CHAPTER 8 Atomic Physics Multi-electron atoms The Pauli exclusion Principle Filling Shells Atomic Structure and the Periodic Table Total Angular Momentum Spin-Orbit Coupling Dimitri Mendeleev (1834 – 1907) What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior. - J. Bronowski Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu

Multi-electron atoms When more than one electron is involved, the potential and the wave function are functions of each electron’s position : Attraction between nucleus and electrons Repulsion between electron pairs Solving the Schrodinger Equation exactly in this case is impossible! But we can approximate the solution as the product of single-particle wave functions: And we’ll approximate each Yi with a Hydrogen wave function.

Probability Distribution Functions Probability densities for some Hydrogen electron states.

Pauli Exclusion Principle What states do the electrons reside in? As in most cases in physics: The electrons in an atom tend to occupy the lowest energy levels available to them. So they’re all in n = 1 states? Nope. Strong absorptions are seen in multi-electron atoms from states with large values of n, so Wolfgang Pauli proposed his famous Exclusion Principle: No two electrons in an atom may have the same set of quantum numbers (n, ℓ, mℓ, ms). It applies to all particles of half-integer spin, such as electrons, which are called fermions.

Fermions vs. bosons Electrons, protons, and neutrons have spin ½ and so are fermions. Bosons have integer spin and can be in the same state. Photons have spin 1 and so are bosons. If states are degenerate, then there can be as many fermions with a given energy as states with that energy.

Atomic Structure Hydrogen: in the ground state. In the absence of a magnetic field, the state ms = +½ is degenerate with the ms = −½ state. So ms can be either +½ or −½. Helium: for the first electron. for the second electron. Helium’s two electrons have anti-aligned (ms = +½ and ms = −½) spins. Lithium: for the first electron. for the third electron. In the absence of a magnetic field, the third electron’s state ms = +½ is degenerate with the ms = −½ state, so it can be either +½ or −½.

Electronic configuration and shells The principle quantum number also has a letter code. n = 1 2 3 4... Letter = K L M N… n = Shells (e.g.: K shell, L shell, etc.) nℓ = Subshells (e.g.: 1s, 2p, 3d) n Hydrogen: 1s Helium: 1s2 Lithium: 1s22s Flourine: 1s22s22p5 etc. # of electrons The list of an atom’s occupied energy levels is called its electronic configuration. The exponent indicates the number of electrons in the subshell. ℓ

Electronic shells and subshells http://godschemistryset.blogspot.com/2011_06_01_archive.html

Filling the shells Filling the 2p subshell 9 10 Boron (B) Carbon (C) Nitrogen (N) Oxygen (O) Fluorine (F) Neon (Ne)

Atomic Structure How many electrons may be in each subshell? For each ℓ: there are (2ℓ + 1) values of mℓ. For each mℓ, there are two values of ms. So there are 2(2ℓ + 1) electrons per subshell. Recall: ℓ = 0 1 2 3 4 5 … letter = s p d f g h … ℓ = 0 (s state) can have two electrons. ℓ = 1 (p state) can have six electrons, and so on. Electrons with higher ℓ values are on average further from the nucleus and so are more shielded from the nuclear charge, so electrons with higher ℓ values lie higher in energy than those with lower ℓ values—even if the value of n is higher. The 4s shell fills before 3d.

Shell-filling order—a geometrical picture http://physics.tutorvista.com/modern-physics/shell-model.html

Filling the shells Wikipedia

Cut-off between metals and nonmetals Filling the shells Cut-off between metals and nonmetals (metals) http://pages.uoregon.edu/ch111/L7.htm

The Periodic Table https://shschemistry1.wikispaces.com/Electron+Configuration,+Ion+Formation,+and+Radioactivity

Electron probability distribution of a closed shell or subshell Closed shells and subshells have spherically symmetric electron probability distributions. Elements with closed shells have the most compact electron distributions and so are also the most stable and most difficult to ionize. They include Helium, Neon, Argon, Krypton, Xenon, and Radon. Elements with one additional electron (beyond a closed shell) are the largest and least stable. Electron probability distribution of a closed shell or subshell

It’s easy to see why closed shells and subshells are spherically symmetrical. s subshell: Subscripts are mℓ values. p subshell: d subshell:

Atoms with closed shells (especially p-shells) are smaller.

