Magnets, molecules and quantum mechanics

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Magnets, molecules and quantum mechanics

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1999 Phys. World 12 (3) 35 (http://iopscience.iop.org/2058-7058/12/3/28) View the table of contents for this issue, or go to the journal homepage for more

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MOLECULAR MATERIALS Experiments on single crystals composed of nanomolecules with magnetic properties are being used to study quantum tunnelling of magnetization and to probe the boundary between quantum and classical mechanics

Magnets, molecules and quantum mechanics Bernard Barbara and Leon Gunther Large-scale quantum behaviour IN 1990 one of us (LG) published an 1 The energy barrier article in Physics World on quantum Can quantum mechanics, which tunnelling in magnetic particles that determines the behaviour of light are a few nanometres in diameter. and matter at the atomic scale and B=0 The article emphasized both the funbelow, manifest itself on a macrodamental challenges - for instance scopic scale? This question, which energy barrier understanding the nature of the was first posed in the early days of transition region between quantum quantum theory in the 1920s, has fasI and classical regimes - as well as the cinated physicists for more than 70 angle technological obstacles to controlling years. The scenario of "Schrodinger's and exploiting quantum coherent cat" - where the cat is described by a 8*0 tunnelling in nanoscopic devices. The wavefunction that is a superposition subject of quantum tunnelling of of two states, one wim die cat alive magnetization (QTM) was in its inand the odier with the cat dead — is a fancy. Since then, however, many famous example that illustrates the angle experimentalists and theorists have conceptual complexity t_hat can arise contributed to this richfield.In 1994, from the intermingling of the quanfor instance, the first workshop on In a single-domain magnetic particle with one easy axis tum and the classical regimes (see the subject was held in Chichilianne, there are two minimum-energy states, corresponding to article by Leggett in Gunther and the two directions in which the magnetic moment of the France (see Gunther and Barbara in particle prefers to point along the easy axis, (a) These two Barbara in further reading). further reading), and interest in quan- states are separated by an energy barrier: this can be The phenomena of superconductum phenomena in magnetic mole- seen if the energy is plotted as a function of the angle tivity, discovered by Kammerlingh between the magnetization and one of the easy-axis cules continues to grow. Onnes in 1911, and superfluidity in directions, (b) If an external magnetic field is applied An exciting type of material has parallel to the easy axis, then the energy of one state helium, discovered several decades recently emerged as the focus of increases, while the other decreases. This reduces the later, are manifestations of quantum attention in the field - molecules energy barrier in one direction and increases it in the mechanics on a macroscopic scale. The barrier disappears when the field exceeds a with magnetic moments that exhibit other. More recently, in the 1980s, quantum certain value called the anisotropyfield,BA. quantum tunnelling in spite of their effects have been observed at scales relatively large size. The focus of this between the microscopic and the article is on one example of such a material, the spin cluster macroscopic. These mesoscopic examples include quantum referred to as "Mnl2-ac". This manganese acetate was the tunnelling of the phase in a Josephson junction, permanent first system to exhibit what is referred to as "thermally currents in small conducting rings and, since the mid- 1990s, assisted resonant tunnelling" (see Friedman et al. and Thomas Bose condensates (see Physics World March 1997 pp29-34). et al. in further reading). Mn 12-ac was actually discovered in These systems, which vary in size from 102 to 105 nm, are rel1980, but it took more than ten years of close collaboration atively complex. Nevertheless, their properties can be debetween chemists and physicists to fully reveal its remarkable scribed with a small number of degrees of freedom defined properties (see Lis in further reading). Quantum tunnelling by an "order parameter". of magnetization has also been observed in the "Fe8" spin Electrons are responsible for atomic magnetism. The basic cluster, so-called because it contains eight iron ions in addi- element is the Bohr magneton, uB = eh/^nmc, where e and m tion to various OH and organic sidegroups (see Sangregorio are the charge and mass on the electron, h is the Planck conetal. in further reading). stant and c is the speed of light. The magnetic moment of an PHYSICS WORLD

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MOLECULAR MATERIALS 2 Manganese-12

A cluster of Mnl2-ac. The large circles represent the two types of atoms of manganese: there are eight Mn3* ions (purple) and four Mn4* ions (blue). The small red circles represent the oxygen atoms. The cluster has a total spin of 10 due to the S = 2 spin of each of the Mn3* ions being antiparallel to the S - 3/2 spin of each of the Mn4+ ions. One can grow crystals of Mnl2-ac, which allows its magnetic properties to be studied in detail.

