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Transition probabilities of the bands of the oxygen systems of the nightglow
Pluncr Space Sci., Vol. 36, No. 9, pp. 869-873. Prmted in Great Britain.
00324633i88 $3.00+0.00 Pergamon Press plc
1988
OF THE BANDS OF THE TRANSITION PROBABILITIES OXYGEN SYSTEMS OF THE NIGHTGLOW D. R. BATES
Department
of Applied
Mathematics
and Theoretical
Physics,
Queen’s University, Belfast
BT7 INN, U.K. (Received 19 May 1988) Abstract-Absolute transition probabilities are given for the bands Herzberg II and Herzberg III systems of molecular oxygen.
INTRODUCTION
With the important exception of the Meinel hydroxyl system (Meinel, 1950a) the nightglow band spectrum is predominantly due to molecular oxygen. From the present viewpoint the oxygen systems may be divided into two classes : those for which only the v’ = 0 progression appears in the nightglow and others. The first class comprises the b’C,+ + X’C; Atmospheric system (Meinel, 1950b) and the u’A~ -+ X3X; Infrared Atmospheric system (Gush and Buijs, 1964). Klotz et ul. (1984) have already calculated their transition probabilities. The other systems of interest are the A%.,+ --+X’C; Herzberg I the A’3A, +a’A,, Chamberlain, the c’C, + X3X,; Herzberg II and the A’ 3A,,+ X3X; Herzberg III systems. The presence of the Herzberg I system in the nightglow was proposed by Dufay (1941) but was not convincingly established until measurements by Chamberlain (1955) on a high resolution spectrum. Chamberlain (1958) also tentatively suggested the presence of the system that now bears his name. Slanger (1978) confirmed the correctness of the suggestion. In the paper in which he gave an account of his discovery of the Herzberg II and III systems in absorption, Herzberg (1953) mooted their possible presence in the nightglow. He found some evidence for the Herzberg II system and commented that identification of the Herzberg III system is more difficult because its bands lie close to those of the Herzberg I system. Conclusive evidence for the presence of the Herzberg II system has been provided by Slanger and Huestis (1981, 1983a). There appears to have been no further observational work on the occurrence of the Herzberg III system in the nightglow. Klotz and Peyerimhoff (1986) have calculated the transition moments and hence the transition probabilities of bands of the v’ = 0 progressions of the four systems just mentioned. More extensive calculations of transition probabilities are reported here. Those on
of the Herzberg
I, Chamberlain,
the Herzberg I, Chamberlain and Herzberg II systems were carried out mainly to allow the intensities and the excitation and quenching rates to be linked ; those on the Herzberg III system to facilitate identifications being made. The transition moments of Klotz and Peyerimhoff (1986) have already been used by Saxon and Slanger (1986) and Cheung et al. (1986) to calculate the molecular oxygen absorption cross-section curve in the Herzberg continuum. There is good agreement with experiment (cf. Fig. 1 of Nicolet and Kennes, 1986).
CALCULATIONS
AND RESULTS
The molecular constants for the X3X;, c’C; and A3C,+ states were taken from Krupenie (1982), those for the A’3A, state were taken from Slanger and Huestis (1983b) and those for the a’d, state were taken from Slanger (1978). The potential energy curves were computed by the Rydberg-Klein-Rees method as described by Vanderslice et al. (1959, 1961) coupled with cubic spline fitting. A comparison was made between obtaining the vibrational eigenfunctions from sturmian basis expansion (Yurtsever, 1982; Yurtsever and Pehlivan, 1986) and obtaining them by solving the differential equation numerically. The latter procedure was judged the more satisfactory and was adopted. Klotz and Peyerimhoff (1986) have given the required transition moments as functions of the distance between the nuclei so that computing the spontaneous transition probabilities A(v’, v”) was straightforward. Tables 1,2, 3 and 4 give the A (v’, v”) arrays for the Herzberg I, Chamberlain, Herzberg II and Herzberg III systems. Table 5 gives for each of these systems the total transition probability out of each v’ level A(u’,Z)
= CA(v’,v”). 0”
(1)
_ ,_, _.
2 3 4 5 6 7 8 9 10 11 12
-
._
3.6-4 2.4-3 8.7-3 2.3-2 5.0-2 9.2-2 1.5-1 2.3-l 3.1-1 3.8-1 4.1-1 3.6- 1 2.1-l
2.6-5 2.0-4 8.9-4 2.9-3 7.7-3 1.8-2 3.8-2 1.5-2 1.4-1 2.5- 1 4.3-l 7.0-t 1.0 0
0
1
1
0
?J’ 0” =
Note: 1.8-4
3 4 5 6 7 8 9 10 11
E 1.8 x 10m4.
