Equation 6 · How Scientific Instruments and Metrology Actually Work
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol d
d occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol λ
the wavelength of the illuminating radiation, and n is the numerical aperture of the imaging system.
Symbol n
n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Denominator: nsinα
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.
What the article says around this equation
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: . where is the wavelength of the illuminating radiation, and n 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…
Read the full surrounding passage
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: . where is the wavelength of the illuminating radiation, and n 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 exploits the far shorter de Broglie wavelength of accelerated electrons — picometre scale at typical accelerating voltages — to push the diffraction limit itself down by more than three orders of magnitude.
Sources cited in the article section
- [8] Imaging single atoms using secondary electrons with an aberration-corrected electron microscope ↗
- [7] Electron ptychography of 2D materials to deep sub-ångström resolution ↗
These citations give research context. Read each source to check which claims it supports.
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