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

Chapter 3 Octupolar Ordering of Γ 8 Ions 45
the ground state moments do not depend on the coupling strengths:
q t→0 = B, and T t→0 = A.
• large-j limit
The first instability occurs now when the coefficient of the q 2 term
changes sign, and pure quadrupolar order sets in at t quad = B 2 j. This
critical line meets the boundary of first-order transitions at the critical
end point j end ≈ 2.75, t end ≈ 177 (Fig. 3.3). For j > j end there are
two phase transitions: the onset of pure quadrupolar order is followed
by the emergence of mixed octupolar–quadrupolar order at t oc . The
lower phase transition is of first order up to the second tricritical point
j tri,2 ≈ 3.75, t tri,2 ≈ 185. For j < j tri,2 , the onset of octupolar order
is reflected in a discontinuity of q (see the right part of Fig. 3.4). For
j > j tri,2 , both transitions are continuous.
We can notice by looking the phase diagram on Fig. 3.3 that the octupolar
transition temperature inside the quadrupolar ordered phase
is less and less affected as we increase the value j, and t oc saturates
at a constant value. Though in this regime, we cannot use Landau
expansion to determine q because the assumption that it is small is not
valid, we may assume that it is near its ground state value B, and use
a low-order expansion in T to obtain in the large-j limit
lim t A 2 exp(Bqj/t oc )
oc = lim
j→∞ j→∞ exp(Bqj/t oc ) + exp(−Bqj/t oc ) = A2 ≈ 246 (3.19)
This is an interesting feature of this octupolar–quadrupolar model because
in familiar phase diagrams of dipolar–quadrupolar models, the
mixed order would be completely suppressed at J quad /J dipole → ∞ (we
will discuss it in detail in Section 4.5). In contrast, we find the finite
saturation value (3.19) as J quad /J oc → ∞. The reason, as we understood
earlier, is that in the Γ 8 subspace the Γ 5 quadrupoles are completely
isotropic. We can also see on Fig. 3.3 that while the octupolar
order induces quadrupoles immediately, it is not true backwards: the
developing of non-zero quadrupolar moments does not mean the appearance
of the octupoles. We will understand this situation in detail
in Section 3.4.