2.4-3 2.1-2 9.5-2 3.1-I 7.9-l 1.7 0 3.0 0 4.5 0 5.6 0 5.3 0 3.3 0 6.6- 1
1.8-4 1.9-3 1.1-2 4.4-2 1.4-1 4.0-I 9.9-l 2.2 0 4.4 0 7.9 0 1.2+1 1.7+ 1
0 1
1
1
0
u’ 0” =
_,__---
2.4-3 1.3-2 3.9-2 8.3-2 1.4-l 2.1-1 2.6- 1 3.0- 1 2.9- 1 2.3-l 1.5-1 6.4-2 1.1-2
2
1.6-2 1.1-I 4.2-l 1.1 0 2.1 0 3.3 0 4.3 0 4.3 0 3.3 0 1.5 0 2.4-l 4.7-2
2
5
5.3-l 1.7 0 2.7 0 2.3 0 1.0 0 6.0-2 2.7- 1 1.1 0 1.4 0 8.4- 1 1.2-1 5.1-2
4
2.2-l 9.7-l 2.1 0 3.0 0 2.9 0 2.6 0 4.2-l 1.5-2 6.4-l 1.4 0 1.2 0 3.5-l
8
1.8 0 1.3 0 2.8-2 5.5-l 1.6 0 4.7-l 2.6-4 4.1-1 7.7- 1 4.2- 1 2.0-2 1.2-1
7 A (u’, u”) in s-l 1.5 0 2.1 0 8.2-l 5.0-4 6.5- 1 1.2 0 7.1-l 5.0-2 1.9-1 6.9- 1 6.2-l 1.3-1
1.0 0 2.3 0 2.1 0 7.4- 1 2.6-4 5.5- 1 1.3 0 1.1 0 3.1-1 1.4-2 4.7 - 1 8.6-l
6
1.8 0 3.9- 1 2.5-l 9.8-1 4.0- 1 1.6-2 5.2-l 6.7- 1 1.9-1 2.1-2 3.5-l 4.5- 1
9
-+ X3X,-
9.7-3 4.2-2 1.0-l 1.7-l 2.2- 1 2.3- 1 2.0-l 1.3-1 6.1-2 1.1-2 2.5-3 3.8-2 9.5-2
3
.-
5.6-2 1.3-1 1.5-1 9.7-2 2.8-2 4.8-6 2.2-2 6.5-2 9.1-2 X.0-2 4.4-2 1.0-2 1.7-4
2.7-2 9.1-2 1.6-1 1.9- 1 1.6- 1 9.0-2 2.8-2 3.7-4 1.5-2 5.5-2 8.9-2 8.9-2 5.2-2
-
5
4
8.7-2 1.3-1 1.0-2 7.1-3 1.0-2 5.4-2 8.0-2 6.5-2 2.7-2 1.9-3 6.1-3 3.0-2 4.7-2
6
_
_
1.0-l 1.5-2 5.0-3 1.9-2 6.2-2 6.1-2 2.4-2 5.7-4 1.1-2 3.6-2 4.7-2 3.3-2 9.1-3
._-
8
9.6-2 1.9-2 1.2-2 5.5-2 4.0-2 4.4-3 6.8-3 3.4-2 4.5-2 2.9-2 6.3-3 5.3-4 1.1-2
7 A (u’, u”) in SC’
_ - ,,,
7.0-2 5.2-5 4.1-2 3.1-2 4.0-4 1.7-2 3.9-2 2.8-2 5.1-3 1.7-3 1.7-2 2.1-2 2.1-2
9
(_,
TABLE2. TRANSITION ARRAYFORCHAMBERLAIN SYSTEM, A’ ‘A, + a ‘4
7.0-2 4.0- 1 1.2 0 2.3 0 3.3 0 3.6 0 2.8 0 1.4 0 2.0-l 1.3-1 1.1 0 2.1 0
3
TABLE1. TRANSITION ARRAYFOR HERZBERG I SYSTEM, A%:
11
1.8-2 2.6-2 7.8-3 1.0-2 2.5-2 4.6-3 4.2-3 2.1-2 2.0-2 4.8-3 6.1-4 9.4-3 1.5-2
4.0-2 1.3-2 3.3-2 X.0-4 1.8-2 3.2-2 1.1-2 2.7-4 1.5-2 2.1-2 1.9-2 4.4-3 4.9-4
1.1 0 1.7-1 7.2- 1 1.5-2 4.0-l 4.9- 1 2.7-2 1.9-I 4.5- 1 1.9-I 1.7-3 1.6-I
1.5 0 3.4-3 1.5- 1 5.0-l 7.7-3 5.2- 1 5.6-l 6.7-2 1.2-l 4.4-l 3.1-1 1.6-2
10
11
10
6.0-3 2.3-2 3.0-4 1.9-2 4.3-3 5.3-3 1.9-2 9.6-3 1.6-6 7.8-3 1.6-2 1.1-2 2.0-3
12
6.2- 1 5.0- 1 2.7-l 1.8-1 5.0-l 3.9-2 1.9-1 4.0-I 9.3-2 4.1-2 2.6-l 2.1-l
12
1.6-3 1.3-2 1.2-3 8.5-3 2.3-3 1.4-2 4.5-3 1.4-3 1.2-2 1.2-2 2.4-3 7.0-4 6.6-3
13
3.0- 1 6.2-l 4.5-3 4.3 - 1 1.1-l 1.4-1 3.7-l 5.8-2 8.6-2 2.8- 1 1.2-l 3.1-3
13
--
3.1-4 5.0-3 1.0-2 1.4-4 9.5-3 3.7-3 2.1-3 1.1-2 5.6-3 5.8-6 5.0-3 8.8-3 4.6-3
14
1.2- 1 5.1-I 1.0- 1 3.0-l 2.6-2 3.3-l 6.4-2 9.1-2 2.6- 1 7.0-2 2.4-2 1.5-l
14
P
P
0
z
_,I_.* .“.^.“
-
1.6-5 1.7-4 9.3-4 3.7-3 1.1-2 3.0--2 6.9-2 1.4-l 2.7-l 4.6- 1 7.1- 1 1.0 0 1.3 0 1.5 0 1.5 0
0
1.1-5 9.3-5 4.4-4 1.5-3 4.2-3 1.0-2 2.3-2 4.8-2 9.6-2 1.8-l 3.3-l 5.6- 1 9.0-l
v’ fJ” =
0 1 2 3 4 5 6 7 8 9 10 11 12
-.--.
