InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.2, pp. 22-23
## Section 1.2.12.2. The cumulant expansion |

A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958; Johnson, 1969). It expresses the function as

Like the moments μ of a distribution, the cumulants κ are descriptive constants. They are related to each other (in the one-dimensional case) by the identity When *it* is substituted for *t*, (1.2.12.5*b*) is the characteristic function, or Fourier transform of (Kendall & Stuart, 1958).

The first two terms in the exponent of (1.2.12.5*a*) can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by .

The Fourier transform of (1.2.12.5*a*) is, by analogy with the left-hand part of (1.2.12.5*b*) (with *t* replaced by ), where the first two terms have been omitted. Expression (1.2.12.6) is similar to (1.2.12.4) except that the entire series is in the exponent. Following Schwarzenbach (1986), (1.2.12.6) can be developed in a Taylor series, which gives

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4), and corresponds to the probability distribution [analogous to (1.2.12.3)]

The relation between the cumulants and the quasimoments are apparent from comparison of (1.2.12.8) and (1.2.12.4):

The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant contributes not only to the coefficient of , but also to higher-order terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a finite truncation of (1.2.12.6), the probability distribution cannot be represented by a finite number of terms. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type.

### References

Johnson, C. K. (1969).*Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermal-motion effects. Acta Cryst.*A

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Kendall, M. G. & Stuart, A. (1958).

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