# Casimir Force

Steven K Lamoreaux (2011), Scholarpedia, 6(10):9746. | doi:10.4249/scholarpedia.9746 | revision #121747 [link to/cite this article] |

**The Casimir force ** is the mutual attraction of two closely spaced, parallel, and uncharged conducting planes that persists even at absolute zero temperature. This force results from a change in the zero point energy of the electromagnetic field between the plates, due to the modification of the field modes as the plates are brought together. For perfect conductors, this force, predicted to exist by H.B.G Casimir in 1948, has magnitude (per unit surface area \( A \))
\[
F(d)/A={\pi^2\over 240}{\hbar c\over d^4}=0.0130 {1\over d^4}\ {\rm dyn\ \mu m^4/cm^2}\ \ \ (1)
\]
where \(d\) is the distance between the plates, \(F(d)/A\) is an effective pressure (force per area), \(\hbar \) is Planck's constant (\(h/2\pi\)), and \(c\) is the velocity of light. It should be noted that in the case of perfect conducting plates, the only fundamental constants that enter are \(c\) and \(\hbar\ .\) The role of the plates in this case is solely to provide a boundary condition on the quantized field.

The notion that boundary conditions can modify the ground state energy of a quantized field has far reaching consequences. This idea is generally considered to be among the final great fundamental discoveries in the formulation of quantum mechanics.

## Contents |

## Historical Evolution

### Generalization of the London-van der Waals Force

Casimir performed the theoretical work leading to the prediction of his eponymous force while he was research director at Philips Laboratories (Casimir 2000). This work, performed with collaborators in the late 1940's, is described in a series of three papers (Casimir and Polder 1948, Casimir 1948, Casimir 1949).

Earlier, Fritz London (1930) computed the attractive force between non-polar atoms with the new quantum theory; before London's calculation, the van der Waals attraction between polar molecules (i.e., those that have a permanent electric dipole moment) was understood, but the origin of the experimentally known van der Waals correction to the gas law for nonpolar gases such as helium, was not. Thus London's derivation represented one of the major early successes of quantum mechanics. This attractive force can be understood as arising from zero point fluctuations of the atoms themselves, and London put forward a simple model to describe the effect based on this: If we imagine two interacting atoms as identical harmonic oscillators, when the two oscillator are separated by a finite distance \(R\) they interact, and the original degenerate eigenfrequency \(\omega_0\) is split, becoming \(\omega_\pm=\omega_0\sqrt{1\pm \kappa}\) where \(\kappa \) is the dipole coupling strength, proportional to \(1/R^3\ .\) If an energy \(\hbar\omega_\pm/2\) is assigned to each frequency, and the total energy for the system at infinite separation is subtracted, the remaining interaction energy is \(E\propto \hbar\omega_0/R^6\) to lowest (square) order in \(\kappa\ .\) This simple picture illustrates the source of the force in the context of the London calculation which is otherwise not at all obvious.

The interatomic potential determined by London varies as the sixth power of the separation between the atoms. However, in Overbeek and Verwey's experimental work with colloids at Philips, it was noticed that the attractive potential between the particles (assumed as resulting from the attraction between the individual atoms that the particles comprise) appeared to fall off faster than \(1/R^6\) at very long distances; Overbeek suggested that perhaps if the finite velocity of light is taken into account, the potential might be modified at large separations as compared to the London calculation which assumed an instantaneous electromagnetic interaction. Overbeek approached Casimir and Polder (1948) who together addressed the problem. Casimir and Polder considered first a simpler problem, a single atom in an electromagnetic cavity with perfectly conducting walls, and calculated the atom's interaction with a cavity wall as a function of distance. Using a full quantum electrodynamic treatment of the field (at zero temperature) in the cavity, they showed that the atom would be attracted to the wall as \(1/R^4\) potential with a surprisingly simple form for large separations:

\[ \Delta E=-{3\over 8\pi}\hbar c {\alpha\over R^4} \] where \(c\) is the velocity of light, and \(\alpha\) is the static electric polarizability. This is the so-called Casimir-Polder force. They next calculated the attractive potential between two atoms, and again found a surprisingly simple result (for two identical atoms): \[ \Delta E=-{23\over 4 \pi}\hbar c {\alpha^2\over R^7}. \]

