Quantum kinetic relativistic theory of linearized
waves in magnetized plasmas
Haidar Al-Naseri ∗ August 2, 2018
Abstract
In this work we have studied linear wave propagation in magnetized plasmas using a fully relativistic kinetic equation of spin-1/2 particles in the long scale approximation. The linearized kinetic equation is very long and complicated, hence we worked with restricted geometries in order to simplify the calculations. The dispersion relation of the rela-tivistic model was calculated and compared with a dispersion relation from a previous work at the semi-relativistic limit.
Moreover, a new mode was discovered that survives in the zero temperature limit. The origin of the mode in the kinetic equation was discussed and derived from a non-relativistic kinetic equation from a previous work.
Contents
1 Introduction 2 2 Vlasov equation 2 3 Linearized theory 4 3.1 Longitudinal waves . . . 6 3.1.1 Semi-relativistic limit . . . 11 3.2 Transverse waves . . . 113.2.1 Long scale limit . . . 15
3.2.2 Non-relativistic limit . . . 17
3.2.3 Short analysis . . . 17
4 Summary 17
∗
1
Introduction
Like solids, liquids and gases, plasma is a state of matter consisting of ions, electrons and neutral atoms. Plasmas can be found in stars, fusion reactors and lightning [1]. At high temperatures and low densities of plasmas, plasmas can be described using classical mechanics. However for high densities and/or low temperatures of the plasmas, the quantum mechanical effects become more important [2].
The interest in quantum plasmas has increased lately due to the the different applications it offers, like quantum wells and astrophysics [3]. Re-cently, a fully relativistic kinetic theory describing spin-1/2 particles has been developed [4], which will be studied in this work. This model is based on the Dirac equation and hence it takes into account the magnetic dipole moment of the spin-1/2 particles. The model uses the Foldy-Wouthuysen transformation [5] that separates particles and anti-particles using an ex-pansion in ¯h. Moreover the model takes into account all orders of v/c, i.e. it is fully relativistic, compared to the semi-relativistic model [6]. However the model is only valid in the long-scale limit and hence it dose not include any particle-dispersive effects.
The aim of this work is to analyze the linear wave propagation in magne-tized plasmas using the fully relativistic model mentioned above. Since this model was recently published, it might offer some new interesting results that have not been discovered yet. However the fully relativistic kinetic equation is very long, thus we consider in this work two restricted geometries in order to simplify things. Firstly, we consider a longitudinal wave propagation par-allel to an external static homogeneous magnetic field. In the second case, we study transverse waves propagating parallel to the magnetic field. In both cases, we calculate the dispersion relation. In the first case we compare the obtained dispersion relation with a dispersion relation from a previous work [6] in the semi-relativistic limit. In the second case, a new mode was discov-ered which survives in the zero temperature limit. However the new mode comes from a non-relativistic term in the fully relativistic kinetic equation. We derive then the new mode from a non-relativistic dispersion relation in a previous work [3].
2
Vlasov equation
The distribution function of a plasma fs in the classical regime for species s is described by the Vlasov equation [2] (collisions are neglected)
∂fs ∂t + v · ∇xfs+ qs ms h E(x, t) + v × B(x, t)i· ∇vfs= 0, (1)
where qs and ms are the charge and mass of the species s respectively.
derivative of fs along the path of one of the particles is zero (collisions are neglected), this is related to the particle conservation [2]. Using the chain rule, the Vlasov equation is given by
d dtfs(x, v, t) = ∂fs ∂t + ∂fs ∂x · dx dt + ∂fs ∂v · dv dt = 0. (2) Since dx/dt = v and dv/dt = (qs/ms) h E(x, t) + v × B(x, t) i is the Lorentz force divided by the mass, hence we have the Vlasov equation.
The classical relativistic Vlasov equation is obtained by using fs= fs(x, p, t),
where p = mγv is the relativistic momentum, here γ = 1/p1 − v2/c2.
