Warren’s method

This equation results from the assumption that the diffraction line has the shape of an error curve [9.1].) Once B has been obtained from Eq. (9-1), it can be inserted into Eq. (3-13) to yield the particle size t. There are several other methods of finding B from Bf compared with Warren s method, they are somewhat more accurate and considerably more intricate. These other methods involve Fourier analysis of the diffraction lines from the unknown and from the standard, and considerable computation [G.30, G.39]. Even the approach involving Eq. (9-1) can be difficult if the line from the standard is a resolved Ka. doublet it is then simpler to use a Kfi line. The Fourier methods automatically take care of the existence of a doublet. 

Scheme 1.16. Warren s method for indirect access to E-alkenes.

Pollard, A.M. (1982). A critical study of multivariate methods as applied to provenance data. In Proceedings of the 22nd Symposium on Archaeometry, University of Bradford, 30th March-3rd April 1982, ed. Aspinall, A. and Warren, S.E., University of Bradford Press, Bradford, pp. 56-66. 

J3—ft. Strictly speaking, it depends on the intensity distribution in a line (Jones, 1938) and the ideal method of obtaining / is by a Fourier analysis of the line shape (Stokes, 1948) in practice it is doubtful whether such elaboration is worth while, and it is usually sufficient to use correction curves given by Jones (1938) for the relation between b/B and jS/JB for different line-shapes, or to use Warren s (1941) relation j32 = J52—6a which gives very similar results (King and Alexander, 1954). 

O Neil, I. A. Lai, J. Y. Q. Wynn, D. Quinucli-dine N-oxide a potential replacement for HMPA. Chem. Commun. 1999, 59-60. Sansbury, F. H. Warren, S. Stereochemically controlled competitive cydisation with phe-nylthio migration in the synthesis of cyclic ethers. Tetrahedron Lett. 1992, 33, 539-542. Herrmann, J. L. Schlessinger, R. H. A novel method of preparing a-substituted hydracry-late and acrylate esters. Tetrahedron Lett. 1973, 2429-2432. 

The chief problem in determining particle size from line breadths is to determine B from the measured breadth fiv/ of the diffraction line. Of the many methods proposed, Warren s is the simplest. The unknown is mixed with a standard which has a particle size greater than 1000 A, and which produces a diffraction line near that line from the unknown which is to be used in the determination. A diffraction pattern is then made of the mixture in either a Debye camera or, preferably, a diffractometer. This pattern will contain sharp lines from the standard and broad lines from the unknown, assumed to consist of very fine particles. Let Bg be the measured breadth, at half-maximum intensity, of the line from the standard. Then B is given, not simply by the difference between B and B, but by the equation... 

The same principle is the fundament of the Warren-Averbach method (cf. Sect. S.2.5.5) for the separation of size and distortion of structural entities. Thus the mathematics is partially identical. 

David M. H. and Warren S. P., Practical methods for purification of anion dyes as their sodium, potassium and lithium salts . AATCC, 1871. (1971)... 

Overall levels of innovation in the area this year are disappointingly low. All the various forms of phosphorus-based olefination continue to be used widely in synthesis and perhaps the relative paucity of new phosphorus chemistry is a reflection on the extent to which these methods have been developed. One area which does continue to develop, and where there is still considerable potential, is the use of phosphorus stabilised anions in enantioselective and asymmetric synthesis. Warren s continuing use of phosphine oxides and Denmark s excellent contributions to this area are especially worthy of mention. 

The first detailed X-ray diffraction (XRD) studies on PEMFC electrodes were performed by Wilson et al. [43] using a Warren-Averbach Fourier transformation method for determining the weighted crystallite sizes. Warren and Averbach s method takes into account not only the peak width but also the shape of the peak. This method is based on a Fourier deconvolution of the measured peaks and the instrument broadening to obtain the true diffraction profile. This method is capable of yielding both crystallite size distribution and lattice microstrain. The particle-size distributions can be determined from the actual shape of the difliaction peaks, with the use of Warren-Averbach analysis. 

Warren GL, Davis JE, Patel S (2008) Origin and control of superlinear polarizability scaling in chemical potential equalization methods. J Chem Phys 128(14) 144110... 

Sistani KR, Warren JG, Lovanh N, Higgins S, Shearer S. 2010. Greenhouse gas emissions from swine effluent applied to soil by different methods. Soil Science Society of America Journal. 74 429-435. 

Thus the theory for both d5 and d1 systems having been outlined, individual results for these configurations may now be considered. Note that the of the adiabatic treatment corresponds to the k of Warren (101) (c.f. Section 4), and the c and s coefficients of that method to the latter author s a and b quantities, for which equivalent, but more cumbrous expressions were given. 

Robert A. Brown is Warren K. Lewis Professor of Chemical Engineering and Provost at the Massachusetts Institute of Technology. He received his B.S. (1973) and M.S. (1975) from the University of Texas, Austin, and his Ph.D. from the University of Minnesota in 1979. His research area is chemical engineering with specialization in fluid mechanics and transport phenomena, crystal growth from the melt, microdefect formation in semiconductors and viscoelastic fluids, bifurcation theory applied to transitions in flow problems, and finite element methods for nonlinear transport problems. He is a member of the National Academy of Engineering, the National Academy of Sciences, and the American Academy of Arts and Sciences. 

Miles, JCS 1931 2532-42 (Formation and characterization of crysts of LA and some other initiating expls) 5)K.S.Warren, PATR 1152 (1942), "Study of the Action of Lead Azide on Copper 6)J.Fleischer J.B. Burtle, USP 2,421,778 (1947) "Initiating Explosives 7)Wm.H.Rinkenbach A.J. Clear, PATR Rev 1(1950), "Standard Laboratory Procedures for Sensitivity, Brisance and Stability of Explosives 8)U.S.Military Specification MIL-L-3055, Amend 1(1952) (Requirements and tests for dextrinated lead azide) 9)J-Bernstein, GLR 51-HI-2332, Pic Arsn (1952) "Hygroscopicity of Dextrinated Lead Azide 10)J.W.Lavitt, PATR 1957 (1953), "An Improved Microscopic Method for the Determination of the Crystal Size Distribution of 2-Micron RDX" 11)F.P. Bowden K.Singh, Nature 172, 378(1953) (Size effects in the initiation and growth of explosives) 12)J.W.C.Taylor, A.T.Thomas... 

Handbook of property estimation methods for chemicals environmental and health sciences/ edited by Robert S. Boethling, Donald Mackay with a foreword by Warren J. Lyman, p. cm. 

The synthetic pathway for a drug or analogue must start with readily available materials and convert them by a series of inexpensive reactions into the target compound. There are no obvious routes as each compound will present a different challenge. The usual approach is to work back from the target structure in a series of steps until cheap commercially available materials are found. This approach is formalized by a method developed by S Warren, which is known as either the disconnection approach or retrosynthetic analysis. In all cases the final pathway should contain a minimum of stages, in order to keep costs to a minimum and overall yields to a maximum. 

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