Casimir and Polder's calculations are laborious and lengthy; Lifshitz (1956), in his paper that presents an extension of Casimir's calculation to realistic materials, comments on the unwieldiness of the calculation. However, the simplicity of the result inspired Casimir to seek a fundamental explanation for the effect. In mentioning surprisingly simple nature of the result to Niels Bohr, according to Casimir (2000), "Bohr mumbled something about zero point energy." This remark lead Casimir to formulate a new approach and the results, obtained previously with much labor, could be obtained in about two printed pages (Casimir 1949). In this new approach, Casimir simply assigned a zero point energy of \(\hbar\omega/2\) to every mode of the cavity, and calculated the net shift in zero point energy by use of the well-known result for the perturbation of cavity mode frequencies:
\[
{\delta \omega\over\omega}=-2\pi {\alpha |E_0(x_0,y_0,z_0)|^2\over \int_V |E_0(x,y,z)|^2 dV}
\]
where \(\alpha\) the electric polarizability of a particle placed at location \((x_0,y_0,z_0)\) in the cavity, and \(\vec E_0(x,y,z)\) is the unperturbed field. The energy is determined by a sum over all cavity modes (which diverges) and the attractive potential is determined by taking the difference between two different configurations. Casimir thereby reduced a full quantum electrodynamic problem to a simpler classical electromagnetism problem. It should be noted that "time" does not explicitly appear in any of these calculation, so it would appear that the term "retarded van der Waals" to described the force modification in the long distance region is a hold-over from the original Overbeek suggestion. In Casimir's calculation, as the distance from the wall, or the distance between two atoms, increases, the contributions from very high frequency modes tends to average to zero, with only the long wavelength modes contributing to a coherent sum.

## The Casimir Force

In the final paper of the series (Casimir 1948), Casimir performed what he calls the "obvious extension" to the force between the cavity walls themselves due to the cavity zero point field, and obtained the famous result for the attractive force between two perfectly conducting plates (mirrors) given in Eq. (1) above. It should be noted that the electron charge does not explicitly appear in this formula. The electron in this case provides the foundation of the perfectly conducting boundary condition, an the value of its charge does not enter in this case.

It should be further noted that there is no reference to a retardation effects in the calculation of the Casimir force, and therefore must be considered as a phenomenon distinctly different from the long-range van der Waals force. There is no characteristic frequency in the problem Casimir force problem, other than perhaps for a good (not perfect) conductor the electrical permittivity diverges as the inverse of the frequency; this suggests that for a metal the retardation distance is infinity. As in the case of the Casimir-Polder force, the modes that contribute to the Casimir force are those with wavelength longer than the plate separation. By simple dimensional analysis, we can determine the form of the Casimir force. Noting that \(\omega=ck\ ,\) where \(k\) has maximum value of order \(k_{max}=\pi/d\ ,\) the *reduction* in zero point field energy between the plates (taken as square, with area \(L^2=A\)) is given approximately by
\[
E=\sum_{\lambda>2d} {\hbar \omega\over 2}\times (nmber\ of\ modes\ at\ \omega) \sim \int_0^{k_{max}} {\hbar c k\over 2} {L^2 k dk\over 4\pi}={\pi \hbar c\over 24} {1\over d^3}L^2.
\]
The derivative with \(d\) then gives the attractive force (field energy decreases with \(d\)), and is in reasonable agreement with Eq. (1). Of course the agreement is not exact as we used an ad hoc criterion to terminate the integral. In practice, all mode contribute but with diminishing magnitude.

In the calculation of the Casimir force it is assumed that the modes are exponentially decaying into the plates' surfaces; it has been recognized that the electromagnetic modes that are responsible for the Casimir force are not free space modes, but evanescent {\it surface modes} of the material {Barton 1979). This suggests that the existence of the Casimir force really doesn't say anything about the vacuum of space itself, but only about the interactions of materials through their own nearby electromagnetic modes.

However, Casimir's result, Eq. (1), strongly suggests a certain reality of electromagnetic zero point energy. When Casimir told Wolfgang Pauli about the attraction of two conducting plates, Pauli rejected the notion as "absolute nonsense." (Casimir 2000). However, Casimir persisted, and Pauli eventually accepted the inevitable conclusion.

