Exchange Effects in Magnetized Quantum Plasmas

Exchange eﬀects in magnetized quantum plasmas

Mariya Iv. Trukhanova∗ and Pavel A. Andreev† Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation. (Dated: July 7, 2021) We apply the many-particle quantum hydrodynamics including the Coulomb exchange interaction to magnetized quantum plasmas. We consider a number of wave phenomenon under influence of the Coulomb exchange interaction. Since the Coulomb exchange interaction affects longitudinal and transverse-longitudinal waves we focus our attention to the Langmuir waves, Trivelpiece-Gould waves, ion-acoustic waves in non-isothermal magnetized plasmas, the dispersion of the longitudinal low-frequency ion-acoustic waves and low-frequencies electromagnetic waves at Te Ti . We obtained the numerical simulation of the dispersion properties of different types of waves.

PACS numbers: 52.30.Ex, 52.35.Dm Keywords: quantum plasmas, exchange interaction, ion-acoustic waves, quantum hydrodynamics

I. INTRODUCTION beam through the plasma had been showed. The extended vorticity evolution equation for the The many-particle quantum hydrodynamics quantum spinning plasma had been derived and the ef- (MPQHD) method had been developed for the sys- fects of new spin forces and spin-spin interaction contri- tems of charged or neutral particles in [1]. General butions on the motion of fermions, evolution of the mag- form of the quantum exchange correlations was derived netic moment density and vorticity generation had been there. The MPQHD for three dimensional spin 1/2 predicted [16]. The spin-orbital corrections to the prop- quantum plasmas with the Coulomb exchange and the agation of the whistler waves in a astrophysical quantum spin-spin exchange interactions was obtained in 2001 in magnetoplasma composed by mobile ions and electrons Ref. [2]. Theory of ultracold quantum gases of neutral had been predicted in [17]. The hydrodynamic model atoms, including derivation and generalization of the including the spin degree of freedom and the electromag- Gross-Pitaevskii equation, was constructed in terms of netic field had been discussed in [18]. The quantum hy- the MPQHD in 2008 [3]. Further development of the drodynamics for the research of many-particles systems MPQHD with exchange interaction for low dimensional had been developed in [3], [19], [20]. and three dimensional Coulomb quantum plasmas has The Coulomb exchange interactions are of great im- been recently performed in Ref. [4]. portance in many systems as well as for magnetic phe- nomena, and have no classical analogy. The Coulomb A quantum mechanics description for systems of N exchange effects had been included in a QHD picture [4] interacting particles is based upon the many-particle for the Coulomb quantum plasmas. To do this, the fun- Schrodinger equation (MPSE) that specifies a wave func- damental equations that determine the dynamics of function in a 3N-dimensional configuration space. As wave tions of three variables, starting from MPSE had been de- processes, processes of information transfer, and other rived. This problem has been solved with the creation of spin transport processes occur in 3D physical space, it a many-particle quantum hydrodynamics method. The becomes necessary to turn to a mathematical method of contribution of the exchange interaction in the dispersion physically observable values that are determined in a 3D of the Langmuir [4] and ion-acoustic waves for three and physical space. The fundamental equations of the mi- two dimensional quantum plasmas had been shown. It croscopic quantum hydrodynamics of fermions in an ex- had been derived that the exchange interaction between ternal electromagnetic field had been derived using the particles with same spin direction and particles with op- many-particle Schrodinger equation [2], [5], [6]. Recently posite spin directions are different. Recently, the kinetic there has been an increased interest in the properties of plasma model containing fermion exchange effects were quantum plasmas [7] - [12]. Dispersion relations for lin- investigated in [21] and the influence of exchange effect on ear waves in amedium formed by electrons and ions and low frequency dynamics, in particular ion acoustic waves arXiv:1405.6294v1 [physics.plasm-ph] 24 May 2014 traversed by a beam of neutrons whose velocity has a was predicted. The generalization of the Vlasov equa- nonzero constant component had been derived by meth- tion to include exchange effects was presented allowing ods of quantum hydrodynamics in [13]. The dispersion for electromagnetic mean fields and the correction to clas- of waves, existed in the plasma in consequence of dy- sical Langmuir waves in plasmas was found in [22]. namic of the magnetic moments had been investigated in [14], [15]. The instabilities at propagation of the neutron The influence of electron-exchange and quantum screening on the collisional entanglement fidelity for the elastic electronion collision was investigated [23]. The ef- fective Shukla− Eliasson potential and the partial wave ∗Electronic address: [email protected] method had been used to obtain the collisional entangle- †Electronic address: [email protected] ment fidelity in quantum plasmas as a function of the 2

