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Phase Transitions

The effect of the heating rate on the phase transition

Aycan Özkan, Bülent Kutlu

Published Mar 10, 2016• 6 citations
Publisher

Abstract

The simple cubic spin-1 Ising model exhibits the ferromagnetic (F)–ferromagnetic (F) phase transition in the low temperature region for the interval 1.40 < d = D/J < 1.48 at k = K/J = –0.5. The degree of the F-F phase transition determines the special point on the (kBT/J, d) phase diagram. In this paper, the critical behavior of the F-F phase transition was investigated for different heating rates using the cellular automaton heating algorithm. The universality class and the type of F-F phase transition were analyzed using the finite-size scaling theory and the power law relations. The results show that the F-F phase transition may be the second order, the first order or the weak first order depending on the heating rate in the interval 1.40 < d < 1.48 for k = –0.5.

DOI
10.1080/01411594.2016.1156678

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The simple cubic spin-1 Ising model exhibits the ferromagnetic (F)–ferromagnetic (F) phase transition in the low temperature region for the interval 1.40 < d = D/J < 1.48 at k = K/J = –0.5. The degree of the F-F phase transition determines the special point on the (kBT/J, d) phase diagram. In this paper, the critical behavior of the F-F phase transition was investigated for different heating rates using the cellular automaton heating algorithm. The universality class and the type of F-F phase transition were analyzed using the finite-size scaling theory and the power law relations. The results show that the F-F phase transition may be the second order, the first order or the weak first order depending on the heating rate in the interval 1.40 < d < 1.48 for k = –0.5.
The fcc spin-1 Ising (BEG) model has a dense ferromagnetic (dfdfdf) ground state instead of the ferromagnetic ground state at low temperature region and exhibits the dense ferromagnetic (dfdfdf) - ferromagnetic (FFF) phase transition for d=D/J=2.9d=D/J=2.9d=D/J=2.9, k=K/J=−0.5k=K/J=-0.5k=K/J=−0.5, ℓ=L/J=0\ell =L/J=0ℓ=L/J=0 and h=H/J=0h=H/J=0h=H/J=0. The critical behavior of the dense ferromagnetic (dfdfdf) - ferromagnetic (FFF) phase transition has been investigated using the cellular automaton cooling and heating algorithms. The universality class and the type of the dense ferromagnetic (dfdfdf) - ferromagnetic (FFF) phase transition have been researched within the framework of the finite - size scaling, the power law relations and the probability distribution. The results show that the dense ferromagnetic- ferromagnetic phase transition is of the second order and the model shows universal second order Ising critical behavior at d=2.9d=2.9d=2.9 parameter value through k=−0.5k=-0.5k=−0.5 line.
The spin-3/2 Ising model on the simple cubic lattice with nearest-neighbour ferromagnetic bilinear interaction (J > 0) is simulated on a cellular automaton by using the cooling algorithm improved from the Creutz cellular automaton. The phase diagrams of the model are constructed in the (D/J, kT/J) and (K/J, kT/J) plane. Comparison of the results are made with those of other methods. The temperature dependence of the order parameters and associated fluctuations are calculated at various of the model parameters and the static critical exponents are estimated within the framework of the finite-size scaling. The results are compatible with the universal Ising critical behavior except for D/J =−3 and K/J =−2.3.
The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=${\ensuremath{\Omega}}^{+}$\ensuremath{\cup}${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$. A finite-size-scaling analysis reveals that the transition remains second order with pure critical exponents for regularly spaced defects, the critical temperature varying with the symmetry of \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, however, makes the transition weakly first order; the transition becomes more abrupt for defects located fully at random, and the long-range order is suppressed when the numbers of defects in ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.
The mixed spin 3–spin 3/2 ferrimagnetic Ising model was simulated using cooling algorithm on cellular automaton (CA). The simulations were carried out in the intervals −4 ≤ DA/J ≤ 8 and −4 ≤ DB/J ≤ 8 for the square lattices with periodic boundary conditions. The ground-state phase diagram of the model has different types of ferrimagnetic phases. Although only the antiferromagnetic nearest-neighbor interaction was contained in the Hamiltonian, the compensation points emerged through DA/J = 2 at kT/J = 0. The values of the critical exponents (ν, α , β and γ) were estimated within the framework of the finite-size scaling theory and power-law relations for the selected DA/J values (−2, 0, 1, 2, and 4). The estimated critical exponent values were in good agreement with the universal values of the two-dimensional Ising model (ν = 1, α = α′ = 0, β = 0.125, β′ = 0.875 and γ = γ′ = 1.75).
The mixed spin 3- spin 3/2 Ising model has been simulated using cooling algorithm on cellular automaton (CA). The simulations have been made in the interval -6<=D<=6 for J=1 for the square lattices with periodic boundary conditions. The ground state phase diagram of the model has different type ferrimagnetic orderings. Through D/J=2 line, compensation points occurs at kT/J=0. The values of the critical exponents ( ν, α, β and γ) are estimated within the framework of the finite-size scaling theory and power law relations for selected D/J values (-2, 0, 1, 2 and 4). The estimated critical exponent values are in a good agreement with their universal values of the two dimensional Ising model.