Separation of particle size and lattice strain in integral breadth measurements

Abstract

1'0 0"8 0'6 i 0"4 0"2 0 0 I I i I ! I I 0"2 0"4 0"6 0"8 bOlb(s) Fig. 1. Plots of be/b(s) versus bD/b(s) Curve 1 : b(s) = b ~ exp [-(bP/l/~bZ~)2]/[1-erf (be/1/TrbO)] Curve 2: b(s) = b v + b D Curve 3 : b(s) = [(bP) 2 + (bD)2] ~ Curve 4: bP/b(s) = 1-[bn/b(s)] 2 where 6i and Dr are the integral breadth strain and particle size, respectively. Equation (1) is too complicated and cannot be handled conveniently to estimate the particle size and strain. We suggest that the equation be/b(s)= 1-[bO/b(s)] 2 (5) is a very good approximation to equation (1) and can be employed without difficulty. We plot be/b(s) versus bD/b(s) from equations (1) and (5) and see that the difference between the two curves is at most 10 ~o. This is illustrated in Fig.1. We analyzed the broadening of cold-worked pure tungsten and Ag-10 ~o In alloy by the Warren-Averbach analysis, by applying equation (5) and also by applying b2(s) = (bO) z + (bP) 2. (6) Equation (6)is obtained when both the particle size and strain profiles are approximated by Gaussian functions. Pure tungsten and Ag-10 YoIn filings were prepared at room temperature (23 °C) and powder pattern peaks were * This investigation was supported by a contract from the Office of Naval Research. recorded on a Norelco diffractometer using the nickel-filtered copper radiation. The Fourier analyses of the powder pattern peaks were carried out by the Warren-Averbach technique and the instrumental correction for the integral breadths was made with the use of a parabolic relationship after Wagner & Aqua (1963). The particle size and strain calculated from the Warren-Averbach analysis, present method and pure Gaussian (considering both strain and particle size distribution as Gaussian) approximation are shown in Table 1. Of the two materials investigated, Ag-10 ~oln is highly faulted and tungsten is free of faults. The particle sizes Df for Ag-10 %In obtained by the present method are about twice as high as those obtained by Warren-Averbach analysis. As shown by Mitra & Halder (1964) and Wagner & Aqua (1965) in the case of h.c.p, metals and by Wagner & Aqua (1963) in the case of f.c.c, and b.c.c, metals, if fault broadening predominates, Dz ~-2De. The particle size Df calculated from Gaussian approximation for both strain and particle size distribution is only slightly smaller than Of. For tungsten filings the integral breadth particle size D~' is also twice as large as the particle size De= D. This is in agreement with the definitions of Dz and/) since Dz = D 2/D > D. As shown above, the approximation AP(L)_exp [-L/De] leads to the relation Dz = 2De. The lattice strains calculated with equation (5) are lower than those obtained with equation (6), and are in rather good agreement with the root-mean-square strains (e z)~ averaged over the dimensions of the coherently diffracting domains , i.e. ef ~ 1.25 (~)~.