Spectral Line Calculator - Transition and Wavelength Analysis

Calculator Inputs

Spectrum Type

Atomic Number (Z)

Lower Level n₁

Upper Level n₂

Radial Velocity (km/s)

Measured Wavelength (nm, optional)

Resolving Power (R = λ/Δλ)

Intrinsic Line Width (pm)

Calibration Offset (nm)

Relative Peak Intensity

Example Data Table

Sample transitions for quick validation and educational reference.

Label	Z	n₂	n₁	Series	Approx. Vacuum Wavelength (nm)	Region
Lyman-α	1	2	1	Lyman	121.567	Ultraviolet
Balmer-α (Hα)	1	3	2	Balmer	656.280	Visible Red
Balmer-β (Hβ)	1	4	2	Balmer	486.133	Visible Blue-Green
Paschen-α	1	4	3	Paschen	1875.626	Infrared
He⁺ 4→3	2	4	3	Paschen-like	468.906	Visible Blue

Formula Used

1) Rydberg relation for hydrogen-like ions:
1 / λ = R × Z² × (1 / n₁² − 1 / n₂²)

2) Frequency:
ν = c / λ

3) Photon energy:
E = hν = hc / λ

4) Wavenumber:
ṽ = 1 / λ

5) Redshift:
z = (λ_obs − λ_rest) / λ_rest

6) Non-relativistic radial velocity estimate:
v ≈ zc

7) Instrument resolution element:
Δλ = λ / R

8) Effective broadening:
FWHM_eff = √(FWHM_intrinsic² + Δλ²)

This calculator uses an ideal hydrogen-like one-electron approximation. It is excellent for teaching, quick checks, and many astrophysical estimates, but not a full multi-electron quantum treatment.

How to Use This Calculator

Choose emission or absorption context.
Enter atomic number Z for a hydrogen-like ion.
Provide lower and upper energy levels, where n₂ must exceed n₁.
Enter radial velocity for a predicted observed wavelength.
Optionally enter a measured wavelength to override the modelled one.
Add resolving power, line width, calibration offset, and relative intensity.
Press Calculate Spectral Line.
Review the result table, graph, and exported report if needed.

FAQs

1) What does this spectral line calculator compute?

It calculates vacuum wavelength, observed wavelength, frequency, wavenumber, photon energy, transition energy gap, redshift, estimated radial velocity, and line broadening metrics for hydrogen-like transitions.

2) Which atoms does the model support?

It is best for hydrogen and hydrogen-like ions with one electron, such as He⁺ or Li²⁺. Multi-electron atoms need more advanced structure models.

3) Why must n₂ be larger than n₁?

A spectral transition between bound states uses a higher initial level and a lower final level for emission. The Rydberg term becomes invalid if n₂ is not greater.

4) What is the difference between rest and observed wavelength?

Rest wavelength comes from the ideal transition formula. Observed wavelength includes motion, redshift, and any calibration offset or directly measured line position you enter.

5) When should I enter a measured wavelength?

Use it when you already identified a line in experimental or telescope data. The calculator then derives redshift and velocity from that measured line position.

6) What does resolving power control?

Resolving power determines the instrument’s wavelength discrimination. Higher values mean smaller resolution elements, sharper lines, and better ability to separate nearby transitions.

7) Is the Doppler velocity fully relativistic?

No. This version uses the common non-relativistic approximation v ≈ zc. It works well for modest speeds but becomes less accurate for very large redshifts.

8) Why is air wavelength sometimes unavailable?

The dry-air conversion is only applied in a practical wavelength range. Outside that range, vacuum values remain the safer and more standard reference.