Magnetic order in quasi-two-dimensional molecular magnets investigated with muon-spin relaxation
Andrew J. Steele, Tom Lancaster, Stephen J. Blundell, Peter J. Baker, Francis L. Pratt, Chris Baines, Marianne M. Conner, Heather I. Southerland, Jamie L. Manson, John A. Schlueter
MMagnetic order in quasi–two-dimensional molecular magnetsinvestigated with muon-spin relaxation
A. J. Steele, T. Lancaster, S. J. Blundell, P. J. Baker, F. L. Pratt, C.Baines, M. M. Conner, H. I. Southerland, J. L. Manson, and J. A. Schlueter Oxford University Department of Physics, Clarendon Laboratory,Parks Road, Oxford. OX1 3PU, United Kingdom ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory,Harwell Science and Innovation Campus, Didcot, Oxfordshire. OX11 0QX, United Kingdom Paul Scherrer Institut, Laboratory for Muon-Spin Spectroscopy, CH-5232 Villigen PSI, Switzerland. Department of Chemistry and Biochemistry, Eastern Washington University, Cheney. WA 99004, USA Materials Science Division, Argonne National Laboratory. Argonne IL 60439, USA (Dated: September 28, 2018)We present the results of a muon-spin relaxation (µ + SR) investigation into magnetic orderingin several families of layered quasi–two-dimensional molecular antiferromagnets based on transitionmetal ions such as S = Cu bridged with organic ligands such as pyrazine. In many of these ma-terials magnetic ordering is difficult to detect with conventional magnetic probes. In contrast, µ + SRallows us to identify ordering temperatures and study the critical behavior close to T N . Combiningthis with measurements of in-plane magnetic exchange J and predictions from quantum Monte Carlosimulations we may assess the degree of isolation of the 2D layers through estimates of the effectiveinter-layer exchange coupling and in-layer correlation lengths at T N . We also identify the likelymetal-ion moment sizes and muon stopping sites in these materials, based on probabilistic analysisof the magnetic structures and of muon–fluorine dipole–dipole coupling in fluorinated materials. PACS numbers: 76.75.+i, 75.50.Xx, 75.10.Jm, 75.50.Ee
I. INTRODUCTION
The S = two-dimensional square-lattice quantumHeisenberg antiferromagnet (2DSLQHA) continues tobe one of the most important theoretical models incondensed matter physics . Experimental realizationsof the 2DSLQHA in crystals also contain an interac-tion between planes, so that the relevant model de-scribing the coupling of electronic spins S i gives riseto the Hamiltonian H = J X h i,j i xy S i · S j + J ⊥ X h i,j i z S i · S j , (1)where J ( J ⊥ ) is the strength of the in- (inter-) planecoupling and the first (second) summation is overneighbors parallel (perpendicular) to the 2D xy -plane.Any 2D model ( J ⊥ = 0) with continuous symmetrywill not show long-range magnetic order (LRO) for T > .However, layered systems approximating 2D models( J ⊥ = 0) will inevitably enjoy some degree of inter-layer coupling and this will lead to magnetic order,albeit at a reduced temperature due to the influenceof quantum fluctuations. Quantum fluctuations arealso predicted to reduce the value of the magnetic mo-ment in the ground state of the 2DSLQHA to around60% of its classical value , and this reduction is oftenseen in the ordered moments of real materials. In lay-ered materials that approximate the 2DSLQHA, themeasurement of the antiferromagnetic ordering tem-perature T N is often problematic due not only to thisreduction of the magnetic moment, but also to short-range correlations that build up in the quasi-2D lay-ers above T N . These correlations lead to a reduction in the size of the entropy change that accompaniesthe phase transition, reducing the size of the anomalyin the measured specific heat . We have shown in anumber of previous cases that muon-spin relaxation(µ + SR) measurements do not suffer from these effectsand therefore represent an effective method for detect-ing magnetic order in complex anisotropic systems .The rich chemistry of molecular materials allowsfor the design and synthesis of a wide variety ofhighly-tunable magnetic model systems . Magneticcenters, exchange paths and the surrounding molec-ular groups can all be systematically modified, al-lowing investigation of their effects on magnetic be-havior. In particular, the existence of different ex-change paths along different spatial directions can re-sult in quasi–low-dimensional magnetic behavior (i.e.systems with magnetic interactions constrained to actin a two-dimensional plane or along a one-dimensionalchain). Such systems have the potential to better ap-proximate low-dimensional models than many tradi-tional inorganic materials. In addition, these molecu-lar materials can have exchange energy scales of order J/k B ∼
10 K which are accessible with typical labo-ratory magnetic fields of B ∼
10 T allowing an addi-tional avenue for their experimental study. This con-trasts with typical inorganic low-dimensional systemswhere the exchange is found to be
J/k B ∼ B ∼ that a small XY -likeanisotropy exists in some molecular materials. Al-though J ⊥ is the decisive energy scale for the mag-netic ordering, this anisotropy has been shown to havean influence on the ordering temperature and deter-mines the shape of the low-field B – T phase diagram a r X i v : . [ c ond - m a t . s t r- e l ] A ug (a) (b)H C N O Figure 1. Bridging ligands used in the compounds de-scribed in this paper: (a) pyrazine (N C H , abbreviatedpyz); and (b) (ii) pyridine- N -oxide (C H NO, abbreviatedpyo). Dashed lines indicate where the ligands bond toother parts of molecular structures. of these systems .Several classes of molecular magnetic materialclosely approximate the 2DSLQHA model and inthis paper we report the results of µ + SR measure-ments performed on several such materials. Thesesystems are self-assembled coordination polymers,based around paramagnetic ions such as Cu , linkedby neutral bridging ligands and coordinating anionmolecules. Our materials are based on combinationsof three different ligands: (i) pyrazine (N C H , ab-breviated pyz) and (ii) pyridine- N -oxide (C H NO,abbreviated pyo), both of which are planar rings; and(iii) the linear bifluoride ion [(HF ) − ], which is boundby strong hydrogen bonds F ... H ... F. The pyz and pyoligands are shown in Fig. 1. Specifically, we investi-gate the molecular system [ M (HF )(pyz) ] X , where M = Cu is the transition metal cation and X − is one of various anions (e.g. BF − , ClO − , PF − etc.).We also report the results of our measurements onother quasi-2D systems. First, [Cu(pyz) (pyo) ] Y ,with Y − = BF − or PF − , in which pyo ligands bridgeCu(pyz) planes. Then, the quasi-2D non-polymericcompounds [Cu(pyo) ] Z , where Z − =BF − , ClO − orPF − are examined. We also investigate materials inwhich either Ni ( S = 1) or Ag ( S = ) form themagnetic species in the quasi-2D planes rather thanCu .This paper is structured as follows. In Sec. II weoutline the µ + SR technique and describe our exper-imental methods. The [Cu(HF )(pyz) ] X family ofmaterials is then discussed in Sec. III, where muondata are used to determine T N and critical parame-ters. Muon–fluorine dipole–dipole oscillations in theparamagnetic regime are found for these materialswhich we use, in conjunction with dipole field sim-ulations, to investigate possible muon sites and con- strain the copper moment. In Sec. IV we explorethe related 2D system [Cu(pyz) (pyo) ] Y . Sec. Vdetails measurements of [Cu(pyo) ] Z . Sec. VI ex-amines a highly two-dimensional silver-based molec-ular material, Ag(pyz) (S O ). Finally, data fromthe [Ni(HF )(pyz) ] X ( X − = PF − , SbF − ) family ofmolecular magnets is presented in Sec. VII. II. EXPERIMENTAL DETAILS
Zero-field (ZF) µ + SR measurements were made onpowder samples of the materials at the ISIS facil-ity, Rutherford Appleton Laboratory, UK using theMuSR and EMU instruments and the Swiss MuonSource (SµS), Paul Scherrer Institut, Switzerland us-ing the General-Purpose Surface-Muon (GPS) instru-ment and Low-Temperature Facility (LTF). For mea-surements at temperatures T ≥ . T < . . + SR experiment , spin-polarized positivemuons are stopped in a target sample. The posi-tive muons are attracted to areas of negative chargedensity and often stop at interstitial positions. Theobserved property of the experiment is the time evo-lution of the muon-spin polarization, the behaviourof which depends on the local magnetic field at themuon site. Each muon decays with an average life-time of 2 . N F ( t ) and N B ( t ) recordthe number of positrons detected in the two detectorsas a function of time following the muon implanta-tion. The quantity of interest is the decay positronasymmetry function, defined as A ( t ) = N F ( t ) − αN B ( t ) N F ( t ) + αN B ( t ) , (2)where α is an experimental calibration constant. Theasymmetry, A ( t ), is proportional to the spin polariza-tion of the muon ensemble.A muon spin will precess around the local mag-netic fields at its stopping site at a frequency ν = γ µ B/ π , where the muon gyromagnetic ratio γ µ =2 π × . − . In the presence of LRO in amaterial we often measure oscillations in A ( t ). Theseresult from a significant number of muons stoppingat sites with a similar internal field, giving rise to a A ( t )( % ) . . . . . . t ( µ s) T < T N T (cid:29) T N Figure 2. Example A ( t ) spectra with fits above ( T = 25 K)and below ( T = 0 .
