sign in sign up
<<
Home , Gyroradius

arXiv:1307.7346v1 [physics.plasm-ph] 28 Jul 2013 aeesue npam hsc.Iseatcascldefini- perpendicular classical of exact particle single Its a physics. for tion in used rameters in nqatmmcaiseetosoe adulevels length Landau magnetic defined obey well a expres- mechanics possessing given quantum the of In exactness the sion. hence zero, usuall be temperature, to undefined assumed ob- have is electrons equivalent Single ion vious. Its 2012). Treumann, and Baumjohann ω v antcfil fstrength of field magnetic version is lcrndfuinregion diffusion dius, Keywords. ([email protected]) to: Correspondence gyroradius The Introduction 1 L the of 3.2 version with Geophysicæ Annales for prepared Manuscript hog h hra eoiy epcieyteperpendicu the respectively velocity, thermal the the through in electron an of level, energy gyroradius Landau the lowest to corresponds which cor- its based, is value value (nonrelativistic) rms the rect which on t respectively speed gyroradius thermal rms mean the of definition the in choice 1 Treumann A. gyroradius R. electron thermal the of theory QM 2018 July 16 Date: 3 2 h yoaisaon o2.%o t m au.Ti fluc- This value. usual rms the within its however, of is, 21.5% tuation to fluctuation amount the gyroradius that the shows the It confirming calculation. thus classical well as purely value expectation mean rep its random-mean-square approach duces quasi-classical The used value. expectation conventionally (rms) clearl its of calculation reproduces point quantum-mechanical quantum-mechanical straight a The view. from re-examined is ticles Abstract. eateto epyisadEvrnetlSine,Muni Sciences, Environmental and Geophysics of Department pc eerhIsiue utinAaeyo cecs Gr Sciences, of Academy Austrian Institute, Research Switzerland Space Bern, Institute, Science Space International ⊥ ce nhg eprtr lsa n enstegyroradius the defines one plasmas temperature high In /ω = ce eB/m v rte eefra lcrnwt ylto with electron an for here written t 2 h vrg teml yoaiso hre par- charged of gyroradius (thermal) average The = lsaprmtr,Temlsra fgyrora- of spread Thermal parameters, Plasma T e ⊥ seaytxbo npam hsc,e.g., physics, plasma on textbook any (see ρ /m 1,2 fapril fcharge of particle a of e n .Baumjohann W. and .A Treumann A. R. . A T E ls copernicus.cls. class X B v ǫ t 2 LLL soeo h udmna pa- fundamental the of one is 2 = T = ⊥ √ 1 2 /m ~ 2 ω “reo”rneof range -“freedom” e ce e 3 n t simplified its and . λ n and m = p v ⊥ 2 π is m ~ ρ /eB na in in s e ro- he = y y - z Austria az, hUiest,Mnc,Germany Munich, University, ch a temperature lar ( ment ae o ovnec nete epn rdopn the dropping or keeping either is in freedom convenience Occasionally for taken gyroradius”. “thermal rms known 9 neemnc ftetemlsedadrsgyrora- rms and speed thermal the dius. introduces of which indeterminacy speed 29% thermal a the of definition the in value h aiso antcfil iethrough line field magnetic a of radius the ii hsyed h bv xrsinfor expression (rms above the random-mean-square yields the this perpe limit In the velocity. to respect particle with dicular distribution classical the of ment orfrt h em itiuin nti aetethermal the velocity case Fermi this the In by distribution. replaced needs Fermi is one speed the to and refer degenerate to become electrons temperature, asdb h unu hrce ftemgei u with flux fields magnetic magnetic the in of quantum character flux property quantum electron the by single caused a being latter rn ocluaeteaeaegyroradius elec- average the the of calculate function distribution to velocity trons the of use makes rn.Oemynt htti does this that note may One trons. unu ehnclmgei egho electrons, of length magnetic mechanical quantum tnnzr temperatures non-zero At nrysae vial n cesbet h ouainat of population spectrum the entire to temperature given accessible the and over available particles states energy of however use distribution make the precision, of greater for should, one picture velocity Fermi the on gyroradius. introduced rms collective be the entering to is correction small vial ttsadtenme fprilsat particles the of between number the competition and well-known states available a is there state energy perature isi endtruhtersnl eoiyi h respecti the in velocity single gyror their through the defined and is temperature, dius zero effectively have particles ~ /m o temperatures For nmn atceter,satn rmtesnl particle single the from starting theory, particle many In ic nytoprilso poiesiscnocp any occupy can spins opposite of particles two only Since v e ⊥ )(6 → T π ⊥ 2 v . N t ⊥ Φ ) T = 1 0 F T / 2 = 3 with , p ⊥ T a , ⊥ = 2 T π . T collective ⊥ ~ 1 ⊥ 2 /e T m ∼ /m F o olcieefc.I defines It effect. collective a not , T e 0 v ⊥ h em eprtr,te the then temperature, Fermi the e i.e. , t 2 hc sntigbttewell- the but nothing is which , ⊥ & oepeiedefinition precise more A . eoiyo eeeaeelec- degenerate of velocity ǫ F T not ⊥ lgtyexceeding slightly . erdc h above the reproduce ρ ǫ v F e, F with , rms h = ρ B T e i p ⊥ ihreplace- with Φ = ftetem- the If . stemo- the as 2 ǫ ǫ λ F F 0 m /m /πλ Fermi The ! ǫ F e √ m 2 ve n- = a- 2 a a ) . 