Multiple small-angle scattering—A review

Small-angle scattering (SAS) is a powerful experimental technique in condensed matter physics for studying structural features of inhomogeneities of colloidal dimensions. So far the technique has been largely exploited to study thin samples for which the single scattering approximation, for the radiation-matrix interaction, holds good. The single scattering approximation is invalid when the thickness of the sample exceeds the scattering mean free path. This situation calls for a guideline to analyse the scattering data having significant contribution from multiple scattering. Since multiple scattering broadens the scattering profile, the beam broadening nature of multiple scattering can also be exploited, by making the sample suitably thick, to study large size inhomogeneities which are otherwise inaccessible to a small-angle scattering set up because of its resolution constraints. The present article presents a review and extension of the theoretical basis for analysing multiple scattering data from the point of view of a recent formalism on multiple small-angle scattering. The formalism is valid for both monodisperse and polydisperse scattering media characterized by the presence of large size inhomogeneities in the matrix. It is shown that multiple scattering from a polydisperse sample can be described by a system of coupled integrodifferential equation. However, multiple scattering from a monodisperse sample can be described by a Fokker-Planck type of equation. These equations have been analysed with an emphasis laid on the nature of the structural information pertaining to the inhomogeneities which is extractable from the multiple scattering profile. When the linear dimension of inhomogeneities becomes comparable to the scattering mean free path of the radiation in the sample, the statistical nature of the medium becomes pronounced. The statistical nature of the medium modulates the scattering profile. The modulation effect could be broadening or narrowing of the profile depending upon the nature of the inhomogeneities and their population distribution. The limiting regimes of validity and the implications of various approximations, frequently used to analyse the scattering data, have been indicated.     

Raman scattering by hot polaritons in NaClO3

Summary By nonlinear mixing of IR and visible radiation,i.e. coherent Raman scattering by polaritons driven by a CO₂ laser, it has been possible to get high-resolution spectra for theF-modes of a NaClO₃ crystal in the region (907÷957) cm⁻¹, where several isotopic modes are present. The obtained independent measurement of the refractive index and absorbance allows an accurate determination of the polariton dispersion curve and its width inq-space. The experimental data confirm the existence of a localized isotopic mode at 931 cm⁻¹ and a strongly damped isotopic mode near 960 cm⁻¹. Finally, an accurate measurement of the dispersion of the second-order nonlinear polarizability in the same energy region has been obtained for the first time. This work was supported by the Italian Consiglio Nazionale delle Ricerche and Ministero della Pubblica Istruzione.     

Electron screening in backward elastic scattering

Abstact: The elastic scattering cross sections, σ (E,θ), for the systems He+Ta and He+W have been measured at θ_lab=165° and E _lab=76.1 keV to 3.988 MeV using targets with a thickness of a few atomic layers. The results are smaller than the results given by the Rutherford scattering law, σ_R(E,θ), due to the effects of electron screening and can be described by σ(E,θ)/σ_R(E,θ)=(1+U_e/E)⁻¹, where U _e is an atomic screening potential energy. The deduced average value, U _e=28 ± 3 keV, is consistent with the Moliére- and Lenz-Jensen-models as well as electron binding energies. Received: 25 May 1998     

Small‐angle Scattering Studies of Adsorption and of Capillary Condensation in Porous Solids

Small‐angle scattering (SAS) studies are reviewed of adsorption and capillary condensation of water, hydrocarbons and halogenated hydrocarbons near room temperature, and of nitrogen at 78 K in some mesoporous solids, mainly silicas. The theory needed for the interpretation of SAS data is briefly covered. Calculations of the scattered intensity I(q) for a model porous medium show that I(q) depends markedly on the film thickness t. Adsorption and capillary condensation of nitrogen at 78 K in mesoporous silicas was studied by use of in situ SANS, and t as function of the relative pressure P/P_s was estimated. Adsorption of N₂ in defects within the silica skeleton at P/P_s<0.1 lead to a significant increase in I(q). Isolated vapor bubbles in capillary condensed nitrogen in a Gelsil^® appeared on adsorption near saturation of the pore system. The kinetics of capillary condensation and of drainage were followed. Power law scattering at low q indicated the formation of ramified clusters of voids on drainage of liquid nitrogen from the xerogel Gelsil^®. Similar clusters were observed on drainage of water from Vycor^® glass. Provided the clusters indicate a percolation process, the desorption branch should not be used for the estimation of a pore size distribution for materials with networked pores. The adsorptive smoothing by benzene was observed of a rough interface in a controlled pore glass.     

