3.1 Excitation spectra
The excitation spectrum corresponding to the ns, np and nf ← A2Σ+ (0, 0) transition is shown in the upper panel of Fig. 2 where unresolved forward-propagating far-infrared emission is monitored. The most prominent band is the nf ← A2Σ+ (0, 0) transition. A small inclusion of a d Rydberg character in the A2Σ+ state makes the f ← A2Σ+ transitions partially allowed in spite of Δl = 2. The nf state belongs to the intermediate case between Hund’s case (b) and (d) [17]. The spectral notation is represented by \({}_{ }{}^{{R}^{\text{'}}-{N}^{\text{'}\text{'}}}{\varDelta N}_{\mathcal{L}}\)where R' (often denoted by N+) is the rotational quantum number of the core, N'' the total angular momentum of A2Σ+ apart from spin, and \(\mathcal{L}\) the quantum number of the orbital angular momentum along the rotational axis. Among nine allowed branches, three Q lines (−2Q2, 0Q0 and 2Q−2) are assignable in the nf (n = 13–15) Rydberg series.
The np (n = 14, 15, and 16) states were also detected with a considerable intensity in the same energy region. In our previous experiment, we could observe np (n = 12–15) peaks while no trace of the np (n = 11) state was detectable. Since two-color multiphoton ionization (MPI) spectroscopy by Anezaki et al [17, 19]. exhibited a considerable line broadening for np (n = 11), we attributed the disappearance of np (n = 11) to the short lifetime (~ 3 ps) due to the strong predissociation. The appearance of np (n = 14 ~ 16) suggests that, in these states, the degree of predissociation is weaker than np (n = 11). The np states in this energy region belong to the intermediate case between Hund’s case (b) and (d). Anezaki et al. [18, 19] observed three allowed branches, 0Q0, 0R1, and 0P−1, in their np (n = 13, v' = 1) ← A2Σ+ (v'' = 1, N'' = 3) MPI spectra, where the strongest 0Q0 line is accompanied by unresolved mixed 0R1 and 0P−1 lines at a few wavenumbers higher side. The np (n = 15, v' = 0) ← A2Σ+ (v'' = 0, N'' = 3) band in Fig. 2 exhibits a sharp line with a weak shoulder in the higher wavenumber side. Judging from the analogous band shape, the line at the low wavenumber side would be attributed to 0Q0. In the np (n = 14 and 16) ← A2Σ+ (v'' = 0) bands in Fig. 2, the peak apparently consists of a single line which would be assigned to 0Q0.
In addition to the excitation of nf and np series, the excitation to the ns series was observable in the vicinity of the (n – 1)f band at higher laser intensities as shown in the lower panel of Fig. 2. The ns (n = 14) ←A2Σ+(0, 0) bands consists of P (ΔN = − 1) and R (ΔN = + 1) branches due to a 2Σ+ - 2Σ+ type transition. The weakness of the ns (n = 14) must be partly due to the strong predissociative character. The ns (n = 15) state is also detected as the same spectral pattern near the 14f band.
The transition wavenumber for the nf, np and ns ← A2Σ+ (vA = 0) transitions is listed in Tables 1, 2 and 3, respectively. The term energy values for these Rydberg states are summarized in Table 4. It is seen that tor the nf Rydberg series, the rotational levels having the same R fall almost in the same energy. It should be also noted that the lowest rotational level (N = 0, J = 0.5) of the 14s state is located ~ 10 cm− 1 below the lowest level (R = 0) of 13f. On the other hand, although only the R = 3 rotational level was observed for the np Rydberg series the these energy values are located ~ 17 cm− 1 higher than the corresponding R = 3 levels of the (n – 1)f series.
3.2 Emission spectra
3.2.1 nf series
The left panel of Fig. 3 shows the dispersed emission spectrum from the nf (n = 13–15) states. The rotational quantum number Nnf is the same as that (NA = 3) of the A2Σ+ state, and \(\mathcal{L}\) = 0 (R' = 3) and 2 (R' = 1) levels are populated through the 0Q0 and −2Q2 excitation lines in Fig. 2, respectively.
