On the magnetization process of ferromagnetic materials

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1 On the mgnetiztion process of ferromgnetic mterils Abstrct Introduction The theory which describes mgnetiztion of ferromgnetic mterils is crucilly importnt both from scientific nd ppliction point of views. A physicl theory will permit correctly tke into ccount ll interctions which re involved in the mgnetiztion process nd to find out the reltionship between the structure nd physicl properties of mgnetic mterils. Applicble mthemticl model could be derived from such the theory where ll energies would be included in the correct wy. This model will give possibility to investigte rel physicl nd structurl properties of mterils from experimentl dt. It is essentil for the synthesis of new mterils with necessry properties. Nowdys there re severl models to describe mgnetiztion of ferromgnetic mterils. ore detiled description nd nlyzes of dvntges nd disdvntges of these models one cn find in works [1-]. In the present pper, there re suggested new theory of mgnetiztion nd n ttempt to derive the pplicble generl mthemticl model to describe mgnetiztion curve for soft nd stiff mgnetic mterils. uch interctions s dipole-dipole interction (dip-dip), n exchnge interction (exch), n nisotropy field nd temperture influence will be included. R. Khchturyn*, V. ekhitrin e-mil: rubenftf@gmil.com Institute for Physicl Reserch Ntionl Acdemy of ciences of Armeni The present rticle sttes tht ferromgnetic smple could be considered s prmgnetic system where mgnetic moments re mgnetic domins. sed on this conclusion nd tking into ccount presence of n nisotropy field the formul which describes mgnetiztion dependence on the externl mgnetic field is derived. Expressions for remnent mgnetiztion nd coercive force re presented. The new prmeter to chrcterize mgnetic stiffness of mteril is introduced. A physicl expression for dynmic mgnetic susceptibility s function of mteril s chrcteristics, externl mgnetic field, nd the temperture is given. Importnt points, s remnent mgnetiztion, nd coercive force, on mgnetiztion curve re considered, s well. The formul for dynmic mgnetic susceptibility is deduced from mgnetiztion field dependence. The new prmeter which chrcterizes the mgnetic stiffness of mteril is introduced nd its temperture dependence is derived. A totl mgnetiztion is considered s sum of hysteretic nd nhysteretic prts of mgnetiztion, like Jiles-Atherton model does [4]. 1. The min ide There re two competing interctions in ferromgnetic mterils: the exch which tends to orient mgnetic moments in the sme direction nd by this mgnetizes the system nd the dip dip which tends to orient mgnetic moments ntiprllel to ech other nd by this demgnetize the system. A relevnt difference between these interctions is tht the exch cts between nerest toms nd its energy is independent of the mgnetic moment of the system. On the contrry, the dip dip energy rises s mgnetic moment increses. Incresing mteril size the dip dip energy cn overcome the exch energy. In this cse, two nd more 1

2 domins structures become more fvorble. It is schemticlly shown in figure 1. As the exch energy is much bigger thn the dip-dip energy between nerest toms mgnetic moments inside domin re firmly connected in the sme direction. The mgnetic shell where the dip-dip energy becomes equl to the exch energy could be ccepted s border of the domin nd by this defining size of the domin [5]. o, fter domin ws completed the next shell of mgnetic moments will recline in the opposite direction nd by this two nd more domins systems exist, figure 1. It should be mentioned tht in relity trnsition from one domin to nother occurs smoothly throw domin wlls region [6]. Eexch Fig.1. Digrm representtion how exch energy nd dipdip energy behve with incresing of size On the ssumption of foregoing we could consider domins like solitry mgnetic prticles which re seprted by domin wlls. For simplicity, we would neglect thickness of domin wlls. The seprted qusiprticles we would cll supermgneton (sm) nlogiclly to R.Hrrin [7-9]. We know tht the exch is compensted by the dip-dip between sms. It mens tht sms mgnetic moments re not rrnged by the exch. o, the problem of ferromgnetic mterils is brought to the problem of prmgnetic mterils where sms ply role of mgnetic moments. eeing tht there re xes of esiest mgnetiztion in ferromgnetic mterils [10] sms re distributed in field of nisotropy ccording to the oltzmnn distribution. For simplicity, we would consider cse of unixil nisotropy. ms with positive projection on ny selected direction long the E dipdip const V nisotropy xis seprted from sms with negtive projection by nisotropy brrier, figure. E E 0 Fig.. ms distribution in n nisotropy field Amount of sms with positive nd negtive projection on ny selected direction could be estimted s: N1 g( ) e d 0 (1) N g( ) e d where g is distribution function, E и E re nisotropy energies of domins t ngles nd.. gnetiztion Generlly, mgnetiztion process could be divided into two stges: on the first stge domin wlls moves, on the second stge domin wlls moves nd domins rotte. ) The first stge of mgnetiztion On the first stge domin wlls move nd dip-did resists to this process. We would strt our considertion from two domin smple, figure. Fig.. Two dipoles system in n externl mgnetic field. The positive direction would be considered s direction of the externl mgnetic field. E H

