290
11 Spintronics
behaves, for H greater than 3 kilogauss, as a free spin in an external magnetic field.
We would expect this same qualitative behavior for many of the 4f rare earths,
no matter what the crystal lattice is. This suggests that if we use 4f rare-earth
spin-systems, it may not be necessary to know anything about the electric-field
environment of the biological tissue.
11.3 Superparamagnetic Iron Oxide
This is what started this discussion of paramagnetic spin-systems. Iron oxide,
whether it is γ − Fe 2 O 3 , called ‘maghemite,’ or, perhaps magnetite, Fe 3 O 4 ([95]
is not clear on this), is ferromagnetic. Because of the small size of the particles
(∼10 nm), their ferromagnetic properties manifest themselves in a single domain,
and such single domain particles can behave magnetically in a manner analogous to
the paramagnetism of moment-bearing atoms [16]. The main distinction is that the
moment of the particle may be 10 5 times the atomic moment, because of the 10 5
atoms ferromagnetically coupled by exchange forces within the single domain. 2,3
Two Spins We’ll make a simple quantum-mechanical calculation of a system of
two electrons coupled through the exchange interaction in a static magnetic field,
H 0 . The Hamiltonian is
H = −gβH 0 ·
S
(1)
+ S
(2)
− 2J exch S
(1)
· S
(2) ,
(11.12)
where gβ = g × 0.0014 GHz/gauss = 2.8 GHz/kgauss, if we take g = 2. J exch is
the exchange energy, with a nominal value of 2.1 × 10 −21 J. Dividing by Plancks
constant, h, gives us the result in frequency units: J exch /h = 2.1 × 10 −21 /6.626 ×
10 −34 = 3169.3 GHz. Hence, the normalized Hamiltonian for the system becomes
H = −2.8H 0 ·
S
(1)
+ S
(2)
− 6338.7S
(1)
· S
(2)
= −2.8H 0
S
(1)
z + S
(2)
z
− 6338.7S
(1)
· S
(2) ,
(11.13)
where we assume that the static field is along the z-direction.
2 Additional References on Superparamagnetic and Ferromagnetic Effects: [6, 23, 25, 49, 67, 75,
76, 116, 119, 124, 136].
3 By a ‘single domain particle,’ we mean a particle that is in a state of uniform magnetization at any
magnetic field[16].
11 Spintronics
behaves, for H greater than 3 kilogauss, as a free spin in an external magnetic field.
We would expect this same qualitative behavior for many of the 4f rare earths,
no matter what the crystal lattice is. This suggests that if we use 4f rare-earth
spin-systems, it may not be necessary to know anything about the electric-field
environment of the biological tissue.
11.3 Superparamagnetic Iron Oxide
This is what started this discussion of paramagnetic spin-systems. Iron oxide,
whether it is γ − Fe 2 O 3 , called ‘maghemite,’ or, perhaps magnetite, Fe 3 O 4 ([95]
is not clear on this), is ferromagnetic. Because of the small size of the particles
(∼10 nm), their ferromagnetic properties manifest themselves in a single domain,
and such single domain particles can behave magnetically in a manner analogous to
the paramagnetism of moment-bearing atoms [16]. The main distinction is that the
moment of the particle may be 10 5 times the atomic moment, because of the 10 5
atoms ferromagnetically coupled by exchange forces within the single domain. 2,3
Two Spins We’ll make a simple quantum-mechanical calculation of a system of
two electrons coupled through the exchange interaction in a static magnetic field,
H 0 . The Hamiltonian is
H = −gβH 0 ·
S
(1)
+ S
(2)
− 2J exch S
(1)
· S
(2) ,
(11.12)
where gβ = g × 0.0014 GHz/gauss = 2.8 GHz/kgauss, if we take g = 2. J exch is
the exchange energy, with a nominal value of 2.1 × 10 −21 J. Dividing by Plancks
constant, h, gives us the result in frequency units: J exch /h = 2.1 × 10 −21 /6.626 ×
10 −34 = 3169.3 GHz. Hence, the normalized Hamiltonian for the system becomes
H = −2.8H 0 ·
S
(1)
+ S
(2)
− 6338.7S
(1)
· S
(2)
= −2.8H 0
S
(1)
z + S
(2)
z
− 6338.7S
(1)
· S
(2) ,
(11.13)
where we assume that the static field is along the z-direction.
2 Additional References on Superparamagnetic and Ferromagnetic Effects: [6, 23, 25, 49, 67, 75,
76, 116, 119, 124, 136].
3 By a ‘single domain particle,’ we mean a particle that is in a state of uniform magnetization at any
magnetic field[16].