Journal of Materials Science and Nanomaterials
Open Access

Our Group organises 3000+ Global Conferenceseries Events every year across USA, Europe & Asia with support from 1000 more scientific Societies and Publishes 700+ Open Access Journals which contains over 50000 eminent personalities, reputed scientists as editorial board members.

Open Access Journals gaining more Readers and Citations
700 Journals and 15,000,000 Readers Each Journal is getting 25,000+ Readers

This Readership is 10 times more when compared to other Subscription Journals (Source: Google Analytics)
  • Research Article   
  • JMSN, Vol 6(1)
  • DOI: 10.4172/jmsn.100031

On the Thermal Conductivity of Magnets near the Point of Phase Transition

Gladkov SO*
Moscow Aviation Institute (National Research University) (MAI) Volokolamsky Shosse, 4. 125993. Moscow, Russia
*Corresponding Author: Gladkov SO, Moscow Aviation Institute (National Research University) (MAI) Volokolamsky Shosse, 125993, Moscow, Russia, Email: sglad51@mail.ru

Received: 21-Jul-2021 / Manuscript No. JMSN-22-15953 / Editor assigned: 26-Jul-2021 / PreQC No. JMSN-22-15953(PQ) / Reviewed: 05-Jan-2022 / QC No. JMSN-21-15953 / Revised: 15-Jan-2022 / Manuscript No. JMSN-22-15953(R) / Accepted Date: 24-Jan-2022 / Published Date: 31-Jan-2022 DOI: 10.4172/jmsn.100031

Abstract

An estimate of the thermal conductivity coefficient of ferromagnetic κ at temperatures lying near the point of phase transition, i.e. at temperature T = Tc −ε , where ε <<Tc . made. The calculations are based on the quasi-classica kinetic equation method, and it shows that

image with a critical index equal image and image where Ha − field of magnetic anisotropy, HE − exchange field.

Keywords

Magnetic phase temperature; Quasi-classical kinetic equation; Thermal conductivity factor.

Introduction

When carefully acquainted with a large number of literary sources dedicated to the description of the ph ysical properties of magnetic substances at temperatures in the area of magnetic phase transition [1- 10], despite the nearly century-long history of this issue, we have not found an analytical calculation of the temperature dependence of the ferromagnetic thermal conductivity coefficient κ (T ) at temperature image is Curie’s temperature, and ε<<Tc . Although the general theory of phase transitions, both macroscopic (ref. [11] – [13]) and microscopic (see [14] – [18]) long been constructed, some questions for some reasons have been overlooked and not covered in the literature. That is why the purpose of this communication is to close this small gap associated with calculating of thermal conductivity in the area of magnetic phase temperature. Experimental measurements devoted to this subject , which is typical of any other physical characteristics in the field) indicate a strong growth of thermal conductivity image at T = Tc the value of which can be calculated analytically, and which, as we shall see, has an end value. Looking ahead, we note that the value of this maximum is determined by the value of the image where κ0 − characteristic thermal conductivity coefficient, Ha− field of magnetic anisotropy, image exchange magnetic field, Jex− energy of exchange interaction, μe− Bore’s magneton.

Hamiltonian interactions at T < Tc

In the theory of phase transitions concerning the unbalanced properties of the substance, the necessary condition is always to know the appropriate time of relaxation.

Specifically, when it comes to thermal conductivity calculation κ need to have an idea of the relaxation time, which in the case of ferromagnets is due to the interaction between magnon and phonon subsystems.

Since we are talking about a gas-kinetic approach, the basic estimated formula in the isotropic case should look according to the general rules of its receipt.

image          (1)

where image group speed of the magnons,

image their dispersion, h − Planck’s constant, k − wave vector of magnon, k − Boltzmann’s constant, image magnon - phonon time of relaxation.

Everywhere we will use the energy system of units, in which we believe that Boltzmann’s constant kB =1 and for the sake of convenience we will put also as the Planck’s constant h =1 . This will not lead to confusion in dimensions (see below).

Because the group speed of the magnon is image it’s always possible to think that d3k = 4π k 2dk , and therefore from (1) we have

image                  (2)

where the lower limit of integration can only be defined after analysis energy and impulse laws for a particular magnon-phonon interaction.

With this goal we need to introduce the energy of magnon – phonon interaction is easy to record for isotropic case considering the crystal symmetry to be cubic.

We can present the relevant interaction in the only form, as

image               (3)

where γ − is dimensionless strict constant order of units, M=(Mi) is the order parameter, i.e. the magnetic moment of the unite volume of the ferromagnet image deformation tensor, u = (ui) − is vector of deformation, V − is the body volume.

By the way, it’s worth noting that interactions like image in crystals with any spatial symmetry it’s can’t be.

