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1DRAFTCollisions between electrons and ionsFelix I. ParraRudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK(This version is of 10 April 2019)1. IntroductionThe Fokker-Planck collision operator can be simplified significantly when we considercollisions between electrons and ions. The simplification is a result of the large massdifference. Assuming that the ion and electron temperatures are of the same order,Te Ti ,(1.1)we find that the thermal speed of the electrons vte is much larger than the thermal speedof the ions vti ,rrrTeTimevti vte ,(1.2)vte memimiwhere me and mi are the mass of electrons and ions, respectively. We will exploit thisdifference in characteristic velocities to study collisions between electrons and ions.2. Simple estimatesBefore attempting a rigorous derivation, we give some simple estimates. We considerelectrons of charge e and mass me colliding with ions with charge Ze and mass mi .Naively, to have a collision, the distance bei between the electron and the ion must besuch that the kinetic energy of the electron is of the order of the potential energy due tothe Coulomb force between the particles,1Ze22me vte Te .24π 0 bei(2.1)Thus, only electrons at a distancebei Ze24π 0 Te(2.2)of ions have significant collisions with those ions.The characteristic collision frequency (the inverse of the time between collisions) canthen be estimated using bei . An electron moves a mean free path λei before it encountersan ion. Since the electron only notices ions at a distance bei , it samples a volume πb2ei λeias it move a mean free path. In this volume, the probability of finding an ion is of orderunity, so it must satisfy ni πb2ei λei 1. From this estimate we obtain the mean free pathλei 1(4π 0 )2 Te2 .ni πb2eiZ 2 e4 ni(2.3)With the mean free path, we obtain the collision frequencyνei vteZ 2 e4 ni .1/2 3/2λei(4π 0 )2 me Te(2.4)

2Felix I. ParraThis estimate ignores the fact that weak collisions between particles separated by theDebye length λD dominate. To include the effect of these weak collisions, we only needto recall that we estimated the effect of weak collisions to be larger than the effect ofcollisions between particles at a distance bei by a Coulomb logarithm ln Λei 1. A largereffect is roughly equivalent to more collisions, that is, it is equivalent to a larger collisionfrequency. Then, we need to multiply (2.4) by a factor of ln Λei to obtainνei Z 2 e4 ni ln Λei1/23/2(4π 0 )2 me Te.(2.5)Similar estimates give us the typical collision frequency of electron-electron collisions andion-ion collisions,νee e4 ne ln Λee1/23/2(4π 0 )2 me Te(2.6)andνii Z 4 e4 ni ln Λii1/23/2(4π 0 )2 mi Ti.(2.7)The estimate that led to (2.5) is not valid for the effect of ion-electron collisions onions. It is true that electrons and ions collide often, but the effect of a single collision onan ion is small. Due to conservation of momentum, if the change of the electron velocityin a collision is ve , the change to the ion velocity isrmememe vi ve vte vti vti .(2.8)mimimiTherefore, a single electron-ion collision only modifies the ion velocity by a small amountof the order of (me /mi )1/2 vti vti . This type of collision that only changes the velocityby a small amount can be thought of as a random walk in velocity space. To achievea total change in the ion velocity of the order of vti , we need a large number Nc ofelectron-ion collisions, 2vtimiNc 1.(2.9) vi meThen, the effective collision frequency of ion-electron collisions is 1/Nc smaller than theelectron-ion collision frequency,1/2νie νeimeZ 2 e4 ni me ln Λei νei .3/2Ncmi(4π 0 )2 mi Te(2.10)Combining equations (2.5), (2.6), (2.7) and (2.10), and assuming Te Ti , Z 1 andne ni , we obtainrmemeνei νie νei .(2.11)νee νei νii mimiThe electrons collide with electrons as often as they collide with ions. Ions collide withions much more rarely, and ions are affected by their collisions with the light electronsonly after a time much longer than the time between collisions with other ions.

