Static neutron scattering function

The easiestexample for establishing the relationship between the structure of a sample and its scattering behaviour is the case of monoatomic liquid [93]. In this section, the static neutron scattering function (static structure factor) will be derived. In a simple homogeneous monoatomic liquid the probability to find a certain atom in volume element d3r at the position r is given by P(r)dir. Since the liquid is homogeneous P(r = and... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The radial distribution function plays an important role in the study of liquid systems. In the first place, g(r) is a physical quantity that can be determined experimentally by a number of techniques, for instance X-ray and neutron scattering (for atomic and molecular fluids), light scattering and imaging techniques (in the case of colloidal liquids and other complex fluids). Second, g(r) can also be determined from theoretical approximations and from computer simulations if the pair interparticle potential is known. Third, from the knowledge of g(r) and of the interparticle interactions, the thermodynamic properties of the system can be obtained. These three aspects are discussed in more detail in the following sections. In addition, let us mention that the static structure is also important in determining physical quantities such as the dynamic an other transport properties. Some theoretical approaches for those quantities use as an input precisely this structural information of the system [15-17,30,31]. 

Knowing the excitation spectrum one can compute the thermodynamic properties. In the local-moment regime they exhibit low-temperature T 7 ) Kondo anomalies that are due to the resonance states. For example, the static magnetic susceptibilty x(T), the specific heat, various transport coefficients and also dynamical quantities (photoemission spectra, dynamical structure function for neutron scattering) have been calculated (Bickers et al. 1985, Cox et al. 1986). An excellent model system for comparison with experimental data are the dilute (La, Ce)Bg alloys because of a fourfold degenerate Fg ground state of cerium (Zirngiebl et al. 1984). 

The characteristics of the self-assemblies based on F(nBA2o%-stat-AAso%)ioo-b- M5 dispersed in aqueous solution were then measured by means of static and dynamic light scattering (SLS and DLS) and small angle neutron scattering (SANS) as a function of the ionization degree a of the AA units in the polymer. The value of a was adjusted by adding the proper amount of NaOH (see Material and Methods ). For a > 20% and in the presence of O.IM NaCl, DLS indicates the presence of two populations of scatterers... 

The static susceptibility x iQfi F) derived from neutron scattering data may be in disagreement with the direct measurement of the static bulk susceptibility, which, by definition, corresponds to the Q=0 value, i.e., (0,0,r). Because of the form factor discussed earlier, part of the electron waveftmctions (the free-electron part) will contribute to the bulk susceptibility, but not to the neutron signal at finite Q (see Liu 1989 for a discussion of this point). In addition, there may be correlations between the localized part of the response function which cause oscillations in a fimction of Q. 

In fig. 43 the width found for YbCuAl by an analysis with just one quasielastic line is plotted as a function of temperature (circles) together with the relaxation rates extracted from NMR data (open circles, MacLaughlin et al. 1982). Whereas the NMR rate shows a strong increase below T = 40 K, this increase is much smaller for the relaxation rates extracted from the neutron scattering data. With new experiments, and an improved analysis, Murani et al. (1985) showed the existence of at least two inelastic magnetic excitations in YbCuAl. They added to fig. 43 the quantity X n X"(( n)> where x is the static bulk susceptibility, represented by x (8> = 0- T) in eq. (5), is the peak... 

For the amphiphilic block copolymer in the non-polar selective solvent, the unpolar blocks form the corona, which provides solubilization and stabilization, while the polar or hydrophilic and functionalized blocks form the core, which is able to dissolve metal compounds due to coordination, followed by the nucleation and growth of metal particles upon reduction. Also the internal structure of block-copolymer micelles, as given by the size of core and corona and the density profile in each domain, has been carefully characterized by static and dynamic light scattering [146] and by small angle neutron scattering using contrast variation techniques [147], The micellar corona has many of the characteristics of a spherical polymer brush. 

---