Echo attenuation, calculation

The first two quantities calculated from S(t, y) depended on measurements of times the second two depend on measurements of amplitudes. From the amplitude of the reflection from the top surface of the cell the impedance, and hence the density, can be found using eqns (8.58) and (8.59). The impedance is plotted in Fig. 9.4(c). Finally, knowing the thickness and the impedance of the cell, the attenuation can be deduced from the amplitude of the echo from the interface between the cell and the substrate the weaker this echo, the greater the attenuation. The attenuation calculated from (8.60), neglecting frequency dependence, is plotted in Fig. 9.4(d). It is also possible to calculate the frequency dependence of the attenuation using (8.70) (Daft et al. 1989). 

Packer and Rees [3] extended the expression derived by Murday and Cotts [7] to include the effects of a droplet size distribution, assuming a log-normal distribution. By curve fitting they were able to determine the principal parameters of such a distribution from the experimental R-values. In the presence of a distribution of sizes, the observed echo attenuation ratio R is expressed in terms of the calculated attenuation of individual droplets, R ... 

Figure 6. Theoretical curves, showing the echo attenuation R versus the time interval 5. Parameters, used in these calculations are A=210 ms, D=1.31-10 9mV, G=2.0T/m.

For a diffusion tensor of axial symmetry with Dj > D., Fig. 1 shows the influence of a diffusion anisotropy on the PFG NMR spin-echo attenuation as simulated in numerical calculations 49). The quantitative analysis shows, however, that for principal tensor elements not too different from each other (i.e., for tensor elements within one order of magnitude), the ini-... 

Fig. 1. Parameter calculations for the PFG NMR spin-echo attenuation due to anisotropic self-diffusion in a diffusion system of axial symmetry with D > Di 49).

Calculations of the spin-echo intensity are complicated by the fact that surface relaxation may play a significant role. A general formalism for calculating PFG spin-echo attenuation for restricted diffusion in isolated pores has recently been proposed that allows for wall relaxation effects. Expressions have been obtained for the cases of diffusion within a sphere, and for planar and cylindrical geometries.These show that diffraction effects are still apparent even when surface relaxation is rapid. Also, the locations of the minima in the spin-echo intensities are not particularly affected by varying the surface relaxation parameter, Analysis of PFG spin-... 

Fig. 26.3. H-difTusion coefficient measured on TSA.28H2O by the PFG-NMR technique , (a) Echo attenuation as a function of applied magnetic field gradient showing separation of the contributions from the mother liquor and the solid, (b) The resultant diffusion coefficient as a function of the reciprocal temperature the diffusion coefficient calculated from the proton conductivity measured by a.c.-impedance spectroscopy is given for comparison (see Chapters 29-31).

Figure 44 shows typical echo attenuation curves recorded with PEO in a solid PHEMA matrix [186]. The good coincidence of the theoretical curves calculated on the basis of Eqs. 76-79 for all accessible values of the experimental parameters k and t for different molecular weights [11] corroborates the validity of the tube/reptation model for linear polymers confined to arti-... 

During the attenuation measurements. Transducer 1 was excited with a narrowband tone burst with center frequency 18 MHz, see Figure 1 for a schematic setup. The amplitude of the sound pressure was measured at Tranducer 2 by means of an amplitude peak detector. A reference amplitude, Are/, was measured outside the object as shown at the right hand side of Figure 1. The object was scanned in the j y-plane and for every position, (x, y), the attenuation, a x, y), was calculated as the quotient (in db) between the amplitude at Transducer 2, A[x, y), and Are/, i.e., a(x,y) = lOlogm Pulse echo measurements and preprocessing... 

This is the factor by which the echo magnetization is attenuated as a result of difhision. More elaborate calculations, which account for phase displacements due to difhision occurring during the application of the gradient pulses yield... 

Each echo has traveled a distance twice the cell length d further than the previous echo and so the velocity can be calculated by measuring the time delay t between successive echoes c = 2d/t. The cell length is determined accurately by calibration with a material of known ultrasonic velocity, e.g. distilled water 2d = cw.tw (where the subscripts refer to water). The attenuation coefficient is determined by measuring the amplitudes of successive echoes A = A0e-2cxd, and comparing them to the values determined for a calibration material. A number of sources of errors have to be taken into account if accurate measurements are to be made, e.g., diffraction and reflection (see below). 

Modulus is calculated (2) from the attenuation coefficient per echo by the relation. 

Figure 2 Pulsed measurement techniques, (a) Pulse-echo measurement. the time taken and energy lost for sound to travel through the emulsion, reflect at a boundary, and return to the transducer is used to calculate velocity and attenuation, respectively, (b) Through-transmission measurement, a second transducer is used to detect the soimd pulse after travelling a known distance through the emulsion.

FIGURE 60.4. Ultrasound pulse-echo pattern obtained at 10 MHz in a polystyrene disk 3 mm thick. The interval between successive reflections indicates the velocity of the longitudinal wave, and the ratio of intensity of any two successive reflections the attenuation. The horizontal scale is 2.00 xs/division. in this material the (longitudinal) speed of sound is 2.14 km/s, the acoustic impedance is 2.25 MRayls (units of 10 kg/(cm -s) and the attenuation coefficient is ca. 12 db/cm. See text below for the calculation [57]. 

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