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

Chapter 3 Octupolar Ordering of Γ 8 Ions 49
0
–1 1
c6
Figure 3.7: Zeeman spectrum of the Γ 8 quartet in H‖(111) magnetic field as a function
of the sixth order crystal field coefficient c 6 . We choose the forth order coefficient as
c 4 = 1.1. The Zeeman splitting consists of a doublet and two singlets at a special ratio of
the crystal field coefficients c 6 /c 4 = −0.2422.
transitions. It can be shown that the relation (3.28) remains valid even at a
tricritical point, in which case sometimes the quadratic term in the equation
of the phase boundary is absent. Besides, a tricritical point has a special
set of critical exponents even in Landau theory. For example, the specific
heat will diverge in accordance with its α t = 1/2 mean-field exponent value.
Therefore, it is not straightforward to see how the relation (3.28) will be
satisfied. We will show this in our model not too far below.
The results shown in Fig. 3.6 were derived for J quad = 0, but this is not an
essential restriction. The relationship (3.28) holds everywhere along the lines
of continuous phase transitions shown in Fig. 3.3. As long as we are dealing
with ordinary second order transitions, Landau theory would be consistent
with all the discontinuities appearing in (3.28) being finite.
Approaching a tricritical point ∆C → ∞, and (3.28) allows several scenarios:
• a H ≠ 0 case
In the expression (3.27) of the parameter a H , the coefficient of g H
vanishes at the tricritical point, while in general cases, parameter y H
is non-zero, therefore a H remains finite. Our expectation is that the
divergence of the ∆C causes the same divergence of ∆∂χ/∂T .
• a H = 0 case
For most of the problems dealt with in this Chapter, the special choice
(3.1) of the Γ 8 basis is inconsequential. However, we may regard the