Forbidden rotational transitions and astrophysics
Graphical abstract
Introduction
Until 1970, the electric dipole selection rules for rotational transitions were thought to be rigorous [1]. For a symmetric top they were ΔJ = ± 1, 0, ΔK = 0, ΔI = 0, and + ↔ −, where J is the rotational quantum number, K is its projection on the symmetry axis, I is the quantum number for the total nuclear spin angular momentum I = ∑ Ii, where i runs over equivalent nuclei (which governs the nuclear spin isomers such as ortho and para), and + and – are the parity. Therefore spontaneous emissions from NH3 metastable rotational levels such as (J, K) = (2, 2) and (3, 3) were thought to be rigorously forbidden and spontaneous emission through its octupole moment were believed to take “longer than the lifetime of the Universe” [2].
The selection rules are rigorous for individual vibrational and rotational wave functions, but they break down when interactions between them are taken into account. Thus rotational transitions violating the ordinary selection rules are weakly allowed and are known as “forbidden rotational transitions.” It was pointed out theoretically [3] that the lifetimes of the (2, 2) and (3, 3) levels of NH3 are ~230 years and ~44 yearsa due to the forbidden (2, 2)s→ (1, 1)s and (3, 3)a → (2, 0)a spontaneous emissions, respectively, eight orders of magnitude shorter than the lifetime of the Universe. Here s and a represent the symmetric (lower) and antisymmetric (upper) inversion levels of NH3 [4]. Within a few years, the forbidden rotational transitions were experimentally demonstrated through observations of the K = 1 ← 2 transition (±1 ← 2, that is, Δk = ± 3 transitions in terms of signed quantum number k) of PH3 [5] and AsH3 [6]. They are an example of the spontaneous breakdown of circular symmetry around the C3 axis in which a rotational wavefunction with signed quantum number k is slightly mixed with that of k ± 3. Until the forbidden rotational transitions were observed, it had been impossible to determine the values of the rotational constant about the symmetry axis C and the centrifugal distortion constants DK, HK etc. [1]. A review of the subject has been published [7].
Footnotea The actual lifetime of the (3, 3)a level is much shorter due to the inversion transition (3, 3)a → (3, 3)s, but this transition does not change the rotational quantum numbers of NH3.
The spontaneous emissions due to forbidden rotational transitions are extremely slow and they are seldom of interest for laboratory spectroscopy. Nevertheless, they are often essential in analyzing the thermalization of interstellar molecules [8], because of the long time scales (~1 year) of molecular collisions and other astrochemical processes. The use of forbidden rotational transitions in molecular astrophysics, i.e. determination of the number density of the gas is discussed in Section 4.
Section snippets
Theory
Watson [9] formulated the general theory of forbidden rotational transitions using a perturbation treatment of the vibration–rotation interaction that results in centrifugal distortion. For symmetric top molecules with C3 symmetry axis discussed in this paper, the effective transition dipole moment for forbidden rotational transition is given aswhere the coordinates z and x are along the C3 axis and perpendicular to it, respectively, is the permanent dipole
Spontaneous emission from metastable levels
The (J, k) rotational level of NH3 was called metastable for k = ± J by Cheung et al. [2], because no transition to a lower level is allowed by the ordinary ΔK = 0 selection rule. Nevertheless forbidden rotational transitions following Δk = ± 3 are weakly allowed in C3 symmetric tops. Since the parity is (–1)K they satisfy the parity selection rule + ↔ –. All selection rules other than the ΔK = 0 rule are followed for the forbidden rotational transitions discussed in this paper.
Number densities of molecules in interstellar space
The spontaneous emission rates Aij and lifetimes of levels, together with observed spectral intensities provide the most direct information on the molecular number densities in interstellar space. Historically, the discovery of the NH3 inversion spectrum in emission [56] provided firm evidence for high density (~104 cm−3) molecular clouds in Sgr B2 in the Galactic center and initiated a new era in the study of star formation. Previous to that discovery, the density of interstellar space had
Final remark
I have calculated the spontaneous emission rates of forbidden rotational transitions and the lifetimes of the upper state of emission. I have also briefly summarized the nature of collisions of interstellar molecules. Together they allow us to calculate the degrees of thermalization of interstellar molecules. So far such a calculation has been published for H3+ [8] and linear molecules [65]. I look forward to continuing such calculations for more molecules.
Declaration of Competing Interest
The author declare that there is no conflict of interest.
Acknowledgements
This paper is dedicated to Stephan Schlemmer as a token of my appreciation of his great invention of action spectroscopy, and in particular its application to the enigmatic spectrum of CH5+ [66], [67]. Shanshan Yu provided lists of rotational energy levels of NH3 and H3O+ and Mitsunori Araki provided the same for CH3CN. Tom Geballe has read and revised the presentation of this paper extensively. Araki also read the manuscript and provided many comments. To all of them I am deeply grateful.
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