Studies of a Large Odd‐Numbered Odd‐Electron Metal Ring: Inelastic Neutron Scattering and Muon Spin Relaxation Spectroscopy of Cr8Mn

Abstract The spin dynamics of Cr8Mn, a nine‐membered antiferromagnetic (AF) molecular nanomagnet, are investigated. Cr8Mn is a rare example of a large odd‐membered AF ring, and has an odd‐number of 3d‐electrons present. Odd‐membered AF rings are unusual and of interest due to the presence of competing exchange interactions that result in frustrated‐spin ground states. The chemical synthesis and structures of two Cr8Mn variants that differ only in their crystal packing are reported. Evidence of spin frustration is investigated by inelastic neutron scattering (INS) and muon spin relaxation spectroscopy (μSR). From INS studies we accurately determine an appropriate microscopic spin Hamiltonian and we show that μSR is sensitive to the ground‐spin‐state crossing from S=1/2 to S=3/2 in Cr8Mn. The estimated width of the muon asymmetry resonance is consistent with the presence of an avoided crossing. The investigation of the internal spin structure of the ground state, through the analysis of spin‐pair correlations and scalar‐spin chirality, shows a non‐collinear spin structure that fluctuates between non‐planar states of opposite chiralities.


Introduction
Paramagnetic cage complexes, sometimes called molecular nanomagnets( MNMs), have been extensivelys tudied [1] due to their interesting physics, and to the possibility of exploiting their magnetic behaviour in technological applications. [2] Cyclic compounds have been investigated since the early1 990s, beginning with studies of an Fe 10 ring, whichs howeds teps in the magnetisation on increasing the appliedm agnetic field. [3] Later work has included studies of the NØel vector [4] tunneling, observation of quantum fluctuationso ft otal spin at avoided crossings, [5] the first measurements of coherence times in molecularm agnets, [6] and the demonstration that these coherence times can be modifiedb yc hemistry. [7] It has also been possible to investigate spin dynamics at the atomics cale in such cyclic metalc ages using single-crystal inelastic neutron scattering (INS) studies. [8] The studiesl isted above all involve even-membered metal rings.W hen the magnetic exchange is antiferromagnetic (AF), as imple picturec an be used to describe the magnetism, with spins on individual metal centres aligneda lternatively up and down.T here are far fewer reports of large odd-numbered metalr ings, in whichs uch as implistic picturem ustf ail, and the physical studies reported are even more limited.E xamples includet he magnetisation studies of a( VO) 7 ring, [9] and ac yclic Fe 9 phosphonate. [10] Our own work has included investigations of aC r 8 Ni ring, which was visualiseda samagnetic Mçbius strip, [11] and later as ar are example of av alence-bond solid. [12] Cr 8 Ni hasa no dd-numbero fm etal ions in the ring, but has an even number of unpaired electrons giving rise to an S = 0s inglet ground state. In some definitions of spin frustration,f or example, that given by Kahn, [13] Cr 8 Ni cannotb ef rustratedd ue to its lack of ground-state degeneracy. In am ore recent work, Schnack has attempted to define frustration by considering whether the ground-state spin structure is bipartite (i.e.,a lternately spin up and down) or not. [14] With this definition Cr 8 Ni is frustrated. We have also proposed an alternative approach in work on slightly distorted Cr 9 rings, [15] in which we define frustration as Type 1-which is the strongest case, in which the ground state is degenerate( Kahn'sd efinition);T ype2-which is essentially Schnack'sd efinition, in which the spin ground state cannotb ed escribed by coupling spin sites classically, but in whichd egeneracy is not required; and Type 3, in which the ground state can be described by coupling spin sites classically,b ut in which competing antiferromagnetic interactions are necessary for the description of the system spin dynamics. The last case would then include an umber of claims of "frustration" from the chemical literaturet hat would not be allowed by either the Kahn or Schnack definitions.
