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Line-strength of the 1.27-μm atmospheric transition of oxygen
Chemical Physics 323 (2006) 243–248 www.elsevier.com/locate/chemphys
Line-strength of the 1.27-lm atmospheric transition of oxygen G. Di Stefano
*
Istituto di Metodologie Inorganiche e dei Plasmi – IMIP, C.N.R., P.O. Box 10, Area della Ricerca di Roma, Via Salaria km 29,300, 00016 Monterotondo Scalo, Roma, Italy Received 6 May 2005; accepted 13 September 2005 Available online 26 October 2005
Abstract The line-strength structure of the molecular oxygen near-IR atmospheric transition a 1 Dg ðv0 ¼ 0Þ X 3 Rg ðv00 ¼ 0Þ is deduced from experiment. Deviations from the expected structure, in terms of theoretical expressions, are put in evidence. As previously shown for the A band [G. Di Stefano, Chem. Phys. 302 (2004) 243], new expressions leading to a satisfactory agreement with the experiment are suited to a defined full (b) case of Hund, as a subsequent one to well-known intermediate (a–b) case. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Line-strength; Gas; Diatomics
1. Introduction The a 1Dg and b 1 Rþ g states of molecular oxygen are of a primary importance in the chemistry and biochemistry of environment. In fact, their chemical and photochemical behavior is a nodal point in oxygenÕs activity, for example in many atmospheric phenomena of widespread interest like ozone equilibria [1,2], day [3] – and night [4] – light, aurorae [5], etc. In a recent study [6] on the line-strength of (0–0) band 3 of O2 ðb 1 Rþ g –X Rg Þ atmospheric transition in the visible, also known as the A band, it has been shown that wellknown theoretical functions [7] were only partially following the experimental outcome. On the contrary, well fitting new functions were deduced for a so-defined full (b) case of Hund, with WatsonÕs expressions [7] attributed to an (a–b) intermediate case not occurring in that system. If this is the case, the question with real systems could be that of finding out which case of Hund applies out of two. In this connection, PH and PD ðb 1 Rþ –X 3 R Þ (0,0) bands, which in a limited set of rotational lines happen to show both line-strength structures, of (a–b) kind at ini*
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tial J values and full (b) at higher [8,9], are worth to be remembered. A further point of interest here is given by the general 3 outline of O2 ðb 1 Rþ g –X Rg Þ line-strength structure as deduced from experiment [6], i.e., with all four branches distributed on two straight lines. This behavior has suggested the idea [6] of no presence of side effects from fine structure rather sizable magnitude, because of the one component – perpendicular [7] – character of transition. This study on the near-IR (0–0) band of O2 ða 1 Dg –X 3 R gÞ atmospheric transition follows that on A band cited above, and, owing to the one component (still perpendicular) character of this transition [10], it is also a due test on what there proposed about fine structure side effects. So, in the following sections both the (a–b) and the full (b) HundÕs cases linestrength functions are given, and compared with the experimental line-strength structure of O2 ða 1 Dg –X 3 R g Þ magnetic dipole transition by checking the branches correlation. Intensity data, by Amiot and Verges [11], had been obtained in a flowing afterglow apparatus coupled with Fourier Transform analysis. Similar techniques, when applied to isoelectronic NF radical by other authors [12], had previously shown in a very good agreement with methods adopted in this laboratory [13], particularly at low J. Data are used here as reported in [10, Table 2], where also a com-
244
G. Di Stefano / Chemical Physics 323 (2006) 243–248
parison with the (a–b) HundÕs case line-strength functions (this paper notation) is carried out together with parametric adjustments. In the next section, the theoretical expression of linestrength for the (a–b) and the full (b) cases of Hund will be discussed. 2. Discussion The basic expression, relating signal intensity of spontaneous emission to transition probability, is the radiation intensity distribution law [14] at a thermal equilibrium in an isotropic medium: Iðe0 v0 J 0 ; e00 v00 J 00 Þ / m4J 0 J 00 S J 0 J 00 exp½F 0 ðJ 0 Þhc=kT ;
ð1Þ
with e and v labeling electronic and vibrational states, and not explicit in the right-hand side. In (1), mJ 0 J 00 is the radiation frequency and F 0 (J 0 ) in Boltzmann exponential factor the rotational energy term of emitting level. The linestrength S J 0 J 00 is the required single particle quantity within signal intensity. Spin forbidden transitions may acquire strength because of spin tilting by rotational perturbation, according to theory [7,15]. This has allowed the use of perturbation techniques in the calculations, and a classification of HundÕs cases amenable to rotational and spin–orbit equilibria. The effective spin-rotation Hamiltonian for 3 R states in HundÕs (b) case is [7]:
be maintained in A band treatment because only the perpendicular share of transition is present [7]. In other words, since the fine structure is obviously limited by X 3 R g0 and X 3 R g1 levels in the ground state, k magnitude has been said [6] to be not relevant for transition mechanisms because of absence of X 3 R g0 level. This argument is suggested here again for the O2 ða 1 Dg –X 3 R g Þ system, being also only featured by perpendicular component [10,16] of transition moment. An identical procedure is thus adopted for deduction of line-strength functions. The O2 ða 1 Dg –X 3 R g Þ transition is featured by 9 branches [14], whose line-strength functions have been calculated [10,16] according to the Watson method. The general expressions can either be found in [16, Table 2], and in [10, Table 1] (in the latter with rovibronic corrections not taken into account here), and are reported in Table 1. In these, WatsonÕs cJ and sJ factors are the mixing coefficients of ground state triplet terms by rotational perturbation. Coefficients relations are: c2J ¼ ðF 2 F 1 Þ=ðF 3 F 1 Þ;
ð4Þ
s2J ¼ ðF 3 F 2 Þ=ðF 3 F 1 Þ.
