Light absorption by a collisional system

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Light absorption by a collisional system

H. van Regemorter

To cite this version:

H. van Regemorter. Light absorption by a collisional system. Journal de Physique, 1987, 48 (8),

pp.1299-1309. �10.1051/jphys:019870048080129900�. �jpa-00210556�

Light absorption by ^a collisional system

H. Van Regemorter

Observatoire de Paris, ⁹²¹⁹⁰ Meudon, ^France

(Requ le 15 décembre 1986, révisé le 18 mars 1987, accepté ^{le 27 avril} 1987)

Résumé.

La théorie des perturbations appliquée à l’étude de l’interaction entre un champ ^{faible et un} système collisionnel binaire permet ^d’obtenir l’expression ^{du taux} d’absorption induit par collisions ^ou le profil spectral d’une transition optiquement permise ^à faible densité. Les fonctions d’onde du système collisionnel ne sont pas affectées par le champ ^{radiatif et} différents développements ^{de ces fonctions sont} utilisés suivant la nature du système: atome-électron ou atome-atome. Des expressions unifiées du profil ^{sont données} qui convergent ^{vers la limite}

impact » pour les petits désaccords 039403C9. L’approche semi-classique ^du problème

collisionnel est donnée pour établir la correspondance ^avec le formalisme de la fonction d’autocorrélation.

Aux grands désaccords ces deux approches ^{sont en défaut et il est} nécessaire d’utiliser un modèle collisionnel

quantique.

Abstract.

Perturbation theory ^is applied ^to the interaction between a binary collisional system ^{and a weak} radiation field to obtain the rate of induced absorption ^{or the line} profile ^{of an} optically allowed transition at low density. The collisional system ^{wave functions are not affected} by the radiation field and appropriate

different expansions of these functions must be used for the two electron-atom or atom-atom systems ^under study. ^Unified expressions ^{of the} profile ^are given which converges ^to the impact ^{limit at small} detunings. ^The

semi-classical approach of the collisional problem ^is given ^to underline the correspondence with the usual autocorrelation formalism. Both approaches break down for large detunings ^{where a} quantum collisional

theory ^{must be used.}

Classification

Physics ^Abstracts

32.70

32.80

1. Introduction.

Most of the problems ⁱⁿ collision induced radiative transition or photons ^assisted scattering phenomena

have been solved within the binary ^model approxi- mation, but many problems ⁱⁿ collisional line

broadening theory, ^which actually ^are genuine ^colli-

sion induced radiative phenomena, ^{have been}

treated in using ^the autocorrelation formalism, ⁱⁿ

order to take account of the simultaneous collisions in time.

At low densities, ^the binary model - for an

optically ^{active atom in} collision with one perturber

-

is valid for the whole profile except the very central part ^{of the} line, ^{at small} detuning åw ^where

simultaneous collisions occur. Moreover when this model breaks down, for Ow of the order of the

impact half width y, Aw Tc is much smaller than

unity ^{when the} density ^is low, ^{the collision duration}

Tc being much smaller than the interval of time between two collisions of the order of y -1: ^the

narrow central part ^{of the} profile ^{has a Lorentz} shape. ^{This is true in} many situations ^{but not in the}

specific ^{case of the} broadening ^of

hydrogenic

lines by charged particles when Tc ^can be very large

even at low density because of the typical very long

range character of the interaction.

For strong ^laser fields, the usual close coupled theory ^of binary atomic collisions has been extended

by Yakovlenko [1] ^and by ^Mies [2] ^{to include the}

radiation field within the binary system. ^{After some} appropriate expansion ^{of the total wave} function of the whole system- coupling ^terms may ^come from the electrostatic or dynamic interactions of the two

body collisions and from V CI) ^the interaction between the collisional complex ^{and the} radiation field.

For the usual weak radiation fields

excluding

the very powerful ^{laser and the} study ^{of non linear}

effects

for ow larger ^{than the} very small Rabi

frequency ow R, first order perturbation theory ^can

be applied ^to the interaction between the binary system ^and light. ^{When the} Hamiltonian of the whole system ^{contains a} strong collisional potential V c ^{and a weak radiative} potential Vw, the so-called

«

distorted wave Bom approximation

^{can be} safely applied ^{and, as we shall see} below, ^a straightforward

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048080129900

expression ^{for an} absorption ^{or emission} probability

can be obtained in terms of the usual collisional

wave functions which are unaffected by the radiation field. Induced radiation occurs during ^the complete

collisional process, but the wave functions of the

binary system ^are given independently of all inter- action with the radiation field. In this sense the collisional problem ^is decoupled ^{from the radiative} problem.

