Probing phonon-rotation coupling in helium nanodroplets: Infrared spectroscopy of CO and its isotopomers

We have recorded the $R\left(0\right){\nu }_{\mathrm{CO}}=1\leftarrow 0$ IR spectrum of CO and its isotopomers in superfluid helium nanodroplets. For droplets with average size $N\gtrsim 2000$ helium atoms, the transition exhibits a Lorentzian shaped linewidth of $0.034{\mathrm{cm}}^{-1}$, indicating a homogeneous broadening mechanism. The rotational constants could be deduced and were found to be reduced to about 60% of the corresponding gas-phase values (63% for the reference ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ species). Accompanying calculations of the pure rotational spectra were carried out using the method of correlated basis functions in combination with diffusion Monte Carlo (CBF/DMC). These calculations show that both the reduction of the rotational $B$ constant and the line broadening can be attributed to phonon-rotation coupling. The reduction in $B$ is confirmed by path integral correlation function calculations for a cluster of 64 ${}^4\mathrm{He}$ atoms, which also reveal a non-negligible effect of finite size on the collective modes. The phonon-rotation coupling strength is seen to depend strongly on the strength and anisotropy of the molecule-helium interaction potential. Comparison with other light rotors shows that this coupling is particularly high for CO. The CBF/DMC analysis shows that the $J=1$ rotational state couples effectively to phonon states, which are only present in large helium droplets or bulk. In particular, they are not present in small clusters with $n⩽20$, thereby accounting for the much narrower linewidths and larger $B$ constant measured for these sizes.

Figure 1

Mass spectra of pure and CO doped ${}^4\mathrm{He}$ droplets taken at different CO partial pressures in the pick-up chamber. The multiplier voltage is 1400 V, corresponding to a current amplification of 3500. The mean size of the droplets is 2600 atoms. Pure He droplets reveal a mass spectrum of lines decreasing in intensity to some hundred amu, reflecting the fragmentation process. For the doped droplets the peaks at 28, 56, 84, … gain in intensity. These peaks result from the ionization of embedded CO monomers, dimers, trimers, etc.

Figure 2

The experimental phonon-roton spectrum $ϵ\left(k\right)$ of bulk ${}^4\mathrm{He}$ (Ref. 4) (full line) and the helium spectrum plus molecular recoil energy, $ϵ\left(k\right)+{\hslash }^2{k}^2∕2M$, (dashed line) are shown together with the energies of the $R\left(0\right)$ transition for CO in the gas phase $\left(2B\right)$ and in ${}^4\mathrm{He}$ droplets $\left(2B_{\mathrm{eff}}\right)$.

Figure 3

The infrared spectrum of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ doped ${}^4\mathrm{He}$ droplets with mean size $\overline{N}=2600$. The solid line shows a Lorentzian fit to the experimental points.

Figure 4

Dependence of the integrated depletion signal of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ doped ${}^4\mathrm{He}$ droplets with mean size $\overline{N}=2600$ on the pick-up pressure. The straight line is a fit using a Poisson function. The maximum for the monomer signal is achieved at $p_o=2\times {10}^{-5}\mathrm{mbar}$.

Figure 5

Absorption spectra of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ in He droplets for different average droplet sizes $(\overline{N}=1200\text{–}3600)$. The line shape and the peak position remain unchanged for droplet sizes above $\overline{N}=2200$. Smaller droplets reveal a blue shift that becomes stronger with decreasing size and an increasingly asymmetric line shape. The dashed line visualizes the shift.

Figure 6

Spectra of ${}^{13}\mathrm{C}{}^{18}\mathrm{O}$, ${}^{13}\mathrm{C}{}^{16}\mathrm{O}$, and ${}^{12}\mathrm{C}{}^{18}\mathrm{O}$ doped He droplets with mean size $\overline{N}=2600$ (points). All spectra can be fitted to Lorentzian functions (solid lines). The resulting linewidths are given in Table . The intensity is given here as a percentage of the beam depletion.

Figure 7

Rotational energies of CO isotopomers plotted versus the ratios of the reduced masses (squares). ${\mu }_i$ is the reduced mass of the isotopomer $i$, with $i=1$ denoting the reference ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ species. ${\omega }_i$ is the measured frequency in He droplets, ${\nu }_i$ the band origin in the gas phase, and $2{\Delta }_z{B}_{\mathrm{eff}}$ the isotope-induced correction (see text). A linear fit allows the magnitude of the shift of the band origin $\alpha$ and the rotational constant $B_{\mathrm{eff}}$ of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ in He droplets to be determined, provided that $\alpha$ is identical for all isotopomers and that the difference in the rotational constants between all isotopomers is controlled by their respective reduced masses and isotope-induced correction only (see text). For comparison, we also show the rotational energies without the correction $2{\Delta }_z{B}_{\mathrm{eff}}$ (large dots). The isotope-induced shift of the intermolecular potentials $2{\Delta }_z{B}_{\mathrm{eff}}$ is opposite for ${}^{13}\mathrm{C}{}^{16}\mathrm{O}$ and ${}^{12}\mathrm{C}{}^{18}\mathrm{O}$ and is significant for the quality of the fit. All units in ${\mathrm{cm}}^{-1}$.