Putting k ffi 0 in (3), well-known expressions [7] are obtained: c2J ¼ J =ð2J þ 1Þ;
ð5Þ
s2J ¼ ðJ þ 1Þ=ð2J þ 1Þ.
where N J S, J and S being, respectively, the total angular momentum and the spin in operator form. B is the rotational constant of the molecule, and c its spin-rotation coupling constant. It has to be reminded that k, which is the fine structure parameter, in HundÕs (b) case is supposed to be small with respect to rotational energy. If this is the case, the eigenvalues can be well approximated by simple forms [7,14]:
Substitution in general expressions of Table 1 leads to the forecast of line-strength functions for O2 (a–X) branches, in the last column of Table 1. New coefficients and formulas, on the other hand, can be obtained remembering that N is also a good quantum number for rotations in the full (b) case of Hund. This is accomplished by a straightforward application of the J () N correspondences J = N + 1, N and N 1 to F1, F2 and F3 terms, respectively [6,8]. With still k ffi 0 in (3), and by application of (5), the new coefficients are readily deduced [6,8]:
F 1 ðJ Þ ¼ F 2 ðJ Þ k þ k=ð2J þ 1Þ ð2B cÞJ ;
c2N ¼ ðN þ 1Þ=ð2N þ 1Þ;
2
H ¼ BN þ cN S þ
2kS 2z ;
ð2Þ
F 2 ðJ Þ ¼ BJ ðJ þ 1Þ þ 2k c; F 3 ðJ Þ ¼ F 2 ðJ Þ k k=ð2J þ 1Þ ð2B cÞðJ þ 1Þ.
ð3Þ
In oxygen, however, this condition is not fulfilled at low rotations since it is B ffi 1.4 cm1 and k ffi 1.985 cm1. In spite of this, the relation k ffi 0 has been suggested [6] to
ð6Þ
s2N ¼ N =ð2N þ 1Þ.
In Table 2, the general expressions for 1 D–3 R magnetic dipole transitions in the full (b) case are reported, together with O2 (a–X) functions in both N and J equivalent forms, according to J () N correspondences and (6).
Table 1 Line-strength functions for the O2 ða 1 Dg –X 3 R g Þ system in the (a–b) case of Hund Type
Branch
HundÕs (a–b) case general expression [10,16]
HundÕs (a–b) case line-strength function
F3 F3 F3 F2 F2 F2 F1 F1 F1
O
c2J ðJ 2ÞðJ 1Þ=½2J c2J ðJ 1ÞðJ þ 2Þð2J þ 1Þ=½2J ðJ þ 1Þ c2J ðJ þ 2ÞðJ þ 3Þ=½2ðJ þ 1Þ (J 2)(J 1)/[2J] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (J + 2)(J + 3)/[2(J + 1)] s2J ðJ 2ÞðJ 1Þ=½2J s2J ðJ 1ÞðJ þ 2Þð2J þ 1Þ=½2J ðJ þ 1Þ s2J ðJ þ 2ÞðJ þ 3Þ=½2ðJ þ 1Þ
(J 2) (J 1)/[2(2J + 1)] (J 1)(J + 2)/[2(J + 1)] J(J + 2)(J + 3)/[2(J + 1)(2J + 1)] (J 2)(J 1)/[2J] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (J + 2)(J + 3)/[2(J + 1)] (J 2)(J 1)(J + 1)/[2J(2J + 1)] (J 1)(J + 2)/[2J] (J + 2)(J + 3)/[2(2J + 1)]
P(J) P Q(J) Q R(J) P P(J) Q Q(J) R R(J) Q P(J) R Q(J) S R(J)
G. Di Stefano / Chemical Physics 323 (2006) 243–248
245
Table 2 Line-strength functions for the O2 ða 1 Dg –X 3 R g Þ system in the full (b) case of Hund expressed both by N and by J variable Type
Branch
HundÕs full (b) case general expression
HundÕs full (b) case line-strength function
F3
Q
P(N)
c2N 1 ðN
F3
P
Q(N)
c2N 1 ðN 2ÞðN þ 1Þð2N 1Þ=½2N ðN 1Þ
F3
Q
F2
P
F2
Q
Q(N)
(N 1)(N + 2)(2N + 1)/[2N(N + 1)]
F2
R
R(N)
(N + 2)(N + 3)/[2(N + 1)]
F1
Q
P(N)
s2N þ1 N ðN 1Þ=½2ðN þ 1Þ
F1
R
Q(N)
s2N þ1 N ðN þ 3Þð2N þ 3Þ=½2ðN þ 1ÞðN þ 2Þ
F1
S
R(N)
s2N þ1 ðN þ 3ÞðN þ 4Þ=½2ðN þ 2Þ
N(N 2)(N 3)/[2(N 1)(2N 1)] (J 1)(J 2)(J+1)/[2J(2J + 1)] (N 2)(N + 1)/[2(N 1)] (J 1)(J + 2)/[2J] (N + 1)(N + 2)/[2(2N 1)] (J + 2)(J + 3)/[2(2J + 1)] (N 2)(N 1)/[2N] (J 2)(J 1)/[2J] (N 1)(N + 2)(2N + 1)/[2N(N + 1)] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (N + 2)(N + 3)/[2(N + 1)] (J + 2)(J + 3)/[2(J + 1)] N(N 1)/[2(2N + 3)] (J 1)(J 2)/[2(2J + 1)] N(N + 3)/[2(N + 2)] (J 1)(J + 2)/[2(J + 1)] (N + 1)(N + 3)(N + 4)/[2(N + 2)(2N + 3)] J(J + 2)(J + 3)/[2(J + 1)(2J + 1)]
3ÞðN 2Þ=½2ðN 1Þ
R(N)
c2N 1 ðN þ 1ÞðN þ 2Þ=½2N
P(N)
(N 2)(N 1)/[2N]
