Abstract
We present a method that we call symbolic asymptotic development (SAD) to obtain joint probability distributions (j.p.d.'s) of phases of structure factors for general even densities of the atomic position vectors. The formula for the triplet and quartet invariant that we obtain in this way reduces to the well known classical formula for the case of a uniform density of the atomic position vectors. For the case of complete knowledge of the atomic vectors it reduces to first order to the exact probability density of the triplet (quartet) phase invariant. Applying this formula to the most general j.p.d. of the atomic vectors we obtain a statistical interpretation of Hauptman's algebraic B(3,0) and B(4,0) formulas. We also give a heuristic derivation of the SAD method. Another method that we shall discuss uses a method called linearization of the invariants that also produces formulas for the triplet phase invariant. This method is based on previous work and is also more laborious to calculate with than the SAD method. It can also give a statistical interpretation of the B(3,0) formula. We show that the formula obtained for the triplet resembles the formula obtained with SAD.
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