Chapter 3. Coulomb collisions

Transcription

1 Chapter 3 Coulomb collisions Coulomb collisions are long-range scattering events between charged particles due to the mutual exchange of the Coulomb force. Where do they occur, and why they are of interest? energy redistribution among particles of the same species or between di erent species equilibrium vs non equilibrium fluid description vs kinetic description slowing-down of fast particles (e.g. charged fusion products, externally injected energetic particles) transport processes (e.g. thermal conduction, viscosity) electric resistivity emission and absorption of radiation Most relevant di erences with respect to collisions in a gas continous and simultaneous : how do we define mean free path, collision frequency, etc.?

2 3 in a magnetised plasma collisions allow transport across B lines, whereas in a gas they inhibit transport Figure 3.1: Series of collisions in a gas and in a plasma In a gas, where molecules move at constant speed between two close-encounter collisions, the mean free path l is defined as a function of number density n and cross-section l = 1 n (3.1) In a plasma, the particle undergoes a continous deflection due to the simultaneous

3 4 Coulomb interaction with many particles. As such, we cannot define the mean free path in the same way as Eq. 3.1, but we define an e ective mean free path as the average distance travelled by particle ending with a 90 o deflection with respect to the original direction. In this chapter we will present a simplified description, which anyhow is able to reproduce all the correct functional dependencies and scaling laws. For rigorous and in-depth description of Coulomb collisions in a plasma please refer to: Sivukhin, D.V. (1966), Coulomb collisions in fully ionized plasma inreview of Plasma Physics (ed. A,M. Leontovich), IV, , Consultants Bureau, New York. adetaileddescriptionofrutherfordcross-sectioncanbefound,forinstance, in: L. Landau and E. Lifshitz, Fisica Teorica, vol. 1 - Meccanica, 19, Ed. Mir, Moscow (1982) M. Bertolotti, T. Papa, D. Sette, Guida alla soluzione di problemi di fisica II, Ed. Veschi/Masson, p. 156 it can be shown that the classical derivation presented in these notes agrees with the quantum mechanical approach: see e.g. L. Landau and E. Lifshitz, Fisica Teorica, vol. 3 - Meccanica Quantistica, 135, p. 655, Ed. Riuniti (1976)

4 3.1 Single binary collision Single binary collision Scattering of a projectile particle P o a target particle F. We recall that in an elastic collision the modulus of the relative velocity is conserved, so that in the centre of mass frame we have a deflection only (see Appenix A). projectile Figure 3.2: binary Coulomb collision Here F is a fixed scattering centre, i.e. its mass M!1. The obtained results can be extended to the general case of finite mass M by substitution of projectile mass m with the reduced mass m r = mm/(m + M), and by changing the projectile velocity v with the relative velocity v r. Particle trajectory is a hyperbola, with focus F and =scatteringangle =anglebetweenhyperbolaaxisandasymptote b = impact parameter

5 3.1 Single binary collision 6 =minimumapproachdistancebetweenprojectileandtarget Recalling the geometric properties of a hyperbola, we can state = FA = b cot 2 (3.2) angular momentum conservation with respect to F bv = v A ) v A = v tan 2 (3.3) energy conservation between P (t = 1) anda where 1 2 mv2 = 1 2 mv2 A + qq ) va 2 = v 2 2qQ 4 " 0 4 " 0 m Inserting Eq.3.3 in 3.5 we obtain ) v 2 A = v 2 b 0 v 2 ) v 2 A = v 2 1 v 2 tan 2 2 = v2 1 cos 2 2 sin 2 2 cos 2 2 = b 0 b b 0 b tan 2 (3.4) (3.5) b 0 = 2qQ 4 " 0 1 mv 2 (3.6) b 0 b tan 2 sin 2 cos 2 It can be seen in Fig.3.4 that =( becomes ) 1 tan 2 2 = b 0 b tan 2 ) cos cos 2 = b 0 b sin 2 ) (3.7) tan = 2b b 0 (3.8) tan 2 = b 0 2b = qq 4 " 0 1 mv 2 b )/2, hence tan =cot /2, and Eq.3.8 (3.9) See Appendix C for a relation between this last equation and Rutherford crosssection.