Atoms with closed shells (especially p-shells) are more stable and hence difficult to ionize. A closed d-shell can also be quite stable for larger values of n

The Periodic Table Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies Electrical conductivity is relatively good Alkaline Earths: Two s electrons in outer subshell Largest atomic radii High electrical conductivity

The Periodic Table Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable configurations occur as the p subshell is filled Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by the s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

The Periodic Table Lanthanides (rare earths): Have the outside 6s2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s2 subshell is complete Difficult to obtain chemical data because they are all radioactive

Additional effects: total angular momentum The total angular-momentum quantum number for the single electron is the sum of the orbital angular momentum and the spin. It can only have the values: depending on their relative directions j and mj are also quantum numbers for single electron atoms. The total and z-components of angular momentum are: For multi-electron atoms, use vector addition of all the electrons’ orbital angular momenta and spins. Then: All of the quantities L, Lz, S, Sz, J, and Jz are quantized.

Spin-Orbit Coupling Nucleus The orbit of the electron produces a magnetic field: But the electron spin also produces a magnetic field proportional to the spin. The spin’s potential energy in the magnetic field of the electron orbit is: Parallel: repulsion Antiparallel: attraction This energy dependence on the interaction of the electrons’ spins and orbital angular momenta is called spin-orbit coupling. It yields fine structure in the energy levels and spectrum.

Total Angular Momentum In terms of j, the selection rules for a single-electron atom become: Δn = anything Δℓ = ±1 Δmj = 0, ±1 Δj = 0, ±1 Use capital letters for the j value. Hydrogen energy-level diagram for n = 2 and n = 3 with spin-orbit splitting. Similar splittings occur for multi-electron atoms.

Many-Electron Atoms Hund’s rules: The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as the above condition is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized. For a two-electron atom: There are LS coupling and jj coupling to combine four angular momenta J.

Hyperfine structure The nucleus also has spin (designated by I), and its magnetic field also contributes to the potential and yields hyperfine structure. The total angular momentum, including nuclear spin, is designated by F. http://upload.wikimedia.org/wikipedia/commons/f/fe/Fine_hyperfine_levels.png

Energy levels of Helium Here, one electron is assumed to remain in the ground state. Parallel spins Anti-parallel spins Parallel spins means greater separation between electrons hence lower energies. In the helium energy level diagram, one electron is presumed to be in the ground state of a helium atom, the 1s state. An electron in an upper state can have spin antiparallel to the ground state electron (S=0, singlet state, parahelium) or parallel to the ground state electron (S=1, triplet state, orthohelium). It is observed that the orthohelium states are lower in energy than the parahelium states. The explanation for this is: The parallel spins make the spin part of the wavefunction symmetric. The total wavefunction for the electrons must be anti-symmetric since they are fermions and must obey the Pauli exclusion principle. This forces the space part of the wavefunction to be anti-symmetric. The wavefunction for the electrons can be written as the product of the space and spin parts of the wavefunction. An anti-symmetric space wavefunction for the two electrons implies a larger average distance between them than a symmetric function of the same type. The probability is the square of the wavefunction, and from a simple functional point of view, the square of an antisymmetric function must go to zero at the origin. So in general, the probability for small separations of the two electrons is smaller than for a symmetric space wavefunction. If the electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy. This effect is sometimes called the "spin-spin interaction" and is addressed byHund's Rule #1 . It is part of the understanding of the ordering of energy levels in multi-electron atoms. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html

Computation of atomic energy levels allows us to determine the atoms present on the sun’s surface from its spectrum. Line Element Wavelength [nm] b -1, 2 Mg 518.4, 517.3 c Fe 495.8 F H 486.1 d 466.8 e 438.4 f 434.0 G Fe i Ca 430.8 g Ca 422.7 h 410.2 396.8 K 393.4 Line Element Wavelength [nm] A -band O2 759.4 - 762.1 B -band 686.7 - 688.4 C H 656.3 a -band 627.6 - 628.7 D -1, 2 Na 589.6, 589.0 E Fe 527.0 http://www.euhou.net/index.php/exercises-mainmenu-13/classroom-experiments-and-activities-mainmenu-186/179-observations-of-various-spectra-with-a-home-made-spectroscope