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Experimental studies of multicritical points in magnetic systems
550 EXPERIMENTAL STUDIES OF M U L T I C R I T I C A L POINTS IN MAGNETIC SYSTEMSt W.P. W O L F Department of Engineering and Applied Science, Yale University, New Haven, CT. 06520, USA (Invited paper) A brief review is given of recent experimental studies of magnetic systems displaying either tricritical or bicritical point behavior. The tricritical systems include the highly anisotropic antiferromagnets FeC12, CsCoCI3.2D20 and Dy3AI50~2, while the bicritical systems include the weakly anisotropic antiferromagnets MnF2, GdAIO3 and NiCI2-6H20. There have also been Monte-Carlo computer studies on both kinds of systems. The results provide significant tests for a number of theoretical predictions and in general good agreement is found. Most of the remaining discrepancies can probably be attributed to experimental difficulties, which are quite severe in studies of this kind.
1. Introduction
A multicritical point (MCP) is a special point on phase boundary at which competitition between two or more types of order leads to the new kind of cooperative behavior [1,2]. They are found in many different physical systems including mixtures of liquid 3He-4He [3], multicomponent fluid mixtures [4], structural phase transitions [5], liquid crystals [6] and ferroelectrics [7], as well as in various magnetic materials, and one reason for their interest is the general similarity in behavior of such widely differing systems. The special appeal of magnetic materials for studies of this kind is the ready access to appropriate fields and temperatures to explore the phase diagrams in the region of possible multicritical points. The direct relation of magnetic spin systems to simple microscopic models is also useful for theoretical comparisons. There have been a number of interesting theoretical predictions [1,2,8-19] for multicritical behavior, but we shall here restrict ourselves to tricritical and bicritical systems, since these are the only kinds of magnetic multicritical systems for which detailed experimental results have been reported. The predictions may be discussed under several headings, even though they mostly stem from the same basic concepts. One class of predictions concerns the shapes of the phase boundaries as the multicritical point is approached [10, 13, 18]. Using renormalization group techniques together with the results of series expansions, one finds phase boundaries such as those shown in fig. 1. The tSupported in part by AROD and by NSF grant DMR 76-23102.
Physica 86--88B (1977) 550-555 ~ North-Holland
h=(H-Ht)IHI~
(a)
m= (M-Mt)IM t
h+= pt ÷c+t~+...~ t =0
~////
, ~ ÷ = a ÷t
pt + c_t÷+ ...
B.
+'..
+P
= bt~U+ ...
rn_ = a_t + - . .
t = (T-T t ]IT t
(b)
H2 ~" ....
t m=( M - M b ) / M b
SF +AF I ~ m + L~m=t ~ ~ P
= w+t
.... ....
-t t = (T-Tb)/T b Fig. 1. Theoretical predictions for phase boundaries of (a) tricritical and (b) bicritical systems. Heavy lines denote first order transitions. ~ = 0 and i = 0 denote the axes of the appropriate scaling fields. The orientation of ~ = 0 is fixed by the geometry of the figure but the slopes of the t = 0 axes are nonuniversal. For tricritical systems, the exponents defining the phase boundaries are given by ~b = 2,/3~ = ~o, = 1, as in classical theory [ 1l, but the amplitudes a + and b are not equal [16, 18]. For bicritical systems, ~b and ~ take on nonclassical but universal values (see section 4) and the phase boundaries must be expressed in terms of the appropriate scaling variables ~ = H 2 - H ~ - pt and {= t + q(H 2 - Hi) [14].
differences from classical mean field (Landau) theory are quite striking and they can readily be tested. The theory also predicts the variation of thermodynamic functions along different paths approaching a multicritical point [9, 13]. These paths are classified according to their relation to the scaling axes which are in turn determined by the form of the phase diagram [3, 9, 14, 15]. In general, one expects singular behavior which can be characterized by one or more critical exponents, though there may also be singular terms involving logarithms which cannot be
551 represented by a simple exponent [17]. At this time there is still no direct evidence for any such logarithmic terms and we shall discuss the experimental results in terms of the more usual exponent laws. It should be noted, however, that even though the log terms are not observed directly, they may well affect the values of fitted critical exponents, which might account for some apparent discrepancies in the analysis of experimental results [20]. Values for various predicted exponents will be given in tables I and II together with the available experimental results. As in the case of ordinary critical points, it is believed that the exponents are not all independent, and by extending the usual scaling hypothesis one can derive a number of relations between them [9, 11]. Scaling theory also provides a basis for constructing equations of state which allow the combination of different experimental measurements into a single curve describing the behavior around the multicritical point [8,9, 11, 19]. There have recently also been some attempts to calculate the detailed form of these scaling functions [18, 19], but so far there have been no detailed comparisons of these results with any magnetic experiments. The theory also makes some general predictions concerning the universality of different types of behavior which can be tested [15]. In particular, one expects to find similar behavior at all multicritical points of a given kind so that, as in the case of ordinary critical points, one should find similar exponents in widely differing materials. We can clearly not discuss all of these predictions here, and further details may be found in the references noted and in the references cited in these papers. H o w e v e r , it may be useful to comment briefly on one limitation of the theory which may affect the interpretation of experiments. As in the case of ordinary critical point theory, the predictions relate principally to the asymptotic limit as the multicritical point is approached and there are presently few indications for the size of higher order terms. The region over which the asymptotic predictions are expected to be valid is therefore undefined and some of the observed discrepancies may be due to nonasymptotic behavior rather than fundamental inadequacies in the theory or the experiments. This problem is particularly
serious in the case of multicritical points since experimental difficulties make it very difficult to obtain unambiguous results very close to the MCP and one is, as always, faced with fitting several parameters over a finite range of variables.