0
0 1 2 3 4 5 6 I 8 9 10 11 12 13 14
01, lJcc =
1.7-4 1.2-3 4.6-3 1.3-2 3.0-2 6.0-2 1.1-I 1.7-l 2.5- 1 3.3-l 3.9-l 3.9- 1 2.9- 1
1
1.9-4 1.7-3 7.7-3 2.5-2 6.1-2 1.3-l 2.2- 1 3.3-l 4.3-l 4.6-I 4.1-I 2.5-l 7.2-2 2.8-3 1.8-l
1
1.2-3 7.1-3 2.3-Z 5.4-2 1.0-i 1.6- 1 2.3-l 2.8- 1 3.1- 1 2.9-l 2.3-l 1.4-l 5.9-2
2
1.2-3 8.3-3 3.1-2 7.8-2 1.5-I 2.3-I 3.0- 1 3.0- 1 2.4- 1 1.3-I 3.3-2 5.1-4 4.2-2 9.6-2 8.2-2
2
1.2-2 5.4-2 1.2-l 1.6-l 1.5-l 8.0-Z 1.5-2 4.1-3 5.1-2 1.0-l 1.1-l 5.8-2 6.5-3 1.2-2 1.6-2
4
2.5-2 8.3-2 1.2-l l.O- 1 3.7-2 1.8-4 2.6-2 7.5-2 8.9-2 5.2-2 7.8-3 4.8-3 3.6-2 5.2-2 2.1-2
5
4.1-2 9.1-2 7.9-2 2.2-2 1.3-3 3.7-2 1.0-2 5.2-2 1.2-2 2.2-3 3.2-2 5.1-2 4.6-2 1.4-2 4.3-5
6
5.2-2 7.0-2 2.2-2 1.4-3 3.5-2 5.3-2 2.5-2 2.3-4 1.6-2 4.4-2 4.0-2 1.2-2 4.9-4 2.0-2 4.2-2
7 A (v’,u’) in s-I 5.4-2 3.4-2 8.7-6 2.5-2 3.9-2 1.2-2 1.5-3 2.5-2 3.7-2 1.7-2 2.0-4 1.0-2 2.1-2 2.1-2 2.9-3
8
5.4-3 2.7-2 7.1-Z 1.3-l 2.0-l 2.4- 1 2.5- 1 2.1-I 1.5-l 6.9-2 1.3-2 2.3-3 4.1-2
3
1.7-2 6.8-2 1.4-l 2.0- 1 2.1-l 1.7-l 1.0-I 3.8-2 2.1-3 9.5-3 4.6-2 8.0-2 7.9-2
4
4.1-2 1.2- 1 1.8-l 1.7-l 1.1-l 3.3-2 4.2-4 1.1-2 5.9-2 9.1-2 8.8-2 5.6-2 1.8-2
5
7
1.1-l 1.4-l 5.6-2 4.7-4 2.8-2 7.5-2 7.8-2 4.1-2 6.0-3 3.0-3 2.5-2 4.4-2 3.8-2
8
1.3-l 7.1-2 1.5-3 3.1-2 7.3-2 5.3-2 1.1-2 2.0-3 2.7-2 4.8-2 4.3-2 1.9-2 1.6-3
A (u’, u”) in se1 1.6-2 1.6-I 1.5- 1 6.5-2 5.5-3 1.1-2 5.5-2 8.8-2 8.1-2 4.4-2 8.7-3 1.1-3 2.0-2
6
TABLE4. TRANSITIONARRAYFORHERZBERGIII SYSTEM,A”4
4.5-3 2.5-Z 7.4-2 1.4-l 2.0-l 2.2-l 1.7-l 8.2-2 1.2-2 7.2-3 6.7-2 1.4-l 1.6-l 1.2-l 4.9-2
3
TABLE3. TRANSITIONARRAYFORHERZBERGIISYSTEM,c’C;
1.2-I 1.9-2 1.x-2 6.3-2 3.6-2 1.1-3 1.5-2 4.4-2 4.4-2 1.9-2 1.6-4 6.0-3 2.1-2
9
-+ X3C;
4.5-2 6.9-3 1.1-2 2.9-2 8.1-3 3.0-3 2.4-2 2.4-2 4.3-3 3.1-3 2.0-2 2.4-2 9.1-3 7.2-8 5.8-3
9
-+ X’ZZ;
9.1-2 1.6-5 4.9-2 3.4-2 6.8-5 2.3-2 4.4-2 2.5-2 2.0-3 5.4-3 2.3-2 2.8-2 1.5-2
10
3.1-2 7.8-5 2.0-2 9.6-3 1.7-3 1.9-2 1.5-2 4.2-4 8.3-3 2.0-2 1.2-2 4.4-4 5.6-3 1.7-2 1.5-2
10
5.8-2 1.5-2 4.3-2 1.3-3 2.1-2 3.7-2 1.2-Z 7.3-4 1.8-2 2.9-2 1.8-2 2.7-3 1.3-3
11
1.8-2 5.1-3 1.4-2 7.6-6 1.3-2 1.1-2 2.2-6 9.6-3 1.5-2 4.2-3 9.6-4 1.1-2 1.2-2 3.4-3 6.1-4
11
3.1-2 3.3-2 1.4-2 1.0-2 3.2-2 7.1-3 3.9-3 2.4-2 2.2-2 5.1-3 8.6-4 1.1-2 1.7-2
12
8.5-3 9.7-3 3.2-3 5.4-3 9.3-3 3.6-S 1.8-3 LO-2 9.2-4 3.5-3 1.1-2 6.8-3 1.7-4 3.0-3 6.6-3
12
1.4-2 3.7-2 3.6-5 2.5-2 9.2-3 3.9-3 2.2-2 1.4-2 2.3-4 7.2-3 1.7-2 1.2-2 2.2-3
13
3.4-3 9.4-3 2.6-5 7.9-3 8.5-4 4.6-3 1.2-3 2.3-4 4.4-3 8.3-3 2.2-3 1.2-4 6.4-3 6.8-3 1.8-3
..~_~_
13
5.1-3 2.1-2 6.8-3 1.7-2 6.1-4 1.8-2 1.0-2 2.5-4 1.2-2 1.5-2 4.9-3 1.1-4 5.4-3
14
1.2-3 6.1-3 2.1-3 3.6-3 1.2-3 5.7-3 2.9-4 3.5-3 5.6-3 5.3-4 2.1-3 5.9-3 2.8-3 1.8-5 3.2-3
14
a g g s. %
m
872
D. R. BATES TABLE 5. TOTAL TRANSITIONPROBABILITIESA(v’Z) OUT OF EACH v’ LEVEL
System u’
Herzberg
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.1+1 1.1+1 1.2+1 1.3+1 1.4+1 1.5+1 1.6+1 1.7+ 1 1.8+1 1.9+1 2.1+1 2.2+ 1
I
Chamberlain Herzbere II A(v’, Z) in ss’ 5.25.86.4-l 7.1-l 7.78.3-l 8.9-l 9.7-l 1.0 1.1 1.2 1.4 1.5
The weighted mean transition probabilities corresponding to the nightglow vibrational distributions reported by Slanger and Huestis (1981) may thence be derived. Their values are 15 s-l for the Herzberg I system, 0.89 ss’ for the Chamberlain system and 1.0 s-’ for the Herzberg II system. The pattern of the Herzberg I transition array is rather similar to that given by Degen (1977). The absolute transition probabilities for this system are greater than supposed before the work of Klotz and Peyerimhoff (1986) and those for the Chamberlain and Herzberg II systems are much greater (cf. Table 3 of Krasnopolsky, 1978). Acknowledgements1 thank Dr Kenneth L. Bell for helpful discussions and for facilitating the use of his code for solving the radial wave equation. I also thank Mrs Norah Scott for carrying the computations through skilfully and the U.S. Air Force for paying her salary under grant AFOSR-85-0202. REFERENCES Chamberlain, J. W. (1955) The ultra violet airglow spectrum. Asfrophys. J. 121, 271. Chamberlain, J. W. (1958) The blue airglow spectrum. Asrrophys. J. 128,713. Cheung, A. S.