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.     

Explanation of the patterns in Mie theory

The far-field scattered light intensity, or the related phase function, for a spherical particle is known to display an overall power-law structure when formulated in terms of the scattering wave vector. Empirically determined patterns in the intensity relating to the particle size and refractive index are known. The cause of the patterns, however, has not been satisfactorily explained. This work applies an exact microphysical model to explain most of the patterns, and specifically, to reveal the physical cause of crossovers from one power-law to another. A unique aspect of this microphysical approach is phasor analysis, which provides a visually based way to examine the angle-dependent wavelet superposition involved in the model. A simple color coding scheme connects the phasors to the interior of the particle, and it is this connection that reveals the meaning of the crossovers.     

Characterization of porous materials by small-angle scattering

Characterization of porous materials by small-angle scattering has been extensively pursued for several years now as the pores are often of mesoscopic size and compatible with the length scale accessible by the technique using both neutrons and X-rays as probing radiation. With the availability of ultra small-angle scattering instruments, one can investigate porous materials in the sub-micron length scale. Because of the increased accessible length scale vis-a-vis the multiple scattering effect, conventional data analysis procedures based on single scattering approximation quite often fail. The limitation of conventional data analysis procedures is also pronounced in the case of thick samples and long wavelength of the probing radiation. Effect of multiple scattering is manifested by broadening the scattering profile. Sample thickness for some technologically important materials is often significantly high, as the experimental samples have to replicate all its essential properties in the bulk material. Larger wavelength of the probing radiation is used in some cases to access large length scale and also to minimize the effect of double Bragg reflections.     

Quantitative evaluation of equatorial small-angle X-ray scattering for cylindrical fibers

Elongated microvoids, internal fibrillar structure, and edge scattering from both surface refraction cause an equatorial streak in small angle X-ray scattering. A model for analyzing the edge scattering of fibers is proposed. Simulation results indicate that the intensity of edge scattering from surface refraction of a cylindrical fiber is strong and makes an important contribution to the equatorial streak. Two factors influence edge scattering intensity. One is the sample-to-detector distance (D); edge scattering intensity increases with increasing D. The equatorial streak becomes weak when D is shortened. The other factor is the refraction index. Edge scattering intensity increases as the real component of the refraction index decreases. In experiment, weak or even no equatorial streaks were found for samples measured in a roughly index-matching fluid. Edge scattering can be eliminated or weakened, and it can be calculated by comparing the intensities of a cylindrical fiber when it is measured in air and in index-matching fluid. The simulation data are basically in agreement with the experimental data.     

Structural properties of C60 in solution

In this paper SANS results on C₆₀ solutions are presented. The data analysis allows to determine the value of the gyration radius of C₆₀ dissolved in CS₂, a solvent capable to guarantee a good contrast for the neutron probe. The study indicates that the gyration radius R_g is independent of concentration.     

小角x射线散射结晶聚合物过渡层厚度的测定

由于结晶聚合物中的过渡层是弥散的，而修正的Porod定律是解决这种问题的有效方法之一. 把相关函数法和Porod定律法求结晶聚合物的过渡层厚度进行了比较，它们的计算结果是一致的. 关键词： 小角x射线散射 结晶聚合物 过渡层厚度 相关函数法 Porod定律     

多次散射的分级计算及数值模拟

本文改进了多次散射的计算方法并提出了一个引理,经简化的数值模拟其结果与巳发表的实验数据相吻合.     

紫外光非视距多次散射信道模型研究

利用蒙特卡罗方法对紫外光非视距传输过程进行了模拟,讨论了传输过程中固体角的确定,仿真分析了系统的角度设置对传输损耗的影响.结果表明,增大收发仰角可以降低系统的传输损耗,并且小角度集中发射与大视场角接收可以降低系统的传输损耗.     

Probing polymer nanocomposite morphology by small angle neutron scattering

Polyamide nanocomposite films were prepared from nanometer-sized silica particles having particle radius of gyration (R _g) of about 66 Å and trimesoyl chloride-m-phenylene diamine-based polyamides having macromolecular units of about 100–140 Å. The nanoscale morphology of the samples was characterized using small angle neutron scattering (SANS). SANS reveals that silica nanoparticles interact well with the polyamide units only at limited silica loading.     

