Exact and approximate analytical solutions of Weiss equation of ferromagnetism and their experimental relevance

TL;DR

This paper presents exact and approximate analytical solutions to the Weiss equation of ferromagnetism using generalized Lambert functions, providing practical formulas for magnetization and temperature dependence relevant for experiments.

Contribution

It introduces new exact solutions based on Lambert functions and develops practical approximate formulas for magnetization in ferromagnetic materials.

Findings

Exact solutions derived using generalized Lambert functions.

Approximate formulas for temperature and field dependence of magnetization.

Discussion on experimental applications, especially for Curie temperature determination.

Abstract

The recent progress in the theory of generalized Lambert functions makes possible to solve exactly the Weiss equation of ferromagnetism. However, this solution is quite inconvenient for practical purposes. Precise approximate analytical solutions are obtained, giving the temperature dependence of the spontaneous magnetization, and also the dependence of the magnetization on both temperature and external magnetic field. The experimental relevance of these results, mainly for the determination of the Curie temperature, is discussed.

Equations

$$M\left(T,H\right)=M_0{B}_S\left(\frac{\overline{\mu }S}{kT}\left(H+\lambda M\right)\right),\overline{\mu }=g{\mu }_B$$

$$M_0=N\overline{\mu }S=NSg{\mu }_B.$$

$$B_S\left(x\right)=\frac{2S+1}{2S}\coth \left(\frac{2S+1}{2S}x\right)-\frac{1}{2S}\coth \left(\frac{1}{2S}x\right)$$

$$B_{1/2}\left(x\right)=\tanh x$$

$$M=M_0\tanh \left(\frac{\overline{\mu }}{2kT}\lambda M\right)$$

$$T_c=\lambda \frac{N{\overline{\mu }}^2}{4k}$$

$$t=\frac{T}{T_c},h=\frac{\overline{\mu }H}{2k{T}_c},m=\frac{M}{M_0}$$

$$m\left(t,h\right)=\tanh \frac{m\left(t,h\right)+h}{t}$$

$$m\left(t,0\right)=m\left(t\right)$$

$$m\left(t\right)=\tanh \frac{m\left(t\right)}{t}$$

$$m\left(t\right)=t\zeta \left(t\right)$$

$$\tanh \zeta \left(t\right)=t\zeta \left(t\right)$$

$${\zeta }^{\prime }\left(t\right)=\frac{\zeta \left(t\right)}{1-t-t^2{\zeta }^2\left(t\right)}$$

$$\zeta \left(t\right)=\zeta \left(t_0\right)+{\zeta }^{\prime }\left(t_0\right)\left(t-t_0\right)+\frac{1}{2}{\zeta }^{\prime \prime }\left(t_0\right){\left(t-t_0\right)}^2+...$$

$$x{e}^x=a.$$

$$e^{x}\frac{\left(x-t_{1}\right)\left(x-t_{2}\right)...\left(x-t_{n}\right)}{\left(x-s_{1}\right)\left(x-s_{2}\right)...\left(x-s_{m}\right)}=a\$$

$$W\left(t_1,...t_n;s_1,...s_m;a\right)$$

$$e^{2\zeta \left(t\right)}\frac{\zeta \left(t\right)-\frac{1}{t}}{\zeta \left(t\right)+\frac{1}{t}}=1$$

$$m\left(t\right)=t\zeta \left(t\right)=\frac{t}{2}W\left(\frac{2}{t};-\frac{2}{t};-1\right)$$

$$x{e}^x+rx=a$$

$$\frac{1}{2}\left(1-e^{-2z}\right)\simeq z\frac{1+az}{1+cz},a=-0.0572,c=1.286,0<z<5$$

$$m\left(t\right)=\tanh \frac{m\left(t\right)}{t}\simeq 1-2\exp \left(-\frac{2m\left(t\right)}{t}\right)$$

