Absorption or emission during a collision : a test case H+ 2

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Absorption or emission during a collision : a test case H+ 2

C. Stehlé, N. Feautrier

To cite this version:

C. Stehlé, N. Feautrier. Absorption or emission during a collision : a test case H+ 2. Journal de Physique, 1986, 47 (6), pp.1015-1020. �10.1051/jphys:019860047060101500�. �jpa-00210277�

Absorption

^or

^emission during

^a

collision :

a test case

H+2

C. Stehlé and N. Feautrier

Département d’Astrophysique Fondamentale,

Observatoire de Paris, Section de Meudon, 5, place ^Jules Janssen, ⁹²¹⁹⁰ Meudon, ^France (Reçu le 27 decembre 1985, accepté ^le 18 février 1986)

Résumé. 2014 La théorie couplée des collisions en présence ^d’un champ ^de rayonnement peut ^{être utilisée} pour décrire les processus d’absorption ^ou d’émission pendant ^une collision. Adoptant ^un formalisme semi-classique

de collision, ^nous pouvons étudier en fonction du temps l’ évolution d’un complexe collisionnel. Le cas de l’hydro- gène atomique perturbé par des protons ^est particulièrement intéressant car le problème collisionnel est soluble exactement pour ^un grand domaine de distances interatomiques. ^L’étude faite dans le cas particulier ^{de la raie} Lyman ^03B1 permet ^de déterminer le temps caractéristique ^de l’absorption ^{ou de} l’émission. Les limites « impact »

et « quasistatiques ^{» sont ensuite} analysées.

Abstract ²⁰¹⁴ The close coupled theory ^of scattering ^{in a radiation field} may be used to describe absorption ^or

emission processes occurring during ^a collision. Using ^a semi-classical description ^{of the collision we} study ^the

evolution of the collisional complex. ^{The case of atomic} hydrogen perturbed by protons ^is particularly interesting

since the exact solution of the dynamical problem is known for a large ^{domain of} internuclear distances. In the

particular ^{case of the} Ly ^{03B1 line we determine the} characteristic time of the absorption ^{or the emission.} The « impact »

and ^« quasistatic » ^{limits are then} analysed

Classification

Physics ^Abstracts

32.70 ^- 32.90

1. Introduction.

The influence of collisions on atomic radiative _pro-

cesses includes a large ^amount of various

phenomena :

line

broadening,

redistribution of

light,

^collision

induced

absorption

^or emission, ^radiative

charge

transfer... The

major

part ^{of these} processes may be

interpreted

^{in term} of radiative interactions

during

a collision.

Recently,

much attention has been

paid

to these radiative collisional reactions and it has been shown that the collisional formalism is well

adapted

for

expressing

^the

properties

of the radiation as

functions of the radiative S-matrix elements and cross

sections

[[1-3]

^and references listed in

[3]].

^{In the case}

of line

broadening

^{the same}

description

is available outside the line _core,

corresponding

^{to the «one}

perturber » approximation

[4] ^which

requires

^the

condition

ð’tc.ðro >

^{1 where}

AT,,

denotes the interval time between two collisions and Aco the

detuning

^from

the line centre.’ The radiative _process is then connected with the

dynamics

^{of the}

binary

system

(atom

⁺

perturber)

^on time scales ðw - 1 smaller than

AT,

[5].

Our _purpose here is to

study

the evolution of the collisional complex ^{in order to determine}

quantita- tively

the characteristic time of the

absorption

^{or the}

emission. We

adopt

^a semi-classical

description

^{of the}

collision which makes the

physical interpretation

easier.

We focus our attention on the Ly a ^line

wings

^of

hydrogen

^atoms

perturbed by

protons but similar discussion _may be

performed

for other

colliding

partners

(A. Spielfiedel,

^{1985, a}

paraitre).