Equa-tion (2) in this regime gives the classical relativistic Vlasov [7] dfs
dt = ∂fs
∂t + v · ∇xfs+ qs(E + v × B) · ∇pfs= 0. (3) Including the quantum mechanical effects, the distribution function now de-pends on the spin f = f (x, v, s, t). We drop the subscript s in f later on in the rapport to simplify the expressions. In the long scale-limit, the non-relativistic quantum mechanical evolution equation for the distribution function f (r, v, s, t) is [8] ∂f ∂t+v·∇xf + " q m E + v × B+µB m∇x s· B + B· ∇s # ·∇vf + 2µB ¯ h s×B·∇sf = 0, (4)
where µB= q¯h/2m is the magnetic moment. Note that we use units where
c = 1 in this work. The new terms that arise in the kinetic equation when the quantum effects are considered are
µB m∇x s · B + B · ∇s 2µB ¯ h s × B,
the first term is the magnetic dipole force while the second one is the spin precession. Going one step further by considering the quantum relativistic kinetic equation, we consider in this work the fully relativistic evolution equation for the distribution function f (x, p, s, t) in the long scale-length limit [4] ∂f ∂t + ( p − µBm∇p " 1 B − p × E + m ! · (s + ∇s) #) · ∇xf + q E + p − µBm∇p 1 B − p × E + m · (s + ∇s) × B ! · ∇pf +µBm ∇x B −p × E) + m · (s + ∇s) ·∇pf + 2µBm ¯ h s × B − p × E + m ·∇sf, (5)
where =pm2+ p2. Equation (5) is a Vlasov-like equation where the
velocity is non-trivially related to the momentum variable, the whole curly bracket in the second term of the equation above is the velocity. The evolu-tion equaevolu-tion above gives us informaevolu-tion about how the particles move in the fields. However, in order to have a closed description we need information about how the motion of the particles are connected to the fields, this is obtained by using Maxwell’s equations
∇ · E = ρf − ∇ · P (6) ∇ · B = 0 (7) ∇ × E = −∂B ∂t (8) ∇ × B = Jf + ∂P ∂t + ∇ × M, (9) where P and M are the polarization and magnetization and Jf and ρf are the free current and charge densities. For the total charge density ρ, we have the following expression
ρ = q Z
dΩf − ∇ · P, (10) where the polarization P is
P = −3µB Z
dΩ ms × p
( + m)f. (11) The total current density is J = Jf + Jp + JM, where Jf is the free
cur-rent density and Jp and JM are the polarization and magnetization current densities Jf = 3q Z dΩ p − µBm∇p 1 B −p × E + m · s f (12) Jp = ∂P ∂t (13) JM = ∇ × M = 3µB∇ × Z dΩm sf. (14)
3
Linearized theory
In this section we will analyze equation (5) by calculating the dispersion relation in the linearized theory to see whether this quantum relativistic model predicts any new effects. In the linearizing of equation (5), we separate the variables into equilibrium quantities (using 0 as subindex) and perturbed quantities (using 1 as subindex)
f = f0+ f1 (15)
E = E1 (16)
where B0 is a constant. Note that we assumed that there is no background E-field, i.e. E0 = 0. Moreover we assume that the equilibrium distribution function is f0= f0(p2, θs), where θsis the spin angle. The evolution equation
for the distribution function up to first order in linearized theory (we remove products of the perturbed quantities) becomes
∂f1 ∂t + p · ∇xf1− µBm h B0· (s + ∇s) i ∇p 1 · ∇xf1+ q (p × B0) · ∇pf1 − qµBm h B0· (s + ∇s) i ∇p 1 × B0 · ∇pf1+ 2µBm ¯ h (s × B0) · ∇sf1= = −qE · ∇pf0− q h p × (B0+ B1) i · ∇pf0 + qµBm B0+ B1− p × E + m · (s + ∇s) ∇p 1 × B0 · ∇pf0 +qµBm h B0·(s+∇s) i ∇p 1 ×B1 ·∇pf0− qµBm ∇p p × E + m · (s + ∇S) × B0 ·∇pf0 −µBm ∇x h B1· (s + ∇s) i · ∇pf0+ µBm ( + m)∇x h (p × E) · (s + ∇s) i · ∇pf0 −2µBm ¯ h s ×B0+ B1− p × E + m · ∇sf0. (18)
Due to the complexity of this equation we will consider restricted geometries in this work. We consider two geometries, in the first case we have longitu-dinal wave propagation parallel to the magnetic field B0, while in the other case we consider transverse wave propagation, but with B0 parallel to E .
