• View
• Edit
• History
• Print

SparseNotes

< Pippard corrections? | Index | Material project, VESTA and Bilbao >

• Following Khomskii Ch. 2 Landau theory of phase transitions. Discussing second order phase transitions

{$$\Phi = \Phi_0 + a(T-T_c)\eta^2 + B\eta^4$$} with {$B>0$} yields a second order (continuous) transition with {$$\Phi_{min} = \begin{cases} \Phi_0 \qquad\qquad\qquad\qquad T>T_c\\ \Phi_0 - \frac {a^2}{2B} (T_c-T)^2\quad T<T_c\end{cases}$$}

• Weak first order transitions arise from

{$$\Phi = \Phi_0 + a(T-T_c)\eta^2 + C\eta^3 + B\eta^4 $$} Two temperatures emerge:

• {$T^*$} below which a new local minimum appears at {$\eta\ne0$}(while the global minimum is still for {$\eta=0$}
• {$T^{**}$}: for {$T^{**}<T<T^*$} two minima exist and below {$T^{**}$} the only minimum is for {$\eta\ne0$}

In between these two temperatures the system will jump to the {$\eta\ne 0$} state depending on cooling history and the two limit undercooling of the {$\eta=0$} and overheating of the {$\eta\ne 0 $} solutions.

• Another type of first order when {$C=0$} and {$B$} is negative but e.g. pressure dependent. Then suppose that (e.g. at high pressure) transitions to finite order occur when

{$$\Phi = \Phi_0 + a(T-T_c)\eta^2 + B\eta^4 + D\eta^6$$} with {$D>0$}. Three minima appear just above {$T_c$} (Fig.2.8): for higher {$T$} the global minimum is at {$\eta=0$}, e.g. the {$\eta\ne0$} solutions might be reached by overheating the low temperature ordered phase. If {$B$} becomes positive reducing pressure, the point where {$B$} changes sign is a tricritical point (Fig 2.9).

• A notable example of tricritical point is in a superconducting phase diagram if one could tune (e.g. by pressure) {$\kappa$}. Just below {$\kappa =1/\sqrt 2$} where type I switches to type II. The latter display second order transitions while the former are first order.
• UN in an antiferromagnet with {$T_N=52$} K. In an external field it shows a metamagnetic transition (the spin order changes above a certain field value). The metamagnetic transition has a tricritical point at {$T=24$} K and {$\mu_0 H=52$} T, i.e. it changes from I to II order at that point (Shrestha et al. Sci Rep. 7, 6642)
• Other examples He3-He4 mixtures, Flux lattice melting, Nematic-SmecticA liquid crystal transition, Cold atom Fermi gas, Metamagnetic transition in FeCl2, Order-disorder structural transition, Ordering of CO on graphite surface