Ultra-Cold Plasma: Ion Motion

Transcription

1 Ultra-Cold Plasma: Ion Motion F. Robicheaux Physics Department, Auburn University Collaborator: James D. Hanson This work supported by the DOE. Discussion w/ experimentalists: Rolston, Roberts, Killian, Gallagher, Gould, Eyler, Raithel PRL 88, (2002) & Phys Plasma 10, 2217 (2003)

2 General References Ultracold Plasmas Come of Age P. Gould & E. Eyler, Physics World, Mar Ultracold Plasmas and Rydberg Gases S. Bergeson & T. Killian, Physics World, Feb Physics World, 03 Magneto-optical trap: cool and trap atoms T = µk n ~ cm 3

3 Cool Xe in MOT T = 10 µk Photoionize e e e e e T i = 10 µk T e = Κ < r 2 > = 340 µm n < 2 X 10 9 cm 3 e e e

4 Schematic of PE seen by electron t 0 = 0 t 1 I 10 ns t 2 I µs Inertia and low temperature of the ions slows expansion of the plasma.

5 Electron Signal vs Time Increasing applied field after 1.8 µs Low density = no bound electrons Higher density of ions to bottom After 2 µs, e bound to space charge (plasma)

6 Fraction of Electrons Trapped vs N for several T e Fewer electrons trapped at higher T e Scale by threshold N, does not depend on T e

7 Plasma Parameters Electron temperature and density: 1 K < T e < 1000 K, n e < 10 9 cm 3 (laser frequency) Ion temperature and density: T i = 10 µk, n i > n e (cooling) Average volume occupied by one electron: 4πa 3 /3 = 1/n a = (3/4πn) 1/3 (laser power) For n e = 10 9 cm 3 : a = 6.2 µm << size of plasma Coulomb coupling parameter: Γ e I <PE>/<KE> Γ e = (e 2 /4πε 0 a)/k B T e For T e = 100 K & n e = 10 9 cm 3 : Γ e = For T e = 1 K & n e = 10 9 cm 3 : Γ e = 2.7

8 Plasma Parameters Avg electron speed: v=(3k B T e /m e ) 1/2, 10 K = 2.1 X 10 4 m/s Size of the plasma: r ~ 350 µm Time for electron to cross plasma: 2 r/v ~ 33 ns Collision thermalization time: 1/τ =n e v[e 4 ln(λ)/4πε 02 v 4 m e2 ] Λ = 4πε 0 3k B T e λ D /e 2 ~ 1/θ min >>1 For T e = 100 K & n e = 10 9 cm 3 : ln(λ) = 6.0 ; τ = µs For T e = 10 K & n e = 10 9 cm 3 : ln(λ) = 2.5 ; τ = µs Ion-electron thermalization (m i /m e ) τ Not important! Evaporation of electrons. How cold do electrons get?!

9 Plasma Parameters Three body recombination rate (e + e + A + e + A * ): Γ = 2 X m 6 s 1 n e n i (ev/k B T e ) 9/2 For T e = 50 K & n e = 10 9 cm 3 : Γ = 10 4 µs 1 For T e = 10 K & n e = 10 9 cm 3 : Γ = 0.1 µs 1 For T e = 1 K & n e = 10 9 cm 3 : Γ = 4000 µs 1 Recombination into states bound by ~2k B T e (size of atom ~ distance between ions at 1 K!) Electron-atom scattering rate = n v σ is proportional to T e -3/2 For T e = 10 K & n e = 10 9 cm 3 : 1.9 X 10 8 Hz

10 Use plasma oscillation to measure t dependence of density Plasma frequency: f p = (e 2 n e /ε 0 m e ) 1/2 /2π For n e = 10 9 cm 3 : f p = 280 MHz RF generator For n e = 10 7 cm 3 : f p = 28 MHz laser e e e e e e e detector

11 Density decrease Why? Thermal pressure from electrons gives expansion Electrons heat ions causes expansion Three body recombination

