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The Fermi-Dirac ( FD) and Bose-Einstein ( BE) integrals were applied to a quantum system to estimate the density of particles and relaxation time in some magnetic alloys at low temperatures. An integral part in the energy equations of vibrations (phonons), spin waves (magnons), and electrons was mathematically treated. Comparison between theoretical and experimental results gave good semi-empirical relations and some physical constants.

The contributions of the Bloch-Grüneisen and Debye integrals family to phonons, photons, magnons and electrons energy in solids were treated in previous parts [

The main problems in the solid state of physics are the relaxation time, the density of particles, and the chemical potential, especially, those they are disorder magnetic alloys like spin glass or Kondo alloys [

The aim of this paper is to calculate some physical variables through a semi-empirical relation by comparing between theoretical and experimental results with the help of Fermi-Dirac and Bose-Einstein Integrals.

Many attempts were made to, mathematically, simplify FD and BE integrals [

where

The Poly-logarithm function Li_{s}(ζ) defined as:

By applying the result of Equation (5) to Equation (4), and made a Series Expansion for the integral at

Because the chemical potential of phonons, magnons, photons equal to zero (for these particles they do not have a conservation law) then from (6):

For bosons that have_{c} when

treatment.

For fermions, and from (3), which subject to a series expansion of the integral at

Fermi energy expresses its fermions energy, but the most calculations of Fermi energy considered as constant or taken at T = 0, and merged with a total energy, but the reality is not so, fermions chemical potential

(

chemical potential can be computed by the use of the Sommerfeld approximation [

(8), the density of fermions will get the exact relation as a function of fermions chemical potential and temperature.

Fermi-Dirac and Bose-Einstein integrals are the cornerstones to calculate the thermal energy and its derivative in all materials (2, 3, 6, 8). This part will shed light on the relaxation time

where

Data analysis of the resistivity as a function of temperature by the least-squares method made it possible to determine all temperature coefficients. From

Data analysis from

Data analysis from

Data analysis from

Comparing between theoretical Equation (10) and experimental Equations (11)-(14), and collect them in a Semi-empirical equation to form a general relaxation time equation, which may be written as follows:

Equation (15) will give the values of relaxation time for all mechanisms. In addition, one could determine a relaxation time by a special method for Kondo effect and spin glass could be applied to relaxation time for these systems. The relaxation time for these regimes may be given as follows [

where, respectively,

Equivalence between internal energy and resistivity and specific heat allows concluding useful semi-empirical relationships, and with the assistance of the integrals of the FD and BE, many problems have been resolved, such as the particles density and relaxation time. It is necessary to collect many experimental results from all other techniques, and then make a comparison between those results to choose the best.

I am greatly indebted to all those whom I analyzed their published crude experimental results. In addition, to Wolfram Mathematica website, Journal of Physical and Chemical Reference Data and National Institute of Standards and Technology for their great efforts to collect experimental data.

Muhammad A.Al-Jalali,Saif A.Mouhammad, (2016) Fermi-Dirac and Bose-Einstein Integrals and Their Applications to Resistivity in Some Magnetic Alloys, Part III. Journal of Applied Mathematics and Physics,04,493-499. doi: 10.4236/jamp.2016.43055