 n . The wavelengths of matter waves λ = 2π /k are larger than the 0
√
average interparticle distance λ > a. Consequently ξ = k/ 3 n0e,3D = ak = 2π a/λ < 1. The second term in formula (28) consists of the product of four multipliers. Each of these multipliers is smaller than 1. Consequently, the second term in formula (28) is always smaller than the first one. We conclude now that the frequency square of three dimensional Langmuir waves is positive. Hence we have a stable spectrum. We have considered high frequency waves. The next step is consideration of the low frequency excitations. Low frequency excitations in unmagnetized electron–ion plasmas are the ion-acoustic waves. We obtain contribution of the Coulomb exchange interaction in their spectrum. The final formula is
2 ζ3D 3 3 3 ωLe ,3D ω3D (k) = kvs,3D 1 − 2/3 2 ϑ3D 4π π n0e,3D vFe ,3D 1
× 1 + (krDe,3D )
2
√
√
1−
ζ3D 3 ϑ3D 4π
3 3
2 ωLe ,3D
,
(31)
π n2/3 v 2 0e,3D Fe,3D
√
where vs,3D = me /mi ϑ3D · vFe,3D /3 is the three dimensional velocity of sound, rDe,3D = ϑ3D vFe,3D /(3ωLe,3D ) is the Debye radius. In formula (31) and similar formulas below we extract contribution of the Fermi pressure. Hence formulas for ion-acoustic waves contain well-known contribution of the pressure multiplied by the factor showing contribution of exchange interaction.
P.A. Andreev / Annals of Physics 350 (2014) 198–210
207
The ion-acoustic wave exists under the following conditions:
v
2 2 Ti k
≪ω ≪ 2
1 3
ϑ v − ζ3D 2 3D Fe
ζ
Consequently, if Z = 1 − ϑ3D 43π 3D
q2 me
3
3√ 3 n0e k2 .
(32)
π
2 ωLe ,3D π n2/3 v 2 0e,3D Fe,3D
3 3
becomes small, then conditions of wave existence
are broken. Hence we cannot obtain formula (31) at small or negative Z . We see that the Coulomb exchange interaction decreases the frequency of the ion-acoustic waves. At large enough contribution of the exchange interaction the condition of the ion-acoustic wave existence is violated. So, the ionacoustic solution disappears. In the long wavelength limit we have
2 ζ3D 3 3 3 ωLe ,3D ω(k) = kvs,3D 1 − . 2/3 2 ϑ3D 4π π n0e,3D vFe ,3D
(33)
In the short wavelength limit we find
ω2 (k) = ωLi2 ,3D .
(34)
From formula (34) we see that the exchange interaction gives no contribution in the ion-acoustic waves in the short wave length limit. Kanazawa et al. present contribution of the Coulomb exchange interaction in the Langmuir wave spectrum in terms of the Green function method (see formula (2.15) of Ref. [5]). Their result coincides with results of Ref. [7]. However their result differs from the spectrum corresponding potential obtained and applied in Refs. [17–19]. Comparing our result given by formula (27) with earlier results presented by (1), (7) we see that there are differences with both of them. We have factor 1/3 in front of the square of Fermi velocity instead of 3/5. This is standard misgiving appearing as a consequence of the use of the Fermi pressure for the equation of state. In terms of describing the exchange interaction contribution we obtain different coefficients in comparison with Refs. [5,7] given by formula (1) 5 ≈ 3.15. And our and [18] given by formula (7). Our coefficient is larger than (1) by the factor √ 3 4
coefficient is smaller than (7) by the factor 1.66. 3.2. Two dimensional quantum plasmas: two dimensional electron gas and ion motion contribution Two dimensional quantum plasmas are systems of electrons and ions being under confinement, so we have plane-like objects. 2D quantum plasmas are surrounded by medium. However, main properties can be obtained considering a 2D layer in empty 3D space. Such objects as 2DEG (two dimensional electron gas) and 2DHG (two dimensional hole gas) are common objects in physics of semiconductors. As application these objects appear as parts of transistors. Consideration of 2DEG and 2DHG corresponds to the description of high frequency excitations. In this section we also include ion dynamics. Spectrum of high frequency longitudinal excitations, which are the Langmuir waves, appears as follows: 2 ω2 = ωLe ,2D − ζ2D
√ π h¯ 2 n0e 2 h¯ 2 k4 β 2π e2 √ n0e k2 + ϑ2D k + , 2 π me m2e 4m2e
(35)
where we have used the two dimensional Langmuir frequency 2 ωLe ,2D =
2π e2 kn0,2D me
∼ k,
which is not a constant, but it is proportional to the wave vector k.
(36)
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P.A. Andreev / Annals of Physics 350 (2014) 198–210
Similarly to the three dimensional spectrum (see formula (27)) different terms in formula (35) have the following meaning: self-consistent Coulomb interaction, the exchange Coulomb interaction, the Fermi pressure, and the quantum Bohm potential. Next we discuss some properties of the 2D Langmuir wave spectrum. Let us consider the exchange interaction with the Fermi pressure for 2D quantum plasmas. To this end we introduce the following dimensionless parameter:
χEF ,2D =
me e2
h¯
2
√
≈
n0e,2D
1016 n0e,2D
.
