---
title: "How the Casimir force gives a micromachined plate two swings"
description: "Correct the ideal-mirror Casimir pressure for real gold, then find the frequency shift, nonlinearity and hysteresis of the Chan et al. torsional oscillator."
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/fluctuational-electrodynamics/casimir-force-and-a-bistable-oscillator/"
last_modified: "2026-10-30"
---

[Fluctuational electrodynamics](https://sandboxphysics.com/stories/fluctuational-electrodynamics/)

L3 · Chapter 6

# How the Casimir force gives a micromachined plate two swings

Sandbox Physics · 2026-10-30

Two uncharged metal bodies brought very close attract each other. This is the Casimir force, and its source is the same as for the heat flow of the earlier chapters: electromagnetic fluctuations between two mirrors. In 2001, Chan and co-workers held a gold-coated sphere about 100 nm above a micromachined torsional plate. The plate’s resonance moved down, its resonance curve leaned over, and at last a single gap gave two stable swings, depending on which side the plate came from.

## Before you begin

Mode energy and zero-point energy from chapter 4; the resonance curve of a driven damped oscillator; Taylor expansion.

- Explain why two plates at equal temperature exchange no net heat but still attract.
- Estimate the Casimir force between gold mirrors from the ideal-mirror formula and reduction factors, and explain the role of the plasma wavelength of gold.
- Find the frequency shift and nonlinear coefficient of a torsional oscillator from the force gradient and higher derivatives, and explain hysteresis and the two swings.

## Zero heat flux, nonzero force

The heat flux of chapter 4 uses only $\Theta$: the zero-point energy $\hbar\omega/2$ is the same on both sides and cancels. The Casimir force is different. It is the momentum flux of the field in the gap, given by the Maxwell stress tensor, and the energy of each mode must then be taken as $\Theta+\hbar\omega/2$. Joulain and co-workers spell out this difference in their Sec. 3.4.

So between two plates at the same temperature the net heat flux is exactly zero while the attraction remains, even at 0 K. Between two ideal mirrors, the gap supports fewer modes than the outside; a narrower gap supports fewer still, the zero-point energy drops, and the plates attract. This is Casimir’s 1948 result.

When does the thermal part matter? Lambrecht and Reynaud give the thermal length $\lambda_T=\hbar c/(k_BT)$, 7.63 µm at 300 K. At gaps much smaller than this the thermal correction is negligible; the gaps in this chapter are around 100 nm, and the force comes almost entirely from zero-point fluctuations.

$$
\frac{F_C}{A}=\frac{\pi^{2}\hbar c}{240\,L^{4}},\qquad \frac{E_C}{A}=\frac{\pi^{2}\hbar c}{720\,L^{3}}
$$

Eqs. (1) and (2) of Lambrecht and Reynaud: the Casimir force and energy per unit area between ideal mirrors. The pressure is 1.30 mPa at 1 µm and 13.0 Pa at 100 nm.

Source: K. Joulain · A. Lambrecht and S. Reynaud

## Real gold: reflection weakens at short range

A real metal reflects well only below its plasma frequency. The plasma wavelength of gold, $\lambda_P$, is 136 nm. At gaps much larger than this the modes involved have low frequencies and gold is nearly an ideal mirror. At gaps comparable to it or smaller, some of the modes involved are too high in frequency for gold to reflect well, and the force weakens.

Lambrecht and Reynaud describe this with the reduction factors $\eta_F$ and $\eta_E$. At long range, $\eta_F$ falls short of 1 by a term proportional to $\lambda_P/L$. At short range, $\eta_F$ is proportional to $L/\lambda_P$ with coefficient 1.193. The course model recomputes these with the plasma model and finds 1.1933 and an energy-to-force ratio of 1.4995, matching their 1.193 and 3/2.

Electron relaxation lowers the result a little more. With the Drude model and a damping-to-plasma-frequency ratio of 0.035/9.0, the force is 1.3% to 1.8% below the plasma model, consistent with their statement of less than 2%. From tabulated optical data for gold they obtain 0.48, 0.81 and 0.96 at 0.1, 0.5 and 3 µm; the course Drude model gives 0.440, 0.793 and 0.948, 8% low at 0.1 µm. The course replaces their measured optical data with a model permittivity. That is the likely cause, but we did not repeat the calculation with optical data to confirm it.

$$
F=\eta_F\,F_C,\qquad \eta_F\simeq1-\frac{8\lambda_P}{3\pi L}\;(L\gg\lambda_P),\qquad \eta_F\simeq1.193\,\frac{L}{\lambda_P},\;\eta_E\simeq\frac{3}{2}\,1.193\,\frac{L}{\lambda_P}\;(L\ll\lambda_P)
$$

Eqs. (4), (5), (14) and (15) of Lambrecht and Reynaud, with $\lambda_P=2\pi c/\omega_P$.