Our new categorisation was based on studies of two chemically similar Cr 9 rings by INS and magnetometry. [15] One had two low-lying S = 1/2 states and the other had an S = 3/2 ground state, and hence they belong to spin frustration classi-ficationsT ype 2a nd Type 3, respectively (we use the convention that capital S referst ot he total spin state of am olecule, while lower-case s referst ot he spin of individual ions). Here we discusst he case of ah eterometallic nine metal ring, [H 2 N i (C 3 H 7 ) 2 ][Cr 8 MnF 9 (O 2 CtBu) 18 ]( 1;C r 8 Mn), which also has both an odd number of metal ions and an odd number of electrons. We describet he synthesis and structure of the compound, and derive the microscopics pin Hamiltonian and the energy spectrum of this molecule by meanso fm agnetisation and INS studies. We then use muon spin relaxation (mSR) spectroscopy to probe the nature of the spin ground state as af unction of appliedm agnetic field and to investigate the crossing of energy levels with different total-spin quantum numbers. mSR spectroscopy hasp reviously been shown to be an ideal tool for studying magnetic phase transitions and spin fluctuations. [16] Finally,w ee xploit the experimentally parameterised Hamiltonian model to investigate the ground state of Cr 8 Mn and identify that the internal spin structure fluctuates between opposite chiralities.
The reaction proceeds to give 1 in ar elatively good yield (42 %, based on Cr), but crystallisation presents an interesting problem.T he green compound can be crystallised readily from ap entane/toluene mixture, giving large hexagonal crystals of compound 1a.T hese crystalsd on ot diffract X-rays well, and the diffractionp attern is only sufficiently resolved to establish the formation of nine-membered metal rings, with these rings all co-planar and the plane of the rings perpendicular to the unique axes of the hexagonal crystals. Av ery similar problem was found for the equivalent Cr 8 Ni ring. [11a] By re-crystallisation of 1a using am ixture of EtOAc/MeCN compound 1 crystallises in P2 1 /n and as tructure of 1b was obtained at 30 K; refinement gave as tructure with an R factor of 8%.T he structure was initially refined with all metal sites as Cr;h owever,t he converged structure showsd istinct metric differences for one position that was then assigned as the Mn site, and the structurer efinedt oc onvergencea gain.
Further evidencef or the localisation of the Mn site comes from the position of the central ammonium cation;t he N1 atom of this cation is displaced from the centre of the ring towards the Mn site, for example, the NÀMn distance is 3.925(6) , compared with the NÀCr6 and NÀCr2 distances, which are 5.967(6) and5 .661(6) , respectively.T herea re NÀ H···F hydrogen bonds to both fluorides bound to the Mn site, with N···F distances of 2.763(6) and 2.860 (6) . The hydrogen bond to afluoride bridging between adivalent metal and atrivalent metal will be stronger than to af luoride bridgingt wo trivalent metals as the electron density on the fluorides hould be greater.T he next shortest N···F contact is 3.130(7),t oF 7.
The metal sites within the nonagon are not all in as ingle plane, with the mean deviation of the Ms ites from the plane being 0.183 . Three sites-Cr3, Cr4 and Cr2-lie very close to the mean plane (deviation < 0.1 ). In each case the two metals between these sites lie with one significantly above the plane and the other significantly below the plane, for example, Cr8 is 0.35 a bove the plane,w hile Cr5 is 0.31 b elow.I n octametallic Cr 7 Mr ings the metal octagoni sp lanar.I nt he octagonalr ings one pivalateo ne ach edgel ies in the plane of the metal ring, with the other either above or below,a lternating aroundt he ring. In 1 this arrangemento fc arboxylates is impossible as there are an odd number of edges.
The packing of the Cr 8 Mn rings is different betweenm onoclinic and the hexagonal form. In compound 1b the rings pack with the mean planeso fh alf the rings at 548 to the others;i n the hexagonalf orm, 1a,the rings are all co-planar.

Magnetic measurements and inelastic neutron scattering
Magnetic measurements were performed on compounds 1a and 1b and no difference was observed between the polymorphs; the magnetic properties are molecular and unaffected by the space group. At both 2a nd 4Kmagnetisation increases with applied magnetic field withoutr eaching saturation within the measured range up to 7T (Figure 2). At 300 K c m T equals 16 cm 3 Kmol À1 ,s lightly less than the calculated value of 18.78 cm 3 Kmol À1 for eight uncoupled s = 3/2 and one s = 5/2 spin with g Cr = 1.96 and g Mn = 2.0. On decreasingt emperature the molecular susceptibility (c m )s teadilyi ncreases before flattening off at around2 5K at av alue of around 0.17 cm 3 Kmol À1 ,i ndicating the presence of as ignificant antiferromagnetic exchange interaction. At lower temperatures, c m increases rapidly confirming the expected non-zeros pin ground state.