(37)
In 2D semiconductor objects n0e,2D ≪ 1016 cm−2 . Consequently, the exchange interaction plays a significant role in collective properties of semiconductors. To this end we present a dimensionless form of formula (35) introducing dimensionless parameters
√ 3/2 Ω2D = ω m/(2π e2 n0e,2d ) and ξ2D = k/ n0e,2D ∼ ak, with a as the average interparticle distance. Hence we have
β 1 1 2 2 Ω2D = ξ 1 − ζ2D √ ξ + ϑ2D Λ2D ξ 2 1 + ξ , 2 4π 2π π 2
(38)
with
Λ2D =
h¯ 2 √ n0e,2D . me2
(39)
For the 2D Langmuir wave spectrum we can present an analysis similar to the three dimensional case described above (see text after formula (30)). This analysis allows us to conclude that spectrum of the 2D Langmuir waves (35) is stable (ω2 > 0). Spectrum of the 2D ion-acoustic waves in presence of the exchange interaction appears as
√ √ ζ2D 2β 2π e2 n0e,2D ω2D (k) = kvs,2D 1 − 2 ϑ2D π 2 mvFe ,2D 1
×
1 + (krDe,2D (k))2 1 −
√ √ ζ2D 2β 2π e2 n0e,2D 2 ϑ2D π 2 mvFe ,2D
,
(40)
√
√
√
with vs,2D = me /mi ϑ2D · vFe,2D /2 is the two dimensional velocity of sound, rDe,2D = ϑ2D vFe,2D / (2ωLe,2D ). The spectrum of the 2D ion-acoustic waves can be written in a more explicit form, which shows dependence on the wave vector k
√ √ ζ2D 2β 2π e2 n0e,2D ω2D (k) = kvs,2D 1 − × 2 ϑ2D π 2 mvFe ,2D
1
1 + kD 1 −
√ √ ζ2D 2β 2π e2 n0e,2D 2 ϑ2D π 2 mvFe ,2D
,
(41)
where D=
√
2 2π
h¯ e2
√
n0e,2D
.
(42)
In the range of low frequency excitations we have similar change of behavior of 2D and 3D ionacoustic waves due to the exchange interaction account. The Coulomb exchange interaction decreases frequency of excitations. It happens till conditions of the ion-acoustic solution existence are broken. Thus we do not have 2D ion-acoustic waves at large exchange interaction.
P.A. Andreev / Annals of Physics 350 (2014) 198–210
209
In the long wavelength limit we have
√ √ ζ2D 2β 2π e2 n0e,2D ω(k) = kvs,2D 1 − . 2 ϑ2D π 2 mvFe ,2D
(43)
In the short wavelength limit we find
ω2 (k) = ωLi2 ,2D ∼ k.
(44)
As in 3D case we find no contribution of the exchange interaction in the short wave length ion-acoustic waves. 4. Conclusions We have briefly described quantum hydrodynamic model for electron–ion quantum plasmas with exchange interaction. We have considered three and two dimensional electron–ion quantum plasmas. We have derived an explicit form of the Coulomb exchange interaction force field for the electron–electron and ion–ion Coulomb interactions. We have applied this model to dispersion properties of electron–ion plasmas tracing contribution of the exchange interaction. We have considered small amplitude linear excitations in the absence of external field. However obtained results open possibilities for consideration of nonlinear effects. Derived force fields for the Coulomb exchange interaction can be used to study various effects in magnetized plasmas as well. Acknowledgment The author thanks Professor L.S. Kuz’menkov for fruitful discussions. References [1] L.S. Kuz’menkov, S.G. Maksimov, Teoret. Mat. Fiz. 118 (1999) 287. [Theor. Math. Phys. 118 (1999) 227]. [2] L.S. Kuz’menkov, S.G. Maksimov, V.V. Fedoseev, Theor. Math. Fiz. 126 (2001) 136. [Theor. Math. Phys. 126 (2001) 110]. [3] A.A. Vlasov, J. Exp. Theor. Phys. 8 (1938) 291; A.A. Vlasov, Sov. Phys. Usp. 10 (1968) 721. [4] D. Bahm, D. Pines, Phys. Rev. 92 (1953) 609. [5] H. Kanazawa, S. Misawa, K. Fujita, Progr. Theoret. Phys. (Kyoto) 23 (1960) 426. [6] H. Kanazawa, M. Watabe, Progr. Them. Phys. 23 (1960) 408. [7] P. Nozieres, D. Pines, Phys. Rev. 111 (1958) 442. [8] P.A. Wolff, Phys. Rev. 92 (1953) 18. [9] D. Pines, Rev. Modern Phys. 28 (1956) 184. [10] R.A. Ferrell, Phys. Rev. 107 (1957) 450. [11] D.F. DuBois, Ann. Physics 8 (1959) 24. [12] P. Burt, Hugo D. Wahlquist, Phys. Rev. 125 (1962) 1785. [13] O. von Roos, J. Zmuidzinas, Phys. Rev. 121 (1961) 941. [14] D. Ter Haar, Rep. Progr. Phys. 24 (1961) 304. [15] V. Sips, Phys. Lett. 25A (1967) 698. [16] V. Sips, M. Sunjic, Z. Phys. 211 (1968) 132. [17] L. Hedin, B.I. Lundqvist, J. Phys. C: Solid State Phys. 4 (1971) 2064. [18] L. Brey, Jed Dempsey, N.F. Johnson, B.I. Halperin, Phys. Rev. B 42 (1990) 1240. [19] N. Crouseilles, P.-A. Hervieux, G. Manfredi, Phys. Rev. B 78 (2008) 155412. [20] A.A. Kugler, J. Stat. Phys. 12 (1975) 35. [21] P. Hohenberpg, W. Kohn, Phys. Rev. 136 (1964) B864. [22] W. Kohn, L.G. Sham, Phys. Rev. 140 (1965) A1133. [23] L.G. Sham, W. Kohn, Phys. Rev. 145 (1966) 561–567. [24] S.A. Khan, Sunia Hassan, J. Appl. Phys. 115 (2014) 204304. [25] M.V. Kuzelev, A.A. Rukhadze, Phys. Usp. 42 (1999) 603. [26] L.S. Kuz’menkov, S.G. Maksimov, V.V. Fedoseev, Russian Phys. J. 43 (2000) 718. [27] L.S. Kuz’menkov, S.G. Maksimov, V.V. Fedoseev, Vestnik Moskov. Univ. Ser. III Fiz. Astronom. (5) (2000) 3. [Moscow Univ. Phys. Bull. (5) (2000) 1]. [28] L.S. Kuz’menkov, S.G. Maksimov, V.V. Fedoseev, Theor. Math. Fiz. 126 (2001) 258. [Theor. Math. Phys. 126 (2001) 212]. [29] F. Haas, G. Manfredi, M. Feix, Phys. Rev. E 62 (2000) 2763. [30] G. Manfredi, F. Haas, Phys. Rev. B 64 (2001) 075316. [31] P.K. Shukla, B. Eliasson, Phys. Usp. 53 (2010) 51.
