ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 8

were accounted for by Jensen and Mackintosh (1990), who showed that
intermediate structures, which they named helifans , could be stabilized
by a magnetic field.
A new field of endeavour has been opened by the fabrication of
multilayers of different species of rare earths and the study of their prop-
erties by Majkrzak, Cable, Kwo, Hong, McWhan, Yafet, Waszczak, and
Vettier (1986), and by Salamon, Sinha, Rhyne, Cunningham, Erwin,
Borchers, and Flynn (1986). The size of the teams working on a num-
ber of these modern projects in rare earth research reflects the technical
complexity of the problems now being tackled, and no doubt also the
collaborative spirit of the age.

1.2 Rare earth atoms
The starting point for the understanding of the magnetism of the rare
earths is the description of the electronic states, particularly of the 4 f
electrons, in the atoms. The wavefunction Ψ( r 1 σ 1 , r 2 σ 2 ,..., r Z σ Z )for
the electrons, which is a function of the space and spin coordinates r
and σ of the Z electrons which constitute the electronic charge cloud ( Z
is the atomic number), is determined for the stationary state of energy
E from the Schrodinger equation

H Ψ= E Ψ , (1 . 2 . 1)

where the non-relativistic Hamiltonian operator is

i ∇

Z Z Z

h 2
2 m

e 2
| r i − r j |

i + 1
2

H − v ext ( r i )

(1 . 2 . 2)

= +

ij i

and, in the case of an atom, the 'external' potential v ext ( r ) is just the
Coulomb potential

Ze 2 /r i due to the nuclear attraction. As is well
known, the diculties in solving this problem reside in the second term,
the Coulomb interaction between the electrons. For heavy atoms, exact
solutions require a prohibitive amount of computation, while any possi-
bility of an exact solution for the electronic states in a metal is clearly
out of the question. It is therefore necessary to replace the Coulomb
interaction by a self-consistent field, which is most satisfactorily deter-
mined by means of the density-functional theory of Hohenberg and Kohn
(1964) and Kohn and Sham (1965).
The first step is to write the Hamiltonian (1.2.2) in the symbolic
form

−

H = T + U + V, (1 . 2 . 3)

incorporating the kinetic energy, the Coulomb repulsion between the
electrons, and the external potential, due to the nucleus in the atom or