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Perturbation of spectral lines by atomic interactions
PERTURBATION OF SPECTRAL LINES BY ATOMIC INTERACTIONS
Frédéric SCHULLER Centre National de la Recherche Scientifique, L.I.M.HP., Bellevue, France
and Wolfgang BEHMENBURG Physikalisches Institut der Universität Düsseldorf, Germany
I
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 12,
flO.
4 (1974) 273--334. NORTH-HOLLANI) PUBLISHING COMPANY
PERTURBATION OF SPECTRAL LINES BY ATOMIC INTERACTIONS Frederic SCHULLER Centre National de ía Recherche Scientijique, L.I.M.HP.. Bellevue, France
and
Wolfgang BEHMENBURG I’hvsikalisches Inst ito t der Unir’ersitdt Di3sseldorf, Germans’ Received February 1974 Contents: 1. Introduction 2. Basic theoretical considerations 2.1. The classical path theory 2.1.1. General expression for the absorption intenSity 2.1.2. Decomposition into individual spectral lines 2.1.3. Phase shift theories 2.2. Quantum theories 3. The derivation of the line profile 3.1. The phase shift theory 3.1 .1. The correlation function 3.1.1.1. Theory for general pressures 3.1.1.2. The quasistatic approximation 3.1.1.3. The impact approximation 3.1.2. The intensity distribution 3.1.2.1. Interatomic potentials 3.1.2.2. The calculation of the line shape 3.1.3. Criteria of validity of the phase shift theory and its approximations 3.1.3.1. The adiabatic assumption 3.1.3.2. The validity range of the quasistatic approximation 3.1.3.3. The validity range of the impact approximation
275 275 275 275 280 281 283 283 283 284 284 285 287 288 288 289 299 299 300 31)1
3.2. Orientation effects 3.2.1. The relaxation of the dipole transition moment in the impact approximation 3.2.2. General expression of the differential cross-section 3.2.3. Approximate solutions of the one-collision problem 3.2.4. Cross sections for the relaxation of orientation and alignment 3.3. The quantum-mechanical impact theory 4. Comparison between theory and experiment 4.1. Line core (halfwidth and shift)-measurements at 9cm~as functions number densities below of number density and temperature 1Q~ 4.2. Line core measurements at number densities above I (Ji9crn3 4.3. Spectral intensity distribution in the line wings 5. Appendix: The calculation of interatomic forces 5.1. Dispersion forces 5.1.1. Dipole—dipole interaction 5.1.2. Higher moment interaction 5.2. Resonance forces 5.3. Overlap forces Acknowledgement References
~Sin~lc orders (~rthis issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 12, No. 4 (1974) 273 334. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check, Single issue price Dfl. 20.00, postage included.
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F. Schuller and W. Behmenburg, Perturbation of spectral liner by atomic interactions
275
1. Introduction The perturbation of atomic spectral lines, emitted or absorbed by gases or gas mixtures, is in general the result of several mechanisms: radiation damping, Doppler effect of thermal motion and collisions. Furthermore, in optically thick layers the line shape will be influenced by additional effects, in particular reabsorption of radiation. The mechanism of perturbation by collisions is very different for various types of interaction. If charged particles take part in the collision process, the line profile will be determined essentially by the interatomic Stark effect. If the particles are neutral, the perturbation of the line is governed by what is called interatomic forces; the latter comprise the well known Van der Waals forces and, in the case of identical particles, resonance forces. During the past ten years a considerable amount of work in the field of neutral atom broadening, both theoretical and experimental, has been published. It is not our intention to give a comprehensive account of all the results which have been presented. Many of them are discussed in the recent review article by Hindmarsh and Farr [77]. Furthermore, there exists a bibliography on spectral line broadening, which covers all the papers published up to 1972 [781. In this article we rather wish to give detailed attention to certain aspects of the subject, which might be of importance for future developements. One important aspect is the problem of determination of interatomic potentials for excited states from line profile measurements. This problem involves careful comparison between experiment and theory with special reference to the assumptions made in the latter. In order to point out those assumptions, it was useful to present the various approximative theories (sections 3.1.1.2, 3) as special cases of the general theory of line broadening (sect. 3.1.1.1), and to derive the validity criteria for the approximations. This was done for both the impact and the quasistatic approximation using realistic potential functions. However, in this paper the semiclassical theory is the main subject of discussion, whereas in the recent paper by Futrelle [79] the stress is put on a fully quantum-mechanical treatment. A further aspect concerns the part taken by orientation effects (sect. 3.2) in the mechanism of broadening and shift of the line. This will be discussed on the basis of Anderson’s theory [31. In this context we point out the connection between line broadening and other relaxation phenomena like, for instance, collisional depolarization of resonance radiation and broadening of double resonance curves (sect. 3.2.4). Relations between the “optical” cross section and those for destruction of orientation and alignment will be derived. Finally, for the purpose of comparison between theory and experiment, a brief account will be presented of several semi-empirical methods for the calculation of interatomic potentials by means of quantum-mechanical perturbation theory (sect. 5).
2. Basic theoretical considerations 2.1. The classical path theory 2.1.1. General expression for the absorption intensity Consider a set of interacting neutral particles which, in principle, could be either atoms or molecules, although in this paper we will focus our attention on atomic spectra. We assume that
276
F Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
these particles form a system of small dimensions, compared to the wavelength of some incident
radiation. (When considering a macroscopic gas or liquid, we decompose it in a great number of independent systems.) The harniltonian of the system will in general depend on the internal coordinates p and momenta p of each particle and on the coordinates R,* and momenta P of the relative motion of their centers of mass. Theories based upon the resolution of the Schroedinger equation of the complete system will be referred to as quantum theories. They will be introduced later. In this section we will assume that the Schroedinger equation depends only parametrically on the relative positions R of the centers of mass of the atoms. We further assume that R is a given function of the time t and call this the classical path approximation. The conditions under which the classical path approximation is valid are essentially the following: i) Only small changes in the internal energy of the particles are allowed, so that the reaction of these changes on the translatory motion may be neglected. ii) The Dc Broglie wavelength associated with the relative motion has to be small compared to the characteristic dimensions of the range of interaction forces. This condition is in general well satisfied except for light atoms and low temperatures. Let H(t) be the hamiltonian of the system in the classical path approximation, where the time t plays the role of a parameter. The variation with t of any operator 0. say, is then described by its Heisenberg representation: QH(
1)
=cutt(t)0q1(t)
(I)
where clt(t) is the unitary evolution operator which satisfies the Schroedinger equation
(2)
ihcll=H(tyl1.
We now consider the absorption problem assuming that the system is submitted to the influence of a monochromatic electromagnetic wave. Following the derivation of Bloom and Margenau [11 we represent this incident radiation classically and consider the electric vector E(t) as a given function of time. Restricting ourselves to dipole absorption, we write the harniltonian of the system in the form: =
11(t)
+
F(t)
(3)
where F(t) is the energy of interaction between the system and the electromagnetic field (electric fieldvector E(t)) as given by the relation:
F(t)
=
-~E(t)~t.
(4)
In this relation p designates the dipole moment operator of the system. which in principle involves all the electrons that may possibly interact with the radiation field. (However, in practice one often considers the situation where one optically active atom is surrounded by one or several perturbers and one only takes into account the electron that makes the optical transition. This ap-
proximation then neglects any participation of the perturbers in the radiative process. *Thc reader should not be confused with the fact, that throughout this article notations r and R are used for the internuclear distance.
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
277
We define a new evolution operator U that includes the influence of the interaction with the electromagnetic field. The relations (1) and (2) are then replaced by the following ones: O~t)= Ut(t)OU(t)
(5)
and ihU= ~(t)U.
(6)
A simple relation between the two quantities 0~(t) and O~(t) can be obtained by perturbation theory. Writing U(t) =Q((t)Ø(t)
(7)
we let the operator ~ take care of the interaction with the electromagnetic field. From (6) ~ obeys the equation iic~= FH~, where by definition FH Q11Fdll.
(8)
This can be solved by the perturbation expansion i2
it =
I
—
~-J”dt’ F’~(t’)+
(i-)
t
r’
fdt’ FH(tI)f dt” F”(t”) +
...
etc.
(9)
By substituting this expansion into the equation for O~,which according to (5) and (7) can be written as ~tOH(t)&
=
(10)
we get a series of terms corresponding to different orders with respect to FH. Up to the first order the result can be expressed as follows: O~(t)=
QH(t)
dtm[OH(t), FH(t’)I,
—
(11)
where the brackets designate the commutator defined by the relation: [A, B1 = AB BA. With Bloom and Margenau we now calculate the absorption by considering the work done by the light wave against the internal forces of our system of interacting particles. We may assume without loss of generality, that the electric vector E of the field defines the z-direction of some space-fixed frame and that its length varies with time according to the equation: —
E(t)
=
E 0cos(wt
+
a).
(12)
In this equation a plays the role of an arbitrary phase. Eqs. (4) and (1 2) yield the following expression for the interaction-hamiltonian: F(t) .~
=
—E0p~cos(wt+ a)
being by definition the projection of p on E.
(13)
278
F Sehuller and W. Behmenburg, Perturbation of spectral line,r bs’ atomic interactions
According to basic quantum mechanics, the required expression for the work done by the dcct~omagneticwave is the lollowing:
W( T) = L0
f (It cos(
wi +
~) —( p~(t))
(14)
where a time interval (0, T) has been considered. Flere the quantum mechanical average of the dipole moment is defined as: K p~(l)) = Tr ~pp~( t) L p being the statistical matrix. ntegrating by parts we get front eq. (1 4): -
tV( T)
cos(wl
=
+
a)Kp ([)~
+
di sin( wt)(p~(I )~.
Ewf
(1 5)
At large T-values the energy increase of the system becomes stationary and we may consider the C1Lt~lntitY ~=
liii~ ~
(16)
T
I,
Obviously the first term on the right of eq. (1 5) gives only a vanishing contribution to ~ so that we may drop it from now on. In order to calculate the second one, we first deterniine p(t) from eqs. (I I ) and (1 3), obtaining: =
p~(t)
+~E,Jdr’ cos( wI + )l p’~(I),
(1 7)
~H (I’).
After substituting this expression into eq. (1 5) and taking the average over all possible values of we get the result:
a.
=
~0WTr
~
dtfdt’ sin(w(i
-
t))[p~t),
p~t’)I}
.
(18)
Now, a simple geometrical argument shows that the following identity holds:
-/dtfdt’
sin(w(t
i~))pH(1~)pFt(1) =
--
=
---
f dt’fdt
sin(w(t
f dtfsin(w(t
-
--
which allows us to transform eq. (1 8) into: T
2w ° Tr iE “hT
(~ f‘ dt r di’ sin(w( t 0
I
))p~(t)p~(t’) ~.
-
-
0
This expression is more conveniently written in the form:
(1 9)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
~W(fl
=~Tr(p(fdt
279
e~tp~(t)f dt e~tp~(t) fdt e_~tp~(t)fdt e~tp~(t))}.(20) —
It constitutes the basic formula of semiclassical pressure-broadening theory. and write Introducing suitable basis functions, we define matrix elements Pif = Piö if and /~t~tf out the trace in eq. (20). This yields: E 0w ~ ~P~(fdt
e
MZif(t)fdte
PH(t)
—
fdt eP~.f(t)fdtePZf~(t))). (21)
We then rewrite the second term in the brackets by interchanging the indices i and f, obtaining ~W(fl
=
L0W ~(p~ — pf)~fdt etp~f(t)~.
(22)
In this expression, the term containing the factor Pf can be interpreted as stimulated emission. Let us now consider the limit of W(T)/T for T-* We may assume that the interaction within the system generates a stationary random process, so the Wiener-Khinchine theorem can be applied [21. This means that the quantity: ~.
I~f(w)=
lim ~
/dt e~tp~f(t)
(23)
is equivalent with the Fourier integral: I(f(w)
=
f
(24)
dr e~~T~tf(r)
involving a correlation function p(r) defined by the equation: p~f(T)= (pHf(t +
r)p~,.(t))~
(25a)
where the symbol ( ~ designates a time average. Replacing the time average by an ensemble average, that is an average over all possible paths followed by the system in the time interval (0, r), we can also write: tP~f(T) =
(i4’,f(r)p~~(0))S.
As can easily be shown from (25a, b), the correlation function has the property mp(f(—r) Using this property we can write eq. (24) as follows: I1f(W)
=
2ReJ’ dr e~(f(r),
which is sometimes more convenient for explicit calculations.
(25b) =
p~(r).
(26)
280
F. Schuller and W. Behrncn burg, Perturbation of spectral lines by atomic interactions
2. 1 .2. Decomposition into indiVidual spectral lutes So far the expressions derived for the absorption energy cover the whole range of co-values as they describe the intensity distribution of the entire spectrum. However, in practice the spectrum will often contain a certain number of isolated lines and the problem then consists in determining the profile of a given line separately. Before studying this problem in general, let us first consider the limiting case of zero interaction (e.g. isolated atom). Matrix elements of ~H are then simply given by:
pZ~fexp(—iwc~t)
P~f(t)
(27)
with w~ = (E~~°~ Ec0))/n, Ecu) and E~°~ being the unperturbed energies. The matrix element =
P~f(O)is of course independent of time. The substitution of (27) into (25a) yields for the
correlation function: -iwc~r) ~~f(~) IPZ~f~2exp(and thus the quantity (24) becomes: IPzjfI2
J~f(W) =
fdr
exp{i(w
(28)
w~)r~= 2~IpZlfI2ö(w
---
—
wv).
(29)
where the well known representation of the delta function has been used. From eq. (22) the absorption power is then given by the expression:
~
7rE2w
IiIfl 7W(T)
21t
~
Co1
--
Pf) PZIf~(W--
wc~).
(30)
Averaging over orientations we may further replace Pzif 12 by~~lp~fI2.Eq. (30) shows the evident fact that in the case of an unperturbed absorption process the spectrum is composed of several infinitely sharp lines corresponding each to a transition i f. We now introduce perturbation but we make the restrictive assumption that the latter might not be strong enough to destroy completely the behaviour of ~(r) as a function oscillating at frequency w~. So we write: -÷
ipif(r)
=
(31)
pf(r)exp(—iw~r)
and I~f(w)=
f
dr ~f(r) exp{i(w
--
w~kr)}
(32)
where ~p’f(r) is supposed to vary slowly compared to the factor exp(—iw~~r)~ Under these circumstances, I,/w) will be very small except in the neighborhood of co = w~.So describes an intensity distribution located at frequencies close to w~ [31 Going back to eq. (22), we see that each term of the sum on the right corresponds to an individual spectral line. With the additional assumptions of neglecting induced emission and of setting p1 = 1, we get as a result that the profile of a perturbed line is just given by ‘if and thus may be calculated from eq. (24) or (26). .
281
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
The above reasoning can easily be extended to the case where the states under consideration are either degenerate or may be treated as such. The result is then, that a sum over degenerate states has to be added in eq. (24) or (26). The situation is quite different when the interaction causes overlapping of lines in such a way that no simple decomposition can be made any more. For this rather difficult case we refer to the literature [4]. 2.1.3. Phase-shift theories Let us again consider the profile of an individual spectral line as given by equation (24) or (26) neglecting any effects due to degeneracy. In many theories the calculation of the corresponding correlation function is performed by introducing adiabatic states. These are defined by a special choice of basis functions assuming that matrix elements of the evolution operator Q1 take the simple form [5] =
~
{_±.fdt’E~(t’)).
(33)
By substitution, the Schroedinger equation than yields the following relation: H~1~ = ~
(34)
which amounts to the solution of an “instantaneous” Schroedinger equation H(t)in) = In)E~(t). 4~fthe following With the evolution operator given by eq. (33), we obtain for the quantity ! expression:
(_~Jdt(Ef(t’) — E~(t’))).
I4~f= (c~jtp°.l).f= p~exp
(35)
It should be noticed that the matrix element = (ilp~ If) is time-dependent since it involves eigenfunctions of the time-dependent “instantaneous” Schroedinger equation. From eq. (33), which of course constitutes an approximation, we derive the correlation function, given by: t+’T
mp, 1(r)
(,4~(t+ r)p~.(t)exp ~—.~-f dt’(Ef(t’) — E.(t’)))
)~.
(36)
In most theories however, only the phase factor is taken into account, whereas the quantity + r)j.L~,(t) is raplaced by a constant value, corresponding to the unperturbed atom [6]. Moreover, this constant factor is in general omitted in the definition of ip(r). As to the phase, it is mostly written in the form: t+’r
..i_f
f
t+r
dt’(E~(t’)— E~(t’))=
+
dt’~w~,~(t’),
(37)
nt where w~)is the frequency of the unperturbed atom, and where ~w~f(t’) designates the frequency
282
F Schuller and W. Behmenburg. Perturbation of spectral lines by atomic interactions
perturbation. The latter involves the interaction potential V, defined by the relation:
L1f(t)
=
E~°~ + ~‘i,f~ t),
(38)
EY~being the energy of the unperturbed atom. By definition: ~w1~ =(Vf
--
V1).
Introducing the phase perturbation:
77(1)
=
fde ~w1f(t)
we write eq. (37) as follows:
I
t+r
TI
dt’(E~.(t’) E1(t)) —
=
wr
+
~( t
+ T)
--
~( I).
Finally, the phase W~~Tcan be removed from the correlation function by shifting the origin of the frequency scale, so that w in eq. (24) now measures the difference w w~. With these modifications the correlation function is given by the following expressions [7 I ‘~: ~,(T)(exp{—i(77(t+r)
(39a)
—77(t))})~~Kexp{----i77(t+T, t)}>~
and ~(T)
=
(39b)
Kexp~—i77(r)}).,
involving respectively a time average and an ensemble average. The correlation function given by eqs. (39a, b) is that of the so-called phase-shift theories, which will be discussed later in more detail. Right now it shall be remarked that any profile predicted by these theories leads to an integrated intensity equal to the intensity of the unperturbed line. As a proof we integrate the expression given by eq. (24) over co in the interval < co < +°°. This yields: —~
2 fdwI(w)
=
fdT~r)
2~(r, 0)
=
2~0)
=
2~lpZifI
(where we have reintroduced the factor ‘~zif12 which had been dropped after eq. (36)). This value is indeed identical to that which one would obtain from eq. (29) in the case of the unperturbed line.
*Fromi~now on we omit the subscripts on ~(r).
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
283
2.2. Quantum theories In the classical path theories discussed so far, one assumes that each atom or molecule is submitted to a given time-dependent perturbation, which, not considering radiation, results from the interaction with the other atoms or molecules of the system. The internal states of all the atoms or molecules are then governed by a time-dependent Schroedinger equation. A more rigorous treatment consists in writing a Schroedinger equation, which applies not only
to the internal states, but also to those of the relative translatory motion of the centers of mass of the atoms or molecules. This equation is of course a stationary one, since, in the absence of radiation, there is no more time-dependent perturbation acting on the system. In fact, as Jablonski pointed out [8], the whole system can be regarded as a single quasimolecule, in which the translatory motion plays essentially the same role as the vibration or rotation in ordinary molecules. As in this picture the translational states are considered as internal ones, we can calculate the absorption energy by applying formula (30), valid for unperturbed systems. But one has to remember that the definition of the states Ii) and If) is now different from the original one, since it contains a translational component. As we shall see later, this leads to some difficulties in the theory because of the continuous spectrum associated with the translatory motion. At the moment we don’t go into details and define a quantum-mechanical absorption intensity from formula (30), by omitting simply the almost constant factor preceding the sum. Thus we write: 1(w)
=
2ir~ (p1
2ö(w —
—
pf)lp1fI
w 1f).
(40)
(Eq. (40) is commonly called the Kubo-formula.) Starting from this expression, quantum-mechanical line-broadening theories have been developed [91. However, in order to be able to make explicit calculations on the profile, one has to solve first a quantum-mechanical collision problem and this constitutes in most cases an extremely difficult task. This may be one reason why no complete quantum-mechanical calculations of line properties have been performed so far in the case of neutral atom perturbation. Moreover, it is not quite clear yet, whether large quantum effects could be expected from such calculations. In one of the next sections we will present, mainly for the sake of completeness, some elementary features of quantum-mechanical line-broadening theory.
3. The derivation of the line profile 3.1. The phase shift theory In section 2. 1 the intensity formula has been derived 1(w)
=
f p(r)exp(iwr)dr
(1)
284
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
with the correlation function ~(T)
=
Kexp[—i77(t
(2)
+ T, 1)] ~
As has been assumed in 2.1, the phaseshift i~is a real quantity, i.e. the perturbations do not cause any amplitude changes of the atomic dipolemoment. 3. 1. 1. The correlation junction
In this section expressions for the correlation function shall be derived for various special cases, without making any specific assumptions regarding the interatomic potential function. 3.1.1.1. Theory for general pressures
At general pressures simultaneous actions of several perturbers on the active atom have to be considered, and the problem of mutual interference of the perturbations arises. The interference is different for different types of interactions. The simplest case is that of Van der Waals (V.d.W.)interaction, where the perturbations and therefore the phase shifts due to different perturbers
simply add like scalars. All other cases, like resonance and exchange interaction are much more complicated to treat. The problem of superposition is, however, avoided at low particle densities, where the collision time is much smaller than the time between the collisions and the perturbations are caused by binary collisions. In the following the theory shall be worked out for noninteracting perturbations, which scalarly add. Then the phase shifts ii~(t+ T, I) ii~the time interval t, ... I + r due to the individual perturbersj are independent and the corretation function becomes: ~T)
=
Kexp[—i
~
77~(t+
T,
1)1),
(3)
where K ~ denotes the time average. If moreover the perturbations are without interaction, then the mean value for the particle! will be independent from all others and ~(r)
=
(exp[—i77(t
+ T,
1)1 )~~‘
(4)
where nq9is the total number of particles in the volumeq9. The time average may be replaced by an
average over the ensemble: with77(T)~77(r,0)
(5)
which is identical with the average over the trajectories. Since the trajectories are determined by
the initial positions and velocities, the average has to be carried out over all initial positions and velocities. We introduce the coordinates R, VR, V4, where R is the internuclear distance and VR and are components of the initial relative velocity in the collision plane. One then shows, that ~r)
~fdFexp
[~~]
where E~m(v~, +v~)+ V~(R)
exp [_iI~w(r(t))dt])~
(6)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
285
is the total energy of the system, m being the reduced mass, Z the partition function and _~r_~i/2l_l dF 1 12m\3”2 R2dRdu v dv
Z
~V
\kT/
R~
~
Furthermore ~w is identical with ~ =
t_~_fdFexp[_.~-](l
I
—
.~-fdF
exp
— (i
E
[—
~
(1
defined on page 282. Rewriting one obtains: —
—
exP[_iJ~w(r(tndt])))”~’
}
exp [—if’ i~w(r(t))dt]
=
(1
~(r) —
-~—
}
fl~V
.
(7)
Then, by making~-+°°, while n is kept constant, one obtains ~(r)
=
(8)
exp[—n~(r)]
with ~(r)
= ~.i/2
~
R2fdvRfdv~v~, exp [_
~
[1 —exp [_if~w(r(t))dt]].
(9) The correlation function (8) with ~(r) given by (9), as reported by Fox and Jacobson [10], describes correctly the total line shape at general pressures in the case of scalar additivity of the perturbations. In earlier theories simplifications with regard to the collision process have been introduced. Such simplifications are useful for the actual computation of the line shape, when comparison with experiment is intended. In the theory of Lindholm [111 and Talman and Anderson [1 21 the collision process is schematically described by perturbers moving on straight lines at constant velocity. Also, in both theories the velocity distribution of the particles is neglected. In addition, in Lindholm’s theory constant frequency perturbation, i.e. linearly increasing phase shift, during the collision time is assumed. In the treatment of Fox and Jacobson, on the other hand, such simplifications are avoided. By expanding the frequency perturbation into a power series in time, however, this treatment is in practice only an improvement of the quasistatic approximation (see next section). 3.1.1.2. The quasistatic approximation 3.1.1.2.1. The general pressure case. It is easy to calculate c~(r)in the quasistatic approximation, where the velocity of the particles is small. Putting VR v~= 0, ~w(r(t)) becomes independent of t, r(t) being replaced by R. One then obtains from (9)
E
2 exp r V kT ][ 1 — exp(—i~w(R)r)1. (10) ~(r) = 4irf dR R 1(RY1 It should be pointed out, that in (10) the Boltzmannfactor exp[—V~(R)/kT] , which describes the influence of the interaction on the occurrence distribution of the internuclear distances, has
286
FSchuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
been neglected widely in the literature. Generally, however, it will influence drastically the far wings of the line [131 . Putting exp[ 17~(r)/kTJ = 1 Margenau’s 1141 result is obtained froni (10). Eq. (1 0) is valid not only for small particle velocities, but also for small correlation times T. Referring to (8), one may show. that when n is large. p(r) is only large for small T. Therefore 1 0) is correct also for large perturber densities. 3. 1.1 .2.2. The nearest neighbour appro.vin/ation. This approximation follows from the quasistatic approximation under the additional assulflptiOfl, that only the nearest perturber contributes to the line shape. This assumption will be fulfilled for sufficiently sniall number densities. In this case a particularly simple expression for the intensity formula is obtained, which is of importance for the determination of interatomic potential functions 11 3 I From the one-particle assumption the product ,i~19in (7) becomes unity. so that one obtains ~(T)
=
I
n~(T).
---
I
11)
Let us replace in eq. (1 0) the upper limit of integration by some quantity R
0 representing the radius of the volume ~79mentioned above, which for convenience may be considered as spherical. Then, using (I) it follows from (11): I(w)
fexp[iwr](I R~)
+00
f
=
dr expfiwTl ~l
2 exp
[-
4~ii[f dR R
R
—f
V(R)
0
.
~
(12)
Then, because of the one-particle assumption, the second term within the brackets becomes unity in the limit of infinitely large R~,.In this limiting case one then obtains:
V~(R)
+00
1(w)
=
4~n
f
dTf dRR2exp[
~T
-~
~w(R))rJ.
113)
Introducing ~w as a new variable, it follows dR(~w) 1(w)47rnJ’d(~w) d(~~P
r
V1t~w)1
[—~-~~—jf
+00
drexp[i(w-
~w)ui
(14)
the integral over ~w being taken over all accessible values of ~w. Since the integral over r is the 6-function by definition, one finally obtains:
dw
2(w)dR(w)exp[— [
2ir 1(w)—47rnR
—j. V(R(w))1 kT
(15)
-
The same result may be derived from elementary probability considerations [I S l~starting from the basic assumption of the quasistatic approximation: The intensity I(w)dw absorbed or emitted in the frequency interval w, ..., co + dw corresponds to the probability ~kW(Rk)dRk, that the
287
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
nearest perturber is at one of those internuclear distances Rk, where it produces the frequency perturbation co: * I(w)dw/2ir=
~W(Rk)dRk.
(16)
For the calculation of W(R)dR it is useful to introduce W~(R,dR), the probability that one particle is in the interval R, ... R + dR, W_(R, dR), the probability, that no particle is in the interval R, ... R + dR and W_(R), the probability, that no particle is at distances smaller than R. W(R)dR may then be expressed by W(R)dR
=
(17)
W~(R,dR)WJR).
W+(R, dR) is found to be W~(R,dR) = 4irn dRR2exp[—V
1(R)/kTl.
(18)
Furthermore,
r
dW (R)
r 2exp[_
WjR+dR)—WjR)~WjR, dR)WJR)+ From (19) W(R) is found by integration:
V~(R)11 kT jj.(l9)
dRWJR)[I_4irndRR
R
WJR)exp[_f
47rndR’R’2exp[_
jj.
(20)
Substituting (20) and (1 8) into (17) one finally arrives at
r V
[~
W(R)dR4irndRR
Since for sufficiently small n according to the one-particle assumption ~ W(R)dR is found to become W(R)dR
=
]],
kT kT 1(R)1jexp L4~f r R dR’R’2exp r V.(R’~fl
2exp[_
(21)
1 becomes unity,
4irn dR R2 exp[—V
1(R)/kT].
(22)
In the special case, that R is a single valued function of w, one obtains from (16) again (1 5). 3.1.1.3. The impact approximation In this approximation it is assumed, that the collision time is small compared to the time between the collisions. This assumption restricts the validity of the approximation to sufficiently small number densities and sufficiently large temperatures. The correlation function for this case shall be derived from the general pressure result (9). For simplification it is assumed, that the trajectories are straight lines, i.e. the influence of the interaction on the trajectories is neglected. ~
should be noted, that in general there are several internuclear distances producing the same frequency perturbation (c.f. p. 294).
288
F Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
The time variation of the frequency perturbation may then be written
~w(t)77(b,
v)’6(t—
t
(23)
0).
Here 77(b, v) is the phase shift caused by the collision with impact parameter b and relative velocity v; ö(t t~)is the delta function; to is defined by r(t0) = b. The time integration then yields —
f z~w(t)dt F
0
77(b,v)
r>t
0
r
(24)
°
,
0
For the calculation of 0(r) the threefold integration in (9) has to be performed. For this purpose one has to transform the variables R, CR and v4 into new variables v, b and t0. Noting that (see
also figure below) 2dRdvRv~dv 3dvbdbdtO R 4-+v one finds nO(r)
= ~I/2n(.~)
5v3 dv exp
[
(25)
f
v~] b db (1 —exp[i77(b, v)1 )fdt 0
rF.