Source: A. Lambrecht and S. Reynaud

## A sphere, a plate and a torsion spring

Experiments use a sphere facing a plate rather than two parallel plates, so nothing has to be aligned. The proximity-force estimate treats the sphere as a stack of thin rings, each contributing as a pair of plates at the local gap; the force on the sphere is $2\pi R$ times the energy per unit area between plates.

The plate of Chan and co-workers is 500 µm across and hangs on two torsion rods, with moment of inertia $I=7.1\times10^{-17}\,\mathrm{kg\,m^2}$, free resonance 2753.47 Hz and quality factor 7150. The sphere has a radius of 100 µm and sits over a point $b=131.0\,\mu\mathrm{m}$ from the rotation axis.

The Casimir force grows as the gap closes, which acts as a spring of negative stiffness and pulls the resonance down. The shift is set by the first derivative of the force. The second and third derivatives make the restoring force nonlinear and the resonance peak leans toward low frequency. In the amplitude equation, one coefficient $\kappa$ sums up the strength of the nonlinearity.

When the lean is strong enough, one drive frequency allows three amplitudes, and the middle one is unstable. A plate approaching from far away stays on the small-amplitude branch; a plate retreating from close in stays on the large-amplitude branch. That is hysteresis.

$$
F_{\mathrm{sphere}}(z)=2\pi R\,\eta_E\,\frac{E_C(z)}{A}\;\xrightarrow{\;\eta_E=1\;}\;\frac{\pi^{3}\hbar c\,R}{360\,z^{3}},\qquad F'(z)=\frac{\pi^{3}\hbar c\,R}{120\,z^{4}}
$$

The proximity-force estimate of Eq. (8) of Lambrecht and Reynaud. The right-hand side is the ideal-mirror case; its derivative is Eq. (4) of Chan et al.

$$
\omega_1^{2}=\omega_0^{2}-\frac{b^{2}F'(z)}{I},\qquad \Delta f\approx-\frac{f_0\,b^{2}F'(z)}{2I\omega_0^{2}}
$$

Eqs. (1) and (2) of Chan et al. Here $I\omega_0^2=2.125\times10^{-8}\,\mathrm{N\,m}$, consistent with their torsional stiffness of $2.1\times10^{-8}\,\mathrm{N\,m\,rad^{-1}}$.

$$
A^{2}\left[\left(\omega-\omega_1-\kappa A^{2}\right)^{2}+\gamma^{2}\right]=\frac{\tau^{2}}{4I^{2}\omega_1^{2}},\qquad \kappa=\frac{3\beta}{8\omega_1}-\frac{5\alpha^{2}}{12\omega_1^{3}}
$$

Eq. (5) of Chan et al. The coefficients $\alpha$ and $\beta$ are proportional to the second and third derivatives of the force, and $\gamma=\omega_0/(2Q)$.

Source: A. Lambrecht and S. Reynaud · H. B. Chan

## Worked example: frequency shift and nonlinearity at 98 nm

Problem. Treat gold as an ideal mirror, as Chan and co-workers did, and find the Casimir force, frequency shift and nonlinear coefficient with the sphere 98 nm from the plate. Compare with $\kappa=-2.8\times10^{8}\,\mathrm{rad^{-2}\,s^{-1}}$ from the caption of their Fig. 4, then say what changes with Drude gold.

Force. $F=\pi^3\hbar cR/(360z^3)$ with $R=100\,\mu\mathrm{m}$ and $z=98\,\mathrm{nm}$ gives 289 pN. The gradient is $F'=3F/z=8.86\times10^{-3}\,\mathrm{N\,m^{-1}}$.

Frequency shift. $b^2F'=(131\times10^{-6})^2\times8.86\times10^{-3}=1.52\times10^{-10}\,\mathrm{N\,m}$. Divided by $2I\omega_0^2=4.25\times10^{-8}\,\mathrm{N\,m}$ this is $3.58\times10^{-3}$, and times 2753.47 Hz the first-order shift is $-9.85\,\mathrm{Hz}$. The square-root formula for $\omega_1$ gives $-9.86\,\mathrm{Hz}$.

Nonlinearity. From the second and third derivatives of the force the model finds $\kappa=-2.80\times10^{8}\,\mathrm{rad^{-2}\,s^{-1}}$, matching the caption. At 141 nm and 116.5 nm it gives $-3.13\times10^{7}$ and $-9.86\times10^{7}$, against the caption’s $-3.1\times10^{7}$ and $-1.0\times10^{8}$.