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Exchange effects in Coulomb quantum plasmas: Dispersion of waves in 2D and 3D quantum plasmas Pavel A. Andreev M. V. Lomonosov Moscow State University, Moscow, Russian Federation
article
info
Article history: Received 9 May 2014 Accepted 10 July 2014 Available online 17 July 2014 Keywords: Quantum plasmas Exchange interaction Ion-acoustic waves Quantum hydrodynamics Non-linear Schrödinger equation
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aop.2014.07.019 0003-4916/© 2014 Elsevier Inc. All rights reserved.
abstract We describe quantum hydrodynamic equations with the Coulomb exchange interaction for three and two dimensional plasmas. Explicit form of the force densities are derived. We present non-linear Schrödinger equations (NLSEs) for the Coulomb quantum plasmas with the exchange interaction. We show contribution of the exchange interaction in the dispersion of the Langmuir, and ionacoustic waves. We consider influence of the spin polarization ratio on strength of the Coulomb exchange interaction. This is important since exchange interaction between particles with same spin direction and particles with opposite spin directions are different. At small particle concentrations n0 ≪ 1025 cm−3 and small polarization the exchange interaction gives small decrease of the Fermi pressure. With increase of polarization role the exchange interaction becomes more important, so that it can overcome the Fermi pressure. The exchange interaction also decreases contribution of the Langmuir frequency. Ion-acoustic waves do not exist in limit of large polarization since the exchange interaction changes the sign of pressure. At large particle concentrations n0 ≫ 1025 cm−3 the Fermi pressure prevails over the exchange interaction for all polarizations. We obtain a similar picture for two dimensional quantum plasmas. © 2014 Elsevier Inc. All rights reserved.
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1. Introduction Exchange effects in quantum plasmas have been considered for a long time. A lot of different approaches, methods and results were obtained in this field. Before we present a new approach to this subject based on the many-particle quantum hydrodynamic (MPQHD) method [1,2] we present a brief review of earlier results. Earlier results are focused on three dimensional electron gas. They do not include dependence on the spin polarization of electron gas. We consider three and two dimensional electron gas and present dependence on its spin polarization. Exchange interaction is a remarkable example of quantum physical effects. The exchange interaction is related to overlapping of the wave functions of interacting particles. Hence it reveals itself as a short range interaction, even when we consider the exchange part of a long range interaction. The Coulomb exchange interaction gives contribution in the spectrum of Langmuir waves in three dimensional degenerate electron gas. This spectrum without exchange contribution was obtained in Refs. [3,4]. In 1960 Kanazawa et al. [5] presented contribution of the Coulomb exchange interaction in the Langmuir wave spectrum in terms of the Green function method [6]. Their result coincides with results of Ref. [7] obtained in 1958 by Nozieres and Pines. However it differs from the results that had been obtained earlier by other authors (see Wolff [8], Pines [9], Ferrell [10], DuBois [11]). Results of Refs. [5,7] for the shift of the frequency square are 2 1ωGFM =−
3
k2
20 (3π 2 n0 )
1
2 3
2 ωLe ∼ n03 .
(1)
Plasma wave dispersion with the exchange interaction was considered in 1962 in terms of the trace of two-particle density matrix (see for instance formula (1) in Ref. [12]). An explicit form of the exchange potential is given by formula (16) of Ref. [12] (see Ref. [13] as well). This potential has the following form:
φex =
1 2
4π e f
1F (k′ , q) ′ dk − 1F |k − k′ |
f (k′ + q, k′ ; ω)
|k − k′ |2
dk′ ,
(2)
where
1F = F (|k + q|) − F (|k|),
(3)
here one has assumed that the one-particle density matrix [14] of electrons in wave-vector space has the following form: R(1) (k, k′ ; s′ , s; t ) = F (|k|) + f (k, k′ ; s′ , s; t )
(4)
f (k + q, k; ω) = Tr(s) [f (k + q, k; s′ , s; ω)],
(5)
and
with s, s are spin coordinates, F (|k|) is the equilibrium Fermi–Dirac distribution, and f is a small perturbation. We see that the exchange potential is presented in a linear approach obtained in terms of distribution function. Contribution of the exchange interaction in the spectrum of longitudinal waves is presented in Ref. [13]. This contribution coincides with results of Refs. [5,7]. In Ref. [12] it was shown that the Coulomb exchange interaction gives contribution in the electromagnetic wave spectrum. It was found despite the fact that the exchange interaction appears as an extra potential in addition to the scalar potential of electromagnetic field (see Eq. (4) of Ref. [12]), one can expect that it gives no trace in the spectrum of transverse waves. Generalization of spectrum (1) was presented in Refs. [15,16]. Hedin and Lundqvist [17] discussed contribution of the Coulomb exchange interaction in density functional theory. Results of Ref. [17] were applied in paper [18] for the study of wide parabolic ′
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quantum wells (see formula (3) for exchange potential). The exchange potential presented in Refs. [17,18] is negative. That reveals an attraction between electrons. Its explicit form is Vex = −0.985
e2
ϵ
n
1 3
1+
0.034 1
aB n 3
ln 1 + 18.376aB n
1 3
,
(6)
where ϵ is the (uniform) static dielectric constant, and aB = ϵ h¯ 2 /(me2 ) is the Bohr radius with contribution of medium by means of the dielectric constant. This formula consists of two parts. The first term is the exchange interaction. The second term is the correlation contribution [18]. Formula (6) shows that the correlations increase contribution of the exchange interaction. This extra contribution is of the order of 50% percent of the exchange interaction. This potential was included in the QHD model at consideration of finite width thin metal films [19]. The exchange part of potential (6) gives the following shift of the 3D Langmuir wave spectrum 2 1ωLDT =−
0.985
k2
12π (n0 ) 23
2 ωLe .
