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Scattering of a weakly bound dimer from a hard wall in one dimension

Xican Zhang, Shina Tan

Abstract

We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the ``angular distribution" of the dissociated pair, where the ``angle" $θ$ depends on the ratio of the velocities of the two outgoing unbound particles.

Scattering of a weakly bound dimer from a hard wall in one dimension

Abstract

We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the ``angular distribution" of the dissociated pair, where the ``angle" depends on the ratio of the velocities of the two outgoing unbound particles.
Paper Structure (6 sections, 49 equations, 7 figures)

This paper contains 6 sections, 49 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Integrable cases
  3. General Cases
  4. Born-Oppenheimer Approximation
  5. High energy collisions
  6. Summary

Figures (7)

  • Figure 1: Dotted curves: the dimer-wall scattering phase shift $\delta$ versus $K/K_\text{th}$ for mass ratios $1$, $5$, $10$, $20$, and $40$. The numbers below the dotted curves are the corresponding mass ratios. Dashed curve: the prediction of Eq. \ref{['BO delta']} based on an approximate treatment of the Born-Oppenheimer approximation for the mass ratio $m_1/m_2=40$. Grey curve: the prediction of the Born-Oppenheimer approximation, based on the numerical solutions of Eq. \ref{['Schrodinger BO']} together with Eqs. \ref{['BO kappa exact']} and \ref{['V eff']} for $m_1/m_2=40$.
  • Figure 2: Dimer-wall scattering length $a_R$ and effective range $r_R$ (in units of the two-body scattering length $a$) versus the natural logarithm of the mass ratio. The red dots and the green squares show the values of $a_R/a$ and $r_R/a$, respectively, according to the numerical solution to Eq. \ref{['integral eq']}. The circles are the predictions of Ref. Lee_2011. The blue dashed line shows the prediction of Eq. \ref{['eq:aR']} based on the Born-Oppenheimer (BO) approximation. The black dashed curve shows the prediction of Eq. \ref{['eq:re']} based on the BO approximation. The vertical dot-dashed lines indicate the integrable cases ($m_1/m_2=1$ or 3), for which there are exact results discussed in Sec. \ref{['sec: integrale cases']}.
  • Figure 3: The reflection coefficient $R$ as functions of $K/K_\text{th}$ for mass ratios $m_1/m_2=1$ to $10$. The numbers below the curves are the mass ratios.
  • Figure 4: The reflection coefficient $R$ as functions of $K/K_\text{th}$ for mass ratios $m_1/m_2=20$, $40$, $75.8$ and $140$. For $m_1/m_2\approx75.8$, $R$ reaches zero at $K/K_\text{th}\approx 1.24$.
  • Figure 5: The minimum value of the reflection coefficient $R_{\text{min}}$ for different mass ratios. The left inset plots $R_\text{min}$ for $1\le m_1/m_2\le 4.3$. The right inset plots $R_\text{min}$ for $70.5\le m_1/m_2\le 81$. We numerically found that $R_{\text{min}}=0$ at a critical mass ratio $m_1/m_2\approx75.8$.
  • ...and 2 more figures