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Collisional transfer of coherence by electric dipole-dipole interaction
W. Gough
To cite this version:
W. Gough. Collisional transfer of coherence by electric dipole-dipole interaction. Journal de Physique, 1983, 44 (3), pp.343-346. �10.1051/jphys:01983004403034300�. �jpa-00209603�
Collisional transfer of coherence by electric dipole-dipole interaction
W. Gough
Department of Physics, University College, P.O. box 78, Cardiff CF1 1XL, U.K.
(Reçu le 22 juillet 1982, accepte le 4 novembre 1982)
Résumé. - En s’appuyant sur les résultats d’une théorie de Chiu, on établit une expression pour la contribution de l’interaction dipôle-dipôle à l’intensité de la fluorescence sensibilisée. On utilise la méthode des opérateurs
tensoriels. On calcule le taux de polarisation dans certains cas particuliers.
Abstract. - An expression is derived for the contribution from dipole-dipole interaction to the intensity of sensi-
tized fluorescence, from the results of a theory by Chiu. Tensor operator methods are used. The degree of polariza-
tion is deduced for certain particular cases.
Classification Physics Abstracts
34.50
1. Introduction. - The phenomenon of sensitized fluorescence has received much attention as a mean
of investigating collisions between excited atoms of a vapour of species A with ground-state atoms of a vapour of a different species B (or A and B may be different isotopes of the same species).
The process can be represented by :
The main difficulty in performing a theoretical analysis is in treating the collision mechanism, but some
use has been made of simplified models [1-3]. Chiu has derived the contribution to the intensity of sensitized
fluorescence, in the general case, from electric dipole-dipole interaction between the atoms [4]. The result, for
white light excitation, when an external magnetic field HZ is applied along the quantization axis Oz is (eqs.
(2. ) and (1. 58))
where F2 is an accumulation of constants, rA and rB are the decay constants of A *(ja) and B*( jb) respectively,
and
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004403034300
344
In (1), the summation is over u, u’, w, v’, Ma, M’al mb, 4, Ma, Ma, Mb and Mb. In (2), the C’s are Clebsch-
Gordan coefficients. f and g (which are ---> and 4 in Chiu’s notation) represent the polarization of the incident and detected radiation. (Instead of g, as used by Chiu, we use g* to allow for the possibility of dealing with circularly or elliptically polarized light.) r(A) and r(Bj) are respectively the radius vectors of the ith electron of atom A and the j th electron of atom B. ru(Ai) is the uth component of r(A) expressed in the spherical basis (§ 2), with u = 0 or ± 1. rv-u(Bj) is similarly defined.
In this communication, we will show that (1) can be expressed very economically by the use of tensor
operators. Whereas (1) may contain many thousands of terms in the summation, the tensor expression contains only 9.
2. Expression for Is in tensor notation. - In deriving an expression for Is from (1) and (2), we absorb constant
factors (for example 2 Ja + 1) into the constants of proportionality, since we will not be concerned with the absolute magnitude of IS, but only its dependence on H, and the direction and polarization of the incident and fluorescent light.
Extensive use will be made of the properties of the Clebsch-Gordan coefficients C( jl j2 j; m1 m2 m)
and the Racah coefficients W(a b c d ; e 1) (see for example [5] chap. 3 and appendix I). Another important relationship is
f ( and other vectors) are expressed in the spherical basis, that is f± 1 =
+ r(fx 1
± i fy) fo =./z- f is complexif the incident light is circularly or elliptically polarized. It is readily shown that (/-J* = ( - 1 )" (/*)x and that
Consider first the excitation matrix element
From (4),
from the Wigner-Eckart theorem.
and using (3), we obtain
Here k = 0, 1 or 2, and q = - k, ..., k.
Now keeping ma and ma (and hence q) fixed, and summing over a (which implies summing over Ma and
a’ too), we obtain
Here we have introduced the tensor E, defined by
The negative sign has been introduced to accord with Happer and Saloman [6].
By a similar analysis, the decay matrix element
can be readily shown to be
We note in passing that the well-known Breit formula for the intensity of resonance fluorescence can be
expressed, from (5) and (6)
where hw zz gi J-lB Hz. J;, j and if refer to the initial, excited and final states respectively. (7) is derived by Happer
and Saloman, with the constant of proportionality specifically expressed.
To proceed with the evaluation of (2), one expresses the last four matrix elements in terms of Clebsch- Gordan coefficients using the Wigner-Eckart theorem, bearing in mind that ru(AJ etc. are components of rank 1
tensors in the spherical basis. The resulting expression contains a product of 8 Clebsch-Gordan coefficients which is summed over u, u’, v, ma, M’a, m, mb, Ma, Mb, k, kl, q and ql, subject to the conditions of non-vanishing
of the Clebsch-Gordan coefficients, namely q = ma - ma = u - u’ = mb - mb = qt. After some manipulation
of the Clebsch-Gordan coefficients, and application of their orthonormality condition, we finally obtain for
the intensity of the sensitized fluorescence
It should be noted that this expression applies to the « weak field » case, as does (7).
