Samuel J. Ling; Jeff Sanny; William Moebs

Learning Objectives

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 = \frac{c}{f} = \frac{h \cdot c}{h \cdot f} = \frac{h c}{Δ E} = \frac{h c}{E_{3} - E_{2}},

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where $Δ E = E_{3} - E_{2}$ is energy carried away by the photon and $h c = 1240 eV \cdot nm$ . 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 Figure 8.18. 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.

Figure 8.18 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 = \pm 1,

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

L = \sqrt{l (l + 1)} ℏ .

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For multi-electron atoms, similar rules apply. To illustrate this rule, consider the observed atomic transitions in hydrogen (H), sodium (Na), and mercury (Hg) (Figure 8.19). 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.

Figure 8.19 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).

The energy-level diagram for hydrogen is similar to sodium, because both atoms have one electron in the outer shell. The valence electron of sodium moves in the electric field of a nucleus shielded by electrons in the inner shells, so it does not experience a simple 1/r Coulomb potential and its total energy depends on both n and l. Interestingly, mercury has two separate energy-level diagrams; these diagrams correspond to two net spin states of its 6s (valence) electrons.

Example 8.6

The Sodium Doublet

The spectrum of sodium is analyzed with a spectrometer. Two closely spaced lines with wavelengths 589.00 nm and 589.59 nm are observed. (a) If the doublet corresponds to the excited (valence) electron that transitions from some excited state down to the 3s state, what was the original electron angular momentum? (b) What is the energy difference between these two excited states?

Strategy

Sodium and hydrogen belong to the same column or chemical group of the periodic table, so sodium is “hydrogen-like.” The outermost electron in sodium is in the 3s (

l = 0

) subshell and can be excited to higher energy levels. As for hydrogen, subsequent transitions to lower energy levels must obey the selection rule:

Δ l = \pm 1 .

We must first determine the quantum number of the initial state that satisfies the selection rule. Then, we can use this number to determine the magnitude of orbital angular momentum of the initial state.

Solution

Allowed transitions must obey the selection rule. If the quantum number of the initial state is $l = 0$ , the transition is forbidden because $Δ l = 0$ . If the quantum number of the initial state is $l = 2, 3, 4$ ,…the transition is forbidden because $Δ l > 1 .$ Therefore, the quantum of the initial state must be $l = 1$ . The orbital angular momentum of the initial state is
$L = \sqrt{l (l + 1)} ℏ = 1.41 ℏ .$
Because the final state for both transitions is the same (3s), the difference in energies of the photons is equal to the difference in energies of the two excited states. Using the equation
$Δ E = h f = h (\frac{c}{λ}),$
we have
$\begin{matrix} Δ E & = h c (\frac{1}{λ_{1}} - \frac{1}{λ_{2}}) \\ = (4.14 \times 10^{−15} eVs) (3.00 \times 10^{8} m/s) \times (\frac{1}{589.00 \times 10^{−9} m} - \frac{1}{589.59 \times 10^{−9} m}) \\ = 2.11 \times 10^{−3} eV . \end{matrix}$

Significance

To understand the difficulty of measuring this energy difference, we compare this difference with the average energy of the two photons emitted in the transition. Given an average wavelength of 589.30 nm, the average energy of the photons is

E = \frac{h c}{λ} = \frac{(4.14 \times 10^{−15} eVs) (3.00 \times 10^{8} m/s)}{589.30 \times 10^{−9} m} = 2.11 eV .

The energy difference $Δ E$ is about 0.1% (1 part in 1000) of this average energy. However, a sensitive spectrometer can measure the difference.

Atomic Fluorescence

Fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon. Once excited, the electron “de-excites” in two ways. The electron can drop back to the ground state, emitting a photon of the same energy that excited it, or it can drop in a series of smaller steps, emitting several low-energy photons. Some of these photons may be in the visible range. Fluorescent dye in clothes can make colors seem brighter in sunlight by converting UV radiation into visible light. Fluorescent lights are more efficient in converting electrical energy into visible light than incandescent filaments (about four times as efficient). Figure 8.20 shows a scorpion illuminated by a UV lamp. Proteins near the surface of the skin emit a characteristic blue light.

The image shows a scorpion hiding in the cracks of rocks, illuminate by a U V lamp. The skin of the scorpion glows blue when illuminated by an ultraviolet light in contrast to the rocks, which glow in violet color.