(7)
This result in 5.24 times more than the earlier result presented by formula (1). A review of local field corrections in electron gas was presented in 1975 by Kugler [20]. Hohenberg–Kohn–Sham local density theory [17,21–23] contains a NLSE for an effective one-body wave function of quasielectrons and quasiholes (see for instance formula (2.2) in Ref. [17]). Sometimes result (6) has been used for consideration of 2D plasmas [24]. It does not seem correct. For instance, if we consider the Fermi pressure of 2D gas it appears in terms of 2D particle concentration and it has a structure, which differs from the structure of 3D Fermi pressure. Similar situation happens with the potential and force field of the Coulomb exchange interaction. It has to be derived for 2D case independently. It should be done in terms of the 2D particle concentration as we do in this paper for 2D plane electron gas. We have not obtained contribution of the correlations in terms of the MPQHD. This paper is dedicated to the Coulomb exchange interaction in the MPQHD. The exchange–correlation potential (6) shows that there is a lot of work for future development of the MPQHD method. It is important to develop a method of derivation of the correlation potential. New approaches to hydrodynamic description of quantum plasmas were considered in 1999 by Kuz’menkov and Maksimov [1] and Kuzelev and Rukhadze [25], where were considered spinless Coulomb quantum plasmas. The self-consistent field approach was the center of attention of these papers. Nevertheless a general form of the exchange interaction for bosons and fermions, in terms of the MPQHD, was derived in Ref. [1]. In 2000–2001 attention shifted towards spin-1/2 quantum plasmas [2,26–28]. The explicit form of the exchange interaction for the Coulomb and the spin–spin interaction was derived in Ref. [28]. Contribution of these interactions in properties of many-electron atoms was described there. During period 2001–2007 most of researches considered spinless quantum plasmas [29,30] (for review see Refs. [31,32]). A great interest to quantum plasmas of spinning particles arose since 2007. Marklund and Brodin drew attention to the field by their papers [33,34], some examples are presented below. Dispersion properties of spin-1/2 quantum plasmas have been under consideration for many years (see Refs. [27,33,35–54]). It was shown that magnetic moment (spin) evolution in quantum plasmas leads to new branches of wave dispersion [36–38,43,44,48,51]. The last of these Ref. [51] shows dispersion of spin wave in the spin-1/2 two dimensional electron gas. Contribution of the exchange interaction in dispersion properties of spin-1/2 quantum plasmas was considered in 2008 (see Ref. [41]). It was demonstrated that the exchange interactions give potential force field depending on the particle concentration and the spin density. In linear approximation the exchange interactions can be combined with the Fermi pressure, so we have an effective shifted Fermi velocity [41]. A description of the spin–plasma wave propagating perpendicular to an external magnetic field was given in 2006 by Vagin et al. (see Ref. [36]), Andreev and Kuz’menkov in 2007 (see Ref. [37]), and Brodin et al. in 2008 (see Ref. [38]). In Refs. [36,38] the wave is considered in terms of the kinetic
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equation, the quantum hydrodynamics was applied for consideration of the wave in Ref. [37]. Spin wave propagating parallel to an external magnetic field was obtained by Misra et al. in 2010 (see Ref. [43]) and Andreev and Kuz’menkov in 2011 (see Ref. [48]). Influence of the spin–orbit interaction on these waves was considered in 2011–2012 (see Refs. [44,48]). Charge of particles creating the electric field is essential for the plasma wave existence. However the spin waves can exist in systems of neutral particles, since the magnetic field plays a main role in spin wave propagation. It was suggested in Ref. [37] that the spin-1/2 quantum plasmas can support spin waves propagating by magnetic field with no electric field contributing in their propagation. Dispersion of such waves was considered in Refs. [37,44]. Three branches of these waves were obtained. Dispersion of one of them depends on the Fermi velocity. Hence the exchange interaction gives contribution in this wave dispersion via the shifted Fermi velocity [41]. Spin waves in systems of neutral particles existing due to the long range spin–spin interaction are considered in Ref. [55] in terms of quantum kinetics. This quantum kinetics was derived in Ref. [55] as direct generalization of the method of many-particle quantum hydrodynamics [1,2,28,44,50,56,57]. This kinetics differs from recent generalization of the Wigner kinetics for spinning particles [58,72–75,59]. All mentioned effects are based on scalar g-factor theories. Effect of tensor g-factor on the spectrum of eigenmodes in spin-1/2 quantum plasmas was considered by Vagin et al. (see Ref. [42]). Let us mention excellent applications of exchange interaction in terms of many-particle quantum hydrodynamics. They were presented in Ref. [60] for the system of neutral quantum particles with the short range interaction. The famous Gross–Pitaevskii equation, for the inhomogeneous non-ideal Bose–Einstein condensates, together with its analog for ultracold fermions, and their generalizations was derived there. Recent achievements in field of quantum plasmas with the exchange interactions can be found in Refs. [61,62], where authors applied the Wigner kinetics [63]. In this paper we consider the contribution of the Coulomb exchange interaction in the spectrum of quantum plasma waves including ion-acoustic waves. Recently a model for classic and quantum plasmas including the finite size of ions was developed (see Ref. [64]). Contribution of the finite size of ions in dispersion of ion-acoustic waves was obtained in Ref. [64]. This paper is organized as follows. Two fluid quantum hydrodynamics for Coulomb plasmas with the exchange interaction is presented in Section 2. In Section 3 we give applications of the developed model to the Langmuir and ion-acoustic waves in two- and three dimensional quantum plasmas. In Section 4 a brief summary of obtained results is presented. 2. Model In this paper we present a set of quantum hydrodynamic equations for spinless Coulomb quantum plasmas without derivation. Method of direct derivation of the quantum hydrodynamic equations from many-particle Schrödinger equation can be found in Ref. [1] (Coulomb interaction with the exchange part for Bose and Fermi particles), [2,28,56] (spin-1/2 charged particles), [57] (charged particles baring electric dipole moment), [65] (spinless semi-relativistic quantum plasmas). Some details of derivation can be also found in Ref. [60] dedicated to quantum neutral particles with exchange interactions. The QHD equations for the electron subsystem in electron–ion quantum plasmas are
∂t ne + ∇(ne ve ) = 0,
(8)
and me ne (∂t + ve ∇)ve + ∇ pe −
h¯ 2 4me
ne ∇
△ne ne
−
(∇ ne )2
2n2e
1 = qe ne Eext + [ve , Bext ] − q2e ne ∇ G(r, r′ )ne (r′ , t )dr′ c
− qe qi ne ∇
G(r, r′ )ni (r′ , t )dr′ + FC ,e ,
(9)
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and the QHD equations for ions are
∂t ni + ∇(ni vi ) = 0,
(10)
and mi ni (∂t + vi ∇)vi + ∇ pi −
h¯ 2 4mi
ni ∇
△ ni ni
1
= qi ni Eext + [vi , Bext ] − q2i ni ∇
− qe qi ni ∇
−
2n2i
G(r, r′ )ni (r′ , t )dr′
c
(∇ ni )2
G(r, r′ )ne (r′ , t )dr′ + FC ,i .