The INS energy spectrum of 1a was measuredw ith an incident neutron wavelength of 5.0 , on the FOCUS spectrometer at 1.5a nd 6.0 K ( Figure 3). Severalm agnetic excitationsa re clearly resolved. Comparison of the spectra at different temperatures indicates aw eak excitation in the shoulder of the elastic scattering line, labelled I.S ubtractiono f1 .5-6.0 Kd ata indi-cates that this excitation comesf rom the ground state (cold) and is centred at approximately 0.42 meV.A ne xcitation centred at 1.22 meV,l abelled II,a lso shows greatest intensity at the base temperature. Twoe xcitationse merge on increasing the sample temperature to 6.0 K, labelled i and ii at 1.7 and 2.0 meV,r espectively.T hese two excitations originate from al ow-lying excited state and involve transitions to further excited states at higher energies.  Further measurements performed on the IN5 spectrometer probedabroader dynamic range. Measurements with a5 .0 setting confirm the spectra from FOCUS with enhanced energy resolution, such that transition I is clearlyr esolved ( Figure 4a). Ah igh-resolution( 8.0 ) instrument setting enables the separation of transition I from the elastic line anda t6 .0 Kt he equivalent excitation is also observed at negative neutron energy transfers.T he temperature dependence of I and II clearly identifies the transitions as cold excitations. With ashorter neutron wavelength of 3.2 a dditional cold excitations labelled III and IV are accessed, with peak centres at 2.5 and 3.6 meV,r espectively.F rom Figure 4a it becomes evident that transition II exhibits as light asymmetry,w hich is not evident for transition I.T he neutron momentum transfer of II has am aximum at 1.2 À1 (Figure 4d), consistent with the intermetallic distance betweenn earestn eighbour metal ions within the Cr 8 Mn ring, relatingt os trong correlationsb etween neighbouring spins within the cluster.
The magnetic data and the INS spectra in Figure 4w ere simulatedu sing am icroscopic spin Hamiltonian [Eq. (2)],i n whichŝ 1 toŝ 8 represent Cr sites with spin 3/2 andŝ 0 represents Mn with spin 5/2. D Cr and D Mn are the axial zero-field parameters for the respective ions, J CrCr is the isotropic exchange interaction between nearest neighbour Cr sites and J CrMn is the isotropic exchange interaction between the Mn site andi ts neighbouring Cr sites.Ĥ The structural similarity of Cr 8 Mn with other Cr-based rings provides aw ell-defined startingp oint. The nearest neighbour Heisenberg exchange couplings, J CrCr and J CrMn ,a re the dominant terms in the Hamiltonian;t here is no justification for including longer range couplings. The magnetic data can be fitted with the parameters: J CrCr = 1.32 meV, J CrMn = 1.28 meV with g Cr = 1.96 and g Mn = 2.0. These parameters also allow us to simulatet he main INS features. However,t oo btain the precise position for the INS transition I and to correctly describe the splitting of transition II,a na dditional exchange parameter needs to be introduced. The simplest choice, which also keeps the overall C 2 symmetry of the Hamiltonian, is to introducet he additional free parameter between Cr ionsŝ 4 andŝ 5 .S lightly increasing the exchange betweent hese two ions by J 4-5 /J CrCr = 1.05 brings the simulated INS peak I into position with the measured resultsa nd reproduces the asymmetry of transition II well. Since the measured transitions are much broader than expectedf rom the instrumentr esolution, the width of the peaks (assumed to be Gaussian) has been determined by fitting the experimental data.
The calculated energieso ft he lowest total spin multiplets of Cr 8 Mn and the observed INS transitions are shown in Figure 5.
As J CrCr is similart oJ CrMn ,C r 8 Mn is characterisedb ya nS = 1/2 ground doublet due to the competition between exchange interactions within the odd-membered antiferromagnetic ring. We note that the additional J 4-5 parameter splits the transitions labelledasII a and II b,reproducing the asymmetry in the measured peak.
The broadening of the measuredI NS peaks hinders the determination of the small anisotropy terms of the Hamiltonian. The single ion D Cr and D Mn values for the axial anisotropy obtained by INS measurements on similarc ompounds (e.g., Cr 8 Zn; [18] D Cr = À28 meV and Cr 7 Mn; [19] D Mn = À3 meV) were included within the Hamiltonian and are compatible with the simulation of INS results.