(26)
Disregarding the velocity distribution by introducing a constant average velocity F, one obtains nO(r)rn~f
2irbdb(1 —exp[i77(b)1)
(27)
which is Lindholm’s [16] result. 3.1.2. The intensity distribution 3.1.2.1. Interatomic potentials Essentially three different types of interatomic interactions have to be distinguished [171: 1) Resonance interaction. This case is typical for identical atoms, of which at least one is in an
excited state:
atom
R
CR
perturber
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
289
(28) V~(r)= _Ca/r3.* 2) Van der Waals-interaction (interaction between instantaneous electrical moments). For dipole—dipole interaction the potential law is of the form 6. (29) 0(r) —C6/r 3) Coulomb- and exchange interaction. This is caused by overlapping electron distributions, if the atoms are at small internuclear distances. Although this type of interaction is quite complicated to treat theoretically, it is useful and Common practice to approximate the corresponding potential law by V
V~(r)= C
2. (30) 12/r’ The Lennard-Jones-(l 2,6)-potential (L.-J.-potential), representing approximately the true potential function for the interaction between neutral atoms in the ground state, may be represented in two different forms: V~(r)= (—C
6
2) 12/r’ 4e[—(a/r)6 + (aIr)12] +
C
(31)
6/r
Vn(r) =
(32)
(e and a being defined in the usual way as depth and zero respectively of the potential). The constants in (31) and (32) are connected by the relations 6
C 6
=
C
4ea
12
=
4ca~~
(33)
The potentials V1(r) and Vf(r) for the initial and final state of the line are different in general. This is the essential reason for the collision broadening effect, which is primarily determined by the difference potential V(r)
Vf(r)
—
J/1(r) = h~w1~.
(34)
Furthermore, if the atomic levels are degenerate, they may split into different sublevels in the quasistatic field of the perturbing atoms. In this case the potentials are different for different substates. A detailed calculation of the potentials is presented in section 5. 3.1.2.2. The calculation of the line shape 3.1.2.2.1. The impact approximation. The general intensity distribution for this case is obtained by substituting (27) into (8) and (8) into (1). The well known dispersion profile is the result [16, 181: W
ReF2+(ReF)2} ir{(w—ImF)
(35)
with the total halfwidth y2ReF *The subscript n standing for any electronic state of the system.
(36)
290
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
and the shift
13ImF
(37)
where F is defined by (26). Neglecting the velocity distribution one finds (38)
FniiYu where a is the complex cross-section U=Ur
—iu1
=
2~fbdb(I —expi77(b)).
(39)
The total halfwidth is then
(40)
~y2ni5ur and the shift
I3niJu.
(41)
In order to calculate the total phase shift 77(h) during a single collision, an inverse power law for the frequency perturbation is assumed z~w(r) Ti
r~
.
(42)
For the straight path-constant velocity model for the collision one then finds
C 77(b)c~~b1P
hv
(43)
with c~= ~/~F(4 (p
—
l))/F(4p).
For various cases of interaction one obtains (‘32
p006
c6r~7r
pl2
c12~-ir.
(44)
The calculation of the cross-section u12~f(l
--cos77(h))bdh
(45)
0
=
27rf’ sin rj(b)bdb
(46)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
291
can then be performed analytically. For p> 3 one obtains Or
=
cos
sin
~c~2/(P_1)F(P
~
p—l
~c~2/~1)F(P
~
p—l
— 3)(Cp)2~1)
p—I
hv
(48)
3)(~)2~1) hv
p—1 —
3.1.2.2.1.1. Van der Waals-interaction. In this case one obtains for halfwidth and shift from (40), (41), (47) and (48) [161: 8.16 U315 (C
(49)
215 n 6/h) 13 = —2.96 U315 (C 3’5 n. 6/h) The ratio of shift to halfwidth becomes
(50)
13/a =—0.357,
(51)
independent of the velocity and the value of C 6, i.e. the interacting system. 3.1.2.2.1.2. Lennard-Jones-interaction. This case has been treated by Hindmarsh [19], Schuller
[201 and Behmenburg Defining a parameter a, characteristic for L.-J.-interaction [21] (52) 6/(c[21]. 516 a (c6C6)~ 12C12) 2 vh and a reduced impact parameter x —ba°~(~-c 5 1C61/Wl)” 6 the total phase shift becomes
(53)
77(x)—2a(—x5 ±x”)
(54)
where the
+
sign holds for C
12> 0 and the — sign for C12 < 0. For halfwidth and shift one obtains 215n (55) = (2ir) 6/h) 13 = (27r)315v+(a)U3’5(C 215n (56) 6/ll) with the broadening functions h~(a)and the shift functions v~(a)defined by 3”5 h~(a)i33’5(C
2/5
h~(a)= 4(~)
/
x dx sin2[a(—x5
±x”)]
(57)
±x~1)].
(58)
2/5
v+(a) =
(~±) J’ xdx sin[2a(—x5
The functions h÷(a)and v~(a)are presented in table 1 and shown graphically in figs. 1 and 2. In fig. 3 the curves of v+(a)/h~(a)are displayed.
292
1’. .S~chuller and W. Pm/omen burg. Perturbation of spectral lines by atomic interactions
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F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
293
5.0
2.0
___-
________
1 - ~< io~ io-2 10-1 100 ~Q Fig. 1. Broadening functions h~(cs),calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.-interaction 1211.
In the limit of pure V.d.W.-interaction (large a, C 12 = 0) h~(a)and v~(a)reduce to the constant values 2.71 and 0.98, respectively. Hence in this case the ratio v~(a)/h÷(a) is independent of3~. temIn perature however, and both there y andwill 13 should with temperature through of the‘y velocity factor the D facgeneral, be an vary additional temperatureonly dependence and 13 through tors h~(a)and v+(a), since a is also velocity dependent. At intermediate a the functions h÷(a)and v+(a) are oscillatory, due to the existence of the minimum in the potential function. It should be remarked, that for the same reason also in atomic beam scattering the total scattering cross-section
as a function of the particle energy displays an oscillatory behaviour (glory extrema). 3.1.2.2.2. The quasistatic approximation. 3.1.2.2.2.1. Van der Waals-interaction. If z~w= —C 6/h is substituted into (10), one obtains, 6R neglecting the Boltzmannfactor
-i
-~
10_i
~
100
,o~
-
~
102
Fig. 2. Shift functions u+(oI), calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.interaction [21]
294
I’~Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
--H 7Q-2
10~I
100
101
~
1-ig. 3. Functions t+(0) — v+(~)/h+(o), calculated on the b~osis1)1 Lindholni’s impact appro\ifllat000 01 phase shitt theory, assoOnhing I -i-interaction 1211 -
~(T)
=+(27r)32(Cor/h)~2(l
Substituting
(59)
i).
(59)
into (8) and performing the Fourier transformation, one obtains, for C
6> 0 1
o.
1(w) ~ (C6)
I 2i
-
w0
32
exp
~
~
]
.
w < w0.
(60)
This result was derived by Margenau [14]. For small n and large 1w w0 the exponential in (60) becomes unity, and the intensity distribution coincides with that obtained from the nearest neighbour approximation [221 0,
w> w10
02 lw 1(w)
~n(C6/h)
w 3’2, 01
w < w 0.
(61)
6/1i into (1 5), neglecting the Boltzmannfactor. This may be Lennard-Jones-interaction. varified by substituting z~w = (‘6R 3.1 .2.2.2.2. The general pressure case has been treated by Bergeon [23] and by Hindmarsh et a]. [24], the nearest neighbour approximation by Behmenburg eta]. [251. Only the results of the second approach shall be presented, for the following reasons. First, on the basis of the nearest neighbour approximation the intensity distribution is obtained in an analytical t’orrn. Therefore the influence of both any realistic potential function and the Boltzmannfactor on the qualitative features of the far wings may easily be studied separately and in detail. Secondly, in certain cases of practical importance the intensity distribution, as obtained from this approximation is exact over a wide frequency range, at sufficiently small perturber densities. Since for a difference potential of the L.-J.-type there are in general two different internuclear
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
295
distances R1 and R2 causing the same frequency perturbation w, the expression for the intensity is, according to (16) and (22), given by I(w)dw
= k1
W(Rk)dRk
= ~
k1
R~dRkexp [_ Vj(Rk)]
(62)
For the evaluation of the intensity formula (62) it is sufficient to consider two cases: C6 > 0, C12> 0. In this case the sum in (62) reduces to a single term in the range R < a, since there R(w) is single-valued. Introducing dimensionless quantities z
=
1lL~w/e;
L
C
=
e~/kT
(63)
(~and; referring to the initial state*) and the abbreviations 1-z
____
1 -~--
____
f 1l+~J1+z;
f21—\/l+z;
V~(Rk)
6fk(z) ÷L12f~(z), vk(L, z) = = —2L one obtains for z < 0 (red wing) 1(z)
~Ii —f(z) I~)exp[—Cu
= ~~flU3~z2
3
k = 1, 2
1(L, z)]
+
C
(64)
1 +f(z) I~~exp[—Cv2(L,z)I } (65)
and for z> 0 (blue wing) 1(z)
~
3
~n~3~z2{I
1 —f(z)I~)exp[—Cv1(L, z)] }.
(66)
In the red wing (z < 0) an infinity occurs at z = —1, which is due to the minimum in V(R); for z < —1 the intensity becomes zero. Due to the collisions, which are neglected in the quasistatic approximation and due to deviations from the one particle approximation, the intensity at z = —1 remains finite, giving rise to a satellite in the close neighborhood of the atomic line [241. It should be noted, that the above infinity in the quasistatic intensity distribution is analogous to the rainbow infinity in the classical differential cross-section of atomic beam scattering. In the blue wing (z> 0) the intensity decreases monotonically with increasing z. Due to the Boltzmannfactor the behaviour of 1(z) is not determined by V(R) alone but in general depends also on V1(R). For the same reason 1(z) will generally be temperature-dependent. The influence of V.(R) on 1(z) is demonstrated by figs. 4 and 5, which show 1(z) calculated for two different values of the quantity L, which characterizes the position of V1(R) relative to that of V(R). If L ~ 1, then the value of the Boltzmannfactor at z = —1 is comparable to unity, and the infinity clearly comes out. If, on the other hand, L 1, then 1(z) decreases rapidly, due to the Boltzmannfactor, already at z> —1, so that the infinity at z = —1 is obliterated. In this case 1(z) is very small in the range z> 0 (blue wing). ~‘-
~o and ~ should not be confused with the quantities appearing in eq. (39) ff.
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
296
~zLo1,4 _-
Lr1.0
Lr0.8~ Lr0
liii
~-
~
1)11(1
Ill
-i0~
-
—
100
z
Fig. 4. Influence of the Boltzmannfactor on the red wing, assuming L.-J.-interaction. Curves calculated from the nearest neighbor approximation of the qua~istatictheory for C 6 > 0, C12 > 0 and C = 1 [251.
C6 > 0, C12 < 0. In this case the sum in (62) reduces to a single term over the whole R-range. Using the abbreviations l_~z
j.;
~
—z
(67)
-z 6f(z) + L’2f2(z) v’1(L, z) = —2L one obtains for the z < 0 (red wing) 1(z) =~~na~~z2 3
~(] +f’(z))~)exp[—Cv(L,
z)]}.
(68)
In the range z > 0 (blue wing) the intensity is zero. 3.1 .2.2.3. Theory for general pressures. This theory has been worked out by Lindholm [Ill, Talman and Anderson [1 21 and by Fox and Jacobson [10]. 3.1.2.2.3.1. Van der Waals-interaction. This case has been calculated through by Lindholm [11] and by Talman and Anderson [12] In the Lindholm treatment, in addition to the straight pathconstant velocity approximation, the frequency shift during the collision time r~is assumed to be constant. This assumption is equivalent with replacing the V.d.W.-potential by a square well potential. Consequently a parameter K was introduced, defined by .
= 2K
b/~7.
(69)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
Lr0~
297
/2
177 ~1
__
,.,,~,.,
io2
z
Fig. 5. Influence of the Boltzmannfactor on the blue wing, assuming L.’J.-interaction. Curves calculated from the nearest neighbor approximation of the quasistatic theory for C 6 > 0, C12 > 0 and C = 1 [25]. i~is
determined by the condition, that for large values of 1w — w01 the intensity distribution agrees with the quasistatic approximation (60). For the representation of the intensity formula the following dimensionless quantities are introduced: /c6C6\ xb~—_—) 2v71
—
—1/5
ic6C6\
3/5
h’47rK(—-—) ~,2 viii
—1/5
vtIc6C6\ Y—~,,—--——) 2K2vh
; ;
(70)
2K/c6 C6 k(w—w0)—(—— U \2 tiP,
1/5
The intensity distribution then becomes I(w)f exp[iky~hij.o(y)Idy
=
=
(71)
‘l’r(Y) + j
f[(l_cos4)(y_x)_sin~~x6+2x]x
dx+f [(1 _cos-~)(x_y)_sin~ x6+2Y]x dx
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
298
~1(r)
_cos~s)[email protected]
= 1t[(1
dv+f [(
_cos~’)xe+sin~’
(x---T’)lx dx.
The problem of calculating the line shape is thus been reduced to that of calculating the dimensionless function ~i(v). Lindholm uses the method to calculate its behaviour for very small values oft’ and for very large values oft’ and then interpolates between the two. He obtains tori’~ I =
(0.795
+
0.577
i)i’ +
0.165
+
0.277 i,
(72)
fori’-~ I 413 +
0.591(1
+
i)v~2.
~(v) = (0.369 + 0.213 i)i’ It may easily be seen, that the v “2-terms, dominating in the expansion t’or v ~ I yields exactly the quasistatic approximation. On the other hand, the t’-terms dominating in the asymptotic expansion, yields the impact approximation. Talman and Anderson treated the general pressure case assuming the straight path-constant ,
velocity model, but, contrary to Lindholm’s treatment, calculated the frequency shift during the collision exactly. The results have bet~npresented and discussed in the article of Traving [26] .It is remarkable, that the intensity distributions in the line core, as computed by Anderson and Talman, differ very little from those obtained by Lindholm. Thus it seems, that the line shape in
the core is rather insensitive to the schematization of the collision process. This might not be so, however, in the wings of the line. 3.1.2.2.3.2. Lennard-Jones-interaction. The application of the L.-J.-potential to the case of the general pressure theory was carried out by Takeo [27], Behmenburg [281 and by Fox and Jacobson [101. Takeo performed the calculations on the basis of the theory of Talman and Anderson, Behmenburg on the basis of Lindholm’s theory. Fox and Jacobson started from the exact expression for the correlation function (eqs. ( 8) and (9)), but, by expanding the difference potential into a power series of time, actually used only an improvement of the quasistatic approximat ion.
On the basis of Lindholm’s theory, with the parameter ~ defined by (52) and the reduced quantities h, k, x and i’ defined by (53) and (70), the following expression for the intensity distribution is found: 1(w)
=
J
exp[ia”5 kv
— a3~
h ~(a,,v)1 dy
(73)
where the function i~i(ct,y) is given by ~i(a,
~
y) y)
=
=
~i~(a, y) +
i ~i~(a,
(74)
y),
f [(1 —f1(a, x)(y—x)—f2(a, x)h0(a, x)+2xlxdv +
1 [(1 —g1(a, x, y))(x—v)—g2(a,
x,
y)h0(a, x)+2y]xdx
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
~
y)
=
+
f[(l
—f
t(a,
x))h0(a,
x)+f2(a,
299
x)(y-x)Ixdx
J’ [(1 —g1(a, x, y)) h0(a, x)+g2(a, x, y)(x—y)] x dx
with the abbreviations 5+x’~’)];
f
5+x~)] ; (75) 2(a,x)sin[2a(—x’ g 6 +x~2)I; g 6 +x’2)I; 1(a, x, y) = cos[2ay(—x 2(a, x, y) = cos[2ay(—x h 2(l — x6)”] 0(a, x) = a’[x’ Because of the assumption of constant frequency shift during the collision time the theory will give only an approximation to the spectral intensity distribution in the wings. The shape of the line core, on the other hand, is believed to come out rather exact. f1(a, x)cos[2a(—x
3.1.3. Criteria of validity of the phase shift theory and its approximations 3.1.3.1. The adiabatic assumption Consider two nearby states k, 1 of the active atom, which, in the absence of the perturbation field, have energies Ek, E 1. If the states are nondegenerate, the energy levels will shift in the presence of the perturbation by amounts Vk(r), J/1(r), assuming the values E~(r)and E(r). The condition necessary and sufficient for a collision being adiabatic is, that the frequency range L~w~ of the Fourier-spectrum of the time dependent perturbation field is small compared to the frequency (E~(r)— E~(r))/h.Since ~ is approximately the inverse of the collision time r~,the criterium for adiabatic collision may be written l/r~
‘~
(76)
{E~(r)— E(r)}/P,.
Approximating r~by b/~(b impact parameter, U mean relative velocity) and denoting L~E= Ek — and V(r) = Vk(r) — V1(r), the relation (76) may be rewritten (replacing r by b) b z~E+i~V(b) — ~‘l.
(77)
h
If one of the states, k say, is degenerate, it becomes splitted in the presence of the perturbation field into sublevels m, m’. In this case the additional condition
~ U/m~(b)
l/r~
—
(78)
Vm(b)}Ih
must be fulfilled for the collision to be adiabatic. The condition (78) may also be regarded as a special case of the condition (76), if one considers k and 1 as belonging to states, which differ only by their space quantum numbers. The condition (78) may be written in a more rigorous manner [29, 30]
~j,
m, m’)~f(Vmi(t)
—
Vm(t))dt~
1.
(79)
300
F. Schuller and W. Behmcn burg. Perturbation of spectral lines by atomic interactions
Herej refers to the total angular momentum quantum number of the unperturbed active atom, and in and rn refer to the interatomic axis. If, in particular. V,0,(r) and V0~,(r)are approximated by functions of the L.-J.-type and assuming the straight path-constant velocity model for the colhisioii, the condition (79) becomes 2ir I
c6~C6 c12z~C12\ 5 b’1 /~ 6 \ b
1
(80)
where
~Co =
Coon’
Co 01
.
~C~2 = Ciom’
(‘i~omi
3.1.3.2. The validity range of the quasistatic approximation A general validity criterium may be derived by Fourier-analysis of the phase disturbed dipole oscillation during the collision [31, 321. Using the method of stationary phase it may be shown. that the spectral intensity distribution may be interpreted as quasistatic in certain frequency ranges, where the condition for stationary phase 2/~(~k)13”2 ~ 1 (81) R(tk)
= ~
I~(tk)I
is fulfilled. In (81) the time
tk
is defined by
(82) where w 00 is the unperturbed frequenc~of the oscillator and w the frequency of the radiation. Fig. 6 shows qualitatively ~w(t) for a realistic difference potential during the collision. Obviously, condition (81) is not fulfilled at values of’ w’, where ~d.’(t) = 0. This is the case for large t, where w’ is very small (i.e. in the line core) and in the neighborhood of the extrema of ~w(t). In order to determine the f’requency ranges, where (81) is fulfilled, R(tk) has to be calculated. For a L.-J.-potential, assuming C6 > 0, (‘2> 0, and the straight path-constant velocity model one obtains [251 2—
R(tk)
=
S(c, v)
=
0.176
a’
c”
I1--c~ -
-
4
II
---t’l
(1
y’2)014+(
I
with
Fig. 6. Time dependence of the frequency perturbation during tile collision. Tile zero of the time scale is chosen to coincide with the time where the internuclear distance equals that of closest approach.
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
c(w’)
=
(i
~
±
~l +~-~-); x’
~~~~chI6(w1);
0< x’ < 1
301
(83)
and a is defined by (52). For wt> 0 (blue wing) and under the additional assumption hw’/e ~ 1 (83) simplifies and one obtains
S(c, x)
=
0.235 a”2(e/hw’)”124f(x’)
f(x’)
14(1 —x’2)”4+(l —x’2Y314.
(84)
From (84) follows the existence of a frequency limit w~,,so that for w’ ~ W’L (81) is fulfilled, i.e. the quasistatic approximation is valid. Denoting by XL an arbitrary value of x, for which — x’L ~ 1 [32] and defining ~ by requiring S(c(wL), xL) = 1, one obtains from (84) WL
0.042 f241’ ‘(xL)a~2~ ‘c/h.
=
(85)
If w’ < 0 (red wing) then S(c(w’), x) becomes very large for Iflw’/I ~ 1 and for Ihw’/l
1,
i.e. (81) is not fulfilled. On the other hand, (81) is fulfilled in the range 0< 6 1(a) < Ihw’/cI <
ó2(a)
< 1
(86)
for systems with sufficiently large values of a: a
~‘
(87)
aO(XL)
where aO(XL) is defined by the requirement S(Cmin, XL) = 1, Cmin being that value of c, for which 51’2(l — 2c)”2 becomes a minimum. c 3.1.3.3. The validity range of the impact approximation In this approximation it is assumed, that the collisions are completed in the correlation time r (see section 3.1 .1 .3). This means, that the results are valid only if r is much greater than the mean duration of a collision
r>>rc.
(88)
Consequently, the impact approximation is valid in the line core in the range Iw’I ~ Iw~I.
(89)
For the estimation of w, Tc may be approximated by bcr/U, where bcr is a critical impact parameter defined
by
‘y = 2ir b~rnU where -y is given by (55). Taking into account, that w
(90) 1/Tc one finds (91)
302
F
3.2. Orientation
eff~’cts
Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
When studying line profiles by means of phase-shift theories, one generally assumes that there is only one potential curve in each, the lower and the upper electronic state of the active atom. Strictly speaking this is certainly not true. since in almost every case at least the upper state is degenerate and thus one should associate different potential curves to different magnetic substates. Of course, this difficulty may be removed by introducing an average potential, for instance
that described by London’s formula [161 - In fact, this rather intuitive procedure is not quite correct, since it neglects any influence of orientational changes, during collisions, on the linebroadening mechanism. However, the simple phase-shift theory with an average potential constitutes in many cases a fair approximation; but when studying more detailed aspects, such as.J-dependence of broadening and shift of fine structure components, one should use a theory that treats degeneracy in a more satisfactory way. We shall call it Anderson’s theory although Anderson’s original work was dealing
only with the case of’ molecular line-broadening [3] . It appears indeed, that his treatment of angular momentum can be applied without any changes to the case of atomic line-broadening as well. One interesting feature of this type of line-broadening theory is its direct relation to the theory of other relaxation phenomena like depolarization by collisions of resonance radiation or, furthermore, collision-broadening of double-resonance curves. Some remarks on these subjects are contained in section 3.2.4. 3.2.1 - The relaxation of the dipole transition moment in the n-npact approximation Following Anderson we introduce the so-called interaction representation by setting:
H(t)=H0 +IIJt)
(92)
and (93) where H0 and Q153 are respectively the Harniltonian and the evolution operator of the unperturbed atoms, and where H~is the Hamiltonian describing the interaction during collisions. The evolution
operator in the interaction picture satisfies the following Schroedinger equation: (94)
ihT=H~T,
with ~
~1~HCQ1O
=
(
exp~J10t~H~exp
~J1l)t)
.
(95)
For the dipole operator we have the relation: (r)T~T,
with ~
expHHor}pzexp{~~~±HOr~.
(96)
This may be written out in terms of’ matrix elements with respect to a set of basis functions representing the unperturbed atomic states. The result is:
F. Schuller and 11/. Behmenburg, Perturbation of spectral lines by atomic interactions
~
= ~
!AZif(T)
(_~w~r)
csttT~fexp
303
(97)
.
Now we make the assumption of non-overlapping lines, already discussed in a previous paragraph. Accordingly, for a given line, corresponding to a transition i -~f, the correlation function must be of the general form (2.31). Consequently we retain in eq. (97) only terms oscillating with frequency which means of course, that a and i~cannot differ from i and f respectively, except by some magnetic quantum numbersM~and Mf.* So we write: ~
/1.f(T) = ~ZMIMf(r)exP{iwIfr}~
(98)
with 1~ZM~Mf(T) =
~
~
T~M,(r)/4~,MP T~,M(r), ‘
I
f
(99)
f
where the upper indices designate the subspaces to which all the magnetic states belong. The state of the system at time r can be related to that at time r + dr by introducing a new evolution operator defined by the relation: T(r ÷dr)
=
T(r) T(r
-+
r + dr).
(100)
With this definition we can express the change of the quantity ~~Mf dr as follows: I d “~zMiMf=
=
‘r+dr~--
‘zM
1Mf’
~
101
if ~zM~M~
E T~M~ (r —~ ‘r + dr)l4,~,M,(r) T~?M(r M~M~
occurring in the interval
-*
r + dr)
— ~ZM~Mf()
I
In the impact approximation the calculation of this quantity can be reduced to a one-collision problem. Let us assume that there is one collision in the interval dr. Then the operator T(r -~ r + dr) takes some value, T(l), say, which depends only on the type of that collision. As we will do in the next section, one may consider all types of collisions and calculate the average effect of one collision on the quantity /.1 ~M0(T). It then turns out that the following relation If holds: 1~M(r) (102) \MiM. ~ TZM,(l)~t’~,M,(r)T~,M(l)\ II I / =A/1 I
/
where A is independent of any magnetic quantum numbers. This important result means, that in the average the initial orientation of the dipole moment is conserved after one collision. (It is understood of course, that the collisions are assumed to come isotropically from all directions.) It is now easy to evaluate the average change of the matrix element J4~M(r)during the inter*
Restricting ourselves to the simplest case we assume, that perturbers are spherically symmetric, M 1 and Mf refer then to the active
atom alone.
304
F. Schuller and h~.Beh menburg, Perturbation of spectral lines by atomic interactions
val dr. Using eqs. (101) and (102) and remembering that the probability for having one collision in dr is equal to dr/Ta (T~= time of mean free path), we obtain: -
dT
—-—(I —A)M’~M(r).
(103)
Let us consider the average of this quantity over the ensemble corresponding to the interval ‘r. Since in the impact approximation collisions are assumed to be uncorrelated, this averaging procedure is independent from the previous one. Thus the result can be regarded as the change during dr of the quantity (/.L’~M(r))eand we get the equation: d(/.L~M(r)~e = —dr
/IMM()e
(104)
withF(l —A)/T~. Hence the relaxation of a component of the absorbing dipole moment is described in the impact approximation by the following relation: zMiMf(T~e
~ZM
1Mf(O)e.
(105)
We will see in the next section that similar relations are obtained for the relaxation of other observables. The correlation function may now be written as:
~
lMi~(0)I2~e~T. i f /
(106)
(Naturally, the exponential factor of eq. (98) has been removed by assuming that frequencies are measured from the center of the unperturbed line.) Usually one introduces an “optical” cross-section, defined by the relation:
F
=
niia
(see eq. (38))
(107)
where n is the number density of perturbers and lithe average velocity. Dropping a constant factor, the correlatian function then takes the form: ~o(r) =
exp(—nUar)
(108)
which corresponds to a Lorentz-profile with the following expressions for the shift and the halfwidth: f3nPImo,
‘y
2nLiReu.
(109)
It might be worthwhile to emphasize, that although degeneracy could be removed during collisions, the impact approximation always leads to a single Lorentz-profile for a given spectral line. 3.2.2. General expression of the differential cross-section
According to eq. (1 02), the calculation of the quantity A, on which depends the relaxation rate
of the absorbing dipole moment, involves an average over all types of collisions. Let a collision be characterized by an impact parameter b and an initial scalar velocity v. Furthermore, all operators being defined with respect to a space-fixed frame, we have to consider the
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
305
orientation of the collision-trajectory with respect to that frame. To do this, we will specify the spatial orientation of the collision trajectory by three Euler angles, for which we use the collectivenotation f~. As a first step in calculating the quantity on the left side of eq. (102), we may now perform the average over all orientations and this will lead to an equation of the following type:
KM~M~ T~Ml(l)i.i~IMi(r)T~iM(l)) =A(b)I4tMM(r).
(110)
(The velocity does not appear explicitly in this equation, since we assume, as usually done, that it may be replaced by some constant average value U.) Obviously, the factor A of eq. (102) represents the average over b of the quantity A (b) defined by eq. (110). Accordingly, the optical cross-section, defined by eq. (107), can be written: ntiaz~F
1-A T~
1 T~
b
and from this it is easily shown, that a is given by: (111)
af2irbdb(l—A(b))/2irbdba(b).
The quantity a(b)
1 —A(b)
is commonly called the differential cross-section. In order to facilitate the calculation of the angular average on the left side of eq. (110), we have to change the definition of the evolution operator, by making it independent of the direction of the trajectory. This will be achieved by introducing a frame of reference that is attached to a given trajectory. We call this the “collision”-frame; it will conveniently be choosen with its z-axis parallel to the incident direction of collision. Its position with respect to the space-fixed frame is thus described by the Euler angles mentioned above. Let R be the operator that rotates one frame into the other: Then there is the following relation between the previous operator T(1) and the new operator T~(l),say, belonging respectively to the space-fixed and the collision frame: (112)
T(l)=RtT~(l)R. The éorresponding matrix elements are given by: Tj~M~’~= I
I
~
p4.’11v7’1’
(
‘-~}~M. (~2)T~t ‘~M~4!” ‘~!“M’(&~~) I
I
1
1
I
I
(113) T~iM(1)
~
= M~’M~”
(~2)T~(1 )M~’M~” D~j,IIM(~2)
where the D’s are the rotation matrices, J being the total angular momentum of the considered electronic state.
306
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
It is now essential to consider the fact, that p5 is a component of an irreducible tensor operator. Then, from the Wigner-Eckart theorem [31, its matrix elements are proportional to some ClebschGordan coefficients, according to the relation: PZM1Mf~)
C(J~lJ1M~0M).