3 m with positive projection on field direction hs mgnetic moment m nd sm with negtive m. In the mgnetic field H the smple will be mgnetized to the vlue. Appernce of mgnetiztion would crete n dditionl nonzero dip dip. o sm's energy is consists of three terms: the energy of mgnetic moments in mgnetic field, energy of dip-dip (which will be further derived from forml 10), nisotropy energy [11-1]. Energies of mgnetic sms with positive nd negtive projections re: U m H cos m cos () U m H cos m cos where is n ngle between m nd H, is n ngle between m nd,, is the distnce between sms, nd their difference is: U U U mh cos m cos () where 0.5 is n nisotropy energy which presumble depends on nisotropy field nd on the ngle between the vector nd n esiest xis of mgnetiztion, m m m is mgnetic moment of one domin in zero mgnetic field. gnetiztion is cused by difference between mgnetic moments in positive nd negtive projection. If mount of sms with positive nd negtive projections re N 1 nd N correspondingly, thn: mh cos m cos dn 1,, e (4) dn,, mh m N Ne (5) 1 mh m N1-N N1A1e mh m N 1+N N1A1e (6) N -N 1 e mh m 1 mh m 1 e mh m E N 1+N tnh (7) ecuse mount of sms is connected with mgnetiztions of smple s: m N 1+N (8) m N1-N is sturtion mgnetiztion nd where mgnetiztion of the smple in the field H. y multiplying eqution (7) on m nd using (8) we get: mh m E tnh (9) b) The second stge of mgnetiztion The energy of the externl mgnetic field is not enough to bring the mgnetiztion of smple to sturtion. For exmple sturted nickel nnorod with 5 n toms, 0.6 ohr per tom nd the 8 lttice constnt 10 sm cretes demgnetize 7 field 10 Oe in the sturted stte. We don't use such kind of externl mgnetic fields to sturte smple. o, dditionl processes must be included in mgnetiztion process. On the second stge, dip-dip could ply both demgnetize nd mgnetize roles. In relity, both processes pper simultneously, but mgnetiztion fctor could be neglected in the beginning. To understnd mgnetiztion process, the dipolr field distribution inside the smple should be tken into ccount. Let's consider the distribution of the induced dipolr field round mgnetic moment more detiled s shown on figure 4. The mgnetic field tht is induced by mgnetic moment in ny point could be clculted in first pproximtion s [1]: n( n) H (10) r

4 where n is unit vector in the direction to the point where nd r is distnce between mgnetic moment nd the point where the field is clculted. 1 b 4 Fig.4. Dipolr field distribution round the mgnetic moment. nd b re lines where mgnetic field chnge their projection sing on mgnetiztion direction. is n ngle between mgnetiztion direction nd lines nd b. 1, re spce regions where dipolr field hs positive projection on mgnetic moment direction nd,4 re spce regions where dipolr field hs negtive projection on mgnetic moment direction. It is seen from formul (10) tht H hs negtive or positive projection on the direction of mgnetic moment in different points. It is not difficult to found points where H chnges its projection sign on direction of the mgnetiztion from formul (10). n n 0 (11) n n cos 0 (1) cos (1) 55 (14) These points belong to the lines nd b which decline under ngles to mgnetiztion direction s shown on figure 4. Lines nd b demrcte spce round mgnetic moment on four regions: 1,,,4. Respectively to it H hs positive projection in ny point which belongs to regions 1 nd with biggest vlue H when r 0 or 180 nd hs negtive projection in ny pint of region nd 4 with biggest vlue H when 90or 70. r y this, dipolr field plys mgnetize role s well. During mgnetiztion process such systems could emerges where dipolr field plys mgnetize role. The exmple of such systems is systems with pores s spheroid with hollows s shown on figure 5. Fig.5. gnetic moment in nd outside of mgnetized ring (exmple of mgnetic moments in the hollow system). Usully the first stge of mgnetiztion is due to the domin wlls motion. After wlls moved mgnetized re forms hollow-like systems nd embrce prts which still hve the negtive projection of mgnetiztion s shown in figure 5. ) b) Fig.6. Formtion of hollow systems during mgnetiztion As shown in figure 5 mgnetic moments inside hollow systems will mgnetize under the influence of externl mgnetic field s well s under the influence of the dipolr field. Then, mgnetiztion process is due to the domin wlls motion nd rottion of the unmgnetized prts s clusters under the influence of the dipolr field inside hollow systems. In the end the whole smple will be mgnetized. And dip-dip plys role of the demgnetiztion fctor only from the outer surfce of mgnetized re [6]. Without hollow-like systems there will be impossible to mgnetize bulk smple becuse dipolr demgnetiztion could be very strong. ecuse of ll written bove the dip-dip energy term in (9) should be replced by m, where is prmeter which depend on difference between demgnetizing nd mgnetizing prts of dip-dip influences, by this depends on externl field nd should depend on surfce nd surfce volume rtio. The higher the vlue of n externl field the lower the vlue of. mh m tnh (15) As the exmples, we will consider twodomin nnorod s it shown on figure 7. 4

