Anhysteresis

Brillouin and Langevin functions

It describes the reversible magnetic transformation in paramagnetic materials in which magnetic material is treated as a net sum of neutral domains while each domain is carrying a dipolar magnetic moment. When an external field ${\displaystyle H}$ is applied, the domains become aligned in the direction of applied field. This is because an equilibrium is established between thermal agitation and the magnetic moments. The resulting magnetization ${\displaystyle M}$ becomes:

${\displaystyle M=Nm_{d}\langle \cos(\theta )\rangle }$

where ${\displaystyle N}$ is the number of domains, ${\displaystyle m_{d}}$ is the magnetic moment and ${\displaystyle \langle \cos(\theta )\rangle }$ is the statistical average of ${\displaystyle m_{d}}$ along the direction of applied magnetic field. The product ${\displaystyle Nm_{d}}$ is the saturation value of ${\displaystyle M_{s}}$ which occurs on full magnetization of the material.^[8]

Hence, it is possible to describe the anhysteretic magnetization (or reduced magnetization) ${\displaystyle M}$ of ferromagnetic materials for every value of externally applied magnetic field by a Langevin function ${\displaystyle L}$.^[9]

${\displaystyle M=M_{s}{\left[{\coth {\biggl (}{H \over a}{\biggr )}}-{\biggl (}{a \over H}{\biggr )}\right]}=M_{s}L{\biggl (}{H \over a}{\biggr )}}$

where the parameter ${\displaystyle a}$ (units A/m) that characterizes the shape of anhysteretic magnetization curve is defined as:

${\displaystyle a={k_{B}T \over \mu _{0}m_{d}}={Nk_{B}T \over \mu _{0}M_{s}}}$ when ${\displaystyle m_{d}={M_{s} \over N}}$

and ${\displaystyle L{\biggl (}{H \over a}{\biggr )}=\operatorname {coth} {{\biggl (}{H \over a}{\biggr )}}-{\biggl (}{a \over H}{\biggr )}}$

To describe this magnetisation for ferromagnetic materials e.g. Iron or Copper, Weiss considered the internal magnetic field (or molecular field) other than externally applied magnetic field ${\displaystyle H}$. This builds the equation for magnetization in Langevin–Weiss model.^[8][9]

${\displaystyle M=M_{s}L{\biggl (}{H+\alpha M \over a}{\biggr )}}$

This ${\displaystyle \alpha }$ is positive for real ferromagnets while negative in value for anhysteretic ferromagnets which was seen in observation by Heisenberg while formulating the value of inter domain coupling coefficient ${\displaystyle \alpha }$:^[8][9]

${\displaystyle \alpha ={2z \over Ng^{2}\mu _{B}^{2}}J_{0}}$

where ${\displaystyle z}$ is the number of nearest neighbors, ${\displaystyle g}$ is the Landé splitting factor, ${\displaystyle \mu _{B}}$ is the Bohr magneton and ${\displaystyle J_{0}}$ is the exchange integral.

A Brillouin function ${\displaystyle B_{J}}$ is simply the limiting case of Langevin function when ${\displaystyle J\rightarrow \infty }$ and using it the anhysteretic magnetization ${\displaystyle M_{an}}$ becomes:^[10][11]

${\displaystyle M_{an}=M_{s}B_{J}(y)}$

where ${\displaystyle B_{J}(y)={{2J+1 \over 2J}\coth {\biggl (}{2J+1 \over 2J}y{\biggr )}}-{1 \over 2J}{\coth {\biggl (}{1 \over 2J}y{\biggr )}}}$ and ${\displaystyle y={\biggl (}{H+\alpha M \over a}{\biggr )}}$

The quantum number ${\displaystyle J}$ is also determined using the magnetic moment ${\displaystyle m_{d}}$ equation which is:

${\displaystyle m_{d}=Jg\mu _{B}}$

At ${\displaystyle 0}$ Kelvins, the average magnetic moment for silicon-iron sheet is ${\displaystyle 18.056\times 10^{-}24\,A/m^{2}}$ which is 1.947 times ${\displaystyle \mu _{B}}$. This gives a value of Landé factor g as 2 and subsequently ${\displaystyle J}$ as 0.9735.^[10]

Preisach model

Irreversible magnetic transformations are studied in this model according to which the ferromagnetic material has zero magnetic moment because of equal distribution of independent domains by having internal magnetizations of either ${\displaystyle h}$ or ${\displaystyle -h}$.^[7]

