Collisional Processes Long range interaction --- between ions/electrons and ions/electrons; Coulomb 1/r Intermediate range interaction --- between ions/electrons and neutral atoms/molecules; Induced dipole 1/r 4 Short range interaction --- between neutrals, 1/r 6
In general, for a two-body collision, A + B Products, the reaction rate per unit volume = n A n B <σ v> AB, where the rate coefficient is < σv > AB = Z 0 σ AB vf(v) dv [cm 3 s -1 ] v = relative velocity between A and B σ AB (v) = reaction cross section; velocity depedent f(v) = velocity distribution function
In thermal equilibrium μ f v dv =4π( 2πkT )3/2 e μv2 /2kT v 2 dv In terms of energy Z < σv > AB =( 8kT )1/2 σ AB(E) E de e E/kT πμ 0 kt kt If the density is high, e.g., in the Earth s atmosphere, three-body collision may become important, A+B+C + Products, the reaction rate per unit volume = k ABC n A n B n C where k ABC is the three-body collisional rate coefficient [cm 6 s -1 ]
Elastic scattering by an inverse- square force, e.g., Rutherford scattering Exact solutions complicated; use the impact tapproximation, i.e., motion in a straight line Assumption: constant velocity during the encounter between the target and the projectile Question: How much momentum is transferred ( Question: How much momentum is transferred ( direction)? Figure from Draine s book b: impact parameter; closest distance v 1 : relative velocity
Impact Approximation Coulomb force Z 1eZ 2 e Z 1Z 2 e 2 F 3 = (b/ cos θ) 2 cos θ = b 2 cos θ Interaction time scale dt = d(b tan θ) v 1 = v b dθ 1 cos 2 θ Total momentum transfer is Z p = F dt = Z 1Z 2 e 2 bv 1 = 2Z 1Z 2 e 2 Z 1Z 2 e 2 b bv 1 b 2 Force at closest distance Z π/2 v 1 Time scale π/2 cos θ dθ
In a collisional ionization, there must be enough momentum transfer. (E=p 2 /2m) Fast moving (1/2) m e v 2 >> E I e So, E 2Z 1Z 2 e 2 ( P ) 2 > 2mE I ( ) 2 > 2mEE I bv 1 b 2 <b 2 max (v) =(2Z 1Z 2 e 2 ) 2 v 2 1 2mE I 2 2 Z 2 4 = 2Z2 p e 4 m e v 2 E I and the ionization cross section becomes 2 4 σ(v) πb 2 max = 2πZ2 p e4 m e v 2 E I This is ok if v.
For v 1 2 min = 2 m e v min = E I < σv > = = Z σ(v)vf(v) ( ) ) dv Z 2πZp 2 e 4 v 4π ( m e )3/2 v 2 e m ev 2 /2kT dv v min m e v 2 E I 2πkT = Z 2 ( 8π p me kt E I )1/2 e4 E e E I/kT For an H atom at level n, E I = 13.6 [ev]/n 2, so for a large n, e.g., n~100, and T~10 4 K, E I (<< kt) in radio frequencies. < σv > 1 E I n 2, so is very large. I
Deflection Timescale Net momentum transfer < d ( P 2 ) > dt Z 1 e - Z 2 e, n [cm -3 ] There must be a range of distance, for which 2 b min = Z 1 Z 2 e 2 /Energy, and b max L D (Debye length) In plasma, the distributions of ions and electrons are correlated because of charge neutrality. Near a proton more electrons than protons the proton is shielded Average charge within a region <Q(L D )> = -e L D e - p + e - kt L D =( 4πn e e 2)1/2 =690T 1/2 n e 4 [ cm 3 ] 1/2 [cm] e - e - e -
< d dt ( P ) 2 > n 2 v 1 ln Λ So Λ is large, b max /b min 20 35 in ISM conditions. For elastic scattering of electrons by ions, weak distant encounters (>> atomic scales) weak distant encounters (>> atomic scales) more important than close encounters.
If an electron comes in in atomic dimensions, the atom is suddenly perturbed transision deexcitation line radiation < σv > 10 = 8.629 10 8 Ω 10 [cm 3 s 1 ] T4 g 1 where Ω 10 is collision strength. (i) almost independent of T for T < T 4 = 10 4 K (ii) 1 < Ω 10 < 10
Ion-Neutral Collisions Interaction potential U(r) = 1 2 α N Z 2 e 2 r 4 r 4 Neutral is polarized. - + Dipole moment ~ P = α N ~ E where α N is polarizability a few a 0. a 0 =Bohrradius h2 m e e 2 =5.292 10 9 [cm]
For such a potential, if b < b 0, the deflection cross section is large. σ = π b 2 02 1/v, and the rate coefficient <σ v> T Usually if T, σ The ion-neutral reactions are important in cool ISM.
Electron-Neutral Collisions In low-ionization ISM (e.g., protoplanetary disks) ions are rare. e -neutral (H 2, He) scattering is important. e -H 2 scattering (1) If E < 0.044 ev pure elastic scattering (2) If E > 0.044 044 ev rotational excitation possible (3) If E > 0.5 ev vibrational excitation possible (4) If E > 11 ev electronic excitation By experiment σ ' 7.3 10 6 E ( 01 ) [cm2 ] 0.01eV 9 T 0.68 3 1 and < σv >' 4.8 10 ( 10 2 K ) [cm s ]
Neutral-Neutral Collisions Repulsive if distance Weakly attractive if distance van der Waals interactions (mutual induced electric dipole) U(r) r 6 Hard sphere OK; radii R ~ 1Ǻ b < R 1 + R 2 σ = π (R 1 + R 2 ) 2 1.2 10-15 [cm 2 ] 1/2 1/2 < σv >=1.81 10 10 T ( 2 1 10 2 K )1/2 ( m H μ )1/2 ( R 1 + R 2 2Å )2 [cm 3 s 1 ]
Collision Gas (hydrogen atoms) root-mean-squared speed For H I regions, Cross sections σ Hard sphere OK for neutral atoms, i.e., physical cross section
Cross sections σ For free e -, p + Conventional unit for cross section
Collision σ v t n v # of collisions = # of particles in the (moving) volume # of collisions per unit time = Time (mean-free time) between 2 consecutive collisions (N=1) = Mean-free path ` = vt ` = 1 collision, i.e.,` n σ
Ex 1 Collisions are indeed very rare. Ex 2 Ex 3