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Atomic Spectroscopy:

Emission, Absorption & Fluorescence

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Hydrogen atom: V=cte/r

(− ℎ28𝜋2𝑚∇2 + 𝑉)𝜓 = 𝐸𝜓

𝜓ሺ𝑟,𝜃,𝜙ሻ= 𝑅ሺ𝑟ሻ.𝛩ሺ𝜃ሻ.Φ(𝜙)

𝐸= 𝑅𝐻𝑍2𝑛2

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Term Symbols

2 1SJL

L = 0, 1, 2, ...S, P, D

J = L + S

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L = | l1 + l2 |, | l1 + l2 - 1 |, ..., | l1 - l2 |

S = | s1 + s2 |, | s1 + s2 - 1 |, ..., | s1 - s2 |

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J = L + S

| L + S |, | L + S - 1 |, ..., | L - S |

Russell-Saunders Coupling

jj – Coupling

ji = li + si

| j1 + j2 |, | j1 + j2 - 1 |, ..., | j1 - j2 |

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Hund’s Rules

The lowest energy term is that which has the greatest spin multiplicity.

For terms that have the same spin multiplicity, the term with the highest orbital angular momentum lies lowest in energy.

spin-orbit coupling (more pronounced for heavier nuclei) splits terms into levels.{ If the unfilled subshell is exactly or more than half full, the level with the highest J value has the lowest energy.{ If the unfilled subshell is less than half full, the level with the lowest J value has the lowest energy.

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𝐵= − 𝐯𝑥𝑬𝑐2

𝐵= 𝒓𝑥𝒑𝑚𝑟𝑐2 |𝐸| = |𝐸|𝑚𝑟𝑐2 𝑳

𝜇= −2𝜋𝜇𝐵𝑔𝑠ℎ 𝑺

∆𝐸= −𝝁.𝑩∝ 𝑳.𝑺

𝐽2 = (𝐿+ 𝑆)2 = 𝐿2 + 𝑆2 + 2𝐿.𝑆

2𝐿.𝑆= 𝐽2 − 𝐿2 − 𝑆2 = 𝐽ሺ𝐽+ 1ሻ− 𝐿ሺ𝐿+ 1ሻ− 𝑆(𝑆+ 1)

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Selection Rules

Helicity of a photon σ = ± 1

(Physically analogous to the angular momentum along the axis of propagation)

Upon absorption or emission of a photon the atom must change the momentum along the zz axis (considered collinear with the axis of propagation) in ± 1

ΔJ = 0; ± 1 (but J ≠ 0 if ΔJ = 0)ΔMJ = 0; ± 1 (but ΔMJ ≠ 0 if ΔJ = 0)

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𝜇𝑥,𝑦,𝑧𝑖𝑗 = න𝜓𝑖∗𝜇𝑥,𝑦,𝑧𝜓𝑗𝑑𝜏 ≠ 0

Spin wave functions are differentially zero (αβ) or one (αα).

ΔS = 0 allowed

ΔS ≠ 0 forbidden

Electric dipole Transition moment

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Grotrian Diagrams

Hydrogen

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