In X-ray spectroscopy, particularly in the context of inner-shell transitions within atoms, spin and screening dublets refer to specific features in the spectrum that arise due to the spin-orbit coupling and screening effects within the atom.

Spin-Orbit Coupling:
- Fine Structure Splitting: Spin-orbit coupling is a relativistic effect where the spin of an electron interacts with its orbital motion around the nucleus. This interaction causes the energy levels associated with different electron configurations (e.g., different angular momentum quantum numbers) to split into multiple closely spaced levels. In X-ray spectroscopy, this splitting results in the observation of closely spaced lines called spin dublets or doublets.
- Example: For instance, in the K-shell (innermost electron shell) of an atom, the spin-orbit interaction splits the energy levels of the Kα and Kβ X-ray emission lines into two closely spaced lines, known as the Kα1, Kα2 doublet and Kβ1, Kβ2 doublet, respectively.
Screening Effect:
- Auger and Coster-Kronig Transitions: The screening effect refers to the shielding of the nuclear charge by inner electrons in multi-electron atoms. This shielding affects the binding energy of outer electrons and hence the energy of emitted X-rays.
- Dublets in Spectra: In X-ray spectra, particularly in Auger electron spectroscopy or X-ray emission spectroscopy, the screening effect can lead to dublets or doublets in the spectral lines. These arise due to transitions where an electron fills the vacancy left by an ejected electron, and the exact energy depends on the effective nuclear charge experienced by the electron.

In summary, spin dublets in X-ray spectra arise from spin-orbit coupling, where fine structure splitting occurs due to interactions between electron spin and orbital angular momentum. Screening dublets, on the other hand, arise from the screening effect in multi-electron atoms, affecting the energies of emitted X-rays or Auger electrons. Both types of dublets provide valuable information about the electronic structure and atomic interactions within the material being studied in X-ray spectroscopy.

In general , the term spin relativity doublet refer to the pair of energy levels which have same n and l values but different j values. In X ray energy level have same n and l but different j values. So, these pairs are called regular dublets. As we know that relativistic formula for total energy is

where R is Rydberg constant 𝛼 is fine structure constant. K is sommerfield azimuthal quantum number and d is sommerfield screening constant. Consider spin relatively doublet (L_II, L_III). For L_II, n= 2 and k =1

For L_III, n= 2, k= 2 we have

Then doublet spin in wave number will be

Neglecting higher term we get

Doublet separation in wave number is proportional to 4th power of (z-d) i.e Δ𝛾 ∝ ( z - d)^4 ...5

Equation 5 is regular doublet law which states that the doublet separation in wave number is approximately proportional to 4th power of effective nuclear charge (z -d). Therefore, it reflects how the splitting between the energy levels due to spin-orbit coupling increases rapidly with increasing atomic number Z and changes in the subshell d of the electron involved.

Here’s an explanation of why Δγ∝(Z−d)^4

Relativistic Spin-Orbit Coupling: In atoms, particularly in heavy atoms where relativistic effects become significant, the spin of an electron interacts with its orbital motion around the nucleus. This interaction leads to the splitting of energy levels that would otherwise be degenerate (i.e., have the same energy in the absence of spin-orbit coupling).
Energy Levels and X-ray Emission: When an inner-shell electron is ejected from an atom (e.g., due to collision with high-energy particles or X-rays), an outer-shell electron can transition to fill the vacancy. The energy of the emitted X-ray photon corresponds to the energy difference between the initial and final states of the electron.
Doublet Formation: Due to spin-orbit coupling, the energy levels of the initial and final electron states split into two closely spaced levels, known as a doublet or dublet. The separation between these levels (in terms of wave number, Δγ is proportional to (Z−d)4.
Dependence on Z and d:
- Z is the atomic number of the atom, which determines the nuclear charge and hence the strength of the electric field experienced by the electron.
- d is related to the quantum number of the inner-shell electron involved in the transition. For instance, in the K-shell (principal quantum number n=1n=1n=1), d could correspond to the azimuthal quantum number (l=0l=0l=0 for the 1s orbital).
Fourth Power Relationship: The fourth power dependence arises from the detailed relativistic calculations involving the spin-orbit coupling strength, which scales with Z and is different for different subshells (d) due to their different radial probability distributions and orbital shapes. This relationship is crucial in X-ray spectroscopy for understanding and analyzing fine structure in X-ray emission and absorption spectra, providing insights into the atomic structure and electronic configurations of materials.

This note is a part of the Physics Repository.

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