(11)
The set of Eqs. (8), (9) and (10), (11) are coupled to each other by means of the last terms in the Euler equations (9) and (11). In the equation set (8)–(11) we assumed that thermal pressure is isotropic: αβ pa = pa δ αβ , where a stands for species of particles. Eqs. (8) and (10) are continuity equations for electrons and ions correspondingly. These equations show conservation of particle number of electrons and ions. Eqs. (9) and (11) are the momentum balance (Euler) equations for electrons and ions. The first terms on the left-hand side of Euler equations ma na (∂t + va ∇)va are the kinematic part. The second terms are the gradient of the thermal pressure or the Fermi pressure for degenerate electrons and ions. It appears as the thermal part of the momentum flux related to distribution of particles on states with different momenta. The third terms are the quantum Bohm potential appearing as the quantum part of the momentum flux [66]. Form of the quantum potential was analyzed in terms of the density matrix in Refs. [67,68]. On the right-hand sides of the Euler equations we present interparticle interaction and interaction of particles with external electromagnetic fields. The first group of terms on the right-hand side of the Euler equations describe interaction with the external electromagnetic fields. The second term in the Euler equation for electrons (9) describes the electron–electron Coulomb interaction. The third term is the Coulomb action of ions on electron motion. The second term in the Euler equation for ions (11) gives the ion–ion Coulomb interaction. The Coulomb field of electrons acting on ions is presented by the third term in Eq. (11). The last terms on the right-hand sides of Eqs. (9) and (11) describe the Coulomb electron–electron and ion–ion exchange interactions FC ,a correspondingly. These two terms are the main subject of this paper. Below we consider their contribution in the plasma wave dispersion. We can rewrite the Euler equations (9) and (11) in terms of the self-consistent electric field
Ea(int) = −qa ∇
G(r, r′ )na (r′ , t )dr′ ,
(12)
where Ea(int) is the electric field created by particles of species a = e, i. Hence Eqs. (9), (11) attain a more familiar form me ne (∂t + ve ∇)ve + ∇ pe −
h¯ 2 4me
ne ∇
△ne ne
−
(∇ ne )2
2n2e
1 = qe ne Eext + Eint + [ve , Bext ] + FC ,e ,
(13)
c
and the Euler equation for ions is mi ni (∂t + vi ∇)vi + ∇ pi −
1
h¯ 2 4mi
ni ∇
△ ni ni
−
2n2i
= qi ni Eext + Eint + [vi , Bext ] + FC ,i . c
(∇ ni )2
(14)
P.A. Andreev / Annals of Physics 350 (2014) 198–210
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Physical meaning of terms in the non-integral Euler equations (13) and (14) is similar to that described above for Eqs. (9) and (11). Interparticle interactions are presented in (13) and (14) in terms of internal electric field satisfying the Maxwell equations. The × Eint = 0 and ∇ Eint = 4π ρ , where ρ3D = a qa na , and ρ2D = Maxwell equations ∇ δ(z ) a qa na,(2D) , with Eint = a Ea(int) . The two dimensional charge density presented here ρ2D = ρ2D (x, y, z ) is explicitly presented as two dimensional layer in the three dimensional physical space. Two dimensional particle concentration is a function of two space coordinates na,(2D) = na,(2D) (x, y) in plane z = 0. The equation of state for degenerate 2D Fermi gas is pa,2D = π h¯ 2 n2a,2D /(2ma ). In 3D case one 5/3
similarly finds pa,3D = (3π 2 )2/3 h¯ 2 na,3D /(5ma ). These equations of state take place for unpolarized fermions at zero temperature pa,ND = pa,ND↑↓ , N = 2 or 3, where subindex ↑↓ means that in each occupied quantum state we have two particles with opposite spins. Hence we have two fermions in each state with energy lower than the Fermi energy εF , fermions of each pair have opposite spins. When system of spin-1/2 fermions is polarized then distribution of fermions looks like one electron in each state with energy lower than 22/3 εF ,3D for 3D mediums, and 2εF ,2D for 2D mediums. For polarized 5/3
systems equations of state appear as pa,3D↑↑ = 22/3 (3π 2 )2/3 h¯ 2 na,3D /(5ma ) for 3D mediums, and
pa,2D↑↑ = 2π h¯ 2 n2a,2D /(2ma ) for 2D mediums, where subindex ↑↑ means that all particles have same spin direction. We may consider partly polarized particles, then we need to introduce the ratio of polarizability η =
|n↑ −n↓ | , n↑ +n↓
with indexes ↑ and ↓ meaning particles with spin up and spin down.