The effective ZFS D S ¼ P i¼0;8 g i D i for the lowest spin multiplets in Cr 8 Mn and Cr 8 Ni rings comparedt ot heir non-frustrated bipartite counterparts [19] (Cr 7 Mn and Cr 7 Ni)r eflects the non-collinear internal spin structure. In the nine-membered metal rings the projection coefficients, [20] g i ,l inking single ion terms with the ZFS of low-energy-spin manifolds, change sign aroundt he ring in contrastt ot he bipartite case, in which sublatticesalign parallel with each other,thus adding constructively.W eh ave calculated D S for the lowest S > 1/2 spin multiplet of some odd-and even-numbered anti-ferromagnetic rings using D Cr and D Mn derived from Cr 7 Mn. [19] The axial anisotropy calculated for the first excited state of Cr 8 Mn (characterised by S = 3/2) is D S = 3/2 = À0.00978 meV,a nd is much smaller than that fort he relatedb ipartite counterparts Cr 7 Ni and Cr 7 Mn in which D S = 3/2 = 0.073855meV and D S = 1 = À0.05962 meV,r espectively.T he projection of single ion anisotropies onto the lowlying spin states was also found [21] to be very small for Cr 8 Ni despite the large Ni single-ion anisotropy term.
The energy level scheme classifies Cr 8 Mn as aT ype II frustrated system. [15] For Cr 8 Mn the bridging ligands are chemically equivalent around the ring, but the presenceo ft he Mn II s = 5/ 2s pin breaks the rotational symmetry,t hus leadingt oa ni solated spin multiplet ground state. Since J CrCr and J MnCr are similar in magnitude the isolated spin ground state is characterised by S = 1/2, thus classifying Cr 8 Mn as aT ype II frustrated system. Other regimes of parameters, such as J CrMn significantly less AF than J CrCr ,w ould yield an S = 5/2 ground state and consequently Type III frustration.
The microscopic Hamiltonian model describing the spin dynamics of Cr 8 Mn was used to investigate the natureo ft he ground state internal spin structure. Figure 6a shows the calculated nearest neighbour spin-pairc orrelations. These are stronger amongst the Cr1-Mn-Cr8 unit duet ot he larger spin momento fM n II (s = 5/2) with respect to Cr III (s = 3/2). This creates ar igid Cr-Mn-Cr spin unit in which the spins are antiparallel and that couples to the remaining chain of Cr ions. Conse-

DE
show expectation values close to zero, indicatinganearly perpendicular arrangementoft he spins. Treating the spins as classical vectors (i.e.,w ith length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s i ðs i þ 1Þ p )r esultsi ns imilar behaviour of the correlationsw ith respect to the quantum spin model.
It is interesting to calculate the effect of an appliedm agnetic field on the ground state (i.e.,i nw hich B is significant but sufficiently less than the S = 1/2 to S = 3/2 crossing field). Figure 6c shows how an external field breaks the spherical symmetry of the isotropic Hamiltonian:d ue the larger magnetic momento fM n II with respect to the Cr III ions, the Mn ion tends to align parallel to the field thus producing an ode on the opposite side of the ring.
One of the classical configurations of minimum energy is shown in Figure 7a,f or the isotropic Heisenberg Hamiltonian.
Due to the rotational invariance, this is only one of the infinite configurations minimising the classical energy.I ti si mportant to note that all the classical spin vectors lie on the same plane, that is, c ijk ¼s i Ás j Âs k ¼ 0 8i; j; k.T his quantity,k nown as scalar chirality for the spins ijk,can be interpreted as ameasure of the solid angle between the three spins. In order to investigate the planarity of the quantum spin state, we have decomposed the ground state spin wave-function ontot he eigenstates of the chirality operatorĉ ¼ P iŝ i Áŝ iþ1 Âŝ iþ2 (usual cyclic boundary conditions are applied). Resultsa re shown in Fig-ure 7b:s imilar to the classical situation, the expectation value y 0 hĵ cy 0 ji vanishesi nt he ground-stated oublet. However,t he quantum ground state results in an equal superposition of chirality eigenstates with opposite eigenvalues. Hence, the spin configuration in the quantum ground state fluctuates between non-planar states corresponding to opposite eigenvalues ofĉ.

