Chapter 8: Atomic physicsmany electrons, the periodic table, and fine structure

8.1Atomic structure and the periodic table

Helium already has two electrons and two protons. With electron–proton attraction and electron–electron repulsion, its Schrödinger equation cannot be solved exactly. The periodic table could be explained, however, if the electrons in an atom were grouped somehow in "closed shells" of 2, 8, and 18 electrons, which are stable. That grouping follows if only one electron can occupy a state defined by four quantum numbers:

The principle applies to all particles of half-integer spin (such as the electron's \(\tfrac12\)), called fermions (Chapter 9). The observed ordering of the periodic table follows from two rules: (1) the electrons in an atom tend to occupy the lowest energy levels available to them; (2) only one electron can be in a state with a given complete set of quantum numbers. (Using hydrogen's quantum numbers for other atoms assumes a hydrogen-like central field for the outer electrons.)

Hydrogen's ground state is \((n,\ell,m_\ell,m_s) = (1,0,0,\pm\tfrac12)\), the two \(m_s\) values being degenerate. Neutral helium has both electrons in \((1,0,0,\pm\tfrac12)\): antialigned spins (↑ and ↓), paired, with total spin zero. The principal quantum number also has letter codes, the shells K, L, M, N for \(n = 1,2,3,4\), and \(n\ell\) labels subshells such as 1s, 2p, 3d, with a superscript for the number of electrons: H is \(1s^1\), He is \(1s^2\), Li is \(1s^22s^1\).

Why does lithium's third electron go into 2s rather than 2p? Semiclassically the 2s state (zero angular momentum) has an orbit passing through the nucleus, while 2p is more nearly circular. A 2p electron sees the \(+3e\) nuclear charge partly screened by the two 1s electrons, \(Z_{\rm eff}\approx+1e\); a 2s electron spends more time near the nucleus and feels \(Z_{\rm eff}>+1e\), a more attractive potential, so 2s lies lower. The third electron has quantum numbers \((2,0,0,\pm\tfrac12)\). For each \(m_\ell\) there are two values of \(m_s\), and for each \(\ell\) there are \(2\ell+1\) values of \(m_\ell\), so

2 for s, 6 for p, 10 for d, and so on. Without external magnetic fields, \(m_\ell\) and \(m_s\) barely affect the energy, so states within a subshell are nearly degenerate; filling proceeds subshell by subshell, and a full subshell is closed or filled. Electrons in outer shells with lower \(\ell\) spend more time inside the inner closed shells (classically, lower \(\ell\) means more elliptical orbits); electrons with higher \(\ell\) are more shielded from \(+Ze\) and lie higher. For a given \(n\) the subshells fill in the order s, p, d, f, …, and this shielding effect is so pronounced that 4s fills before 3d, even though it has a larger \(n\). The resulting order (Table 8.1) is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d: subshells fill in order of increasing \(n+\ell\), and for equal \(n+\ell\), increasing \(n\). Some important variations from this order produce the rare-earth lanthanides and actinides.

Groups and periods. The vertical columns (groups) have similar chemical and physical properties, because they have the same valence electron structure: the same number of electrons in an \(\ell\) orbit, forming similar chemical bonds. The horizontal rows (periods) correspond to filling subshells: in the fourth row, 4s fills first with 2 electrons, then 3d with 10, then 4p with 6, for 18 elements. When a single electron is added to a new p or d subshell, the ionization energy drops markedly, showing the shell structure; atomic radii are smallest when subshells are filled. The inert gases (He, Ne, Ar, Kr, Xe, Rn) with closed subshells are monatomic gases at room temperature. The alkalis (Li, Na, K, Rb, Cs, Fr) have valence +1 and are easily ionized. The alkaline earths (Be, Mg, Ca, Sr, …) have 2 s electrons in their outer subshell, valence +2. The halogens (F, Cl, Br, I, …) have valence −1, easily gaining an electron, and form strong ionic bonds with the alkalis, as in NaCl. (For the transition metals, lanthanides, and actinides, see the textbook.) Conceptual Example 8.2: copper (\(3d^{10}4s^1\)) and silver (\(4d^{10}5s^1\)) have the highest electrical conductivities because each has one very weakly bound electron outside a closed-subshell core, practically free to move through the metal.

