Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 Plenary paper Random fields and phase transitions in model magnetic systems R.J. Birgeneau* Department of Physics, Massachusetts Institute of Technology, 6-123, Cambridge, MA 02139, USA Abstract Random fields occur in a wide variety of physical systems varying from type II superconductors to two-component fluids in a random medium. However, only in model magnetic systems have systematic studies as a function of both temperature and random-field strength been possible. In this article we review recent neutron and magnetic X-ray scattering studies of the magnetic ordering processes in the antiferromagnets Mn Zn F, Fe Zn F and Fe Co TiO in an applied magnetic field. These systems should all represent realizations of the three-dimensional random-field Ising model which is the simplest version of the random-field problem in models with discrete symmetry. In all cases on field cooling (FC) the systems evolve continuously from a high-temperature paramagnetic state to a low-temperature antiferromagnetic domain state. However, on cooling to low temperatures in zero field and then applying a field (ZFC) long-range order (LRO) is obtained. On subsequent heating in the three systems the LRO vanishes continuously with a rounded power-law behavior which has been labelled trompe l'oeil critical behavior. The width of the transition region scales as H. Reconsideration of indirect ZFC specific-heat measurements shows that the observed peaks, previously attributed to equilibrium critical fluctuations, instead arise entirely from a LRO contribution, scaling like dM/d¹, to the measured quantity. Here M is the staggered magnetization. These results thus reconcile scattering and bulk property measurements of random-field Ising systems. 1998 Elsevier Science B.V. All rights reserved. Keywords: Random fields; Antiferromagnetism; Phase transitions; Metastability 1. Introduction Here h One of the major unsolved problems in the study of G are the random fields, which typically are as- sumed to average to zero but with a finite variance. phase transitions is the behavior of systems with both Systems with critical behavior which can be modelled by quenched disorder and competing interactions or fields. this Hamiltonian include absorbed monolayers on The random-field model, proposed originally by Larkin a square substrate, many structural order-disorder in 1970 [1] to model the defect pinning of vortices in type transitions, and certain Jahn-Teller systems, all with II superconductors, has proven to be a useful paradigm impurities which pin the order parameter, as well as two for this class of problems. The Ising version of this model, component fluids in a random medium. the random-field Ising model (RFIM), most simply en- Initially, much of the theoretical research on the RFIM capsulates the essential physics of the problem in systems dealt with the problem of the lower critical dimension, d with discrete symmetry. The RFIM has been the focus of , the dimension at which, in equilibrium, the random-field intense study over the past two decades. The Hamil- fluctuations drive the transition temperature to zero. The tonian for such systems is pioneering theoretical work of Imry and Ma [2] first discussed the competition between the gain in the statisti- H" J cal random-field energy which occurs when a region GHSXGSXH! hGSXG. (1) 6GH7 G follows its weak random field and the loss in the Ising interaction energy at the walls of the domain, for the case of compact domain formation in the d-dimensional * Corresponding author. Tel.: #1 617 253 8900; fax: RFIM. Using simple phenomenological arguments it was #1 617 253 8901; e-mail: robertjb@mit.edu. concluded that d "2 [2]. This competition between the 0304-8853/98/$19.00 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 9 9 8 - 0 2 R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 domain-wall energy and the random-field energy lies at 2. Neutron scattering the heart of the RFIM and has formed the basis of much of the subsequent theoretical work. The value of Neutron scattering has played a central role in the the lower critical dimension became controversial after elucidation of the phenomenological behavior of RFIM arguments based on perturbation theory suggested that systems. This is because neutrons couple directly to the d "3. After almost a decade of considerable debate spin; they are able to probe the antiferromagnetic spin a consensus emerged that d "2. This view has been correlations on length scales from &1 to &1000 A> and supported, in particular, by the rigorous proof of Imbrie for energies varying from microvolts to millivolts. Begin- [3, 4] that the three-dimensional (3D) RFIM is ordered ning with the pioneering studies in 1980 by Cowley and at ¹"0. coworkers of RFIM effects in Co\VZnVF [18, 19], Initially, systematic experimental studies of RFIM a series of neutron scattering experiments have been systems seemed to be prohibitively difficult. However, performed on