r electron-exchange parameter, Fermi energy, plasmon en- 1 3 √ = q n E +E + [v , B ] +24/3q2 3 3 n ∇n . ergy and collision energy. The electron-exchange effects i i ext int c i ext i π i i on the charge capture process had been investigated in (4) degenerate quantum plasmas [24]. It had been showed The second terms on the left sides of Eq. (3) and (4) that the electron-exchange effect enhances the charge are the gradient of the thermal pressure or the Fermi capture radius and the charge capture cross section in pressure for degenerate electrons and ions. It appears as semiconductor quantum plasmas. the thermal part of the momentum flux related to distri- This paper is organized as follows. In Sec. (II) bution of particles on states with different momentum. we present the quantum hydrodynamics equations taken The third terms are the quantum Bohm potential ap- into account the Coulomb exchange interactions in a self- pearing as the quantum part of the momentum flux. In consistent field approximation. In Sec. (III) we pre- the right-hand sides of the Euler equations we present sented the dispersion characteristics of high frequency interparticle interaction and interaction of particles with Langmuir waves and low frequency waves with the ac- external electromagnetic fields. The first group of terms count of Coulomb exchange interactions. In Sect. (IV) in the right-hand side of the Euler equations describe in- we show the influence of exchange interactions on the teraction with the external electromagnetic fields. The dispersion of ion-acoustic waves in non-isothermal mag- force fields of Coulomb exchange interaction of electrons netized plasmas. In Sec. (V) we show that exchange and positrons (the second terms on the right side of (3) interactions leads to the existence of a dispersion char- and (4)) are obtained for fully polarized systems of iden- acteristics of the Trivelpiece-Gould waves. In Sec. (VI) tical particles. For polarized systems of electrons or ions and Sec. (VII) we show the influence of exchange interac- 2/3 2 5/3 equations of state appears pa,3D↑↑ = 2 ¯h na,3D/5ma tions on the dispersion of the longitudinal low-frequency for 3D mediums, where subindex ↑↑ means that all par- ion-acoustic waves and low-frequencies electromagnetic ticles have same spin direction. The ratio of polariz- waves at Te Ti in the magnetized plasma. ability η = |n↑−n↓| , with indexes ↑ and ↓ means par- n↑+n↓ ticles with spin up and spin down. In general case of partially polarized system of particles we can write II. MODEL 2 2/3 2 5/3 pa,3Dm = ϑ3D(3π ) ¯h na,3D/(5ma) for partially polarized systems, that means that part of states contain two We apply the equations for the system of charged par- particle with opposite spins and other occupied states ticles in the external magnetic field [4]. For a 3D system contain one particle with same spin direction [4] of particles the continuity equations and the momentum balance equations for electrons and ions may be written 1 ϑ = [(1 + η)5/3 + (1 − η)5/3], (5) down in terms of electrical intensity of the field that is 3D 2 created by charges of the particle system. Thus the continuity equations for ions and electrons are For partially polarized particles the force fields reap- q √ 2 3 3 3 pear as FC,a(3D) = ζ3Dqa π na∇na, with