3.1 Longitudinal waves
We define the wave vector k = k⊥ex+ kzezwithout loss of generality. In this
geometry we consider k = kzez, B1 = 0 and E1 = E1exp(ikz −iωt)ez, where
ω is the frequency of the E-field. The fully relativistic evolution equation, equation (5), becomes in this geometry
∂f1 ∂t + p m · ∇xf1− µBm h B0· (s + ∇s)i∇p1 · ∇xf1+ q m(p × B0) · ∇pf1 − qµBmhB0· (s + ∇s)i∇p1 × B0 · ∇pf1+ 2µB ¯ h (s × B0) · ∇sf1= = −qE · ∇pf0− q m p × B0 · ∇pf0+ qµBm " B0− p × E 2m ! · (s + ∇s) # × ∇p 1 × B0 · ∇pf0− qµB ∇p p × E 2m · (s + ∇S) × B0 · ∇pf0 + µB 2m∇x h (p × E) · (s + ∇s) i · ∇pf0− 2µB ¯ h " s × B0− p × E 2m !# · ∇sf0. (19) Note that we assumed that the velocities of the particles are relatively low that we can approximate γ to 1 in = mγ in equation (5), for ∇p(1) we
evaluated the derivative first then took the limit → m. We use also a plane-wave ansatz of the perturbed quantities f1(x, p, s, t) = ef1(p, s) exp[i(kzz −
wt)]. For the momentum p, we express it in cylindrical coordinates: p = p⊥cos ϕpex+ p⊥sin ϕpey+ pzez while the spin s is expressed in spherical
co-ordinates: s = sin θscos ϕsex+ sin θssin ϕsey+ cos θsez. We expand ef1(p, s)
in eigenfunctions of the right-hand side operators in the same way as in reference [3] e f1(p, s) = ∞ X n=−∞ ∞ X n0=−∞ gnn0(p⊥, pz, θs)ψn(ϕp, p⊥)√1 2πexp(in 0ϕ s), (20) where ψn(ϕp, p⊥) = 1 √ 2πexp " i nϕp− k⊥p⊥ mωce sin ϕp # =√1 2π ∞ X l=−∞ Jl k⊥p⊥ mωce ei(n−l)ϕp, (21)
where Jl is the Bessel function of first kind and ωce = qB0/m is the electron
cyclotron frequency. Since we are considering only longitudinal waves (k⊥=
0), then
ψn(ϕp, p⊥) =
1 √
We use the following relations to simplify the calculations q m(p × B0) · ∇pf1 = −ωce ∂f1 ∂ϕp (23a) 2µBm m¯h (s × B0) · ∇sf1= −ωcg ∂f1 ∂ϕs , (23b)
where ωcg = 2µBB0/¯h is the spin precession frequency. With these relations
and the plane-wave ansatz, we can express the left hand side of equation (19) as ( − iω +ipzkz m 1 +µBB0 m (cos θs− sin θs ∂ ∂θs ) − ωce " 1 +µBB0 m cos θs− sin θs ∂ ∂θs # ∂ ∂ϕp − ωcg ∂ ∂ϕs ) f1= RHS, (24)
where RHS is the right hand side of equation (19). Equation (24) is hard to solve analytically. However, if we rearrange the terms in equation (24) as
−iω +ipzkz m − ωce ∂ ∂ϕp − ωcg ∂ ∂ϕs f1 = RHS −µBB0 m cos θs− sin θs ∂ ∂θs ipzkz m − ωce f1. (25)
To solve this equation using the summation of eigenfunctions in equation (20), we assume that µBB0/m 1, hence the second term in the right-hand
side of equation (25) is small relative the left-hand side of the same equation. We can then make an expansion of the perturbed distribution function f1 up to first order using perturbation theory
f1 = f10+ f11, (26)
where f10 and f11 are the zeroth and first order terms in the perturbation
expansion respectively. The zeroth order term in the perturbation expansion is given by −iω +ipzkz m − ωce ∂ ∂ϕp − ωcg ∂ ∂ϕs f10= RHS. (27)
While the first order term in the perturbation expansion is given by
−iω +ipzkz m − ωce ∂ ∂ϕp − ωcg ∂ ∂ϕs f11 = −µBB0 m cos θs− sin θs ∂ ∂θs ipzkz m − ωce ∂ ∂ϕp f10. (28)
In solving the zeroth order term in perturbation, we use the eigenfunction expansion equation (20). Equation (27) is now
∞ X n=−∞ ∞ X n0=−∞ −i ω −ipzkz m + ωcen + ωcgn 0 gnn0 1 2πe i(nϕp+n0ϕs)ei(kzz−ωt) = −qE∂f0 ∂pz −kzµBEp⊥ 4m sin θs+ cos θs ∂ ∂θS ei(ϕs−ϕp)− e−i(ϕs−ϕp)∂f0 ∂pz +µBEp⊥ 2¯hm ei(ϕs−ϕp)+ e−i(ϕs−ϕp)∂f0 ∂θs −qµBB0E 4m
ei(ϕs−ϕp)+ e−i(ϕs−ϕp)sin θ
s+ cos θs ∂ ∂θS ∂f0 ∂p⊥ . (29)
Multiplying both sides by e−i(lϕp+l 0
ϕs)/2π for arbitrary values of l and l0
and integrating over ϕs and ϕp, the resulting equation is
