Kinetic Processes in Recombining

H3+ plasmas

Rainer JohnsenUniversity of Pittsburgh

THEO MURPHY INTERNATIONAL SCIENTIFIC MEETING ONChemistry, astronomy and physics of H3

+

February 2012 The Kavli Royal Society International Centre

See also my poster: “Old and new recombination and spectroscopic studies in H3

+ afterglows”

What is the source of H3 (D3)emissions in discharges/afterglows?

•Dabrowski & Herzberg: H3+ recombination ( Coll. Rad. ?)

•Miderski an Gellene: H5+ + e- → H3

*+ H2 (line broadening)

•Amano (D2 plasmas): D5+ + e- → D3

*+ D2 (some lines broadened, some not)

Why does H3+ recombine faster in afterglows than in storage rings (theory)?

•Neutral assisted three-body recombination? Mechanism?

•Collisional radiative recombination?

•Presence of H5+ ?

How does it all fit together?

A quick look at a few afterglow studies.H3

+ ions recombine faster in afterglows than in storage rings, more so at low temperatures.

Corrected for three-body recombination

An important study that changed the history of H3+ recombination.

Amano: Afterglow of a hollow cathode discharge in pure [H2]~ 5x1016 cm-3 Monitored decay of H3

+(v=1) by optical absorption

Tgas = 77 KTe= 110 K ??

Amano’s recombination plot looks perfect. Compatible with binary recombinationConclusion: a(110 K)~4.3x10-7 cm3/s

Note: High initial ne= 5x1011 cm-3

But: 1. The H3

+ ions should have converted to H5+ (which recombines 20 times faster!)

2. Collisional radiative recombination (CRR) should have been a big effect

Same as above but 3-body H3+ to H5+10 times slower

This uses the rate of Hiraoka & Kebarle,

(H3+ +2H2 H5+ + H2k3= 1e-28 cm6/s

(H3+ +2H2 H5+ + H2k3= 1e-29 cm6/s

This uses 10% of rate of Hiraoka & Kebarle,

3 2 2 5 2H H H H H

5 2 3 2 2H H H H H

At 77 K the reaction:

is fast!

It’s inverse

is slow!

Something must have broken up H5+

First problem

Same as above but 3-body H3+ to H5+10 times slower

This uses the rate of Hiraoka & Kebarle

(H3+ +2H2 H5+ + H2k3= 1e-28 cm6/s

(H3+ +2H2 H5+ + H2k3= 1e-29 cm6/s

This uses 10% of the rate of Hiraoka & Kebarle

3 2 2 5 2H H H H H

5 2 3 2 2H H H H H

At 77 K the reaction:

is fast!

It’s inverse

is slow!

Something must have broken up H5+ !?

First problemExpected H5

+ effect

Expected H5+ effect

Tentative answer:The plasma contained vibrationally excited H2

3 2 2 5 2H H H H H

The reaction

equilibrates, but at a temperature higher than the kinetic temperature

3 3 5 5

3 5

( ) ( )

1eff f H f H

f f

a a a

The recombination coefficient for the mixture

is much higher! f5=4% would double the rate!

H5+ is a weakly bound ion.

Binding energy ~ 0.35 eV

5 2 3 2 2( )H H vib H H H

0 10 20 30 40 50 600.00E+00

5.00E-12

1.00E-11

1.50E-11

2.00E-11

Amano, H2 afterglow

datawithout heating

time (microsec)

1/[H

3+]

Second problemCRR should have had a big effect. Why did it not?

CRR at fixed Te=110 K

The “Stevefelt* formula” gives the “effective binary” CRR rate coefficient as:

10 0.63 9 2.18 0.37 3 1CRR e

9 4.5e e e e1.55 10 6 103.8 10 cm sT T Tn na

collisions radiation correction term

At low Te and high ne, the first term dominates.

Looks very simple!But the electron temperature is not really an independent, known variable!

