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σ* LUMO+1 π* LUMO What is a near-edge spectrum? Molecular orbital approach - transitions to boundstate molecular orbitals. S1s →π* S1s → σ* I. J. Pickering and G. N. George GEOL 498.3/898.3 Spectral linewidths Two components contribute to the spectral linewidth – the core-hole lifetime and the optical resolution. Core-hole lifetime. Heisenberg’s uncertainty principal states that: 1 ∆E∆t ≥ h 2 Thus, comparing high and low energy edges, we expect the higher energy edge to have shorter core hole lifetimes (∆t) and correspondingly broader experimental linewidth (∆E) (assuming that the spectroscopic resolution is not limiting). This adds a Lorentzian component to the lineshape. I. J. Pickering and G. N. George GEOL 498.3/898.3 Spectral linewidths Example – aqueous solution of molybdate [MoO 4 ] 2- measured at the K-edge (1s excitation) and the L I edge (2s excitation). These are very similar ground states, and no significant differences in the nature of the near-edge transitions are expected. The spectra have been offset by 20008.70 eV and 2869.95 eV, respectively. The K edge is has a much shorter core-hole lifetime than the L I edge, and has corresponding broader linewidths. L I edge K edge I. J. Pickering and G. N. George GEOL 498.3/898.3 S OH O
Spectrometer Resolution Spectral linewidths In a modern EXAFS beamline this is usually only a function of the monochromator. Each monochromator material has an inherent energy resolution - the Darwin width of the crystal. This adds a Gaussian component to the overall experimental lineshape function. The experimental lineshape is expected to be approximated by a convolution of a Gaussian and a Lorenztian due to monochromator and lifetime broadening, respectively. This is known as a Voigt lineshape function – in practice it can be approximated by the sum of a Gaussian and Lorenztian – a pseudo Voigt lineshape function. I. J. Pickering and G. N. George GEOL 498.3/898.3 Near-edge Spectra We can write Fermi’s Golden Rule as: µ ∝ ∑ 2 i( k⋅r ) i ( e⋅ p) e ψ f ψ If we use a series expansion of the exponential, and examine just the first term, we get what is called the “dipole-allowed” transitions. These are the most intense transitions observed, and can be thought of as being stimulated by an oscillating electric field. ψ i - the initial state wavefunction ψ f - the final state wavefunction e - the X-<strong>ray</strong> electric vector p - the electron momentum vector k - the X-<strong>ray</strong> forward propagation vector r - the transition operator (x, y or z) in the molecular axis system I. J. Pickering and G. N. George GEOL 498.3/898.3 Dipole and Quadrupole Transitions Dipole transitions are described by: ∑ µ D ∝ ψ i ( e⋅ p) ψ f These are the most intense transitions observed, and can be thought of as being stimulated by an oscillating electric field, and have ∆l= ±1. Including the next term in the series expansion gives “quadrupole transitions”, which have ∆l= ±2, and these are described by: ∑ µ Q ∝ ψ i ( e⋅ p)( k ⋅r) ψ f Quadrupole transitions are of low intensity and can be thought of as being stimulated by the electric field gradient, which is significant due to the short wavelength of the X-radiation being used. I. J. Pickering and G. N. George GEOL 498.3/898.3 2 2
Page 1 and 2: Synchrotron X-ray Absorption Spectr
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Page 7 and 8: Dipole and Quadrupole transitions C
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Page 11 and 12: Cu K near-edge spectra of digonal a
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