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Variation of the critical behavior in metamagnets
PHYSICS LETTERS
Volume SOA, number 5
30 December 1974
VARIATION OF THE CRITICAL BEHAVIOR IN METAMAGNETS J.M. KINCAID and E.G.D. COHEN The Rockefeller University, New York, New York 10021, USA Received 19 November 19 74 From a study of a simple mean field model for metamagnets we predict that new types of critical behavior may be observed in FeBrz and other similar systems.
Recently there has been much interest in the critical properties of anisotropic antiferromagnets such as FeCl,, FeBr, [l] and DAG [2] which, due to their unusual magnetic properties, are known as metamagnets. The study of these systems has centered primarily around the tricritical point; other types of possible critical behavior seem to have been ignored. The purpose of this letter is to call attention to the following possibilities: (1) that the critical point in FeBr, which has been called a tricritical point [3] may in fact be a critical endpoint; (2) near this critical endpoint there may be a critical point within the ordered antiferromagnetic phase - a bicritical endpoint; (3) there may exist a metamagnetic system which, under pressure, would exhibit a fourth order critical point. The basic of our suggestions (l), (2) and (3) lies in the properties of a mean field treatment of a spin l/2 Ising model with antiferromagnetic nearest neighbor (nn) and ferromagnetic next-nearest neighbor (nnn) 1:xchange constants J and J’, respectively. The Hamiltonian for this model is: H=JgSiSi-
JtnTnSiSj-@
5Si, 1
where the N Ising spins (Si = 1, i= 1, . . .N) are distributed over two equivalent sublattices each having N/2 spins; p is the magnetic moment per spin; B is the external magnetic field; Xhn and Znnn denote sums over nn and nnn lattice sites, respectively. Although this model has been studied before [4], the variety and nature of all critical points which can occur has not. We find that the phase behavior of the model depends critically on the value of E =z’J/zJ, where z(z’) are the number of nn (nnn) of a lattice site. When E > 3/S, the phase diagram contains a tricritical point (TCP) with critical exponents [S] : (Y~=O, a; = l/2,
(bi
"Ol-
Magnettc held,6
mt
Magnettzatlon,M
Fig. 1. The phase diagram in the T-B and T-M planes. When E > 3/5 the phase diagrams (a) and (b) exhibit a tricritical point (TCP). When 0 < E < 3/5 the phase diagrams (c) and (d) exhibit a bicritical endpoint (BCE) and critical endpoint (CE). The solid lines represent lines of critical points (A-points), the dashed lines and dotted lines represent coexistence curves. m or M denotes the magnetization, reduced by the saturated magnetization; the subscript t refers to the TCP, while P and AF to paramagnetic - or antiferromagnetic phases respectively.
13t=1/4,yt=y;=1,6t=5,a,=a~=o,pu+=~”_=1, “/~=7~+=O,y~_=l,6~+=l,6,_=2[6,7].(Seefig. la, b.) When 0 < e < 3/S, there is no tricritical point. Instead there appear a critical endpoint (CE) [6] and a bicritical endpoint (BCE) [6], each having classical criticalexponents:a=a’=O,fl=1/2,y=y’=1,6=3. (See fig. 1c, d.) The bicritical endpoint is especially interesting in that it is a critical point within the order317
Volume 50A, number 5
PHYSICS LETTERS
ed antiferromagnetic phase. When ~=3/.5, the BCE and CE merge with the TCP forming a fourth order critical point [8] at T* =$ TN, l.lBzO.614 kTN, where k is Boltzmann’s constant and TN is the NCel temperature. The critical exponents at this point are: 01 = 0, cr’=2/3,~=1/6,y=y’=1,6=7,crU=a~=0,/3,+=1, pu_=1/2,yu=y~+=o,y~_=1,6u+=1,6u_=3. The value of E depends on the molecular composition and crystal structure of the metamagnet and therefore should depend on the external pressure, p. Indeed, recent experiments by Vettier, Alberts and Bloch [3] indicate that changes in the hydrostatic pressure result in changes in the location of the Feel, tricritical point in the T-B plane. These changes are well accounted for qualitatively by the mean field theory sketched above, if one assumes that E is a decreasing function of p. Their data on FeBr, [3] seems to indicate that their so-called tricritical point may well be a critical endpoint. This is suggested (i) by the presence of a kink in the phase boundary, which is absent when a tricritical point occurs (compare fig. la and 1b); (ii) by their low value of T,/T, = 0.33, while the mean field theory would predict Tt > T#’ ; (iii) by the results of Jacobs and Lawrence [I] indicating that E = 0.28 at p = 0, which is well below the range E > 3/S which the mean field theory requires for the existence of a tricritical point. A further experimental investigation of this region would clearly be of interest. If the mean field predictions presented here are borne our, it would appear to be the first observation of magnetic critical and bicritical endpoints. The fourth order critical point at E = 3/5 probably
318
30 December 1974
cannot be observed in either FeCl, or FeBr, - except, perhaps, under stress: fitting the mean field theory to the data of Vettier et al., we find that for all p, E 3 3/5 for FeCl, and E < 3/5 for FeBr,. One might expect, however, that such a critical point could be observed in those metamagnetic materials for which the temperature of either the TCP or CE is near T *. The authors are grateful to G. Stell and J. Hoye for many helpful discussions.