6 3.1 Single binary collision Meaning of b 0 parameter (Landau distance) Let s simply rewrite Eq.3.6 in this way b 0 = qq mv 2 4 " 0 2, (3.10) which cleary indicates that b 0 is the distance at which the initial kinetic energy of the projectile is equal to the electrostatic potential energy. The projectile can reach this distance in the case of head-on collision only, when the impact parameter b is zero. Figure 3.3: Scattering angle as a function of impact parameter for a given b, collisionsbecome harder withincreasingb 0 for the general formula, make the substitution mv 2! m r v 2 r small-deflection collisions are much more frequent than large-deflection collisions (dn / bdb). we can arbritrarily define a threshold and distinghish collisions in b<b 0 : large-deflection collisions b>b 0 : small-deflection collisions

7 3.1 Single binary collision 8 aquantummechanicalapproachindicatesthatweshouldreplace b 0! max(b 0, debroglie) also the plasma screening implies that a particle interacts via the Coulomb force only with particles within its Debye sphere, so that b< D Large-angle deflections in an ideal plasma are typically the result of a series of smallangle deflections. Let c be the frequency of large-angle deflection as a result of many small-angle deflections. Let cr be the frequency of large-angle deflection as a result of a single deflection. Then it can be shown (see, e.g., Pucella-Segre, pp ) that c D =2ln =2ln, (3.11) cr b min where the Coulomb logarithm ln (see hereafter) is typically in the range Hence the main contribution comes from collisions with small-angle deflection, whereas hard Coulomb collisions are rare events with respect to these. The impact parameter can assume values in the range max(b 0, debroglie) =b min apple b apple b max = D (3.12) large-angle defl. small-angle defl. no Coulomb interaction Figure 3.4: deflection type as a function of impact parameter

8 3.2 Simultaneous multiple collisions in a plasma Simultaneous multiple collisions in a plasma Let s study the cumulative e ect of many Coulomb collisions in the small-angle regime. In small angle approximation, Eq. 3.9 becomes tan 2 ' 2 = b 0 2b ) ' b 0 b (3.13) the variation of v in the perpendicular direction is therefore v? = v? ' v ' v b 0 b (3.14) After a large number N of collisions we have that the average deflection is almost θ Figure 3.5: small-angle deflection approximation zero NX i=1 v?i ' 0 (3.15) because this corresponds to just the most probable deflection with respect to the instantaneous velocity direction. We are instead interested in assessing the cumulative e ect of many collosions with respect to a given initial direction. Hence we study the spread in velocity directions given by NX i=1 which increases with the number of collisions. ( v?i ) 2 ' N( v? ) 2 ' Nv 2 b2 0 b 2, (3.16) Focussing our attention to a given time interval t, the number of collisions for a particle with velocity v with impact parameter within b and b + db is dn(b) =n 2 bdbv t, (3.17)

9 3.2 Simultaneous multiple collisions in a plasma 10 where n is the number density of scattering centres. The corresponding cumulative velocity spread is NX i=1 ( v?i ) 2 ' n 2 bdbv 3 t b2 0 b 2, (3.18) and integrating over all possible impact parameters we obtain where ln bmax b min NX ( v?i ) 2 ' 2 nv 3 tb 2 0 i=1 Z bmax b min db b =2 nv3 tb 2 0 ln b max b min, (3.19) =ln isthecoulomblogarithm,whichdependsonseveralplasma parameters, but it is anyhow in the range 5 to 20 for the plasmas considered in this course Collision time and collision frequency We define the collision time as the time interval necessary for a particle to be deflected (on average) by 90 o,i.e. whenthecumulativespreadvelocityisofthe order of v 2. Hence, using Eq.3.19, we solve for in NX ( v?i ) 2 ' 2 nv 3 b 2 0 ln ' v 2 (3.20) i=1 and we obtain the collision frequency as. =1 2 nvb 2 0 ln (3.21) which can be written using 3.6 as = 1 =2 nvb2 0 ln, (3.22) = 1 2 " 2 0 q 2 Q 2 n ln, (3.23) m 2 v3 Let s rewrite this last equation for the general case of a particle of species scattering o particles of species with number density n : = 1 2 " 2 0 q q m 2 v3 r n ln, (3.24)

