Spectral Line Physics Atomic Structure and Energy Levels Atomic Transition Rates Molecular Structure and Transitions 1

Quantum Numbers ( n, principal quantum number. Defines the distance of the electron from the nucleus in the Bohr model. l, the azimuthal quantum number. l takes on the integral values 0, 1, 2,..., n-2, n-1. Defines angular momentum. m, the magnetic quantum number. m takes on the integral values -l, -(l-1),..., -1, 0, 1,..., (l-1), l. s, the spin quantum number. This describes the spin of the electron, and is either +1/2 or -1/2. 2

3

Quantum Numbers Pauli exclusion principle: no two electrons have the same set of the 4 quantum numbers n, l, m, s There are 2n 2 possible states for an electron with principal quantum number n (statistical weight). The n=1 levels can contain only 2 electrons. This level is called the 1s orbit or the K shell (shells with n=1,2,3,4,5,6,7 are called the K, L, M, N, O, P, Q shells, respectively). An orbit, or shell, containing the maximum number 2n 2 electrons forms a closed shell. 4

Energy levels for n = 3 5

Quantum Numbers for Atoms l =0, 1, 2, 3,4  s, p, d, f, g total orbital ang. mmt. for multi- electron systems L =0, 1, 2, 3,4  S, P, D, F, G S is total spin J=L+S is the total angular mmt. 6 (2J+1) degenerate levels for each J unless magnetic field applied (Zeeman splitting) or electric field applied (Stark effect).

Spectroscopic Notation The atomic level is described as n 2S+1 L J where S, n, and J are the quantum numbers, and L is the term (S,P,D,F,G, etc.). 2S+1 is the multiplicity. Ground state of Boron: 1s 2 2s 2 2p 1 configuration, with 2 e- in the n=1 level (l=0), 2 e- in the n=2, l=0 s orbital, and last e- in the 2p orbital. The ground state of Boron has a 2 P 1/2 term. Closed shells always have a 1 S 0 term. 7

Selection Rules (Electric Dipole): Permitted and Forbidden Transitions ΔL = 0, +/- 1 Δl = 1 ΔJ = 0, +/- 1, except that J=0 -> J=0 is forbidden. ΔS = 0 ΔM = 0, +/- 1, except that M=0 -> M=0 is forbidden if ΔJ=0. As the atoms become more complex, strict L-S coupling fails and selection rules weaken 8

Multiplets Transitions arising from a one term to another term give rise to a multiplet. The multiplicity of a term is given by 2S+1. S=0 is a singlet term; S=1/2 is a doublet term; S=1 is a triplet term; S=3/2 is a quartet term... Alkali metals (S=1/2) form doublets (Li, Na, K...). Ions with 2 e- in outer shell (He I, Ca I, Mg I) form singlets or triplets. 9

Neutral Sodium Grotrian Diagram 10 Outer e- n=3 Energy depends on l Δl rule applies Na D line from 3p spin difference

Neutral Helium Grotrian Diagram 11 one e- in 1s state e- spin interactions in multi e- cases: L-S coupling total spin=0,1 1 S singlet 3 S triplet e- stuck in high E level is metastable

M Degeneracy Broken By Magnetic Fields - Zeeman Effect Normal Zeeman effect operates in a singlet state and results in three lines: lines with ΔM = 0, the π components, are unshifted, polarized parallel to the field; lines with ΔM = +/- 1, the σ components, are shifted by +/- 4.7 X g λ 2 B, where g = Lande g factor, λ = wavelength, and B = strength of the magnetic field in Gauss. g = 1 + (J(J+1) + S(S+1) - L(L+1))/2J(J+1) 12

Hyperfine Structure Coupling between the magnetic moment of electron and the nuclear magnetic moment Quantum number I = net nuclear spin Construct F=I+J, for J-I,J-I+1,... J+I-1, J+I. 2 S 1/2 ground state of H has J=1/2, I=1/2 (because the spin of the proton is ½). F=1 corresponds to parallel spins for p and e- F=0 to anti-parallel spins (lower energy) Energy difference: 1420 MHz or 21 cm. 13

Typical Energies of Interaction Central potential (configurations) 4 eV Electrostatic interaction (L-S coupling, terms) 0.4 eV Spin-orbit interaction (e- magnetic field and e- magnetic moment; fine structure) to eV Hyperfine structure (nuclear spin, isotope) to eV 14

Transition Probabilities (Mihalas Section 4.2) Recall Einstein coefficients for b-b transitions Express actual cross section with oscillator strength f ij f ij from QM calculation (based on volume coincidence of wave functions of two states) or laboratory measurements Tabulated with the statistical weight: log gf 15

Molecular Structure (Robinson 2007; Tennyson 2005) Molecules exist in cool atmospheres Consider simple diatomic molecule where differences create dipole moment: ex. CO Quantum numbers rule the rotation rates, vibrational states, electronic states Interaction with light causes transitions between states (similarly to atomic bound- bound transitions) 16

Rotational States: 2 axes 17

Vibrational transition: With Rotational Level Change +/-1 18

Vibration-Rotation Transitions 19

20

21

Molecular bands Transition between two electron energy levels splits into series of vibrational transitions, each of which splits into several rotation transitions. Result is molecular band in spectrum in which large numbers of lines collect at the band head at the short wavelength end