Spatial resonance in ferromagnetics

Accepted Manuscript Spatial resonance in ferromagnetics Yu.D. Zavorotnev, O.Yu. Popova, K.V. Gumennyk PII: DOI: Reference:

S0304-8853(16)31195-7 http://dx.doi.org/10.1016/j.jmmm.2017.06.133 MAGMA 62929

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

23 June 2016 29 December 2016 30 June 2017

Please cite this article as: Yu.D. Zavorotnev, O.Yu. Popova, K.V. Gumennyk, Spatial resonance in ferromagnetics, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.06.133

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1 Spatial resonance in ferromagnetics Yu.D. Zavorotneva, O.Yu. Popovab, K.V. Gumennyka a

Donetsk Institute for Physics and Engineering

72 R.Luxemburg St., 83114 Donetsk, Ukraine b

Donetsk National Technical University

2 Shibankova square, 85300 Krasnoarmeysk, Ukraine

Abstract

Phenomenological Landau theory is used to study an interaction between structural and magnetic subsystems in a crystal in the vicinity of phase-transition lines. It is demonstrated that the twist-induced severe plastic deformation results in a forced onset of inhomogeneous distribution in the magnetic subsystem. A nonsinusoidal spatial modulation of magnetic and structural order parameter moduli is found to arise near the lines of magnetic or structural phase transition. The obtained results permit to construct various distributions of ferromagnetic vector in crystals under twist-induced severe plastic deformation.

Keywords: twist-induced severe plastic deformations, ferromagnetic, order parameter, spatial oscillation, phase transition, harmonics

1. Introduction An interaction between two or more order parameters (OP) presents a substantial interest since the knowledge of corresponding mechanisms permits to control the behavior of one of the OPs by changing the others. A classical example of this is magnetostriction. An inverse effect is also of substantial interest, when the structural OP determines the behavior of the magnetic one. A systematic study of such effects was commenced in [1], where the phenomenological Landau theory was applied to interaction between two OPs. The corresponding phase diagrams were plotted in a space of coefficients of non-equilibrium thermodynamic potential (NTDP). This approach was further developed in [2]. The majority of succeeding works was carried out within the approximation of OPs of constant moduli, which is valid at a sufficient distance from the line of phase transition (PT). It was demonstrated in [3] that in the vicinity of PT line one must account for the change of OP modulus. The change of moduli of magnetic OP in the spiral crystalline phase results in a forced transformation of the structural OP [4]. When applying *

Corresponding autor. Tel. +38 062 311 05 51; fax +38 062 337 90 18 E-mail address: [email protected] (Yu.D.Zavorotnev)

2 theoretical results to real compounds, a question of the priority of OPs arises, i.e one must decide which of the OPs is the governing one which of them is being governed. This problem does not arise in the case of external action such as e.g. a uniform or otherwise compression. Refs. [5,6] describe the behavior of magnetic and structural OPs under a twist-induced severe plastic deformation (TISPD) when the axis of deformation is normal to the easy axis. It has been shown that variations in temperature and the torsional moment allow controlling the form of spatial distribution of the magnetic OP modulus. As shown in [7] a uniaxial spatially-nonuniform temporally-constant pressure when applied simultaneously with the TISPD may lead to a ferromagnetic spatial resonance, when the oscillation range of the magnetic OP modulus increases by several times and some additional distortions appear. Ref. [8] is devoted to investigation of the effect of TISPD applied along the easy axis of ferromagnetics. The authors find temperature dependences of possible spatial distributions of the magnetic OP in the vicinity of the first and second-order phase transitions. In this situation an onset of spatial resonance is possible when an additional spatially-nonuniform temporally-constant pressure is applied. The present work is aimed at establishing the necessary conditions for emergence of such resonance. 2. Theoretical model

Suppose that a model ferromagnetic is subjected to a twist-induced severe plastic deformation directed along the easy Z-axis. Let us designate by M the twisting moment and by k the propagation vector directed along the OZ-axis. Suppose that along the Y-axis is applied a

spatially-nonuniform temporally-constant pressure (SNTCP), which can be written as

Py 1   5 cos  tz   . Here  5  1 determines the relative magnitude of the variable component. Let us employ Landau phenomenological theory to find the distribution of moduli of magnetic and structural order parameters. It shall be supposed, that the emerging spiral crystal structure is symmetrical, i.e. similarly to magnetic superstructure [9], the spatial spiral structure can be described by a linear combination of Lifshitz invariants. In such a case the density of the nonequlibrium thermodynamical potential (DNETP) can written as 