The theory ^{leads to basic} expressions ^{for the line} profile ^which have been used in the past ⁱⁿ particular by ^Jablonski [3] ^{and Mies} [4] ^{for atom-atom colli-}

sions and by the Meudon Group ^{in the} early

seventies for electron-atom collisions (see ^{Van Re-}

gemorter ^{et al.} [5] ^{referred to as} paper I). ^The

formalism is quite ^different from the usual line

broadening theory originally developed by ^Anderson [6] ^{and much} improved ^{later on - see} Baranger [7]

and Allard and Kielkopf [8]. ^The semi-classical

approach given ^below emphasizes ^the difference of

starting point ^{of view.}

On purpose ^{we are} using ^the concept ^of

binary

collision model » and not the concept ^of « quasi

molecular model ». The first includes the second as a

typical ^{case when, for slow atom-atom} collisions, ^the

total wave functions can be expanded ^{in terms of a}

very few

electronic molecular functions » as de- fined in the Born Oppenheimer ^{treatment of}

diatomic molecules

see paper I.

Moreover

the quasi ^{molecular model » has been}

very often associated with the Born Oppenheimer approximation together ^{with the use} of the Franck Condon principle, ^see Szudy ^and Baylis [9]. ^If they

have given many reliable results for far wings studies, ^these approximations break down in a very

large region ^{of the} profile ^where rotational coupling

or Coriolis interactions, mixing ^{the Born} Op- penheimer ^states asymptotically, play ^an important

role as shown in particular by ^Julienne [10].

Within the binary model, ^with appropriate expan- sions in terms of

electronic molecular functions

or of

unperturbed ^{atomic functions}

^{of the}

^colli-

sional complex

^wave function, according the mag- nitude of the velocity ^{of the relative} motion, proper account is taken of all the collisional dynamic, ⁱⁿ particular ^{of the non} adiabatic and inelastic effects.

As already ^{shown this model} gives ^{a continuous} picture ^{of the} absorption ^{or emission} profiles ^from

the

quasi ^static

regime ^{to the} impact regime ^near

the line centre. Moreover it gives ^{the ties between}

the case of slow atomic perturber ^{and electron line} broadening, ^see paper I and Peach [11].

The aim of the present is twofold. Firstly ^to give ⁱⁿ chapter ^{2 the basic} justification ^{of the} quantum expressions ^{for the collision induced} absorption

rates in the case of weak radiation field which are

used in paper ^I. Secondly ^to give ⁱⁿ chapter ^{3 a}

semiclassical translation of the collisional aspects ^of

the theory ^{in order to} give ^{a better} understanding ^of

the correspondence between the binary ^collision

model and the usual line broadening theory. ^The impact ^{limit is} given ⁱⁿ chapter ^{4 to show how the} theory ^{takes account} of inelastic effects.

Emphasis ^is given ^{on the} transition region ^between

very large Aw ^{where a} binary ^molecular quantum

approach, [2] and paper I, ^must be used and the small Aw where the usual autocorrelation approach

is valid. Parallel treatments are applied ^{to collisions}

with electron and with slow atomic perturbers.

2. Radiation of a collisional complex.

The Hamiltonian of two particles A and B in the presence of ^a radiation field is

where HR = - 2h: ^V2 describes the relative motion of the colliding particles, H Cl) ^{is the free field} Hamiltonian, ^whose eigenfunctions I nw) ^correspond

to n photons, V c is the electrostatic interaction between the atoms and V Cl) the interaction of the field with the collisional complex.

In absence of the radiation field, ^the colliding system ^must be characterized by ^{its initial condition} iki where i stands for the quantum numbers of the

separated ^{atoms A and B} (in ^the following ^we

suppose ^{that atom B} stays ^{in a} spherisymmetrical ground state) ^and where ki = /.t vi h ^{is the momen-}

tum of the relative motion.

For simplicity, ^we first consider the case when the

wave function of the colliding system ^{A + B can be} expanded ^{in terms of the} unperturbed ^{wave function}

where the cp a(r

rA, rB) represent ^the product ^of

the atomic wave functions and y ^{the wave function}

of the relative motion of the two atoms. B can be

replaced by ^a structureless proton ^or electron. More details on the notations used are given in paper I. It will be shown below that the general expressions given ^{in the} present chapter ^remain valid when

expansion (2) ^{must be} replaced by ^{a limited} expan- sion in terms of electronic molecular functions

as

it is the case for low energy ^atom-atom collisions.

In the presence of a radiation field of ni photons ^of frequency w ^and polarization ^{E the} initial condition of the total system ^is characterized by i k; ni and the total wave function can be written in the form

with s

0 or ± 1 for the study ^{of one} photon

transition.