Referring the emission patterns for the lower nf Rydberg states [16], we assigned the newly detected emission around 88 µm after the excitation of 13f to the transition from the laser-prepared 13f state down to the 12g Rydberg state, 13f → 12g, as illustrated in the right panel. This assignment is straightforward because no lower states other than the 12g state exist with the energy gap corresponding to ~ 88 µm. No transitions to the nearby levels such as 13f → 12d, 13p, 13s were detected.
The emission wavelength from 13f (R = 1, \(\mathcal{L}\) = 2) (trace (a)) coincides with that from 13f (R = 3, \(\mathcal{L}\) = 0) (trace (b)) within our energy resolution. This is explainable by considering (1) the rotational selection rule for the f \(\leftrightarrow\) g transition and (2) the energy level structure of the f and g states.
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Fujita and Morita [20] reported that the ng ← 4f transition was governed exclusively by the single rotational line corresponding to the ΔR = 0 transition. Therefor the single line in trace (a) and (b) must be assigned to 13f (R = 1, \(\mathcal{L}\) = 2) → 12g (R = 1) and 13f (R = 3, \(\mathcal{L}\) = 0) → 12g (R = 3), respectively.
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The g state can be treated as a pure case (d) coupling and the rotational energy is given with rotational quantum number R by
E rot = BR(R + 1),
which is independent of the quantum number \(\mathcal{L}\). The energy terms of the f state are also well approximated with the above equation. Since the rotational constant (B ~ 1.97 cm–1) is not significantly different between the f and g states, the above two emission lines, more generally speaking, nf → (n – 1)g ΔR = 0 transitions appear almost at the same position.
In a similar manner, the emission around 112 µm in trace (c) and (d) after the excitation of 14f is assigned to undoubtedly the transition from the laser-prepared 14f state down to the 13g state, 14f → 13g. Then the emission around 88 µm should be assigned to the cascade 13g → 12f transition. Again no transitions from the laser prepared 14f state down to the nearby s, p and d levels were detected.
Analogous consideration may lead to the expectation that, when the 15 f state is excited, the initial transition should be 15f → 14g as marked in the dotted arrow, though the corresponding wavelength is ~ 140 µm is out of the sensitivity range of the detector used. The appearance of the cascade emission around 112 and 88 µm, which is the 14g → 13f and 13f → 12g transition, respectively, certified the 15f → 14g transition as an initial optical step.
In addition to a similar level structure, the absolute term values for the same quantum number R are very close each other for the nf and ng states. For instance, the term values for 12g (v = 1, R = 2) and 12f (v = 1, R = 2) is reported to be 76134.5 and 76133.5 cm− 1, respectively [21], being the energy difference Δ12f−12g ~ 1.0 cm− 1. The term values for 15g (v = 1, R = 2) and 15f (v = 1, R = 2) is 76589.5 and 76588.7 cm− 1, respectively, being the energy difference Δ15f−15g ~ 0.8 cm− 1. Such identical energy structure causes the same emission wavelength of 14g → 13f (trace (e)) and 14f → 13g (trace (d)) as well as 13g → 12f (trace (d)) and 13f → 12g (trace (b)).
3.2.2 np series
The left panel of Fig. 4 shows the dispersed emission spectrum from the np (n = 14 and 15) states. If we assume that the 0Q0 branch is responsible for the excitation as mentioned earlier, the upper levels are designated as 14, 15p (R = 3, N = 3, \(\mathcal{L}\) = 0).
