Sections · Spectra

Vol. Science · Astronomy · The lines

Spectra
of hydrogen

A hydrogen atom does not glow in a smear. The electron sits on a ladder of levels, numbered n = 1, 2, 3, …, and a jump from an upper level n₂ to a lower level n₁ takes one photon. The vacuum wavelength is the Rydberg formula, 1/λ = R_H (1/n₁² − 1/n₂²). The constant on this desk is R_H = 1.09678×10⁷ m⁻¹. That is the 2022 CODATA Rydberg constant, 10973731.568157 m⁻¹, multiplied by the proton–electron mass ratio 1836.152673426 over one plus that ratio, then rounded to six figures. The unrounded product is 10967758.340 m⁻¹. Frequency is c/λ and the photon energy is hc/λ, with h, c, and e the exact SI defining constants on the same CODATA wallet. NIST’s unresolved Hα is 656.280 nm in air and 656.463 nm in vacuum. Lyman-alpha is 121.56701 nm, already vacuum. The formula’s Hα is 656.467 nm. The few thousandths of a nanometre are the rounding, plus fine structure, which one Bohr jump does not include. The same (1+z) that the redshift desk uses walks Lyman-alpha into the visible. The continuum under an absorption line is the blackbody desk. Where those lines sit among stars is the HR diagram and the stars desk.

Educational calculator. A classroom Bohr atom. Not a line list, not a measured spectrum, and not an ephemeris. The hue on the strip is a sketch so a line can be found. It is not a CIE color match. Air wavelengths are shorter than vacuum. The redshift desk quotes NIST air values for Hα and Hβ. This desk’s marks are the vacuum formula.

Interactive · Rydberg line, series, redshift

Balmer series · vacuum formula · emission

Selected line
Series limit
400–700 nm

Atom · rest frame · the photon on the strip is the one that moves

Opens on Hα, n = 3 to 2, in emission, at z = 0. The formula wavelength is 656.467 nm in vacuum.

Line · lower level n₁, upper level n₂

n₁

n₂

Appearance

Redshift · λ observed = λ rest × (1+z)

Balmer · Hα · emission

656.467 nm

Vacuum · visible · z = 0.00

Vacuum656.467 nm
Frequency4.5668×10¹⁴ Hz
Energy1.8887 eV
SeriesBalmer · n₁ = 2
BandVisible
Observed656.467 nm
At the telescope1.8887 eV
NIST vacuum656.463 nm
NIST air656.280 nm

Hα is the jump from n = 3 to n = 2. The formula returns 656.467 nm in vacuum. NIST’s unresolved center, from 97492.304 − 82259.158 cm⁻¹, is 656.463 nm in vacuum. The customary air wavelength is 656.280 nm.

At z = 0 the mark sits where the atom emits it. Raise z and every line of the series moves to a longer wavelength by the same factor, 1+z.

Emission draws bright lines on a dark field, the nebula case. Absorption draws dark lines on a schematic continuum, the stellar case. The continuum is not a Planck curve.

The sliders need JavaScript. The formula is still 1/λ = R_H (1/n₁² − 1/n₂²) with R_H = 1.09678×10^7 m⁻¹. That constant is 10973731.568157 × 1836.152673426 / 1837.152673426, rounded to six figures. Hα (3→2) is 656.467 nm in vacuum. Hβ (4→2) is 486.272 nm. Lyman-alpha (2→1) is 121.568 nm. NIST ASD unresolved centers: Lyα 121.56701 nm vacuum; Hα levels 82259.158 and 97492.304 cm⁻¹ give 656.463 nm vacuum, and the customary air value is 656.280 nm (ASD observed air 656.279 nm); Hβ levels 82259.158 and 102823.904 cm⁻¹ give 486.269 nm vacuum, customary air 486.132 nm. The visible window is 400–700 nm. λ_obs = λ_rest (1+z). Lyman-alpha reaches 400 nm near z = 2.29. Not a line list.