In the case of absorption s

^{0 or -1. If now a}

refers to states of the system associated to n;

photons, ^and {3 ^{to states} associated to n; -1 photons

Under the action of the two potentials Vc ⁺ V w , ^the transition probability per unit of time between two states of the total system i kl ni

f kf n f is

with

in which a plane ^{wave for the relative motion is}

associated to the final state of density p (6f). ^For absorption nf = n; or ni -1. Energy conservation im lies

hw with e

E 1 h ² E

implies )e - §i p ei ^{= hw with 6} e

^E + ! k 2, ^E being

2 tt ^{k , bemg}

the energy ^{of the two} separated ^atoms.

In the important ^{case of weak radiation field, the}

«

distorted wave Bom approximation (DWB)

see Newton [12]

^{can be} applied ^{to the} theory ^of scattering ^{when two kinds of} interactions are present,

Vc which may be strong ^and Vw ^{which is} supposed ^to

be weak. The transition probability ^{due to the}

interaction with the electromagnetic ^{field is}

where ik - I ni - 1 ) and t/J + [ ni) ^{are now} solutions of the total Hamiltonian (1) ^with V, only. ^{The wave}

functions of the colliding system ^are independent ^of

all interactions with the radiation field.

Using (4) ^{it is now} unnecessary ^to specify ^the photon ^states. With collisional coupling only

ik- Ini - l and Q + ] ni ) ^{can be} replaced by

corresponding respectively ^{to the}

outgoing

^wave solution .p + ^at ei ^{and to the} « incoming » ^wave solution qi - at g f of the collision problem (B leaving

A and state f). ^Formula (6) is similar to the formula for radiative spontaneous emission of an electron- atom system of Percival and Seaton [13] ^{which was}

obtained as a generalization ^{of the} Bremstrahlung

emission of an electron in a central field.

But we need the absorption ^rate when A goes from state a

ja _mja ^{to b}

ib _mjb (or eventually ^the

total rate from ja ^to jb, remembering that B does not

change state) ^{of a radiation field of} wavelength ^close by w ab, whatever are the initial and final conditions of the system ^{A +} B, collisions occurring ^{at random.}

In the case of the partial ^rate from ja _mja ^to ib mjb only ^{the terms} ja mja ^and ib _mjb ^are retained in

expansions (7) ^and (8). ^The transition probability iki

fkf for ^{the total} system ^{while the atom A} goes

from ja _mja ^to jb MjbiS

Additional sums over mi, ^and mja ^inside the modulus

give ^{the total rate from} ja ^to jb.

With one term in (7) ^and (8) ^the partial ^wave expansions give

with

where the radial functions of the relative motion have the asymptotic ^forms

Usual notations in collision physics ^{are used} [5, 11, 13]. ^For simplicity each channel will be charac- terized by ^y

jmj ^{1m. Other quantum} numbers and

parity ^{will not} be written explicitely. ^{I is the} angular

momentum of the relative motion of the two particles

and m its projection on !pace ^{field Oz. S is the usual}

scattering ^matrix.

In the case of atom-atom collisions, the relative

velocity ^is much smaller than the electronic vel- ocities. The unperturbed functions 0,,(r) in ^expan-

sion (2) or Oy (rfl) ^{in (12) must be} replaced by space fixed electronic molecular functions .0a (rR) ^and O,y (rR) ^{which tend to cfJ a (r)} and cP ’Y (rR) ^{when the}

internuclear distance R -> oo. These functions as

defined by ^Mies [2] ^are built from the molecular- fixed electronic wave functions Ojn (rR) generated by ^the electronic Hamiltonian at R fixed

by ^frame transformation

Dj(R) being ^the Wigner ^rotation matrix and 1Ii the

projection of j ^on the molecular axis.

Evidently the radial function F and the S matrix, defined by ^the asymptotic ^forms (13), will be very sensitive to an appropriate quasi ^molecular approach

of the collision.

The collisional problem ^{has to} be solved twice for two systems of different energies. The energy of each is conserved during ^the collision 6i = 6a ^and ef

6b. Expansions (7) ^and (8) ^{for atom-atom colli-}

sion are limited to different subspaces including ⁱⁿ

one case the ground ^{state sublevels} ja _mja and ^{a few}

adjacent ^{fine structure levels,} in the other case the

same sublevels corresponding ^{to atom A in its ex-}

cited state. Usually transitions between the two manifolds of states do not occur in the absence of radiation.

