What determines the morphology of a decomposing alloy? Besides the well-established effect of the nucleation barrier, we demonstrate that, in a concentrated multicomponent Ni(Al,Cr) alloy, the details of the diffusion mechanism strongly affect the kinetic pathway of precipitation. Our argument is based on the combined use of atomic-scale observations, using three-dimensional atom-probe tomography (3D APT), lattice kinetic Monte Carlo simulations and the theory of diffusion. By an optimized choice of thermodynamic and kinetic parameters, we first reproduce the 3D APT observations, in particular the early-stage transient occurrence of coagulated precipitates. We then modify the kinetic correlations among the atomic fluxes in the simulation, without altering the thermodynamic driving force for phase separation, by changing the vacancy–solute interactions, resulting in a suppression of coagulation. Such changes can only be quantitatively accounted for with non-zero values for the off-diagonal terms of the Onsager matrix, at variance with classical models.

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The mechanism of morphogenesis in a phase-separating concentrated multicomponent alloy.

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Mobius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.

Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.

Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications

Atomistic study on the structure and thermodynamic properties of Cr7C3, Mn7C3, Fe7C3

Kinetic pathways for phase separation: An atomic-scale study in Ni–Al–Cr alloys

The present work develops a unified and concise solution for inverse lattice problems. Also, a uniformly sampled arithmetic Fourier transform is presented in this work which uses Ramanujan's sum rule.

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Multidimensional inverse lattice problem and a uniformly sampled arithmetic Fourier transform

This paper develops the Mobius inversion formula for the Gaussian integers and Eisenstein's integers, and gives two applications. The first application is to the two-dimensional arithmetic Fourier transform (AFT), which is suitable for parallel processing. The second application is to two-dimensional inverse lattice problems, and is illustrated with the recovery of interatomic potentials from the cohesive energy for monolayer graphite. The paper demonstrates the potential application in the physical science of integral domains other than the standard integers.

https://iopscience.iop.org/article/10.1088/0305-4470/29/17/029/pdf

Algebraic rings of integers and some 2D lattice problems in physics

The generalized Mobius function and Mobius inversion formula are applied to a multiplicative semigroup. A general mathematical method based on this Mobius inversion is presented to solve inversion problems of expansions with unequally weighted terms. By this method, all the inverse lattice problems in physics can be solved concisely. The solutions of four inverse lattice problems: the Fibonacci structure, the square lattice structure, the bcc and the hcp lattice structures are given. These are difficult to be solved by other methods. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Chen Zhaodou

Papers

Multidimensional inverse lattice problem and a uniformly sampled arithmetic Fourier transform

Algebraic rings of integers and some 2D lattice problems in physics

Möbius inversion on a multiplicative semigroup and inverse lattice problems in physics