$$2\exp \left(-\frac{2m\left(t\right)}{t}\right)=1-m\left(t\right)$$

$$\frac{m}{t}=u,m=ut$$

$$\exp \left(-2u\right)=-\frac{t}{2}\left(u-\frac{1}{t}\right)$$

$$\exp \left(-cx\right)=a\left(x-r\right)$$

$$x=r+\frac{1}{c}W\left(\frac{c}{a}\exp \left(-cr\right)\right)$$

$$c=2,a=-\frac{t}{2},r=\frac{1}{t}$$

$$u=\frac{1}{t}+\frac{1}{2}W\left(-\frac{4}{t}\exp \left(-\frac{2}{t}\right)\right)$$

$$m=ut=1+\frac{t}{2}W\left(-\frac{4}{t}\exp \left(-\frac{2}{t}\right)\right)$$

Full text

1 Introduction

The Weiss theory of ferromagnetism is  an exactly solvable model ”that allows one to study thermodynamic functions in detail, in particular their properties near the critical temperature”[1]. The only element which hampers the possibility of calculating analytically any thermodynamic aspect is the fact that the equation of state, which is a transcendental equation involving the magnetization, the temperature and the external magnetic field, cannot be solved analytically. Recently, this obstacle has been removed, as, due to work of Mezö, Baricz [2] and Mugnaini [3], the equation of state was in principle solved, using generalized Lambert functions.

However, these solutions are quite cumbersome and difficult to handle, at least in the context of experimental magnetism. This is why it is still important to have, besides the exact solutions, some simpler analytical approximations, more convenient in practical situations. In the last decades, several analytical approximations have been obtained for the Brillouin and Langevin functions or their inverses, with applications in ferromagnetism [16], polymers (strong polymer deformation and flow) [20], [21], [18], [19] or solar energy conversion (daily clearness index) [22], [17]. The main goal of this paper is to give new and precise examples of such analytical approximations.

The structure of this paper is as follows. In Section 2, we shall expose the main elements of the Weiss theory and the attempts of solving - exactly or approximately - the equation of state. The exact solutions, in terms of generalized Lambert functions, are presented. In Section 3, we obtain an approximate but very precise expression of the magnetization as a function of temperature, in zero magnetic field, involving elementary functions only. In Section 4, a similar result is obtained, for the case of a non-zero external magnetic field. In Section 5, the experimental relevance of these results is discussed. The last section is devoted to conclusions.

2 An outline of Weiss theory of ferromagnetism

The main equation of this theory is the equation of state [4], [5], [6], [7]:

[TABLE]

describing the dependence of magnetization $M$ on temperature $T$ and on external magnetic field $H$. Also,

[TABLE]

So, $M_{0}$ is the maximum value of the spontaneous magnetization.

The parameter $\lambda$ is the molecular field parameter, introduced by Pierre Weiss in 1907, and $B_{S}$ is the Brillouin function:

[TABLE]

If $S=1/2:$

[TABLE]

So, if $S=1/2$ and the external field is zero, $H=0,$ eq. (1) becomes:

[TABLE]

This equation has a non-zero solution, and, equivalently, the system has a spontaneous magnetization $M\neq 0,$ if the temperature $T$ is under a critical value, namely under the critical temperature $T_{c},$ given by the following relation ([4], eqs. (6.18), (6.19)):

[TABLE]

It is convenient to introduce the reduced parameters:

[TABLE]

as the equation of state (1) can be written now in a simpler form:

[TABLE]

If we write the reduced magnetization in zero magnetic field as

[TABLE]

the equation (1) becomes:

[TABLE]

Because eq. (10) is sometimes called Weiss equation, we shall also adopt this terminology in the present paper. It is a transcendental equation, as its solution cannot be written as a finite combination of elementary functions.

There are several attempts of solving, exactly or approximately, this equation. An interesting approach is due to Siewert and Essing [8]; it is based on the theory of complex functions, mainly on the theory of singular integral equations [9]. It has been systematically applied by Siewert and his co-workers in order to solve a large number of transcendental equations, involving hyperbolic (or trigonometric) and algebraic functions. However, the results obtained in this way are not explicit and can be hardly used in practical calculations.

