Historical research guide · 0995 · 2010
Casimir force at a knife’s edge
Noah Graham · Alexander Shpunt · Thorsten Emig · Sahand Jamal Rahi · Robert L. Jaffe · Mehran Kardar
In one minute
The Casimir force between two mirrors is simple to state and very hard to compute for any shape but the simplest. Graham, Shpunt, Emig, Rahi, Jaffe and Kardar add a new exactly solvable shape to that short list: the parabolic cylinder, a mirror curved like the blunt nose of a wedge. Because the wave equation separates in parabolic coordinates, the scattering of vacuum fluctuations off such a mirror can be written in closed form, and fed into the scattering formalism this group built it gives the Casimir energy between the cylinder and a flat plate as a function of the gap, the tilt and the sharpness of the tip. Two limits carry the paper. Bring a blunt cylinder close and the old proximity-force rule of thumb becomes exact. Sharpen the tip to nothing and you have a knife’s edge, where that rule predicts no force at all while the true energy falls off as one over the gap squared. The authors then propose the table-top test: a thin metal disc held edge-on above a plate.
Connection
Chapter 2 holds that the vacuum is a real medium whose energy depends on the shape of the boundaries you place in it, and this paper makes that dependence exact for the hardest case, a sharp edge. Chapter 6’s Casimir power cells pull precisely that lever — geometry chosen so the vacuum pushes unevenly — so an exact edge calculation is a design tool, not a curiosity.
From the source
The parabolic cylinder joins infinite plates, circular cylinders and spheres as a geometry in which the Casimir interaction can be computed exactly, because the Helmholtz equation separates in parabolic cylinder coordinates and the scattering amplitudes of a perfectly reflecting parabolic mirror can be written in closed form as ratios of parabolic cylinder functions for Dirichlet and for Neumann boundary conditions alike.Equations (1), (2), (7), (8) and (11)
For a blunt cylinder held close to the plate the proximity force approximation becomes exact: the numerical Casimir energy is 0.9961 of the proximity-force value at a separation of a quarter of the tip radius, and the same proximity-force expression applies to a circular cylinder of equal radius.Equation (15); Figure 2
In the opposite limit, where the tip radius vanishes and the parabolic cylinder becomes a semi-infinite plate — a knife’s edge facing a mirror — the proximity force approximation and the dilute-limit perturbative expansion both give exactly zero, while the true Casimir energy per unit length is minus 0.0067415 times the reduced Planck constant times the speed of light divided by the separation squared; the Dirichlet part of that number, minus 0.0060485, agrees with the independent world-line computation of Gies and Klingmuller.Equation (16); comparison with Reference 13
Tilting the edge changes the energy through a single angle-dependent coefficient, and as the half-plane turns parallel to the mirror the result separates into the ordinary parallel-plate law plus an edge correction of about minus 0.0009, made of a Dirichlet part near minus 0.0025 and a Neumann part near minus 0.0034; the whole calculation extends to finite temperature by summing Matsubara frequencies, and in the high-temperature limit only the lowest mode survives, leaving an energy per unit length of minus 0.0472 times the temperature divided by the separation.Equation (17); Figure 3; the finite-temperature paragraph
The authors propose a measurable version of the edge geometry: a thin metal film disc held perpendicular to a plate, whose energy in an edge proximity approximation grows as the square root of the disc radius divided by twice the cube of the separation, unlike the sphere-plate case which goes as radius over separation squared. For a disc of 100 micrometre radius the present 0.1 piconewton force sensitivity corresponds to a separation near 350 nanometres, comfortably above the 130 nanometre plasma wavelength and 25 nanometre skin depth that set the limits of the perfect-mirror treatment, and the micromechanical torsion oscillators already used for Casimir measurements are readily adaptable to it.Final two paragraphs, the edge proximity force approximation
Because the result comes from scattering amplitudes rather than from pairwise summation, the same machinery extends to a parabolic cylinder facing another cylinder, to several objects at once, and to an object placed inside a parabolic cylinder; the open extension is finite conductivity, where the matrix of scattering amplitudes is no longer diagonal and the calculation becomes substantially harder.Penultimate paragraph and the closing sentences on finite conductivity
The way in
https://doi.org/10.1103/physrevd.81.061701How to cite it
Noah Graham, Alexander Shpunt, Thorsten Emig, Sahand Jamal Rahi, Robert L. Jaffe, Mehran Kardar (2010) Casimir force at a knife’s edge doi:10.1103/physrevd.81.061701