Strange relaxation and metastable behaviours of the Ising ferromagnetic thick cubic shell

TL;DR

This work investigates how the thickness $\Delta$ of a thick Ising ferromagnetic cubic shell affects both equilibrium and nonequilibrium magnetic behavior using Monte Carlo Metropolis dynamics. The authors show that the pseudo-critical temperature $T_c^p$ increases with $\Delta$ and is well described by $T_c^p(\Delta)=a\tanh(b\Delta)+c$, with $a=2.98\pm0.19$, $b=0.321\pm0.026$, $c=1.43\pm0.19$, approaching the 3D Ising value for large $\Delta$. In the nonequilibrium regime, the relaxation time $\tau_{relax}$ decreases with thickness and exhibits three regimes: rapid fall, plateau, and linear decrease, indicating a rich thickness-dependent relaxation dynamics. Metastable lifetimes $\tau_{meta}$ and reversal times $\tau_{rev}$ vary non-monotonically with $\Delta$, with a thickness range $\Delta\approx 3$–$5$ that maximizes metastability; these results highlight geometry as a control parameter for magnetic time scales and phase behavior, with possible relevance to ferromagnetic samples containing cavities.

Abstract

We have studied the equilibrium and nonequilibrium behaviours of the Ising ferromagnetic thick cubic shell by Monte Carlo simulation. Our goal is to find the dependence of the responses on the thickness of the shell. In the equilibrium results, we found that the pseudo-critical temperature of ferro-para phase transition of thick cubic shell increases with the increase of the thickness following a hyperbolic tangent relation. In the nonequilibrium studies, the relaxation time has been found to decrease with the increase of the thickness of the cubic shell. Here three different regimes are found, namely rapid fall, plateau and linear region. The metastable behaviour has been studied also as another kind of non-equilibrium response. The metastable lifetime has been studied as function of the thickness of the cubic shell. A non-monotonic variation of metastable lifetime with the thickness of the shell is observed. A specified thickness for longest-lived metastability has been identified.

Figures (7)

• Figure 1: A schematic cross section of a cubic shell of size $L$ and thickness $\Delta$. Total lattice sites $N_s = L^3 - (L-2\Delta)^3$. Here we have demonstrated a cubic shell of $L=10$ having thickness $\Delta = 3$. Inside the shell there exists non-magnetic hollow space.
• Figure 2: Thermodynamic quantities i.e., (a) equilibrium magnetisation $M$ and (b) susceptibility $\chi$ are studied as the functions of temperature ($T$) and demonstrated for Ising ferromagnetic cubic shell of size $L=50$ having different thickness ($\Delta$).
• Figure 3: The variation of pseudo-critical temperature ($T_c^p(\Delta)$) with thickness $\Delta$ of the cubic shell $T_c^p(\Delta)$ and studied as a '$\tanh$' function of $\Delta$ i.e., $T_c^p(\Delta)=a\times \tanh(b~\Delta) + c$. The best fitted line is shown in solid green line. The fitted parameters : $a = 2.98 \pm 0.19$, $b = 0.321 \pm 0.026$ and $c = 1.43 \pm 0.19$.
• Figure 4: (a) The decay of magnetisation $m(t)$ with time ($t$) in the Ising ferromagnetic thick cubic shell, where external field $h_{ext} = 0$ and temperature ($T$) of the system fixed at $T=1.10T_c^p(\Delta)$ (paramagnetic phase). Relaxation of magnetisation are plotted for the different values of shell-thickness $\Delta$. Results are obtained by averaging over 8000-25000 random samples. (b) The semi-logarithmic plot of decay of magnetisation for selected values of shell-thickness. $\Delta=$ 1, 2, 5, 20 and 25. The data are fitted to a exponential function $f(x)=a\exp(-cx)$, represented by solid lines. Here $f(x)$ stands for $m(t)$ and $x$ represents $t$. The exponential nature of magnetic relaxation is evident (equation-4). It may be noted here that all data points (in (a)) are not shown in (b).
• Figure 5: Dependence of Relaxation time $\tau_{\rm relax}$ on the thickness of shell $\Delta$. The systems are kept at $T=f \times T_c^p (\Delta)$. We have considered $f=1.10$ and $1.15$ for all values of the thickness($\Delta$) of the shell. Three distinct regimes are identified, namely rapid fall($\Delta=$ 1-6), plateau ($\Delta=$ 6-20) and linear region ($\Delta=$ 20-25).