∂ n + ∇(n v ) = 0, (1) 4/3 4/3 t e e e ζ3D = (1 + η) − (1 − η) (6) and We should mention that coefficient ζ3D ∼ η is proportional to spin polarization. Limit cases of ζ3D are 4/3 ∂tni + ∇(nivi) = 0. (2) ζ3D(0) = 0, ζ3D(1) = 2 . Considering two electrons one finds that full wave func- The equation of motion for electrons is tion 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 2 2 ! ¯h 4ne (∇ne) variables. In opposite case of anti-parallel spins one has mene(∂t + ve∇)ve + ∇pe − ne∇ − 2 4me ne 2ne anti-symmetry of wave function on spin variables and symmetry of wave function on space variables. Systems of unpolarized electrons then average numbers of electrons with different direction of spins equal to 1 r 3 √ 4/3 2 3 3 = qene Eext +Eint + [ve, Bext] +2 qe ne∇ne, each other, we find that average number of particles for c π a chosen with parallel and anti-parallel spins is the same. (3) Consequently we have that average exchange interaction and the Euler equation for ions equals to zero. In partly polarized systems the numbers of particles with different spin are not the same. In this case a contribution of the average exchange interaction 2 2 ! ¯h 4ni (∇ni) appears. At full measure it reveals in fully polarized sys- mini(∂t + vi∇)vi + ∇pi − ni∇ − 2 4mi ni 2ni tem then all electrons have same direction of spins. In 3 accordance with the previous discussion we find that ex- where the dispersion of quantum Langmuir waves with change interaction, for this configuration, gives attractive the account of Coulomb exchange interactions (9) is pre- contribution in the force field. sented at Fig. (1). In general case of of partially polarized system of particles with spin up and spin down, the dispersion of quantum Langmuir waves as a function of III. APPLICATIONS ratio of polarizability with account of Coulomb exchange interactions (9) is presented at Fig. (2). In this section we consider small perturbations of equi- We see that the first term in Eq. (9) is proportional to librium state describing by nonzero particle concentra- the electron equilibrium concentration ∼ n0e and grows 1/3 tion n0, and zero velocity field v0 = 0 and electric field faster then the second term ∼ n0e . The third term E0 = 0. has an intermediate rate of grow being proportional to 2/3 Assuming that perturbations are monochromatic ∼ n0e . The Coulomb exchange interaction is larger than 24 −3 the Fermi pressure when ∼ n0e ≤ 10 cm , this sit-  δne   NAe  uation is realized in metals and semiconductors, but  δni   NAi  in the astrophysical objects like white drafts ∼ n0e ≤  δv   V  28 −3  e  =  Ae  e−ıωt+ıkr, (7) 10 cm , the Fermi pressure is larger than the Coulomb  δv   V   i   Ai  exchange interaction.  δE   EA  We have considered high frequency waves. Nest step δB BA is consideration of the low frequency excitations v we get a set of linear algebraic equations relatively to NA u r 2 ζ3D 3 3 3 ωLe,3D and VA. Condition of existence of nonzero solutions for ω (k) = kv u1 − × 3D s,3Dt 2/3 ϑ3D 4π π 2 amplitudes of perturbations gives us a dispersion equa- n0e,3DvF e,3D tion in the form of

1.8 ´ 1015 1 × s , (10) q ω2 2 ζ3D 3 3 3 Le,3D 1 + (krDe,3D) 1 − 2/3 15 ϑ3D 4π π 2 1.6 ´ 10 n0e,3D vF e,3D L s √ 15 1 p 1.4 ´ 10 H where vs,3D = me/mi ϑ3D · vF e,3D/3 is the Ω √three dimensional velocity of sound, rDe,3D = 1.2 ´ 1015 ϑ3DvF e,3D/(3ωLe,3D) is the Debye radius. In formula (9) and similar formulas below we extract contribution 15 of the Fermi pressure. Hence formulas for ion-acoustic 1.0 ´ 10 waves contains well-known contribution of the pressure 6 7 7 7 0 5.0 ´ 10 1.0 ´ 10 1.5 ´ 10 2.0 ´ 10 multiplied by factor showing contribution of exchange in- k 1 sm teraction.

