Today: Monday 2 August 2021 , 9:39 am


advertisment
search




Geometrical frustration

Last updated 4 Month , 19 Day 36 Views

Advertisement
In this page talks about ( Geometrical frustration ) It was sent to us on 15/03/2021 and was presented on 15/03/2021 and the last update on this page on 15/03/2021

Your Comment


Enter code
  In condensed matter physics, the term geometrical frustration (or in short: frustrationThe psychological side of this problem is treated in a different article, frustration) refers to a phenomenon, where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favoring rather simple, but different structures) lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets.
The term frustration, in the context of magnetic systems, has been introduced by Gerard Toulouse (1977). Indeed, frustrated magnetic systems had been studied even before. 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. Related features occur in magnets with competing interactions, where both ferromagnetic as well as antiferromagnetic couplings between pairs of spins or magnetic moments are present, with the type of interaction depending on the separation distance of the spins. In that case commensurability, such as helical spin arrangements may result, as had been discussed originally, especially, by A. Yoshimori, T. A. Kaplan, R. J. Elliott, and others, starting in 1959, to describe experimental findings on rare-earth metals. A renewed interest in such spin systems with frustrated or competing interactions arose about two decades later, beginning in the 1970s, in the context of spin glasses and spatially modulated magnetic superstructures. In spin glasses, frustration is augmented by stochastic disorder in the interactions, as may occur, experimentally, in non-stoichiometric magnetic alloys. Carefully analyzed spin models with frustration include the Sherrington–Kirkpatrick model, describing spin glasses, and the ANNNI model, describing commensurability magnetic superstructures.

Magnetic ordering


Geometrical frustration is an important feature in magnetism, where it stems from the relative arrangement of spins. A simple 2D example is shown in Figure 1. 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. Once the first two spins align antiparallel, the third one is frustrated because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy.
Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration. If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel. There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and between 3 and 4) are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated.
Geometrical frustration is also possible if the spins are arranged in a non-collinear way. If we consider a tetrahedron with a spin on each vertex pointing along the easy axis (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away. 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. There are three different equivalent arrangements with two spins out and two in, so the ground state is three-fold degenerate.

Mathematical definition


The mathematical definition is simple (and analogous to the so-called Wilson loop in quantum chromodynamics): One considers for example expressions ("total energies" or "Hamiltonians") of the form
\mathcal H=\sum_G -I_{k_\nu , k_\mu\,\,S_{k_\nu\cdot S_{k_\mu\,,
where G is the graph considered, whereas the quantities are the so-called "exchange energies" between nearest-neighbours, which (in the energy units considered) assume the values ±1 (mathematically, this is a signed graph), while the are inner products of scalar or vectorial spins or pseudo-spins. If the graph G has quadratic or triangular faces P, the so-called "plaquette variables" PW, "loop-products" of the following kind, appear:
P_W=I_{1,2\,I_{2,3\,I_{3,4\,I_{4,1 and P_W=I_{1,2\,I_{2,3\,I_{3,1\,, respectively,
which are also called "frustration products". One has to perform a sum over these products, summed over all plaquettes. The result for a single plaquette is either +1 or −1. In the last-mentioned case the plaquette is "geometrically frustrated".
It can be shown that the result has a simple gauge invariance: it does not change – nor do other measurable quantities, e.g. the "total energy" \mathcal H – even if locally the exchange integrals and the spins are simultaneously modified as follows:
I_{i,k\to\varepsilon_i I_{i,k\varepsilon_k ,\quad S_i\to\varepsilon_i S_i ,\quad S_k\to \varepsilon_k S_k\,.
Here the numbers εi and εk are arbitrary signs, i.e. +1 or −1, so that the modified structure may look totally random.

Water ice

frameleftFigure 5: Scheme of water ice molecules
Although most previous and current research on frustration focuses on spin systems, the phenomenon was first studied in ordinary ice. In 1936 Giauque and Stout published The Entropy of Water and the Third Law of Thermodynamics. 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. The entropy was calculated by integrating the heat capacity and adding the latent heat contributions; the low temperature measurements were extrapolated to zero, using Debye's then recently derived formula. The resulting entropy, S1 = 44.28 cal/(K·mol) = 185.3 J/(mol·K) was compared to the theoretical result from statistical mechanics of an ideal gas, S2 = 45.10 cal/(K·mol) = 188.7 J/(mol·K). The two values differ by S0 = 0.82 Â± 0.05 cal/(K·mol) = 3.4 J/(mol·K). This result was then explained by Linus Pauling to an excellent approximation, who showed that ice possesses a finite entropy (estimated as 0.81 cal/(K·mol) or 3.4 J/(mol·K)) at zero temperature due to the configurational disorder intrinsic to the protons in ice.
In the hexagonal or cubic ice phase the oxygen ions form a tetrahedral structure with an O–O bond length 2.76 Ã… (276 pm), while the O–H bond length measures only 0.96 Ã… (96 pm). Every oxygen (white) ion is surrounded by four hydrogen ions (black) and each hydrogen ion is surrounded by 2 oxygen ions, as shown in Figure 5. Maintaining the internal H2O molecule structure, the minimum energy position of a proton is not half-way between two adjacent oxygen ions. There are two equivalent positions a hydrogen may occupy on the line of the O–O bond, a far and a near position. Thus a rule leads to the frustration of positions of the proton for a ground state configuration: for each oxygen two of the neighboring protons must reside in the far position and two of them in the near position, so-called ‘ice rules’. Pauling proposed that the open tetrahedral structure of ice affords many equivalent states satisfying the ice rules.
Pauling went on to compute the configurational entropy in the following way: consider one mole of ice, consisting of N O2− and 2N protons. Each O–O bond has two positions for a proton, leading to 22N possible configurations. However, among the 16 possible configurations associated with each oxygen, only 6 are energetically favorable, maintaining the H2O molecule constraint. Then an upper bound of the numbers that the ground state can take is estimated as Ω 
 
Comments

There are no Comments yet

last seen
Most vists