Theory of Phase Transitions in Mixed Crystals Rb1−x(NH4)xH2PO4

Eiko MATSUSHITA 0 TakeoMATsuBARA 0 0 Department of Physics, Kyoto University , Kyoto 606 Within the cluster approximation, phase transitions in the mixed crystals of hydrogen bonded fer roelectrics and antiferroelectrics Rbl-ANH4)xH2P04 are discussed on the basis of a pseudospin model for the proton system. The phase diagram of this system in the temperaturecomposition plane is derived theoretically and compared with the experimental results in good agreement. - 2. Formulation of cluster theory. 4 H4= V(SlS2+S2S3+S3S4+S4S1)+ U(SIS3+S2S4)- ~ hiSi, i=l where the mean field at the i-th site hi is given by La Fig. 1. Ordered proton configurations. (a) Ferroelectric RDP. (b) Antiferroelectric ADP. 4V+2U~ 1Zl.1Zl1Zllzi ~l2JczjCZL 1Zf 1Zl al2J [21121 t - o + ,' , +- }ZJ 1Zl-4V+2U + ' 0 1Zf 1Zl -2Ual2J V= ~ (co'+2cn. - V-+- V+g1, -U-+-U-g2. Energy Ei 4V+2U-~hi 4V+2U+~hi - h,- h2- ha+h. - h,- hz+ ha- h. -h,+h2-ha-h. h,- h2- ha- h. h, + h2+ ha- h. h, + hz- ha+ h. h,- h2+ ha+ h. - h, + h2+ ha+ h. -4 V +2U -h,+hz-ha+h. -4 V +2U +h,-.hz+ha-h. -2U -h,-h2+ha+h. -2U - h, + h2+ha- h. -2U +h,+hz-ha-h. -2U +h,- hz- ha+h. Boltzmann Factor a=exp( -co/ks T), b=exp( -c1lkB T) /i=exp( - hJkB T). (i=1,2,3,4) With the use of Table I it is straightforward to evaluate the partition function of the four proton cluster + ab(fdzfaf4-1+ /1-1/2-1/3-1/4 + /dzf3 -1/4 + /1-1/2-1/af4-1 + /d2 -1/af4+ /1-1/zf3 -1/4-1+ /1-1/zfaf4+ /d2- 1/3 -1/4-1) +a(f1-1/zf3-1/4+ /d2 -1/af4-1)+ /dzf3- 1/4 -1+ /1-1/2-1/af4 + /d2- 1/3- 1/4+ /1-1/zfaf4- 1 from which the thermal average of Sl is calculated as +ab(fl-l/z- 1/3- l/4- Iddd4- 1+Il- l/z- l/d4- l - Idd3- l/4 +Il- l/d3-:- l/4- l - Idz- l/d4+ Il- l/dd4- Idz- 1/3- l/4- l ) + a(fl-l/d3-1/4- Idz- l/d4- l )+ Il- l/z- l/d4- Idd3- l/4- l +11-1Idd4-1_ Idz-113-114]. Upon eliminating <Sl) from (213) and (214), one obtains an equation which is also to be derived from (i=1, 2, 3, 4) has such a property. 3. Relation to previous theories Then the coupled equations (2 13) and (2 14) become /=/1 = /3= /2-1= /4-1. (5)= 2 sinh(2h/kBT)[ab+a cosh(2h/kB T)] 2+a3 b4 +4ab cosh(2h/kB T)+a cosh(4h/kBT) , 1 W/ks T=exp( -Icol/ks T)-Z; ks TeF = Ieol/ In 2 , /=/1=/2=/3-1=/4- 1. Then the coupled equations are (33) and (5)- 2 sinh(2h/kBT)[ab+cosh(2h/ksT)] - 1+ a+ a3 b4 +4ab cosh(2h/kB T)+cosh(4h/ks T) which may be transcribed as (311) 4_ Free energy of mixed crystals 2h1/kB T=X1 , 2ha/kB T=xa, W/hT=).. Then (4-5) may be put in the following form: :f=F!k T=l).( 2+ B 2 Y1 Ya 2)~ln[a COSh(X1+xa)+cosh(x1-xa)+l] 2 cosh(x1-).Y1)cosh(Xa~).ya) .. Thus it holds for both cases that and discrimination between ferro and antiferroelectric orders can be made by the difference in (51)=<5a) (51)= -<5a) for ferroelectric phase, for antiferroelectric phase. Y=YI-Y3. (4'11) Neutral: Antiferro: 1 c;[ =In 2+ P-T/\(Y12+Y32)+ P-ln[cosh(xl- /\Yl )cosh(x3- /\Y3)] Yi=tanh(xi-/\Yi) . (i=1 or 3) Then we have the following expansion of c;[: The coefficients in (420) up to the 4-th order are aj (l+IIP] (2+aj) , (P+ +Po) (1 ~611 )4 - ~Pj 2ft2~IIL) [2+ (1 + II )3(1- (2; aj))J, A( TO and B( T):;;;'O, A(T):;;;'O and B(TO. It is evident from (44) that phase transitions to ferroelectric and antiferroelectricphases take place when No. of NH. Po instead of (422). when E(T)<O Fig. 5. TC"x phase diagram with the parameter ,1=0.20. PE : Paraelectric phase. FE: Ferroelectric phase. AFE : Antiferroelectric phase. 5. Discussion (...truncated)