2. Experimental problems From the phase diagrams shown in fig. 1 it can be seen that the experimental location of either a tricriticai or a bicritical point involves finding the point at which a first order discontinuity goes over into a second order singularity. In practice, this can be quite difficult, since the corresponding susceptibility close to the first order transition diverges as the MCP is approached, thereby masking the vanishing discontinuity. This problem is complicated by demagnetizing effects, especially in irregularly shaped samples, and by the inevitable "rounding" which is found in all real crystals. There have been several different approaches to overcome this problem, and we shall mention these in the discussion of specific materials in sections 3 and 4. H o w e v e r , it is probably important to remark at this point that none of the presently used techniques is entirely satisfactory and that one may expect a significant improvement in the overall comparison of theory and experiment if this problem could be resolved. The second major problem, which affects particularly bicritical point studies, concerns the alignment of the field relative to the preferred easy-axis of the magnetic eystem. This has been discussed by Rohrer [21] and by Fisher [15] and it remains as a major source of uncertainty in present day studies, even though alignments as close as 10 -4 deg. have been achieved [22]. The solution to this problem is probably to find materials in which the alignment is not so critical. A third, and perhaps trivial difficulty which confronts the experimentalist, especially in the study of tricritical systems, is the bewildering variety of notations which have been used by theorists to express their predictions. An attempt was made by Griffiths [12] to propose a unified notation for tricritical points, but unfortunately he did not explicitly consider all of the cases encountered experimentally and a
552 known at this time. In addition to the measurements in the immediate vicinity of the tricritical point, the neutron experiments also yielded values for the critical exponent at different points along the second order phase boundary [25]. The results were found to be consistent with the hypothesis of smoothness, which predicts that the critical exponents should be the same for different paths of approach towards the critical line.
different notation based on Riedel's original scaling ideas [9] is still in use [3]. We shall here use Grifliths' notation supplemented by Riedel's where necessary. Useful summaries of definitions have also been given in refs. 23 and 24. For bicritical points the notation problem is not so acute, perhaps because there has not yet been so much work in this field. 3. Experimental results for tricritical systems Detailed experiments have been reported for three materials and there have also been computer studies using Monte-Carlo techniques.
b) CsCoCI3.2D,O. This is a somewhat more complicated four sublattice antiferromagnet of which only relatively small irregular crystals are available [24]. The only tricritical point studies so far have used neutron scattering [24, 27, 28], which again allows masking of sample edges. The results in table I can be seen to be in relatively poor quantitative agreement with theory, though the general behavior was as expected. It is not clear at this time whether the disagreement lies outside the experimental uncertainties, which are quite large for this material. The measurements for CsCoC13.2D20 have also been tested for scaling and good data collapsing was found. See also ref. 28. As in the case of FeC12, the results along the critical line above the tricritical point were found to be consistent with the smoothness hypothesis.
a) FeCl2. This is a well known and simple metamagnet which unfortunately suffers from the disadvantage of poor mechanical and chemical properties. This makes it hard to control demagnetizing effects and the tricritical studies have therefore used neutron scattering [25] and optical techniques [26] to measure the magnetization, since these methods permit shielding of the sample edges. In the neutron experiments it is of course possible to measure also the antiferromagnetic order parameter and the critical fluctuations close to the phase boundaries. The published results are summarized in table I. It can be seen that there is general agreement, but there is also some disagreement with theory and in one case, between the different experiments. The reasons for these differences are not
Table
I. E x p e r i m e n t a l
and theoretical
~,
c) Dy3AIsO~2. Dysprosium aluminum garnet (DAG) is an Ising-like six sublattice cubic an-
t r i c r i t i c a l e x p o n e n t s ~a~
/3,
/3~
0.36
0.19
0.36
1.11 -+0.11
--
--
-2
--
--
--
0.65 -+0.05
0.7 +-0.4
0.7 -+0.4
0.65 -+0.20
- 1.3
3 -+ I
2 -+ 0.2
0.3 -+0.1
0.15 -+0.02
0.36 -+0.05
- 1
- 1
0.98 -+0.05
1.01 -+0.07
2.1 -+0.25
1.95 -+0.11
--
--
--
1
1
1
1
2
1/2
I/4
I/2
~a~ N o t a t i o n
based
o n r e f . 12. S e e a l s o r e f s . 23 a n d 24.