f10= −iqE ω − kzpz/m ∂f0 ∂pz −ikzµBEp⊥ 4m ei(ϕs−ϕp) ω + ∆ωc− kzpz/m − e i(ϕp−ϕs) ω − ∆ωc− kzpz/m ! × sin θs+cos θs ∂ ∂θS ∂f0 ∂pz +iµBEp⊥ 2¯hm ei(ϕs−ϕp) ω + ∆ωc− kzpz/m + e i(ϕp−ϕs) ω − ∆ωc− kzpz/m ∂f0 ∂θs −iqµBB0E 4m ei(ϕs−ϕp) ω + ∆ωc− kzpz/m + e i(ϕp−ϕs) ω − ∆ωc− kzpz/m ! sin θs+cos θs ∂ ∂θS ∂f0 ∂p⊥ , (30)
where ∆ωc= ωcg− ωce. We have now the zeroth order of f1 in the
pertur-bation expansion. Next step is to use the expression for f10, equation (30), to calculate f11 in equation (28). Using the same procedure of calculation
as for f10, we get the following expression for f1
f1 = 1 2π h g00+ g1−1ei(ϕp−ϕs)+ g−11ei(ϕs−ϕp) i . (31) where g00= − iqE ω − kzpz/m 1 + kzpzµBB0 m2(ω − k zpz/m) cos θs− sin θs ∂ ∂θS ∂f0 ∂pz , (32) and
g±1∓1 = 1 ω ∓ ∆ωc− kzpz/m ( sin θs+ cos θs ∂ ∂θS ± ikzµBEp⊥ 4m ∂f0 ∂pz −iqµBB0E 4m ∂f0 ∂p⊥ ± µBB0(kzpz/m ∓ ωc) m(ω ∓ ∆ωc− kzpz/m) × cos θs cos θs ∂ ∂θS − sin θs ∂ 2 ∂θ2 s ! ikzµBEp⊥ 4m ∂f0 ∂pz ∓iqµBB0E 4m ∂f0 ∂p⊥ ! +iµBEp⊥ 2m¯h 1 + µBB0(kzpz/m ∓ ωc) m(ω ∓ ∆ωc− kzpz/m) cos θs− sin θs ∂ ∂θs ! ∂f0 ∂θs ) . (33) Equation (31) gives the first order approximation of the relativistic distri-bution function. Comparing equation (31) with equation (20), one can see that n and n0 in the eigenfunction expansion gave no contribution except for n = −1, 0, 1 and n0 = −1, 0, 1. In g00, the first term comes from f10 while
the second term is from f11. In g1−1and g−11, the terms that come from f11
are those with square of (ω − ∆ωc− kzpz/m) and (ω + ∆ωc− kzpz/m) in
the denominators respectively, the other terms are from f10.
To calculate the total current density J, we use equations (12)–(14) where we only use the perturbed distribution function f1 in the integrals. For the longitudinal waves, the current density has only the z-component, one can easily check it by using Ampère’s law. Thus the magnetization current density is zero in our case. Calculating firstly the free current density to first order in linearized theory
Jf = q Z dΩpz m " 1 +µBB0 m cos θs− sin θs ∂ ∂θS # g00 2π. (34) For the background distribution function f0(p2, θs), we assume that it has a
Maxwellian distribution and a spin-dependent part f0(p2, θs) = e−(−m)/m 2v2 t h eµBB0/kBT(1 + cos θ s) + e−µBB0/kBT(1 − cos θs) i , (35) where T is the temperature and kB is the Boltzmann constant. For more information about the thermodynamical equilibrium in the background dis-tribution function see reference [8]. Note that even though the derivation of the background distribution function in reference [8] was non-relativistic, the result should work in our case. We normalize the distribution function by dividing it with the normalizing factor Nm =R dΩf0(p2, s). In the
integra-tion over p, we make a first order Taylor expansion of ; the argument of the exponential part of f0(p2, s) becomes: ( − m)/m2v2th≈ −p2/m2v2
vth is the thermal velocity. The normalized distribution function ˆf0(p2, s) is now ˆ f0(p2, s) = 1 Nm e−p2/m2v2th h eµBB0/kBT(1 + cos θ s) + e−µBB0/kBT(1 − cos θs) i , (36) where Nm = 8π(πm2vth2 )3/2cosh(µBB0/kBT ) is the normalization factor.
We make a Taylor expansion of the denominators in equation (32), (ω − kzpz/m), otherwise the integration over pz would be very hard to be solved
analytically. The result we have for the free current density is
Jf = iq2En ωm " 1 +3k 2 zvth2 2ω2 1 + µ2 BB20 m2 ! +µBB0 m 1 + 3k2 zv2th 2ω2 ! tanhµBB0 kBT # . (37)
Similarly, the polarization current density is calculated to
Jp= iµ2 BωEn 4m " v2 thkz2 2 1 ω2 + + 1 ω2 − ! +3v 4 thk4z 4 1 ω4 + + 1 ω4 − ! +qB0 m 1 ω+ − 1 ω− ! +qB0v 2 thk 2 z 2m 1 ω3 + − 1 ω3 − !# −iµ 2 BωEn 2¯h " v2 th 2 1 ω+ − 1 ω− ! +v 4 thkz2 4 1 ω3 + − 1 ω3 − !# tanhµBB0 kBT , (38)
where ω± = ω ± ∆ωc, this notation is used to simplify the calculations.