5 Let s compre processes of mgnetiztion of such rod when the externl mgnetic field is directed long longitude (x xis) nd when the externl mgnetic field is directed long width (y or z xes). Fig.7. Two domin nnorod. When the nnorod is mgnetized long x xis the dipolr field of one domin will mgnetize the second domin becuse it belongs to the region of the spce where dipolr field hs positive projection on mgnetiztion direction. The digrm of the process is shown in figure 8. y this dipolr field will help to mgnetize the system. x y z H x After the externl mgnetic field ws bolished dip-dip tends to demgnetize the smple. ecuse of this sms would turn from positive projection to negtive through nisotropy brrier. ut not ll sms cn overcome nisotropy brrier nd the prt of them would remin with positive projection. o, fter the externl mgnetic field ws bolished ferromgnetic smple would steel remin in mgnetize condition. This mgnetiztion is clled remnent mgnetiztion. It is represented in figure 10. In order to get expression for remnent mgnetiztion it is necessry to put H=0 into (15): m R R tnh (16) where R R, R is remnent mgnetiztion. The remnent mgnetiztion stte is schemticlly represented in figure 10. mh E b Fig.8. The nnorod is mgnetized long longitude. When the nnorod is mgnetized long z xis one domin will direct the second domin in the opposite direction becuse it belongs to the region of the spce where dipolr field hs negtive projection on mgnetiztion direction s shown in figure 9. In this cse 1. z Fig.9. The nnorod is mgnetized long width. In bulk smples the both cses tke plce. It hs to be mentioned tht demgnetize field influence is bigger; in other cse spontneous mgnetiztion will emerge.. Remnent mgnetiztion nd Coercive force H Fig.10. ms with positive projection (blue) nd negtive projection (red) on field direction ) in stte of sturtion mgnetiztion; b) in stte of remnent mgnetiztion As is known coercive force is mgnetic field which should be pplied to the smple to demgnetize it. o to get n expression of coercive force one need to put =0 into (15): HC T (17) m It is seen tht coercive force depends on the field direction (ngle between field nd nisotropy xis). 4. gnetic stiffness 0 The next prmeter cn chrcterize mgnetic stiffness of ferromgnetic mterils: m R E k e (18) As is seen from (5) k shows the rtio between the mount of sms which overcme nisotropy brrier to the mount of sms which 0 5

6 remin with positive projection fter the externl mgnetic field ws bolished, figure 10. Vlues of k could chnge in the rnge from 0 to 1. The bigger the vlue of k the softer mgnetic mteril is nd vice verse. For exmple, in cse of strong nisotropy no of sms re ble to overcome nisotropy brrier, it mens tht there re no sms with negtive projection ( k 0 ) nd R rectngulr-like hysteresis loop, figure 11 ). In the cse when ll sms were ble to overcome nisotropy brrier, the mount of sms with negtive projection is equl to the mount of positive projection ( k 1) nd consequently R 0, figure 11 c). ) b ) c) Fig.11. Three possible cse of sms distribution in stte of remnent mgnetiztion: ) 0; b) k 0;1 ; k 1. k Cses ) nd c) corresponds to mgneto stiff mteril nd mgneto soft mterils correspondingly 5. Dynmic mgnetic susceptibility A low which describes mgnetiztion dependence on externl mgnetic field gives possibility to find out lw for mgnetic susceptibility: (19) d d m H tnh dh dh kt d d m1 4 dh dh 1 m H E ch d d m1 dh dh mh ktch m kt (0) It should be noted tht decreses with incresing of H thus the term d 0. dh This term is responsible for high vlue of mgnetic susceptibility of ferromgnetic mterils. This term is responsible for lrge vlues of mgnetic susceptibility of ferromgnetic mterils. It is seen tht mgnetic susceptibility depends on the externl mgnetic field, the dipolr interction, the temperture of the smple nd the nisotropy energy, s it ws expected. In cse of high temperture or smll externl mgnetic field tnh x x ; 1 from (15) one cn get: d m dh m 6. Hysteretic prt of mgnetiztion (1) The nisotropy field influence on the rottion of mgnetic moments nd it reflects on mgnetiztion curve. In order to estimte n influence of n nisotropy field on mgnetiztion curve one cn compre mgnetiztion of the smple with nisotropy field nd without it [4]: () hyst hyst tot n m H tnh m H tnh () where tot is the totl mgnetiztion, n is n nhysteretic mgnetiztion which is due to the energy of mgnetic moment in n externl mgnetic field nd is the hysteretic mgnetiztion which is due to losses processes (pinning, nisotropy etc.) Grphiclly () is shown on figure 1. It could be ssumed tht the re under the H curve corresponds to energy loses hyst during mgnetiztion nd demgnetiztion of the ferromgnetic. Fig.1. Dshed is hysteresis, dshed with dots is the nhysteresis mgnetiztions, blue nd red lines re difference between hysteretic nd nhysteretic mgnetiztions. hys 6

