Spontaneous symmetry breaking and inversion-line spectroscopy in gas mixtures

According to quantum mechanics, chiral molecules, that is, molecules that rotate the polarization of light, should not exist. The simplest molecules which can be chiral have four or more atoms with two arrangements of minimal potential energy that are equivalent up to a parity operation. Chiral molecules correspond to states localized in one potential energy minimum and can not be stationary states of the Schrödinger equation. A possible solution of the paradox can be founded on the idea of spontaneous symmetry breaking. This idea was behind work we did previously involving a localization phase transition: at low pressure, the molecules are delocalized between the two minima of the potential energy while at higher pressure they become localized in one minimum due to the intermolecular dipole-dipole interactions. Evidence for such a transition is provided by measurements of the inversion spectrum of ammonia and deuterated ammonia at different pressures. A previously proposed model gives a satisfactory account of the empirical results without free parameters. In this paper, we extend this model to gas mixtures. We find that also in these systems a phase transition takes place at a critical pressure which depends on the composition of the mixture. Moreover, we derive formulas giving the dependence of the inversion frequencies on the pressure. These predictions are susceptible to experimental test.

Figure 1

Molecular structures of ammonia (${\mathrm{NH}}_3$, left) and deuterated disulfane (${\mathrm{D}}_2{\mathrm{S}}_2$, right) in one of their two localized states.

Figure 2

Critical pressure $P_{\mathrm{cr}}$ as a function of the chiral species fraction $x$ for ${\mathrm{NH}}_3$-He (bottom) and ${\mathrm{D}}_2{\mathrm{S}}_2$-He (top) mixtures at $T=300$ K. Note that pressure units are atm in the case of ${\mathrm{NH}}_3$-He and natm in the case of ${\mathrm{D}}_2{\mathrm{S}}_2$-He.

Figure 3

Coefficients $a_1/\sqrt{N_1}$ and $a_2/\sqrt{N_2}$ defining the molecular states ${\psi }_1,{\psi }_2$ of a two-species mixture as a function of $P/P_{\mathrm{cr}}$. The coefficients have been obtained by solving numerically Eq. (19) in the case of a ${\mathrm{NH}}_3\text{−}{\mathrm{ND}}_3$ mixture (${\mathrm{ND}}_3$ is species $i=2$) with $x_2=0.1$ at temperature $T=300\mathrm{K}$. The coefficients bifurcating at $P=P_{\mathrm{cr}}$ and tending for $P\gg P_{\mathrm{cr}}$ to $1/\sqrt{2}$ (dashed line) describe two degenerate molecular states, named $c_{LL}$ and $c_{RR}$, with molecules of both species in a chiral configuration of type $L$ or $R$. Notice further bifurcations appearing at $P\simeq 4P_{\mathrm{cr}}$, they correspond to stationary states of higher energy.

Figure 4

Parameters $q_1,q_2$ encoding the difference between the delocalized symmetric state $d_{++}$ and the chiral states $c_{LL}$ and $c_{RR}$ of Table 2 evaluated numerically for the same mixture of Fig. 3.

Figure 5

Energies $\hslash {\overline{\omega }}_{\pm }$ of the excitation normal modes as a function of pressure $P$ in the case of a ${\mathrm{NH}}_3\text{−}{\mathrm{ND}}_3$ mixture (${\mathrm{ND}}_3$ is species $i=2$) with $x_2=0.1$ at $T=300\mathrm{K}$. The horizontal dotted line is the asymptotic value $\hslash {\overline{\omega }}_{+}^{\mathrm{as}}$. The dashed lines which vanish at $P_{\mathrm{cr}}^{\left(1\right)}=1.69\mathrm{atm}$ and $P_{\mathrm{cr}}^{\left(2\right)}=0.11\mathrm{atm}$ correspond to the inversion frequencies of pure species. The critical pressure of the mixture, the point where $\hslash {\overline{\omega }}_{-}$ vanishes, is $P_{\mathrm{cr}}=0.70\mathrm{atm}$. Energy is measured in ${\text{cm}}^{-1}$.

Figure 6

As in Fig. 5 for $x_2=0.0001$. For such small value of $x_2$ the critical pressure of the mixture is $P_{\mathrm{cr}}\simeq P_{\mathrm{cr}}^{\left(1\right)}$ and we have $\hslash {\overline{\omega }}_{+}^{\mathrm{as}}\simeq \Delta E_{i_{-}}$. Inset is a zoomed-in view of the region $P\simeq P_{\mathrm{cr}}$.

Figure 7

Intensities $I_{\pm }$ of the absorption coefficient as a function of pressure $P$ for the mixture of Fig. 5. Energy is measured in ${\text{cm}}^{-1}$ and intensities, which have dimensions energy divided by length, are given in ${\text{cm}}^{-2}$.

Figure 8

As in Fig. 7 for the mixture of Fig. 6.

Figure 9

Eigenvalues $E_k$, $k=1,...,2^N$, divided by $N$, of the $N$-body Hamiltonian of Eq. (1) with ${\mathfrak{g}}_{ij}=G/N$ and $N=13$ calculated numerically for different values of $G$ (dots). The solid lines are the single-molecule energies $e_i=\mathcal{E}\left({\psi }_{{\lambda }_i}\right)/N$ associated with the mean-field stationary states ${\psi }_{{\lambda }_i}$, $i=1,...,4$.

Figure 10

Energy gap $E_2-E_1$ of the $N$-body Hamiltonian (1) with ${\mathfrak{g}}_{ij}=G/N$ and $N=3$, 5, 7, 9, 11, 13 (dashed lines from top to bottom) as a function of the ratio $G/\Delta E$. The solid line represents $\hslash \overline{\nu }/\Delta E$, where $\overline{\nu }$ is the inversion doubling frequency predicted by Eq. (B2).

Figure 11

Plot of the universal function of Eq. (C1). Points are the experimental results for ${\mathrm{NH}}_3$ (circles) or ${\mathrm{ND}}_3$ (squares) available from Refs. [9, 10, 11], respectively. The experimental data have been scaled using the values of $\Delta E$ and $P_{\mathrm{cr}}$ from Table 1.