~ R e f . 25. ~°~ R e f . 26. ~Refs. 24 a n d 28. f~ R e f s . 31 a n d 32. ~f~ A f t e r r e f . 10. S e e a l s o r e f s . 3 a n d 23.
2
_+ 0.02
--
/3*,
1.13 -+0.14
I]2
2
/3,
1.03 -+0.05
T h e o r y ~f~
.
/3,
--
--
.
4,
Opticai~c~
D y ~ A I ~ O , 2 : H [ I { 1 1 0 } (c~
.
,%
--
+_ 0 . 0 4
.
vo
F e C I 2 N e u t r o n s ~b~
CsC°C13"2D20*e~
1
/3
_+0 . 0 4
553 uniform fields, so that the entire H - H s - T phase diagram can be investigated [34].
tiferromagnet, and early experiments with fields applied along {111} indicated significant discrepancies from tricritical point theory [29]. It was subsequently recognized that the symmetry of this material allows an "induced staggered field" if HxH~-Iz~O, so that for H[I{Ill} one actually observes a "wing critical point" [30]. Later experiments with HH{110}, located a true tricritical point and the results [31,32] in table I can be seen to be in very good agreement with the theory. The measurements were also found to be consistent with scaling [33]. The method used for these studies was based on the difference in the transient response of domain states in the first order region from that of the homogeneous antiferromagnetic or paramagnetic states. So far, only measurements of the magnetization have been reported and it would now be interesting to study the antiferromagnetic order parameter.
4. Experimental results for bicritical systems Results for bicritical systems are much more limited and in particular there have not yet been any neutron studies of the order parameter. There are again measurements on three materials and one computer study. a) MnF2. This is one of the best known examples of a Heisenberg-like (n = 3) antiferromagnet with weak anisotropy. Its bicritical point at Hb ~ 11.8 T and Tb ~ 65 K is somewhat hard to study, especially since extremely accurate alignment of the sample relative to the field is required. In the experiments of King and Rohrer [22] the alignment was estimated to be better than 10-4deg., a nontrivial achievement. The phase boundaries were located using both differential susceptibility and NMR measurements, but in both cases there were some questions of interpretation which increase the overall uncertainty of the analysis. In addition to the cross-over exponent d~, the results give values for Q = m±/mll, the ratio of two amplitudes describing the second order phase boundaries and q, the slope of the optimal scaling axis f = 0 (see fig. lb). The fitted values shown in table II can be seen to be in quite good agreement with the theory, though the overall uncertainties are still quite large.
d) Computer studies. These are free of the usual experimental problems of demagnetizing effects and impurities, but they suffer severe limitations due to finite sample size. The largest system studied so far had 20 × 20 x 20 spins, for which extrapolation to the N = ~ limit is still quite hard. H o w e v e r , the results [34] were generally consistent with the theory and with larger computers one should be able to make quite detailed tricritical studies in this way. One particular advantage of computer studies is that it is easy to apply "staggered" as well as Table
II. Experimental
and theoretical
1.29 1.26
n = 3 : M n F 2 (a~
1.25
T h e o r y ~b~
_+0.015 1.15 _+ 0 . 0 8
n = 2: G d A I O ~ "~
1.25 _+ 0 . 0 7
NiCI~.6H20 ~
1.175
T h e o r y ~b)
~' R e f . 22. T w o ~"~ R e f s . 1 3 - 1 5 .
- 1.18
_+0.015 estimates
for phase
bicritical exponents
1.25 1.75 2.5
1.38 1.06 - 1.35
(a~ __ja~ 1
boundaries.
~o R e f . 21. T w o a l i g n m e n t s - 0 . 1 4 d e g . a n d - 0 . 0 8 d e g . ~ F i t s a s s u m i n g ~ol/oJll -= 1. ~' R e f . 36.
--
--
0.85
0.40
_+0.07
_+0.09
0.92 _+0.03 _
-0.15 _
0.84
0.33
_+0.05
_+0.06
554 b) GdAl03. This is an o r t h o r h o m b i c antiferr o m a g n e t [21] and one would e x p e c t its bicritical point to be described by a t w o - c o m p o n e n t order p a r a m e t e r (n = 2). The t h e o r y for n = 2 bicritical points predicts tol[toTr-= 1 and this was used in fitting the data. F r o m previous w o r k one e x p e c t s GdAIO3 to be not as sensitive to orientation as MnF2, but e x p e r i m e n t s with estimated misalignments of only 0.14deg. and 0.08deg. still s h o w e d significant differences [21]. H o w ever, the results in table II are in generally g o o d a g r e e m e n t with the t h e o r y , and they certainly confirm that & > 1, the classical value. Analysis of the magnetization leads to a value of /3 (corrected for misalignment) w h i c h is in g o o d a g r e e m e n t with t h e o r y , and an estimate of the susceptibility e x p o n e n t ~ w h i c h a p p e a r s to be rather low, t h o u g h the uncertainties are quite large. M o r e r e c e n t e x p e r i m e n t s on GdAIO3 [35] have studied the variation of the phase boundaries as a f u n c t i o n of the orientation of the applied field and f o u n d e v i d e n c e for a predicted line of bicritical points for one particular plane. T h e c o r r e s p o n d i n g bicritical e x p o n e n t s have not yet been reported. c) NiCI2"6H20. This is a monoclinic crystal which has rather m o r e c o m p l i c a t e d properties and so far only the phase b o u n d a r y has been studied near the bicritical point [36]. The results were f o u n d to be consistent with the t h e o r y f o r an n = 2 bicritical point, but no detailed analysis has been reported. d) Computer studies. M o n t e - C a r l o studies of bicriticai points are even m o r e difficult than those on tricriticai points b e c a u s e the three dimensional nature of the spins leads to m u c h slower c o n v e r g e n c e o f the statistical iteration process. So far, only preliminary results have been r e p o r t e d [37]. T h e s e are consistent with the theory, but more w o r k is clearly needed.
