328
12 Carbon-Nanotube Reinforced Polymers
environment of the iron ion. It is this information that is lacking when we consider
electron-paramagnetic spin systems in CNT composites, and is the subject of current
research.
12.7.3 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 ([79]
is not clear on this), is ferromagnetic. Because of the small size of the particles
(∼ 10nm), 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 [80]. 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. 1
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) ,
(12.18)
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) ,
(12.19)
where we assume that the static field is along the z-direction. The eigenspectrum
of (12.19) is plotted as a function of H 0 in Fig. 12.10. The left-hand figure shows
all four solutions, and the right-hand the bottom three eigenvalues. The two parallel
branches have a constant separation of 6338.7, which is exactly 2J exch , where J exch
is the exchange energy. It is important to note that the transition (resonant) frequency
1 A ‘single-domain particle’ is a particle that is in a state of uniform magnetization at any magnetic
field [80].
12 Carbon-Nanotube Reinforced Polymers
environment of the iron ion. It is this information that is lacking when we consider
electron-paramagnetic spin systems in CNT composites, and is the subject of current
research.
12.7.3 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 ([79]
is not clear on this), is ferromagnetic. Because of the small size of the particles
(∼ 10nm), 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 [80]. 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. 1
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) ,
(12.18)
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) ,
(12.19)
where we assume that the static field is along the z-direction. The eigenspectrum
of (12.19) is plotted as a function of H 0 in Fig. 12.10. The left-hand figure shows
all four solutions, and the right-hand the bottom three eigenvalues. The two parallel
branches have a constant separation of 6338.7, which is exactly 2J exch , where J exch
is the exchange energy. It is important to note that the transition (resonant) frequency
1 A ‘single-domain particle’ is a particle that is in a state of uniform magnetization at any magnetic
field [80].