A critical force constant of the structural phase transition in quartz

Long wavelength optical phonons of quartz were analyzed by a Born-Von Karman model not previously used. It was found that only one force constant associated with the turning of the Si-O bonds has a critical effect on the soft-mode frequency and the α-β transition in quartz. The square of the soft-mode frequency was found to depend linearly on this force constant which has the temperature dependence K(T)= -5.33+225.3x10-4(851-T)2/3 in units of 104 dyn/cm2.

Phase transition in the countdown problem

We present a combinatorial decision problem, inspired by the celebrated quiz show called Countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit.

Determination of the crystal and magnetic structure of the DyCrO4-scheelite polymorph by neutron diffraction

Neutron diffraction data of DyCrO4 oxide, prepared at 4 GPa and 833 K from the ambient pressure zircon-type, reveal that crystallize with the scheelite-type structure, space group I41/a. Accompanying this structural phase transition induced by pressure the magnetic properties change dramatically from ferromagnetism in the case of zircon to antiferromagnetism for the scheelite polymorph with a T N= 19 K. The analysis of the neutron diffraction data obtained at 1.2 K has been used to determine the magnetic structure of this DyCrO4-scheelite oxide which can be described with a k = [0, 0, 0] as propagation vector, where the Dy and Cr moments are lying in the ab-plane of the scheelite structure. The ordered magnetic moments are 10 µB and 1 µB for Dy+3 and Cr+5 respectively

Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed