A MODEL FOR THERMODYNAMICS OF SECOND, POSSIBLE THIRD AND FOURTH ORDER PHASE TRANSITIONS

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1 Afrian Jurnal f Physis Vl., 010. A MODEL FOR THERMODYNAMICS OF SECOND, POSSIBLE THIRD AND FOURTH ORDER PHASE TRANSITIONS 1 E. C. Ekuma, C. M. I. Okye, and G. C. Asmba 1 Deartment f Physis, University f Prt Harurt, Nigeria Deartment f Physis and Astrnmy, University f Nigeria, Nigeria 1 Present Address & Crresnding Authr: Deartment f Physis and Astrnmy, Luisiana State University, Batn Ruge, LA 7080, USA; anaeamee@yah.m Abstrat The thermdynamis f rders f hase transitins have been investigated by identifying the rders thrugh the analyti ntinuatin f the funtinal f the free energy using, Ehrenfest thery. We develed the Euler-Lagrange equatins fr the rder arameter and the vetr tential. By regnizing that near the hase bundary, that, there exist disntinuities in thermdynami bservables, we derived Ginzburg-Landau free energies fr studying ntinuus hase transitin exliitly in terms f their rders. The free energy is fund t bey a generalized trend by its exansin in the Ehrenfest sense f lassifiatin f rder f hase transitin. We seulate that there are hase transitins f rders greater than tw as there is n knwn hysial reasn why suh transitins shuld nt exist in nature. Keywrds: Ehrenfest thermdynamis, ntinuus hase transitins, free energy, thermdynami variables. Afrian Jurnal f Physis Vl.,. 11-1, (010) ISSN: PRINT: CD ROM: ONLINE:

2 Afrian Jurnal f Physis Vl., INTRODUCTION The thery f hase transitin is generally mliated due t the infinite number f system arameters that sum u t give the verall iture f the intrinsi rerties f materials at mar-sale. The relatinshi between ertain instabilities and materials has been the subjet f intensive study bth exerimentally and theretially, with large vlume f literature already available in leading sientifi jurnals. Even at this, the nature f the hase transitin urring in these unnventinal materials, fr e.g., suerndutrs has nt been well understd. The aggregatin f large number f interating artiles reveal nvel and interesting henmena that annt be understd by mere study f nly ne r few artiles. Phase transitin is ne f suh henmena that exhibit these rerties that we enunter in ur daily life [1]. Phase transitin is a thermdynami and statistial mehanial net defined simly as a hmgenus system, existene with the same hysial rerties but in different states. It is the transfrmatin f a thermdynami system frm ne hase r state f matter t anther []. In nature, a number f systems under ertain nditins (nstraints) underg hase transitin(s). The mst familiar examle is water with transitin between ie, liquid and var. Hwever, hase transitin als ur in many ther hysial systems suh as: 1) transfrmatin f a aramagneti t rdered magneti states; ) nrmal t suerfluid transitin in helium; ) the transitin f a nrmal metal t a suernduting hase; ) the insulatr-suerndutr transitin, et [,,5]. T desribe these systems either quantitatively r qualitatively, dee knwledge f thermdynamis and statistial hysis is required. Hwever, ur resent interest will be limited t its thermdynamial nature. Thermdynami nets rvide the general framewrk fr the desritin f hases and hase transitins. Within this net, hase transitin is an abrut hange f ne hase int anther due t the variatin f the (extensive) arameters f the system like the temerature T, ressure P, the magneti field H, et [,6,7]. Hwever, fr a given set f fixed arameters (sine thermdynami systems are defined fr fixed external nditins, e.g. T, V, et) there always exist a funtin knwn as the thermdynami tential having a minimum in the state where the system is in thermdynami equilibrium [6]. Fr instane, the Gibb s tential, within the Gibb s stability nditins, aids the study f the main rerties f hase transitin(s). 1