random antiferromagnets in a field. The an important breakthrough occurred when Fishman most detailed experiments have been carried out on the and Aharony [5] pointed out that the application of diluted antiferromagnets Mn\VZnVF and Fe\VZnVF an external magnetic field along the easy axis of a which are, respectively, weakly and strongly anisotropic random antiferromagnet generates a term in the Hamil- two-sublattice Ising antiferromagnets. tonian which behaves like a random field that couples Generally, the Ising component of the neutron scatter- linearly to the order parameter. Random antiferro- ing cross section in a 3D RFIM system may be written as magnets are ideally suited to the study of the RFIM because the strength of the random field may be con- A B tinuously varied simply by adjusting the applied field. S(Q)" # #C (q), (2) ( #q) #q A variety of experimental techniques have been applied to these systems including neutron and magnetic X- ray where q"Q!Q$. The function represents any long- scattering, optical birefringence and Faraday rotation, range magnetic-order component. The second term cor- dilatometry, AC susceptibility, SQUID magnetometry, responds to the longitudinal dynamic susceptibility. The and NMR techniques. Until recently, the scatt- Lorentzian-squared term arises from static fluctuations ering and thermodynamic measurements seemed to due to the quenched random-field. Written in this form, yield results which were not easily reconciled with A is the integrated intensity for these fluctuations. From each other. However, recent measurements, including the fluctuation dissipation theorem, the structure factor especially synchrotron magnetic X-ray scattering stud- S(Q) of Eq. (2) may be written as the sum of two terms, ies, have elucidated further the behavior of RFIM S(Q)"¹ (Q)#¹ (Q), where " H1 21 H2 and systems and, specifically, seem to have resolved this " H[1  H2!1 21 H2], are the so-called discon- conundrum. nected and connected susceptibilities, respectively. In As we shall discuss in this review, experiments a nonrandom system, for ¹'¹ , H1 21 H2"0 and reveal that nonequilibrium effects play an essential S(0)&¹ &t\A. However, in random systems  is no role in the behavior of RFIM systems. On the one hand, longer zero and a new exponent is defined, &t\AN. In these make both experiment and theory for the RFIM Section 6, we will discuss experimental measurements of much more difficult; on the other hand, they make the the equilibrium critical fluctuations in the high-temper- physics much richer and complex. We now have a very ature paramagnetic region which yield values for and N, detailed empirical description of the equilibrium and together with the correlation length exponent , where hysteretic behavior in a number of model RFIM systems. 1/ &t\J. In Eq. (2), we identify the thermal fluctuations, However, there is, in our view, no satisfactory theory given by parameter B with (Q) and the random-field for the behavior in the nonequilibrium regime and espe- fluctuations, A, with (Q). In this section we will focus cially for the transition from metastability to equilibrium on the evolution of the RFIM from the high-temperature behavior. equilibrium paramagnetic state to the low-temperature In this paper we shall review recent studies of three metastable regime. different model RFIM systems Mn Zn F [6-10], A detailed set of neutron experiments in Fe Zn F [11-15] and Fe Co TiO [16, 17] in Mn Zn F at various fields are reported in Ref. an applied magnetic field. Neutron scattering experi- [6-8]. We show in Fig. 1 data obtained at 5.0 T using ments are reviewed in Section 2. The time dependence is a triple-axis configuration with energy resolution 20 eV discussed in Section 3. Magnetic X-ray scattering half-width at half-maximum (HWHM). It was anticip- measurements are reviewed in Section 4. Direct and in- ated that the high-energy resolution would eliminate the direct specific-heat measurements are discussed in Lorentzian term in Eq. (2) which is dynamic in character Section 5. The critical behavior in the equilibrium para- and isolate the Lorentzian squared term which originates magnetic state is reviewed in Section 6. Conclusions and from the static-ordered moments induced by the random our overall perspective are given in Section 7. field. R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 3 Fig. 2. The temperature dependence of the inverse corre- lation length, , close to the metastability boundary in Mn Zn F for data taken with H"5.0 T with FC and ZFC procedures [6-8]. achieved near but below ¹ (H) equals the corresponding FC value to within the errors. We shall discuss the detailed behavior in the transition region in Section 4. The critical behavior in the equilibrium region above ¹ Fig. 1. The neutron scattering observed as a function of wave (H) will be discussed in Section 6. Results essentially identical to these are obtained in vector Q"(1, , 0) in Mn Zn F in a field of 5.0 T, with both Fe a two-axis configuration. At 