12 Density drops like n~n/(v 0 t) 3 v 0 decreases with T e Mechanism? Recombination ruled out Thermal pressure from electrons Electrons heat ions

13 Straight line is conservation of energy: all of the electron thermal energy into radial ion motion. Rules out thermalization Why plasma expands so fast for T e < 50 K? Complicated plasma? Γ e ~1/3 at E e = 15 K

14 Thermalization e-e collisions rotates v 1 v 2 and leaves v 1 +v 2 unchanged on time scale τ = ns e-ion collisions change electron s direction & transfer thermal energy to ion on time scale τ e-i =τ (m i /m e )~100 µs Evaporation of electrons cools plasma. How cold does plasma get during expansion? Theory usually by Fokker-Planck equation; Monte Carlo approximation of e-e and e-ion collision dσ/dcos(θ)=(e 4 /8πε 02 µ 2 v 4 )/(1 cos(θ)+ ) 2

15 Thermalization (implementation) If the differential cross section is chosen to be dσ/dcos(θ)=(e 4 /8πε 02 µ 2 v 4 )/(1 cos(θ)+ ) 2 with = 2 e -1 /Λ 2 then the <δθ 2 > will be correct. Use this differential cross section to give a total cross section and a scattering rate. Pick 2 electrons determine if a scattering occurs using a random number 0<x<1. If a scattering occurs, then rotate the relative velocity vector using a random number 0<y<1 and 0<ϕ<2π y = 1 σ cos( θ ) dσ d cos( θ ) d cos( θ ) -1

16 Fokker-Planck/Monte Carlo Comparison Uniform spatial distribution

17 Fokker-Planck/Monte Carlo Comparison

18 Fokker-Planck/Monte Carlo (e-ion) Uniform spatial distribution

19 Velocity Distribution Central 3%

20 Velocity Distribution Central 3%

21 Velocity Distribution Central 3%

22 Velocity Distribution Central 3%

23 Velocity Distribution Central 3%

24 Velocity Distribution Central 3%

25 Velocity Distribution Central 3%

26 T e vs t at different radii in plasma

27 Effect of ion motion ion mass 100X real real ion mass

28 Early Time Electron Flux 60 K, 0.4 X 10 9 cm -1, 340 µm 18 mv/cm applied field Convolved calculation 200 ns (2 ticks) Shifted expt 150 ns (1.5 ticks) Scaled expt 15% Changes within extpl uncertainties Correct ratio peak/tail Correct decay > 1 ms Dashed line (unpub expt J Roberts & S Rolston [Thanks!]) Solid line calc (Monte Carlo code)

29 Flux of escaping electrons

30 Number of remaining electrons

31 Average Density

32 Plasma Expansion (ion motion) The ion expansion is apparent even in the crudest approximations. Treat the expansion using the same ideas in Ion Acoustic Waves. r r qi r e V(r) n e(r, t) ni (r, t) C exp e k B Te r k B Te r V(r, t) = ln[ n (r, t) ] i + cons e r r k B T r e r E(r, t) = - r ni (r, t) e n (r, t) i

33 Acceleration vs r Inverse logic: n e = α exp[ev(r)/k B T e ] ev(r) = k B T e ln(n e )+const = k B T e ln(n i )+const Solid model; dotted 15 K; dashed 30 K; dot-dash 75 K

34 Gaussian Expansion The equations for the ion density and velocity flow: r n(r, t) r r r r + [n(r, t) v(r, t)] = 0 t r r v(r, t) r r r r r k B Te (t) r r + [v(r, t) ]v(r, t) = - ln n(r, t) t m i [ ] At this level, an initial Gaussian density remains Gaussian: r n(r, r r v(r, t) t) β(t) = N π r = γ (t)r 3/ 2 exp[ - β (t) r 2 ]

35 Gaussian Expansion The equations for the ion density and velocity flow: dβ (t) = - 2 β (t) γ (t) dt dγ (t) 2 k B Te (t) β (t) + γ (t) = 2 dt m 3 2 k B T e (0) = 3 2 k B T At large t, γ ~ 1/t and β ~ 1/t 2 e (t) i m i 2 γ (t) β (t) This is ballistic expansion with the asymptotic value of γ 2 /β fixed by energy conservation. + E Ryd

