Landau theory for spin transition and ordering phenomena in Fe(II) compounds

We analyze spin transitions in molecular crystals with the help of Landau theory. Experimental observations show that a pure spin transition may be described as an isostructural transition with a totally symmetric order parameter. For long-range ordering of different spin states, a model is developed in which the spin transition is coupled with a structural transformation. The resulting phase diagrams are in good qualitative agreement with experiments.

Figure 1

Transition curves for an isostructural transformation according to Eq. (3). The upper half-plane $(x>0)$ corresponds to the HS phase, the lower half-plane $(x<0)$ to the LS phase. The smooth crossover from HS to LS $({\alpha }_2>0)$ changes to a first-order transformation $({\alpha }_2⩽0)$ at $({\alpha }_2=0)$. The hysteresis region is bracketed by arrows and stability limits are shown as circles.

Figure 2

Phase diagram for an isostructural spin transition. The gray area between the spinodal lines (dashed) marks the region of coexistence of phases; i.e., domains for which the Landau potential [Eq. (1)] has more than one minimum. The bold line ${\alpha }_1=0$, ${\alpha }_2⩽0$ ending at the critical point ${\alpha }_2={\alpha }_1=0$ delineates the first-order transition regime. The different orientations of the ${\alpha }_2$ and ${\alpha }_1$ axes in the $(p,T)$ plane [Eq. (2)] correspond to: (a) $m_1<0$, $n_1>0$, $m_2>0$, $n_2>0$, (b) $m_1<0$, $n_1>0$, $m_2<0$, $n_2<0$, (c) $m_1<0$, $n_1<0$, $m_2>0$, $n_2<0$.

Figure 3

Experimental hysteresis widths as a function of pressure, (a) $\mathrm{Fe}{\left(\mathrm{btr}\right)}_2{\left(\mathrm{NCS}\right)}_2\bullet {\mathrm{H}}_2\mathrm{O}$ (Ref. 6), (b) $\mathrm{Fe}{\left(\mathrm{phy}\right)}_2{\left({\mathrm{BF}}_4\right)}_2$ (Ref. 9), and (c) $\mathrm{Fe}{\left(\mathrm{PM}\text{−}\mathrm{PEA}\right)}_2{\left(\mathrm{NCS}\right)}_2$ (Ref. 10). The gray zones emphasize the convergence of the domain of phase coexistence towards the critical point.

Figure 4

The bold solid line corresponds to normal spin crossover behavior according to Eq. (3) and Fig. 1 $({\alpha }_2>0)$. The dashed and thin solid lines indicate parabolas $m{\alpha }_1+n{\alpha }_2+k+{\lambda }_{1}x+{\lambda }_2{x}^2=0$ for negative and zero linear-quadratic coupling, respectively. Inside of the parabolas the condition for the appearance of an IP is fulfilled. The bold dashed line, which has physical meaning only within the hatched area $(y^2>0)$, describes the spin transition curve in the IP.

Figure 5

Generic phase diagram for a two-step spin transition. The orientation of the $({\alpha }_1,{\alpha }_2)$ axes in the $(p,T)$ plane follows from Eq. (2) with the conditions $m_1<0,n_1>0,m_2>0,n_2>0$. The spinodal and second-order lines have been calculated as described in the Appendix $(n=2,m=-0.5,k=-10,{\lambda }_2=45,{\beta }_2=80)$. In the light gray areas the free energy has two minima, in the dark gray areas it has three. Critical points are denoted as follows: (1) ${\mathrm{A}}^{+}=({\alpha }_1^{A+},{\alpha }_2^{A+})$ and $A^{-}=({\alpha }_1^{A-},{\alpha }_2^{A-})$ separate regions of first and second order HS-IP and LS-IP transitions, (2) in the region delineated by $B=(0,{\alpha }_2^B=-k∕n)$, $B^{\prime }=(0,{\alpha }_2^{B^{\prime }}=2k∕({\lambda }_2-2n))$, $A^{+}$ and $A^{-}$ the IP is the stable phase, (3) $C=[0,{\alpha }_2^C=-k{\lambda }_2∕(n{\lambda }_2-2{\beta }_2)]$ terminates the region of metastability of the IP, (4) $D=[0,{\alpha }_2^D=k∕(\sqrt{\left({\beta }_2\right)}-n)]$ is the triple point. The vertical dash-dotted lines indicate the isobaric regimes discussed in the text.

Figure 6

Temperature dependence of the OPs $x$ and $y^2$ corresponding approximately to the isobaric regime 3 of Fig. 5. Panels (a) and (b) represent the experimental transition curve and the intensity of superstructure reflections observed for $\left[\mathrm{Fe}{\left(2\text{−}\mathrm{pic}\right)}_3\right]{\mathrm{Cl}}_2\bullet \mathrm{Et}\mathrm{O}\mathrm{H}$, panels (c) and (d) show the curves calculated from the Landau model, the rotation of $({\alpha }_1,{\alpha }_2)$ axes relative to $p$ and $T$ is neglected $(m_1=-1,n_1=0,m_2=0,n_2=1,{\beta }_2=1)$.