44.0 K the results are independent \VZnVF [11-15] and Fe Co TiO [16, 17] in an applied magnetic field. The former is of the previous history but at 43.4 K they are dependent on the a highly anisotropic diluted Ising magnet whereas the history [6-8]. latter is a mixture of Ising (FeTiO) and X½ (CoTiO) Thermal expansion measurements which were carried magnets. Thus, the behavior shown in Figs. 1 and 2 is out on the same system show a hysteretic peak at generic depending only on the overall 3D Ising symmetry ¹ and not on the microscopic details. We will discuss (5.0 T) 43.7 K [20, 21]. The samples used in the neu- tron and thermal expansion experiments were cut from current models for this hysteretic behavior, especially in adjacent sections of the same boule and their H"0 Ne´el the transition region, in Section 5. temperatures coincided to within the errors (¹,(0)" 46.0 K). As shown in Fig. 1 at 44.0 K S(q) is the same for the FC and ZFC procedures. The correlation length 3. Time dependence " \ is &200 A>. However, at 43.4 K the spin config- uration explicitly depends on the history of the sample. The statement that d "2 requires that the FC domain The FC correlation length at 43.4 K is &550 A> whereas state we observe in these 3D RFIM magnets is the ZFC system has long-range order (LRO). a nonequilibrium state and that in equilibrium one ex- The results for the inverse correlation length from fits pects true LRO. This, in turn, suggests that the FC to the H"5.0 T neutron data are shown in Fig. 2. For domains should expand as a function of time. Theories the ZFC data the profiles were fit to the sum of a resolu- which assume an instantaneous quench from the para- tion-limited Gaussian and a Lorentzian squared. The magnetic phase into the `ordered' region predict that the double-arrow in Fig. 2 denotes the transition region. domains will grow logarithmically with time and further This will be discussed in more detail in Section 4. that they cannot then contract unless one crosses the These data illustrate the essential behavior of RFIM phase boundary [22-24]. The latter is in agreement with systems. Above a certain temperature, which we label experiment. The quenched domain size is predicted to ¹ scale like (H), equilibrium is obtained. Below ¹ (H) the phys- ical state of the system depends on the history of the J¹ sample. For FC measurements the domains grow pro- & ln(t/ ), (3) H gressively in size but saturate at a finite value. However, for ZFC measurements LRO obtains up to ¹ (H). The where is a microscopic time which cannot be shorter loss of LRO on heating in the ZFC procedure occurs that K/J&10\ s. The linear dependence on ¹ in with accompanying Lorentzian-squared fluctuations Eq. (3) is not observed in field cooling; rather saturates where the maximum length of these fluctuations which is at relatively high temperatures; presumably this reflects 4 R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 the difference between field-cooling and the instan- tropy and consequent broad domain walls in that system. taneous quench procedure assumed in deriving Eq. (3). They predicted that the logarithmic growth in time of In FeVZn\VF at low temperatures does indeed scale the domains should be observable in a narrow domain- like H\ [25]. However, in Co Zn F the exponent wall system such as Fe\VZnVF. They further pre- is 3.6$0.4 [18, 19] while in Mn Zn F it is 3.4$0.4 dicted that the time dependence should be generalized [6-8] although recent experiments at small fields on this to ln(t/ )\R. latter system give a value closer to 2 [14]. The order of Recently, Feng and co-workers [13] have studied the magnitude of is given correctly by Eq. (3). time dependence of the FC domain size in the highly The most important qualitative feature of Eq. (3) is anisotropic diluted 3D Ising magnet Fe Zn F. They that should increase logarithmically with time. By rap- have, in addition, measured the time dependence of the idly quenching a sample through the phase boundary it is excess magnetization M "M$!!M8$! where M$! possible to measure for times varying between &10 and M8$! are the magnetizations at a given field and and &10 s - enough to test Eq. (3) with "10\ s. temperature after field cooling and zero-field cooling, Birgeneau et al. [6-8] report an experiment carried out respectively. Heuristically, M  should provide a measure for H"7.0 T in Mn Zn F where the sample was of the excess magnetization in the FC domain walls; quenched from 0.4 K above the phase boundary to 0.4 K a phenomenological argument suggests that M  should below the boundary. As may be seen from Fig. 3, the scale like \ possibly raised to a power depending on ratio of domain sizes at 5.4;10 and 6.0;10 s was the fractal nature of the domain walls [26, 27]. measured to be 1.01$0.03 compared with a minimum Fig. 4 shows a comparison of the inverse correlation ratio of 1.14 predicted by the logarithmic law, Eq. (3). length obtained directly using neutron scattering tech- Thus, logarithmic expansion with time