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singlet which, in common with all singlet states, carries no moment. The
first term in (1.2.24) therefore gives no contribution to the susceptibility,
but the mixing of the
1 > excited doublet into the ground state by
the field gives a Van Vleck susceptibility at low temperatures which, if
we neglect the exchange, has the form
2 g 2 µ 2 B M α
N
V
χ =
,
(1 . 5 . 16)
where M α =
2 is the square of the matrix element of the
component of J in the field direction, and ∆ is the energy separation
between the ground state and the first excited state. Since M α is zero
when the field is applied along the c -axis, no moment is initially gen-
erated on the hexagonal sites, as confirmed by the neutron diffraction
measurements of Lebech and Rainford (1971), whereas the susceptibility
in the basal plane is large. An applied field in the c -direction changes
the relative energies of the crystal-field levels however, and at 4.2 K a
field of 32 tesla induces a first-order metamagnetic transition to a phase
with a large moment (McEwen et al. 1973), as shown in Fig. 7.13. This
is believed to be due to the crossing of the ground state by the second
excited state, as illustrated in Fig. 7.12.
If the exchange is included in the mean-field approximation, the
q -dependent susceptibility becomes, in analogy with (1.5.13),
|
<
±
1
|
J α |
0 >
|
2 M α −J
( q ) 1
N
V
χ MF ( q )= g 2 µ 2 B
.
(1 . 5 . 17)
From this expression, it is apparent that the susceptibility diverges, cor-
responding to spontaneous ordering, if
( q ) M α
2
J
1 .
(1 . 5 . 18)
The magnetic behaviour of such a singlet ground-state system is there-
fore determined by the balance between the exchange and the crystal
field. If the exchange is strong enough, magnetic ordering results; oth-
erwise paramagnetism persists down to the absolute zero. In Pr, the
crystal-field splitting is strong enough to preclude magnetic order, but
the exchange is over 90% of that required for antiferromagnetism. We
shall return to the consequences of this fine balance in Chapter 7.
The remaining close-packed light rare earths Ce, Nd, and Sm, which
are amenable to experimental study (radioactive Pm is very intractable),
all have an odd number of 4 f electrons and thus, according to Kramers'
theorem , crystal-field levels with even degeneracy and a magnetic mo-
ment. The crystal fields cannot therefore suppress magnetic ordering,
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