(Bates, Byron et al pointed this out long time ago)

* J. Stevefelt, J. Boulmer, and J-F. Delpech, Phys. Rev. A 12, 1246 (1975)

Energy released per recombined ion:Ionization potential of the state corresponding to the bottleneck

Here: n~ 8, corresponding to ~ 13.6/ 64 ~ 0.2 eV

kT

dE n-changing collisions

radiation

CRR heats the electron gas

0 10 20 30 40 50 601.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

kij A(i)

Quantum # i (or n)ra

te [s

-̂1]

ne = 1011 Te =77 K

collisions

radiation

“bottleneck”

free

9 9/2 3 1 3

9 2 3/2

1/2 3/2

3.8 10 [ ]

3 33.2 10 ( ) ( )2 2

23 ln[ ( ) ] (Coulomblogarithm)

2 3 3( )2 2

CRRe e

e ione e e ion

ion

e e

e gas ee coll e ion

gas

coll

dUT n E eVs cm

dt

dUn kT kT kT

dt m

n kT

dU mn f kT kTdt m

f e gas collision frequency

d

Electron heating and cooling in CRR recombination.

U= internal energy of the electron gas

Heat input from CRR

Heat transfer to ions

Heat transfer to neutrals

Energy released per recombined ion

In steady state : heat input=heat loss

0 10 20 30 40 50 600.00E+00

5.00E-12

1.00E-11

1.50E-11

2.00E-11

Amano, H2 afterglow

with heatingdatawithout heating

time (microsec)

1/[H

3+]

If one takes the heating into account, CRR looks “binary”

The rate-limit is the rate of “electron cooling” in electron-ion collisionsThe electrons must cool before they can recombine!

Fixed Te

Variable Te

Conclusion 1:

•Recombination in a low-temperature (~100 K) pure H2 afterglow (ne> ~1011occurs mainly by CRR and H5

+ recombination.

•The experiments don’t prove or disprove that H3+

binary recombination is “fast”.

•The H3 emission and absorption spectra most like arise from CRR and H5+

The next question is how the line shapes and their rotational dependence are to be understood. We assume that D 3 is supplied to each state through two distinct paths:a direct supply from the dissociative recombination of D5

+ and a cascade from the upper states. The molecules cascaded down from the upper states have less kinetic energy, resulting in smaller Doppler widths.

* Amano, T. and Chan, M-C. (2000) Infrared absorption spectroscopy of D3: an investigation into the formation mechanism of triatomic hydrogenic species Phil. Trans. R. Soc. Lond. A358, 2457-2470

We found that no substantial increase of the absorption intensity was achieved at liquid nitrogen temperature. If the formation process is the dissociative recombination of D5

+ , a much more conspicuous temperature dependence is likely to show up.

This largely agrees with the conclusion drawn from spectroscopic observations.Two quotes from Amano and Chan (2000)*

But they had some doubts:

Recombination is enhanced by helium.The binary recombination coefficient is obtained from the [He] → 0 limit

Afterglows in mixtures of He, Ar, and H2

Very extensive data from the Prague group

FALP and AISA data from PragueDependence on [H2]

Not OK

OK

Why not extrapolate to [H2] =0?

The H3+ formation becomes rate-limiting!

Ar+ + H2→ArH++ H and ArH++ H2→Ar+ H3+ (10-9 cm3/s) will take ~ 10 msec at [H2] = 1011 cm-3

Recombination of an ion with a= 10-7 cm3/s (at ne= 1010 cm-3) takes time 1/( a ne)=1 msec. (It would work if a actually were much smaller but does not rule out a= 10-7 cm3/s )

[H2] must be high enough to produce H3+

much faster than it is lost by recombination.

