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Emission and absorption lines as jumps between energy levels, hydrogen's series and Rydberg's formula, energy-level diagrams, and Doppler shifts.
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By the end of this lesson you will be able to find the wavelengths of spectral lines from energy levels, apply Rydberg's formula, and read Doppler shifts.
From the last two lessons you know that photons carry energy $hf = 1240/\lambda$ eV and that one photon interacts with one electron. From chemistry you know that different elements give flames different colors. This lesson explains those colors: each atom emits and absorbs only particular photons, and the pattern identifies it.
| Term | What it means |
|---|---|
| Line spectrum | Light at only a few sharp wavelengths, characteristic of an element. |
| Energy level | One of the discrete energies an atom's electrons can have. |
| Ground state | The lowest energy level. |
| Emission line | A bright line from photons given off as electrons drop between levels. |
| Absorption line | A dark line where a gas removes photons matching its level gaps. |
| Rydberg constant | $R = 1.097 \times 10^7$ m⁻¹, the constant in hydrogen's line formula. |
| Redshift | $z = \Delta\lambda/\lambda$, the stretch of a spectrum from a source moving away. |
Heat a thin gas and it glows at a few sharp wavelengths, its emission spectrum. Shine white light through the same gas, cooler, and dark lines appear at exactly those wavelengths, its absorption spectrum. The explanation is that an atom's electrons can have only certain energies, its energy levels, and each line is a photon carrying the difference between two of them:
$$hf = \frac{hc}{\lambda} = E_{\text{upper}} - E_{\text{lower}}.$$
For hydrogen, the wavelengths fit a simple rule found in the 1880s by Johann Balmer and Johannes Rydberg:
$$\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \qquad R = 1.097 \times 10^7\ \text{m}^{-1},$$
with $n_1 = 2$ giving the visible Balmer series. Lesson 13 shows where the rule comes from.
Another way: picture
Picture an atom's levels as the rungs of a ladder with uneven spacing. An electron can stand only on rungs, never between them. Stepping down releases a photon whose energy is the height of the step; stepping up needs a photon of exactly that height. Every element has its own ladder, so every element has its own set of lines, a fingerprint written in light.
Another way: steps
In a hot thin gas, collisions knock electrons into higher levels, and they fall back, each fall emitting a photon of a definite energy: bright lines on a dark background. Neon signs, sodium streetlights and the red glow of a hydrogen discharge tube are emission spectra.
When white light passes through a cooler gas, atoms absorb photons that exactly match a gap between levels, lifting electrons up. Those wavelengths are missing from the light that emerges: dark lines on a rainbow. The absorbed energy is re-emitted, but in all directions, so little of it continues along the beam.
In 1814 Joseph von Fraunhofer, making precision lenses, noticed hundreds of dark lines in the spectrum of sunlight. Decades later Gustav Kirchhoff and Robert Bunsen showed that each line matched an emission line of an element heated in a flame: sodium, iron, calcium, hydrogen. The Sun's cooler outer layers were absorbing light from the hotter surface below.
That discovery let astronomers find what stars are made of without visiting them. Helium was discovered first in the Sun, in 1868, from a yellow line matching no known element; it was found on Earth only in 1895. Cecilia Payne, working at Harvard in 1925, used spectra to show that stars are mostly hydrogen.
Hydrogen's lines fall into families that end on the same lower level. The Lyman series, ending on level 1, lies in the ultraviolet, from $121.6$ nm down to $91.2$ nm. The Balmer series, ending on level 2, is visible: red $656.3$ nm, blue-green $486.1$ nm, blue $434.0$ nm, violet $410.2$ nm, crowding toward $364.6$ nm. The Paschen series, ending on level 3, is in the infrared.
Rydberg's formula fits every line to a part in ten thousand with one constant. Such a simple pattern demanded an explanation; Niels Bohr supplied one in 1913, treated in unit 4, and quantum mechanics a fuller one in the 1920s.
An energy-level diagram draws each level as a horizontal line at its energy, usually in eV, with zero at the top for an electron just free of the atom. Levels are negative: an electron in the atom is bound, and it takes energy to free it. The ground state is the lowest line.
A transition is an arrow between two lines. Its length is the photon's energy: long arrows mean energetic, short-wavelength photons; short arrows mean long wavelengths. An atom in its ground state can absorb only photons that carry it exactly to a higher level, or enough to free the electron entirely, which is a continuous range above the ionization energy.
Light from a source moving away from us arrives stretched to longer wavelengths, a redshift; from an approaching source, compressed to shorter ones, a blueshift. For speeds much less than $c$, the fractional shift is $z = \Delta\lambda/\lambda = v/c$.
Because a spectrum has many lines in a fixed pattern, astronomers can recognize hydrogen's lines even when shifted and read off the speed. Vesto Slipher at Lowell Observatory in Flagstaff found in the 1910s that most galaxies are moving away, and Edwin Hubble at Mount Wilson in California showed in 1929 that the farther away they are, the faster: the universe is expanding.
Checking an answer. Larger gaps must give shorter wavelengths. Every Balmer line must lie between $364.6$ and $656.3$ nm. Photon energies must be positive. And absorption from the ground state of hydrogen must need at least $10.2$ eV, far into the ultraviolet.