Molecular computers Might a molecular spin cluster serve as a computer element, as several researchers have proposed? Allowing for a distance of 5 nm between neighbouring spins, a disc with an area of 100 cm2 could hold a staggering 50 000 gigabytes of spin clusters! Presumably, the states m=±S would be used to store a classical bit. However, quantum tunnelling renders these two states unstable, even at absolute zero temperature (see Gunther 1990 in further reading). At 1.5 K, the relaxation time of Mnl2-ac (~ 108 seconds or three years) would not be long enough for computer elements, even if the problems of refrigeration could be solved. And even if a material with an acceptable relaxation time (at least 15 years) at room temperature could be found, the problem of reading and writing the bits would have to be solved. Next, consider a quantum computer, which stores information as a superposition of two basis states in a quantum bit or "qubit" (see Physics World March 1998 pp33-57). For the spin cluster, these two basis states could be the two lowest spin states resulting from the splitting through tunnelling of the doubly degenerate ground state, m = ±S, into an anti-crossing at zero field, similar to the anti-crossing shown infigure 5. The problem with a system such as Mnl2-ac is that these states are decoherently mixed by various interactions, such as the hyperfine interaction with the nuclei. Interestingly, it is these interactions that apparently render quantum tunnelling observable in the first place, as pointed out in the main text.

atom is given approximately by the product of \iB/k and the Such particles consist of a single domain and we can define a sum of the orbital angular momentum and twice the spin "collective moment", M, referred to simply as the moment of angular momentum of the electrons. the particle. This is die order parameter of the system. Its The dominant interaction between two electron spins is the magnitude can range from several Bohr magnetons (for partiexchange interaction. This interaction, discovered by Heisen- cles only a few angstroms in diameter) to of the order of 10 u.B berg, is an effect of the Pauli exclusion principle. One result of for particles with dimensions of tens of nanometres. Particles this principle is that two spins that are parallel tend to be fur- larger than this tend to exhibit domain structure. In thermal equilibrium, according to classical physics, the ther apart than two spins that are antiparallel. As a consequence, a pair of electrons with parallel spins has a lower direction of the magnetic moment of such a single-domain electrostatic potential energy than a pair with antiparallel nanoparticle is constantly fluctuating, with the most likely spins. Such an interaction is said to be "ferromagnetic". The direction being along the easy axis. Suppose that the moment exchange energy is usually much larger than the energy asso- is initially oriented along one of the two easy directions. It can ciated with other magnetic interactions, such as the "dipole reverse by using thermal energy to pass over the energy barrier. energy" and the "anisotropy energy". In addition to this "thermal activation", which is a classical Dipole coupling, which accounts for the coupling between process, a quantum process is possible - quantum tunnelling two compass needles, is electromagnetic in origin and has a of the moment from one side of the barrier to the other. In long range. It tends to cause the magnetic moments on the principle, no thermal energy is needed because quantum tunsurface of a magnetic material to lie in antiparallel direc- nelling leads the system from one state to another with the tions (although the moments all remain parallel to the sur- same energy. The process is analogous to quantum tunnelling face). In large enough samples, typically with dimensions of an alpha particle in a radioactive nucleus. The first indilarger than a few tens of nanometres, dipole coupling causes cations of quantum tunnelling in macroscopic magnetism, neighbouring magnetic domains to point in antiparallel or observed by one of us (BB) and colleagues in the Laboratoire Louis Neel in Grenoble, France, were seen in the late 1970s. perpendicular directions. The anisotropy energy tends to align the magnetization along certain crystal axes. In particular, this energy has min- New materials and measurement techniques ima along certain directions, with energy barriers separating There have been recent developments in various disciplines them. Since all physical systems seek to be in the state of low- that have led to great progress in die study of macroscopic est energy, the moments tend to line up along these directions, tunnelling in magnetism. In materials science, magnetic materials have been produced as isolated aggregates, as deposits which are referred to as "easy axes of magnetization". For simplicity, we will assume that the system has only one of aggregates, as carbon nanotubes and nanocages filled easy axis, so there is an energy barrier, U, between the two with magnetic material, as electro-deposits of magnetic mastates in which the moment is aligned with the easy axis (figure terial in nanoporous polycarbonate membranes and as dis1). For sufficiently small ferromagnetic particles the exchange persals in polymers. Molecular chemistry has produced energy dominates and all the moments point in one direction. molecules with giant spins, and colloidal chemistry has used PHYSICS WOULD

