Electromagnetism at the nanoscale
Cut gold into spheres a few dozen nanometres across and it stops looking like gold: a suspension glows ruby red, the colour of medieval stained glass. Silver turns yellow; larger particles turn purple and blue. The reason is how light, an electromagnetic wave, drives the electrons inside a particle far smaller than the wave itself.
01
Light is an oscillating electric and magnetic field. Visible light has wavelengths of 400 to 700 nanometres, and the colour you see is the wavelength. A nanoparticle is typically 10 to 100 nm across; a human hair is about 80 000 nm.
So a small particle never sees the shape of the wave. At any instant it sits in an almost uniform electric field that flips direction some 600 trillion times a second. This is the quasi-static limit: what matters next is how the particle’s own charges respond.
The particle spans 7.5% of a wavelength: at any instant it feels one uniform field, pointing up, then down, hundreds of trillions of times a second.
02
A metal is a sea of free electrons around fixed positive ions. The light’s field pushes the electron cloud to one side, leaving negative charge on one surface and positive charge on the other, and these pull the cloud back. Like a mass on a spring, the cloud has a frequency of its own.
When the light’s colour matches it, the electrons swing far more than at any other colour: a localized surface plasmon resonance, where the metal’s permittivity reaches ε = −2εmedium. Gold gets there in green light, so green is absorbed and the red that passes makes the vial red. Silver resonates in the violet and looks yellow. Raise the refractive index around it and the resonance moves to the red: biosensors detect molecules this way.
Map the field around a sphere03
As a particle grows, the field across it is no longer uniform: the resonance shifts to the red and broadens, and new ways of sloshing appear, such as a quadrupole with charge piled up in four places. Mie theory, Gustav Mie’s exact 1908 solution of Maxwell’s equations for a sphere, contains all of them.
Small particles mostly absorb light and turn it into heat, which photothermal therapies use. Large ones mostly scatter it, which is why they shine in dark-field microscopes.
Compute cross-sections of any sphere04
Stretch the sphere into a rod and the electrons can slosh two ways. Across the rod the resonance stays near green. Along it the charges sit farther apart, their pull weakens, and the resonance moves to the red and into the near infrared, where living tissue is almost transparent.
Chemists tune a rod’s colour simply by its length-to-width ratio.
Spheroid cross-sections05
Bring two particles close and their charges talk to each other. With the light polarised along the pair, opposite charges face each other across the gap and pile up: the field there grows far beyond the incoming light, in a volume of a few cubic nanometres.
Such hot spots let surface-enhanced Raman spectroscopy (SERS) detect single molecules. Turn the polarisation across the pair and the hot spot is gone.
Here each sphere is a single dipole; the two-sphere calculator solves the pair exactly.
Two spheres, exactly06
Real particles sit on surfaces, in focused beams, next to molecules and in arrays. Each calculator solves one of these situations exactly.