Here we have that instead p0 = p↑ + p↓ , with p↑ = p↓ = p0 /2 for η = 0, we find p = p˜ ↑ + p˜ ↓ , with p˜ ↑ = 25/3 p↑ = 22/3 p0 and p˜ ↓ = 0 for η = 1 in 3D case. Similarly we have p0 = p↑ + p↓ , with p↑ = p↓ = p0 /2 at η = 0, we obtain for η = 1 p = p˜ ↑ + p˜ ↓ , with p˜ ↑ = 4p↑ = 2p0 and p˜ ↓ = 0 for 2D case. In the general case of partially polarized system of particles we can write 5/3
pa,3D⇕ = ϑ3D (3π 2 )2/3 h¯ 2 na,3D /(5ma ) for 3D mediums, and pa,2D⇕ = ϑ2D π h¯ 2 n2a,2D /(2ma ) for 2D mediums, with
ϑ3D =
1 2
[(1 + η)5/3 + (1 − η)5/3 ],
(15)
and
ϑ2D = 1 + η2 ,
(16)
where ⇕ stands for partially polarized systems, that means that part of states contains two particles with opposite spins and other occupied states contain one particle with same spin direction. Considering two electrons one finds that full wave function is anti-symmetric. If one has two electrons with parallel spins one has that wave function is symmetric on spin variables, so it should be anti-symmetric on space variables. In the opposite case of anti-parallel spins one has anti-symmetry of wave function on spin variables and symmetry of wave function on space variables. Considering energy of two electron Coulomb interaction one finds it has two parts: the classic like part C and the exchange part A. For the parallel (anti-parallel) spins one obtains E↑↑ = C − A (E↑↓ = C + A). Parallel (anti-parallel) configuration of spins decreases (increases) energy of the Coulomb interaction. Systems of unpolarized electrons then the average numbers of electrons with different directions of spins equal to each other, we find that the average number of particles with parallel and anti-parallel spins is the same. Consequently we have that average exchange interaction equals to zero. In partly polarized systems the number of particles with different spins are not the same. In this case a contribution of the average exchange interaction appears. At full measure it reveals in fully polarized system then all electrons have same direction of spins. In accordance with the previous discussion we find that exchange interaction, for this configuration, gives attractive contribution in the force field.
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This result allows us to find the force field of the Coulomb exchange interaction
FC ,a(3D) =
3 24/3 q2a
= 6q2a
6
3
π
3√ 3 na ∇ na
π
na ∇ n1a /3 .
(17)
We obtain the force field of exchange interaction for 2D quantum plasmas in the following form: FC ,a(2D) = 23/2
√
2π
24 arsh1
π
2
q2a
√
na ∇ na
√ β √ = 8 π 2 q2a na ∇ na π where we introduce β ≡ 24 arsh1 = 21.153.
(18)
The force fields (17) and (18) are obtained for fully polarized systems of identicalparticles. For partially polarized particles the force fields reappear as FC ,a(3D) = ζ3D q2a FC ,a(2D) = ζ2D
√
2π
√ 2
β q π2 a
√
3 3 3
π
na ∇ na and
na ∇ na , with
ζ3D = (1 + η)4/3 − (1 − η)4/3
(19)
ζ2D = (1 + η)3/2 − (1 − η)3/2 .
(20)
and We should mention that coefficients ζ3D ∼ η and ζ2D ∼ η are proportional to spin polarization. Limit cases of ζ3D and ζ2D are ζ3D (0) = 0, ζ3D (1) = 24/3 , ζ2D (0) = 0 and ζ2D (1) = 23/2 . If we do not apply a magnetic field to systems under consideration we need to have systems, where self-organization of equilibrium state leads to formation of population levels with uncompensated spin, such that it is in ferromagnetic domains. Otherwise we have no contribution of the Coulomb exchange interaction. Force fields of exchange interaction (17) and (18) are potential fields. Thus they do not give contribution in dispersion of transverse waves. They affect longitudinal waves and waves with complex polarization: longitudinal–transverse waves. Consequently electromagnetic waves are not affected by the Coulomb exchange interaction in 3D and 2D mediums in the absence of external fields. Force fields of the Coulomb exchange interaction can be presented as a product of the particle concentration on the gradient of a function when it gives no contribution in equations of the vorticity evolution (see Refs. [51,69–71]). Many-particle quantum hydrodynamic equations can be represented in form of the non-linear Schrödinger equation (NLSE). Let us consider evolution of electrons at motionless ions. We introduce the wave function in the medium defined in terms of hydrodynamic variables
Φ=
√
n exp ı
mS
h¯
,
(21)
where S is the potential of velocity field. Let us mention that the NLSE can be derived for eddy-free motion of electrons. Definition (21) can be √ √applied for three dimensional and low dimensional systems of particles |Φ3D | = n3D , and |Φ2D | = n2D , with [n3D ] = cm−3 , [n2D ] = cm−2 . To derive the NLSE we need to differentiate function (21) with respect to time. After some calculations we find the NLSE for electrons in three dimensional quantum plasmas ıh¯ ∂t Φ3D =
−
h¯ 2 △ 2me
2/3
+ ϑ3D
(3π 2 )2/3 h¯ 2 ne
ıh¯ ∂t Φ2D =
−
h¯ 2 △ 2me
+ ϑ2D
−3
2me
For 2DEGs we obtain
π h¯ 2 me
n−
2β
√
2π
π2
ζ
3
3
π
2 2D qe
ζ3D q2 n1/3 + qe ϕ Φ3D .
√
(22)
n + qe ϕ Φ2D .
