11.3 Superparamagnetic Iron Oxide
295
H = −2.8H 0
S
(1)
z +S
(2)
z +S
(3)
z
−6338.7
S
(1)
· S
(2)
+S
(1)
· S
(3)
+S
(2)
· S
(3)
(11.23)
where the various three-particle spin-matrices are obtained by taking three-fold leftand right-direct products of the Pauli spin matrices, s, with the two-dimensional
identity matrix, I 2 :
S (1) = s ⊗ I 2 ⊗ I 2 S (2) = I 2 ⊗ s ⊗ I 2 S (3) = I 2 ⊗ I 2 ⊗ s .
(11.24)
The results are:
S
(1)
x =
1
2
0 4 I 4
I 4 0 4
S
(2)
x =
S
(1)
x
0 4
0 4 S
(1)
x
S
(3)
x =
S
(2)
x
0 4
0 4 S
(2)
x
S
(1)
y =
1
2
0 4 −jI 4
jI 4 0 4
S
(2)
y =
S
(1)
y
0 4
0 4 S
(1)
y
S
(3)
y =
S
(2)
y
0 4
0 4 S
(2)
y
S
(1)
z =
1
2
I 4 0 4
0 4 −I 4
S
(2)
z =
S
(1)
z
0 4
0 4 S
(1)
z
S
(3)
z =
S
(2)
z
0 4
0 4 S
(2)
z
(11.25)
where the primed,’, submatrices refer to the corresponding 4×4 matrices in (11.16),
and I 4 , 0 4 are the four-dimensional identity and null-matrices, respectively.
Using the results of (11.25), we easily compute the dot-product matrices:
S
(1)
· S
(2)
=
1
4
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 −1 0 2 0 0 0
0 0 0 −1 0 2 0 0
0 0 2 0 −1 0 0 0
0 0 0 2 0 −1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(1)
· S
(3)
=
1
4
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0 0 0 0 0 0
0 −1 0 0 2 0 0 0
0 0 1 0 0 0 0 0
0 0 0 −1 0 0 2 0
0 2 0 0 −1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 2 0 0 −1 0
0 0 0 0 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