Thermal Motion and Spectral Lines

Atoms and molecules in a gas move at speeds that depend on temperature. These motions cause emitted or absorbed radiation to experience slight Doppler shifts. Because the atoms are moving in many directions with a range of speeds, the resulting spectral line is broadened. This effect is called Doppler broadening or thermal broadening. Understanding the width of spectral lines is essential in astrophysics and plasma diagnostics, where the line shape reveals temperature and other properties of the emitting gas.

Mathematical Background

The distribution of particle velocities in a gas at thermal equilibrium is described by the Maxwell–Boltzmann distribution. Each velocity component along the line of sight leads to a small Doppler shift $\mathrm{\backslash Delta\; \backslash lambda}$ of the emitted or absorbed wavelength $\mathrm{\backslash lambda\backslash \_0}$. Integrating over the distribution yields a Gaussian line profile whose full width at half maximum (FWHM) is

$\mathrm{\backslash Delta\; \backslash lambda\backslash \_\{FWHM\}}=2\sqrt{\mathrm{\backslash ln}2}\mathrm{\backslash lambda\backslash \_0}\frac{\sqrt{2\mathrm{k\backslash \_B}T}}{c}$

where $\mathrm{k\backslash \_B}$ is Boltzmann’s constant, $T$ is temperature, $m$ is the atomic mass in kilograms, and $c$ is the speed of light. This formula assumes nonrelativistic speeds and neglects natural or collisional broadening.

Uses in Astronomy

Astronomers analyze the widths of spectral lines in stellar atmospheres to determine temperatures. Hotter stars produce broader lines because the atoms move faster. In nebulae and interstellar clouds, Doppler broadening reveals kinetic temperatures and turbulent motions. With high-resolution spectra, scientists can even distinguish thermal broadening from other effects such as rotation or magnetic splitting.

Laboratory Spectroscopy

In laboratory plasmas or gas-discharge lamps, Doppler broadening sets a fundamental limit on spectral resolution. Accurate temperature measurements often rely on fitting observed line profiles to a Doppler-broadened Gaussian. Understanding how the width depends on atomic mass and temperature helps researchers design experiments and interpret data.

Using the Calculator

Enter the central wavelength in nanometers, the atomic mass in atomic mass units, and the temperature in kelvin. When you press the compute button, the script converts the mass to kilograms and evaluates the equation for $\mathrm{\backslash Delta\; \backslash lambda\backslash \_\{FWHM\}}$. The result appears in nanometers alongside the Doppler width expressed as a relative fraction of $\mathrm{\backslash lambda\backslash \_0}$. This output provides a quick way to see how much thermal motion broadens your spectral line.

Effect of Atomic Mass

Lighter atoms move faster at the same temperature, yielding larger Doppler broadening. For example, hydrogen exhibits wider lines than heavier elements like iron at identical temperatures. This dependence allows astrophysicists to interpret complex spectra where multiple species contribute overlapping lines.

Temperature Sensitivity

Doppler width increases with the square root of temperature. Doubling the temperature does not double the width, but it increases by a factor of about 1.4. By carefully measuring line widths, scientists can determine temperatures with impressive precision, especially when other broadening mechanisms are small.

Beyond Simple Gases

Real spectra often include additional broadening sources such as natural lifetime broadening, pressure broadening from collisions, and Stark or Zeeman effects due to electric and magnetic fields. Nonetheless, Doppler broadening frequently dominates in low-pressure gases and astrophysical plasmas, making it a cornerstone in spectral analysis.

Historical Insights

The Doppler effect was first described by Christian Doppler in 1842 for sound waves and later applied to light. Spectral line broadening due to thermal motion was recognized soon after the discovery of the Maxwell–Boltzmann distribution. With the advent of lasers and precision spectroscopy, accurate models of Doppler broadening became essential tools in modern physics and chemistry laboratories.

Conclusion

Doppler broadening reflects the microscopic motion of atoms, linking macroscopic temperature to observable spectral features. By calculating the FWHM for a given wavelength, atomic mass, and temperature, this calculator helps you understand how thermal energy shapes the appearance of spectral lines in astronomy, plasma physics, and countless other applications.