Given the complexity of this transition system, several line-strength correlations between branches will be shown in the next section, with theoretical models above checked by the experiment. 3. Results In this section, the experimental values of the branches line-strength, as deduced by use of (1) from the data of Table 2 of [10] with referred temperature of 325.2 K (ffi226 cm1), are shown in correlation. Comparisons with respective theoretical functions of either (a–b) and full (b) HundÕs cases (this paper notations) are also made, in order to find out which one applies better to the O2 ða1 Dg –X 3 R gÞ real system. In all figures, one branch is taken for reference, by superposing its experimental values and theoretical
function(s), as best way to compare correlations. When both (a–b) and full (b) HundÕs cases are shown, functions of reference branch are superposed by a small relative shift in the ordinate scale of arbitrary units. To begin with, in Fig. 1 the experimental line-strengths of PP, QQ, and RR unperturbed branches (of F2 kind) are shown with theoretical functions, which are obviously identical in (a–b) and full (b) HundÕs cases (see Tables 1 and 2). It is clearly seen that the experiment is very close to theoretical prevision at lower J lines, which are featured by better statistics. Some further datum has been plotted, also in following figures, for comparison. It has to be added besides that no previous examples have been seen, to my knowledge, of strongly diverging trends in the branches line-strength of diatomics forbidden transitions [17].
(a-b)
arb.unit
arb.unit
(b)
0
5
10
15
20
25
J P
Q
R
Fig. 1. Experimental line-strengths of P (j), Q (s) (reference), and R (d) systems, with theoretical functions (solid lines).
0
5
10
15
20
25
J Fig. 2. Experimental line-strengths of QP (s) (reference), and QR (d) branches, with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
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G. Di Stefano / Chemical Physics 323 (2006) 243–248
In Fig. 2, the theoretical and experimental line-strengths of QP and QR branches are reported. A correlation interval of DJ 8 units either in full (b) HundÕs case line-strength functions and experiment, and of DJ 6 in (a–b) HundÕs case, can be noted. In Fig. 3, the experimental line-strengths of RQ and PQ branches are shown to correlate like respective full (b) HundÕs case line-strength functions, i.e., DJ 1 unit with P Q higher. In (a–b) HundÕs case, the line-strength functions are reversed, i.e., RQ is higher (see Tables 1 and 2). This discrepancy with data has been illustrated in [10], and not
surmounted by parametric adjustments, authors acknowledge. In this connection, it is worth underlining that a similar functional exchange, still making PQ to be higher than RQ, has been shown [6] to occur also in the A band. In Fig. 4, the QR and SR scenario is qualitatively similar to that of Fig. 3, though one branch is clearly featured by a worse statistics. In any case, the experimental values of QR branch seem to lie higher than those of SR, as in respective full (b) HundÕs case line-strength functions. Like in the branches of Fig. 3, line-strength functions in (a–b) HundÕs case are reversed, i.e., SR is higher (see Tables 1 and 2).
(a-b)
arb.unit
arb.unit
(b)
0
5
10
15
20
0
25
5
10
J
15
20
25
J Fig. 5. Experimental line-strengths of QP (d) (reference), and SR (s) branches. Respective (a–b) and full (b) HandÕs case theoretical functions (solid lines) are indicated.
arb.unit
arb.unit
Fig. 3. Experimental line-strengths of RQ (h) (reference), and PQ (j) branches, with respective full (b) HandÕs case theoretical functions (solid lines). These functions in (a–b) HandÕs case are exchanged.
