Moment analysis of absorption profiles in terms of anisotropic interatomic potentials

I. Quon~ Spectmc.

Rod&.

Transfer.

Vol.15,pp.211-222. Pergamon Press 197.5. Printed inGreat Britain

MOMENT ANALYSIS OF ABSORPTION PROFILES IN TERMS OF ANISOTROPIC INTERATOMIC POTENTIALS E.

LEBOUCHERand

F.

SCHUL~R

Laboratoire de Spectroscopic Atomique, DRP Universitt Paris VI and LIMHP, CNRS, 92 Bellevue, France (Received 17 May 1974)

Abstract-We derive general expressions for the first three moments of atomic absorption profiles. In the special case of isolated lines, we show that orientation effects yield a major contribution to the second-order moment. Numerical calculations using Baylis’ adiabatic potentials have been carried out for the resonance doublet 7800/7947Aof Rb perturbed by Ne. I. INTRODUCTION THEMOMENTmethod,

as a tool for analyzing theoretically emission or absorption profiles, has been widely discussed in the literature, especially in the field of molecular band spectra. More recently, JACOBSON(~) has applied this method to the case of isolated atomic lines in order to draw some information on interatomic potentials from optical line broadening data. In this paper, we first introduce some more general expressions, which are not necessarily restricted to the case of isolated lines. The latter case will be discussed in a more detailed fashion, assuming that, in general, the interatomic potential of the excited state is orientation-dependent. It is shown that the expression of the second-order moment is different from that obtained by using an average potential and ignoring orientation effects. II. GENERAL

EXPRESSIONS FOR THE MOMENTS OF AN ATOMIC SPECTRUM THE CLASSICAL PATH THEORY

IN

We consider an optical atom surrounded by several perturbers, assuming that the system is submitted from time t = 0 on to monochromatic radiation of frequency o. Assuming, furthermore, that the perturbers move along classical trajectories, so that the interaction becomes a given function of time, BLOOMand MARGENAU”’derived the general expression for the absorption profile I(w) resulting from this situation. Using the same methods, we focused our attention on the spectral moments defined as follows:

M,,=

I

oa

(w - oo)“I(o) dw,

(1)

where w. can be any frequency inside the spectrum. However, in practice one will often identify w0 with the unperturbed frequency of some given spectral line. The first three moments are then given by the following expressions:

M,,= Trp(pzH(t)[HH(f),/-~~(t)l)e M,

(2)

= fi-‘Trp(/~z~(t)[H”(t>, [HN(t), /JZ”(~)]~>~

- woTrp(p*H(t)rHn(t), PFLZW(t)l)r7 217

(3)

218

E.LEBOLJCHERand

F. SCHULLER

Mz= h-‘Trp(pzH(t)[H”(t), [HH(t), [HH(th ~~“(t)lll)t -

Trp(pzH(t)[SiH(t),

kzH(t)l)t

- 200h-‘Trp(CLP(t)[H”(t),

[HH(th

kH(t)ll)t

+ oo’Trp(pz”(t)[H”(t), pzH(t)l)t.

(4)

Here (i) HH(t) is the Heisenberg representation of the total hamiltonian; (ii) FZH(t) is the projection on the z-direction of a space-fixed frame of the dipole-moment of the system in the Heisenberg representation; (iii) p is the statistical matrix at time t = 0. The averages in the formulas (2), (3), (4) are computed over time t. The first three moments, as expressed above, are a generalization of JACOBSON'S moments.“) In our case, the spectrum to which the moments refer could be either a band or a line, with the restriction, however, that its extension in frequency, Aw say, is very small compared to some suitably defined central frequency. In our expressions, stimulated emission is contained and the influence of atomic interactions on the dipole-moment operator is taken into account. The latter is sometimes referred to as the “induced’ moment.‘” In a full quantum-mechanical theory, that is without the classical-path assumption, the moments are given, as shown by FUTRELLE,(~) by the following expressions:

MO= (I.+ ), M,

(3

= h-*(pL2/-t)- w&-b),

Mz = h-‘(@,‘r_c) - 2ooh-‘(pL*p)

(6) + wo’(@p),

(7)

where ( ) denotes the quantum-mechanical average and L is the Liouville operator defined by its action on an arbitrary operator o according to the relation Lo = [H, 01.

If we compare equations (2), (3), and (4) with the equations (5), (6), (7), we see that we have similar expressions for the moments M. and Ml.However, concerning the moment M,, there is in the classical-path theory one additional term which does not appear in the quantum theory. This term, which contains the second derivative of H(t), will be shown to be negligible for an isolated spectral line, due to the fact that the perturbation will be assumed to be small with respect to the unperturbed energy.

IILTHEMOMENTS

OF AN ISOLATED SPECTRAL LINE ATGENERALPRESSURES

We consider now the special case of an isolated line corresponding to a transition between two atomic levels i -3 f. We will take into consideration effects related to the degeneracy of these levels in the unperturbed state. However, we neglect stimulated emission and assume furthermore that the dipole-moment operator can be replaced by its unperturbed value. The latter approximation is commonly used in the literature, although it is not quite clear yet whether it is entirely justified. In order to write out the traces in expressions (2), (3), (4), we choose a complete set of functions

Ii) = laiJiMi),

If) = IcrrJ,M,),

Moment analysis of absorption profiles in terms of nnisotropic interatomic potentials