Microphase separation in weakly charged hydrophobic polyelectrolytes

Aqueous solutions of a well-defined poly(N-isopropylacrylamide-co-sodium 2-acrylamido-methylpropanesulfonate) (NIPAM/NaAMPS in a 95/5 molar ratio) have been investigated by means of small-angle neutron scattering (SANS) and rheological experiments as a function of temperature ( 25^° C T 60^° C) and polymer concentration ( 0.5wt% C 12wt%). The solutions remain optically transparent and isotropic over the whole temperature range, in contrast with the homopolyNIPAM which precipitates above its lower critical solution temperature (LCST = 32^° C). Upon addition of salt, the systems undergo a micro-macrophase separation. At temperatures above 45^° C, the SANS spectra exhibit a sharp peak at a scattering wave vector, q _max, which increases slightly with temperature. At high temperature ( T∼ 60^° C), the scattered intensity follows a power law I(q) ∼q ^-4 in the asymptotic regime, characteristic of two-density media with sharp interfaces, and q _max is found to vary with polymer concentration as q _max∼C ^0.22. Estimates of the typical sizes give values between 40 ? and 200 ?. These results provide a strong evidence of a thermally induced microphase separation, which is corroborated by the very sharp increases of the viscosity (over 2 decades) and of the stress relaxation time of the solutions, occurring in the temperature range where the scattering peak is observed. The results are discussed and compared with the theoretical models proposed for weakly charged polyelectrolytes in a poor solvent. Received 1 October 2001     

Electromagnetic waves in a randomly inhomogeneous Josephson junction

Spectrum modification and damping of Josephson plasma waves induced by random inhomogeneities of the critical current through the superconductor contact and the averaged Green function of such excitations are analyzed. In the self-consistent approximation that makes it possible to take into account multiple wave scattering on the inhomogeneities, the frequency and damping of averaged waves, as well as position ν _m and peak width Δν of the Fourier transform imaginary part of the averaged Green function, are determined as functions of wavevector k. The evolution of such functions with the variation of the correlation radius and the relative r.m.s. fluctuations of inhomogeneities is studied. The inhomogeneity-induced wave frequency decrease observed in the long wavelength spectral region qualitatively agrees with the ν _m behavior. It is established that in the case of “long-range” inhomogeneities, the linear dependence of damping on k changes to the inversely proportional one, and damping tends to zero as k → 0, while Δν at small k attains its maximal values due to nonuniform broadening. In the presence of “short-range” inhomogeneities, the wave damping and Δν are found to be similar functions of k. The results are compared to the numerical calculation data.     

Al-Zn-Mg-Cu-Li合金时效过程微结构演化的小角x射线散射研究

应用小角x射线散射技术分析了Al-Zn-Mg-Cu-Li合金在130，150和160℃温度时效24 h析出粒子的微结构参数的变化情况. 粒子的半径随着时效温度的增高而增加，它的比内表面积和体积百分数随着时效温度的增高而减小. 对Porod曲线q³J(q)-q²的分析表明,析出粒子与基体之间有明显的界面. 关键词： 小角x射线散射 Al-Zn-Mg-Cu-Li合金 时效 析出粒子     

Glassy dynamics of simulated polymer melts: Coherent scattering and van Hove correlation functions

