Shape Fourier

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

The integral describes the spatial amplitude modulation of the excited magnetization. It represents the excitation or slice profile, g(z), of the pulse in real space. As drops to zero for t outside the pulse, the integration limits can be extended to infinity whereupon it is seen that the excitation profile is the Fourier transfonn of the pulse shape envelope ... 

In electron-spin-echo-detected EPR spectroscopy, spectral infomiation may, in principle, be obtained from a Fourier transfomiation of the second half of the echo shape, since it represents the FID of the refocused magnetizations, however, now recorded with much reduced deadtime problems. For the inhomogeneously broadened EPR lines considered here, however, the FID and therefore also the spin echo, show little structure. For this reason, the amplitude of tire echo is used as the main source of infomiation in ESE experiments. Recording the intensity of the two-pulse or tliree-pulse echo amplitude as a function of the external magnetic field defines electron-spm-echo- (ESE-)... 

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

This result, when substituted into the expressions for C(t), yields expressions identieal to those given for the three eases treated above with one modifieation. The translational motion average need no longer be eonsidered in eaeh C(t) instead, the earlier expressions for C(t) must eaeh be multiplied by a faetor exp(- co2t2kT/(2me2)) that embodies the translationally averaged Doppler shift. The speetral line shape funetion I(co) ean then be obtained for eaeh C(t) by simply Fourier transforming ... 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

Fig. la-c. Theoretical 2H NMR line shapes for axially symmetric FGT (r = 0) in rigid solids, cf. Equ. (1). a Line shapes for the two NMR transitions b 2H spectrum (Pake diagram) in absorption mode as obtained by Fourier transform methods c 2H spectrum in derivative mode as obtained by wide line methods... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Figure 3.6 The first set of Fourier transformations across <2 yields signals in V2, with absorption and dispersion compronents corresponding to real and imaginary parts. The second FT across /, yields signals in V, with absorption (i.e., real) and dispersion (i.e., imaginary) components quadrants (a), (b), (c), and (d) represent four different combinations of real and imaginary components and four different line shapes. These line shaptes normally are visible in phase-sensitive 2D plots.

Another resolution-enhancement procedure used is convolution difference (Campbell et ai, 1973). This suppresses the ridges from the cross-peaks and weakens the peaks on the diagonal. Alternatively, we can use a shaping function that involves production of pseudoechoes. This makes the envelope of the time-domain signal symmetrical about its midpoint, so the dispersionmode contributions in both halves are equal and opposite in sign (Bax et ai, 1979,1981). Fourier transformation of the pseudoecho produces signals... 

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

So far no hypotheses are required concerning the true shape of the peak profile. Flowever, in order to avoid or reduce the difficulties related to the overlapping of the peaks, the experimental noise, the resolution of the data and the separation peak-background, the approach most frequently used fits by means of a least squared method the diffraction peaks using some suitable functions that allow the analytical Fourier transform, as, for example, Voigt or pseudo-Voigt functions (4) which are the more often used. 

Many other filter functions can be designed, e.g. an exponential or a trapezoidal function, or a band pass filter. As a rule exponential and trapezoidal filters perform better than cut-off filters, because an abrupt truncation of the Fourier coefficients may introduce artifacts, such as the annoying appearance of periodicities on the signal. The problem of choosing filter shapes is discussed in more detail by Lam and Isenhour [11] with references to a more thorough mathematical treatment of the subject. The expression for a band-pass filter is H v) = 1 for v j < v < else... 

Particle shapes influence properties such as surface area, bulk density, flow, and so on. A number of methods are available for describing shape from simpler qualitative descriptions, through property ratios, to techniques that employ fast Fourier transformations to describe the projected perimeter of the particle. The measurement of the shape and the relevance of the data obtained are generally the two difficulties associated with particle shape. Fortunately, in the processing of materials physically unlike those in chemical processing, shape is perhaps is less significant and is more often than not inherently accounted for in the nominal diameter. 

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