Page 126 - Principles and Applications of NanoMEMS Physics

114 Chapter 3

§ E ·
v c > min ¨ ¨ ¸ ¸ . (90)
© p ¹

0
In particular, if v ≠ , then it is possible for the fluid to flow free of
c
excitations, i.e., without friction/dissipation, as long as v < v . This is the
c
so-called Landau’s criterion for superfluidity. This condition is maintained
as long as v is less than the speed of sound. This insight, led Landau to
propose that the low energy excitations of the superfluid ground state should
consist of two types of particles, namely, phonons and “rotons.” Phonons
being quantized sound waves, with an energy dispersion,

E = Sp , (91)

where S is the speed of sound and p the momentum, and rotons being
quantized rotational motion (vortices), with an energy dispersion,

E = ∆ + (p − p ) 2 2m . (92)
0 eff

At temperatures above absolute zero, the fluid will be excited by thermal
energy. Therefore, it will be possible for some of the thermally excited fluid
particles to achieve velocities greater than v and will, consequently,
c
experience friction. Under these circumstances, the fluid will be composed
of these normal particles and superfluid particles, resulting in a mass current
given by,
G G G
j = ρ v + ρ v , (93)
n n s s

where ρ and v are the mass density and velocity of the normal fluid, and
n
n
ρ and v those of the superfluid. If one assumes that the whole fluid flows
s s G G G
with velocity v = v = v , then the total mass current may be written as,
n s
G G G
j = (ρ + ρ )v = ρ v . (94)
n s

One of the fundamental properties of a superfluid derives from the fact
that, since it possesses no excited particles, its momentum doesn’t change
and, consequently, it can’t exert a force on a body immersed in it. Flow with
this property, denoted “potential flow,” is mathematically characterized by
the equation,