Rydberg atoms

Rydberg atoms are highly excited atoms with their outermost electron at a high energy level, very near ionization; the electron sits far outside a core of charge \(+e\), just like hydrogen. They are gigantic, up to 100,000 times larger than normal atoms, and surprisingly long-lived (up to a second), because the selection rules make decay difficult, especially for high \(\ell\). They are observed in radio astronomy with \(n\) near 400 and made in the laboratory, usually near \(n = 30\), with tunable lasers. Because of their high degeneracy, a weak electric field shifts their levels strongly (the Stark effect, discovered by Johannes Stark in 1913), and can easily ionize them.

8.2Total angular momentum

An atom has orbital angular momentum and spin angular momentum due to one or more of its electrons, which combine into a total angular momentum. The interaction between the orbital and spin angular momenta splits energy levels into doublets even without an external magnetic field. Consider first single-electron atoms, with one electron outside an inert core (such as the alkalis). The total angular momentum is

Since \(L\), \(L_z\), \(S\), \(S_z\) are quantized, so are \(J\) and \(J_z\). With \(m_\ell\) an integer and \(m_s\) a half-integer, \(m_j\) is a half-integer. With \(L = \sqrt{\ell(\ell+1)}\hbar\) and \(S = \sqrt{s(s+1)}\hbar\),

The vectors add vectorially; for \(\ell = 1\), \(j = \tfrac32\) or \(\tfrac12\). The notation is \(nL_j\): the principal quantum number, then the orbital angular momentum as an uppercase letter (S, P, D, …), then \(j\) as a subscript.

Spin-orbit coupling. In the electron's frame, the orbiting proton produces an internal magnetic field \(\vec B_{\rm internal}\propto\vec L\), and the spin's magnetic moment \(\vec\mu_s\propto-\vec S\) has a dipole potential energy

where \(\alpha\) is the angle between \(\vec S\) and \(\vec L\). For \(j = \ell+\tfrac12\), \(\cos\alpha>0\); for \(j = \ell-\tfrac12\), \(\cos\alpha\lt 0\), which has lower energy. The same applies in an external magnetic field. Therefore \(j\) and \(m_j\) are "better" quantum numbers than \(m_\ell\) and \(m_s\): more directly related to a physical observable, since a state of definite energy has a definite \(J_z\) but not definite \(L_z\) and \(S_z\). The wave functions depend on \(n\), \(\ell\), \(j\), and \(m_j\), and 2P splits into \(2P_{3/2}\) and the lower \(2P_{1/2}\). \(\vec L\) and \(\vec S\) precess around \(\vec J\); in an external field \(\vec J\) precesses about \(z\), and only \(J_z\) can be known, since the uncertainty principle forbids knowing \(J_x\) and \(J_y\) at the same time.

Selection rules. With \(\Delta\ell = \pm1\), \(\Delta j = \pm1\) or 0. The allowed transitions for a single-electron atom are

since \(m_j = m_\ell+m_s\) and \(m_s\) is not affected. For hydrogen, the H\(_\alpha\) line (\(n = 3\to2\)) acquires fine structure, because the spin-orbit interaction splits each \(\ell\ne0\) state. In sodium, with a single \(3s^1\) electron outside an inert core, the levels resemble hydrogen's \(n\ge3\) levels, but the strong attraction of small-\(\ell\) electrons pulls those levels considerably lower: the 3s level of sodium is at −5.12 eV, while 5f and 6f nearly coincide with hydrogen's. The spin-orbit splitting of \(3P_{3/2}\) and \(3P_{1/2}\), \(2.1\times10^{-3}\) eV, splits the 3p → 3s (≈ 2.1 eV) line into the famous yellow sodium doublet at 589.0 and 589.6 nm. Example 8.3: for a small splitting, \(|\Delta\lambda| = \frac{\lambda^2}{hc}|\Delta E| = \frac{(589.3\ \text{nm})^2(2\times10^{-3}\ \text{eV})}{1.240\times10^3\ \text{eV·nm}} = 0.6\) nm, in agreement with experiment. Example 8.8: the spin-orbit splitting \(\Delta E = g_s\frac{e\hbar}{2m}B = \frac{e\hbar}{m}B\) gives an internal field \(B = m\Delta E/e\hbar = 18\) T: internal magnetic fields are large.

