Spectra from Bohr's levels · printable packet

Counting lines from a ladder of levels, hydrogen's series and their limits, lines of one-electron ions, the isotope shift, and identifying lines from their wavelengths.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to predict hydrogen's and ions' spectral lines, find series limits, and identify a line's upper level from its wavelength.

2. What you already have

From the last lesson you have Bohr's levels, $E_n = -13.6/n^2$ eV, and how a jump between two gives one spectral line. From lesson 7 you know the observed series and Rydberg's formula. This lesson works the other way, predicting whole spectra from the level ladder: how many lines, where each series ends, and how the lines move for other nuclei.

3. Words for this lesson

4. A ladder of levels makes a pattern of lines

Term	What it means
Spectral series	The lines produced by drops that all end on the same lower level.
Series limit	The shortest-wavelength line of a series, from $n = \infty$.
Cascade	A sequence of downward jumps through intermediate levels.
Lyman, Balmer, Paschen	Hydrogen's series ending on levels 1, 2 and 3.
Reduced mass	$\mu = m_eM/(m_e + M)$, correcting for the nucleus's own motion.
Isotope shift	The small change in line wavelengths between isotopes of an element.
Selection rule	A restriction on which jumps can emit a photon efficiently.

An atom excited to level $n$ can return to the ground state by one jump or by several. Every possible pair of levels gives a line, so a sample with atoms reaching $n$ can show $n(n - 1)/2$ distinct lines. Grouping the lines by their lower level $m$ gives the series:

$$\Delta E = 13.6\left(\frac{1}{m^2} - \frac{1}{n^2}\right)\ \text{eV}, \qquad n = m + 1, m + 2, \ldots$$

The first line of each series, from $m + 1$, has the longest wavelength; the lines crowd together toward the series limit, a drop from $n = \infty$ with energy $13.6/m^2$ eV. For one-electron ions every energy scales by $Z^2$, so He⁺ lines are four times as energetic as hydrogen's between the same levels.

Another way: picture

Picture the energy levels as the rungs of a ladder that crowd closer toward the top. A ball dropped from a high rung can bounce down one rung at a time or fall straight to the ground, and every possible fall between two rungs has its own length. Drops that land on the same rung form a family, and because the rungs crowd together at the top, so do the family's lines.

Another way: steps

List the levels involved and their energies $-13.6Z^2/n^2$ eV.
For a line, subtract upper minus lower and convert with $\lambda = 1240/\Delta E$.
Group lines by lower level into series.
Series limit: $13.6Z^2/m^2$ eV; first line: the drop from $m + 1$.
Count lines as pairs of levels, $n(n - 1)/2$.

5. Counting the lines

With levels 1 to 4 available, the possible jumps are $4 \to 3$, $4 \to 2$, $4 \to 1$, $3 \to 2$, $3 \to 1$ and $2 \to 1$: six lines. In general, $n$ levels have $n(n - 1)/2$ pairs. A single atom follows only one path down, emitting one, two or three photons, but a sample of many atoms takes every path and shows every line.

The relative brightness of the lines depends on how likely each jump is, which Bohr's model cannot predict. Quantum mechanics shows some jumps are strongly favored and others nearly forbidden, according to selection rules on the angular momentum quantum numbers met in lesson 16.

6. Series and their limits

The Lyman series, ending on level 1, runs from $121.6$ nm down to its limit at $91.2$ nm, all in the ultraviolet. The Balmer series, ending on level 2, runs from $656.3$ nm to $364.7$ nm, its first few lines visible. Paschen, ending on 3, runs from $1875$ nm to $820$ nm in the infrared; Brackett and Pfund lie still further out.

Beyond each series limit lies a continuum: photons with more than the limit energy can free the electron entirely from that level, with the excess going to its kinetic energy. In absorption, hydrogen gas in its ground state is opaque to all light shorter than $91.2$ nm, which is why the far ultraviolet from distant stars is largely blocked by interstellar hydrogen.

7. Overlapping series

Series from different lower levels can overlap in wavelength. Paschen's limit, $820$ nm, lies at shorter wavelength than Brackett's first line, $4052$ nm, but Brackett's limit at $1459$ nm falls inside the range covered by Paschen's lines. In hydrogen the Lyman and Balmer series do not overlap, which made the Balmer series so easy to recognize first.