(114)
Substituting the expressions (113) and (11 4) into eq. (110) one can carry out the averaging procedure. This rather lengthy calculation, based on the Clebsch-Gordan series shall not be developed here. It leads to the following result for the differential cross-section:
~
1
u(b)
i
C(JflJ.;MfMM.)C(JflJ.;M~M’M~)Tit(l)MM,Tf(l)MM
(115)
aIIM
which is called Anderson’s formula. According to this equation, the collision frame plays from now on the role of a fixed frame and any reference to the original space-fixed frame has disappeared. Calling in the future the collision-frame “fixed”, we will speak of a “moving” frame when choosing the quantization axis parallel to the line joining the two atoms. Those will be the two frames used in our further calculations. 3.2.3. Approximate solutions of the one-collision problem
Anderson’s formula (11 5) reduces the calculation of the optical cross-section to a one-collision probleni. Let T(t) designate the evolution operator during one collision (defined with respect to the collision-frame) and let t = and ~ +oo correspond respectively to the times before and after the collision. Then we may assume, that T~(l)is given by T(+oo) whereas T(--°°)equals unity. In order to determine T(t) we could, in principle, start from eq. (94) but this amounts to solving a great number of coupled differential equations. (The fact that Anderson’s formula involves particular matrix elements connecting states that differ only by their magnetic quantum numbers, does not change anything on the situation as far as the solution of the system given by eq. (94) is concerned.) Thus, in practice, eq. (94) is of not very much use and we introduce a different approach for determining the operator T(t). Suppose for a moment, that we are considering a state n which is non-degenerate (corresponding for instance to J = 0). Then we make the adiabatic approximation and obtain from eq. (2.33) the following expression for the diagonal elements of T(t): —-°°
T~(t)
exp~—~fdt’
V~(t’)}
CC
(116)
where V~,is the interatomic potential introduced on p. 282. This expression obviously satisfies the following equation: ~hTn
CC
V0T~.
(117)
We now assume that in the more general case, where n corresponds to a degenerate state, it may still be possible to define some quantity similar to our previous V,~.This of course will be an M-dependent quantity and we shall designate it as V~lM.Matrix elements of T(t) belonging to the
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
307
state n should then satisfy the equation: ihT~IM=
~
(118)
VJ~I,MI,T7~JIIM.
M”
We shall see soon, how in practice the quantity V~IM?lcan be defined more explicitly. Right now we want to mention that V~,Mllcan be interpreted as a matrix element of some sort of an “effective” interaction hamiltonian according to TTn
— tijneff~
VMIMII
=
5~11c
)M’M”
However, it is in general not allowed to identify H eff with H~.The latter procedure would yield only the first order energy which, as is well known, vanishes in the case of Van der Waals interaction. (Even in the case of resonance interaction although it is finite there, it does not represent the main effect [341.) Eq. (118) corresponds to a certain type of adiabatic approximation in the sense that it involves only one electronic state n. This is true, no matter what further approximations will be used to solve the system which is still (2J + 1)-dimensional. One of those latter approximations shall be discussed now. Giving T(t) the meaning of an operator acting in the subspace of the state n, we may write eq. (118) in the form: (119)
ihT=VT. If this were an equation between scalar quantities, the solution would obviously be: T
ex~{_~-fdt’ V(t’)}
=
exp{—P(t)}.
(120)
In our case, where operators or matrices are considered, eq. (120) constitutes what is called the “sudden”- or “scalar’ ‘-approximation. * Matrix elements of P(+oo) are: (121)
Mhfdt~TM0M(t).
To get matrix elements of T(+oo), we first define a matrix that diagonalizes the matrix P(+oo), according to: ~
A~IMI0~f”M”
AM?IIM
=
~M’M
exp { —~}
and this then leads to the expression: T~,M(+oo)=
E AMIMII
eXp(—~MU)AM,lM.
(122)
There is an opposite to the “sudden”-approximation which must also be considered. For that *As is well known this amounts to neglecting the effect of the Dyson time ordering operator.
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
308
purpose we introduce a “moving” frame (already mentioned in the previous paragraph), having its z-direction parallel to the interatomic axis. Let D~be the matrix describing the rotation of the fixed frame (which is here the “collision”-frame) into the moving frame, and let us define new quantities by the relations: T~oM
=
~
=
~
(123)
D~I0M0TMI.M
and MIM
~
(1 24)
V~J~0~I00?D~J000AI.
The matrix , which describes the interaction energy with respect to the interatomic axis, may be assumed to be diagonal for symmetry reasons. Introducing again the effective interaction hamiltonian, which in the new frame may be designated by ~ we would write the elements of e as follows: — (..~,fleff-~
M
‘“c
‘M
Then, with these quantities, eq. (118) transforms into: ih
~‘M
=
~
+
ih
~
~
(125)
Suppose now, that the second term on the right of this equation is negligible. Then the solution will be given by: r~(t) 6MIMexP(----
~ J dt’e~(t’)}
.
(126)
This obviously corresponds to an adiabatic approximation. But now the word “adiabatic” is used in a very restrictive sense since it applies to magnetic sublevels 114. To avoid confusions one has to bear in mind that there are two types of adiabatic approximations: The first one, concerning the electronic level nasa whole, is always made tacitly when eq. (118) is used; the second one, concerning magnetic sublevels (with respect to the interatomic axis) corresponds to a special kind of solution of that equation. However, from now on only eq. (1 26) will be referred to as the adiabatic approximation. Matrix elements of T(+oo) in the adiabatic approximation are easily obtained from eqs. (1 23) and (126) which yield [351: T~,M~
= ~1’M
exp
-~
J
dt e~(t)} -
(127)
The quantities given by the expressions (1 22) and (1 27) obviously depend on the impact parameter b. It can be shown by more detailed considerations that these expressions are both limiting ones, corresponding respectively to large values of b in the case of the “sudden”-approximation and to small ones in the case of the adiabatic approximation.
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
309
In either case the differential cross-section can now be calculated from Anderson’s formula, by letting n correspond to the initial respectively the final state. The question now arises how to define the M-dependent interatomic potential. One possibility is to calculate both, V7~?Mand EM by second-order perturbation theory. This has been done recently in the case of dispersion forces [30, 34]. However, if repulsive forces are to be included, such direct calculations lead to much more difficult problems. An alternative approach would be to define a priori the interaction energy in the moving frame as a function of the interatomic distance, writing for instance: (l28a) in the case of a London-type potential, and EIMI =
_C~/R6+
(128b)
C~IyR12
in the case of a Lennard-Jones potential. From these expressions matrix elements of V can easily be calculated by inversing formula (124). Formulae expressing elements of the matrix P(+oo) in terms of the IMI’5 have been derived for straight-line trajectories defined by the equation: R(t) = (b2 + u2t2)112. The results obtained for several values off 1 andff are listed below [361*: i)J~= 0:
P’0 ~i-fe~dt;
Jf= 1:
P~1=P~,.1 =~J°~[e~ +—~-~(e~ ~e~)]dt, +OOr
pf00
=
± h ~r
I[Co — R2
~C
—~‘
E~jdt
~
0
~ “~
—
J~b2~-~-(e~ —e~)dt,
P~.,=P~,, -~ —i
P~
0=P~,=P~1 =~=o. +00
~ j
=
I
j
f
= 1~ 2
pi,f
=
1/2 1/2
=
1/2 —1/2
=
~:
P312 3/2
= ~i_
~
—1/2—1/2
pi,f Jf
pi,f pi,f
r J
E1/2
~ di’
J00
[e~/2 + ~
=
—1/2 1/2
=
1 p 3/2
3/2
sl
(e~ — c312)]dt,
*In the isotropic case these expressions reduce to those given by eqs. (29) and (31), p. 289.
310
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
1/2 =
1/2-1/2
= ~i/
2~3/2
pf
= ~~I/2
=p~’
3/2 1/2
ICC ~
=
3/2
=pf
3/2—3/2
~
~—3/2
1/2
=pf 1/2—1/2
1/2 3/2
(e~,2 — ~3/2
)j di’,
~~3±jP~
=pf —1/2 1/2
~
(e~/2 =p~’
—1/2 3/2
=0
=pf
—3/2 3/2
—3/2—1/2
We further set:
-
+=
± C ~J
if
~M
‘~dt
‘129
~M
Here we want to make a remark on how matrix elements could be calculated in a systematic way by making use of the symmetry properties of the effective interaction hamiltonian. Expressing the latter as a sum over irreducible tensor components, we write
x
~ A°~T,~~0 (internuclear frame),
= =
(collision frame).
A~D~Tkq
~ kq
C
Then, according to the Wigner-Eckart theorem, matrix elements are given by E’/l
=
(x~~
1~”~I’M
~ A~C(JkJ;M0M)
=
(H~~)MIM ~A~D~ 0C(JkJ;MqM’),
kq
J being the angular part of n. Substituting into this the following identity [33] 1/2
D~0=
(~~-~)
~kq~’
0)
with sin ~
CC
b/R
one finally obtains 1/2 =
~ (~-j)
3~ 0)C(JkJ;MqM’). A~Ykq(I
The coefficientsA~maybe calculated from: A~
2k CC
+
I
Z~e7~C(JkJ;M0M).
For symmetry reasons we have k = pair [42] so that for the cases of interest in this paper only the values k = 0,2 are relevant. Special results are given in the following table:
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Special values of
as functions of
311
44).
~/23/201
0
~(i/2)
2
—
~j(2c~’~ +
1(~(32) + ~(3’~2))
J~(4%~_~~%2))
—
,(~(1)
1))
—
Calculating differential cross-sections from Anderson’s formula, we first consider a transition = -~ Jf corresponding for instance to the 2P,,2 component of the alkali doublets. We call ~,
this the isotropic case since it does not involve any orientation effects. In this case eq. (118) can be solved exactly and the result for the differential cross-section is given by: a(b)
=
1
—
exp{—i(~~,2 — ~11,2)}.
(130)
This is just Lindholm’s formula which is strictly valid only in the isotropic case. In the more general anisotropic case we have to consider either the “sudden” or the “adiabatic” approximation when calculating the differential cross-section. For a transition J1 = ~- Jf = 2P corresponding to the 312 -component of the alkali-doublets, we get the following results: “sudden”-approximation: -+
a(b) = 1
—
~-[exp~—(~,2
—
P~/2)}+ exp{—(z~,2— P~12)}];
(l3la)
“adiabatic”-approximation: a(b)
1.
(131b)
Taking as another example the transition results are: “sudden”-approximation a(b)
~
=
0 -~ Jf
=
1 (e.g. intercombination line of Hg), the
1 —4[exp{—(P~1+P~,—P~)}+exp-[—(P~,—P~,—P~)}+exp{—(P~ —P~)}I;(l32a)
“adiabatic’‘-approximation a(b)
=
1
+~-
exp{—i(r~— i~)}.
(132b)
Let us now ask the question what would happen if one or the other of these approximations were used alone for the calculation of the total cross-section. Obviously the adiabatic approximation would lead to an infinite result since a(b) as given by eqs. (l3lb) and (l32b) does not go to zero for b —~ As to the “sudden” approximation, it breaks down indeed for small b-values. But in that region the function a(b) has a rapidly oscillating behaviour and only its average value counts. Since the latter equals unity for both, the “sudden” and the “adiabatic” approximation, almost no modification will be introduced in the result when replacing at small b’s one approximation by the other. Therefore we can expect that the expressions (131 a) and (1 32a) lead to reasonable values for the total cross-section. This conclusion can be confirmed by more quantitative arguments [37]. As has already been mentioned, in many calculations on line-broadening and shift one disre°°.
F Schuller and W. Behmenburg. Perturbation of spectral lines by atomic interactions
312
gards the anisotropy of the potential, which is replaced by an average over 111-values. From a standpoint of pure theory such a procedure is certainly incorrect, but one may ask however, whether it introduces large errors in the calculated cross-sections. Defining an average phase-shift, one shows from the invariance of a trace, that the following relation holds: ~M
~MM
~
CC~L ~
(133)
=~
With this value of the phase-shift, Lindholm’s formula yields the following expressions for the differential cross-sections: For the transition J1 = -~~ CC
u(b) = I
—
exp{—-~-(~/2 +
and for the transition J1 u(b)
0
=
~i/2
-*
--
2P’112)}
(134)
I~= I:
I —exp{---4(2P~1+P~ —-3P~3)}.
(135)
Comparing these expressions to eqs. (131a) and (l32a), we see that the more correct procedure consists in first defining cross-sections for different M-transitions and averaging afterwards, instead of averaging before defining a ~ross-section. As an example consider the case df dispersion f’orces (eq. (1 28a)). Then the differential crosssections given by eqs. (l3la) and (l32a) take respectively the form: t)) a(b) 1 -- ~(exp(—A1/b~) + exp(--A2/b CC
and
u(b)
5)+ exp(—A/bt) + exp(—A~/b5)). 1/b The corresponding total cross-sections are given by: =
I
—-
-~(exp(—A’
-
CC
A215
~
+
A215
.
and ~
-~~)
A7”5
+
A ‘2/5
--iTe13~0F(~)(~~_-
+
A ‘2/5
—--—--~- -)
whereas the isotropic theory leads to the results: -
IA 1
CC
+A2
—i~e’~~°F(~)
2/5
and a =
.
_iTeI3~10F(~)
A’~ +A’
(_~~_
+A~
2/5
-~-
In practice, th6 values obtained from the isotropic theory are only slightly different from the previous ones. On the other hand, the ratio: r = halfwidths/shift
=
Re a/Im a
remains rigorously unchanged since the A’s are real. The situation might be quite different and the influence of anisotropy much stronger if repulsive forces are taken into account. This, at least, is the result of a recent calculation using a potential [37]. With respect to comparison of theory and experiment (section 4) we want to
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
313
present the results for halfwidth and shift in the case of L.-J.-interaction, as obtained from the sudden approximation. Defining the quantities p
C~/C~,;
=
p’
=
(136)
~y’= C?2/C~2
[c6C~(p — 1)] 11/6
2iih [c12C2(p’ — 1)] 5/6 1
=
a2
(i initial state)
6/C~
C12/C12
=
1 a1
= C°
[c6C~(p(5÷y)/6
—
1)] 11/6
2iih [c12C~2(p’(ll +y’)/12—
1)15/6
— 1 [c6C~(p(l + 5y)/6 — 1)] 11/6 a3—--—2v7-) [c,2C2(p’(l + 1 ly’)/12 — 1)15/6 one has the following expressions ~
315~315 [~ (p ~
n(2~)
—
1)12/5
x 4[h÷(a,) + h~(a2)+ h÷(a3)]
—
1)]
x 4[v+(a,) + v~(a
(137)
2)+ v~(a3)]
where the functions h~(a)and v~(a)are given by (57) and (58). We note, that line core data contain information both on the isotropic and anisotropic part of the potentials. As far as the anisotropy is concerned, this information may be compared to that obtained from depolarization and double resonance measurements. For this purpose the relation between the corresponding cross-sections shall be discussed in the next section. 3.2.4. Cross-sections for the relaxation of orientation and alignment Some of the best known cases, where anisotropy of the potentials manifests itself, are relaxation phenomena like depolarization of resonance radiation by collisions, broadening of double resonance curves, etc. Although these topics are not within the scope of our article, we will write down some of the expressions of the corresponding cross-sections and compare them to those obtained in the case of optical line-broadening. In many situations this procedure might be useful for extracting potential parameters from experimental data. When studying the relaxation phenomena mentioned above, one considers an atom which is brought to an excited state by absorption of resonance radiation. Let the corresponding wavefunction be written as:
‘i,E
C~4)UM
Mf
where i and f refer as usual to the initial and final state respectively and where u designates an unperturbed wavefunction. Now, if several initial states are involved with equal statistical probabilities, the state of the system posterior to the excitation can be described by the following quantity:
314
I’~Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
Mf 0~c~ Mf c~
‘~M.Mf = ~
which is called the density matrix. Considering the time evolution of the density matrix under the influence of absorption, reemission and collisions, its new value ~oM~M~(t) will describe a stationary state if the total rate of change per unit time is zero. Concerning the relationship, for various experimental setups, between the elements and P~z~Mf(t)of the density matrix and the polarization properties of the absorbed and reemifte~Iradiation respectively, we refer to more detailed surveys [38, 391. Here we are considering only the evolution of the density matrix and more specifically that part of it that is due to collisions. Let (dpM~M(t)/dt)COflbe the corresponding rate of change per unit time averaged over all types of collisions. Then we are looking for a relation similar to eq. (104) which had been obtained in the case of a dipole transition. There is, however, a difference between the latter case and that of the density matrix, coming from the fact that, according to eq. (114) ~zM-M~ is proportional to a single Clebsch-Gordan coefficient, whereas PM Mf(t) has to be represel!lted by the expansion [40, 411: PM’M ~‘
~
(t)
JM
~
p~(t)C(Jf.JJf;M.MMf),
(138)
involving several of those coefficients. (We disregard any influence of nuclear spin, assuming that we are dealing with even isotopes.) Thus we expect that a result analogous to eq. (104) will be obtained for each of the f-values appearing in eq. (138). This will be given by the equation (dp~/dt)~ 011 —F~p~,
(139)
which defines a set of relaxation constants according to different terms in the development of the density matrix. After introducing in the usual way total and differential cross-sections by the relations: (140) and ajf
(141)
27rbdba~(b),
we can derive an expression for a~(b)by carrying out an averaging procedure similar to that introduced previously in the case of the optical cross-section. The result is given by: a~(b)CC1
—
2Jf+ 1
1
all ~ M C(f~J~M2MM1)
~
IT~lJfMl>(ffM2IT~t~JfM~>.
(142)
In most studies, both experimental and theoretical, one is dealing with a level ff = 1. Then the possible values off are 0, 1 and 2, according to a scalar, a dipole and a quadrupole component. The case 1 is referred to as “orientation” and the case J = 2 as “alignment” [411. For the fCC
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
315
corresponding cross-sections we will use the notations aOR and aAL respectively. Applying the same methods as in the optical case, we may now derive approximate expressions for these quantities [37, 42, 43]. For the “sudden”-approximation we obtain the result: 2boR + sin coR) (143a) aOR(b) = .(sin2a + sin with aOR
1 COR
e~)~-~ di’,
~f(eo_
+00
=
/ (e~— ~) ~l
—
boR
=
~f(eo—
Ci)(l
—
2~)dt,
b2\ R~)dt
and further aAL(b)
aOR(b).
(143b)
-~
The “adiabatic” approximation leads to values that are independent of the impact parameter. These are given by: O 0R(b)=3
(l44a)
and
—
— 4
3
aAL~u)—S5aOR
It should be noticed that with the “sudden”-approximation the average value of aOR(b) in the oscillatory region equals unity, whereas the adiabatic approximation gives UOR = One may therefore expect that by using the “sudden”-approximation alone one underestimates the values of the total cross-sections aOR and ~jAL~ Thus, there is at this point a slight difference with respect to the optical case [441. It may further be remarked that both approximations lead to a value of the ratio aOR/aAL that is equal to ~ However, the experimental value seems to be much smaller; possibly, contributions from a region where neither approximation is valid, may not be quite negligible. We now introduce for comparison the optical cross-section, or more precisely the real part of it which describes broadening. Restricting ourselves to the “sudden”-approximation, we get from eq. (132a) the following expression: ~-.
.
Re[a0~(b)]
=
4[sin2aopT + Sin bopT + sin COPT
with aOPT
=
—
1
(145)
316
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
bopT
=
CC
~
~+00
1J
~
+
~
—
(e~— e~)
dt
]
— ~1~2
dt.
Considering the ratio of the total cross-sections calculated from eqs. (145) and (l43a) respectively, one obtains for a R6-potential an expression given by [37, 431 Re(UQPT)/aOR = 0.44[q215 + (q + ~)2/5 + (q + ~)2/5 1 where q, a parameter first introduced by Rebane [451 is defined as follows: ,
q
=
(C~- ~)/(C~
Ci).
--
Typical experimental values of the ratio Re(u 0~1)/°oR are in the range of 1.5 2.5 [42]. From these values we can determine q and thus, in principle, the anisotropy of the potential, assuming that C~— C~is known from line broadening measurements. However, as we have shown in the previous paragraph, a repulsive term should be included in
the potential and then, of course, the ratio Re(aOPT)/aOR has to be calculated numerically as a function of the potential parameters. Comparing again the result to experimental data, some indications concerning the magnitude of the anisotropy parameters for the repulsive term may be obtained, as has been shown for a Lennard-Jones 6-12 potential [37, 43] They constitute probably the only information on that subject available up to now. .
3.3. The quantum-mechanical impact theory As has been mentioned previously, quantum theories don’t seem to play an important part so far in the field of neutral atom broadening and therefore we treat them very briefly. In fact, we shall restrict ourselves to the case of the impact theory, for which a quantum-mechanical version has already been given in early days by Lindholm. We derive Lindholm’s formulae for broadening and shift, by using a method due to Baranger [91. Let us start from the general formula (2.40) and neglect stimulated emission. Using the integral expression for the delta-function, we write:
f2
E pi’~~~’
drexp{i(w
-—
w~f)r}
fdrexp(iwr)Ø(r)
(146)
with ~(r)
=
~?p1IIx~fIexp(—iw1fr).
(147)
We assume for simplicity, that our system is that of one active atom at a space-fixed position, surrounded by N moving perturbers.
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
317
We now introduce a Born-Oppenheimer type approximation which consists in writing the total wave-function of the system as a product of the unperturbed electronic wavefunction pel and a translational wavefunction x~.The transition moment is then given by the following expression: CC
M~x~’~ IXtr)
where~i~(~p~IpI~p~) is the transition moment of the unperturbed active atom. For the frequencies we have the relation: w~1~ = + where w~corresponds to the change in translational energy between the initial and final state of the system (w~~ being the unperturbed frequency). Dropping a constant factor I,i(~)I2 and taking the origin of w at frequency w~,we write the correlation function as follows: 2exp—iw~r. (148) ~(r) = ~ P~XX~ This expression shows, that within our approximations, the only origin of line-broadening lies in the overlap effect of the translational wavefunctions. It should be noticed that x 1 and Xf are by no means orthogonal, since they correspond to different interatomic potentials in the translational Schroedinger equation, depending on the electronic state of the active atom. As a further strong simplication, we now consider the case of binary collisions: Assuming that the perturbers are all independent from each other, we write Xtr as a product of one-perturber wavefunctions, that is XtT X(1)X(2) x~ and further w~ w~ + w(2) + + We can then define an one-perturber correlation function: ...
Ip(r)
~p(I)I(X(1)lX(1))I2
exp~—iw~~r},
(149)
so that i~(r)is given by: çb(r)
[p(r)1’~’.
(150)
As will be shown below, p(r) differs from unity by of an the infinitesimal quantity which is intV’ enclosing theonly atoms system. Thus we may write: versely proportional to the volume p(r)= 1 —g(r)/CV (151) .
Introducing the number density n by the relation N for 0(r) the following result: 0(r)
lim [1—g(r)/Q9]’~’=
~—ng(-r)
nCV, and letting ~Vgo to infinity, we obtain (152)
00
From now on we will specify our states x by a momentum parameter k, which is related to the total energy of the translational motion by the equation: h2k2/2mE.
(153)
(In our picture, where the active atom is assumed to be fixed in space, m represents the mass of the perturber; more generally one may base the calculations on the relative motion of the two atoms and interpret m as the reduced mass of the pair.) We then write ~p(r) in the form:
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
318
~p(r)
=
~
(154)
Pk~k(T),
k
with ~
I(XikIXfk)I2exP(_~(E’
—
E)r)
(155)
,
the primes corresponding to the upper state of the transition. As is well known, in the limiting case of a continuous energy spectrum, matrix elements can not
be evaluated by space integration, since the integrands would not go to zero at large distances. In order to get convergent integrals, we have to introduce the following transformation: 171X~k~
~Xfk’’Xik~ ~kk’
(Xfk’ -~
,
~
(156)
where ~ V is the difference potential between the upper and the lower state and c is an infinitesimal quantity which ensures that there is no divergence at F’ — E. Eq. (1 56) is a consequence of the so-called Lippmann-Schwinger equation. It can be made plausible by multiplying both sides by F’ -— F and letting e go to zero. This yields: (F’ E)(xfkvIxIk) (Xfk~J~VIXIk~, (157) =
which is certainly true, since F’ and F are cigenvalues of the hamiltonians Hf (upper state) and lI~(lower state) respectively and ~V = Hf — H 1. Furthermore eq. (156) is verified for ~V 0, since in that case Xf -~ X~. Most simply the results of the impact theory appear when considering the first derivative of ~k(T) which is given by: dpk =
—
dT
~I~(E’— F)(xIk Xfk’XXfk’
—
Xik~P
—
(E’
—
F)r
~k’
This will be transformed by using eq. (1 56) and the complex conjugate of eq. (1 57). The result is: dlpk dr
= —
i ~(X.kI~VIX ) /1 fk
i — —
i~k’
I
(XlkI~VIXfk~XXfk~l~VIXIk) exp — ---(F F’ F ic -—-------
—---
---
--
E)r
--
As we are dealing with a continuous energy spectrum, the summation over k’ amounts to the calculation2)fk’2 of andk’. integral, is to obtained bythe replacing formally the sign ~k’ by the symbol Thus, which one has evaluate following expression: (CV/27r dpk CC
i
iQ9 (X~kI~VIXf~ —
00
k’2dk’
(XjkI~VIXfkPXXfk~I~VIXlk) —--—~~-~--exP(_~~(E’ — E)r~ (158) .
To do this in a general way would be an extremely difficult task. However, in the particular case of very large correlation times one can go one step further by using the following argument: For r tending towards infinity, the oscillations of the exponential in the integrand of the second term on the right of eq. (158) become increasingly fast. So there will be no contribution to the in-
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
319
tegral unless one of the other factors shows a comparably fast variation. This happens near k’ = k because of the denominator F’ — F — ie. As a consequence, the range of integration could be reduced to a small intervall ~k’ around k and then all factors are practically constants except those which have just been mentioned. This again means that our integral may be replaced by the expression: m —~
k(xjkl~VlxfkXxfkL~VIXjk)j
+00
d(E —E)
exp{—i(E’ — E)r//f} E’ —F—ic
(to show the validity of this expression, the qualitative argument given above can easily be supported by a more rigorous calculation). The new integral is shown to be equal zero and hence we get the result: dlpk
i
-~- _~~(xikL~VIxfk),
for larger.
(159)
Now, as Baranger has pointed out, assuming r very large is equivalent with making the impact approximation. Then, by comparing r to some characteristic “collision”-time, a validity criterium for this approximation can be established. For this and other more detailed discussions we want to refer to Baranger’s work. By integrating eq. (1 59) and by using the value y,(O) = 1, which follows from eq. (155), we see that in the impact approximation lpk(r) is given by the expression: lPk(T)
1 —~-(XIkII~VIXfJC)T.
(160)
This expression can be transformed further by using some of the results of quantum-mechanical collision theory. Let us introduce polar coordinates r, 0 to describe the position of the perturber with respect to that of the active atom; in the case of spherically symmetric interaction potentials the azimuthal angle does not need to be specified since obviously there is then a symmetry of revolution around the incident direction 0 0, Any wavefunction can now be expressed in terms of Legendre functions by writing: CC
Xk(r, 0)0~aklP,(cos0)uk,(r),
1=0,1,2,...
(161)
where 1 plays the part of an angular momentum quantum number. In the case of normally behaved interatomic potentials one can introduce some characteristic phases 5k1 which are defined by the asymptotic expression of the radial wavefunction Ukl. The latter can be shown to be of the form: uk,(r)-+ ~—
sin(kr— 4irl+ ak,)’
rlarge.
As one can further show, the coefficient ak, can be expressed in terms of those phases cording to the relation:
(162) ~kl’
ac-
F Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
320
5)z(21+ 1)i’exp(i~kl),
(163)
where ~Z is a normalizing factor equal to CV112 CV being the volume introduced previously, in which the perturber is enclosed. Going back to eq. (1 60), we first express the matrix element on the right by means of an integral over CV, writing: (XjkXfkf9XjkXfk(Vf
(164)
V.),
—
and it is understood that CV goes to infinity at the end of the calculation. Now the Xk’~satisfy the translational Schroedinger equation which is of the form: h2 ~--~
~
V2~~ + VXk =Fxk
~2
~FCC
~
k2)
--~
(165)
-‘in
and from this the integrand in eq. (164) can be rewritten as follows: /12
2xfk XfkVX~f). (x~V transform the right-hand side of eq.
(166) It is now convenient to (1 64) into an integral over the boundary surface of CV, which can be choosen as a large sphere of radius R. Using Green’s theorem, we write: XjkXfk(~’f — V
1)
=
~—
~
(167)
Expressing the
Xk’S
by means of eqs. (161) and (163) and integrating over 0, we get the relation:
fdCVx~xfk(V
2 ~ 1— V1)
=
(21+1 )exp{--i(öfkl—~Ikl)}R2(uik1~~ufkl_ufkl~_uik1) R (168)
h
Now R is assumed to go to infinity and consequently we can replace in eq. (168) the asymptotic expressions, given by eq. (1 62). This yields: i a R2 ~,ulkl ~
Ufkl
ufkl
—
a
t1ikl
)
~
1 ~
ar
by their
sin(~,,. 1
—
yr
UkI’S
(169)
~ikl)
rR
Thus we obtain for the integral the following final expression:
fdCVx~xfk(Vf
—
2i(öfkl V~)CC_i
~
-~—~ (21+
l)(l
—
—
ölkl)}).
(170)
exp{—
According to eq. (164), this represents the quantity (x1kI~VIxfk). With the following definition: cx
1~I~VIxf~)~-q,
(171)
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
321
eq. (160) takes the form: ~Pk(T)=
(172)
1 —qr/CV,
from which we get, according to equation (154): 1 —~—(q)r
p(r)CC
(173)
g(r)(q>r.
and
Here the brackets stand for a Boltzmann average. As a result, the correlation-function, defined by eq. (152), is given by: 0(r)
exp{—n(q)r}.
