Physicists in the US may have ended a decade-old debate about how the Casimir force – which affects objects separated by tiny distances – should be calculated for two metal objects. They say that the so-called Drude model, which treats metal as a collection of billiard-ball-like positive ions and electrons, wins out over the “plasma model”, which assumes the electrons move in a fixed lattice of positive ions. Understanding how to determine the force could play an important role in the design of micrometre- and nanometre-sized machines.

The Casimir force was first predicted in 1948 by Dutch physicist Hendrik Casimir, who considered what happens when two uncharged, perfectly conducting metal plates are placed opposite one another in a vacuum. According to quantum mechanics, the energy of an electromagnetic field in a vacuum is not zero but continuously fluctuates around a certain mean value. However, resonance means that only certain wavelengths will exist between two plates separated by a particular distance.

What Casimir worked out was that the radiation pressure of the field outside the plates will tend to be slightly greater than that between the plates, which will therefore be attracted to one another. As it is so tiny, the Casimir force proved extremely difficult to measure and it was not until 1997 that Steve Lamoreaux, then at the University of Washington in the US, provided the first firm experimental confirmation of Casimir’s theory. Although Lamoreaux and others have since made better measurements, an important mystery remained regarding how the Casimir force should be calculated for realistic objects.

Larger gap, weaker force

Although successful Casimir measurements have been made between two gold surfaces, the problem is that gold is not a perfect conductor – which means that electromagnetic radiation can penetrate a finite distance into the metal. The gap between the surfaces is effectively greater and the force weaker than if the metal were a perfect conductor, explains Thorsten Emig of the University of Paris Sud, an expert on the Casimir force who was not involved in this latest work.

Both the plasma and Drude models are good at describing how short-wavelength light interacts with the metal surfaces – and can therefore be used to calculate the Casimir force at relatively short separations of less than about 1 µm. At larger separations, however, the models differ. The plasma model predicts that the “static transverse” electric mode of the electromagnetic field within the gap contributes to the Casimir force, whereas the Drude model says that it does not. Unfortunately, physicists had not been able to use one apparatus to measure the Casimir force over a large enough range of distances to decide which model works best at all separations.

Drude works best

Lamoreaux, who is now at Yale University, has joined forces with Hong Tang and colleagues to measure the Casimir force over the widest range of distances to date – from 100 nm to 2 µm. In doing so, the team is the first to show that the Drude model works best at both long and short distances.

While Casimir originally formulated his theory for parallel plates, actually measuring the force in this way is tricky because it is very difficult to align the plates well enough to perform the experiment. Lamoreaux’s breakthrough in 1997 involved measuring the force between a metal plate and a metal sphere – an arrangement that does not require precise alignment. His latest experiment involves measuring the force between a gold-covered sphere of radius 4 mm and an extremely thin membrane of silicon nitride that is also coated with gold. The membrane is just a few hundred nanometres thick and the gold coating is 200 nm think. An important feature of the resulting gold surface is that it is flat to within 3 nm throughout the entire membrane, which is a square with sides measuring 1 mm.

Vibrating membrane

The membrane is stretched drum-like across a silicon frame, which is vibrated using a piezoelectric actuator. A measurement is made by bringing the sphere to within about 1 µm of the gold surface while monitoring the vibrations of the membrane using a fibre interferometer. The presence of the Casimir force can be detected by its effect on how the membrane vibrates, with the force measured by varying the separation from between about 100 nm to 2 µm.

In theory, all points on a metallic surface should be at the same electrical potential, but in practice the molecules adsorbed on the surface make the potential vary such that it can affect force measurements – particularly at relatively large separations. To allow for this effect, the team raster-scanned the sphere across several membranes to measure the surface potential as a function of position. This allowed the researchers to select a membrane with the smallest variation for their Casimir measurements. Information from the scan is also used to correct for spatial variations during the measurement.

As well as showing that the Drude model is best at describing the Casimir force, the research also reveals the important role that variations in potential across the surfaces play in Casimir measurements. Indeed, the team suggests that an important next step in Casimir measurements will be to map the potential variations on the surface of the sphere. If successful, this could allow measurements to be made at even greater separations, which is something that Lamoreaux sees as an important next step in our understanding of the Casimir force.

The experiment is described in Physical Review Letters.