3.2 The ground state
3.3.3 Linewidths and lifetimes
All1Πuand1Σ+u levels lie above the dissociation energies of at least two electronic states, as listed in Tab. 3.3, and may predissociate. The observed linewidths of these levels, or equiv- alent lifetimes, show significant and haphazard vibrational, rotational and isotopomeric dependence; suggesting the widespread presence of indirect predissociation. The varia- tion of measured linewidths for 1Πu levels below 105 000 cm−1 is plotted in Fig. 3.7. This wanders over more then three orders of magnitude as well as demonstrating isotopomeric dependence.
Many experimentalists have attended to the study of N2 predissociation because of
its curious nature. The quantitative observation of predissociating levels is usually ac- complished by one of three techniques: the measurement of predissociative broadening
Atomic Products De Ref. Progenitors 4S+4S 79844 [134] X1Σ+ g 4S+2D 99113.5 [111] C′3Π u 4S+2P 108729.1 [111] 2D+2D 118294 [134] C3Πu,b1Πu, 23Σu 2D+2P 128652.5 [111] b′1Σ+ u
Table 3.3: Asymptotic energies, De, of the lowest four dissociation channels, relative to the
bottom of the N2 X1Σ+g potential-energy curve. Dissociation limits, D0, for14N2, 14N15N, and 15N
2may be found by subtracting the energy ofX(v= 0, J = 0); 1175.7, 1156.1, and 1136.1 cm−1;
respectively. The first ionisation limit occurs at 125 667.5 cm−1 above the neutral ground state.
Also listed are some of the molecular electronic-states that dissociate to a particular limit.
absorption resonances, the detection of dissociation products, or in time resolved mea- surement of a decaying excited state. The first and last methods are directly related by Eq. (2.79), and all must allow for the component of the observed linewidths (lifetimes) that arises from radiative decay. Because of the variable nature of N2 predissociation, the
probability of dissociative decay varies across the spectrum, from completely dominating to entirely negligible.
Photographic recording of spectra has permitted the qualitative assessment of N2 pre-
dissociation by noting the appearance of diffuse absorption features, along with an absence of corresponding emission [13, 111]. In particular, it was noted by Carroll and Collins [13] that features corresponding to b1Πu(v′= 0,2,3,4)←X1Σ+g(v′′= 0) appeared diffusely in their absorption measurements of 14N2, in line with the lack of observed emission from
these excited levels. In most cases, linewidths may not be determined precisely from pho- tographic spectra because of the nonlinear response of photographic emulsion. Lewiset al.
[104] did, however, deduce the 3.3 cm−1FWHM broadening evident in absorption spectra pertaining to the excited level b1Πu(v= 3).
The photoelectrically-detected absorption-spectra of Stark et al.[155], and new mea- surements discussed in Chap. 4, allow for the quantitative measurement of resonance linewidths for several14N2bands between 100 000 and 118 000 cm−1. The experimental res-
olution of these experiments permit the detection of broadening greater than 0.1 cm−1FWHM. Similar synchrotron-based measurements are discussed in Chap. 5 which have been made at higher energies, and slightly higher resolution, using a Fourier transform spectrometer. The group of Ubachs et al. examined the predissociative properties of 14N2, 14N15N,
and 15N2 by employing two methods [46, 129, 147, 150, 152, 169, 170, 171, 172, 177, 179].
First, when examining the most predissociated levels the low-bandwidth of the frequency- multiplied laser used by this group, combined with supersonic cooling of the target N2,
enabled the measurement of absorption linewidths as narrow as 0.01 cm−1FWHM. In a second experiment, rotational levels of a particular excited-state were pumped by a pulse of XUV radiation, then subsequently ionised by a precisely delayed ultraviolet pulse, and the ionisation products detected. In this way, a decay time constant was observed di- rectly. This method is suitable for the most weakly predissociated, and longest lived, excited levels; in which case, the de-excitation will occur by a mixture of dissociation and fluorescence. The two experimental techniques employed by Ubachs et al. are complemen- tary and exclusive; short-lived states decay too quickly to be measured by the pump-probe technique, and long-lived states are too narrow to be deconvoluted from the instrumental bandwidth.
§3.4 3Πu states 65 14N2 14N15N 15N2 b1Π u v= 0−14 0,1,5,6, 0−9 c31Πu 0−5 0 0,1 o31Πu 0−3,5 0,1 b′1Σ+ u 1,4−9,11−22 1,5,6 c′ 41Σ+u 0−2,4−6 1 0,1 c′ 51Σ + u 1
Table 3.4: Excited vibrational levels of N2 1Πu and 1Σ+u states for which some linewidth infor-
mation is available. This data originates from many sources which are listed in the text.
