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

Abstract

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.

Correlation between subgrains and coherently scattering domains

“…Nanocrystalline Pd pallets were prepared by inert-gas condensation and subsequent compaction (Krill and Birringer, 1998). Dark field TEM micrographs were evaluated quantitatively.…”

“…There is ample experimental evidence that the log-normal size distribution function, f ( x ), given by the median m and the variance σ , can describe crystallite size distribution in a wide range of bulk or loose powder materials (Valiev et al , 1994; Terwilliger and Chiang, 1995; Krill and Birringer, 1998; Ungár et al , 1999; Langford et al , 2000; Scardi and Leoni, 2002). Hinds (1982) has shown that with m and σ the arithmetic-, the area-, and the volume-weighted mean crystallite diameters are where k =0.5, 2.5, and 3.5 in the case of arithmetic-, area-, and volume-weighted mean and j stands for these different averages, respectively.…”

X-ray diffraction analyses of 3D MgO-based replicas of diatom microshells synthesized by a low-temperature gas/solid displacement reaction

“…The size parameters and the effect of crystallite-size distributions on X-ray diffraction line profiles have recently been discussed in detail by Langford et al (2000). It has been shown by Krill and Birringer (1998) and Langford et al (2000) that by assuming lognormal size distribution functions f ( x ) for the case of spherical crystallites, the median and the variance of f ( x ), m , and σ 2 are directly related to the volume-weighted and the area-weighted column lengths, ✩ L vol ✪ and ✩ L area ✪, respectively:

Figure 3.Williamson-Hall plots obtained using the integral breaths of the diffraction profiles of MgO in five specimens after varied times at 700 °C.

✩ L area ✪=(2∕3) m ∙ exp ((5∕3) ln 2 ω ) and ✩ L vol ✪=(3∕4) m exp (7∕2 ln 2 ω ), where the normalized lognormal crystallite size distribution is given as: f ( x )=[√2 π ( ω ) x ] −1 exp {−[ ln ( x ∕ m )] 2 ∕2( ω ) 2 } and σ 2 = m 2 [ exp ( ω 2 ∕2)]∙[ exp ( ω 2 )−1].…”

“…The median and the variance of the crystallite-size distribution were obtained using the approach described in Krill and Birringer (1998). The results of the line profile analyses are listed in Table I.…”