Repulsive Casimir force for electromagnetic fields with mixed boundary conditions

1. Repulsive Casimir force for electromagnetic fields with mixed boundary conditions L.P. Teo and S.C. Lim Multimedia University 29, November, 2008

2. In 1948, Casimir [1] postulated the existence of a force exerted between separate objects, which is due to the resonance of all pervasive energy fields in the intervening space between the objects.

3. For a pair of infinitely conducting parallel plates separated by a distance a, Casimir [1] found that the pressure exerted on the plates is given by It gives rise to an attractive force which is negligibly small at macroscopic scale. However, at the scale of 10nm, the pressure is approximately 1 atm = 101.3 kPa.

4. By definition, the Casimir energy is the sum of ground state energies of a quantum field: This mode sum definition leads to a divergent sum that needs to be renormalized. However, there is still no consensus on the renormalization procedure even in the simple case such as a rectangular cavity.

5. In 2004, Cavalcanti [2] proposed a new geometric setup – piston – for which an unambiguous (renormalization independent) Casimir force can be calculated. Since then, Casimir piston has attracted a lots of attention [3-18]. A three dimensional piston

6. It was found that for either massless scalar field with Dirichlet or Neumann boundary conditions, or electromagnetic field with perfect electric conductor conditions or perfect magnetic conductor conditions, the Casimir force acting on the piston is always attractive tending to pull the piston to the nearer end. Although the temperature will change the strength of the Casimir force, but it does not affect the attractive nature of the force [19]. In nanotechnology, this might lead to undesirable effect such as the collapse of nano devices known as stiction or adhesion.

7. As a result, it is desirable to investigate the scenarios that would change the attractive nature of the Casimir force. In nanotechnology, there has been some research done on the possible application of the Casimir effect in levitation devices. In1974, T. H. Boyer showed that changing the electric permittivity and magnetic permeability of the parallel plates can change the attractive nature of the Casimir force. In particular, he showed that if one plate is infinitely conducting and the other is infinitely permeable, then the Casimir force becomes repulsive and its magnitude is 7/8 times of that obtained by Casimir.

8. Our present work generalizes Boyer’s setting to a piston. We assume that a perfectly conducting piston is moving freely inside a closed rectangular cavity with infinitely permeable walls. This setup was proposed in [20] before but no exact formulas for the Casimir force was calculated. Moreover, we also study in detail the thermal effect. By duality, our results also hold for an infinitely permeable piston moving freely inside a closed rectangular cavity with perfectly conducting walls.

9. We consider the case where the piston has rectangular cross section  = [0, L2]  [0, L3] moving freely inside the rectangular cavity [0, L1]  . Denote by x1 = a the position of the piston.

10. The eigenfrequencies for TE modes and TM modes are given respectively by

13. This is obviously a positive decreasing function of a. Therefore, the Casimir force acting on the piston has positive sign when a < L1/2 and has negative sign when a > L1/2. In other words, the Casimir force always tends to restore the piston to the equilibrium position a = L1/2. Moreover, the magnitude of the Casimir force decreases as the piston moves towards its equilibrium position.

14. On the other hand, when T , the Casimir force is dominated by a term linear in T corresponding to those terms with l = 0. The remaining terms decay exponentially as T .To restore the constants ħ, c and kB into the expression for the Casimir force, we need to replace T by kBT/ (ħc) everywhere and multiply the overall expression by ħc. Therefore the high temperature expansion of the Casimir force is the same as the small- ħ expansion of the Casimir force. A term with order Tj will be accompanied with the term ħ1j. The leading term of the Casimir force is linear in T implies that this leading term is independent of ħ, and the remaining terms go to zero if we formally let ħ goes to 0. As a result, the Casimir force has a classical limit (or high temperature limit) which is given by

15. For small a, the leading term of the classical Casimir force is which is equal to 3/4 times the classical term of the Casimir force acting on a pair of perfectly conducting parallel plates.

16. At zero temperature, the Casimir force is given by

17. For small a, it’s leading behavior is given by In particular, for small plate separation, the leading term of the Casimir force is which is 7/8 times the Casimir force acting on a pair of perfectly conducting infinite parallel plates, in consistent with the result of Boyer for parallel plates.

20. References: [1] H.B.G. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Nederland. Akad. Wetensch. B 51 (1948), 793--795. [2] R. M. Cavalcanti, Casimir force on a piston, Phys. Rev. D 69 (2004), 065015. [3] S. A. Fulling, L. Kaplan, K. Kirsten, Z. H. Liu and K. A. Milton, Vacuum Stress and Closed Paths in Rectangles, Pistons, and Pistols, preprint arXiv:0806.2468. [5] M. P. Hertzberg, R. L. Jaffe, M. Kardar, A. Scardicchio, Attractive Casimir forces in a closed geometry, Phys. Rev. Lett. 95 (2005), 250402. [6] M. P. Hertzberg, R. L. Jaffe, M. Kardar, A. Scardicchio, Casimir forces in a piston geometry at zero and finite temperatures, Phys. Rev. D 76 (2007), 045016. [7] V. N. Marachevsky, One loop boundary effects: techniques and applications, preprint arXiv: hep-th/0512221 (2005). [8] G. Barton, Casimir piston and cylinder, perturbatively, Phys. Rev. D 73 (2006), 065018. [9] V. N. Marachevsky, Casimir energy of two plates inside a cylinder, Phys. Rev. D 75 (2007), 085019. [10] A. Edery, Casimir piston for massless scalar fields in three dimensions, Phys. Rev. D 75 (2007), 105012. [11] A. Edery and I. Macdonald, Cancellation of nonrenormalizable hypersurface divergences and the d-dimensional Casimir piston, J. High Energy Phys. 9 (2007), 0709:005. [12] X. H. Zhai and X. Z.Li, Casimir pistons with hybrid boundary conditions, Phys. Rev. D 76 (2007), 047704.

21. [13] V. N. Marachevsky, Casimir interaction: pistons and cavity, J. Phys. A: Math. and Theor. 41 (2008), 164007.[14] A. Edery, V. N. Marachevsky, The perfect magnetic conductor (PMC) Casimir piston in d+1 dimensions, Phys. Rev. D 78, 025021 (2008).[15] H. Cheng, The Casimir force on a piston in the spacetime with extra compactified dimensions, Phys. Lett. B 668, 72-77.[16] S. C. Lim and L. P. Teo, Three dimensional Casimir piston for massive scalar fields, preprint arXiv: hep-th:0807.3613.[17] S. C. Lim and L. P. Teo, Casimir piston at zero and finite temperature, preprint arXiv: hep-th:0808.0047.[18] X. H. Zhai, Y. Y. Zhang and X. Z. Li, Casimir Pistons for Massive Scalar Fields, preprint arXiv: hep-th:0808.0062.[19] S. C. Lim and L. P. Teo, Casimir piston at zero and finite temperature, preprint arXiv: hep-th:0808.0047.[20] S. A. Fulling, L. Kaplan and J. H. Wilson, Vacuum energy and repulsive Casimir forces in quantum star graphs, Phys. Rev A 76 (2007), 012118.