Another approach, invented and re-invented by several authors [10], [11], puts the transcendental equation in a differential form and writes its solution as a series expansion. If we define the function $\zeta\left(t\right)$ by

[TABLE]

the Weiss equation (10) takes the form:

[TABLE]

We easily find that $\zeta^{\prime}\left(t\right)$ can be expressed in terms of the function $\zeta\left(t\right)$ and of $t$:

[TABLE]

As any higher order derivative of $\zeta\left(t\right)$ has the same property, we can write down the series expansion of $\zeta\left(t\right)$ near an arbitrary point $t_{0}$ as:

[TABLE]

This approach has two inconconveniences: (1) the radius of convergence of the series is quite small, and the expansion must be repeated in the neighborhood of several points $t_{0}$; (2) we have to check that the series exists in the neighborhood of $t_{0}$ (this is not the case, for instance, if $t_{0}$ is an essential singularity of $\zeta\left(t\right)$ ).

Recently, a crucial progress was achieved in solving a large class of transcendental equations, namely using generalized Lambert functions [2], [3], with applications to magnetism and other branches of physics [12], [13]. Let us mention that the Lambert function $W\left(a\right)$ is the solution in $x$ of the transcendental equation:

[TABLE]

In other words, the Lambert function is the inverse of the function appearing in the left hand side of (15). Similarily, the solution in $x$ of the transcendental equation:

[TABLE]

is the generalized Lambert function

[TABLE]

One of the beneficiaries of the progress made in the study of generalized Lambert functions is the Weiss equation. Actually, eq. (12) takes the form:

[TABLE]

whose solution can be written as [13]:

[TABLE]

where $W\left(\frac{2}{t};-\frac{2}{t};-1\right)$ is a particular case of (17), namely of $W\left(t;s;a\right)$. Clearly, the parameters $t,\ s$ in $W\left(t;s;a\right)$ are particular cases of the parameters $t_{1},...t_{n},\ s_{1},...s_{m}$ from (16), (17) and have nothing to do with any physical quantity, like temperature or spin. At its turn, $W\left(t;s;a\right)$ can be expressed in terms of a simpler, but similar function, the $r-$Lambert function, $W_{r}.\ W_{r}\left(a\right)$ is defined as the solution of the equation:

[TABLE]

and can be expressed as a series expansion in terms of Mezö - Baricz polynomials, $M_{k}^{\left(n\right)}\left(1/\left(r+1\right)\right)$ [2].

So, even if eq. (19) provides a direct and exact solution of the Weiss equation for $h=0$ and $S=1/2$ (e.g. the dependence of the spontaneous magnetization of a magnetic monodomain system with $S=1/2$ versus temperature), it is quite unconvenient for practical applications. This is why a simpler analytical approximation of (19) might still be useful.

An interesting approach in finding approximate analytical solutions of transcendental equations is to approximate the non-algebraic functions involved in such equations with algebraic ones. In this way, the equation becomes algebraic [14], [11]. For instance, eq. (45) of [14] suggests the following approximation for the exponential function:

[TABLE]

Even if such an approximation might give useful results when applied to certain quantum mechanical problems, it is inappropriate for the Weiss equation, as it cannot reproduce the critical behavior of the magnetization on a large enough interval.

In the following section, we shall propose a precise approximate analytical solution of the Weiss equation, inspired from the standard ways of obtaining physically convenient solutions of the Schroedinger equation.

