A theory is presented for describing the effect on the transverse NMR relaxation rate of microscopic spatial inhomogeneities in the static magnetic field. The theory applies when the inhomogeneities are weak in magnitude and the nuclear spins diffuse a significant distance in comparison with a length scale characterizing the inhomogeneities. It is shown that the relaxation rate is determined by a temporal correlation function and depends quadratically on the magnitude of the inhomogeneities. For the case of unrestricted diffusion, a simple algebraic approximation for the temporal correlation function is derived. The theory is illustrated by applying it to a model of randomly distributed magnetized spheres. The theory is also used to fit experimental data for the dependence of the relaxation rate on the interecho time for a Carr-Purcell-Meiboom-Gill pulse sequence. Key words: relaxation rate; magnetic inhomogeneities; brain; blood; diffusion Some biological tissues, when placed in a uniform external magnetic field, generate a secondary inhomogeneous field that can significantly increase the transverse NMR relaxation rate (1,2). The inhomogeneous field may be due to intrinsic structures, such as iron-rich cells in the brain (3) and deoxygenated red blood cells (4), or it may be induced by administering a contrast agent (5). A variety of applications of this effect to MR imaging have been studied (1,2,5-9).The purpose of this article is to provide a quantitative theoretical description for the relaxation rate shift caused by static microscopic field inhomogeneities that are weak in magnitude. We assume that the nuclear spins (usually water protons) diffuse with an effective diffusion constant D. Our theory is most useful if the relaxation rate shift is less than D/L 2 , where L is a length scale characterizing the inhomogeneities. Our results are thus complementary to those of static dephasing and weak diffusion approaches (10 -13).As an illustration, the theory is first applied to a system of randomly distributed, uniformly magnetized spheres. We compare our predictions with the results of other theoretical methods, numerical simulations, and experiments. Then, to demonstrate its application to biological systems, we use the theory to analyze experimental measurements made by Ye and Allen (17) for in vitro rat liver. The theory is applied to fit the dependence of the relaxation rate on the interecho time of a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. We show that the theory gives a better fit than the commonly used Luz-Meiboom chemical exchange formula (18). MODELWe consider a system with a total static magnetic field magnitude at a position r ofwhere B 0 represents the uniform part of the field and B represents the inhomogeneous part. In practice, B is often highly irregular, and hence we formally treat it as a random quantity with an associated probability distribution. Averages over this distribution are indicated with angle brackets ͗. . .͘ fld . Without loss of generality, we may takeLet us make the n...

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Purpose – This study aims to explore the extent to which social capital plays a role in firm development, internationalization and growth, in the context of an emerging market, Brazil. The study aims to provide a new context and perspective on the role social capital plays in fostering growth and internationalization among small and medium enterprises (SMEs) in an emerging market. Design/methodology/approach – This is an exploratory study. First, empirical data are drawn from a commodity industry, the Brazilian Stone industry, which is a competitive, well-established sector in the country. Using a previously validated questionnaire from the World Bank, the extent of social capital possessed and used by the firms in this industry is correlated to their growth and performance. Based on the exploratory results, a set of research propositions are developed that point the way to questions that are important and interesting to further understand the role of social capital in this context. Findings – The results point to fairly low levels of social capital among Brazilian SMEs in this sector and relatively low levels of awareness of the potential opportunities to exploit social capital to further development and internationalization. Research limitations/implications – Social capital matters for firm growth and performance, but the extent of development of social capital in this context is fairly low. The limitations include the sample size and the homogeneity of the sample, which restricts generalizability. Practical implications – Building and exploiting social capital is a void that currently exists in the commodity sector in Brazil. Developing this can lead to more positive firm performance and growth, especially as the institutional context in Brazil continues to develop. Originality/value – The paper offers a unique context, as well as a new perspective on the role of firm social capital, by using an emerging market and a commodity industry that has been rarely studied in the literature.

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