Oguchi Approximation of a mixed spin-2 and spin-5/2 Blume-Capel Ising ferrimagnetic system

Transcription

1 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October 14 1 Oguch pproxmaton of a mxed spn- and spn-5/ lume-capel Isng ferrmagnetc system Hadey K Mohamad Department of Physcs, l-muthanna Unversty,Samawa55, Iraq bstract- y usng the Oguch approxmaton(o), we study a mxed spn- and spn-5/ lume-capel Isng ferrmagnet wth dfferent sngle-on ansotropes on a smple cubc lattce It has been carred out Oguch method and calculated the free energy of a mxed spn ferrmagnetc model from the partton functon y mnmzng the free energy, t has been obtaned the equlbrum magnetzatons and the compensaton temperatures The exstence and dependence of a compensaton temperature on the crystal feld s manly nvestgated Index Terms- Mxed-spn lume-capel Isng model; Oguch approxmaton; Ferrmagnet; Crystal feld; Compensaton ponts R I INTRODUCTION ecently, a lot of efforts has been drected to the ferrmagnetc materals, specally due to ther great potental for technologcal applcatons [1,] In a ferrmagnetc materal two nequvalent moments, nteractng antferromagnetcally, can gve rse to a zero spontaneous magnetzaton below ts crtcal temperature[3] Then, a specal pont whch s called spn compensaton temperature can appear at a temperature below the ( T c ) crtcal one, where the sublattce magnetzatons cancel exactly each other[4-8] T Kaneyosh et al have proposed that a dluted mxed spn- and spn-5/ ferrmagnetc Isng system on honeycomb lattce can explan the characterstc temperature dependence of magnetzaton observed at low temperatures n the molecular-based magnetc materal, Fe Fe ( CO4 ) 3[ N( n Cn H n1) 4, n 3 5] [9] Y Nakamura et al observed that the effect of external magnetc felds on the lattce possbly leads to a compensaton transton[1] The authors of [11], n ths respect, nvestgated the magnetc propertes of a mxed spn- and spn-5/ Isng ferrmagnetc system n the phase dagram and n the compensaton temperature It s worth to note that a mxed-spn Isng ferrmagnetc system on a smple cubc lattce n whch the two mxng sublattces have spns two ( 1/, 3/, 5/, 1, ) and spns fve-half( ), has not been examned wthn Oguch approxmaton(o) So, nstead of usng the decouplng approxmaton, we have ntroduced the concept of a correlaton behavour to understand the corporatve effects exhbted by such systems We found that the system consdered exhbts the compensaton phenomenon at low temperatures Our results show the possblty of many compensaton ponts at low temperatures dependng on certan values of ansotropes near by the boundares of order-phase transtons n the ground-state structure of the system II THEORY The mxed-spn ferrmagnetc Isng model, whch s S S consdered, conssts of three-dmensonal sublattces and S, 1, S 1/, 3/, 5/ wth spns and respectvely In ths research, the system s descrbed by the Oguch approxmaton[1], that: H JS S D ( S ) D ( s ) ( h S h S ) wth, (1) h J( z 1) m h J( z 1) m, where, and J J s the exchange nteracton between spns at stes and z s the number of nearest neghbourng m spns and the sublattce magnetzatons and m are the S S thermal averages of and, respectvely, e, m S m S, and Takng the egenvalues of the Hamltonan(1), one can obtan the sublattce magnetzatons D, D per ste (ppendx I) are the ansotropes, e, the crystal felds, actng on the spn- and spn-5/ respectvely The knowledge of the partton functon(z) allows to express the relatons for thermodynamc quanttes Then, the free energy of the model s defned as[13], F kt ln Z ; Z e, H where F s the free energy of H gven by relaton (1), K 1T The sublattce magnetzaton per ste s obtaned by mnmzng the free energy (Eq) whch s gven by ppendx 1 It s worth notng that the ferrmagnetc case shows that the sgns of sublattce magnetzatons are dfferent, and there ()

2 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October 14 may be a compensaton pont at whch the total longtudnal magnetzaton per ste[11,14], that s equal to zero 1 M ( m m ) III RESULTS ND DISCUSSIONS In ths work, we examne the magnetc propertes concernng spn compensaton temperatures of the threedmensonal mxed spn- and spn-5/ ferrmagnetc lume- Capel Isng model Frst, let us study the phase dagram of the mxed-spn ferrmagnetc system wth dfferent crystal felds through whch, we nterest to consder the characterstc magnetc propertes of the system 3 M D/IJI= Fg1 Thermal varatons of the total magnetzatons M for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6,when 1, wth varous values of

