ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 16

where C is the Curie constant . The second-order non-diagonal term in
(1.2.24) gives a paramagnetic contribution to χ which is independent of
temperature, provided that the thermal population of the excited states
is negligible. This Van Vleck paramagnetism is very small in the heavy
rare earths, but in the first half of the series it is given by

2 µ 2 B ( L +1) S
3( J +1)∆

N
V

χ V = , (1 . 2 . 33)

which may be significant, since

χ V
χ

2( L +1) S
g 2 J ( J +1) 2

k B T
∆

αk B T
∆

= = , (1 . 2 . 34)

where, from Table 1.1, α takes the modest value of 0.19 for Pr, but
is 12 for Sm. Since ∆ is only 1450 K, the Van Vleck paramagnetism
in Sm is significant even at rather low temperatures. In trivalent Eu,
J = 0 in the ground state and the paramagnetic susceptibility is due
entirely to the mixing of the excited states into the ground state by the
field, and to the thermal excitation of these states. However, Eu metal
is divalent and the 8 S 7 / 2 ionic susceptibility follows Curie's law very
closely. The Van Vleck paramagnetism arising from the mixing of states
of different J will not play a significant role in our later discussion, but
the analogous phenomenon of the mixing of states of different M J , split
by the crystalline electric field in the metal, will be of central importance.

1.3 The metallic state
When a large number of rare earth atoms are assembled to form a solid,
the 4 f electrons generally remain localized, in a sense which will be made
more precise later, so that their magnetic properties closely resemble
those in the free atoms. The external 5 d and 6 s electrons, on the other
hand, become delocalized into Bloch states , extending throughout the
metal and constituting the conduction-electron gas. The conduction
electrons themselves make only a modest contribution to the magnetic
moment, but by mediating the magnetic interactions they play a crucial
role in determining the characteristic magnetic properties in the solid
state. An understanding of the magnetism therefore requires a detailed
description of the conduction electron gas, and this section is concerned
with our theoretical and experimental knowledge of the Bloch states,
and their influence on the structural properties of the metals. Some of
these structural properties of the rare earth metals are collected in Table
1.2, from which it may be seen that the room-temperature structures are
all close-packed, with a coordination number of 12, with the exception
of Eu, which is bcc. The remaining elements all form hexagonal phases,