(a-b) (b)
0
5
10
15
20
25
J Fig. 4. Experimental line-strengths of QR (j) (reference), and SR (h) branches with respective full (b) HandÕs case theoretical functions (solid lines). These functions in (a–b) HandÕs case are exchanged.
0
5
10
J
15
20
25
Fig. 6. Experimental line-strengths of SR (h) and PQ (j) (reference) branches with respective (a–b) and full (b) HandÕs case functions (solid lines).
In Fig. 5, the line-strengths of QP and SR branches are reported. The difference between SR line-strength functions of (a–b) and full (b) HundÕs cases is dramatic at very initial J values, with the full (b) HundÕs case (DJ 6 units) in a high accord with the experiment. In Fig. 6, the experimental and theoretical line-strengths of PQ and SR branches are shown. A clear difference, still diverging at initial J (see Fig. 5), exists between (a–b) and full (b) HundÕs cases line-strength functions, with full (b) very close to experiment.
(a-b)
(a-b)
0
(b)
247
arb.unit
G. Di Stefano / Chemical Physics 323 (2006) 243–248
5
10
(b)
15
20
25
J
arb.unit
Fig. 9. Experimental line-strengths of QQ (s) (reference) and RQ (d) with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
0
5
10
15
20
25
J
Fig. 7. Experimental line-strengths of QR (j) (reference) and RQ (h) branches with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
(a-b)
arb.unit
(b)
In Fig. 7, the experimental and theoretical line-strengths of QR and RQ branches are reported. Full (b) HundÕs case is in a complete agreement with the experiment. In Fig. 8, the experimental and theoretical line-strengths of OP and SR branches are reported. A clear difference between SR functions of (a–b) and full (b) HundÕs cases is dramatic at initial J values, like in Fig. 5, with full (b) HundÕs case (DJ 6 units) well fitting the experiment. No other cases of meaningful differences between theory and experiment, as well as between (a–b) and full (b) HundÕs cases line-strength functions, have been found: as given examples, the RQ–SR, OP–PQ, QP–RQ, OP–QR, Q Q–RQ systems can be quoted. The latter is shown in Fig. 9, where a clear choice between (a–b) and full (b) HundÕs cases is beyond resolution. According to these results, the full (b) HundÕs case linestrength functions, as deduced by use of (3) putting the fine structure parameter k ffi 0, plainly show more suitable for a description of this near-IR atmospheric transition, and more in general of the radiative behaviour of molecular oxygen in its lowest states, probably. However, more data are desirable, and new experiments on oxygenÕs linestrength should be welcome. 4. Conclusions
0
5
10
15
20
25
J
Fig. 8. Experimental line-strengths of QP (h) (reference) and SR (j) branches with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
Intensity data of near-IR atmospheric forbidden transition of the oxygen molecule have been studied with the same quantitative method (of the line-strength correlation of branches) recently applied to those of the visible atmospheric forbidden transition. Conclusions are similar, together with an extension of the reference theory leading to new line-strength functions. Data of this kind are sparse, though clearly helpful in such crucial areas as atmospheric remote sensing. These quantitative techniques may allow
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G. Di Stefano / Chemical Physics 323 (2006) 243–248
observations of a very high quality for diatomics, and a satisfactory interpretation of available data. References [1] R.L. Miller, A.G. Suits, P.L. Houston, R. Toumi, J.A. Mack, A.M. Wodtke, Science 265 (1994) 1831. [2] R.J. Sica, R.P. Lowe, J. Geophys. Res. 98 (1993) 1051. [3] L. Wallace, J.W. Chamberlain, Planet. Space Sci. 2 (1958) 60. [4] M.R. Torr, D.G. Torr, R.R. Laher, J. Geophys. Res. 90 (1985) 8525. [5] R.L. Gattinger, A.J. Vallance-Jones, J. Geophys. Res. 81 (1976) 4789. [6] G. Di Stefano, Chem. Phys. 302 (2004) 243. [7] J.K.G. Watson, Can. J. Phys. 46 (1968) 1637.
[8] G. Di Stefano, M. Lenzi, G. Piciacchia, A. Ricci, Chem. Phys. 165 (1992) 201. [9] G. Di Stefano, M. Lenzi, A. Ricci, Chem. Phys. 246 (1999) 267. [10] V.P. Bellary, T.K. Balasubramaniam, J. Mol. Spectrosc. 126 (1987) 436. [11] C. Amiot, J. Verges, Can. J. Phys. 59 (1981) 1391. [12] M. Vervloet et, J.K.G. Watson, Can. J. Phys. 64 (1986) 1529. [13] G. Di Stefano, M. Lenzi, G. Piciacchia, A. Ricci, J. Chem. Phys. 107 (1997) 2752. [14] G. Herzberg, Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules, Van Nostrand, Princeton, 1950. [15] H. Lefebvre-Brion, R.W. Field, Perturbations in the Spectra of Diatomic Molecules, Academic Press, New York, 1986. [16] T.K. Balasubramaniam, V.P. Bellary, Acta Phys. Hung. 63 (1988) 249. [17] G. Di Stefano, Trends Chem. Phys. 10 (2002) 63.
