Ph.D. THESIS Multipolar ordering in f-electron systems

Chapter 3 Octupolar Ordering of Γ 8 Ions 43
500
para phase
t
quadrupole
order
octupole and quadrupole order
0
j
10
Figure 3.3: The mean-field phase diagram of the zero-field quadrupolar-octupolar model
(3.10) in the quadrupolar coupling–temperature plane (j = J quad /J oc and t = T k B /J oc ))
[2]. The dashed and continuous lines signify first and second order phase transitions,
respectively. Observe the regime of first-order transitions bounded by two tricritical points
(marked by black dots).
Phase diagram
In this Landau mean-field approximation the critical temperatures of a continuous
phase transition are defined as the changing of sign in the coefficient
of either of the quadratic terms. Below the critical temperature, for small j
quadrupolar coupling constant values, a mixed octupolar-quadrupolar, while
for large j values pure quadrupolar order develops at first. In the case of
intermediate j coupling values there is a regime where the transition is first
order. Fig. 3.3 shows the mean-field phase diagram of the model in the
quadrupolar coupling-temperature (j − t) plane, and it shows the features
mentioned above.
Let us examine the cases of small and large j values separately.
• weak-j limit
The critical temperature determined from the vanishing of the coefficient
of T 2 term is t oc = A 2 /2. At t oc , octupolar moment appears as
the primary order parameter, but there is also induced quadrupolar
order. Minimizing F with respect to q, we get
q = BA2 T 2
4t(jB 2 − t)
(3.15)
Substituting this expression of the quadrupolar moment into the energy
expansion (3.14), we can see that the terms which contains the