Isomorphisms

The crystal in Fig. 5.34 illustrates the concept of isomorphism, the possibility that two different motifs fit into the same crystal. For polymers isomorphism of repeating units is most important. Isomorphism of complete chains is seldom possible. Polyfvinyl fluoride) that fits into a crystal of poly(vinylidene fluoride) provides a rare exception. The repeating-unit isomorphism can be separated into three types Type 1 occurs when both homopolymers have die same crystal structure, and a smooth change of the lattice parameters occurs on changing the concentration. An example of type 1 repeating-unit isomorphism is poly(vinylidene fluoride-co-vinyl fluoride). Type 2 is also called isodimorphism and occurs if the homopolymers have different crystal structures. A change in structure occurs at an intermediate concentration. Type 3 occurs if one homopolymer does not crystallize by itself, but participates in the crystal of the other. 

The uncertainty of the propylene copolymer data and the surprisingly smaller effect of the larger side-groups is due to a partial rejection of the side-groups to the crystal surface. 

Titanium nitride is completely and mutually soluble with nitrides of Groups rV and V see Fig. 11.6). It is isomorphous with TiC as carbon can substitute for nitrogen to form a binary solid solution, titanium carbonitride, Ti(CN), over a wide range of composition. The properties of TiCN are comparable to those of TiC and TiN (see Ch. 4, Sec. 6.0 and Fig. 4.8). 

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Tutton salts The isomorphous salts M2 SO4, M S04,6H20 where M is an alkali metal and M is a diposilive transition metal. 

We are particularly concerned with isomorphisms and homomorphisms, in which one of the groups mvolved is a matrix group. In this circumstance the matrix group is said to be a repre.sentation of the other group. The elements of a matrix group are square matrices, all of the same dimension. The successive application of two... 

Obviously, we have chosen the name C2 (M) for the molecular syimnetry group of PH because this group is isomorphic to C... 

Rigid linear molecules are a special case in which an extended MS group, rather than the MS group, is isomorphic to the point group of the equilibrium structure see chapter 17 of [1]. 

Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

Chandler D and Wolynes P 1979 Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids J. Chem. Rhys. 70 2914... 

This plays a critical role in the method of isomorphous replacement, which we discuss below. 

A teclmique that employs principles similar to those of isomorphous replacement is multiple-wavelength anomalous diffraction (MAD) [27]. The expression for the atomic scattering factor in equation (B1.8.2h) is strictly accurate only if the x-ray wavelength is well away from any characteristic absorption edge of the element, in which case the atomic scattering factor is real and Filiki) = Fthkl V- Since the diffracted... 

In this section we look briefly at the problem of including quantum mechanical effects in computer simulations. We shall only examine tire simplest technique, which exploits an isomorphism between a quantum system of atoms and a classical system of ring polymers, each of which represents a path integral of the kind discussed in [193]. For more details on work in this area, see [22, 194] and particularly [195, 196, 197]. 

This is better understood with a picture see figure B3.3.11. The discretized path-integral is isomorphic to the classical partition fiinction of a system of ring polymers each having P atoms. Each atom in a given ring corresponds to a different imaginary tune point p =. . . P. represents tire interatomic interactions... 

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

Arsenic(V) acid, H3ASO4 (strictly, tetraoxoarsenic(V) acid) is obtained when arsenic is oxidised with concentrated nitric acid or when arsenic(V) oxide is dissolved in water. It is a moderately strong acid which, like phosphoric V) acid, is tribasic arsenates V) in general resemble phosphates(V) and are often isomorphous with them. 

The chromates of the alkali metals and of magnesium and calcium are soluble in water the other chromates are insoluble. The chromate ion is yellow, but some insoluble chromates are red (for example silver chromate, Ag2Cr04). Chromates are often isomorph-ous with sulphates, which suggests that the chromate ion, CrO has a tetrahedral structure similar to that of the sulphate ion, SO4 Chromates may be prepared by oxidising chromium(III) salts the oxidation can be carried out by fusion with sodium peroxide, or by adding sodium peroxide to a solution of the chromium(IIl) salt. The use of sodium peroxide ensures an alkaline solution otherwise, under acid conditions, the chromate ion is converted into the orange-coloured dichromate ion ... 

This state exists as a manganate(V), the blue MnO Na3Mn04 lOHjO is isomorphous with Na3V04 (p. 374). 

Ferrate(VI) has powerful oxidising properties, for example ammonia is oxidised to nitrogen. Potassium ferratefVI) is isomorphous with potassium chromatefVI), and both anions are tetrahedral. 

Figure 2 40. To illustrate the isomorphism problem, phenylalanine is simplified to a core without representing the substituents. Then every core atom is numbered arbitrarily (first line). On this basis, the substituents of the molecule can be permuted without changing the constitution (second line). Each permutation can be represented through a permutation group (third line). Thus the first line of the mapping characterizes the numbering of the atoms before changing the numbering, and the second line characterizes the numbering afterwards. In the initial structure (/) the two lines are identical. Then, for example, the substituent number 6 takes the place of substituent number 4 in the second permutation (P2), when compared with the reference molecule.

Redundant, isomorphic structures have to be eliminated by the computer before it produces a result. The determination of whether structures are isomorphic or not stems from a mathematical operation called permutation the structures are isomorphic if they can be interconverted by permutation (Eq. (6) see Section 2.8.7). The permutation P3 is identical to P2 if a mathematical operation (P ) is applied. This procedure is described in the example using atom 4 of P3 (compare Figure 2-40, third line). In permutation P3 atom 4 takes the place of atom 5 of the reference structure but place 5 in P2. To replace atom 4 in P2 at position 5, both have to be interchanged, which is expressed by writing the number 4 at the position of 5 in P. Applying this to all the other substituents, the result is a new permutation P which is identical to P]. 

Thus, the mathematical operation with all combinations of permutations shows the isomorphism of the structures. 

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