(174)
This leads to a Lorentzian profile characterized by a shift and a half-width respectively equal to: f3—nIm(q>,
y—n2Re(q).
With the definition of q from eq. (171) the explicit relations for shift and half-width are: 2(~fk,
13CC —fl’7T-~~ K’-~-~(2l+ l)sin[
7n27r-(-E(21+
—
~ik1)]~
l)(1 _cos[2(~fkl—6Ikl)]))
.
(175)
These relations may be called the quantum-mechanical Lindholm formulae since they look very similar to the corresponding expressions of the quasiclassical impact theory. Furthermore, it can be shown, by using for instance the WKB-method, that the classical expressions are obtained as a limiting case of formulae (175). However only few attempts have been made to calculate shift and broadening in the general quantum-mechanical case [20, 46]. In order to do so, one would have to determine the phases ~~klby solving the Schroedinger equation of the translatory motion. This makes the situation
rather hopeless, except possibly for very simple and crude models without real physical significance. 4. Comparison between theory and experiment One basic difficulty arises when comparison between theory and experiment of neutral atom line broadening is intended: In order to simplify line profile calculations, one has to rely on approximations to the theory, independent of any assumptions regarding the interaction potential. The range of applicability of such approximative theories is not always clearly defined. It would therefore cause difficulties to quote exactly the error of the intensity at a certain frequency as calculated on the basis of the approximation and on the assumption of some potential function. As a consequence, the reliability of potential parameters, as derived from experiment on the basis of some approximative theory, would in general be restricted. This would be true at least in cases, where the number n of linearly independent observables equals the number m of unknown parameters of the potential function. On the other hand, the reliability would be large in cases, where n m. ~‘
322
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Table 2 Spectral lines perturbed by foreign gases investigated experimentally for comparison with line broadening theory line
transition
Kr 7601 A
Hg2537A Cs 8943 A Cs 4593 A
4p5(2P)SsIS] _4p5(2P 312)5p[~] 3P 6’S0--6 1 2S 2P,, 62S112 62P 2 6 112- 7 1/2
K4047A
425112_52p112
perturber
ref.
noblegas
47
noble gas noble gas lie, Ar
37,48—55,13 56—59 56-59, 60
Kr
61
For comparison with theory those systems investigated experimentally were selected, for which the assumptions of the theory are fulfilled or approximately fulfilled. In the classical path theories so far only the general elastic case has been treated for realistic potentials. This means, that the applicability of these treatments is restricted to cases, where the contribution of inelastic collisions (e.g. inducing transitions between fine- and hyperfine levels or quenching) to broadening may be neglected. From this point of view the systems listed in table 2 have been selected for comparison between theory and experiment. 3 as func-
4.1. Line (halfwidth andand shift)-measnrements at number densities below lO’~cm tions core of number density temperature Core measurements at low number densities for comparison with theory were carried out at various absorption and emission lines, perturbed by noble gases (table 3). The experiments were performed with pressures below I atm. at temperatures between 80 °Kand about 400 °K.Halfwidths and shifts were found to be linear functions of the perturber density n. The experimental values for y/n and 13/n are listed in table 3. Under the experimental conditions quoted above the impact approximation of line broadening theory is valid and the experimental results may be interpreted on the basis of this approximation. In terms of Lindholm’s impact approximation of the phase shift theory (sect. 3.1), however, the observed blue shifts with He as perturber and the different values for the ratio 13/y (table 3) cannot be explained, if a potential law C 6 is assumed. An interpretation, on the other hand, is 6/r possible in terms of the more realistic L.-J.-potential. The L.-J.-parameters C 6, C12 may then be deterniined by adjustment from the values of y/n and 13/n measured at fixed temperature. The validity of this interpretation can be tested by examining whether for a certain system a single pair of L.-J.-parameters describes consistently all meas’ired line shape data. Experimental consistency checks have been performed for the systems Kr 7601/noble gas and Hg 2537/noble gas. The results will be discussed below for each system separately. Kr 7601/noble gas In table 4 are listed the values for the ratios of the halfwidths 7295/780 and shifts 13295/1380 measured at 295 °Kand 80 °K.The theoretical values were calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.-interaction (eqs. (3.55), (3.56)). The comparison with the experimental values yields fairly good agreement, except for the case of Ar
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Table 3
323
3. -i/n and 13/n in
Halfwidths and shifts of various systems measured at number densities below 101 9 cnc units of icr2°cml/cnc3 ref.
‘y/n
jIm
80
47
1.38
+0.108
+0.156
295
47
2.57
+0.283
+0.222
80
47
0.683
—0.227
—0.664
295
47
1.14
—0.2l~
—0.371
80
47
1.18
—0.32i
—0.545
295
47
2.44
—0.73~
—0.602
Kr
80 295
47 47
1.304 1.93~
—0.44~ —0.652
—0.684 —0.678
He
293
48
0.88
+0.048
+0.055
Ne Ar Kr
293 293 293
48 37
0.56 1.16
—0.084 —0.26
—0.151 —0.227
37
0.78
—0.20
—0.256
Xe
293
37
1.12
—0.25
—0.222
He Ar
400 400
60
8.8 ±0.5
60
6.7
+1.5 ±0.1 —1.63
—0.243
line
perturber
Kr76O1A 4p5(2pai 2)5s[~I 2
He
..4p5(2p3,2)sP[~]2
Ne
(emission) Ar
198 Hg 61S
2537 A
T(°K)
3P
0 —6 1 (absorption)
Cs4593A 2Sl, 2P1/2 (absorption) 6 27
Cs4555A 62S1/2—72P3/2 (absorption)
K 4047 A
--
He
380
60
Ar
380
60
Kr
470
61
±0.5
±0.05
6.9 ±0.4 5.8
+0.73 ±0.05 —1.55
±0.3
±0.06
6.6
—1.84
425
±0.2
1/252p1/2
±0.05
(absorption)
Table 4 Comparison between observed and predicted temperature dependence of shift (jI) and broadening (‘y) for the Kr A 7601 line perturbed by inert gases [47]
perturber
He Ne Ar Kr
Y295/780
13295/1380 -
observed
predicted
observed
predicted
2.62 0.93
2.16 0.84
1.87 1.67
1.73 1.70
2.29
1.79 1.43
2.07 1.48
1.64 1.43
1.47
+0.171
+0.106 —0.267
—0.279
324
F Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
as perturber. The agreement is much better than in the case of pure V.d.W.-interaction, where the theoretical ratio ‘y295/’yso equals that of 13295/1380, and assumes the constant value 1.43, indepently of the nature of the perturber. The existent deviations have to be discussed with regard to the assumptions of the above approximations: I) The perturber passes the active atom on a straight path at constant velocity; 2) The contribution of inelastic collisions and orientation effects to broadening is negligible. The influence of the deviations of the trajectories from the straight path on broadening and shift was studied by Schuller [20]. Assuming pure V.d.W.-interaction, however, the expected temperature effect on broadening, due to those deviations, is in disagreement with observation: The halfwidth is expected to decrease with temperature, whereas an increase is observed (table 4). Thus it appears, that, although the influence of curved trajectories may not be underestimated, the assumption of’ a repulsive term in the potential function seems to be necessary for the interpretation of the experimental results. The contribution of inelastic collisions to broadening is probably negligible: Although cross-sections for collisions inducing transitions between fine structure levels are not known for the systems under discussion, they are expected to be small due to the comparatively large energy distance of the levels [131. On the other hand-, the contribution of M-mixing collisions to broadening and shift may not be neglected, if a L.-J.-interaction is assumed [43] Hg 2537/noble gas For the system Hg 2537/Ar the L.-J.-parameters, obtained from low density measurements of halfwidth and shift, cannot account for the asymmetries of the profiles observed both in the line core and the line wings [13, 50, 51. 531 : The profiles, calculated on the basis of Lindholm’s phase shift theory for general pressures (eq. (3.73)) using the L.-J.-parameters obtained in the above way, display “blue” asymmetry even at very low pressures [131, whereas the measured profiles show “red” asymmetry. Discussing the assumptions of Lindholrn’s phase shift theory it may not be excluded, that for this system the path curvature influences considerably broadening and shift, thus introducing errors in the determination of the L.-J.-parameters from those quantities. The contribution of f-mixing collisions to broadening will be negligible, since the cross-sections for quenching of the Hg 2537 A-emission line by noble gases is very small compared to the cross-sections for broadening (table 5). The effects of M-mixing collisions however, may be considerable, as was pointed out by Butaux, Schuller and Lennuier [49] . These authors have therefore interpreted the line core data in terms of splitted potentials I/t(r) for the Hg 63P1 state. For the systems Hg 2537/noble gas
line
Table 5 Comparison of cross-sections for phase shift collisions (oel) and inelastic collisions (ojfl~l) 0e1 ref. °inel ref. perturber (ltYbo cm2) (lO~~ cm2)
Hg 2537 A 615 3P 0—6 1 Cs 2Sij 8943 A2 P 6 2—6 112
He
68
48
0.00
62
Ne Ar
92 260
48 37
1.04 1.38
62 62
He Ar Ne
99 336 174
13 13
5.7 x l0~ 5 1.9 xX i0~ 1.6 1fF
63 63 63
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
325
the four parameters C’~,C?2, C~,C~2occurring in the two L.-J.-potentials V1(r) and V~(r)were obtained semi-empirically on the basis of Anderson’s theory (sect. 3.2) as follows: Using the ratio C?/C~calculated by means of eq. (5.5) and values of C~,C~2given in the literature [17], the four parameters C?~C?2, C~,Cl2 were determined from measured halfwidths, shifts and depolarization cross-sections [641, with the use of eq. (3.46) (and a corresponding equation for the case of depolarization [43]). The validity of this interpretation may be tested by comparing the L.-J.-potentials obtained in the above way with those derived from measurements of the far wings (sect. 4.3). K 4047/Kr The validity of the interpretation of the line core data for this system will be discussed in sect. 4.3. 3 4.2.Extensive Line coremeasurements measurementsatatthe number densitiescomponents above 1019 cm fine structure of the Cs-principal series perturbed by noble gases were reported by Chen et al. [56—59] and at the Hg 2537 A line perturbed by noble gases by Füchtbauer et al. [50], Granier [541 and Behmenburg [13]. The measurements were performed in absorption with noble gas pressures up to 200 atm at temperatures between 300 and 1500 °K.Halfwidth shift 1~and asymmetry ratio a,* were observed to be in general nonlinear functions of the perturber density n. This is in line with the predictions of the theory for general pressures [13] which has to be applied under the experimental conditions quoted above. Detailed comparison with the phase shift theory has been restricted to the Cs line 62S 2P 1,2 -~ 6 1,2 A 8943 A, since only in this case2Pthere are no contributions to broadening due to M-mixing and f-mixing collisions: 1) Since the 1~2state is an isotropic one, there is no V.d.W.splitting in this state in the spherically symmetric field of the noble gas atom; 2) The measured 2P cross-sections for transitions 112 -~ are very small for all noble gases (table 5). Comparison was reported for the systems Cs 8943/Ar and Cs 8943/Ne by Behmenburg [13, 28] and for Cs 8943/Ar by Jacobson et al. [651. Behmenburg pointed out the improvement of qualitative agreement between theory and experiment by using a L.-J.-potential instead of a V.d.W.-potential in the calculation. As an example, the results for the system Cs 8943/Ar are shown: figs. 7, 8, 9 show the functions y(n), /3(n) and a(n) measured and calculated on the basis of Lindholm’s general pressure theory (eq. (3.73)). The constants C6 and a, used for the calculations, were obtained from the values 7/n and 13/n, measured at low number densities, by adjustment. The parameter i~ entering the theory (eq. (3.69)) was determined by adjustment to data measured at higher densities. The qualitative behaviour of the observed functions is seen to be well accounted for by the calculated ones. In case of a(n) the agreement of theory and experiment is even quantitative. The deviations of experimental from theoretical functions ‘y(n) and /3(n) at higher densities might be an indication, that the additivity assumption in the general pressure theory is not fully valid. Recently Jacobson [661 suggested to check the validity of this assumption by systematically studying the effect of mixtures of two different foreign gases on the line shape. Atakan and Jacobson [65] compared the experimental data of the Cs 8943/Ar system with those calculated on the basis of the exact correlation function (eqs. (3.8), (3.9)) of the phase shift ~,
*The asymmetry ratio a is defined by the ratio of “red” halve halfwidth to the “blue” halve halfwidth.
326
F Schuller and
W.
Behmenburg, Perturbation of spectral lines by atomic interactions
DC
21cm3) Argon Numb., D.r6lty (70 Fig. 7. Comparison of theory and experiment in the pressure broadening of the Cs absorption line A 8943 A by argon (Behmenburg [131). —o—o----o— experiment (Ch’en et al. [56—59]); Lindholm’s theory assuming L.-J.-interaction (cs = 10, = 1.04, C 6)i~—.—-Lindholm’s theory assuming V.d.W.-interaction (rs = ~, = 1.7, 6 = 6.95 X i0’~ erg cm C 58 erg cm6). 6 = 6.95 X 1fF
,~
: iii -20 //
d
II
(I)
-10
2 Argon Number Density [10 2~cm 3j Fig. 8. Comparison of theory and experiment in the pressure shift of the Cs absorption line A 8943 A by argon (Behmenburg [13]). —o--o— experiment (Ch’en et al. [56—591); —~ Lindholm’s theory assuming L.-J.-interaction (~= 10, ,c = 1.04, C 6); —.---.-— Lindholm’s theory assuming V.d.W.-interaction(a = o~, = 1.7, C 58 erg cm6). 6 = 6.95 X 10 erg cm 6 = 6.95 X lIT
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
327
21cm-3J Argon Numb.r D.nslly [10 Fig. 9. Comparison of theory and experiment in the asymmetry of the Cs absorption line A 8943 A perturbed by argon (Behmenburg [13]). —0—0— experiment (Ch’en et al [56—591); Lindholm’s theory assuming L.-J.-interaction (a = 10, = 1.04, C 6); Lindholm’s theory assuming V.d.W.-interaction(a = ~, = 1.7, C 6). 6 = 6.95 x 10”58 erg cm 6 = 6.95 X l0”58 erg cm
—.—.—
theory. The comparison was extended to the total shape of the line core, up to about 3 halfwidths from the line center. Furthermore, the sensitivity of the line core parameters (7, j3, a) to the potential parameters was investigated. The L.-J.-parameters obtained, differ, however, considerably from those derived on the basis of Lindholm’s treatment (table 6). This seems to indicate, that the straight path-constant velocity assumption in this treatment is inadequate in this case. 4.3. Spectral intensity distribution in the line wings Wing measurements for comparison with theory were carried out at the absorption line K A 4047 A perturbed by Kr [61]and at the absorption line Hg A 2537 A perturbed by Ar [25]. The foreign gas pressures used were below 1 atm. at temperatures between 400 °Kand 700 °K. K A 4047/Kr For the interpretation of the red wing, the intensity distribution to be expected from the general pressure phase shift theory was approximated by folding the quasistatic profile into a dispersion profile accounting for the effects of the collisions: The quasistatic profile was calculated by Table 6 Lennard-Jones-parameters derived from measured line profiles on the basis of the phase shift theory line
perturber
ref.
C
2.96
6 58 erg cm6)
C1 (1001 2 erg cm12)
Cs8943A
Ar
13
(1IJ” 6.95
62S1,
Ar
65
2.25
0.46
Ne
13
1.59
9.60
2P1,2
(absorption) 2—6
328
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
10’
_______
i~~~--~-..-~
1102
2P Fig. marsch 10. etComparison al.for. [61]). of theory experiment; experiment of2P calculations redL.-J.-interaction, wing bywe of the the method K absorption ~that of Hindmarsch line A 4047 andthat Farr A perturbed [24] .assuming bywith krypton .-interaction. (Hindcounted optimum means was taken of Bergeon’s adjustment toline The be••... equal L.-J.-parameters theory to toand the the [23] measured inverse assuming from ofthe profile; the this collision adjustment note, time. are Fig. in the theexcellent 10 halfwidth observed shows agreement the red of theoretical the satellite dispersion isL.-J profile well those profile acobat collision-induced tained from core transitions data. This is remarkable, one would think, the cross-section for collisions. Furthermore we feel, 112 that -~ the 312 influence is effects notsince quite of negligible path curvature on degree. broadening to that formay phase notshift be negligible either in this case. Possibly these compensate tocompared some Nevertheless, as a further consistency check of the present interpretation measurements of the temperature dependence of the red wing would be highly desirable. Hg A 2537/Ar In this case no satellite is observed in the red wing (fig. 11). This is in line with the theoretical expectation, since due to the Boltzmannfactor the intensity in the far red wing should decrease monotonically, if L 1 (sect. 3.1.2.2.2.2). 3P For the interpretation of the observed wing shape, as well as of the line core, the splitting of the 1 -level of the Hg atom has to be taken into account. The theoretical profile was calculated on the basis of the nearest neighbor approximation of the quasistatic theory (eqs. 311(3.67), (3.68)) assuming, that the far red wing is exclusively formed by transitions X~0 —~A 0.* This assumption is well justified both by experimental results and theoretical considerations [67] In order to obtain optimum agreement with the experimental profile it is necessary to assume C]2 C]2 < 0. For the L.-J.-parameters obtained in this way the validity ~—--—-
~‘
-
—
*Notation of Herzberg [68].
__
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
329
io-3~
U
C
-io~ 1J (cm Fig. 11. Interpretation of the red wing of the Hg absorption line A 2537 A perturbed by argon in terms of L.-J.-interaction (Behmenburg et al [25]). ..... experiment; —nearest neighbor a~proximationof the quasistatic theory (C~= 6.41 X 10”60 erg cm6 C] 2 = —3.21 ~ 1 2) 10_i 4 erg cm ~2x102
~
criteria of the quasistatic theory (sect. 3.1.3) are well fulfilled*. Fig. 11 demonstrates the remaining differences between theoretical and experimental wing shape existing at optimum adjustment. This indicates, that the L.-J.-type of potential function is, in this case, only a rough approximation to the true potential function.
5. Appendix: The calculation of interatomic forces It cannot be the purpose of this article to present a comprehensive survey on quantal calculations of interatomic forces as reported in the literature. Rather a brief account will be given on several semi-empirical methods, currently used for the interpretation of collision broadening of spectral lines by neutral atoms. For more detailed information on the theory of intermolecular forces Margenau and Kestner’s recent monograph [69] should be consulted, which also contains an extensive bibliography of papers up to 1966.
*Unfortunately they cannot be compared as yet with those obtained from line core data, since these were derived under the assumption c]
2 — C]2 > 0.
330
F. Schuller and W. Behmenburg, Perturbation of spectral lines bv atomic interactions
5. 1. Dispersion
krcc.s’
London [701 was the first, who calculated on the basis of perturbation theory the long range, attractive part of the potential between neutral atoms. He expanded the classical electrostatic energy between two neutral atoms into a power series of R~(R internuclear distance) and applied this as perturbation on the system. 5. 1 .1. Dipole—-dipole interaction We consider the system consisting of an active atom and a perturbing one. The active atom is assumed to be in a certain electronic state k (energy Ek) with total angular momentum quantum number J and space quantum number M, the perturbing atom in the electronic ground state k’ (energy Eke). This situation is often realized in experiments on neutral atom broadening. Assuming
that both atoms are unlike, the first order energy of the system is zero. The expression for the second order energy becomes [301
~EkJM (2)
—
1 3 h~e2
~k
~ (E~-EK)(Ek
~K~k
~p(x,J~k~s~) . M)
(1)
with ~p(x,J,M)~[l+3C2(xlf;M0M)] where the C(...) are the Clebsch-Gordan coefficients; ~ ~ the oscillator strengths for the transitions k i~,k’ -÷ ii’, respectively. The summation has to be taken over all intermediate states K and of the atoms of the system. Obviously the expression (1) contains explicitely the J- and M-dependence of the second order energy. The M-dependence shows that the interaction is anisotropic in cases with J ~ 0,-k. By averaging over M the function ~pbecomes unity and (1) becomes identical with London’s result [70]. In many cases the lacking knowledge of the f-values rules out the exact calculation of absolute values of the second order interaction energy. It is, however, often useful to know the ratios -~
,~‘
EJ 1/EJ2 or EM1/EM2 of the interaction energies in states with different J- orM-values, respectively. In these cases the radial parts of the wave function occurring in f~ are equal for both states and may therefore be put in front of the summation sign in (I). Furthermore the energies may be replaced by some average value. If one, in addition, assumes for the general case of atoms with more than one valence electron the radial parts of the wavefunctions to be the same for both
electrons and the coupling of angular momenta to be of the LS-type, one obtains [37] —a
C6kJM
+
b(L,J, S) C(J2J; MOM),
with a1, r
~
2L
~ ,-,-~ (2L—l)(2L+3)
1
\‘~
j
1/2
where W(...) is the Racah-coefficient.
W(SLJ2-JL)
(2)
F. Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
331
The expression (1) may also be greatly simplified if the perturbing atom has a much greater separation of its energy states than the active atom. This situation arises for alkali- or alkalineearth resonance lines perturbed by noble gases. The term (Ek+Ek_EK —EK) in the denominator may then be set approximatively equal to Ek EK and the summation separated into two summations, each involving only one atom. If furthermore the J- and M-dependence is neglected, one obtains —
~~=e2a
E (kIrIK)12
(3)
,
K*k
where a is the polarizability of the perturber. In case of a one electron system the sum over the dipole matrix elements may be approximated (Coulomb approximation) by [151
~ I
2~~ —31(11-1)],
2 [5j~* 0 p—
(4)
where a 0 is the radius of the first Bohr orbit, n* the effective principal quantum number of the state k and 1 the orbital angular momentum quantum number. With respect to the various approximations the C6-values calculated from (3) and (4) may be in error to within 10%. The uncertainty may be somewhat larger for heavy elements and for atoms with more than one valence electron. These errors, however, compensate to some degree, if the difference C~— C~ of the constants for the initial and final state of the line, which occurs in line broadening theory, is calculated. 5.1 .2. Higher moment interaction Comparison between the C6-values, calculated on the basis of London’s formula, and experimental values, derived from atomic beam scattering and line broadening experiments [19, 71] strongly indicate the effectiveness of higher moment interaction in addition to dipole—dipoleinteraction. Its contribution may be estimated using the model of isotropic, three dimensional harmonic oscillator for the interacting atoms [72]. Recently calculations of the dipole—quadrupole interaction constants C8 have been carried through for the systems alkali—noble gas on the basis of the Coulomb approximation [73]. The values for the ratio C8/C6 in the ground state interaction were found to deviate at most by 20% from those obtained by means of the oscillator model. More recently Baylis [74] developed a semi-empirical method to calculate complete potential functions for the systems alkali—noble gas. In the electrostatic part of the interaction was included the interaction of the induced dipole of the noble gas atom with all multipoles of the alkali. The noble gas atom was treated as a sphere of radius r0 which acts like a point polarizable dipole as long as both alkali valence electron and alkali core are outside, but give constant interaction whenever the alkali electron is inside the sphere. The value of r0 is chosen by adjusting the well depth of the ground state potential to agree with experimental scattering data. Furthermore, the contribution of the spin—orbit coupling in the alkali to the interaction was included. 5.2. Resonance forces In case of identical particles the first order perturbation energy does not vanish, if one of them is in an excited state. For the root mean square value over all M one obtains [18]
F. Schuller and hi. Behmenburg, Perturbation of spectral lines by atomic interactions
332
=
2hf 2J+ 1 I x/~e R3 8irmw 2Jg+l
5)
0
where f and w
0 are the oscillator strength and angular frequency, respectively, of the resonance line and Jg belongs to the ground state. 5.3. Overlap forces
In the past much work has been done in calculating, both ab initio and semi-empirically, short range interaction of binary systems in the ground state. Extensive reviews have been given in this field by Hirschfelder [1 7, 751. On the other hand, little effort has been undertaken so far to in-
vestigate short range interaction of systems in the excited state. Roueff [76], in an attempt to estimate the order of magnitude treated the repulsive part of interaction as essentially arising from the scattering of the valence electron of the radiating atom by the perturbing one. A more rigorous treatment was proposed by Baylis (1969), who used pseudopotentials for representing the effect of the Pauli exclusion principle on overlapping electron states. This method 2P,,was applied to alkali-rare gas diatomic systems, composed of an alkali in its 2~ ground state or 2,~2first excited states and a noble gas atom in its ‘S-ground state. The results show that the repulsive part of the interaction is strongly anisotropic. This fact may be explained qualitatively 2~-state the alkali electronbyis sketches mainly of athe electron symmetric probability su distributions (fig.B2~-state,the 12). In the X alkali electron is mainly in a pu orbital, in spherically orbital. In the which overlaps the noble gas at larger internuclear separations than the su orbital. In the A2fl state the alkali electronic wave function has pir character with a node along the internuclear axis and the noble gas can approach the alkali rather closely before the repulsive interaction dominates. ~,
alkal, atom
x2[
A2ff
noble atom
gas
Q
2
B2E
Fig. 12. Schematic representation of the electron distribution for the X2E, A2fl and B22 states ofan alkali—noble gas diatomic system.
Including also dispersion forces (sect. 5.1) Baylis was able to calculate complete potentials for the systems mentioned above. Well depths, equilibrium internuclear separations and reduced curvatures were tabulated. The comparison of the potential parameters determined from scattering *
Notation of Herzberg
1681.
F. Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
333
data show relatively good agreement. The method should thus be applicable also to other Systems, like alkaline earth plus noble gas, which are of interest in neutral atom line broadening. Acknowledgement The authors take pleasure in acknowledging the financial support of the “Gesellschaft der Freunde und Förderer der Universität Düsseldorf” during the preparation of the manuscript. Thanks are also due to Mr. M. Backer for recomputing numerically the broadening- and shift functions listed in table 1.
References [11 S. Bloom and H. Margenau, Phys. Rev. 90(1953) 791. [21 J. van Kranendonk, Thesis, Amsterdam 1953. [31P.W. Anderson, Phys. Rev. 76 (1949) 647. [4] M. Baranger, Phys. Rev. 111 (1958) 494. [5]H. Margenau and M. Lewis, Rev. Mod. Phys. 31(1959) 569. [6] S.Y. Ch’en and M. Takeo, Rev. Mod. Phys. 29 (1957) 20. [71G. Traving, in: Plasma Diagnostics, ed. W. Lochte-Holtgreven (North-Holland, Amsterdam, 1968) p. 66. [8] A. Jablonski, Phys. Rev. 68 (1945) 78. 191 M. Baranger, Phys. Rev. 111 (1958) 481. [101 R.L. Fox and H.C. Jacobson, Phys. Rev. 188 (1969) 232. [11] E. Lindholm, Ark. Mat. Astr. Fys. 32(1945) Al. [12]J.D. Talman and P.W. Anderson, Proc. Conf. Broadening of Spectral Lines (1956) p. 29. [13] W. Behmenbuig, Habilitationsschrift, SaarbrUcken 1970, published in Annales Universitatis Saraviensis, Heft 10, 1973. [141H. Margenau, Phys. Rev. 40(1932) 387. [15] A. Unsold, Physik der Sternatmosphären (2nd. ed., Springer, Berlin, 1955). [16] E. Lindholm, Thesis, Uppsaila 1942. [17] J.O. Hirschfelder, C.F. Curtis, and R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954). [18] H.M. Foley, Phys. Rev. 69(1946) 616. [19] W.R. Hindmarsh, A.D. Petford and G. Smith, Proc. Roy. Soc. A 297 (1967) 296. [20] F. Schuller, Thesis, Paris 1962. [21] W. Behmenburg, J.Q.S.R.T. 4 (1964) 177. [22] H.G. Kuhn, Phil. Mag. 18 (1934) 987. [23] R. Bergeon, Thesis, Paris 1960. [24] W.R. Hindmarsh andi.M. Fair, J. Phys. B2 (1969) 1388. [25] J. Losen and W. Behmenburg, Z. Naturf. 28a (1973) 1620. [261 -G. Traving, in: Mitteilungen des Astronomischen Geseilschaft, Sonderheft 1, Hamburg 1959. [27] M. Takeo, Phys. Rev. Al (1970) 1143. [28] W. Behmenburg, Z. Astrophys. 69(1968) 368. [29] L. Spitzes, Phys. Rev. 58 (1940) 348. [30] F. Schuiler and B. Oksengorn, J. Phys. 30(1969) 531. [31] T. Holstein, Phys. Rev. 79 (1950) 744. [32] 1.1. Sobelman, Fortschr. Phys. 5 (1957) 175. [33] M.E. Rose, Elementary Theory of Angular Momentum (Wiley, N.Y., 1957). [34]F.W. Byron and H.M. Foley, Phys. Rev. 134 (1964) A625. [35] A. Ben Reuven, J. Chem. Phys. 26(1965) 2037. [36]F. Schuiler, J.Q.S.R.T. 11(1971)1725. [37]J. Butaux, F. Schuiler and R. Lennuier, J. Phys. 33 (1972) 635 and 35 (1974) 361. [38] R. Seiwert, in: Ergebnisse des exakten Naturwissenschaften 41(1968)143. [39] J.P. Barrat, D. Casalta, J.L. Cojan and J. Hamel, J. Phys. 27 (1966) 608.