Whereas the first part of this paper dealt with the relaxation in the β-regime, this part investigates the final relaxation (α-relaxation) of a simulated polymer melt consisting of short non-entangled chains in the supercooled state above the critical temperature of ideal mode-coupling theory (MCT). The temperature range covers the onset of a two-step relaxation behaviour down to a temperature merely 2% above . We monitor the incoherent intermediate scattering function as well as the coherent intermediate scattering function of both a single chain and the melt over a wide range of wave numbers q. Upon approaching the coherent α-relaxation time of the melt increases strongly close to the maximum q _max of the collective static structure factor S_q and roughly follows the shape of S_q for q q _max. For smaller q-values corresponding to the radius of gyration the relaxation time exhibits another maximum. The temperature dependence of the relaxation times is well described by a power law with a q-dependent exponent in an intermediate temperature range. Deviations are found very close to and far above , the onset of which depends on q. The time-temperature superposition principle of MCT is clearly borne out in the whole range of reciprocal vectors. An analysis of the α-decay by the Kohlrausch-Williams-Watts (KWW) function reveals that the collective KWW stretching exponent and KWW relaxation time show a modulation with S_q. Furthermore, both incoherent and coherent KWW times approach the large-q prediction of MCT already for q > q _max. At small q, a q^-3 power law is found for the coherent chain KWW times similar to that of recent experiments.     

Distribution of Particles Which Produces a “Smart” Material

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ³ is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function , where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D. MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30. PACS: 03.04.Kf.     

Canonical interpretation of the <Emphasis Type="Italic">η</Emphasis><Subscript>2</Subscript>(1870)

We argue that the mass, production, total decay width, and decay pattern of the η ₂(1870) do not appear to contradict with the picture of it as being the conventional 2 ¹ D ₂ q[`(q)]q\bar{q} state. The possibility of the η <sub>2</sub>(1870) being a mixture of the conventional q[`(q)]q\bar{q} and a hybrid is also discussed.     

Electron yield per ion charge-state correction for an ion collector with unsuppressed secondary electron emission

The electron yield per ion charge-state γ/q was measured for emission of electrons from clean polycrystalline gold induced due to impact of Ta ^q+ (11≤q≤41) ions with kinetic energy per chargeE _i/q from 15 keV/q to 150 keV/q. The dependence of γ on angle of incidence was analyzed with use of relation γ(ϑ)=γ₀ cos^−f ϑ. The fitting of experimental data gives a range of γ₀/q from 1 to 1.75 for Ta¹³⁺ and from 1.5 to 1.73 for Ta³⁹⁺. The dependence of γ₀/q onq andE _i is discussed with respect to measurement of ion currents emitted from laser-produced plasmas with an ion collector with unsuppressed secondary electron emission. This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy, and by grant A1010819 from the Grant Agency of the Academy of Sciences of the Czech Republic.     

Studies on pore morphology of titanium and its oxide by small angle neutron scattering

Titanium metal bodies have been prepared from the sintered powder compacts of TiO₂ by a novel molten salt electrochemical approach, known as FFC Cambridge process. The phase and compositional characterizations of both Ti and TiO₂ have been carried out by X-ray diffraction. The pore morphologies of sintered TiO₂ pellet and the metallic Ti pellet, obtained after electrochemical reduction have been studied by SANS over a scattering wave vector q range of 0.003–3.5 nm⁻¹ using a double crystal diffractometer and a pin-hole collimated SANS instrument. In the case of reduced metal pellet, average pore size was found to be larger than that of the oxide pellet as the voids left behind after the oxygen atoms left the oxide matrix, could not coalesce.      

Saturation of Coulomb sum rules in the <Superscript>6</Superscript>Li case

The Coulomb sums S _L(q) of the ⁶Li nucleus have been obtained from electron scattering measurements at 3-momentum transfers q = 1.125–1.625 fm⁻¹. It is found that at q > 1.35 fm⁻¹ the Coulomb sum of the nucleus becomes saturated: S _L(q) = 1 .     