It is helpful to interpret the terms in the sum (8) as the product of 5 factors : (a) ( - 1)q E (’) 17 W(Ja ja 1 k ;1 ja)
represents the optical excitation of atoms A; (b) (rA + iqw A) - 1 the evolution of atoms A in the magnetic field ; (c) W(Ja ja 1 k 1 ja) W(2 Ilk; ;1 1) W(Jb jb 1 k 1 jib) the collisional transfer; (d) (f B + iq(OB) - ’ the evo-
lution of atoms B in the field; and (e) U(’) W(Jb jb 1 k;1 jb) the detection of fluorescence from atoms B. The collisional transfer factor (c) is proportional to the cross-section for the transfer of population (k = 0), orien-
tation (k = 1) or alignment (k = 2). Thus the ratios of these three cross-sections (i.e. g° : g 1 : g2 in the notation of Cheron [3]), can be deduced.
3. Consideration of particular cases. - Equations (7) and (8) can be conveniently expressed
In table I, Fr(k) and FS(k) are given for particular
values of J and j, for the case where the excited states
all have the same quantum number j, and the ground-
states all have the same quantum number J. Fr(O)
and Fs(O) have arbitrarily been put equal to 1. We see,
as is well known, that to observe the Hanle effect,
one must excite with, and detect, circularly (or ellipti- cally) polarized light in the case where J = j = t.
To evaluate I, and IS, one ascertains the values of
E,,Ik) and U(k) for any particular directions and states of polarization of the incident and fluorescent light.
E,(-) is closely related to the quantity a(,k) given by
346
Table I.
Baker and Gough [7] for linearly or circularly polariz- ed, or unpolarized, light. In fact E(k) (- 1)k+q ackq.
The U (k) ) are the same as the q with ø, cjJ and a
referring to the fluorescent radiation.
For example, Chiu [4] considers the case in zero
field, where the fluorescent light travels along Oz, and the incident radiation travels along Ox, and is polarized parallel to Oy. It is readily shown that the degrees of polarization of resonance and sensitized
fluorescence are respectively and
If these are applied to the cases given in table I, it is readily verified that P, and P,, are,
as expected, the same as given by Chiu. The depen-
dence of Ps upon an applied magnetic field Hz also
agrees with Chiu.
4. Conclusion. - It should be realized that the electrostatic interaction between two atoms will be
more complicated than dipole-dipole. The potential
energy of interaction between two non-overlapping arbitrary charge distributions I and II can be expanded [8] as a series in terms of the interaction between the 2L moments of I with the 2’ moments of II. For neutral
charge distributions, the leading term in the expansion
is dipole-dipole. The remaining terms are neglected
in Chiu’s analysis.
Consequently, Chiu’s theory gives completely
wrong values for the absolute collision cross-sections where the Born approximation is not valid. Likewise,
it is not to be expected that the theory gives accurate
results for the transfer of polarization. It does, however, give the contribution to this transfer arising from
weak collisions in which the dipole-dipole interaction is dominant, and does at least give a rough value
of the transfer of polarization to be expected. The
transfer rate constants have also been calculated, by numerical integration, by Carrington, Stacey and Cooper [9] for resonant dipole-dipole interaction between two atoms for 4 different combinations of J
and j.
The main purpose of this communication is, however, to show the power of tensor algebra in analysing problems of this kind in atomic collisions,
and to express in simple closed form the contribution from the leading term in the multipole-multipole expansion.
Acknowledgment. - The author wishes to thank Dr. A. R. Lee for valuable discussion and comments.
References
[1] GOUGH, W., Proc. Phys. Soc. 90 (1967) 287.
[2] CHÉRON, B., 4th E.G.A.S. Conference, Amsterdam (unpublished), 1972.
[3] CHÉRON, B., J. Physique 36 (1975) 17.
[4] CHIU, L. Y. C., Phys. Rev. A 5 (1972) 2053.
[5] ROSE, M. E., Elementary theory of angular momentum (Wiley) 1957.
[6] HAPPER, W. and SALOMAN, E. B., Phys. Rev. 160 (1967)
22.
[7] BAKER, R. S. and GOUGH, W., J. Phys. B 8 (1975) 552.
[8] ROSE, M. E., J. Math. and Phys. 37 (1958) 215.
[9] CARRINGTON, C. G., STACEY, D. N. and COOPER, J.
J. Phys. B 6 (1973) 417.
