Scattering wavevector, calculation

Figure 7. Top to bottom Fresnel reflectivity, penetration length, evanescent wave intensity and reflected beam plane calculations for a model system. The parameter bu = 2/uk i Q], where is the scattering wavevector, Qc is the critical angle wavevector, and p is the linear absorption coefficient. [From Elements of Modem X-ray Physics by Als-Neilson and DesMorrow, with permission from the editors at John Wiley and Sons.]...

Figure 15 Plots of the SANS intensity / vs. wavevector 0 (symbols) fora copolymer gel of /V-isopropylactylamide and sodium acrylate (95 5 mol.%) swelled in heavy water at various temperatures 7"= 32 (1) 36 (2) 40 (3) 44 (4), 46 (5) 48 (6) and 50 °C (7). Solid cun/es show the theoretical neutron scattering curves calculated using the Borue-Erukhimovich theory. ...

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

This is the generalized three-dimensional scattering relationship for the response just above the surface at x to an oscillatory pressure just above the surface at x, due to Rayleigh wave excitation, in the case where the y component of the wavevector is constant. The three-dimensional scattering function can now be calculated. 

In order to calculate V(x, z) for a spherical lens in the presence of a crack a summation must be made over ky. For each value of ky a double summation is first made over kx and k x. A wave is considered to be incident with components of wavevector k x and ky. It is then transmitted by the crack with components of wavevector kx and ky, and reflected with components —kx and ky transmitted and reflected waves may be summed in the same integration. If the axis of the lens is displaced a distance x from the crack, the resulting phase change is kxx for the incident wave and —kxx for the scattered wave. Then, by extension of eqn (12.2),... 

Fig. 2.54 Neutron reflectivity profile for a symmetric PS-dPMMA diblock (Mw 30 kg moP1) as a function of incident wavevector (Russell 1990). The inset shows the scattering length density (b/V, the neutron scattering length per unit volume) profile normal to the film surface that was used to calculate the reflectivity profile shown as the solid line, This is typical of a block copolymer film containing a multilayer stack, with lamellae parallel to the surface.

Use the data in Table 4.3 to calculate the zero wavevector hmit of the scattering function 5 (0) and the mean-square concentration fluctuation S4>) ) at a 50 A scale (at q = 2nj50 = 0.126 A ) assuming the Ornstein-... 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

Fig. 22 Steady state incoherent intermediate scattering functions (z) measured in the vorticity direction as functions of accumulated strain jf for various shear rates y data from molecular dynamics simulations of a supercooled binary Lenard-Jones mixture below the glass transition ate taken from [91]. These collapse onto a yield scaling function at long times. The wavevector is q = 3.55/R (at the peak of Sq). The quiescent curve, shifted to agree with that at the highest y, shows ageing dynamics at longer times outside the plotted window. The apparent yielding master function from simulation is compared to those calculated in ISHSM for glassy states at or close to the transition (separation parameters s as labeled) and at nearby wave vectors (as labeled). ISHSM curves were chosen to match the plateau value fq, while strain parameters yc = 0.083 at = 0 solid line) and y, = 0.116 at e = 10 dashed line) were used from [45]...

Figure 29. Time-of-flight spectra for a series of temperatures at the specular angle (45°) for NaCl(001), transformed to energy transfer distributions. The laige peak at zero energy transfer is the coherent elastic scattering (specular beam) which diminishes in intensity with temperature according to the Debye-Waller factor. The solid curves fitting the multiphonon foot of the elastic peaks are calculated from the Manson theory as described in the text. The incident helium wavevector for these experiments was 9.25 A , corresponding to an incident energy of 44 meV. (Reproduced from Fig. 1 of Ref. 96, with permission.)...

Fig. 40. (a) Temperature dependence of the longitudinal acoustic-phonon frequencies of Smo Y jsS in the [111] direction for four different values of the wavevector q (see Mook et al. 1981). (b) Temperature dependence of the bulk modulus Cg of Sm Y 25S measured by Bril-iouin scattering. Cg continues to soften upon cooling below 200 K, uniike the behavior of the phonon mode frequencies for qaO.l (flg. 40a). (c) Temperature dependence of the charge relaxation rate derived from the experimental data in figs. 40a and 40b (open circles) and calculated from theory (Schmidt and Miiller-Hartmann 1985) (solid line). The theoretical curve has been matched at 300 K to the experimental value. 

The differential intensity calculated in that way is related to a point in the diffraction pattern corresponding to wavevector q. Usually, all multipliers are excluded, although they can be taken into account when necessary (for example cos 2 9), and the scattering intensity 7(q) is expressed in relative, electron units ... 

Note that in the hypothetical incompressible limit the form of both the scattering functions and spinodal condition are much simpler than the rigorous expressions of Eqs. (6.4)-(6.6). Although the IRPA can usually be fitted to low wavevector experimental scattering data, and an apparent chi-parameter thereby extracted, the literal use of the IRPA for the calculation of thermodynamic properties and phase stability is generally expected to represent a poor approximation due to the importance of density-fluctuation-induced compressibility or equation-of-state effects [2,65,66]. The latter are non-universal, and are expected to increase in importance as the structural and/or intermolecular potential asymmetries characteristic rrf the blend molecules increase. 

Where k is the wavevector of the gamma ray and is the mean-square displacement of the Mossbauer nucleus in the direction of the gamma ray. The exponent 2W s the Debye-Waller factor, which is also relevant to X-ray and neutron scattering line intensities. Thus a measurement of the recoil-free fraction at a particular temperature /(T) yields the value of at this temperature, which can be compared with theoretical calculations of... 

Lattice vibrations can be measured experimentally by means of classical vibration spectroscopic techniques (infrared and Raman) or neutron inelastic scattering. However, only the latter technique allows one to measure the full spectrum in a range of k vectors, whereas with infrared and Raman spectroscopy, only lattice vibrations at r k = 0) are usually detected (the second-order spectra, corresponding to nonzero wavevector k 0 are demanding). The calculations of the vibrational frequencies only at F point require the solution of only one equation... 

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