Collisions between electrons and ions33. Electron-ion collision operatorWepions. We perform an expansionp start with the effect on electrons of collisions within me /mi 1. We first consider the lowest order in me /mi 1, and later we keephigher order terms.p3.1. Electron-ion collision to lowest order in me /mi 1The electron-ion Fokker-Planck collision operator is#)(Z"γeife (v)fi (v0 )3 00Cei [fe , fi ] v · g g g · v fe (v) v0 fi (v ) d v .memm e {z } i {z } fi fe / me Te(3.1) fi fe / mi TiTo find the order of magnitude estimate, we have used v fe fe /vte and v0 fi fi /vti .0The term m 1i fe (v) v 0 fi (v ) is then negligible. Moreover, we findg {z}v {z}v0 ' v, vte(3.2) vtileading tov 2 I vv.(3.3) g g g ' v v v v3RWiththis result, and using fi (v0 ) d3 v 0 ni , equation (3.1) becomes to lowest order inpme /mi 1 2 v I vvγei ni v ·· v fe .(3.4)Cei [fe , fi ] ' Lei [fe ] m2ev3This approximate operator is known as Lorentz collision operator or pitch-angle scatteringcollision operator.To understand the pitch-angle scattering operator, we rewrite it using the sphericalcoordinates {v, α, β} in velocity space, shown in figure 1. In the orthonormal basis {v̂ v/v, α̂ v α/ v α , β̂ v β/ v β }, the gradient and divergence with respect to thevelocity of general functions f and Γ are v f v v f f f f1 f1 f v α v β v̂ α̂ β̂ v α β vv αv sin α β(3.5)and 1 (J Γ · v v) (J Γ · v α) (J Γ · v β)J v α β 1 1 1 2v 2 Γ · v̂ (sin α Γ · α̂) Γ · β̂ ,v vv sin α αv sin α β v · Γ (3.6)where J det[ v/ (v, α, β)] [ v v · ( v α v β)] 1 v 2 sin α is the determinantof the Jacobian of the transformation v(v, α, β). Using (3.5) and (3.6), equation (3.4)becomes γei ni1 fe1 2 feLei [fe ] 2 3sin α .(3.7)me v sin α α αsin2 α β 2The Lorentz operator diffuses the distribution function in α and β, but leaves its structurein v unchanged. The reason for this lack of diffusion in v is that electrons do not changethe magnitude of its velocity when they collide with heavy ions. According to (2.8), thevelocity of the ion barely changes in a collision with an electron. Then, the ion kinetic

4Felix I. Parraẑv̂ˆv ˆ ŷx̂Figure 1. Spherical coordinates {v, α, β} in velocity space. The orthonormal basis{v̂ v/v, α̂ v α/ v α , β̂ v β/ v β } is also sketched.energy does not change, and since the total kinetic energy of both the electron and the ionis conserved, the kinetic energy of the electron is the same before and after the collision.Thus, only the direction of the electron velocity changes after a collision, leading to thediffusion in α and β seen in (3.7).The Lorentz operator tends to make the electron distribution function isotropic, thatis, it tends to give a function fe that is only a function of v and not of α or β. To show thisproperty, we prove that the Lorentz operator satisfies its own H-theorem. The entropyproduction due to the Lorentz operator isZLσ̇ei ln fe Lei [fe ].(3.8)Using (3.4) and integrating by parts, we obtainZγei niv 2 I vvLσ̇ei fe v ln fe ·· v ln fe d3 v2mev3Z2fev · v ln feγei nid3 v lnf v vem2evv2" 2 2 #Zγei nife fe1 fe d3 v 0.m2ev3 αsin2 α β(3.9)LLThus, the entropy grows until σ̇ei 0. The entropy production σ̇eivanishes only when v ln fe is proportional to v, that is, when fe (v) is only a function of the velocity magnitude. Interestingly, we did not need to add the entropy production of the ions due toelectron-ion collisions to show that the entropy increases. The isotropizationp process isthen independent of the ion distribution function because to this order in me /mi 1,the ions seem just stationary particles compared to the fast electrons.3.2. Electron-ion collision to first order inpme /mi 1We have argued in (2.11) that the electron-ion collisions are much more frequent thanother types of collisions. Thus, it is usual to have an electron distribution function that

