Many-electron atoms
low-energy states, optical transitions

The following exercises focus on the optical excitations of many-electron atoms / ions.

They must be solved based on some (usually approximate) quantum-mechanical treatment (Hartree-Fock approximation to Schrödinger's wave equation, and consequent effective-Z approximation...).

The nuclear charge is assumed to be Z q_e.

a₀ = ℏ²/(m_e e²) is the Bohr radius and
E_Ha = e²/a₀ is the Hartree energy.
Here m_e is the electron mass and e² = q_e² /(4πε₀) is the electromagnetic characteristic coupling constant.

For many-electron atoms reduced-mass corrections are irrelevant and conceptually meaningless.

Relativistic effects (e.g. spin-orbit coupling) are increasingly important especially for heavy atoms, since they scale roughly as Z⁴. As a consequence, external magnetic fields can be usually considered as "weak", in most of the following problems.

1. The emission spectrum of Cs shows 3 lines with the following wavelengths: λ₁=1002.710 nm, λ₂=1012.619 nm, and λ₃=1012.637 nm, corresponding to the 4f²F → 5d²D transitions. Draw a line spectrum with all involved levels and dipole-allowed transitions, assuming that the splitting of the 4f level is smaller and inverted. Evaluate the spin-orbit splitting of the 4f and 5d levels. Evaluate also the spin-orbit coupling parameters ξ_4f and ξ_5d. Optionally, estimate the effective Z^eff-SO for these 4f and 5d levels (as defined in this page).
   RESULT: ΔE^s-o_4f=-22.5 μeV, ΔE^s-o_5d=12.1 meV; ξ_4f=-6.43 μeV, ξ_5d=4.84 meV.
2. Compute the energy change of all the states in the previous exercise if and atomic Cs sample is placed in a uniform magnetic field of 7 T. How would the three relevant lines change under these conditions?
3. [written test 19 July 2012] The excitation energy of the 1s²2s²2p3s (¹P₁) state of carbon equals 7.685 eV; that of 1s²2s²2p² (¹D₂) equals 1.264 eV. Compute the wavelength emitted in the electromagnetic transition between these levels. Show how a uniform magnetic field modifies these levels, indicating the electric-dipole-allowed transitions in a diagram. Compute the maximum and minimum wavelengths for these transitions, for a magnetic-field intensity 0.5 T.
   RESULT: λ_{0 field} = h c/ΔE = 193.092 nm. As both states are spin-0 (also called spin-singlet) states, all couplings between the field and electronic moments involve the orbital angular momentum, not spin. Therefore the g-factors are unity for both states. With equal g-factors, both levels split into equally-spaced sublevels with different M_J=M_L, with a spacing between adjacent levels of μ_BB. As a consequence, we are in a regular-Zeeman condition, with 3 split lines only, and λ_max = h c/(ΔE - μ_BB) = 193.093 nm; λ_min = h c/(ΔE + μ_BB) = 193.091 nm.
4. The lowest excited level of Be (Z=4) corresponds to an electronic configuration [He] 2s2p. Which ^2S+1[L]_J terms are originated? What is their energy ordering according to Hund's rules?
   RESULT: Clearly L = 1; S = 0 or 1. Taking the angular momentum composition rules into account, and Hund's rules for ordering, we obtain: ³P₀, ³P₁, ³P₂, ¹P₁.
5. The fine-structure components of the lowest term of the iron atom have the following excitation energies (in cm^-1 units):
   ⁵D₄ 0
   ⁵D₃ 415.932
   ⁵D₂ 704.004
   ⁵D₁ 888.129
   ⁵D₀ 978.072
   Use these experimental energies to check the validity of the Landé interval rule.
6. Examine a ³S₁→³P₂ emission line in the spectrum of He in the presence of a field |B|=0.16 T. Draw the level structure and the dipole-allowed transitions. Knowing that for B=0 the wavelength of this line is 706.5 nm, predict the number of lines that should be observed and compute the maximum and minimum wavelengths in this multiplet.
   RESULT: Here, both S and L are nonzero in the the final state, therefore the g-factors acquire nontrivial values, and we obtain an anomalous Zeeman pattern of splittings. λ_max=706.507 nm, λ_min=706.493 nm.