334
F~Schuller and W. Behmenburg Perturbation of spectral lines by atomic interactions
[40] M.l. Dyakonov, Soviet Phys. JETP 20(1965)1448. [411 A. Omont, J. Phys. 26(1965) 26. [42] J.P. Faroux, Thesis, Paris 1969. [43] J. Butaux, Thesis, Paris 1972. [44] F. Schuller and B. Oksengorn, C.R. Acad. Sci. (Paris) 267 (1968) 116. [45] V.N. Rebane, Optics and Spectroscopy 26(1969) 371. [46] R.J. Hood and G.P. Reck, J. Chem. Phys. 56 (1972) 4053. [47] J.M. Vaughan and G. Smith, Phys. Rev. 166 (1968) 17. [48] J. Butaux and R. Lennuier, CR. Acad. Sci. (Paris) 261 (1965) 671. [49] Chr. Fiichtbauer, C. Joos and 0. Dinkelacker, Ann. Phys. 71(1923)201. [50] H. Kuhn, Proc. Roy. Soc. 158 (1937) A212. [511 A. Michels and H. De Kluiver, Physica 22(1956) 919. [52] A. Michels, H. De Kluiver and CA. Ten Seldam, Physica 25 (1959) 1321. [53] R. Granier and J. Granier, C.R. Acad. Sd. (Paris) 260B (1965) 92. [54] R. Granier, Ann. de Phys. 4 (1969) 383. [55]F. Schuller and G. Attal, C.R. Acad. Sci. (Paris) 262B (1966) 807. [56]S.Y. Ch’en and R.O. Garrett, Phys. Rev. 144 (1966) 59. [57] R.O. Garrett and S.Y. Ch’en, Phys. Rev. 144 (1966) 66. [58] S.Y. Ch’en, E.C. Looi and R.O. Garrett, Phys. Rev. 155 (1967) 38. [59] R.O. Garrett, S.Y. Ch’en and E.C. Looi, Phys. Rev. 156 (1967) 48. [60] F. Rostas and J.L. Lemaire, J. Phys. B4 (1971) 555. [61] D.G. McCartan and W.R. Hindmarsh, J. Phys. B2 (1969) 1396. [62] J.P. Faroux and J. Brossel, CR. Acad. Sci. (Paris) 261 (1965) 3092; 263B (1966) 612; 264B (1967) 1452. [63] M. Czajkowski, D.A. Mc Gihis and L. Krause, Can. J. Phys. 44 (1966) 91. [64] CA. Piketty-Rives, F. Grosset&e and J. Brossel, CR. Acad. Sci. (Paris) 258 (1964) 1189. [65] AK. Atakan and H.C. Jacobson, J.Q.S.R.T. 12(1972) 289. [66]H.C. Jacobson, Phys. Rev. AS (1972) 989. [67]W. Behmenburg, Z. Naturf. 27a (1972) 31. [68]G. Herzberg, Spectra of Diatomic Molecules (2nd. ed., Van Nostrand, London, 1950). [69]H. Margenau and N.R. Kestner, Theory of Intermolecular Forces (Pergamon, Oxford, 1969). [70]F. London, Z. Phys. 63 (1930) 245. [71]G.D. Mahan,J. Chem. Phys. 48 (1968) 950. [72]H. Margenau, J. Chem. Phys. 6 (1938) 896. [73] W.D. Davison, J. Phys. Bl (1968) 139. [74] WE. Baylis, J. Chem. Phys. 51(1969) 2665. [75] J.O. Hirschfelder, Intermolecular Forces, in: Adv. Chem. Phys. 12 (lnterscience Publishers, New York, 1967). [76] E. Roueff, Astron. Astrophys. 7 (1970) 4. [77] W. Hindmarsh and J.M. Farr, Collision Broadening of Spectral Lines by Neutral Atoms, in: Progr. Quant. Electr. 2, eds. J.H. Sanders and S. Stenhoim (Pergamon, 1972). [78] Bibliography on Atomic Line Shapes and Shifts, ed. W.L. Wiese (NBS Washington, D.C., 1972). [79] R.P. Futrelle, Phys. Rev. AS (1972) 2162.
Frédéric SCHULLER Centre National de la Recherche Scientifique, L.I.M.HP., Bellevue, France
and Wolfgang BEHMENBURG Physikalisches Institut der Universität Düsseldorf, Germany
I
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 12,
flO.
4 (1974) 273--334. NORTH-HOLLANI) PUBLISHING COMPANY
PERTURBATION OF SPECTRAL LINES BY ATOMIC INTERACTIONS Frederic SCHULLER Centre National de ía Recherche Scientijique, L.I.M.HP.. Bellevue, France
and
Wolfgang BEHMENBURG I’hvsikalisches Inst ito t der Unir’ersitdt Di3sseldorf, Germans’ Received February 1974 Contents: 1. Introduction 2. Basic theoretical considerations 2.1. The classical path theory 2.1.1. General expression for the absorption intenSity 2.1.2. Decomposition into individual spectral lines 2.1.3. Phase shift theories 2.2. Quantum theories 3. The derivation of the line profile 3.1. The phase shift theory 3.1 .1. The correlation function 3.1.1.1. Theory for general pressures 3.1.1.2. The quasistatic approximation 3.1.1.3. The impact approximation 3.1.2. The intensity distribution 3.1.2.1. Interatomic potentials 3.1.2.2. The calculation of the line shape 3.1.3. Criteria of validity of the phase shift theory and its approximations 3.1.3.1. The adiabatic assumption 3.1.3.2. The validity range of the quasistatic approximation 3.1.3.3. The validity range of the impact approximation
275 275 275 275 280 281 283 283 283 284 284 285 287 288 288 289 299 299 300 31)1
3.2. Orientation effects 3.2.1. The relaxation of the dipole transition moment in the impact approximation 3.2.2. General expression of the differential cross-section 3.2.3. Approximate solutions of the one-collision problem 3.2.4. Cross sections for the relaxation of orientation and alignment 3.3. The quantum-mechanical impact theory 4. Comparison between theory and experiment 4.1. Line core (halfwidth and shift)-measurements at 9cm~as functions number densities below of number density and temperature 1Q~ 4.2. Line core measurements at number densities above I (Ji9crn3 4.3. Spectral intensity distribution in the line wings 5. Appendix: The calculation of interatomic forces 5.1. Dispersion forces 5.1.1. Dipole—dipole interaction 5.1.2. Higher moment interaction 5.2. Resonance forces 5.3. Overlap forces Acknowledgement References
~Sin~lc orders (~rthis issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 12, No. 4 (1974) 273 334. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check, Single issue price Dfl. 20.00, postage included.
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F. Schuller and W. Behmenburg, Perturbation of spectral liner by atomic interactions
275
1. Introduction The perturbation of atomic spectral lines, emitted or absorbed by gases or gas mixtures, is in general the result of several mechanisms: radiation damping, Doppler effect of thermal motion and collisions. Furthermore, in optically thick layers the line shape will be influenced by additional effects, in particular reabsorption of radiation. The mechanism of perturbation by collisions is very different for various types of interaction. If charged particles take part in the collision process, the line profile will be determined essentially by the interatomic Stark effect. If the particles are neutral, the perturbation of the line is governed by what is called interatomic forces; the latter comprise the well known Van der Waals forces and, in the case of identical particles, resonance forces. During the past ten years a considerable amount of work in the field of neutral atom broadening, both theoretical and experimental, has been published. It is not our intention to give a comprehensive account of all the results which have been presented. Many of them are discussed in the recent review article by Hindmarsh and Farr [77]. Furthermore, there exists a bibliography on spectral line broadening, which covers all the papers published up to 1972 [781. In this article we rather wish to give detailed attention to certain aspects of the subject, which might be of importance for future developements. One important aspect is the problem of determination of interatomic potentials for excited states from line profile measurements. This problem involves careful comparison between experiment and theory with special reference to the assumptions made in the latter. In order to point out those assumptions, it was useful to present the various approximative theories (sections 3.1.1.2, 3) as special cases of the general theory of line broadening (sect. 3.1.1.1), and to derive the validity criteria for the approximations. This was done for both the impact and the quasistatic approximation using realistic potential functions. However, in this paper the semiclassical theory is the main subject of discussion, whereas in the recent paper by Futrelle [79] the stress is put on a fully quantum-mechanical treatment. A further aspect concerns the part taken by orientation effects (sect. 3.2) in the mechanism of broadening and shift of the line. This will be discussed on the basis of Anderson’s theory [31. In this context we point out the connection between line broadening and other relaxation phenomena like, for instance, collisional depolarization of resonance radiation and broadening of double resonance curves (sect. 3.2.4). Relations between the “optical” cross section and those for destruction of orientation and alignment will be derived. Finally, for the purpose of comparison between theory and experiment, a brief account will be presented of several semi-empirical methods for the calculation of interatomic potentials by means of quantum-mechanical perturbation theory (sect. 5).
2. Basic theoretical considerations 2.1. The classical path theory 2.1.1. General expression for the absorption intensity Consider a set of interacting neutral particles which, in principle, could be either atoms or molecules, although in this paper we will focus our attention on atomic spectra. We assume that
276
F Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
these particles form a system of small dimensions, compared to the wavelength of some incident
radiation. (When considering a macroscopic gas or liquid, we decompose it in a great number of independent systems.) The harniltonian of the system will in general depend on the internal coordinates p and momenta p of each particle and on the coordinates R,* and momenta P of the relative motion of their centers of mass. Theories based upon the resolution of the Schroedinger equation of the complete system will be referred to as quantum theories. They will be introduced later. In this section we will assume that the Schroedinger equation depends only parametrically on the relative positions R of the centers of mass of the atoms. We further assume that R is a given function of the time t and call this the classical path approximation. The conditions under which the classical path approximation is valid are essentially the following: i) Only small changes in the internal energy of the particles are allowed, so that the reaction of these changes on the translatory motion may be neglected. ii) The Dc Broglie wavelength associated with the relative motion has to be small compared to the characteristic dimensions of the range of interaction forces. This condition is in general well satisfied except for light atoms and low temperatures. Let H(t) be the hamiltonian of the system in the classical path approximation, where the time t plays the role of a parameter. The variation with t of any operator 0. say, is then described by its Heisenberg representation: QH(
1)
=cutt(t)0q1(t)
(I)
where clt(t) is the unitary evolution operator which satisfies the Schroedinger equation
(2)
ihcll=H(tyl1.
We now consider the absorption problem assuming that the system is submitted to the influence of a monochromatic electromagnetic wave. Following the derivation of Bloom and Margenau [11 we represent this incident radiation classically and consider the electric vector E(t) as a given function of time. Restricting ourselves to dipole absorption, we write the harniltonian of the system in the form: =
11(t)
+
F(t)
(3)
where F(t) is the energy of interaction between the system and the electromagnetic field (electric fieldvector E(t)) as given by the relation:
F(t)
=
-~E(t)~t.
(4)
In this relation p designates the dipole moment operator of the system. which in principle involves all the electrons that may possibly interact with the radiation field. (However, in practice one often considers the situation where one optically active atom is surrounded by one or several perturbers and one only takes into account the electron that makes the optical transition. This ap-
proximation then neglects any participation of the perturbers in the radiative process. *Thc reader should not be confused with the fact, that throughout this article notations r and R are used for the internuclear distance.
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
277
We define a new evolution operator U that includes the influence of the interaction with the electromagnetic field. The relations (1) and (2) are then replaced by the following ones: O~t)= Ut(t)OU(t)
(5)
and ihU= ~(t)U.
(6)
A simple relation between the two quantities 0~(t) and O~(t) can be obtained by perturbation theory. Writing U(t) =Q((t)Ø(t)
(7)
we let the operator ~ take care of the interaction with the electromagnetic field. From (6) ~ obeys the equation iic~= FH~, where by definition FH Q11Fdll.
(8)
This can be solved by the perturbation expansion i2
it =
I
—
~-J”dt’ F’~(t’)+
(i-)
t
r’
fdt’ FH(tI)f dt” F”(t”) +
...
etc.
(9)
By substituting this expansion into the equation for O~,which according to (5) and (7) can be written as ~tOH(t)&
=
(10)
we get a series of terms corresponding to different orders with respect to FH. Up to the first order the result can be expressed as follows: O~(t)=
QH(t)
dtm[OH(t), FH(t’)I,
—
(11)
where the brackets designate the commutator defined by the relation: [A, B1 = AB BA. With Bloom and Margenau we now calculate the absorption by considering the work done by the light wave against the internal forces of our system of interacting particles. We may assume without loss of generality, that the electric vector E of the field defines the z-direction of some space-fixed frame and that its length varies with time according to the equation: —
E(t)
=
E 0cos(wt
+
a).
(12)
In this equation a plays the role of an arbitrary phase. Eqs. (4) and (1 2) yield the following expression for the interaction-hamiltonian: F(t) .~
=
—E0p~cos(wt+ a)
being by definition the projection of p on E.
(13)
278
F Sehuller and W. Behmenburg, Perturbation of spectral line,r bs’ atomic interactions
According to basic quantum mechanics, the required expression for the work done by the dcct~omagneticwave is the lollowing:
W( T) = L0
f (It cos(
wi +
~) —( p~(t))
(14)
where a time interval (0, T) has been considered. Flere the quantum mechanical average of the dipole moment is defined as: K p~(l)) = Tr ~pp~( t) L p being the statistical matrix. ntegrating by parts we get front eq. (1 4): -
tV( T)
cos(wl
=
+
a)Kp ([)~
+
di sin( wt)(p~(I )~.
Ewf
(1 5)
At large T-values the energy increase of the system becomes stationary and we may consider the C1Lt~lntitY ~=
liii~ ~
(16)
T
I,
Obviously the first term on the right of eq. (1 5) gives only a vanishing contribution to ~ so that we may drop it from now on. In order to calculate the second one, we first deterniine p(t) from eqs. (I I ) and (1 3), obtaining: =
p~(t)
+~E,Jdr’ cos( wI + )l p’~(I),
(1 7)
~H (I’).
After substituting this expression into eq. (1 5) and taking the average over all possible values of we get the result:
a.
=
~0WTr
~
dtfdt’ sin(w(i
-
t))[p~t),
p~t’)I}
.
(18)
Now, a simple geometrical argument shows that the following identity holds:
-/dtfdt’
sin(w(t
i~))pH(1~)pFt(1) =
--
=
---
f dt’fdt
sin(w(t
f dtfsin(w(t
-
--
which allows us to transform eq. (1 8) into: T
2w ° Tr iE “hT
(~ f‘ dt r di’ sin(w( t 0
I
))p~(t)p~(t’) ~.
-
-
0
This expression is more conveniently written in the form:
(1 9)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
~W(fl
=~Tr(p(fdt
279
e~tp~(t)f dt e~tp~(t) fdt e_~tp~(t)fdt e~tp~(t))}.(20) —
It constitutes the basic formula of semiclassical pressure-broadening theory. and write Introducing suitable basis functions, we define matrix elements Pif = Piö if and /~t~tf out the trace in eq. (20). This yields: E 0w ~ ~P~(fdt
e
MZif(t)fdte
PH(t)
—
fdt eP~.f(t)fdtePZf~(t))). (21)
We then rewrite the second term in the brackets by interchanging the indices i and f, obtaining ~W(fl
=
L0W ~(p~ — pf)~fdt etp~f(t)~.
(22)
In this expression, the term containing the factor Pf can be interpreted as stimulated emission. Let us now consider the limit of W(T)/T for T-* We may assume that the interaction within the system generates a stationary random process, so the Wiener-Khinchine theorem can be applied [21. This means that the quantity: ~.
I~f(w)=
lim ~
/dt e~tp~f(t)
(23)
is equivalent with the Fourier integral: I(f(w)
=
f
(24)
dr e~~T~tf(r)
involving a correlation function p(r) defined by the equation: p~f(T)= (pHf(t +
r)p~,.(t))~
(25a)
where the symbol ( ~ designates a time average. Replacing the time average by an ensemble average, that is an average over all possible paths followed by the system in the time interval (0, r), we can also write: tP~f(T) =
(i4’,f(r)p~~(0))S.
As can easily be shown from (25a, b), the correlation function has the property mp(f(—r) Using this property we can write eq. (24) as follows: I1f(W)
=
2ReJ’ dr e~(f(r),
which is sometimes more convenient for explicit calculations.
(25b) =
p~(r).
(26)
280
F. Schuller and W. Behrncn burg, Perturbation of spectral lines by atomic interactions
2. 1 .2. Decomposition into indiVidual spectral lutes So far the expressions derived for the absorption energy cover the whole range of co-values as they describe the intensity distribution of the entire spectrum. However, in practice the spectrum will often contain a certain number of isolated lines and the problem then consists in determining the profile of a given line separately. Before studying this problem in general, let us first consider the limiting case of zero interaction (e.g. isolated atom). Matrix elements of ~H are then simply given by:
pZ~fexp(—iwc~t)
P~f(t)
(27)
with w~ = (E~~°~ Ec0))/n, Ecu) and E~°~ being the unperturbed energies. The matrix element =
P~f(O)is of course independent of time. The substitution of (27) into (25a) yields for the
correlation function: -iwc~r) ~~f(~) IPZ~f~2exp(and thus the quantity (24) becomes: IPzjfI2
J~f(W) =
fdr
exp{i(w
(28)
w~)r~= 2~IpZlfI2ö(w
---
—
wv).
(29)
where the well known representation of the delta function has been used. From eq. (22) the absorption power is then given by the expression:
~
7rE2w
IiIfl 7W(T)
21t
~
Co1
--
Pf) PZIf~(W--
wc~).
(30)
Averaging over orientations we may further replace Pzif 12 by~~lp~fI2.Eq. (30) shows the evident fact that in the case of an unperturbed absorption process the spectrum is composed of several infinitely sharp lines corresponding each to a transition i f. We now introduce perturbation but we make the restrictive assumption that the latter might not be strong enough to destroy completely the behaviour of ~(r) as a function oscillating at frequency w~. So we write: -÷
ipif(r)
=
(31)
pf(r)exp(—iw~r)
and I~f(w)=
f
dr ~f(r) exp{i(w
--
w~kr)}
(32)
where ~p’f(r) is supposed to vary slowly compared to the factor exp(—iw~~r)~ Under these circumstances, I,/w) will be very small except in the neighborhood of co = w~.So describes an intensity distribution located at frequencies close to w~ [31 Going back to eq. (22), we see that each term of the sum on the right corresponds to an individual spectral line. With the additional assumptions of neglecting induced emission and of setting p1 = 1, we get as a result that the profile of a perturbed line is just given by ‘if and thus may be calculated from eq. (24) or (26). .
281
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
The above reasoning can easily be extended to the case where the states under consideration are either degenerate or may be treated as such. The result is then, that a sum over degenerate states has to be added in eq. (24) or (26). The situation is quite different when the interaction causes overlapping of lines in such a way that no simple decomposition can be made any more. For this rather difficult case we refer to the literature [4]. 2.1.3. Phase-shift theories Let us again consider the profile of an individual spectral line as given by equation (24) or (26) neglecting any effects due to degeneracy. In many theories the calculation of the corresponding correlation function is performed by introducing adiabatic states. These are defined by a special choice of basis functions assuming that matrix elements of the evolution operator Q1 take the simple form [5] =
~
{_±.fdt’E~(t’)).
(33)
By substitution, the Schroedinger equation than yields the following relation: H~1~ = ~
(34)
which amounts to the solution of an “instantaneous” Schroedinger equation H(t)in) = In)E~(t). 4~fthe following With the evolution operator given by eq. (33), we obtain for the quantity ! expression:
(_~Jdt(Ef(t’) — E~(t’))).
I4~f= (c~jtp°.l).f= p~exp
(35)
It should be noticed that the matrix element = (ilp~ If) is time-dependent since it involves eigenfunctions of the time-dependent “instantaneous” Schroedinger equation. From eq. (33), which of course constitutes an approximation, we derive the correlation function, given by: t+’T
mp, 1(r)
(,4~(t+ r)p~.(t)exp ~—.~-f dt’(Ef(t’) — E.(t’)))
)~.
(36)
In most theories however, only the phase factor is taken into account, whereas the quantity + r)j.L~,(t) is raplaced by a constant value, corresponding to the unperturbed atom [6]. Moreover, this constant factor is in general omitted in the definition of ip(r). As to the phase, it is mostly written in the form: t+’r
..i_f
f
t+r
dt’(E~(t’)— E~(t’))=
+
dt’~w~,~(t’),
(37)
nt where w~)is the frequency of the unperturbed atom, and where ~w~f(t’) designates the frequency
282
F Schuller and W. Behmenburg. Perturbation of spectral lines by atomic interactions
perturbation. The latter involves the interaction potential V, defined by the relation:
L1f(t)
=
E~°~ + ~‘i,f~ t),
(38)
EY~being the energy of the unperturbed atom. By definition: ~w1~ =(Vf
--
V1).
Introducing the phase perturbation:
77(1)
=
fde ~w1f(t)
we write eq. (37) as follows:
I
t+r
TI
dt’(E~.(t’) E1(t)) —
=
wr
+
~( t
+ T)
--
~( I).
Finally, the phase W~~Tcan be removed from the correlation function by shifting the origin of the frequency scale, so that w in eq. (24) now measures the difference w w~. With these modifications the correlation function is given by the following expressions [7 I ‘~: ~,(T)(exp{—i(77(t+r)
(39a)
—77(t))})~~Kexp{----i77(t+T, t)}>~
and ~(T)
=
(39b)
Kexp~—i77(r)}).,
involving respectively a time average and an ensemble average. The correlation function given by eqs. (39a, b) is that of the so-called phase-shift theories, which will be discussed later in more detail. Right now it shall be remarked that any profile predicted by these theories leads to an integrated intensity equal to the intensity of the unperturbed line. As a proof we integrate the expression given by eq. (24) over co in the interval < co < +°°. This yields: —~
2 fdwI(w)
=
fdT~r)
2~(r, 0)
=
2~0)
=
2~lpZifI
(where we have reintroduced the factor ‘~zif12 which had been dropped after eq. (36)). This value is indeed identical to that which one would obtain from eq. (29) in the case of the unperturbed line.
*Fromi~now on we omit the subscripts on ~(r).
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
283
2.2. Quantum theories In the classical path theories discussed so far, one assumes that each atom or molecule is submitted to a given time-dependent perturbation, which, not considering radiation, results from the interaction with the other atoms or molecules of the system. The internal states of all the atoms or molecules are then governed by a time-dependent Schroedinger equation. A more rigorous treatment consists in writing a Schroedinger equation, which applies not only
to the internal states, but also to those of the relative translatory motion of the centers of mass of the atoms or molecules. This equation is of course a stationary one, since, in the absence of radiation, there is no more time-dependent perturbation acting on the system. In fact, as Jablonski pointed out [8], the whole system can be regarded as a single quasimolecule, in which the translatory motion plays essentially the same role as the vibration or rotation in ordinary molecules. As in this picture the translational states are considered as internal ones, we can calculate the absorption energy by applying formula (30), valid for unperturbed systems. But one has to remember that the definition of the states Ii) and If) is now different from the original one, since it contains a translational component. As we shall see later, this leads to some difficulties in the theory because of the continuous spectrum associated with the translatory motion. At the moment we don’t go into details and define a quantum-mechanical absorption intensity from formula (30), by omitting simply the almost constant factor preceding the sum. Thus we write: 1(w)
=
2ir~ (p1
2ö(w —
—
pf)lp1fI
w 1f).
(40)
(Eq. (40) is commonly called the Kubo-formula.) Starting from this expression, quantum-mechanical line-broadening theories have been developed [91. However, in order to be able to make explicit calculations on the profile, one has to solve first a quantum-mechanical collision problem and this constitutes in most cases an extremely difficult task. This may be one reason why no complete quantum-mechanical calculations of line properties have been performed so far in the case of neutral atom perturbation. Moreover, it is not quite clear yet, whether large quantum effects could be expected from such calculations. In one of the next sections we will present, mainly for the sake of completeness, some elementary features of quantum-mechanical line-broadening theory.
3. The derivation of the line profile 3.1. The phase shift theory In section 2. 1 the intensity formula has been derived 1(w)
=
f p(r)exp(iwr)dr
(1)
284
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
with the correlation function ~(T)
=
Kexp[—i77(t
(2)
+ T, 1)] ~
As has been assumed in 2.1, the phaseshift i~is a real quantity, i.e. the perturbations do not cause any amplitude changes of the atomic dipolemoment. 3. 1. 1. The correlation junction
In this section expressions for the correlation function shall be derived for various special cases, without making any specific assumptions regarding the interatomic potential function. 3.1.1.1. Theory for general pressures
At general pressures simultaneous actions of several perturbers on the active atom have to be considered, and the problem of mutual interference of the perturbations arises. The interference is different for different types of interactions. The simplest case is that of Van der Waals (V.d.W.)interaction, where the perturbations and therefore the phase shifts due to different perturbers
simply add like scalars. All other cases, like resonance and exchange interaction are much more complicated to treat. The problem of superposition is, however, avoided at low particle densities, where the collision time is much smaller than the time between the collisions and the perturbations are caused by binary collisions. In the following the theory shall be worked out for noninteracting perturbations, which scalarly add. Then the phase shifts ii~(t+ T, I) ii~the time interval t, ... I + r due to the individual perturbersj are independent and the corretation function becomes: ~T)
=
Kexp[—i
~
77~(t+
T,
1)1),
(3)
where K ~ denotes the time average. If moreover the perturbations are without interaction, then the mean value for the particle! will be independent from all others and ~(r)
=
(exp[—i77(t
+ T,
1)1 )~~‘
(4)
where nq9is the total number of particles in the volumeq9. The time average may be replaced by an
average over the ensemble: with77(T)~77(r,0)
(5)
which is identical with the average over the trajectories. Since the trajectories are determined by
the initial positions and velocities, the average has to be carried out over all initial positions and velocities. We introduce the coordinates R, VR, V4, where R is the internuclear distance and VR and are components of the initial relative velocity in the collision plane. One then shows, that ~r)
~fdFexp
[~~]
where E~m(v~, +v~)+ V~(R)
exp [_iI~w(r(t))dt])~
(6)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
285
is the total energy of the system, m being the reduced mass, Z the partition function and _~r_~i/2l_l dF 1 12m\3”2 R2dRdu v dv
Z
~V
\kT/
R~
~
Furthermore ~w is identical with ~ =
t_~_fdFexp[_.~-](l
I
—
.~-fdF
exp
— (i
E
[—
~
(1
defined on page 282. Rewriting one obtains: —
—
exP[_iJ~w(r(tndt])))”~’
}
exp [—if’ i~w(r(t))dt]
=
(1
~(r) —
-~—
}
fl~V
.
(7)
Then, by making~-+°°, while n is kept constant, one obtains ~(r)
=
(8)
exp[—n~(r)]
with ~(r)
= ~.i/2
~
R2fdvRfdv~v~, exp [_
~
[1 —exp [_if~w(r(t))dt]].
(9) The correlation function (8) with ~(r) given by (9), as reported by Fox and Jacobson [10], describes correctly the total line shape at general pressures in the case of scalar additivity of the perturbations. In earlier theories simplifications with regard to the collision process have been introduced. Such simplifications are useful for the actual computation of the line shape, when comparison with experiment is intended. In the theory of Lindholm [111 and Talman and Anderson [1 21 the collision process is schematically described by perturbers moving on straight lines at constant velocity. Also, in both theories the velocity distribution of the particles is neglected. In addition, in Lindholm’s theory constant frequency perturbation, i.e. linearly increasing phase shift, during the collision time is assumed. In the treatment of Fox and Jacobson, on the other hand, such simplifications are avoided. By expanding the frequency perturbation into a power series in time, however, this treatment is in practice only an improvement of the quasistatic approximation (see next section). 3.1.1.2. The quasistatic approximation 3.1.1.2.1. The general pressure case. It is easy to calculate c~(r)in the quasistatic approximation, where the velocity of the particles is small. Putting VR v~= 0, ~w(r(t)) becomes independent of t, r(t) being replaced by R. One then obtains from (9)
E
2 exp r V kT ][ 1 — exp(—i~w(R)r)1. (10) ~(r) = 4irf dR R 1(RY1 It should be pointed out, that in (10) the Boltzmannfactor exp[—V~(R)/kT] , which describes the influence of the interaction on the occurrence distribution of the internuclear distances, has
286
FSchuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
been neglected widely in the literature. Generally, however, it will influence drastically the far wings of the line [131 . Putting exp[ 17~(r)/kTJ = 1 Margenau’s 1141 result is obtained froni (10). Eq. (1 0) is valid not only for small particle velocities, but also for small correlation times T. Referring to (8), one may show. that when n is large. p(r) is only large for small T. Therefore 1 0) is correct also for large perturber densities. 3. 1.1 .2.2. The nearest neighbour appro.vin/ation. This approximation follows from the quasistatic approximation under the additional assulflptiOfl, that only the nearest perturber contributes to the line shape. This assumption will be fulfilled for sufficiently sniall number densities. In this case a particularly simple expression for the intensity formula is obtained, which is of importance for the determination of interatomic potential functions 11 3 I From the one-particle assumption the product ,i~19in (7) becomes unity. so that one obtains ~(T)
=
I
n~(T).
---
I
11)
Let us replace in eq. (1 0) the upper limit of integration by some quantity R
0 representing the radius of the volume ~79mentioned above, which for convenience may be considered as spherical. Then, using (I) it follows from (11): I(w)
fexp[iwr](I R~)
+00
f
=
dr expfiwTl ~l
2 exp
[-
4~ii[f dR R
R
—f
V(R)
0
.
~
(12)
Then, because of the one-particle assumption, the second term within the brackets becomes unity in the limit of infinitely large R~,.In this limiting case one then obtains:
V~(R)
+00
1(w)
=
4~n
f
dTf dRR2exp[
~T
-~
~w(R))rJ.