^Hereafter

we shall assume a

complete degeneracy

^{of the H levels}

which is

justified

^for

large detunings

^{and we make}

the usual no

quenching approximation.

^{In the}

dipolar approximation

for the electrostatic interaction, ^the

solution of the

dynamical problem

^{is exact}

[6]

^without

the traditional adiabatic

assumption

^or any numerical computation. ^It follows that the

study

of the time

dependence

of the radiative process is easy. ^In addition the

long

range of ^{H -} H+ interaction

(linear

^Stark

effect) yields

the collision duration _{T, very}

large

^so

that the

probability

^of

absorption

^{or emission}

during

the collision is

particularly important leading

^to

possible

consequences for redistribution of

light [7].

After a short

description

^{of the} method, ^{we recall}

briefly

^the determination of exact wave functions of ^an

hydrogen

^atom perturbed

by charged particles

^and

we

give

^an

expression

^{for the «} cumulated »

absorption

or emission

probability

^{at time T, i.e. the}

probability

that a radiative transition occurs in the time interval between the

beginning

of the collisions and the time T

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047060101500

1016

when the collision is not

necessarily

achieved At the end of the collision (T -+

oo)

^this

probability

^{is related}

to the total

absorption

^{or emission rate. Hence our}

purpose is to

analyse

^which part of the collision has

given ^the

major

contribution to this rate. This allows

a rough determination of AT, ^the duration of the interaction between the radiation field and the col-

liding

partners ^{which cannot} exceed the collisional duration. The

large

variations of AT with the

detuning

are

responsible

of the different behaviours of the

absorption

^{or the emission rate : the two}

limiting

^cases

(small

^and

large detunings)

^are

analysed

2. Method.

Following

Yakovlenko

[1],

^we

adopt

^{a molecular}

description

of the collision and treat the radiative collisional reaction as a transition between two

adiabatic states

Ii> and If>

^{of the}

complete (colliding particles

^{A and P +} monochromatic field

g)

system.

For

simplicity,

^{we assume} that P is structureless but the same method can be

applied

^{in the}

general

^case.

The method is

given

^{here for}

absorption

^{but similar}

conclusions would be drawn for emission. This two state model

gives

^the

principal

features of the _process,

particularly

^{in the}

H’

case as can be seen in the next section. The reaction can be described

by :

The total Hamiltonian can be written :

where

HA

is the Hamiltonian of A,

UAP

^the interaction between A and P,

H.

the Hamiltonian of the free radiation held and

V AP-g

^the interaction between the «

quasimolecule »

^{AP and the field In the absence}

of the radiative field, ^{we define the adiabatic}

quasi-

molecular states

(A

⁺

P);

^and

(A

⁺

P)f.

^{These states}

have

energies Ei(R) = hcoi

⁺

U;(R)

^and

Ef(R) = nWf

⁺

Uf(R) respectively, hcoi

^and

hcof being

^the

energies

^{of the involved atomic states for A and}

U;(R)

^and

Uf(R)

^{the molecular}

potentials

(

i I UAP i) and f I UAp I f >. Ui

^and

Uf

^are functions of time t

throught

^{the relation}

R(t).

Denoting by ) I n. >

^the

eigenfunctions

^{of the free}

field with energy nO) hco, ^we shall take the

products

^of

wave

junctions I i > =- (A

⁺

^P);

^*

^{nw ) and} I f > =-

I ^(A

⁺

^P)f

^*

n. - I >

^{as a basis for the}

complete

system. ^{The total wave} function is

expanded

^{in terms}

of this basis :

where

6;(t)

⁼

Ei(t)

^{+ nhw and}

Sf(/)

⁼

Ef(t)

⁺

^{(n - 1)}

^1iro

are the

energies of I i > and I f >.