It’s connected due to the physical fact that any type of interaction must satisfy the rule of invariance in relation to the time inversion operation, that image , where t − is the time.

Because the vector of magnetism is an axial vector in relation to the replacement of inversion time t →−t i.e. image , this indicates that this type of interaction is not possible, although it should be noted that this type of interaction is not prohibited by the laws of energy and momentum.

To rewrite the interaction (3) in the language of operators of birth and destruction of magnons and phonons, it must be presented in a secondary – quantization form. To do this, we should remember that in accordance with Holstein-Primakov’s transformations, we have the right to write down that

image                       (4)

image is spontaneous magnetism in the Curie’s temperature area, S − is the spin of atom image is the operator of the birth (destruction) of the magnon at the local point with a radius - vector r. The expression for the dimensionless value 0 δ are given below.

In the k − presentation the operators of berth (destruction) of magnon we can presented as a Fourier’s row

image                           (5)

where N − full number of atoms in the magnet.

The vector of deformation is also easily carried out with the help of a simple Fourier - decomposition

image

where αe − is an polarization vector of phonon, index a is specify to three possible polarization branches: two acoustic (longitudinal and transversal) and one optical, ρ − is the density of ferromagnet, q − is a wave vector of phonon image is an operator of berth (destruction) of phonon with the wave vector q and polarization a, image is phonon dispersion, where index α =1,2,3 .

For the acoustical phonons the dispersion is image where ct,l − is accordingly the longitudinal and transversal speed of the sound. Single designation will be used everywhere below cs . For optical phonons, we will believe that image

According to (6) the components of the tensor deformations can then be presented as a form of

image

After substitution (5) and (7) in definition (3) as a result of simple rule-adjusted changes

image

we come to the interaction in the form

image

Where the abbreviation c.c. means a complex - conjugated value and indices 1,2 at birth and destruction operators mean a reduction i.e. image At the same the amplitude of interaction is defined

image

It is clear that in the absence of interaction the main Hamiltonian can be recorded as

image

where k ε and (q) α ω are determined above.

Calculating the relaxation time in the field of phase transition temperature

We can use the quasi-classical kinetic equation (see, for example, refs. [26] – [29]) in the form

image

where nk − is an collision integral, nk − function distribution of

magnons, fq − is an equilibrium distribution function of the phonons. The phonons we are assuming an thermostat. At high temperatures in the region of Curie’s temperature it can presented in following

image

Before moving on to specific calculations, it is worth noting that

the equation (11) will be fairly ̧ if the condition is met image

where image is the magnon – phonon relaxation time.

As we shall see later (see formula (20)), this condition is fulfilled near the phase point and it is quite possible to use the equation (11).

According to the interaction (9) the integral of collisions will look like this image

where the presence of delta function automatically takes into account the law of energy conservation. According to (13) the relaxation time in the “tau - approximation” will be

image

where the equilibrium function distribution of magnons at the temperatures near the Curie’s point is

image

As a result, instead of (14) we get

image

A simple analysis of energy and momentum laws, taking into account the rules of transition from summation to integration

image

allows us to get the following interim expression

image

where the parameters are

image

In fact, simple but rather cumbersome calculations of the integrals featured in (19) lead to the next final result for the time of relaxation of interest to us

image

where q1,2 identified in (20), and the wave vector of the magnon should be “squeezed” in the segment

image

Expression analysis (21) shows that only the large values of the magnon wave vector will make the greatest contribution to the thermal conductivity coefficient. This means that the last term in the expr. (21) will be the most significant. Leaving only it and substituting the result in the general definition (2), we get

image

Due to the convergence of this integral, we can put the lower limit of integration to zero, and the upper limit - infinity.

Assuming that at a point c T = T over an extremely short period of time τ , which is significantly less than all relaxation times, an instant magnetic order appears, then we have the right to write down the following expression for the thermal conductivity factor

image

where the coefficient

image

Formulae (24), (25) give an answer to the question posed at the beginning of the article, and the dimensionless parameter 0 δ , which appears in the definition M0 (see formula (4)) according to (24) is

image

It would seem that when c T = T we have a small but finable value of the order parameter. However, it should be understood that we are talking about temperatures just below the point of phase transition, that is, it should be considered c T = T −ε that, and the field of magnetic anisotropy behaves like a

image

That is, near the point of Curie begins to form the energy of magnetic anisotropy and thermal conductivity should increase sharply, which is in accordance with the general principles of the theory of phase transitions. Its maximum value can be estimated as

image

The thermal conductivity of the paramagnetic at T > Tc

It is quite clear that in this range of temperatures, not much far from the melting point, the magnetic moment in the absence of an external magnetic field is zero, and therefore no magnetic interactions should not be about, as well as energy of magnetic anisotropy. This means that heat transfer in the paramagnetic phase can only be carried out thanks to acoustic phonons, even if there is no crystalline order. This is clearly a burden of the fact that there is always a heterogeneous density of ρ (r) material, which within the theory of elastic deformation can be presented as a ratio image the unperturbed density of the material.

