A critical evaluation of x-ray small angle scattering parameters by transmission electron microscopy: GP zones in Al alloys

Harkness, S.D.; Gould, R. W.; Hren, J. J.

doi:10.1080/14786436908217767

Cited by 55 publications

(8 citation statements)

References 8 publications

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“…Several simple methods approximate the function D,(R) with a special two-parameter function (for example the so-called log normal distribution): Roess & Shull (1947), Hosemann (1951), Mittelbach & Porod (1965), Mittelbach (1965), Harkness, Gould & Hren (1969), Neilson (1973), Plestil & Baldrian (1976).…”

Section: Numerical Methods -Linear Modelsmentioning

confidence: 99%

Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Glatter¹

1980

J Appl Crystallogr

294

166

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Small‐angle scattering data from polydisperse systems can be evaluated under the assumption that all particles have the same shape and that the size distribution depends only on one linear size parameter R. The shape of the particles is assumed to be known a priori. The corresponding size distribution function for the number of particles Dn(R) or for the volume Dv(R) can be computed from the smeared, unsmoothed scattering data by the indirect transformation method restricting the range of definition of the D(R) functions to a finite range Rmin≤Rmax. Rmin may be equal to zero, Rmax is limited by the sampling theorem of the Fourier transformation. The resolution in real space is given by the distance of the knots of the B spline functions approximating the size distribution function. The propagated statistical error band in real space can be computed using the inverted matrix of the normal equations. The method gives satisfactory results even in those cases where the shape of the particles is not known exactly, and is superior to analytical methods if the termination effect is critical.

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Section: Numerical Methods -Linear Modelsmentioning

confidence: 99%

Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Glatter¹

1980

J Appl Crystallogr

294

166

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“…Using small-angle X-ray scattering (SAXS), Harkness et al (1969) evaluated two ratios of moments of g( D) ( k D 3 l /k D 2 l and k D 7 l /k D 5 l ) in samples containing spherical Guinier± Preston (GP) zones. These ratios were then used to determine the parameters of a log-normal approximation for the size distribution of the GP zones.…”

Section: Known or Assumed Distribution Formmentioning

confidence: 99%

Estimating grain-size distributions in nanocrystalline materials from X-ray diffraction profile analysis

Birringer¹

1998

Philosophical Magazine A

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It is well known that the Fourier analysis of X-ray di raction peak pro® les (as implemented by Warren and Averbach) can accurately determine the areaweighted average grain size of a ® ne-grained sample. Less well known is the fact that this method simultaneously yields a volume-weighted average grain size. Under certain circumstances, knowledge of these two weighted average grain sizes is su cient to permit reliable estimation of the grain-size distribution, even when the distribution cannot be calculated directly from the Fourier coe cients, as is usually the case. We demonstrate this for a nanocrystalline Pd sample prepared by inert-gas condensation; average grain sizes and the grain-size distribution are estimated by X-ray di raction pro® le analysis and compared with the same quantities measured directly by transmission electron microscopy (TEM). Very good agreement between the resulting average grain sizes is achieved only after compensating for the fact that di raction pro® le analysis directly yields an average unit-cell column length rather than an average grain diameter. The agreement between the grain-size distributions determined by pro® le analysis and TEM was good for crystallite sizes larger than the area-weighted average grain size but deteriorated for smaller sizes due to the volume dependence of the di raction peak intensity.

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“…These have included direct inversion of data with methods for extrapolating beyond the realm of observations (Brill & Schmidt, 1968;Federova, 1977) and the representation of p(R) by a simple model function, e.g. a log-normal function, described by only two parameters (Harkness, Gould & Hren, 1969;Shuin, 1977) or by a set of P basis functions hk(R) with P__ M (Schelten & Hendricks, 1978):…”

Section: Ij-= I S;(r)p(r) Dr J= 1 Mmentioning

confidence: 99%

A new method for the determination of particle size distributions from small-angle neutron scattering measurements

Potton¹,

Daniell²,

Rainford³

1988

J Appl Cryst

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The determination of a particle size distribution p(R) from small-angle scattering data is an example of a practical linear inverse problem for which there is no unique solution. It is shown (i) how the maximum entropy algorithm may be used to extract a unique solution which is the most uniform function compatible with the data set and is necessarily positive, and (ii) how the Backus-Gilbert [Geophys. J. R. Astron. Soe. (1968), 16, 169-205] method may be used to judge the significance of features in distributions derived from a data set of given statistical accuracy. As an illustration, void size distributions are presented for a sample of irradiated stainless steel which was successively annealed at increasing temperatures.

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A critical evaluation of x-ray small angle scattering parameters by transmission electron microscopy: GP zones in Al alloys

Cited by 55 publications

References 8 publications

Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method

Estimating grain-size distributions in nanocrystalline materials from X-ray diffraction profile analysis

A new method for the determination of particle size distributions from small-angle neutron scattering measurements

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