Energy conservation says the photon carries exactly what the atom loses, and the sharpness of the lines says the atom's energies are discrete. That discreteness, quantization, is the experimental fact; classical physics, in which an orbiting electron could have any energy, cannot explain it.
Rydberg's formula is empirical, a fit to measurements that happens to be exact for hydrogen. It works only for atoms with one electron; for others the levels must be measured or calculated with quantum mechanics, and diagrams list them directly.
Spectroscopy identifies substances everywhere. Forensic laboratories burn tiny samples and read the element lines; environmental agencies measure lead and mercury in water by the light they absorb; steelmakers check alloys with handheld spectrometers that spark the metal and read its lines in seconds.
Astronomers use the same technique on light billions of years old. The James Webb Space Telescope, operated from the Space Telescope Science Institute in Baltimore, reads the absorption lines of water, carbon dioxide and methane in the atmospheres of planets around other stars as the planets pass in front of them.
Spectral lines are narrow because the levels are well defined, but not infinitely narrow. An electron that stays in an upper level only a short time, about $10^{-8}$ s for many transitions, has a slightly uncertain energy, as unit 3's uncertainty relation explains, and its line has a small natural width.
Hot gases add more width: atoms moving toward and away from the observer Doppler shift their light, broadening each line in proportion to the square root of the temperature. Astronomers measure a star's surface temperature from this broadening, and atomic clocks cool their atoms nearly to absolute zero to make the lines as sharp as possible.
A laser is an atomic spectrum put to work. Its medium, a gas, crystal or semiconductor, is pumped so that more atoms sit in an upper level than in the lower one below it. A photon whose energy matches the gap then does something new: instead of being absorbed, it stimulates an excited atom to drop and emit a second photon identical to the first, same energy, direction and phase. Mirrors at the ends of the medium send the photons back and forth, and the cascade builds into a beam.
The beam's color is set by the gap. The red helium–neon laser uses a $1.96$ eV gap in neon, giving $633$ nm. The green of a laser pointer comes from an infrared laser at $1064$ nm whose light is doubled in frequency by a crystal. The laser diodes in barcode scanners and fiber-optic links use gaps in semiconductors, chosen by the mix of elements. Theodore Maiman built the first laser, from a ruby crystal, at Hughes Research Laboratories in Malibu, California, in 1960.
At Lowell Observatory in Flagstaff, Arizona, Vesto Slipher measured the spectra of spiral nebulae from 1912 on and found their lines shifted, most toward the red, by amounts meaning speeds of hundreds of kilometers per second. Edwin Hubble, at Mount Wilson Observatory near Los Angeles, measured their distances and in 1929 found the speeds grew in proportion to distance.
A galaxy whose hydrogen line at $656.3$ nm appears at $662.9$ nm has $z = 0.01$ and is receding at about $3000$ km/s, roughly $140$ million light-years away. The same measurement, repeated on millions of galaxies by surveys such as the Sloan Digital Sky Survey based at Apache Point in New Mexico, maps the universe in three dimensions.
For decades many American cities lit their streets with low-pressure sodium lamps, whose light is almost all in sodium's $589$ nm line. The color was a harsh yellow, but the efficiency was high, and astronomers loved it: a single narrow line is easy to filter out of a telescope's view.
Tucson and Flagstaff, near major observatories, kept sodium lighting for that reason. As cities switch to white LEDs, whose light spreads across the whole spectrum, observatories cannot filter it, and both cities now require shielded, warm-toned LEDs that keep the sky dark. Spectral lines shape even city lighting codes.
It is tempting to think an atom absorbs any photon with enough energy to lift an electron, keeping the leftover. For transitions between bound levels this is wrong: a photon is absorbed only if its energy matches a gap exactly, and otherwise passes through. That is why cool gases are transparent except at a few sharp wavelengths.
The exception is ionization: a photon with more than the ionization energy can free the electron and give the excess to its kinetic energy, since a free electron can have any energy.
Sodium's strongest line comes from a drop of $2.10$ eV. Find its wavelength.
$\lambda = \dfrac{1240}{2.10} = 590\ \text{nm}$
$hc/\Delta E$.
Place it in the spectrum.
$\text{yellow-orange}$
Between $570$ and $600$ nm.
Find its frequency.
$f = \dfrac{3.00 \times 10^8}{590 \times 10^{-9}} = 5.08 \times 10^{14}\ \text{Hz}$
$c/\lambda$.
Predict the absorption line of cool sodium vapor.
$590\ \text{nm, dark}$
The same gap, run upward.
Explain the color of old streetlights.
$\text{nearly all their light in this one line}$
Low-pressure sodium lamps.
Find $1/\lambda$ for hydrogen's drop from level 4 to level 2.
$\dfrac{1}{\lambda} = 1.097 \times 10^7\left(\dfrac{1}{4} - \dfrac{1}{16}\right)$
Rydberg's formula.
Simplify the bracket.
$\dfrac{1}{4} - \dfrac{1}{16} = \dfrac{3}{16} = 0.1875$
Common denominator.