This means that to calculate the only time in this case, the phonon - the phonon relaxation, we can come from an isotropic Hamiltonian species (see. [27])

image

where θ − some characteristic energy, in order of magnitude corresponding to Debai’s energy θD . Substituting here (7) with account (8), we find image

image

Interaction amplitude is

image

According to (30) the integral of the collisions will be

image

Because

image

that of (33) we have

image

where ω =ω (q) .

Taking into account the laws of preservation and after the transition from summation to integration under the rule (18), taking into account the explicit expression for the amplitude of interaction (31) and taking into account the properties of the delta - functions we get as a result of integration

image

image

Substituting (34) in the overall expression for the thermal conductivity ratio

image

For a numerical estimate κ it can be assume that image where a − is some average interatomic distance. Substituting in (35) specific numerical values

image

we get from here image If put in (35) T = Tc and compare with (28), we’ll see that

image

Therefore, the jump in thermal conductivity in the transition from the paramagnetic phase to the magnetic will be the magnitude of

image

Ratios (36), (37) allow us to build dependence κ (T ) in a wide interval of temperature up to melting temperature, which is reflected in (Figure 1).

journal-materials-science-nanomaterials

Figure 1: Schematic illustration of temperature dependence of ferromagnets in wide diapason of temperature, including the Curie’s point.

Conclusion

In conclusion, it is worth paying attention to some of the results above.

1. Using a quasi-classical kinetic equation, the thermal conductivity ratio of ferromagnetic in the vicinity of phase temperature is calculated;

A critical staid growth index has been found and it’s shown that

image

A comparison of the phonon thermal conductivity in the paramagnetic phase T > Tc with thermal conductivity in the magnetic phase at T ≤ Tc and calculated the jump in thermal conductivity at the point T = Tc

A graphic interpretation of the result in a wide range of temperatures is given.

References

  1. Sinoviev VE, Abelsky SS, Sandakova MI (1974) The thermal properties of iron and solid silicon solutions in it near Curie point. J Exper Theore Phys 66: 354-359.
  2. Ilyinov SA, Taluz SG, Sinoviev VE, Bautin SP (1984) Measuring the temperature in subsecond heating mode. High Temp Phys 22: 709-714.
  3. Belov KP, Zaitseva MA, Ped’ko AV (1959) Magnetic Properties of Gadolinium Oxides. Soviet Physics JETP 36:1672-1679.
  4. Google Scholar        

  5. Iwase A, Hamatani Y, Mukumoto Y, Ishikawa N, et.al. (2003) Anomalous shift of Curie temperature in iron-nickel Invar alloys by high-energy heavy ion irradiation. Nucl Instrum Methods Phys Res B 209: 323-328.
  6. Indexed at   Google Scholar    Crossref

  7. Ginsburg VL, Landau LD (1950) On the theory of superconductivity. J Exp Theor Phys 20:1064-1072.
  8. Indexed at    Google Scholar    Crossref

  9. SO Gladkov (2019) On a Nonlinear Effect in the Superconductivity Theory. Solid State Phys 61:1955-1959.
  10. Indexed at   Google Scholar    Crossref

  11. Fong YF, Hao-Kun Li, Zhao R (2019) Phonon teat transfer across a vacuum through quantum fluctuations. Nature 576: 243-247
  12. Indexed at   Google Scholar    Crossref

  13. Schwab K, Henriksen EA, Worlock JM, ML Roukes (2000) Measurement of the quantum of the thermal conductance. Letters to Nature 404: 974-976.
  14. Indexed at Google Scholar             Crossref

  15. Jou D, Casas-Vasquez J (2015) Heat transfer and thermodynamics: A foundational problem in classical thermodynamics and in contemporary non-equilibrium thermodynamics. J Non-Equil Thermody 11:131-136.
  16. Indexed at Google Scholar Crossref

  17. Gladkov SO, Kaganov MI (1981) To the Theory of Relaxation of Nuclear Spins in Ferromagnets. J Exp Theor Phys 80:1577-1585.
  18. Google Scholar

  19. Gladkov SO (1982) Relaxation in Ferromagnetic Metals. J Exp Theor Phys 83: 806-809.
  20. Google Scholar

Citation: Gladkov SO (2022) On the Thermal Conductivity of Magnets near the Point of Phase Transition. J Mater Sci Nanomater 6: 031. DOI: 10.4172/jmsn.100031

Copyright: © 2022 Gladkov SO. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Top