Collision-induced line-shape effects limiting the accuracy in Doppler-limited spectroscopy of ${\mathrm{H}}_2$

Recent advances in theoretical calculations of ${\mathrm{H}}_2$ dissociation energies and ultra-accurate measurements of ${\mathrm{H}}_2$ transition frequencies give possibilities not only for testing QED and relativistic effects, but also for searching for physics beyond the standard model. In this paper we show that at the level of ${10}^{-4}{\mathrm{cm}}^{-1}$ the uncertainty of the Doppler-limited ${\mathrm{H}}_2$ line position determination is dominated by collisional line-shape effects. We question the paradigm that the unperturbed transition energy can be determined from linear extrapolation of the line shift to zero pressure.

Figure 1

Shown on top is the simplest scheme of the zero-pressure line position extrapolation, which is adopted in our analysis. The bottom shows the uncertainty of the zero-pressure line position (with respect to the line position uncertainty at $p_2$) as a function of $p_1$ (with respect to $p_2$). The minimum of this function is indicated by the crossing of the gray lines. The symbols show the lowest pressure from the measurements of the ${\mathrm{H}}_2 Q$(1) (2-0) line [20] (blue dot), the ${\mathrm{H}}_2 O$(2) (2-0) line [18] (green diamond), the ${\mathrm{H}}_2 S$(3) (3-0) line [17] (red square), and the ${\mathrm{D}}_2 S$(1) (1-0) line [13] (orange star).

Figure 2

Shown on top is the nonlinear behavior of the line position as a function of pressure for the case of the ${\mathrm{H}}_2 Q$(1) fundamental line at $T=296$ K. The green dashed line shows the linear fit to two shifts at $p_2=1$ atm and $p_1=0.46p_2$ and its extrapolation to zero pressure gives the systematic error of the unperturbed line position. The bottom shows how this error scales with $p_2$, assuming that $p_1=0.46p_2$.

Figure 3

Comparison of the ${\mathrm{H}}_2$ (3-0) $S$(3) line position determined from experimental spectra with different approaches; see the text for details. Zero is shifted to the recommended value $12559.74939{\mathrm{cm}}^{-1}$. The theoretical value calculated by Komasa et al. [2] is also depicted.

Figure 4

Systematic contributions to the line position uncertainty budget illustrating the limitations of current Doppler-limited spectroscopy. In practice, the green and yellow bars can be reduced by at least one order of magnitude, since the pressure and temperature can be determined more accurately than the assumed $1\%$ for pressure and 1 K for temperature. It can be clearly seen that the measurements targeted on the ${10}^{-6}{\mathrm{cm}}^{-1}$ are limited by the collisional line-shape effects.