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MOLECULAR M A T - " " micelles as microreactors for making all sorts of new mag- 3 The staircase phenomena in manganesenetic nanoparticles. Naturally occurring biological systems have given us ferritin, and biochemistry has provided us with artificial analogues. Improved measurement techniques — with magnetometers based on micro-superconducting quantum interference devices (micro-SQUIDs) — have made it possible to probe quantum tunnelling effects in the magnetic properties of single nanoparticles. At Grenoble, BB and colleagues have studied both single crystals of identical molecules with large spins, such as Mn 12-ac, as well as individual mesoscopic magnetic particles (see Thomas et al. and Wernsdorfer et al). In addition, a number of other groups - notably those at the City University of New York and the Universities of Barcelona and Zaragosa in Spain - have made important and significant contributions -1.2 to the study of molecular spin clusters. There have also been -2 - 1 0 1 suggestions that the clusters could be used in molecular clusmagnetic field (tesla) ters (see box). Hysteresis loops of a monocrystalline grain of Mn 12-ac measured in a Manganese acetate (Mnl2-ac)

longitudinal field at different temperatures. Each point along the hysteresis loop, taken anticlockwise from the topright,is obtained as follows: initially the crystal is cooled from above the blocking temperature in a field of 5 T, thus placing all the spins in the m = +10 state. The field is reduced from one value to the next over a period of 10 s and, after a pause of 400 s for stabilization, the magnetization is measured. As the applied field is lowered, the magnetization decreases in a series of steps, separated by plateaus. The steps can also be seen on the upward part of the hysteresis loop. As the temperature is lowered, there is a decrease in the transition rate due to reduced thermal activation and reduced quantum tunnelling (due to a decreased occupancy of the levels close to the energy barrier). A larger applied field is therefore necessary to produce a significant transition rate and the loop widens. (From Thomas et al.)

In the early investigations of quantum tunnelling in magnetism, the only samples that were available consisted of systems of many mesoscopic spins, such as aggregates of isolated particles, domain walls in alloys or liquid suspensions. The broad distribution of particle sizes and shapes, with a consequent distribution of anisotropy energies, made it difficult to interpret the results of experiments. The great advantage of Mn 12-ac, Fe8 and similar systems is that it is possible to grow crystals of identical magnetic clusters. Mn 12-ac, which is an abbreviation for the much more complex formula [Mn^O^CF^COO^HgO),,.], is a molecule experiments. As Louis Neel discovered in 1949, the moment with tetragonal symmetry. The 12 manganese ions are cou- reverses direction very rapidly above the blocking temperapled by strong antiferromagnetic interactions through over- ture, a phenomenon known as "superparamagnetism". lap of electron orbitals via the oxygen atoms (figure 2). As a result, the Mn 12-ac molecule can be represented by two Applied fields sub-ensembles of spins: one contains eight parallel spins, each Two types of experimental studies have been carried out in with S=2 and, therefore, a net spin of 16; the other sub- which the magnetization (the total magnetic moment of a ensemble contains four parallel spins with 5 = 3 / 2 and a net sample) is measured. Hysteresis involves monitoring the magspin of six. The net spins of the two sub-ensembles are netization while an applied field, parallel to the easy axis, is antiparallel, so the total spin, S, of the molecule is 10. The swept continuously from a large value in one direction to a corresponding magnetic moment is M=gS\iB ~ 20uB, where large value in the opposite direction and then back. The g~ 2 is the Lande g-factor. curve of magnetization versus applied field is called a "hysEach cluster occupies a volume of about 1 nm3, with mole- teresis loop". The second type of study involves monitoring cules of water and acetate separating the clusters from one the relaxation over time of the magnetization in afixedfield. another. As a result, the clusters occupy only about 5% of the To appreciate the hysteresis results, it is important to note total volume of the crystal. The large distance between neigh- that a magnetic moment, M, has an energy in the presence of bouring molecules prevents any exchange interaction and an appliedfield,B. This energy is associated with the torque allows only weak dipolar interactions. Furthermore, while the that tries to align the moment with the field and has a value anisotropy energy barrier is 6 meY the dipole interaction -MBcosQ, where 0 is the angle between die vectors M and B. energy between neighbouring spins is estimated to be about Thus when the two vectors are parallel, die energy is -MB, and 0.03 meV Thus we can regard the spins as independent of when diey are antiparallel it is +MB. In addition, the applied each other to afirstapproximation. The design of such an el- field reduces the energy barrier, which disappears above a egant structure reflects the great power of modern chemistry. value of the applied field known as die anisotropy field, BA. In the absence of an appliedfield,the mean time for a large The spins reverse direction spontaneously above BA> irrespectspin to reverse from one easy direction to another over the ive of temperature. In Mn 12-ac, die anisotropy field is 9.6 T energy barrier is proportional to the Boltzmann factor, when thefieldis parallel to the easy axes (i.e. longitudinal). exp [U/kT~\, where Uis the height of the energy barrier, k is the The monocrystals of Mn 12-ac used in Grenoble were Boltzmann constant and Tis temperature. As a result, one can grown by Roberta Sessoli, Dante Gatteschi and co-workers at define a "blocking temperature", 7~B, above which the rate of the University of Florence in Italy. The largest of die crystals, spin reversal is much faster than the measurement timescale. about 0.07 mm in volume, are sufficient to permit precise Mn 12-ac has a blocking temperature of about 3 K in these measurement of the magnetization using a conventional PHYSICS WORLD

