DIDÁCTICA DE LA ERE

individual rotational lines may be invisible due to poor experimental resolution. In these cases, a line-by-line comparison of experimental and model resonances will be fruitless and a model of an entire band must be considered instead. Such a band model would also suit those applications of synthetic spectra which are not interested in the properties of individual rotational lines. In what follows, quantities relating to the excited and ground states are labelled with single and double primes, respectively.

Total dissociation cross sections may be calculated for all possible rotational transi- tions according Eq. (2.176). These are combined into an absorption band model once each rotational transition has been weighted by its respective ground state fractional pop- ulation. These weights, αJ′′, are calculated according to Eq. (2.106) and depend on the assumed temperature. For very high temperatures it may be necessary to consider multiple ground-state vibrational levels in Eq. (2.106). The combined cross section is then

σ(ν) = X J′_J′′

αJ′′σ_J′_J′′(ν), (2.177)

where only rotational transitions that are permitted by the selection rules of Sec. 2.8.2 need be considered.

The cross section in Eq. (2.177) may be directly used by theoretical radiative-transfer models, or for comparison with an experimental transmission spectrum, in which case it must be further transformed according to

I(ν) =I0(ν)

Z ∞

−∞

e−σ(ν′)NF(ν₋ν′) 2π~_cdν′_{. (2.178)}

Here the column density, N; instrument function F(ν); and background intensity, I0(ν)

must be known a priori or determined as fitting parameters.

2.13

Rydberg states

A detailed discussion of electronic wavefunctions is beyond the scope of this thesis. How- ever, in the case ofRydberg states a qualitative, and sometimes quantitative, understanding of the observed spectra may be obtained without any explicit electronic analysis. In regard to what follows, Mulliken [121] has written a review discussing many of the properties of Rydberg states and aspects of the theory of quantum defects are discussed in the collection edited by Jungen [71].

For a neutral molecule, Rydberg states may be approximated as the product of a positive ionic core, Ψ+(r), and a single Rydberg-electron with an orbital wavefunction,

ψ(r), with the majority of its radial extent beyond that of the core region. In comparison, the definition of a valence state requires all electrons to be near the molecular core, R.

2 ˚A, so that all electrons in a valence state are spatially correlated.

The wavefunction of Rydberg state may be written, for some complete set of coordi- nates r,

Ψ(r) = Ψ+(r)ψ(r). (2.179)

In this case the Rydberg electron will not be sensitive to the detailed structure of the ionic wavefunction and instead will move in a roughly-spherical attractive potential. Its properties will then resemble those of an electron bound to a hydrogen nucleus. The quality of this approximation will improve as the mean orbital radius of the Rydberg

electron increases, and its correlation with the core electrons is reduced. The function Ψ+_{(r) is expressed, as for a neutral molecule, in terms of molecular-frame coordinates and}

a rotation with respect to the laboratory frame; whereas, Rydberg orbitals are expressed in terms of non-rotating coordinates, as is done for atoms. Then Eq. (2.179) may be written

|EJ_i=

E+J+Ω+ ERnlλ

. (2.180)

Here, the core state, with +-superscripts, is written in Hund’s case (c) form, but may correspond to other limiting Hund’s cases. The Rydberg electron is written in terms of principal and orbital angular-moment quantum numbers, n and l, as in the atomic case. The total energy and angular-momentum are divided between the core state and Rydberg electron, and E+ is equivalent to the ionisation energy of the neutral molecule. Because there is always some correlation between the Rydberg electron and the non- sphericity of the core, the projection of Rydberg angular-momentum on the internuclear axis may be quantised according to the quantum numberλ. The goodness’ ofland λare mutually exclusive and depend on the validity of the wavefunction separation expressed by Eq. (2.179). Frequently, an intermediate case occurs where neither l norλare strictly defined, but both form convenient labels and so are included in Eq. (2.180).

The energy levels of an experimental series of Rydberg states of increasing n, based on a common ionic core, and with common l, may be fit to the following form converging on the ionisation energy,

En=E+−Ry/(n−δn)2, (2.181)

whereRy is the Rydberg constant and thequantum defect,δ, may be treated as a fitting parameter. The Rydberg constant is slightly mass dependent and in the case of 14N2 has

a value of 109 735 cm−1.

The parameter δn is roughly analogous to the phase shift of scattering theory if the Rydberg electron is considered to be mostly remote from the ionic core, but periodically scattering from it. Alternatively, the inner part of the radial Rydberg-wavefunction will always deviate from that of a hydrogen orbital because of correlations with the core wave- function, and the details of the interactions occurring in this region are encoded in δn. For Rydberg series with l > 0, δn is observed to approach a constant, nonzero, value for large n. This trend is consistent with the radial charge density distribution of analytic hydrogenic wavefunctions [143, p. 117] which are concentrated at increasing radius as n

increases, apart from the case ofl= 0 orbitals, for which there is no centrifugal repulsion. The asymptotically stable value,δn→δ, may be exploited during the extrapolation of Eq. (2.181) to unobserved energy levels. Further approximate properties of high Rydberg states may be predicted from δ alone [90, pp. 128, 570, 668; Chap. 8], and are discussed here briefly.