113)
Introducing ~w as a new variable, it follows dR(~w) 1(w)47rnJ’d(~w) d(~~P
r
V1t~w)1
[—~-~~—jf
+00
drexp[i(w-
~w)ui
(14)
the integral over ~w being taken over all accessible values of ~w. Since the integral over r is the 6-function by definition, one finally obtains:
dw
2(w)dR(w)exp[— [
2ir 1(w)—47rnR
—j. V(R(w))1 kT
(15)
-
The same result may be derived from elementary probability considerations [I S l~starting from the basic assumption of the quasistatic approximation: The intensity I(w)dw absorbed or emitted in the frequency interval w, ..., co + dw corresponds to the probability ~kW(Rk)dRk, that the
287
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
nearest perturber is at one of those internuclear distances Rk, where it produces the frequency perturbation co: * I(w)dw/2ir=
~W(Rk)dRk.
(16)
For the calculation of W(R)dR it is useful to introduce W~(R,dR), the probability that one particle is in the interval R, ... R + dR, W_(R, dR), the probability, that no particle is in the interval R, ... R + dR and W_(R), the probability, that no particle is at distances smaller than R. W(R)dR may then be expressed by W(R)dR
=
(17)
W~(R,dR)WJR).
W+(R, dR) is found to be W~(R,dR) = 4irn dRR2exp[—V
1(R)/kTl.
(18)
Furthermore,
r
dW (R)
r 2exp[_
WjR+dR)—WjR)~WjR, dR)WJR)+ From (19) W(R) is found by integration:
V~(R)11 kT jj.(l9)
dRWJR)[I_4irndRR
R
WJR)exp[_f
47rndR’R’2exp[_
jj.
(20)
Substituting (20) and (1 8) into (17) one finally arrives at
r V
[~
W(R)dR4irndRR
Since for sufficiently small n according to the one-particle assumption ~ W(R)dR is found to become W(R)dR
=
]],
kT kT 1(R)1jexp L4~f r R dR’R’2exp r V.(R’~fl
2exp[_
(21)
1 becomes unity,
4irn dR R2 exp[—V
1(R)/kT].
(22)
In the special case, that R is a single valued function of w, one obtains from (16) again (1 5). 3.1.1.3. The impact approximation In this approximation it is assumed, that the collision time is small compared to the time between the collisions. This assumption restricts the validity of the approximation to sufficiently small number densities and sufficiently large temperatures. The correlation function for this case shall be derived from the general pressure result (9). For simplification it is assumed, that the trajectories are straight lines, i.e. the influence of the interaction on the trajectories is neglected. ~
should be noted, that in general there are several internuclear distances producing the same frequency perturbation (c.f. p. 294).
288
F Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
The time variation of the frequency perturbation may then be written
~w(t)77(b,
v)’6(t—
t
(23)
0).
Here 77(b, v) is the phase shift caused by the collision with impact parameter b and relative velocity v; ö(t t~)is the delta function; to is defined by r(t0) = b. The time integration then yields —
f z~w(t)dt F
0
77(b,v)
r>t
0
r
(24)
°
,
0
For the calculation of 0(r) the threefold integration in (9) has to be performed. For this purpose one has to transform the variables R, CR and v4 into new variables v, b and t0. Noting that (see
also figure below) 2dRdvRv~dv 3dvbdbdtO R 4-+v one finds nO(r)
= ~I/2n(.~)
5v3 dv exp
[
(25)
f
v~] b db (1 —exp[i77(b, v)1 )fdt 0
rF.
(26)
Disregarding the velocity distribution by introducing a constant average velocity F, one obtains nO(r)rn~f
2irbdb(1 —exp[i77(b)1)
(27)
which is Lindholm’s [16] result. 3.1.2. The intensity distribution 3.1.2.1. Interatomic potentials Essentially three different types of interatomic interactions have to be distinguished [171: 1) Resonance interaction. This case is typical for identical atoms, of which at least one is in an
excited state:
atom
R
CR
perturber
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
289
(28) V~(r)= _Ca/r3.* 2) Van der Waals-interaction (interaction between instantaneous electrical moments). For dipole—dipole interaction the potential law is of the form 6. (29) 0(r) —C6/r 3) Coulomb- and exchange interaction. This is caused by overlapping electron distributions, if the atoms are at small internuclear distances. Although this type of interaction is quite complicated to treat theoretically, it is useful and Common practice to approximate the corresponding potential law by V
V~(r)= C
2. (30) 12/r’ The Lennard-Jones-(l 2,6)-potential (L.-J.-potential), representing approximately the true potential function for the interaction between neutral atoms in the ground state, may be represented in two different forms: V~(r)= (—C
6
2) 12/r’ 4e[—(a/r)6 + (aIr)12] +
C
(31)
6/r
Vn(r) =
(32)
(e and a being defined in the usual way as depth and zero respectively of the potential). The constants in (31) and (32) are connected by the relations 6
C 6
=
C
4ea
12
=
4ca~~
(33)
The potentials V1(r) and Vf(r) for the initial and final state of the line are different in general. This is the essential reason for the collision broadening effect, which is primarily determined by the difference potential V(r)
Vf(r)
—
J/1(r) = h~w1~.
(34)
Furthermore, if the atomic levels are degenerate, they may split into different sublevels in the quasistatic field of the perturbing atoms. In this case the potentials are different for different substates. A detailed calculation of the potentials is presented in section 5. 3.1.2.2. The calculation of the line shape 3.1.2.2.1. The impact approximation. The general intensity distribution for this case is obtained by substituting (27) into (8) and (8) into (1). The well known dispersion profile is the result [16, 181: W
ReF2+(ReF)2} ir{(w—ImF)
(35)
with the total halfwidth y2ReF *The subscript n standing for any electronic state of the system.
(36)
290
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
and the shift
13ImF
(37)
where F is defined by (26). Neglecting the velocity distribution one finds (38)
FniiYu where a is the complex cross-section U=Ur
—iu1
=
2~fbdb(I —expi77(b)).
(39)
The total halfwidth is then
(40)
~y2ni5ur and the shift
I3niJu.
(41)
In order to calculate the total phase shift 77(h) during a single collision, an inverse power law for the frequency perturbation is assumed z~w(r) Ti
r~
.
(42)
For the straight path-constant velocity model for the collision one then finds
C 77(b)c~~b1P
hv
(43)
with c~= ~/~F(4 (p
—
l))/F(4p).
For various cases of interaction one obtains (‘32
p006
c6r~7r
pl2
c12~-ir.
(44)
The calculation of the cross-section u12~f(l
--cos77(h))bdh
(45)
0
=
27rf’ sin rj(b)bdb
(46)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
291
can then be performed analytically. For p> 3 one obtains Or
=
cos
sin
~c~2/(P_1)F(P
~
p—l
~c~2/~1)F(P
~
p—l
— 3)(Cp)2~1)
p—I
hv
(48)
3)(~)2~1) hv
p—1 —
3.1.2.2.1.1. Van der Waals-interaction. In this case one obtains for halfwidth and shift from (40), (41), (47) and (48) [161: 8.16 U315 (C
(49)
215 n 6/h) 13 = —2.96 U315 (C 3’5 n. 6/h) The ratio of shift to halfwidth becomes
(50)
13/a =—0.357,
(51)
independent of the velocity and the value of C 6, i.e. the interacting system. 3.1.2.2.1.2. Lennard-Jones-interaction. This case has been treated by Hindmarsh [19], Schuller
[201 and Behmenburg Defining a parameter a, characteristic for L.-J.-interaction [21] (52) 6/(c[21]. 516 a (c6C6)~ 12C12) 2 vh and a reduced impact parameter x —ba°~(~-c 5 1C61/Wl)” 6 the total phase shift becomes
(53)
77(x)—2a(—x5 ±x”)
(54)
where the
+
sign holds for C
12> 0 and the — sign for C12 < 0. For halfwidth and shift one obtains 215n (55) = (2ir) 6/h) 13 = (27r)315v+(a)U3’5(C 215n (56) 6/ll) with the broadening functions h~(a)and the shift functions v~(a)defined by 3”5 h~(a)i33’5(C
2/5
h~(a)= 4(~)
/
x dx sin2[a(—x5
±x”)]
(57)
±x~1)].
(58)
2/5
v+(a) =
(~±) J’ xdx sin[2a(—x5
The functions h÷(a)and v~(a)are presented in table 1 and shown graphically in figs. 1 and 2. In fig. 3 the curves of v+(a)/h~(a)are displayed.
292
1’. .S~chuller and W. Pm/omen burg. Perturbation of spectral lines by atomic interactions
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F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
293
5.0
2.0
___-
________
1 - ~< io~ io-2 10-1 100 ~Q Fig. 1. Broadening functions h~(cs),calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.-interaction 1211.
In the limit of pure V.d.W.-interaction (large a, C 12 = 0) h~(a)and v~(a)reduce to the constant values 2.71 and 0.98, respectively. Hence in this case the ratio v~(a)/h÷(a) is independent of3~. temIn perature however, and both there y andwill 13 should with temperature through of the‘y velocity factor the D facgeneral, be an vary additional temperatureonly dependence and 13 through tors h~(a)and v+(a), since a is also velocity dependent. At intermediate a the functions h÷(a)and v+(a) are oscillatory, due to the existence of the minimum in the potential function. It should be remarked, that for the same reason also in atomic beam scattering the total scattering cross-section
as a function of the particle energy displays an oscillatory behaviour (glory extrema). 3.1.2.2.2. The quasistatic approximation. 3.1.2.2.2.1. Van der Waals-interaction. If z~w= —C 6/h is substituted into (10), one obtains, 6R neglecting the Boltzmannfactor
-i
-~
10_i
~
100
,o~
-
~
102
Fig. 2. Shift functions u+(oI), calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.interaction [21]
294
I’~Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
--H 7Q-2
10~I
100
101
~
1-ig. 3. Functions t+(0) — v+(~)/h+(o), calculated on the b~osis1)1 Lindholni’s impact appro\ifllat000 01 phase shitt theory, assoOnhing I -i-interaction 1211 -
~(T)
=+(27r)32(Cor/h)~2(l
Substituting
(59)
i).
(59)
into (8) and performing the Fourier transformation, one obtains, for C
6> 0 1
o.
1(w) ~ (C6)
I 2i
-
w0
32
exp
~
~
]
.
w < w0.
(60)
This result was derived by Margenau [14]. For small n and large 1w w0 the exponential in (60) becomes unity, and the intensity distribution coincides with that obtained from the nearest neighbour approximation [221 0,
w> w10
02 lw 1(w)
~n(C6/h)
w 3’2, 01
w < w 0.
(61)
6/1i into (1 5), neglecting the Boltzmannfactor. This may be Lennard-Jones-interaction. varified by substituting z~w = (‘6R 3.1 .2.2.2.2. The general pressure case has been treated by Bergeon [23] and by Hindmarsh et a]. [24], the nearest neighbour approximation by Behmenburg eta]. [251. Only the results of the second approach shall be presented, for the following reasons. First, on the basis of the nearest neighbour approximation the intensity distribution is obtained in an analytical t’orrn. Therefore the influence of both any realistic potential function and the Boltzmannfactor on the qualitative features of the far wings may easily be studied separately and in detail. Secondly, in certain cases of practical importance the intensity distribution, as obtained from this approximation is exact over a wide frequency range, at sufficiently small perturber densities. Since for a difference potential of the L.-J.-type there are in general two different internuclear
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
295
distances R1 and R2 causing the same frequency perturbation w, the expression for the intensity is, according to (16) and (22), given by I(w)dw
= k1
W(Rk)dRk
= ~
k1
R~dRkexp [_ Vj(Rk)]
(62)
For the evaluation of the intensity formula (62) it is sufficient to consider two cases: C6 > 0, C12> 0. In this case the sum in (62) reduces to a single term in the range R < a, since there R(w) is single-valued. Introducing dimensionless quantities z
=
1lL~w/e;
L
C
=
e~/kT
(63)
(~and; referring to the initial state*) and the abbreviations 1-z
____
1 -~--
____
f 1l+~J1+z;
f21—\/l+z;
V~(Rk)
6fk(z) ÷L12f~(z), vk(L, z) = = —2L one obtains for z < 0 (red wing) 1(z)
~Ii —f(z) I~)exp[—Cu
= ~~flU3~z2
3
k = 1, 2
1(L, z)]
+
C
(64)
1 +f(z) I~~exp[—Cv2(L,z)I } (65)
and for z> 0 (blue wing) 1(z)
~
3
~n~3~z2{I
1 —f(z)I~)exp[—Cv1(L, z)] }.
(66)
In the red wing (z < 0) an infinity occurs at z = —1, which is due to the minimum in V(R); for z < —1 the intensity becomes zero. Due to the collisions, which are neglected in the quasistatic approximation and due to deviations from the one particle approximation, the intensity at z = —1 remains finite, giving rise to a satellite in the close neighborhood of the atomic line [241. It should be noted, that the above infinity in the quasistatic intensity distribution is analogous to the rainbow infinity in the classical differential cross-section of atomic beam scattering. In the blue wing (z> 0) the intensity decreases monotonically with increasing z. Due to the Boltzmannfactor the behaviour of 1(z) is not determined by V(R) alone but in general depends also on V1(R). For the same reason 1(z) will generally be temperature-dependent. The influence of V.(R) on 1(z) is demonstrated by figs. 4 and 5, which show 1(z) calculated for two different values of the quantity L, which characterizes the position of V1(R) relative to that of V(R). If L ~ 1, then the value of the Boltzmannfactor at z = —1 is comparable to unity, and the infinity clearly comes out. If, on the other hand, L 1, then 1(z) decreases rapidly, due to the Boltzmannfactor, already at z> —1, so that the infinity at z = —1 is obliterated. In this case 1(z) is very small in the range z> 0 (blue wing). ~‘-
~o and ~ should not be confused with the quantities appearing in eq. (39) ff.
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
296
~zLo1,4 _-
Lr1.0
Lr0.8~ Lr0
liii
~-
~
1)11(1
Ill
-i0~
-
—
100
z
Fig. 4. Influence of the Boltzmannfactor on the red wing, assuming L.-J.-interaction. Curves calculated from the nearest neighbor approximation of the qua~istatictheory for C 6 > 0, C12 > 0 and C = 1 [251.
C6 > 0, C12 < 0. In this case the sum in (62) reduces to a single term over the whole R-range. Using the abbreviations l_~z
j.;
~
—z
(67)
-z 6f(z) + L’2f2(z) v’1(L, z) = —2L one obtains for the z < 0 (red wing) 1(z) =~~na~~z2 3
~(] +f’(z))~)exp[—Cv(L,
z)]}.
(68)
In the range z > 0 (blue wing) the intensity is zero. 3.1 .2.2.3. Theory for general pressures. This theory has been worked out by Lindholm [Ill, Talman and Anderson [1 21 and by Fox and Jacobson [10]. 3.1.2.2.3.1. Van der Waals-interaction. This case has been calculated through by Lindholm [11] and by Talman and Anderson [12] In the Lindholm treatment, in addition to the straight pathconstant velocity approximation, the frequency shift during the collision time r~is assumed to be constant. This assumption is equivalent with replacing the V.d.W.-potential by a square well potential. Consequently a parameter K was introduced, defined by .
= 2K
b/~7.
(69)
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
Lr0~
297
/2
177 ~1
__
,.,,~,.,
io2
z
Fig. 5. Influence of the Boltzmannfactor on the blue wing, assuming L.’J.-interaction. Curves calculated from the nearest neighbor approximation of the quasistatic theory for C 6 > 0, C12 > 0 and C = 1 [25]. i~is
determined by the condition, that for large values of 1w — w01 the intensity distribution agrees with the quasistatic approximation (60). For the representation of the intensity formula the following dimensionless quantities are introduced: /c6C6\ xb~—_—) 2v71
—
—1/5
ic6C6\
3/5
h’47rK(—-—) ~,2 viii
—1/5
vtIc6C6\ Y—~,,—--——) 2K2vh
; ;
(70)
2K/c6 C6 k(w—w0)—(—— U \2 tiP,
1/5
The intensity distribution then becomes I(w)f exp[iky~hij.o(y)Idy
=
=
(71)
‘l’r(Y) + j
f[(l_cos4)(y_x)_sin~~x6+2x]x
dx+f [(1 _cos-~)(x_y)_sin~ x6+2Y]x dx
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
298
~1(r)
_cos~s)[email protected]
= 1t[(1
dv+f [(
_cos~’)xe+sin~’
(x---T’)lx dx.
The problem of calculating the line shape is thus been reduced to that of calculating the dimensionless function ~i(v). Lindholm uses the method to calculate its behaviour for very small values oft’ and for very large values oft’ and then interpolates between the two. He obtains tori’~ I =
(0.795
+
0.577
i)i’ +
0.165
+
0.277 i,
(72)
fori’-~ I 413 +
0.591(1
+
i)v~2.
~(v) = (0.369 + 0.213 i)i’ It may easily be seen, that the v “2-terms, dominating in the expansion t’or v ~ I yields exactly the quasistatic approximation. On the other hand, the t’-terms dominating in the asymptotic expansion, yields the impact approximation. Talman and Anderson treated the general pressure case assuming the straight path-constant ,
velocity model, but, contrary to Lindholm’s treatment, calculated the frequency shift during the collision exactly. The results have bet~npresented and discussed in the article of Traving [26] .It is remarkable, that the intensity distributions in the line core, as computed by Anderson and Talman, differ very little from those obtained by Lindholm. Thus it seems, that the line shape in
the core is rather insensitive to the schematization of the collision process. This might not be so, however, in the wings of the line. 3.1.2.2.3.2. Lennard-Jones-interaction. The application of the L.-J.-potential to the case of the general pressure theory was carried out by Takeo [27], Behmenburg [281 and by Fox and Jacobson [101. Takeo performed the calculations on the basis of the theory of Talman and Anderson, Behmenburg on the basis of Lindholm’s theory. Fox and Jacobson started from the exact expression for the correlation function (eqs. ( 8) and (9)), but, by expanding the difference potential into a power series of time, actually used only an improvement of the quasistatic approximat ion.
On the basis of Lindholm’s theory, with the parameter ~ defined by (52) and the reduced quantities h, k, x and i’ defined by (53) and (70), the following expression for the intensity distribution is found: 1(w)
=
J
exp[ia”5 kv
— a3~
h ~(a,,v)1 dy
(73)
where the function i~i(ct,y) is given by ~i(a,
~
y) y)
=
=
~i~(a, y) +
i ~i~(a,
(74)
y),
f [(1 —f1(a, x)(y—x)—f2(a, x)h0(a, x)+2xlxdv +
1 [(1 —g1(a, x, y))(x—v)—g2(a,
x,
y)h0(a, x)+2y]xdx
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
~
y)
=
+
f[(l
—f
t(a,
x))h0(a,
x)+f2(a,
299
x)(y-x)Ixdx
J’ [(1 —g1(a, x, y)) h0(a, x)+g2(a, x, y)(x—y)] x dx
with the abbreviations 5+x’~’)];
f
5+x~)] ; (75) 2(a,x)sin[2a(—x’ g 6 +x~2)I; g 6 +x’2)I; 1(a, x, y) = cos[2ay(—x 2(a, x, y) = cos[2ay(—x h 2(l — x6)”] 0(a, x) = a’[x’ Because of the assumption of constant frequency shift during the collision time the theory will give only an approximation to the spectral intensity distribution in the wings. The shape of the line core, on the other hand, is believed to come out rather exact. f1(a, x)cos[2a(—x
3.1.3. Criteria of validity of the phase shift theory and its approximations 3.1.3.1. The adiabatic assumption Consider two nearby states k, 1 of the active atom, which, in the absence of the perturbation field, have energies Ek, E 1. If the states are nondegenerate, the energy levels will shift in the presence of the perturbation by amounts Vk(r), J/1(r), assuming the values E~(r)and E(r). The condition necessary and sufficient for a collision being adiabatic is, that the frequency range L~w~ of the Fourier-spectrum of the time dependent perturbation field is small compared to the frequency (E~(r)— E~(r))/h.Since ~ is approximately the inverse of the collision time r~,the criterium for adiabatic collision may be written l/r~
‘~
(76)
{E~(r)— E(r)}/P,.
Approximating r~by b/~(b impact parameter, U mean relative velocity) and denoting L~E= Ek — and V(r) = Vk(r) — V1(r), the relation (76) may be rewritten (replacing r by b) b z~E+i~V(b) — ~‘l.
(77)
h
If one of the states, k say, is degenerate, it becomes splitted in the presence of the perturbation field into sublevels m, m’. In this case the additional condition
~ U/m~(b)
l/r~
—
(78)
Vm(b)}Ih
must be fulfilled for the collision to be adiabatic. The condition (78) may also be regarded as a special case of the condition (76), if one considers k and 1 as belonging to states, which differ only by their space quantum numbers. The condition (78) may be written in a more rigorous manner [29, 30]
~j,
m, m’)~f(Vmi(t)
—
Vm(t))dt~
1.
(79)
300
F. Schuller and W. Behmcn burg. Perturbation of spectral lines by atomic interactions
Herej refers to the total angular momentum quantum number of the unperturbed active atom, and in and rn refer to the interatomic axis. If, in particular. V,0,(r) and V0~,(r)are approximated by functions of the L.-J.-type and assuming the straight path-constant velocity model for the colhisioii, the condition (79) becomes 2ir I
c6~C6 c12z~C12\ 5 b’1 /~ 6 \ b
1
(80)
where
~Co =
Coon’
Co 01
.
~C~2 = Ciom’
(‘i~omi
3.1.3.2. The validity range of the quasistatic approximation A general validity criterium may be derived by Fourier-analysis of the phase disturbed dipole oscillation during the collision [31, 321. Using the method of stationary phase it may be shown. that the spectral intensity distribution may be interpreted as quasistatic in certain frequency ranges, where the condition for stationary phase 2/~(~k)13”2 ~ 1 (81) R(tk)
= ~
I~(tk)I
is fulfilled. In (81) the time
tk
is defined by
(82) where w 00 is the unperturbed frequenc~of the oscillator and w the frequency of the radiation. Fig. 6 shows qualitatively ~w(t) for a realistic difference potential during the collision. Obviously, condition (81) is not fulfilled at values of’ w’, where ~d.’(t) = 0. This is the case for large t, where w’ is very small (i.e. in the line core) and in the neighborhood of the extrema of ~w(t). In order to determine the f’requency ranges, where (81) is fulfilled, R(tk) has to be calculated. For a L.-J.-potential, assuming C6 > 0, (‘2> 0, and the straight path-constant velocity model one obtains [251 2—
R(tk)
=
S(c, v)
=
0.176
a’
c”
I1--c~ -
-
4
II
---t’l
(1
y’2)014+(
I
with
Fig. 6. Time dependence of the frequency perturbation during tile collision. Tile zero of the time scale is chosen to coincide with the time where the internuclear distance equals that of closest approach.
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
c(w’)
=
(i
~
±
~l +~-~-); x’
~~~~chI6(w1);
0< x’ < 1
301
(83)
and a is defined by (52). For wt> 0 (blue wing) and under the additional assumption hw’/e ~ 1 (83) simplifies and one obtains
S(c, x)
=
0.235 a”2(e/hw’)”124f(x’)
f(x’)
14(1 —x’2)”4+(l —x’2Y314.
(84)
From (84) follows the existence of a frequency limit w~,,so that for w’ ~ W’L (81) is fulfilled, i.e. the quasistatic approximation is valid. Denoting by XL an arbitrary value of x, for which — x’L ~ 1 [32] and defining ~ by requiring S(c(wL), xL) = 1, one obtains from (84) WL
0.042 f241’ ‘(xL)a~2~ ‘c/h.
=
(85)
If w’ < 0 (red wing) then S(c(w’), x) becomes very large for Iflw’/I ~ 1 and for Ihw’/l
1,
i.e. (81) is not fulfilled. On the other hand, (81) is fulfilled in the range 0< 6 1(a) < Ihw’/cI <
ó2(a)
< 1
(86)
for systems with sufficiently large values of a: a
~‘
(87)
aO(XL)
where aO(XL) is defined by the requirement S(Cmin, XL) = 1, Cmin being that value of c, for which 51’2(l — 2c)”2 becomes a minimum. c 3.1.3.3. The validity range of the impact approximation In this approximation it is assumed, that the collisions are completed in the correlation time r (see section 3.1 .1 .3). This means, that the results are valid only if r is much greater than the mean duration of a collision
r>>rc.
(88)
Consequently, the impact approximation is valid in the line core in the range Iw’I ~ Iw~I.
(89)
For the estimation of w, Tc may be approximated by bcr/U, where bcr is a critical impact parameter defined
by
‘y = 2ir b~rnU where -y is given by (55). Taking into account, that w
(90) 1/Tc one finds (91)
302
F
3.2. Orientation
eff~’cts
Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
When studying line profiles by means of phase-shift theories, one generally assumes that there is only one potential curve in each, the lower and the upper electronic state of the active atom. Strictly speaking this is certainly not true. since in almost every case at least the upper state is degenerate and thus one should associate different potential curves to different magnetic substates. Of course, this difficulty may be removed by introducing an average potential, for instance
that described by London’s formula [161 - In fact, this rather intuitive procedure is not quite correct, since it neglects any influence of orientational changes, during collisions, on the linebroadening mechanism. However, the simple phase-shift theory with an average potential constitutes in many cases a fair approximation; but when studying more detailed aspects, such as.J-dependence of broadening and shift of fine structure components, one should use a theory that treats degeneracy in a more satisfactory way. We shall call it Anderson’s theory although Anderson’s original work was dealing
only with the case of’ molecular line-broadening [3] . It appears indeed, that his treatment of angular momentum can be applied without any changes to the case of atomic line-broadening as well. One interesting feature of this type of line-broadening theory is its direct relation to the theory of other relaxation phenomena like depolarization by collisions of resonance radiation or, furthermore, collision-broadening of double-resonance curves. Some remarks on these subjects are contained in section 3.2.4. 3.2.1 - The relaxation of the dipole transition moment in the n-npact approximation Following Anderson we introduce the so-called interaction representation by setting:
H(t)=H0 +IIJt)
(92)
and (93) where H0 and Q153 are respectively the Harniltonian and the evolution operator of the unperturbed atoms, and where H~is the Hamiltonian describing the interaction during collisions. The evolution
operator in the interaction picture satisfies the following Schroedinger equation: (94)
ihT=H~T,
with ~
~1~HCQ1O
=
(
exp~J10t~H~exp
~J1l)t)
.
(95)
For the dipole operator we have the relation: (r)T~T,
with ~
expHHor}pzexp{~~~±HOr~.
(96)
This may be written out in terms of’ matrix elements with respect to a set of basis functions representing the unperturbed atomic states. The result is:
F. Schuller and 11/. Behmenburg, Perturbation of spectral lines by atomic interactions
~
= ~
!AZif(T)
(_~w~r)
csttT~fexp
303
(97)
.
Now we make the assumption of non-overlapping lines, already discussed in a previous paragraph. Accordingly, for a given line, corresponding to a transition i -~f, the correlation function must be of the general form (2.31). Consequently we retain in eq. (97) only terms oscillating with frequency which means of course, that a and i~cannot differ from i and f respectively, except by some magnetic quantum numbersM~and Mf.* So we write: ~
/1.f(T) = ~ZMIMf(r)exP{iwIfr}~
(98)
with 1~ZM~Mf(T) =
~
~
T~M,(r)/4~,MP T~,M(r), ‘
I
f
(99)
f
where the upper indices designate the subspaces to which all the magnetic states belong. The state of the system at time r can be related to that at time r + dr by introducing a new evolution operator defined by the relation: T(r ÷dr)
=
T(r) T(r
-+
r + dr).
(100)
With this definition we can express the change of the quantity ~~Mf dr as follows: I d “~zMiMf=
=
‘r+dr~--
‘zM
1Mf’
~
101
if ~zM~M~
E T~M~ (r —~ ‘r + dr)l4,~,M,(r) T~?M(r M~M~
occurring in the interval
-*
r + dr)
— ~ZM~Mf()
I
In the impact approximation the calculation of this quantity can be reduced to a one-collision problem. Let us assume that there is one collision in the interval dr. Then the operator T(r -~ r + dr) takes some value, T(l), say, which depends only on the type of that collision. As we will do in the next section, one may consider all types of collisions and calculate the average effect of one collision on the quantity /.1 ~M0(T). It then turns out that the following relation If holds: 1~M(r) (102) \MiM. ~ TZM,(l)~t’~,M,(r)T~,M(l)\ II I / =A/1 I
/
where A is independent of any magnetic quantum numbers. This important result means, that in the average the initial orientation of the dipole moment is conserved after one collision. (It is understood of course, that the collisions are assumed to come isotropically from all directions.) It is now easy to evaluate the average change of the matrix element J4~M(r)during the inter*
Restricting ourselves to the simplest case we assume, that perturbers are spherically symmetric, M 1 and Mf refer then to the active
atom alone.
304
F. Schuller and h~.Beh menburg, Perturbation of spectral lines by atomic interactions
val dr. Using eqs. (101) and (102) and remembering that the probability for having one collision in dr is equal to dr/Ta (T~= time of mean free path), we obtain: -
dT
—-—(I —A)M’~M(r).
(103)
Let us consider the average of this quantity over the ensemble corresponding to the interval ‘r. Since in the impact approximation collisions are assumed to be uncorrelated, this averaging procedure is independent from the previous one. Thus the result can be regarded as the change during dr of the quantity (/.L’~M(r))eand we get the equation: d(/.L~M(r)~e = —dr
/IMM()e
(104)
withF(l —A)/T~. Hence the relaxation of a component of the absorbing dipole moment is described in the impact approximation by the following relation: zMiMf(T~e
~ZM
1Mf(O)e.