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Mm ETUI AP MATTDIAI C

SQUID magnetometer. Hysteresis loops were measured at 4 Quantum relaxation in manganese-12 various temperatures below the blocking temperature of 3 K (figure 3). When thefieldis reduced to zero from a large value, io9 the remnant magnetization is not much smaller than its saturated value in die presence of a large applied field. Thus the spins are strongly blocked along the easy direction. When thefieldis applied in the opposite direction, the spins I io7 are expected to remainfixeduntil they suddenly reverse at the coo anisotropyfieldBA = 9.6 T. However, the reversal occurs for IO5 much smaller fields. Moreover, it occurs in a series of steps separated by 0.45 T. This "staircase" behaviour is completely unexpected in the classical description given previously and gives die impression mat the magnetization vector of die £ IO 3 crystal is quantized. This leads us into a completely new a; region of physics in which quantum effects are observed at a human timescale. This behaviour also suggests that, along a -1.5 -1.0 -0.5 -2.5 -2.0 0.5 plateau, the relaxation time for the magnetization is much magnetic field (tesla) greater than die measurement time (about 10 minutes for each data point). During a jump, on the omer hand, die relax- The variation of the relaxation time of the magnetization (red points) as a function of applied field measured at 1.9 K (which corresponds to the green ation times are much shorter than on die plateau. hysteresis loop in figure 3). Initially, the crystal is cooled in a field of 5 T. The This idea is confirmed by direct measurements of the relax- field is then reduced rapidly to the desired value and kept fixed while the ation time, t, in various constant appliedfields.The relaxation relaxation of the magnetization is monitored. The minima seen in the figure time decreases as the applied field is increased, as a conse- occur at the same values of applied field as the steps in the hysteresis curves quence of the lowering of die energy barrier. It also shows shown infigure3. The arrows are located at the fields corresponding to the level crossings (see text and figure 5). The blue curve is afitto the data. The narrow minima at certain values of the appliedfield(figure 4). green curve is the classical prediction. (From Thomas etal.) These are die same values at which steps appear in die hysteresis loops. The minima are superimposed on a continuous variation of the relaxation time widi temperature according to can be approximated by 2SD/g\i^. the classical law for thermal activation, i°ztx-p\U(B)/kT~\, The energy levels can be plotted as a function of die applied field (figure 5). Consider two energy levels, E_m and where U(B) is afield-dependentenergy barrier. EnMl. As the appliedfieldis increased, one energy will increase and the otiier will decrease so t_hat, at a certain value of the Theoretical discussion To understand bodi me jumps in the hysteresis loops and die appliedfield,the two levels will have the same energy. This is minima in die relaxation time at certain values of die applied called a level crossing. The values of die field at these crossfield, we will use the following expression for the anisotropy ings can be easily shown to be B-Bn-nD/g\xB = nBA/2S, which is independent of m. There are actually several level energy of a single molecule of Mn 12-ac crossings for each value of n (and therefore each £„). MoreE D S ? S B where Sz is die ^-component of the spin vector, S, D is a so- over, die crossings are equally spaced - by BA/2S— 0.48 T. called anisotropy constant and the easy axis is along die z- This is very similar to the spacing of 0.45 T between die axis. This expression is based on symmetry arguments as well jumps in the hysteresis loops, which suggests diat die two pheas numerous experiments carried out on Mn 12-ac. The first nomena are linked, with the difference being due to terms in term, -DS^, is the intrinsic anisotropy energy and accounts the anisotropy energy that are quartic in Sz. for the easy axis. The second term, —g\xBS'B, reflects die The jumps in die hysteresis loops represent a gready energy of the magnetic moment in the applied field. enhanced transition rate between one level and another at a In the experiments described above, die appliedfield,B, is level crossing: for a given n, jumps will occur at various values along die £-axis, so the energy depends on only one dynam- of m, ranging from —10, —9, ..., until the top of the energy ical variable, Sc The possible observable values of Sv its barrier (which will occur at m < 0). Jumps have also been "quantum numbers", are m — —S, —S+ 1, ..., S+ 1, S. Eachobserved at level crossings in the hysteresis loops of otiier value £j = m corresponds to an energy level and a quantum systems, such as magnetically doped semiconductors. Howstate of me spin. ever, there is no energy barrier in these systems, so that the At zero applied field, we note tiiat the second term is zero reversalfora single spin is extremely fast. As thefieldchanges, and diat the anisotropy energy is a minimum when m = ±S the ensemble of spins evolves while in a state of quasi-static (that is, when the spin points along one of die two easy-axis thermal equilibrium and the width of the steps is propordirections), and a maximum when m = 0. The energy barrier, tional to the temperature. Results from Mn 12-ac suggest diat a "resonant tunnelling" U, in zero field is therefore given by DS2, the difference between the minimum and maximum energies. In thermal process is taking place between two levels widi equal energy. equilibrium, die spin statefluctuatesamong diese 25+ 1 =21 However, for tunnelling to take place, the energy term must states, widi die m - ±S states having die highest probability of contain a term with a spin component in either the x ory being occupied. As die field is increased, die maximum in direction, or both. It can be shown mat due to the tetragonal the anisotropy energy shifts to lower values of m. At die symmetry of the Mn 12-ac lattice, the lowest-order term that anisotropyfield,die energy is a maximum for die lowest two can exist is proportional to the fourth power of Sx and Sr If values, namely m = —S and m = —S+l. The anisotropy field the appliedfieldhas a component in die x ory direction, diis 38