1 2

Fz2 

2 4

Fz4 

3 6

Fz6 

1 2

q2 

2 4

q4 

3 6

q 6   1q 2 Fz2 

2   q y  q y   qx  s  qx   2 M  qx  qy    3 M       4 Py q y (1   5 cos(tz ))   z  z   z     z   2

r

(1)

3 where  i ,  i ,  i are phenomenological constants, Fz , q denote, correspondingly, the magnetic OP and the modulus of the structural OP, Py is the y-component of the temporally-constant pressure. The terms containing spatial derivatives describe a long-period crystalline structure generated by the TISPD; they include a multiplier proportional to the torsional moment. In other words, the superstructure does not emerge at zero TISPD. As shown in [5,8], the difference of powers r  s  4 . Here we shall put r  6 , s  2 . Potential (1) must account for the elastic and magnetoelastic interaction. The corresponding variables can be eliminated by establishing their equilibrium values as functions of the structural and magnetic OPs using the state equations. Substituting the obtained relations into (1) yields the necessary DNETP, which formally coincides with the DNETP written without the account for elastic and magnetoelastic interaction. The new constants are the functions of temperature and pressure. In what follows we shall presume that the procedure of elimination has been performed. Upon the abandonment of the assumption of the constant moduli of irreducible vectors we arrive at the following system of Euler’s equations 2 q y  s  qx 2  M   2M r  q x 1   2 q 2   3 q 4  2 1 Fz2  0  3 2  z z   2qy q  s   2 M r x   4 Py (1   5 cos(t * z ))  q y 1   2 q 2   3 q 4  2 1 Fz2  0 2 3 M 2 z z  2 4 1   2 Fz   3 Fz  2 1q 2  0  









(2)

The use of the known analytical approximate methods of the solution of differentialalgebraic equations does not permit to find harmonics of sufficiently high order. For this reason we employed MathCad 15 to perform the numerical analysis which permitted to reveal some aliquant harmonics, which are responsible for a number of peculiar effects. 3. Results and discussion

The present analysis of system (2) is an extension of the study carried out without the account for the constant pressure (Ref. [8]). For this reason we used numerical values of coefficients analogous to those reported in [8]. 1) 1  0, 2  0, 3  0, 1  0,  2  0,  3  0,  1  0,  2  0,  3  0, 1  1 . Such a set of coefficients describes the magnetic and structural second-order phase transition at various temperatures. It follows from Fig.1 of Ref. [8] that the resonance phenomena can arise in two cases. In the first case, the vectors of propagation of the disturbing pressure and the fundamental

4 almost sine-shaped spatial oscillation q y are nearly equal. In the second case, the magnitudes of propagation vectors of the noise term and the applied constant pressure (ACP) approximately coincide at certain points in space. Fig. 1a shows the resonance case, when the vectors of propagation of the fundamental oscillation q y (z ) and spatially-uniform pressure almost coincide at low values of the torsional moment. The periods of fundamental oscillations q x (z ) and q y (z ) are smaller as compared to the case with no resonance. When Py increases, the amplitude of the oscillatory component

q x (z ) is also reduced. Only the y-component of the structural OP is substantially changed. The other component q x (z ) ceases to be sine-shaped and becomes a saw-tooth one (Fig.1b). The change of the modulus of the structural OP is determined by the noise term. The said modulus is almost sine-shaped with a minor superimposed antiphase two-way amplitude modulation with different maximum deviations. An analogous behavior is characteristic of the modulus of the magnetic OP Fz (z ) (Fig.1c). Fourier analysis demonstrates that only the first 50 harmonics are essential for the description of evolution of q(z ) and Fz (z ) (Fig. 1d). The most substantial of them are grouped near the 20th and the 40th harmonics. In Fig. 1d the amplitude of the fundamental harmonic is reduced by a factor of 200. At higher values of M , the resonance processes become more pronounced (Fig. 2a, 2b) and the frequency modulation is added to the amplitude one which results in the emergence of a greater number of harmonics (Fig.2с). In the second case the resonance is attained by shifting the phase of perturbation by π. At low values of M , the amplitude of oscillations q x (z ) is drastically reduced and the noise term almost vanishes, whereas oscillations remain quasi-sine-shaped. At large M ’s the spatial resonance is observed in the case of the coincidence of oscillation phases of the spatiallynonuniform pressure and the noise term (Fig.3a). Substantial harmonics group near the 8th, the 13th, the 18th and the 22nd ones (Fig.3b). 2) 1  0,  2  0, 3  0, 1  0,  2  0,  3  0,  1  0,  2  0,  3  0 . In this case we shall have the first-order structural and magnetic phase transition. At the absence of the spatiallynonuniform pressure the projections of the structural and magnetic OPs are characterized by a fundamental oscillation with hypermodulation. The oscillations of the moduli of both OPs are almost sine-shaped [8]. If a spatially-nonuniform pressure is applied whose propagation vector is almost coincident with the noise oscillation, a single-sided amplitude modulation arises (Fig.4a) due to the emergence of sets of harmonics near the 12th , 45th , 65th , 85th and 105th ones (Fig.4b). The maximum amplitude is attained in regions where the perturbation and the noise oscillation are in antiphase. When the spatially-nonuniform pressure is increased, the range of