Line-strength of the 1.27-lm atmospheric transition of oxygen G. Di Stefano
*
Istituto di Metodologie Inorganiche e dei Plasmi – IMIP, C.N.R., P.O. Box 10, Area della Ricerca di Roma, Via Salaria km 29,300, 00016 Monterotondo Scalo, Roma, Italy Received 6 May 2005; accepted 13 September 2005 Available online 26 October 2005
Abstract The line-strength structure of the molecular oxygen near-IR atmospheric transition a 1 Dg ðv0 ¼ 0Þ X 3 Rg ðv00 ¼ 0Þ is deduced from experiment. Deviations from the expected structure, in terms of theoretical expressions, are put in evidence. As previously shown for the A band [G. Di Stefano, Chem. Phys. 302 (2004) 243], new expressions leading to a satisfactory agreement with the experiment are suited to a defined full (b) case of Hund, as a subsequent one to well-known intermediate (a–b) case. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Line-strength; Gas; Diatomics
1. Introduction The a 1Dg and b 1 Rþ g states of molecular oxygen are of a primary importance in the chemistry and biochemistry of environment. In fact, their chemical and photochemical behavior is a nodal point in oxygenÕs activity, for example in many atmospheric phenomena of widespread interest like ozone equilibria [1,2], day [3] – and night [4] – light, aurorae [5], etc. In a recent study [6] on the line-strength of (0–0) band 3 of O2 ðb 1 Rþ g –X Rg Þ atmospheric transition in the visible, also known as the A band, it has been shown that wellknown theoretical functions [7] were only partially following the experimental outcome. On the contrary, well fitting new functions were deduced for a so-defined full (b) case of Hund, with WatsonÕs expressions [7] attributed to an (a–b) intermediate case not occurring in that system. If this is the case, the question with real systems could be that of finding out which case of Hund applies out of two. In this connection, PH and PD ðb 1 Rþ –X 3 R Þ (0,0) bands, which in a limited set of rotational lines happen to show both line-strength structures, of (a–b) kind at ini*
Tel.: +39 06 90672215; fax: +39 06 90672238. E-mail address: [email protected].
0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.09.021
tial J values and full (b) at higher [8,9], are worth to be remembered. A further point of interest here is given by the general 3 outline of O2 ðb 1 Rþ g –X Rg Þ line-strength structure as deduced from experiment [6], i.e., with all four branches distributed on two straight lines. This behavior has suggested the idea [6] of no presence of side effects from fine structure rather sizable magnitude, because of the one component – perpendicular [7] – character of transition. This study on the near-IR (0–0) band of O2 ða 1 Dg –X 3 R gÞ atmospheric transition follows that on A band cited above, and, owing to the one component (still perpendicular) character of this transition [10], it is also a due test on what there proposed about fine structure side effects. So, in the following sections both the (a–b) and the full (b) HundÕs cases linestrength functions are given, and compared with the experimental line-strength structure of O2 ða 1 Dg –X 3 R g Þ magnetic dipole transition by checking the branches correlation. Intensity data, by Amiot and Verges [11], had been obtained in a flowing afterglow apparatus coupled with Fourier Transform analysis. Similar techniques, when applied to isoelectronic NF radical by other authors [12], had previously shown in a very good agreement with methods adopted in this laboratory [13], particularly at low J. Data are used here as reported in [10, Table 2], where also a com-
244
G. Di Stefano / Chemical Physics 323 (2006) 243–248
parison with the (a–b) HundÕs case line-strength functions (this paper notation) is carried out together with parametric adjustments. In the next section, the theoretical expression of linestrength for the (a–b) and the full (b) cases of Hund will be discussed. 2. Discussion The basic expression, relating signal intensity of spontaneous emission to transition probability, is the radiation intensity distribution law [14] at a thermal equilibrium in an isotropic medium: Iðe0 v0 J 0 ; e00 v00 J 00 Þ / m4J 0 J 00 S J 0 J 00 exp½F 0 ðJ 0 Þhc=kT ;
ð1Þ
with e and v labeling electronic and vibrational states, and not explicit in the right-hand side. In (1), mJ 0 J 00 is the radiation frequency and F 0 (J 0 ) in Boltzmann exponential factor the rotational energy term of emitting level. The linestrength S J 0 J 00 is the required single particle quantity within signal intensity. Spin forbidden transitions may acquire strength because of spin tilting by rotational perturbation, according to theory [7,15]. This has allowed the use of perturbation techniques in the calculations, and a classification of HundÕs cases amenable to rotational and spin–orbit equilibria. The effective spin-rotation Hamiltonian for 3 R states in HundÕs (b) case is [7]:
be maintained in A band treatment because only the perpendicular share of transition is present [7]. In other words, since the fine structure is obviously limited by X 3 R g0 and X 3 R g1 levels in the ground state, k magnitude has been said [6] to be not relevant for transition mechanisms because of absence of X 3 R g0 level. This argument is suggested here again for the O2 ða 1 Dg –X 3 R g Þ system, being also only featured by perpendicular component [10,16] of transition moment. An identical procedure is thus adopted for deduction of line-strength functions. The O2 ða 1 Dg –X 3 R g Þ transition is featured by 9 branches [14], whose line-strength functions have been calculated [10,16] according to the Watson method. The general expressions can either be found in [16, Table 2], and in [10, Table 1] (in the latter with rovibronic corrections not taken into account here), and are reported in Table 1. In these, WatsonÕs cJ and sJ factors are the mixing coefficients of ground state triplet terms by rotational perturbation. Coefficients relations are: c2J ¼ ðF 2 F 1 Þ=ðF 3 F 1 Þ;
ð4Þ
s2J ¼ ðF 3 F 2 Þ=ðF 3 F 1 Þ.
Putting k ffi 0 in (3), well-known expressions [7] are obtained: c2J ¼ J =ð2J þ 1Þ;
ð5Þ
s2J ¼ ðJ þ 1Þ=ð2J þ 1Þ.
where N J S, J and S being, respectively, the total angular momentum and the spin in operator form. B is the rotational constant of the molecule, and c its spin-rotation coupling constant. It has to be reminded that k, which is the fine structure parameter, in HundÕs (b) case is supposed to be small with respect to rotational energy. If this is the case, the eigenvalues can be well approximated by simple forms [7,14]:
Substitution in general expressions of Table 1 leads to the forecast of line-strength functions for O2 (a–X) branches, in the last column of Table 1. New coefficients and formulas, on the other hand, can be obtained remembering that N is also a good quantum number for rotations in the full (b) case of Hund. This is accomplished by a straightforward application of the J () N correspondences J = N + 1, N and N 1 to F1, F2 and F3 terms, respectively [6,8]. With still k ffi 0 in (3), and by application of (5), the new coefficients are readily deduced [6,8]:
F 1 ðJ Þ ¼ F 2 ðJ Þ k þ k=ð2J þ 1Þ ð2B cÞJ ;
c2N ¼ ðN þ 1Þ=ð2N þ 1Þ;
2
H ¼ BN þ cN S þ
2kS 2z ;
ð2Þ
F 2 ðJ Þ ¼ BJ ðJ þ 1Þ þ 2k c; F 3 ðJ Þ ¼ F 2 ðJ Þ k k=ð2J þ 1Þ ð2B cÞðJ þ 1Þ.
ð3Þ
In oxygen, however, this condition is not fulfilled at low rotations since it is B ffi 1.4 cm1 and k ffi 1.985 cm1. In spite of this, the relation k ffi 0 has been suggested [6] to
ð6Þ
s2N ¼ N =ð2N þ 1Þ.
In Table 2, the general expressions for 1 D–3 R magnetic dipole transitions in the full (b) case are reported, together with O2 (a–X) functions in both N and J equivalent forms, according to J () N correspondences and (6).
Table 1 Line-strength functions for the O2 ða 1 Dg –X 3 R g Þ system in the (a–b) case of Hund Type
Branch
HundÕs (a–b) case general expression [10,16]
HundÕs (a–b) case line-strength function
F3 F3 F3 F2 F2 F2 F1 F1 F1
O
c2J ðJ 2ÞðJ 1Þ=½2J c2J ðJ 1ÞðJ þ 2Þð2J þ 1Þ=½2J ðJ þ 1Þ c2J ðJ þ 2ÞðJ þ 3Þ=½2ðJ þ 1Þ (J 2)(J 1)/[2J] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (J + 2)(J + 3)/[2(J + 1)] s2J ðJ 2ÞðJ 1Þ=½2J s2J ðJ 1ÞðJ þ 2Þð2J þ 1Þ=½2J ðJ þ 1Þ s2J ðJ þ 2ÞðJ þ 3Þ=½2ðJ þ 1Þ
(J 2) (J 1)/[2(2J + 1)] (J 1)(J + 2)/[2(J + 1)] J(J + 2)(J + 3)/[2(J + 1)(2J + 1)] (J 2)(J 1)/[2J] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (J + 2)(J + 3)/[2(J + 1)] (J 2)(J 1)(J + 1)/[2J(2J + 1)] (J 1)(J + 2)/[2J] (J + 2)(J + 3)/[2(2J + 1)]
P(J) P Q(J) Q R(J) P P(J) Q Q(J) R R(J) Q P(J) R Q(J) S R(J)
G. Di Stefano / Chemical Physics 323 (2006) 243–248
245
Table 2 Line-strength functions for the O2 ða 1 Dg –X 3 R g Þ system in the full (b) case of Hund expressed both by N and by J variable Type
Branch
HundÕs full (b) case general expression
HundÕs full (b) case line-strength function
F3
Q
P(N)
c2N 1 ðN
F3
P
Q(N)
c2N 1 ðN 2ÞðN þ 1Þð2N 1Þ=½2N ðN 1Þ
F3
Q
F2
P
F2
Q
Q(N)
(N 1)(N + 2)(2N + 1)/[2N(N + 1)]
F2
R
R(N)
(N + 2)(N + 3)/[2(N + 1)]
F1
Q
P(N)
s2N þ1 N ðN 1Þ=½2ðN þ 1Þ
F1
R
Q(N)
s2N þ1 N ðN þ 3Þð2N þ 3Þ=½2ðN þ 1ÞðN þ 2Þ
F1
S
R(N)
s2N þ1 ðN þ 3ÞðN þ 4Þ=½2ðN þ 2Þ
N(N 2)(N 3)/[2(N 1)(2N 1)] (J 1)(J 2)(J+1)/[2J(2J + 1)] (N 2)(N + 1)/[2(N 1)] (J 1)(J + 2)/[2J] (N + 1)(N + 2)/[2(2N 1)] (J + 2)(J + 3)/[2(2J + 1)] (N 2)(N 1)/[2N] (J 2)(J 1)/[2J] (N 1)(N + 2)(2N + 1)/[2N(N + 1)] (J 1)(J + 2)(2J + 1)/[2J(J + 1)] (N + 2)(N + 3)/[2(N + 1)] (J + 2)(J + 3)/[2(J + 1)] N(N 1)/[2(2N + 3)] (J 1)(J 2)/[2(2J + 1)] N(N + 3)/[2(N + 2)] (J 1)(J + 2)/[2(J + 1)] (N + 1)(N + 3)(N + 4)/[2(N + 2)(2N + 3)] J(J + 2)(J + 3)/[2(J + 1)(2J + 1)]
3ÞðN 2Þ=½2ðN 1Þ
R(N)
c2N 1 ðN þ 1ÞðN þ 2Þ=½2N
P(N)
(N 2)(N 1)/[2N]
