Spectral Line Physics

Atomic Structure and Energy Levels Atomic Transition Rates

Molecular Structure and Transitions

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Quantum Numbers(http://www.ess.sunysb.edu/fwalter/AST341/qn.html)

• 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.

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Quantum Numbers

• Pauli exclusion principle: no two electrons have the same set of the 4 quantum numbers n, l, m, s

• There are 2n2 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 2n2 electrons forms a closed shell.

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Energy levels for n = 3

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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.

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(2J+1) degenerate levels for each J unless magnetic field applied (Zeeman splitting) orelectric field applied (Stark effect).

Spectroscopic Notation

• The atomic level is described as n 2S+1LJ 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: 1s22s22p1 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 2P1/2 term.

• Closed shells always have a 1S0 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• http://physics.nist.gov/Pubs/AtSpec/node17.html

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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.

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Neutral Sodium Grotrian Diagram

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• Outer e- n=3• Energy

depends on l• Δl rule applies• Na D line

from 3p spin difference

Neutral Helium Grotrian Diagram

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• one e- in 1s state• e- spin interactions

in multi e- cases: L-S coupling

• total spin=0,11S singlet3S 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 10-13g λ2B, 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. • 2S1/2 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.

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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) 10-4 to 10-1 eV

• Hyperfine structure (nuclear spin, isotope) 10-7 to 10-4 eV

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Transition Probabilities(Mihalas Section 4.2)

• Recall Einstein coefficients for b-b transitions• Express actual cross section with oscillator

strength fij

• fij from QM calculation (based on volume coincidence of wave functions of two states) or laboratory measurements

• Tabulated with the statistical weight: log gf• http://www.nist.gov/pml/data/atomspec.cfm

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€

σtot =πe2

mc

⎛

⎝ ⎜

⎞

⎠ ⎟f ij = Bij

hν ij4π

⎛

⎝ ⎜

⎞

⎠ ⎟

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)

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Rotational States: 2 axes

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Vibrational transition:With Rotational Level Change +/-1

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Vibration-Rotation Transitions

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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

• http://www.nist.gov/pml/data/molspecdata.cfmhttp://spec.jpl.nasa.gov/ftp/pub/catalog/catform.html

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