5 oscillation minima decreases and that of maxima increases (Fig.4c). If the propagation vector of spatially-nonuniform pressure approximately coincides with that of the fundamental oscillation, the resonance sets in only for y-components of the structural and magnetic OPs (Fig. 4d). At the same time the spatial distribution of OP moduli is the same as in the case of zero SNP [8]. The case of the approximate coincidence of propagation vectors of perturbation and fundamental oscillation is not considered since no steady solution of system (2) can be obtained under resonance conditions. 3) 1  0, 2  0, 3  0, 1  0,  2  0,  3  0,  1  0,  2  0,  3  0 . In this case only the first-order magnetic PT takes place. As shown in [8], the oscillating character of the OP moduli is completely determined by the noise term. Application of SNP resonant with the fundamental oscillation results in the emergence of a double-sided sine antiphase amplitude modulation of the terms q x (z ) and q y (z ) . Since the difference of oscillation phases varies in space there is singlesided amplitude modulation of moduli of both OPs (Fig.5a). The maximum of the modulating oscillations is attained in the vicinity of points where the phases of the SNP and the fundamental oscillation coincide. The minimum is reached at the points where the oscillations are in antiphase. Fourier-analysis of OP moduli upon the elimination of the noise term shows the emergence of sets of substantial harmonics near the 3rd , 7th and 14th ones (Fig.5b). The noise term generates a set of harmonics near the 9th one. In case the resonance with the noise term occurs at large values of the torsional moment, application of SNP results in formation of a longperiod double-sided antiphase amplitude modulation of the moduli of the structural and magnetic OPs (Fig.5c). Since the periods of the fundamental oscillation and the amplitude modulation are substantially different the Fourier analysis was not performed. 4) Soft materials characterized by large values of propagation vector. According to [8], this

condition

is

satisfied

if

 2   3 .

In

the

case

of

1  0;  2  0;  3  0; 1  0;  2  0;  3  0;  1  0;  2  0;  3  0 (the second-order structural and magnetic phase transition), application of SNP results in hypermodulation of the magnetic OP. In regions of approximate coincidence of propagation vector of SNP and that of the noise oscillation, the frequency of hypermodulation drastically increases (Fig.6). 5) The last case we consider is the hard materials with small values of the propagation vector k , when  2   3 . Here we analyze the situation when at certain value of M the magnetic OP changes its sign [8]. The signs of parameters of the nonequilibrium thermodynamic potential are the

following:

1  0;  2  0;  3  0; 1  0;  2  0;  3  0;  1  0;  2  0;  3  0 . The

resonance is possible at approximately equal propagation vectors of the perturbation force and

6 the noise component. Since the noise term is not sine-shaped with varying period, there exist separate areas of the approximate coincidence of the values of propagation vectors. In these regions, the amplitude of the noise of the structural OP is sharply reduced (Fig.7a), and the curve describing the magnetic OP becomes continuous (Fig.7b).

4. Conclusion

An interaction between structural and magnetic subsystems in a crystal in the vicinity of phase transitions lines is being analyzed. It is demonstrated that application of the twist-induced severe plastic deformation and the temporally-constant spatially-sinusoidal pressure in the vicinity of temperatures of magnetic and structural phase transitions results in the appearance order parameters, whose moduli are described by complicated functions of the spatial coordinates. Non-sinusoidal spatial amplitude as well as the frequency modulation of the moduli of magnetic and structural order parameters arises. The properties of modulation are studied with the help of Fourier analysis. The conditions are established for the appearance of spatial resonance of moduli of interacting OPs. It may be concluded that application of twist-induced severe plastic deformation and a temporally-constant spatially-sinusoidal pressure permits to govern the distribution of the modulus of ferromagnetic vector in a crystal at resonant situations. This is due to interaction between the structural and magnetic sublattices under changing deformation and temperature.