Figure 8

Cluster size dependence of the $a$-type and $b$-type $R\left(0\right)$ transitions for ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$, ${}^{13}\mathrm{C}{}^{16}\mathrm{O}$, and ${}^{12}\mathrm{C}{}^{18}\mathrm{O}$ in small helium clusters (Refs 31, 32) is shown together with the corresponding $R\left(0\right)$ transition frequencies in large droplets measured in this work. $N$ is the number of He atoms. The solid squares refer to reliable assignments, the hollow squares to tentative assignments of $a$-type frequencies, respectively, from the work of Tang and McKellar (Refs. 31, 32). The $b$-type frequencies are indicated by solid circles, with hollow circles similarly indicating tentative assignments. The horizontal dotted lines represent the transition frequency in large droplets reported in this work.

Figure 9

Comparison of helium interaction potentials for CO, HCN, and acetylene (HCCH). Also shown for comparison is the potential of the much heavier OCS molecule with helium. $r=\mid \mathbf{r}\mid$ where $\mathbf{r}$ is the position of the ${}^4\mathrm{He}$ atom in the center of mass frame of CO. $\theta$ is the angle between the CO symmetry axis and $\mathbf{r}$. The orientation of the molecule is defined by the atomic symbols. Contour lines are shown in steps of 5 K.

Figure 10

(Color online) The anisotropic component of the CO-helium pair distribution function $g(r,\cos \theta )$, i.e., the pair distribution function without the spherical contribution in a Legendre expansion, $g(r,\cos \theta )-g_0\left(r\right)$.

Figure 11

Comparison of the $\ell =1$ components $g_1\left(r\right)$ of the Legendre expansion of the pair distribution function $g(r,\cos \theta )$ between helium and HCN and CO, respectively, as obtained from DMC simulation in bulk helium. Note that for HCCH, $g_1\left(r\right)$ vanishes due to symmetry (Ref. 30).

Figure 12

The CBF/DMC self-energy momentum density ${\sigma }_1(\omega ,k)$ (scaled by the gas phase value of $B$) is shown for the $J=0\to 1$ transition of CO, HCN, and HCCH (upper panel). Note that, for HCCH, this transition is only allowed in the rovibrational spectrum. The lower panel compares ${\sigma }_1(\omega ,k)$ of CO with that for a hypothetical symmetric CO molecule, for which only even $\ell$ components of the pair density distribution are retained in calculation of ${\sigma }_1\left(k\right)$.

Figure 13

The renormalization of the rotational levels in He droplets is shown as a function of the cluster size $N$ for CO from this work and compared with experimental and theoretical values for several different molecules. We show $E(J=1)∕E{(J=1)}_0$ instead of $B_{\mathit{eff}}∕B_0$, since experimental values of $B$ are sometimes but not always fit together with the distortion constant $D$, which modifies the fit values. Experiment: ${\mathrm{CO}}_2$ values from Refs. 93, 94, ${\mathrm{N}}_2\mathrm{O}$ values from Refs. 92, 94, OCS values from Refs. 90, 91. Theory: ${\mathrm{CO}}_2$ from Refs. 89, 95, ${\mathrm{N}}_2\mathrm{O}$ from Refs. 87, 95, OCS from Refs. 38, 86, 88, 95.

Figure 14

The $R\left(0\right)$ absorption line profile of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ in large He droplets $(\overline{N}=2600)$, compared with the corresponding profile measured in $\mathrm{CO}\text{−}{\left(\mathrm{He}\right)}_N$ small clusters $(N=2\text{–}20)$ (Refs. 30, 32). A fit of the Lorentzian-shaped large droplet spectrum yields a FWHM linewidth of $0.034{\mathrm{cm}}^{-1}$. The linewidth of $0.0033{\mathrm{cm}}^{-1}$ for small clusters was taken from Refs. 31, 32. This small cluster value is limited by the spectral width of the laser source. For better comparison of the line shape, the spectral frequency of CO in small clusters has been shifted to the transition frequency observed here in large droplets