of the FC domain niques and M  determined by SQUID measurements radius is excluded. A series of additional field cooling or field lowering experiments in Mn Zn F and Mn Zn F confirm this conclusion [6-8]. This apparent contradiction with theory was ad- dressed by Natterman and Vilfan [26]. They concluded that the absence of measurable ln(t/ ) domain expansion behavior in Mn\VZnVF was due to the weak aniso- Fig. 4. Comparison of the time dependence of the inverse cor- relation length, , as measured by neutron scattering and the excess magnetization from domain walls M  as measured by the SQUID for three temperatures below the transition in Fe Zn F in a field of 5.5 T. The M  values have been Fig. 3. Scattering profiles in Mn Zn F in a field rescaled according to the neutron scattering . Both techniques H"7.0 T. The LRO-state data were obtained by the ZFC show frozen dynamics at low temperatures. At higher temper- procedure. The FC data were obtained by first raising the atures, the two techniques also agree, suggesting that the fractal temperature of the ZFC state to 41.4 K, and then cooling it properties of the domains do not change noticeably while the rapidly to 40.6 K. The solid lines are guides to the eye [6-8]. average size of the domain grows with time [13]. R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 5 in Fe Zn F, taken at the same temperatures at H" occur just below ¹ (H). They have been interpreted by 5.5 T. At ¹"6.8 K, both techniques show that the mag- some groups as signalling an equilibrium phase netic domains are frozen. At ¹"21.5 and 27.1 K, transition with RFIM critical behavior [28-30]. How- and M  undergo a similar percentage of decay over the ever, it is difficult to reconcile this claimed critical behav- same period of time. Detailed fits show that the decay is ior with the fact that the system is manifestly not in consistent with the simple logarithmic form ln(t/ ), i.e., equilibrium and with the absence of a divergent magnetic "1 although the uncertainties in the exponent are correlation length (cf. Fig. 2). large. More generally, from detailed studies of M  it is We discuss first results in Mn Zn F [9, 10]. The found that is typically much less than 1 at low fields, salient features of the disordering process are shown in increasing to 1 at higher fields [13]. Figs. 5 and 6. Representative scans through the (1 0 0) In brief, then, the experiments of Feng et al. [13] magnetic Bragg peak are shown in Fig. 5 at a series of confirm the predictions of Natterman and Vilfan [26]. temperatures for H"6.0 T. The system was initially Specifically, in narrow domain-wall systems logarithmic cooled into the X½ phase and then warmed with the field expansion of the FC metastable domains is indeed dir- fixed at H"6.0 T. This is equivalent to the ZFC process ectly observed. The measurements also confirm the close since the X½ phase has true LRO. The peak is well-fit by association between the FC domain-wall radius and the the resolution function, which is a Lorentzian, for all excess magnetization M "M$!!M8$!. temperatures up to ¹ (H), the metastability boundary. This corresponds to a domain size in excess of 20 000 A>. Fig. 6 shows the (1 0 0) peak intensity versus temper- 4. Magnetic X-ray scattering ature for a series of such runs at different fields in two different samples. A remarkable feature of these data is Neutron scattering studies of the ZFC behavior are that the behavior is universal for all fields studied. In compromised because the diffraction is severely affected particular, at low temperatures there is a linear dimin- by extinction. Thus it is not possible to measure quanti- ution with increasing temperature of the order para- tatively the temperature evolution of the order meter, crossing over to a power-law-like decay with parameter, at least outside of the transition region. An exponent, "0.2$0.05, near ¹ (H). Further, a careful alternative experimental approach made possible by the advent of dedicated synchrotron sources is to study the magnetic correlations using magnetic X-ray scattering. This technique is complementary to neutron scattering with several important strengths. First, the small cross section results in extinction-free scattering, so that the order parameter may be reliably determined. This is of particular importance in the work discussed here. Sec- ond, the small penetration depth of X-rays, typically on the order of 2 m, may reduce the effect of concentration gradients. Third, the high reciprocal space resolution allows large length scales to be probed. Fourth, the relatively poor energy resolution (&10 eV) ensures in- tegration over all relevant thermal fluctuations. Fifth, though not pertinent to this work, the contributions due to the orbital and spin magnetic moments may be distin- guished through polarization analysis. Extensive studies of each of Mn\VZnVF [9, 10], Fe\VZnVF [11, 12] and Fe\VCoVTiO [16, 17] in an applied field have been carried out using magnetic X-ray scattering techniques. The