35 K) the magnetic transition temper-ature T N = 1 . )(pyz) ]BF . Note the ap-proximately equal initial asymmetry, and the oscillationsand ‘ -tail’ observed in the ordered phase. The slow oscil-lation observed for T > T N is due to muon–fluorine dipole–dipole oscillations. coherent precession of the ensemble of muon spins.Since the spins precess in local magnetic fields directedperpendicularly to the spin polarization direction, wewould expect that, for a powder sample with a staticmagnetic field distribution, of the total spin com-ponents should precess and the remaining shouldbe non-relaxing. The non-relaxing third of muon-spin components give rise to the so-called -tail in A ( t ), whose presence therefore provides additional ev-idence for a static field distribution in powder sample.Taken together, these effects provide an unambigu-ous method for sensitively identifying a transition toLRO. An example of typical spectra above and be-low the magnetic ordering temperature is shown for[Cu(HF )(pyz) ]BF in Fig. 2; the oscillations above T N are characteristic of a quantum-entangled Fµ state(see Sec. III D). III. [Cu(HF )(pyz) ] X The synthesis of the [ M (HF )(pyz) ] X system represented the first example of the use of a bifluoridebuilding block to make a three-dimensional coordi-nation polymer. This class of materials possesses ahighly stable structure due to the exceptional strengthof the bifluoride hydrogen bonds. The structure of the[ M (HF )(pyz) ] X system comprises infinite 2D[ M (pyz) ] sheets which lie in the ab plane, with bi-fluoride ions (HF ) − above and below the metal ions,acting as bridges between the planes to form a pseu-docubic network. The X − anions occupy the body-center positions within each cubic pore. An examplestructure, for [Cu(HF )(pyz) ]PF , is shown in Fig.3. Samples are produced in polycrystalline form viaaqueous chemical reactions between MX salts andstoichiometric amounts of ligands. Preparation de-tails for the compounds are reported in Refs. 14–16.In this section we consider those materials where H C N F P Cu abc
Figure 3. The structure of [Cu(HF )(pyz) ]PF , as an ex-ample of the [ M (HF )(pyz) ] X series. Copper ions arejoined in a 2D square lattice by pyrazine ligands to formCu(pyz) sheets; the 2D layers are joined in the third di-mension by HF − groups, making a pseudocubic 3D struc-ture; and this structure is stabilised by a PF − anion atthe center of each cubic pore. For clarity, hydrogen atomsattached to pyrazine rings have been omitted, and onlyone PF − anion is shown. the M cations are Cu S = centers. It isthought that the magnetic behavior of these mate-rial results from the 3 d x − y orbital of the Cu at thecenter of each CuN F octahedron lying in the CuN plane so that the spin exchange interactions betweenneighboring Cu ions occur through the s -bondedpyz ligands . The interplane exchange through theHF bridges connecting two Cu ions should be veryweak as these bridges lie on the 4-fold rotational axisof the Cu 3 d x − y magnetic orbital, resulting in lim-ited overlap with the fluorine p z orbitals. Thereforeto a first approximation, the magnetic properties of[ M (HF )(pyz) ] X can be described in terms of a 2Dsquare lattice.Measurements for X − = BF − , ClO − and SbF − were made using the MuSR spectrometer at ISIS,whilst PF , AsF , NbF and TaF were measured us-ing GPS at PSI. A. Long-range magnetic order
The main result of our measurements on these sys-tems is that below a critical temperature T N , oscilla-tions in the asymmetry spectra A ( t ) are observed attwo distinct frequencies, for all materials in the series.This shows unambiguously that each of these materi- A ( t )( % ) (a)[Cu(HF )(pyz) ]BF ν ( M H z ) (b) λ ( M H z ) (c) A ( t )( % ) t ( µ s) (d)[Cu(HF )(pyz) ]ClO ν ( M H z ) T (K) (e) λ ( M H z ) T (K) (f)0.3 K0.3 K Figure 4. Data and the results of fits to Eq. (3) for [Cu(HF )(pyz) ] X magnets with tetrahedral anions X − . From leftto right: (a) and (d) show sample asymmetry spectra A ( t ) for T < T N along with a fit to Eq. (3); (b) and (e) showfrequencies as a function of temperature [no data points are shown for the second line because this frequency ν was heldin fixed proportion to the first, ν (see text)]; and (c) and (f) show relaxation rates λ i as a function of temperature. Inthe ν ( T ) plot, error bars are included on the points but in most cases they are smaller than the marker being used. Thesolid line representing ν in (b) and (c) corresponds to the filled circles in the third column of graphs [(c) and (f)] forthat component’s relaxation, λ , whilst the dashed line and unfilled circles correspond to ν and λ , respectively. Thefilled triangles correspond to the fast relaxation λ . als undergoes a transition to a state of LRO. Exampleasymmetry spectra are shown in the left-hand columnof Fig. 4 and Fig. 5. They were found to be best fittedwith a relaxation function A ( t ) = A (cid:2) p e − λ t cos(2 πν t + φ )+ p e − λ t cos(2 πP ν t + φ ) + p e − λ t (cid:3) + A bg e − λ bg t , (3)where A represents the contribution from thosemuons which stop inside the sample and A bg accountsfor a relaxing background signal due to those muonsthat stop in the silver sample holder or cryostat tails,or with their spin parallel to the local field. Of thosemuons which stop in the sample, p indicates theweighting of the component in an oscillating state withfrequency ν ; p is the weighting of a lower-frequencyoscillating state with frequency ν ; and p representsthe weighting of a component with a large relaxationrate λ . All parameters were initially left free to vary.The second frequency was found to vary with temper-ature in fixed proportion to ν via ν = P ν for eachmaterial. The parameter P was identified by fittingthe lowest-temperature A ( t ) spectra where Eq. (3)would be expected to most accurately describe thedata, and subsequently held fixed during the fittingprocedure. Phase factors φ i were also found to benecessary in some cases to obtain a reliable fit. Theparameters resulting from these fits are listed in Ta-ble I, and data with fits are shown in Figs. 4 and 5.We also note here that the discontinuous nature of the change in all fitted parameters and the form ofthe spectra at T N strongly suggest that these materi-als are magnetically ordered throughout their bulk.The frequencies and relaxation rates as a functionof temperature extracted from these fits are shown inthe central column of Figs. 4 and 5. The muon preces-sion frequency, which is proportional to the internalfield in the material, can be considered an effectiveorder parameter for the system. Consequently, fittingextracted frequencies as a function of temperature tothe phenomenological function ν ( T ) = ν (0) (cid:20) − (cid:18) TT N (cid:19) α (cid:21) β , (4)allows an estimate of the critical temperature and theexponent β to be extracted. Our results fit well with aprevious observation that the compounds divide nat-urally into two classes: those with tetrahedral anions X − = BF − , ClO − and those with octahedral anions X − = A F − . The tetrahedral compounds have lowertransition temperatures T N (cid:46) T N (cid:38) .This difference has been explained in terms ofdifferences in the crystal structure between the twosets of compounds. Firstly, the octahedral anions arelarger than their tetrahedral counterparts. Secondly,the pyrazine rings are tilted by differing amounts withrespect to the normal to the 2D layers: those in the A ( t )( % ) (a)[Cu(HF )(pyz) ]PF ν ( M H z ) (b) λ ( M H z ) (c) A ( t )( % ) (d)[Cu(HF )(pyz) ]AsF ν ( M H z ) (e) λ ( M H z ) (f) A ( t )( % ) (g)[Cu(HF )(pyz) ]SbF ν ( M H z ) (h) λ ( M H z ) (i) A ( t )( % ) (j)[Cu(HF )(pyz) ]NbF ν ( M H z ) (k) λ ( M H z ) (l) A ( t )( % ) t ( µ s) (m)[Cu(HF )(pyz) ]TaF ν ( M H z ) T (K) (n) λ ( M H z ) T (K) (o)1.6 K1.6 K0.4 K2.0 K2.3 K Figure 5. Example data and fits for [Cu(HF )(pyz) ] X magnets with octahedral anions X − . From left to right: (a),(d), (g), (j) and (m) show sample asymmetry spectra A ( t ) for T < T N along with a fit to Eq. (3); (b), (e), (h), (k) and(n) show frequencies as a function of temperature [no data points are shown for the second line because this frequency ν was held in fixed proportion to the first, ν (see text)]; and (c), (f), (i), (l) and (o) show relaxation rates λ i as afunction of temperature. In the ν ( T ) plot, error bars are included on the points but in most cases they are smaller thanthe marker being used. The solid line representing ν in (b), (e), (h), (k) and (n) corresponds to the filled circles in thethird column of graphs [(c), (f), (i), (l) and (o)] for that component’s relaxation, λ , whilst the dashed line and unfilledcircles correspond to ν and λ , respectively. octahedral compounds are significantly more upright.Since the Cu 3 d x − y orbitals point along the pyrazinedirections, these tilting angles might be expected,to first order, to make little difference to nearest- neighbor exchange because such rotation is about asymmetry axis as viewed from the copper site. How-ever, it may be that the different direction of the de-localized orbitals above and below the rings through X ν (MHz) ν (MHz) λ (MHz) p p p φ ( ◦ ) φ ( ◦ ) T N (K) β α J/k B (K) | J ⊥ /J | BF . . − . . . . × − ClO . . − . . . . × − PF . . − . . . . × − AsF . . − . . . . × − SbF . . − − . . . . × − NbF . . − . . . . . . . . )(pyz) ] X family. The first parameters shown relateto fits to Eq. (3), which allow us to derive frequencies at T = 0, ν i ; probabilities of stopping in the various classes ofstopping site, p i , in percent; and phases associated with fitting the oscillating components φ i . Then, the temperaturedependence of ν i is fitted with Eq. (4), extracting values for the Néel temperature, T N , critical exponent β and parameter α . Finally, the quoted J/k B is obtained from pulsed-field experiments , and the ratio of inter- to in-plane coupling, J ⊥ /J ,is obtained by combining T N and J with formulae extracted from quantum Monte Carlo simulations (see Sec. III B, andRef. 7). Dashes in the λ column for the SbF and TaF compounds indicate that no fast-relaxing component was usedto fit those data. Dashes in the J/k B and J ⊥ /J columns for NbF and TaF indicates a lack of pulsed-field data forthese materials. which exchange probably occurs, possibly in conjunc-tion with hybridization with the anion orbitals, resultin an altered next-nearest neighbor or higher-order in-teractions, changing the transition temperature.Within the tetrahedral compounds, the difference inthe weighting of the oscillatory component ( p + p )in X − = BF − and ClO − probably results from thedifficulty in fitting the fast-relaxing component. Evenwith little change in the size of the oscillations, anyerror assigning the magnitude of this component willaffect the proportion of the A ( t ) signal attributed tothem. This difficulty is partly due to the resolution-limited nature of ISIS arising from the pulsed beamstructure. In the octahedral compounds, we foundthat X − = SbF − and TaF − did not have a resolvablefast-relaxing component, and consequently p was setto zero during the fitting procedure. This is reflectedby dashes in the p and λ columns in Table I.The fact that two oscillatory frequencies are ob-served points to the existence of at least two mag-netically distinct classes of muon site. In general wefind that p ≈ p for these materials, making the prob-ability of occupying the sites giving rise to magneticprecession approximately equal. The weightings p , were found to be significantly less than the weighting p relating to the fast-relaxing site. This, in com-bination with the magnitude of the fast relaxation λ ( T = 0) (cid:38)
10 MHz, suggests that this term shouldnot be identified with the -tail which results frommuons with spins parallel to their local field. (If thatwere the case then we would expect ( p + p ) /p = 2,which we do not observe.) It is likely that each ofthe components, p , p and p , therefore reflect theoccurence of a separate class of muon site in this sys-tem. We investigate the possible positions of thesethree classes of site in Sec. III E.The temperature evolution of the relaxation rates λ i is shown in the right-hand columns of Figs. 4and 5. In the fast-fluctuation limit, the relaxationrates are expected to vary as λ ∝ ∆ τ , where∆ = q γ h ( B − B ) i is the second moment of the local magnetic field distribution (whose mean is B )in frequency units, and τ is the correlation time. Inall measured materials, the relaxation rate λ , cor-responding to the higher oscillation frequency, startsat a small value at low temperature and increases as T N is approached from below. This is the expectedtemperature-dependent behavior and most likely re-flects a contribution from critical slowing down of fluc-tuations near T N (described e.g. in Ref. 18). In con-trast, the relaxation rate λ (associated with the lowerfrequency) starts with a higher magnitude at low tem-perature and decreases smoothly as the temperature isincreased. This is also the case for the relaxation rate λ of the fast-relaxing component. This smooth de-crease of these relaxation rates with temperature hasbeen observed previously in magnetic materials and seems to roughly track the magnitude of the localfield. It is possible that muon sites responsible for λ and λ lie further from the 2D planes than those sitesgiving rise to λ , and are thus less sensitive to 2Dfluctuations, reducing the influence of any variationin τ . The temperature evolution of λ and λ mightthen be expected to be dominated by the magnitudeof ∆, which scales with the size of the local field andwould therefore decrease as the magnetic transition isapproached from below.The need for nonzero phases φ i has been identifiedin previous studies of molecular magnets , butnever satisfactorily explained. One possible explana-tion for these might be that the muon experiences de-layed state formation. However, we can rule out thesimplest model of this as the phases appear not to cor-relate with ν i . Such a correlation would be expectedsince a delay of t before entering the precessing statewould give rise to a component of the relaxation func-tion a i ( t ) = cos [2 πν i ( t + t )] = cos (2 πν i t + φ i ), with φ i ∝ ν i , which is not observed. This does not com-pletely rule out delayed state formation, as t couldbe a function of temperature (although this seems un-likely at these temperatures). Nonzero phases are alsosometimes observed when attempting to fit data withcosinusoidal relaxation functions from systems havingincommensurate magnetic structures. The phase thenemerges as an artifact of fitting, as a cosine with a π phase shift approximates the zeroth-order Bessel func-tion of the first kind J ( ωt ) which is obtained fromµ + SR of an incommensurately-ordered system .The Bessel function arises because the distribution offields seen by muons at sites is asymmetric. However,attempts to fit the data with a pair of damped Besselfunctions produced consistently worse fits than fitsto Eq. (3), suggesting that a simple incommensuratestructure is not a satisfactory explanation. It is alsopossible that several further magnetically-inequivalentmuon sites exist, resulting in multiple, closely-spacedfrequencies which give the spectra a more complexcharacter which is not reflected in the fitting function.The simpler relaxation function would then obtain abetter fit if the phase were allowed to vary. This hasbeen observed , for example, in LiCrO . A final pos-sibility is that the distribution of fields at muon sites isasymmetric for another reason, perhaps arising froma complex magnetic structure. This may give rise toa Fourier transform which is only able to be fittedwith phase-shifted cosines. However, the mechanismby which this would occur is unclear. B. Parametrizing exchange anisotropy
The extent to which these systems approximate the2DSLQHA can be quantified by comparing the tran-sition temperature T N to the exchange parameter J .The temperature T N can be extracted using µ + SR,whilst J can be obtained reliably from pulsed-fieldmagnetization measurements , heat capacity or mag-netic susceptibility.Mean-field theory predicts a simple relationship forthe ratio of the transition temperature T N and theexchange J given by k B T N J = 23 zS ( S + 1) , (5)where k B is Boltzmann’s constant, z is the number ofnearest neighbors and S is the spin of the magneticions. In the pseudocubic [Cu(HF )(pyz) ] X systems, S = and z = 6, and Eq. (5) yields k B T N /J = 3.However, the reduced dimensionality increases theprevalence of quantum fluctuations, depressing thetransition temperature and in [Cu(HF )(pyz) ]BF ,we find k B T N /J ≈ .
25, which is indicative of largeexchange anisotropy.Combining the experimental measures of T N and J with the results of quantum Monte Carlo (QMC) sim-ulations allows us to deduce the exchange anisotropy J ⊥ /J in the system . Specifically, QMC simulations for 2DSLQHA where 10 − ≤ J ⊥ /J ≤ J ⊥ J = e b − π ρ s /T N , (6)where ρ s is the spin stiffness and b is a numerical con-stant. For S = , the appropriate parameters are ρ s /J = 0 .
183 and b = 2 .
43. This expression allows abetter estimate of k B T N /J in a 3D magnet: evaluat-ing for J ⊥ /J = 1 yields k B T N /J = 0 .
95. This is lowerthan the crude mean-field estimate because mean-fieldtheory takes no account of fluctuations. Estimates of J for our materials, from pulsed magnetic field stud-ies except where noted, along with calculated J ⊥ /J ratios, are shown in the summary tables throughoutthis paper.Another method of parametrizing the exchangeanisotropy is to consider the predicted correlationlength of two-dimensional correlations in the layersat the temperature at which we observe the onsetof LRO. The larger this length, the better isolatedthe layers can be supposed to be. This can be es-timated by combining an analytic expression for thecorrelation length, in a pure 2DSLQHA , ξ , withquantum Monte Carlo simulations to obtain an ex-pression appropriate for 1 ≤ ξ /a ≤ , ξ a = 0 . . J/k B T " − . (cid:18) k B TJ (cid:19) + O (cid:18) k B TJ (cid:19) , (7)where a is the square lattice constant, and T is thetemperature. This formula yields ξ ( T N ) ≈ . a forthe mean-field model ( k B T N /J = 3), and ξ ( T N ) ≈ a for k B T N /J = 0 .
95 from quantum Monte Carlo simu-lations (i.e. Eq. (6) with J ⊥ /J = 1). By comparison,in [Cu(HF )(pyz) ]BF Eq. (7) gives ξ ( T N ) ≈ a ,showing a dramatic increase in the size of correlatedregions which build up in the quasi-2D layers beforethe onset of LRO. C. Nonmonotonic field dependence of T N Although we expect the interplane exchange cou-pling J ⊥ to have a large amount of control of the ther-modynamic properties of these materials, it may bethe case that single-ion anisotropies are also respon-sible for deviations in the behaviour of our materialsfrom the predictions of the 2DSLQHAF model. Inparticular, these anisotropies has been demonstratedto show a crossover to magnetic behaviour consistentwith the 2D XY model . It was recently reported that [Cu(HF )(pyz) ]BF exhibits an unusual non-monotonic dependence of T N as a function of appliedmagnetic field B [see Fig. 6(d)]. This behavior was ex-plained as resulting from the small XY -like anisotropyof the spin system in these systems. The physics of theunusual field-dependence then arises due to the dualeffect of B on the spins, both suppressing the ampli-tude of the order parameter by polarizing the spinsalong a given direction, and also reducing the phasefluctuations by changing the order parameter phasespace from a sphere to a circle. A more detailed ex-planation for the behavior reveals that the energyscales of the physics are controlled by a Kosterlitz–Thouless-like mechanism, along with the interlayer ex-change interaction J ⊥ . A ( t )( a r b . ) (a) B = 2 T T = 2 . A ( t )( a r b . ) . . . . t ( µ s) (b) B = 2 T T = 1 . . . . σ ( M H z ) . . . T (K) (c) B = 2 T B ( T ) . . T (K) (d) HCtheory µ + SR Figure 6. Sample TF µ + SR data measured for [Cu(HF )(pyz) ]BF in an applied field of 2 T are shown in (a) and(b). Data are shown in the ‘rotating reference frame’, rotating at γ µ × . σ with T , showinga magnetic transition at 1.98 K in 2 T. (d) The B – T phase diagram from Ref. 12 showing the nonmonotonic behaviourat low applied magnetic field. In the key, HC is heat capacity, theory represents the results of computational modelling,and µ + SR shows our results from TF measurements (see main text).