Pauli's viewpoint is illuminated by his *Lectures on Physics, vol. 4* (Pauli 1973) where he states that, "For radiation the zero point \(E_0\) is not important," (p. 75). He had considered the effects of a real electromagnetic zero point energy ( Appendix, p. 115), and came to the conclusion that its gravitational effect would be so enormous that the radius of the universe, "Would not even reach the moon." His calculation, which was never published, cuts off the zero point field for wavelengths shorter than the classical electron radius \(r_e\ .\) The total energy density is given by the sum of the zero point energy of each mode, \(\hbar k/2c\ ,\) and the mode density is proportional to \(2\times 4\pi^2k^2dk\) (where the factor 2 is required for the two polarization states of each mode). Thus the energy density should be
\[
\rho_E= {\hbar\over 2c}\int_0^{k_{max}} 2k^3 d k=\hbar k^4/c
\]
and, taking \(k_{max}=2\pi/r_e\) where \(r_e=2.82\times 10^{-13}\) cm is the classical electron radius, gives \(\rho_E\sim 10^{33}\) eV/cm\(^3\ .\) This is about \(10^{36}\) times larger than the presently accepted value of the *dark energy* component of the universe, \(10^{-4}\) eV/cm\(^3\ .\) Taking the cutoff as the classical radius of the electron is artificial; a more natural cutoff would correspond to the Planck length, \(\ell_p=\sqrt{\hbar G/c^3}=1.6\times 10^{-33}\) cm, yielding an energy density \(\rho_E\sim 10^{114}\) ev/cm\(^3\ ,\) or about 118 orders of magnitude larger than the observed background energy. It has been suggested that fermionic fields (which have negative zero point energy) might cancel the positive bosonic field zero point energy, but the level of fine tuning required is beyond imagination, especially noting that there are on the order of \(10^{80}\) particles in the universe, placing the scale of the required accuracy into perspective.

### Source of the Casimir Force and Lifshitz's Extension to Real Material

Some light can be shed on this conundrum by considering Lifshitz's extension of Casimir's calculation to realistic materials. In the Lifshitz calculation, fields are treated as completely classical (no zero point energy) and the field source terms are the quantum polarization fluctuations in the material. If we assume that these fluctuations can be assigned a spectral density, at each frequency there is a zero point energy of \(\hbar\omega/2\) that persists even at absolute zero. These material fluctuations, at finite temperature, are responsible for Johnson noise, for example, well-known in electronics, and follows from the quantum fluctuation-dissipation theorem (Callen and Welton ). If the real and imaginary parts of the electric permittivity \(\epsilon(\omega)\) are given for the material(s) that the two flat plates comprise, the allowed modes between the plates can be determined; the imaginary part of \(\epsilon\) represent dissipation and thus serves as the source term for the electromagnetic fields between the plates. The point is that the Casimir and related forces can be understood without reference to fluctuation of the vacuum itself. For example, Casimir and Polder could have performed their calculation using a real material with dissipation for the cavity walls. The field excitations would then follow the material fluctuations, with \(\hbar\omega/2\) being the apparent zero point energy of each cavity mode, but without field quantization. The question is analogous to Planck's treatment of black body radiation, or the photoelectric effect, and whether proof for the existence of photons is provided: in either of these cases, one cannot decide between quantization of the field or quantization of the field/matter interaction. In Callen and Welton's view, the vacuum represents a dissipative medium; in this context, the quantization of the oscillators occurs in the material, and the vacuum contributes to the imaginary part of the electrical permittivity.

Thus we are still left with the question of whether the vacuum of free space contains electromagnetic zero point energy. Results from observational cosmology suggest that it does not, due to the enormous amount of energy of the zero point. We might reformulate this question as to one of whether space is simply a map or is an object in its own right? Modern observation implies the existence of a cosmological constant that imbibes space with an intrinsic property. However, given the enormous energy associated with the electromagnetic zero point, we should probably conclude that it does not contribute to the cosmological constant, but is an artifact of the way electromagnetic fields interact with matter; the interaction is *such that* the field can be treated as having a zero point energy. Of course, this argument is not entirely satisfactory, and the question remains open.

### Thermal Correction

The Casimir force can be treated for finite temperature (Lifshitz 1956) by making the substitution \[ {1\over 2}\hbar\omega\rightarrow {1\over 2}\hbar\omega+ {\hbar\omega\over e^{\hbar\omega/k_bT}-1}={\hbar\omega\over 2}\coth{\hbar\omega\over 2k_bT} \] to account for the thermal excitation of the modes, where \(k_b\) is Boltzmann's constant and \(T\) is the absolute temperature. In the limit of very high temperature, the denominator can be expanded, and the net energy contribution per mode with frequency \(\omega\) is simply proportional to \(k_bT\) instead of \(\hbar\omega/2\ .\) In this limit, the force is no longer proportional to \(\hbar\) and is therefore analogous to the Rayleigh-Jeans limit of the black body spectrum. In this limit, the thermal Casimir attractive force is \[ F(d)={2.4 k_bT\over 4\pi d^3}\ \ \ (2) \] for perfect conductors.

