Physics · Atomic and quantum physics · Grade 11

Atomic structure and spectra

Why does a neon sign glow red and a sodium lamp yellow?

The situation

In the evening, advertising signs in the streets of Tashkent glow in different colours: a neon tube is red-orange, while other gases give blue, pink or yellow light. The yellowish glow of some street lamps comes from sodium vapour. If you look at this light through a diffraction grating, instead of a rainbow you see separate coloured lines on a black background. Where do these lines come from, and why can scientists use them to find the composition of the Sun and distant stars?

Topic

Atomic structure and spectra. Neon sign and the spectra of stars

Key ideas

Line spectrum
Bohr model
Rydberg formula

Formulas

E_{n} = - \frac{13.6 eV}{n^{2}}

Bohr model: the allowed energies of the electron in a hydrogen atom; $n = 1, 2, 3, \dots$

h ν = \frac{h c}{λ} = E_{n_{2}} - E_{n_{1}}

When an electron drops from a higher level to a lower one, a photon is emitted whose energy equals the difference in energies.

\frac{1}{λ} = R_{H} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}), R_{H} = 1.0968 \cdot 10^{7} m^{- 1}

The Rydberg formula; $n_{1} = 2$ is the Balmer series (the visible lines).

r_{n} = n^{2} r_{1}, r_{1} \approx 0.053 nm

The radius of the Bohr orbits is proportional to $n^{2}$ .

d \sin θ = m λ

Diffraction grating: $d = 1 / 600$ mm; in the first order ( $m = 1$ ) each colour is deflected through its own angle.

Theory

Types of spectra and spectral analysis

An electric discharge in a rarefied gas excites the atoms. An excited atom emits light only at particular wavelengths — in a spectroscope separate lines are seen on a black background (a line spectrum). Hot dense bodies (a glowing coal, a lamp filament, the Sun’s photosphere) give a continuous spectrum, while molecules give a banded spectrum made of dense crowds of lines. The lines of each element are a unique “fingerprint”: neon is rich in red-orange lines, sodium has a yellow double line at 589.0 and 589.6 nm, and mercury has strong blue (435.8 nm) and green (546.1 nm) lines. Spectral analysis based on this (Kirchhoff and Bunsen, 1859–1860) detects even tiny impurities in a substance: the composition of an alloy in metallurgy, heavy metals in water and air in ecology, the composition of stars in astrophysics. In a spectroscope the light is spread out by a diffraction grating ( $d \sin θ = m λ$ ) or by a prism — through dispersion. In the laboratory (for example, with a UM-2 monochromator) the scale is first calibrated against known lines, and then the lines of a mercury lamp are measured — do this with the cursor in the “Mercury” tube.

Bohr's postulates

N. Bohr (1913): 1) an electron in an atom can only be in stationary orbits and does not radiate in them; in the allowed orbits the angular momentum is quantised: $m v r = n ℏ$ ; 2) when an electron passes from one stationary state to another, it emits or absorbs a single photon of energy $h ν = E_{n_{2}} - E_{n_{1}}$ . From this follow $E_{n} = - 13.6 / n^{2}$ eV and $r_{n} = n^{2} r_{1}$ for hydrogen. Transitions to $n_{1} = 1$ (Lyman) give ultraviolet, to $n_{1} = 2$ (Balmer) visible, and to $n_{1} = 3$ (Paschen) infrared light. A. Sommerfeld (1916) also quantised elliptical orbits: $\oint p d q = n h$ . In one-electron hydrogen-like ions (He⁺, Li²⁺) $E_{n} = - 13.6 Z^{2} / n^{2}$ eV. The $n \to 4$ transitions of He⁺ — the Pickering series — were first found in the spectra of stars and thought to belong to hydrogen; Bohr showed that they belong to He⁺, and the small difference from the Balmer lines comes from the mass of the nucleus: $R = R_{\infty} / (1 + m_{e} / M)$ . In the “Bohr atom”, choose He⁺ and compare the 6 → 4 transition with Hα.