Ampère’s law gives us the relation
−iωE = −4πJ. (39)
Using equation (39) together with the expressions of the polarization current density, equation (38), and the free current density, equation (37), we have then the dispersion relation
ω2 ( 1 − ¯h 2ω2 p 32m2 " vthk2 2z 1 ω2 + + 1 ω2 − +3v 4 thkz4 2 1 ω4 + + 1 ω4 − # −ω 2 p¯hµBB0 8m2 " 1 ω+ − 1 ω− +v 2 thkz2 2 1 ω3 + − 1 ω3 − # +¯hω 2 pv2th 16m " 1 ω+ − 1 ω− + v 2 thkz2 2 1 ω3 + − 1 ω3 − # tanh(µBB0 kBT ) ) = ωp2 " 1 +3k 2 zv2th 2ω2 1 +µ 2 BB20 m2 +µBB0 m 1 +3k 2 zv2th 2ω2 tanh(µBB0 kBT ) # . (40)
Equation (40) includes the spin effects of the electrons and is a generalization of the dispersion relation for the classical longitudinal waves in magnetized
plasmas. If one neglects the spin effects, equation (40) becomes
ω2 = ω2p+3k
2v2
th
2 , (41)
which is the classical dispersion relation for Langmuir waves. 3.1.1 Semi-relativistic limit
In this section we compare the dispersion relation, equation (40), with the dispersion relation in the relativistic limit in reference [6]. In the semi-relativistic limit, µBB0/m is small enough that it can be neglected. Thus
the dispersion relation, equation (40), becomes in this limit
ω2 ( 1 − ¯h 2ω2 p 32m2 " vth2 k2z 1 ω+2 + 1 ω−2 +3v 4 thk4z 2 1 ω+4 + 1 ω−4 # +¯hω 2 pvth2 16m " 1 ω+ − 1 ω− +v 2 thk2z 2 1 ω3+ − 1 ω−3 # tanh(µBB0 kBT ) ) = ωp2 1 +3k 2 zvth2 2ω2 ! . (42)
This dispersion relation is identical with the one obtained in reference [3] except for an overall −4π2 that is missing in the polarization current. More-over one neglected terms with 1/ω+ since that term is small compared with 1/ω− and the Landau damping-term appearing in reference [3] is neglected
in our dispersion relation.
3.2 Transverse waves
In this geometry we have k = k⊥ex, B0 = B0ez, E = E1exp(ikz − iωt)ez
and B1= B1exp(ikz − iωt)ey. As in the longitudinal case, we use the
plane-wave ansatz f1(x, p, s, t) = ef1(p, s) exp[i(k⊥x − ωt)]. The momentum p is
expressed in cylindrical coordinates while the spin s is expressed in spherical coordinates. fe1(p, s) is expanded in the eigenfunctions of the right hand side operators using equation (20), but this time we need to use the full equation, equation (21), for ψn(ϕp, p⊥). We use here the same assumption
that we have low velocities of the particles as in the longitudinal case. With the relations in equation (23) and the plane-wave ansatz, we can express the left hand side LHS of equation (18) as
( − iω +ip⊥k⊥ m cos ϕp 1 +µBB0 m (cos θs− sin θs ∂ ∂θs ) − ωce 1 +µBB0 m (cos θs− sin θs ∂ ∂θs ) ∂ ∂ϕp − ωcg ∂ ∂ϕs ) f1= RHS. (43)
We have again a complicated equation and need to use the perturbation theory to obtain an analytical solution. We use equation (26) where we assume that µBB0/m 1 and expand in the perturbation theory up to first
order, the zeroth order term in perturbation is −iω +ip⊥k⊥ m cos ϕp− ωce ∂ ∂ϕp − ωcg ∂ ∂ϕs f10= RHS, (44)
while the first order term in perturbation is −iω +ip⊥k⊥ m cos ϕp− ωce ∂ ∂ϕp − ωcg ∂ ∂ϕs f11 = −µBB0 m cos θs− sin θs ∂ ∂θs ip⊥k⊥ m cos ϕp− ωce ∂ ∂ϕp f10. (45)
