Page 77 - Principles and Applications of NanoMEMS Physics

64 Chapter 2

F 0 π 2 c = 1
Cas = − . (52)
A 240 z 4

For planar parallel metallic plates with an area A = 1cm and separated a
2
distance z = 5 . 0 µ m , the Casimir force is 2 × 10 − 6 N .
Many experiments measuring the Casimir force under various conditions,
such as effecting normal displacement between a sphere and a smooth planar
metal and between parallel metallic surfaces, as well as, effecting lateral
displacement between a sphere and a sinusoidally corrugated surface, have
been performed [89-95]. A good recent review of experiments and theory for
Casimir forces has been published by Bordag, Mohideen, and Mostepanenko
[89].
Since the Casimir energy/force is a sensitive function of the boundary
conditions, corrections to the ideal expression (52) have been introduced to
account for certain deviations. For example, for the sphere-plate geometry,
the zero-temperature Casimir force is given by,

π 3 c =
F 0 () z = − R
360 z
Cas _Sphere− Plate 3 , (53)
where R is the radius of curvature of the spherical surface.
To include the finite conductivity of the metallic boundaries, two
approaches have been advanced. In one, the force is modified as [96, 97],

ª § · 2 º
, 0 σ 0 c 72 c
F ()= Fz ()« − 41z + ¨ ¸ »
Cas Cas _ Sphere −Plate ¨ ¸ , (54)
« zω 5 zω »
¬ p © p ¹ ¼
where ω is the metal plasma frequency [64]. In the other, obtained by
p
Lifshitz [98], the correction is ingrained in the derivation of the Casimir
force, and is given by,

­ ) 2 2 pξ z º −1 ½
° « ( ª s + p ) 2 e c − » 1 + °
°
°
2
z
F , 0 σ () = − R= z dz ' ∞ ∞ p ξ 3 dpd ξ × ° ( « ¬ s − p » ¼ ° ,
®
¾
π c 3 ³ 0 ³³ 1 ° ( ª s + pε ) 2 2 pξ z º −1 °
Cas
0
°« 2 e c − » 1 °
° ( « ¬ s − pε ) » ¼ °
¿
¯
(55)