Muon spin relaxation
In a mSR experiment spin-polarised positive muons are implanted into as ample. [22] Muons have am ean lifetime of 2.2 ms. Each muon decays into ap ositron and two neutrinos. Due to the parity-violating nature of the decay,t he positroni s preferentially emitted in the direction of the muons pin at the momento fd ecay and hence following the positrond ecay distribution as af unction of time allows the determination of the time-dependence of the muon spin polarisation. The measured quantity of the experiment is the asymmetry A(t) = (N F ÀaN B )/ (N F + aN B ), whichi sp roportionalt ot he average polarisation of the muon ensemble. Here N F (N B )i st he number of positrons recorded in detectors placed forward (backward) relative to the initial muon spin polarisation and a is ac alibration constant. If the fieldd istribution probedb yt he muon ensemble has non-zero width, or if it fluctuatesi nt ime, then the muon spins will be depolarised. In large appliedl ongitudinal fields, the muon spin will be decoupled from the electron and nuclear spins in the system, and the initial muon polarisation at implantation is preservedt hroughout the measurement. However,a tc ertain values of the magnetic field, an avoided crossing of energy levels may occur, which leads to al oss of muon polarisation. [22,23] Muon spin relaxation was used to study the Cr 8 Mn spin ground state as af unction of applied magnetic field. Equation (2) predicts ac hange of ground state at below 4T (Figure 9), therefore we investigated the spin dynamics at this region. Avoided crossings have been identified in Cr 7 Ni rings by using torque magnetometry,a nd single-crystal inelastic neutron scattering has shown that the molecular spin at the avoidedc rossing oscillates coherently. [5] In Cr 7 Ni the critical field to achieve ac hange of ground state is > 10 T. The more modest field required for Cr 8 Mn allowsu st op erform mSR experiments at the crossingp oint.
Previously we reported the first observation of a S = 0!S = 1g round-statec rossing with muons. [24] Although it is possible that muons might only be sensitive to the "switching on" of am agnetic state in crossings involving the S = 0!S = 1t ransition, it should be expected that, as the Cr 8 Mn system is driven through the S = 1/2!S = 3/2 level crossing, the dynamic magnetic field fluctuationst hat accompanyt he transition should cause rapid depolarisation of muon spins in paramagnetic stoppings tates (i.e.,m uon states that involveh yperfine coupling to the electronic spin) and that this should lead to ad ip of the asymmetry versusa pplied magnetic field, allowing us to detect the crossing.
The initial asymmetry A(t = 0) measured as af unctiono fa pplied magnetic field B at T = 100 mK is showni nF igure 8, in .Due to the larger magnetic momento ft he Mn ion (grey arrow) if compared to the Cr ions,t he neighbouringC ri ons are locked in an almost collinear, antiparallel configuration.Moreover,anode is induced on the opposite side of the ring. All the spins belongtot he same plane,a sc an be checked by computingt he scalarc hirality c ijk on each set of three spins i, j and k.b )Decomposition of the spin Hamiltonian ground state y 0 ji onto the eigenstates 0 j of the chiralityo perator,ĉ 0 j ¼ c j 0 j .T he ground state results in an equal superposition of states with opposite chirality eigenvalues. Hence,i nc ontrast with the classicals ituation (in which the chirality is zero, indicating aplanar spin configuration), the ground state fluctuates between states with opposite chirality.
Chem. Eur.J.2016, 22,1 779 -1788 www.chemeurj.org which we see that the dominant contribution is as low,p eriodic undulation with appliedf ield. This behaviour of A(t = 0) may be accounted for by noting that the application of B deflects the decay positrons with respectt ot he detector array and changes the effective value of a in the definition of A(t)( see above), compared to its value determinedi nz ero field, causing the general trend of an increase in A(t = 0) with appliedf ield (right lower inset, Figure 8).
In addition, the size of the incoming muon beam at the sample varies with B leading to broad undulations, with minimae xpected at the focusing fields of 1.0T ,2 .5 Ta nd 4T [22,23,25] (as shown in left inset, Figure 8, the amplitude of this undulation is reduced at high fields). This contribution, with its expectedm inimum around 4Tmakes unambiguously identify-ing the precise field range of al evel crossingf rom the mSR difficult. However,a round3 .5 Tw ef ind that A(t = 0) levels off, before rising sharply from around 3.7 T. Althoughi ti sd ifficult to precisely account for the background (resulting from the effects of the magnetic field on the muonb eam spot and positron trajectories), this feature suggestsalevel crossing in Cr 8 Mn. Indeed, the peak is larger than the expected variation in background in this field range. If we interpret the data in this way then, assuming linear variation of A(t = 0) either side of the crossing, we estimate this occurs with ac entre at B cross = 3.73(5) T. Since we expect am inimum in the asymmetry as af unction of the applied field, in correspondence of the level crossing, [22] superimposing this minimum on the increasei n A(t)c aused by the field-dependent change in a would then result in the behaviour we observe. Assumingt his model, then as ubtraction of an estimated background contribution (grey line, bottom-right inset, Figure 8) leads to the resonance showni nt he upper panel of the right inset, Figure 8, which has af ull width half maximum of around0 .5 T. The precise width of ar esonance derived in this way is highly dependent on the model used to fit the background,b ut is generally much larger than that expectedf or ap ure level crossing involvingstates of different symmetries.