References [l] I.S. Jacobs and P.E. Lawrence, Phys. Rev. 164 (1967) 866; R.J. Birgeneau, W.B. Yelon, E. Cohen and J. Makovsky, Phys. Rev. B.5 (1972) 2607,261s; J.A. Griffin, S. Schatterly, Y. Farge, M. Regis and M. Fontana, preprint (1974). [2] D. Landau, B. Keen, B. Schneider and W.P. Wolf, Phys. Rev., B3 (1971) 2310; W.P. Wolf, B. Schneider, D. Landau and B. Keen, Phys. Rev. B5 (1972) 4472; M. Blume, L. Corliss, J. Hastings and E. Schiller, Phys. Rev. Lett. 32 (1974) 544. (31 C. Vettier, H. Alberts and D. Bloch, Phys. Rev. Lett. 31 (1973) 1414. [4] K. Motizuki, .I. Phys. Sot. Japan 14 (1959) 759; C.J. Gorter and Tineke Van Peski-Tinbergen, Physica 22 (1956); R. Bidaux, P. Carrara and B. Vivet, J. Phys. Chem. Solids 28 (1967) 2453. (51 R. Griffiths, Phys. Rev. B7 (1973) 545. [6] J. Kin&d, Ph.D. thesis, Rockefeller University (1974). [7] R. Bausch, Z. Physik 254 (1972) 81. [8] T. Chang, G. Tuthill and H. Stanley, Phys. Rev. B9 (1974) 4882.
Volume SOA, number 5
30 December 1974
VARIATION OF THE CRITICAL BEHAVIOR IN METAMAGNETS J.M. KINCAID and E.G.D. COHEN The Rockefeller University, New York, New York 10021, USA Received 19 November 19 74 From a study of a simple mean field model for metamagnets we predict that new types of critical behavior may be observed in FeBrz and other similar systems.
Recently there has been much interest in the critical properties of anisotropic antiferromagnets such as FeCl,, FeBr, [l] and DAG [2] which, due to their unusual magnetic properties, are known as metamagnets. The study of these systems has centered primarily around the tricritical point; other types of possible critical behavior seem to have been ignored. The purpose of this letter is to call attention to the following possibilities: (1) that the critical point in FeBr, which has been called a tricritical point [3] may in fact be a critical endpoint; (2) near this critical endpoint there may be a critical point within the ordered antiferromagnetic phase - a bicritical endpoint; (3) there may exist a metamagnetic system which, under pressure, would exhibit a fourth order critical point. The basic of our suggestions (l), (2) and (3) lies in the properties of a mean field treatment of a spin l/2 Ising model with antiferromagnetic nearest neighbor (nn) and ferromagnetic next-nearest neighbor (nnn) 1:xchange constants J and J’, respectively. The Hamiltonian for this model is: H=JgSiSi-
JtnTnSiSj-@
5Si, 1
where the N Ising spins (Si = 1, i= 1, . . .N) are distributed over two equivalent sublattices each having N/2 spins; p is the magnetic moment per spin; B is the external magnetic field; Xhn and Znnn denote sums over nn and nnn lattice sites, respectively. Although this model has been studied before [4], the variety and nature of all critical points which can occur has not. We find that the phase behavior of the model depends critically on the value of E =z’J/zJ, where z(z’) are the number of nn (nnn) of a lattice site. When E > 3/S, the phase diagram contains a tricritical point (TCP) with critical exponents [S] : (Y~=O, a; = l/2,
(bi
"Ol-
Magnettc held,6
mt
Magnettzatlon,M
Fig. 1. The phase diagram in the T-B and T-M planes. When E > 3/5 the phase diagrams (a) and (b) exhibit a tricritical point (TCP). When 0 < E < 3/5 the phase diagrams (c) and (d) exhibit a bicritical endpoint (BCE) and critical endpoint (CE). The solid lines represent lines of critical points (A-points), the dashed lines and dotted lines represent coexistence curves. m or M denotes the magnetization, reduced by the saturated magnetization; the subscript t refers to the TCP, while P and AF to paramagnetic - or antiferromagnetic phases respectively.