The measurement of the B – T phase diagram in[Cu(HF )(pyz) ]BF reported in Ref. 12 was made byobserving a small anomaly in specific heat. In or-der to test whether the phase boundary could be de-termined using muons, we carried out transverse-field(TF) µ + SR measurements using the LTF instrumentat SµS. In these measurements, the field is applied per-pendicular to the initial muon spin direction, causing aprecession of the muon-spins in the sum of the appliedand internal field directed perpendicular to the muon-spin orientation. Example TF spectra measured in afield of 2 T are shown in Fig. 6 (a) and (b). We findthat the spectra are well described by a function A ( t ) = A (0)e − σ t / cos(2 πνt + φ ) , (8)where the phase factor depends on the details ofthe detector geometry, and σ is proportional to thesecond moment of the internal field distribution via σ = γ h B i . Upon cooling through T N we see a largeincrease in σ , as shown in Fig. 6 (c). This approxi-mately resembles an order parameter, and we identifythe discontinuity at the onset of the increase with T N by fitting σ with the above- T N relaxation adding inquadrature to the additional relaxation present be-low the transition. The resulting point at T N ( B =2 T) = 1 . T = 1 . B = 1 . ± . + SR measurements possiblylie slightly lower in T than both that predicted by the-ory, and the line predicted on the basis of the specificheat measurements. The theoretical calculations use J/k B = 5 . J ⊥ /J = 2 . × − , whilst our esti-mates suggest J/k B = 6 . J ⊥ /J = 0 . × − .Performing these calculations for a purely 2D sys-tem results in the entire curve shifting to the left ,and consequently the leftward shift of our data pointsis consistent with our finding of increased exchangeanisotropy. It is clear that the TF µ + SR techniquemay be used in future to measure the B – T phase dia-gram and enjoys some of the same advantages it has inZF over specific heat and susceptibility in anisotropicsystems. D. Muon response for
T > T N Above T N , the character of the measured spectrachanges considerably and we observe lower-frequencyoscillations characteristic of the dipole–dipole interac-tion between muons and fluorine nuclei . The Cu electronic moments, which dominate the spectra for T < T N , are disordered in the paramagnetic regimeand fluctuate very rapidly on the muon time scale.They are therefore motionally narrowed from the spec-tra, leaving the muon sensitive to the quasi-static nu-clear magnetic moments.A muon and nucleus interact via the two-spinHamiltonianˆ H = X i>j µ γ i γ j (cid:126) π r [ S i · S j − S i · ˆ r ) ( S j · ˆ r )] , (9)where the spins S i,j with gyromagnetic ratios γ i,j areseparated by the vector r . This gives rise to a preces-sion of the muon spin, and the muon-spin polarization M X r µ–F (nm) p (%) σ (MHz) T (K)Cu BF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M (HF )(pyz) ] X , extractedfrom fitting data to Eq. (12). In addition to separationby metal ion, Cu compounds are grouped by the temper-ature at which the measurement was made: those com-pounds measured over a range of temperatures appear inboth sections of the table. along a quantization axis z varies with time as D z ( t ) = 1 N *X m,n |h m | σ q | n i| e i ω m,n t + q , (10)where N is the number of spin states, | m i and | n i areeigenstates of the total Hamiltonian ˆ H , σ q is the Paulispin matrix corresponding to the direction q , and hi q represents an appropriately-weighted powder average.The vibrational frequency of the muon–fluorine bondexceeds by orders of magnitude both the frequenciesobservable in a µ + SR experiment, and the frequencyappropriate to the dipolar coupling in Eq. (9); thebond length probed via these entangled states is thustime-averaged over thermal fluctuations. Fluorine isan especially strong candidate for this type of interac-tion firstly because it is highly electronegative causingthe positive muon to stop close to fluorine ions, andsecondly because its nuclei are 100% F, which has I = .Data were fitted to a relaxation function A ( t ) = A ( p e − λ F–µ t D z ( t ) + p e − σ t ) + A bg e − λ bg t , (11)where the amplitude fraction p ≈
70% reflects themuons stopping in a site or set of sites near to afluorine nucleus, which result in the observed oscil-lations D z ( t ); the weak relaxation of the muon spinsis crudely modelled by a decaying exponential. Thefraction p ≈
30% describes those muons stopping ina class of sites primarily influenced by the randomly-orientated fields from other nuclear moments, givingrise to a Gaussian relaxation with σ ≈ . D z ( t ) functions wereattempted, including that resulting from a simple Fµ A ( t )( % ) t ( µ s) |↑↑i , |↓↓i √ ( |↑↓i − |↓↑i ) √ ( |↑↓i + |↓↑i ) Figure 7. Data taken at T = 15 K (cid:29) T N = 1 . )(pyz) ]BF , showing Fµ oscillations along witha fit to Eq. (11). The inset shows the energy levels presentin a simple system of two S = spins, along with theallowed transitions. bond (previously observed in some polymers ) andthe better-known FµF complex comprising a muonand two fluorine nuclei in linear symmetric configu-ration, which is seen in many alkali fluorides . Thislatter model was also modified to include the possi-bilities of asymmetric and nonlinear bonds. Previ-ous measurements made in the paramagnetic regimeof [Cu(HF )(pyz) ]ClO suggested that the muonstopped close to a single fluorine in the HF group andalso interacted with the more distant proton. This in-teraction is dominated by the F–µ coupling and, forour fitting, the observed muon–fluorine dipole–dipoleoscillations were found to be well described by a sin-gle Fµ interaction damped by a phenomenological re-laxation factor. For such Fµ entanglement, the timeevolution of the polarization is described by D z ( t ) = 16 X j =1 u j cos ( ω j t ) , (12)where u = 2, u = 1 and u = 2. The frequen-cies ω j = jω d /
2, where ω d = µ γ µ γ F (cid:126) / π r , in which γ F = 2 π × . × MHz T − is the gyromagnetic ra-tio of a F nucleus , and r is the muon–fluorine sep-aration. These three frequencies arise from the threetransitions between the three energy levels present ina system of two entangled S = particles (see in-set to Fig. 7). The fact that the relaxation func-tion is similar in all materials in the series, includ-ing [Cu(HF )(pyz) ]ClO which is the only compoundstudied without fluorine in its anion, (the only differ-ence being a slight lengthening of the µ–F bond, andwith no significant change in oscillating fraction) sug-gests that the muon site giving rise to the Fµ oscilla-tions in all systems is near the HF bridging ligand.The temperature evolution of the Fµ signal wasstudied for T ≤
300 K in [Cu(HF )(pyz) ]BF and[Cu(HF )(pyz) ]ClO . In both cases, the dipole–0 A ( t )( % ) [Cu(HF )(pyz) ]BF (a) . . .
35 0 . .
25 0 . . ν d (MHz) (b) T ( K ) . . . (c) λ F − µ λ A ( t )( % ) [Cu(HF )(pyz) ]ClO (d) . . (e) T ( K ) . . . (f) λ F − µ λ A ( t )( % ) t ( µ s)[Cu(HF )(pyz) ]SbF (g) . . r F − µ (nm) (h) T ( K ) . . . λ (MHz) (i) λ F − µ Figure 8. (a) Muon–fluorine dipole–dipole oscillations in [Cu(HF )(pyz) ]BF over a temperature range 2 ≤ T ≤
300 K.Asymmetry spectra are displaced vertically so as to approximately align with the temperature scale on plots (b) and (c).Ticks on the y -axis of (a) denote 1% asymmetry. Plot (b) shows the fitted value for r F–µ as a function of temperature.The line shown is a fit to the low- T points with a T scaling law, Eq. (14). The upper x -axis shows values of thedipole frequency, ν d , which correspond to the lower x -axis values of r F–µ . Plot (c) shows fitted relaxation rates λ F–µ and λ , referring to the relaxation of the Fµ function D z ( t ) in Eq. (11), and the pure relaxation in Eq. (13). Shadedregions indicate temperatures where Q = 2 π λ F–µ /ω d >
2, roughly parametrizing the disappearance of the oscillations.(d), (e) and (f) follow (a), (b) and (c), but show data for [Cu(HF )(pyz) ]ClO . (g), (h) and (i) similarly, but for[Cu(HF )(pyz) ]SbF over the range 5 ≤ T ≤
250 K. The 5 K A ( t ) plot is omitted because the background is raisedsubstantially by approach to the transition to LRO. material r (nm) a (10 − nm K − )[Cu(HF )(pyz) ]BF . . )(pyz) ]ClO . . . . T scaling law, as Eq. (14). dipole oscillations disappear gradually in a tempera-ture range 150 (cid:46) T (cid:46)
250 K, with oscillations totallyabsent in the center of this range, followed by reap-pearing as temperature is increased further. Plots of A ( t ) spectra at a variety of temperatures are shownin Fig. 8 (a) and (d). The data were initially fit-ted to Eq. (11), with all parameters left free to vary.The temperature-evolution of the muon–fluorine bondlength, r F–µ , can be seen in Fig. 8 (b) and (e). Thespectra were also fitted with A ( t ) = A ( p e − λ t + p e − σ t ) + A bg e − λ bg t , (13)a sum of an exponential and a Gaussian relaxation,which might be expected to describe the data in theregion where the oscillations vanish. Both this relax-ation and that extracted from Eq. (11) are plotted inFig. 8 (c) and (f), labelled λ and λ F–µ respectively.This bond length appears to grow and then shrinkby nearly 20% over the 100 K range where the os-cillations fade from the spectra and reappear. Thisvariation is significantly larger than any variation incrystal lattice parameters which would be expected.Since the oscillations visibly disappear from the mea-sured spectra, results from fitting with an oscillatoryrelaxation function are artifacts of the fitting proce-dure: since the frequencies scale with 1 /r , increasingbond length together with the associated relaxationrate fits the data with a suppressed oscillatory signal.This can be approximately quantified by examiningthe ratio Q = 2 π λ F–µ /ω d , where a large value indi-cates that the function relaxes significantly before asingle Fµ oscillation is completed. The shaded re-gions in Fig. 8 show where Q >
2, which acts as anapproximate bound on where the parametrization inEq. (11) would be expected to fail. In the low- T re-gion where Q <
2, the bond lengths appear to scaleroughly as T , which has previously been observed influoropolymers . Parameters extracted from fittingto r F–µ = aT + r (14)are shown in Table III.The observation in these two samples of Fµ oscilla-tions which disappear and then reappear is puzzling.While we have not identified a definitive mechanism,we can probably rule out an electronically mediatedeffect since, for T (cid:29) T N , the Cu moment fluctuationswill be outside the muon time-window. An explana-tion could involve nearby nuclear moments, possiblyinfluenced by a thermally-driven structural distortionor instability. A similar study of [Cu(HF )(pyz) ]SbF is shownin Fig. 8 (g), (h) and (i). In this material, the os-cillations appear not to vanish over the temperaturerange studied, though we cannot rule out a brief dis-appearance at T ≈
200 K. Instead, the oscillationsshow an apparently monotonic increase in dampingwith temperature, and the fitted bond length doesnot follow Eq. (14). The shaded region in Fig. 8 (h)and (i) has no upper bound, though we cannot ruleout a constraint at
T >
250 K. The pure relaxation λ is omitted because there is no region where the Fµoscillations are sufficiently damped for Eq. (13) to bea good parametrization. E. Muon site determination
Combining the data measured above and below thetransition in these materials allows us to attempt toconstruct a self-consistent picture of possible muonsites. The observed dipole–dipole observations above T N suggest that at least one muon stopping site isnear a fluorine ion. We consider three classes of prob-able muon site: Class I sites near the fluorine ions inthe HF groups, Class II sites near the pyrazine rings,and Class III sites near the anions at the centre ofthe pseudocubic pores. Comparison of Tables I andII show that the dominant amplitude component for T > T N arises from dipole–dipole oscillatory compo-nent p e − λt D z ( t ) and from the fast-relaxing compo-nent p e − λ t for T < T N , and that these are compa-rable in amplitude. It is plausible therefore, to sug-gest that these two signals correspond to contribu-tions from the same Class I muon sites near the HF groups. Moreover, the analysis of the T > T N spec-tra in the previous section implies that this site lies r µ–F ≈ .