The time

dependent Schrodinger equation gives

^the

following

system ^of

equations

^{for the}

amplitudes ai(t)

^and

af(t) :

with the initial conditions

a;( - oo)

^{= 1 and}

af( - oo)

^{= 0.}

In the

dipolar approximation, V(t)

^{can be}

expressed

in terms of the

dipole

^moment

d(t)

^{of the} quasimole-

cule at the distance

R(t)

^{and the}

amplitude

&w ^{of the}

radiative field

V(t) = -

^{1i - 1 d}

E..

^{In the weak field} limit, ^the system

(3)

^{is solved in the} first order pertur- bation

approximation leading

^{to the}

following

expres- sion for the

probability I aif(7) I’

^{to be in state}

^{I f)}

at time T :

Substituting

^the

expression

^of

Bf

^and

8;,

^{we obtain}

the usual

expression :

When T goes ^to

infinity, 1 aif(T) 12 ^gives

^the

probability

of

absorption

^{for a}

given impact

parameter p and a

given

^relative

velocity

^v.

This _very

simple

^adiabatic model breaks down

generally

^{due to} strong ^{non adiabatic effects}

during

the collision. The

colliding

system ^{H + H+ forms an}

exception

^as

long

^as the electrostatic interaction poten- tial is dominated

by

^its

dipolar

part

corresponding

to the

perturbing

^electric microfield in

plasma physics.

Considering

^an

hydrogen

^atom

perturbed by

^{a static}

electric field, ^the

eigenvectors belonging

^{to a fixed}

principal

quantum ^{number n are the well known} Stark states

nil i2 >.

^{These states}

quantified along

the field axis are

analogous

^to adiabatic molecular

states for atom-atom systems. ^{When the}

perturbed

proton moves

along

^its

trajectory (assumed rectilinear),

the electric field rotates

leading

^{to a} rotational

coupling

among the adiabatic Stark states. As was first

pointed

out

by

Lisitsa and Sholin

[6]

^the symmetry

properties

of the

hydrogen

^atom

corresponding

^to the rotation group

04 [8]

^{can be used to obtain exact wave functions}

unn’n" and ^exact energy levels

Enn’n"( t)

^{of the atom in a}

rotating

electric field This exact

diagonalization

^of

the Hamiltonian in the

rotating

system ^{is a conse-}

quence of the same

dependence

^{on R} of the electro- static interaction

potential

and the rotational

coupling.

The construction of the ^wave functions Unn’n" is

explain-

ed in details in

[6]

^{and we}

give

^here

only

^{the method}

using

^{the same notations.}

The result is more

easily

^{obtained in the}

rotating

coordinate system : ^{at each moment t of the} collision,

we define the ^x axis directed along the intemuclear axis and the z axis normal to the collision plane.

The transformation from the laboratory ^{to the}

rotating

axis at time t

corresponds

^to successive rotations

given by

the three Euler

angles (00, 00, V/0) corresponding

to the initial orientation of the intemuclear axis and the

angle 1/I(t)

of rotation around the ^z axis

during

the collision

(4/(0)

⁼

0).

^{For a}

given impact

parameter

p and

velocity

v, the

eigenfunctions

unn,n" ^are obtained from the

parabolic

^{wave functions} unÏ1Î2

(corresponding

to the x axis of

quantization) by

^{means of}

simple

rotations :

Pt

^and

P2

^are

given by

^the relations :

a ⁼

3/2

^ne2

ao/1i

^{measures the}

strength

of the electro- static interaction

potential

^{for an}

hydrogen

^{atom in}

state n. For the energy

Enn’n,,(t)

^{we obtain :}

where a

= 1 + b

^and

^hwn

is the energy of the

atomic state n.

Using the j I nn’n" >

^{diabatic states as}

a basis for the radiative

problem,

^{one can see imme-}

diately

^{that the} transition results in ^{a sum} of two state processes described

by :

This

clearly

shows that the method described above may be used

In the H + H + case, it is

interesting

^{to remark that}

the matrix elements of the

dipole

^moment operator ^d in

the I nn’n" >

^{basis are}

independent

^of time t, ^{so that}

the time

dependence

of the radiative

coupling V(t)

comes from the rotation of d relative to the radiative field

C..