For one-electron ions the series shift to shorter wavelengths by a factor of $Z^2$, and the He⁺ series ending on level 4 includes lines within a whisker of hydrogen's Balmer lines. In the 1890s, before Bohr, these lines were seen in hot stars and mistaken for a new hydrogen series.

8. The nucleus moves too

Bohr's formulas assume an infinitely heavy nucleus. In reality the electron and nucleus both circle their common center of mass, and the electron's mass should be replaced by the reduced mass $\mu = m_eM/(m_e + M)$. For hydrogen, $\mu$ is $0.99946m_e$, lowering every level's magnitude by about five parts in ten thousand.

For deuterium, with a nucleus twice as heavy, $\mu$ is $0.99973m_e$. Its levels are deeper by $0.000272$ of themselves, and its lines are shorter by the same fraction: the red Balmer line moves from $656.47$ to $656.29$ nm. Harold Urey found these faint companion lines in 1931 at Columbia University and so discovered deuterium, winning the 1934 Nobel Prize in Chemistry.

9. The method, step by step, and how to check it

Checking an answer. Every line of a series must lie between its first line and its limit. Lyman lines must be ultraviolet, Balmer mostly visible, Paschen infrared. Upper levels found from wavelengths must come out as whole numbers. And ion lines must be $Z^2$ times as energetic as hydrogen's for the same levels.

10. Why each step is allowed

Treating each line as a single jump between two levels follows from energy conservation and the photon picture: an atom emits one photon per jump. Cascades happen because an excited atom can reach the ground state through intermediate levels, each jump being a separate emission.

The $Z^2$ scaling assumes one electron. Atoms with several electrons have levels that depend on more than $n$, because inner electrons shield the nucleus unequally for different orbits. That failure, and others, are the subject of the next lesson.

11. Solving for levels from a spectrum

Given a measured line and its series, the upper level follows from rearranging: $1/n^2 = 1/m^2 - \Delta E/13.6$. Astronomers do exactly this to identify lines in nebulae and stars, where the hydrogen lines appear alongside those of other elements. A whole-number answer confirms the identification; a non-integer one says the line belongs to something else.

In the gas around hot young stars, electrons recombine with protons into high levels and cascade down, lighting up the whole ladder. Radio astronomers even detect recombination lines between levels near $n = 100$, at wavelengths of centimeters, from hydrogen in distant galaxies.

12. Energy-level diagrams as tools

An energy-level diagram turns spectral problems into pictures. Draw the levels as horizontal lines at their energies, drawn to scale, and every emission line becomes a downward arrow whose length is the photon's energy. Long arrows are ultraviolet, short ones infrared.

Laser designers, astronomers and chemists all think in these diagrams. A three-level laser, for instance, pumps atoms from the ground state to a high level, from which they drop quickly to a long-lived middle level; the laser line is the arrow from the middle level back down. The same picture, drawn for hydrogen, holds every line in this lesson.

13. Hydrogen in the laboratory

Hydrogen discharge tubes, glowing pink from the combined Balmer lines, are standard equipment in college physics laboratories. Students view them through diffraction gratings, measure the angles of the red, blue-green and violet lines, and compute their wavelengths, recovering Rydberg's constant to a fraction of a percent.

Precision measurements go much further. Laser spectroscopy of the $1s$ to $2s$ transition in hydrogen has reached fifteen significant figures, making it one of the most precisely measured quantities in physics and a stringent test of quantum electrodynamics. Small discrepancies in such measurements led, in 2010, to the "proton radius puzzle," since resolved by better experiments.

14. Muonic and exotic atoms

Bohr's formulas work for any two charges bound by the Coulomb force, not just an electron and a proton. Replace the electron by a muon, $207$ times heavier, and the reduced mass grows about $186$ times. The orbits shrink by that factor and the energies grow by it: muonic hydrogen's ground state lies at about $-2.5$ keV, and its lines are X-rays.

Because a muon orbits so close to the nucleus, its energy levels are sensitive to the nucleus's size. Measurements of muonic hydrogen's spectrum at the Paul Scherrer Institute in 2010 gave a proton radius slightly smaller than electron experiments had found, starting the proton radius puzzle; later electron measurements, including one at Jefferson Lab in Virginia, came into agreement with the muonic value. Positronium, an electron bound to a positron, has a reduced mass of half the electron's, so its levels are half as deep as hydrogen's, just as Bohr's scaling predicts.

