Difference frequencies

A RIKES experunent is essentially identical to that of CW CARS, except the probe laser need not be tunable. The probe beam is linearly polarized at 0° (—>), while the polarization of the tunable pump beam is controlled by a linear polarizer and a quarter waveplate. The pump and probe beams, whose frequency difference must match the Raman frequency, are overlapped in the sample (just as in CARS). The strong pump beam propagating tlirough a nonlinear medium induces an anisotropic change in the refractive mdices seen by tlie weaker probe wave, which alters the polarization of a probe beam [96]. The signal field is polarized orthogonally to the probe laser and any altered polarization may be detected as an increase in intensity transmitted tlirough a crossed polarizer. When the pump beam is Imearly polarized at 45° y), contributions... 

The tenn slow in this case means that the exchange rate is much smaller than the frequency differences in the spectrum, so the lines in the spectrum are not significantly broadened. Flowever, the exchange rate is still comparable with the spin-lattice relaxation times in the system. Exchange, which has many mathematical similarities to dipolar relaxation, can be observed in a NOESY-type experiment (sometimes called EXSY). The rates are measured from a series of EXSY spectra, or by perfonning modified spin-lattice relaxation experiments, such as those pioneered by Floflfman and Eorsen [20]. 

A variation on the transit time method is the frequency-difference or sing-around method. In this technique, pulses are transmitted between two pairs of diagonally mounted transducers. The receipt of a pulse is used to trigger the next pulse. Alternatively this can be done using one pair of transducers where each acts alternately as transmitter and receiver. The frequency of pulses in each loop is given by... 

In practice is a small number and the sing-around frequencies are scaled up for display. In one example, for a pipe 1 m in diameter and water flowing at 2 m/s, the frequency difference is 1.4 Hz (10). Frequency difference transit time meters provide greater resolution than normal transit time ultrasonic meters. The greatest appHcation is in sizes from 100 mm to 1 m diameter. 

Bulk-wave piezoelectric quartz crystal sensors indirecdy measure mass changes of the coating on the surface of the sensing device. This change in mass causes changes in the resonant frequency of the device, and measurements ate based on frequency differences. 

It is convenient to reference the chemical shift to a standard such as tetramethylsilane [TMS, (C//j)4Si] rather than to the proton fC. Thus, a frequency difference (Hz) is measured for a proton or a carbon-13 nucleus of a sample from the H or C resonance of TMS. This value is divided by the absolute value of the Larmor frequency of the standard (e.g. 400 MHz for the protons and 100 MHz for the carbon-13 nuclei of TMS when using a 400 MHz spectrometer), which itself is proportional to the strength Bg of the magnetic field. The chemical shift is therefore given in parts per million (ppm, 5 scale, Sh for protons, 5c for carbon-13 nuclei), because a frequency difference in Hz is divided by a frequency in MHz, these values being in a proportion of 1 1O. ... 

First-order spectra (mulliplels) are observed when the eoupling constant is small compared with the frequency difference of chemical shifts between the coupling nuclei This is referred to as an A n spin system, where nucleus A has the smaller and nucleus X has the considerably larger chemical shift. An AX system (Fig. 1.4) consists of an T doublet and an X doublet with the common coupling constant J x The chemical shifts are measured from the centres of eaeh doublet to the reference resonance. 

The third common level is often invoked in simplified interpretations of the quantum mechanical theory. In this simplified interpretation, the Raman spectrum is seen as a photon absorption-photon emission process. A molecule in a lower level k absorbs a photon of incident radiation and undergoes a transition to the third common level r. The molecules in r return instantaneously to a lower level n emitting light of frequency differing from the laser frequency by —>< . This is the frequency for the Stokes process. The frequency for the anti-Stokes process would be + < . As the population of an upper level n is less than level k the intensity of the Stokes lines would be expected to be greater than the intensity of the anti-Stokes lines. This approach is inconsistent with the quantum mechanical treatment in which the third common level is introduced as a mathematical expedient and is not involved directly in the scattering process (9). 

