Doppler Broadening Calculator

Overview: Doppler Broadening and Thermal Motion

Atoms and molecules in a gas are in constant random motion. Because of the Doppler effect, radiation emitted or absorbed by these moving particles is slightly shifted in wavelength depending on their line-of-sight velocity. In a thermal gas, particles move with a wide range of speeds and directions, so instead of a perfectly sharp spectral line, you observe a broadened profile. This effect is called Doppler broadening or thermal broadening.

Quantifying this line width is essential in astrophysics, plasma physics, and laboratory spectroscopy. From the width, you can infer temperatures, distinguish between different broadening mechanisms, and evaluate whether your spectrograph has sufficient resolving power.

Mathematical Background and Formula

For a gas in thermal equilibrium at temperature T, the particle velocities follow the Maxwell–Boltzmann distribution. Considering only the component of velocity along the line of sight and applying the (nonrelativistic) Doppler shift, one obtains a Gaussian line profile for the spectral line intensity as a function of wavelength.

The full width at half maximum (FWHM) of this Gaussian, expressed in wavelength units, is given by

where:

• ${\lambda }_0$ is the central (rest) wavelength of the transition.
• $k_B$ is Boltzmann’s constant.
• $T$ is the temperature in kelvin.
• $m$ is the atomic or ionic mass in kilograms.
• $c$ is the speed of light in vacuum.

The calculator expects the wavelength in nanometers and the atomic mass in atomic mass units (amu). Internally, it converts

where $A$ is the mass in amu. The final FWHM is returned in nanometers, and the tool also reports the relative Doppler width $\frac{{\mathrm{\Delta \lambda }}_{\text{FWHM}}}{{\lambda }_0}$, which is dimensionless.

How to Use the Calculator

To compute the thermal Doppler broadening of a spectral line:

1. Enter the central wavelength ${\lambda }_0$ in nanometers. Optical and near-UV lines typically lie between about 200 nm and 1,000 nm.
2. Enter the atomic mass in atomic mass units (amu). For example, hydrogen is about 1 amu, helium about 4 amu, and iron about 56 amu.
3. Enter the temperature in kelvin. Typical stellar photospheres range from ~3,000 K to ~20,000 K, while laboratory plasmas can span from a few hundred to tens of thousands of kelvin.
4. Click the compute button to evaluate the FWHM using the formula above.

The output includes:

• Doppler FWHM in nanometers.
• Relative width, giving a sense of how broad the line is compared to its central wavelength.

Interpreting the Results

The Doppler width increases with temperature and decreases with particle mass. A larger FWHM can indicate:

• A hotter gas (faster thermal motion), or
• A lighter species dominating the emission or absorption.

For a given temperature, hydrogen lines will be much broader than iron lines. If you compare your computed thermal width to the width actually measured in a spectrum, you can:

• Estimate how much of the observed width is due to thermal Doppler broadening.
• Identify cases where other mechanisms (pressure, turbulence, rotation, instrument resolution) must be contributing because the observed width is significantly larger than the thermal prediction.

Because the output includes the relative width, you can quickly relate it to instrumental resolving power. A spectrograph with resolving power $R=\frac{\lambda }{\mathrm{\Delta \lambda }}$ must have $R$ significantly larger than $\frac{{\lambda }_0}{{\mathrm{\Delta \lambda }}_{\text{FWHM}}}$ to fully resolve the Doppler-broadened line.

Worked Example: Hydrogen Line in a Stellar Atmosphere

Consider the Hα hydrogen Balmer line at a central wavelength ${\lambda }_0=\; 656.3$ nm in a stellar photosphere at $T=\; 10,000$ K. Hydrogen has an atomic mass of about 1 amu.

Using the calculator, enter:

• Central wavelength: 656.3 nm
• Atomic mass: 1 amu
• Temperature: 10000 K

The tool converts 1 amu to $1.6605\; \times \; 10$⁻²⁷ kg, substitutes all constants, and evaluates the FWHM. A typical result is on the order of ~0.02 nm (exact value depends on the constants used). The corresponding relative width is then roughly

relative width ≈ 0.02 / 656.3 ≈ 3 × 10⁻⁵

Interpreting this:

• The line is extremely narrow in absolute terms, but still wide enough to be resolved by high-resolution spectrographs.
• If your observed Hα profile is substantially broader than this, additional broadening (e.g., Stark broadening in hot, dense plasmas or instrumental effects) must be significant.

Effect of Atomic Mass and Temperature

The Doppler width scales approximately as $\sqrt{\frac{T}{m}}$:

• At fixed temperature, lighter elements (small $m$) give larger widths.
• At fixed species, increasing temperature widens the line.

The table below shows qualitative trends for an optical line near 500 nm:

Use the calculator to plug in the exact wavelengths and conditions relevant to your application for more precise numbers.

Applications in Astronomy and Laboratory Spectroscopy

Astronomy: In stellar atmospheres, Doppler broadening provides a direct probe of thermal motions. Combined with models of line formation, it helps constrain effective temperatures and turbulent velocities. In nebulae and interstellar clouds, measured widths of emission lines (e.g., hydrogen recombination lines) distinguish between thermal and non-thermal motions.

Laboratory plasmas and gas-discharge lamps: In fusion-relevant plasmas, Doppler-broadened impurity lines are used to estimate ion temperatures. In precision spectroscopy experiments, thermal broadening sets a lower limit to the achievable line width unless special cooling or beam techniques are used.

Assumptions and Limitations

This Doppler broadening calculator is based on a simplified physical model. Keep the following assumptions and limitations in mind when interpreting results:

• Nonrelativistic velocities: The formula assumes particle speeds are much smaller than the speed of light. For ultra-hot or relativistic plasmas, a relativistic Doppler treatment is required.
• Purely thermal motion: Only random thermal velocities are included. Bulk flows, turbulence, rotation, winds, or expansion can add additional broadening or asymmetries.
• Gaussian line shape: The line profile is assumed to be purely Gaussian, as expected for Doppler broadening alone.
• No other broadening mechanisms: Natural (radiative), collisional/pressure (Stark, van der Waals), Zeeman, and instrumental broadening are neglected. In dense or strongly magnetized environments, these can dominate.
• Optically thin approximation: The model effectively assumes that reabsorption and radiative transfer effects do not strongly distort the profile.
• Single species: The calculation applies to a single atomic or ionic species at a single temperature. Mixtures or multi-temperature components require more detailed modeling.

Because of these limitations, the computed FWHM should be treated as the thermal Doppler contribution to the line width. Compare it to observed widths and, if needed, combine it with other broadening terms using appropriate line-shape theory.

Further Reading and References

For more in-depth treatments of Doppler broadening and spectral line formation, standard references include:

• Mihalas, D., Stellar Atmospheres.
• Rybicki, G. B., & Lightman, A. P., Radiative Processes in Astrophysics.
• Huber, M. C. E., et al., Atomic and Molecular Spectroscopy.

These sources discuss how Doppler broadening interacts with other line-shape mechanisms and how to interpret observed profiles in more complex environments.