FC results so-obtained agree in detail with those measured using neutron scattering techniques. We will not review those results here. Instead, we will focus on the ZFC transition behavior. As dis- cussed previously, bulk thermodynamic measurements of properties such as the thermal expansion [14] and the Fig. 5. Representative transverse X-ray scans in temperature derivative of the uniform magnetization Mn [11, 12, 27] or optical birefringence [28-30] show  Zn F taken on warming from the X½ phase at H"6.0 T. These data are offset by two counts/s for each suc- anomalies on warming from the low-temperature ZFC cessive temperature. The data are well fit by a Lorentzian resolu- state. As we shall discuss in Section 5, these anomalies tion function of constant width "4;10\ r.l.u. [9, 10]. 6 R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 Fig. 7. Scaling of the ZFC data in the transition region in Mn Zn F. The data are plotted as a function of the scaling variable (¹!¹ (H))/H; the relative intensity is then the only free parameter and is adjusted to optimize the data collapse [9, 10]. Fig. 6. Order parameter squared versus temperature after ZFC (H"3.5 and 5.0 T) and FHXY (H"6.0, 6.5 and 7.0 T) in two different samples of Mn Zn F. The solid lines through the H"0 T data are the results of power-law fits I&(¹,!¹)@ with 2 "0.70. The solid lines through the HO0 data are the results of fits to a Gaussian rounded power law, Eq. (4), with 2 "0.40 and 8$!(H)"0.0034H (K/¹) [9, 10]. inspection of the data in the transition region reveals that the transition is not a true power law, but rather it is smeared out. The solid lines in Fig. 6 represent the results of fits to a power law with "0.2 and a Gaussian distri- bution of transition temperatures; 1 I(¹, H)"  t !¹ @ ( t 8$!(H) ;exp ! t !¹ (H) 8$!(H)  dt , (4) from the fits one finds that to within the errors 8$!(H)&H. Fig. 8. (a) The ZFC order parameter squared in Fe With the knowledge that the width of the transition  Zn F as measured at the (1 0 0) position with X-rays for five fields and region scales as H, one can construct a scaling plot for H"0 T. For HO0, the data are well described by a power- these data. That is, by measuring the temperature in units law-like behavior with a broadened transition region. The H, all of the HO0 data should collapse onto a single broadening is modelled by a Gaussian distribution of transition curve. Fig. 7 displays the data of Fig. 6 as a function of temperatures of width 8$!(H)JAH, Eq. (4). (b) The HO0 the variable (¹!¹ data of (a) replotted as a function of the temperature interval (H))/H. The relative intensity is then the only free parameter, and is adjusted to optimize away from ¹ (H) as measured in units of H. This illustrates the rounding of the transition which is attributed to nonequilibrium the data collapse. The data do indeed collapse onto effects arising from extreme critical slowing down and the uni- a single function, shown in the figure as the solid line. The versal scaling behavior of the trompe l'oeil critical phenomena. scaling function is a power law with exponent "0.2 The inset shows the phase boundary of Fe with a transition region rounded as H. This has been  Zn F as deter- mined from the X-ray fits [11, 12]. R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 7 rounded like H can also describe the data. Currently there is no real theory for the observed trompe l'oeil behavior. 5. Scattering, direct and indirect heat capacity data One of the most important experimental issues for the RFIM has been how one reconciles X-ray and neutron scattering data with direct and indirect heat capacity results. Recent work seems to have resolved this conun- drum [11, 12]. We show in Fig. 10 results of neutron and X-ray scattering measurements on an identical sample of Fe Zn F in a field of 6.1 T. The X-ray and neutron thermometer scales were normalized at H"0 T (¹," (0)"36.8 K), and no further temperature-scale correc- tion was made in comparing the neutron and X-ray data at 6.1 T. Fig. 10 contains several important results. First, it shows that neutron and magnetic X-ray scattering yield identical results for M in the transition region. Second, the ZFC fluctuation correlation length is a max- imum at ¹ (H), and this length equals the corresponding FC value at all temperatures *¹ (H). This was also seen earlier in measurements in Mn Zn F (Fig. 2). Con- comitantly, the critical scattering amplitude measured at Fig. 9. RFIM behavior of Fe Co TiO in a magnetic field. (1, !0.003, 0) is a maximum at ¹ The upper panel shows the (1, 1, 1.5) Bragg intensity after ZFC. (H). Fig. 11 shows the results of SQUID and neutron The solid lines for HO0 are the results of fits to the trompe l'oeil model, Eq. (4). The H"0 line is the result of a fit to a simple measurements in Fe Zn F at 5 T [11, 12]. The power law with SQUID data were taken on a small piece cut from the ,"0.36(3). The inset shows the (1, 1, 1.5) intensity for ¹"15 K as a function of field for X-rays and neutrons. The H dependence in the X-ray cross section orig- inates in a subtle coupling term which scales like MM  where M is the magnetization, M the staggered magnetization and  the staggered charge density. The bottom panel shows the FC inverse correlation length 2 for varied fields [16, 17]. labelled trompe l'oeil critical behavior since the behavior simulates ordinary critical phenomena at a second-order phase transition but, in fact, represents escape from meta- stability [9, 10]. A similar set of experiments and analysis was carried out in Fe Zn F [11, 12]. The results are shown in Fig. 8. Again one finds that 8$!