Resonances in the Time Domain

It is clear that a core-hole represents a very interesting example of an unstable state in the continuum. It is, however, also rather complicated [150]. A simpler system with similar characteristics is a doubly excited state in few-body systems, as helium. Here, it is possible [151-153] to simulate the whole sequence of events that take place when the interaction with a short light pulse first creates a wave packet in the continuum, including doubly excited states, and the metastable components subsequently decay on a timescale that is comparable to the characteristic time evolution of the electronic wave packet itself. On the experimental side, techniques for such studies are emerging. Mauritsson et al. [154] studied recently the time evolution of a bound wave packet in He, created by an ultra-short (350 as) pulse and monitored by an IR probe pulse, and Gilbertson et al. [155] demonstrated that they could monitor and control helium autoionization. Below, we describe how a simulation of a possible pump-probe experiment, targeting resonance states in helium, can be made. 

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Most methods assume an exponential decay for the resonances in the time domain giving rise to Lorentzian lineshapes in the frequency domain. This assumption is only valid for ideal experimental conditions. Under real experimental circumstances multi-exponential relaxation, imperfect shimming, susceptibility variations and residual eddy current usually lead to non-ideal... 

While in the frequency domain all the spectroscopic information regarding vibrational frequencies and relaxation processes is obtained from the positions and widths of the Raman resonances, in the time domain this information is obtained from coherent oscillations and the decay of the time-dependent CARS signal, respectively. In principle, time- and frequency-domain experiments are related to each other by Fourier transform and carry the same information. However, in contrast to the driven motion of molecular vibrations in frequency-multiplexed CARS detection, time-resolved CARS allows recording the Raman free induction decay (RFID) with the decay time T2, i.e., the free evolution of the molecular system is observed. While the non-resonant contribution dephases instantaneously, the resonant contribution of RFID decays within hundreds of femtoseconds in the condensed phase. Time-resolved CARS with femtosecond excitation, therefore, allows the separation of nonresonant and vibrationally resonant signals [151]. 

The relationship between how long we observe the signal from a particular resonance in the time domain and how narrow the resonance... 

Electron magnetic resonance in the time domain has been greatly facilitated by the introduction of novel resonance structures and better computational tools, such as the increasingly widespread use of density-matrix formalism. This second volume in our series, devoted both to instrumentation and computation, addresses applications and advances in the analysis of spin relaxation time measiuements. 

Confirmation analysis In most cases, the occurrence of dynamic resonance can be quickly confirmed. When monitoring phase and amplitude, resonance is indicated by a 180° phase shift as the rotor passes through the resonant zone. Figure 44.44 illustrates a dynamic resonance at 500 rpm, which shows a dramatic amplitude increase in the frequency-domain display. This is confirmed by the 180° phase shift in the time-domain plot. Note that the peak at 1200 rpm is not resonance. The absence of a phase shift, coupled with the apparent modulations in the FFT, discount the possibility that this peak is resonance-related. 

Nuclear Resonance Scattering Using Synchrotron Radiation (Mossbauer Spectroscopy in the Time Domain)... 

Equation (9.1) documents that quadmpole splittings A q exhibit quantum-beat spectra with period H/IuAEq superimposed over the time dependence of the nuclear decay exp(—f/t) with mean decay time t = 141 ns for Fe. In Fig. 9.2, quadmpole splittings A q = 0 and 2 mm s in the energy domain (conventional MS) are compared with those in the time domain (MS using synchrotron radiation) [7]. The QBs in the time domain spectmm for A q = 2 mm s are the result of the interference between the radiation scattered by different nuclear resonances. Consequently, their frequencies correspond to the energetic differences between these resonances. 

We can design a controller by specifying an upper limit on the value of Mpco. The smaller the system damping ratio C the larger the value of Mpco is, and the more overshoot, or underdamping we expect in the time domain response. Needless to say, excessive resonance is undesirable. 

A major limitation of CW double resonance methods is the sensitivity of the intensities of the transitions to the relative rates of spin relaxation processes. For that reason the peak intensities often convey little quantitative information about the numbers of spins involved and, in extreme cases, may be undetectable. This limitation can be especially severe for liquid samples where several relaxation pathways may have about the same rates. The situation is somewhat better in solids, especially at low temperatures, where some pathways are effectively frozen out. Fortunately, fewer limitations occur when pulsed radio and microwave fields are employed. In that case one can better adapt the excitation and detection timing to the rates of relaxation that are intrinsic to the sample.50 There are now several versions of pulsed ENDOR and other double resonance methods. Some of these methods also make it possible to separate in the time domain overlapping transitions that have different relaxation behavior, thereby improving the resolution of the spectrum. 

Every NMR experiment must have a preparation sequence (inducing the nuclei to resonate) and detection capability (finding out what happened). Two-dimensional NMR spectroscopy adds two more domains between preparation and detection. These are an indirect evolution time, q, and a mixing sequence (see Figure 3.15). The two dimensions of two-dimensional NMR spectroscopy are those of time. In one time domain, FIDs containing frequency and intensity information about the observed nuclei is collected. The second time dimension refers to the time that elapses between some perturbation of the system and the onset of data collection in the time domain. The second time period is varied, and a series of FID responses are collected for each of the variations. 

Figure 8.2.10 (A) Pulse sequence used for selective excitation of each of the four samples in turn. The RF pulses are frequency-selective and applied at different resonant offsets via phase modulation in the time domain. (B) Normal 1H spectrum of (a) the four samples, and the resulting sub-spectra of 0.5 M (b) 1-propanol, (c) 2-propanol, (d) acetic acid and (e) ethanol in D2O. Since the four spectra are acquired within the relaxation time 77 eff, which is typically set to be3Ti, there is an increase in efficiency by a factor of four. Reprinted with permission from Hou, T., Smith, J., MacNamara, E., Macnaughton, M. and Raftery, D., Anal. Chem., 73, 2541-2546 (2001). Copyright (2001) American Chemical Society...

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

Bayesian probability theory157 can also be applied to the problem of NMR parameter estimation this approach incorporates prior knowledge of the NMR parameters and is particularly useful at short aquisition times158 and when the FID contains few data points.159 Bayesian analysis gives more precise estimates of the NMR parameters than do methods based on the discrete Fourier transform (DFT).160 The amplitudes can be estimated independently of the phase, frequency and decay constants of the resonances.161 For the usual method of quadrature detection, it is appropriate to apply this technique to the two quadrature signals in the time domain.162-164... 

All of the analysis in this chapter will be carried out from the time-dependent theoretical point of view. This method of analysis gives results which are identical with those of the Franck-Condon analysis. However, thinking in the time domain offers a new point of view from which the unity between electronic and resonance Raman spectroscopy and the interpretation of the spectroscopic effects of multiple mode distortions can more readily be seen. Thus, before we discuss specific molecules, we will introduce the important aspects of time-dependent theory. 

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