(23)
P.A. Andreev / Annals of Physics 350 (2014) 198–210
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Eqs. (22) and (23) contain potential of the electric field ϕ presenting the external and internal electric fields: E = −∇ϕ . Considering dynamics of two or more species we should present the wave functions in the medium for each species and derive NLSEs for each species either. 3. Applications In this section we consider small perturbations of equilibrium state described by nonzero particle concentration n0 , zero velocity field v0 = 0 and electric field E0 = 0. Assuming that perturbations are monochromatic
δn NA δ v = VA e−ıωt +ıkr , δE EA
(24)
we get a set of linear algebraic equations relatively to NA and VA . The condition of existence of nonzero solutions for amplitudes of perturbations gives us a dispersion equation. 3.1. Three dimensional quantum plasmas In classic plasmas the Langmuir waves have the following spectrum: 2 ω2 = ωLe +
γT me
k2 ,
(25)
with the three dimensional Langmuir frequency 2 ωLe ,3D =
4π e2 n0,3D me
,
(26)
T is the temperature, γ is the adiabatic index. We consider quantum plasmas, so we are interested in the low temperature properties, then carriers are degenerated. Hence we have contribution of the Fermi pressure instead of the temperature. Spectrum of the Langmuir waves in 3D quantum plasmas with the Coulomb exchange interaction appears as
ω =ω 2
2 Le,3D
− ζ3D
3
2/3
3 e2 √ (3π 2 )2/3 h¯ 2 n0e 2 h¯ 2 k4 3 k + . n0e k2 + ϑ3D π me 3m2e 4m2e
(27)
The first term in formula (27) is the 3D Langmuir frequency existing due to the Coulomb interaction in the self-consistent field approximation. The second term describes the Coulomb exchange interaction between electrons. The third term appears from the Fermi pressure. The last term describes contribution of the quantum Bohm potential. Let us consider a dimensionless form of the spectrum of 3D Langmuir waves. Introducing dimen√ sionless frequency Ω = ω/ωLe,3D and wave vector ξ = k/ 3 n0e,3D we obtain
Ω 2 = 1 − ζ3D
1
3
4π
3
π
ξ 2 + ϑ3D
(3π 2 )2/3 3
+
1 16π
ξ 2 Λ3D ξ 2 ,
(28)
where we also have a dimensionless parameter
Λ3D =
h¯ 2 √ 3 n0e , me e2
(29)
√
depending on fundamental constants h¯ , e, me , and the equilibrium concentration of electrons 3 n0e,3D . The contribution of the exchange interaction can be considered as a shift of the Fermi pressure since both of them are proportional to ξ 2 . On the other hand, the term describing the exchange interaction
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is proportional to the square of the Langmuir frequency. From this point of view we see that exchange interaction gives a shift of the Langmuir frequency and this shift is proportional to square of the wave vector ξ 2 . So exchange interaction is considerable in the short wave length limit. Increase of the particle concentration allows to increase maximal wave vector of wave propagation in the medium. Hence, in short wave length limit the first two terms reveal same dependence on the equilibrium particle concentration. For whole range of wave vectors we see that the first term ∼ n0e,3D 1/3
grows faster than the second term ∼ n0e,3D with the increase of the particle concentration. The third 2/3 term has an intermediate rate of growth being proportional to n0e,3D . However, in the short wave 2/3
length limit the third and fourth terms increase rather faster like n0e,3D k2 and k4 correspondingly. Finally formula (28) shows that the third and fourth terms can be rather large at high densities (particle concentrations) and large wave vectors. Let us estimate contribution of the exchange interaction in comparison with other terms in dispersion dependence of the Langmuir waves. Comparing exchange interaction with the Fermi pressure we should consider the ratio of the second term to the third term
χEF =
me e2
√
h2 3
¯
=
n0e,3D
1
Λ3D
≈
3
1025 n0e,3D
.
(30)
Exchange interaction prevails over the Fermi pressure than χEF > 1, that corresponds to n0e,3D < 1025 cm−3 . Hence the Coulomb exchange interaction is larger than the Fermi pressure in metals n0e,3D ∼ 1022 cm−3 and semiconductors n0e,3D ∼ 1018 cm−3 . Considering extreme astrophysical objects like white dwarfs n0e,3D ∼ 1028 cm−3 we find that Fermi pressure is larger than the Coulomb exchange interaction. The ratio between the Fermi pressure and the exchange interaction does not depend on the wave vector k. We see that at large concentrations the role of the Coulomb exchange interaction is negligible in comparison with the Fermi Pressure even for fully polarized fermions. At small concentrations the role of the exchange interaction increases, but it never overcomes the self-consistent part of the Coulomb interaction presented by the square of Langmuir frequency. Hence the spectrum of the Langmuir waves is stable. Let us consider this problem closer. It is useful to introduce the average 1 interparticle distance a = √ 3 n . The wavelengths of matter waves λ = 2π /k are larger than the 0
√
average interparticle distance λ > a. Consequently ξ = k/ 3 n0e,3D = ak = 2π a/λ < 1. The second term in formula (28) consists of the product of four multipliers. Each of these multipliers is smaller than 1. Consequently, the second term in formula (28) is always smaller than the first one. We conclude now that the frequency square of three dimensional Langmuir waves is positive. Hence we have a stable spectrum. We have considered high frequency waves. The next step is consideration of the low frequency excitations. Low frequency excitations in unmagnetized electron–ion plasmas are the ion-acoustic waves. We obtain contribution of the Coulomb exchange interaction in their spectrum. The final formula is
2 ζ3D 3 3 3 ωLe ,3D ω3D (k) = kvs,3D 1 − 2/3 2 ϑ3D 4π π n0e,3D vFe ,3D 1
× 1 + (krDe,3D )
2
√
√
1−
ζ3D 3 ϑ3D 4π
3 3
2 ωLe ,3D
,
(31)
π n2/3 v 2 0e,3D Fe,3D
√
where vs,3D = me /mi ϑ3D · vFe,3D /3 is the three dimensional velocity of sound, rDe,3D = ϑ3D vFe,3D /(3ωLe,3D ) is the Debye radius. In formula (31) and similar formulas below we extract contribution of the Fermi pressure. Hence formulas for ion-acoustic waves contain well-known contribution of the pressure multiplied by the factor showing contribution of exchange interaction.
P.A. Andreev / Annals of Physics 350 (2014) 198–210
207
The ion-acoustic wave exists under the following conditions:
v
2 2 Ti k
≪ω ≪ 2
1 3
ϑ v − ζ3D 2 3D Fe
ζ
Consequently, if Z = 1 − ϑ3D 43π 3D
q2 me
3
3√ 3 n0e k2 .