5. Other systems T h e r e are m a n y other magnetic materials which s h o w field i n d u c e d phase transitions and f o r s o m e of these one would e x p e c t to find multicritical point b e h a v i o r similar to that discussed above. In s o m e of these cases one might hope to find situations which are less sensitive to s o m e of the experimental p r o b l e m s w h i c h have limited e x p e r i m e n t s so far.
In s o m e cases one m a y e x p e c t to find new kinds of behavior. F o r example, results w h i c h h a v e been r e p o r t e d for D y P O 4 [38-40], FeBr2 [41] and M n O u n d e r pressure [42] all s h o w f e a t u r e s w h i c h do not fit obviously, at this time, into the simple types of multicritical points illustrated in fig. 1. T h e r e are also predictions of tetracritical b e h a v i o r [13, 42-44], and the onset of spiral order (Lifshitz point) [45-47] w h i c h have not yet been studied e x p e r i m e n t a l l y in any detail. It would seem clear that there is still a considerable a m o u n t of experimental and theoretical w o r k to be d o n e in this field. It is a pleasure to thank R.J. Birgeneau, R.B. Grifliths, N. G i o r d a n o , D. J a s n o w , D.P. L a n d a u , H. M e y e r and H. R o h r e r for their help in preparing this paper. I am also grateful to A . L . M . B o n g a a r t s f o r sending me preprints o f his w o r k .
References [I] L.D. Landau, Phys. Z. Sowjetunion I1 (1937) 26. Reprinted in Collected Papers of L.D. Landau, D. ter Haar, ed. (Pergamon, London, 1965) p. 193. [2] See for example, A. Aharony, Physica 86-88B (1977) 545, Proceedings of International Conference on Magnetism, Amsterdam. 1976 and references cited therein. [3] See for example, E.K. Riedel, H. Meyer and R,P. Behringer, J. Low Temp. Physics 22 (1976) 369. G. Ahlers in The Physics of Liquid and Solid Helium, Part I, K.H. Bennemann and J.B. Ketterson, eds. (Wiley, New York, 1976) p. 85 and references cited in these papers. [4] See for example, B. Widom, J. Phys. Chem. 77 (1973) 2196. [5] See for example, C.W. Garland, D.E. Bruins and T.J. Greytak, Phys. Rev. BI2 (1975) 2759. [6] P.H. Keyes, H.T. Weston and W.B. Daniels, Phys. Rev. Lett. 31 (1973) 628. [7] P.S. Peercy, Phys. Rev. Len. 35 (1975) 1581. [8] R.B. Griffiths, Phys. Rev. Len. 24 (1970) 715. [9] E.K. Riedel, Phys. Rev. Len. 28 (1972) 675. [10] E.K. Riedel and F.J. Wegner, Phys. Rev. Lett. 29 (1972) 349. [11] A. Hankey, H.E. Stanley and T.S. Chang, Phys. Rev. Lett. 29 (1972) 278. [121 R.B. Grifliths, Phys. Rev. B7 (1973) 545. [13] M.E. Fisher and D.R. Nelson, Phys. Rev. Lett. 32 (1974) 1350. [14] M.E. Fisher, Phys. Rev. Lett. 34 (1975) 1634. [15] M.E. Fisher, Proceedings of 20th Conference on Magnetism and Magnetic Materials, AlP Conf. Proc. 24 (1975) 273, and references cited therein. [16] V.J. Emery, Phys. Rev. 1311 (1975) 3397. [17] M.J. Stephen, E. Abrahams and J.P. Straley, Phys. Rev. B12 (1975) 256, and references cited therein.