The Rydberg formula, terms and measurement

Balmer lines: Hα 656.3 nm (3 → 2), Hβ 486.1 nm (4 → 2), Hγ 434.0 nm (5 → 2), Hδ 410.2 nm (6 → 2). The wavenumber of each line is the difference of two spectral terms $T (n) = R / n^{2}$ : $1 / λ = T (n_{1}) - T (n_{2})$ . That is why the difference of the wavenumbers of two lines often gives yet another line — the Ritz combination principle (1908): for example, $1 / λ_{H β} - 1 / λ_{H α}$ is the 4 → 3 line of the Paschen series (1875 nm). The Rydberg formula gives the wavelength in a vacuum; tables give the value in air, which is about 0.03 % shorter. Lab work: measure the lines of the hydrogen tube with the cursor and calculate $R = 1 / [λ (1 / 4 - 1 / n_{2}^{2})]$ for each — all the lines should give the same $R$ . In a grating spectroscope the wavelength is found from the angle of deflection of the line: $λ = d \sin θ$ .

Absorption spectrum: what does the Sun tell us?

A hot dense body (the Sun’s photosphere) gives a continuous spectrum. The cooler gas above it absorbs exactly the wavelengths it can emit itself — black Fraunhofer lines appear in the rainbow: hydrogen, sodium, magnesium, iron, calcium. The A and B lines come from oxygen in the Earth’s atmosphere. In a total eclipse the photosphere is covered and the bright lines of the chromosphere are seen — this is how helium was discovered in 1868. Model: the line intensities are approximate, and their widths are set by the resolution of the spectroscope; the Bohr model is correct only for hydrogen-like atoms; the Franck–Hertz curve is calculated from a simplified model (the spread of electron energies is given by a Gaussian distribution).

Franck–Hertz experiment

J. Franck and G. Hertz (1914) accelerated electrons in a tube filled with mercury vapour and measured the anode current. While the electron energy is less than 4.9 eV, collisions with mercury atoms are elastic: the heavy atom gains almost no energy. When the energy reaches 4.9 eV, the electron raises the atom to the first excited state and gives up its energy — now it cannot overcome the retarding voltage $U_{r}$ , does not reach the anode, and the current drops sharply. If $U$ rises by another 4.9 V, the electron collides twice, and so on: the distance between the current maxima is always ≈ 4.9 V. The excited mercury atoms emit ultraviolet light with $λ = h c / E_{1} \approx 254$ nm. This is direct proof of Bohr’s postulates: an atom takes up energy only in definite portions. In a neon tube the spacing of the maxima is ≈ 19 V, the excited atoms give off orange light, and bright layers can be seen in the tube. The first maximum is shifted by the contact potential difference, which is why the spacings are measured. Franck and Hertz received the Nobel Prize in 1925.

Luminescence and the limits of Bohr’s theory

If a substance becomes excited by absorbing light and then emits light itself — this is photoluminescence. Stokes’ rule: the wavelength of the emitted light is longer than that of the absorbed light — part of the energy goes into heat. If the glow dies away within ≈ 10⁻⁹–10⁻⁸ s after the lighting stops — fluorescence; if it lasts from milliseconds to hours because of metastable states — phosphorescence (paints that glow in the dark). In a daylight lamp, mercury atoms mainly emit ultraviolet light at 253.7 nm — the same transition as in the Franck–Hertz experiment — and the phosphor on the tube wall converts it into visible light. Shortcomings of Bohr’s theory: it gives correctly only the spectra of atoms with one electron; it cannot explain line intensities, many-electron atoms (even helium), fine structure or chemical bonding; and it gives no reason why an electron moving in a circle with acceleration does not radiate. Quantum mechanics answered these questions (the “Orbitals of the hydrogen atom” experiment).

Did you know?

Helium was found first not on Earth but on the Sun: during the total solar eclipse of 18 August 1868, Pierre Janssen observed a yellow line at 587.6 nm in the spectrum of the gas around the Sun. Norman Lockyer decided it was a new element and named it from the Greek “helios” — the Sun. On Earth, helium was isolated by William Ramsay in 1895.
Robert Bunsen and Gustav Kirchhoff used spectral analysis to discover the elements caesium in 1860 and rubidium in 1861. Their names come from the colour of their spectral lines: Latin caesius — “sky blue”, rubidus — “deep red”.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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The photoelectric effect — Grade 11
Radioactivity, half-life — Grade 11
Nuclear reactions — Grade 11
Orbitals of the hydrogen atom — Grade 11
Thermal radiation and Planck — Grade 11
Photon momentum and the Compton effect — Grade 11
De Broglie waves — Grade 11
Rutherford's experiment — Grade 11
The atomic nucleus and binding — Grade 11