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QM Theory of the Thermal Electron Gyroradius

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QM Theory of the Thermal Electron Gyroradius Manuscript prepared for Annales Geophysicæ with version 3.2 of the LATEX class copernicus.cls. Date: 16 July 2018 QM theory of the thermal electron gyroradius R. A. Treumann1,2 and W. Baumjohann3 1Department of Geophysics and Environmental Sciences, Munich University, Munich, Germany 2International Space Science Institute, Bern, Switzerland 3Space Research Institute, Austrian Academy of Sciences, Graz, Austria 1 2 Abstract. The average (thermal) gyroradius of charged par- lar temperature T⊥ = 2 mevt⊥. A more precise definition ticles is re-examined from a quantum-mechanical point of makes use of the velocity distribution function of the elec- view. The straight quantum-mechanical calculation clearly trons to calculate the average gyroradius ρe as the mo- reproduces its conventionally used random-mean-square ment of the classical distribution with respecth toi the perpen- (rms) expectation value. The quasi-classical approach repro- dicular particle velocity. In the random-mean-square (rms) duces its mean expectation value as well thus confirming the limit this yields the above expression for ρe,rms with replace- purely classical calculation. It shows that the fluctuations in ment v v = 2T /m , which is nothing but the well- ⊥ → t⊥ ⊥ e the gyroradius amount to 21.5% of its rms value. This fluc- known rms “thermalp gyroradius”. Occasionally freedom is tuation is, however, within the usual √2-“freedom” range of taken for convenience in either keeping or dropping the √2 choice in the definition of the rms gyroradius respectively the value in the definition of the thermal speed which introduces mean thermal speed on which the rms value is based, its cor- a 29% indeterminacy of the thermal speed and rms gyrora- 2 rect (nonrelativistic) value vt =2T⊥/me and its simplified dius. 2 version vt = T⊥/me. For temperatures T 0, i.e. T . ǫ , with ǫ Fermi ⊥ ∼ ⊥ F F Keywords. Plasma parameters, Thermal spread of gyrora- temperature, electrons become degenerate and one needs dius, Electron diffusion region to refer to the Fermi distribution. In this case the thermal speed is replaced by the Fermi velocity vF = 2ǫF /me = 2 1/3 (~/me)(6π N) ,a collective velocity of degeneratep elec- trons. One may note that this does not reproduce the above 1 Introduction quantum mechanical magnetic length of electrons, λm! The latter being a single electron property in magnetic fields The gyroradius ρ of a particle of charge e and mass m in a arXiv:1307.7346v1 [physics.plasm-ph] 28 Jul 2013 caused by the quantum character of the magnetic flux with magnetic field of strength B is one of the fundamental pa- flux quantum Φ =2π~/e, not a collective effect. It defines rameters used in plasma physics. Its exact classical defini- 0 the radius of a magnetic field line through B =Φ /πλ2 . tion for a single particle of perpendicular velocity v is ρ = 0 m ⊥ e At non-zero temperatures T & ǫ slightly exceeding ǫ a v /ω written here for an electron with cyclotron frequency ⊥ F F ⊥ ce small correction is to be introduced on the Fermi velocity ω = eB/m (see any textbook on plasma physics, e.g., ce e entering the collective rms gyroradius. Baumjohann and Treumann, 2012). Its ion equivalent is ob- In many particle theory, starting from the single particle vious. Single electrons have undefined temperature, usually picture one should, for greater precision, make use however assumed to be zero, hence the exactness of the given expres- of the distribution of particles over the entire spectrum of sion. In quantum mechanics electrons obey Landau levels energy states available and accessible to the population at a possessing a well defined magnetic length λ = 2π~/eB m given temperature T . which corresponds to the gyroradius of an electronp in the ⊥ Since only two particles of opposite spins can occupy any lowest Landau energy level, ǫ = 1 ~ω . LLL 2 ce energy state there is a well-known competition between the In high temperature plasmas one defines the gyroradius available states and the number of particles at . If the tem- through the thermal velocity, respectively the perpendicu- T⊥ perature T⊥ . TF , with TF the Fermi temperature, then the Correspondence to: R. A. Treumann particles have effectively zero temperature, and the gyrora- ([email protected]) dius is defined through their single velocity in the respective 2 R. A. Treumann and W. Baumjohann: Gyroradius state. This case is encountered in high density compounds at written here for the anisotropic case, with z the fugacity 3 which depends on the chemical potentials µk,⊥ in parallel 2 3 −1 meT⊥ 11 2 −3 and perpendicular direction, and βk,⊥ = T are the inverse Ne 4.6 10 T m (1) k,⊥ ∼ 2π2~2 ≈ × ⊥,[eV] temperatures (in energy units).1 At sufficiently large tem- peratures the unity in the denominator is neglected, a very with temperature measured in eV. (One may note that this well justified approximation which still takes into account the number is remarkably close to the conditions encountered in non-continuousenergy distribution over discrete Landau lev- the solar chromosphere!). When this condition holds, quan- els thus maintaining the quantum character of electrons. The tum effects become important; it implies that the interparticle fugacities enter into the normalization constant now. This is distance the case far away from Eq. (1) which for plasmas interests 1 3 − 3 2 2 us here. Under these conditions the expectation value of the Ne λT , λT = 2π~ /mT⊥ (2) ≃ (average) perpendicular energy of the electrons (i.e.the per- equals the thermal de Broglie wavelength λT , here written pendicular electron pressure) is calculated from the integral with T = T⊥. At higher temperatures and lower densities ∞ ∞ the average gyroradius should be calculated by adding up −βkǫk −β⊥ǫ⊥(ℓ,σ) Nǫ⊥ dp e ǫ⊥(ℓ,σ)e (5) h i∝ Z k all electrons in the available states. By assuming a classi- σX=± cal distribution function this is circumvented and reduced to −∞ ℓ=0 a simple integration yielding the above thermal rms value The spin contributionin the perpendicularenergy either com- ρe,rms = √2meT⊥/ωce. Here we present the exact quantum pensates for the half Landau energy level or completes it to mechanical calculation which naturally confirms the thermal the first order level. Thus the sum splits into two terms which rms gyroradius result but, in addition, yields the expectation both are geometric progressions which can immediately be value ρe of the gyroradius and provides its width of uncer- done. The final result, using the normalization of the inte- tainty,h i.ei its thermal spread. gral to the average density of particles and dividing through N = N thus yields for the average energy h i 2 −β ~ωce 2β⊥(~ωce) e ⊥ ~ 2 Quantum mechanical treatment 1 −β⊥ ωce (6) ǫ⊥ = ~ 2 1 2 e h i (1 e−β⊥ ωce ) − − Knowing the spectrum of quantum mechanical states of an At the assumed large temperatures the exponentials must be electron in magnetic field there are two ways of calculating expanded to first order yielding the very well known and ex- the thermal gyroradius at given temperature T⊥. Either the pected classical result that the average energy is the temper- gyroradius is calculated by averaging it over the distribution −1 ature, ǫ⊥ = β⊥ T⊥. Hence, taking its root and inserting – which is its expectation value ρ , would be the exact way h i ≡ h ei into the gyroradius we find what is expected in this case: to do it –, or one calculates the energy expectation value ǫ⊥ h i ρ = ρ2 = 2m ǫ /eB, ǫ = T (7) which provides the expectation value of the squared gyrora- e,rms h ei eh ⊥i h ⊥i ⊥ dius ρ2 whose root can also be taken to represent the ther- p p h ei This is the root-mean-squaregyroradius, a well known quan- mal gyroradius. This latter value will be calculated first. tity. At lower temperatures T ǫ , still by far exceeding ⊥ ≫ F Fermi energy. the former expression for ǫ⊥ has to be used 2.1 Root-mean squared gyroradius in this expression. h i The energy levels of an electron in a homogeneous magnetic 2.2 Expectation value field (the case of interest here as a magnetic inhomogeneity provides only higher order corrections) has been calculated However, the correct gyroradius is not its root mean square long ago (Landau, 1930; Landau & Lifschitz, 1965). Since but the expectation value ρ . This is substantially more dif- the parallel dynamics of an electron is not of interest here, it ficult to calculate. There areh i two ways of finding the expec- can be taken classically. Then the energy of an electron in the tation value. Either one makes use of the Landau solution magnetic field becomes of the electron wave function and refers to the Wigner dis- tribution of quantum mechanical states. This, for our pur- ǫ = ǫ +ǫ (ℓ,σ)= ǫ +~ω ℓ+ 1 +σ , ℓ N (3) e k ⊥ k ce 2 ∈ poses would be overdoing the problem substantially. We do 2 1 1One may notice that this is the only compatible with statisti- with ǫk = pk/2me, quantum number ℓ, and σ = 2 the two spin directions. The average distribution of electrons± over cal mechanics anisotropic form of the Fermi distribution. It shows these ℓ perpendicular energy states is given by the Fermi dis- that both, the chemical potential and inverse temperature are vector tribution quantities, i.e. in relativistic terms they are 4-vectors, a conclusion which has been much debate about in relativistic thermodynamics 1 (for references see Nakamura, 2006) but here comes out naturally β⊥µ⊥+βkµk (4) nF = −1 , z =e h i 1+z exp(βkǫk +β⊥ǫ⊥) without much effort.
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