324
12 Carbon-Nanotube Reinforced Polymers
dρ mn
dt
=
j
¯
h
k
(ρ mk H kn − H mk ρ kn ) +
pq
R mn,pq
ρ pq − ρ
(T )
pq
,
(12.9)
where ρ mn is the density matrix connecting energy states u m and u n of the
unperturbed system, R mn,pq are real numbers that account for spin-lattice relaxation, and the superscript, T , denotes the thermal equilibrium density matrix.
H jk = H 0jk + H 1jk (t), where H 0jk is the unperturbed, time-independent spinHamiltonian associated with the crystalline field, and H 1jk (t) = ghβ [H(t) · S] jk
is the time-dependent perturbation. Here g is a constant, hβ the Bohr magneton,
h Planck’s constant, ¯
h = h/2π , H(t) the time-dependent (rf) magnetic field, and
S = S x a x + S y a y + S z a z is the vector spin operator.
Because {u m } is an orthonormal system of eigenstates of H 0 , it follows
immediately that H 0mm = E m , and all off-diagonal elements of H 0mn vanish.
Furthermore, in order to get a linear (i.e., first-order in H(t)) response for the overall
system, we must set the diagonal terms of (12.9) to their thermal equilibrium values,
ρ mn (t) = ρ
(T )
mn , and solve the off-diagonal terms to first-order in H(t):
dρ mn
dt
=
jω 0mn −
1
τ mn
ρ mn +
j
¯
h
ρ
(T )
mm − ρ
(T )
nn
H 1mn (t) ,
(12.10)
where ω 0mn =
E n − E m
¯
h
, and the relaxation times, τ mn , replace the R mn,pq of
(12.9).
For a sinusoidally time-varying field, we have H 1mn (t)=
gβh
2
He jωt +H ∗ e −jωt
·
S mn . If we assume solutions of (12.10) of the form ρ mn = A mn e jωt + B mn e −jωt ,
then the coefficients of the positive-frequency terms, A mn , and negative-frequency
terms, B mn , are given by
A mn =
(j/ ¯
h)
ρ
(T )
mm − ρ
(T )
nn
τ mn gβh/2
1 − j (−ω + ω 0mn ) τ mn
S mn · H
B mn =
(j/ ¯
h)
ρ
(T )
mm − ρ
(T )
nn
τ mn gβh/2
1 − j (ω + ω 0mn ) τ mn
S mn · H .
(12.11)
The magnetic dipole-moment operator for each spin is gβhS, which means that
the average dipole-moment for each spin is m = Tr [ρgβhS], where Tr is the trace
of an operator (sum of the diagonal elements of its matrix representation). The
macroscopic dipole-moment per unit volume, M, is obtained by multiplying m by
the number density, N, of spins. Upon evaluating the trace, we find
M = γ
2
j
N
(T )
j
− N
(T )
k
τ jk
12 Carbon-Nanotube Reinforced Polymers
dρ mn
dt
=
j
¯
h
k
(ρ mk H kn − H mk ρ kn ) +
pq
R mn,pq
ρ pq − ρ
(T )
pq
,
(12.9)
where ρ mn is the density matrix connecting energy states u m and u n of the
unperturbed system, R mn,pq are real numbers that account for spin-lattice relaxation, and the superscript, T , denotes the thermal equilibrium density matrix.
H jk = H 0jk + H 1jk (t), where H 0jk is the unperturbed, time-independent spinHamiltonian associated with the crystalline field, and H 1jk (t) = ghβ [H(t) · S] jk
is the time-dependent perturbation. Here g is a constant, hβ the Bohr magneton,
h Planck’s constant, ¯
h = h/2π , H(t) the time-dependent (rf) magnetic field, and
S = S x a x + S y a y + S z a z is the vector spin operator.
Because {u m } is an orthonormal system of eigenstates of H 0 , it follows
immediately that H 0mm = E m , and all off-diagonal elements of H 0mn vanish.
Furthermore, in order to get a linear (i.e., first-order in H(t)) response for the overall
system, we must set the diagonal terms of (12.9) to their thermal equilibrium values,
ρ mn (t) = ρ
(T )
mn , and solve the off-diagonal terms to first-order in H(t):
dρ mn
dt
=
jω 0mn −
1
τ mn
ρ mn +
j
¯
h
ρ
(T )
mm − ρ
(T )
nn
H 1mn (t) ,
(12.10)
where ω 0mn =
E n − E m
¯
h
, and the relaxation times, τ mn , replace the R mn,pq of
(12.9).
For a sinusoidally time-varying field, we have H 1mn (t)=
gβh
2
He jωt +H ∗ e −jωt
·
S mn . If we assume solutions of (12.10) of the form ρ mn = A mn e jωt + B mn e −jωt ,
then the coefficients of the positive-frequency terms, A mn , and negative-frequency
terms, B mn , are given by
A mn =
(j/ ¯
h)
ρ
(T )
mm − ρ
(T )
nn
τ mn gβh/2
1 − j (−ω + ω 0mn ) τ mn
S mn · H
B mn =
(j/ ¯
h)
ρ
(T )
mm − ρ
(T )
nn
τ mn gβh/2
1 − j (ω + ω 0mn ) τ mn
S mn · H .
(12.11)
The magnetic dipole-moment operator for each spin is gβhS, which means that
the average dipole-moment for each spin is m = Tr [ρgβhS], where Tr is the trace
of an operator (sum of the diagonal elements of its matrix representation). The
macroscopic dipole-moment per unit volume, M, is obtained by multiplying m by
the number density, N, of spins. Upon evaluating the trace, we find
M = γ
2
j
N
(T )
j
− N
(T )
k
τ jk