Impact broadening, shifting, and asymmetry of the $D_1$ and $D_2$ lines of alkali-metal atoms colliding with noble-gas atoms

The Anderson Talman theory of spectral line broadening is used together with potential energy curves calculated at the spin-orbit multi-reference configuration interaction level to compute broadening, shifting, and asymmetry coefficients of the $D_1$ and $D_2$ lines of alkali-metal atoms M, as they collide with noble gas atoms N, where $M=\mathrm{K},$ Rb, and Cs, and $N=\mathrm{He}$, Ne, and Ar. Our calculated coefficients are compared to experimental results for a variety of temperatures. In all cases general agreement is observed for the broadening coefficients, while significant disagreement is observed for the shifting coefficients. We also compare our $\mathrm{K}+\mathrm{He}$ broadening and shifting results with fully quantum-mechanical calculations that employ the Baranger theory of collisional line broadening, and we compare our results with other semiclassical calculations. As with the comparison to experiment, closer agreement is observed for the broadening coefficients while the shifting coefficients exhibit significant disagreement. We use the natural variation between the difference potentials of the nine $M+N$ pairs to explore the relationship between potential and line shape as determined by Anderson-Talman theory and develop a picture for the mechanism that underlies the general agreement between theoretical and experimental results on the broadening coefficient and the general disagreement on shifting coefficients.

Figure 1

The $\Delta V_{{\Pi }_{1/2}}$ difference potentials for all $M+N$ combinations.

Figure 2

The $\Delta V_{{\Pi }_{3/2}}$ difference potentials for all $M+N$ combinations.

Figure 3

The $\Delta V_{{\Sigma }_{1/2}}$ difference potentials for all $M+N$ combinations.

Figure 4

The $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Sigma }_{1/2}}$ difference potential is plotted along with the corresponding integrands ${\alpha }_{\mathrm{int}}$ and ${\beta }_{\mathrm{int}}$ given by Eqs. (12) and (13), and $\theta (v,b)$ given by Eq. (9) calculated at $\overline{v}(T=500$ K). Note that ab initio calculations were not performed for $R<1.6\AA$ and the $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Sigma }_{1/2}}$ DP is linearly extended for values of $R<1.6\AA$.

Figure 5

Integrands ${\alpha }_{\mathrm{int}}$ and ${\beta }_{\mathrm{int}}$ given by Eqs. (12) and (13) computed using the $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Pi }_{3/2}}$ DP at $\overline{v}\left(T=1000\mathrm{K}\right)$ are plotted on the bottom. The integrals $2\pi \overline{v}{\int }_0^{b}d{b}^{\prime }{\alpha }_{\mathrm{int}}\left(b^{\prime }\right)$ and $2\pi \overline{v}{\int }_0^{b}d{b}^{\prime }{\beta }_{\mathrm{int}}\left(b^{\prime }\right)$ are plotted on the top and closely follow the effective hard-sphere values until $b=b_0$.

Figure 6

The integrand ${\alpha }_{\mathrm{int}}$ given by Eqs. (12) is computed using the $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Sigma }_{1/2}}$ DP for several different values of $\overline{v}\left(\mathrm{T}\right)$ and corresponds to the window of interruption at $b\approx 3.75\AA$ in Fig. 4.

Figure 7

Broadening and shifting coefficients computed using the $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Sigma }_{1/2}}$ DP. The $\alpha$ average and $\beta$ average results are an average over the Maxwell speed distribution while the $\alpha$ mean and $\beta$ mean results are computed using $\overline{v}\left(T\right)$. One cycle of the oscillation of $\alpha$ mean about $\alpha$ average begins with a local maximum at $T=875$ K, followed by a local minimum at $T=1000$ K, and ends with a local maximum at $T=1150$ K, and corresponds to the oscillations of ${\alpha }_{\mathrm{int}}$ into and then back out of the window of interruption in Fig. 6.

Figure 8

Predicted broadening (half-width) coefficients for the $D_1$ line of all $M+N$ combinations.

Figure 9

Predicted broadening (half-width) coefficients for the $D_2$ line of all $M+N$ combinations.

Figure 10

The $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Pi }_{1/2}}$, $\Delta V_{{\Pi }_{3/2}}$, and $\Delta V_{{\Sigma }_{1/2}}$ DPs are plotted on the bottom and the corresponding broadening coefficients $\alpha \left(\mathrm{T}\right)$ are plotted on the top. The $A{}^2{\Pi }_{3/2}$ and $B{}^2{\Sigma }_{1/2}^{+}$ broadening coefficients are plotted here separately. Their weighted average in Eq. (11) is used to compute the $D_2$ broadening coefficient.

Figure 11

The integrand ${\alpha }_{\mathrm{int}}(v,b)$ given by Eq. (12) is calculated using the $\mathrm{Cs}+\mathrm{He}$ $A{}^2{\Pi }_{1/2}$ DP and plotted for several different temperatures. As the temperature increases from 100 K in the bottom panel to 2000 K in the top panel, the value of $b_0$ drops from $7.2\AA$ down to $4.1\AA$. As $b_0$ decreases it crosses through a window of interruption corresponding to the maximum in $\Delta V_{{\Pi }_{1/2}}$ at $R=5.9\AA$ and becomes somewhat ambiguous at 1000 K.

Figure 12

Predicted shifting coefficients for the $D_1$ line of all $M+N$ combinations.

Figure 13

Predicted shifting coefficients for the $D_2$ line of all $M+N$ combinations.

Figure 14

The $\mathrm{Cs}+\mathrm{He}$ $\Delta V_{{\Pi }_{1/2}}$, $\Delta V_{{\Pi }_{3/2}}$, and $\Delta V_{{\Sigma }_{1/2}}$ DPs are plotted on the bottom and the corresponding shifting coefficients $\beta \left(T\right)$ are plotted on the top. The $A{}^2{\Pi }_{3/2}$ and $B{}^2{\Sigma }_{1/2}^{+}$ shifting coefficients are plotted here separately. Their weighted average in Eq. (11) is used to compute the $D_2$ shifting coefficient.

Figure 15

The shifting coefficient $\beta \left(T\right)$ computed using the $M+\mathrm{Ar}$ $\Delta V_{{\Sigma }_{1/2}}$ DPs. Negative values of $\beta \left(T\right)$ are caused by a very shallow well, approximately 0.5 ${\mathrm{cm}}^{-1}$ deep, in the $\Delta V_{{\Sigma }_{1/2}}$ DPs shown in the inset in Fig. 3 and illustrate the sensitivity of the shifting coefficients to the PECs.

Figure 16

Predicted asymmetry coefficients for the $D_1$ line of all $M+N$ combinations.

Figure 17

Predicted asymmetry coefficients for the $D_2$ line of all $M+N$ combinations.

Figure 18

A comparison of broadening coefficients computed using the semiclassical AT theory with other theoretical calculations. Loper [22] and Loper and Weeks [23] compute broadening coefficients using the quantum-mechanical Baranger theory with the same PECs used for the AT calculations. Note that radial derivative coupling is ignored for the Loper $D_1$ coefficients. Allard et al. [12] compute $\alpha \left(T\right)$ using the dipole autocorrelation formulation, and Mullamphy et al. [19] compute $\alpha \left(T\right)$ using quantum-mechanical Barringer theory. Both Allard et al. [12] and Mullamphy et al. [19] employ different PECs than those used for the AT calculations.

Figure 19

A comparison of Shifting coefficients computed using the semiclassical AT theory with other theoretical calculations. The Loper [22], Loper and Weeks [23], and Mullamphy et al. [19] calculations are described in Fig. 18.