Interference between vibrational modes

FIGURE 2.2. Energy diagram showing interference between vibrational modes. Interference arises because VSFS is a coherent process. The multiple paths are analogous to those in the double slit experiment. 

Having considered only the single mode case so far, we can also derive an expression of x"(copr, t) for a multimode system in a similar fashion. In the twomode case, for instance, %"(apr, x) can be divided into three terms, each of which corresponds to interference between the vibrational processes of the two modes. It should be noted here that within the same approximations as used above, the density matrix of the two modes during the time interval x can be expressed as a product of each mode s matrix. 

The peak shift data in Fig. 17 show oscillatory character, as is our first two examples (I2 and LH1). This arises from vibrational wavepacket motion. In addition, the very fast drop in peak shift to about 65% of the initial value in -20 fs results from the interference between the wavepackets created in different intramoleculear modes. This conclusion follows directly from obtaining the frequencies and relative coupling strengths of the intramolecular modes from transient grating studies of IR144, carried out in the same solvents (data not shown). Thus, by visual inspection of Fig. 17, an answer to a long-standing question—What fraction of the spectral width arises from intra- and intermolecular motion —is immediately apparent. 

Recent experimental studies on interference effects in solution, and on collisional vibrational energy transfer between molecules in solution, provide some insight into the molecular time scales of these relaxation events. For example [171], the time scale for transfer of population to die vibrational modes in liquid CH3OH is on thd order of 5 to 15ps [172], Further, studies of the preparation of coherent superpositions of states in solution show that phase coherences of molecules exist in solution for time scales greater than 100 fs [173, 174], -- i... 

Since the non-linear susceptibility is generally complex, each resonant term in the summation is associated with a relative phase, y , which describes the interference between overlapping vibrational modes. The resonant macroscopic susceptibility associated with a particular vibrational mode v, Xr, is related to the microscopic susceptibility also called the molecular hyperpolarizability, fiy, in the following way... 

The interference between different vibrations (including those of different molecules) resulting from the coherent nature of the experiment makes the analysis of VSFS spectra considerably more complicated than that of spectra recorded with linear spectroscopic techniques. However, this complexity can be exploited to provide orientational information if a complete analysis of the VSF spectrum is employed taking into account the phase relationships of the contributing vibrational modes to the sum-frequency response [15,16]. In the analysis it is possible to constrain the average orientation of the molecules at the surface by relating the macroscopic second-order susceptibility, Xs g of the system to the molecular hyperpolaiizabilities, of the individual... 

It was quantitatively interpreted (Rice et al., 1977) as originating from bond alternation phase oscillations (in contrast to the bond alternation amplitude oscillations mentioned in subsection 4.8.2D). The vibrational absorption lines labeled 2 to 10 are directly related to the Ag Raman lines of TCNQ. The broad peak above 1600 cm originates from the single electron transition across the gap, and the indented line shape of mode 2 is a consequence of Fano interference between the single electron continuum and the phonon mode. The line intensities are determined by the respective electron - vibration coupling constants. 

The second argument is based on the fact that interference occurs only between indistinguishable states. All molecules in the beam are in different states. There are 174 different vibrational modes and also the rotational modes are populated at various energies with a Boltzmann distribution. The chance of having two subsequent molecules in exactly the same state of all internal modes is vanishingly small. Therefore, interference in our experiments really is a single particle phenomenon. 

It was common opinion that water is a major hurdle for THz technology [17]. Until recently, almost all successful experiments set up have been conducted using either dry or low water content samples. Several tentative attempts, however, have been made to examine DNA vibrational modes in liquid over the spectral region 10-25 cmT. The results demonstrate that there is little interference between the spectral features of the probed sample and the water background, except for the band located at 18.6 cm where the water absorption is clearly predominant [18]. The extreme sensitivity of THz-rays to water can be turned into an advantage in distinguishing between healthy and cancerous tissue [4], because cancerous tissue tends to have higher water content than healthy tissue. 

S(co) is a frequency-dependent phase for the nonresonant background, mainly from substrate. Here, Sy is the relative phase angle for each vibrational mode. Normally, one assumes that these resonance modes are in phase (Sy = 0) or out of phase Sy = it), which can give rise to constructive or destructive interference between them. Furthermore, the phase difference. By — S, between the resonance mode and nonresonant background will significantly affect the shape of the SFG spectra when the nonresonant background cannot be ignored. The Lorentzian-type function is usually employed to analyze the SFG spectra. 

The spatial resolution of the CI SEM mode depends mainly on the electron-probe size, the size of the excitation volume, which is related to the electron-beam penetration range in the material (see the articles on SEM and EPMA), and the minority carrier diffusion. The spatial resolution also may be afiFected by the signal-to-noise ratio, mechanical vibrations, and electromagnetic interference. In practice, the spatial resolution is determined basically by the size of the excitation volume, and will be between about 0.1 and 1 pm ... 

---