Otherwise, H3+ is not the dominant ions

2

2 3

3

Ar H ArH H

ArH H Ar H

H e products

[H2]=1x 1012 cm3/s

[H2]=1x 1011 cm3/s

eH3

+

eH3

+

Enough

Not enough

Production:

Loss:

Conclusion 2

1. There is good evidence for the dependence a(H3+) on [He]

2. But not for a dependence of a(H3+)* on [H2] below 1012 cm-3**

Notes: • Possible loophole: H3

+ para/ortho ratio depends on [H2], and para and ortho recombine with different rates (not likely at 300 K)

** This does not exclude a possible dependence of a(H3+) on [H2] at higher densities

How do we explain the dependence on [He]?

The “classical” neutral-stabilized mechanism is too slow

• Energy transfer in e- He collision is inefficient.• Momentum transfer is more efficient

• Look at angular momentum mixing “l-mixing”

Observed

kT

~ - 4 kT

l-mixing

Binary capture

autoionization

Low l High l

Third-body effects can enhance recombination only if they stabilize states that can still be re-ionized. (by collisions or autoionization)At low temperatures, low n states cannot be ionized and their stabilization by third bodies is not effective

Rydberg states

3-body capture Collisional ionization

Third-body- assisted recombination mechanisms

continuum

bound

*3 3 ( , )e H M H n l M

*/2 33

3

[ ( )] ( )[ ]

nE kTth

e

H n K n n eH n

2 1/2( / (2 )th eh m kT

Quantitative model of collisional dissociative recombination with l-mixing:

High Rydbergs (n>12) are formed very fast by three-body capture:

Their concentration is estimated from the Saha equilibrium

max

min

( ) ( )n

sn

K n na

Irreversible destruction of these states enhances the recombination rate coefficient by:

Irreversible destruction involves:•l-mixing, either by electrons or atoms, into low l-states that predissociate•Chemical reactions with neutral molecules (H2)

2 5 3, 04.4 [ / ]mix e ek v a n cm s

, ,' 'e mix e e mixk v

5 3, 2.7

13.1 10 [ / ]mix Hek cm sn

Note:The rates are for l-mixing from a given l to any other l’. The rates from a given l to a particular l’ other than l are smaller by 1(n2-1)

Dutta, S.K., Feldbaum, D., Walz-Flannigan, A., Guest, J.R. and Raithel, G. 2001, "High-angular-momentum states in cold Rydberg gases", Physical Review Letters, vol. 86, no. 18, pp. 3993-3996.

Hickman, A.P. 1978, "Theory of angular momentum mixing in Rydberg-atom-rare-gas collisions", Physical Review A, vol. 18, no. 4, pp. 1339-1342.

l-mixing rates

By electrons:

By helium:

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

[He] 1017 cm-3

aeff

[107

cm3/]

Observed dependence of the H3+ recombination coefficient at T=300 K on the

experimental helium density. Squares and triangles; data from Glosik (2009). Cross: from Leu at al, 1973. The line indicates the density dependence expected from the model described in the text.

Model vs. data

3 3

3 3

3

*( , )

*( , ) *( , )

*( , ) ??

rotational capture

H e H p nl mixingH p n He H l n He

H l n stabilized

Model of Glosik…. Greene, Kookouline

300 K

He-assisted Model of Glosik…. Greene, Kookouline

Rotational capture of an electron into n>40 , followed by l-mixing with helium, and eventual stabilization

Comments on the Glosik et al He-assisted model

•The assumed l-mixing rates for high n (>40) are too large

•States with n>40 are in Saha equlibrium, no further l-mixing is needed

It may work for rotational capture into lower Rydberg states

But: It can contribute only if these resonances do not dissociate, as assumed

However, if they actually do dissociate, that should leave a trace in the storage-ring data.So, let’s look

A. Petrignani et al , PHYSICAL REVIEW A 83, 032711 (2011)

Low energy peaks in storage-ring data

Lifetime graph for rotational resonances [blue for paraH3+ (1,1) to (2,1)]

Low-energy structureof the storage-ring data

There seems to be a correlation between the peaks and valleys

Note: The “A” peak has no counterpart. Maybe it comes from rotational excitation (2,1) to (3,1)

The “A” peak does not seem to change when para enriched H2 is used in the ISR

Conclusion Speculation

•H3+ recombination around 77 K is largely due to para H3

+

•Two rotational resonances contribute (1,1) to (2,1) and (2,1) to (3,1)

•The storage-ring thermal rate at 77 K may be a bit large (too much para in (2,1)?)

