Casimir effect

In 1948 Dutch physicist Hendrik B. G. Casimir of Philips Research Labs predicted that two uncharged parallel metal plates will have an attractive force pressing them together. This force is only measurable when the distance between the two plates is extremely small, on the order of several atomic diameters. This attraction is called the Casimir effect.

The Casimir effect is caused by the fact that space is filled with vacuum fluctuations, virtual particle-antiparticle pairs that continually form out of nothing and then vanish back into nothing an instant later. The gap between the two plates restricts the range of wavelengths possible for these virtual particles, and so fewer of them are present within this space. This results in a lower energy density between the two plates than is present in open space; in essence, there is "less than nothing" between the two plates, creating negative energy and pressure, which pulls the plates together.

The narrower the gap, the more restricted the wavelength of the virtual particles, the more negative the energy and pressure, the more restricted the vacuum modes and the smaller the vacuum energy density, and thus the stronger is the attractive force.

Similarly, fluctuations in the electronic structure of molecules cause transient dipoles which lead to the Van der Waals force.

The Casimir effect has recently been measured by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen of the University of California at Riverside and his colleague reduced Planck constant (sometimes known as the Dirac constant),

<math>c<math> is the speed of light,

<math>\pi<math> is Archimedes's constant, the ratio of the circumference of a circle to its diameter, and

<math>d<math> is the distance between the two plates.

This shows that the Casimir force per unit area <math>F_c / A<math> is very small.

The calculation shows that the force happens to be proportional to the sum <math>1+2+3+4+5+\dots<math> where the numbers <math>1,2,3,4,5,\dots<math> represent the frequencies of standing waves between the plates; each possible standing wave behaves as a quantum harmonic oscillator whose ground state energy equal to <math>\hbar\omega/2<math> contributes to the total potential energy; the force then equals minus the derivative of the potential energy with respect to the distance.

The series (the sum of integers) is divergent and needs to be regularized. A useful tool is provided by the Riemann zeta function because the sum can be formally written as <math>\zeta(-1)<math> which equals <math>-1/12<math>. Although it may sound strange (and even though more rigorous ways to obtain the same result exist), the correct result for the sum of positive integers is <math>-1/12<math>. The same sum also appears in string theory.

It has since been shown (see Ref.1) that, with materials of certain permittivity and permeability, the Casimir effect can be repulsive instead of attractive.

References

1. Repulsive Casimir forces, O. Kenneth, I. Klich, A. Mann and M. Revzen, Department of Physics, Technion - Israel Institute of Technology, Haifa, February 2002