FIG. 1: The ﬁgure shows the dispersionH L characteristic of the quantum Langmuir wave frequency ω versus the wave vector k, which is described by the equation (9). The green branch of dispersion is presented the classical high frequency Langmuir wave, the red and blue branches characterize the Coulomb exchange interactions and quantum Bohm potential inﬂuence, 21 −3 where n0e ' 10 cm , η = 1.

2 2 4πe n0,3D ωLe,3D = . (8) me

r 3 e2 √ 2 2 3 3 2 FIG. 2: The ﬁgure shows the dispersion characteristic of the ω = ωLe,3D − ζ3D n0ek π me quantum Langmuir wave frequency ω as a function of the wave vector k and the ratio of polarizability η, which is described by the equation (9). The Coulomb exchange interactions and 2 2/3 2 2/3 2 4 (3π ) ¯h n0e 2 ¯h k quantum Bohm potential are taken into account and n0e ' 21 −3 + ϑ3D 2 k + 2 , (9) 10 cm . 3me 4me 4

In the long wavelength limit we have 1.2 ´ 109 v u ζ 3 r 3 ω2 ω(k) = kv u1 − 3D 3 Le,3D . (11) 1.0 ´ 109 s,3Dt 2/3 ϑ3D 4π π 2 n0e,3DvF e,3D 8.0 ´ 108 L s In the short wavelength limit we ﬁnd 1 6.0 ´ 108 H

2 2 Ω ω (k) = ωLi,3D. (12) 4.0 ´ 108

IV. ION-ACOUSTIC WAVES IN 2.0 ´ 108 NON-ISOTHERMAL MAGNETIZED PLASMAS 0 0 10 000 20 000 30 000 40 000 50 000 60 000 k 1 sm 8 ´ 108

FIG. 4: The ﬁgure shows the dispersionH L characteristic of the long wavelength ion-acoustic wave frequency ω versus the wave 6 ´ 108 vector k, which is described by the equation (15). The red

L mode shows the dispersion characteristic of the long wave- s

8 1 length ion-acoustic wave, where the thermal speed is defined 4 ´ 10 H by the Fermi pressure and the blue mode presents the Coulomb Ω exchange interactions influence for η = 1, the green mode for 8 η = 0.5 and the black mode for η = 0.2. System parameters 2 ´ 10 5 are assumed to be as follows: B0 ' 5 · 10 G - the uniform 18 −3 magnetic field, n0 ' 10 cm - the equilibrium density. 0 0 2000 4000 6000 8000 10 000 k 1 sm 1 ´ 1012 FIG. 3: The figure shows the dispersion characteristic of the H L long wavelength ion-acoustic wave frequency ω versus the wave 8 ´ 1011 vector k, which is described by the equation (14). The red L mode shows the dispersion characteristic of the long wave- 11 s 6 ´ 10 1