Figure 8.20 A scorpion glows blue under a UV lamp. (credit: Ken Bosma)

X-rays

The study of atomic energy transitions enables us to understand X-rays and X-ray technology. Like all electromagnetic radiation, X-rays are made of photons. X-ray photons are produced when electrons in the outermost shells of an atom drop to the inner shells. (Hydrogen atoms do not emit X-rays, because the electron energy levels are too closely spaced together to permit the emission of high-frequency radiation.) Transitions of this kind are normally forbidden because the lower states are already filled. However, if an inner shell has a vacancy (an inner electron is missing, perhaps from being knocked away by a high-speed electron), an electron from one of the outer shells can drop in energy to fill the vacancy. The energy gap for such a transition is relatively large, so wavelength of the radiated X-ray photon is relatively short.

X-rays can also be produced by bombarding a metal target with high-energy electrons, as shown in Figure 8.21. In the figure, electrons are boiled off a filament and accelerated by an electric field into a tungsten target. According to the classical theory of electromagnetism, any charged particle that accelerates emits radiation. Thus, when the electron strikes the tungsten target, and suddenly slows down, the electron emits braking radiation. (Braking radiation refers to radiation produced by any charged particle that is slowed by a medium.) In this case, braking radiation contains a continuous range of frequencies, because the electrons will collide with the target atoms in slightly different ways.

Braking radiation is not the only type of radiation produced in this interaction. In some cases, an electron collides with another inner-shell electron of a target atom, and knocks the electron out of the atom—billiard ball style. The empty state is filled when an electron in a higher shell drops into the state (drop in energy level) and emits an X-ray photon.

A sketch of an x ray tube. A heated filament at one end acts as a cathode that emits an electron beam. The electrons accelerate in a gap toward a tungsten target mounted on an anode. X rays are emitted from the target.

Figure 8.21 A sketch of an X-ray tube. X-rays are emitted from the tungsten target.

Historically, X-ray spectral lines were labeled with letters (K, L, M, N, …). These letters correspond to the atomic shells ( $n = 1, 2, 3, 4, …$ ). X-rays produced by a transition from any higher shell to the K ( $n = 1$ ) shell are labeled as K X-rays. X-rays produced in a transition from the L ( $n = 2$ ) shell are called $K_{α}$ X-rays; X-rays produced in a transition from the M ( $n = 3$ ) shell are called $K_{β}$ X-rays; X-rays produced in a transition from the N ( $n = 4$ ) shell are called $K_{γ}$ X-rays; and so forth. Transitions from higher shells to L and M shells are labeled similarly. These transitions are represented by an energy-level diagram in Figure 8.22.

Different energy levels are shown in the form of horizontal lines. The line at the bottom is labeled as energy level n equal to one, or the K shell. Above this line, another horizontal line is labeled as energy level for n equal or the L shell. Similarly, other lines are shown for the M and N shells. As we move from bottom to the top, the distance between the lines decreases. The transitions are shown as arrows from one line down to a lower line and are labeled. Transitions from n=2, 3, 4, and 5 to the n=1 level form the K series and are, in order, the K sub alpha, K sub beta, K sub gamma, and K sub delta lines. Transitions from n= 3, 4, and 5 to the n=2 level form the L series and are, in order, the L sub alpha, L sub beta, and L sub gamma lines. Transitions from n= 4 and 5 to the n=3 level form the M series and are, in order, the M sub alpha and L sub beta lines. The transition fro n=5 to the n=4 level is also shown and labeled as N sub alpha.

Figure 8.22 X-ray transitions in an atom.

The distribution of X-ray wavelengths produced by striking metal with a beam of electrons is given in Figure 8.23. X-ray transitions in the target metal appear as peaks on top of the braking radiation curve. Photon frequencies corresponding to the spikes in the X-ray distribution are called characteristic frequencies, because they can be used to identify the target metal. The sharp cutoff wavelength (just below the $K_{γ}$ peak) corresponds to an electron that loses all of its energy to a single photon. Radiation of shorter wavelengths is forbidden by the conservation of energy.

A graph of X-ray intensity versus wavelength in nanometers is shown. The wavelength scale is logarithmic and its range is 0.01 nanometers to just past 1.0 nanometers. The curve starts from a point a little more than half way between 0.01 and 0.1 n m and increases. Before the frequency attains its maximum value at approximately 0.1 n m, three sharp peaks, labeled K sub alpha, K sub gamma, and K sub alpha are formed, after which the X-ray intensity decreases gradually. Two sharp peaks are seen about half way between 0.1 and 1.0, labeled L sub beta and L sub alpha. Another peak, at a wavelength longer than 1.0 n m, is labeled M sub alpha.

Figure 8.23 X-ray spectrum from a silver target. The peaks correspond to characteristic frequencies of X-rays emitted by silver when struck by an electron beam.

Example 8.7