Theory of magnetic deflagration in crystals of molecular magnets

Theory of magnetic deflagration (avalanches) in crystals of molecular magnets has been developed. The phenomenon resembles the burning of a chemical substance, with the Zeeman energy playing the role of the chemical energy. Nondestructive reversible character of magnetic deflagration, as well as the possibility to continuously tune the flammability of the crystal by changing the magnetic field, makes molecular magnets an attractive toy system for a detailed study of the burning process. Besides simplicity, new features, as compared to the chemical burning, include possibility of quantum decay of metastable spin states and strong temperature dependence of the heat capacity and thermal conductivity. We obtain analytical and numerical solutions for criteria of the ignition of magnetic deflagration and compute the ignition rate and the speed of the developed deflagration front.

Figure 1

Energy of a molecular magnet as function of $s_z$. Quantum energy levels are shown by black circles for $S=4$.

Figure 2

The dependence of the magnetic field that ignites deflagration on the temperature of the crystal of ${\mathrm{Mn}}_{12}$. Dimensionless parameters are given by Eqs. (3, 29).

Figure 3

Plot of $\delta \left({\theta }_{\max }\right)$ for $d=1,2,3$ that allows one to obtain the ignition threshold from the maximum of these curves.

Figure 4

Numerical results for temperature and heat-release profiles at the ignition threshold, $\delta ={\delta }_c$, in the model with bias-field gradient.

Figure 5

Dependence of the ignition threshold on the bias-field gradient, see Eq. (49).

Figure 6

Numerical results for (a) ${\delta }_c$ and (b) $x_{\max ,c}$ vs the temperature-bias parameter ${\theta }_1$ for the $1d$ model with different temperatures at the ends. $x=0$ corresponds to the center of the sample. The dashed line in (a) is the asymptote ${\delta }_c\cong {(\mid {\theta }_1\mid +2\ln 2)}^2∕16+1∕4$ at $\mid {\theta }_1\mid ⪢1$.

Figure 7

(Color online) Ignition rate ${\Gamma }_{\mathrm{ig}}$ vs $1∕\delta$ in the $1d$ model with different temperatures at the ends. Parameter ${\theta }_1$ is given by Eq. (22).

Figure 8

(Color online) Ignition rate ${\Gamma }_{\mathrm{ig}}$ vs $1∕\delta$ in the $1d$ model beyond the explosive approximation with equal temperatures at the ends.

Figure 9

Ignition rate ${\Gamma }_{\mathrm{ig}}$ vs $1∕\delta$ in the $1d$ model with uniform initial condition $\theta ={\theta }_1<0$ followed by a rapid temperature rise to $\theta =0$ at the ends of the sample.

Figure 10

Ignition rate ${\Gamma }_{\mathrm{ig}}$ vs ${\theta }_1$ in the semi-infinite $1d$ model, obtained numerically from Eq. (70).

Figure 11

(Color online) Analytical results for the deflagration front in molecular magnets in the high-barrier case, $\tilde{v}=0.3$ (i.e., ${\nu }_f=1∕{\tilde{v}}^2\simeq 11$).

Figure 12

(Color online) Numerical results for the reduced deflagration speed $\tilde{v}$ vs the flame’s Arrhenius exponent $W_f$ for different temperature dependences $C_{\mathrm{ph}}\propto T^{\alpha }$ and $\kappa \propto T^{-\beta }$. Dashed lines are the $W_f⪢1$ asymptotes $\tilde{v}\cong \sqrt{(\alpha +1)∕W_f}$.

Figure 13

(Color online) Ignition and deflagration in the uniform $1d$ model for $\alpha =3$, $W_f=10$, ${\tilde{T}}_0=0.4$, and $\tilde{R}=1000$.

Figure 14

(Color online) Time dependence of the average metastable population in the uniform $1d$ model for $\alpha =3$, $W_f=10$, $\tilde{R}=1000$, and different ${\tilde{T}}_0$.

Figure 15

(Color online) Ignition and deflagration in the $1d$ model with temperature gradient. The parameters are $\alpha =3$, $W_f=10$, ${\tilde{T}}_0=0.4$, and $\tilde{L}=2\tilde{R}=1000$. Initial state is the stationary state in the absence of heat release with $T=T_0$ at the left end and $T=0$ at the right end. Ignition occurs after the ignition time ${\tilde{\tau }}_{\mathrm{ig}}\approx 3000$ near the hot left end $(\tilde{x}=0)$, and then the deflagration front propagates toward the right cold end.

Figure 16

Time dependence of the average metastable population $\tilde{n}$ in the $1d$ model with temperature gradient for $\alpha =3$, $W_f=10$, $\tilde{R}=1000$, and different ${\tilde{T}}_0$. The slope of $\tilde{n}\left(\tau \right)$ at the steps is proportional to the front speed $\tilde{v}$.

Figure 17

(Color online) (a) Variation of the metastable population $\tilde{n}$ during ignition and deflagration in the $1d$ model with the temperature quickly raised up to ${\tilde{T}}_0$ at both edges, whereas the initial temperature is zero. The parameters are $\alpha =3$, $W_f=10$, ${\tilde{T}}_0=0.5$, and $\tilde{L}=2\tilde{R}=1000$. Ignition occurs after the ignition time ${\tilde{\tau }}_{\mathrm{ig}}\approx 4500$ near the edges and then the deflagration fronts propagate toward the center and meet there. (b) Temperature variation during ignition and deflagration in the same case as above. One can see how the heat propagates into the sample, and the width of the region with elevated temperature near the edges increases.