36 Thermal Equilibrium Electrons are assumed to be in thermal equilibrium; use the results from Monte Carlo runs to estimate initial T e and number of escaped electrons 2 V(r) = - e ε 0 ni (r) α exp ev(r) k B T e Fluid equation for ions with no self pressure Conservation of energy put in explicitly: determines T e

37 Thermal Equilibrium Problem There are more ions than electrons in the plasma because some electrons escape. V(r) = N e/(4 π ε 0 r) at large r The Maxwell-Boltzmann expression for the electron density goes to small constant as r B. This gives a small fraction of B electrons outside of the plasma!!! Thermal equilibrium is not established outside of the plasma. Artificially have T go smoothly to 0 at a finite distance.

38 2 Thermal Equilibrium Method There are several methods that can be used to solve U(r) = - e ε 2 0 [ n (r) n (r)] n (r) = C exp i e Iteration: Guess n e, Compute U, Compute new n e by adding only small fraction of exp(u/k B T e ), REPEAT Newton s method: Guess U 0 and compute n e,0, write U = U 0 + δu and only keep first order terms in the equation, solve the linear differential equation for δu, REPEAT e U(r) k B T e

39 Density vs r

40 Comparison with Monte Carlo

41 Comparison with Monte Carlo Electron T vs t starting with <n> = 10 9 cm -3, T e = 100 K, <r 2 > 1/2 = 340 µm Solid = Monte Carlo, Dashed = Thermal Approximation

42 Acceleration vs r Inverse logic: n e = α exp[ev(r)/k B T e ] ev(r) = k B T e ln(n e )+const = k B T e ln(n i )+const Solid model; dotted 15 K; dashed 30 K; dot-dash 75 K

43 Results asymptotic expansion + experiment; Solid conservation of energy All thermal energy converted into ion expansion energy Recombination needed

44 Ion Expansion

45 Ion Expansion

46 Acceleration vs r Solid model; dotted 15 K; dashed 30 K; dot-dash 75 K

47 Ion Expansion 10 9 cm -3, 100 K, t = 4.6 µs

48

49 Ion Acoustic Waves

50 Ion Acoustic Waves ω 2 = (k B T e /m i ) k 2

51 Scaling of Ion Acoustic Waves Solid line 10 µs, Dashed line 12 µs n i = A exp(-βr 2 ) [1 + b cos(kr)] β= 1.5/(340 µm) 2 b = 0.2 k = 6 π/340 µm T e = 100 K

52 Fixing the Ion Fluid Equation All plasmas expanding into vacuum develop singularities at finite time. The singularities are simply phase space effects. Can fix the fluid equation by not using it! Have ion particles distributed in r so that matches the initial density function. The particles move F = m a. Compute the ion density from the ion particles. Compute the E-field from the density.

53 e detector At late times ramp E-field to measure number of Rydberg (highly excited) atoms made E-field plates laser e e e e e e E

54 Prompt Plasma Exp Rydberg atoms More deeply bound atoms require larger field to strip electron from atom Rydberg atoms account for extra expansion energy?

55 Excess plasma expansion energy roughly equals energy in Rydberg atoms Electron-ion recombination source of extra energy Recombination mechanism?

56 Rydberg atom distribution at various times 10 K corresponds to princ quant # ~ 125 Rydberg distribution at late times is strange # of atoms decreases then increases? Mechanism?

57 Electron signal vs time for Rydberg atoms made in plasma Large t means large field More recombination at higher density---ok More deeply bound at higher density---??? More deeply bound at lower T e ---???

58 Three Body Recombination? Measured recombination increases with density---ok Measured Rydberg distribution gives more deeply bound as T e decreases---??? Measured Rydberg distribution gives more deeply bound as electron density increases---??? Number of recombined atoms has strange time dependence---???

