Precessing Ball Solitons as Self-Organizing Systems during a Phase Transition in a Ferromagnet

Transcription

1 Applied Mathematics,, 4, Published Online Octobe ( Pecessing Ball Solitons as Self-Oganiing Systems duing a Phase Tansition in a Feomagnet V V Niet Joint Institute fo Nuclea Reseach, Dubna, Russia niet@jinu Received July, ; evised August, ; accepted August 8, Copyight V V Niet This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited ABSTRACT Pecessing ball solitons (PBS) in a feomagnet duing the fist ode phase tansition is induced by a magnetic field diected along the axis of anisotopy, while the action of the peiodic field pependicula to the main magnetic field has been analyed Unde these conditions, the chaacteistics of aising equilibium PBS ae uniquely detemined by the fequency of the peiodic field, but the solitons with othe fequencies ae impossible Fo such stuctue, the entopy incease connected with dissipation is compensated by the decease of the entopy due to the extenal peiodic field It is shown that the equilibium PBS ae essentially the self-oganiing systems that can aise spotaneously in a metastable state of feomagnet Keywods: Feomagnet; Magnetic Field; Peiodic Magnetic Field; Fist-Ode Phase Tansition; Pecessing Ball Soliton; Dissipative Magnetic Stuctues; Self-Oganiing Systems Intoduction It is known that a vey lage numbe of phenomena and pocesses which exist in natue can be epoduced atificially, and can be classified as self-oganiing systems o dissipative stuctues [-4] Even life itself in all its foms is an example of such a stuctue Pocesses belonging to this categoy, which can be demonstated on a laboatoy table ae, fo example, the chemical Zhabotinskiy eaction [,6] and Bena effect [7] Some of them ae peiodic stuctues in space o time Othes ae moe complex But the common featue of all these pocesses is that the loss of enegy of the system associated with the dissipation is fully offset by the influx of enegy fom extenal souces, ie, inflow of entopy due to the dissipation equal to the negative flow of entopy due to the coupling to an extenal souce Anothe case of self-oganied systems, occuing away fom the equilibium state of the system is pesented in this pape Kinetics of the phase tansition in a uni-axis feomagnet unde the action of a magnetic field along the easy axis has been consideed in [8] In such cases, in the metastable state of the cystal, pecessing magnetic solitons (PBS) with the symmety of the ellipsoid can spontaneously aise The pecession fequency and, coespondingly, the amplitude, sie and enegy of PBS ae chaacteied by a continuous spectum The PBS can gow and tansfom into the macoscopic domain of a new phase state This tansfomation is elated to the enegy dissipation and, as a esult of this pocess, the PBS quickly disappea In the pesent pape we conside the conditions when, in addition to the main magnetic field that povides phase tansition, a peiodic magnetic field pependicula to the easy axis opeates In such a case, only solitons with the pecession fequency equal to the fequency of the magnetic field could aise In this case, thee is a stabiliation of PBS, ie a feee of them, and, most impotantly, fully compensation of changes in entopy, and thus above conditions fo the existence of self-oganiing systems ae fulfilled The next chapte is a theoetical intoduction to the physics of pecessing magnetic solitons in phase tansition with the additional action of a peiodic magnetic field The last chapte pesents seveal examples of equilibium PBS, which ae self-oganiing systems, showing thei main chaacteistics and the conditions of thei occuence Equations fo PBS In the given aticle, to analye magnetic solitons in a feomagnet at the fist-ode tansition in the pesence of Copyight SciRes