(H)&H. This once more enables one to construct a scaling plot as shown in the bottom panel of Fig. 8. The best fit of Eq. (4) to the scaled data yields "0.15$0.05. As shown in Fig. 9, similar results are obtained in Fe Co TiO al- though in that case the data are more complicated be- cause of a novel coupling between the magnetization, the staggered magnetization and the staggered charge den- sity which affects the magnetic superlattice X-ray cross Fig. 10. Top panel: X-ray and neutron magnetic intensities in section [16, 17]. Fe We should emphasize that Eq. (4) is not unique. In-  Zn F for H"6.1 T. The solid line is the result of a fit to the rounded power law of Eq. (4) with ¹ deed, as pointed out by Wong [31] and Hill et al. [32] (6.1 T)"25.54 K (dashed line) and (6.1 T)"0.9 K. Bottom panel: Inverse cor- a model assuming a mean-field first-order transition relation length versus temperature on FC and ZFC [11, 12]. 8 R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 hypothesized that the thermal contribution to d(¹M)/d¹, which is determined by short-range effects, would not be very different for FC and ZFC measure- ments. Therefore, as a first approximation, the ZFC d(¹M)/d¹, should equal the FC result augmented by the dM/d¹ contribution. The solid line in the bottom panel of Fig. 11 is the result of such an analysis. In this case, only the amplitude of the dM/d¹ contribution (dashed line) has been varied, and there has been no further adjustment in the temperature scale. Clearly this simple model describes the ZFC d(¹M)/d¹ data very well. Sim- ilar agreement is obtained at all other fields; in each case the adjustment of the temperature scales to match the peak temperatures is well within the temperature uncer- tainties. The evident good agreement is compelling evid- ence that the basic model is correct. In order to test these ideas further, direct heat-capacity measurements were carried out by Feng and Ramirez both on samples of Fe Zn F taken from the same boule as the samples used in the above work, and on crystals of Mn Zn F [11, 12, 15]. The above model necessitates that the direct heat capacity, which is sensitive to the local spin configuration and not to the LRO, should show little hysteresis. In the top panel of Fig. 11. Top panel: neutron scattering intensity at (1, 0, 0) (LRO) and (1, !0.003, 0) at H"5.0 T. The solid line is the Fig. 12, direct heat-capacity data taken on Fe Zn F result of a fit to the trompe l'oeil rounded power-law form, at H"1.5 T, and 5.5 T are displayed. These data were Eq. (4). The small arrow at the top indicated the 0.3 K shift in taken using a semiadiabatic technique. The time scale for the neutron data temperature scale. Bottom panel: FC and ZFC each datum point is &20 s. There is no observable data for d(¹M)/d¹ at H"5.0 T [11, 12]. hysteresis in either data set. This demonstrates that the FC and ZFC thermal fluctuations are closely similar. neutron sample. They agree in detail with previous re- Identical results are obtained in Mn Zn F [15]. It sults by Lederman and co-workers [27]. The neutron is clear that the ZFC peak of indirect heat-capacity data are analogous to those shown in Fig. 10 at 6.1 T. methods is not indicative of the true heat capacity. The temperature derivative of the SQUID magnetization Finally, these ideas have been tested on published shows a sharp peak on ZFC, but not FC. The temper- birefringence data, as shown in the bottom panels in ature scales of the neutron and magnetization data have Fig. 12. In Fig. 12b, we show data of Ferreira et al. again been normalized at H"0 T. The net accuracy of [28-30] taken on a sample of Fe Zn F of very this normalization is $0.3 K. In plotting the data at 5 T high quality. In this case, the solid line is the X-ray in Fig. 11, the neutron temperature scale has been shifted scaling function with the width taken from the H scaling by 0.3 K, as indicated by the small arrow at the top of the law of Fig. 8. The FC data are used for the background figure. This is based on the physically compelling argu- arising from the noncritical fluctuations. ¹ (H) was ad- ment that d(¹M)/d¹ should have its maximum at the justed slightly from the value determined in the X-ray same temperature as that at which the correlation length experiments on Fe Zn F. Again the agreement is is a maximum. In any case, the temperature-scale shift good. Finally, in Fig. 12c similar