(105)
We will see in the next section that similar relations are obtained for the relaxation of other observables. The correlation function may now be written as:
~
lMi~(0)I2~e~T. i f /
(106)
(Naturally, the exponential factor of eq. (98) has been removed by assuming that frequencies are measured from the center of the unperturbed line.) Usually one introduces an “optical” cross-section, defined by the relation:
F
=
niia
(see eq. (38))
(107)
where n is the number density of perturbers and lithe average velocity. Dropping a constant factor, the correlatian function then takes the form: ~o(r) =
exp(—nUar)
(108)
which corresponds to a Lorentz-profile with the following expressions for the shift and the halfwidth: f3nPImo,
‘y
2nLiReu.
(109)
It might be worthwhile to emphasize, that although degeneracy could be removed during collisions, the impact approximation always leads to a single Lorentz-profile for a given spectral line. 3.2.2. General expression of the differential cross-section
According to eq. (1 02), the calculation of the quantity A, on which depends the relaxation rate
of the absorbing dipole moment, involves an average over all types of collisions. Let a collision be characterized by an impact parameter b and an initial scalar velocity v. Furthermore, all operators being defined with respect to a space-fixed frame, we have to consider the
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
305
orientation of the collision-trajectory with respect to that frame. To do this, we will specify the spatial orientation of the collision trajectory by three Euler angles, for which we use the collectivenotation f~. As a first step in calculating the quantity on the left side of eq. (102), we may now perform the average over all orientations and this will lead to an equation of the following type:
KM~M~ T~Ml(l)i.i~IMi(r)T~iM(l)) =A(b)I4tMM(r).
(110)
(The velocity does not appear explicitly in this equation, since we assume, as usually done, that it may be replaced by some constant average value U.) Obviously, the factor A of eq. (102) represents the average over b of the quantity A (b) defined by eq. (110). Accordingly, the optical cross-section, defined by eq. (107), can be written: ntiaz~F
1-A T~
1 T~
b
and from this it is easily shown, that a is given by: (111)
af2irbdb(l—A(b))/2irbdba(b).
The quantity a(b)
1 —A(b)
is commonly called the differential cross-section. In order to facilitate the calculation of the angular average on the left side of eq. (110), we have to change the definition of the evolution operator, by making it independent of the direction of the trajectory. This will be achieved by introducing a frame of reference that is attached to a given trajectory. We call this the “collision”-frame; it will conveniently be choosen with its z-axis parallel to the incident direction of collision. Its position with respect to the space-fixed frame is thus described by the Euler angles mentioned above. Let R be the operator that rotates one frame into the other: Then there is the following relation between the previous operator T(1) and the new operator T~(l),say, belonging respectively to the space-fixed and the collision frame: (112)
T(l)=RtT~(l)R. The éorresponding matrix elements are given by: Tj~M~’~= I
I
~
p4.’11v7’1’
(
‘-~}~M. (~2)T~t ‘~M~4!” ‘~!“M’(&~~) I
I
1
1
I
I
(113) T~iM(1)
~
= M~’M~”
(~2)T~(1 )M~’M~” D~j,IIM(~2)
where the D’s are the rotation matrices, J being the total angular momentum of the considered electronic state.
306
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
It is now essential to consider the fact, that p5 is a component of an irreducible tensor operator. Then, from the Wigner-Eckart theorem [31, its matrix elements are proportional to some ClebschGordan coefficients, according to the relation: PZM1Mf~)
C(J~lJ1M~0M).
(114)
Substituting the expressions (113) and (11 4) into eq. (110) one can carry out the averaging procedure. This rather lengthy calculation, based on the Clebsch-Gordan series shall not be developed here. It leads to the following result for the differential cross-section:
~
1
u(b)
i
C(JflJ.;MfMM.)C(JflJ.;M~M’M~)Tit(l)MM,Tf(l)MM
(115)
aIIM
which is called Anderson’s formula. According to this equation, the collision frame plays from now on the role of a fixed frame and any reference to the original space-fixed frame has disappeared. Calling in the future the collision-frame “fixed”, we will speak of a “moving” frame when choosing the quantization axis parallel to the line joining the two atoms. Those will be the two frames used in our further calculations. 3.2.3. Approximate solutions of the one-collision problem
Anderson’s formula (11 5) reduces the calculation of the optical cross-section to a one-collision probleni. Let T(t) designate the evolution operator during one collision (defined with respect to the collision-frame) and let t = and ~ +oo correspond respectively to the times before and after the collision. Then we may assume, that T~(l)is given by T(+oo) whereas T(--°°)equals unity. In order to determine T(t) we could, in principle, start from eq. (94) but this amounts to solving a great number of coupled differential equations. (The fact that Anderson’s formula involves particular matrix elements connecting states that differ only by their magnetic quantum numbers, does not change anything on the situation as far as the solution of the system given by eq. (94) is concerned.) Thus, in practice, eq. (94) is of not very much use and we introduce a different approach for determining the operator T(t). Suppose for a moment, that we are considering a state n which is non-degenerate (corresponding for instance to J = 0). Then we make the adiabatic approximation and obtain from eq. (2.33) the following expression for the diagonal elements of T(t): —-°°
T~(t)
exp~—~fdt’
V~(t’)}
CC
(116)
where V~,is the interatomic potential introduced on p. 282. This expression obviously satisfies the following equation: ~hTn
CC
V0T~.
(117)
We now assume that in the more general case, where n corresponds to a degenerate state, it may still be possible to define some quantity similar to our previous V,~.This of course will be an M-dependent quantity and we shall designate it as V~lM.Matrix elements of T(t) belonging to the
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
307
state n should then satisfy the equation: ihT~IM=
~
(118)
VJ~I,MI,T7~JIIM.
M”
We shall see soon, how in practice the quantity V~IM?lcan be defined more explicitly. Right now we want to mention that V~,Mllcan be interpreted as a matrix element of some sort of an “effective” interaction hamiltonian according to TTn
— tijneff~
VMIMII
=
5~11c
)M’M”
However, it is in general not allowed to identify H eff with H~.The latter procedure would yield only the first order energy which, as is well known, vanishes in the case of Van der Waals interaction. (Even in the case of resonance interaction although it is finite there, it does not represent the main effect [341.) Eq. (118) corresponds to a certain type of adiabatic approximation in the sense that it involves only one electronic state n. This is true, no matter what further approximations will be used to solve the system which is still (2J + 1)-dimensional. One of those latter approximations shall be discussed now. Giving T(t) the meaning of an operator acting in the subspace of the state n, we may write eq. (118) in the form: (119)
ihT=VT. If this were an equation between scalar quantities, the solution would obviously be: T
ex~{_~-fdt’ V(t’)}
=
exp{—P(t)}.
(120)
In our case, where operators or matrices are considered, eq. (120) constitutes what is called the “sudden”- or “scalar’ ‘-approximation. * Matrix elements of P(+oo) are: (121)
Mhfdt~TM0M(t).
To get matrix elements of T(+oo), we first define a matrix that diagonalizes the matrix P(+oo), according to: ~
A~IMI0~f”M”
AM?IIM
=
~M’M
exp { —~}
and this then leads to the expression: T~,M(+oo)=
E AMIMII
eXp(—~MU)AM,lM.
(122)
There is an opposite to the “sudden”-approximation which must also be considered. For that *As is well known this amounts to neglecting the effect of the Dyson time ordering operator.
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
308
purpose we introduce a “moving” frame (already mentioned in the previous paragraph), having its z-direction parallel to the interatomic axis. Let D~be the matrix describing the rotation of the fixed frame (which is here the “collision”-frame) into the moving frame, and let us define new quantities by the relations: T~oM
=
~
=
~
(123)
D~I0M0TMI.M
and MIM
~
(1 24)
V~J~0~I00?D~J000AI.
The matrix , which describes the interaction energy with respect to the interatomic axis, may be assumed to be diagonal for symmetry reasons. Introducing again the effective interaction hamiltonian, which in the new frame may be designated by ~ we would write the elements of e as follows: — (..~,fleff-~
M
‘“c
‘M
Then, with these quantities, eq. (118) transforms into: ih
~‘M
=
~
+
ih
~
~
(125)
Suppose now, that the second term on the right of this equation is negligible. Then the solution will be given by: r~(t) 6MIMexP(----
~ J dt’e~(t’)}
.
(126)
This obviously corresponds to an adiabatic approximation. But now the word “adiabatic” is used in a very restrictive sense since it applies to magnetic sublevels 114. To avoid confusions one has to bear in mind that there are two types of adiabatic approximations: The first one, concerning the electronic level nasa whole, is always made tacitly when eq. (118) is used; the second one, concerning magnetic sublevels (with respect to the interatomic axis) corresponds to a special kind of solution of that equation. However, from now on only eq. (1 26) will be referred to as the adiabatic approximation. Matrix elements of T(+oo) in the adiabatic approximation are easily obtained from eqs. (1 23) and (126) which yield [351: T~,M~
= ~1’M
exp
-~
J
dt e~(t)} -
(127)
The quantities given by the expressions (1 22) and (1 27) obviously depend on the impact parameter b. It can be shown by more detailed considerations that these expressions are both limiting ones, corresponding respectively to large values of b in the case of the “sudden”-approximation and to small ones in the case of the adiabatic approximation.
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
309
In either case the differential cross-section can now be calculated from Anderson’s formula, by letting n correspond to the initial respectively the final state. The question now arises how to define the M-dependent interatomic potential. One possibility is to calculate both, V7~?Mand EM by second-order perturbation theory. This has been done recently in the case of dispersion forces [30, 34]. However, if repulsive forces are to be included, such direct calculations lead to much more difficult problems. An alternative approach would be to define a priori the interaction energy in the moving frame as a function of the interatomic distance, writing for instance: (l28a) in the case of a London-type potential, and EIMI =
_C~/R6+
(128b)
C~IyR12
in the case of a Lennard-Jones potential. From these expressions matrix elements of V can easily be calculated by inversing formula (124). Formulae expressing elements of the matrix P(+oo) in terms of the IMI’5 have been derived for straight-line trajectories defined by the equation: R(t) = (b2 + u2t2)112. The results obtained for several values off 1 andff are listed below [361*: i)J~= 0:
P’0 ~i-fe~dt;
Jf= 1:
P~1=P~,.1 =~J°~[e~ +—~-~(e~ ~e~)]dt, +OOr
pf00
=
± h ~r
I[Co — R2
~C
—~‘
E~jdt
~
0
~ “~
—
J~b2~-~-(e~ —e~)dt,
P~.,=P~,, -~ —i
P~
0=P~,=P~1 =~=o. +00
~ j
=
I
j
f
= 1~ 2
pi,f
=
1/2 1/2
=
1/2 —1/2
=
~:
P312 3/2
= ~i_
~
—1/2—1/2
pi,f Jf
pi,f pi,f
r J
E1/2
~ di’
J00
[e~/2 + ~
=
—1/2 1/2
=
1 p 3/2
3/2
sl
(e~ — c312)]dt,
*In the isotropic case these expressions reduce to those given by eqs. (29) and (31), p. 289.
310
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
1/2 =
1/2-1/2
= ~i/
2~3/2
pf
= ~~I/2
=p~’
3/2 1/2
ICC ~
=
3/2
=pf
3/2—3/2
~
~—3/2
1/2
=pf 1/2—1/2
1/2 3/2
(e~,2 — ~3/2
)j di’,
~~3±jP~
=pf —1/2 1/2
~
(e~/2 =p~’
—1/2 3/2
=0
=pf
—3/2 3/2
—3/2—1/2
We further set:
-
+=
± C ~J
if
~M
‘~dt
‘129
~M
Here we want to make a remark on how matrix elements could be calculated in a systematic way by making use of the symmetry properties of the effective interaction hamiltonian. Expressing the latter as a sum over irreducible tensor components, we write
x
~ A°~T,~~0 (internuclear frame),
= =
(collision frame).
A~D~Tkq
~ kq
C
Then, according to the Wigner-Eckart theorem, matrix elements are given by E’/l
=
(x~~
1~”~I’M
~ A~C(JkJ;M0M)
=
(H~~)MIM ~A~D~ 0C(JkJ;MqM’),
kq
J being the angular part of n. Substituting into this the following identity [33] 1/2
D~0=
(~~-~)
~kq~’
0)
with sin ~
CC
b/R
one finally obtains 1/2 =
~ (~-j)
3~ 0)C(JkJ;MqM’). A~Ykq(I
The coefficientsA~maybe calculated from: A~
2k CC
+
I
Z~e7~C(JkJ;M0M).
For symmetry reasons we have k = pair [42] so that for the cases of interest in this paper only the values k = 0,2 are relevant. Special results are given in the following table:
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Special values of
as functions of
311
44).
~/23/201
0
~(i/2)
2
—
~j(2c~’~ +
1(~(32) + ~(3’~2))
J~(4%~_~~%2))
—
,(~(1)
1))
—
Calculating differential cross-sections from Anderson’s formula, we first consider a transition = -~ Jf corresponding for instance to the 2P,,2 component of the alkali doublets. We call ~,
this the isotropic case since it does not involve any orientation effects. In this case eq. (118) can be solved exactly and the result for the differential cross-section is given by: a(b)
=
1
—
exp{—i(~~,2 — ~11,2)}.
(130)
This is just Lindholm’s formula which is strictly valid only in the isotropic case. In the more general anisotropic case we have to consider either the “sudden” or the “adiabatic” approximation when calculating the differential cross-section. For a transition J1 = ~- Jf = 2P corresponding to the 312 -component of the alkali-doublets, we get the following results: “sudden”-approximation: -+
a(b) = 1
—
~-[exp~—(~,2
—
P~/2)}+ exp{—(z~,2— P~12)}];
(l3la)
“adiabatic”-approximation: a(b)
1.
(131b)
Taking as another example the transition results are: “sudden”-approximation a(b)
~
=
0 -~ Jf
=
1 (e.g. intercombination line of Hg), the
1 —4[exp{—(P~1+P~,—P~)}+exp-[—(P~,—P~,—P~)}+exp{—(P~ —P~)}I;(l32a)
“adiabatic’‘-approximation a(b)
=
1
+~-
exp{—i(r~— i~)}.
(132b)
Let us now ask the question what would happen if one or the other of these approximations were used alone for the calculation of the total cross-section. Obviously the adiabatic approximation would lead to an infinite result since a(b) as given by eqs. (l3lb) and (l32b) does not go to zero for b —~ As to the “sudden” approximation, it breaks down indeed for small b-values. But in that region the function a(b) has a rapidly oscillating behaviour and only its average value counts. Since the latter equals unity for both, the “sudden” and the “adiabatic” approximation, almost no modification will be introduced in the result when replacing at small b’s one approximation by the other. Therefore we can expect that the expressions (131 a) and (1 32a) lead to reasonable values for the total cross-section. This conclusion can be confirmed by more quantitative arguments [37]. As has already been mentioned, in many calculations on line-broadening and shift one disre°°.
F Schuller and W. Behmenburg. Perturbation of spectral lines by atomic interactions
312
gards the anisotropy of the potential, which is replaced by an average over 111-values. From a standpoint of pure theory such a procedure is certainly incorrect, but one may ask however, whether it introduces large errors in the calculated cross-sections. Defining an average phase-shift, one shows from the invariance of a trace, that the following relation holds: ~M
~MM
~
CC~L ~
(133)
=~
With this value of the phase-shift, Lindholm’s formula yields the following expressions for the differential cross-sections: For the transition J1 = -~~ CC
u(b) = I
—
exp{—-~-(~/2 +
and for the transition J1 u(b)
0
=
~i/2
-*
--
2P’112)}
(134)
I~= I:
I —exp{---4(2P~1+P~ —-3P~3)}.
(135)
Comparing these expressions to eqs. (131a) and (l32a), we see that the more correct procedure consists in first defining cross-sections for different M-transitions and averaging afterwards, instead of averaging before defining a ~ross-section. As an example consider the case df dispersion f’orces (eq. (1 28a)). Then the differential crosssections given by eqs. (l3la) and (l32a) take respectively the form: t)) a(b) 1 -- ~(exp(—A1/b~) + exp(--A2/b CC
and
u(b)
5)+ exp(—A/bt) + exp(—A~/b5)). 1/b The corresponding total cross-sections are given by: =
I
—-
-~(exp(—A’
-
CC
A215
~
+
A215
.
and ~
-~~)
A7”5
+
A ‘2/5
--iTe13~0F(~)(~~_-
+
A ‘2/5
—--—--~- -)
whereas the isotropic theory leads to the results: -
IA 1
CC
+A2
—i~e’~~°F(~)
2/5
and a =
.
_iTeI3~10F(~)
A’~ +A’
(_~~_
+A~
2/5
-~-
In practice, th6 values obtained from the isotropic theory are only slightly different from the previous ones. On the other hand, the ratio: r = halfwidths/shift
=
Re a/Im a
remains rigorously unchanged since the A’s are real. The situation might be quite different and the influence of anisotropy much stronger if repulsive forces are taken into account. This, at least, is the result of a recent calculation using a potential [37]. With respect to comparison of theory and experiment (section 4) we want to
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
313
present the results for halfwidth and shift in the case of L.-J.-interaction, as obtained from the sudden approximation. Defining the quantities p
C~/C~,;
=
p’
=
(136)
~y’= C?2/C~2
[c6C~(p — 1)] 11/6
2iih [c12C2(p’ — 1)] 5/6 1
=
a2
(i initial state)
6/C~
C12/C12
=
1 a1
= C°
[c6C~(p(5÷y)/6
—
1)] 11/6
2iih [c12C~2(p’(ll +y’)/12—
1)15/6
— 1 [c6C~(p(l + 5y)/6 — 1)] 11/6 a3—--—2v7-) [c,2C2(p’(l + 1 ly’)/12 — 1)15/6 one has the following expressions ~
315~315 [~ (p ~
n(2~)
—
1)12/5
x 4[h÷(a,) + h~(a2)+ h÷(a3)]
—
1)]
x 4[v+(a,) + v~(a
(137)
2)+ v~(a3)]
where the functions h~(a)and v~(a)are given by (57) and (58). We note, that line core data contain information both on the isotropic and anisotropic part of the potentials. As far as the anisotropy is concerned, this information may be compared to that obtained from depolarization and double resonance measurements. For this purpose the relation between the corresponding cross-sections shall be discussed in the next section. 3.2.4. Cross-sections for the relaxation of orientation and alignment Some of the best known cases, where anisotropy of the potentials manifests itself, are relaxation phenomena like depolarization of resonance radiation by collisions, broadening of double resonance curves, etc. Although these topics are not within the scope of our article, we will write down some of the expressions of the corresponding cross-sections and compare them to those obtained in the case of optical line-broadening. In many situations this procedure might be useful for extracting potential parameters from experimental data. When studying the relaxation phenomena mentioned above, one considers an atom which is brought to an excited state by absorption of resonance radiation. Let the corresponding wavefunction be written as:
‘i,E
C~4)UM
Mf
where i and f refer as usual to the initial and final state respectively and where u designates an unperturbed wavefunction. Now, if several initial states are involved with equal statistical probabilities, the state of the system posterior to the excitation can be described by the following quantity:
314
I’~Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
Mf 0~c~ Mf c~
‘~M.Mf = ~
which is called the density matrix. Considering the time evolution of the density matrix under the influence of absorption, reemission and collisions, its new value ~oM~M~(t) will describe a stationary state if the total rate of change per unit time is zero. Concerning the relationship, for various experimental setups, between the elements and P~z~Mf(t)of the density matrix and the polarization properties of the absorbed and reemifte~Iradiation respectively, we refer to more detailed surveys [38, 391. Here we are considering only the evolution of the density matrix and more specifically that part of it that is due to collisions. Let (dpM~M(t)/dt)COflbe the corresponding rate of change per unit time averaged over all types of collisions. Then we are looking for a relation similar to eq. (104) which had been obtained in the case of a dipole transition. There is, however, a difference between the latter case and that of the density matrix, coming from the fact that, according to eq. (114) ~zM-M~ is proportional to a single Clebsch-Gordan coefficient, whereas PM Mf(t) has to be represel!lted by the expansion [40, 411: PM’M ~‘
~
(t)
JM
~
p~(t)C(Jf.JJf;M.MMf),
(138)
involving several of those coefficients. (We disregard any influence of nuclear spin, assuming that we are dealing with even isotopes.) Thus we expect that a result analogous to eq. (104) will be obtained for each of the f-values appearing in eq. (138). This will be given by the equation (dp~/dt)~ 011 —F~p~,
(139)
which defines a set of relaxation constants according to different terms in the development of the density matrix. After introducing in the usual way total and differential cross-sections by the relations: (140) and ajf
(141)
27rbdba~(b),
we can derive an expression for a~(b)by carrying out an averaging procedure similar to that introduced previously in the case of the optical cross-section. The result is given by: a~(b)CC1
—
2Jf+ 1
1
all ~ M C(f~J~M2MM1)
~
IT~lJfMl>(ffM2IT~t~JfM~>.
(142)
In most studies, both experimental and theoretical, one is dealing with a level ff = 1. Then the possible values off are 0, 1 and 2, according to a scalar, a dipole and a quadrupole component. The case 1 is referred to as “orientation” and the case J = 2 as “alignment” [411. For the fCC
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
315
corresponding cross-sections we will use the notations aOR and aAL respectively. Applying the same methods as in the optical case, we may now derive approximate expressions for these quantities [37, 42, 43]. For the “sudden”-approximation we obtain the result: 2boR + sin coR) (143a) aOR(b) = .(sin2a + sin with aOR
1 COR
e~)~-~ di’,
~f(eo_
+00
=
/ (e~— ~) ~l
—
boR
=
~f(eo—
Ci)(l
—
2~)dt,
b2\ R~)dt
and further aAL(b)
aOR(b).
(143b)
-~
The “adiabatic” approximation leads to values that are independent of the impact parameter. These are given by: O 0R(b)=3
(l44a)
and
—
— 4
3
aAL~u)—S5aOR
It should be noticed that with the “sudden”-approximation the average value of aOR(b) in the oscillatory region equals unity, whereas the adiabatic approximation gives UOR = One may therefore expect that by using the “sudden”-approximation alone one underestimates the values of the total cross-sections aOR and ~jAL~ Thus, there is at this point a slight difference with respect to the optical case [441. It may further be remarked that both approximations lead to a value of the ratio aOR/aAL that is equal to ~ However, the experimental value seems to be much smaller; possibly, contributions from a region where neither approximation is valid, may not be quite negligible. We now introduce for comparison the optical cross-section, or more precisely the real part of it which describes broadening. Restricting ourselves to the “sudden”-approximation, we get from eq. (132a) the following expression: ~-.
.
Re[a0~(b)]
=
4[sin2aopT + Sin bopT + sin COPT
with aOPT
=
—
1
(145)
316
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
bopT
=
CC
~
~+00
1J
~
+
~
—
(e~— e~)
dt
]
— ~1~2
dt.
Considering the ratio of the total cross-sections calculated from eqs. (145) and (l43a) respectively, one obtains for a R6-potential an expression given by [37, 431 Re(UQPT)/aOR = 0.44[q215 + (q + ~)2/5 + (q + ~)2/5 1 where q, a parameter first introduced by Rebane [451 is defined as follows: ,
q
=
(C~- ~)/(C~
Ci).
--
Typical experimental values of the ratio Re(u 0~1)/°oR are in the range of 1.5 2.5 [42]. From these values we can determine q and thus, in principle, the anisotropy of the potential, assuming that C~— C~is known from line broadening measurements. However, as we have shown in the previous paragraph, a repulsive term should be included in
the potential and then, of course, the ratio Re(aOPT)/aOR has to be calculated numerically as a function of the potential parameters. Comparing again the result to experimental data, some indications concerning the magnitude of the anisotropy parameters for the repulsive term may be obtained, as has been shown for a Lennard-Jones 6-12 potential [37, 43] They constitute probably the only information on that subject available up to now. .
3.3. The quantum-mechanical impact theory As has been mentioned previously, quantum theories don’t seem to play an important part so far in the field of neutral atom broadening and therefore we treat them very briefly. In fact, we shall restrict ourselves to the case of the impact theory, for which a quantum-mechanical version has already been given in early days by Lindholm. We derive Lindholm’s formulae for broadening and shift, by using a method due to Baranger [91. Let us start from the general formula (2.40) and neglect stimulated emission. Using the integral expression for the delta-function, we write:
f2
E pi’~~~’
drexp{i(w
-—
w~f)r}
fdrexp(iwr)Ø(r)
(146)
with ~(r)
=
~?p1IIx~fIexp(—iw1fr).
(147)
We assume for simplicity, that our system is that of one active atom at a space-fixed position, surrounded by N moving perturbers.
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
317
We now introduce a Born-Oppenheimer type approximation which consists in writing the total wave-function of the system as a product of the unperturbed electronic wavefunction pel and a translational wavefunction x~.The transition moment is then given by the following expression: CC
M~x~’~ IXtr)
where~i~(~p~IpI~p~) is the transition moment of the unperturbed active atom. For the frequencies we have the relation: w~1~ = + where w~corresponds to the change in translational energy between the initial and final state of the system (w~~ being the unperturbed frequency). Dropping a constant factor I,i(~)I2 and taking the origin of w at frequency w~,we write the correlation function as follows: 2exp—iw~r. (148) ~(r) = ~ P~XX~ This expression shows, that within our approximations, the only origin of line-broadening lies in the overlap effect of the translational wavefunctions. It should be noticed that x 1 and Xf are by no means orthogonal, since they correspond to different interatomic potentials in the translational Schroedinger equation, depending on the electronic state of the active atom. As a further strong simplication, we now consider the case of binary collisions: Assuming that the perturbers are all independent from each other, we write Xtr as a product of one-perturber wavefunctions, that is XtT X(1)X(2) x~ and further w~ w~ + w(2) + + We can then define an one-perturber correlation function: ...
Ip(r)
~p(I)I(X(1)lX(1))I2
exp~—iw~~r},
(149)
so that i~(r)is given by: çb(r)
[p(r)1’~’.
(150)
As will be shown below, p(r) differs from unity by of an the infinitesimal quantity which is intV’ enclosing theonly atoms system. Thus we may write: versely proportional to the volume p(r)= 1 —g(r)/CV (151) .
Introducing the number density n by the relation N for 0(r) the following result: 0(r)
lim [1—g(r)/Q9]’~’=
~—ng(-r)
nCV, and letting ~Vgo to infinity, we obtain (152)
00
From now on we will specify our states x by a momentum parameter k, which is related to the total energy of the translational motion by the equation: h2k2/2mE.
(153)
(In our picture, where the active atom is assumed to be fixed in space, m represents the mass of the perturber; more generally one may base the calculations on the relative motion of the two atoms and interpret m as the reduced mass of the pair.) We then write ~p(r) in the form:
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
318
~p(r)
=
~
(154)
Pk~k(T),
k
with ~
I(XikIXfk)I2exP(_~(E’
—
E)r)
(155)
,
the primes corresponding to the upper state of the transition. As is well known, in the limiting case of a continuous energy spectrum, matrix elements can not
be evaluated by space integration, since the integrands would not go to zero at large distances. In order to get convergent integrals, we have to introduce the following transformation: 171X~k~
~Xfk’’Xik~ ~kk’
(Xfk’ -~
,
~
(156)
where ~ V is the difference potential between the upper and the lower state and c is an infinitesimal quantity which ensures that there is no divergence at F’ — E. Eq. (1 56) is a consequence of the so-called Lippmann-Schwinger equation. It can be made plausible by multiplying both sides by F’ -— F and letting e go to zero. This yields: (F’ E)(xfkvIxIk) (Xfk~J~VIXIk~, (157) =
which is certainly true, since F’ and F are cigenvalues of the hamiltonians Hf (upper state) and lI~(lower state) respectively and ~V = Hf — H 1. Furthermore eq. (156) is verified for ~V 0, since in that case Xf -~ X~. Most simply the results of the impact theory appear when considering the first derivative of ~k(T) which is given by: dpk =
—
dT
~I~(E’— F)(xIk Xfk’XXfk’
—
Xik~P
—
(E’
—
F)r
~k’
This will be transformed by using eq. (1 56) and the complex conjugate of eq. (1 57). The result is: dlpk dr
= —
i ~(X.kI~VIX ) /1 fk
i — —
i~k’
I
(XlkI~VIXfk~XXfk~l~VIXIk) exp — ---(F F’ F ic -—-------
—---
---
--
E)r
--
As we are dealing with a continuous energy spectrum, the summation over k’ amounts to the calculation2)fk’2 of andk’. integral, is to obtained bythe replacing formally the sign ~k’ by the symbol Thus, which one has evaluate following expression: (CV/27r dpk CC
i
iQ9 (X~kI~VIXf~ —
00
k’2dk’
(XjkI~VIXfkPXXfk~I~VIXlk) —--—~~-~--exP(_~~(E’ — E)r~ (158) .
To do this in a general way would be an extremely difficult task. However, in the particular case of very large correlation times one can go one step further by using the following argument: For r tending towards infinity, the oscillations of the exponential in the integrand of the second term on the right of eq. (158) become increasingly fast. So there will be no contribution to the in-
F. Schuller and W. Behmenburg, Perturbation of spectrallines by atomic interactions
319
tegral unless one of the other factors shows a comparably fast variation. This happens near k’ = k because of the denominator F’ — F — ie. As a consequence, the range of integration could be reduced to a small intervall ~k’ around k and then all factors are practically constants except those which have just been mentioned. This again means that our integral may be replaced by the expression: m —~
k(xjkl~VlxfkXxfkL~VIXjk)j
+00
d(E —E)
exp{—i(E’ — E)r//f} E’ —F—ic
(to show the validity of this expression, the qualitative argument given above can easily be supported by a more rigorous calculation). The new integral is shown to be equal zero and hence we get the result: dlpk
i
-~- _~~(xikL~VIxfk),
for larger.