2 V V NIETZ 79 peiodic magnetic field, as in [8], we use the Landau- Lifshit equation [9] with a Gilbet fom of dissipative tem: m B W m m m () t m t and coesponding expession fo the density of themodynamic potential (as in [8]) K m m W m X Y m mh 4πM Z π mh m M Hee H, H is the peiodic field diected pependicula to the Z-axis, K, ; m is a non-dimensional vecto of feomagnetism equal in the absolute value to ; m mx imy, in initial state: m Futhe, we conside the PBS in a flat plate pependicula to the Z-axis, use the following dimensionless values: and note:, B K t x K X, y K Y, K Z ; h H K 4π M K, D 4π M K If the added peiodic field is e i () H Kh () and to pesent the expession fo magnetic component in the view,, e i, m p (4) (hee we estict ouselves to the case of moveless solitons, but the phase of pecession of magnetic moments diffes fom the phase of peiodic field and depends on a adius, ie ), we obtain the following equa- tions fo m, p : m m m m m m Dm h D m hm m cos m m () m m m h sin (6) * * mm mm i Fom (), we obtain the expessions fo the density of PBS enegy elative to the initial state, togethe with the enegy of inteaction with the extenal field (see, fo example [8]): Dm e, m hdm m m h m cos hd and fo the change of this enegy connected with dissipation and the action of the extenal peiodic field: h e, m m m m sin The Equations ()-(8) constitute a complete desciption of PBS, including thei time tansfomation Howeve, in the given pape we conside only the equilibium state of PBS inside the feomagnet, ie when the decease of enegy caused by dissipation is compensated by enegy flow fom the extenal peiodic field, ie e, Futhemoe, in this case, m, Theefoe, fo the equilibium state of PBS we have the following expessions: m, change m m h sin e, change m m h sin and the elation: (7) (8) (9) () acsin a m, () whee a h Coespondingly, the equation fo equilibium PBS is Copyight SciRes

3 8 V V NIETZ the following: d d a m m d d m m m m d m a m d m Dm h D hm m a m () In ou case, equilibium PBS ae self-oganiing systems o dissipative stuctues [-] that can aise spontaneously in a metastable state duing the fist-ode phase tansition to the stable equilibium state, ie to m Fo such PBS the dissipative stuctues, the entopy incease connected with dissipation is compensated by the decease of the entopy that due the extenal peiodic field It can be expessed as follows: whee diss d diss h, () d d d h m d d T (4) Chaacteistics of Equilibium PBS It follows fom the foegoing that at the action of a peiodic magnetic field only the PBS with a fequency specified by this field may aise spontaneously This is diffeent fom the phase tansition discussed in [8], whee the fequency and, accodingly, the configuation of aising PBS ae not defined In Figues -, the configuations of the PBS fo seveal fequencies of pecession at k, h 998 ae pesented Fo each fequency, thee is the solution of the Equation () with coesponding PBS that pecesses in substatum of unifom pecession of the bulk cystal Beside such solution, thee is the solution that coesponds to homogeneous pecession of magnetic moments, without soliton Such a solution at is shown in Figue Note that the maximum fequency of PBS at h 998 without peiodic field equals es and coesponds to magnetic esonance in a metastable state In Figue and in the following examples, the same paametes as in the [8] aticle ev ( M 8 Oe, K, Oe Oecm Oecm ) ae used and futhemoe, H Oe, 4 In Figues 4 and, the fequency dependencies fo main paametes of PBS at h 998 ae pesented: the enegy Es 4πMK e d (in coep p p spondence with (7)); the amplitude sm im p Figue Configuations of solutions of the () equation fo h = 998, h : numbe is at, and numbe is at 8 Numbe is the solution fo homogeneous pecession, without PBS, at p 4 4 Figue The same as in Figue, fo the following fe- quencies: numbe is if ω, numbe 4 is fo, and numbe is fo p Figue The same as in Figue, fo the following fequencies: numbe 6 is fo ω, numbe 7 is fo, and numbe 8 is fo Copyight SciRes

4 V V NIETZ 8 p sm Figue 4 Fequency dependencies of enegy (full cicles), amplitude p sm (full squaes), and adius (cosses) of PBS fo h 998, h Hee and futhe, the utmost values of paametes of equilibium PBS ae noted by a dotted line Figue Fequency dependencies of enegy and amplitude p of PBS fo h 998, h 6 sm (hee p is the amplitude of unifom pecession); and adius of PBS is As can be seen fom (), if psm, the equilibium state of PBS is possible only unde the condition: p sm p sm h () Fequency ange of equilibium PBS inceases with the incease of peiodic field amplitude Fo h 998, if h, 4, condition psm h coesponds to max If h 6, then max 7 (see in Figues 4 and ) The possibility of equilibium PBS is connected with the fact, that duing thei pecession, the magnetic moments lag by phase behind pecession of peiodic field In the Figues 6 and 7, the fequency dependency of the degee Figue 6 The fequency dependencies of the angle (full cicles) if h 998, h, and coesponding exponential facto at degee 4 - T 9 K Figue 7 The same fequency dependency if the angle as in Figue 6 in the ange nea angle of such a delay in cente of PBS, ie, is shown Besides, in Figue 6 the cuve of coesponding exponential facto, that defines tempeatue dependency of the PBS pobability, is pesented The pobability of PBS oigin deceases shaply at the incease of PBS enegy If the magnetic field h is enough lage, the equilibium PBS with amplitude m m ae possible It is obviously in such cases, the condition fo equilibium PBS can be witten as: h (6) In Figues 8 and 9, two examples of such case fo h 99 ae pesented As seen in the second example, in Figue 9, the equilibium PBS ae possible with the negative enegy and at E s, ie in the bifucation point b The configuation of such equilibium PBS is shown in Figue exp(-e s /) Copyight SciRes

