The small‐angle x‐ray intensities, scattered by four of the most typical natural coals, are analyzed assuming that coals behave as polydisperse distributions of cubic particles having a layered structure with an interlayer spacing partly dependent on particle sizes. The particle distributions are determined by a rather fast numerical algorithm, and in the considered momentum‐transfer range, they appear independent of the interlayer spacing. The model reproduces quite well only the intensities which are neither particularly structured nor fractal at very small momentum transfers. Micropore, mesopore, and macropore relative sizes are estimated starting from the obtained particle distributions.

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*h*is defined as follows: $h\u2261(4\pi /\lambda )sin(\theta /2),$ where λ and θ denote the photon wavelength and the scattering angle, respectively.

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In the three‐dimensional space $(\u2261R3),$ 𝒟 obeys to the condition $2\u2a7dD\u2a7d3.$ Well‐known geometrical surfaces are fractal surfaces characterized by the value $D\u2009=\u20092,$ while surfaces

*dense*in $R3$ are characterized by $D\u2009=\u20093.$14.

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*L*, we mean the function $\psi (h,L)$ such that $\u222b0\u221e\u2009dh\u2009\psi (h,L)\phi (h,L\u2032)\u2009=\u2009\delta (L\u2212L\u2032),$ where $\phi (h,L\u2032)$ is the form factor of a similarly shaped particle of size $L\u2032.$18.

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The 47 particles of the two sets contain a number of layers specified by the following list: 1, 2, 3, 5, 6, 7, 8, 11, 14, 17, 21, 23, 28, 31, 35, 43, 53, 69, 89, 113, 139, 149, 163, 181, 199, 223, 239, 251, 281, 293, 311, 317, 337, 347, 349, 359, 373, 383, 389, 397, 449, 457, 593, 691, 773, 1009, 1307.

20.

In order to better estimate how the aforesaid numerical difficulties increase both with 𝒩 and with the particle sizes, we note that the double‐precision compilation, i.e., 15 instead of 30 significant digits, becomes quite satisfactory when one excludes the three larger particles of the list reported in Ref. 19 and one confines to a particle size number 𝒩 smaller than 25. However, one should notice that in the two cases the CPU time, required for a single complete optimization run (to be later defined), never exceeded 5 s and few tenths of second, respectively.

21.

We are grateful to Professor Paul W. Schmidt for having confirmed to us that words ‘circles’ and ‘squares’ have to be interchanged in the caption of Fig. 2 of Ref. 8, so that the intensity showing an inflection refers to PSOC‐248 instead of PSOC‐93. Consequently, the attribution reported in Ref. 5 must be accordingly changed.

22.

The isotropy has been checked by one of us (A. B.) with a pinhole camera on a Beulah sample and for different orientations of the latter. A.B. thanks Professor J. B. Cohen for having let him carry through the measurements at the Department of Material Science and Engineering at the Northwestern University. We are also very grateful to Professor P. W. Schmidt for having generously made a Beulah sample available to S. C.

23.

Besides, we have attempted to reproduce the peak in a more empirical way choosing subsets of the 47 particles, made up only of those particles which present maxima and minima close to the observed ones. In this way we hoped that a constructive mechanism of interference could be enforced. Other runs have been made by fixing the populations of some of these particles, so that the peak was to a large extent reproduced at the outset, and then optimizing the populations of the remaining particles. In no cases we have been able to find a peak.

24.

We stress that in Eq. (21) one sums upon all particle sizes and not only on the sizes considered in sets I and II. These sizes in fact are not distributed with the same density among the possible ones and thus deviations can still take place.

25.

26.

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1990

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