Atoms Get Weird: Bohr's Model & Spectral Lines

Pass light through a prism, or through a diffraction grating ruled with thousands of fine grooves, and its wavelengths fan out into a spectrum. A hot, dense object, such as a lamp filament or the surface of the Sun, gives a continuous spectrum, the smooth blackbody rainbow of chapter 3. A hot, thin gas gives an emission spectrum, also called a line spectrum: a few bright lines at sharply defined wavelengths, with darkness between them. Each element has its own pattern. Sodium glows with a pair of yellow lines at 589.0 and 589.6 nm, the colour of old sodium street lamps, and lithium turns a flame crimson. Measuring such patterns is spectroscopy, and it let chemists name the elements in a sample with a flame and a prism.

The pattern also works in reverse. In 1814 the Bavarian lens maker Joseph von Fraunhofer mapped more than 500 fine dark lines crossing the Sun's spectrum. In 1859 Gustav Kirchhoff and Robert Bunsen at Heidelberg found their origin: when sunlight passed through a sodium flame, the dark D lines grew darker still. A cool gas in front of a hotter source absorbs precisely the wavelengths it emits when hot. The result is an absorption spectrum, dark lines on a bright continuous background at the same wavelengths as the element's emission lines, so emission and absorption are two readings of one fingerprint. With it the pair soon discovered two new metals, cesium and rubidium, named from the Latin for the sky-blue and deep-red colours of their lines.

Comte had written in 1835 that although we might learn the shapes, distances and motions of the stars, we could never study their chemical composition. Barely twenty-five years later Kirchhoff matched the Sun's dark lines, one by one, to laboratory spectra of sodium, iron, magnesium, calcium and more, showing that the cooler solar atmosphere contains them. Fraunhofer's C and F lines turned out to be hydrogen's. Today astronomers read the composition of stars, nebulae, distant galaxies and even the atmospheres of planets orbiting other stars. Shifted lines reveal motion too, through the Doppler effect.

A barcode is not a recipe, though, because the strength of a line depends on temperature as well as on abundance. In 1925 Cecilia Payne, a doctoral student at Harvard College Observatory, combined measured line strengths with Meghnad Saha's 1920 theory of how hot gases ionize, and concluded that stars are made overwhelmingly of hydrogen and helium. The astronomer Henry Norris Russell told her the result was clearly impossible, and she hedged it in print; four years later Russell reached the same conclusion by another route. The Sun is roughly three-quarters hydrogen and one-quarter helium by mass. Why temperature matters so much, Bohr's atom will explain in Part 6.

By 1909 physicists knew that atoms contain lightweight, negatively charged electrons, discovered by J. J. Thomson in 1897, but not how the balancing positive charge is arranged; Thomson guessed it was spread through the atom like a pudding. That year Hans Geiger and Ernest Marsden, in Ernest Rutherford's laboratory in Manchester, fired alpha particles, the fast helium nuclei shot out by radioactive atoms, at thin metal foils. Most sailed straight through, but about one in 8,000 striking a platinum foil bounced back by more than 90°. Rutherford later compared it to firing a 15-inch shell at a piece of tissue paper and having it come back and hit you.

In 1911 Rutherford explained why. All of an atom's positive charge and nearly all its mass sit in a nucleus between 10⁻¹⁵ and 10⁻¹⁴ metres across, tens of thousands of times smaller than the atom itself: a pea on the centre spot of a stadium. The natural guess was a miniature solar system, with electrons circling the nucleus like planets round the Sun, held by electric attraction instead of gravity.

Classical physics condemned that picture at once. A charge moving in a circle is accelerating, and Maxwell's electromagnetism says that accelerating charges radiate electromagnetic waves, which carry energy away. An orbiting electron would spiral inward and strike the nucleus in about 1.6×10⁻¹¹ seconds, and as its orbit shrank its circling frequency would rise smoothly, broadcasting a continuous smear of light, never a crisp line. Classical theory cannot even set an atom's size, because the electron's charge and mass and the strength of the electric force cannot be combined into a length. Yet every hydrogen atom is stable, identical and about a tenth of a nanometre across, and it shines only in sharp lines.