219

where ty and J are some principal and angular momentum quantum numbers respectively, and where the magnetic quantum number M refers to the space-fixed frame introduced above in connection with the definition of CL,. From now on, we take for tie the unperturbed frequency wif ; furthermore, we shall consider ali energy perturbations as small compared to fioj,. We now want to reduce our problem to a binary one and thus make the assumption that the contributions of the various perturbers to the interaction hamiltonian are all additive and that they are uncorrelated with respect to each other. Introducing the binary potentials V&M,,,V$+, which can be interpreted in the usual way as matrix elements of some ‘effective’ hamiltonian~~ we obtain the following results:

As stated above, pLiMfetc. are the unperturbed transition moments between states Ii}and /,f}.The averages that appear in equations (g), (91,(10) are taken over al1initial positions P of one perturber with respect to the active atom. The binary potential can be made diagonal in M if the internu&~eardirection is taken as the quanti~ati~n axis. In this case, it will be referred to as the adiabatic potential V%$, It can further be analyzed in terms of irreducibie tensor components, which leads to the following relation:‘” v Ma MM

1 =

T

A ";fC(JkJ; MOM),

0 6 k 6 ZJ,

k even.

(11)

Here J, M stand for .?&,&&,respectively. The coeffcients Ak are functions of the internuclear distance only. More explicitly, one has the relations

if one assumes that the initial state Ii>is isotropic (k equal 0 or 112)and that the final state If>has a quantum number Jr s 3/2. Considering the transformation of irreducibIe tensor components under rotations, we obtain for the binary potential in the fixed frame the relation vgM,

Z

x

k

QSRT Vol. 15 No. 3-B

A :fI%LXTM; M’qM),

k = 0,2.

(13)

220

E.

LEBOUCHER

and F.

SCHULLER

Here D$, represents an element of the matrix associated with the rotation of the fixed frame into the frame defined by the internuclear axis. We shall now express the potentials in equations (8), (9) (10) by means of equation (13) and we shall average over all possible orientations of the internuclear axis. Using the Wigner-Eckart theorem

we obtain for the first three moments

the following result:

(16)

x [N((Aof -A;)*),.

The averages ( )r involve the Boltzmann

(A: -AA),

+ A+A:~),

+ N(N-

l)(Ao’ -

&):I.

(17)

factor exp

?(A;

=hl-exp(-$$))x

-Ai)dr

etc.,

where v is the total volume occupied by the system. As we see, the anisotropy of the potential in the excited state leads to a contribution that appears in the moment of second order. Its value depends linearly on the perturber density. IV. NUMERICAL

RESULTS

As a numerical example, we tried to evaluate the moments for the case of the zP,,2 and zP1,2 components of the resonance doublet 7800/7947 A of Rb perturbed by Ne. From equation (12), the coefficients Ak as functions of the adiabatic potentials are given for the zP3,2component by

For the ‘Pliz component,

the coefficients

Ak are A; = V$!$;;“, (011 L/Z A: = V,&;/2 .

Moment analysis of absorption profiles in terms of anisotropic interatomic potentials

221

Using for the adiabatic potentials the data obtained theoretically by BAYLIS,@’we determined the averages in equations (15), (16), (17) by numerical integration. The results of our calculations, made at different temperatures, are shown in the figures. In Fig. 1, the first moment, which varies M%!! MO

is expressed

in KK

Fig. 1. The evolution of the moment M, with density n of the perturbers. The curves are plotted for T = 350°Kand T = 650°K and for the two components 2P,,,, 2P,,2 of the resonance doublet 7800/79478, of Rb perturbed by Ne.

Fig. 2. The evolution of the moment Mz with density n of perturbers. The curves are plotted for T = 350°K and T = 650°Kand for the two components 2P,,2, *PII of the resonance doublet 7800/7947Aof Rb perturbed by Ne.

222

E. LEBOUCHERand F. &HULLER

Fig. 3. The evolution of the integrals:

with temperature.

linearly with perturber density, is represented for two temperatures T = 350°K and T = 650°K. In Fig. 2, the second moment is plotted, for the same temperatures, as function of the perturber density. Up to densities of about 1020cm--3for the 2P,,2 component and lo*’ cm-j for the 2P1,2 component, the variation is essentially linear; at densities higher than 1022cm-3 for the zP,,z component and 10z4crne3 for the ‘PIIz component, the quadratic term is dominant. In Fig. 3, we studied the evolution of the two quantities yI = ((Ao(-AA)‘), and yz = (l/S)(Af), with temperature for the 2P,i2 component. In the temperature range considered, both variations are represented by fairly straight lines, which can be extrapolated so that they meet at the same point on the abscissa. As a consequence, the ratio between the anisotropic and the isotropic term in the linear part of Mz is independent of temperature. Furthermore, this ratio is slightly larger than unity. The latter fact signifies that, if anisotropy is taken into account, the values of M2 are considerably modified, at least in the region n < 1020cmm3. From this result, we conclude that it would not be quite correct to use orientation-dependent potentials in connection with an isotropic theory by introducing, say, the average value of those potentials. On the other hand, it might be possible, as suggested by JACOBSON:‘) to determine potential parameters from experimental values of the moments, at least in simple cases like that of the Lennard-Jones model. The same method can easily be extended to the case of anisotropic potentials. However, we do not consider this problem here since for absorption spectra no reliable experimental data on moments have as yet been obtained. REFERENCES 1. 2. 3. 4. 5. 6.

H. C. JACOBSON, Phys. Rev. 4, 1363 (1971); Phys. Rev. 4, 1368 (1971). S. BLOOMand H. MARGENAU,90,791 (1953). .I. GRANIERand R. GRANIER,JQSRT 13, 473 (1973). R. P. FUTRELLE,Phys. Rev. 5, 2162 (1972). J. P. FAROUX,Thesis, Paris (1969). W. E. BAYLIS,J. &em. Phys. 51, 2665 (1969).