Lorentzian broadening

In this model, the symbols have the following meaning. ( )R(a3) ao(l)HD(a3) is the optical rotation produced by the vapour and depends on the number of absorption lengths ao and on the lineshape function D(o)) which takes the form of a Doppler-broadened dispersion curve for magnetic and electric field induced rotation ( )r will depend on the strength of the field and D(oa) on the direction and type of field (see table 2). The transmitted intensity 1 - Ij exp[-aoG(o))] where the lineshape function G(o)) for a single spectral component can usually be accurately described by a Doppler-broadened Lorentzian curve. Finally the terms B and C in equation (53) represent respectively the finite extinction ratio of the polarisers and a laser independent... 

Fig. 23. Comparison between experimental (dots) and theoretical (lull curve) 3dXPS spectrum for CeRuj. To describe the spin-orbit splitting of the 3d level in the calculated spectra, the appropriate energy separation and weights have been superimposed. An inelastic background (dashed curve) was also added as well as broadening (Lorentzian) of one 1.8eV (full width at half maximum, FWHM).

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

Figure Cl.5.2. Fluorescence excitation spectra (cps = counts per second) of pentacene in /i-teriDhenyl at 1.5 K. (A) Broad scan of the inhomogeneously broadened electronic origin. The spikes are repeatable features each due to a different single molecule. The laser detuning is relative to the line centre at 592.321 nm. (B) Expansion of a 2 GHz region of this scan showing several single molecules. (C) Low-power scan of a single molecule at 592.407 nm showing the lifetime-limited width of 7.8 MHz and a Lorentzian fit. Reprinted with pennission from Moemer [198]. Copyright 1994 American Association for the Advancement of Science.

The full width at half height of these Lorentzian peaks is 2/x. One says that the individual peaks have been pressure or eollisionally broadened. 

In these Lorentzian lines, the parameter x deseribes the kinetie deeay lifetime of the moleeule. One says that the speetral lines have been lifetime or Heisenberg broadened by an amount proportional to 1/x. The latter terminology arises beeause the finite lifetime of the moleeular states ean be viewed as produeing, via the Heisenberg uneertainty relation AEAt >fe, states whose energy is "uneertain" to within an amount AE. 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

In the slow exchange limit, where 22 > A2, two Lorentzians, centered at cb + A and bb — A, respectively, are observed with width 2/Tf + 20. The exchange imposed by the molecular motion thus causes an extra broadening of the lines observed in absence of motion. 

Both lines are broadened independently and solely by adiabatic phase shift as in Lorentz and Weisskopf theories. They are Lorentzians of width (1 — cosa) and frequency shift (sin a). In general off-diagonal elements of f are not zero though they are less than diagonal elements. Consequently, the spectrum may collapse even in the adiabatic case when A 1/tc. However, adiabatic collapse is hardly ever achieved in the gas phase where l/rc > l/t0 > jS since A > 1/tc > j8 and hence only the resolved doublet limit is available. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

We consider the same atom as in Case 1, with a valence electron at an orbital energy of = 12.0 eV above the bottom of the sp band, when the atom is far from the surface. This level is narrow, like a delta function. When approaching the surface the adsorbate level broadens into a Lorentzian shape for the same reasons as described above, and falls in energy to a new position at 10.3 eV. From Eq. (73) for Wa(e) we see that the maximum occurs for e = -i- A(e), i.e. when the line described... 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

Fig. 2.7 Dependence of the experimental line width Cexp on the effective absorber thickness t for Lorentzian lines and inhomogenously broadened lines with quasi-Gaussian shape (from [9])...

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

In many cases, the actual width of a Mossbauer line has strong contributions from inhomogeneous broadening due to the distribution of unresolved hyperfine splitting in the source or absorber. Often a Gaussian distribution of Lorentzians,... 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

Model Mixed Gaussian and Lorentzian Peaks. Even if one of the distributions must be modeled by a Gaussian and the other by a Lorentzian while the instrumental broadening is already eliminated, a solution has been deduced (Ruland [124], 1965). 

---