Low-Lying Superfluid States in a Rotating Annulus

Abstract

The low-lying states of rotating liquid He II in an annulus (${R}_{1}<r<{R}_{2}$) are studied with the model of a classical inviscid fluid. An exact hydrodynamic solution is obtained with the method of images for a system consisting of rectilinear vortices with circulation $\ensuremath{\kappa}$ combined with circulation ${\ensuremath{\Gamma}}_{1}$ about the inner cylinder. The energy and angular momentum are calculated, both for an arbitrary configuration of vortices and for the particular configuration of a symmetric ring of $l$ vortices. If ${\ensuremath{\Gamma}}_{1}$ is treated as a variational parameter, the critical angular velocity ${\ensuremath{\Omega}}_{0}$ for the appearance of vortices in a narrow annulus is $(\frac{\ensuremath{\kappa}}{\ensuremath{\pi}{d}^{2}})\mathrm{ln}(\frac{2d}{\ensuremath{\pi}a})$, where $d$ is the width of the annulus and $a$ is the radius of the vortex core. For $\ensuremath{\Omega}<{\ensuremath{\Omega}}_{0}$, the equilibrium state is an irrotational (vortex-free) flow with quantized circulation ${n}_{\ensuremath{\kappa}}(n=1,2,\ensuremath{\cdots})$; these levels are equally spaced, and a given quantum state represents the lowest free energy only in a narrow angular-velocity interval of $\frac{\ensuremath{\kappa}}{2\ensuremath{\pi}{R}^{2}}$, where $R$ is the mean radius of the annulus. The maximum quantum number of irrotational circulation is $\frac{2\ensuremath{\pi}{R}^{2}{\ensuremath{\Omega}}_{0}}{\ensuremath{\kappa}}=2{(\frac{R}{d})}^{2}\mathrm{ln}(\frac{2d}{\ensuremath{\pi}a})\ensuremath{\gg}1$. For $\ensuremath{\Omega}>{\ensuremath{\Omega}}_{0}$, the vortices lie on the circumference of a ring midway between the walls, and the number of vortices increases rapidly with $\ensuremath{\Omega}$. If ${\ensuremath{\Gamma}}_{1}$ is constrained to vanish identically, the critical angular velocity ${\ensuremath{\Omega}}_{c}$ for the appearance of vortices in a narrow annulus is of order $\frac{\ensuremath{\kappa}}{2\ensuremath{\pi}\mathrm{Rd}}$; this is equivalent to Feynman's critical velocity ${v}_{c}=O(\frac{\ensuremath{\hbar}}{\mathrm{md}})$ for singly quantized vortices with $\ensuremath{\kappa}=\frac{h}{m}$. In the opposite limit of a wide annulus (${R}_{1}\ensuremath{\ll}{R}_{2}$), the equilibrium state is shown to agree with Vinen's earlier calculations.