(32)
π
2 ωLe ,3D π n2/3 v 2 0e,3D Fe,3D
3 3
becomes small, then conditions of wave existence
are broken. Hence we cannot obtain formula (31) at small or negative Z . We see that the Coulomb exchange interaction decreases the frequency of the ion-acoustic waves. At large enough contribution of the exchange interaction the condition of the ion-acoustic wave existence is violated. So, the ionacoustic solution disappears. In the long wavelength limit we have
2 ζ3D 3 3 3 ωLe ,3D ω(k) = kvs,3D 1 − . 2/3 2 ϑ3D 4π π n0e,3D vFe ,3D
(33)
In the short wavelength limit we find
ω2 (k) = ωLi2 ,3D .
(34)
From formula (34) we see that the exchange interaction gives no contribution in the ion-acoustic waves in the short wave length limit. Kanazawa et al. present contribution of the Coulomb exchange interaction in the Langmuir wave spectrum in terms of the Green function method (see formula (2.15) of Ref. [5]). Their result coincides with results of Ref. [7]. However their result differs from the spectrum corresponding potential obtained and applied in Refs. [17–19]. Comparing our result given by formula (27) with earlier results presented by (1), (7) we see that there are differences with both of them. We have factor 1/3 in front of the square of Fermi velocity instead of 3/5. This is standard misgiving appearing as a consequence of the use of the Fermi pressure for the equation of state. In terms of describing the exchange interaction contribution we obtain different coefficients in comparison with Refs. [5,7] given by formula (1) 5 ≈ 3.15. And our and [18] given by formula (7). Our coefficient is larger than (1) by the factor √ 3 4
coefficient is smaller than (7) by the factor 1.66. 3.2. Two dimensional quantum plasmas: two dimensional electron gas and ion motion contribution Two dimensional quantum plasmas are systems of electrons and ions being under confinement, so we have plane-like objects. 2D quantum plasmas are surrounded by medium. However, main properties can be obtained considering a 2D layer in empty 3D space. Such objects as 2DEG (two dimensional electron gas) and 2DHG (two dimensional hole gas) are common objects in physics of semiconductors. As application these objects appear as parts of transistors. Consideration of 2DEG and 2DHG corresponds to the description of high frequency excitations. In this section we also include ion dynamics. Spectrum of high frequency longitudinal excitations, which are the Langmuir waves, appears as follows: 2 ω2 = ωLe ,2D − ζ2D
√ π h¯ 2 n0e 2 h¯ 2 k4 β 2π e2 √ n0e k2 + ϑ2D k + , 2 π me m2e 4m2e
(35)
where we have used the two dimensional Langmuir frequency 2 ωLe ,2D =
2π e2 kn0,2D me
∼ k,
which is not a constant, but it is proportional to the wave vector k.
(36)
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Similarly to the three dimensional spectrum (see formula (27)) different terms in formula (35) have the following meaning: self-consistent Coulomb interaction, the exchange Coulomb interaction, the Fermi pressure, and the quantum Bohm potential. Next we discuss some properties of the 2D Langmuir wave spectrum. Let us consider the exchange interaction with the Fermi pressure for 2D quantum plasmas. To this end we introduce the following dimensionless parameter:
χEF ,2D =
me e2
h¯
2
√
≈
n0e,2D
1016 n0e,2D
.
(37)
In 2D semiconductor objects n0e,2D ≪ 1016 cm−2 . Consequently, the exchange interaction plays a significant role in collective properties of semiconductors. To this end we present a dimensionless form of formula (35) introducing dimensionless parameters
√ 3/2 Ω2D = ω m/(2π e2 n0e,2d ) and ξ2D = k/ n0e,2D ∼ ak, with a as the average interparticle distance. Hence we have
β 1 1 2 2 Ω2D = ξ 1 − ζ2D √ ξ + ϑ2D Λ2D ξ 2 1 + ξ , 2 4π 2π π 2
(38)
with
Λ2D =
h¯ 2 √ n0e,2D . me2
(39)
For the 2D Langmuir wave spectrum we can present an analysis similar to the three dimensional case described above (see text after formula (30)). This analysis allows us to conclude that spectrum of the 2D Langmuir waves (35) is stable (ω2 > 0). Spectrum of the 2D ion-acoustic waves in presence of the exchange interaction appears as
√ √ ζ2D 2β 2π e2 n0e,2D ω2D (k) = kvs,2D 1 − 2 ϑ2D π 2 mvFe ,2D 1
×
1 + (krDe,2D (k))2 1 −
√ √ ζ2D 2β 2π e2 n0e,2D 2 ϑ2D π 2 mvFe ,2D
,
(40)
√
√
√
with vs,2D = me /mi ϑ2D · vFe,2D /2 is the two dimensional velocity of sound, rDe,2D = ϑ2D vFe,2D / (2ωLe,2D ). The spectrum of the 2D ion-acoustic waves can be written in a more explicit form, which shows dependence on the wave vector k
√ √ ζ2D 2β 2π e2 n0e,2D ω2D (k) = kvs,2D 1 − × 2 ϑ2D π 2 mvFe ,2D
1
1 + kD 1 −
√ √ ζ2D 2β 2π e2 n0e,2D 2 ϑ2D π 2 mvFe ,2D
,
(41)
where D=
√
2 2π
h¯ e2
√
n0e,2D
.
(42)
In the range of low frequency excitations we have similar change of behavior of 2D and 3D ionacoustic waves due to the exchange interaction account. The Coulomb exchange interaction decreases frequency of excitations. It happens till conditions of the ion-acoustic solution existence are broken. Thus we do not have 2D ion-acoustic waves at large exchange interaction.
P.A. Andreev / Annals of Physics 350 (2014) 198–210
209
In the long wavelength limit we have
√ √ ζ2D 2β 2π e2 n0e,2D ω(k) = kvs,2D 1 − . 2 ϑ2D π 2 mvFe ,2D
(43)
In the short wavelength limit we find
ω2 (k) = ωLi2 ,2D ∼ k.
(44)
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