3 Afrian Jurnal f Physis Vl., 010. A hase may exist in an equilibrium ntat line r their differene may vanish as the transitin int is arahed. A nsequene f this, tw markedly thermdynami behavirs may exist: 1) the seemingly well-knwn first rder hase transitin; ) the (ntinuus) higher rder hase transitins. Hwever, the rder and the main features f the hase transitin are deendent n the way the exhange f stability between the ssible hases artiiating in the frmatin f the hase is rrelated. Anther very imrtant ntrlling arameter eseially in higher rder (ntinuus) hase transitin is the ritial state. It is a hmgeneus thermdynami state urring at higher rder transitin int. i.e., the ritial int where T T and the hases are marked by indistinguishability [8]. It must be nted that n arahing (within and near) the ritial int (state), the hases are ntinuus as the distintive features f the hases gradually disaear [9]. Within the lality f the ritial ints (r lines), the henmena taking lae are alled ritial henmena. This is true as the ritial behavir near (and at) the ritial line is a seial subjet f the ritial thermdynamis. This is als true as the hases in the ritial state and in the near-t-ritial states have almst the same stability. A tyial examle is the ara-t-ferrmagneti transitin where belw T, the lw temerature (ferrmagneti) hase is stable and has a nnzer sntaneus magnetizatin m in a zer magneti field ( H 0). But when the temerature arahes T frm the T T side ( T T T ; 0) the stability f the ferrmagneti hase is redued and at T T aramagneti ( m 0) hase bemes stable. The rdered (ferrmagneti) hase annt exist fr T T either as stable r metastable state (Stner riterin) and this again is a feature f higher rder hase transitin [10,11]. It must be nted that in the ntext f ratinal thermdynamis, the likelihd f the aearane f the ritial ints is related t the existene f nn-analyti singularities f the thermdynami funtins. Fr instane, the send derivative f the thermdynami tential i.e. the susetibility may shw either simle disntinuities r wer and lgarithmi singularities at the transitin int [10].. STATEMENT OF PROBLEM Phase transitin must be a very general henmenn harateristi f the basi rerties f many-bdy rblems. At the hase transitin int, thermdynami funtins beme singular whih lead t many unusual rerties f the material knwn as ritial henmena. This rvides us with infrmatin abut the real nature f the system whih is nt therwise aarent. 1

4 Afrian Jurnal f Physis Vl., 010. Large vlume f literature bth theretial and exerimental findings f the rders f hase transitins as utlined abve abund in many sientifi jurnals and mngrahs. Mst f them even at the first rder hase transitin disagree n sme vital ints. In the higher rder hase transitin, there is n yet a general nsensus n the nature and rder f these transitins [see fr examle,,1]. Thus, there is the need t ritially investigate these rders f hases eminent in sme materials. Thus, the aim f resent study is t investigate theretially the thermdynami behavir f rders f hase transitins in materials. Mst imrtantly, we aim t devel mdel equatins that will hel in the exliit lassifiatin f rders f hase transitins. We will start frm the investigatin f send rder hase transitin and we will use the symmetri nature f the funtinal f the free energy t devel a Landau-like thery fr third and furth rder hase transitins within the Ehrenfest lassifiatin f rder f hase transitins. Fr larity, ertainly, higher rder hase transitins exist sine, fr a hase transitin t be a send rder, the relatin T C H, T T dt 0 must hld. Hwever, as bserved in many materials and systems, fr instane, Na x CO yh O and BaKBiO [1-18]; antiferrmagneti Blume-Cael mdel [19], Lattie and ntinuum gauge theries [0]; Chiral mdel [1]; Sin glasses []; DNA under mehanial strain []; Ferrmagneti and antiferrmagneti sin mdels with temerature driven transitins []; Invar tyes allys [5]; and many ther materials and systems, this nservatin law is vilated. Of urse, the Beerezinskii-Ksterlitz-Thuless transitin is f infinite rder [10]. Thus, we assert that, the nn-detetin f hase transitins f rders stritly greater than tw might have been due t the errneusly, hasty generalizatin that all that dearts frm hase transitin f rder tw an always be exlained in terms f thermdynami flutuatin [6]. 0. THEORETICAL FRAMEWORK MEAN FIELD THEORY Mean field theries are vital tls fr studying mlex systems like the suerndutrs [7-8]. The starting int is the Landau thery whih was riginally designed t desribe systems lse t a hase transitin where the rder arameter, is small [9-0]. The Landau thery, thugh henmenlgial 1