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Mm rrm an luaTrnta levels — to be observed. In Fe8, on the other hand, the intrinsic transverse anisotropy of the crystal is large enough for groundstate tunnelling to be observed in the absence of an applied transverse field. Future challenges

Experiments on Mnl2 and similar nanomolecules have taught us a great deal about the quantum aspects of magnetism. The unusual behaviour observed in Mn 12 can be explained in terms of a process called thermally assisted 4> resonant tunnelling. However, the passage from quantum to classical behaviour is not sudden, as one might have expected. Instead, the magnetization jumps in the hysteresis curves m=+10 gradually become less pronounced and the depths of the minima in the relaxation time decrease as the system continuously approaches classical behaviour. applied field (tesla) However, to fully understand these materials we must Energy levels as a function of the applied field, B, for the 21 spins levels in a understand the interactions between the individual spin clusMnl2-ac cluster. The levels are labelled by the quantum number m = -10. -9, ters in die crystal. The quantitative disagreement between .... 9,10. In the absence of an appliedfield,the level m = 0 has the highest theory and experiment with respect to the size and width of energy and Em = £.m. However, when a field is applied, the energies of levels with m < 0 increase, while the energies of levels with m > 0 decrease. This the jumps gives evidence that the spins cannot be tunnelling means that the energies of different levels, say m = +9 and m = -10, can in isolation. Moreover, theoretical studies indicate that if the cross at certain values of the applied field. The levels cross at fields given by Bn = nD/gp,B, where n is an integer. For Mnl2-ac, the gap between crossings, spins are entirely isolated, the requirement that a spin be very D/gu,B is about 0.48 T. The inset displays the details at a level crossing where close to a level crossing before tunnelling can take place is simthe transverse interaction that produces tunnelling turns the crossing into an ply too stringent to allow tunnelling to be observed. Instead, anti-crossing (in red). Suppose that the spin is on the lower level and to the left of the anti-crossing as the field is swept through Bn. A fast sweep rate favours hyperfine interactions are needed to bring the levels into coincidence for long enough periods of time, as pointed out by the spin jumping onto the upper level, and remaining in the same spin state. On the other hand, a slow sweep rate favours the spin remaining at the initial Nikolai Prokof'ev of the Kurchakov Institute in Moscow and level, so that the spin quantum tunnels from one spin state to another. Philip Stamp of the University of British Columbia in (See L Gunther 1997 Spin tunnelling in a swept magnetic field Europhys. Lett. Canada. They have also shown that dipole interactions be3 9 1 ; erratum 40 233). tween clusters lead to a relaxation of the magnetization that is not exponential at short times — a prediction that has been will also produce tunnelling. Indeed, experiments have shown confirmed experimentally. The mysterious magnetic properties of nanomolecules will that the presence of such a component enhances the minima in the relaxation rate. Theoretical analysis shows that signi- keep physicists busy for years to come. ficant tunnelling occurs only between two states of nearly equal energy, corresponding to the system being very close to Further reading a