Journal of Materials Science and Nanomaterials
Open Access

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

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 is Curie’s temperature, and ε<<T_c . 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 at T = T_c 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 where κ₀ − characteristic thermal conductivity coefficient, H_a− field of magnetic anisotropy, exchange magnetic field, J_ex− energy of exchange interaction, μ_e− Bore’s magneton.

Hamiltonian interactions at T < T_c

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.

          (1)

where group speed of the magnons,

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

Everywhere we will use the energy system of units, in which we believe that Boltzmann’s constant k_B =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 it’s always possible to think that d³k = 4π k ²dk , and therefore from (1) we have

                  (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

               (3)

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

By the way, it’s worth noting that interactions like 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 , 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. , 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

                       (4)

is spontaneous magnetism in the Curie’s temperature area, S − is the spin of atom 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

                           (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

where α_e − 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 is an operator of berth (destruction) of phonon with the wave vector q and polarization a, is phonon dispersion, where index α =1,2,3 .

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

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

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

we come to the interaction in the form

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

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

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

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

magnons, f_q − 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

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

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

where 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

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

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

As a result, instead of (14) we get

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

allows us to get the following interim expression

where the parameters are

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

where q_1,2 identified in (20), and the wave vector of the magnon should be “squeezed” in the segment

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

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

where the coefficient

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

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

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

The thermal conductivity of the paramagnetic at T > T_c

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 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])

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

Interaction amplitude is

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

Because

that of (33) we have

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

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

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

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

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

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).

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

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

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

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

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