11 nm from an F in the HF groups. Theremainder of the signal (the oscillating fraction below T N and the Gaussian relaxation above) can also beidentified, suggesting that the sites uncoupled fromfluorine nuclei (Classes II and/or III) result in themagnetic oscillations observed for T < T N .We note further that the evidence from Fµ oscil-lations makes the occurence of Class III muon sitesunlikely. The fact that spectra observed for T > T N in the X − = ClO − material are nearly identical tothose in all other compounds, in which X contains flu-orine, suggest that the muons do not stop near the an-ions. Moreover, as discussed in Sec. IV and V below,we observe no Fµ oscillations in [Cu(pyz) (pyo) ] X ( Y − = BF − , PF − ) or [Cu(pyo) ](BF ) , suggest-ing that muons do not stop preferentially near thesefluorine-rich anions either. We therefore rule out theexistence of Class III muon sites and propose that themagnetic oscillations measured for T < T N most prob-ably arise due to Class II sites found near the pyrazineligands.Below T N , the measured muon precession frequen-cies allow us to determine the magnetic field at theseClass II muon sites via ν = γ µ B/ π . Simulating themagnetic field inside the crystal therefore allows us to2 . . . near anionsBF ClO PF near Fnear Onear C or N . . . f ( ν )( M H z − ) near HF . . . ν (MHz) near pyz ν B F ν P F ν B F ν P F Figure 9. Probability density functions of muon precessionfrequencies at positions close to likely muon stopping sitesin [Cu(HF )(pyz) ] X with Cu moments µ = µ B . Thegraphs show dipole fields near the fluorine or oxygen atomsin the negative anions; near the fluorine atoms in the bi-fluoride ligands; and near the carbon and nitrogen atoms,as a proxy for proximity to the pyrazine ring. The type ofline indicates the compound for which the calculation wasperformed. The muon site is constrained to be close toparticular atoms, indicated by line color. The shaded ar-eas indicate ranges of fitted frequencies as T →
0; two fre-quencies ν BF , represent those observed where X − = BF − ,and ν PF , those observed in the X − = PF − analogue. compare these B -fields with those predicted for likelymagnetic structures and may permit us to constrainthe ordered moment. For the case of our ZF measure-ments in the antiferromagnetic state, the local mag-netic field at the muon site B local is given by B local = B dipole + B hyperfine , (15)where B dipole is the dipolar field from magnetic ionslocated within a large sphere centred on the muonsite and B hyperfine the contact hyperfine field causedby any spin density overlapping with the muon wave-function. This spin density is difficult to estimate ac-curately, particularly in complex molecular systems,but it is probable for insulating materials such as thesethat the spin density on the copper ion is well localisedand so we ignore the hyperfine contribution in our . . . . f ( ν / µ )( µ B · M H z − ) ν/µ (MHz · µ − )(a) g ( µ | { ν i } )( µ − B ) . . . µ ( µ B )(b) Figure 10. Probability density functions for muons inthe putative oscillating sites near the pyrazine rings in[Cu(HF )(pyz) ]BF (a) for muon precession frequency ν assuming that the moment on the copper site µ = µ B ,created fom a histogram of dipole fields evaluated usingEq. (16) at points satisfying the constraints detailed inthe text; and (b) for moment on the copper sites giventhe frequencies actually observed, evaluated using the pdfsin (a) and Eq. (18). Lines represent trial magnetic struc-tures. All exhibit antiferromagnetic coupling through boththe bifluoride and pyrazine exchange paths, whilst (i) hascopper moments pointing at 45 ◦ to the pyrazine grid, (ii)has moments along one of the pyrazine grid directions ( a or b ), and (iii) has copper moments pointing along thebifluoride axis ( c ). analysis. The dipole field B dipole is a function of thecoordinate of the muon site r µ , and comprises a vec-tor sum of the fields from each of the magnetic ionsin the crystal approximated as a point dipole, so that B dipole ( r µ ) = µ X i µ π r [3( ˆ µ i · ˆ r )ˆ r − ˆ µ i ] , (16)where r = r i − r µ is the relative position of the muonand the i th ion with magnetic moment µ i = µ ˆ µ i , and i is an index implying summation over all of the ionswhich make up the crystal.Although these materials are known to be anti-ferromagnetic from their negative Curie–Weiss tem-peratures and zero spontaneous magnetization at lowtemperatures , their magnetic structures are un-known. Dipole field simulations were therefore per-formed for a variety of trial magnetic structures with µ = µ B . We analyse the results of these calculationsusing a probabilistic method. We begin by allow-ing the possibility that the magnetic precession sig-nal could arise from any of the possible classes ofmuon site identified above. Random positions in the3unit cell were generated and dipole fields calculatedat these. To prevent candidate sites lying too close toatoms we constrain all sites such that r µ– A > . A is any atom. Possible Class I muon siteswere identified with r µ–F = r ± .
01 nm (where r isthe muon–fluorine distance established from Fµ os-cillations) and possible Class II sites were selectedwith the constraint that 0 . ≤ r µ–C,N ≤ .
12 nm.The predicted probability density function (pdf) ofmuon precession frequencies (resulting from the mag-nitudes of the calculated fields) are plotted in Fig. 9,with the observed frequencies superimposed. Resultsare shown for a trial magnetic structure compris-ing copper spins lying in the plane of the pyrazinelayers and at 45 ◦ to the directions of the pyrazinechains, and with spins arranged antiferromagneticallyboth along those chains and along the HF groups.This candidate structure is motivated by analogy with[Cu(pyz) ](ClO ) , which also comprises Cu ions inlayers of 2D pyrazine lattices , and with the parentphases of the cuprate superconductors , which arealso two-dimensional Heisenberg systems of S = Cu ions. Other magnetic structures investigatedgive qualitatively similar results. From Fig. 9 it isclear that the only sites with significant probabilitydensity near to the observed frequencies are those ly-ing near the anions (i.e. Class III sites) which we haveargued are not compatible with our data. The moreplausible muon sites correspond to higher frequenciesthan those observed. Our conclusion is that it is likelythat the Cu moments are rather smaller than the µ B assumed in the initial calculation.If we accept that the muon sites giving rise to mag-netic precession are near the pyrazine groups then wemay use this calculation to constrain the size of thecopper moment. Since ν is obtained from experiment,what we would like to know is g ( µ | ν ), the pdf of coppermoment µ given the observed ν . This can be obtainedfrom our calculated f ( ν/µ ) using Bayes’ theorem ,which yields g ( µ | ν ) = µ f ( ν/µ ) ´ µ max µ f ( ν/µ ) d µ , (17)where we have assumed a prior probability for the cop-per moment that is uniform between zero and µ max .We take µ max = 2 µ B , although our results are in-sensitive to the precise value of µ max as long as itis reasonably large. When multiple frequencies { ν i } are present in the spectra, it is necessary to multiplytheir probabilities of observation in order to obtainthe chance of their simultaneous observation, so weevaluate g ( µ |{ ν i } ) ∝ Y i ˆ ν i +∆ ν i ν i − ∆ ν i f ( ν i /µ ) d ν i , (18)where ∆ ν i is the error on the fitted frequency. Re-sults are shown in Fig. 10, along with the dipole fieldpdfs which gave rise to them. By inspection of thepdfs, the copper moment is likely to be µ (cid:46) . µ B . B C N O F Cu a bc
Figure 11. Structure of [Cu(pyz) (pyo) ](BF ) . Cop-per ions lie in 2D square layers, bound by pyrazine rings.Pyridine- N -oxide ligands protrude from the coppers ina direction approximately perpendicular to these layers.Tetrafluoroboride ions fill the pores remaining in the struc-ture. Ion sizes are schematic; copper ions are shown twiceas large for emphasis, and hydrogens have been omittedfor clarity. The dipole field simulations results also lend weightto our contention that the oscillatory signal cannotarise from the sites that also lead to the Fµ compo-nent above- T N . If this were the case then the mostlikely moment on the copper would be µ Cu (cid:46) . µ B ,which seems unreasonably small. We note that mo-ment sizes of µ (cid:46) . µ B were also observed for the2DSLQHA system La CuO (a recent estimate fromneutron diffraction gave [0 . ± . µ B ), despite thepredictions of 0 . µ B from spin wave theory and Quan-tum Monte Carlo . It was suggested in that case that disorder might play a role in reducing the mo-ment sizes; an additional possible mechanism for thissuppression is ring exchange .One limitation of this analysis is that the mecha-nism for magnetic coupling of copper ions through thepyrazine rings is postulated to be via spin exchange,in which small magnetic polarisations are induced onintervening atoms . Density functional theory calcu-lations estimate that these are small, with the nitro-4 A ( t )( % ) (a)[Cu(pyz) (pyo) ](BF ) . . . . ν ( M H z ) (b) λ ( M H z ) (c) A ( t )( % ) . . . . t ( µ s) (d)[Cu(pyz) (pyo) ](PF ) . . . . ν ( M H z ) T (K) (e) λ ( M H z ) T (K) (f)0.1 K0.2 K Figure 12. Example data and fits for [Cu(pyz) (pyo) ] Y . From left to right: (a) and (d) show sample asymmetryspectra A ( t ) for T < T N along with a fit to Eq. (19); (b) and (e) show the frequency ν as a function of temperature; and(c) and (f) show relaxation rates λ i as a function of temperature. The relaxation rates λ , associated with the oscillation,and λ , the fast-relaxing initial component, do not vary significantly with T and are not shown. Only a slight trend in λ in the Y − = PF − material [graph (f)] is observed. gen and carbon moments estimated at µ C ≈ . µ B and µ N ≈ . µ B , respectively . However, their ef-fect may be non-negligible: they may be significantlycloser to the muon site than a copper moment, anddipole fields fall off rapidly, as 1 /r . Further, sincemuch of the electron density in a pyrazine ring is de-localised in π -orbitals, the moments may not be point-like, as assumed in our dipole field calculations. Fur-ther, this may lead to overlap of spin density at themuon site and result in a nonzero hyperfine field. IV. [Cu(pyz) (pyo) ] Y In this section, we report the magnetic behaviorof another family of molecular systems which showsquasi-2D magnetism, but for which the interlayergroups are very different and arranged in a com-pletely different structure, resulting in a 2D coordi-nation polymer. This system is [Cu(pyz) (pyo) ] Y ,where Y − = BF − , PF − . As with the previous case, S = Cu ions are bound in a 2D square lattice of[Cu(pyz) ] sheets lying in the ab -plane. Pyridine- N -oxide (pyo) ligands [shown in Fig. 1 (b)] protrude fromthe copper ions along the c -direction, perpendicularto the ab -plane in the Y − = PF − material, but mak-ing an angle β − ≈ ◦ with the normal in Y − =BF − . The anions then fill the pores remaining in thestructure. The structure of [Cu(pyz) (pyo) ](BF ) isshown in Fig. 11.In a typical synthesis, an aqueous solution of Cu Y hydrate ( Y − = BF − or PF − ) was combined with anethanol solution that contained a mixture of pyrazine and pyridine- N -oxide or 4-phenylpyridine- N -oxide.Deep blue-green solutions were obtained in each case,and when allowed to slowly evaporate at room temper-ature for a few weeks, dark green plates were recoveredin high yield. Crystal quality could be improved by se-quential dilution and collection of multiple batches ofcrystals from the original mother liquor. The relativeamounts of pyz and pyo were optimized in order toprevent formation of compounds such as Cu Y (pyz) or [Cu(pyo) ] Y .Samples were measured in the LTF apparatus atSµS. Example data measured on [Cu(pyz) (pyo) ] Y are shown in Fig. 12, where we observe oscillations in A ( t ) at a single frequency below T N . Data were fittedto a relaxation function A ( t ) = A (cid:0) p cos(2 π ν t )e − λ t + p e − λ t + p e − λ t (cid:1) + A bg . (19)The small amplitude fraction p <
10% for both sam-ples refers to muons stopping in a site or set of siteswith a narrow distribution of quasi-static local mag-netic fields, giving rise to the oscillations; p ≈
50% isthe fraction of muons stopping in a class of sites givingrise to a large relaxation rate 30 (cid:46) λ (cid:46)
60 MHz and p ≈
50% represents the fraction of muons stoppingin sites with a small relaxation rate λ ≈ φ = 0, andit is thus omitted from this expression. Frequenciesobtained from fitting the data to Eq. (19) were thenmodelled with Eq. (4). The results of these fits areshown Fig. 12, and Table IV.Our results show that [Cu(pyz) (pyo) ](BF ) hasa transition temperature T N = 1 . ± . ν ( T =50) = 1 . ± . ν show a significant trend in the temperature region0 . ≤ T ≤ . β = 0 . ± .