The clue had been waiting since 1885, when Johann Balmer, a mathematics teacher at a girls' school in Basel, found that hydrogen's four visible lines, at about 656, 486, 434 and 410 nm, obey a simple formula: λ = B·n²/(n² − 4), where B is 364.56 nm and n is 3, 4, 5 or 6. The formula predicted a fifth line near 397 nm, which, a colleague told him, had already been observed, along with further lines photographed in the spectra of white stars. Around 1890 the Swedish physicist Johannes Rydberg recast such patterns in the form used today, 1/λ = R × (1/n₁² − 1/n₂²), where n₁ and n₂ are whole numbers and R is the Rydberg constant. It worked superbly, and nobody knew why whole numbers should rule the light of an atom.

Niels Bohr, a young Danish theorist, spent four months of 1912 in Rutherford's laboratory and left convinced of two things: the nucleus was real, and classical physics could not govern the electrons around it. Early in 1913 a Copenhagen colleague, Hans Marius Hansen, asked how his ideas would account for spectral lines. 'As soon as I saw Balmer's formula,' Bohr later recalled, 'the whole thing was immediately clear to me.' His paper, published in July 1913 when he was 27, rests on three bold rules.

First, stationary states: the electron can occupy only certain allowed orbits, and while it stays in one it does not radiate, whatever Maxwell says, so each state has a fixed energy. Second, quantization: in the form textbooks keep, which Bohr also stated, the electron's angular momentum, its mass times its speed times its orbital radius, must be a whole-number multiple of the reduced Planck constant ħ = h/2π: L = m·v·r = nħ, with n = 1, 2, 3 and so on. Third, quantum jumps: light is emitted or absorbed only when the electron jumps between stationary states, with a frequency f fixed by the rule that h·f equals the difference between the two states' energies. Bohr himself distrusted Einstein's light quanta until the mid-1920s, but in today's language each jump emits or absorbs a single photon: chapter 4's photon wired into the atom. The rule cuts the classical link between the frequency of the light and the motion of the electron.

Now turn the rules loose on hydrogen, one electron bound to one proton. Two conditions hold at once: the proton's electric pull must supply exactly the inward force that keeps the electron on its circle, and the angular momentum must equal nħ. Together they give radii that grow as the square of n: rₙ = n²a₀, where a₀, the Bohr radius, is built from ħ, the electron's mass and charge and the electric constant ε₀, and equals 0.0529 nm. Planck's constant has supplied the missing length, and it is the right one. The next orbits have radii of 0.212 and 0.476 nm, and in the lowest the electron moves at 2.19×10⁶ metres per second, about the speed of light divided by 137.

Adding the kinetic energy to the electric potential energy gives each state's energy: Eₙ = −13.6 eV/n². The ladder reads −13.6 eV for n = 1, then −3.40, −1.51, −0.85, −0.54 and −0.38 eV, its rungs crowding together toward zero. The lowest rung is the ground state, where an undisturbed atom stays indefinitely. Higher rungs are excited states, reached by excitation: absorbing a photon of exactly the right energy, or a hard enough collision. Excited states are fleeting, typically lasting billionths of a second before the electron drops and a photon departs. Give a ground-state atom 13.6 eV or more and the electron escapes altogether. That is ionization, and 13.6 eV is hydrogen's ionization energy; above zero, a free electron may have any energy at all.

In 1914 James Franck and Gustav Hertz caught such rungs in the act: electrons fired through mercury vapour lost energy only in chunks of about 4.9 eV, mercury's first excitation energy, and the vapour then emitted ultraviolet light at 253.7 nm, whose photons carry just that energy. They at first mistook the 4.9 eV for mercury's ionization energy, but the discovery earned them the 1925 Nobel Prize in Physics.

Bohr's third rule turns the ladder into a spectrum. When the electron drops from an upper rung n₂ to a lower rung n₁, the photon carries off 13.6 eV × (1/n₁² − 1/n₂²). A photon's energy is h·c/λ, so dividing by h·c gives Rydberg's formula and explains his constant: R = 13.6 eV ÷ (h·c) ≈ 1.097×10⁷ per metre. Bohr built R from nothing but the electron's mass and charge and Planck's constant, joined in today's SI form by ε₀ and the speed of light. In 1913, in the frequency form then used, his value came out at 3.1×10¹⁵ per second against a measured 3.290×10¹⁵, agreement within the uncertainties of the day.

