Critical scattering

Here we consider the concentration fluctuations in the homogeneous phase and also the manner in which these are reflected in measured scattering functions. 

0A and denoting the free energy densities of the one component phases. Being in close contact, neighboring blocks, interact with each other across the interfaces and we have to inquire about the related interfacial energy. We know that it must vanish for equal concentrations and increase with the concentration difference, independent of the direction of change. The simplest expression with such properties is the quadratic term 

The equation can be further simplified if a hnearization approximation is used. Clearly the state with a uniform concentration, 

This is a useful result. It relates the Gibbs free energy of a given fluctuation state to two parameters only, namely the integral or mean values of and 

We now turn to scattering experiments. They may generally be regarded as carrying out a Fourier analysis, in our case a Fourier analysis of the concentration fluctuations in the mixture. We therefore represent 64 r) as a sum of wave-like modulations with amplitudes 4 k 

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The effect of critical scattering on the resistance was evaluated assuming that the holes reside in a simple parabolic band (Omiya et al. 2000). The corresponding contribution to the resistivity has to take into account the presence of correlation between neighboring spins, (Si Sj) (Sf)Sij (Dietl 1994),... 

In both these special cases, A6/A2 is independent of disorder.[Haas et al., 1997] A variation of A6/A2 can, therefore, only be expected for a2 / a6 and Ag6 A22A66- However, since A6/A2 depends only on j/2 — ir6 the variation remains modest even for Tc —> 0. The inset of Fig. 1 gives an example of the variation of A6/A2 with F and So. The main panel shows the transition temperature normalized to its clean limit value as function of the scattering rate, normalized to the critical scattering rate for pure. v-wavc scattering Tcrit = 0.882 7co, for three values of the scattering phase shift. In the case of... 

M. F. Collins, Magnetic Critical Scattering , Oxford University Press, 1989, p. 16. 

Fig. 29. The crystal structure of K2CUF4 and the section of the reciprocal lattice. The critical scattering ridge appears along the broken lines.

This conversion procedure must be attempted for the longitudinal as well as for the transverse transport. The results for p (7) in (TMTSFljPF at ambient pressure is given in Fig. 14 where a cross-over from a superlinear to a linear (or sublinear) power law temperature dependence is observed in the vicinity of 80 K. A detailed analysis based on the Fermi liquid theory shows that a law is apparently well satisfied below 50 K down to the vicinity of the spin density-wave transition where the resistivity is dominated by critical scattering effects [75]. 

Shirane G, Axe JD (1971) Acoustic-phonon instability and critical scattering in NbsSn. Phys Rev Lett 27 1803-1806... 

It is easy to see that the above treatment of the critical scattering and correlations in lerms of the wavevector dependent susceptibility goes through as previously, i.e. we still have y, = 1, i>t = 1/2, = 0 as in the standard Landau theory. But the behavior of the specific heat changes, since [cf. eq. (46)]... 

Note that the density of states exponents ore independent of direction and considerably weaker than the b-axis critical scattering exponents and hence they should not contribute to the experimentally observed b-axis exponents. 

Critical Scattering Exponents Density of Stotes Exponents Collective Mode Pinning Percolation... 

S ii(q = 0) describes the critical scattering along the coexistence curve, while Scoii(q = 0) describes the critical scattering for T > Tc and A i = 0 (i.e., at critical composition < > = crit). This fact that for the description of the response function, different expressions need to be used above and below the phase transition, in order to take into account the spontaneous symmetry breaking that appears in the thermodynamic limit between the coexisting phases, is well known [105] but has not been considered in some of the work on polymer mixtures [101-103, 107]. 

The analysis of the polarised neutron experiment on K2NaCrFe (Fig. 9) has been completed< >. The aspherical form factor component/4(0 was shown to be particularly sensitive to the covalent spin distribution, and the shape of fi Q) supported the inference for eg spin polarization drawn from the comparison of powder neutron diffraction data and resonance data (Section VI. B). A preliminary report of the form factor determination for both tetra-hedrally and octahedrally coordinated Fe + in YaFesOis by polarized neutrons has been given ). The form factors are not the same and that for tetrahedral Fe + is contracted relative to that for octahedral Fe +, which follows closely the calculated free-ion curve. The ligand forward peak (Section VI. B) has been directly observed ) in a measurement of the critical scattering of neutrons by ferromagnetic K2CUF4. 

Experimental measurements of the spinodal curve require bridging of the metastable region of the miscibility gap, i.e. traversing this region without phase separation. A nice technique developed for this purpose is the Pulse Induced Critical Scattering (PICS) [64]. 

KI2 Kiepen, F. and Borehard, W., Pressure-pulse-induced critical scattering of ohgostyrene in n-... 

KI4 Kiepen, F. and Borehard, W., Determination of binodal and spinodal points by means of the pressure pulse-induced critical scattering method, Integr. Fundam. Polym. Sci. Technol., Elsevier, London, 2, 227, 1988. 

WEL Wells, P.A., Loos, Th.W. de, and Kleintjens, L.A., Pressure pttlsed induced critical scattering spinodal and binodal curves for the system polystyrene + methylcyclohexane (experimental data by Th.W. de Loos), Fluid Phase Equil, 83, 383,1993. 

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