10, where the large uncer-tainty results in part from the difficulty in fitting the A ( t ) data in the critical region.Our results for [Cu(pyz) (pyo) ](PF ) show thatthe transition temperature is slightly higher at T N =1 . ± .
02 K and the oscillations occur at a lowerfrequency of ν ( T = 0) = 1 . ± .
03 MHz. The re-laxation rates λ and λ also decrease in magnitude astemperature is increased, settling on roughly constantvalues λ ≈ . λ ≈
15 MHz for
T > T N .No other quantities show a significant trend in thetemperature region 0 . ≤ T ≤ . β = 0 . ± . (pyz), where there is also a rel-atively small precessing fraction of muons and littlevariation in relaxation rates as T N is approached frombelow . V. [Cu(pyo) ] Z The next example is not a coordination polymer,but instead forms a three-dimensional structure ofpacked molecular groups. The molecular magnet[Cu(pyo) ] Z , where Z − = BF − , ClO − , PF − , com-prises Cu ions on a slightly distorted cubic lattice,located in [Cu(pyo) ] complexes, and surrounded byoctahedra of oxygen atoms . The structure is shownin 13. This approximately cubic structure, whicharises from the molecules’ packing, might suggest thata three-dimensional model of magnetism would be ap-propriate. In fact, although the observed bulk prop-erties of [ M (pyo) ] X where M = Co , Ni orFe are largely isotropic, but the copper analoguesdisplay quasi–low-dimensional, S = Heisenberg an-tiferromagnetism . Weakening of superexchange incertain directions, and thus the lowering of the sys- ab c B C N O F Cu
Figure 13. Structure of [Cu(pyo) ](BF ) , viewed alongthe three-fold ( c -) axis, after Ref. 44. Copper ions aresurrounded by octahedra of six oxygens, each part of apyridine- N -oxide ligand; [Cu(pyo) ] complexes space-pack with BF − stabilising the structure. Ion sizes areschematic; copper ions are shown twice as large for em-phasis, and hydrogens have been omitted for clarity. Asindicated in the top left, the a and b directions lie in theplane of the paper, separated by γ = 120 ◦ , whilst the c direction is out of the page. tems’ effective dimensionality, is attributed to length-ening of the superexchange pathways resulting fromJahn–Teller distortion of the Cu–O octahedra, whichis observed in structural and EPR measurements .At high temperatures, ( T (cid:38)
100 K), these distortionsare expected to be dynamic but, as T is reduced (to ≈
50 K), they freeze out. The anion Z − determinesthe nature of the static Jahn–Teller elongation. The Z − = BF − material displays ferrodistortive order-ing which, in combination with the antiferromagneticexchange, gives rise to 2D Heisenberg antiferromag-netic behavior . By contrast, Z − = ClO − , NO − (neither of which is investigated here) display antifer-rodistortive ordering , which gives rise to quasi-1DHeisenberg antiferromagnetism . All of the samplesinvestigated were measured in the LTF spectrometerat SµS.6 A ( t )( % ) . . . . . t ( µ s) (a)[Cu(pyo) ](BF ) . . ν ( M H z ) . . T (K) (b) . . . . . λ ( M H z ) . . T (K) (c)0.15 K1 K Figure 14. Example data and fits for [Cu(pyo) ](BF ) . From left to right: (a) shows sample asymmetry spectra A ( t )for T < T N and T > T N , along with fits to Eq. (19) and Eq. (20), respectively; (b) shows frequency as a functionof temperature; and (c) shows relaxation rate λ as a function of temperature; λ was held fixed during the fittingprocedure, and λ persists for T > T N . In the ν ( T ) plot, error bars are included on the points but in most cases theyare smaller than the marker being used. In the Z − = BF − compound, below a temperature T N , a single oscillating frequency is observed, indi-cating a transition to a state of long-range magneticorder. Example data above and below the transition,along with fits, are shown in Fig. 14 (a). Data werefitted to Eq. (19), and the frequencies extracted fromthe procedure fitted as a function of temperature toEq. (4)as shown in Fig. 14 (b). This procedure iden-tifies a transition temperature T N = 0 . ± .
005 K.The only other parameter found to vary significantlyin the range 20 mK ≤ T ≤ T N was λ , shownin as shown in Fig. 14 (c). The fitted parametersare shown in Table IV. We may compare this withthe result of an earlier low-temperature specific heatstudy which found a very small λ -point anomaly at T N = 0 . ± .
01 K, slightly lower than our result. Fit-ting the magnetic component of the heat capacity withthe predictions from a two-dimensional Heisenbergantiferromagnet gives
J/k B = − . ± .
02 K, andsimilar analysis of the magnetic susceptibility yields J/k B = − . ± .
03 K. Using these values, togetherwith the muon estimate of T N and Eq. (6), allows us toestimate the inter-plane coupling, J ⊥ /J = 0 . ± . T N from heat capacity results inan estimate J ⊥ /J = 0 . ± . T N < T ≤ A ( t ) = A ( p e − λt + p e − σ t ) + A bg , (20)comprising an initial fast-relaxing component with λ ≈
20 MHz, and a Gaussian relaxation with σ ≈ . Z − = ClO − material also shows evidence for amagnetic transition, although in this case we do notobserve oscillations in the muon asymmetry. Insteadwe measure a discontinuous change in the relaxationwhich seems to point towards an ordering transition.Example asymmetry spectra are shown in Fig. 15 (a)and data at all measured temperatures are well de- scribed with the relaxation function A ( t ) = A e − λt + A bg . (21)Evidence for a magnetic transition comes from thetemperature evolution of λ [Fig. 15(b)], where wesee that the relaxation decreases with increasing tem-perature until it settles at T ≈ . λ ≈ . T N = 0 . ± .
01 K.The final member of this family studied, Z − = PF − ,shows no evidence for a magnetic transition over therange of temperatures studied, 0 .
02 K ≤ T ≤ and ClO compounds and it therefore seems likelythat the paramagnetic state persists to the lowest tem-perature measured. VI. Ag(pyz) (S O ) The examples so far have used Cu (3d ) as themagnetic species. An alternative strategy is to employAg (4d ) which also carries an S = moment. Thisidea has led to the synthesis of Ag(pyz) (S O ), whichcomprises square sheets of [Ag(pyz) ] units spacedwith S O − anions . Each silver ion lies at the cen-tre of an elongated (AgN O ) octahedron, where theAg–N bonds are significantly shorter than the Ag–O.Preparation details can be found in Ref. 49.Measurements were made using the GPS instru-ment at SµS. Example muon data, along with fits tovarious parameters, are shown in Fig. 17. Asymmetryoscillations are visible in spectra taken below a tran-sition temperature T N . The data were fitted with arelaxation function A ( t ) = A (cid:0) p cos(2 π ν t + φ )e − λ t + p e − λ t (cid:1) + A bg , (22)7 A ( t )( % ) t ( µ s) (a)[Cu(pyo) ](ClO ) λ ( M H z ) . . . T (K) (b)20 mK 1 K Figure 15. Example data and fits for [Cu(pyo) ](ClO ) . Representative A ( t ) spectra for T < T N and T > T N , alongwith fits to Eq. (21), are shown in (a), whilst the value of the relaxation rate, λ , as a function of temperature is shownin (b). comprising a single damped oscillatory component, aslow-relaxing component, and a static background sig-nal. The onset of increased relaxation λ as the tran-sition is approached from below leads to large statisti-cal errors on fitted values, as is evident in Fig. 17 (c).The relaxation λ decreases with increasing tempera-ture. Fitting to Eq. (4) allows the critical parameters β = 0 . ± .