The tensorial

components dk

^{of the}

dipole

moment are known in the molecule fixed frame and the electric field is defined in the

laboratory

system.

Denoting by

ep ^the

polarization

^vector of the radiative field and ek the unit ^vectors in the

rotating

frame,

V(t)

^is

given by :

The time

dependence

of Y is hence

given by

^{the term}

exp[ip(t)]. Using expression (8)

^for

Ef

^and

Ei,

the

probability 1 af(1) p

^{to be in state}

I f) =- I nf nf’ ? >

at time T is obtained from

(4)

^{after an}

angular

average relative to the orientation

(00, 00, 1/10)

of the collision axis : ^-

A summation relative to p ^{has to be}

performed

^{in the case}

of unpolarized

^{radiation fields.}

In the case of the Lyman ^a transition,

expression (10)

^{takes the form :}

where nf and nf

take the values ±

1 j2.

The time

dependence of 1 aif(T) 12

^is

just given by :

with s ⁼ ± 1, ±

(o-f

±

1).

^A

change

^of

sign

^{for both}

Aco and s leads to the same result for

A(T),

^{thus we}

discuss

only

^{the case ð.úJ} > 0. At the end of the collision when T tends to

infinity, A(oo) corresponds

^{to the}

quantity

^2nA

([6] expression (24))

^obtained

by

^Lisitsa

and Sholin

using

^an autocorrelation function method The SIN part ^{of the}

exponential

ⁱⁿ

(12)

^{does not}

contribute to

A(oo)

^{so that the}

only interesting

part of the time

dependence

^{of the} radiative transition is

given by :

1018

P(T)

^{can also be studied} in function of R

or 0

using

the

following

relations : and

This collisional

description

^{of the}

absorption

process allows a very simple

interpretation

^{of the}

quasistatic

limit : when the two

potential

^curves

&i(R)

^and

&f(R)

intersect at

Ro

^the transition

probability

^{may be}

dominated

by

^the contribution of a

region

^{close to}

Ro.

Neglecting

^{the time}

dependence

^{of the}

dipole

^matrix element, ^the

equality Uf - U;

^{= h Am at}

Ro

^cor-

responds

^{to the}

stationary phase

ⁱⁿ

(5).

^A calculation of this

integral by

^{the saddle}

point

^{method leads to the}

usual Landau formula

[9]. By integration

^relative

to p and

averaging

^{over the} Maxwellian distribution of velocities, ^{we obtain the}

quasistatic

formula. A

study

^of the evolution of the system ^{with the time T} allows to

precise

^the

validity

of this limit in the ^caste of

large detunings

and also the

impact

limit for small

detunings.

3. Results.

3.1 LARGE DETUNINGS 4m. - We are

only

^concerned

here with the values of s

leading

^to

stationary

phase (or ^Franck

Condon) points

^{for a}

given

An, ^the oppo- site

sign

^values

giving exponentially decreasing

^con-

tributions to the transition

probability.

Figure

^{la shows} the characteristic features of P in function of R

during

^the collision,

Ro

denotes the Franck Condon

point

^{At first, it} is clear that the transition is

mainly

^{localized near}

Ro.

Large ^variations

of the

phase 4>( t)

of the COS term for other values of R leads to oscillations in P, but these oscillations do not contribute ^{on an} _average to the final

probability.

Fig. ^{la. -} Variations of the transition rate P(7J ^{in units}

of P(oo) ^vs. the intemuclear distance R (atomic units).

8 = 1 + erf = 56.1 ; ^the stationary phase point ^is R0 ~ ^{31.4 ua.}

The arrows refer to increasing ^times along ^the trajectory.

(t2 = (R2 - p2)

^{x 12 484} where t, R, p ^are expressed ⁱⁿ

a.u.).