length ion-acoustic wave, where the thermal speed is deﬁned H

by the Fermi pressure, the orange mode presents the Coulomb Ω exchange interactions influence for η = 1, the green mode for 4 ´ 1011 the case of partially polarized system η = 0.8, the black mode for η = 0.5 and the blue mode presents the Coulomb exchange 2 ´ 1011 interactions influence for η = 0.2. System parameters are as- 5 sumed to be as follows: B0 ' 5·10 G - the uniform magnetic 18 −3 0 field, n0 ' 10 cm - the equilibrium density. 0 5.0 ´ 106 1.0 ´ 107 1.5 ´ 107 2.0 ´ 107 k 1 sm Lets discuss the ion-acoustic waves in non-isothermal 2 2 FIG. 5: The figure shows the dispersion characteristic of the magnetized plasmas with Te >> Ti ≈ 0 at Ωe ωLi H L 2 short wavelength quantum Langmuir wave frequency ω versus Ωi [25]. In these conditions the ion-acoustic waves exist giving two branches of wave dispersion. One branch is at the wave vector k, which is described by the equation (9). The green mode shows the dispersion characteristic of quantum ω < Ωi and another one at ω > Ωi. ion-acoustic waves which are described by equation (14), the Corresponding dispersion equation appears as red mode shows the dispersion characteristic of classic ion- k2 ω2 k2ω2 ω2 acoustic waves which occur from (14) if the term proportional ⊥ Li z Li Le to ¯h2 is neglected. The blue mode shows the dispersion char- 1 − 2 2 2 − 2 2 + 2 2 = 0, (13) k (ω − Ωi ) k ω k vT e acteristic of the short wavelength wave where the Coulomb 2 exchange interactions are dominate. where vT e - is the thermal electron velocity. Using the 2 2 approximation ωLi Ωi and θ 6= 0 we find ω2 + Ω2(1 + ω2 /k2v2) 2 Li i Li s where ω+ = 2 2 2 , (14) 1 + ωLi/k vs and ω2 Ω2 cos2 θ e2 r 3 √ 2 Li i 2 3 3 ω− = 2 2 2 2 2 , (15) υs = −ξ3D n0e ωLi + Ωi (1 + ωLi/k vs ) mi π 5

2 2/3 2 2/3 2 2 (3π ) ¯h noe ¯h k VI. THE LONGITUDINAL LOW-FREQUENCY + ϑ3D + . (16) ION-ACOUSTIC WAVES IN MAGNETIZED 3memi 4memi PLASMAS contains the Coulomb exchange interaction between electrons (the first term). The second term appears from the Fermi pressure. and the third term describes contribu- In absence of external magnetic field longitudinal oscil- tion of the quantum Bohm potential lations exists in electron-ion plasmas with hot electrons Solutions of the equation (13) for the different ratio and cold ions [25]. They are weakly damping ion-acoustic of polarizability are presented by two branches of the oscillations [27]. Dispersion of these waves is dispersion at Fig. (3), (4) and (5). kvs When the distribution of the wave is parallel to the ωs(k) = , (22) p1 + k2r2 background magnetic field θ = 0 the ion-acoustic modes D have the form of p 2 where rD = Te/4πe n0 is the Debay radius, vs = 2 2 p ω = Ωi (17) Te/mi. This solution corresponds to phase velocities of waves, which are intermediate in compare with other pa- and rameters of plasmas vi ω/k ve with the thermal ve- 2 r 2 2/3 2 2/3 2 2 ! e 3 √ (3π ) ¯h noe ¯h k p p 2 3 3 locities of electrons ve = Te/me and ions vi = Ti/mi. ω = −ξ3D n0e+ϑ3D + × mi π 3memi 4memi If equilibrium state of a medium reveals distribution (18) of electrons with non-zero magnetization, as it happens in ferromagnetic domains, the Coulomb exchange inter- ω2 k2 action gives considerable contribution in spectrum of the × Li longitudinal waves [4]. q 2 2/3 2 2/3 2 2 2 e2 3 3 √ (3π ) h¯ noe h¯ k2 ω + k (−ξ3D 3 n0e + ϑ3D + ) It is well-known that an external magnetic field af- Li mi π 3memi 4memi fects the ion-acoustic waves at ωs(k) ≤ Ωi and kρi ≤ 1. Presence of external magnetic field creates or increases V. THE TRIVELPIECE-GOULD WAVES difference in occupation of spin-up and spin-down states. Consequently, account of the Coulomb exchange interac- Trivelpiece-Gould wave [25], [26] is a longitudinal wave tion in magnetized plasmas is even more important than appearing along with the Langmuir wave at propagation in plasmas with no external magnetic field. Under conditions v | ω/k | v we find the of waves at an angle to external magnetic field kx 6= 0 i k e following dispersion equation for the longitudinal low- and kz 6= 0. It is a low-frequency wave with ω | Ωe |. 2 2 2 frequency ion-acoustic waves It exists at long wavelengths limit 1, kz vT e ω . All of these conditions give the following dispersion equation 2 2 2 2 2 2 2 2 2 ω k ω k v ωLe ωLi ωLi 2 ωLi 2 1 − Le z + Le x T e = 0 (19) 1 + + − cos θ − sin θ = 0, (23) 2 2 2 2 2 Ω2 k2v2 ω2 ω2 − Ω2 ω k k vT e Ωe e s i