Because the \(coth\) function has poles at \(\omega_n=2\pi i n k_bT/\hbar\ ,\) the integral over frequency in the calculation of the force is replaced by a sum over the residues at the poles. In the high temperature limit, only the \(n=0\) terms contribute, resulting in Eq. (2) above. If we take a maximum characteristic mode frequency as \(c/d\ ,\) the high temperature limit occurs when \(d\gg {\hbar c\over 2\pi k_bT} .\) For \(T=300\) K, the high temperature limit is valid when \(d\gg 1\ \mu\)m.

The thermal correction to the Casimir force has a long and illustrious history. The issues finally came to the forefront of the filed when, in 2000, M. Bostrom and Bo Sernlius showed conclusively that there was an lapse in previous calculations of the finite conductivity correction when combined with the non-zero temperature correction.

At large separations, the dominant contribution comes from the pole at \(n=0\ ,\) corresponding to zero (Matsubara) frequency. For perfect conductors, both the \(TE\) (electric field parallel to the surface) and \(TM\) (magnetic field parallel to the surface) modes contribute equally. In particular, the full contribution of the \(TE\) mode implied that a quasistatic magnetic field will not penetrate into the plates. For a real metal, e.g., assuming a Drude model electric permittivity, the contribution of the \(n=0\ TE\) electromagnetic mode vanishes simply because a real conductor does not pose a boundary condition on a static magnetic field. This results in a large correction to the force at distances greater than about 1 micron (at 300 K), and the force is reduced by a full factor of two in the limit of large distance as compared to the perfect conducting case.

The Bostrom/Sernelius analysis has led to some controversy, mostly because it appears to disagree with experiments that show a larger force than expected for the Drude model. However, recent careful experiments have uncovered a number of systematic effects that all increase the measured force. These effect include surface roughness, mechanical vibrations, and surface electrostatic patch potentials, among others. Sushkov et al. (2011) have conducted measurements to distances (7 microns) where the finite temperature contribution dominates the force, with results that support the Bostrom/Sernelius analysis.

## References

- Casimir, H.B.G. (1948). On the Attraction between Two Perfectly Conducting Plates.
*Proc. K. Ned. Adad. Wet.*51: 793. - Casimir, H.B.G (1949). Sur les forces van der Waals-London.
*Jour. de Chimie Physique*46: 407. - Casimir, H.B.G. (2000). On the history of the so-called Casimir effect.
*Comments on Modern Physics*5-6: 175. - Casimir, H.B.G and Polder, D. (1948). The influence of retardation of the London-van der Waals forces.
*Phys. Rev.*73: 360. - Callen, H.B and Welton, T.A. (1951). Irreversibility and generalized noise.
*Phys. Rev.*83: 34-50. - Lifshitz, E.M. (1956). The theory of molecular attractive forces between solids.
*Sov. Phys. JETP*2: 73-83. - Pauli, Wolfgang (1973). Pauli Lectures on Physics, Volume 4: Statistical Mechanics. MIT Press, Cambridge, MA.
- Bostrom, M. (2000). Thermal effects on the Casimir force in the 0.1-5 \(\mu\)m range.
*Phys. Rev. Lett.*84: 4757-4760. - London, F. (1930). On the Theory and System of Molecular Forces.
*Z. Phys.*63: 245. - Barton, G. (1979). Some Surface Effects in the Hydrodynamical Model of Metals.
*Rep. Prog. Phys.*42: 963. - Sushkov, A.O. et al. (2011), Observation of the thermal Casimir force, Nature Physics Volume 7, 230–233, doi:10.1038/nphys1909.

## Further reading

- Milonni, Peter W. (1994). The Quantum Vacuum. Academic Press, San Diego. ISBN 0-12-498080-5..

- Barrera, Ruben G. and Reynaud, Serge (2006), Focus on Casimir Forces, New J. Phys. 8, doi: 10.1088/1367-2630/8/10/E05

- Vladimir Mostepanenko, N. N. Trunov, and R. L. Znajek (Translator), The Casimir Effect and Its Applications, (Oxford Science Publications), Oxford University Press, USA (1997), ISBN-10: 0198539983, ISBN-13: 978-0198539988

- V. Adrian Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge University Press (2005), ISBN-10: 0521839068, ISBN-13: 978-0521839068

- Kimball A. Milton, The Casimir Effect, World Scientific Pub Co Inc; 1st edition (October 2001), ISBN-10: 9810243979, ISBN-13: 978-9810243975

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