-C., Yoshino, K., Parkinson, W. H., Guberman, S. L. and Freeman, D. E. (1986) Absorption crosssection measurements of oxygen in the wave length region 195-241 nm of the Herzberg continuum. Planet. Space. Sci. 34, 1007. Degen, V. (1977) Nightglow emission rates of the O2 Herzberg bands. J. geophys. Res. 82, 2437. Dufay, J. (1941) A possible interpolation of certain night sky emissions in the ultra violet region. C.R. Acad. Sci. Paris 213, 284. Gush, H. P. and Buijs, H. L. (1964) The near infrared spec-
1 1
1
0 0 0 0 0
3.0-I 4.1-l 5.1-l 6.2-l 7.2-l 8.3-l 9.4-l 1.1 1.2 1.3 1.5 1.6 1.7 1.9 2.0
0 0 0 0 0 0 0 0
Herzberz
6.97.3-l 7.68.28.6-l 9.1-I 9.61.0 1.1 1.2 1.2 1.4 1.5
III
1 1 1
1 0 0 0 0 0 0
trum of the night airglow observed from high altitude. Can. J. Phys. 42, 1037. Herzberg, G. (1953) Forbidden transitions in diatomic molecules-III. New ‘Z; + ‘Z; and ‘A, + ‘Z; absorption bands of the oxygen molecule. Can. J. Phys. 31, 657. Klotz, R., Marian, C. M., Peyerimhoff, S. D., Hess, B. A. and Buenker, R. J. (1984) Calculation of spin-forbidden radiative transitions using correlated wave functions : lifetimes of 6’Z+, a’A states in 02, S, and SO. Chem. Phys. 89, 223. Klotz, R. and Peyerimhoff, S. D. (1986) Theoretical study of the intensity of the spin or dipole forbidden transitions between the c’Z;, At3A, A3Ez and X%6, a’$, b’Ei states of OZ. Molec. Phys. 57, 573. Krasnopolsky, V. A. (1981) Excitation of oxygen emissions in nightglow of terrestrial planets. Planet. Space Sci. 29, 925. Krupenie, P. H. (1972) The spectrum of molecular oxygen. J. phys. Chem. (Re$ Data) 1,423. Meinel, A. B. (195Oa) OH emission bands in the spectrum ofthe night sky-I. Astrophys. J. 111,555; -11. Astrophys. J. 112, 120. Meinel, A. B. (1950b) 0, emission bands in the infrared spectrum of the night sky. Astrophys. J. 112,464. Nicolet, M. and Kennes, R. (1986) Aeronomic problems of the molecular oxygen photodissociation-I. The 0, Herzberg continuum. Planet. Space. Sci. 34, 1043. Saxon, R. P. and Slanger, T. G. (1986) Molecular oxygen absorption continua at 195-300 nm and O2 radiative lifetimes. J. geophys. Res. 91, 9877. Slanger, T. G. (1978) Generation of O2 (c’Z;, C’A,, A ‘XT,‘, from oxygen atom recombination. J. them. Phys. 69,4779. Slanger, T. G. and Huestis, D. L. (1981) 0, (c’C; + X’Z;) emission in the terrestrial nightglow. J. geophys. Res. 86, 3551. Slanger, T. G. and Huestis, D. L. (1983a) The rotationally resolved 34W3800 A terrestrial nightglow. J. geophys. Res. 88,4137.
Oxygen Slanger, T. G. and Hue&s, D. L. (1983b) vibrational analysis of the 0 (A “A, + them. Phys. l&2274. Vanderslice, J. T., Mason, E. A., Maisch, pintott, E. R. (1959) Ground state of RydberggKlein-Rees method. J. molec. Errata (1961) 5, 83.
transition
Rotational and a ‘Ag) system. J. W. G. and Liphydrogen by the Spectrosc. 3, 17.
probabilities
873
Yurtsever, E. (1982) FranckkCondon integrals over a sturmian basis. An application to photoelectron spectra of H, and N,. Yurtsever, E. and Pehlivan, M. (1986) One dimensional vibrational eigenvalue problem with numerical potentials. Computer Phys. Communic. 39,43 1.
