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

Chapter 3 Octupolar Ordering of Γ 8 Ions 47
+ 2 exp (Bjq/t) cosh (y H H/t)] . (3.22)
Here g H and y H are the two parameters of the Zeeman splitting scheme of
the Γ 8 subspace (see Appendix B). The overall shape of the phase boundary
in the t–H plane can be obtained by expanding free energy F(T , q, H) in
powers of T , and identifying the coefficient of the T 2 -term
α(H, t) = 1 2 −
A2
2g H H · sinh (g H H/t)
cosh (g H H/t) + cosh (y H H/t) . (3.23)
Solving α(H, t) = 0 gives a line of continuous transitions in the t–H plane
(right part of Fig. 3.5).
The free energy expansion with respect to the order parameters and the
magnetic field is
F(T , q, H) ≈ F(T , q, H = 0)
+ g2 H − yH
2 BjqH 2 − g2 H
4t 2
12t 4 H2 A 2 BjqT 2
+ g2 H + 3y 2 H
48t 3 A 2 T 2 H 2 . (3.24)
The first term in the second line describes field-induced Γ 5 quadrupoles.
The absence of T H term from the free energy expansion gives that sharp
octupolar phase transition is possible even in magnetic field. Note that this
is a property of the model and H‖(111). In general, time reversal invariance
would allow the existence of T H term.
Following the same calculation as we did in the H = 0 case, for the
secondary order parameter we get
B
q ≈ −
4t(t − B 2 j) · (
A 2 T 2 + (gH 2 − yH)H 2 2) . (3.25)
Replacing this into the free energy (3.24) and minimizing it with respect to
T , we get the primary order parameter in the field
T
≈
=
√ 8t(B2 j − t)
( ) A · √
2
B 2 A 4 j 2 − t
√
8t(B2 j − t)
·
B 2 A 4 j
+ A2
24
(A 2 − 8B 2 j)gH 2 + 3A 2 yH
2 H
t 2 (B 2 j − t)
2
√
(t oc (H = 0) − t) − a H H 2 . (3.26)
The second term of the product is the reduced temperature in magnetic field.
Replacing t = t oc (H = 0) = A 2 /2 into the expression of parameter a H , it