5Collisions between electrons and ionspis isotropic to lowest order in me /mi 1,fe (v) fe0 (v) fe1 (v) . . . {z } q{z }isotropic memi(3.10)fe0If this is the case,pwe need to continue the expansion of the electron-ion collision operatorto next order in me /mi 1. For g g g, instead of the lowest order approximationin (3.3), we keep the next order correction to find g g g M(g) M(v v0 ) ' M(v) v0 · v M(v) v v v v0 · v v v v. (3.11)Substituting this result and the expansion in (3.10) into (3.1), we obtain(Z ": 0 due to isotropy γeifi (v0 ) v · v v· f(v) v v v · v fe1 (v)Cei [fe , fi ] ' vv e0meme #) f (v)e0003 0 v · v v v v · v fe0 (v) v v v · v0 fi (v ) d v .mi(3.12)Using1 fe0v,v vZZZfi (v0 ) d3 v 0 ni ,fi (v0 )v0 d3 v 0 ni ui , v0 fi (v0 ) d3 v 0 0, v fe0 (v) and(3.13)(3.14)the electron-ion collision operator in (3.12) can be rewritten asγei ni v ·Cei [fe , fi ] 'm2e!1 fe0 v v v · v fe1 ui · v v v v · v .v v(3.15)One further useful manipulation is:0 *· I v v. v v v v · v v ( v v vv v · v) v vv v(3.16)With this result equation (3.15) finally becomes" #γei niui fe0Cei [fe , fi ] ' v · v v v · v fe1 m2ev v" # γei niv · ui fe0v · ui fe0 v · v v v · v fe1 Lei fe1 .m2ev vv v(3.17)The electron-electron collisions are usually as frequent as the electron-ion collisions (see(2.11)), and as a result, it is usually the case that the lowest order electron distributionfunction is not only isotropic, but also Maxwellian, 3/2 me v 2mefe0 (v) fM e (v) neexp .(3.18)2πTe2TeIf this is the case, equation (3.17) becomes me v · uiCei [fe , fi ] ' Lei fe1 fM e .Te(3.19)

6Felix I. ParraWe have seen that the Lorentz operator tends to make the distribution functionisotropic. Thus, the collision operator in (3.17) will givev · ui fe0.(3.20)v vwhere ge1 (v) is isotropic. When calculating the total distribution function fe ' fe0 (v) fe1 (v), we can absorb the isotropic correction ge1 (v) into the lowest order isotropic distribution function fe0 (v), leading tofe1 (v) ge1 (v) fe0v · ui fe0' fe0 (v) ( v ui v)' fe0 ( v ui ).(3.21)v v vThen, the electron-ion collisions tend to give an electron distribution function that isisotropic around the average velocity of the ions ui .We proceed to calculate the collisional friction force and the collisional energy exchangeusing (3.17).fe (v) ' fe0 (v) 3.2.1. Electron-ion collisional friction forceThe collisional force on the electrons isZFei me v Cei [fe , fi ] d3 v.(3.22)Substituting equation (3.17) into this expression, and integrating by parts, we find Zγei ni*·I v · f ui fe0 d3 v vFei v vv e1 vmev v Zγei niui fe0 v v v · v fe1 d3 v.(3.23)mev vIntegrating by parts the first term in the integral, we find Z γei niv 2 I vv fe0Fei fe1 2v v v ·ud3 v.imev4 v(3.24)To simplify the integral further, we use that fe0 (v) is isotropic, and we take the integral inthe spherical coordinates sketched in figure 1. Since v v[sin α(cos β x̂ sin β ŷ) cos α ẑ],the integral over the angles α and β givesZ πZ 2πv21v2dα(3.25)dβ sin αvv (x̂x̂ ŷŷ ẑẑ) I.4π 0330Then,Z v 2 I vv fe0 38π fe08πfe0 (0)dv Idv I.v4 v3 v30Using this expression, and employing h v i v · v v · v v 2v v v v [ v · ( v v)] v v · v vvv2 3 v·v22v v 3 v 3,vvvvZ(3.26)(3.27)equation (3.24) becomesγei niFei me 8πfe0 (0)ui 23Z v3fe1 d v .v3(3.28)