The close coupling solutions of each system ^allows for all mixing ^of molecular electronic states during

the collision due to curve crossing, spin orbit inter- action and Coriolis coupling. These solutions are

independent ^{of the} strength ^of the radiation field as

long ^{as the DWB} approximation ^{is valid.}

At low density, ^{the atoms} spend ^most of the time

as free particles ^and occasionally collide and form a

transitory quasi molecule. Initial conditions at t

oo are simply given by ^{the usual} thermodynam-

ical equilibrium conditions of free particles. ^Each

collisional _process must be considered as complete

from t = - oo to t = + oo.

With the usual normalization (13) ^{of the wave} function, ^the density ^{of final states is} given by

The number of binary systems per unit of volume is

assuming ^a Maxwellian distribution f (ki). ^After

averages and ^{sums over} initial and final conditions of the total system, ^{the rate of} absorption per unit of volume is

Using ^the partial ^wave expansions (10) ^and (11) ^it

is easy ^to obtain this rate in the form

if

with

in terms of the « light ^induced scattering ^{matrix »}

elements

KCIJ(ja Mj. -+ ib mjb) is ^{the « two body » rate coef-}

ficient. The total rate KCIJ(ja -+ jb) is ^{obtained after}

summation over mja ^and mjb under the modulus in

(20). When the level ja ^{is well} separated ^{from all} neighbouring levels j; ^{of atom} A, quenching ^{can be} neglected ^and one can assume ii

ja. ^{When, in} addition, the collisions insure an isotropic ^distri-

bution of the sublevels j a mja ^{- a very good approxi-}

mation in the case of weak radiation field

the

expressions ^above simplify ^as

with

or

At a given frequency ^w, energy conservation insures

The interaction potential ^with the radiation field is

V CI)

de where d is the electronic part ^{of the}

dipole operator ^{of the} binary system

^d

^{er of}

atom A -, e is the polarization ^vector and 6. ^{is the}

monochromatic field strength ^{such that}

g; = 2 7T líw c - 1 cP where 0 is the photon ^flux cm- 2 s-1.

The absorption coefficient per unit of length ^is

defined by [14]

The weak field DWB approximation ^{is valid when}

the light induced matrix Sw’ elements, ^{which are now} defined by (21), ^are much smaller than unity.

Explicit expressions ^{of S w’ or of the} absorption

coefficient Kw in terms of radial functions F or of the collision S matrix elements have been given ^else-

where [5, ^11, 15]. ^{In the} present paper ^we shall be concerned only ^{now with the} equivalent semiclassical formalism for which explicit expressions ^{will be} given ^below.

The quantum approach ^{has been} applied indepen- dently ^{to two different} classes of problems. ^For

electron-atom systems, elaborate calculations have been made of the electron contribution to the

hydrogen ^line wings Lyman ^{a and} Hf3 ^{[16, 17, 18].}

This work has been reviewed by ^Peach [11] ^and

Lisitsa [19].

For atom-atom systems, ^{a series} of papers have been devoted ^to collision induced transitions

th-

ese transitions being asymptotically ^{allowed or for-}

bidden in the absence of collision. Detailed studies have been made for the transition 0 (1S ) ^{+ Ar +}

h v - 0 (’D) ^{+ Ar} [10] ^{and for} Sr (’S) ^{+ Ar +} h v ->

Sr (1P) + Ar [14, 20] ^{in which the close} coupled theory ^{of collision in a radiation field} [2] ^{is used to}

calculated the absorption profile ^{as well as the} polarization ^{of the} (1P ) fluorescence following wing

excitation by ^a polarized light.

3. The semi-classical approximation.

The semi-classical treatment of the collisional prob-

lem is given ^{here to} simplify ^the quantum ^formalism given ^{above and to} emphasize ^some analogies ^and

some differences with the usual treatment in line

broadening theory. The role of inelastic effects in collision induced radiative transitions is underlined.

For Ni ^{atoms A in state i} at t = 2013 oo, ^to

NB v; 7Tk2(2ii + 1 ) ^the frequency ^{of arrival of a} perturber ^of angular ^momentum li, corresponds classically ^the frequency NB Vi ² wp dp.

The « two body » ^rate coefficient, ^as defined in formula (19) ^can be written in the form

r

The semi-classical expression ^{of the} light ^induced scattering ^{matrix elements is :}

P;if (P )

IS’ (a -> b ) I 2 ^{is the} probability ^{of induced} scattering ^{for a classical} trajectory ^of impact par- ameter p, ^or the probability ^of absorption ^{a - b for} given ^initial conditions of the scattering problem.

