11.3 Superparamagnetic Iron Oxide
291
The Pauli spin-matrices for a single electron in a z-directed magnetic field are
s x =
1
2
0 1
1 0
s y =
1
2
0 −j
j 0
s z =
1
2
1 0
0 −1
,
(11.14)
and the eigenstates of s z are
1
0
,
0
1
, with the first one corresponding to ‘spin
up’ (parallel to the magnetic field), and the second to ‘spin down’ (antiparallel to
the magnetic field).
Since we have two coupled spins, we must work in the four-dimensional directproduct space of the operators of (11.14):
S
(1)
x S
(2)
x =
1
4
⎡
⎢
⎢
⎣
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
⎤
⎥
⎥
⎦
S
(1)
y S
(2)
y =
1
4
⎡
⎢
⎢
⎣
0 0 0 −1
0 0 1 0
0 1 0 0
−1 0 0 0
⎤
⎥
⎥
⎦
S
(1)
z S
(2)
z =
1
4
⎡
⎢
⎢
⎣
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
⎤
⎥
⎥
⎦
S
(1)
· S
(2)
= S
(1)
x S
(2)
x + S
(1)
y S
(2)
y + S
(1)
z S
(2)
z
=
1
4
⎡
⎢
⎢
⎣
1 0 0 0
0 −1 2 0
0 2 −1 0
0 0 0 1
⎤
⎥
⎥
⎦ .
(11.15)
The four-dimensional representations of S x , S y , S z are obtained by taking the
left- and right-direct products of the single-electron Pauli spin-matrices, (11.14),
with the two-dimensional identity matrix. The results are