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A Bethe lattice with coordination number z = 3. In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by Hans Bethe in 1935.
Sodium chloride crystal lattice. The concept of lattice energy was originally applied to the formation of compounds with structures like rocksalt and sphalerite where the ions occupy high-symmetry crystal lattice sites. In the case of NaCl, lattice energy is the energy change of the reaction Na + (g) + Cl − (g) → NaCl (s)
For example, NO 2 − is a strong-field ligand and produces a large Δ. The octahedral ion [Fe(NO 2 ) 6 ] 3− , which has 5 d -electrons, would have the octahedral splitting diagram shown at right with all five electrons in the t 2 g level.
Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry.
However, in real materials there are deviations from this in some metals where the unit cell is distorted in one direction but the structure still retains the hcp space group—remarkable all the elements have a ratio of lattice parameters c/a < 1.633 (best are Mg and Co and worst Be with c/a ~ 1.568).
Using the above assumptions, let us examine a homopolymerization reaction starting from a single monomer with z-functional groups with a fraction p of all possible bonds already having been formed. The polymer we create follows the form of a Cayley tree or Bethe lattice – known from the field of statistical mechanics.
For most infinite lattice graphs, p c cannot be calculated exactly, though in some cases p c there is an exact value. For example: for the square lattice ℤ 2 in two dimensions, p c = 1 / 2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, [6] see ...
The effect is largest for cations with high charge and low C.N. (especially when r+/r- approaches the lower limit of the polyhedral stability). Generally, smaller elements fulfill the rule better. [6] As one example, Pauling considered the three mineral forms of titanium dioxide, each with a coordination number of 6 for the Ti 4+ cations.