5 Afrian Jurnal f Physis Vl., 010. wrks well in systems with lng rrelatin length (knwn as herent length in suerndutr). The general sheme f the Landau mean field thery whih is the fundatin f the refined frm suitable fr disussin f higher rder hase transitins is the exansin f the free energy (whih must be well defined) in a Taylr series u t the minimum relevant rder [5,6]. The sntaneus value(s) f the rder arameter is btained by minimizatin f the free energy with reset t the rder arameter, keeing the thermdynami arameters nstant. The Ginzburg-Landau thery [10,1] thugh henmenlgial, it inrrates all the quantum infrmatin needed fr quantitative desritin f hase transitin (eseially with minimal lal flutuatin). It has its fundatin in the general Landau thery [] fr ndensed matter whih lays its remises in that, the rder arameter is small and unifrm near the transitin temerature, T and yields a lt f infrmatin abut hase transitins [6,6]. We will adt a theretial arah based n the lassifiatin f rder f hase transitins by Ehrenfest [18,6] whih is in terms f the rdered state free energy near the hase bundary and the funtinal f the system in terms f the lal free energy. The validity f using Ehrenfest lassifiatin f rder f hase transitin has been questined by many authrs [-]. The neglet f Ehrenfest lassifiatin has errneusly led t the lassifiatin f rder f hase transitins int first rder and ntinuus hase transitins. Hwever, as has been shwn by Hilfer [5], rewriting the singular art f the lal free energy within a restrited urve thrugh the ritial int in terms f the finite differene qutient and analyti ntinuatin in allws ne t lassify ntinuus hase transitins reisely arding t their rders. Arding t lassifiatin f rder f hase transitins by Ehrenfest, the transitin in general an be f any rder [18,5]. In a send rder hase transitin, the seifi heat and the susetibility, whih are the send derivatives f the free energy (with reset t temerature and magneti field, resetively) are disntinuus at the transitin line [6]. Often, this disntinuity is relaed by a weak (r in general lgarithmi singularity). Thus, ne uld view the Ehrenfest definitin f an rder f hase transitin as ne where the lwer derivatives f the free energy are ntinuus at the transitin but, the higher derivatives are disntinuus. Fr a tyial third rder hase transitin, all send rder derivatives are ntinuus and all furth rder derivatives are singular at the transitin. The third rder derivatives are either disntinuus r a weakly lgarithmi singular. The equilibrium thermdynamis is mletely determined by the funtinal (f the free energy), FT,, where is the lal rder arameter [10,7-8]. Hwever, F must be invariant under the symmetry gru say, G f the 15

6 Afrian Jurnal f Physis Vl., 010. disrdered hase in rder t minimize the ttal energy [6]. Examles f suh behavir are mmnly fund in magneti dmains in ferrmagneti [5] where the Ising (N = 1) Hamiltnian has been alied []. In general, F is a very mliated funtinal f but sine in the viinity f ritial line, is essentially zer fr T T, we will fllw William Bragg s thery [7] and exand F in a Taylr series in. T make t be satially unifrm in equilibrium in the rdered hase, we essentially fr all ases redefine the rder arameter. This suggests that F be exressed in terms f a lal free energy density, f T, x whih is a funtin f the field rder arameter, x at the int x nly [7]. After arse graining [6,7], in its simlest frm, based n lattie frmulatin f Ginzburg-Landau thery [11,6], F is given by, 1 F d x f T x d x x d d, (1) where is a henmenlgial nstants knwn as the uling nstant r the gradient term and f T, x is the lal free energy density, whih is well thery knwn in the. It must be asserted that Eq. (1) fails in the shrt - limit. Hwever, we make haste t add that the shrt -limit is never a rblem as mst system underging higher rder hase transitins in the viinity f the ritial ints are ntrlled by lng- flutuatins and as suh, Eq.(1) is adequate fr ur rblem s lng as the shrt- is signifiantly suressed [9-0]. This is amlished by the s-alled hard utff mehanism where exitatin f wave-number greater than a utff a, (where a is a length f rder f the same range as the inter-artile interatins) are restrited in the artitin trae [6]. Eq. (1) will be generalized t suit the thermdynami theries f higher (>) rder hase transitins that will be develed, whih is ttally different frm thse develed fr send rder. After arse-graining and renrmalizing, fllwing Kumar and Saxena [1]; Farid et al. [], the free energy an be generalized t Landau-like frm as: F e d r a t b A A () i d 1 i 1 1 whih in terms f a weight funtin, fr > [6] is: d, F T d r a t b ; () 16

