XAS fundaments

Question 1

What is the multiplet effect and what is the order of magnitude of multiplet splitting?

Answer 1

• After a core electron vacancy is created, the system can relax into various final states characterized by different combinations between the remaining unpaired electron in the core with the unpaired electrons in the outer shell
• The observed multiplet splitting can thus be attributed to these differences in energy levels arising from various configurations influenced by both spin and orbital interactions
• For 3d Transition Metals: Multiplet splittings are often observed in the range of 0.5 to 2 eV. For 4d and 5d Transition Metals: The multiplet splitting can be larger, often exceeding 2 eV, due to stronger spin-orbit coupling effects in heavier elements.
• Factors Influencing Splitting:
  1. Spin-Orbit Coupling: Heavier transition metals exhibit stronger spin-orbit coupling, leading to larger multiplet splittings.
  2. Crystal Field Effects: The symmetry and strength of the ligand field can affect the extent of splitting.
  3. Electron Correlation: Strong electron-electron interactions can also contribute to the observed multiplet structures.

Question 2

Define bandwidth of the incoming beam and what implications does it have in the quality of the spectrum measured.

Answer 2

• Bandwidth is defined as the width of a wavelength range that transmits incident energy through a filter, typically measured as the FWHM of a spectral line
• Is has a dependency on the photon energy, as the light is not spread uniformly by the monochromator
• As bandwidth increases, the amount of thermal noise captured also increases. This is because noise power is proportional to bandwidth; thus, wider bandwidths can include more noise, which lowers the SNR
• The Nyquist–Shannon sampling theorem states that the sample rate must be at least twice the bandwidth of the signal. Therefore, we record the spectrum in stepwidths that are half of the bandwidth.

Question 3

Why is the S/N so bad for the M and N edges measured?

Answer 3

The weak signal is due to the bad overlap between 3p and 4d orbitals, for M edges, and between 4p and 5d orbitals, for N edges → probability of transition is smaller

Question 4

What information brings the X-ray absorption spectrum?

Answer 4

• Energy region around the ionization potential: the absorption edge. Right below the absorption edge are the energies related to core electron transitions into bound states, which are called resonances. The resonances provide information about the electronic structure of the sample, i.e., the frontier molecular orbitals or density of states.
• Differently from ultraviolet-visible spectroscopy that probes valence electrons transitions giving the total density of states, X-ray absorption spectroscopy probes only transitions from core-levels with well defined orbital angular momentum. Also, the transitions might be dipole allowed: the orbital angular momentum quantum number of the initial state should change by one (∆l=±1 selection rule).
• The binding energy of the core electron is related to the respective nuclear charge, and therefore, the absorption edges of different elements can be exactly distinguished in energy (element specific)
• The energy region between the resonances and the ionization potential is often associated with higher level transitions into empty orbitals that are usually diffuse, rarely providing any information about the electronic structure of the system of interest.

More accurate structural information is found in an even higher energy region, above the ionization potential, by extended X-ray absorption fine structure (EXAFS) analysis

Question 5

define cross-section in the context of XAS and give a typical value.

Answer 5

The absorption cross section, typically denoted as σ, is expressed in units of cm², but it should be interpreted as a probability of absorption rather than a physical area. It quantifies a molecule’s ability to absorb a photon of a particular wavelength and polarization.
Typical cross sections of resonant excitation at the L2,3-edge of 3d transition metals is of 10 Mbarn (1x10^-17 cm² per atom

1 barn = 1e-28 m2 = 1e-24 cm2

Question 6

Why is the chemical shift at the metal absorption edge energy linearly correlated to the oxydation state of the metal?

Answer 6

• In general, as the oxidation state increases, there is a decrease in the number of valence electrons, which reduces the screening of the nuclear charge experienced by the core electrons. This leads to a higher effective nuclear charge for the remaining electrons → the core electrons become more tightly bound due to the stronger nuclear charge, requiring more energy to excite them to unoccupied states or into the continuum
• However, the mathematical correlation between the effective nuclear charge at the core electrons and the energy of the absorption edge in X-ray absorption spectroscopy is described by Moseley’s law
  E = A (Z - σ)²
  A: constant; σ is a screening constant that follows a Coulomb interaction (V = k q₁ q₂/r²); (Z - σ) = Z_eff
• Nevertheless, experimental data consistently show a near-linear relationship between the metal edge absorption energy and the formal oxidation state. This might be because of many effects:
  1. Core-hole effect: The creation of a core hole during the X-ray absorption process significantly affects the electronic structure → contraction of the 3d orbitals → contraction is more pronounced for higher oxidation states
  2. Final state effects: The energy shift is not solely determined by the ground state electronic configuration but also by the final state reached after core excitation
  3. Coulomb interactions: changes in classical Coulomb interactions between the core hole and the valence electrons
  4. Limited range of oxidation states: The observed linearity may be an approximation valid within the limited range of accessible oxidation states for a given element

Question 7

Give a summary of the selection rules in XAS.

Answer 7

Selection rules arise because X-ray transition, like other optical transitions are usually electric dipole transitions. Dipole transitions are the primary and most intense transitions in XAS, Quadrupole transitions are generally much weaker than dipole transitions but can still be observed in XAS, they occur when ∆l = ±2

Question 8

What is a typical broadening due to lifetime of core holes in XAS?

Answer 8

For the L3 edge of 3d transition metals, the lifetime broadening is approximately 0.2 eV in energy terms. This corresponds to a lifetime of around 3.3 femtoseconds (10^-15 s).
For the M3 and N3 edges of heavier elements, the broadening could be in the order of a few (1-5) eV.

uncertainty principle relationship between energy and time: ΔE⋅Δt∼ℏ
ℏ = 6.56×10⁻¹⁶ eV⋅s