3 A precise approximate analytical solution of Weiss equation

In order to find a convenient solution of the Weiss equation (10) on the interval $\left(0,1\right)$, we shall find out firstly the behavior of $m\left(t\right)$ for $m\left(t\gtrsim 0\right)$ and $m\left(t\lesssim 1\right).$

So, for $t\gtrsim 0:$

[TABLE]

which can be written as:

[TABLE]

If we put:

[TABLE]

eq. (23) can be written as:

[TABLE]

The most general form of an equation which can be solved using the Lambert W\function is similar to (25):

[TABLE]

and its solution is given by:

[TABLE]

So, comparing (25) and (26), we have to identify:

[TABLE]

and we find the solution of eq. (25) as:

[TABLE]

Consequently:

[TABLE]

For $t\gtrsim 0$:

[TABLE]

where the higher order terms in the expression of $W$ can be neglected [15]. Hence, we get:

[TABLE]

So, $m\left(t\right)$ cannot be written as a series expansion near the origin, as $t=0$ is an essential singular point for this function. This mathematical aspect is unimportant for the physicist, as the correct low temperature dependence of the magnetization is not exponential, but power-like; it obeys the well-known ”$3/2$ Bloch law” (see also the eqs. (10), (11) in Ch. 15 of [5], where the previous result, (32) of this preprint, was obtained under the implicit assumption that $m\left(0\right)$ is finite).

Near the critical temperature $t_{c}=1,$ the magnetization behaves as

[TABLE]

So, for the whole interval $\left(0,1\right),$ it seems convenient to write the magnetization in the form:

[TABLE]

where $P\left(t\right)$ can be approximate, as precisely as we want, with a polynomial. As

[TABLE]

we can presume for $P\left(t\right)$ the following form:

[TABLE]

[TABLE]

The coefficients $a_{1},\ ...\ a_{7}$ can be obtained by solving a linear system of 7 equations, obtained for 7 couples of values $\left(t_{0},\ m\left(t_{0}\right)\right).$ The numerical value of $m\left(t_{0}\right)$ for a given $t_{0}$ can be obtained solving numerically the equation:

[TABLE]

using (for instance) the FindRoot command of Mathematica. The aforementioned pairs of values $\left(t_{0},m\left(t_{0}\right)\right)$ might be those included in the following list:

[TABLE]

{0.2 -$>$ 0.9999091217152326‘}

{0.4 -$>$ 0.9856238716346567‘}

{0.6 -$>$ 0.9073323166453099‘}

{0.8 -$>$ 0.7104117834878704‘}

{0.85 -$>$ 0.6295014763911393‘}

{0.9 -$>$ 0.525429512658009‘}

{0.95 -$>$ 0.37948520667808994‘}

[TABLE]

Solving the system of 7 linear equations generated by eq. (36) applied to each from the 7 above pairs, the polynomial $P\left(t\right)$ can be expressed as:

[TABLE]

[TABLE]

Finally, the reduced magnetization is:

[TABLE]

[TABLE]

[TABLE]

Let us remind ourselves that, looking for the form (34) of the solution of eq. (10), we applied actually an approach systematically used in order to find a physically acceptable solution of the Scroedinger equation. A wellknown example of such an approach might be the following: when we are solving the Schroedinger equation in a Coulombian field, we are looking for a solution which ”behaves well” near origin and at the infinity. It is, in fact, similar to what we have done here, isolating the behaviour of $m\left(t\right)$ near $t=0$ and $t=1.$

The plot of the reduced magnetization $m\left(t\right)$ given by eq. (39) is given in Fig. 1.

If $m\left(t\right)$ satifies (with a good approximation) the equation:

[TABLE]

it will satisfy (with an equally good approximation) the equation:

[TABLE]

The function $f\left(t\right)=m\left(t\right)\coth\left(m\left(t\right)/t\right),$ with $m\left(t\right)$given by eq. (39) and the function $g\left(t\right)=1$ are plotted in Fig. 2, for comparison.

It is easy to see that the approximation of $f\left(t\right)$ by $g\left(t\right)$ is excellent. The largest relative deviation, less than $3\cdot 10^{-3}$, corresponds to a region of little physical interest, being far away from the critical point of the transition temperature, where the superposition with experimental values should be the most relevant.

In applications to experimental physics involving temperatures of hundread of Kelvins, the exponential factor in the expression of magnetization plays no role and instead of (34) we can look for a solution of the Weiss equation (10) of the form:

[TABLE]

where $Q_{1/2}\left(t\right)$ is a polynomial (of the same degree as $P_{1/2}\left(t\right)$).