7 Afrian Jurnal f Physis Vl., 010. where the integer is the rder f the hase transitin, a a 1 T T, b is a sitive nstant and is the arriate uling nstant. Eq. () r () is the funtinal f the free energy that ntains all the thermdynami infrmatin (f any rder) f systems underging hase transitin(s).. ANALYSIS AND DISCUSSION As we inted ut befre, the funtinal f the free energy maniulated thrugh the lal free energy ntains all the thermdynami infrmatin thrugh the rder arameter and vetr tential f any system underging hase transitin. In that light, we will analyze Eq. () r () fr the varius: nd, rd and th rders f hase transitin. We will aly variatinal rinile (withut and with bundary nditin) with reset t the rder arameter and the vetr tential, resetively, t the funtinal f the free energy..1 Funtinal f the Free Energy in Absene f Gradient Term and Zer Field If we neglet the uling nstant and in the absene f any alied field, the lal free energy fr nd, rd, and th rders f hase transitin an resetively, be btained as: f at b 6 f at b () 6 8 f at b The minimizatin f Eq. () redues fr nd rder hase transitin as: at a T f 0 a 1 ; t b ; T T d b b T 0 ; T T (5a) Fr rd rder hase transitin, we have, 17

8 Afrian Jurnal f Physis Vl., at a T df 5 1 ; T T at 6b 0 ; b b T d 0 ; T T (5b) Then, fr th rder hase transitin as, at a T df ; T T at 6b 0 ; b b T d 0 ; T T (5) On using Eq. (5) in Eq. () resetively, fr the varius rders f hase transitins we are investigating here, we an rewrite the lal free energy fr nd, rd, and th rders f hase transitins resetively as: f f a T 1 b T a T 1 7b T (6) f 7a 56b T 1 T By regnizing the definitin f the rder arameter fr the varius hase transitins, we an redefine Eq. (6) in terms f the rder arameter as, 18

9 Afrian Jurnal f Physis Vl., 010. f t b; at b ; t 0 ( t 0; t 0) b 6 f t ; at b ; t 0 ( t 0; t 0) (7) b 8 f t ; at b ; t 0 ( t 0; t 0) Als, we an use Eq. (6) t investigate the varius thermdynami quantities f interest. The entry, S F T and the seifi heat, C T F T are resetively given as, a T a S 1 ; C T; bt T bt fr nd rder (8a) S a T 8 at T C 9 bt T 9bT T 1 ; 1 ; fr rd rder (8b) and S 7 a T 81 at T C 6 bt T 6 bt T 1 ; 1 ; fr th rder (8). Funtinal f the Free Energy with Gradient Term and Alied Field Let us investigate the effet f the aliatin f field and the inrratin f the gradient term n the thermdynami behavir f the system. T ahieve this, we minimize the free energy f the funtinal f Eqs. ( r ). First, with reset t the rder arameter and sendly, with reset t the tential field. The minimizatin f the Eq. () r () with reset t the rder arameter fr nd, rd, and th rders f hase transitin gives the fllwing Euler-Lagrange i equatin f the free energy fr the rder arameter: e. 19

