SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 260

been discussed in detail by Freeman (1972). First we assume that the
Coulomb interaction in eqn (5.7.5) is strongly shielded, so that it can
be replaced by a δ -function. Next, using plane waves for the Bloch

functions, we obtain

d r |

1) I ( n k ,n k ) j ( q = k − k )

2 e −i q · r ,

( g − ≈ ∝ φ 4 f ( r ) | (5 . 7 . 15)

which is the form factor of the 4 f -electron density, approximately the
same as the local moment density (4.1.15). In this simplified model,
where the conduction electrons are assumed to be free-electron-like, the
interband exchange integrals, in which n = n , are obtained by adding

reciprocal-lattice vectors to q in eqn (5.7.15). In this model, we obtain

a rigid band-splitting, independent of k , between the spin-down and
spin-up bands, of magnitude

τ

∆= ε n k ↓ − ε n k ↑ =2 J z j ( 0 ) . (5 . 7 . 16)

Since j ( 0 ) has the same sign as ( g − 1), it is positive in the heavy rare

earth metals. If ( ε ) is the density of electronic states per ion and per

spin state in the paramagnetic phase, the shifts of the spin-up and spin-
down bands lead to an excess number of spin-up electrons proportional
to

N

ε F + 2

ε F )= 1

N ( N ( ε ) dε, (5 . 7 . 17)

∆

ε F − 2

when the small modification of the density of states due to the interband
coupling is neglected, so that ε F is close to the Fermi energy ε F of

the non-magnetic system. In combination with eqn (5.7.14), this result
predicts a (positive) augmentation of the ferromagnetic moment of the
4 f electrons, due to the conduction electrons, of magnitude

µ z c . el . = µ B N ( ε F )∆ , (5 . 7 . 18)

when k B T ε F . The total moment per ion may then be expressed in

terms of an effective g -factor:

µ z = gµ B J z + µ z c . el . =( g +∆ g ) µ B J z , (5 . 7 . 19 a )

where

∆ g =2 j ( 0 ) N ( ε F ) . (5 . 7 . 19 b )

In the metals, the effective exchange integral j ( 0 )is 0 . 1eV,

leading to an exchange splitting ∆ which, in Gd for example, is nearly 1
eV. This relatively large splitting has the consequence that

∼ ( g − 1) ×

N ( ε F )may