On the other hand:Something “problematic”* may be happening in the storage rings

*Petrignani et al , PHYSICAL REVIEW A 83, 032711 (2011)

“Our [afterglow] measurements indicate that the de-ionization coefficient in the range (1.5 -2.5 x 10-7 cm3/s may be appropriate for modeling H3

+ plasmas of reasonably high densities and perhaps in some planetary atmospheres.It is an entirely different question which recombination coefficient should be used in environments of very low density, e.g. in the interstellar medium. Here, the true binary recombination coefficient at low temperatures is needed. It is conceivable that the recombination cross sections measured in ion storage rings are close to the true values, but this is far from obvious, since the presence of electric fields in the interaction region may also lead to l-mixing effects”

“A working hypothesis concerning the low-energy discrepancy between theory and experiment in Fig. 5a is that something problematic must be occurring in the treatment or the detection of the very highest Rydberg states in theory or experiment.”

“………….. in this case what should be explored is the possibility that either the Rydberg states are destroyed or the angular momentum quantum numbers are changed by external fields in the storage ring”

“Presently no rate coefficient measurement with a confirmed temperature below 300 K exists“.

1995: T. Gougousi, M.F. Golde, and R. Johnsen, Int. J. Mass Spectr. 149, 131 (1995)

2011: A. Petrignani et al., Phys. Rev. A 83, 032711 (2011)

We have made tremendous progress, but the H3+ enigma is still with us

Extra slides

Direct “intervention” in the DR of H3+ by third bodies, helium in particular

The very extensive studies by the Prague group leave little doubt that the presence of helium in the afterglow enhances recombination

Helium has little effect on the ioncomposition of the plasma

Apparently, it changes H3+ Rydbergstates by mixing angular momentumquantum numbers.

Only approximate models are available

From H3+ para/ortho ms.

Why does the three-body rate decrease below~150 K?

The same is found for both ortho and para ions

The reason may be that the tree-body mechanism involves low –n RydbergsStabilization comes at the expense of collisional ionization which slows down at low T

The plasma contains free electrons of positive energy

But also Rydbergs in Saha equilibrium for E < ~-4kT

0 20 40 60 80 100 1201.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

ionization time

ionization time

n

time [s]

T=300 K, ne=1e11

Johnsen, Huang, Biondi JCP 65, 1539 (1976)

3 2 5H H M H M

Equilibrium constant vs T

Example:T=200 K, [H2] ~ 1014 cm-3

[H5+]/ [H3

+]~ 4%

About 50% of recombination loss would be due to H5

+ !

Worse at lower T and higher [H2]

3 2 2 5 2H H H H H 3 2 5 2H H H H

Hiraoka &KebarleJCP 63, 746 ( 1975)

Johnsen, Huang, Biondi JCP 65, 1539 (1976)

But: The chemical equilibrium is approached fairly slowly. One needs to make a model the afterglow! The relevant rates are fairly well known.

50 100 150 200 250 300 3501.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

CRR rate coefficient

1.00E+091.00E+101.00E+111.00E+12

Te [K]

a CRR

ne [cm-3]

9 4.5 10 0.63 9 2.18 0.37 3 1CRR e e e e e3.8 10 1.55 10 6 10 cm sT n T T na

[cm3/s]

Collisional radiative recombination (CRR) heats the electron gas:

When n drops below a critical value (“bottleneck”).radiative transitions dominate.

kT

Delta E

3-body capture re-ionizationFree electrons

Bound electronsn-changing collisions

*3 3H e e H e