Intercalation in Layered Materials pp 233-234 | Cite as

# Structural Correlations in Transition Metal GICs

## Abstract

We present here a discussion on the use of scattering measurements to provide information on structural correlations with particular reference to intercalate reflections in GICs. The real space assembly of atoms, known as the specimen function \(R(\overrightarrow r )\), which is the target for a scattering experiment (using x-rays, neutrons, or electrons), can be described by the product of a pair correlation function and a shape function,^{1} written as \(R(\overrightarrow r ) = \left\langle {\rho (\overrightarrow r )\rho (0)} \right\rangle S(\overrightarrow r )\). For crystalline specimens, the pair correlation function can be replaced by the convolution of \(F(\overrightarrow r )\) with \(G(\overrightarrow r )\) so that the specimen function^{2} is given by \(R(\overrightarrow r ) = [F(\overrightarrow r )*G(\overrightarrow r )]S(\overrightarrow r )\) where \({\text{F}}(\overrightarrow r )\) is the scattering density within a unit cell, \(G(\overrightarrow r )\) is the sum over δ functions on sites of the real lattice, and \(S(\overrightarrow r )\) is the crystal shape function, defined as \(S(\overrightarrow r )\) inside the specimen and \(S(\overrightarrow r ) = 0\) outside, and “*” denotes the convolution of the indicated functions. Taking the Fourier transform of the specimen function, we obtain for 3-dimensional crystals^{2} the relation \(\overline R (\overrightarrow q ) = [\overline F (\overrightarrow q )\overline G (\overrightarrow q )]*\overline S (\overrightarrow q )\) where \(\overline F (\overrightarrow q )\) is the structure amplitude and \(\overline G (\overrightarrow q )\) and \(\overline S (\overrightarrow q )\) are respectively the Fourier transforms of \(G(\overrightarrow r )\) and \(S(\overrightarrow r )\). However, what we measure is \({\left| {\overline R (\overrightarrow q )} \right|^2}\) which to a good approximation^{1} is given by \({\left| {\overline R (\overrightarrow q )} \right|^2} = {\left| {\overline F (\overrightarrow q )\overline G (\overrightarrow q )} \right|^2}*{\left| {\overline S (\overrightarrow q )} \right|^2}\). Since the Fourier transform ‘inverts’ dimensions, large crystals give very narrow linewidths and small crystals give very broad linewidths. Hence, the FWHM linewidth in a given direction in \(\overrightarrow k \)-space for scattering from aggregates of crystallites is the reciprocal of the average crystal dimension in that \(\overrightarrow k \)
direction. If the linewidth of a reflection is governed by the specimen and not by the resolution function of the experimental set-up, then careful linewidth analysis on several reflections can provide a wealth of information on crystallite sizes and correlation lengths ξ in defective samples.

## Keywords

Select Area Electron Diffraction Pattern Pair Correlation Function Area Electron Diffraction Pattern Structure Amplitude Real Lattice## References

- 1.J. Cowley,
*Diffraction Physics*, ( North-Holland, New York ) 1984.Google Scholar - 2.L. Reimer,
*Transmission Electron Microscopy*, (Springer—Verlag) 1984.Google Scholar