qaB0 Dispersion dependence of the Trivelpiece-Gould wave where Ωa = - are the the electron or ion cyclotron mac in the classical magnetic field appears as [26] frequency. Increasing of the Coulomb exchange interaction in 2 2 2 ωLe cos θ compare with the Fermi pressure can break condition of ω = 2 2 . (20) ωLe sin θ the ion-acoustic wave existence vi | ω/kk | ve. 1 + 2 Ωe Getting into account the fact that in the problem un- It looks like thermal velocity does not affect this spec- der consideration the second term in equation (23) much trum. If it is correct we can mention that the exchange in- smaller than the third term we find next solution teraction does not influence it either. In the quantum ex- 1 1q ω2(k, θ) = (ω2 + Ω2) ± (ω2 + Ω2)2 − 4ω2Ω2 cos2 θ. ternal magnetic field the thermal velocity effect might be 2 s i 2 s i s i important. This corresponds to a regime of very strong (24) magnetic field in which the external field strength ap- We have also neglected small anisotropic terms. proaches or exceeds the quantum critical magnetic field, In presence of an external magnetic field, there are two 13 B0 ≥ 4.4·10 G. Evidently, sinceh ¯Ωe >> Te there must longitudinal low-frequency oscillations instead of the ion- exist the frequency acoustic wave. At k → 0 we find from formula (24) 2 2 meΩe tan θ ω = × (21) ω(k, θ) = kvs cos θ, (25) ¯h and 2 r 2 2/3 2 2/3 2 2 ! e 3 √ (3π ) ¯h noe ¯h k 3 3 2 2 2 × −ξ3D n0e + ϑ3D 2 + 2 . k vs sin θ me π 3me 4me ω(k, θ) = Ωi 1 + 2 . (26) 2Ωi 6

The condition for instability is thus that the nega- 2 11 tive term of vs dominates over all the others. When 2 ´ 10 the Coulomb exchange interactions is larger than Fermi 23 −3 pressure n0 ≤ 10 sm , the solution (25) is instability. 0 The solution (26) is presented at Fig. (6) as a function 11 2 of η. -2 ´ 10 N In opposite limit of small wavelengths k → ∞, or in 11 other terms krD 1, from formula (24) we obtain -4 ´ 10 1 1q 11 ω2 (θ) = (ω2 + Ω2) ± (ω2 + Ω2)2 − 4ω2 Ω2 cos2 θ. -6 ´ 10 ± 2 Li i 2 Li i Li i (27) 0 2 ´ 106 4 ´ 106 6 ´ 106 8 ´ 106 1 ´ 107 2 2 Formula (27) is obtained at k ρi 1. It can be ap- Ω 1 s 2 Ti ωLi plied at T Ω2 1. Under condition ωLi Ωi formula e i FIG. 7: The ﬁgure shows the corresponding classical re- H L (27) does not work. Formula (24) can be applied in the fractive index (30) in the limit of the long wavelength limit long wavelength limit krD 1 if condition ωLi Ωi is krD 1, where the Coulomb exchange interactions are dom- 2 2 satisﬁed. inate γex > υF e. System parameters are assumed to be as 15 −3 3 7 follows: n0 ' 10 sm , B0 = 5 · 10 G. The blue branch is 8 ´ 10 2 2 the index c /γex. 6 ´ 107 2.5 ´ 108 4 ´ 107 L s 8 1 2 ´ 107 H 2.0 ´ 10 Ω L