Using the same procedure as in the longitudinal case, the distribution func-tion to the zeroth order f10 is
f10= ∞ X ρ,β=−∞ Jβ k⊥p⊥ mωce eiϕp(β−ρ)hA +B + Ce±iϕsi, (46) where A = −iqE∂f0 ∂pz Jρ ω − ωceρ (47) B =X +,−
−isin θs+ cos θs
∂ ∂θS ∂f0 ∂p⊥ " qEµBB0 4m(ω − ωceρ ± ωcgρ) Jρ∓1 ± µBB1k⊥ 4(ω − ωceρ ± ωcgρ) (Jρ±1+ Jρ∓1) − µBEk⊥p⊥ 8m(ω − ωceρ ± ωcgρ) ( ± Jρ∓ Jρ∓2) # (48) C =X +,− iµB ¯ h ∂f0 ∂θs " B1 Jρ (ω − ωceρ ± ωcgρ) +Ep⊥ 2m Jρ∓1 (ω − ωceρ ± ωcgρ) # , (49)
where Jρis the Bessel function of first kind where we omitted the argument
term of the distribution function f11is f11= µBB0ωce m cos θs− sin θs ∂ ∂θS X∞ τ,β,α,ρ=−∞ eiϕp(β−ρ)ρJ βJαJρ+α−τ× X +,− " A (ω − ωceρ) + B + C (ω − ωceρ ± ωcgρ) e±iϕs # , (50)
where the argument of all Bessel functions is k⊥p⊥/mωce, but we omit it
to simplify the expression. As in the longitudinal case, we assume that the background distribution function f0 is given by equation (36). The free
current density Jf is calculated using equation (12). We carried out the
integration over the spin angles, pz and ϕp and deduce
Jf = iEωp2 4πmkBT ( 1+µBB0 m tanh µBB0 kBT X∞ β=−∞ 1 ω − ωceβ Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBTJ2 β +µBB0ωce m tanhµBB0 kBT +µBB0 m X∞ τ,β,α=−∞ β (ω − ωceτ )(ω − ωceβ) × Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBTJ βJαJτJβ+α−τ ) . (51)
Similarly, the polarization and magnetization current density, Jp and Jm, are calculated to Jp= iωω2 p¯hE 16πm3k BT Z ∞ 0 dp⊥p2⊥e−p 2 ⊥/2mkBT ∞ X β=−∞ X +,− Jβ ( p⊥ kBT " ±µBB0 4m Jβ (ω − ωceβ ± ωcg) − ¯hk 2 ⊥ 8mω(ω − ωceβ ± ωcg) Jβ+ Jβ±2 − ¯hk⊥p⊥ 16m2(ω − ω ceβ ± ωcg) Jβ±1− Jβ∓1 # ∓1 2tanh µBB0 kBT " −k⊥ ω Jβ±1 (ω − ωceβ ± ωcg) + p⊥ 2m Jβ (ω − ωceβ ± ωcg) #) (52) JM = iω2 p¯hk⊥E 8πmkBT Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBT ∞ X β=−∞ X +,− Jβ ( − p⊥ mkBT " ±µBB0 4m Jβ∓1 (ω − ωceβ ± ωcg) − ¯hk 2 ⊥ 8mω(ω − ωceβ ± ωcg) Jβ±1+ Jβ∓1 − ¯hk⊥p⊥ 16m2(ω − ω ceβ ± ωcg) Jβ− Jβ∓2 # ± 1 2mtanh µBB0 kBT " −k⊥ ω Jβ (ω − ωceβ ± ωcg) + p⊥ 2m Jβ∓1 (ω − ωceβ ± ωcg) #) . (53)
Note that the first order term f11 survives the integration only in the free current density. To obtain the dispersion relation, we use Ampère’s law where we get the relation
iωE 1 −k 2 ⊥ ω2 ! = 4πJ. (54)
The dispersion relation for the transverse propagation is
ω2 1 − k 2 ⊥ ω2 − ω2 p¯h 4m3k BT Z ∞ 0 dp⊥p2⊥e−p 2 ⊥/2mkBT ∞ X β=−∞ X +,− Jβ ( p⊥ kBT × " ±µBB0 4m Jβ (ω − ωceβ ± ωcg) − ¯hk 2 ⊥ 8mω(ω − ωceβ ± ωcg) Jβ+ Jβ±2 − ¯hk⊥p⊥ 16m2(ω − ω ceβ ± ωcg) Jβ±1−Jβ∓1 # ∓1 2tanh µBB0 kBT " −k⊥ ω Jβ±1 (ω − ωceβ ± ωcg) + p⊥ 2m Jβ (ω − ωceβ ± ωcg) #) + ω 2 p¯hk⊥ 2m2k BT ω Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBTX +,− ∞ X β=−∞ Jβ× ( p⊥ kBT " ±µBB0 4m Jβ∓1 (ω − ωceβ ± ωcg) − ¯hk 2 ⊥ 8mω(ω − ωceβ ± ωcg) Jβ±1+ Jβ∓1 − ¯hk⊥p⊥ 16m2(ω − ω ceβ ± ωcg) Jβ− Jβ∓2 # ∓1 2tanh µBB0 kBT " −k⊥ ω Jβ (ω − ωceβ ± ωcg) + p⊥ 2m Jβ∓1 (ω − ωceβ ± ωcg) #)! = ω 2 p mkBT ( 1 + µBB0 m tanh µBB0 kBT X∞ β=−∞ 1 1 − ωce ω β Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBTJ2 β +µBB0ωce mω tanhµBB0 kBT +µBB0 m X∞ τ,βα=−∞ β (1 −ωce ω τ )(1 − ωce ω β) × Z ∞ 0 dp⊥p⊥e−p 2 ⊥/2mkBTJ βJαJτJβ+α−τ ) . (55)