10 Afrian Jurnal f Physis Vl., 010. a t b 0; fr nd rder (9a) at b 0, fr rd rder (9b) at b 0, fr rd rder (9) d d After saling, and using the fat that dx d d d, we an re-write Eq. (9) as: d d and 0; fr nd rder (10a) 5 0 ; fr rd rder (10b) 7 5 0; fr th rder (10) Aart frm Eq. (10a), Eqs. (10b & 10) has n lsed frm slutins. The rresnding Euler-Lagrange equatins in terms f the vetr tential, A an be btained by minimizing Eq. ( r ) rresnding t the variatin f F, T : F, t, A F, t, A A F, t, A t btain: A A 0 A A 0 (11a) (11b) 6 A A 0 10 (11) whih is nthing but the Lndn equatin fr the behavir f the system in nd, rd, and th rders hase transitin regime resetively. In the Meissner state at the deth, the gradient term an be negleted t btain Euler-Lagrange equatin fr the vetr tential in terms f as,

11 Afrian Jurnal f Physis Vl., 010. e m e m (1a) (1b) e 6 6 m (1) Observe that the enetratin deth, λ is inversely related t the rder arameter. Thus, it an be suggested that near T, all temerature deendene in mes frm the temerature deendene f the rder arameter. Defining the enetratin deth in this way and nting that the seifi heat fllws frm the generalized saling laws [10], A G G 1 1 ; 1 1 A 1 ; A 1 1 G 1 0 we an write the seifi heat as: T A C a As a nsequene f Eq. (1), we an derive a generalized relatin f the seifi heat with the enetratin deth, fr any rder f hase transitin ( u 1 ) [1] as: (1) (1) C bt 1 u u T u (15) where is the rder f hase transitin, b is a sitive nstant, is the uling nstant. 5. CONCLUSION We have develed mdel equatins fr investigating thermdynamis f ntinuus hase transitin. The develed mdel equatins enable us t study 11

12 Afrian Jurnal f Physis Vl., 010. the rders f the hase transitin exliitly. The free energy is fund t bey a generalized trend by its exansin in the Ehrenfest sense f lassifiatin f rder f hase transitin and by bserving its symmetri nature. We seulate that there are hase transitin f rders greater than tw as there is n knwn hysial reasn why suh transitins shuld nt exist in nature sine they ertainly exist in a number f theretial mdels like QCD, lattie field thery and statistial hysis. At least, higher rder hase transitins ( ) is tenuus at best and their nn-detetin might have been due t the errneusly hasty generalizatin that all deartures frm hase transitin f rder tw an always be exlained in terms f thermdynami flutuatin. Certain thermdynamial relatins well-knwn fr systems underging send rder hase transitins are revered fr bth third and furth rders f hase transitin. We derived Ginzburg-Landau free energies, develed Euler- Lagrange equatins fr the rder arameter and the vetr tential. ACKNOWLEDGEMENT One f the authrs Chinedu Ekuma, wishes t thank the Gvernment f Ebnyi State, Nigeria fr Oversea Shlarshi (Award n: EBSG/SSB/FSA/00/VOL. VIII/09). Insightful disussins frm Prf. Dila Bagayk, Prf. Mark Jarrell, and Dr. Maliki are areiated. REFERENCES [1] Lidmar, J. and Walin, M. (1998), Phys. Rev. B 58, 5. [] Gitterman, M and Halern, V. (00), Phase Transitin: A Brief Aunt with Mdern Aliatins, Wrld Sientifi Pub. Ltd. [] Uzunv, D.I. (00), Intrdutin t Thery f Critial Phenmenn, Wrld Sientifi Pub. Ltd. [] Onuki, A. (00), Phase Transitin Dynamis, Cambridge Press. [5] Yemans, J.M. (199), Statistis Mehanis f Phase Transitin, OxfrdUniversity Press, Lndn. [6] Stanley, H.E. (1971), Intrdutin t Phase Transitin and Critial Phenmena, Clarendn Press, Lndn. 1