Evaluate the inverse wavelength.
$\dfrac{1}{\lambda} = 2.057 \times 10^6\ \text{m}^{-1}$
Per meter.
Invert for the wavelength.
$\lambda = 486.2\ \text{nm}$
Blue-green.
Find the photon energy.
$E = \dfrac{1240}{486.2} = 2.55\ \text{eV}$
In eV.
Check against the level energies.
$-\dfrac{13.6}{16} + \dfrac{13.6}{4} = -0.85 + 3.40 = 2.55\ \text{eV}$
Agrees with $E_n = -13.6/n^2$, derived in lesson 13.
Hydrogen at room temperature is in its ground state, $-13.6$ eV. Find the smallest photon it can absorb.
$\Delta E = -3.40 + 13.6 = 10.2\ \text{eV}$
To level 2.
Find its wavelength.
$\lambda = \dfrac{1240}{10.2} = 121.6\ \text{nm}$
Lyman alpha, ultraviolet.
Decide whether visible light is absorbed.
$1.6\text{–}3.3\ \text{eV} < 10.2\ \text{eV}: \text{no}$
Cold hydrogen is transparent.
Find the photon that ionizes it.
$13.6\ \text{eV}: \ \lambda = 91.2\ \text{nm}$
Any shorter wavelength also works.
A $12.09$ eV photon arrives. Find where it takes the electron.
$-13.6 + 12.09 = -1.51\ \text{eV} = E_3$
Level 3 exactly.
Find the lines emitted as it falls back.
$3 \to 1: 102.6\ \text{nm}; \quad 3 \to 2: 656.3\ \text{nm}; \quad 2 \to 1: 121.6\ \text{nm}$
Either directly or in two steps.
Explain why hot stars show Balmer absorption.
$\text{enough atoms already in level 2}$
Heat populates the upper level.
Write the wavelength formula.
$\lambda = \dfrac{hc}{\Delta E}$
Photon energy equals the gap.
Substitute the gap.
$\lambda = \dfrac{1240}{1.96}$
In nm.
Evaluate the wavelength.
An atom has energy levels at $-8$ eV and $-4$ eV. An electron drops from the higher level to the lower one. What energy does the emitted photon carry?
Complete the worked solution: hydrogen's levels are $E_n = -13.6/n^2$ eV. For hydrogen gas to absorb light and lift an electron from level $2$ to level $4$, find the photon energy needed in eV, its wavelength in nm, and its frequency in units of $10^{14}$ Hz. Use $hc = 1240$ eV·nm and $c = 3.00 \times 10^8$ m/s.
Subtract the level energies.
$\Delta E = -\dfrac{13.6}{4^2} + \dfrac{13.6}{2^2} =$ d
Upper minus lower.
Convert the gap to a wavelength.
$\lambda = \dfrac{1240}{\Delta E} =$ l
In nm.
Convert to a frequency.
$f = \dfrac{c}{\lambda} =$ f
In $10^{14}$ Hz.
Check that emission matches.
$\text{the same line appears bright when the electron falls back}$
Absorption and emission lines coincide.
Match each kind of spectrum or series to its origin.
| electrons dropping between levels in a hot thin gas | photons of matching energy removed by a cool gas | light from a hot dense body | hydrogen transitions ending on level 2 | |
|---|---|---|---|---|
| emission lines | ||||
| absorption lines | ||||
| continuous spectrum | ||||
| Balmer series |
Hydrogen emits a Balmer line when an electron drops from level $4$ to level $2$. With $R = 1.097 \times 10^7$ m⁻¹ and $hc = 1240$ eV·nm, fill in $1/\lambda$ in units of $10^6$ m⁻¹, the wavelength in nm, and the photon energy in eV.
| value | |
|---|---|
| 1/λ (10⁶ m⁻¹) | |
| wavelength (nm) | |
| photon energy (eV) |
Hydrogen's Lyman series comes from electrons dropping to level $1$. With $R = 10.97$ μm⁻¹, write $1/\lambda$ in μm⁻¹ for a line starting on level $n$, as a formula in $n$.
Answer:
An energy-level diagram for hydrogen shows a transition in which the atom loses $3.40$ eV. With $hc = 1240$ eV·nm, at what wavelength does it emit, in nm?
Answer: nm
An astronomer at Lowell Observatory in Flagstaff, Arizona, finds hydrogen's red line, emitted at $656.3$ nm, in a galaxy's spectrum at $689.12$ nm. With $c = 3.00 \times 10^5$ km/s, how fast is the galaxy moving away, in km/s?
Answer: km/s away from us
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Hydrogen's Lyman series comes from electrons dropping to level $1$. With $R = 10.97$ μm⁻¹, write $1/\lambda$ in μm⁻¹ for a line starting on level $n$, as a formula in $n$.
Answer:
You can interpret atomic spectra. Explain to someone why a cool gas absorbs exactly the wavelengths it emits when hot.
21. Your turn: a helium–neon laser's line comes from a $1.96$ eV drop. Find its wavelength., step 3
$\lambda = 633\ \text{nm}$
The classic red laser line.