Brownian and Néel relaxation times in magnetic particle dynamics

As seen in Fig. 2a,b the Néel relaxation time is more than 4 orders of magnitude larger than the Brownian relaxation time when a 40 mT magnetic field is suddenly turned off. We note that 〈x〉 is proportional to the magnetization. These time series are consistent with τ
b
= 3.05 µs and τ
n
= 81 ms from the formulas. From this, one might be tempted to conclude that Néel relaxation can be ignored for these particle parameters. However, if a sufficiently strong magnetic field is turned on, this will not be the case. Fig. 2c,d show the relaxation times when a 20 mT magnetic field is suddenly applied. Now the Néel relaxation time is only 2 orders of magnitude larger than the Brownian relaxation time. When a 40 mT magnetic field is suddenly applied, Fig. 2e,f, the Néel relaxation time is comparable to the Brownian relaxation time. For even larger magnetic field strengths the Néel relaxation time can be orders of magnitude less than the Brownian relaxation time. To demonstrate this more generally, Fig. 3 shows a surface plot of the Néel relaxation time τ
n
as a function of α and α
K
. It is seen that as the anisotropy constant K increases, τ
n
becomes more sensitive to changes in B, changing by many orders of magnitude for large α
k
. Note that for K = 0, τ
n
changes by less than 2 orders of magnitude. This is also the case for the Brownian relaxation time τ
b
, which is given in Fig. 3, if τ
n
/τ
n0
is replaced with τ
b
/tτ
b0
and α
K
is taken to be zero, where τ
B0
= 3ηV
h
over kT. Using the parameter values of Fig. 2, it is noted that τ
n0
= 13.8 ns, τ
B0
= 3.05 µs, α
K
= (0.00202 m
³
/J) × K, and α = (479 T
⁻¹
) × B.