In this case, observation of N2 photoabsorption was achieved by direct measurement of
the absorption-attenuated laser intensity, measured relative to a reference intensity fur- nished by a beam splitter. The achieved laser bandwidth was 0.01 cm−1FWHM and
the target was cooled by supersonic expansion. Natural linewidths of the excited levels
b′1Σ+u(v = 8,11), o31Πu(v = 2,3), and b1Πu(v = 12) were observed. The narrowest of these, b1Π
u(v= 12), was found to have a linewidth of only 0.06 cm−1FWHM.
Shemansky and coworkers [1, 64, 108, 109, 139] have observed the dispersed emission from a number of electronically-excited levels of N2. These rotationally-resolved mea-
surements of line strength provide an estimate of the branching ratio between emissive and dissociative decay pathways, once combined with a detailed knowledge of the optical oscillator-strengths and electronic-excitation rates of the observed levels.
The distribution of atomic products following the dissociation of a particular excited level is also of interest. In most cases a single predissociation channel dominates, but near the dissociation energies listed in Tab. 3.3 significant branching may occur to multiple channels. The work of Cosby et al. [23, 52, 53, 181, 182, 183] includes measurements of rotationally-resolved predissociation branching-ratios for a number of1Πu and1Σ+u states excited from an intermediate electronic-state, a′′1Σ+g.
Several other experiments have determined predissociation linewidths for particular excited-levels by using optical [125] or electron scattering [55, 73, 173, 191] techniques.
Table 3.4 lists all levels for which knowledge of linewidths has been obtained. The precision of this information, and the rotational levels to which it applies, vary widely according to the particular experimental source. Many further levels have upper bounds on their linewidths, imposed by the instrumental resolutions of various measurements in which no broadening was detected.
3.4
3Π
ustates
Identities for the electronic states responsible for the predissociation of N2 singlet levels
were first considered by Dressler [30] and Carroll and Collins [13]. Both works cited the states C3Πu and C′3Πu, with potential-energy curves for these plotted in Fig. 3.8 using a diabatic representation.
At all energies relevant to N2 electric-dipole-allowed transitions, C′3Πu is unbound and provides the most likely ultimate dissociation channel for the predissociation of 1Πu states. The C3Π
u state is strongly electronically-coupled to C′3Πu [16] and a second spin-orbit coupling to the isoconfigurational b1Π
u could explain the observed variability of 1Πu predissociation. That is, those 1Πu levels which are accidentally degenerate with
C3Πu levels will be indirectly coupled to the unboundC′3Πu and appear broadened, or with reduced lifetimes, in observed spectra. Lewis et al.[104] successfully confirmed the
1.0 1.5 2.0 2.5 Internuclear Distance (Å) 9.0 10.0 11.0 12.0 Potential Energy (10 4 cm − 1 ) 18 16 8 7 2 1 0 0 1 2 3 45 3 2 1 0 0 1 C C’ F III G N(4S)+N(2D) N(4S)+N(2P) N(2D)+N(2D) Figure 3.8: Some 3Π
u states of N2, after Lewiset al. [105]. Dashed: Adiabatic potential-energy
curve of III3Π
u calculatedab initio by Partridge [128]. Solid: Diabatic potential-energy curves
ofC3Π
u, C′3Πu,F33Πu, andG33Πu, deduced by Lewis et al.[105]. Term energies of all known
vibrational levels for these states are plotted as horizontal lines.
mechanism postulated by Dressler [30] and Carroll and Collins [13] by reproducing the observed predissociation ofb1Π
u(v= 0−6) andc31Πu(v= 0) levels for14N2,14N15N, and 15N
2by means of a coupled-channels model that includedC3ΠuandC′3Πu. This analysis deduced values for the matrix elements governing electronic-coupling between 3Πu states, and the spin-orbit interactions that mixes these with the1Πu states.
Severalab initio investigations have been made of the valence statesC3Π
u andC′3Πu [9, 25, 33, 42, 122, 128], including the calculation of adiabatic potential-energy curves. The electronic configuration of the diabaticC3Πu is stronglyR-dependent, as can be seen from the unusual kink in its potential-energy curve near 1.4 ˚A in Fig. 3.8. Below this point, the principal configuration listed in Tab. 3.1 is strongly bound, whereas for large-R
the configuration changes, and is associated with a much larger equilibrium internuclear- distance. A diabatic formulation ofC3Πu where it is separated into two strongly-coupled states would likely remove most effects of configuration interaction. Similarly, the maxi- mum in the potential-energy of C′3Πu, near 2 ˚A in Fig. 3.8, suggests that this state could alternatively be devolved into two diabatic states, one of which is purely dissociative.