02 and T N = 7 . ± . J is too large to be deter-mined with pulsed fields : M ( B ) does not saturatein fields up to 64 T. However, fitting χ ( T ) data al-lows an estimate of the exchange J/k B ≈
53 K (andthus, in conjunction with g measured by EPR, thesaturation field B c is estimated to be 160 T). Thus, k B T N /J = 0 . ± . | J ⊥ /J | ∼ − , but thisvery small ratio of ordering temperature to exchangestrength is outside the range in which the equationis known to yield accurate results. The alternativemethod of parametrizing the low dimensionality interms of correlation length at the Néel temperature(see Sec. III B) yields ξ ( T N ) /a = 1000 ± A ( t )( % ) t ( µ s)[Cu(pyo) ](PF ) Figure 16. Sample asymmetry spectrum for[Cu(pyo) ](PF ) measured at T = 0 .
02 K. Thespectra remain indistinguishable from this across therange of temperatures examined, 0 .
02 K ≤ T ≤ VII. [Ni(HF )(pyz) ] X In order to investigate the influence of a dif-ferent spin state on the magnetic cation in the[ M (HF )(pyz) ] X architecture the [Ni(HF )(pyz) ] X ( X − = PF − , SbF − ) system has been synthesised .These materials are isostructural with the copper fam-ily discussed in Sec. III, but contain S = 1 Ni cations.Data were taken using the GPS spectrometer atPSI. Example data are shown in Fig. 18. We observeoscillations at two frequencies below the materials’ re-spective ordering temperatures. Data were fitted witha relaxation function A ( t ) = A (cid:2) p e − λ t cos(2 πν t ) + p e − λ t cos(2 πP ν t )+ p e − λ t (cid:3) + A bg e − λ bg t . (23)Of those muons which stop in the sample, p ≈
25% indicates the fraction of the signal correspond-ing to the low-frequency oscillating state with ν ( T =0) ≈ . p ≈
10% corresponds to muonsstopping in the high-frequency oscillating state with ν ( T = 0) ≈ . p ≈
65% representsmuons stopping in a site with a large relaxation rate λ ( T = 0) ≈
70 MHz. The frequencies were ob-served to scale with one-another, and consequentlythe second frequency was held in fixed proportion ν = P ν during the fitting procedure. The onlyother parameter which changes significantly in valuebelow T N is λ , which decreases with a trend qualita-tively similar to that of the frequencies. Fitting theextracted frequencies to Eq. (4) allows the transitiontemperature T N = 12 . ± .
03 K and critical exponent β = 0 . ± .
04 to be extracted. In contrast to thecopper family studied in Sec. III, the relation λ ∝ ν holds true, suggesting that a field distribution whosewidth diminishes with increasing temperature is re-sponsible for the variation in λ , and that dynamicsare relatively unimportant in determining the muonresponse. This is shown graphically in the inset toFig. 18 (f), where a plot of frequency against relax-ation rate lies on top of a line representing a λ = ν relationship. The phase φ required in previous fits8 A ( t )( % ) t ( µ s) (a)Ag(pyz) (S O ) . . . . . . . ν ( M H z ) T (K) (b) λ ( M H z ) T (K) (c)1.7 K Figure 17. Example data and fits for Ag(pyz) (S O ). From left to right: (a) shows sample asymmetry spectra A ( t )for T < T N along with a fit to Eq. (22); (b) shows the frequency as a function of temperature; and (c) shows relaxationrates λ i as a function of temperature. Filled circles show the relaxation rate λ , which relaxes the oscillation. The filledtriangles correspond to the relaxation λ .material ν (MHz) p p p T N (K) β α J/k B (K) | J ⊥ /J | [Cu(pyz) (pyo) ](BF ) . <
10 50 50 1 . . . (pyo) ](PF ) . <
10 50 50 1 . . . . × − [Cu(pyo) ](BF ) . . . . . † . (S O ) 2 . . . . ∼ − Table IV. Fitted parameters for molecular magnets in the family [Cu(pyz) (pyo) ] Y , [Cu(pyo) ](BF ) andAg(pyz) (S O ). The first parameters shown relate to fits to Eq. (19) (for the first three rows) or Eq. (22) [forAg(pyz) (S O )]. This allows us to derive frequencies at T = 0, ν i ; and probabilities of stopping in the various classesof stopping site, p i , in percent. Then, the temperature dependence of ν i is fitted with Eq. (4), extracting values forthe Néel temperature, T N , critical exponent β and parameter α . Finally, the quoted J/k B is obtained from pulsed-fieldexperiments The asterisk (*) indicates the value of J was obtained using g ab = 2 . † ) indicates that the value of J is extracted from heat capacity and susceptibility fromRef. 45.). The ratio of inter- to in-plane coupling, J ⊥ /J , is obtained by combining T N and J with formulae extracted fromquantum Monte Carlo simulations (see Sec. III B, and Ref. 7). The dash in the p column for Ag(pyz) (S O ) reflectsthe fact that there is no third component in Eq. (22). Dashes in the J/k B and J ⊥ /J columns for [Cu(pyz) (pyo) ](BF ) indicate a lack of pulsed-field data for this material. (e.g. Eq. (3)) is not necessesary in fitting these spec-tra, and is set to zero. Fitted parameters are shownin Table V.Data for the X − = PF − compound was subject tosimilar analysis, fitting spectra below T N to Eq. (23),this time with p ≈ ν ( T = 0) ≈ . p ≈ ν ( T = 0) ≈ . p ≈ λ ( T = 0) ≈
100 MHz. The phase φ again provedunnecessary. These spectra do not show as sharp atransition as the X − = SbF − compound, with the os-cillating fraction of the signal decaying rather beforethe appearance of spectra whose different characterindicates clearly that the sample is above T N . Theavailable data do not allow reliable extraction of crit-ical parameters, but we estimate 5 . ≤ T N ≤ . . ≤ β ≤ .
4. Fitted values are shown in Ta-ble V.Another method to locate the transition is to ob-serve a transition in the amplitude of the muon spectraat late times to observe the transition as a functionof temperature from zero in the unordered state tothe ‘ -tail’ characteristic of LRO, described in Sec. II.Spectra were fitted with the simple relaxation function A ( t > A bg e − λ bg t , and then the amplitudes ob- tained fitted with a Fermi-like step function A ( t > , T ) = A + A − A e ( T − T mid ) /w + 1 , (24)which provides a method of modelling a smooth tran-sition between A = A ( T < T N ) and A = A ( T >T N ). The fitted amplitudes and Fermi function areshown in Fig. 19 (c). The fitted mid-point T mid =6 . ± . w = 0 . ± . T N , and so one would ex-pect that T N lies at the lower end of this transition.Thus, the µ + SR analysis suggests T N = 6 . ± . T mid − w ± w ). This is consistent with the esti-mate from ν i ( T ) and the value T N = 6 . .Members of the M = Ni family exhibit Fµ oscilla-tions rather like their copper counterparts, with a sim-ilar fraction of the muons in sites giving rise to dipole–dipole interactions. The results of these fits are shownalong with those from Cu compounds in Table II. Be-cause the nickel data were measured at temperaturesdifferent from those of the copper compounds and, asdescribed in Sec. III D, the muon–fluorine bond lengthin the compound is sensitive to changes in tempera-9 A ( t )( % ) (a)[Ni(HF )(pyz) ]PF ν ( M H z ) (b) λ ( M H z ) (c) A ( t )( % ) t ( µ s) (d)[Ni(HF )(pyz) ]SbF ν ( M H z ) T (K) (e) λ ( M H z ) T (K) (f) λ ( M H z ) ν (MHz)1.6 K1.8 K Figure 18. Example data and fits for M = Ni magnets. From left to right: (a) and (d) show sample asymmetry spectra A ( t ) for T < T N along with a fit to Eq. (3); (b) and (e) show frequencies as a function of temperature [no data pointsare shown for the second line because this frequency ν was held in fixed proportion to the first, ν (see text)]; and (c)and (f) show relaxation rate λ as a function of temperature. In the ν ( T ) plot, error bars are included on the points butin most cases they are smaller than the marker being used. The inset to (f) shows ν ≡ ν plotted against λ ≡ λ on alog–log scale. The black line shows λ = ν . X T N (K) ν (MHz) ν (MHz) λ (MHz) p p p β α PF . . . . . . . . . . M = Ni magnets. Errors shown are statistical uncertainties on fitting and thus representlower bounds. Errors on Ni...PF could not be estimated due to the fitting procedure (see text). ture, care must be taken when comparing these val-ues to those of the Cu family. Linearly interpolatingthe bond lengths for [Cu(HF )(pyz) ]SbF at 9 K and29 K to find an approximate value of the bond lengthat 19 K yields r µ–F ( T = 19 K) = 0 . ± . )(pyz) ]SbF , r µ–F ( T =19 K) = 0 . ± . )(pyz) ] X systems, the dimensionalityof these Ni variants is ambiguous. Susceptibilitydata fit acceptably to a number of models, and abinitio theoretical calculations are suggestive of one-dimensional behavior, dominated by the exchangealong the bifluoride bridges. This is discussed morefully in Ref. 16. VIII. DISCUSSION
Fig. 20 collects the results from this paper andshows how isolation between two-dimensional layers varies over a variety of systems; those presented inthis paper, molecular materials studied elsewhere, andinorganic materials. The primary axis is the experi-mental ratio T N /J . Also shown are the ratios of thein-plane and inter-plane exchange interactions, J ⊥ /J ,and the correlation length at the transition, ξ ( T N ) /a ,extracted from fits to quantum Monte Carlo simu-lations. Of the materials in this paper, the leastanisotropic is [Cu(pyo) ](BF ) , whose low transitiontemperature is caused by a small exchange constantrather than particularly high exchange anisotropy.The members of the [Cu(HF )(pyz) ] X family withoctahedral anions have k B T N /J ≈ .
33, rather morethan the k B T N /J ≈ .
