Wilson statistics

In a very traditional village game, popular over the period of Lent (usually on the pigñata day, the first Sunday of Lent), a young player, suitably blindfolded and armed with a long cudgel, tries to hit a pot (the pigñata) located some distance away, in order to win the sweetmeats contained inside. To break the pot they take random steps, and at each step they try to hit the pot with the cudgel. Is it possible to guess the distance of the player from their starting position after n random steps? Is it possible to guess the direction of the vectorial resultant of the n steps? A very simple analysis of the problem suggests that the distance after n steps may be estimated but the direction of the resultant step cannot, because a preferred privileged orientation does not exist. The situation is very similar to structure factor statistics. Each of the N atoms in the unit cell provides the vectorial contribution . . . fj = fj exp(2πih · rj) = fj exp(i θj). . . to the structure factor; this is equivalent to a vectorial step of the pigñata player. The modulus of the atomic contribution, like the amplitude of the step in the pigñata game, is known (because the chemical composition of the molecules in the unit cell is supposed to be known), but the phase θj (corresponding to the direction of the step) remains unknown; indeed we do not know the position rj of the j th atom. The analogy with the pigñata game suggests that some information on the moduli of the structure factors can be obtained via a suitable statistical approach, while no phase information can be obtained using this approach. This chapter deals just with this statistical approach and owing to the relevant contributions of A. J. C. Wilson, we call this chapter Wilson statistics.