Applying the same approach, outlined after eq. (36), we get the following expression for $Q_{1/2}\left(t\right):$

[TABLE]

[TABLE]

The precision of this simpler expression is still higher then the previous one: the error is less than $10^{-3},$ for $t\simeq 0.1.$

Even more interesting is the fact that this approach can be applied to any Brillouin function $B_{S}$ defined in (3), not only for $S=1/2.$ For an arbitrary $S,$ the Weiss equation in zero magnetic field is given by the formula:

[TABLE]

and the reduced magnetization can be approximated by:

[TABLE]

For instance

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

with a maximum relative deviation smaller than $1.5\cdot 10^{-3},$ also for $t\simeq 0.1;$ for larger values of $t$, the approximation is much better.

4 The magnetization in the presence of an external magnetic field

The results of [2] can be easily used in order to obtain an explicit solution of the more general equation of state (8), written in reduced parameters. Actually, according to [13], this explicit solution is:

[TABLE]

For reasons explained in the previous section, this solution is quite inconvenient for practical purposes. The main goal of the present section is to obtain an alternative expression for $m\left(t,h\right)$, using simpler functions - actually, a series expansion in powers of $h$, whose coefficients are known, being expressed as elementary functions of $t.$

With:

[TABLE]

the equation of state (8) becomes:

[TABLE]

A closed form of $m\left(t,0\right)$ was obtained in the previous section, eq. (34), so an accurate analytic approximation of the function $\zeta\left(t,0\right)$ is known, at least down to enough low $t$. Differentiating (52) with respect to $h,$ and putting

[TABLE]

we get:

[TABLE]

and:

[TABLE]

Consequently:

[TABLE]

So, the value of the previous expression is known, as the function $\zeta\left(t,0\right)$ is known, e.g. via $m(t,0)$ provided by (39). This property of the derivative remains valid in any order; consequently, we can write

[TABLE]

In this way, we obtain a series expansion for $m\left(t,h\right)$, according to (51), valid for small values of $h.$ It should have in mind, however, that we did not investigate the convergence of this series.

5 Discussion on experimental implications

From the experimental point of view, a direct expression of the magnetization versus temperature and field might be of high interest for getting material dependent parameters of ferromagnetic materials (magnetic moment and spontaneous magnetization, Curie temperature, molecular field constant and exchange integral, etc.) through a suitable fitting of the experimental data. It is worth to mention that case of ferrimagnetic (or even no completely compensated antiferomagnets, e.g. due to structural/interfacial/surfacial defects) can be treated in a similar manner by a simple association of a finite magnetic moment to a formula unit. Some real (and very often defect) magnetic systems could present an enough high magnetic disorder temperature, in order to be completely analyzed within the temperature range of sensitive magnetometers (e.g. SQUIDS). Mainly these are the cases where a complete estimation of the material parameters can become effective by fitting the experimental data obtained over a narrower range of field and temperature via the above discussed theoretical expressions.

A proof of concept on how this procedure can be experimentally implemented for magnetization measurements versus temperature will be provided for one of the above analyzed cases, corresponding to a solid state system with $S=1/2$ (a quenching of the orbital magnetic moment is assumed). To note that the local magnetic moment of the magnetic entity (atom, molecule, formula unit) can be verified from the saturation magnetization at very low temperatures which also approaches the spontaneous magnetization $M_{0}$. Hence, the experimental function $m(t,h)$ can be obtained if $T,H$ and $T_{c}$ are known. The first two are experimental parameters, whereas the last one can be obtained via specific magnetization measurements versus temperature, e.g. as the temperature where the spontaneous magnetization drops to zero. Very often, such measurements have to be done at temperatures much higher than the room temperature, involving therefore additional higher temperature options for usual magnetometers. Instead, the Curie temperature can be obtained by fitting the experimental dependence $M(T)$, as collected under a low enough applied magnetic field, over a much reduced range of temperatures. The aforementioned general relation (51), written under the form $m(t,h)=t\zeta\left(t,h\right)$ should be used in this respect.