13 Afrian Jurnal f Physis Vl., 010. [7] Ma, S.K. (1976), Mdern Thery f Critial Phenmena, Lndn Amsterdam DnMills, Ontari, Benjamin. [8] Gibbs, J.W. (198), The Clleted Wrks f J.Willard Gibbs in tw Vlumes, Lngmans, Green and C, NY. [9] Sahdev, S. (1998), Quantum Phase Transitins, Cambridge University Press. [10] Ekuma, C. E. (010), M.S Thesis, Deartment f Physis and Astrnmy, University f Nigeria, Nigeria. [11] Sinai, Y.G. (198), Thery f Phase Transitins: Rigrus Results, Pergamn Press. [1] Wannier, G.H. (1966), Statistial Physis, Wiley NY. [1] Buquet, F.; R. A. Fisher; N. E. Phillis; D. G. Hinks and J. D. Jrgensen, (001). Phys. Rev. Lett. 87, [1] Yang, H. D.; J.-Y. Lin; H. H. Li; F. H. Hsu; C.-J. Liu; S.-C. Li; R.-C. Yu and C.-Q. Jin. (001). Phys. Rev. Lett. 87, [15] Yang. H. D and J.-Y. Lin. (001). J. Phys. Chem. Slid. 6, [16] Lin, J.-Y.; P. L. H; H. L. Huang; P. H. Lin; Y. L. Zhang; R. -C. Yu; C. Q. Jin and H. D. Yang (00). Phys. Rev. B 67, [17] Hall D; Gdrih, R.G; Grenier, C.G; Kumar, P; Chaarala, M. and Nrtn, M.L. (000). Phil. Mag. B 80, 61. [18] Ekuma, et al. Manusrit in rearatin. [19] Wang, Y.L. and Kimel, J.D. (1991), J. Al.Phys. 69, [0] Grss, D.J. and Witten, E. (1980). Phys. Rev. D 1, 6. [1] Camstrini, M; Rssi, P. and Viar, E. (1995). Phys. Rev. D 5, 95 [] Crisanti, A; Riz, T. and Temesvari, T. (00). Eur. Phys. J. B, 0. [] Rudnik, J. and Bruinsma, R. (00). Phys. Rev. E 65, [] Kanaya, K. and Kaya, S. (1995). Phys. Rev. D 51, 0. [5] Kumar, P; Hall, D. and Gdrih, R.G. (1999). Phys. Rev. Letts 8, 5. [6] Ekuma, C. E., G. C. Asmba and C. M. I. Okye. (010). Physia B, 05 (9), [7] Kristian F. and Asle S. (00), Suerndutivity Physis and Aliatins, Jhn Wiley and Sns, Ltd. [8] Kittel, C. (1996), Intrdutin t Slid State Physis, 7 th Editin, Jhn Wiley and Sns, In. [9] Alexandrv, A.S. (00), Thery f Suerndutivity frm Weak t Strng Culing, Cey, J.M.D; Tilley, D.R. and Vij, D.R. (Eds)., Institute f Physis Publishing, Bristl. [0] Yukhnvskii, I.R. (1987), Phase Transitins f the Send Order - Clletive Variables Methd, Wrld Sientifi. [1] Kamerlingh Onnes, H. (1911). Leiden Cmm. 10 b. 1

14 Afrian Jurnal f Physis Vl., 010. [] Landau, L. D. and E. M. Lifshitz (1980), Statistial Physis, Pergamn, Oxfrd, third editin. [] Cnigli, A. and Zanetti, M. (1990). Phys. Rev. B, 687. [] Wu, X.Z; Kadanff, L.P; libhaber, A and San, M. (1990). Phys. Rev. Lett. 6, 10. [5] Hilfer, R. (199). Phys. Rev. Lett. 68, [6] Huang, K. (1987). Statistial Mehanis, nd Editin, Jhn Wiley and Sns In. [7] Chaikin, P.M. and Lubensky, T.C. (000). Priniles f ndensed matter hysis Cambridge University Press. [8] Shmidt, V.V. (1997). The Physis f Suerndutrs: Intrdutin t Fundamentals and Aliatins, Muller, P. and Ustinv, A.V. (Eds.), Sringer Publishers. [9] Simn, B. (001). Phase Transitin and Clletive Phenmena1 Cambridge Leture Ntes. [0] Alexandrv, A.S. (00). Thery f Suerndutivity frm Weak t Strng Culing, Cey, J.M.D; Tilley, D.R. and Vij, D.R. (Eds)., Institute f Physis Publishing, Bristl. [1] Kumar, P. and Sexana, (00), Phil. Mag. B 8, [] Farid, F; Yu, Y; Sexana, A. and Kumar, P. (005), Phys. Rev. B 71,