5 8 V V NIETZ m m m - - Figue 8 The fequency dependencies of enegy and am- plitude fo equilibium PBS if h 99, h Fo compaison, these dependencies fo h case ae shown too (by open cicles fo enegy and open squaes fo the amplitude) The enegy E = is fo the initial homogeneous metastable state Figue Configuation of equilibium PBS at Es Hee h 99, h, 4 - b exp(-ie s I/) m m Figue 9 Fequency dependencies of enegy and amplitude fo equilibium PBS if h 99, h Fo compaison, these dependencies fo h case ae shown too The pobability of the equilibium PBS nea the bifucation point whee Es inceases shaply is shown in Figue The enegy mev coesponds to tempeatue about 9 K 4 Conclusions Peiodic magnetic field acting pependicula to the axis of easy magnetiation, ie pependicula to the main magnetic field, leads to a significant change in the natue of the pecessing magnetic solitons in the fist ode tansition, in compaison with [8] ) Nea the bounday of the existence of the metastable state, only PBS with pecession fequency equals Figue Fequency dependencies of the enegy, β angle in cente of PBS (noted by asteisks) and exponential tempeatue facto fo equilibium PBS at h 99, h The bifucation point is noted as b the fequency of the extenal field can occu The fequency of the extenal field specifies uniquely of othe chaacteistics of PBS (amplitude, configuation, enegy) too ) Aising PBS ae equilibium, ie educing thei enegy due to dissipative pocesses is fully compensated by the influx of enegy fom an extenal peiodic field ) We can conside the equilibium PBS as dissipative stuctues o self-oganiing systems [-], when the incease of entopy connected with enegy dissipation is compensated by a negative flow of entopy due to the action of an extenal peiodic field 4) At the phase tansition in a feomagnet, the dissipative stuctues in the fom of equilibium PBS can be oiginated not only in the pesence of the bifucation point, but also in a moe geneal case: when the enegy of PBS state is small enough Copyight SciRes

6 V V NIETZ 8 ) The compensation of enegy and entopy is accompanied by the fact that PBS duing the pecession ae delayed by phase behind the pecession of a peiodic field The magnitude of this delay depends on the adius and the maximum delay is at the cente of PBS 6) On the side of the positive values of fequencies, amplitude and fequency of pecession of equilibium PBS ae defined by the following elations: m if and h if m m h m, m m REFERENCES [] P Glansdoff and I Pigogine, Themodynamic Theoy of Stuctue, Stability and Fluctuations, Wiley and Sons, London, 97 [] G Nicolis and I Pigogine, Self-Oganiation in Nonequilibium Systems Fom Dissipative Stuctues to Ode though Fluctuations, Wiley and Sons, London, New Yok, 977 [] H Haken, Advanced Synegetics, Instability Hieachies of Self-Oganiing Systems and Devices, Spinge-Velag, Belin, Heidelbeg, New Yok, Tokyo, 98 [4] D Kondepudi and I Pigogine, Moden Themodynamics: Fom Heat Engines to Dissipative Stuctues, Wiley and Sons, Chicheste, 998 [] A M Zhabotinskiy, Oscillatoy Pocesses in Biological and Chemical Systems, Moscow, 967 (in Russian) [6] A M Zhabotinskiy, Biophysics, Vol 9, 964, p 6 (in Russian) [7] S Chandasekha, Hydodynamic and Hydomagnetic Stability, Oxfod, 96 [8] V Niet, Kinetics of Pecessing Ball Solitons in Feomagnetic at the Fist-Ode Tansition, Euopean Physical Jounal B, Vol 8,, p [9] L D Landau and E M Lifshit, Physikalische Zeitschift de Sowjetunion, Vol 6, 9, p Copyight SciRes