Qi£ (t ) corresponds ^{to the} part ^{of the full} scattering

state (B arriving ^{on A in state i at} infinity) ^which

involves A in state a. qi - (t ) corresponds ^{to the} part of the full scattering ^state (B leaving ^{A in state f at} infinity) which involves A in state b.

in which the U a (t ) ^and Ui (t) ^are given ^{from the}

solutions of systems ^of differential equations ^{of the}

whole scattering problem. ^{With the} expansion

appropriate ^to define the asymptotic conditions at t

oo or +oo when atom A is in a given ^state

jmj, ^{V being the} electrostatic interaction Vc ^of expression (1), ^the resolution of the Schroedinger equation gives

In the specific ^{case of} electron-atom collisions the

cP a ^are simply ^the unperturbed ^{atomic wave} function cP a (rl). Expansions (2) ^and (29) ^must include many open channels, the electrostatic potential V c giving

rise to important inelastic effects.

In the case of atom-atom collisions, the cP a (rR) given by (15) ^are the space fixed electronic molecular functions. Expansion (29) may usually be limited to the fine structure levels jmj ^{for given LS. When}

solved in this representation

corresponding ^{to the}

Hund coupling ^{scheme e}

^which diagonalizes ^the asymptotic Hamiltonian HA ⁺ HB, strong coupling

terms occur during the collision. At small _r, in the molecular region, where Hund case a (or b) ^are valid, ^the Vaa, ^{are built from the} diagonal ^Born- Oppenheimer potentials by unitary transformation.

Coupling ^{terms due to the} R (t ) dependence ^{of the} cP a ^are neglected ⁱⁿ (30). They usually vary ^as

R - 6 - compared ^{to the} R - 2 or R - 3 behaviour of the

.

The implementation ^{of this} semi-classical close

coupled theory ^{has been discussed} by ^Delos [21] ^and

in many papers ^on line broadening ^{and fine structure}

transitions in atom-atom collisions [22-24].

Straight paths have been used in many cal-

culations, ^{but the results} may be very much improved

in taking ^{account of} trajectory ^effects [25]. ^The

systems ^of equation (30) ^{are obtained as the} high angular ^momentum limit of the quantum coupled equations ⁱⁿ assuming ^{a common} turning point.

Two systems ^of equations (30) ^{have to be solved}

for the same given impact parameter ^p

1 k- 1.

Indeed the usual assumption of weak transfer of energy implies ki - ka, kf - kb ^and ki - kf ^from (24).

Conservation of the total angular ^momentum J ^{= I} + j ^{in the} collisional processes and the dipole

selection rule OJ

0 ± 1 for the radiative _process leads to li i -- If ^{in the classical limit of} high angular

momentum. This corresponds ^{to the} semiclassical treatment of the close coupled theory ^{of Mies} [2] ⁱⁿ

which the radiation field gives additional couplings

in a unique system ^of equation.

Initial conditions i (or f) ^{are such that} U; (- oo ) = Sia ^and Uib (+ 00) = Stb. ^{When the}

system ^{is known to be in} eigenstate i of the unper- turbed Hamiltonian at t

oo, the transition ampli-

tude is effectively given by the scalar product uia(t) _ I> a (r ) I U (t, - 00 ) I I> i (r »

where U(t, - oo ) is the time evolution operator ⁱⁿ the interaction picture. U+ (t ) = U (t, - oo ) ^and

U- (t )

U(t, + oo ) ^are unitary. ^{With U+} (- oo ) _ 1, ^U+ (+ oo ) is the classical analogue of the S matrix.

With U- (+ oo )

1, U- (- oo ) ^{= S *.}

For given ^initial conditions, operators ^{U+ and} U-, corresponding ^to Vc (t ) ^{of one} perturber, gener- ate the wave

functions ^{qi + (t )} and qi - (t ) ^{at time t.}

I U a (t ) 12 or 1 Uib (t ) I ^{give the} probability ^{to have}

the atom in state a (or b).

The scattering qi a+ (t ) ^and Qg (t) ^not only ^take

account of non adiabatic effects like rotational

coupling ^{or eventual curve} crossing but also of

possible ^{fine structure} excitation in atom-atom col- lisions. They ^{allow for} important inelastic effects in electron atom system. They ^{are not to be confused}

with the « collision smeared states >> oa(t) ^and 4Jb(t) introduced by ^{Bloom and} Marguenau [26-28]

usually ^{used in the} autocorrelation function formal- isms which are formally solutions of equations ^of

type (30) in which V corresponds ^{to all} perturbers.

a

ja mja and b

^ib mjb being ^{the states} involved in the radiative process. All inelastic effects ^are neg- lected at this stage ^{of the formalism.}

Sometimes good results may be obtained using

different models [29, 30] which have been derived from a suggestion ^made by ^Nikitin [31] according

which each stage of the collision can be described by

a given ^Hund coupling scheme. Instead of solving

system (30), the evolution operator ^{for t} > t2 ^{can be} written in the form

U(t, - 00 U,, (t, t2) paii

x U ä (t2, t1) P aii (t1) U a (t1, - 00 ) (31) where t1 and t2 ^{are the times at which the} system passes abruptly ^{from one} representation ^{to the other} by transformation P, ^each separate ^evolution operator being diagonal ^{in a} given time interval [32, 33].