From comparison of Fig. 4 with Fig. 3, we tentatively assigned the emission line around 88 µm (trace (a)) to the 13f → 12g transition as illustrated in the right panel, though the possibility of 13g→ 12f appearing at the same wavelength can’t be ruled out. Then the peak around 67 µm is assigned to the cascade 12g → 11f transition. Two possibilities are considered for the population transfer from 14p down to 13f. First, the 13f state is generated form 14p by collisions. Second, the optical transition, 14p → 13f, occurs in spite of its forbidden character (Δl = 2). The weak f – d mixing, namely, the inclusion of the d character in the f state, can drive the p → f transition. In this sense, the direct optical transition, 14p→ 13g, would be unlikely to occur because the g Rydberg state is considered to be isolated. The 14p (R = 3, \(\mathcal{L}\) = 0) → 13f (R = 3, \(\mathcal{L}\) = 0) transition corresponds to the emission wavelength of ~ 580 µm. We could not argue if the optical transition, 14p → 14s, is occurring or not, because this transition is expected to appear at ~ 200 µm.
Accepting the discussion above, the assignment for the 15p state (trace (b)) is rather straightforward; 112 and 88 µm emission lines are undoubtedly 14f → 13g and the subsequent cascade 13g → 12f transition, respectively. The absence of the emission around 67 µm, 12f → 11g, would be due to the small population in the 12f state. It should be noted that the emission around 112 µm is absent in trace (a). This means that 14f, which is located ~ 30 cm− 1 higher than 14p, is not produced by collisions.
3.2.3 ns series
Figure 5(a) shows the comparison of the emission from 14s (N = 2, J = 2.5; red trace) and from 13f (R = 2, N = 2, \(\mathcal{L}\) = 0; black trace). As mentioned in 3.2.1, the first step from the nf Rydberg states is restricted to the optical transition down to the nearest (n – 1) g Rydberg states. Thus the peak at 88 µm is certainly assignable to the 13f → 12g transition. The emission from 14s apparently coincides with that from 13f. Because no trace of the 14s → 12f emission expected at around 105 µm is observed, the population in the 14s state is almost exclusively transported to the 13f state, or possibly 13g state. As listed in Table 4, the lower rotational levels of the14s state is located energetically lower than the 13f (R = 0) state, which indicates the occurrence of the upward population transfer channel. We could not argue the existence of the most probable optical transition, 14s → 13p, is occurring or not, because this transition is expected to appear at ~ 182 µm. For the lower ns (n = 9, 10 and 11) states, the ns → (n – 1)p transitions were observed strongly [12].
A similar emission pattern was recorded for the excitation of the 15s (N = 2, J = 2.5) level as shown in Fig. 5(b). Upon the excitation of 15s, the major emission bands are 14f →13g at 112 µm and the subsequent 13g →12f transition at 88 µm. As mentioned in 3.2.2, the upward transfer process is considered to be induced by collisions.
3.3 Mechanism of the generation of far-infrared radiation
Figure 6 illustrates a schematic diagram for the generation of directional emission driven by BBR. A part of the population in the ground state (State I) is excited to the high Rydberg state (State II). The Rydberg states can interact with BBR at room temperature and a part of the population in State II is transferred to the lower Rydberg state (State 3). Because the population inversion is formed between State II and III, the corresponding wavelength is amplified through a stimulated emission process along the incident laser radiation.
In the presence of thermal radiation, BBR-induced transition rate \({K}_{{n}^{\text{'}},n}\) is given by
$${K}_{{n}^{\text{'}},n}=\stackrel{-}{n}{A}_{{n}^{\text{'}},n}$$
,
where \({A}_{{n}^{\text{'}},n}\) and \(\stackrel{-}{n}\) are Einstein coefficient and a photon occupation number, respectively [22]. The photon occupation number \(\stackrel{-}{n}\) is defined as
$$\stackrel{-}{n}= \frac{1}{\text{e}\text{x}\text{p}\left(h{\nu }_{{n}^{\text{'}},n}/kT\right)-1}.$$
As shown in Fig. 3, the initial energy dissipation pathway from 14f is limited to the transition down to 13g at ~ 112 µm; no trace of the 14f → 12g transition at ~ 49 µm was seen. The photon occupation number for 14f → 13g and 14f → 12g is 1.9 and 0.6, respectively. The BBR transition rate with Δn = 1 is ~ 3 times faster than that with Δn = 2, suggesting that the transitions at longer wavelength is preferentially amplified.