Fig. ^{1 b. - Same as} figure ^{1 a,} but relative to the transition rate Ps(t)/Ps( (0).

It is

interesting

^to compare this behaviour with the

stationary

phase results in order to obtain the

validity

conditions of it The transition is assumed to be localized at

Ro corresponding

^{to time} to ^and

angle 1/J( to).

At

Ro :

Retaining

^{the two} lowest order terms of the

expansion

of

0(t)

^{near +} to, ^we obtain the

following approximate expression :

The variations of the exact and

approximate phase 4>8(t)

in function of t are shown in

figure

^2.

They

^are

Fig. ^{2. - Time} variations of the phase 0(t) ^{= wt -} et/J( t) along ^the trajectory s ^{= 1} + Qf ⁼ 56.1. Units are rds for the

phase ^and (103 ua) for the time. The broken curve shows the series expansion «Ps(t) ^{in the} vicinity ^{of the} stationary phase points ± to ⁼ ± ^{344.8 au}

both

symmetric

^with

regard

^{to t =} 0, very similar in the

vicinity

^{of t =}

:t to; but they clearly

differ for other values of t.

Assuming

^{that the two} contributions at t ⁼ ± to ^{do not} interfere, ^{the use of}

expression (15)

for

os(t)

^{leads to the}

stationary

phase

expression PS(T)

^for

P(T).

^{When T tends to}

infinity, Ps(7)

is

given by

^a very

simple expression :

The variation of PS in function of R

(Fig.

1 b) presents the same

general

^{behaviour as} P,

particularly

^{in the}

vicinity

^of

Ro.

^{But, as}

expected

from the variation of

os(t)

^{with t, the}

periods

of oscillations are very dif- ferent Nevertheless, this is of little

importance

^since

the final result

P( oo)

^{is not affected}

by

these oscillations.

The variations

of P(oo)

^and

PS(oo)

in function of the

impact

parameter ^{p axe}

compared

ⁱⁿ

figure

^{3 for a}

given

^{Am and}

velocity

^{v. The} agreement ^is

quite good

as long ^{as the}

impact

parameter p is lower than

Ro.

For p >

Ro,

the Franck Condon

point

is located near or inside the classical forbidden

region

^{and the saddle}

point

method becomes invalid

One can see that the

validity

conditions of the quasi-

static limit is

directly

given

by

^{the use} of the saddle

point

method for

calculating

^the

integral

ⁱⁿ

(3).

^At first, ^{if AT} represents ^the time interval near ± to

giving

the main contribution to the

absorption,

^the

corresponding

^{variation of}

os(t)

ⁱⁿ

(15)

^{has to be of the}

order of n

leading

^{to the}

equality :

Fig. ^{3. -} Variations of P(0) ^{= 0.25} P(oo) ^{vs. the} impact parameter p (full curve) compared with the result of the saddle point ^method

(expression

^{(16), broken}

curve)

^and

with the mean value of this expression where the oscillations of SIN2 are weighted ^to 1/2 (dotted curve); ^{8 = 1} + a,.

Units are (10- 27 s - 1) ^{for P and} atomic units for _p.

This characteristic time of the radiative _process

varying

^{as Aw- I becomes} very small for large ^detun-

ings

and this allows an

interpretation

^{in terms of}

«

quasistatic

^{limit ».}

Obviously,

^the

perturber

^moves

along

^its

trajectory

^{and the} expression ^« static » has not the

meaning

^{of «} static ions ».

For the use of the saddle

point

method, ^the contri- bution of values of t outside the interval AT has to be _very small,

corresponding

^{to many} oscillations of

Ø(1)

^{as well for} T -> 0 as for T --> oo. This first condi- tion is realized if

Then the rotation of the

dipole during

the collision becomes

negligible

^{and the}

stationary

phase

point Ro given by

the relation Aco -- 3

Ii /me Rg

^is

independent

of _p and v. The second condition _expresses the neces-

sity

^{for the}

stationary

phase

region

^{to be}

sufficiently

distant from the

turning point

(t ⁼ 0).