We can compare this dispersion relation with the one for the longitudinal case, equation (40). We let k⊥and kzbe zero and the two dispersion relations
(55) is then ω2 " 1 −ω 2 p¯hµBB0 8m2 1 ω − ωce+ ωcg − 1 ω − ωce− ωcg +ω 2 p¯hvth2 16m tanh µBB0 kBT × 1 ω − ωce+ ωcg − 1 ω − ωce− ωcg # = ωp2 1 +µBB0 m tanh µBB0 kBT ! . (56) Equation (40) has the same expression in this limit. Thus we have both dispersion relations in the two geometries to approach the same result in the very high frequency limit as expected.
3.2.1 Long scale limit
The integrals in equation (55) have no simple analytical solutions. We make here an approximation of the argument of the Bessel functions in the integral k⊥p⊥/mωce 1. This approximation enable us to make a Taylor expansion
of the Bessel functions, we expand the Bessel function up to the second order in k⊥p⊥/mωce. The terms that we kept from the summation over the Bessel
functions are J0, J±1 and J−2, which become after the Taylor expansion
J0 = 1 − k⊥2p2⊥ 4m2ω2 ce J±1= ± k⊥p⊥ 2mωce J−2= k2⊥p2⊥ 8m2ω2 ce . (57)
ω2 1 +X +,− ω2p¯h 8m2 ( µBB0 " ∓ 1 ω ± ωcg ∓ k 2 ⊥ ω2 ce v2 th 2 1 ω − ωce± ωcg + 1 ω + ωce± ωcg − 2 ω ± ωcg ! + k 2 ⊥ ωωce 1 ω ∓ ωce± ωcg − 1 ω ± ωcg !# +¯hk 4 ⊥v2th 4ωω2 ce " 1 2(ω + 2ωce± ωcg) − 1 ω ∓ ωce± ωcg + 2 ω ± ωcg − 1 ω ± ωce± ωcg + 1 2(ω ± ωcg) # +¯hk 2 ⊥ 4ωce " ±v 2 th 2 2 ω ± ωcg − 1 ω ± ωce± ωcg − 1 ω ∓ ωce± ωcg ! −k 2 ⊥ ω2 1 ω − ωce± ωcg − 1 ω + ωce± ωcg !#) +X +,− ω2p¯h 4m tanh µBB0 kBT ( ± v 2 th 4 + k2 ⊥ ω2 ! × " 1 ω ± ωcg + k 2 ⊥ ω2 ce v2 th 2 1 ω − ωce± ωcg + 1 ω + ωce± ωcg − 2 ω ± ωcg !# +k 2 ⊥vth2 4ωωce 1 ω ± ωce± ωcg − 1 ω ∓ ωce± ωcg !)! = k⊥2 +X +,− ωp2 " 1 − k 2 ⊥ ω2 ce v2 th 2 1 − 1 2(1 ±ωce ω ) ! ±µ 2 BB20k2⊥vth2 4m2ωω ce 1 (1 ∓ωce ω ) 2 # +X +,− ω2 pµBB0 m tanh µBB0 kBT " 1−k 2 ⊥ ω2 ce vth2 2 1− 1 2(1 ±ωce ω ) ! ±k 2 ⊥v2th ωωce 1 (1 ∓ ωce ω )2 # . (58)
In the limit of low temperatures, we let T → 0, the dispersion relation becomes then ω2 1 +X +,− ωp2¯h 8m2 ( µBB0 " ∓ 1 ω ± ωcg + k 2 ⊥ ωωce 1 ω ∓ ωce± ωcg − 1 ω ± ωcg !# − ¯hk 4 ⊥ 4ω2ω ce 1 ω − ωce± ωcg − 1 ω + ωce± ωcg !) X +,− ±ω 2 p¯hk2⊥ 4mω2 tanh(µBB0/kBT ) ω ± ωcg ! = k2⊥+ ω2p 1 +µBB0 m tanh( µBB0 kBT ) ! . (59)
If the background B-field (B0) is weak, i.e. µBB0/m 1, the dispersion
relation becomes even simpler
ω2= k⊥2 + ωp2+ X +,− ωp2¯h2k⊥4 32m2ω ce 1 ω − ωce± ωcg − 1 ω + ωce± ωcg ! . (60)
3.2.2 Non-relativistic limit
Equation (60) offers an interesting result since the quantum mechanical term on the right hand side of the equation is non-zero in the zero-temperature limit. This term has its origin in the (µBm
∇x
h
B1· (s + ∇s)
i
· ∇pf0)-term
in the fully relativistic kinetic equation (5), which is not a relativistic term. Thus it means that the new interesting term in the dispersion relation can be found in a non-relativistic dispersion relation of quantum plasmas.