Figure captions Fig.1. Resonance at small values of torsional moment M . a) Approximate coincidence of propagation vectors of the fundamental oscillation q y (z ) and the spatially nonuniform pressure, b) a plot of the component q x (z ) , c) a plot of the modulus of magnetic OP Fz (z ) , d) distribution of

the

Fourier

amplitudes

WF

of

function

7

Fz (z ) .

Fig.2. Resonance at large values of torsional moment M . a) a plot of q y (z ) , b) aplot of

Fz (z ) ,

c)

distribution

of

the

Fourier

amplitudes

WF

of

function

8

Fz (z ) .

Fig.3. M is large. a) a plot of Fz (z ) at an approximate coincidence of propagation vectors of the noise and fundamental oscillation of the structural OP, b) distribution of Fourier

WF

amplitudes

of

function

Fz (z ) .

Fig.4. Propagation vectors of the SNP and the noise oscillation are almost coincident. a) a plot of Fz (z ) (small SNP), b) distribution of Fourier amplitudes WF of function Fz (z ) , c) a plot of Fz (z ) (large SNP), d) propagation vector of SNP approximately coincides with corresponding vector

of

fundamental

oscillation;

plot

of

q y (z )

and

the

perturbation

9 oscillation.

Fig.5. SNP is in resonance with the fundamental oscillation. a) a plot of Fz (z ) (small SNP), b) distribution of Fourier amplitudes WF of function Fz (z ) , c) a plot of Fz (z ) at a

10 resonance

with

the

Fig.6.

noise

term

Plot

(large

torsional

moment).

Fz (z ) .

of

Soft

materials. Fig.7. Hard materials. Vectors of propagation of SNP and the noise term are similar in magnitude.

a)

a

plot

of

q(z ) ,

b)

a

plot

of

11

Fz (z ) .

References 1. Yu.M. Gufan, E.S. Larin, Fiz. Tverd. Tela 22 (1980) 270. Matched ISSN: 0367-3294. 2. V.D. Buchel’nikov, A.N. Vasil’ev, V.V. Koledov, S.V. Taskaev, V.V. Khovailo, V.G. Shavrov, Physics-Uspekhi 49 (2006) 871. Matched ISSN : 1468-4780. 3. Yu.D. Zavorotnev, L. I. Medvedeva, Low Temp. Phys. 34 (2008) 131. Matched ISSN : 1063-777X. 4. Yu.D. Zavorotnev, Low Temp. Phys. 39 (2013) 133. Matched ISSN : 1063-777X. 5. Yu.D. Zavorotnev, E.H. Pashinskaya, V.N.Varjuchin, O.Yu.Popova, JMMM 349 (2014) 244. 6. Yu.D. Zavorotnev, E.H. Pashinskaya, V.N.Varjuchin. Bulletin of the Russian Academy of Sciences: Physics, 78 (2014) 781. Matched ISSN : 1062-8738. 7. Yu.D. Zavorotnev, E.H. Pashinskaya, Low Temp. Phys. 40 (2014) 967 Matched ISSN : 1063-777X. 8. Yu.D. Zavorotnev, E.G. Pashinskaya, Physics of the Solid State 58 (2016) 665. Matched ISSN : 1063-7834. 9.

I.E.

Dzyaloshinskii,

http://dx.doi.org/10.1063/1.3056990.

Soviet

Physics

JETP

19

(1964)

960.

DOI:

12 Combined effect of the severe plastic deformation by twisting and temporarily-constant space-varied pressure in a ferromagnetic crystal is studied. The behavior of the moduli of magnetic and structural order parameters is considered. The analysis is carried out in different temperature ranges within the frameworks of Landau phenomenological theory. The magnetic and structural phase transitions can be of the first and second order. The temperatures of the phase transitions are at a small distance. Two types of the spatial resonance are possible. In the first case, the propagation vectors of the noise term and the applied temporarily-constant pressure coincide. In the second case, there exists the coincidence of the vectors of the fundamental oscillation of the structural order parameter and the constant pressure. Governing of the distribution function of the changes in the moduli of the magnetic and the structural order parameters by means of the temperature and the applied pressure is possible.