The increasing centrifugal barrier experienced by electrons of successively higher-l

rapidly reduces their overlap with the core region. However, the shape of the core region wavefunction for constantland increasingnremains largely unchanged, merely scaling in magnitude according to _∼(n₋δn)−3/2.

If there is a perturbative matrix element operating between a Rydberg state and va- lence state, then an approximate factorisation may be made,

§2.13 Rydberg states 51

The first factor describes any interaction between the Rydberg-core and valence states and will be constant with regard to the state of the Rydberg electron. The contribution of the Rydberg electronic wavefunction to the second factor will be zero except for the innermost part where it overlaps with the valence wavefunction. Then, in consideration of the above approximate scaling of the Rydberg radial wavefunctions,

hcore+nlλ_|Hpert_|valence_{i ∝∼}(n₋δn)−3/2. (2.183) An application of this relation may be made to the decrease in absorption strength with increasing nobserved for many Rydberg series, in which case the electric-dipole operator of Eq. (2.67) must be substituted forHpert_.

Equation (2.183) may be extended to the case of two mutually-perturbing Rydberg states. When these are built upon different cores, or the same core with different l, their interaction energy scales as, approximately

hcore+_i niliλi|Hpert|core+j njljλji ∝_∼(ni−δi)−3/2(nj−δj)−3/2. (2.184) Rydberg states built upon a common core, and with common l and λ, but having different values of n; are assumed to be completely noninteracting. For the case where both states have high principal quantum numbers, their wavefunctions will be hydrogenic and orthogonal, prohibiting electronic interactions; and the Rydberg electrons will be completely independent of the nuclear-coordinates and so these states will not be mixed by rotational or nonadiabatic perturbations.

An important case occurs for two Rydberg states,i and j, with common ion core, n, and l; but where λi−λj = 1. Under these circumstances it must be that Λi−Λj = 1 and the two states differ by only one electron orbital, namely the Rydberg electron. All conditions are then satisfied for the existence of a rotational perturbation between the two states and the magnitude of the interaction may be simply estimated by considering the

pure precession approximation, with detailed discussions given by Hougen [56, Sec. 4.3] and Lefebvre-Brion and Field [90, p. 327]. The relevant off-diagonal matrix element may be derived from Eq. (2.47) where, because the configurations of the perturbed states differ only in the Rydberg electron,L± is equivalent tol±. Then,

hi_|Hrot(R)_|j_i=DJ(Ωi+ 1)core+nl(λ+ 1) − 1 2µR2(J +_l−_+J−_l+₎ JΩicore +_nlλE (2.185) = −~ 2 2µR2 [J(J+ 1)−Ω(Ω + 1)] 1/2_{[l(l+ 1)} −λ(λ+ 1)]1/2. (2.186) Equation (2.185) may be applied to the important p-complex that is partially com- prised of the cn1Πu and cn′+11Σ+u Rydberg states of N2 [12]. These two states have the

same ionic core complemented by the addition of an npπ and npσ Rydberg electron, re- spectively. Thenpπ configuration is doubly degenerate, with alternative orbitals labelled

npπ+ and npπ−, having orbital angular-momentum which is aligned and anti-aligned, respectively, with the internuclear axis. This degeneracy is similar to that discussed in Sec. 2.1 with respect to the doubling of states with Λ>0. The orthogonal linear combina- tions 2−1/2(npπ++npπ−) and 2−1/2(npπ+−npπ−) must be formed in order to generate states of definite e/f parity. According to the selection rules of rotational perturbations, nonzero matrix elements may only exist between these and a non-degenerate npσ orbital

of the same parity.

For the case of cn1Πu and cn′+11Σ+u, it is the e-parity levels that are affected and Eq. (2.185) may be expanded according to

hcn1Πu|Hrot(R)|c′n+11Σ+ui= D J1 core+npπ − 1 2µR2(J +_l−_+J−_l+₎ J0 core +_npσE (2.187) = −~ 2µR2 p J(J+ 1)_·2−1/2 npπ+ l+ npσ + npπ− l− npσ . (2.188) Then, evaluating the remaining matrix elements by means of Eq. (2.40) ,

hcn1Πu|Hrot(R)|c′n+11Σ+ui= −

~2

2µR2

p

Chapter 3

Spectroscopy of N

₂

Good reviews of N2 spectroscopy up to their publication are given in the exhaustive works

of Lofthus and Krupenie [111] and K. P. Huber and G. Herzberg [72]. A further brief summary of the difficulties encountered in the study of those N2 excited states observed in

the XUV is given by Carroll and Hagim [15]. While the subsequent study of N2 has been

far from dormant, the overall picture of the ground and excited states remains largely unchanged. This chapter falls far short of an exhaustive review of the N2 spectrum and

the means by which it has been observed. Instead, an outline is provided here, in terms of a selection of experimental and theoretical investigations which have proved most useful to this thesis.