Group the lines by the rung they land on and the barcode sorts itself into series. Drops to n = 1 make the Lyman series, all ultraviolet, from 121.6 nm down to a series limit at 91.2 nm. Drops to n = 2 make the Balmer series, the only one reaching the visible: 656 nm red from n = 3, 486 nm blue-green from n = 4, 434 nm blue-violet from n = 5 and 410 nm violet from n = 6, with more lines crowding toward a limit at 365 nm in the ultraviolet. Drops to n = 3 make the infrared Paschen series, found by Friedrich Paschen in 1908, from 1,876 nm down to 820 nm.

Absorption is the same rule run backwards, and it settles the puzzle from Part 2. An atom absorbs a photon only if its energy exactly matches a gap from the atom's present rung to a higher one, or is at least the energy needed to ionize it. In cool atomic hydrogen, such as the thin gas between the stars, essentially every atom is in the ground state, so the gas absorbs only ultraviolet Lyman lines and is transparent to visible light. Balmer absorption needs atoms already in n = 2, which requires a gas at thousands of degrees. That is why Balmer lines are strongest in stars with surfaces near 10,000 K: cooler stars hold too few atoms in n = 2, and hotter ones have ionized most of their hydrogen.

In everyday speech a quantum leap is a giant advance, which gets the physics backwards. A quantum leap is the smallest change an atom can make: a jump between allowed states, with no allowed states in between. The electron that makes hydrogen's red line drops from a Bohr orbit of 0.476 nm to one of 0.212 nm, less than a third of a nanometre, releasing 1.89 eV, about 3.0×10⁻¹⁹ joules. Lifting a one-gram paperclip by one millimetre would take some thirty trillion such leaps. What makes the leap remarkable is discreteness, not size: measure the energy of hydrogen's electron and you find −3.40 eV or −1.51 eV, never anything in between.

Bohr's rules say only what goes into a jump and what comes out, not how or when it happens. Rutherford spotted the gap at once, asking Bohr in March 1913 how an electron decides what frequency to emit, as if it knew beforehand where it would stop. In quantum mechanics, which jump an excited atom makes, and when, is a matter of probability, and chapter 10 shows that the change itself unfolds smoothly.

The successes were stunning. Bohr's model gave hydrogen's size, its ionization energy, every line of every series, and the Rydberg constant from fundamental constants. It even explained why laboratory tubes showed about a dozen Balmer lines while some stars showed 33: high orbits are so swollen that in a laboratory gas they would overlap neighbouring atoms, whereas thin stellar atmospheres leave room for them. The model fits any ion with a single electron, with the nuclear charge Z multiplying every energy by Z². So Bohr identified a puzzling series that Edward Pickering had reported in the star ζ Puppis in 1896 as the light of helium with one electron removed. When Alfred Fowler objected that the wavelengths were off by a hair, Bohr let the nucleus move too, both bodies circling their common centre of mass, and the agreement became nearly perfect. Einstein reportedly called it an enormous achievement.

The failures were just as instructive. The model could not handle helium, with only two electrons, let alone heavier atoms. Arnold Sommerfeld's refinements of 1916, with elliptical orbits and relativity, could not fix that, nor predict how bright each line should be. Bohr also gave hydrogen's ground state one unit of angular momentum, ħ, where quantum mechanics gives zero. Deepest of all, the orbits are not real paths: as chapter 8 will show, the uncertainty principle forbids an electron a sharply defined position and momentum at the same time. In 1925 and 1926 the quantum mechanics of Heisenberg, Schrödinger and Born replaced orbits with orbitals, clouds of probability, and recovered Bohr's energies without his improvised rules. Bohr, who won the 1922 Nobel Prize in Physics for this work, had built a brilliant bridge: the right energies for hydrogen, the wrong picture of the atom.