Drude gold. At the same 98 nm the force is only 149 pN, the shift $-4.28\,\mathrm{Hz}$ and $\kappa=-9.1\times10^{7}$. This does not mean the experiment was wrong. Chan and co-workers obtained their distances by fitting the frequency shift with the ideal-mirror formula (closest approach 85.9 nm), and they state that they did not attempt a precise comparison including finite conductivity and roughness. With Drude gold, the same shifts appear at smaller gaps.

Hysteresis. With the drive fixed at 2748 Hz and a free swing of 39 nm, the plate approaches and then retreats. For ideal mirrors the model finds two swings between 117.25 and 120.25 nm, differing by up to 5.71 times; for Drude gold, between 95.0 and 98.25 nm, differing by up to 6.21 times. Chan and co-workers write that the swings differ by up to a factor of 6.

Source: H. B. Chan · A. Lambrecht and S. Reynaud

## Try it in the Lab

### [Pull the resonance down](https://sandboxphysics.com/labs/casimir-torsional-oscillator/)

1. Predict first: as the sphere moves from 3.3 µm to 141 nm, by how many hertz will the resonance drop?
2. Choose the Sphere far away preset, then Resonance pulled down, and read the resonance shift and the Casimir force. Open the Resonance shift for three mirror models chart and compare ideal mirrors with gold.
3. Explain the shift with the force gradient, and explain why ideal mirrors give a larger shift with the plasma wavelength of gold.

### [One gap, two swings](https://sandboxphysics.com/labs/casimir-torsional-oscillator/)

1. Predict first: with the drive fixed at 2748 Hz, the sphere moves slowly in and then slowly out. Is the swing the same both times it passes 97 nm?
2. Choose the Hysteresis scan preset and compare the approach and retreat curves in the Swing while the gap is scanned chart. Then choose Two stable swings and Ideal mirrors and compare where the two-swing region sits.
3. Explain hysteresis with the leaning resonance curve and its unstable middle solution. Finally choose Snap to contact and say what happens when the negative stiffness beats the torsion spring.

### [Let the boundary move: from attraction to photon pairs](https://sandboxphysics.com/labs/dynamical-casimir-photon-pairs/)

1. Predict first: static mirrors only attract. If a pump modulates one boundary quickly, what will the detectors see? Looking at one output alone, can you tell it from thermal light?
2. In the dynamical Casimir Lab, set the pump area to zero, then raise it step by step. Compare the output with thermal light of equal brightness, then rotate the shared receiver phase and watch how the fluctuations the two outputs share change.
3. Explain how the static Casimir force and these photon pairs start from the same vacuum fluctuations: a still boundary only changes the zero-point energy, while a boundary modulated by a pump turns the fluctuations into photon pairs, with the energy supplied by the pump.

## Exercises with explained answers

1. What is the ideal-mirror Casimir pressure at 10 nm? Use the short-range slope of Lambrecht and Reynaud to estimate the reduction factor for gold there and the actual pressure.

   Show the answer

   The pressure scales as $L^{-4}$: 13.0 Pa at 100 nm becomes $1.30\times10^{5}\,\mathrm{Pa}$ at 10 nm, about 1.3 atmospheres. With $L/\lambda_P=10/136=0.074$, $\eta_F\approx1.193\times0.074=0.088$ and the actual pressure is about $1.1\times10^{4}\,\mathrm{Pa}$. This is only an asymptotic estimate, and whether a local permittivity still holds at 10 nm was not tested.

2. The torsional stiffness is $I\omega_0^2=2.125\times10^{-8}\,\mathrm{N\,m}$. For ideal mirrors, how close must the sphere be for the Casimir force gradient to cancel the spring, $b^2F'=I\omega_0^2$?

   Show the answer

   $F'=2.125\times10^{-8}/(131\times10^{-6})^2=1.238\,\mathrm{N\,m^{-1}}$. From $F'=\pi^3\hbar cR/(120z^4)$, $z^4=6.60\times10^{-31}\,\mathrm{m^4}$ and $z=28.5\,\mathrm{nm}$. Any closer and the plate snaps in. With Drude gold the distance is 17.0 nm.

3. In the hysteresis scan, why does the plate stay on the small-amplitude branch while approaching and on the large-amplitude branch while retreating, and not the other way round?

