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The analysis of temperature-dependent EXAFS spectra based on the cumulant expansion is critically reviewed, seeking for accurate relations between EXAFS parameters and physical properties of crystals. The treatment is based on the distinction between the real and effective distribution of distances, and is divided into three logical steps. (a) The connection between lattice dynamics and cumulants C_n^* of the real distribution is studied and the extent of the usual approximations are numerically evaluated. Atomic vibrations perpendicular to the bond direction make the EXAFS thermal expansion larger than the crystallographic one; the difference is connected to a shift of the effective pair potential rather than to its asymmetry. Peculiar information on lattice dynamics of crystals can be obtained from accurate EXAFS measurements and their cumulant analysis. (b) The differences between cumulants of the real and effective distribution (C_n^* and C_n, respectively) are calculated for various physically realistic distributions. The largest discrepancy concerns the first cumulant: C_1^* measures the thermal expansion of the interatomic bond, while C_1 is a better estimate of the crystallographic thermal expansion. (c) The convergence properties of the cumulant series are discussed and some phenomenological procedures are suggested to monitor and possibly work out the connected failures of the cumulant method. Benefits and risks of the use of an effective pair potential are at last debated.

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