555 [18] D.R. Nelson and J. Rudnick, Phys. Rev. Lett. 35 (1975) 178. [19] D.R. Nelson, Proceedings of 21st Conference on Magnetism and Magnetic Materials, AlP Conf. Proc. 29 (1976) 450, and references cited therein. [20] See for example, W.B. Yelon, D.E. Cox, P.J. Kortman and W.B. Daniels, Phys. Rev. B9 (1974) 4843. [21] H. Rohrer, Phys. Rev. Lett. 34 (1975) 1638, and references cited therein. 122] A.R. King and H. Rohrer, Proceedings of 21st Conference on Magnetism and Magnetic Materials, AlP Conf. Proc. 29 (1975) 420; ibid p. 456. [23] J.M. Kincaid and E.G.D. Cohen, Physics Reports 22C (1975) 57. [24] A.L.M. Bongaarts, thesis, University of Eindhoven (1975), unpublished. [25] R.J. Birgeneau, G. Shirane, M. Blume and W.C. Koehler, Phys. Rev. Lett. 33 (1974) 1098, Proceedings of the 20th Conference on Magnetism and Magnetic Materials, AlP Conf. Proc. 24 (1975) 258, and to be published. [26] J.A. Griffin and S.E. Schnatterly, Phys. Rev. Lett. 33 (1974) 1576 and AIP Conf. Proc. 24 (1975) 195. [27] A.L.M. Bongaarts, W.J.M. de Jonge and P. van der Leeden, Phys. Rev. Lett. 37 (1976) 1007. [28] A.L.M. Bongaarts and W.J.M. de Jonge, Proceedings of International Conference on Magnetism, Amsterdam, 1976, Physica 86-88B (1977) 595, 671. [29] W.P. Wolf, D.P. Landau, B.E. Keen and B. Schneider, Phys. Rev. B5 (1972) 4472. [30] M. Blume, L.M. Corliss, J.M. Hastings and E. Schiller, Phys. Rev. Lett. 32 (1974) 544. [31] N. Giordano and W.P. Wolf, Phys. Rev. Lett. 35 (1975) 799. [32] N. Giordano, Phys. Rev. B14 (1976) 2927.
[33] N. Giordano and W.P. Wolf, Proceedings of 21st Conference on Magnetism and Magnetic Materials, AIP Conf. Proc. 29 (1976) 459. [34] D.P. Landau, Phys. Rev. BI4 (1976) 4054. [35] H. Rohrer, B. Derighetti and Ch. Gerber, Proceedings of International Conference on Magnetism, Amsterdam, 1976, Physica 86-88B (1977) 597. [36] N.F. Oliveira, Jr., A. Paduhan Filho and S.F. Salinas, Phys. Lett. 55A (1975) 293. [37] D.P. Landau and K. Binder, Proceedings of 21st Conference on Magnetism and Magnetic Materials, AIP Conf. Proc. 29 (1976) 461. [38] J.E. Battison, A. Kasten, M.J.M. Leask and J.B. Lowry, Solid State Comm. 17 (1975) 1363. [39] P.J. Becket and I.R. Jahn, J. Phys. C: Solid State Phys. 9 (1976) L505. [40] M. R~gis, J. FerrY, Y. Farge and I.R. Jahn, Proceedings International Conference on Magnetism, Amsterdam, 1976, Physica 86-88B (1977) 599. [41] J.M. Kincaid and E.G.D. Cohen, Phys. Lett. 50A (1974) 317, and references cited therein. [42] P. Bak, S. Krinsky and D. Mukamel, Phys. Rev. Lett. 36 (1976) 829, and references cited therein. [43] D. Mukamel, Phys. Rev. Bl4 (1976) 1303. [44] A.D. Bruce and A. Aharony, Phys. Rev. BII (1975) 478. [45] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678. [46] M. Droz and M.D. Coutinho-Filho, AIP Conf. Proc. 29 (1976) 465. [47] J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Proceedings of the Joint MMM-Intermag Conference, 1976, to be published.
1. Introduction
A multicritical point (MCP) is a special point on phase boundary at which competitition between two or more types of order leads to the new kind of cooperative behavior [1,2]. They are found in many different physical systems including mixtures of liquid 3He-4He [3], multicomponent fluid mixtures [4], structural phase transitions [5], liquid crystals [6] and ferroelectrics [7], as well as in various magnetic materials, and one reason for their interest is the general similarity in behavior of such widely differing systems. The special appeal of magnetic materials for studies of this kind is the ready access to appropriate fields and temperatures to explore the phase diagrams in the region of possible multicritical points. The direct relation of magnetic spin systems to simple microscopic models is also useful for theoretical comparisons. There have been a number of interesting theoretical predictions [1,2,8-19] for multicritical behavior, but we shall here restrict ourselves to tricritical and bicritical systems, since these are the only kinds of magnetic multicritical systems for which detailed experimental results have been reported. The predictions may be discussed under several headings, even though they mostly stem from the same basic concepts. One class of predictions concerns the shapes of the phase boundaries as the multicritical point is approached [10, 13, 18]. Using renormalization group techniques together with the results of series expansions, one finds phase boundaries such as those shown in fig. 1. The tSupported in part by AROD and by NSF grant DMR 76-23102.
Physica 86--88B (1977) 550-555 ~ North-Holland
h=(H-Ht)IHI~
(a)
m= (M-Mt)IM t
h+= pt ÷c+t~+...~ t =0
~////
, ~ ÷ = a ÷t
pt + c_t÷+ ...
