9.2 X-ray series The X-rays emitted in transitions between inner-shell energy levels will have energies corresponding to the difference in binding energy of the electrons in the two shells concerned. The binding energy of an electron in a given shell of quantum number, n , may be expressed using a hydrogenic model: E n = R ( Z - σ n ) 2 n 2 (211) Where R is Rydberg’s constant. σ n is a screening factor that accounts for the effect of the other electrons. For the K-shell ( n = 1) there are 2 electrons. As the ejected electron moves outwards the remaining electron provides a spherically symmetric shell around the nucleus of Z protons and reduces the effective nuclear charge to ( Z - 1). The other electrons in higher shells also make a contribution to the screening. The total screening factor, σ k is then approximately 2. It is difficult to calculate screening factors, although good estimates can be made using atomic structure calculations. Usually we reply on experimental (empirical) values for σ . (The screening factors also depend on the angular momentum of the states involved.) Transitions from higher shells to a vacancy in the K-shell give rise to a series of lines. The wavenumber ( ν = 1 /λ ) of these lines will be given by the differences in the binding energies: K-series: ν K = R ( Z - σ K ) 2 1 2 - ( Z - σ i ) 2 n 2 i (212) Where n i = 2 , 3 , 4 etc. In general: ν X = R ( Z - σ i ) 2 n 2 i - ( Z - σ j ) 2 n 2 j (213) With n i , n j integers and n i < n j . K-series L-series M-series n=4 n=3 n=2 n=1 K L M N abg abg abg Figure 37: Origin of X-ray series from vacancies created in inner shells. The longest wavelength series member is labelled α , with successive lines denoted β, γ etc. 57
Atomic Physics, P. Ewart 9 X-Rays: transitions involving inner shell electrons 9.3 Fine structure of X-ray spectra A single vacancy in an otherwise full shell has the properties of a single electron in an otherwise empty shell. The X-ray energy levels therefore resemble those of hydrogen or alkali atoms. The energy levels are split into terms and the terms are split by spin-orbit interactions giving “fine structure”. The energy splitting due to fine structure can be written Δ E fs = 5 . 8 Z 4 n 3 l ( l + 1) (214) The levels are labelled by quantum numbers ( n, l, s, j ). The “ Z 4 ” factor results in very large “fine structure” splitting for heavy elements (large Z ) eg. 10 7 cm - 1 in Uranium! This structure was relatively easy to measure and for a long time such measurements gave the most accurate values of α , the fine structure constant. The X-ray lines have a multiplet structure governed by selection rules: Δ l = ± 1 , Δ j = 0 , ± 1 (215) e.g. the K α line becomes a doublet K α 1 , K α 2 . 9.4 X-ray absorption Absorption spectra in the visible or UV-range of the spectrum consist of a series of discrete lines whose wavelengths coverage to a series limit; the ionization limit. The strength of the lines decreases also towards the ionization limit. For shorter wavelengths than this limit the absorption is spectrally continuous and continues to decrease in strength as the absorbed wavelengths get shorter.
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