However, in order to achieve this task, a series of experimental conditions should be fulfilled. As a first condition, the system might respect the overall Brillouin dependence of magnetization versus temperature, that is, size or anisotropy related magnetization relaxation phenomena (e.g. superparamagnetic behavior) should be avoided. A second condition is related to the removal of magnetic domains in the system, condition which might be fulfilled by measuring the magnetization versus temperature under convenient constant applied magnetic fields. In the case of quite soft magnetic materials the measurements can be done in low enough magnetic fields (e.g. of magnetic induction as low as $0.1T$), in order to keep only the term proportional to $h$ in relation (51). Accordingly, the following experimental steps are proposed. The $M(H)$ curve at the lowest temperature (e.g. $2K$) should be initially obtained for estimating the saturation field and the saturation magnetization $M_{0}$. Then, the dependence $M(T)$ can be obtained under an applied field higher than the saturation field and up to the maximum temperature of the device (e.g. 400K for a SQUID). According to Figs. 1 and 2, the most reliable values for $T_{c}$ might be obtained if the maximum experimental temperature will overpass at least $60\%$ from $T_{c}$. That is, for a maximum measuring temperature of $400K$, a Curie temperature of about $700K$ may be determined with acceptable accuracy.

The procedure of determining the transition temperature is the following. The experimental ratio $M(T,H)/M_{0}$ will be represented versus $t=T/T_{c}$ for a set of $T_{c}$ values (e.g. $500K,\ 550K,\ 600K,\ 750K,$

$800K$). In this way, the experimental representations of $m(t,h)$ versus t will be obtained for different values of $T_{c}$. The theoretical representations of $m(t,h)$ via the relation $m(t,h)=t\zeta\left(t,h\right)$ with $\zeta\left(t,h\right)$ given by eq. (51) will be obtained for the same set of $T_{c}$ values. In a first approximation, $T_{c}$ is estimated as that temperature value in the considered set corresponding to the best superposition between the experimental and the theoretical $m(t,h)$ curves. The procedure can be repeated for a new set of $T_{c}$ values (with a better temperature discretization, e.g. of $10K$) around the first approximation until the desired precision of $T_{c}$ is obtained.

It is also to mention that actually there are experimental techniques providing parameters proportional to the spontaneous magnetization. Among them, the local techniques of ${}^{57}Fe$ Mossbauer spectroscopy is extremely powerful in following the temperature variation of the $Fe$ magnetic moment on each $Fe$ phase (the hyperfine magnetic field, $B_{hf}$ as specific parameter is proportional in certain circumstances to the magnitude of the $Fe$ magnetic moment and hence with the spontaneous magnetization attributed to that $Fe$ phase). Being a local techniques seeing directly the magnitude of the magnetic moment, it does not involve any applied magnetic field and therefore, the simplest equation (51) can be used as reflecting the experimental temperature dependence of the ratio $m(t)=B_{hf}(T)/B_{hf}(T=0).$

6 Conclusions

A proof of concept for direct series-expansion solutions of the Weiss equation of ferromagnetism is provided. Even the concept is exemplified in case of non-zero magnetic field for a magnetic system with $S=1/2$, it works in principle also for systems with enhanced $S$ values, with the mention that in that case the Brillouin function will replace the $\tanh$ function, with direct implications on its derivatives. Also the spontaneous magnetization and Curie temperataure will depend specifically on $S$, which might become an additional fitting parameter in the final expression of magnetization. The general solution of the reduced magnetization in applied field is always expressed as function of the temperature dependent solution of the spontaneous magnetization in the absence of the applied field. Specific procedures to use the developed expressions in case of magnetization or Mossbauer spectroscopy measurements are also mentioned.

Acknowledgement 1

The authors acknowledge the financial support of IFIN-HH - ANCSI project PN 16 42 01 01/2016 and to the IFIN-HH - JINR Dubna grant 04-4-1121-2015/17 (VB) and PN16-480102 (VK).