13t=1/4,yt=y;=1,6t=5,a,=a~=o,pu+=~”_=1, “/~=7~+=O,y~_=l,6~+=l,6,_=2[6,7].(Seefig. la, b.) When 0 < e < 3/S, there is no tricritical point. Instead there appear a critical endpoint (CE) [6] and a bicritical endpoint (BCE) [6], each having classical criticalexponents:a=a’=O,fl=1/2,y=y’=1,6=3. (See fig. 1c, d.) The bicritical endpoint is especially interesting in that it is a critical point within the order317
Volume 50A, number 5
PHYSICS LETTERS
ed antiferromagnetic phase. When ~=3/.5, the BCE and CE merge with the TCP forming a fourth order critical point [8] at T* =$ TN, l.lBzO.614 kTN, where k is Boltzmann’s constant and TN is the NCel temperature. The critical exponents at this point are: 01 = 0, cr’=2/3,~=1/6,y=y’=1,6=7,crU=a~=0,/3,+=1, pu_=1/2,yu=y~+=o,y~_=1,6u+=1,6u_=3. The value of E depends on the molecular composition and crystal structure of the metamagnet and therefore should depend on the external pressure, p. Indeed, recent experiments by Vettier, Alberts and Bloch [3] indicate that changes in the hydrostatic pressure result in changes in the location of the Feel, tricritical point in the T-B plane. These changes are well accounted for qualitatively by the mean field theory sketched above, if one assumes that E is a decreasing function of p. Their data on FeBr, [3] seems to indicate that their so-called tricritical point may well be a critical endpoint. This is suggested (i) by the presence of a kink in the phase boundary, which is absent when a tricritical point occurs (compare fig. la and 1b); (ii) by their low value of T,/T, = 0.33, while the mean field theory would predict Tt > T#’ ; (iii) by the results of Jacobs and Lawrence [I] indicating that E = 0.28 at p = 0, which is well below the range E > 3/S which the mean field theory requires for the existence of a tricritical point. A further experimental investigation of this region would clearly be of interest. If the mean field predictions presented here are borne our, it would appear to be the first observation of magnetic critical and bicritical endpoints. The fourth order critical point at E = 3/5 probably
318
30 December 1974
cannot be observed in either FeCl, or FeBr, - except, perhaps, under stress: fitting the mean field theory to the data of Vettier et al., we find that for all p, E 3 3/5 for FeCl, and E < 3/5 for FeBr,. One might expect, however, that such a critical point could be observed in those metamagnetic materials for which the temperature of either the TCP or CE is near T *. The authors are grateful to G. Stell and J. Hoye for many helpful discussions.
References [l] I.S. Jacobs and P.E. Lawrence, Phys. Rev. 164 (1967) 866; R.J. Birgeneau, W.B. Yelon, E. Cohen and J. Makovsky, Phys. Rev. B.5 (1972) 2607,261s; J.A. Griffin, S. Schatterly, Y. Farge, M. Regis and M. Fontana, preprint (1974). [2] D. Landau, B. Keen, B. Schneider and W.P. Wolf, Phys. Rev., B3 (1971) 2310; W.P. Wolf, B. Schneider, D. Landau and B. Keen, Phys. Rev. B5 (1972) 4472; M. Blume, L. Corliss, J. Hastings and E. Schiller, Phys. Rev. Lett. 32 (1974) 544. (31 C. Vettier, H. Alberts and D. Bloch, Phys. Rev. Lett. 31 (1973) 1414. [4] K. Motizuki, .I. Phys. Sot. Japan 14 (1959) 759; C.J. Gorter and Tineke Van Peski-Tinbergen, Physica 22 (1956); R. Bidaux, P. Carrara and B. Vivet, J. Phys. Chem. Solids 28 (1967) 2453. (51 R. Griffiths, Phys. Rev. B7 (1973) 545. [6] J. Kin&d, Ph.D. thesis, Rockefeller University (1974). [7] R. Bausch, Z. Physik 254 (1972) 81. [8] T. Chang, G. Tuthill and H. Stanley, Phys. Rev. B9 (1974) 4882.