3 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October m 15 1 D/IJI= m Fg Thermal varatons of the sublattce magnetzatons for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6,when Fg(1), and Fg() stand for the total magnetzaton and sublattce magnetzatons versus the absolute temperature for the 1 values of, respectvely s shown n the fgures, the system may exhbt one compensaton pont n the phase dagrams dependng on the values of crystal J 5 J 5 felds( ), n the range We report an nterestng feature of compensaton temperatures for 1, and for the value The result shown n the Fg(3) s consstent wth that derved from Fg() s s seen from the fgure, n the regon where the system may 1, wth varous values of show a compensaton pont, the spn- sublattce magnetzaton s more ordered than the spn-5/ sublattce magnetzaton below the compensaton temperature s the system s heated up, the drecton of resdual magnetzaton may swtch That s to say, due to entropy some spns can flp ther drectons[] Thus, the spn-5/ sublattce magnetzaton becomes more ordered than the spn- sublattce magnetzaton for temperatures above the compensaton temperature So there s an ntermedate temperature such that the cancellaton s complete( M )[11,14]

4 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October m m Fg3 Thermal varatons of the sublattce magnetzatons for the mxed spn lume-capel Isng ferrmagnet wth the D 1 coordnaton number z=6,when D, and 5 m 15 1 D/IJI= m -5-3

5 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October 14 5 Fg4 Thermal varatons of the sublattce magnetzatons for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6,when, wth varous values of m, m Fg(4) expresses the characterstc behavours of as a functon of k T for dfferent values of 5,, 15, 1, 5, where the system may exhbt three or many compensaton temperatures, when 5, and Our results are obtaned n a mcroscopc model descrbng the magnetc behavors of ths mxed-spn system that shows remarkable nsght compared to[9,1] It s worth notng that we have ntroduced the concept of a correlaton behavour s s seen n Fgs((4),(5)), when the value of the negatve crystal feld on - atoms s relatvely large, a compensaton pont may be exsted So, the possblty of many compensaton ponts s found when the negatve crystal felds actng on -atoms ncrease The author of [9] nvestgated the mxed spn-5/ and spn- Isng model on a honeycomb lattce, whch may exhbt a compensaton pont when the value of the postve sngle-on ansotropy, e, the crystal feld on -atoms(spn-) s relatvely large n the layered system on the bass of MC smulaton Whereas, the effectvefeld approxmaton predcts the exstence of a compensaton temperature, but only for relatvely small transverse felds, n the absence of crystal feld[1] One can compare our results wth that ones publshed n refs[9,1] 4-1 M -5 D/IJI= Fg5The temperature dependences of the total magnetzaton M for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6, when, wth varous values of

6 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October M Fg6 close vew of the temperature dependence of the total magnetzaton M for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6, when, and 5 4 M D/IJI= Fg7 The temperature dependences of the total magnetzatons M for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton number z=6,when, wth varous values of

