Line intensities and spectra

Line strength

The integrated absorption cross section of a line is (Tellinghuisen 2011)

\[ S = \frac{2\pi^2\,\nu}{3\varepsilon_0 h c}\;\frac{s_{J'J''}}{2J''+1}\;\left|\langle v'J'|\,\mu_e(R)\,|v''J''\rangle\right|^2\; f(v'',J''), \tag{1} \]

where s = J″ for P lines and s = J″ + 1 for R lines is the Hönl–London factor, μₑ(R) is the electronic transition moment and f is the fractional population of the lower level,

\[ f = \frac{g_\text{ns}(J'')\,(2J''+1)\,e^{-c_2 E''/T}}{Q(T)},\qquad Q(T) = \sum_{v'',J''} g_\text{ns}(J'')\,(2J''+1)\,e^{-c_2E''/T}, \tag{2} \]

with g_ns the nuclear-spin statistical weight and c₂ = hc/k_B = 1.4388 cm K. The transition moment is that of Tellinghuisen (2011), |μₑ(R)|² = 21.5 exp[−1.18 (R − 3.65)²] / R² D², with R in Å.

The matrix element in Eq. (1) is computed with the J-dependent vibrational wavefunctions of both states, so the variation of the intensity with J within a band (Herman–Wallis effect) is included. Only f depends on temperature. The line list therefore stores the temperature-independent quantity S₀ = S Q(T) exp(c₂E″/T), from which S is obtained for any T.

Figure 1. Vibrational wavefunctions of X v″ = 0 and of B v′ = 5 and v′ = 32, drawn at their energies, and the transition moment μₑ(R) (lower panel). The band strength is proportional to the square of the overlap ⟨v′|μₑ|v″⟩. The v″ = 0 wavefunction is localised near R = 2.67 Å (shaded). The overlap is largest for upper levels whose inner turning point lies in this region, where the upper wavefunction has its largest amplitude; for v″ = 0 this is the case near v′ = 30, at 500–540 nm.

Upper levels are included up to the last bound level of each J′. Absorption to the continuum above the B-state dissociation limit is computed separately by the Python package (i2spec.continuum).

Absorption spectrum

At temperature T each line is Doppler broadened to a Gaussian with full width at half maximum

\[ \Delta\nu_D = \nu\,\sqrt{\frac{8k_BT\ln 2}{mc^2}} \approx 4.5\times10^{-8}\,\nu\,\sqrt{T/\mathrm{K}} \quad (^{127}\mathrm{I}_2), \tag{3} \]

about 430 MHz at 532 nm and 293 K. The absorption cross section is

\[ \sigma(\nu) = \sum_i S_i\,g(\nu-\nu_i), \tag{4} \]

with g a normalised Gaussian (a Voigt profile, including pressure broadening, in the Python package). A cell of length L transmits

\[ \mathcal T(\nu) = \exp\left[-\sigma(\nu)\,N\,L\right], \tag{5} \]

where the number density N is determined by the iodine vapour pressure at the cold-finger temperature (Tellinghuisen 2011, Eq. 8).

Figure 2. Absorption cross section near 532 nm at 20 °C (upper panel) and transmission of a 10 cm cell with the cold finger at 20 °C (lower panel). Each peak is a single rovibronic line; the strongest are labelled.

The Python package also computes sub-Doppler spectra: Lamb dips, crossover resonances between components that share a level, and the third-harmonic line shape to which the BIPM frequencies refer.