Many-electron atoms

Hund's rules, applied to two electrons outside a closed shell (as in He and the alkaline earths):

1. The total spin angular momentum \(S\) should be maximized to the extent possible without violating the Pauli exclusion principle.
2. Insofar as rule 1 is not violated, \(L\) should also be maximized.
3. For atoms having subshells less than half full, \(J\) should be minimized.

For example, the first five electrons in a d subshell should all have the same \(m_s\), which requires different \(m_\ell\) (from −2, −1, 0, 1, 2); by rule 2, the first two should have \(m_\ell = 2, 1\) or \(-2, -1\). Physically, maximum \(S\) occurs because the electrons repel each other and want to be as far apart as possible: electrons with the same \(m_s\) must have different \(m_\ell\), normally meaning different spatial distributions. Maximum \(L\) means the electrons revolve in the same direction, staying far apart; antialigned \(\vec L_1\) and \(\vec L_2\) would pass each other more often. Besides spin-orbit, there are spin-spin and orbital-orbital interactions, and the nuclear spin produces the much smaller hyperfine structure.

For two electrons, \(\vec J = \vec L_1+\vec L_2+\vec S_1+\vec S_2\), combined in one of two schemes. In LS coupling (Russell–Saunders), appropriate for most atoms when the magnetic field is weak, \(\vec L = \vec L_1+\vec L_2\), \(\vec S = \vec S_1+\vec S_2\), and then \(\vec J = \vec L+\vec S\). Two electrons give \(S = 0\) (antiparallel spins) or 1 (parallel). For a given \(L\), \(J\) runs from \(|L-S|\) to \(L+S\): \(2S+1\) values when \(L\ge S\), fewer when \(L\lt S\). The value \(2S+1\) is the multiplicity, and the notation becomes

the spectroscopic or term symbol: singlet states (\(S = 0\)) and triplet states (\(S = 1\)) for two electrons, while a single electron (\(s = \tfrac12\)) gives a doublet, \(2s+1 = 2\). For one electron in 4p and one in 4d (Table 8.2) the terms are \(4^1P_1, 4^1D_2, 4^1F_3\) and \(4^3P_{0,1,2}, 4^3D_{1,2,3}, 4^3F_{2,3,4}\), split successively by spin-spin correlation, the residual electrostatic energy, and spin-orbit energy. Magnesium (two electrons, one in 3s) has separate singlet and triplet level schemes, because the LS-coupling selection rules forbid transitions between them:

jj coupling predominates for heavier elements, where the large nuclear charge makes spin-orbit interactions as strong as the forces between the individual \(\vec S_i\) and between the individual \(\vec L_i\): \(\vec J_1 = \vec L_1+\vec S_1\), \(\vec J_2 = \vec L_2+\vec S_2\), and then \(\vec J = \sum_i\vec J_i\).

Conceptual Example 8.4: helium's ground state has both electrons in 1s, so \(L = 0\); in the same subshell the spins must be antialigned, so \(S = 0\), \(J = 0\): \(1^1S_0\). Example 8.5: for \(\ell_1 = 1\), \(\ell_2 = 2\), \(L = 1,2,3\), \(S = 0,1\), and \(J = 0\) to 4. Example 8.6: with \(L = 3\), \(S = 1\) and minimum \(J\), \(\vec L\) and \(\vec S\) are antialigned and precess about \(\vec J\), which precesses about \(z\). Example 8.7: the lowest excited states of helium are \(1s^12s^1\) (\(L = 0\): \(^1S_0\) or \(^3S_1\), with \(^3S_1\) lowest) and \(1s^12p^1\) (\(L = 1\): \(^1P_1\) or \(^3P_{0,1,2}\)); the ground state \(^1S_0\) lies some 20 eV below them. Example 8.9: carbon has two 2p electrons outside closed 2s². Naively \(L = 0,1,2\) and \(S = 0,1\), but the Pauli principle forbids \(^3S_1\) and \(^3D_{1,2,3}\) (both electrons would share \(m_s\) and \(m_\ell\)), and antisymmetry of the wave function (Chapter 9) forbids \(^1P_1\). The allowed terms are \(^1S_0\), \(^1D_2\), and \(^3P_{0,1,2}\); by Hund's rules the ground state is \(^3P_0\), with low-lying excited states \(^3P_1\), \(^3P_2\), \(^1D_2\), and \(^1S_0\).