00324633i88 $3.00+0.00 Pergamon Press plc
1988
OF THE BANDS OF THE TRANSITION PROBABILITIES OXYGEN SYSTEMS OF THE NIGHTGLOW D. R. BATES
Department
of Applied
Mathematics
and Theoretical
Physics,
Queen’s University, Belfast
BT7 INN, U.K. (Received 19 May 1988) Abstract-Absolute transition probabilities are given for the bands Herzberg II and Herzberg III systems of molecular oxygen.
INTRODUCTION
With the important exception of the Meinel hydroxyl system (Meinel, 1950a) the nightglow band spectrum is predominantly due to molecular oxygen. From the present viewpoint the oxygen systems may be divided into two classes : those for which only the v’ = 0 progression appears in the nightglow and others. The first class comprises the b’C,+ + X’C; Atmospheric system (Meinel, 1950b) and the u’A~ -+ X3X; Infrared Atmospheric system (Gush and Buijs, 1964). Klotz et ul. (1984) have already calculated their transition probabilities. The other systems of interest are the A%.,+ --+X’C; Herzberg I the A’3A, +a’A,, Chamberlain, the c’C, + X3X,; Herzberg II and the A’ 3A,,+ X3X; Herzberg III systems. The presence of the Herzberg I system in the nightglow was proposed by Dufay (1941) but was not convincingly established until measurements by Chamberlain (1955) on a high resolution spectrum. Chamberlain (1958) also tentatively suggested the presence of the system that now bears his name. Slanger (1978) confirmed the correctness of the suggestion. In the paper in which he gave an account of his discovery of the Herzberg II and III systems in absorption, Herzberg (1953) mooted their possible presence in the nightglow. He found some evidence for the Herzberg II system and commented that identification of the Herzberg III system is more difficult because its bands lie close to those of the Herzberg I system. Conclusive evidence for the presence of the Herzberg II system has been provided by Slanger and Huestis (1981, 1983a). There appears to have been no further observational work on the occurrence of the Herzberg III system in the nightglow. Klotz and Peyerimhoff (1986) have calculated the transition moments and hence the transition probabilities of bands of the v’ = 0 progressions of the four systems just mentioned. More extensive calculations of transition probabilities are reported here. Those on
of the Herzberg
I, Chamberlain,
the Herzberg I, Chamberlain and Herzberg II systems were carried out mainly to allow the intensities and the excitation and quenching rates to be linked ; those on the Herzberg III system to facilitate identifications being made. The transition moments of Klotz and Peyerimhoff (1986) have already been used by Saxon and Slanger (1986) and Cheung et al. (1986) to calculate the molecular oxygen absorption cross-section curve in the Herzberg continuum. There is good agreement with experiment (cf. Fig. 1 of Nicolet and Kennes, 1986).
CALCULATIONS
AND RESULTS
The molecular constants for the X3X;, c’C; and A3C,+ states were taken from Krupenie (1982), those for the A’3A, state were taken from Slanger and Huestis (1983b) and those for the a’d, state were taken from Slanger (1978). The potential energy curves were computed by the Rydberg-Klein-Rees method as described by Vanderslice et al. (1959, 1961) coupled with cubic spline fitting. A comparison was made between obtaining the vibrational eigenfunctions from sturmian basis expansion (Yurtsever, 1982; Yurtsever and Pehlivan, 1986) and obtaining them by solving the differential equation numerically. The latter procedure was judged the more satisfactory and was adopted. Klotz and Peyerimhoff (1986) have given the required transition moments as functions of the distance between the nuclei so that computing the spontaneous transition probabilities A(v’, v”) was straightforward. Tables 1,2, 3 and 4 give the A (v’, v”) arrays for the Herzberg I, Chamberlain, Herzberg II and Herzberg III systems. Table 5 gives for each of these systems the total transition probability out of each v’ level A(u’,Z)
= CA(v’,v”). 0”
(1)
_ ,_, _.
2 3 4 5 6 7 8 9 10 11 12
-
._
3.6-4 2.4-3 8.7-3 2.3-2 5.0-2 9.2-2 1.5-1 2.3-l 3.1-1 3.8-1 4.1-1 3.6- 1 2.1-l
2.6-5 2.0-4 8.9-4 2.9-3 7.7-3 1.8-2 3.8-2 1.5-2 1.4-1 2.5- 1 4.3-l 7.0-t 1.0 0
0
1
1
0
?J’ 0” =
Note: 1.8-4
3 4 5 6 7 8 9 10 11
E 1.8 x 10m4.