(159)
Now, as Baranger has pointed out, assuming r very large is equivalent with making the impact approximation. Then, by comparing r to some characteristic “collision”-time, a validity criterium for this approximation can be established. For this and other more detailed discussions we want to refer to Baranger’s work. By integrating eq. (1 59) and by using the value y,(O) = 1, which follows from eq. (155), we see that in the impact approximation lpk(r) is given by the expression: lPk(T)
1 —~-(XIkII~VIXfJC)T.
(160)
This expression can be transformed further by using some of the results of quantum-mechanical collision theory. Let us introduce polar coordinates r, 0 to describe the position of the perturber with respect to that of the active atom; in the case of spherically symmetric interaction potentials the azimuthal angle does not need to be specified since obviously there is then a symmetry of revolution around the incident direction 0 0, Any wavefunction can now be expressed in terms of Legendre functions by writing: CC
Xk(r, 0)0~aklP,(cos0)uk,(r),
1=0,1,2,...
(161)
where 1 plays the part of an angular momentum quantum number. In the case of normally behaved interatomic potentials one can introduce some characteristic phases 5k1 which are defined by the asymptotic expression of the radial wavefunction Ukl. The latter can be shown to be of the form: uk,(r)-+ ~—
sin(kr— 4irl+ ak,)’
rlarge.
As one can further show, the coefficient ak, can be expressed in terms of those phases cording to the relation:
(162) ~kl’
ac-
F Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
320
5)z(21+ 1)i’exp(i~kl),
(163)
where ~Z is a normalizing factor equal to CV112 CV being the volume introduced previously, in which the perturber is enclosed. Going back to eq. (1 60), we first express the matrix element on the right by means of an integral over CV, writing: (XjkXfkf9XjkXfk(Vf
(164)
V.),
—
and it is understood that CV goes to infinity at the end of the calculation. Now the Xk’~satisfy the translational Schroedinger equation which is of the form: h2 ~--~
~
V2~~ + VXk =Fxk
~2
~FCC
~
k2)
--~
(165)
-‘in
and from this the integrand in eq. (164) can be rewritten as follows: /12
2xfk XfkVX~f). (x~V transform the right-hand side of eq.
(166) It is now convenient to (1 64) into an integral over the boundary surface of CV, which can be choosen as a large sphere of radius R. Using Green’s theorem, we write: XjkXfk(~’f — V
1)
=
~—
~
(167)
Expressing the
Xk’S
by means of eqs. (161) and (163) and integrating over 0, we get the relation:
fdCVx~xfk(V
2 ~ 1— V1)
=
(21+1 )exp{--i(öfkl—~Ikl)}R2(uik1~~ufkl_ufkl~_uik1) R (168)
h
Now R is assumed to go to infinity and consequently we can replace in eq. (168) the asymptotic expressions, given by eq. (1 62). This yields: i a R2 ~,ulkl ~
Ufkl
ufkl
—
a
t1ikl
)
~
1 ~
ar
by their
sin(~,,. 1
—
yr
UkI’S
(169)
~ikl)
rR
Thus we obtain for the integral the following final expression:
fdCVx~xfk(Vf
—
2i(öfkl V~)CC_i
~
-~—~ (21+
l)(l
—
—
ölkl)}).
(170)
exp{—
According to eq. (164), this represents the quantity (x1kI~VIxfk). With the following definition: cx
1~I~VIxf~)~-q,
(171)
F. Schuller and W. Behmen burg, Perturbation of spectral lines by atomic interactions
321
eq. (160) takes the form: ~Pk(T)=
(172)
1 —qr/CV,
from which we get, according to equation (154): 1 —~—(q)r
p(r)CC
(173)
g(r)(q>r.
and
Here the brackets stand for a Boltzmann average. As a result, the correlation-function, defined by eq. (152), is given by: 0(r)
exp{—n(q)r}.
(174)
This leads to a Lorentzian profile characterized by a shift and a half-width respectively equal to: f3—nIm(q>,
y—n2Re(q).
With the definition of q from eq. (171) the explicit relations for shift and half-width are: 2(~fk,
13CC —fl’7T-~~ K’-~-~(2l+ l)sin[
7n27r-(-E(21+
—
~ik1)]~
l)(1 _cos[2(~fkl—6Ikl)]))
.
(175)
These relations may be called the quantum-mechanical Lindholm formulae since they look very similar to the corresponding expressions of the quasiclassical impact theory. Furthermore, it can be shown, by using for instance the WKB-method, that the classical expressions are obtained as a limiting case of formulae (175). However only few attempts have been made to calculate shift and broadening in the general quantum-mechanical case [20, 46]. In order to do so, one would have to determine the phases ~~klby solving the Schroedinger equation of the translatory motion. This makes the situation
rather hopeless, except possibly for very simple and crude models without real physical significance. 4. Comparison between theory and experiment One basic difficulty arises when comparison between theory and experiment of neutral atom line broadening is intended: In order to simplify line profile calculations, one has to rely on approximations to the theory, independent of any assumptions regarding the interaction potential. The range of applicability of such approximative theories is not always clearly defined. It would therefore cause difficulties to quote exactly the error of the intensity at a certain frequency as calculated on the basis of the approximation and on the assumption of some potential function. As a consequence, the reliability of potential parameters, as derived from experiment on the basis of some approximative theory, would in general be restricted. This would be true at least in cases, where the number n of linearly independent observables equals the number m of unknown parameters of the potential function. On the other hand, the reliability would be large in cases, where n m. ~‘
322
F Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Table 2 Spectral lines perturbed by foreign gases investigated experimentally for comparison with line broadening theory line
transition
Kr 7601 A
Hg2537A Cs 8943 A Cs 4593 A
4p5(2P)SsIS] _4p5(2P 312)5p[~] 3P 6’S0--6 1 2S 2P,, 62S112 62P 2 6 112- 7 1/2
K4047A
425112_52p112
perturber
ref.
noblegas
47
noble gas noble gas lie, Ar
37,48—55,13 56—59 56-59, 60
Kr
61
For comparison with theory those systems investigated experimentally were selected, for which the assumptions of the theory are fulfilled or approximately fulfilled. In the classical path theories so far only the general elastic case has been treated for realistic potentials. This means, that the applicability of these treatments is restricted to cases, where the contribution of inelastic collisions (e.g. inducing transitions between fine- and hyperfine levels or quenching) to broadening may be neglected. From this point of view the systems listed in table 2 have been selected for comparison between theory and experiment. 3 as func-
4.1. Line (halfwidth andand shift)-measnrements at number densities below lO’~cm tions core of number density temperature Core measurements at low number densities for comparison with theory were carried out at various absorption and emission lines, perturbed by noble gases (table 3). The experiments were performed with pressures below I atm. at temperatures between 80 °Kand about 400 °K.Halfwidths and shifts were found to be linear functions of the perturber density n. The experimental values for y/n and 13/n are listed in table 3. Under the experimental conditions quoted above the impact approximation of line broadening theory is valid and the experimental results may be interpreted on the basis of this approximation. In terms of Lindholm’s impact approximation of the phase shift theory (sect. 3.1), however, the observed blue shifts with He as perturber and the different values for the ratio 13/y (table 3) cannot be explained, if a potential law C 6 is assumed. An interpretation, on the other hand, is 6/r possible in terms of the more realistic L.-J.-potential. The L.-J.-parameters C 6, C12 may then be deterniined by adjustment from the values of y/n and 13/n measured at fixed temperature. The validity of this interpretation can be tested by examining whether for a certain system a single pair of L.-J.-parameters describes consistently all meas’ired line shape data. Experimental consistency checks have been performed for the systems Kr 7601/noble gas and Hg 2537/noble gas. The results will be discussed below for each system separately. Kr 7601/noble gas In table 4 are listed the values for the ratios of the halfwidths 7295/780 and shifts 13295/1380 measured at 295 °Kand 80 °K.The theoretical values were calculated on the basis of Lindholm’s impact approximation of phase shift theory, assuming L.-J.-interaction (eqs. (3.55), (3.56)). The comparison with the experimental values yields fairly good agreement, except for the case of Ar
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions Table 3
323
3. -i/n and 13/n in
Halfwidths and shifts of various systems measured at number densities below 101 9 cnc units of icr2°cml/cnc3 ref.
‘y/n
jIm
80
47
1.38
+0.108
+0.156
295
47
2.57
+0.283
+0.222
80
47
0.683
—0.227
—0.664
295
47
1.14
—0.2l~
—0.371
80
47
1.18
—0.32i
—0.545
295
47
2.44
—0.73~
—0.602
Kr
80 295
47 47
1.304 1.93~
—0.44~ —0.652
—0.684 —0.678
He
293
48
0.88
+0.048
+0.055
Ne Ar Kr
293 293 293
48 37
0.56 1.16
—0.084 —0.26
—0.151 —0.227
37
0.78
—0.20
—0.256
Xe
293
37
1.12
—0.25
—0.222
He Ar
400 400
60
8.8 ±0.5
60
6.7
+1.5 ±0.1 —1.63
—0.243
line
perturber
Kr76O1A 4p5(2pai 2)5s[~I 2
He
..4p5(2p3,2)sP[~]2
Ne
(emission) Ar
198 Hg 61S
2537 A
T(°K)
3P
0 —6 1 (absorption)
Cs4593A 2Sl, 2P1/2 (absorption) 6 27
Cs4555A 62S1/2—72P3/2 (absorption)
K 4047 A
--
He
380
60
Ar
380
60
Kr
470
61
±0.5
±0.05
6.9 ±0.4 5.8
+0.73 ±0.05 —1.55
±0.3
±0.06
6.6
—1.84
425
±0.2
1/252p1/2
±0.05
(absorption)
Table 4 Comparison between observed and predicted temperature dependence of shift (jI) and broadening (‘y) for the Kr A 7601 line perturbed by inert gases [47]
perturber
He Ne Ar Kr
Y295/780
13295/1380 -
observed
predicted
observed
predicted
2.62 0.93
2.16 0.84
1.87 1.67
1.73 1.70
2.29
1.79 1.43
2.07 1.48
1.64 1.43
1.47
+0.171
+0.106 —0.267
—0.279
324
F Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
as perturber. The agreement is much better than in the case of pure V.d.W.-interaction, where the theoretical ratio ‘y295/’yso equals that of 13295/1380, and assumes the constant value 1.43, indepently of the nature of the perturber. The existent deviations have to be discussed with regard to the assumptions of the above approximations: I) The perturber passes the active atom on a straight path at constant velocity; 2) The contribution of inelastic collisions and orientation effects to broadening is negligible. The influence of the deviations of the trajectories from the straight path on broadening and shift was studied by Schuller [20]. Assuming pure V.d.W.-interaction, however, the expected temperature effect on broadening, due to those deviations, is in disagreement with observation: The halfwidth is expected to decrease with temperature, whereas an increase is observed (table 4). Thus it appears, that, although the influence of curved trajectories may not be underestimated, the assumption of’ a repulsive term in the potential function seems to be necessary for the interpretation of the experimental results. The contribution of inelastic collisions to broadening is probably negligible: Although cross-sections for collisions inducing transitions between fine structure levels are not known for the systems under discussion, they are expected to be small due to the comparatively large energy distance of the levels [131. On the other hand-, the contribution of M-mixing collisions to broadening and shift may not be neglected, if a L.-J.-interaction is assumed [43] Hg 2537/noble gas For the system Hg 2537/Ar the L.-J.-parameters, obtained from low density measurements of halfwidth and shift, cannot account for the asymmetries of the profiles observed both in the line core and the line wings [13, 50, 51. 531 : The profiles, calculated on the basis of Lindholm’s phase shift theory for general pressures (eq. (3.73)) using the L.-J.-parameters obtained in the above way, display “blue” asymmetry even at very low pressures [131, whereas the measured profiles show “red” asymmetry. Discussing the assumptions of Lindholrn’s phase shift theory it may not be excluded, that for this system the path curvature influences considerably broadening and shift, thus introducing errors in the determination of the L.-J.-parameters from those quantities. The contribution of f-mixing collisions to broadening will be negligible, since the cross-sections for quenching of the Hg 2537 A-emission line by noble gases is very small compared to the cross-sections for broadening (table 5). The effects of M-mixing collisions however, may be considerable, as was pointed out by Butaux, Schuller and Lennuier [49] . These authors have therefore interpreted the line core data in terms of splitted potentials I/t(r) for the Hg 63P1 state. For the systems Hg 2537/noble gas
line
Table 5 Comparison of cross-sections for phase shift collisions (oel) and inelastic collisions (ojfl~l) 0e1 ref. °inel ref. perturber (ltYbo cm2) (lO~~ cm2)
Hg 2537 A 615 3P 0—6 1 Cs 2Sij 8943 A2 P 6 2—6 112
He
68
48
0.00
62
Ne Ar
92 260
48 37
1.04 1.38
62 62
He Ar Ne
99 336 174
13 13
5.7 x l0~ 5 1.9 xX i0~ 1.6 1fF
63 63 63
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
325
the four parameters C’~,C?2, C~,C~2occurring in the two L.-J.-potentials V1(r) and V~(r)were obtained semi-empirically on the basis of Anderson’s theory (sect. 3.2) as follows: Using the ratio C?/C~calculated by means of eq. (5.5) and values of C~,C~2given in the literature [17], the four parameters C?~C?2, C~,Cl2 were determined from measured halfwidths, shifts and depolarization cross-sections [641, with the use of eq. (3.46) (and a corresponding equation for the case of depolarization [43]). The validity of this interpretation may be tested by comparing the L.-J.-potentials obtained in the above way with those derived from measurements of the far wings (sect. 4.3). K 4047/Kr The validity of the interpretation of the line core data for this system will be discussed in sect. 4.3. 3 4.2.Extensive Line coremeasurements measurementsatatthe number densitiescomponents above 1019 cm fine structure of the Cs-principal series perturbed by noble gases were reported by Chen et al. [56—59] and at the Hg 2537 A line perturbed by noble gases by Füchtbauer et al. [50], Granier [541 and Behmenburg [13]. The measurements were performed in absorption with noble gas pressures up to 200 atm at temperatures between 300 and 1500 °K.Halfwidth shift 1~and asymmetry ratio a,* were observed to be in general nonlinear functions of the perturber density n. This is in line with the predictions of the theory for general pressures [13] which has to be applied under the experimental conditions quoted above. Detailed comparison with the phase shift theory has been restricted to the Cs line 62S 2P 1,2 -~ 6 1,2 A 8943 A, since only in this case2Pthere are no contributions to broadening due to M-mixing and f-mixing collisions: 1) Since the 1~2state is an isotropic one, there is no V.d.W.splitting in this state in the spherically symmetric field of the noble gas atom; 2) The measured 2P cross-sections for transitions 112 -~ are very small for all noble gases (table 5). Comparison was reported for the systems Cs 8943/Ar and Cs 8943/Ne by Behmenburg [13, 28] and for Cs 8943/Ar by Jacobson et al. [651. Behmenburg pointed out the improvement of qualitative agreement between theory and experiment by using a L.-J.-potential instead of a V.d.W.-potential in the calculation. As an example, the results for the system Cs 8943/Ar are shown: figs. 7, 8, 9 show the functions y(n), /3(n) and a(n) measured and calculated on the basis of Lindholm’s general pressure theory (eq. (3.73)). The constants C6 and a, used for the calculations, were obtained from the values 7/n and 13/n, measured at low number densities, by adjustment. The parameter i~ entering the theory (eq. (3.69)) was determined by adjustment to data measured at higher densities. The qualitative behaviour of the observed functions is seen to be well accounted for by the calculated ones. In case of a(n) the agreement of theory and experiment is even quantitative. The deviations of experimental from theoretical functions ‘y(n) and /3(n) at higher densities might be an indication, that the additivity assumption in the general pressure theory is not fully valid. Recently Jacobson [661 suggested to check the validity of this assumption by systematically studying the effect of mixtures of two different foreign gases on the line shape. Atakan and Jacobson [65] compared the experimental data of the Cs 8943/Ar system with those calculated on the basis of the exact correlation function (eqs. (3.8), (3.9)) of the phase shift ~,
*The asymmetry ratio a is defined by the ratio of “red” halve halfwidth to the “blue” halve halfwidth.
326
F Schuller and
W.
Behmenburg, Perturbation of spectral lines by atomic interactions
DC
21cm3) Argon Numb., D.r6lty (70 Fig. 7. Comparison of theory and experiment in the pressure broadening of the Cs absorption line A 8943 A by argon (Behmenburg [131). —o—o----o— experiment (Ch’en et al. [56—59]); Lindholm’s theory assuming L.-J.-interaction (cs = 10, = 1.04, C 6)i~—.—-Lindholm’s theory assuming V.d.W.-interaction (rs = ~, = 1.7, 6 = 6.95 X i0’~ erg cm C 58 erg cm6). 6 = 6.95 X 1fF
,~
: iii -20 //
d
II
(I)
-10
2 Argon Number Density [10 2~cm 3j Fig. 8. Comparison of theory and experiment in the pressure shift of the Cs absorption line A 8943 A by argon (Behmenburg [13]). —o--o— experiment (Ch’en et al. [56—591); —~ Lindholm’s theory assuming L.-J.-interaction (~= 10, ,c = 1.04, C 6); —.---.-— Lindholm’s theory assuming V.d.W.-interaction(a = o~, = 1.7, C 58 erg cm6). 6 = 6.95 X 10 erg cm 6 = 6.95 X lIT
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
327
21cm-3J Argon Numb.r D.nslly [10 Fig. 9. Comparison of theory and experiment in the asymmetry of the Cs absorption line A 8943 A perturbed by argon (Behmenburg [13]). —0—0— experiment (Ch’en et al [56—591); Lindholm’s theory assuming L.-J.-interaction (a = 10, = 1.04, C 6); Lindholm’s theory assuming V.d.W.-interaction(a = ~, = 1.7, C 6). 6 = 6.95 x 10”58 erg cm 6 = 6.95 X l0”58 erg cm
—.—.—
theory. The comparison was extended to the total shape of the line core, up to about 3 halfwidths from the line center. Furthermore, the sensitivity of the line core parameters (7, j3, a) to the potential parameters was investigated. The L.-J.-parameters obtained, differ, however, considerably from those derived on the basis of Lindholm’s treatment (table 6). This seems to indicate, that the straight path-constant velocity assumption in this treatment is inadequate in this case. 4.3. Spectral intensity distribution in the line wings Wing measurements for comparison with theory were carried out at the absorption line K A 4047 A perturbed by Kr [61]and at the absorption line Hg A 2537 A perturbed by Ar [25]. The foreign gas pressures used were below 1 atm. at temperatures between 400 °Kand 700 °K. K A 4047/Kr For the interpretation of the red wing, the intensity distribution to be expected from the general pressure phase shift theory was approximated by folding the quasistatic profile into a dispersion profile accounting for the effects of the collisions: The quasistatic profile was calculated by Table 6 Lennard-Jones-parameters derived from measured line profiles on the basis of the phase shift theory line
perturber
ref.
C
2.96
6 58 erg cm6)
C1 (1001 2 erg cm12)
Cs8943A
Ar
13
(1IJ” 6.95
62S1,
Ar
65
2.25
0.46
Ne
13
1.59
9.60
2P1,2
(absorption) 2—6
328
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
10’
_______
i~~~--~-..-~
1102
2P Fig. marsch 10. etComparison al.for. [61]). of theory experiment; experiment of2P calculations redL.-J.-interaction, wing bywe of the the method K absorption ~that of Hindmarsch line A 4047 andthat Farr A perturbed [24] .assuming bywith krypton .-interaction. (Hindcounted optimum means was taken of Bergeon’s adjustment toline The be••... equal L.-J.-parameters theory to toand the the [23] measured inverse assuming from ofthe profile; the this collision adjustment note, time. are Fig. in the theexcellent 10 halfwidth observed shows agreement the red of theoretical the satellite dispersion isL.-J profile well those profile acobat collision-induced tained from core transitions data. This is remarkable, one would think, the cross-section for collisions. Furthermore we feel, 112 that -~ the 312 influence is effects notsince quite of negligible path curvature on degree. broadening to that formay phase notshift be negligible either in this case. Possibly these compensate tocompared some Nevertheless, as a further consistency check of the present interpretation measurements of the temperature dependence of the red wing would be highly desirable. Hg A 2537/Ar In this case no satellite is observed in the red wing (fig. 11). This is in line with the theoretical expectation, since due to the Boltzmannfactor the intensity in the far red wing should decrease monotonically, if L 1 (sect. 3.1.2.2.2.2). 3P For the interpretation of the observed wing shape, as well as of the line core, the splitting of the 1 -level of the Hg atom has to be taken into account. The theoretical profile was calculated on the basis of the nearest neighbor approximation of the quasistatic theory (eqs. 311(3.67), (3.68)) assuming, that the far red wing is exclusively formed by transitions X~0 —~A 0.* This assumption is well justified both by experimental results and theoretical considerations [67] In order to obtain optimum agreement with the experimental profile it is necessary to assume C]2 C]2 < 0. For the L.-J.-parameters obtained in this way the validity ~—--—-
~‘
-
—
*Notation of Herzberg [68].
__
F. Schuller and W. Behmenburg, Perturbation of spectral lines by atomic interactions
329
io-3~
U
C
-io~ 1J (cm Fig. 11. Interpretation of the red wing of the Hg absorption line A 2537 A perturbed by argon in terms of L.-J.-interaction (Behmenburg et al [25]). ..... experiment; —nearest neighbor a~proximationof the quasistatic theory (C~= 6.41 X 10”60 erg cm6 C] 2 = —3.21 ~ 1 2) 10_i 4 erg cm ~2x102
~
criteria of the quasistatic theory (sect. 3.1.3) are well fulfilled*. Fig. 11 demonstrates the remaining differences between theoretical and experimental wing shape existing at optimum adjustment. This indicates, that the L.-J.-type of potential function is, in this case, only a rough approximation to the true potential function.
5. Appendix: The calculation of interatomic forces It cannot be the purpose of this article to present a comprehensive survey on quantal calculations of interatomic forces as reported in the literature. Rather a brief account will be given on several semi-empirical methods, currently used for the interpretation of collision broadening of spectral lines by neutral atoms. For more detailed information on the theory of intermolecular forces Margenau and Kestner’s recent monograph [69] should be consulted, which also contains an extensive bibliography of papers up to 1966.
*Unfortunately they cannot be compared as yet with those obtained from line core data, since these were derived under the assumption c]
2 — C]2 > 0.
330
F. Schuller and W. Behmenburg, Perturbation of spectral lines bv atomic interactions
5. 1. Dispersion
krcc.s’
London [701 was the first, who calculated on the basis of perturbation theory the long range, attractive part of the potential between neutral atoms. He expanded the classical electrostatic energy between two neutral atoms into a power series of R~(R internuclear distance) and applied this as perturbation on the system. 5. 1 .1. Dipole—-dipole interaction We consider the system consisting of an active atom and a perturbing one. The active atom is assumed to be in a certain electronic state k (energy Ek) with total angular momentum quantum number J and space quantum number M, the perturbing atom in the electronic ground state k’ (energy Eke). This situation is often realized in experiments on neutral atom broadening. Assuming
that both atoms are unlike, the first order energy of the system is zero. The expression for the second order energy becomes [301
~EkJM (2)
—
1 3 h~e2
~k
~ (E~-EK)(Ek
~K~k
~p(x,J~k~s~) . M)
(1)
with ~p(x,J,M)~[l+3C2(xlf;M0M)] where the C(...) are the Clebsch-Gordan coefficients; ~ ~ the oscillator strengths for the transitions k i~,k’ -÷ ii’, respectively. The summation has to be taken over all intermediate states K and of the atoms of the system. Obviously the expression (1) contains explicitely the J- and M-dependence of the second order energy. The M-dependence shows that the interaction is anisotropic in cases with J ~ 0,-k. By averaging over M the function ~pbecomes unity and (1) becomes identical with London’s result [70]. In many cases the lacking knowledge of the f-values rules out the exact calculation of absolute values of the second order interaction energy. It is, however, often useful to know the ratios -~
,~‘
EJ 1/EJ2 or EM1/EM2 of the interaction energies in states with different J- orM-values, respectively. In these cases the radial parts of the wave function occurring in f~ are equal for both states and may therefore be put in front of the summation sign in (I). Furthermore the energies may be replaced by some average value. If one, in addition, assumes for the general case of atoms with more than one valence electron the radial parts of the wavefunctions to be the same for both
electrons and the coupling of angular momenta to be of the LS-type, one obtains [37] —a
C6kJM
+
b(L,J, S) C(J2J; MOM),
with a1, r
~
2L
~ ,-,-~ (2L—l)(2L+3)
1
\‘~
j
1/2
where W(...) is the Racah-coefficient.
W(SLJ2-JL)
(2)
F. Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
331
The expression (1) may also be greatly simplified if the perturbing atom has a much greater separation of its energy states than the active atom. This situation arises for alkali- or alkalineearth resonance lines perturbed by noble gases. The term (Ek+Ek_EK —EK) in the denominator may then be set approximatively equal to Ek EK and the summation separated into two summations, each involving only one atom. If furthermore the J- and M-dependence is neglected, one obtains —
~~=e2a
E (kIrIK)12
(3)
,
K*k
where a is the polarizability of the perturber. In case of a one electron system the sum over the dipole matrix elements may be approximated (Coulomb approximation) by [151
~ I
2~~ —31(11-1)],
2 [5j~* 0 p—
(4)
where a 0 is the radius of the first Bohr orbit, n* the effective principal quantum number of the state k and 1 the orbital angular momentum quantum number. With respect to the various approximations the C6-values calculated from (3) and (4) may be in error to within 10%. The uncertainty may be somewhat larger for heavy elements and for atoms with more than one valence electron. These errors, however, compensate to some degree, if the difference C~— C~ of the constants for the initial and final state of the line, which occurs in line broadening theory, is calculated. 5.1 .2. Higher moment interaction Comparison between the C6-values, calculated on the basis of London’s formula, and experimental values, derived from atomic beam scattering and line broadening experiments [19, 71] strongly indicate the effectiveness of higher moment interaction in addition to dipole—dipoleinteraction. Its contribution may be estimated using the model of isotropic, three dimensional harmonic oscillator for the interacting atoms [72]. Recently calculations of the dipole—quadrupole interaction constants C8 have been carried through for the systems alkali—noble gas on the basis of the Coulomb approximation [73]. The values for the ratio C8/C6 in the ground state interaction were found to deviate at most by 20% from those obtained by means of the oscillator model. More recently Baylis [74] developed a semi-empirical method to calculate complete potential functions for the systems alkali—noble gas. In the electrostatic part of the interaction was included the interaction of the induced dipole of the noble gas atom with all multipoles of the alkali. The noble gas atom was treated as a sphere of radius r0 which acts like a point polarizable dipole as long as both alkali valence electron and alkali core are outside, but give constant interaction whenever the alkali electron is inside the sphere. The value of r0 is chosen by adjusting the well depth of the ground state potential to agree with experimental scattering data. Furthermore, the contribution of the spin—orbit coupling in the alkali to the interaction was included. 5.2. Resonance forces In case of identical particles the first order perturbation energy does not vanish, if one of them is in an excited state. For the root mean square value over all M one obtains [18]
F. Schuller and hi. Behmenburg, Perturbation of spectral lines by atomic interactions
332
=
2hf 2J+ 1 I x/~e R3 8irmw 2Jg+l
5)
0
where f and w
0 are the oscillator strength and angular frequency, respectively, of the resonance line and Jg belongs to the ground state. 5.3. Overlap forces
In the past much work has been done in calculating, both ab initio and semi-empirically, short range interaction of binary systems in the ground state. Extensive reviews have been given in this field by Hirschfelder [1 7, 751. On the other hand, little effort has been undertaken so far to in-
vestigate short range interaction of systems in the excited state. Roueff [76], in an attempt to estimate the order of magnitude treated the repulsive part of interaction as essentially arising from the scattering of the valence electron of the radiating atom by the perturbing one. A more rigorous treatment was proposed by Baylis (1969), who used pseudopotentials for representing the effect of the Pauli exclusion principle on overlapping electron states. This method 2P,,was applied to alkali-rare gas diatomic systems, composed of an alkali in its 2~ ground state or 2,~2first excited states and a noble gas atom in its ‘S-ground state. The results show that the repulsive part of the interaction is strongly anisotropic. This fact may be explained qualitatively 2~-state the alkali electronbyis sketches mainly of athe electron symmetric probability su distributions (fig.B2~-state,the 12). In the X alkali electron is mainly in a pu orbital, in spherically orbital. In the which overlaps the noble gas at larger internuclear separations than the su orbital. In the A2fl state the alkali electronic wave function has pir character with a node along the internuclear axis and the noble gas can approach the alkali rather closely before the repulsive interaction dominates. ~,
alkal, atom
x2[
A2ff
noble atom
gas
Q
2
B2E
Fig. 12. Schematic representation of the electron distribution for the X2E, A2fl and B22 states ofan alkali—noble gas diatomic system.
Including also dispersion forces (sect. 5.1) Baylis was able to calculate complete potentials for the systems mentioned above. Well depths, equilibrium internuclear separations and reduced curvatures were tabulated. The comparison of the potential parameters determined from scattering *
Notation of Herzberg
1681.
F. Schuller and W. Behmenburg, Perturbation ofspectral lines by atomic interactions
333
data show relatively good agreement. The method should thus be applicable also to other Systems, like alkaline earth plus noble gas, which are of interest in neutral atom line broadening. Acknowledgement The authors take pleasure in acknowledging the financial support of the “Gesellschaft der Freunde und Förderer der Universität Düsseldorf” during the preparation of the manuscript. Thanks are also due to Mr. M. Backer for recomputing numerically the broadening- and shift functions listed in table 1.
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