Raman scattering by hot and thermal polaritons in crystal quartz

Summary Nonlinear mixing of IR and visible radiation,i.e. coherent Raman scattering by polaritons driven by a CO₂ laser, has been used to obtain the dispersion curve and its width inq-space of the polariton associated to theE-phonon at 1065 cm⁻¹ in crystal quartz. It is shown in this paper that a direct method to determine indipendently, with high precision, the refractive index and absorbance of a crystal can be obtained in this way. The results are compared with accurate data obtained from Raman scattering by polaritions in thermal equilibrium and very good agreement is found between the two measurements. It is finally shown that nonlinear-mixing techniques turn out to be completely consistent with the simple picture of scattering of light by hot polaritons. This work was supported by the Italian Consiglio Nazionale delle Ricerche del Ministero della Pubblica Istruzione.     

Dynamics of liquid silane

Neutron scattering of cold neutrons from liquid silane at 137° K and 98°K is explained on the basis of a simple model. The rotational diffusion constant,D _r, and the delay time,τ ₀, after which rotational diffusion may be said to occur are derived on the basis of this model. At 137° K we getD _r=0.22×10¹³ sec⁻¹ andτ ₀=0.68×10⁻¹³ sec. At 98°KD _r (=0.06×10¹³ sec⁻¹) is down by a factor of more than three butτ ₀=(0.54×10⁻¹³ sec) shows only a small change. By comparison with data on liquid CH₄ it is concluded that the law of corresponding states is not applicable for describing rotational dynamics of CH₄ and SiH₄. Rotational motions in SiH₄ are more hindered than in CH₄ at the same reduced temperature.     

Small angle neutron scattering study of pore microstructure in ceria compacts

Ceria powders were prepared by gel combustion process using cerium nitrate and hitherto unexplored amino acids such as aspartic acid, arginine and valine as fuels. The powders have been characterized by X-ray and laser diffraction. Cold pressed compacts of these powders have been sintered at 1250°C for 2 h. Internal pore microstructure of the sintered compacts has been investigated by small angle neutron scattering (SANS) over a scattering wave vector q range of 0.003–0.17 nm⁻¹. The SANS profiles indicate surface fractal morphology of the pore space with fractal dimensionality lying between 2.70 and 2.76.      

SU(3) classification of p-wave ηπ and π systems

An exotic meson, the π₁(1400) with J ^PC = 1^{- +}, has been seen to decay into a p-wave ηπ system. If this decay conserves flavor SU(3), then it can be shown that this exotic meson must be a four-quark state ( qˉq + qˉq) belonging to a flavor ˉ10 representation of SU(3). In contrast, the π₁(1600) with a substantial decay mode into π is likely to be a member of a flavor octet. Received: 27 March 2002 / Accepted: 8 August 2002 / Published online: 10 December 2002 RID="b" ID="b"e-mail: klempt@iskp.uni-bonn.de Communicated by V. Vento     

Radio-wave backscattering from artificial ionospheric turbulence

Conclusions The feasibility of investigating the inhomogeneous structure of the perturbed region obtained with the use of a high-power ionospheric station located at a small distance from the heater station is demonstrated. It is shown that the test wave signals backscattered by the perturbed region in the 5–10 MHz frequency range have appreciable amplitudes. The use of several frequencies of test waves and computations of the trajectories for determining the position of the scattering volume permits investigation of the perturbed region in a large height range. The experiments showed that the region occupied by the artificial inhomogeneities extends to more than 50 km below the level of reflection of the heater wave. The characteristic dimensions of the scattering inhomogeneities are estimated (l ~ 40–90 m,l ~ 2–2.5 km.).Scientific-Research Institute of Radiophysics. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 28, No. 5, pp. 579–584, May, 1985.     

On the theory of multiple scattering

In this paper a kinetic equation is derived for the distribution function in the variable q=2 sin(ϑ/2) for the case of a scattering cross section of general form under the assumption that the region of multiple scattering (the diffusion region) is small. The limits of the kinetic equation are discussed, with no restrictions imposed on the scattering angles. It is found that the equation has a solution in the form of an integral. Finally, it is established that the solution is applicable over the entire range of angles, from 0 to 180°. Zh. éksp. Teor. Fiz. 116, 418–435 (August 1999)