This method has been applied ^for depolarization

cross sections calculations in Na-He collisions [29]

with a simple model in which the quasi-molecular

passes from Hund ^case (e) ^at long ^{distance to Hund}

case (a) ^{and back to Hund case} (e). ^{The same} procedure ^{has been} applied ^to absorption profile

and redistribution calculations in Sr + Ar collisions

[34], giving good ^results compared ^to the full quan- tum study of Julienne and Mies [14, 35].

Let us now consider the absorption profile ^{of a}

continuous radiation between two degenerate ^levels ja

ib. The monochromatic energy flow Qhw ^must be replaced by IpwE ^{dw dpw, the radiation energy}

incident on unit area per unit time of polarization ^E

and wave vector p. in the solid angle do. (using ^the

notations of Landau and Lifshitz [36]). ^The prob- ability ^of absorption per interval of dw and of

do. ^from (27) ^is

For simplicity ^{let us} suppose that the incident radiation is isotropic ^and unpolarized. After inte-

gration ^over dfp. ^{and summation over E, with} I w

⁸ 7T’I WE ^the probability per interval of dlù is

in which we have used expansions (27). ^{For sim-} plicity ^{we have assumed} that B is an atom in a

j

^{0 state or a} structureless particle. a

_{ja mja} ^and

b

ib mjb. ^{Other quantum numbers are not written} explicitely.

In electron-atom collisions the 0 ⁱⁿ (33) ^are unperturbed atomic functions and the dipole ^mo-

ment is independent of the time. In atom-atom

collisions for Aw not too large

^{in the near} wing ^or

in the core of an allowed transition, ^{the main} contribution to the time integral ⁱⁿ (33) ^is given ^for large ^{values of} R (t ) for which the R dependence ^of

the dipole matrix elements can be neglected. ^Actual- ly ^this dependence ^is usually neglected ^all along ^the profile ^{even at} large Aw [8, 9] and it has been shown

recently [20] ^{that it} gives negligible effects in the case

of

optical» asymptotically ^allowed transitions for which the molecular transition moment varies as

d(R) = d(oo )[1 - CR-3].

When this approximation is valid the dipole ^matrix

elements of the unperturbed system ^can be factorized in (33). Expanding ^the dipole ^{moment of atom A in} spherical components, ^d

^er

E d1u, ^{one obtains}

A

where B is the well known Einstein coefficient for absorption ^{of the} unperturbed ^atom. Without any external

perturbation ^U+ (t ) = 1 and U- (t ) ^{= 1, i}

^{a and f}

b and it is easy ^to check that after integration ^{over dw}

one obtains the usual absorption ^rate

Iw _{R(i - i- I}

with

Similarly, ^the

profile

^{of the rate} coefficient in (26) ^is given by

r

This formula can also be obtained in a different way from the full quantum expressions (19) ^and (20)

and using ^the expansions (12) ^{in terms of} radial functions F. Following ^{Tran Minh et al.} [16], ^but taking

account of the incoming ^and outgoing ^{wave forms} of y+ and qi-, ^{one finds the classical} limits of the quantum overlap integrals

which _{appear in} (37).

For kT bigger ^{than all} the Ei - Ea, Na ^can be factorized in (37). Solving the collisional problem ^{in the}

semi-classical approximation, levels i and a (or ^{f and} b) ^are closely spaced ^{and a mean} velocity

/t?j ^{va is} associated to a common trajectory ^of impact parameter p. The profile ^{of the rate related to level} ja ^is

using ^for simplicity ^{the usual} notation for the angular average [8].

Dropping ^the constant ^factor I CJJ ’Bc-l, ^{which is the}

rate of absorption ^{of the} unperturbed ^{atom A, one}

finds the usual normalized profile F(Aw). ^For comparison ^{with the} autocorrelation formalism, ^the profile F (A w ) may be written as a Fourier transform

Defining ^{the time evolution operator}

and summing ^over i and f in (39) ^one easily ^finds

the usual _average over an ensemble of one perturber

time evolution operators.