7 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October 14 7 Fg(5) may show a characterstc feature for the temperature dependences of M n the system for dfferent values of crystal felds actng on -atoms, wth a fxed one of Fg(6) expresses a close vew of the compensaton behavour derved from Fg(5), for the mxed spn lume-capel Isng ferrmagnet wth the coordnaton D number z=6, when D 5, and Now, let us examne the system n the range 4 J s s seen from the Fg(7), the present system may clearly exhbt many compensaton ponts for the M D 1 5 curves labeled D, at T, wth dfferent values of, whch s classfed after Neel as the W-type[1] One can compare our results wth those obtaned n the mxed spn- and spn-5/ systems[9,15], n whch the models show one compensaton temperature, respectvely lso, we fnd that the transton (or crtcal) temperature depends strongly on the values of ansotropes D, D actng on the -atoms and -atoms, respectvely, where the transton temperature can be determned by requrng that the sublattce magnetzatons, m (ppendx I) tend to zero contnuously as the temperature s close to the Cure pont So, we observe that, for the values of crystal felds on -atoms, the transton temperature may be decreased However, by usng the Oguch J m approxmaton, t has been obtaned clear and characterstc behavors of the molecular-based magnet Fe Fe ( CO4 ) 3[ N( n Cn H n1) 4, n 3 5] [3, 9,1] IV CONCLUSION We have nvestgated the compensaton phenomenon of a mxed-spn lume-capel Isng ferrmagnetc system composed S S 5/ of and, n the applcaton of crystal feld In ths work, we have appled O to the study of a smple cubc lattce to nvestgate nterestng features, one can observe Fgs(4,5,7) Partcularly, t has been studed the effect of sngleon ansotropy,e, the crystal feld on an exstence of compensaton temperature So, our results predct the possblty of many compensaton ponts at low temperatures dependng on the negatve values of crystal felds(fgs(6,7)) However, as far as we know, up to now no studes have been made concernng the spn compensaton phenomenon of the mxed spn- and spn-5/ lume-capel Isng model on the bass of O Fnally, we can conclude that the crystal feld plays a relevant role n the exstence of the compensaton temperature and t s useful to note that the approxmaton presented can be used to obtan mportant results for three-dmensonal lattces It s worth to note that we found unusual behavours wll stmulate expermental and theoretcal works on the system whch s consdered PPENDIX I y usng the Oguch pproxmaton, the sublattce magnetzatons of the mxed-spn lume-capel Isng systems are gven: m=((a1snh(g1)+b1snh(g)+bsnh(g3)+c1snh(g4)+csnh(g5))+(exp(-3βd))(asnh(h1) -a3snh(h)+b3snh(h3)-b4snh(h4)+c3snh(h5)+c4snh(h6))) / (a1cosh(g1)+exp(- 5βJ)+b1cosh(g)+bcosh(g3)+c1cosh(g4)+ccosh(g5)+(exp(-3βD))(acosh(h1) +a3cosh(h)+b3cosh(h3)+b4cosh(h4)+c3cosh(h5)+c4cosh(h6))+(exp(-4βjd ))(cosh(h7)+cosh(h8)+cosh(h9))), m=(5)(5(a1snh(x1)+asnh(x)+a3snh(x3)+snh(x4))+(3exp(-4βd))(b1snh(y1)- bsnh(y)+b3snh(y3)+b4snh(y4)+snh(y5))+(exp(-6βd))(c1snh(z1)-csnh(z)+c3snh(z3) -4snh(z4)+snh(z5))) / (a1cosh(x1)+exp(-5βj)+acosh(x)+a3cosh(x3)+cosh(x4)+(exp(-4βd ))(b1cosh(y1)+bcosh(y)+b3cosh(y3)+b4cosh(y4)+cosh(y5)) +(exp(-6βd))(c1cosh(z1)+ccosh(z)+c3cosh(z3)+c4cosh(z4)+cosh(z5))), where, x1=5βj(z-1)m ; x=375βj(z-1)m ; x3=15βj(z-1)m ; x4=5βj(z-1)m ; y1=4βj(z-1)m ; y=βj(z-1)m ; y3=75βj(z-1)m ; y4=5βj(z-1)m ; y5=15βj(z-1)m ; z1=3βj(z-1)m ; z=βj(z-1)m ; z3=175βj(z-1)m ; z4=75βj(z-1)m ; z5=5βj(z-1)m ; x11=5βj(z-1)m ; x=375βj(z-1)m ; x33=15βj(z-1)m ; x44=5βj(z-1)m ; y11=4βj(z-1)m ; y=βj(z-1)m ; y33=75βj(z-1)m ; y44=5βj(z-1)m ; y55=15βj(z-1)m ; z11=3βj(z-1)m ; z=βj(z-1)m ; z33=175βj(z-1)m ; z44=75βj(z-1)m ; z55=5βj(z-1)m ; g1=4βj(z-1)m ; g=3βj(z-1)m ; g3=8βj(z-1)m ; g4=4βj(z-1)m ; g5=16βj(z-1)m ; h1=3βj(z-1)m ; h=βj(z-1)m ; h3=βj(z-1)m ; h4=βj(z-1)m ; h5=14βj(z-1)m ; h6=6βj(z-1)m ; h7=βj(z-1)m ; h8=1βj(z-1)m ; h9=4βj(z-1)m ; a1=exp(5βj) ; a=exp(5βj) ; a3=exp(-5βj) ; b1=exp(3βj) ; b=exp(-3βj) ; b3=exp(15βj) ; b4=exp(-15βj) ; c1=exp(βj) ; c=exp(-βj) ; c3=exp(5βj) ; c4=exp(-5βj) ; d1=(βj(z-1)m+5βj(z-1)m) ;

