Editing Superparamagnetism (section)

== Effect of a magnetic field ==

[[Image:Langevin function.png|300px|thumb|right|Langevin function (red line), compared with <math display="inline">\tanh\left(\frac{1}{3}x\right)</math> (blue line).]]

When an external magnetic field ''H'' is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped [[increasing function]]. This function is quite complicated but for some simple cases:
# If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (''T''<sub>B</sub> &lt; ''T'' ≲ ''KV''/(10 ''k''<sub>B</sub>)), then the magnetization of the assembly is
#: <math>M(H) \approx n \mu \tanh\left(\frac{\mu_0 H \mu}{k_\text{B} T}\right)</math>.
# If all the particles are identical and the temperature is high enough (''T'' ≳ ''KV''/''k''<sub>B</sub>), then, irrespective of the orientations of the easy axes:
#: <math>M(H) \approx n \mu L\left(\frac{\mu_0 H \mu}{k_\text{B} T}\right)</math>

In the above equations:
* ''n'' is the density of nanoparticles in the sample
* <math display="inline">\mu_0</math> is the [[magnetic permeability]] of vacuum
* <math display="inline">\mu</math> is the magnetic moment of a nanoparticle
* <math display="inline">L(x) = \frac{1}{\tanh(x)} - \frac{1}{x}</math> is the [[Langevin function]]

The initial slope of the <math>M(H)</math> function is the magnetic susceptibility of the sample <math>\chi</math>:
: <math>\chi = \begin{cases}
  \displaystyle \frac{n \mu_0 \mu^2}{k_\text{B} T} & \text{for the 1st case} \\
  \displaystyle \frac{n \mu_0 \mu^2}{3k_\text{B} T} & \text{for the 2nd case}
\end{cases}</math>

The latter susceptibility is also valid for all temperatures <math>T > T_\text{B}</math> if the easy axes of the nanoparticles are randomly oriented.

It can be seen from these equations that large nanoparticles have a larger ''μ'' and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.

=== Time dependence of the magnetization ===

There is no time-dependence of the magnetization when the nanoparticles are either completely blocked (<math>T \ll T_\text{B}</math>) or completely superparamagnetic (<math>T \gg T_\text{B}</math>). There is, however, a narrow window around <math>T_\text{B}</math> where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility<ref>{{cite journal |title=Superparamagnetism and relaxation effects in granular Ni-SiO<sub>2</sub> and Ni-Al<sub>2</sub>O<sub>3</sub> films  |first1=J. I. |last1=Gittleman |first2=B. |last2=Abeles |first3=S. |last3=Bozowski |journal=[[Physical Review B]] |volume=9 |issue = 9|pages=3891&ndash;3897 |year=1974 |doi=10.1103/PhysRevB.9.3891|bibcode = 1974PhRvB...9.3891G }}</ref> is:

: <math>\chi(\omega) = \frac{\chi_\text{sp} + i \omega \tau \chi_\text{b}}{1 + i\omega\tau}</math>

where
* <math display="inline">\frac{\omega}{2\pi}</math> is the frequency of the applied field
* <math display="inline">\chi_\text{sp} = \frac{n \mu_0 \mu^2}{3k_\text{B} T}</math> is the susceptibility in the superparamagnetic state
* <math display="inline">\chi_\text{b} = \frac{n \mu_0 \mu^2}{3KV}</math> is the susceptibility in the blocked state
* <math display="inline">\tau = \frac{\tau_\text{N}}{2}</math> is the relaxation time of the assembly

From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:

:<math>\tau \frac{\mathrm{d}M}{\mathrm{d}t} + M = \tau \chi_\text{b} \frac{\mathrm{d}H}{\mathrm{d}t} + \chi_\text{sp} H</math>