International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.2, pp. 258-259

Section 3.2.3.3. Granularity, microabsorption and surface roughness

P. W. Stephensa*

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA
Correspondence e-mail: peter.stephens@stonybrook.edu

3.2.3.3. Granularity, microabsorption and surface roughness

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The above discussion of absorption in a powder sample is premised on the assumption that the powder sample is sufficiently fine that the beam traverses a large number of grains, i.e., that [\mu\ell\ll 1], where μ is the absorption coefficient in the solid material and [\ell] is a characteristic linear grain dimension. In that case, the effective linear absorption constant is [\bar \mu\!= \!\mu\rho_{\rm powder}/\!\rho_{\rm solid}], independent of the path that the radiation follows through the sample. However, if [\mu\ell] is not very small, the simple picture of an average absorption length in the sample breaks down, leading to changes in the diffracted intensity.

There have been several treatments of microabsorption due to porosity and surface roughness. Rather than reviewing the literature, this section summarizes work of Hermann & Ermrich (1987[link]) on a flat sample in the symmetrical reflection geometry discussed above. The single-phase sample is taken to be a collection of random polyhedra of average chord length [\ell] with packing fraction α0. The surface has the same lateral correlation length, but the average packing fraction rises from zero at the surface with characteristic dimension ts, i.e., α(z) = α0[1 − exp(−z/ts)], where z is the distance from the surface into the sample. ts = 0 represents an abrupt termination of the bulk porous material.

Hermann & Ermrich find that the diffracted intensity is reduced from the fine powder limit by a factor of 1 − PbulkPsurface, where [\eqalign{ P_{\rm bulk} & \simeq 2 \mu \ell (1-\alpha_0)^2, \cr P_{\rm surface} & \simeq {{2 \mu \alpha_0 t_s}\over{\sin \theta}} \left( 1-{{t_s}\over{\ell (1-\alpha_0) \sin \theta}} \right).}]If surface roughness is small, there is no dependence on diffraction angle 2θ, and the only effect is an overall reduction of diffracted intensity. Excess surface roughness (ts > 0), as might be produced by allowing a granular powder to settle against a flat surface, leads to additional diminution of diffracted intensity at low angles. The expression for Psurface given above is to leading order in 1/sin θ, and is therefore not valid for small grazing angles of incidence. Note that both the surface and bulk correction factors approach zero as [\mu\ell\rightarrow 0]. This analysis is in general agreement with analysis and measurements performed by Suortti (1972[link]).

An additional effect of microabsorption in multi-component powders has been considered by Brindley (1945[link]). The principle is illustrated in Fig. 3.2.3[link]. An X-ray propagating through the sample (depicted by the broken line) suffers absorption according to the weighted average [\bar\mu], but an X-ray diffracted in a grain of phase a has travelled further in that phase than average (solid line). If its path in the diffracting grain is x and the total path through the sample is L, absorption will reduce its intensity by a factor[\exp({-\mu_a x})\exp[{-\overline{\mu}(L-x)}]. ]This leads to a weaker observed reflection from a phase a with larger absorption constant μa if the particle size is an appreciable fraction of [1/(\mu_a - \bar \mu)].

[Figure 3.2.3]

Figure 3.2.3 | top | pdf |

Sketch illustrating an X-ray propagating through a mixture of two kinds of particles.

Averaging over the particle volume, we see that the observed intensity of phase a will be lowered from its value in a fine powder by a factor[ \tau = V_a^{-1} \textstyle\int \exp[{-(\mu_a-\overline{\mu})x}]\, {\rm d}V_a, ]where the integral is taken over one representative grain of phase a. This is exactly the integral of equation (3.2.15)[link], and if one assumes monodisperse spherical particles, it is tabulated in International Tables for Crystallography, Volume C, Section 6.3.3.2[link] and elsewhere, with [\mu_a - \bar \mu] replacing μ. One hitch is that the tabulations are for μ > 0, whereas [\mu_a - \bar \mu\,\lt\, 0] for the less absorbing phases in a mixture. Brindley handles this with a series expansion and provides a table of τ versus [(\mu_a - \bar \mu)R].

By way of illustration, consider a mixture of equal weight fractions of corundum, magnetite and zircon powders, analysed with Cu Kα radiation. Fig. 3.2.4[link] shows the weight fractions that would be measured from such a sample as a function of particle radius (assumed equal for all three phases). In this case, microabsorption biases the result by about 10% for particles of 10 µm diameter, and misses by a factor of two if the diameter is 70 µm.

[Figure 3.2.4]

Figure 3.2.4 | top | pdf |

Weight fractions that would be measured from a mixture of equal parts of three phases with Cu Kα radiation.

In considering microabsorption, there are several important points to note. This entire analysis is only valid for [\mu R \ll 1], since otherwise the concept of an average absorption length becomes ill defined. That means that the Brindley analysis breaks down precisely when it becomes significant. Of course, the Brindley assumption of monodisperse spherical particles is unrealistic for most real samples. Both the surface roughness and Brindley corrections depend on diffraction angle as well as μR, but, as noted above, for moderate absorption, the angle dependence appears as an effective Debye–Waller factor. The parameter R is the radius of the grain of material, which is commonly very much larger than the size of a coherently diffracting region; a Scherrer analysis of particle size through line broadening does not provide a useful estimate of R. Most importantly, unless one has an accurate independent measurement of the size and shape of the grains, microabsorption corrections should be taken as a sign of impending trouble, not a quantitative correction.

References

Brindley, G. W. (1945). The effect of grain or particle size on x-ray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by x-ray methods. Philos. Mag. 36, 347–369.Google Scholar
Hermann, H. & Ermrich, M. (1987). Microabsorption of X-ray intensity in randomly packed powder specimens. Acta Cryst. A43, 401–405.Google Scholar
Suortti, P. (1972). Effects of porosity and surface roughness on the X-ray intensity reflected from a powder specimen. J. Appl. Cryst. 5, 325–331.Google Scholar








































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