B.
+'..
+P
= bt~U+ ...
rn_ = a_t + - . .
t = (T-T t ]IT t
(b)
H2 ~" ....
t m=( M - M b ) / M b
SF +AF I ~ m + L~m=t ~ ~ P
= w+t
.... ....
-t t = (T-Tb)/T b Fig. 1. Theoretical predictions for phase boundaries of (a) tricritical and (b) bicritical systems. Heavy lines denote first order transitions. ~ = 0 and i = 0 denote the axes of the appropriate scaling fields. The orientation of ~ = 0 is fixed by the geometry of the figure but the slopes of the t = 0 axes are nonuniversal. For tricritical systems, the exponents defining the phase boundaries are given by ~b = 2,/3~ = ~o, = 1, as in classical theory [ 1l, but the amplitudes a + and b are not equal [16, 18]. For bicritical systems, ~b and ~ take on nonclassical but universal values (see section 4) and the phase boundaries must be expressed in terms of the appropriate scaling variables ~ = H 2 - H ~ - pt and {= t + q(H 2 - Hi) [14].
differences from classical mean field (Landau) theory are quite striking and they can readily be tested. The theory also predicts the variation of thermodynamic functions along different paths approaching a multicritical point [9, 13]. These paths are classified according to their relation to the scaling axes which are in turn determined by the form of the phase diagram [3, 9, 14, 15]. In general, one expects singular behavior which can be characterized by one or more critical exponents, though there may also be singular terms involving logarithms which cannot be
551 represented by a simple exponent [17]. At this time there is still no direct evidence for any such logarithmic terms and we shall discuss the experimental results in terms of the more usual exponent laws. It should be noted, however, that even though the log terms are not observed directly, they may well affect the values of fitted critical exponents, which might account for some apparent discrepancies in the analysis of experimental results [20]. Values for various predicted exponents will be given in tables I and II together with the available experimental results. As in the case of ordinary critical points, it is believed that the exponents are not all independent, and by extending the usual scaling hypothesis one can derive a number of relations between them [9, 11]. Scaling theory also provides a basis for constructing equations of state which allow the combination of different experimental measurements into a single curve describing the behavior around the multicritical point [8,9, 11, 19]. There have recently also been some attempts to calculate the detailed form of these scaling functions [18, 19], but so far there have been no detailed comparisons of these results with any magnetic experiments. The theory also makes some general predictions concerning the universality of different types of behavior which can be tested [15]. In particular, one expects to find similar behavior at all multicritical points of a given kind so that, as in the case of ordinary critical points, one should find similar exponents in widely differing materials. We can clearly not discuss all of these predictions here, and further details may be found in the references noted and in the references cited in these papers. H o w e v e r , it may be useful to comment briefly on one limitation of the theory which may affect the interpretation of experiments. As in the case of ordinary critical point theory, the predictions relate principally to the asymptotic limit as the multicritical point is approached and there are presently few indications for the size of higher order terms. The region over which the asymptotic predictions are expected to be valid is therefore undefined and some of the observed discrepancies may be due to nonasymptotic behavior rather than fundamental inadequacies in the theory or the experiments. This problem is particularly
serious in the case of multicritical points since experimental difficulties make it very difficult to obtain unambiguous results very close to the MCP and one is, as always, faced with fitting several parameters over a finite range of variables.
2. Experimental problems From the phase diagrams shown in fig. 1 it can be seen that the experimental location of either a tricriticai or a bicritical point involves finding the point at which a first order discontinuity goes over into a second order singularity. In practice, this can be quite difficult, since the corresponding susceptibility close to the first order transition diverges as the MCP is approached, thereby masking the vanishing discontinuity. This problem is complicated by demagnetizing effects, especially in irregularly shaped samples, and by the inevitable "rounding" which is found in all real crystals. There have been several different approaches to overcome this problem, and we shall mention these in the discussion of specific materials in sections 3 and 4. H o w e v e r , it is probably important to remark at this point that none of the presently used techniques is entirely satisfactory and that one may expect a significant improvement in the overall comparison of theory and experiment if this problem could be resolved. The second major problem, which affects particularly bicritical point studies, concerns the alignment of the field relative to the preferred easy-axis of the magnetic eystem. This has been discussed by Rohrer [21] and by Fisher [15] and it remains as a major source of uncertainty in present day studies, even though alignments as close as 10 -4 deg. have been achieved [22]. The solution to this problem is probably to find materials in which the alignment is not so critical. A third, and perhaps trivial difficulty which confronts the experimentalist, especially in the study of tricritical systems, is the bewildering variety of notations which have been used by theorists to express their predictions. An attempt was made by Griffiths [12] to propose a unified notation for tricritical points, but unfortunately he did not explicitly consider all of the cases encountered experimentally and a
552 known at this time. In addition to the measurements in the immediate vicinity of the tricritical point, the neutron experiments also yielded values for the critical exponent at different points along the second order phase boundary [25]. The results were found to be consistent with the hypothesis of smoothness, which predicts that the critical exponents should be the same for different paths of approach towards the critical line.
