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By the end of this section, you will be able to:
  • Describe the absorption and emission of radiation in terms of atomic energy levels and energy differences
  • Use quantum numbers to estimate the energy, frequency, and wavelength of photons produced by atomic transitions in multi-electron atoms
  • Explain radiation concepts in the context of atomic fluorescence and X-rays

The study of atomic spectra provides most of our knowledge about atoms. In modern science, atomic spectra are used to identify species of atoms in a range of objects, from distant galaxies to blood samples at a crime scene.

The theoretical basis of atomic spectroscopy is the transition of electrons between energy levels in atoms. For example, if an electron in a hydrogen atom makes a transition from the n = 3 to the n = 2 shell, the atom emits a photon with a wavelength

λ = c f = h · c h · f = h c Δ E = h c E 3 E 2 ,

where Δ E = E 3 E 2 is energy carried away by the photon and h c = 1940 eV · nm . After this radiation passes through a spectrometer, it appears as a sharp spectral line on a screen. The Bohr model of this process is shown in [link] . If the electron later absorbs a photon with energy Δ E , the electron returns to the n = 3 shell. (We examined the Bohr model earlier, in Photons and Matter Waves .)

The hydrogen atom is represented as a proton in the nucleus, charge plus e, and an electron in a circular orbit around the nucleus. Three orbits, labeled n =1, n = 2, and n = 3 in order of increasing radius, are shown. An arrow indicates an electron transitioning from the outer to the middle orbit, and a wave labeled delta E equals h f is shown near the transition, leaving the atom.
An electron transition from the n = 3 to the n = 2 shell of a hydrogen atom.

To understand atomic transitions in multi-electron atoms, it is necessary to consider many effects, including the Coulomb repulsion between electrons and internal magnetic interactions (spin-orbit and spin-spin couplings). Fortunately, many properties of these systems can be understood by neglecting interactions between electrons and representing each electron by its own single-particle wave function ψ n l m .

Atomic transitions must obey selection rules    . These rules follow from principles of quantum mechanics and symmetry. Selection rules classify transitions as either allowed or forbidden. (Forbidden transitions do occur, but the probability of the typical forbidden transition is very small.) For a hydrogen-like atom, atomic transitions that involve electromagnetic interactions (the emission and absorption of photons) obey the following selection rule:

Δ l = ± 1 ,

where l is associated with the magnitude of orbital angular momentum,

L = l ( l + 1 ) .

For multi-electron atoms, similar rules apply. To illustrate this rule, consider the observed atomic transitions in hydrogen (H), sodium (Na), and mercury (Hg) ( [link] ). The horizontal lines in this diagram correspond to atomic energy levels, and the transitions allowed by this selection rule are shown by lines drawn between these levels. The energies of these states are on the order of a few electron volts, and photons emitted in transitions are in the visible range. Technically, atomic transitions can violate the selection rule, but such transitions are uncommon.

The energy level diagrams for hydrogen, sodium and mercury are shown as horizontal lines. The horizontal lines in this diagram correspond to atomic energy levels, and the transitions are shown by arrows drawn between these levels. Lines belonging to the same subshell (s, p, d, etc) are drawn in a column, and the different subshells are drawn next to each other in columns labeled by the subshell letter. The vertical direction represents the energy in e V. Figure a is the hydrogen spectrum. Columns for the s, p, d and f subshells are shown. The n=1 level has only one subshell, the 1 s state, with energy -13.6 e V. The n=2 level has states in the s and p subshells, with energy -3.4 e V. The n=3 level has states in the s, p and d subshells, with energy -1.5 e V. The n=4 level has states in the s, p, d, and f subshells, with energy -0.85 e V. An infinite number of energy exist for all n to infinity, getting closer and closer together. Several transitions are shown, from the s states at higher n to the p states at n=2, from the p states at higher n to the 1 s state, from the d states at higher n to the 2 p state, and from the f states at higher n to the 2 d state. Figure b is the sodium spectrum, with the energies of the hydrogen n=2 through n=6 states shown to the left for reference. The energy scale is from -5.0 to 0 e V. Columns for the s, p d, and f states are shown. The spacing between the levels is more complex than for hydrogen: the 3 s, 3 p, and 3 d levels have different energies: 3 s is a little below -5 e V, 3 p at about -3 e V, and 3 d at around -1.5 e V. Other states at the same subshell are likewise split. Transitions are shown as for hydrogen, going to lower n and changing subshell by one, f to d, d to p, s to p, etcetera. Figure c is the mercury spectrum. The energy scale is -10.0 to 0 e V. The s, p, d, f states are shown for the two net spin states of the 6 s electrons. As in the case of sodium, the states with different quantum numbers l (that is, different subshells) but the same quantum number n have different energies. In addition, we see the states split further. The one of the 6 p states (the so-called triplet state) splits into three lines which have energies that are close but clearly distinguishable, and the 7 p state for this net spin state also splits into three lines.
Energy-level diagrams for (a) hydrogen, (b) sodium, and (c) mercury. For comparison, hydrogen energy levels are shown in the sodium diagram.

The hydrogen atom has the simplest energy-level diagram. If we neglect electron spin, all states with the same value of n have the same total energy. However, spin-orbit coupling splits the n = 2 states into two angular momentum states ( s and p ) of slightly different energies. (These levels are not vertically displaced, because the energy splitting is too small to show up in this diagram.) Likewise, spin-orbit coupling splits the n = 3 states into three angular momentum states ( s , p , and d ).

Practice Key Terms 5

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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