Abstract
We propose an extension of the axial next nearest neighbour Ising (ANNNI) model to a general number of interactions between spins. We apply this to the calculation of stacking fault energies in magnesium—particularly challenging due to the long-ranged screening of the pseudopotential by the free electron gas. We employ both density functional theory (DFT) using highest possible precision, and generalized pseudopotential theory (GPT) in the form of an analytic, long ranged, oscillating pair potential. At the level of first neighbours, the Ising model is reasonably accurate, but higher order terms are required. In fact, our ‘ ANNNI model’ is slow to converge—an inevitable feature of the free electron-like electronic structure. In consequence, the convergence and internal consistency of the ANNNI model is problematic within the most precise implementation of DFT. The GPT shows the convergence and internal consistency of the DFT bandstructure approach with electron temperature, but does not lead to loss of precision. The GPT is as accurate as a full implementation of DFT but carries the additional benefit that damping of the oscillations in the ANNNI model parameters are achieved without entailing error in stacking fault energies. We trace this to the logarithmic singularity of the Lindhard function.
Highlights
It is a 100 years since Wilhelm Lenz suggested a problem in magnetism to his student Ernst Ising, resulting in the famous eponymous model [1] which has become a paradigm in statistical mechanics
If the parameters of the axial next nearest neighbour Ising (ANNNI) model differ in sign the opportunity for competing interactions arises resulting in complexity in the phase diagram [2] and this has been suggested as an origin of the many high temperature modulated phases in binary alloys [9], first discovered in the same year as the Ising model by Johannson & Linde [10]
This is because N + 1 total energies furnish us with the ANNNI model parameters J up to JN and these in turn lead to four stacking fault energies
Summary
It is a 100 years since Wilhelm Lenz suggested a problem in magnetism to his student Ernst Ising, resulting in the famous eponymous model [1] which has become a paradigm in statistical mechanics. It is curious that Mg alloys, having the lowest density among structural metals, are not as widely used as their low cost and high strength to weight ratio would suggest. One reason for this is difficulty in forging, as there is only a narrow temperature range in which this is possible; and their tendency during metallurgical processing to develop deleterious crystallographic textures. From the viewpoint of fundamental theory, what singles out Mg in the present context is its electronic structure being free electron-like This has the benefit that an analytic pair potential exists that emerges out of the density functional theory (DFT) [11,12].
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