2.4-3 2.1-2 9.5-2 3.1-I 7.9-l 1.7 0 3.0 0 4.5 0 5.6 0 5.3 0 3.3 0 6.6- 1
1.8-4 1.9-3 1.1-2 4.4-2 1.4-1 4.0-I 9.9-l 2.2 0 4.4 0 7.9 0 1.2+1 1.7+ 1
0 1
1
1
0
u’ 0” =
_,__---
2.4-3 1.3-2 3.9-2 8.3-2 1.4-l 2.1-1 2.6- 1 3.0- 1 2.9- 1 2.3-l 1.5-1 6.4-2 1.1-2
2
1.6-2 1.1-I 4.2-l 1.1 0 2.1 0 3.3 0 4.3 0 4.3 0 3.3 0 1.5 0 2.4-l 4.7-2
2
5
5.3-l 1.7 0 2.7 0 2.3 0 1.0 0 6.0-2 2.7- 1 1.1 0 1.4 0 8.4- 1 1.2-1 5.1-2
4
2.2-l 9.7-l 2.1 0 3.0 0 2.9 0 2.6 0 4.2-l 1.5-2 6.4-l 1.4 0 1.2 0 3.5-l
8
1.8 0 1.3 0 2.8-2 5.5-l 1.6 0 4.7-l 2.6-4 4.1-1 7.7- 1 4.2- 1 2.0-2 1.2-1
7 A (u’, u”) in s-l 1.5 0 2.1 0 8.2-l 5.0-4 6.5- 1 1.2 0 7.1-l 5.0-2 1.9-1 6.9- 1 6.2-l 1.3-1
1.0 0 2.3 0 2.1 0 7.4- 1 2.6-4 5.5- 1 1.3 0 1.1 0 3.1-1 1.4-2 4.7 - 1 8.6-l
6
1.8 0 3.9- 1 2.5-l 9.8-1 4.0- 1 1.6-2 5.2-l 6.7- 1 1.9-1 2.1-2 3.5-l 4.5- 1
9
-+ X3X,-
9.7-3 4.2-2 1.0-l 1.7-l 2.2- 1 2.3- 1 2.0-l 1.3-1 6.1-2 1.1-2 2.5-3 3.8-2 9.5-2
3
.-
5.6-2 1.3-1 1.5-1 9.7-2 2.8-2 4.8-6 2.2-2 6.5-2 9.1-2 X.0-2 4.4-2 1.0-2 1.7-4
2.7-2 9.1-2 1.6-1 1.9- 1 1.6- 1 9.0-2 2.8-2 3.7-4 1.5-2 5.5-2 8.9-2 8.9-2 5.2-2
-
5
4
8.7-2 1.3-1 1.0-2 7.1-3 1.0-2 5.4-2 8.0-2 6.5-2 2.7-2 1.9-3 6.1-3 3.0-2 4.7-2
6
_
_
1.0-l 1.5-2 5.0-3 1.9-2 6.2-2 6.1-2 2.4-2 5.7-4 1.1-2 3.6-2 4.7-2 3.3-2 9.1-3
._-
8
9.6-2 1.9-2 1.2-2 5.5-2 4.0-2 4.4-3 6.8-3 3.4-2 4.5-2 2.9-2 6.3-3 5.3-4 1.1-2
7 A (u’, u”) in SC’
_ - ,,,
7.0-2 5.2-5 4.1-2 3.1-2 4.0-4 1.7-2 3.9-2 2.8-2 5.1-3 1.7-3 1.7-2 2.1-2 2.1-2
9
(_,
TABLE2. TRANSITION ARRAYFORCHAMBERLAIN SYSTEM, A’ ‘A, + a ‘4
7.0-2 4.0- 1 1.2 0 2.3 0 3.3 0 3.6 0 2.8 0 1.4 0 2.0-l 1.3-1 1.1 0 2.1 0
3
TABLE1. TRANSITION ARRAYFOR HERZBERG I SYSTEM, A%:
11
1.8-2 2.6-2 7.8-3 1.0-2 2.5-2 4.6-3 4.2-3 2.1-2 2.0-2 4.8-3 6.1-4 9.4-3 1.5-2
4.0-2 1.3-2 3.3-2 X.0-4 1.8-2 3.2-2 1.1-2 2.7-4 1.5-2 2.1-2 1.9-2 4.4-3 4.9-4
1.1 0 1.7-1 7.2- 1 1.5-2 4.0-l 4.9- 1 2.7-2 1.9-I 4.5- 1 1.9-I 1.7-3 1.6-I
1.5 0 3.4-3 1.5- 1 5.0-l 7.7-3 5.2- 1 5.6-l 6.7-2 1.2-l 4.4-l 3.1-1 1.6-2
10
11
10
6.0-3 2.3-2 3.0-4 1.9-2 4.3-3 5.3-3 1.9-2 9.6-3 1.6-6 7.8-3 1.6-2 1.1-2 2.0-3
12
6.2- 1 5.0- 1 2.7-l 1.8-1 5.0-l 3.9-2 1.9-1 4.0-I 9.3-2 4.1-2 2.6-l 2.1-l
12
1.6-3 1.3-2 1.2-3 8.5-3 2.3-3 1.4-2 4.5-3 1.4-3 1.2-2 1.2-2 2.4-3 7.0-4 6.6-3
13
3.0- 1 6.2-l 4.5-3 4.3 - 1 1.1-l 1.4-1 3.7-l 5.8-2 8.6-2 2.8- 1 1.2-l 3.1-3
13
--
3.1-4 5.0-3 1.0-2 1.4-4 9.5-3 3.7-3 2.1-3 1.1-2 5.6-3 5.8-6 5.0-3 8.8-3 4.6-3
14
1.2- 1 5.1-I 1.0- 1 3.0-l 2.6-2 3.3-l 6.4-2 9.1-2 2.6- 1 7.0-2 2.4-2 1.5-l
14
P
P
0
z
_,I_.* .“.^.“
-
1.6-5 1.7-4 9.3-4 3.7-3 1.1-2 3.0--2 6.9-2 1.4-l 2.7-l 4.6- 1 7.1- 1 1.0 0 1.3 0 1.5 0 1.5 0
0
1.1-5 9.3-5 4.4-4 1.5-3 4.2-3 1.0-2 2.3-2 4.8-2 9.6-2 1.8-l 3.3-l 5.6- 1 9.0-l
v’ fJ” =
0 1 2 3 4 5 6 7 8 9 10 11 12
-.--.