Small angle neutron scattering studies on the interaction of cationic surfactants with bovine serum albumin

The structure of the protein-surfactant complex of bovine serum albumin (BSA) and cationic surfactants has been studied by small angle neutron scattering. At low concentrations, the CTAB monomers are observed to bind to the protein leading to an increase in its size. On the other hand at high concentrations, surfactant molecules aggregate along the unfolded polypeptide chain of the protein resulting in the formation of a fractal structure representing a necklace model of micelle-like clusters randomly distributed along the polypeptide chain. The fractal dimension as well as the size and number of micelles attached to the complex have been determined.      

Small angle neutron scattering study on the aggregation behaviour of PEO-PPO-PEO copolymers in the presence of a hydrophobic diol

Small angle neutron scattering (SANS) measurements on aqueous solutions of four polyethylene oxide-polypropylene oxide-polyethylene oxide block copolymers (commercially known as Pluronic®)F88, P85, F127 and P123 in the presence of hydrophobic C14Diol (also known as Surfynol® 104) reveal information on micellization, micellar size and micellar transitions. While most hydrophilic F88 (with least PPO/PEO ratio) remained unimers in water at 30°C, other copolymers formed micellar solutions. Surfynol® 104 is sparingly soluble in water to only about ～0.1 wt%, but on addition to pluronic solution, it gets incorporated in the micellar region of block copolymer which leads to increase in aggregation number and transformation of spherical to ellipsoidal micelles. The added diol-induced micellization in F88, though hydrophilic copolymers F88 and F127 did not show any appreciable micellar growth or shape changes as observed for P85 and P123 (which are comparatively more hydrophobic). The SANS results on copolymer pairs with same molecular weight PPO but different % PEO (viz. F88 and P85, F127 and P123) and with same molecular weight PEO but different PPO (F88 and F127) reveal that the copolymer with large PPO/PEO ratio facilitate micellar transition in the presence of diol. An increase in temperature and presence of added electrolyte (sodium chloride) in the solution further enhances these effects. The micellar parameters for these systems were found out using available software and are reported.     

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

The theory of small-angle scattering is reviewed with special attention paid to the anomalous scattering and multiphase systems. A general equation is derived that describes the scattering of a multiphase system as a sum of scattering functions of each of the phases, as if it scattered alone in a two-phase system, and interphase interference scattering functions. These scattering functions depend only on the spatial distribution of the phase boundaries, but not on the scattering density. Contrast variation techniques are most rewarding when the scattering density of only one phase can be varied. For anomalous small-angle X-ray scattering (ASAXS), this means the most favourable is the case in which resonant atoms are contained in one phase only. The general equation involves n(p ? 1) unknown partial atomic number density differences, where p is the number of phases and n the number of the different atom types in the sample. These partial atomic number density differences can be found if a suitable structure model is applied to calculate the phase scattering functions. Then, the phase compositions and densities can be calculated by solving a system of linear equations incorporating the atom number conservation law. The partial structure factors formalism is also reviewed. Corresponding equations for a system of n types of atoms and p phases are derived. The number of independent partial structure factors is p(p ? 1)/2 and depends on the number of phases, but not on the number of the types of the atoms in the sample, as in the case of wide-angle scattering.     

一种基于多重散射的光学Hash函数

本文提出了一种基于光与多重散射介质相互作用的光学Hash函数构造方法.该方法创新性地利用多重散射介质对相干调制光的天然随机散射作用,实现了对调制光的"混淆"和"扩散",从而满足了Hash函数的核心功能要求:高安全强度的单向编码/加密.所设计的光电混合系统能有效地模拟Hash函数中的"压缩函数",结合具有特征提取功能的S...     

富勒烯-PVP聚合物链团结构的中子小角散射实验研究

应用中子小角散射技术研究了水溶液中富勒烯-PVP聚合物的链团结构及其大小以及它们在不 同富勒烯含量下的变化.结果表明：当加入富勒烯后，不论是PVP单体分子链还是大分子链团 ，其相关长度与纯PVP溶液相比均变小，且大分子链团的变化更为明显；在不同富勒烯含量 情况下，高富勒烯含量的富勒烯-PVP分子链团的体积更小. 关键词： 中子小角散射 富勒烯 PVP聚合物     

Renormalized cluster expansion for multiple scattering in disordered systems

We study wave propagation in a disordered system of scatterers and derive a renormalized cluster expansion for the optical potential or self-energy of the average wave. We show that in the problem of multiple scattering a repetitive structure of Ornstein-Zernike type may be detected. We derive exact expressions for two elementary constituents of the renormalized scattering series, called the reaction field operator and the short-range connector. These expressions involve sums of integrals of a product of a chain correlation function and a nodal connector. We expect that approximate calculation of the reaction field operator and the short-range connector allows one to find a good approximation to the self-energy, even for high density of scatterers. The theory applies to a wide variety of systems.     

