$$d \;\approx\; \frac{0.61\,\lambda}{n\sin\alpha}$$

Equation 6 · Scientific Methods

How Scientific Instruments and Metrology Actually Work

This equation gives an approximation: it relates the quantities while allowing an approximation.

An electron microscope is a useful worked example precisely because almost the entire modern instrument exists to correct for a limitation baked into its own optics. The achievable resolution of any imaging system with a circular aperture is bounded, in the simplest diffraction-limited approximation, by a form of the Rayleigh criterion: $d \;\approx\; \frac{0.61\,\lambda}{n\sin\alpha}$. where $\lambda$ is the wavelength of the illuminating radiation, and n$\sin$$\alpha$ is the numerical aperture of the imaging system. For visible light this caps optical microscopy at roughly 200 nanometres — far too coarse to resolve individual atoms, whose spacing in a solid is typically a few tenths of a nanometre. Electron microscopy…

Meanings in this article

• $\lambda$: the wavelength of the illuminating radiation, and n$\sin$$\alpha$ is the numerical aperture of the imaging system.
• $\alpha$: the numerical aperture of the imaging system.

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