8.3Anomalous Zeeman effect

In Section 7.4, the normal Zeeman effect, the splitting of a spectral line into three components in an external field \(\vec B_{\rm ext}\), was understood from the interaction \(\vec\mu_\ell\cdot\vec B_{\rm ext}\) of the orbital magnetic moment. Often, however, more than three closely spaced lines were observed: the "anomalous" Zeeman effect. The magnetic moment depends also on the spin magnetic moment \(\vec\mu_s\). If \(\vec B_{\rm ext}\) is small compared with the internal field (say \(B_{\rm ext}\lt 0.1\) T), \(\vec L\) and \(\vec S\) precess about \(\vec J\) quickly, while \(\vec J\) precesses slowly about \(\vec B_{\rm ext}\). The total magnetic moment is

so \(-\vec\mu\) and \(\vec J\) point in the same direction only when \(S = 0\). Because \(\vec\mu\) precesses rapidly about \(\vec J\), we first average \(\vec\mu\) over that precession, \(\vec\mu_{\rm av} = (\vec\mu\cdot\vec J)\vec J/(\vec J\cdot\vec J)\), and then find its energy \(V = -\vec\mu_{\rm av}\cdot\vec B\) (Problem 31). Using \(\vec S\cdot\vec J = \tfrac12(J^2+S^2-L^2)\) and \(\vec L\cdot\vec J = \tfrac12(J^2+L^2-S^2)\), with \(J^2 = J(J+1)\hbar^2\) and \(J_z = m_J\hbar\),

where \(g\) is the dimensionless Landé g factor. The field splits each state \(J\) into \(2J+1\) equally spaced levels, one for each \(m_J\) from \(-J\) to \(J\). In addition to the earlier selection rules, \(\Delta m_J = 0,\pm1\), but \(m_J = 0\to m_J = 0\) is forbidden when \(\Delta J = 0\).

Conceptual Example 8.10: for \(^1D_2\to{}^1P_1\), \(S = 0\) and \(J = L\), so \(g = 1\) for both states (as always for \(S = 0\)). The \(^1D_2\) state splits into five equally spaced levels and \(^1P_1\) into three; nine transitions are allowed, but they have only three different energies. The normal Zeeman effect is observed whenever \(S = 0\). The anomalous effect is a direct result of spin. For sodium's \(^2P\to{}^2S\) lines, the spin-orbit interaction splits \(^2P_{3/2}\) and \(^2P_{1/2}\), with \(^2P_{1/2}\) lower, while \(^2S_{1/2}\) is not split (\(\ell = L = 0\)). The Landé factors are

all different, so the three states split by different amounts: there are four different transition energies for \(^2P_{1/2}\to{}^2S_{1/2}\) and six for \(^2P_{3/2}\to{}^2S_{1/2}\). If the field is increased, \(\vec L\) and \(\vec S\) precess too rapidly about \(\vec B_{\rm ext}\), the averaging breaks down, and the analysis no longer applies: the Paschen–Back effect.

This completes Chapter 8. Chapter 9 turns to statistical physics.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 8.
• W. Pauli, "Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren," Z. Phys. 31, 765 (1925).
• H. E. White, Introduction to Atomic Spectra (McGraw-Hill, 1934).
• A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database (ionization energies), National Institute of Standards and Technology.
• T. F. Gallagher, Rydberg Atoms (Cambridge University Press, 1994).