Morphology of carbon nanotubes prepared via chemical vapour deposition technique using acetylene: A small angle neutron scattering investigation

Small angle neutron scattering (SANS) has been utilized to study the morphology of the multi-walled carbon nanotubes prepared by chemical vapour deposition of acetylene. The effects of various synthesis parameters like temperature, catalyst concentration and catalyst support on the size distribution of the nanotubes are investigated. Distribution of nanotube radii in two length scales has been observed. The number density of the smaller diameter tubes was found more in number compared to the bigger one for all the cases studied. No prominent scaling of the structure factor was observed for the different synthesis conditions.      

α粒子散射实验的理论模拟

本文在α粒子一次散射理论的基础上,进一步考虑了第二个原子对散射概率的贡献,建立了散射角与各散射参数间的关系.通过改变α粒子的入射能量以及两原子的位置参量,较好地模拟了各种情况下的散射概率随散射角的分布情况.     

Small angle neutron scattering study of two nonionic surfactants in water micellar solutions

Two classic nonionic surfactants — C₁₄E₇ (heptaethylene glycol monotetradecyl ether) and C₁₀E₇ (heptaethylene glycol monodecyl ether) were investigated in heavy water solution for concentration c = 0.17% (dilute regime) at different temperatures in the range t = 10–35°C by small angle neutron scattering (SANS) method. In the case of C₁₄E₇ surfactant — for all temperatures at c = 0.17% there are two axial ellipsoidal micelles with longer axis 15 nm at 10°C and 49.5 nm at 35°C in investigated solutions. For C₁₀E₇ surfactant at the same concentration of solution and temperature — two axial ellipsoidal micelles were observed, too. The longer axis is equal to 7.5 nm at 10°C, 9 nm at 20°C and at 35°C this axis is equal to 12 nm. Micelles of C₁₀E₇ nonionic surfactant are smaller than those of C₁₄E₇ surfactant in the same experimental conditions.      

On the influence of anisotropy on the magnetic small-angle neutron scattering of superparamagnetic particles

This paper presents a calculation of the magnetic small-angle neutron scattering cross-section resulting from a dilute ensemble of superparamagnetic particles exhibiting uniaxial magnetic anisotropy. We focus on the two experimentally relevant scattering geometries in which the incident neutron beam is perpendicular or parallel to an applied magnetic field, and we discuss several orientations of the anisotropy axes with respect to the field. Magnetic anisotropy has no influence on the magnetic small-angle neutron scattering when the particles are mobile, as is the case e.g. in ferrofluids, but, when the particles are embedded in a rigid non-magnetic matrix and the orientations of the anisotropy axes are fixed, significant deviations compared to the case of negligible anisotropy are expected. For the particluar situation in which the anisotropy axes are parallel to the applied field, closed-form expressions suggest that an effective anisotropy energy or anisotropy-energy distribution can be determined from experimental scattering data. Received 8 November 2001     

基于多重散射强度和偏振特征的舰船尾流气泡激光探测方法

研究多重散射效应对舰船尾流气泡群光散射强度和偏振特征的影响是舰船光尾流探测以及新型光自导鱼雷研究的基础. 基于矢量Monte Carlo方法建立了舰船尾流气泡群激光后向探测仿真模型, 重点研究了尾流气泡群的多重散射机理,分析了多重散射效应、尾流气泡群密度对回波信号强度和偏振特征的影响规律. 基于粒子碰撞重要性抽样的基本思想, 在传统能量接收方法的基础上, 提出了回波光子偏振贡献接收方法和回波信号偏振信息统计方法, 解决了小视场系统光子返回概率低无法形成回波能量的难题. 构建了模拟尾流气泡群激光散射强度和偏振探测实验平台, 从实验的角度验证了模拟结果的准确性. 实验和模拟结果的一致性表明, 利用回波强度、偏振信息可表征气泡群距离、密度信息, 从而可对舰船尾流特别是低密度尾流进行高精度的探测和辨识. 关键词： Monte Carlo 偏振 多重散射 气泡