0
0 1 2 3 4 5 6 I 8 9 10 11 12 13 14
01, lJcc =
1.7-4 1.2-3 4.6-3 1.3-2 3.0-2 6.0-2 1.1-I 1.7-l 2.5- 1 3.3-l 3.9-l 3.9- 1 2.9- 1
1
1.9-4 1.7-3 7.7-3 2.5-2 6.1-2 1.3-l 2.2- 1 3.3-l 4.3-l 4.6-I 4.1-I 2.5-l 7.2-2 2.8-3 1.8-l
1
1.2-3 7.1-3 2.3-Z 5.4-2 1.0-i 1.6- 1 2.3-l 2.8- 1 3.1- 1 2.9-l 2.3-l 1.4-l 5.9-2
2
1.2-3 8.3-3 3.1-2 7.8-2 1.5-I 2.3-I 3.0- 1 3.0- 1 2.4- 1 1.3-I 3.3-2 5.1-4 4.2-2 9.6-2 8.2-2
2
1.2-2 5.4-2 1.2-l 1.6-l 1.5-l 8.0-Z 1.5-2 4.1-3 5.1-2 1.0-l 1.1-l 5.8-2 6.5-3 1.2-2 1.6-2
4
2.5-2 8.3-2 1.2-l l.O- 1 3.7-2 1.8-4 2.6-2 7.5-2 8.9-2 5.2-2 7.8-3 4.8-3 3.6-2 5.2-2 2.1-2
5
4.1-2 9.1-2 7.9-2 2.2-2 1.3-3 3.7-2 1.0-2 5.2-2 1.2-2 2.2-3 3.2-2 5.1-2 4.6-2 1.4-2 4.3-5
6
5.2-2 7.0-2 2.2-2 1.4-3 3.5-2 5.3-2 2.5-2 2.3-4 1.6-2 4.4-2 4.0-2 1.2-2 4.9-4 2.0-2 4.2-2
7 A (v’,u’) in s-I 5.4-2 3.4-2 8.7-6 2.5-2 3.9-2 1.2-2 1.5-3 2.5-2 3.7-2 1.7-2 2.0-4 1.0-2 2.1-2 2.1-2 2.9-3
8
5.4-3 2.7-2 7.1-Z 1.3-l 2.0-l 2.4- 1 2.5- 1 2.1-I 1.5-l 6.9-2 1.3-2 2.3-3 4.1-2
3
1.7-2 6.8-2 1.4-l 2.0- 1 2.1-l 1.7-l 1.0-I 3.8-2 2.1-3 9.5-3 4.6-2 8.0-2 7.9-2
4
4.1-2 1.2- 1 1.8-l 1.7-l 1.1-l 3.3-2 4.2-4 1.1-2 5.9-2 9.1-2 8.8-2 5.6-2 1.8-2
5
7
1.1-l 1.4-l 5.6-2 4.7-4 2.8-2 7.5-2 7.8-2 4.1-2 6.0-3 3.0-3 2.5-2 4.4-2 3.8-2
8
1.3-l 7.1-2 1.5-3 3.1-2 7.3-2 5.3-2 1.1-2 2.0-3 2.7-2 4.8-2 4.3-2 1.9-2 1.6-3
A (u’, u”) in se1 1.6-2 1.6-I 1.5- 1 6.5-2 5.5-3 1.1-2 5.5-2 8.8-2 8.1-2 4.4-2 8.7-3 1.1-3 2.0-2
6
TABLE4. TRANSITIONARRAYFORHERZBERGIII SYSTEM,A”4
4.5-3 2.5-Z 7.4-2 1.4-l 2.0-l 2.2-l 1.7-l 8.2-2 1.2-2 7.2-3 6.7-2 1.4-l 1.6-l 1.2-l 4.9-2
3
TABLE3. TRANSITIONARRAYFORHERZBERGIISYSTEM,c’C;
1.2-I 1.9-2 1.x-2 6.3-2 3.6-2 1.1-3 1.5-2 4.4-2 4.4-2 1.9-2 1.6-4 6.0-3 2.1-2
9
-+ X3C;
4.5-2 6.9-3 1.1-2 2.9-2 8.1-3 3.0-3 2.4-2 2.4-2 4.3-3 3.1-3 2.0-2 2.4-2 9.1-3 7.2-8 5.8-3
9
-+ X’ZZ;
9.1-2 1.6-5 4.9-2 3.4-2 6.8-5 2.3-2 4.4-2 2.5-2 2.0-3 5.4-3 2.3-2 2.8-2 1.5-2
10
3.1-2 7.8-5 2.0-2 9.6-3 1.7-3 1.9-2 1.5-2 4.2-4 8.3-3 2.0-2 1.2-2 4.4-4 5.6-3 1.7-2 1.5-2
10
5.8-2 1.5-2 4.3-2 1.3-3 2.1-2 3.7-2 1.2-Z 7.3-4 1.8-2 2.9-2 1.8-2 2.7-3 1.3-3
11
1.8-2 5.1-3 1.4-2 7.6-6 1.3-2 1.1-2 2.2-6 9.6-3 1.5-2 4.2-3 9.6-4 1.1-2 1.2-2 3.4-3 6.1-4
11
3.1-2 3.3-2 1.4-2 1.0-2 3.2-2 7.1-3 3.9-3 2.4-2 2.2-2 5.1-3 8.6-4 1.1-2 1.7-2
12
8.5-3 9.7-3 3.2-3 5.4-3 9.3-3 3.6-S 1.8-3 LO-2 9.2-4 3.5-3 1.1-2 6.8-3 1.7-4 3.0-3 6.6-3
12
1.4-2 3.7-2 3.6-5 2.5-2 9.2-3 3.9-3 2.2-2 1.4-2 2.3-4 7.2-3 1.7-2 1.2-2 2.2-3
13
3.4-3 9.4-3 2.6-5 7.9-3 8.5-4 4.6-3 1.2-3 2.3-4 4.4-3 8.3-3 2.2-3 1.2-4 6.4-3 6.8-3 1.8-3
..~_~_
13
5.1-3 2.1-2 6.8-3 1.7-2 6.1-4 1.8-2 1.0-2 2.5-4 1.2-2 1.5-2 4.9-3 1.1-4 5.4-3
14
1.2-3 6.1-3 2.1-3 3.6-3 1.2-3 5.7-3 2.9-4 3.5-3 5.6-3 5.3-4 2.1-3 5.9-3 2.8-3 1.8-5 3.2-3
14
a g g s. %
m
872
D. R. BATES TABLE 5. TOTAL TRANSITIONPROBABILITIESA(v’Z) OUT OF EACH v’ LEVEL
System u’
Herzberg
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.1+1 1.1+1 1.2+1 1.3+1 1.4+1 1.5+1 1.6+1 1.7+ 1 1.8+1 1.9+1 2.1+1 2.2+ 1
I
Chamberlain Herzbere II A(v’, Z) in ss’ 5.25.86.4-l 7.1-l 7.78.3-l 8.9-l 9.7-l 1.0 1.1 1.2 1.4 1.5
The weighted mean transition probabilities corresponding to the nightglow vibrational distributions reported by Slanger and Huestis (1981) may thence be derived. Their values are 15 s-l for the Herzberg I system, 0.89 ss’ for the Chamberlain system and 1.0 s-’ for the Herzberg II system. The pattern of the Herzberg I transition array is rather similar to that given by Degen (1977). The absolute transition probabilities for this system are greater than supposed before the work of Klotz and Peyerimhoff (1986) and those for the Chamberlain and Herzberg II systems are much greater (cf. Table 3 of Krasnopolsky, 1978). Acknowledgements1 thank Dr Kenneth L. Bell for helpful discussions and for facilitating the use of his code for solving the radial wave equation. I also thank Mrs Norah Scott for carrying the computations through skilfully and the U.S. Air Force for paying her salary under grant AFOSR-85-0202. REFERENCES Chamberlain, J. W. (1955) The ultra violet airglow spectrum. Asfrophys. J. 121, 271. Chamberlain, J. W. (1958) The blue airglow spectrum. Asrrophys. J. 128,713. Cheung, A. S.