$s$-wave scattering for deep potentials with attractive tails falling off faster than $-1/r^2$

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the $s$-wave phase shift ${\delta }_0$. It exposes a universal dependence of ${\delta }_0\left(E\right)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length $a$, or equivalently, the noninteger part ${\Delta }_{\mathrm{th}}$ of the threshold quantum number $n_{\mathrm{th}}$. The formula accurately reproduces ${\delta }_0\left(E\right)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\delta }_0\left(E\right)$ and demonstrate the validity of the formula for attractive potential tails proportional to $1/r^6$ or to $1/r^4$, and also for a mixed potential tail consisting of a $1/r^4$ term together with a non-negligible $1/r^6$ contribution.

Figure 1

The ratio $A_{\mathrm{s}}/A_{\mathrm{c}}$ of the amplitudes defined in Eqs. (5) is plotted for an inverse-power tail (15) with $\alpha =6$ together with its low-energy behavior (19) (dotted line) and its asymptotic high-energy value (dot-dashed line).

Figure 2

The universal phases ${\phi }_{\mathrm{s}}$ and ${\phi }_{\mathrm{c}}$ defined in Eqs. (5) are plotted for an inverse-power tail (15) with $\alpha =6$ together with their low-energy behavior (20) (dotted lines) and their high-energy behavior (23) (dot-dashed line).

Figure 3

$s$-wave phase shifts as given by Eq. (14) for a potential with a $1/r^6$ tail, plotted against the dimensionless product $k{\beta }_6$. The solid lines show the phase shifts for ${\Delta }_{\mathrm{th}}=0,0.01,0.1,0.5,0.99$ (from bottom to top). For the lowest three values of ${\Delta }_{\mathrm{th}}$ the plots are repeated with a shift of $\pi$, which would correspond to one more bound state in a potential well (dashed lines). The dot-dashed line represents the phase shift of the Lennard-Jones potential (29) with $B_{\mathrm{LJ}}={10}^5$. The difference between the exact phase shifts and the prediction of Eq. (14), with ${\Delta }_{\mathrm{th}}=0.820557$, is less than ${10}^{-6}\pi$.

Figure 4

The difference of the exact $s$-wave phase shift in a Lennard-Jones potential with $B_{\mathrm{LJ}}={10}^5$ to the phase shift obtained from Eq. (14) with ${\Delta }_{\mathrm{th}}=0.820557$ is plotted against the scaled energy. The abscissa covers a range $0\le k{\beta }_6\lesssim 20$, which reaches far beyond the range of Fig. 3.

Figure 5

The ratio $A_{\mathrm{s}}/A_{\mathrm{c}}$ of the amplitudes defined in Eqs. (5) is plotted for an inverse-power tail (15) with $\alpha =4$ together with its low-energy behavior (19) (dotted line) and its asymptotic high-energy value (dot-dashed line).

Figure 6

The universal phases ${\phi }_{\mathrm{s}}$ and ${\phi }_{\mathrm{c}}$ defined in Eqs. (5) are plotted for an inverse-power tail (15) with $\alpha =4$ together with their low-energy behavior (20) (dotted lines) and their high-energy behavior (23) (dot-dashed line).

Figure 7

$s$-wave phase shifts as given by Eq. (14) corresponding to Fig. 3, but for a potential with a $1/r^4$ tail. The solid lines show the phase shifts for ${\Delta }_{\mathrm{th}}=0,0.01,0.1,0.5,0.99$ (from bottom to top). The dot-dashed line represents the phase shift of the ($12|4$)-Lennard-Jones potential (31) with $B_{\mathrm{LJ}}={10}^4$. The difference between the exact phase shifts and the prediction of Eq. (14), with ${\Delta }_{\mathrm{th}}=0.881418$, is less than ${10}^{-6}\pi$.

Figure 8

The difference of the exact $s$-wave phase shift in a 12-4 Lennard-Jones potential with $B_{\mathrm{LJ}}={10}^4$ to the phase shift obtained from Eq. (14) with ${\Delta }_{\mathrm{th}}=0.8814184$ is plotted against the scaled energy. The abscissa covers a range $0\le k{\beta }_4\lesssim 100$, which reaches far beyond the range of Fig. 7.

Figure 9

The ratio $A_{\mathrm{s}}/A_{\mathrm{c}}$ of the amplitudes defined in Eqs. (5) for the mixed tail (32) with ${\beta }_4={\beta }_6\equiv \overline{\beta }$. The shaded area is bounded by the corresponding results for the single-power tails already shown in Figs. 1 and 5.

Figure 10

Phases ${\phi }_{\mathrm{s}}$ and ${\phi }_{\mathrm{c}}$ for the mixed tail (32) with ${\beta }_4={\beta }_6\equiv \overline{\beta }$. The short-dashed line shows the phase ${\xi }_{\mathrm{t}}$ defined by Eq. (13).

Figure 11

Phase shift ${\delta }_0$ as function of $k\overline{\beta }$ for the model potential (38). The exact values (blue solid line) are reproduced within $3\times {10}^{-6}\pi$ by formula (14) with the tail functions for the mixed tail (32); see Figs. 9 and 10. In contrast, using the tail functions of the single-power $1/r^4$ case (red dashed line) is insufficient beyond the immediate near-threshold regime.