Geometrical frustration
Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950.
In that case commensurability, such as helical spin arrangements may result, as had been discussed originally, especially, by A. Yoshimori,[4] T. A. Kaplan,[5] R. J. Elliott,[6] and others, starting in 1959, to describe experimental findings on rare-earth metals.
In spin glasses, frustration is augmented by stochastic disorder in the interactions, as may occur experimentally in non-stoichiometric magnetic alloys.
Three magnetic ions reside on the corners of a triangle with antiferromagnetic interactions between them; the energy is minimized when each spin is aligned opposite to neighbors.
With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins.
The net magnetic moment points upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out (i.e. are antiferromagnetically aligned), as do forwards and backwards.
If the graph G has quadratic or triangular faces P, the so-called "plaquette variables" PW, "loop-products" of the following kind, appear: which are also called "frustration products".
Although most previous and current research on frustration focuses on spin systems, the phenomenon was first studied in ordinary ice.
Heat Capacity of Ice from 15 K to 273 K, reporting calorimeter measurements on water through the freezing and vaporization transitions up to the high temperature gas phase.
Maintaining the internal H2O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions.
Due to the strong crystal field in the material, each of the magnetic ions can be represented by an Ising ground state doublet with a large moment.
Currently the spin ice model has been approximately realized by real materials, most notably the rare earth pyrochlores Ho2Ti2O7, Dy2Ti2O7, and Ho2Sn2O7.
The word frustration was initially introduced to describe a system's inability to simultaneously minimize the competing interaction energy between its components.
The frustration of a spin glass is understood within the framework of the RKKY model, in which the interaction property, either ferromagnetic or anti-ferromagnetic, is dependent on the distance of the two magnetic ions.
With the help of lithography techniques, it is possible to fabricate sub-micrometer size magnetic islands whose geometric arrangement reproduces the frustration found in naturally occurring spin ice materials.
Recently R. F. Wang et al. reported[13] the discovery of an artificial geometrically frustrated magnet composed of arrays of lithographically fabricated single-domain ferromagnetic islands.
These artificially frustrated ferromagnets can exhibit unique magnetic properties when studying their global response to an external field using Magneto-Optical Kerr Effect.
[14] In particular, a non-monotonic angular dependence of the square lattice coercivity is found to be related to disorder in the artificial spin ice system.
A common feature of all these systems is that, even with simple local rules, they present a large set of, often complex, structural realizations.
Geometric frustration plays a role in fields of condensed matter, ranging from clusters and amorphous solids to complex fluids.
Two-dimensional examples are helpful in order to get some understanding about the origin of the competition between local rules and geometry in the large.
A well known solution is provided by the triangular tiling with a total compatibility between the local and global rules: the system is said to be "unfrustrated".
The stability of metals is a longstanding question of solid state physics, which can only be understood in the quantum mechanical framework by properly taking into account the interaction between the positively charged ions and the valence and conduction electrons.
It is nevertheless possible to use a very simplified picture of metallic bonding and only keeps an isotropic type of interactions, leading to structures which can be represented as densely packed spheres.
Twenty irregular tetrahedra pack with a common vertex in such a way that the twelve outer vertices form a regular icosahedron.
There are one hundred and twenty vertices which all belong to the hypersphere S3 with radius equal to the golden ratio (φ = 1 + √5/2) if the edges are of unit length.
