Vector models

Guida R and Zinn-Justin J 1998 Critical exponents of the A/-vector model J. Phys. A Mathematical and General 31 8103-21... 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

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 model has not been studied very intensely so far in particular, none of the features which are characteristic for amphiphilic systems have been recovered yet. However, it is close enough to the successful vector models and simple enough that it might be a promising candidate for off-lattice simulations of idealized amphiphilic systems in the future. 

JSji and j2 were vectors and j their resultant, the triangle whose sides are ju j2, and j3 would not dose unless j were limited by Eq. (7-43) hence, the name triangle condition, and hence, also, the analogy between quantum mechanics and the vector model. 

A pictorial representation of the Tg-S mixing process follows from Fig. 6. Just as in normal n.m.r. or e.s.r. spectroscopy, precession can be represented by a vector model. When placed in an external magnetic field the two unpaired electrons of the radical pair 1 and 2 will precess... 

Parameter vector Independent variables Output vector Model Equation... 

Khakhar, D. V., and Ottino, J. M., A note on the linear vector model of Olbricht, Rallison, and Leal as applied to the breakup of slender axisymmetric drops. J. Non-Newtonian Fluid Mech 21,127-131 (1986b). 

Guth, S. L. et al. (1980). Vector model for normal and dichromatic color vision. Journal of the Optical Society of America 70 197-212. 

FIGURE 6.1 The vector model used to calculate dipole moments from bond moments. 

If polarized light passes through a medium that exhibits optical rotation, the motion along one of the circular vectors is slower than that of the other. The resultant vector is thus displaced from the original vector by some angle, . Figure 16.5 shows the vector model in which the phase difference is [Pg.588]

To verify the theory of PIP, a computer program using C language was developed. It can be used to directly calculate the excitation profiles by PIPs or any other RF pulses. The calculation is based on the Bloch vector model for a non-interacting spin-1/2 system, where the spin-lattice relaxation during the pulse is neglected. The basic idea of the program is discussed as follows. 

Holme, T. A., and Levine, R. D. (1988), An Algebraic Hamiltonian for Electronic Nuclear Degrees of Freedom Based on the Vector Model, Inti. J. Quant. Chem. 34, 457. 

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

Fig. 6. Vector model coupling of spin and orbital angular momenta for which L = 2, S = 1...

Fig. 5. Vector model coupling of the spin angular momenta of two, three, and. four electrons...

This kind of representation for a single nuclear spin is absolutely equivalent to the classic vector model. 

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