59 Ion Rydberg Particles The ion particles are useful for treating TBR. First determine whether a recombination happened during that time step, then determine which ion captured an electron and into which state. The probability that a recombination occurred: P = δt C T -9/2 0 r 2 n i (r) n 2 e (r) The r position of the recombination is determined by a random number 0<x<1 (the ion particle closest to r) x = δt C T P -9/2 r 0 ~ r 2 n i ( ~ r) n 2 e ( ~ r) dr d ~ r

60 Ion Rydberg Particles The probability that the atom forms in the n-manifold, ν, is proportional to ν 6. To a good approximation the n- state is from the random number 0<y<1 ν = [y ν *7 + (n 0-1/2) 7 ] 1/7-1/2 where ν* = (27.2 ev/4 k B T e ) 1/2 and n 0 is a small number (e.g. 6). The energy from TBR is added into the electron thermal energy and the number of electrons is reduced by 1.

61 Results asymptotic expansion + experiment; Solid conservation of energy; Dotted include three body recombination Too much recombination; wrong distribution of atoms; E=0?

62 Processes with Rydberg Particles A Rydberg particle does not have force on center of mass. Moves with the speed of the ion when TBR occurred. Two processes need to be followed: radiative decay and electron-rydberg scattering. Probability for photon emission computed P = A tot δt; photon is emitted if random number 0<x<1 is less than P. The final state is found from a second random number 0<y<1. A ni -1 n tot = An n y An n /A i = f i f tot n = 1 n = 1 f f

63 Processes with Rydberg Particles The transitions due to electron collisions are computed using a Monte Carlo technique. A transition occurs if a random number 0<x<1 is less than the scattering probability P = <v σ> n e (r) δt for that BE and T e. The final energy of the atom is computed from a random number 0<y<1 using the differential rate y = E f - d v σ de de / - d v σ de de Energy lost/gained by the atom is added/removed from the electron plasma. If E f > 0, an electron is added to the plasma.

64 Results asymptotic expansion + experiment; Solid conservation of energy; Dotted include three body recombination; Dashed also e-rydberg scattering Geometric collision rate ~ 40 MHz Fewer recombinations because each gives more energy More deeply bound for lower T e More deeply bound for larger n e

65 Self Consistency, Γ e = PE/KE Size of Rydberg/a = Γ e /2 Simple plasma Γ e << 1 Plasma expansion decreases T e faster than a increases => increase Γ e Recombination increases a and increases T e => decrease Γ e Can estimate Γ e ~ 1/6 for these parameters

66 Rydberg Distribution vs T e Experiment Killian et al Calculation Solid E e = 100 K Dashed E e = 200 K

67 Number of Rydberg atoms vs t Experiment Killian et al T e = 6 K Calculation T e = 6 K n e = 10 9 cm -3 At low T e recombination gives Rydberg atoms that heat plasma. Removes Rydberg bound by less than (2-4)k B T e

68 Summary of Results Classic Plasma & Atomic Physics Recombination mechanism: three body recombination plus subsequent electron-rydberg scattering to get atomic distribution & extra energy/recombination More deeply bound atoms for lower T e because cold plasmas expand more slowly at low T e Coulomb coupling parameter does not get large no need to invoke exotic atomic or plasma processes Rydberg plasma transition energy from electron- Rydberg scattering (Virginia experiments)?????????

69 Predictions All plasma dynamics should vary smoothly with initial electron energy; even as E 0 and below Velocity of Rydberg atom depends on velocity of ion when recombination occurs; should be relationship between velocity of Rydberg atom and binding energy For spherical plasmas, temperature will rapidly increase or decrease to give Γ e ~ 1/5; this gives an estimated electron temperature k B T e ~ 5e 2 /4πε 0 a Rydberg atoms are present in Rydberg plasma experiments but too diffuse in binding E to measure???

70 Future Magnetic fields affect recombination and plasma motion (anti-hydrogen expts, UMich, ) Precise comparisons between experiments and simulations Test of assumptions local heating from RF, size of atoms in molecular dynamics, Different geometries cylinders or sheets Other plasma phenomena ion acoustic waves, expansion into vacuum,