Anisotropic quantum scattering in two dimensions

We study the quantum scattering in two spatial dimensions without the usual partial-wave formalism. The analysis beyond the partial-wave approximation allows a quantitative treatment of the anisotropic scattering with a strong coupling of different angular momenta nonvanishing even at the zero-energy limit. High efficiency of our method is demonstrated for the two-dimensional (2D) scattering on the cylindrical potential with the elliptical base and dipole-dipole collisions in the plane. We reproduce the result for the 2D scattering of polarized dipoles in binary collisions obtained recently by Ticknor [Phys. Rev. A 84, 032702 (2011)] and explore the 2D collisions of unpolarized dipoles.

Figure 1

The dependence of real (a) and imaginary (b) part of the scattering amplitude on the momentum $q$ for the potential barrier (20) with circular base $a\left(\varphi \right)=a_0=1$ and $U_0={10}^4$. Calculations were carried out for ${\varphi }_q=0$ with the parameters: $M={10}^2$, $N={10}^5$, and ${\rho }_N=15$. The quantities $f$ and $q$ are given in the units $\hslash =\mu =1$.

Figure 2

The dependence of the differential cross section on the momentum $q$ for the potential barrier (20) with circular base $a\left(\varphi \right)=a_0=1$ and $U_0={10}^4$. Calculations were carried out for ${\varphi }_q=0$ with the parameters: $M={10}^2$, $N={10}^5$, and ${\rho }_N=15$. The quantities $d\sigma /d\Omega$ and $q$ are given in the units $\hslash =\mu =1$.

Figure 3

The dependence of the differential cross section on the momentum $q$ and scattering angle $\varphi$ in the case of potential barrier (20) with $U_0={10}^3$ and circular base $a\left(\varphi \right)=a_0=1$. Calculations were carried out for ${\varphi }_q=0$ with the parameters: $M={10}^2$, $N={10}^5$, and ${\rho }_N=15$. The quantities $d\sigma /d\Omega$ and $q$ are given in the units $\hslash =\mu =1$.

Figure 4

The dependence of the differential cross section on the momentum $q$ and scattering angle $\varphi$ in the case of potential barrier (20) with $U_0={10}^3$ and elliptic base with $a(\pi /2)=1.1$ and $a\left(0\right)=1$. Calculations were carried out for ${\varphi }_q=0$ with the parameters: $M={10}^2$, $N={10}^5$, and ${\rho }_N=15$. The quantities $d\sigma /d\Omega$ and $q$ are given in the units $\hslash =\mu =1$.

Figure 5

The same as in Fig. 4 but for the case of strong anisotropy of the scatterer (20) with $a(\pi /2)=2$ and $a\left(0\right)=1$. The quantities $d\sigma /d\Omega$ and $q$ are given in the units $\hslash =\mu =1$.

Figure 6

Collision in the plane $XY$ of two arbitrarily oriented dipoles ${\mathit{d}}_1$ and ${\mathit{d}}_2$.

Figure 7

A comparison of the total cross section (in the units of ${\sigma }_{SC}$) with the result of C. Ticknor [17] calculated for potential (27) at $D=1,Dq=10$.

Figure 8

The total cross sections $\sigma$ in the units of ${\sigma }_{SC}$ as a function of the dipole tilt angle $\alpha =\gamma$ and the rotational angle $\beta$ calculated for potential (23) at $D=1,Dq=10$.

Figure 9

The dependence of the dipole-dipole potential $U(\mathit{\rho },{\mathit{d}}_1,{\mathit{d}}_2)$ (23) in the units of $E_D={\hslash }^6/{\mu }^3{d}^4$ ($d=d_1=d_2$) on the tilt angle $\alpha$ for two extreme mutual orientations $\beta =0$ and $\pi$ of the dipole polarization planes $Z{d}_1$ and $Z{d}_2$.

Figure 10

The total cross section $\sigma$ (solid line), bosonic cross section ${\sigma }_g$ (dashed line), and fermionic cross section ${\sigma }_u$ (dash-dot line) calculated for potential (23) as a function of the angle $\beta$ for the fixed $\alpha =\gamma =0.2\pi$ at $D=1,Dq=10$ (in the units of ${\sigma }_{SC}$).

Figure 11

The total cross sections $\sigma$ in the units of ${\sigma }_{SC}$ as a function of the dipole tilt angles $\alpha$ and $\gamma$ calculated for potential (23) at $D=1,Dq=10$. The rotational angle $\beta$ is equal to $\pi /2$.