8 s

0 1.5 ´ 10 1 H

-2 ´ 107 1.0 ´ 108 Ω

0 200 400 600 800 5.0 ´ 107 k 1 sm

0 FIG. 6: The ﬁgure shows the dispersionH L characteristic of the 0 100 200 300 400 low-frequency oscillations in the long wavelengths k → ∞ k 1 sm 18 −3 limit, which is described by the equation (26), n0 ' 10 sm , 4 B0 = 5 · 10 G. The red branch characterizes the total polar- FIG. 8: The ﬁgure shows the dispersionH L characteristic of the ized system η = 1, the blue mode η = 0.5 and green mode slow and fast magneto-sonic waves, which is described by the η = 0.2 present the dispersion properties of partially polarized equation (24) in the long wavelength limit krD 1, n0 ' systems. 23 −3 4 10 sm , B0 = 5 · 10 G. The red branch characterizes the dispersion of total polarized system η = 1, the blue mode η = 0.3 and green mode η = 0.2 present the dispersion properties Under conditions kvs Ωi and krD 1, we obtain the following solutions from formula (24) of partially polarized systems.

ω(k) = kvs, (28) Using the deﬁnition (16) the Coulomb exchange pres- and sure can be important for the slow and fast magneto-sonic wave, see Fig. (8). ω(k, θ) = Ωi cos θ. (29) The Coulomb exchange interaction in (28) is larger 25 −3 VII. THE LOW-FREQUENCY than the Fermi pressure when ∼ n0e ≤ 10 cm , this situation is realized in metals and semiconductors, but ELECTROMAGNETIC OSCILLATIONS IN THE MAGNETIZED PLASMA in the astrophysical objects like white drafts ∼ n0e ≤ 1028cm−3, the Fermi pressure is larger than the Coulomb exchange interaction. Here we discuss low-frequencies electromagnetic waves at T T under inﬂuence of the Coulomb exchange At k → 0 (kvs Ωi) larger of solutions (24) presented e i interaction. by formula (28) getting to Ωi. We can also present corresponding refractive index Dispersion equation existing in the case under consideration is rather huge. Thus we do not present it here. 2 2 2 2 c ω − Ωi Nevertheless, we present description of limit cases. In low N = 2 2 2 2 . (30) υs ω − Ωi cos θ frequency limit, it corresponds to the long wavelength 7 limit k → 0, we have the dispersion of the slow and fast magneto-sonic waves

ퟕ 푩ퟎ = ퟓ ∙ ퟏퟎ 푮 ퟐ 흊푨

ퟒ ퟐ 푩ퟎ = ퟓ ∙ ퟏퟎ 푮 흊푨

푩ퟎ = ퟏퟎ푮

ퟐ 흊푭풆

ퟐ 휸풆풙

FIG. 9: A figure illustrating regions of importance in parameter space for various quantum plasma effects of the longitudinal wave propagating perpendicular to the external magnetic field (36). The orange, green and black branches describe the Alfven regime.

ω = kvA cos θ (31) For the wave propagating parallel to the external magnetic ﬁeld cos θ ' 0 the dispersion low (32) has the form and ( vA, ω = kv±, (32) ω± = k (35) vs. where the Alfven velocity is Lets consider the longitudinal wave propagating per- Ωi B0 pendicular to the external magnetic ﬁeld. The dispersion vA = c = √ (33) ωLi 4πn0mi low (32) takes the form and e2 r 3 √ 1 1q ω2 = k2 υ2 + υ2 − ξ 3 3 n (36) v2 = (v2 + v2) ± (v2 + v2)2 − 4v2 v2 cos2 θ (34) A i 3D m π 0e ± 2 A s 2 A s A s i 8