In normal magnetization of the magnetic material, the magnetization value ${\displaystyle M}$ depends on externally applied field ${\displaystyle H}$ at present time ${\displaystyle t_{0}}$ and at an earlier time ${\displaystyle (t<t_{0})}$ also. In anhysteretic magnetization when ${\displaystyle M}$ reaches a saturation value, it can only reduce to ${\displaystyle M_{an}}$ which will depend on ${\displaystyle N}$ that is the demagnetization factor. Every material which has different ${\displaystyle N}$ will result in different levels of ${\displaystyle M_{an}}$ upon degaussing or applying ${\displaystyle H}$. Thus for some Preisach distributions, ${\displaystyle M_{an}}$ is independent of ${\displaystyle N}$ while for others it is highly dependent on ${\displaystyle N}$. Also a new magnetization curve ${\displaystyle M_{an}(H)}$ is observed.^[12]

The internal field ${\displaystyle H_{i}}$ governing the Preisach operators is measured as:

${\displaystyle H_{i}=H-NM}$

A relative permeability for anhysteretic magnetization ${\displaystyle \mu _{an}(N)}$ is also observed in the Preisach model that is different from relative permeability for normal magnetization ${\displaystyle \mu }$.^[12]

${\displaystyle \mu ={1 \over \mu _{0}}{{\biggl (}{dB \over dH}{\biggr )}}}$

${\displaystyle \mu _{an}(N)={[\exp(N\mu _{an(0)}-1] \over N}}$

Jiles–Atherton model

The JA model is different for isotropic and anisotropic materials such that the net anhysteretic magnetization depends on sum of isotropic and anisotropic anhysteretic magnetizations. It is experimentally obtained by reducing the constant externally applied magnetic field ${\displaystyle H}$ which is applied with a varying external magnetic field. This varying field keeps changing between minimum and maximum as its range steadily decreases until a point is reached when it aligns with ${\displaystyle H}$. At such a point, anhysteresis (or hysteresis free) ferromagnetic material would have been achieved. ^[7][13]

Isotropic anhysteretic magnetisation is defined as:^[13]

${\displaystyle M_{an(iso)}=M_{s}{\left[{\coth {\biggl (}{H \over a}{\biggr )}}-{\biggl (}{a \over H}{\biggr )}\right]}}$ which is similar to the one given by Langevin function.

Anisotropic anhysteretic magnetization is defined as:^[13]

${\displaystyle M_{an(aniso)}=M_{s}{\int \limits _{0}^{\pi }{e^{E(1)+E(2) \over 2}\sin(\theta )\cdot \cos(\theta )\cdot d(\theta )} \over \int \limits _{0}^{\pi }{e^{E(1)+E(2) \over 2}\sin(\theta )\cdot d(\theta )}}}$

Total anhysteretic magnetization in JA model thus becomes:

${\displaystyle M_{an(tot)}=(1-t)M_{an(iso)}+tM_{an(aniso)}}$

The magnetisation ${\displaystyle M}$ of ferromagnetic materials as a function of externally applied field ${\displaystyle H}$ is dependent on ${\displaystyle M_{an}(H)}$ which is the anhysteretic magnetisation as a function of ${\displaystyle H}$.^[7]

${\displaystyle {dM \over dH}={1 \over 1+c}{{M_{an}(H)-M} \over {\delta k-\alpha ({M_{an}(H)-M})}}+{c \over 1+c}{dM_{an}(H) \over dH}}$

where ${\displaystyle M_{an}(H)=M_{s}{\biggl (}\coth {\biggl (}{{H+\alpha M} \over \alpha _{J}}{\biggr )}-{\alpha _{J} \over {H+\alpha M}}{\biggr )}}$

Also it is possible to know the anhysteretic differential susceptibility ${\displaystyle \chi _{an}}$ as well as the magnetic moment ${\displaystyle m_{d}}$ of the ferromagnetic material for every value ${\displaystyle H_{a}}$ of the externally applied field ${\displaystyle H}$. For this, linearized approximation of ${\displaystyle M_{an}(H)}$ would be needed to be taken as externally varying magnetic field approaches smaller but constant external magnetic field ${\displaystyle H_{a}}$^[7]

${\displaystyle H\approx H_{a}}$

${\displaystyle M_{an}(H)\approx M_{s}{\biggl (}\operatorname {coth} {\biggl (}{H_{a} \over a}{\biggr )}-{a \over H_{a}}{\biggr )}}$

Using linearized approximation we get, ${\displaystyle M_{an}^{a}(H)={M_{s} \over 3}{\mu _{0}m_{1} \over k_{B}T}H_{a}}$

${\displaystyle \chi _{an}^{a}={M_{an}^{a} \over H_{a}}}$

and ${\displaystyle m_{1}={3k_{B}T\chi _{an}^{a} \over \mu _{0}M_{s}}}$