304
11 Spintronics
E(GHz)
2J exch
4J exch
exch
6J
8J exch
10J exch
36448.1
17431.7
−7923.5
49125.7
55464.5
−39617.5
3
5
7
9
11
S
2 =0
S
2 =6
S
2 =12
S
2
=20
S
2
=30
β =0
β=2
β=4
β=6
β=8
β=10
F
E
D
C
B
A
S 2 =2
Fig. 11.7 Energy-levels (in GHz) for the system comprising two Fe
(3+) ions connected through
the exchange interaction J exch . The numbers just inside the vertical line indicate the number of
energy levels associated with that branch of the spectrum; the spacing between consecutive pairs
of levels is equal to 2.8H 0 , where H 0 is in kGauss. The letters, A-F, correspond to the physical
arrangement of the ten electrons in the coupled system, as shown in Table 11.1. S 2 are the
eigenvalues of the magnitude-squared spin-operator, S 2 , and the β-values indicate the number of
Bohr magnetons in the various arrangements of parallel spins
Table 11.1 Spin
arrangements for Fig. 11.7.
The bullets, •, correspond to
spins that are parallel to the
z−axis, and the circles, ◦, to
spins that are anti-parallel
A • • • • • • • • •• S = 5 : 2S + 1 = 11
B • • • • • • • • •◦ S = 4 : 2S + 1 = 9
C • • • • • • • • ◦◦ S = 3 : 2S + 1 = 7
D • • • • • • • ◦ ◦◦ S = 2 : 2S + 1 = 5
E • • • • • • ◦ ◦ ◦◦ S = 1 : 2S + 1 = 3
F • • • • • ◦ ◦ ◦ ◦◦ S = 0 : 2S + 1 = 1
The eigenvalues (in GHz) corresponding to the lowest eleven energy levels (the
‘A-section’ spectrum of Fig. 11.10) are listed here in the order, E 1 to E 11 , left-toright, for H 0 = 1 kGauss:
h = 1.000000000000000
A-section spectrum
-39662.7967 -39661.1857 -39580.6189 -39558.0700 -39525.8847 -39453.1469
-39436.4784 -39306.8435 -39284.5805 -39098.4235 -39070.4491