Ising interaction

It is actually an oversimplication to assume that the spins in the new block must all have the same orientation. It is sufficient that the new block has two possible states. If an effective Ising interaction exists between the blocks, the relationship between the coupling constants is more complicated, but analytic... 

A next question could be is the interaction energy pair-wise additive, i.e. is the interaction energy a sum of paim>ise interactions ... 

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... 

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... 

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF... 

Most potentiometric electrodes are selective for only the free, uncomplexed analyte and do not respond to complexed forms of the analyte. Solution conditions, therefore, must be carefully controlled if the purpose of the analysis is to determine the analyte s total concentration. On the other hand, this selectivity provides a significant advantage over other quantitative methods of analysis when it is necessary to determine the concentration of free ions. For example, calcium is present in urine both as free Ca + ions and as protein-bound Ca + ions. If a urine sample is analyzed by atomic absorption spectroscopy, the signal is proportional to the total concentration of Ca +, since both free and bound calcium are atomized. Analysis with a Ca + ISE, however, gives a signal that is a function of only free Ca + ions since the protein-bound ions cannot interact with the electrode s membrane. 

Low sulfate selectivity of the ion-selective electrodes (ISE) based on lipophilic quaternary ammonium salts (QAS) is usually explained by unfavorable ratio of sulfate hydration and solvation energies. We have been shown that another reason does exist as well namely, low efficiency of sulfate-QAS cation interaction caused by steric hindrance for simultaneous approach of two QAS cations, containing four long-chain hydrocarbon substituents, to sulfate ion. 

B. Bergersen, Z. Racz. Dynamical generation of long-range interactions Random Levy flights in the kinetic Ising and spherical models. Phys Rev Lett 67 3047-3050, 1991. 

More recently suggested models for bulk systems treat oil, water and amphiphiles on equal footing and place them all on lattice sites. They are thus basically lattice models for ternary fluids, which are generalized to capture the essential properties of the amphiphiles. Oil, water, and amphiphiles are represented by Ising spins 5 = -1,0 and +1. If one considers all possible nearest-neighbor interactions between these three types of particle, one obtains a total number of three independent interaction parameters, and... 

After this short intermezzo, we turn back to introduce the last class of lattice models for amphiphiles, the vector models. Like the three-component model, they are based on the three state Ising model for ternary fluids however, they extend it in such a way that they account for the orientations of the amphiphiles explicitly amphiphiles (sites with 5 = 0) are given an additional degree of freedom a vector with length unity, which is sometimes constrained to point in one of the nearest neighbor directions, and sometimes completely free. It is set to zero on sites which are not occupied by amphiphiles. A possible interaction term which accounts for the peculiarity of the amphiphiles reads... 

The simplest model is the lattice-gas or Ising model. The whole space is divided into a lattice of N sites, and on each site two different states are possible a crystalline state denoted by the variable 5, = 1 and a gaseous state by Sj = -. The variable s denotes the degree of crystalline order. The cohesion of nearest-neighboring solid atoms leads to the following interaction energy... 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

The parameters of the semi-infinite alloy Ising Hamiltonian are the configurationally independent part of the alloy internal energy Eq, the on-site energies the interatomic pair interactions and generally, interatomic interactions of higher order. 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

It should be noted again that ISEs sense the activity, rather than the concentration of ions in solution. The term activity is used to denote the effective (active) concentration of the ion. The difference between concentration and activity arises because of ionic interactions (with oppositely charged ions) that reduce the effective concentration of the ion. The activity of an ion i in solution is related to its concentration, c by... 

A significant development in the methodology of potentiometry that paved the way for its utility in bioanalysis was the discovery of the ion selective electrode (ISE). Conceptually, the ISE involves the measurement of a membrane potential. The response of the electrochemical cell is therefore based on an interaction between the membrane and the analyte that alters the potential across the membrane. The selectivity of the potential response to the analyte depends on the specificity of the membrane interaction for the analyte. 

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