different notation based on Riedel's original scaling ideas [9] is still in use [3]. We shall here use Grifliths' notation supplemented by Riedel's where necessary. Useful summaries of definitions have also been given in refs. 23 and 24. For bicritical points the notation problem is not so acute, perhaps because there has not yet been so much work in this field. 3. Experimental results for tricritical systems Detailed experiments have been reported for three materials and there have also been computer studies using Monte-Carlo techniques.
b) CsCoCI3.2D,O. This is a somewhat more complicated four sublattice antiferromagnet of which only relatively small irregular crystals are available [24]. The only tricritical point studies so far have used neutron scattering [24, 27, 28], which again allows masking of sample edges. The results in table I can be seen to be in relatively poor quantitative agreement with theory, though the general behavior was as expected. It is not clear at this time whether the disagreement lies outside the experimental uncertainties, which are quite large for this material. The measurements for CsCoC13.2D20 have also been tested for scaling and good data collapsing was found. See also ref. 28. As in the case of FeC12, the results along the critical line above the tricritical point were found to be consistent with the smoothness hypothesis.
a) FeCl2. This is a well known and simple metamagnet which unfortunately suffers from the disadvantage of poor mechanical and chemical properties. This makes it hard to control demagnetizing effects and the tricritical studies have therefore used neutron scattering [25] and optical techniques [26] to measure the magnetization, since these methods permit shielding of the sample edges. In the neutron experiments it is of course possible to measure also the antiferromagnetic order parameter and the critical fluctuations close to the phase boundaries. The published results are summarized in table I. It can be seen that there is general agreement, but there is also some disagreement with theory and in one case, between the different experiments. The reasons for these differences are not
Table
I. E x p e r i m e n t a l
and theoretical
~,
c) Dy3AIsO~2. Dysprosium aluminum garnet (DAG) is an Ising-like six sublattice cubic an-
t r i c r i t i c a l e x p o n e n t s ~a~
/3,
/3~
0.36
0.19
0.36
1.11 -+0.11
--
--
-2
--
--
--
0.65 -+0.05
0.7 +-0.4
0.7 -+0.4
0.65 -+0.20
- 1.3
3 -+ I
2 -+ 0.2
0.3 -+0.1
0.15 -+0.02
0.36 -+0.05
- 1
- 1
0.98 -+0.05
1.01 -+0.07
2.1 -+0.25
1.95 -+0.11
--
--
--
1
1
1
1
2
1/2
I/4
I/2
~a~ N o t a t i o n
based
o n r e f . 12. S e e a l s o r e f s . 23 a n d 24.
~ R e f . 25. ~°~ R e f . 26. ~Refs. 24 a n d 28. f~ R e f s . 31 a n d 32. ~f~ A f t e r r e f . 10. S e e a l s o r e f s . 3 a n d 23.
2
_+ 0.02
--
/3*,
1.13 -+0.14
I]2
2
/3,
1.03 -+0.05
T h e o r y ~f~
.
/3,
--
--
.
4,
Opticai~c~
D y ~ A I ~ O , 2 : H [ I { 1 1 0 } (c~
.
,%
--
+_ 0 . 0 4
.
vo
F e C I 2 N e u t r o n s ~b~
CsC°C13"2D20*e~
1
/3
_+0 . 0 4
553 uniform fields, so that the entire H - H s - T phase diagram can be investigated [34].
tiferromagnet, and early experiments with fields applied along {111} indicated significant discrepancies from tricritical point theory [29]. It was subsequently recognized that the symmetry of this material allows an "induced staggered field" if HxH~-Iz~O, so that for H[I{Ill} one actually observes a "wing critical point" [30]. Later experiments with HH{110}, located a true tricritical point and the results [31,32] in table I can be seen to be in very good agreement with the theory. The measurements were also found to be consistent with scaling [33]. The method used for these studies was based on the difference in the transient response of domain states in the first order region from that of the homogeneous antiferromagnetic or paramagnetic states. So far, only measurements of the magnetization have been reported and it would now be interesting to study the antiferromagnetic order parameter.
4. Experimental results for bicritical systems Results for bicritical systems are much more limited and in particular there have not yet been any neutron studies of the order parameter. There are again measurements on three materials and one computer study. a) MnF2. This is one of the best known examples of a Heisenberg-like (n = 3) antiferromagnet with weak anisotropy. Its bicritical point at Hb ~ 11.8 T and Tb ~ 65 K is somewhat hard to study, especially since extremely accurate alignment of the sample relative to the field is required. In the experiments of King and Rohrer [22] the alignment was estimated to be better than 10-4deg., a nontrivial achievement. The phase boundaries were located using both differential susceptibility and NMR measurements, but in both cases there were some questions of interpretation which increase the overall uncertainty of the analysis. In addition to the cross-over exponent d~, the results give values for Q = m±/mll, the ratio of two amplitudes describing the second order phase boundaries and q, the slope of the optimal scaling axis f = 0 (see fig. lb). The fitted values shown in table II can be seen to be in quite good agreement with the theory, though the overall uncertainties are still quite large.