On the other hand, the autocorrelation formalism

[8, 37] gives

which, ^{at low} density NB, only ^{differs from the}

«

binary ^{model » formula} (42) ^{in a} very small region

around the line centre. In a typical ^low density experiment [14] ^the binary model is valid for

Aw -- 1 cm- 1 although ^the measurements of the

profile extend up ^{to ±} 100 cm - 1 !

The autocorrelation approach giving (43) ^is

known to be based on the

factorization approxi-

mation » [7] ^{of the} density ^matrix p

p A p P in which the density matrix pup of the perturbers ^takes

its equilibrium value and is time independent. ^This assumption meaning ^{that the} perturber ^bath

^even

when only ^{the nearest} perturber plays ^a role in the

line wings

^{does not} change during ^the collision, ^is evidently compatible ^{with the use of a} straight path

in a semiclassical calculation.

It is generally ^{assumed that the} factorization

approximation ^{leads to} negligible ^{errors as} long ^as

h Aw is much smaller than kT. This is confirmed by

the identity ^of (42) ^and (43) ^{in a} large ^{domain of the} profile.

When Aw increases, allowance of trajectory ^effects

may improve ^the semiclassical calculations but for

large ^{Aw close} correlations in the classically ^for-

bidden region ^{of R are} important and the classical

approach completely breaks down as well as the usual autocorrelation approach. ^For these values of 3w the quantum binary model, which is sketched out in chapter ^{1 and takes} fully ^{account of} binary correlations, ^is entirely ^valid.

This is confirmed by ^recent elaborate semiclassical calculations with allowance of trajectory ^effects.

Compared ^{to a full} quantum calculations, good

results are obtained for the core and the near wings

of optical transitions like 4s-4p ^and 4s-5p ^{in K + Ne} collisions, ^{but cannot} be obtained in the far wings

where the transition occurs near the turning point [38].

The same is true in the very different ^case of the electron contribution of hydrogen ^line wings ^{in low} density plasmas. ^{In Stark} broadening ^the interaction

potential falls of like R-2 and analytical expressions

can be found for both the quantum [17, ^39, 40] ^and

the semiclassical [41] expression ^{of the} overlap integrals (38) ^and of the line profile. The semiclassi- cal approach gives good results in the case of Lyman

a for AA 10 A but, ^{when AA} increases, ^both assumptions ^{of a} pure dipole interaction and of a

straight path ^{break down for AA}

25 A.

4. The impact approximation.

At small detuning, ^{when 0 w -1 is much larger than}

the collision duration _Tc

Rc v- 1 ^where 7?c ^{is of the}

order of the range of the interaction potential, ^the

radiative process mainly ^{occurs in the} asymptotic

region where the radial functions have the

forms (13).

As shown in paper I, ^the overlap integrals ^{of the}

radial functions have the impact ^limit

Within the semi-classical picture, ^{for t}

Tc ; the coefficient U a (t ) ^and Ui (t) ^{have their} asymptotic

values corresponding ^{to t = ± oo and the} integrals

appearing ⁱⁿ (37) have their impact limit forms

with S = U+ (+ 00 ).

Then it is _easy to derive the impact ^{limit of the} probability (33) and of the absorption ^rate (37).

This rate of absorption ^{in the interval dw can be} split into three terms [5] :

i) ^{One which} involves the relaxation of the initial level ja only

ii) One which involves the relaxation of the final level ib only

iii) ^The interference term, which is obtained when j;

ja and if

ib

The first contribution (46) ^{is obtained} for Boltzmann population ratio between level ja ^{and all} neighbouring levels ji.

The sum of the three contributions gives

in terms of the rate of absorption ^{of the} unperturbed ^{atom and of the}

profile

F (A w )

^y (2 ^ir A co 2)-l

where y is identical to the known expression ^of Baranger [7].

which includes the

quenching

^inelastic contributions to the profile. Starting ^{from a different} point ^of view, Baranger ^{has obtained} (51) ^{from the} impact limit of his autocorrelation function given by

which gives ^{a Lorentz} profile where the half width

y /2 ^{and the shift are} given by ^{the real} part ^{and the} imaginary part ^{of the} exponent. ^{The first is} equival-

ent to (51). ^{One must note} that « inelastic » terms

only appear ^{at the last} stage ^{of the} argument, ^when

taking ^{account of the} unitary ^{of the S matrix in} (52).