Considering σzz of the conductivity tensor in reference [3], using the same geometry as in our case and making the same expansion of the Bessel functions, the dispersion relation of the non-relativistic kinetic equation is then ω2= k⊥2 + ωp2− X +,− ωp2¯h2k⊥4 16m2ω ce 1 ω − ωce± ωcg − 1 ω + ωce± ωcg ! . (61) 3.2.3 Short analysis
The dispersion relation equation (60) can be analyzed further by considering the limit of high frequency, ω ωce + ωcg. Taking the first order Taylor
expansion of the nominators in equation (60), we get
ω2 = k2⊥+ ωp2+ ω
2 p¯h2k2⊥
16m2ω2. (62)
This is a new wave-mode. Even though we considered the long scale approx-imation where the argument of the Bessel functions is assumed to be small, this mode can be found for relatively small k⊥p⊥/mωce, which makes the
approximation still valid.
A further analysis can be done to get a more general dispersion relation than equation (62) that takes into account more terms in the Bessel functions in equation (55).
4
Summary
In this work we have used in linearized theory, the fully relativistic kinetic equation of spin-1/2 particles in the long scale approximation, equation (5). Since equation (5) is fairly complicated, we chose to restrict the geometry. In the first case we chose to consider longitudinal propagation parallel to the background B-field, while in the second case we considered transverse prop-agation that is perpendicular to the background B-field. In both cases, we considered magnetized plasmas with a homogeneous background distribu-tion funcdistribu-tion, equadistribu-tion (36). In solving the perturbed distribudistribu-tion funcdistribu-tion f1, we used perturbation theory, equation (26), and solved up to first order
in perturbation in both cases. However the first order term in perturbation f11 contributed only in the free current Jf in the transverse case.
In the longitudinal case, the major result was the dispersion relation equation (40), which in the limit of weak background B-field matches the dispersion relation in the semi-relativistic case in reference [6] (we disre-garded here the overall factor −4π2 and the Landau damping-term). In the transverse case, the major result was the dispersion relation, equation (55), which has no simple analytical solution thus we made the long scale approx-imation. In this approximation we assumed that the argument of the Bessel functions is small, k⊥p⊥/mωce 1, thus we could make a second order
Taylor expansion around k⊥p⊥/mωce = 0. In this approximation we got
the dispersion relation equation (60). A more detailed analysis of equation (55) might be necessary to do in order to obtain a more general dispersion relation. However the calculations we have done that resulted in equation (60) have shown that the low frequent waves might be strongly affected by the quantum effects that follow from the magnetic moments of the electrons. Acknowledgements I would thank Gert Brodin and Robin Ekman for supporting me during this work. I would thank also Jens Zamanian for the helpful comments that helped to improve this work.
References
[1] Princeton Plasma Physics laboratory, A collaborative national center for fusion & plasma physics (2018).
[2] J. Zamanian, Umea Universitet. diva2:511354 (2012) [3] J. Lundin and G. Brodin, Phys. Rev. E 82, 056407 (2010) .
[4] R. Ekman, F. A. Asenjo, and J. Zamanian, Phys. Rev. E 96, 023207 (2017).
[5] L. L. Foldy and S. A. Wothuysen, Phys. Rev. 78, 29 (1950).
[6] F. A. Asenjo, J. Zamanian, M. Marklund, G. Brodin, and P. Johansson, New J. Phys. 14, 073042 (2012).
[7] H. S. Thomas, Waves in Plasmas, American institute of physics (1992). [8] J. Zamanian, M. Marklund, and G. Brodin, New J. Phys. 12, 043019