8 Internatonal Journal of Scentfc and Research Publcatons, Volume 4, Issue 1, October 14 8 d=(βj(z-1)m-5βj(z-1)m) ; d3=(βj(z-1)m+15βj(z-1)m) ; d4=(βj(z-1)m-15βj(z-1)m) ; d5=(βj(z-1)m+5βj(z- 1)m) ; d6=(βj(z-1)m-5βj(z-1)m) ; d7=(βj(z-1)m+5βj(z-1)m) ; d8=(βj(z-1)m-5βj(z-1)m) ; d9=(βj(z-1)m+15βj(z-1)m) ; d1=(βj(z-1)m-15βj(z-1)m) ; d11=(βj(z-1)m+5βj(z-1)m) ; d1=(βj(z-1)m-5βj(z-1)m) ; d13=(5βj(z-1)m) ; d14=(15βj(z-1)m) ; d15=(5βj(z-1)m) ; k1=β (5J+4D+65D) ; k=β (-5J+4D+65D) ; k3=β (3J+4D+5D) ; k4=β (-3J+4D+5D) ; k5=β (J+4D+5D) ; k6=β (-J+4D+5D) ; k7=β (5J+D+65D) ; k8=β (-5J+D+65D) ; k9=β (15J+D+5D) ; k1=β (-15J+D+5D) ; k11=β (5J+D+5D) ; k1=β (- 5J+D+5D) ; k13=(65βd) ; k14=(5βd) ; k15=(5βd) ; The free energy of the model s defned as : F kt ln Z, that, Z=(exp(k1)cosh(d1)+exp(k)cosh(d)+exp(k3)cosh(d3)+exp(k4)cosh(d4) +exp(k5)cosh(d5)+exp(k6)cosh(d6)+exp(k7)cosh(d7)+exp(k8)cosh(d8)+exp(k9)cosh(d9)+exp(k1)cosh(d1)+exp(k11)cosh(d11)+ex p(k1)exp(d1)+exp(k13)cosh(d13)+exp(k14)cosh(d14)+exp(k15)cosh(d15)) REFERENCES [1] Ekz C, Strecka J, Jascur M(9) Mxed spn-1/ and spn-1 Isng model wth unaxal and baxal sngle-on ansotropy on ethe lattce, Cent Eur J Phys,Vol, Pp1-17 [] Dakhama, enayad N() On the exstence of compensaton temperature n d mxed-spn Isng ferrmagnetc:an exactly solvable model, J Magn Magn Mater, Vol13, Pp [3] Kaneyosh T,Nakamura Y,Shn S(1998) dluted mxed spn- and spn- 5/ ferrmagnetc Isng system; a study of a molecular-based magnet, Jphys:CondensMatter, Vol1, Pp [4] uenda G M, Lendo J (1997) Monte Carlo smulaton of a mxed spn- and spn-1/ Isng ferrmagnetc system, Jphys:Condens Matter, Vol9, Pp [5] uenda G M, Machado E(1998) Knetcs of a mxed Isng ferrmagnetc system, PhysRevE, Vol58(), Pp [6] Otmaa J, Entng I G(6) seres study of a mxed-spn S=(1/,1) ferrmagnetc Isng model, J Phys:CondensMatter,Vol18, Pp [7] obak, Jurcsn M(1996) Ferrmagnetsm n dluted mxed-spn twodmensonal Isng models, JMagnMagnMater, Vol163, Pp9-98 [8] obak, Jurcsn M(1997) theoretcal study of the dluted mxed spn-1 and spn-3/ Isng ferrmagnet, phys, Vol33, Pp [9] Nakamura Y() Exstence of a compensaton temperature of a mxed spn- and spn-5/ Isng ferrmagnetc system on a layered honeycomb lattce, PhysRev, Vol6(17), Pp [1] Nakamura Y, Shn S, Kaneyosh T() The effects of transverse feld on the magnetc propertes n a dluted mxed spn- and spn-5/ Isng system, Physca Vol84, Pp [11] Cruz Flho J S da, Godoy M, de rruda S(13) Phase dagram of the mxed spn- and spn-5/ Isng system wth two dfferent sngle-on ansotropes, Physca, Vol39, Pp [1] obak, Pokorny V, Dely J(1) nomalous behavour of the magnetc susceptblty of the mxed spn-1 and spn-1/ ansotropc Hesenberg model n the Oguch approxmaton, J Phys: Conference Seres, 1 (do:1188/ ///1) [13] Dely J, Strecka J, Canova L(7) Phase dagram of the spn-1 ansotropc Hesenberg model wth a snge-on ansotropy,condmatstatvol, Pp1- [14] Devren, Keskn M,Canko O(9) Magnetc propertes of an antferromagnetc and ferrmagnetc mxed spn-1/ and spn-5/ Isng model n the longtudnal magnetc feld wthn the effectve-feld approxmaton, Physca, Vol388, Pp [15] Keskn M, Ertas M(9) Exstence of a dynamc compensaton temperature of a mxed spn- and spn-5/ Isng ferrmagnetc system n an oscllatng feld, PhysRev E, Vol 8, Pp6114 UTHORS Frst uthor Hadey K Mohamad, Sc, M Sc, PhD, Department of Physcs, College of Scence, l-muthanna Unversty, Samawa, Iraq, PHONE: e-mal:hadeyk@yahoocom