10 3.2 Simultaneous multiple collisions in a plasma 11 where m = m m /(m + m )isthereducedmass,andv r is a characteristic relative velocity for the two considered species. We will consider in the following the electron-electron collisions, the electron-ion collisions, and the ion-ion collisions. Table presents the parameters corresponding the various cases. We assume that the plasma is fully ionised, and that the two species, ions and electrons, are in thermal equilibrium within themselves, so that the typical velocities are the respective thermal velocities v th,e = p 3k B T e /m e and v th,i = p 3k B T i /m i.inordertoestimatethecharacteristicrelativevelocityv r, we observe that in general v e v i. For instance, we have ee ' 1 2 " 2 0 e 4 m 1/2 e (3k B T e ) 3/2 n e ln ee (3.25) ' 4, n e [m 3 ] (T e [ev]) 3/2 ln ee Hz (3.26) ' 1, n e [m 3 ] (T e [kev]) 3/2 ln ee Hz (3.27) ee = 1 ee ' 7, (T e[kev]) 3/2 n e [m 3 ]ln ee s (3.28) Table 3.1: typical parameters for the collision frequency for a fully ionised plasma collision charge m v r density e! e e 4 m e /2 v th,e n e = Zn i e! i e 4 Z 2 m e v th,e n i i! i e 4 Z 4 m i /2 v th,i n i Comparing the various collision frequencies we can state that r 3/2 ee : ei : ii = Z : Z 2 me : Z 4 Te (3.29) m i T i so that typically the dominating frequency is the ei collision frequency. In the special, but very important case, of fully ionised Hydrogen plasma (Z = 1) in thermal equilibrium (T e = T i ) we have ee : ei : ii =1:1:1/43 (3.30)

11 3.2 Simultaneous multiple collisions in a plasma 12 so that ee and ei collisions occur at the same frequency, much larger than the ii collision frequency. Finally we can estimate the characteristic electron mean free path (for 90 o deflection) as e ' v e ei ' 2, (T e [kev]) 2 n e [m 3 ]Z ln ei m (3.31) Table reports the electron mean free path for three Hydrogen plasmas of interest. In all three cases, the collisions are important, since the plasma confinement time is much larger than the collision time. It is important to notice that in the case of a tokamak, the electron mean free path is much larger than the linear dimensions of the plasma, but collisions are still important due to the fact that particle trajectories are winding around the machine. Table 3.2: typical electron mean free path for H plasma (ln ' 10). n e [m 3 ] T e [kev] e[m] ionosphere tokamak ICF burning plasma , Energy exchange characteristic times Coulomb collisions lead to a re-distribution of energy among particles. As such, ee collisions make the electron distribution to relax to a Maxwellian velocity distribution, for which a single parameter, i.e. the temperature T e,canbeusedtodescribe the whole distribution. The same happens for the ion species via ii collisions. Energy exchange and thermal equilibrium between electrons and ions is achieved via ei collisions. All these processes evolve on a di erent timescale, which we want to estimate. Let s consider collisions between and species. The characteristic time E for

12 3.2 Simultaneous multiple collisions in a plasma 13 energy exchange can be defined as E ' N coll, (3.32) where N coll is the number of collision needed to transfer an amount of energy E, and is the collision time. From a head-on elastic collision, we can obtain the order of magnitude of energy transfer in a single collision < E> ' E head on = 4 m /m (1 + m /m ) 2 E (3.33) So that N coll ' E/ < E>,andinsertingthisinEq.3.32weobtain E ' (1 + m /m ) 2 4 m /m (3.34) The relevant energy exchange times are Using these equations and Eq.3.29, we have (T e ' T i ) E ee ' ee = 1 ee (3.35) ei E m i ' ei = 1 m i 4m e ee 4m e (3.36) ii E ' ii = 1 ii (3.37) E ee : E ii : E ei =1: r 1 mi : 1 m i, (3.38) Z 3 m e Z 4m e which typically means that E ee E ii E ei. Hence electron and ions relax to amaxwellianmuchfasterthantheyreachthermalequilibriumbetweenthetwo species. In cases where there are heating or cooling processes happening on a timescale shorter than E ei,butlongerthan E ee and E ii,wecanconsidertheplasmaasmadeup by two species, each one in thermal equilibrium, but having a di erent temperature, i.e. T e 6= T i. In this case, we can model the plasma as a single fluid with two temperatures, andthevolumetricrateofenergyexchangereads de e!i dt = 3 n 2 ek B (T e T i ) ei E (3.39)

