12.7 Modeling Paramagnetic Effects in Carbon Nanotubes
331
of room (or body) temperature, and a magnetic field of a few kGauss, the exponent,
(E 4 − E 1 )/kT is of the order of 10 −3 , which means that the term in parenthesis is
approximately equal to 3 × ¯
hω 0 /kT , so that the absorption coefficient in (12.24) is
approximately equal to
A(ω) ≈
9μ 0 γ 2
4Z
e
−E 1 /kT ¯
hω 0
kT
τ/ ¯
h
1 + (ω 0 − ω) 2 τ 2
=
μ 0 g 2 h 2 (3β) 2
4Z
e −E 1 /kT
kT
ω 0 τ
1 + (ω 0 − ω) 2 τ 2 ,
(12.25)
where β is the Bohr magneton (the magnetic dipole of a single spin). Therefore, we
can conclude from (12.25) that the exchange interaction causes individual spins
to align themselves parallel to each other, thereby producing an atomic system
of spin-1/2, but with an equivalent dipole three times that of a single spin-1/2
particle. As indicated in (12.24), the effect of the AC field is to cause this single
macrospin system to transit from spin-up to spin-down, with all individual spins
remaining parallel to each other. In contrast, three noninteracting spins would have
an equivalent absorption coefficient of
A(ω) =
3μ 0 γ 2
4Z
e
−E 1 /kT
− e
−E 2 /kT
τ/ ¯
h
1 + (ω 0 − ω) 2 τ 2
≈
3μ 0 γ 2
4Z
e −E 1 /kT
kT
ω 0 τ
1 + (ω 0 − ω) 2 τ 2 ,
(12.26)
which is much smaller than (12.25). We can imagine what happens when 10 5
particles interact under exchange effects. This confirms, once again, that the
exchange interaction associated with ferromagnetic single-domain particles gives
rise to the notion of ‘superparamagnetism.’
This model of superparamagnetism results from the large value of J exch , because
that isolates the upper energy levels of Figs. 12.10 and 12.11 from the lower levels
for all (reasonable) values of H 0 . The upper energy levels may be degenerate, as in
Fig. 12.11 for the three-spin problem, but the lower levels are always nondegenerate,
and have the same number of equal intervals as the number of spins. Furthermore,
the large value of the exchange energy ensures that the upper levels will be virtually
unpopulated compared to the lower levels. These facts are crucial to the theory.
331
of room (or body) temperature, and a magnetic field of a few kGauss, the exponent,
(E 4 − E 1 )/kT is of the order of 10 −3 , which means that the term in parenthesis is
approximately equal to 3 × ¯
hω 0 /kT , so that the absorption coefficient in (12.24) is
approximately equal to
A(ω) ≈
9μ 0 γ 2
4Z
e
−E 1 /kT ¯
hω 0
kT
τ/ ¯
h
1 + (ω 0 − ω) 2 τ 2
=
μ 0 g 2 h 2 (3β) 2
4Z
e −E 1 /kT
kT
ω 0 τ
1 + (ω 0 − ω) 2 τ 2 ,
(12.25)
where β is the Bohr magneton (the magnetic dipole of a single spin). Therefore, we
can conclude from (12.25) that the exchange interaction causes individual spins
to align themselves parallel to each other, thereby producing an atomic system
of spin-1/2, but with an equivalent dipole three times that of a single spin-1/2
particle. As indicated in (12.24), the effect of the AC field is to cause this single
macrospin system to transit from spin-up to spin-down, with all individual spins
remaining parallel to each other. In contrast, three noninteracting spins would have
an equivalent absorption coefficient of
A(ω) =
3μ 0 γ 2
4Z
e
−E 1 /kT
− e
−E 2 /kT
τ/ ¯
h
1 + (ω 0 − ω) 2 τ 2
≈
3μ 0 γ 2
4Z
e −E 1 /kT
kT
ω 0 τ
1 + (ω 0 − ω) 2 τ 2 ,
(12.26)
which is much smaller than (12.25). We can imagine what happens when 10 5
particles interact under exchange effects. This confirms, once again, that the
exchange interaction associated with ferromagnetic single-domain particles gives
rise to the notion of ‘superparamagnetism.’
This model of superparamagnetism results from the large value of J exch , because
that isolates the upper energy levels of Figs. 12.10 and 12.11 from the lower levels
for all (reasonable) values of H 0 . The upper energy levels may be degenerate, as in
Fig. 12.11 for the three-spin problem, but the lower levels are always nondegenerate,
and have the same number of equal intervals as the number of spins. Furthermore,
the large value of the exchange energy ensures that the upper levels will be virtually
unpopulated compared to the lower levels. These facts are crucial to the theory.