Absorption lineshape broadening

IvX(co) represents an absorption lineshape broadened by Jg(x), whereas 7V/1(oj) represents an emission lineshape broadened by J x). 

The Time Dependent Processes Seetion uses time-dependent perturbation theory, eombined with the elassieal eleetrie and magnetie fields that arise due to the interaetion of photons with the nuelei and eleetrons of a moleeule, to derive expressions for the rates of transitions among atomie or moleeular eleetronie, vibrational, and rotational states indueed by photon absorption or emission. Sourees of line broadening and time eorrelation funetion treatments of absorption lineshapes are briefly introdueed. Finally, transitions indueed by eollisions rather than by eleetromagnetie fields are briefly treated to provide an introduetion to the subjeet of theoretieal ehemieal dynamies. 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

Equations (12.60) and (12,61) are expressions for the low temperature (i.e. ksT < hota) electronic absorption lineshape. The frequency dependence originates from the individual transition peaks, that in reality are broadened by intramolecular and intermolecular interactions and may overlap, and from the Franck-Condon envelope... 

Lifetime broadening is ubiquitous, because all excited states decay. The observed absorption lineshape is seldom Lorentzian, however, since other mechanisms than lifetime broadening usually dominate the actual lineshape. 

Fig. S a Valence band spectra of Gd C82 (grey) and C82 (black) measured with Al Ka x-rays, b Symbols Gd 4f photoemission after subtraction of the empty C82 C 2s/2p spectrum. The vertical lines are individual components of atomic calculations for a 4f> multiplet, and the solid curve is their broadened sum. c Gd-N4>5 x-ray absorption spectrum (Gd 4d-4f excitations) of Gd C82. The complex lineshape comes from the widely spaced multiplet components resulting from the strong Coulomb interaction between the single hole in the 4d shell and the eight electrons present in the 4f shell in the x-ray absorption final state [see Fig. lc]. The arrows represent the two photon energies used for the data shown in panel d. d Resonant photoemission data of the valence band region of Gd C82 recorded off (hv=137 eV) and on (hv=149 eV) the Gd 4d-4f giant resonance...

For a Voigt function that is almost Lorentzian, the extent of Gaussian broadening can be visualized by plotting the dispersion of the lineshape, D f) against the absorption, A(f).76,77 For a pure Lorentzian lineshape, a circle is obtained. Hence, the extent of the departure from this circular shape indicates the extent of the Gaussian broadening. 

A high-resolution spectrum of the clock transition is shown in Fig. 2. The clock-laser power was reduced to 30 nW to avoid saturation broadening. The fit with a lorentzian curve results in a linewidth of 170 Hz (FWHM), corresponding to a fractional resolution bv/v of 1.3 10-13. A spectral window of 200 Hz width contains 50% of all excitations. According to our present experimental control of the ion temperature, electromagnetic fields and vacuum conditions, no significant Doppler, Zeeman, Stark or collisional broadening of the absorption spectrum of the ion is expected beyond the level of 1 Hz. The linewidth is determined by the frequency instability of the laser and the lineshape is not exactly lorentzian... 

Therefore, the absorption line is massively inhomogeneously broadened at low temperature. An inhomogeneous lineshape can be used to determine the static or quasistatic frequency spread of oscillators due to a distribution of environments, but it provides no dynamical information whatsoever [94, 95]. As T is increased to 300 K, the absorption linewidth decreases and increases. At 300 K, the lineshape is nearly homogeneously broadened and dominated by vibrational dephasing, because fast dephasing wipes out effects of inhomogeneous environments, a well known phenomenon termed motional narrowing [95]. 

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