Thus, it is not strictly correct to interpret the frequency difference between adsorbed and gas phase C-O in terms of chemisorption bond strength only. 

In our tip-enhanced near-field CARS microscopy, two mode-locked pulsed lasers (pulse duration 5ps, spectral width 4cm ) were used for excitation of CARS polarization [21]. The sample was a DNA network nanostructure of poly(dA-dT)-poly(dA-dT) [24]. The frequency difference of the two excitation lasers (cOi — CO2) was set at 1337 cm, corresponding to the ring stretching mode of diazole. After the on-resonant imaging, CO2 was changed such that the frequency difference corresponded to none of the Raman-active vibration of the sample ( off-resonant ). The CARS images at the on- and off- resonant frequencies are illustrated in Figure 2.8a and b, respectively. 

Let us now derive the equations that relate the spatial information to the signal behavior. As we have seen previously, a spin at position r possesses a Larmor frequency co(r) = y B(r) = v( Bo + g r). It is convenient to subtract the reference value, given by the average field, oi0 = v B0. so that we obtain the frequency difference relative to an (arbitrarily chosen) position r= 0 ... 

Another possibility is that the vibrational frequency difference increases the cross-gliding rate, and therefore the deformation-hardening rate. In this case, when the temperature becomes high enough, dislocation climb causes rapid enough recovery to cancel the deformation-hardening rate. 

The equations describing linear, adiabatic stellar oscillations are known to be Hermitian (Chandrasekhar 1964). This property of the equations is used to relate the differences between the structure of the Sun and a known reference solar model to the differences in the frequencies of the Sun and the model by known kernels. Thus by determining the differences between solar models and the Sun by inverting the frequency differences between the models and the Sun we can determine whether or not mixing took place in the Sun. 

The terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

Taking into consideration relations (3.2.4), expressions (3.2.6) for absorption lines allowed in infrared spectra (A/ 0) are the same as those obtained formerly by an alternative method which involves no sublattice concept.81 The determination of Davydov splittings as squared frequency differences (3.2.6) results in their independence from the static frequency renormalization. For structures with [Pg.62]

Similar changes in the CO stretch frequency have been observed experimentally. As mentioned early in this chapter the IR spectrum of carbon monoxymyoglobin contains three main CO stretching bands (A0 1965 cnT1, A1 x 1949 cnT1, and A3 x 1933 cm-1). That we obtain different CO frequency shifts for different His64 conformations supports the interpretation that the distal His determines the A states [9b]. Our results can, moreover, aid interpretation of the peaks in the spectrum. First, it should be noted that our zero frequency (i.e. the conformation where the CO is not influenced by the protein environment) corresponds to the A0 state. Thus, our computed frequency shifts are related to the frequency differences between each peak of the IR spectrum and the A0... 

Finally, when the exchange rate is much faster than the reciprocal frequency difference the line is sharpened and no kinetic information can be obtained. The position of the line in the spectrum is determined by the values of vG and vHG weighted by the fractions of the free and bound guest. 

The vibrational spectrum of methylguanine-methylcytosine (GC) complex consists of 99 normal modes frequencies. Differently from the AT base pair, in the GC complex the normal modes of the two bases are coupled together, thus an analysis of the shift relatively to the isolated bases is extremely complicated. This stronger coupling can possibly he ascribed to the presence of three h-bonds, rather than two as in AT. However, we tentatively discuss some significant shifts. 

Fig. 4. The HNCO-TROSY experiment for recording solely interresidual 1HN, 15N, 13C correlations in 13C/15N/2H labelled proteins. All 90° (180°) pulses for the 13C and 13C spins are applied with a strength of 2/ /l5 (p/ /3), where 2 is the frequency difference between the centres of the 13C and 13Ca regions. All 13Ca pulses are applied off-resonance with phase modulation by Q. A = 1/(4/hn) Tn = l/(4/NC ) S = gradient + field recovery delay 0 < k < TN/z2,max- Phase cycling i = y 4>2 = x, — x + States-TPPI 03 = x 0rec = x, — x.

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