15. In the world: reading nebulae

At Kitt Peak National Observatory near Tucson, Arizona, spectrographs record the light of glowing nebulae such as the Orion Nebula, where ultraviolet light from hot young stars ionizes hydrogen. Protons and electrons recombine into high levels and cascade down, emitting every Balmer line from $656.5$ nm toward the limit.

Astronomers identify each line by solving for its upper level and checking that the answer is a whole number. The ratios of the Balmer lines' brightnesses, compared with the predictions of the cascade, reveal how much dust lies between us and the nebula, since dust dims blue light more than red.

16. In the world: deuterium and the Big Bang

Nearly all the deuterium in the universe was made in the first few minutes after the Big Bang, and the amount depends sensitively on how dense ordinary matter was then. Astronomers measure it by finding deuterium's absorption lines beside hydrogen's in the light of distant quasars, shifted by exactly the reduced-mass fraction.

Observers using the Keck telescopes on Mauna Kea in Hawaii measured the ratio at about $25$ deuterium atoms per million hydrogen atoms. The value pins down the density of ordinary matter in the universe and agrees with the independent measurement from the cosmic microwave background, a striking success for both.

17. Excited atoms can cascade

It is tempting to think an atom excited to level $n$ must return by one jump, emitting a single photon of energy $E_n - E_1$. It can do that, but it can also cascade through intermediate levels, emitting several smaller photons whose energies add to the same total. A sample of atoms does both, which is why it shows all $n(n - 1)/2$ lines.

A related error is to think the series limit is the longest wavelength of a series. It is the shortest: drops from very high levels release the most energy.

18. All the lines from level 4

19. Balmer's limit and first line

20. Identifying a line

21. Your turn: find the wavelength of the He⁺ line from level 4 to level 3.

22. Guided practice

A sample of hydrogen atoms is excited so that some reach level $n = 4$ and none go higher. As they cascade down to the ground state in every possible way, how many distinct spectral lines can appear?

23. Guided practice

Complete the worked solution: hydrogen's H-beta line has wavelength $486.27$ nm. Deuterium's nucleus is twice as heavy, which raises the electron's reduced mass, and so every level's energy, by $0.000272$ of itself. Find how much shorter deuterium's line is in nm, and its wavelength in nm, and the ratio of the two wavelengths.

24. Guided practice

25. Practice

Hydrogen's Brackett series ends on level $m = 4$. With $E_n = -13.6/n^2$ eV and $hc = 1240$ eV·nm, fill in the energy of the series-limit photon in eV, the limit wavelength in nm, and the wavelength of the series' first line, the drop from $5$, in nm.

26. Practice

For hydrogen lines ending on level $m = 4$, write the photon energy, in eV, as a formula in the starting level $n$.

27. Practice

Singly ionized helium, He⁺, has one electron and nuclear charge $Z = 2$. Its electron drops from level $4$ to level $3$. With $hc = 1240$ eV·nm, what wavelength does it emit, in nm?

28. Somewhere new

An astronomer at Kitt Peak National Observatory in Arizona records a hydrogen line at $486.3$ nm in a glowing nebula and identifies it as a Balmer line. With $E_n = -13.6/n^2$ eV and $hc = 1240$ eV·nm, from which level $n$ did the electron drop?

29. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

30. Test question

For hydrogen lines ending on level $m = 4$, write the photon energy, in eV, as a formula in the starting level $n$.

31. What you can do now

You can predict spectra from energy levels. Explain to someone why hydrogen excited to level 4 can show six different lines.

Working for the steps left to you

	value
limit energy (eV)
limit wavelength (nm)
first-line wavelength (nm)

21. Your turn: find the wavelength of the He⁺ line from level 4 to level 3., step 3

$\lambda = \dfrac{1240}{2.644} = 469\ \text{nm}$

	drops to n = 1, ultraviolet	drops to n = 2, mostly visible	drops to n = 3, infrared	a drop from n = ∞
Lyman series
Balmer series
Paschen series
series limit