2 2/3 2 2/3 2 2 ! " ! # (3π ) ¯h noe ¯h k 2 q √ 2 2 e 3 3 3 h¯ k 2 2 2 +ϑ3D + , " 2 −ξ3D m π n0e + 4m m + sin θvA Ωi # 3memi 4memi i e i × 1− ! , 2 q √ 2 2 2 2 e 3 3 3 h¯ k 1 2k vA −ξ3D m π n0e + 4m m ϑ = [(1 + η)5/3 + (1 − η)5/3], (37) i e i 3D 2

where NA = c/vA, Ns = c/vs. The dispersion of quan- 4/3 4/3 ξ3D = [(1 + η) − (1 − η) ]. tum slow magneto-sonic waves in the small wavelength limit k → ∞ is presented at Fig. (11). The figure (9) shows regions of importance in parameter space for various quantum plasma effects η = 1. 5.6 ´ 107 2 2/3 The Fermi-pressure ∼ υF e ∼ noe becomes important when the Fermi temperature approaches the thermo- dynamic temperature, when the plasma concentration 5.6 ´ 107 25 −3 n0 ≥ 10 cm . The Alfven mode described by the first

2 L term υ on the right side of (36). The eﬀects due to the c A 7

5.6 ´ 10 1 magnetic pressure depend on the magnetic ﬁeld strength H

2 −1 Ω υA ∼ n0 . The magnetic pressure effects can be impor- 20 −3 tant in regimes of n0 ≤ 10 cm and external magnetic 5.6 ´ 107 4 field B0 ' 5 · 10 G. The quantum regime correspond to lower temperatures. The Coulomb exchange interactions 2 ´ 7 proportional to ∼ γex ∼ n0e can be important in regimes 5.6 10 20 24 −3 6 6 6 6 7 of 10 ≤ n0 ≤ 10 cm . The Coulomb exchange force 0 2 ´ 10 4 ´ 10 6 ´ 10 8 ´ 10 1 ´ 10 does not provide a stabilizing mechanism. The exchange k 1 cm q √ pressure ∼ γ2 = ξ 3 3 3 n e2/m is the negative pres- ex 3D π 0e i FIG. 11: The figure shows the dispersionH L characteristic of the sure term and therefore the source of the instability. But slow magneto-sonic waves, which is described by the equation 7 for the high magnitude magnetic field B0 ∼ 10 G, the (39) in the small wavelength limit k → ∞, where the exchange instability is stabilized. effects are taken into account. The red branch of dispersion represents the classical low, the blue branch include the dispersion characteristic of quantum slow magneto-sonic waves which occurs due to Bohm quantum potential and the green mode takes account of Coulomb exchange interactions.

VIII. CONCLUSIONS

We have briefly described quantum hydrodynamic model for the magnetized quantum plasmas. In our work we consider the electron-electron and ion-ion Coulomb exchange interactions. Using QHD equations with FIG. 10: The 3D figure shows the dispersion characteristic of Coulomb exchange force field we analyzed elementary ex- the magneto-sonic waves, which is described by the equation citations in various physical systems in a linear approxi- (36), where the exchange effects are taken into account. The mation. We investigated the dispersion properties of the 2 3D figure represents the wave frequency ω (k, η) as a function ion-acoustic waves in non-isothermal magnetized plas- of the wave vector k and the ratio of polarizability η. The mas, the Trivelpiece-Gould waves, the longitudinal low- Coulomb exchange interactions and quantum Bohm potential are taken into account and n ' 1023cm−3. frequency ion-acoustic waves, the magneto-sonic waves, 0e high-frequency and low-frequency electron sound tracing contribution of the exchange interaction. We described Let us consider the small wavelength limit k → ∞. In the different regimes and showed that the exchanges in- this regime the refractive index and frequency of smallest teractions can lead to instability. of three solutions appear as

2 2 2 2 2 Ωi N = (2NA + Ns sin θ) 2 2 2 , (38) Ωi cos θ − ω Acknowledgments and The authors thank Professor L. S. Kuz’menkov for ω(k, θ) = Ωi cos θ× (39) fruitful discussions. 9

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