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Encyclopedia of Physical Science and Technology EN014A-653 July 28, 2001 20:55

10 Rare Earth Elements and Materials

FIGURE 4 Crystal structures exhibited by the rare earth elements. (a) Hexagonal close packed (hcp), (b) cubic close
packed (ccp), (c) double hexagonal close packed (dhcp), and (d) the complex structure of Sm. [From Gschneider,
K. A., Jr. (1961). Crystallography of the rare-earth metals. In “The Rare Earths” (F. H. Spedding and A. H. Daane,
eds.), p. 190, Wiley, New York.]

Sm and Tm, are involved. Although these latter elements mines the magnetic properties. Here we invoke principles
are trivalent in the metallic state, they show a strong ten- first discovered by the early spectroscopists, which are
dency to divalency in many compounds. called Hund’s rules. We regard the 4 f levels as 14 boxes
As mentioned, the rare earths exhibit five different crys- labeled uniquely according to the quantum numbers as
tal structures at room temperature. Three of these are com- below: l = 3, m l =+3, +2, +1, 0, −1, −2, −3.
mon metallic structures, hexagonal close packed (hcp),
cubic close packed (ccp), also called face-centered cu-
bic fcc, and body-centered cubic. The first two are pic- m l =+3 +2 +1 0 −1 −2 −3
tured in Fig. 4. The bcc structure is not pictured but con- s=+ 1 2
sists of cubes of atoms surrounding another atom inserted
into the geometric center or body-center of the cube. The m l =+3 +2 +1 0 −1 −2 −3
1
two structures, unique to the rare earths are the double- s=− 2
hexagonal close packed structure (dhcp) found for La, Pr,
Nd and Pm and the complex structure found for Sm. Both
of these are variants of the hcp structure. Their occurrence We also need Pauli’s exclusion principle which states
for the early members of the series can be explained by that each box can contain only one electron. Hund’s
postulating that the 4 f electrons, which have relatively first rule requires that electrons be added to available
large radial extensions for the early elements, participate boxes so as to maximize the total spin of the sys-
in the metallic bonding. tem, S = i s i . Hund’s second rule insists that, consistent
Finally, the physical properties that have attracted the with the first rule, the total magnetic quantum number
most attention are the great variety of magnetic prop- M L = i m l i also be maximized. It is also standard to de-
erties exhibited by the rare earths. The origin of mag- fine M L (max) ≡ L, the total orbital angular momentum
netism in these elements lies in their electronic structure. quantum number. With these ideas in mind it is easy to
The rare earths, as mentioned earlier, are characterized write down the total spin and orbital quantum numbers for
n
by partially filled 4 f levels, and it is the detailed man- all possible 4 f configurations as shown in the following
ner in which these 4 f levels are occupied that deter- table.