Scattering Hypervolume of Spin-Polarized Fermions

TL;DR

This paper introduces the three-body scattering hypervolume $D_F$ for spin-polarized fermions, analyzing its properties, asymptotic behavior, and effects on energy, pressure, and recombination rates in ultracold Fermi gases.

Contribution

It defines and characterizes the three-body scattering hypervolume $D_F$ for spin-polarized fermions, extending the concept from bosons and analyzing its physical implications.

Findings

Derived asymptotic wave function expansions for three fermions.

Calculated energy and pressure shifts due to $D_F$ in Fermi gases.

Estimated three-body recombination rates at finite temperatures.

Abstract

We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume $D_F$. The scattering hypervolume $D$ was first defined for identical bosons in 2008 by one of us. It is the three-body analog of the two-body scattering length. We solve the three-body Schr\"{o}dinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which $D$ has the dimension of length to the fourth power, here the $D_F$ we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\hbar\mathbf{k}_1$, $\hbar\mathbf{k}_2$ and $\hbar\mathbf{k}_3$ in a large periodic cubic box. The energy shift due to $D_F$ is proportional to $D_F/\Omega^2$, where $\Omega$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero $D_F$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.