7 7. Results nd discussions There ws shown tht mgnetic domins could be considered s qusi prticles with mgnetic moments. These qusi prticles don't prticipte in exch. There were considered the wy of mgnetiztion: domin wlls motion nd rottion of domins. sed on this ssumption there ws elicited the nlyticl excretion to describe dependence of mgnetiztion of ferromgnetic mteril on n externl mgnetic field (15). It is importnt tht ll energies tht tke plce in the process of mgnetiztion re included dditively. It mens tht dditionl energies like pinning effects could be esily dded in the formul (15). There were derived expressions for remnent mgnetiztion (16) nd coercive force (17) s specil points on ( H ) curve. It is seen tht temperture dependence of remnent mgnetiztion bers exponentil chrcter nd depend on dip-dip in smple nd on n nisotropy brrier. It is lso seen how coercive force depend on nisotropy brrier nd mgnetic moment of domin. A new prmeter which chrcterizes mgnetic stiffness of mteril nd its temperture dependence is introduced (18). From ( H ) function there ws derived n expression for mgnetic susceptibility nd its dependence on temperture, field, dip-dip mong domins nd on n nisotropy energy (0). It is shown tht in extreme cses, like high temperture or low field, mgnetiztion depends on field linerly, nd mgnetic susceptibility is field independent like in the prmgnetic cse. There ws shown how mgnetic susceptibility of ferromgnetic cn reches so high vlues. It is essentil to note tht expressions (0) (1) re pplicble t ll temperture regions. An influence of n nisotropy field on the mgnetiztion of ferromgnetic ws considered () nd shown in figure 1. Further, it could be compred with results of J-A model [4] nd necessry prmeters like k could be found. Literture 1. D.C Jiles, X. Fng, W. Zhng, Hndbook of Advnced gnetic terils, (006). F.Liorzou,. Phelps, nd D.L. Atherton, croscopic models of mgnetiztion, IEEE trnsctions on mgnetics, vol.6, No.., (000). ergey E. Zirk, Yuriy L. oroz, Robert G. Hrrison, nd Krzysztof Chwstek, On the physicl spects of the Jiles-Atherton hysteresis models, J. Appl. Phys. 11, (01) 4. D.C. Jiles nd D.L. Atherton, Theory of ferromgnetic hysteresis, J. gn. gn. ter. 61, 48, (1986) 5. J. Frnkel nd J. Dorfmn; pontneous nd Induced gnetistion in Ferromgnetic odies, (190) 6. L. Lndu, E. Lifshits, On the theory of the dispersion of mgnetic permebillity in ferromgnetic bodies, Phys. Zeitsch. der ow. 8, pp , (195) 7. R.G. Hrrison, Physicl model of spin ferromgnetism, (00) 8. R.G. Hrrison, Vrible-Domin-ize theory of spin ferromgnetism, (004) 9. R.G. Hrrison, Physicl theory of ferromgnetic first-order return curves, IEEE TRANITION ON AGNETIC, Vol. 45 NO, 4, (009) 10. L. D. Lndu nd E.. Lifshitz, Course of Theoreticl Physics, Vol. 8, (005) 11. C. Kittel, Theory of the structure of ferromgnetic domins in films nd smll prticles, Phys.Rev. vol.70, NO. 11 nd 1, (1946) 1. L. D. Lndu nd E.. Lifshitz, Course of Theoreticl Physics, Vol. : The Clssicl Theory of Fields (Nuk, oscow, 1988; Pergmon, Oxford, 1975) 7

arxiv: v3 [cond-mat.mtrl-sci] 15 Jul 2016

arxiv: v3 [cond-mat.mtrl-sci] 15 Jul 2016 On the mgnetiztion process in ferromgnetic mterils Ruben Khchturyn nd Vhrm ekhitrin Institute for Physicl Reserch, NA of Armeni, Ashtrk, Armeni (Dted: July 18, 2016) rxiv:1506.01805v [cond-mt.mtrl-sci]