How to read a spectrum

Discrete lines, then a shift.

Why the lines are discrete. A hydrogen atom has one proton and one electron. NASA’s Spectroscopy 101 puts the classroom point directly: the electron moves from one level to another, not to a place between them, and the photon has to carry that exact energy. Short wavelength means a bigger jump. In the visible, NASA lists hydrogen at about 410 nm, 434 nm, 486 nm, and 656 nm. Those are Hδ, Hγ, Hβ, and Hα, the Balmer jumps that end on n = 2. This page computes them from the formula instead of rounding to the nearest nanometre. Lyman jumps end on n = 1 and land in the ultraviolet. Paschen jumps end on n = 3 and land in the infrared.

Emission and absorption. The same jump runs both ways. Dropping down emits the photon. Climbing up absorbs it. NRAO’s Essential Radio Astronomy primer states Kirchhoff’s account in those two pictures: a continuous spectrum seen through a cooler gas shows dark absorption lines, and the same gas seen off to the side shows bright emission lines on a dark background. NASA, ESA, and CSA draw the third piece, the continuous spectrum of a hot dense body, which is the blackbody desk. The toggle here is only the drawing. It does not change the wavelength.

Hα in a nebula. An H II region is gas ionized by hot stars. Peimbert, Peimbert, and Delgado-Inglada, in their 2017 PASP tutorial, describe the bright permitted lines of hydrogen as recombination lines: a free electron is captured and then steps down, emitting a photon at each step. They list Hα at 6563 Å, Hβ at 4861 Å, and Hγ at 4340 Å, which are the air wavelengths rounded to an angstrom, not a replacement for the NIST figures above. The pink of many emission nebulae is that Hα. The emission button is that picture: bright lines, little continuum.

Balmer lines in an A star. A stellar spectrum is usually the other picture, dark lines cut into a bright continuum. Astronomy Picture of the Day notes that the spectral types began as a way to rank the strength of hydrogen absorption lines. The letters that survived, OBAFGKM, now run with temperature. NASA’s Space Math spectral-classification sheet groups Vega (α Lyrae) with the stars whose spectra are dominated by thick hydrogen lines, and identifies that group as A-type. The stars desk says the same of A stars in one sentence. Vega’s temperature and luminosity are on the HR diagram. Balmer absorption starts from n = 2, so those lines are strong only when enough atoms already have an electron there. This page does not compute that population. It computes the wavelength of the jump.

The shift. A receding source stretches every wavelength by the same factor. λ_obs = λ_rest (1+z), which is the definition on the redshift desk. Lyman-alpha starts at 121.568 nm on this formula, ultraviolet. It crosses 400 nm, the blue edge of the NASA Imagine classroom window, near z = 2.29, and it has left the red edge by z = 5. The atom in the diagram does not stretch. The photon on the strip does. At z = 0 the gap between 656.28 nm in air and 656.46 nm in vacuum is the refractive index of air, about 0.18 nm. It is not a Doppler shift. A school spectrometer reads the air number. The formula, and a wavelength reduced to vacuum, read the other one.

This series · formula wavelengths, shifted by the slider

Balmer series at z = 0.00
Line	Jump	Vacuum, rest	Observed	Band
Hα	3 → 2	656.467 nm	656.467 nm	Visible

