Absorption spectral lineshape

Fig. 10.10 Absorption spectral lineshapes calculated as the Fourier transform of the Kubo relaxation function <)> (). (A) (indicated in arbitrary time units) is varied, while a is fixed at 1 reciprocal time unit (B) is fixed at 1 time unit and a is varied as indicated. To use the full Fourier transform (Eq. 10.70), < (r) is treated as an even function of time (Fig. 10.1 lA)...

For all of the cases considered earlier, a C(t) function is subjected to Fourier transformation to obtain a spectral lineshape function 1(G)), which then provides the essential ingredient for computing the net rate of photon absorption. In this Fourier transform process, the variable 0) is assumed to be the frequency of the electromagnetic field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to 1(G)) is ... 

These two delta functions correspond to absorption and emission of radiation at frequency c o, respectively, with spectral lineshapes exhibiting zero full width at half maximum (fwhm). Such uninterrupted molecular rotation, in which the dipole correlation function (8.17) maintains perfect sinusoidal coherence for an indefinite period of time, produces no broadening in the lineshape function I co). 

As a final detail, we allow for broadening (e.g., Doppler broadening, which determines the lasing lineshape in He/Ne lasers) of the two-level transition. Assume that the spectral lineshape function is gf(v), normalized so that J g v)dv = 1. Then the negative absorption coefficient in Eq. 9.7 at the lineshape center frequency Vq should be multiplied by (vo), which will be inversely proportional to the lineshape fwhm Av since (v) is normalized the product givojAv will be about unity. This implies that the lasing criterion in Eq. 9.12 should be replaced by [1]... 

If the off-diagonal matrix elements that describe the coherence between a ground state and an excited electronic state decay exponentially with time, the homogeneous absorption line should have a Lorentzian shape (Figs. 10.6 and 10.7 Eqs. (2.70) and (10.35)). More generally, as we discussed in Sect. 10.6, the spectral lineshape is the Fourier transform of the relaxation function ... 

We have seen in Section 7.5.3 that the power spectrum of a given system is closely associated with the absorption lineshape in that system. The analysis presented above indicates that the spectral lineshape of a stochastically modulated oscillator assumes qualitatively different forms depending on the amplitude and timescale of the modulation. We will return to these issues in Chapter 18. 

It is quite important for the spectra measured to represent the pure absorption mode with neither saturation nor transient effects. They should be measured in a homogenous external magnetic field B0 in order to obtain purely Lorentzian lineshapes for individual spectral transitions. The signal-to-noise ratio should be as high as possible and experimental... 

Figure 3.13 Spectral patterns for a paramagnetic ion in a polycrystalline specimen with (a) an axially symmetric and (b) a rhombic g-tensor, shown as absorption (top) and first derivative (bottom) lineshapes. Solid curves indicate idealised line shapes, whereas the dashed curves represent typical experimental patterns.

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... 

We would also like to mention the informational approach, which determines the most probable absorption lineshapes given a limited amount of information, that is, a finite number of spectral moments. The method is based on minimizing the informational entropy. ... 

Aniotropb fluctuatbn Iheshapes. Rancourt and Daniels (1984) showed that the i+ and T values that arise from typical interaction fields, with typical values of fo, give dynamic lineshapes of sextets having broad and asymmetric absorption lines similar to the observed ones in many nanoparticle systems. This suggests that the often extracted HFDs are artefacts of an incorrect spectral analysis. One difficulty that inhibited the... 

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

A 2D NMR experiment can lead to a data set that is either phase modulated or amplitude modulated as a function of fj, depending on the particular experiment and coherence pathways selected. A regular ID spectrum consists of absorption A(p) and dispersion peaks corresponding to the real and imaginary parts of the spectral lines, respectively. In 2D experiments, phase modulation in fj results in twisted 2D real lineshapes as a result of the Fourier transformation of bi-exponential time domain... 

A very important result of the theory of quantum dynamics is the connection between the time evolution in a given spectral region and the absorption lineshape into the same region. That such a cormection exists is to be expected, because the time evolution is detennined by the distribution of initial amplitudes among exact eigenstates according to Eq. (2.6), while the absorption process, in principle, prepares these initial amplitudes in the spectral region of interest. 

To see this connection in more detail we extend the model of Figs 9.1 and Eq. (9.2) to include two discrete states, the ground state g) and an excited state lx), and a continuum of states I/) that may represent the ground state dressed by environmental or radiation field states. We assume that l ) is the only excited state in the relevant spectral region that is radiatively coupled to the ground state g) so it can be initially prepared as explained in Section 9.2.4. In the subspace that encompasses the state [s ) and the continuum Z), the former plays the same role as state 11) in Fig. 9.1. We now focus on the excitation from g to s specifically we pose the question What is the corresponding absorption lineshape ... 

In Section 9.3 we have seen that there is in principle a close relationship between an absorption lineshape and the underlying dynamics of a molecule excited to the corresponding spectral region. The discussion in the previous section however has taught us that life is less simple In many systems the absorption lineshape is an average over many individual molecules that experience different local environments,... 

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... 

Laser-induced fluorescence also offers the possibility of looking at the solvation structure in its isolated form, that is, short-range order without the continuum contribution. This has attracted keen interest in supersonic molecular beam studies of van der Waals complexes (27-28). The successive additions of n solvating molecules B form a solvated species AB. Spectral shifts and lineshape changes in the absorption of A as it becomes AB, or in the laser-induced fluorescence decay of those consecutively... 

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