As defined in chapter 3, ^{in the case of} absorption by ^a continuous radiation I., ^{the line} profile ^{is the}

ratio of the rate of collision induced absorption per interval of dcw , Nja NBdK-(ja ’ jb ), with ^dKw given by (39), ^{to the rate} absorption N ja I. c-’B(ja - ib)

of atom A. This gives

where the frequency dependent ^width y (Aw ) ^is given ^{in terms of the} overlap integrals ⁱⁿ (39). ^{At the} impact ^limit y(Aw)

^y given by (51).

The binary ^model obviously ^{breaks down some-}

where near the line centre when A w is of the order of

a few _{y :} the two body system ^{is not an isolated} system anymore. But it breaks down long ^{before the} impact approximation, because T c is much smaller than y -1. Therefore, ^{near the line centre the} profile

has the Lorentz shape given by the usual line

broadening theory which allows for the overlap ⁱⁿ

time of weak interactions and (53) ^can be modified in this way.

5. Conclusion.

The binary theory described in paper I and in the present paper proceeds ^{from a different} point ^of

view than the usual autocorrelation function for- malism. The collisional complex ^{A + B in a} quantum isolated system with known initial conditions, ^on

which the induced absorption ^{rate can be} easily averaged. ^{In the} semi-classical approach ^{the wave}

function of the colliding system ^at all time t is built up from its known simple ^{form at t}

^oo.

Within the usual formalism all perturbers ^{P of a} given ^{volume are} included in the system, ^but nearly

all

unified » theories are based on an approxi-

mation according ^{which the} density matrix of the total system ^{at all times t can be} factorized as

p (t )

P A (t) p P (t ). ^{As known this}

factorization

approximation ^{is not valid} in the line wings ^{as soon}

as the atom and the perturber subsystems ^are highly

correlated.

Recently many efforts have been made ^to built a

many-body density operator including ^all binary

correlations [42-45]. Physically ^this corresponds ^to

the evolution of the state of a gas ^at time t from a

completely uncorrelated state at time t

oo.

Proper ^{account of} two-body correlations implies ^a

correct collisional quantum approach.

A semi-classical description makes the physical interpretation easier. It is given ⁱⁿ chapter ^{3 as the}

usual high angular ^{momentum limit of the} quantum

theory, ^formulae (26) ^and (27) corresponding ^to (19)

and (21). ^{But it can} be derived in a different _way.

From the theory ^of approximate method of calcu- lation of the evolution operator [46], ^{when one} part

Vw (t) ^{of the total} Hamiltonian H = Ho + V c (t) ⁺ V Cù (t) ^can be considered as a perturbation, expression (27) ^{of the} light ^induced scattering ^matrix

Sw can be obtained as the semi-classical analogue ^of

the DWB approximation given ⁱⁿ chapter ^2.

In the case of atomic perturbers, ^{the correct} expansion ^{of the} colliding system ^wave function in terms of electronic molecular functions must be

used, ^the two-body ^{time evolution} operator ^{U or the} collisional S-matrices being very sensitive ^to short range correlations effects, ^{even at small} detuning.

From chapter ^{4 one can} say that, ^{for low} density plasmas, ^the binary ^model is valid for small Aw until the impact approximation ^{is valid and before the}

occurrence of simultaneous collisions in time.

On the contrary far in the wings, ^for large ^Aw - depending ^{of course on} the range of interaction,

the weight ^{of the} perturber, ^{and the} temperature

the radiative process ^occurs mainly ^{in the} quasi-

molecular region. As well known the factorization

approximation breaks down as soon as I A w I ^{is of}

the order of kTIh. But this is also the limit of

validity ^{of the} impact parameter semi-classical ap-

proximation using ^a straight trajectory.

In this region ^a quantum approach ^{must be} used,

the basis of which are given ⁱⁿ chapter 2. ^This theory, valid in the specific ^case of weak radiation fields has been detailed in paper I [5] ^and compared

to the alternative binary approach of the « close

coupled theory of collision in the presence of ^a radiation field » of Yakovlenko [1] ^{and Mies} [2].

We have shown that in the intermediate region, ⁱⁿ using ^a semi-classical approximation for the collisio- nal process, that the ^same results should be expected

in following ^the binary ^model approach ^{or the}

autocorrelation approach. Contrary ^{to a well known}

statement in line broadening theory, according ^to

which it is simpler ^to calculate the Fourier transform of the profile ^{than the} profile ^{itself, we feel that}

there is a huge ^{domain of} validity ^{of a} binary ^model theory. Emphasis ^{has been} given ^{here on line} broadening studies but this method of treatment of

light interaction with collisional systems may be

applied ^to many other phenomena ^{like some} prob-

lems of redistribution of light [14, 47].

Acknowledgments.

The author whishes to thank N. Feautrier, ^Ch.

Stehlé and A. Spielfiedel ^for helpful discussions.

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