Literacy · the words on the readout

Rydberg formula: 1/λ = R_H (1/n₁² − 1/n₂²), λ in vacuum. R_H here is 1.09678×10⁷ m⁻¹. The lower level names the series: 1 Lyman, 2 Balmer, 3 Paschen, 4 Brackett, 5 Pfund. NIST’s compendium prints the first three as Lα, Hα, and Pα. A lower level above 5 is labeled by n₁ and is not given another name on this page.
Vacuum and air: NIST ASD tabulates vacuum wavelengths below 200 nm and air wavelengths from 200 nm to 2000 nm. λ_vac in nanometres is 10⁷ divided by the wavenumber in cm⁻¹. Hα’s unresolved levels, 82259.158 and 97492.304 cm⁻¹, differ by 15233.146 cm⁻¹, which is 656.463 nm in vacuum. The customary air figure is 656.280 nm. The formula’s 656.467 nm is the vacuum Bohr value.
Band: Ultraviolet, visible, or infrared, using the NASA Imagine classroom window of about 400–700 nm, the same cut as the EM-spectrum, blackbody, and redshift desks. The edges are a convention. The band on the readout follows the observed wavelength, so a redshift can carry an ultraviolet line into the window and out the other side.
Fine structure: The real Hα line is a tight group of components, not one jump between principal numbers. NIST ASD lists them separately, and also lists an unresolved center for the blend. This calculator is the unresolved Bohr jump. It is the right tool for seeing where a series sits. It is the wrong tool for splitting the components.
What the two drawings are: The strip is the spectrum a telescope records, rest wavelengths times (1+z). Emission is bright lines on black. Absorption is dark lines on a painted continuum, and that paint is a finder, not a Planck curve and not a stellar model. The ladder is the atom at rest: E_n = −hcR_H / n², so the ground level on this page is −13.598 eV. That binding energy is the rounded constant. It is not a quoted laboratory ionization energy. The arrow points down for emission and up for absorption.

Sources · the constants and the anchors

Cited, not invented

NIST CODATA — Rydberg constant — 2022 recommended value, R_∞ = 10973731.568157 m⁻¹. The 2022 CODATA wallet (NIST SP 959, May 2024) prints the same figure, the proton–electron mass ratio m_p/m_e = 1836.152673426, and the exact SI defining constants c = 299792458 m/s, h = 6.62607015×10⁻³⁴ J s, and e = 1.602176634×10⁻¹⁹ C. This page forms R_H = R_∞ × (m_p/m_e) / (m_p/m_e + 1) = 10967758.340 m⁻¹ and rounds to six significant figures, 1.09678×10⁷ m⁻¹. Photon energy in eV is hc/(λ e).
NIST Atomic Spectra Database — H I, unresolved centers. Lyα, levels 0 to 82259.158 cm⁻¹, observed and Ritz wavelength 121.56701 nm, vacuum because the line is below 200 nm. Hα, levels 82259.158 and 97492.304 cm⁻¹, observed air 656.279 nm, Ritz air 656.2819 nm. Hβ, levels 82259.158 and 102823.904 cm⁻¹, observed air 486.135 nm, Ritz air 486.1333 nm. The database notes that a line labeled only by principal numbers is the unresolved blend, pertinent to solar-corona observations, not one fine-structure component.
NIST ASD — spectral lines help — a Ritz wavenumber is the difference of the level energies. λ_vac in nm is 10⁷ / σ with σ in cm⁻¹. Air wavelengths are that vacuum wavelength divided by the refractive index of standard air (Peck and Reeder 1972, the formula ASD uses). The default table is vacuum for σ ≥ 50,000 cm⁻¹, air between 5,000 and 50,000 cm⁻¹, and vacuum again below 5,000 cm⁻¹. This page does not reimplement the refractive-index formula. It quotes the vacuum wavelength from the level difference, and the air wavelength from the sources that already print it. For Hα, 10⁷ / 15233.146 = 656.463 nm, which is 656.46 nm to 0.01 nm. The customary air value 656.280 nm is the 656.28 nm in the same comparison.
NIST — Atomic spectroscopy compendium, hydrogenic series — customary unresolved wavelengths. Below 2000 Å the value is vacuum; above 2000 Å it is air. Lyα 1215.67 Å, Hβ 4861.32 Å, Hα 6562.80 Å, and the same table’s Hγ 4340.46 Å, Hδ 4101.73 Å, and Pα 18751.0 Å. This desk uses 121.567 nm, 486.132 nm, and 656.280 nm from that list, matching the redshift desk. The compendium names Lα, Hα, and Pα. Brackett (n₁ = 4) and Pfund (n₁ = 5) are the next names in that counting. They are labels for the lower level, not extra data.
NASA Imagine the Universe — Wavelength, frequency, and energy — classroom bands. Optical is about 400–700 nm, ultraviolet about 10–400 nm, infrared about 700 nm to 1 mm. Boundaries are conventions. The same chart is the visible slice on the EM spectrum desk.
NASA Science — Spectroscopy 101 — an electron absorbs or emits only the photon that matches a jump between levels. In the visible, hydrogen is listed at 410 nm, 434 nm, 486 nm, and 656 nm. A star’s absorption spectrum and a nebula’s emission spectrum are the two readings of that fingerprint.
National Radio Astronomy Observatory — Essential Radio Astronomy, absorption and emission lines — Kirchhoff’s laws as used here: a continuum viewed through cooler gas shows dark absorption lines; the same gas viewed away from the continuum shows bright emission lines on a dark background. The lines of a given gas sit at the same wavelengths either way.
NASA / ESA / CSA — Types of spectra — continuous, emission, and absorption. The same plate is cited on the blackbody desk for the continuum and for absorption lines as dark cuts in it.
Peimbert, Peimbert, and Delgado-Inglada, PASP 129, 082001 (2017) — a tutorial on H II regions and planetary nebulae. Recombination lines form when a free electron is captured and descends. They list H I lines Hα at 6563 Å, Hβ at 4861 Å, and Hγ at 4340 Å. Those are rounded air wavelengths in the tutorial, not the NIST centers this calculator is checked against.
NASA APOD, 18 April 2004 — Stellar spectral types — the classification began as a ranking of hydrogen absorption-line strength. The sequence that remained is OBAFGKM, read today as a temperature sequence. NASA GSFC Space Math, spectral classification — Vega (α Lyrae) is grouped with the spectra dominated by thick hydrogen lines and identified as an A-type star.

