Isomorphous systems

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. 

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. 

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... 

Fig. 6. Effect of alloying elements on the phase diagram of titanium (a) a-stabilized system, (b) P-isomorphous system, and (c) P-eutectoid system.

In the nickel—carbon and cobalt—carbon systems, the nickel carbide (3 1) [12012-02-1], Ni C, and cobalt carbide (3 1) [12011-59-5] C03C, are isomorphous with Fe C and exist only at low temperatures. The manganese—carbon system contains manganese carbide (3 1) [12121 -90-3] Mn C, isomorphous with Fe C, and manganese carbide (23 6) [12266-65-8] isomorphous with chromium carbide (23 6) [12105-81 -6] These... 

Consider, for definiteness, a set of otherwise identical lowest-level components of a system, so that the hierarchy is a tree of constant depth. Since we assume that the components are all identical, the only distinction among the various nodes of the hierarchy consists of the structure of the subtrees. Now suppose we have a tree T that consists of /3 subtrees branching out from the root at the top level. We need to determine the number of different interactions that can occur on each level, independent of the structure of each subtree i.e. isomorphic copies of trees do not contribute to our count. We therefore need to find the number of nonisomorphic subtrees. We can do this recursively. 

One of the most important parameters that defines the structure and stability of inorganic crystals is their stoichiometry - the quantitative relationship between the anions and the cations [134]. Oxygen and fluorine ions, O2 and F, have very similar ionic radii of 1.36 and 1.33 A, respectively. The steric similarity enables isomorphic substitution of oxygen and fluorine ions in the anionic sub-lattice as well as the combination of complex fluoride, oxyfluoride and some oxide compounds in the same system. On the other hand, tantalum or niobium, which are the central atoms in the fluoride and oxyfluoride complexes, have identical ionic radii equal to 0.66 A. Several other cations of transition metals are also sterically similar or even identical to tantalum and niobium, which allows for certain isomorphic substitutions in the cation sublattice. 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

Isentropic change. 75 lsochore, 44, 337 Isolated system, 37 Isomorphous mixture, 417 Isopiestic change, 44, 337 Isopneuma, 442 Isosteres, 442... 

Another important type of condensation polymer are the linear polyesters, such as poly (ethylene terephthalate) (PET) and poly (butylene terephthalate) (PBT). Copolymers of polyesters and PA have been studied in detail, and it has been shown that random copolyesteramides have a low structural order and a low melting temperature. This is even the case for structurally similar systems such as when the group between the ester unit is the same as that between the amide unit, as in caprolactam-caprolactone copolymers (Fig. 3.10).22 Esters and amide units have different cell structures and the structures are not therefore isomorphous. If block copolymers are formed of ester and amide segments, then two melting temperatures are present. 

There seem to be many binary metallic systems in which there are phases of this sort. In the sodium-lead system there are two such phases. One of them, based on the ideal structure Na3Pb, extends from 27 to 30 atomic percent lead, with its maximum at about 28 atomic percent lead and the other, corresponding to the ideal composition NaPb3, extends from 68 to 72 atomic percent lead, with maximum at about 70 atomic percent. The intensities of X-ray reflection have verified that in the second of these phases sodium atoms occupy the positions 0, 0, 0, and the other three positions in the unit cell are occupied by lead atoms isomorphously replaced to some extent by sodium atoms (Zintl Harder, 1931). These two phases are interesting in that the ranges of stability do not include the pure compounds Na8Pb and NaPb3. 

Figure 16, Relation between symmetry in (a) the single space and (b) the double space of a system whose molecular symmetry group has a a plane in the single space, and is isomorphic with C2v in the double space.

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