   Show the answer

   The 2748 Hz drive lies below the far-away resonance, and $\kappa<0$ bends the peak toward low frequency, so the drive sits on exactly the side the peak leans over. As the sphere approaches, the resonance moves down toward the drive; the plate starts on the small-amplitude solution and follows it through the two-swing region until that branch ends, then jumps up. On retreat the plate starts on the large-amplitude solution; the leaning peak extends that branch to drive frequencies farther from resonance, that is to larger gaps, before it falls back. At 97.25 nm the model gives a 7.2 nm swing on approach and 36 nm on retreat for Drude gold.

4. Two gold plates are both at 300 K with a 100 nm gap. What are the net heat flux and the Casimir force between them? What changes if both are cooled to 4 K?

   Show the answer

   The net heat flux is zero, because $\Theta$ is the same on both sides of every channel. The Casimir force is not zero: it comes from $\Theta+\hbar\omega/2$, and the zero-point part does not depend on temperature. Since 100 nm is far below the 7.63 µm thermal length at 300 K, the force comes almost entirely from the zero-point part; at 4 K it is nearly unchanged, and the net heat flux is still zero.

[Previous: Why near-field heat transfer grows as the inverse square of the gap](https://sandboxphysics.com/stories/fluctuational-electrodynamics/tunnelling-channels-and-the-inverse-square-law/) [Next: Reading Shchegrov et al. (2000) on why silicon carbide looks different near and far](https://sandboxphysics.com/stories/fluctuational-electrodynamics/reading-shchegrov-2000/)

## Sources and what we recomputed

1. [K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati and J.-J. Greffet, Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and Casimir forces revisited in the near field, arXiv:physics/0504068v2; Surface Science Reports 57, 59–112 (2005)](https://arxiv.org/abs/physics/0504068v2)

   Version 2, 24 April 2005. Sec. 2: surface waves, the Drude permittivity (Eq. 21) and the SiC oscillator (Eq. 22). Sec. 3.2: the fluctuation–dissipation theorem and the mean oscillator energy (Eqs. 25 to 30). Secs. 4.1 and 4.2: the emitted energy density (Eqs. 44 to 48), emissivity as the half-sum of the two transmission factors and its link to Kirchhoff’s law, and Fig. 6 at 100 µm, 1 µm and 100 nm above SiC. Sec. 7: heat flux between two half-spaces (Eqs. 89 to 93), the growth as the inverse square of the gap, and Figs. 17 and 18, which they reproduce from another paper. Sec. 8: the Casimir force from the same fluctuating fields. The evanescent term of Eq. 45 as printed carries an extra factor of two; we fixed the prefactor from Eq. 48. The prefactor of Eq. 93 is garbled in the text version we read, so we quote only its inverse-square dependence on the gap.

2. [H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop and F. Capasso, Nonlinear micromechanical Casimir oscillator, arXiv:quant-ph/0109046v2; Physical Review Letters 87, 211801 (2001)](https://arxiv.org/abs/quant-ph/0109046v2)

   Version 2, 21 September 2001. The device: a 500 µm polysilicon plate, gold-coated, on two torsional rods, with a gold-coated sphere of radius 100 µm. Eq. (1): the equation of motion with the Casimir force expanded to third order. Eqs. (2) and (3): the shifted resonance and the nonlinear coefficients. Eq. (4): the perfect-mirror force gradient used for their fit, with the closest distance 85.9 nm and lever arm 131.0 µm from the fit. Eq. (5): the amplitude equation. Fig. 4 caption: Q of 7150, the four distances, the four values of the nonlinear coefficient, a 39 nm swing and a fixed drive at 2748 Hz in panel (b). Text: in the bistable region the two swings differ by up to a factor of 6. We recomputed the coefficients and the hysteresis; we did not digitise the data.

3. [A. Lambrecht and S. Reynaud, Casimir force between metallic mirrors, arXiv:quant-ph/9907105v2; European Physical Journal D 8, 309–318 (2000)](https://arxiv.org/abs/quant-ph/9907105v2)

   Version 2. Eqs. (1) and (2): the Casimir energy and force between perfect mirrors. Eqs. (3) to (6): the plasma wavelength and the reduction factors for force and energy. Eq. (7): the thermal length, a few µm at room temperature. Eq. (8): the proximity-force estimate for a sphere and a plane. Eqs. (14) and (15): the short-distance slopes. Sec. V and table (20): gold with a plasma frequency of 9.0 eV (plasma wavelength 136 nm) and damping 35 meV, and reduction factors at 0.1, 0.5 and 3 µm from tabulated optical data. Text: relaxation changes the force by less than 2 %. We recomputed the asymptotes and the table with model permittivities.