Given the complexity of this transition system, several line-strength correlations between branches will be shown in the next section, with theoretical models above checked by the experiment. 3. Results In this section, the experimental values of the branches line-strength, as deduced by use of (1) from the data of Table 2 of [10] with referred temperature of 325.2 K (ffi226 cm1), are shown in correlation. Comparisons with respective theoretical functions of either (a–b) and full (b) HundÕs cases (this paper notations) are also made, in order to find out which one applies better to the O2 ða1 Dg –X 3 R gÞ real system. In all figures, one branch is taken for reference, by superposing its experimental values and theoretical
function(s), as best way to compare correlations. When both (a–b) and full (b) HundÕs cases are shown, functions of reference branch are superposed by a small relative shift in the ordinate scale of arbitrary units. To begin with, in Fig. 1 the experimental line-strengths of PP, QQ, and RR unperturbed branches (of F2 kind) are shown with theoretical functions, which are obviously identical in (a–b) and full (b) HundÕs cases (see Tables 1 and 2). It is clearly seen that the experiment is very close to theoretical prevision at lower J lines, which are featured by better statistics. Some further datum has been plotted, also in following figures, for comparison. It has to be added besides that no previous examples have been seen, to my knowledge, of strongly diverging trends in the branches line-strength of diatomics forbidden transitions [17].
(a-b)
arb.unit
arb.unit
(b)
0
5
10
15
20
25
J P
Q
R
Fig. 1. Experimental line-strengths of P (j), Q (s) (reference), and R (d) systems, with theoretical functions (solid lines).
0
5
10
15
20
25
J Fig. 2. Experimental line-strengths of QP (s) (reference), and QR (d) branches, with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
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G. Di Stefano / Chemical Physics 323 (2006) 243–248
In Fig. 2, the theoretical and experimental line-strengths of QP and QR branches are reported. A correlation interval of DJ 8 units either in full (b) HundÕs case line-strength functions and experiment, and of DJ 6 in (a–b) HundÕs case, can be noted. In Fig. 3, the experimental line-strengths of RQ and PQ branches are shown to correlate like respective full (b) HundÕs case line-strength functions, i.e., DJ 1 unit with P Q higher. In (a–b) HundÕs case, the line-strength functions are reversed, i.e., RQ is higher (see Tables 1 and 2). This discrepancy with data has been illustrated in [10], and not
surmounted by parametric adjustments, authors acknowledge. In this connection, it is worth underlining that a similar functional exchange, still making PQ to be higher than RQ, has been shown [6] to occur also in the A band. In Fig. 4, the QR and SR scenario is qualitatively similar to that of Fig. 3, though one branch is clearly featured by a worse statistics. In any case, the experimental values of QR branch seem to lie higher than those of SR, as in respective full (b) HundÕs case line-strength functions. Like in the branches of Fig. 3, line-strength functions in (a–b) HundÕs case are reversed, i.e., SR is higher (see Tables 1 and 2).
(a-b)
arb.unit
arb.unit
(b)
0
5
10
15
20
0
25
5
10
J
15
20
25
J Fig. 5. Experimental line-strengths of QP (d) (reference), and SR (s) branches. Respective (a–b) and full (b) HandÕs case theoretical functions (solid lines) are indicated.
arb.unit
arb.unit
Fig. 3. Experimental line-strengths of RQ (h) (reference), and PQ (j) branches, with respective full (b) HandÕs case theoretical functions (solid lines). These functions in (a–b) HandÕs case are exchanged.