The direct detection of 3Πu levels is hindered by the electric-dipole selection rule re- garding multiplicity, which suppresses their optical accessibility from the ground state. Additionally, the considerable predissociation linewidths of many 3Π
u levels would pre- vent their discrimination in spectra. Those vibrational-levels that have nonetheless been observed are plotted in Fig. 3.8. Some of these measurements [45, 69, 76, 77, 112, 113, 173] were achieved following electron excitation, in which case the electric-dipole selection rules do not apply; and further levels have been observed at high-resolution following forbidden photoabsorption from the ground-state [98, 146, 148, 166]. Observation of the latter is made possible by the presence of spin-orbit perturbations with singlet levels. Certain opti- cal transitions between mutually excited triplet states are allowed and have been observed in emission and absorption [11, 16, 48, 86, 87, 98, 136].
In some cases, the presence of triplet levels may be deduced indirectly where perturba- tions in the energy levels or linewidths of singlet states indicate the presence of an invisible perturber [98, 151]. An example of this is shown in Fig. 3.9, whereby the reduced term values of15N
§3.4 3Πu states 67
0
5
10
15
20
J2.0
1.5
1.0
0.5
0.0
0.5
1.01.5
2.0
Re du ced te rm s ( cm 1 )Figure 3.9: Reduced term values of15N
2c31Πu(v= 1), as measured by Sprengerset al.[151]. A
third order polynomial has been subtracted from the deduced term values in order to highlight a of series level crossings withC3Π
u(v= 14).
between the pairs of rotational-levels, J = 3−4, 6−8, and 10−11. These are, in fact, due to the Ω = 0, 1, and 2 sublevels ofC3Πu(v = 14) [98]. Similarly structured perturbations occur elsewhere in the N2 spectrum, appearing in some cases as multiple peaks in the
rotational dependence of observed linewidths.
Further triplet states,F33Πu, and G33Πu, are known to exist with energies similar to
the observable1Πu and 1Σ+u levels, and may be characterised as Rydberg states. Specifi- cally, F33Πu is a configurational analogue of o31Πu, which differs only in the relative sign of the spin functions of the open-shell orbitals 3σg and 3pπu. Consequently, the potential energy curve ofF33Πu, plotted in Fig. 3.8, is very similar to that ofo31Πu, differing prin-
cipally by an ∼880 cm−1 R-independent energy shift of the former relative to the latter. Similarly related are the configurational analogues G33Πu and c31Πu, with an energetic separation of ∼630 cm−1. Ab initio calculations of the Rydberg 3Π
u states have been performed [24, 92].
Electronic interactions are configurationally permitted between all combinations of
F33Πu, G33Πu, C3Πu, and C′3Πu; and are likely to be large given the strength of cou- pling between isoconfigurational1Π
ustates. New determinations of the Rydberg–Rydberg interaction betweenF33Πu andG33Πu and Rydberg–valence couplings mixing these with C′3Πu are discussed in Sec. 6.6.1.
The Ω = 0 and 2 3Πu-substates are forbidden to interact with the Ω = 1 1Πu states in the case of a nonrotating molecule. However, the triplet substates become mixed with increasing J by the S-uncoupling operator of Sec. 2.5.2, and are then universally free to induce perturbations.
The predissociation of1Σ+
u levels has also been attributed to the3Πu states [98, 183]. This effect is even less direct than previously, and involves the 1Σ+
u ∼ 1Πu rotational- interaction discussed above as a further intermediate step towards dissociation.
Less is known of the 3Σ+
u states of N2 situated above 100 000 cm−1 than for the 3Πu case, but with regard to the perturbation of optically-observable levels, the Rydberg state
D3Σ+u is probably the most significant of the lowest energies. This is the configurational analogue of c′41Σ+u and there likely exists a significant off-diagonal matrix element of the spin-orbit operator which mixes these levels. Additionally, D3Σ+u and G33Πu form the lowest-n members of a Rydbergp-complex convergent on the ground state of the N2 ion,
along with c′41Σ+u and c31Πu. Significant rotational coupling of these two states is then
likely.
Various low-resolution observations of D3Σ+
on its v= 0−3 levels [45, 77, 173], including a determination of an extended D3Σ+u(v= 0) lifetime [81]. High-resolution measurements [39, 69, 74, 146] have led to molecular parameters for v= 0 and 1 in14N2, andv= 1 for15N2. Kanamoriet al. [74] observed line
broadening ofD3Σ+u(v= 1)←E3Σ+g[1]14N2 transitions with a quadratic dependence on J. They attributed this to predissociation of D3Σ+u(v = 1) mediated by a heterogeneous coupling. A new study of D3Σ+
u and its interactions with the 3Πu states is discussed in Sec. 6.6.4.
The first unbound state of 3Σ+u symmetry, labelled “2” in Tab. 3.3, has a sufficiently low dissociation energy that it may influence the predissociation of all1Π
u and1Σ+u levels. This is particularly likely where the potential-energy curve of 23Σ
u crosses those of b1Πu and b′1Σ+u near 1.7 ˚A [173, Fig. 1].