Scattering from a multiphase system

Abstract

347major difference between them is that one is based on a narrow sublattice d band (V-V) neglecting the cation-anion-cation interactions, and the other implies a wider band due to a large degree of hybridization between the oxygen 2p and vanadium 3d orbitals. Whichever it may be, the doubling in size of the unit cell from the tetragonal to the monoclinic symmetries results in a reduction by one-half of the size of the Brillouin zone and the opening of an energy gap. The energy gap is a function of the monoclinic distortion, and as described in the previous section, this distortion seems larger in the non-stoichiometric VO2.07 than in VOz.00. Although little can be said about the transport mechanism in the low-temperature phase because of the lack of knowledge regarding the details of the nature of the defects involved, the experimental data with specimens of well-characterized chemical composition (see, for instance, Kimizuka et al., 1970) strongly suggest that the degree of monoclinic distortion has an essential bearing on the transport behaviour of vanadium dioxide. Precise measurement of the lattice parameters of the low-temperature phase is needed as well as further investigations of the lattice vibrational spectra throughout the low-and hightemperature phases.References ADLER, D. (1968) J. AppL Cryst. (1971). 4, 347Scattering from a Multiphase System The theory of small-angle X-ray scattering or light scattering from a system of N randomly distributed, internally homogeneous phases is presented. The special case of three components yields a correlation function which is the sum of two exponential terms whose associated 'correlation lengths' are complicated functions of all the interphase surfaces and volume fractions and cannot, therefore, be interpreted in a simple way. From the correlation function, three independent parameters are obtained which in principle may be used to determine the three interphase surfaces $2~, $23 and $31, provided the volume fractions of the phases are known.

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

“…Equations (49) and (50) are mathematically equivalent to (8) and can be also presented in the form of (10). Since 49) and (50) show that there are p(p-1)/2 independent () Ŝ functions which is exactly the number of the linearly independent "stick probability functions" [39,41]. We should also 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 …”

“…Small-angle scattering of three phase systems were treated first by Peterlin [38]. Multiphase systems were considered in terms of "stick probability functions" [39][40][41]. The main concern was the evaluation of the correlation function for three phase system.…”

Determination of surface areas for supported-metal catalysts from small-angle scattering

“…We can eliminate P~3~ ) and p~3) from ),(3)(r) with the help of (5), (10) and (11). Then (Debye, Anderson & Brumberger, 1957;Goodisman & Brumberger, 1971), V being the total illuminated volume, we finally obtain…”

“…The joint probability Pij is then just the product of the independent probabilities for finding the two ends in phases i and j. Thus Pu(r)=~p~p; (Goodisman & Brumberger, 1971). Then…”

“…Several analyses of the small-angle X-ray scattering (SAXS) of catalysts have yielded relationships which allow the determination of this area from SAXS measurements (Goodisman & Brumberger, 1971;Renouprez & Imelik, 1973). Renouprez, Hoang-Van & Compagnon (1974) have used a model which incorporates the assumptions that the support remains unchanged upon deposition of the metal, that the metal-support contact area $31 is negligibly small compared to other surfaces, and that the void-support area $21 consequently remains essentially constant as metal is added.…”