d) Computer studies. These are free of the usual experimental problems of demagnetizing effects and impurities, but they suffer severe limitations due to finite sample size. The largest system studied so far had 20 × 20 x 20 spins, for which extrapolation to the N = ~ limit is still quite hard. H o w e v e r , the results [34] were generally consistent with the theory and with larger computers one should be able to make quite detailed tricritical studies in this way. One particular advantage of computer studies is that it is easy to apply "staggered" as well as Table
II. Experimental
and theoretical
1.29 1.26
n = 3 : M n F 2 (a~
1.25
T h e o r y ~b~
_+0.015 1.15 _+ 0 . 0 8
n = 2: G d A I O ~ "~
1.25 _+ 0 . 0 7
NiCI~.6H20 ~
1.175
T h e o r y ~b)
~' R e f . 22. T w o ~"~ R e f s . 1 3 - 1 5 .
- 1.18
_+0.015 estimates
for phase
bicritical exponents
1.25 1.75 2.5
1.38 1.06 - 1.35
(a~ __ja~ 1
boundaries.
~o R e f . 21. T w o a l i g n m e n t s - 0 . 1 4 d e g . a n d - 0 . 0 8 d e g . ~ F i t s a s s u m i n g ~ol/oJll -= 1. ~' R e f . 36.
--
--
0.85
0.40
_+0.07
_+0.09
0.92 _+0.03 _
-0.15 _
0.84
0.33
_+0.05
_+0.06
554 b) GdAl03. This is an o r t h o r h o m b i c antiferr o m a g n e t [21] and one would e x p e c t its bicritical point to be described by a t w o - c o m p o n e n t order p a r a m e t e r (n = 2). The t h e o r y for n = 2 bicritical points predicts tol[toTr-= 1 and this was used in fitting the data. F r o m previous w o r k one e x p e c t s GdAIO3 to be not as sensitive to orientation as MnF2, but e x p e r i m e n t s with estimated misalignments of only 0.14deg. and 0.08deg. still s h o w e d significant differences [21]. H o w ever, the results in table II are in generally g o o d a g r e e m e n t with the t h e o r y , and they certainly confirm that & > 1, the classical value. Analysis of the magnetization leads to a value of /3 (corrected for misalignment) w h i c h is in g o o d a g r e e m e n t with t h e o r y , and an estimate of the susceptibility e x p o n e n t ~ w h i c h a p p e a r s to be rather low, t h o u g h the uncertainties are quite large. M o r e r e c e n t e x p e r i m e n t s on GdAIO3 [35] have studied the variation of the phase boundaries as a f u n c t i o n of the orientation of the applied field and f o u n d e v i d e n c e for a predicted line of bicritical points for one particular plane. T h e c o r r e s p o n d i n g bicritical e x p o n e n t s have not yet been reported. c) NiCI2"6H20. This is a monoclinic crystal which has rather m o r e c o m p l i c a t e d properties and so far only the phase b o u n d a r y has been studied near the bicritical point [36]. The results were f o u n d to be consistent with the t h e o r y f o r an n = 2 bicritical point, but no detailed analysis has been reported. d) Computer studies. M o n t e - C a r l o studies of bicriticai points are even m o r e difficult than those on tricriticai points b e c a u s e the three dimensional nature of the spins leads to m u c h slower c o n v e r g e n c e o f the statistical iteration process. So far, only preliminary results have been r e p o r t e d [37]. T h e s e are consistent with the theory, but more w o r k is clearly needed.
5. Other systems T h e r e are m a n y other magnetic materials which s h o w field i n d u c e d phase transitions and f o r s o m e of these one would e x p e c t to find multicritical point b e h a v i o r similar to that discussed above. In s o m e of these cases one might hope to find situations which are less sensitive to s o m e of the experimental p r o b l e m s w h i c h have limited e x p e r i m e n t s so far.
In s o m e cases one m a y e x p e c t to find new kinds of behavior. F o r example, results w h i c h h a v e been r e p o r t e d for D y P O 4 [38-40], FeBr2 [41] and M n O u n d e r pressure [42] all s h o w f e a t u r e s w h i c h do not fit obviously, at this time, into the simple types of multicritical points illustrated in fig. 1. T h e r e are also predictions of tetracritical b e h a v i o r [13, 42-44], and the onset of spiral order (Lifshitz point) [45-47] w h i c h have not yet been studied e x p e r i m e n t a l l y in any detail. It would seem clear that there is still a considerable a m o u n t of experimental and theoretical w o r k to be d o n e in this field. It is a pleasure to thank R.J. Birgeneau, R.B. Grifliths, N. G i o r d a n o , D. J a s n o w , D.P. L a n d a u , H. M e y e r and H. R o h r e r for their help in preparing this paper. I am also grateful to A . L . M . B o n g a a r t s f o r sending me preprints o f his w o r k .
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