-C., Yoshino, K., Parkinson, W. H., Guberman, S. L. and Freeman, D. E. (1986) Absorption crosssection measurements of oxygen in the wave length region 195-241 nm of the Herzberg continuum. Planet. Space. Sci. 34, 1007. Degen, V. (1977) Nightglow emission rates of the O2 Herzberg bands. J. geophys. Res. 82, 2437. Dufay, J. (1941) A possible interpolation of certain night sky emissions in the ultra violet region. C.R. Acad. Sci. Paris 213, 284. Gush, H. P. and Buijs, H. L. (1964) The near infrared spec-
1 1
1
0 0 0 0 0
3.0-I 4.1-l 5.1-l 6.2-l 7.2-l 8.3-l 9.4-l 1.1 1.2 1.3 1.5 1.6 1.7 1.9 2.0
0 0 0 0 0 0 0 0
Herzberz
6.97.3-l 7.68.28.6-l 9.1-I 9.61.0 1.1 1.2 1.2 1.4 1.5
III
1 1 1
1 0 0 0 0 0 0
trum of the night airglow observed from high altitude. Can. J. Phys. 42, 1037. Herzberg, G. (1953) Forbidden transitions in diatomic molecules-III. New ‘Z; + ‘Z; and ‘A, + ‘Z; absorption bands of the oxygen molecule. Can. J. Phys. 31, 657. Klotz, R., Marian, C. M., Peyerimhoff, S. D., Hess, B. A. and Buenker, R. J. (1984) Calculation of spin-forbidden radiative transitions using correlated wave functions : lifetimes of 6’Z+, a’A states in 02, S, and SO. Chem. Phys. 89, 223. Klotz, R. and Peyerimhoff, S. D. (1986) Theoretical study of the intensity of the spin or dipole forbidden transitions between the c’Z;, At3A, A3Ez and X%6, a’$, b’Ei states of OZ. Molec. Phys. 57, 573. Krasnopolsky, V. A. (1981) Excitation of oxygen emissions in nightglow of terrestrial planets. Planet. Space Sci. 29, 925. Krupenie, P. H. (1972) The spectrum of molecular oxygen. J. phys. Chem. (Re$ Data) 1,423. Meinel, A. B. (195Oa) OH emission bands in the spectrum ofthe night sky-I. Astrophys. J. 111,555; -11. Astrophys. J. 112, 120. Meinel, A. B. (1950b) 0, emission bands in the infrared spectrum of the night sky. Astrophys. J. 112,464. Nicolet, M. and Kennes, R. (1986) Aeronomic problems of the molecular oxygen photodissociation-I. The 0, Herzberg continuum. Planet. Space. Sci. 34, 1043. Saxon, R. P. and Slanger, T. G. (1986) Molecular oxygen absorption continua at 195-300 nm and O2 radiative lifetimes. J. geophys. Res. 91, 9877. Slanger, T. G. (1978) Generation of O2 (c’Z;, C’A,, A ‘XT,‘, from oxygen atom recombination. J. them. Phys. 69,4779. Slanger, T. G. and Huestis, D. L. (1981) 0, (c’C; + X’Z;) emission in the terrestrial nightglow. J. geophys. Res. 86, 3551. Slanger, T. G. and Huestis, D. L. (1983a) The rotationally resolved 34W3800 A terrestrial nightglow. J. geophys. Res. 88,4137.
Oxygen Slanger, T. G. and Hue&s, D. L. (1983b) vibrational analysis of the 0 (A “A, + them. Phys. l&2274. Vanderslice, J. T., Mason, E. A., Maisch, pintott, E. R. (1959) Ground state of RydberggKlein-Rees method. J. molec. Errata (1961) 5, 83.
transition
Rotational and a ‘Ag) system. J. W. G. and Liphydrogen by the Spectrosc. 3, 17.
probabilities
873
Yurtsever, E. (1982) FranckkCondon integrals over a sturmian basis. An application to photoelectron spectra of H, and N,. Yurtsever, E. and Pehlivan, M. (1986) One dimensional vibrational eigenvalue problem with numerical potentials. Computer Phys. Communic. 39,43 1.