level crossing and therefore to an applied field being very J R Friedman etal. 1996 Macroscopic measurement of resonant close to one of the discrete values Bn. magnetization tunnelling in high-spin molecules Phys. Rev. Lett. 763830 Suppose that the spin in an Mnl2 cluster is in an initial state L Gunther 1990 Quantum tunnelling of magnetisation Physics World close to a level crossing: two processes can "help" it to the other December pp28-34 side of the barrier - thermal activation or resonant tunnelling. L Gunther and B Barbara (ed) 1995 Proceedings of first NATO workshop on The tunnelling ratefromvarious initial states m is proportional Quantum Tunnellingof Magnetization: QTM 94 (Kluwer, Dordrecht) to the product of two factors, the probability for the initial state J Hernandez et al. 1996 Field tuning of thermally activated magnetic quantum to be occupied and the tunnelling rate from that initial state. tunnelling in Mnl2-Ac molecules Europhys. Lett. 35 301 Thefirstof these decreases exponentially with energy accord- T Us 1980 Preparation, structure, and the magnetic properties of a ing to the Boltzmann distribution, being proportional to dodecanuclear mixed valence manganese carboxylite Acta Crystallog. B36 2042 exp [—£/A;T], where E is the energy of the level. Thus, at zero N V Prokof ev and P C E Stamp 1998 Low-temperature quantum relaxation in a applied field, the occupation probability for the m = 0 state is system of magnetic nanomolecules Phys. Rev. Lett. 80 5794 lower than the probability for the m - +S state by a factor SSangregorio etal. 1997 Quantum tunnelling of the magnetization in an iron cxp[DS2/kT]. This factor is equal to exp[-22] ~ 3 x KT10 cluster nanomagnet Phys. Rev. Lett. 78 4645 for Mnl2-ac at 3 K. However, the height and width of the R Sessoli etal. 1993 Magnetic bistability in a metal-ion cluster Nature 365141 energy barrier both decrease as the energy of the initial state is P C E Stamp, E M Chudnovsky and B Barbara 1992 Quantum tunneling of increased, thereby increasing the tunnelling rate. magnetization in solids Int. J. Mod. Phys. B61355 The end result is that the "thermally assisted tunnelling" L Thomas et al. 1996 Macroscopic quantum tunnelling of magnetization in a rate in Mnl2-ac is a maximum for the m = ±3 levels in the single crystal of nanomagnets Nature 383145 absence of an appliedfield.In general, for Mn 12-ac it is only WWernsdorfer eta/. 1997 Macroscopic quantum tunnelling of magnetization possible to observe tunnelling for level crossings that are close of single ferromagnetic nanoparticles of barium ferrite Phys. Rev. Lett. 79 4014 to the top of the energy barrier. However, the introduction of a transversefieldincreases the tunnelling rate, and it is thought Bernard Barbara is in the Laboratoire de Magnetisme, Louis Neel, CNRS-BP166 that a transverse field of 6 T would allow ground-state tun- Grenoble, France. Leon Gunttter is in the Department of Physics and Astronomy, nelling - that is, tunnelling between the m = -lO to m= +10 Tufts University, Medford, 02155 MA, US PHVSICS WORLD

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PHASE TRANSITIONS A Multinational Journal Editor-in-Chief A.M. Glazer, Clarendon Laboratory, Oxford, UK Phase Transitions is the only journal devoted exclusively to this fast growing subject. It provides a focus for papers on most aspects of phase transitions in condensed matter. Subscription Information ISSN: 0141-1594 • 4 issues per volume • Gordon and Breach Science Publications

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