← Back to article

Equation 6 · How Scientific Instruments and Metrology Actually Work

What does this equation mean?

d  ≈  0.61 λnsin⁡αd \;\approx\; \frac{0.61\,\lambda}{n\sin\alpha}

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.

Read it piece by piece

dd

Symbol d

d occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Understand this part →

λ\lambda

Symbol λ

the wavelength of the illuminating radiation, and nsin⁡\sinα\alpha is the numerical aperture of the imaging system.

Understand this part →

nn

Symbol n

n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Understand this part →

α\alpha

Symbol α

the numerical aperture of the imaging system.

Understand this part →

fraction

fraction

Divide the expression above the line by the one below it.

Understand this part →

See an illustrated explanation →
≈

≈

Approximately equal to; the equality is not exact.

Understand this part →

0.61 λ0.61\,\lambda

Numerator: 0.61λ

The complete quantity above the fraction bar.

Understand this part →

nsin⁡αn\sin\alpha

Denominator: nsinα

The complete quantity below the fraction bar; it must be nonzero for this division.

Understand this part →

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: d  ≈  0.61 λnsin⁡αd \;\approx\; \frac{0.61\,\lambda}{n\sin\alpha}. where λ\lambda is the wavelength of the illuminating radiation, and nsin⁡\sinα\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…
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: d  ≈  0.61 λnsin⁡αd \;\approx\; \frac{0.61\,\lambda}{n\sin\alpha}. where λ\lambda is the wavelength of the illuminating radiation, and nsin⁡\sinα\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 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.

Read the equation in its article →

Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

Return to How Scientific Instruments and Metrology Actually Work

See this formula across 1 published context →

Browse the mathematical compendium →