Experimental Methods and Data Analysis for Fluctuation Microscopy
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-1 (k, (I(k,Q,r))2
(1)
where ( ) denotes averaging over the image position coordinate r. To understand in a qualitative way why the variance is sensitive to medium-range order (MRO), consider two samples: one a homogeneous random assortment of atoms with no 155 Mat. Res. Soc. Symp. Proc. Vol. 589 ©2001 Materials Research Society
MRO and the other a heterogeneous material with small, randomly-oriented ordered clusters. Set Q so the mesoscopic volume is approximately the same size as those clusters. All of the mesoscopic volumes of the random sample will be statistically the same, so the diffracted intensity from each volume will be the same, and the image will have a low variance. In the heterogeneous sample, some of the clusters will be oriented near a Bragg condition and diffract strongly and others will not. These differences in diffracted intensity lead to an image with large variance. In general, a large image variance indicates some form of MRO. By varying the imaging conditions k and Q we can obtain information about the character of any MRO present. Varying k at constant Q is called variable coherence microscopy and gives information about the structure and degree of ordering inside any ordered regions. Varying Q at constant k is called variable resolution microscopy and gives information about the size of the ordered regions. Only variable coherence microscopy has been experimentally implemented so far.
The quantitative information fluctuation microscopy provides is complicated and still incompletely understood, but we know that it depends on the four-body pair-pair correlation function g 4 (rT,r2, r, 0), where r1 and r 2 define one pair lengths, r is the distance between pairs, and 0 is their relative angle [3]. It has been shown that the pair-pair correlation function contains more information about MRO than g 2 (r) [4]. Fluctuation microscopy has been used to show that thin films of amorphous semiconductors contain more MRO than can be explained by the continuous random network, and that that MRO is reduced on thermal annealing [1] (and for hydrogenated amorphous silicon by exposure to light [5]). This has lead to the development of the paracrystalline theory of the structure of as-deposited amorphous thin films [6]. In this paper we focus on recent advances in the variable coherence experimental method and data analysis. RECOVERING THE TRUE ELECTRON STATISTICS The advent of charge-coupled device (CCD) cameras and other linear devices for recording electron images has made statistical techniques such as fluctuation microscopy possible. However, care must still be taken when analyzing such images. For fluctuation microscopy, we must carefully correct for the modulation transfer function (MTF) of our CCD camera. The MTF is the reciprocal-space function which connects the measured electron intensity with the intensity incident on the device: Imeasured(&)
= MTF(r")Iincident(K)
(2)
where n is the Fourier transform coordinate of the image, not a scattering vector. Since Parseval's theorem connects the
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