The Brillouin and Langevin functions are a pair of
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
that appear when studying an idealized
paramagnetic
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, d ...
material in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
.
Brillouin function
The Brillouin function
[C. Kittel, '']Introduction to Solid State Physics
''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption ...
'' (8th ed.), pages 303-4 is a special function defined by the following equation:
:
The function is usually applied (see below) in the context where
'' is a real variable and
is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as
and -1 as
.
The function is best known for arising in the calculation of the
magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
of an ideal
paramagnet
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...
. In particular, it describes the dependency of the magnetization
on the applied
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
and the
total angular momentum quantum number
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's s ...
J of the microscopic
magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
s of the material. The magnetization is given by:
[
:
where
* is the number of atoms per unit volume,
* the g-factor,
* the ]Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mathrm ...
,
* is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy :
::
* is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and the temperature.
Note that in the SI system of units given in Tesla stands for the magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, , where is the auxiliary magnetic field given in A/m and is the permeability of vacuum
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
.
:
Takacs proposed the following approximation to the inverse of the Brillouin function:
:
where the constants and are defined to be
:
:
Langevin function
In the classical limit, the moments can be continuously aligned in the field and can assume all values (). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin
Paul Langevin (; ; 23 January 1872 – 19 December 1946) was a French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the ''Comité de vigilance des intellectuels antifascistes'', an ant ...
:
:
For small values of , the Langevin function can be approximated by a truncation of its Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
:
:
An alternative, better behaved approximation can be derived from the
Lambert's continued fraction expansion of :
:
For small enough , both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from catastrophic cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
For example, if there are two studs, one L_ ...
for where .
The inverse Langevin function is defined on the open interval (−1, 1). For small values of , it can be approximated by a truncation of its Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
:
and by the Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...
:
Since this function has no closed form, it is useful to have approximations valid for arbitrary values of . One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen:
:
This has a maximum relative error of 4.9% at the vicinity of . Greater accuracy can be achieved by using the formula given by R. Jedynak:
:
valid for . The maximal relative error for this approximation is 1.5% at the vicinity of x = 0.85. Even greater accuracy can be achieved by using the formula given by M. Kröger:
:
The maximal relative error for this approximation is less than 0.28%. More accurate approximation was reported by R. Petrosyan:
:
valid for . The maximal relative error for the above formula is less than 0.18%.[
New approximation given by R. Jedynak,] is the best reported approximant at complexity 11:
valid for . Its maximum relative error is less than 0.076%.[
Current state-of-the-art diagram of the approximants to the inverse Langevin function
presents the figure below. It is valid for the rational/Padé approximants,][
A recently published paper by R. Jedynak,] provides a series of the optimal approximants to the inverse Langevin function. The table below reports the results with correct asymptotic behaviors,.[
Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1)][
Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns,] where Matlab code is also given to generate the spline-based approximant and to compare many of the previously proposed approximants in all the function domain.
High-temperature limit
When i.e. when is small, the expression of the magnetization can be approximated by the Curie's law
For many paramagnetic materials, the magnetization of the material is directly proportional to an applied magnetic field, for sufficiently high temperatures and small fields. However, if the material is heated, this proportionality is reduced. For ...
:
:
where is a constant. One can note that is the effective number of Bohr magnetons.
High-field limit
When , the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field:
:
References
{{reflist
Magnetism