(a-b) (b)
0
5
10
15
20
25
J Fig. 4. Experimental line-strengths of QR (j) (reference), and SR (h) branches with respective full (b) HandÕs case theoretical functions (solid lines). These functions in (a–b) HandÕs case are exchanged.
0
5
10
J
15
20
25
Fig. 6. Experimental line-strengths of SR (h) and PQ (j) (reference) branches with respective (a–b) and full (b) HandÕs case functions (solid lines).
In Fig. 5, the line-strengths of QP and SR branches are reported. The difference between SR line-strength functions of (a–b) and full (b) HundÕs cases is dramatic at very initial J values, with the full (b) HundÕs case (DJ 6 units) in a high accord with the experiment. In Fig. 6, the experimental and theoretical line-strengths of PQ and SR branches are shown. A clear difference, still diverging at initial J (see Fig. 5), exists between (a–b) and full (b) HundÕs cases line-strength functions, with full (b) very close to experiment.
(a-b)
(a-b)
0
(b)
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G. Di Stefano / Chemical Physics 323 (2006) 243–248
5
10
(b)
15
20
25
J
arb.unit
Fig. 9. Experimental line-strengths of QQ (s) (reference) and RQ (d) with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
0
5
10
15
20
25
J
Fig. 7. Experimental line-strengths of QR (j) (reference) and RQ (h) branches with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
(a-b)
arb.unit
(b)
In Fig. 7, the experimental and theoretical line-strengths of QR and RQ branches are reported. Full (b) HundÕs case is in a complete agreement with the experiment. In Fig. 8, the experimental and theoretical line-strengths of OP and SR branches are reported. A clear difference between SR functions of (a–b) and full (b) HundÕs cases is dramatic at initial J values, like in Fig. 5, with full (b) HundÕs case (DJ 6 units) well fitting the experiment. No other cases of meaningful differences between theory and experiment, as well as between (a–b) and full (b) HundÕs cases line-strength functions, have been found: as given examples, the RQ–SR, OP–PQ, QP–RQ, OP–QR, Q Q–RQ systems can be quoted. The latter is shown in Fig. 9, where a clear choice between (a–b) and full (b) HundÕs cases is beyond resolution. According to these results, the full (b) HundÕs case linestrength functions, as deduced by use of (3) putting the fine structure parameter k ffi 0, plainly show more suitable for a description of this near-IR atmospheric transition, and more in general of the radiative behaviour of molecular oxygen in its lowest states, probably. However, more data are desirable, and new experiments on oxygenÕs linestrength should be welcome. 4. Conclusions
0
5
10
15
20
25
J
Fig. 8. Experimental line-strengths of QP (h) (reference) and SR (j) branches with respective (a–b) and full (b) HandÕs case theoretical functions (solid lines).
Intensity data of near-IR atmospheric forbidden transition of the oxygen molecule have been studied with the same quantitative method (of the line-strength correlation of branches) recently applied to those of the visible atmospheric forbidden transition. Conclusions are similar, together with an extension of the reference theory leading to new line-strength functions. Data of this kind are sparse, though clearly helpful in such crucial areas as atmospheric remote sensing. These quantitative techniques may allow
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observations of a very high quality for diatomics, and a satisfactory interpretation of available data. References [1] R.L. Miller, A.G. Suits, P.L. Houston, R. Toumi, J.A. Mack, A.M. Wodtke, Science 265 (1994) 1831. [2] R.J. Sica, R.P. Lowe, J. Geophys. Res. 98 (1993) 1051. [3] L. Wallace, J.W. Chamberlain, Planet. Space Sci. 2 (1958) 60. [4] M.R. Torr, D.G. Torr, R.R. Laher, J. Geophys. Res. 90 (1985) 8525. [5] R.L. Gattinger, A.J. Vallance-Jones, J. Geophys. Res. 81 (1976) 4789. [6] G. Di Stefano, Chem. Phys. 302 (2004) 243. [7] J.K.G. Watson, Can. J. Phys. 46 (1968) 1637.
[8] G. Di Stefano, M. Lenzi, G. Piciacchia, A. Ricci, Chem. Phys. 165 (1992) 201. [9] G. Di Stefano, M. Lenzi, A. Ricci, Chem. Phys. 246 (1999) 267. [10] V.P. Bellary, T.K. Balasubramaniam, J. Mol. Spectrosc. 126 (1987) 436. [11] C. Amiot, J. Verges, Can. J. Phys. 59 (1981) 1391. [12] M. Vervloet et, J.K.G. Watson, Can. J. Phys. 64 (1986) 1529. [13] G. Di Stefano, M. Lenzi, G. Piciacchia, A. Ricci, J. Chem. Phys. 107 (1997) 2752. [14] G. Herzberg, Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules, Van Nostrand, Princeton, 1950. [15] H. Lefebvre-Brion, R.W. Field, Perturbations in the Spectra of Diatomic Molecules, Academic Press, New York, 1986. [16] T.K. Balasubramaniam, V.P. Bellary, Acta Phys. Hung. 63 (1988) 249. [17] G. Di Stefano, Trends Chem. Phys. 10 (2002) 63.