data and analysis for is within the combined temperature uncertainties, and Fe Zn F are shown [11, 12, 28-30]. In this case, the its omission has no important effect on the overall coefficient of the H width is fitted at H"4 T. Once argument. more the model describes the ZFC birefringence data In Refs. [11, 12] it was hypothesized that for indirect well. heat-capacity techniques for RFIM systems there may be The above analysis removes one of the major stumbl- a term of the form dM/d¹. The argument was based on ing blocks in understanding the phenomenology of general phenomenological considerations for probes RFIM systems. In general, it is now clear that there are where the response is determined by short-range spin- three important temperatures: ¹ (H)'¹ (H)'¹,(H). spin correlations. In fact, such a term has been hy- ¹ (H) is the onset temperature for metastability ef- pothesized previously by many authors for different fects. ¹ (H) is the midpoint of the transition region of kinds of measurements in other kinds of systems and has the diminution of the order parameter after ZFC. ¹,(H) been observed in a number of cases. It was further is the equilibrium Ne´el temperature. So far no true R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 9 which modifies the hyperscaling relation to (d! ) " 2! . Arrhenius' law then implies that the characteristic relaxation time of the system close to the transition, i.e., the time to flip a correlated volume, is +  exp(C F), (5) where  and C depend on the random-field strength. The `activated dynamics' of Eq. (5) result in extreme critical slowing down and the system cannot equilibrate on ex- perimentally accessible time scales, as PR. ¹ (H) then is the temperature above ¹,(H) at which exceeds laboratory measurement time scales. The ZFC trompe l'oeil critical behavior then may be understood as follows: just as extreme critical slowing down prevents the attainment of equilibrium on field cooling, activated dynamics with the concomitant ex- treme critical slowing down will also operate on warming the ZFC state. Thus, as the temperature is increased to ¹:¹,(H), the system will fall out of equilibrium, at least on experimentally relevant time scales, and the correla- tion length will saturate, as is observed experimentally. The system will not be able to relax fully until it has passed through the critical region to reach the high- temperature equilibrium phase. This leads to a qualitat- ive picture of the process in which the LRO is destroyed Fig. 12. (a) Heat capacity of Fe Zn F. There is no evidence through `flipping' of domains with successively larger of hysteresis on FC and ZFC, nor is there any sign of critical heat capacity in the ZFC data. (b) Optical birefringence data sizes. The maximum rate of change occurs at the center of taken from Ferreira et al. [29] for Fe the trompe l'oeil Gaussian distribution which is when the  Zn F. The solid line is the FC data plus the contribution from the dM size of the volume being flipped becomes approximately  /d¹ term. (c) Similar results and analysis for Fe equal to the FC domain size at that temperature. As  Zn F (Refs. [28-30]). In each panel, the open symbols are ZFC data and the closed found experimentally, the rounding of the transition may symbols are FC results [11, 12]. then be understood as a finite-size effect, in which the growth of the correlation length in the transition region equilibrium phase transition in the bulk has been ob- is limited by the random fields to the FC domain size served in RFIM systems. A transition to LRO on FC has [6-12]. been observed in certain samples in the near-surface region (&2 m) using X-ray scattering techniques [9, 10, 14]. However, the interpretation of these experi- 6. Equilibrium critical behavior ments is still problematical so we do not discuss them explicitly here. Because of the onset of metastability at ¹ (H), which How does one understand the breakdown of equilib- is well above the equilibrium phase transition temper- rium at ¹ (H)? A phenomenological model has been ature ¹,(H), studies of the asymptotic equilibrium criti- introduced by Villain [33] and Fisher [34]. The Villain- cal behavior in RFIM systems appear not to be possible. Fisher picture is as follows. For a given correlated vol- Further, the behavior well above ¹,(H) is complicated ume, there is only one thermally populated minimum. by an anticipated crossover from random exchange to Thermal fluctuations in this minimum are small because random-field Ising critical behavior. Thus, at best crude of the steep curvature in the free energy. However, on estimates of the 3D RFIM critical exponents can be rare occasions, with probability ¹/ F two minima occur obtained experimentally. Such estimates are nevertheless within a correlated volume which differ in energy only by valuable since the critical behavior of the 3D RFIM is order ¹. In such cases, the equilibrium fluctuations may expected to be radically different from that of the uniform be thermally activated