Ising ferromagnets

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Fig. 5. Optimized temperature sets for the two-dimensional Ising ferromagnet. The initial temperature set with 20 temperature points is determined by a geometric progression for the temperature interval [0.1,10]. After feedback of the local diffusivity the temperature points accumulate near the critical temperature Tc = 2.269 of the phase transition dashed line). Similar to the ensemble optimization in energy space the feedback of the local diffusivity relocates resources towards the bottleneck of the simulation...

Fig. 18. Snapshot pictures of a nearest-neighbor Ising ferromagnet on the square lattice with bulk field H = 0 and boundary fields H] — Hi. — 0 (this model is isomorphic to the c(2x2) ordering at coverage 0 = 0.5) at the temperatures T — 0.95 Tc (a), T = Tc (b) and T = 1.05 Ft (c), for a L x M system with L — 24, M — 288 and two free boundaries of length M, while periodic boundary conditions arc used along the strip. Up spins (adatoms on sublattice 1) are shown in black, down spins (adatoms on sublattice 2) are shown in white. Domain formation at T 5 Tc can be clearly recognized. From Albano ei a . (1989b).

Fig. 65. (a) Boundary conditions used to impose at tilled interface in an Ising ferromagnet antiperiodic (APBC) in the z-direction, periodic in the y-direction (PBC), and screw periodic boundary conditions (SPBC) in the x-direction. 

When 2s g > (s + Sgg), the binary alloy corresponds to an Ising ferromagnet (J> 0) and the system splits into two phases one rich in A and the other rich in component B below the critical temperature T. On the other hand, when 2s g < (Sy + Sgg), the system corresponds to an antiferromagnet the ordered phase below the critical temperature has A and B atoms occupying alternate sites. 

Fig. 2. Molecular magnetic fields in the Ising ferromagnets TbES and DyES in the V-c plane.

The critical exponents a, j8, y, 5, are expressed in the perturbation expansion in terms of the parameter e=d -d, and the expansion coefficients depend upon an order parameter dimensionality. In the diluted systems d = 6, in tricritical points and in three-dimensional dipolar-coupled Ising ferromagnets d = 3. Among lanthanide compounds uniaxial and... 

Fig. 26. Forced magnetostriction in the LiTbF4 Ising ferromagnet (7J =2.885 K) in the longitudinal field solid curves represent experimental data obtained at 4.2 K (1) and 1.6 K (2). The...

Fig. 15. Schematic plot of the free energy Fof a spin glass as a function of a phase-space coordinate which measures the projection of the considered state on a particular ordered state. The inset shows the situation in an Ising ferromagnet for comparison.

This evaluation of Z by summing all possible conformations is a gigantic task and can only be accomplished for very short chains. Other methods must be used. Matrix multiplication and a treatment which was used for the Ising ferromagnet can be used. Let us consider a chain with n bonds, each in two rotational states (a or p). The statistical matrices... 

In models of Ising ferromagnets a common method to stabilize one interface between spin-up and spin-down phases is the use of antiperiodic boundary conditions in one direction of the system, unit vector in the z-direction (perpendicular to the interface that is stabilized), and L is the linear dimension of the system perpendicular to the interface. In the x,y-directions parallel to the interface, where one then uses linear dimensions which may be chosen different from Li, standard periodic boundary conditions are used ([Pg.391]

F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91 (1969)... 

Chayes, J. T., Chayes, L., Fisher D. S. and Spencer, T. (1989). Correlation Length Boimds for Disordered Ising Ferromagnets, Commun. Math. Phys. 120, pp. 501-523. 

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