Position-dependent phase shift

This is the position-dependent phase shift expressed in terms of the x and y components of the magnetization. Here, we assume that the I spin chemical shift is on-resonance, so that ordinary chemical-shift evolution can be ignored during the gradient. Now we can look at the effect of a gradient on the spherical operators by expressing them in terms of the Cartesian operators ... 

For several gradients applied at different points during the pulse sequence, the position-dependent phase shift accumulates ... 

This time dependent phase shift leads to a frequency modulation that is proportional to the time derivative of the self induced phase shift fused silica with its positive Kerr coefficient rio = 2.5 x 10 16 cm2/W [28] the leading edges of the pulses are creating extra frequencies shifted to the red ( jvi(t) < 0) while the trailing edges causes blue shifted frequencies to emerge. Self-phase modulation modifies the envelope function according to... 

Equation 14 consists of an unmodulated part with amplitude 1 - U2, the basic frequencies and cop with amplitudes kJl, and the combination frequencies < and w+ with amplitudes k 4, and inverted phase. To compute the frequency-domain spectram, first the unmodulated part is subtracted, as it gives a dominant peak at zero frequency for the usual case of small k values. A cosine Fourier transform (FT) of the time trace results in a spectrum that contains the two nuclear frequencies, w and cop, with positive intensity, and their sum and difference frequencies, a>+ and m, with negative intensity. If the initial part of the time-domain trace is missing, then the spectrum can be severely distorted by frequency-dependent phase shifts and it may be best to FT the time-domain trace and compute the magnitude spectrum. 

In Raman spectroscopy the intensity of scattered radiation depends not only on the polarizability and concentration of the analyte molecules, but also on the optical properties of the sample and the adjustment of the instrument. Absolute Raman intensities are not, therefore, inherently a very accurate measure of concentration. These intensities are, of course, useful for quantification under well-defined experimental conditions and for well characterized samples otherwise relative intensities should be used instead. Raman bands of the major component, the solvent, or another component of known concentration can be used as internal standards. For isotropic phases, intensity ratios of Raman bands of the analyte and the reference compound depend linearly on the concentration ratio over a wide concentration range and are, therefore, very well-suited for quantification. Changes of temperature and the refractive index of the sample can, however, influence Raman intensities, and the band positions can be shifted by different solvation at higher concentrations or... 

Fig. 1.3 Effect of a pulsed magnetic field zero. During the gradient pulse, the field gradient of strength g on the phase of a signal becomes position-dependent and a phase shift contribution originating from spins at position is accumulated that is proportional to t. Prior to the gradient pulse, all spins position t and time t. After the gradient pulse...

That is, the phase shift depends on the initial position x0, the initial velocity vx0 and the initial acceleration ax0. Higher order terms vanish if the flow field is stationary on the time scale of the NMR experiment (i.e., time-dependent accelerations do not occur in this case). For a gradient pulse of duration t and strength Gx the total phase shift is [see Figure 2.9.4(a)]... 

For many systems and flow fields, the position dependent term dominates the phase shift while the velocity dependent term is still significantly larger than the acceleration dependent term. As a direct consequence, terms with a higher order of t than the term including the parameter to be evaluated may often be neglected,... 

For phase encoding of a velocity component, a gradient pulse sequence similar to the sequence in Figure 2.9.4(b) can be used. Two gradient pulses of duration t and of the same magnitude of the amplitude Gx but with opposite sign are applied. The position dependent term of the phase shift vanishes whereas the velocity dependent term provides a finite phase shift proportional to the velocity component [see Figure 2.9.4(b)] ... 

Fig. 32 Dependence of cosine of A and tangent of where A and are ellipsometric angles related to the change of amplitude and phase shift of the incident polarized light, on the wavelength of the incident polarized light collected from PDMAEMA- -PAA brushes immersed in solutions of a constant ionic strength 0.001 M) and pH ranging from 3.52 to 9.50. The data in figures a and b (c and d) have been collected at the position 1 (4). For clarity the data for cos(A) and tan(V ) collected at pH > 3.53 were shifted vertically by - 0.2 relative to each previous set...

If the specimen is moved away from the focal position, then this will cause a phase shift that depends on 6. If the wavenumber in the coupling fluid is k = 2n/Xo, then the z component of the wavevector is kz = k cos 6. Defocusing the specimen by an amount z causes a phase delay of 2zkz, or 2zk cos 0 (the factor of two arises because both the incident wave and the reflected wave suffer a change in path length). Expressing this phase delay as the complex exponential of a phase angle, the response of the microscope with a defocus z is... 

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