over the free-energy barrier. To or random exchange 3D Ising model, including especially achieve such a reversal of a block of spins of size R, an unusually rapid divergence of the disconnected sus- a free-energy barrier of height (R)+RF must be over- ceptibility [35, 36]. come. RF is the random-field energy of a volume RB (the Feng et al. [14] have carried out a study of the FC surface tension vanishes as ¹P¹ ) and is the exponent behavior in Fe Zn F in fields of 5 and 6 T using 10 R.J. Birgeneau / Journal of Magnetism and Magnetic Materials 177-181 (1998) 1-11 two-axis neutron scattering techniques. From decon- 7. Conclusions volutions of the neutron scattering spectra they are able to extract the inverse correlation length , the connected We now have a fairly complete empirical description susceptibility &B/  and the disconnected susceptibil- of the equilibrium and nonequilibrium properties of ity &A/ . The results so-obtained at 5 T are shown several model RFIM systems. There appear to be no in Fig. 13. From fits of power laws, &(¹!¹,(H))J, to major contradictions between different classes of the data they find "1.5$0.3. With ¹,(H) fixed at measurements on these materials. The application of the value determined from fits to the data, they then synchrotron magnetic X-ray scattering techniques to find for the connected susceptibility "2.6$0.5 and for this problem has clarified the behavior at large length the disconnected susceptibility N"5.7$1. The large er- scales and has provided the first reliable measurements rors reflect the strong dependence of the exponents on of the order parameter, especially in the transition region. the choice of ¹,(H). Cowley et al. [6-8] have carried out We also now have well-developed heuristic ideas which a similar analysis on their data on Mn Zn F. They seem to describe successfully the overall dynamic behav- find "1.4$0.3 and N"4.5$1. Thus the experiments ior. However, a detailed quantitative theory for the in Fe Zn F and Mn Zn F yield similar, if equilibrium and nonequilibrium properties of the RFIM poorly determined, values for the equilibrium RFIM as well as the transition between these two behaviors is exponents. The values for , and N are consistent within lacking. Clearly this represents an important challenge the combined experimental and theoretical errors with to all physicists interested in the behavior of systems most current theoretical predictions for the respective with quenched disorder. Further experiments, espe- critical exponents of the 3D RFIM [35, 36]. Interesting- cially on the equilibrium critical behavior, are also ly, the values for and agree within the errors with required. those of a pure Ising model in &1.6 dimensions [37], These experiments illustrate clearly the role of model consistent with the dimensional reduction argument of magnetic systems in condensed matter and statistical Villain [33] and Fisher [34]. physics. As we noted in the Introduction to this paper, random-field effects are ubiquitous in condensed-matter materials. However, typically these systems are quite complicated and it is difficult to isolate those effects which arise purely from the random fields. Further, most often the random fields arise from quenched impurities so that one cannot easily vary the strength of the field in a given sample in order to deduce the quantitative de- pendence of the effects on the strength of the random field. By contrast, following the suggestion of Fishman and Aharony [5], it has been possible to explore thor- oughly the physics of random fields using as `laborator- ies' simple diluted or otherwise random two-sublattice Ising antiferromagnets. We anticipate that the field of Magnetism and such model magnetic systems will con- tinue to play a central role in condensed-matter physics overall for the indefinite future. First of all, we would like to thank the International Union of Pure and Applied Physics for honoring us with the IUPAP Magnetism Award 1997. This award, of course, honors not just ourselves but all of our collabor- ators. In addition, it recognizes our basic approach of using simple magnetic materials to elucidate the funda- mental behavior of complex many-body systems. We would like to thank our collaborators in this RFIM work including A. Aharony, R.A. Cowley, Q. Feng, Q.J. Harris, J.P. Hill, A. Ito, A.P. Ramirez, G. Shirane, T.R. Fig. 13. Inverse correlation length , disconnected susceptibility Thurston and H. Yoshizawa. We would also like to thank , and connected susceptibility as measured by neutron scattering for Fe J. Als-Nielsen, Y. Endoh, M. Greven, H.J. Guggenheim,  Zn F at 5 T. The dotted line shows the metastability temperature. ¹ M.T. Hutchings, M.A. Kastner, K. Yamada and many , is the temperature at which the solid-line fit for reaches zero. The estimated critical exponents others for their outstanding contributions overall to our are given [14]. studies of model magnetic systems. The research at MIT R.J. 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