Putting AT 4 to

and

assuming

the relation

(18)

realized, ^{we obtain :}

This relation is

closely

^{related to} the usual condition for the

quasistatic approximation

in the line

wings

theories

[10, 11] :

the time of interest åW-l has to be small

compared

^to the collision duration p/v

Ro/v.

At the

stationary

phase

point

^{Aw = 3}

n /me Ro leading

to the

inequality

^{3 h}

Aco/m,, v’ >>

^1.

In the limit of this

binary

model, ^the

quasistatic

limit is obtained for protons ^{as soon} as

I AA

I > ¹ A

at E = 1 eV. For electrons in the same conditions, ^the

criterion would

give

^the

relation I A A I >>

^{100 A which}

is out of the

validity

limit of the use of classical tra-

jectories

^and

dipolar potential.

If the

preceding

conditions are realized, ^{it is}

possible

to

integrate analytically

^the

probability PS( (0)

^over p

(with

^{the mean} value 0.5 for the SIN2

term).

^After

averaging

^over the Maxwellian distribution of the kinetic energy of relative motion, ^we obtain the usual Holtsmark

expression

^{for the}

m-dependent

^{rate of}

transition

which leads to a

profile varying

^{as N I1m-SI2.}

3.2 SMALL DETUNINGS 4b£V. - The

typical

^variation

of the

probability

^P in function of R for I1À = 10-3 A

given

ⁱⁿ

figure

⁴ shows that the transition _may occur

throughout

the collision and is not localized at any distance. The

phase stationary points

^{tend to}

infinity.

For a

dipolar potential,

the collision time is infinite but the

Debye screening

^{at distance} pD due to other

charge

particles

diminishes the potential range ^so that the collision duration is

given by

^{A T a} i

PDIV-

The contribution of

large impact

parameters becomes

predominant corresponding

^{to s = ± 1 in}

(12)

^{so that}

we

point

^{out the}

importance

of the rotation of the

1020

Fig.

^{4. - Same as} figure ^{1a with e = 1 +} af ⁼ 28.56.

dipole (p-term

ⁱⁿ

(11))

^which

gives

^the

major

^contri-

bution. If we introduce the time of interest

Aco- 1,

this time is _very

large compared

^to r, for small values of Aco

according

^to the relation

Aw. ’rc

^{1. In}

parti-

cular, ^{when this}

inequality

^is realized for the half width

AW1/2

^{we obtain the usual}

^validity

^condition

of the

impact

^{limit :}

AW1/2.PD/V

^1. This condition is verified for protons ^{for N} 1014 cm- 3 at 104 K.

In the

impact

limit, ^the

predominance

^{of distant}

collisions allows a second order

perturbational approach previously

^{described for}

Hn

^lines

[12]

^and

leading

^{to for the} _Lyman ^{a line}

where 0 denotes the temperature ^{and N the} proton

density.

4. Conclusion.

We have

presented

^{in this paper a} method for the

study

^{of the}

dynamics

^{of radiative} collisional reactions.

Using

^the

particular

^case

of H + H+, we

^{have shown}

that the characteristic time for the

absorption

^{or the}

emission _may be related to ð.w- 1 : this

quantitative study

^{allows a}

physical interpretation

^{of the so called}

«time of interest ». Furthermore we have

proved

^that

the

quasistatic

^{limit is valid for} protons

for I A A

> ¹ A

and is never valid for electrons. This condition is more

drastic than it is

usually

^asserted [7]. ^We

emphasize

that this limit is

independent

^{of the}

perturber density

as

long

^as simultaneous strong collisions do not occur

(N

¹⁰¹⁴

cm-3).

^Some conclusions

given

here may be extended to more

complicated

systems.

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