25 shown by their counter-parts with tetrahedral anions (as has been notedpreviously ). This makes the latter comparableto highly 2D molecular systems [Cu(pyz) ](ClO ) and CuF (H O) (pyz), and the cuprate parent com-pound La CuO . Below this, [Cu(pyo) (pyz) ](PF ) exhibits a ratio k B T N /J ≈ . CuO Cl exhibits k B T N /J = 0 . ± . ξ ( T N ) /a = 280 ± (S O ), has k B T N /J = 0 . ± . A ( t > µ s ) ( % ) T (K) µ + S R ∆ T N S H C ∆ T N Figure 19. The asymmetry at late times A ( t > µ s) mea-sured in Ni(HF )(pyz) PF along with a fit to Eq. (24).Overlaid are regions corresponding to the range of valuesfound by fitting the frequency of oscillations as a functionof temperature with Eq. (4) (5 . ≤ T N ≤ . λ -like anomaly in specific heat capacity(SHC) measurements with a representative error of 0 . . ≤ T N ≤ . plying ξ ( T N ) /a = 1000 ± (S O ) being the best realization foundto date.Another method of examining the dimensionalityof these systems is to consider their behavior in thecritical region. The critical exponent β is a quantityfrequently extracted in studies of magnetic materials,and it is often used to make inferences about the di-mensionality of the system under study. In the criticalregion near a magnetic transition, an order parameterΦ, identical to the (staggered) magnetization, wouldbe expected to vary asΦ( T ) = Φ (cid:18) − TT N (cid:19) β . (25)In simple, isotropic cases, the value of β depends onthe dimensionality of the system, d , and that of theorder parameter, D . For example, in the 3D Heisen-berg model ( d = 3, D = 3), β = 0 . d = 2, D = 1), β = . Sincethe muon precession frequency is proportional to thelocal field, it is also proportional to the moment onthe magnetic ions in a crystal, and can be used as aneffective order parameter. However, Eq. (25) wouldonly be expected to hold true in the critical region.The extent of the critical region (defined as that re-gion where simple mean-field theory does not apply)can be parametrized by the Ginzburg temperature T G , ξ/a J ⊥ /J k B T N /J . . . . . . )(pyz) ]BF [Cu(HF )(pyz) ]ClO [Cu(HF )(pyz) ]PF [Cu(HF )(pyz) ]AsF [Cu(HF )(pyz) ]SbF [Cu(pyo) ](BF ) [Cu(pyo) (pyz) ](PF ) Ag(pyz) (S O )[Cu(pyz) ](ClO ) CuF (H O) (pyz)(5BAP) CuBr La CuO Sr CuO Cl − − − . − − − from this workother molecularinorganic systems Figure 20. Quantification of the two-dimensionality of thematerials in this paper and comparison with other no-table 2DSLQHA systems. Filled circles show materialsinvestigated in this paper; open circles show other molec-ular materials; and filled triangles show inorganic systems.The low dimensionality is parametrized firstly with the di-rectly experimental ratio T N /J , and then with predictionsquantum Monte Carlo simulations for both the exchangeanisotropy, J ⊥ /J [Eq. (6)], and the correlation length of anideal 2D Heisenberg antiferromagnet with the measured J at a temperature T N [Eq. (7)]. The greying-out of the axisfor J ⊥ /J indicates where Eq. (6) is extrapolated beyondthe range for which it was originally derived . Values of T N /J for the other materials were evaluated from Refs. 50–52. which is related to the transition temperature T c by | T G − T c | T c = "(cid:18) ξa (cid:19) d (cid:18) ∆ Ck B (cid:19) d − , (26)where d is the dimensionality, ξ is the correlationlength and ∆ C is the discontinuity in the heat ca-1 . . . . . β . . . k B T N /J from this work[Cu(HF )(pyz) ] X other molecularLandau3D Heisenberg3D XY3D Ising2D Ising Figure 21. Critical exponent β , as commonly ex-tracted from Eq. (4), plotted against the experimen-tal ratio k B T N /J , indicative of exchange anisotropy.Bright filled circles indicate [Cu(HF )(pyz) ] X materi-als examined in Sec. III, whilst darker circles indicateother materials studied in this work: [Cu(pyz) (pyo) ]BF (Sec. IV); [Cu(pyo) ](BF ) (Sec. V); Ag(pyz) (S O )(Sec. VI). Open circles indicate other molecular materials:CuF (H O) (pyz) (Ref. 54); [Cu(pyz) ](ClO ) (Ref. 8). pacity. Quantum Monte Carlo simulations suggest that ∆ C/k B ≈ J ⊥ /J . It follows that for d = 3, where∆ C/k B ≈ | T G − T c | /T c ≈ ( ξ/a ) − , giv-ing rise to a narrow critical region. In two dimen-sions, we have | T G − T c | /T c ≈ ( ξ/a ) − (∆ C/k B ) − .Anisotropic materials with small J ⊥ /J only show asmall heat capacity discontinuity, while ξ/a grows ac-cording to Eq. (7). This leads to a | T G − T c | /T c oforder 1 for our materials, that is, a larger critical re-gion for 2D (as compared to 3D) systems.The large critical region in these materials allowsmeaningful critical parameters to be extracted frommuon data. The simplest method of doing so is tofit the data to Eq. (4), as we have throughout thisstudy; alternatively, critical scaling plots can be used(e.g. Ref. 18), which we have performed, finding theresults are unchanged within error. Since β mightbe expected to give an indication of the dimensional-ity of the hydrodynamical fluctuations in these ma-terials, a comparison between extracted β and ex-change anisotropy parametrized by k B T N /J is shownin Fig. 21. Members of the [Cu(HF )(pyz) ] X familyshow some correlation between the critical exponentand the effective dimensionality but overall, the rela-tionship is weak. This is probably because β , which isnot a Hamiltonian parameter, is not simply a functionof the dimensionality of the interactions, but probesthe nature of the critical dynamics (including prop-agating and diffusive modes) which could differ sub-stantially between systems. IX. CONCLUSIONS
We have presented a systematic study of muon-spinrelaxation measurements on several families of quasitwo-dimensional molecular antiferromagnet, compris-ing ligands of pyrazine, bifluoride and pyridine- N -oxide; and the magnetic metal cations Cu , Ag and Ni . In each case µ + SR has been shown tobe sensitive to the transition temperature T N , whichis often difficult to unambiguously identify with spe-cific heat and magnetic susceptibility measurements.We have combined these measurements with predic-tions of quantum Monte Carlo calculations to identifythe extent to which each is a good realization of the2DSLQHA model. The critical parameters derivedfrom following the temperature evolution of the µ + SRprecession frequencies do not show a strong correla-tion with the degree of isolation of the 2D magneticlayers.The analysis of magnetic ordering in zero appliedfield in terms of inter-layer coupling J ⊥ presented heredoes not take into account the effect of single-ion–typeanisotropy on the magnetic order. This has been sug-gested to be important close to T N in several examplesof 2D molecular magnet where it causes a crossoverto XY -like behavior. In fact, its influence is con-firmed in the nonmonotonic B – T phase diagram seenin [Cu(HF )(pyz) ]BF . It is likely that this is onefactor that determines the ordering temperature of asystem, although, as shown in Ref. 11, it is a smallereffect than the interlayer coupling parametrized by J ⊥ . The future synthesis of single crystal samplesof these materials will allow the measurement of thesingle-ion anisotropies for the materials studied here.The presence of muon–fluorine dipole–dipole oscil-lations allows the determination of some muon sites inthese materials, although it appears from our resultsthat these are not those that lead to magnetic oscil-lations. However, the Fµ signal has been shown to beuseful in identifying transitions at temperatures wellabove the magnetic ordering transition, which appearto have a structural origin. The fluorine oscillationshamper the study of dynamic fluctuations above T N ,which often appear as a residual relaxation on top ofthe dominant nuclear relaxation. It may be possiblein future to use RF radiation to decouple the influ-ence of the fluorine from the muon ensemble to allowmuons to probe the dynamics.The muon-spin precession signal, upon which muchof the analysis presented here is based, is seen moststrongly in the materials containing Cu and is moreheavily relaxed in the Ni materials. This is likelydue to the larger spin value in the Ni-containing ma-terials. This is borne out by measurements on pyz-based materials containing Mn and Fe ions , whereno oscillations are observed, despite the presence ofmagnetic order shown unambiguously by other tech-niques. In the case of Mn-containing materials mag-netic order is found with µ + SR through a change inrelative amplitudes of relaxing signals due to a differ-erence in the nature of the relaxation on either side of2the transition. It is likely, therefore that muon studiesof molecular magnetic materials containing ions withsmall spin quantum numbers will be most fruitful inthe future.Finally, the temperature dependence of the relax-ation rates in these materials has been shown to bequite complex, reflecting the variety of muon sites inthese systems. In favourable cases these data could beused to probe critical behavior, such as critical slow-ing down, although the unambiguous identification ofsuch behavior may be problematic.Despite these limitations on the use of µ + SR in ex-amining molecular magnetic systems of the type stud-ied here, it is worth stressing that the technique stillappears uniquely powerful in providing insights intothe magnetic behavior of these materials and will cer-tainly be useful in the future as a wealth of new sys-tems are synthesised and the goal of microscopicallyengineering such materials is approached.
ACKNOWLEDGMENTS
This work was partly supported by the Engineer-ing and Physical Sciences Research Council, UK. Ex- periments at the ISIS Pulsed Neutron and MuonSource were supported by a beamtime allocation fromthe Science and Technology Facilities Council. Fur-ther experiments were performed at the Swiss MuonSource, Paul Scherrer Institute, Villigen, Switzerland.This research project has been supported by the Eu-ropean Commission under the 7 th Framework Pro-gramme through the ‘Research Infrastructures’ actionof the ‘Capacities’ Programme, Contract No: CP-CSA_INFRA-2008-1.1.1 Number 226507-NMI3. Thework at EWU was supported by the National ScienceFoundation under grant no. DMR-1005825. Worksupported by UChicago Argonne, LLC, Operator ofArgonne National Laboratory (‘Argonne’). Argonne,a US Department of Energy Office of Science labo-ratory, is operated under Contract No. DE-AC02-06CH11357. 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