Scattering length of Andreev reflection from quantized vortices in ${}^3\mathrm{He}\text{−}B$

Andreev reflection of thermal quasiparticles from quantized vortices is an important technique to visualize quantum turbulence in low temperature ${}^3\mathrm{He}\text{−}B$. We revisit a problem of Andreev reflection from the isolated, rectilinear vortex line. For quasiparticle excitations whose impact parameters, defined as distances of the closest approach to the vortex core, do not exceed some arbitrary value $b$, we calculate exactly the reflected fraction of the total flux of excitations incident upon the vortex in the direction orthogonal to the vortex line. We then define and calculate exactly, as a function of $b$, the scattering length, that is the scattering cross section per unit length of the vortex line. We also define and calculate the scattering lengths for the flux of energy carried by thermal excitations, and for the net energy flux resulting from a (small) temperature gradient, and analyze the dependence of these scattering lengths on temperature.

Figure 1

Schematic illustration of the Andreev reflection by a rectilinear quantized vortex; small solid (red) circles and thick red lines are quasiparticles and their trajectories, small open (blue) circles and thick dashed (blue) lines are quasiholes and their trajectories, $y_0$ is the $y$ coordinate of an incident excitation. In regions “$+$” ($y<0$) and “$-$” ($y>0$), $\mathbf{p}·\mathbf{v}>0$ and $\mathbf{p}·\mathbf{v}<0$, respectively. The vertical dashed green line shows the interval of the $y$ coordinates such that the impact parameters $|y_0|$ do not exceed $b$.

Figure 2

(a) The reflected fraction of the total flux for impact parameters such that they do not exceed an arbitrary value $b$ and (b) the scattering length in units of the scale $b_0$ as functions of the nondimensional impact parameter $z=b/b_0$. Dashed (red and blue) lines show asymptotics (18) and (19) for large and small values of the impact parameter, respectively.

Figure 3

(a) The reflected fraction of the total energy flux and (b) the scattering lengths normalized by the scale $b_0$ as functions of the nondimensional impact parameter $z=b/b_0$ for temperature $T=0.25T_c$. Curves, from top to bottom, correspond to the particle (“number”) flux, energy flux, and the net energy flux resulting from temperature gradient $[F$, $F_E$, and $F_T$ in (a), and $K/b_0$, $K_E/b_0$, and $K_T/b_0$ in (b)]. Inset shows the behavior of scattering length for $0\le z\le 5$.

Figure 4

(a) Temperature dependence of the nondimensional scattering lengths $K_E/b_0$ and $K_T/b_0$ for two values of the impact parameter. In each pair of curves, the upper (black) and the lower (red) lines correspond to $K_E$ and $K_T$, respectively. (b) Temperature dependence of the dimensional scattering lengths $K$, $K_E$, and $K_T$ $\left[\mu \mathrm{m}\right]$ for 0 bar pressure. In each group of three curves, the top (blue), middle (black), and bottom (red) lines show the behavior of $K$, $K_E$, and $K_T$, respectively.