Page 132 - Principles and Applications of NanoMEMS Physics

120 Chapter 3

2 π =
Γ = ∆ φ = N , (120)
m
that is, it would be quantized. Thus, a change in potential of π2 = m would
bring it to the point of departure, due to its non-single-valuedness, yet would
allow a non-zero vorticity due to its finiteness. The quantum nature of a
superfluid contained in a rotating vessel manifests, therefore, in that its
circulation becomes quantized. One remarkable aspect of a rotating vessel
containing a superfluid pertains to the shape of its meniscus. In particular,
from the fact that a normal fluid in a vessel of area A rotating at an angular
velocity Ω has a circulation 2Ω A , and that a superfluid on the same vessel
would have a circulation νΓ A , where ν is the density of vortices per unit
area, one finds, equating circulations, that the Ω = Γ ν 2 . This signifies, that

although the superfluid would not necessarily be rotating, due to the
appearance of vortices, the shape of its meniscus will be the same as that of a
normal fluid rotating at an angular velocity Ω . In other words, one can
simulate the effect of rotation on a normal fluid by a population of vortices.
The fact that the circulation of a superfluid contained in a rotating vessel
is quantized means that the vessel must reach a certain minimum angular
velocity, the critical angular velocity, Ω , and rotational energy before the
c
vortices begin to be created. From the ratio of vortex energy to vortex
angular momentum it can be shown that,

Ω = = , (121)
c 2
mR

where R is the vessel radius. Figure 3-21 shows a picture of vortices in a
superfluid.

Figure 3-21. Observation of vortex lattices. The examples shown contains approximately
80, vortices. The vortices have “crystallized” in a triangular pattern. Reprinted with
permission from [156]. Copyright 2001 AAAS.