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. R. A. Treumann and W. Baumjohann: Gyroradius 3 not need the quantum mechanical probability distribution for 3 Discussion the simple estimate we envisage here. Afterwards we would anyway have to combine the Wigner function with a suitable The expected result of the quantum mechanical calculation momentum distribution. The second and simpler way is to of the electron gyroradius of a hot plasma is that the root- refer to the above Fermi distribution and directly using the mean-square value of the gyroradius is reproduced by quan- energy distribution as above. Under the same conditions as tum mechanics in the high temperature limit which clearly in the calculation of the rms value, this procedure circum- completely justifies its use not adding anything new except vents the use of the wave function being interested only in for the confirmation of well-known facts which can be gen- the energy levels. It, however, requires the solution of the erated in much simpler way by using the Boltzmann calcu- integral-sum lation. For lower temperatures we obtained a slightly more precise expression for the average energy which should be ∞ used in the definition of the thermal gyroradius. The expec- ∞ ǫ (ℓ,σ) −βkǫk ⊥ −β⊥ǫ⊥(ℓ,σ) tation value of the gyroradius is different from the root mean Nρe dpke e (8) h i∝ Z p ωce square value, which is also known. Its value is lower given −∞ σ=X±,ℓ=0 by the Gaussian mean. Usually this value is not used in any The sum, the integral contains, cannot anymore be done in classical calculation. The difference between both values is closed form as there is no known way to tackle the sum- within the commonly used √2 freedom for choosing the rms mation over the root quantum index ℓ in a non-geometric thermal speed. progression. (We may note in passing that for a non-moving The above calculation is based on keeping the discrete dis- plasma calculating the average velocity moment would lead tribution of all plasma electrons over Landau levels. Since to a null result. However, in calculating the expectation value these are equally spaced up to infinity, the energy remains of the gyroradius, the perpendicular velocity, being a radius discontinuous even at large temperatures. The correction is, vector in perpendicular velocity space, is always positive. however, unimportant. This is reflected by the summation over positive quantum It is sometimes claimed that the discrete distribution of numbers ℓ only. It and the gyroradius are positive definite electron energies is reflected in the dielectric function of a quantities which both do not average out.) One thus needs plasma which leads to collective cyclotron harmonics, i.e. to deviate from summing and to approximate the sum by an Bernstein modes. However, Landau levels have nothing in integral, which slightly overestimates the final result. Trans- common with cyclotron harmonics. They are single particle forming the energy into a continuous variable, which implies energy levels occupied at most by two electrons of oppositely that the summation index becomes continuous, then simply directed spins, while cyclotron harmonics are eigenmodes of leaves us with a Gaussian integral of which we know that the the magnetized plasma, channels which allow for propaga- mean value and the rms values are related by the classical tion of electro(magnetic) signals. The discrete energy dis- formula tribution of the electrons naturally disappears in the caclu- ation of the expectation values and is not reflected in any cy- clotron harmonic waves. The average gyroradius appearing ρ . 0.886 ρ2 =0.886ρ (9) e e e,rms in the arguments of the modified Bessel functions in the di- h i ph i electric expression of the harmonics in the combination k⊥ρ the classical result for the mean, where by the . sign we is a mere scaling factor of wavelengths related to the perpen- indicated that the integral yields an upper limit for the expec- dicular thermalspeed. It cannotbe taken as definition of a gy- tation value of the gyroradius. roradius, which is a particle property. In fact, calculation of the dielectric function from quantum mechanics accounting 2.3 Fluctuation for the Landau energy distribution is very difficult. In clas- sical theory one integrates the Vlasov equation with respect The above estimates permit determining the thermal fluctu- to time along classical particle orbits v(x,t),x(t) which, in ation (thermal spread) of the gyroradius. This fluctuation is quantum mechanics, is forbidden as no orbits exist. Orbits the difference between the mean square and squared expec- are replaced by probability distributions in space varying tation values. With the above results we find that the thermal with time. The calculation therefore needs to be performed fluctuation of the gyroradius amounts to via a Feynman path integral approach which has not yet been done. It is, however, reasonably expected that it will, in the 2 2 2 2 ∆ ρe ρe ρe 21.5% ρe (10) classical limit reproduce the classical dielectric function and h i ≈ h i−h i ≈ h i the harmonic structure of Bernstein and cyclotron modes. √ This number falls into the 2 range of “freedom” in choos- Acknowledgements. This research was part of an occasional Visit- ing the thermal speed that is commonly used in the def- ing Scientist Programme in 2006/2007 at ISSI, Bern. RT thankfully inition of the rms gyroradius as either vt = 2T/me or recognizes the assistance of the ISSI librarians, Andrea Fischer and vt = T/me. p Irmela Schweizer. p 4 R. A. Treumann and W. Baumjohann: Gyroradius

References

Baumjohann, W. & Treumann, R. A.: Basic Space Plasma Physics, Revised Edition, Imperial College Press, London 2012. Harris, E. G.: On a plasma sheath separating regions of oppositely directed magnetic fields, Nuovo Cim. 23, 115-121, 1962. Landau L. D., Diamagnetismus der Metalle, Zeitschrift f¨ur Physik 64, 629-637 (1930). Landau L. D. & Lifshitz E. M., Quantum Mechanics (Pergamon Press, New York, 1965) ch. 15. Nakamura, T. K., Covariant thermodynamics of an object with finite volume, Phys. Lett. A 352, 175-177, doi: 10.1016/j.physleta.2005.11.070, 2006. Sonnerup, B. U. O.:¨ Magnetic field reconnection, in: Solar system plasma physics, Vol. III., pp. 45-108, edited by: Lanzerotti, L. T., Kennel, C. F., and Parker, E. N., North-Holland Publ. Comp., Amsterdam-New York, 1979.