13 3.2 Simultaneous multiple collisions in a plasma 14 Practical formulas for energy exchange rate are ee E ' 1, Te 3/2 [kev] s (3.40) n e [m 3 ]ln ee E ei ' A Z where A is the ion mass number, and Z the atomic number. Te 3/2 [kev] s (3.41) n e [m 3 ]ln ei Table compares the thermal equilibrium times for equimolar DT plasma. It is worthwhile to notice that the equilibrium times are typically longer than other chacteristic times for these plasmas, so that in several cases we can assume T e 6= T i. Table 3.3: typical thermal equilibrium times for equimolar DT plasma (Z = 1;A = 2,5; ln ' 10). ei E n e [m 3 ] T e =1keV T e =10keV tokamak ,025 s 0,8s mid range density ns 800 ns ICF burning plasma ,25 ps 8ps

14 3.3 Appendix A: elastic collision properties Appendix A: elastic collision properties In an elastic collision the modulus of relative velocity is conserved. Hence in the CoM frame the elastic collision is just a deflection. Let s consider two point masses m 1 and m 2 moving respectively with velocity v 1 and v 2 in the laboratory frame. The kinetic energy of the two colliding masses can be written (König theorem) T = 1 2 (m 1 + m 2 ) v 2 c m 1 v 2 1c m 2 v 2 2c, (3.42) where v c is the CoM velocity, and v 1c and v 2c the point mass velocities with respect to the CoM: v c = m 1 m 1 + m 2 v 1 + m 2 m 2 m 1 + m 2 v 2 (3.43) v 1c = v 1 v c = (v 1 v 2 )= v r (3.44) m 1 + m 2 m 1 + m 2 m 1 m 1 v 2c = v 2 v c = (v 2 v 1 )= v r (3.45) m 1 + m 2 m 1 + m 2 m m 1 v1c m 2 v2c 2 = 1 m 1 m 2 m 2 vr m 2 m 1 m 1 vr 2 (3.46) 2 m 1 + m 2 m 1 + m 2 2 m 1 + m 2 m 1 + m 2 = 1 m 1 m 2 vr 2 = 1 2 m 1 + m 2 2 m r vr 2, (3.47) where v r is the relative velocity, and m r is the reduced mass. Hence Eq becomes T = 1 2 (m 1 + m 2 ) v 2 c m r v 2 r (3.48) Since T is conserved (elastic collision), and v c does not change (internal forces only), v r is conserved. Therefore we have a deflection of relative velocity only.

15 3.4 Appendix B: geometry of Rutherford scattering Appendix B: geometry of Rutherford scattering Figure 3.6: Geometric properties of a hyperbola

16 3.5 Appendix C: Rutherford scattering cross-section Appendix C: Rutherford scattering cross-section The di erential cross section for axially simmetric interactions can be written d d = 2 bdb 2 sin d = b sin db d (3.49) scattering center Figure 3.7: Scattering diagram (source: Wikipedia) Di erentiating Eq.3.9 we obtain Hence we have 1 d cos 2 2 = b 0 2b db ) d = 4sin2 2 db (3.50) 2 b 2 0 d d = b b 0 1 = b sin 4sin 2 sin b 0 4sin b 0 1 b 0 = 2tan 2sin cos 4sin (3.51) (3.52) = b sin qq 1 = " 0 m 2 v 4 sin 4 2 (3.53), (3.54) which is the Rutherford cross section.

17 NOTATION Symbol Definition Units or Value c Speed of Light m s 1 k B Boltzmann s constant 1, JK 1 h Planck s constant 6, Js N A Avogadro s number 6, mol e Elementary charge 1, C 0 Vacuum permittivity 8, Fm 1 µ 0 Vacuum permeability 1, Hm 1 Z Atomic number A Atomic mass g mol 1 m e Electron mass 0,511 MeV/c 2 9, kg m p Proton mass 938,27 MeV/c 2 1, kg m p /m e mass ratio eV correspondsto anenergyof1, J atemperatureof11600k aradiationwavelengthof1,24 µm 1