Classroom Bohr atom. Not a line list. Hα on the formula is 656.467 nm in vacuum. NIST’s unresolved vacuum wavelength is 656.463 nm, and the customary air wavelength is 656.280 nm. Lyman-alpha on the formula is 121.568 nm against NIST’s 121.56701 nm. Hβ is 486.272 nm against 486.269 nm. The visible window is 400–700 nm. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The shift

n₁ and n₂	Integers. n₁ runs from 1 to 8. n₂ runs from n₁+1 to 20. The lower level names the series. Lyα, Hα, Hβ, Hγ, Paα, and Brα are buttons for six jumps.
Emission or absorption	Emission is bright lines on a dark field. Absorption is dark lines on a schematic continuum. The wavelength does not change.
Window	90–2000 nm is a logarithmic strip. The visible zoom is 370–750 nm, with the 400–700 nm classroom window marked on both.
Redshift	z from 0 to 5. Observed wavelength is the formula wavelength times (1+z). “Lyα at 400 nm” is the z that lands the formula Lyman-alpha on the blue edge. “Lyα at Hα” lands it on the formula Hα wavelength.

Vacuum, frequency, energy	The rest-frame photon from the formula. Frequency is c/λ. Energy is hc/λ in eV.
Observed	The same photon after (1+z). The band follows the observed wavelength.
NIST	Shown for Lyα, Hα, and Hβ. Vacuum is 10⁷ over the level difference. Air is the customary unresolved wavelength. Other jumps have no NIST row on this page, because this page does not invent one.

Strip	Left and Right step n₂. Up and Down step n₁. The focus ring is the accent color.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Spectra
of hydrogen

Discrete lines, then a shift.

Cited, not invented

Redshift

Blackbody

HR diagram

Stars

EM spectrum

Solar map

Spectra desk

Inputs

Readout

Keyboard

Spectraof hydrogen

Discrete lines, then a shift.

Cited, not invented

Redshift

Blackbody

HR diagram

Stars

EM spectrum

Solar map

Spectra
of hydrogen