We also expect am inimum in relaxation rate at the crossing field, contributed by those muons in diamagnetic stopping states (i.e.,m uon states coupledt ot he electronics pin through ad ipolari nteraction). This reflects the increased magnetic fluctuations at the level crossingp roducing am otionally narrowed responsea tt hese muon stopping states and on resonancet he apparent relaxation amplitude becomes reduced. Our results suggest such ac ontribution to the mSR signal comesf rom av ery small relaxingc omponent A rel ,t he amplitude of which is also strongly affected by the beam spot size effects, with maximao bserveda tt he focusingf ields (left inset, Figure 8). After subtracting the same periodically modulated background as assumed above,t here is some evidencef or the expected minimum at the crossing field,w ith as imilar width to that seen in the paramagnetic contribution. We also note ac hange in relaxation rate of this component as we pass through the crossingr egion,w hich is also indicative of the crossing.
If the states at the level crossing becomem ixed due to symmetry lowering, quantum oscillationso ft he electronic spin may occur resultingi na na voided state crossing. In general, muon spin-flipping can occur if there is significant amplitude in the spectrald ensity of the electronic spin fluctuations, evaluated at the frequency corresponding to the muon Zeeman splitting in the local field. In this case of at ransition around 4Tand ad iamagnetic muon state, we require spectral density in the region of 1 meV.W eh ave shown previouslyt hat the muon is sensitivet ot he dynamics of the electronic spin in am olecular nanomagnet, but, crucially,t hat on the muon timescale it is the dephasing of these electronic spins by the disordered nuclear moments in the materialt hat causes the muon-spin relaxation. [26] It is likely that the stochastic field caused by nuclear moments plays ar ole in any crossing process that we detect and the electronic spin fluctuations probedw ill be incoherent. Since the dephasing of the elec-  tronic spins probably represents the shortest timescale in the process, we would expect the transitionprobability of the electronic spins to be determinedb yt he sum of instantaneous probabilities and therefore to be more complex than the Landau-Zener prediction.R egardless of the precise quantum oscillation mechanism, the width of ar esonance in applied field provides an indication of the avoided crossing energy gap. The large resonance width of 0.5 Tsuggests the presence of an avoided crossingi nt he order of af ew tens of meV.D ue to uncertaintiesi nt he field dependence of the background contributions to the measurement, ar eliable estimate of the avoidedc rossinge nergy gap is not possible here. However al arge resonance width, on the order suggesteda bove,i ndicates the presence of an avoided crossing.
The Hamiltonian in Equation (2) with the parameters determined from INS and magnetisation results does not produce an avoided crossing. The inclusion of aD zyaloshinski-Moriya interaction (DMI) term within the spin Hamiltonian can produce an avoided crossing in theo rdero fs everal tens of meV as suggested by mSR. For instance, au niform DMI parallel to Z (normalt ot he plane of the Cr 8 Mn ring), with coupling constant of J/50, leads to an anti-crossing of about 0.05 meV.F or simplicity,w eh ave included only the Z component of aD MI, which is also the most important in producing al evel repulsion. We finally note that, while the introductiono faDMI within the microscopicm odel is the easiestw ay to induce asizeable level repulsion, other mechanisms, such as avariance in the single-ion anisotropy terms around the ring, would also create as mall avoided crossing.

Conclusions
The two Cr 8 Mn variants (1a and 1b)d iffer in the way molecules are packed within the unit cell. The magnetic properties of the samples are found to be identical confirming that the magnetism of the crystalline samples is of molecular origin. A detaileds pectroscopici nvestigation was pursued on 1a.I NS measurements precisely characterise the low-energy spin dynamics of Cr 8 Mn, showinga nS = 1/2 ground state. The results are accurately reproduced by am icroscopics pin Hamiltonian with Heisenberg antiferromagnetic nearestn eighbour exchange interactions, the magnitude of which is similart oo ther double-carboxylate-and single-fluoride-bridgedC r III rings. The heterometallic nature of the ring breaksi ts translational symmetry resulting in an isolated S = 1/2 ground state. Calculations of the spin-pair correlations within the ground state give the internal spin structure of Cr 8 Mn and it is found that the region aroundt he Mn site has an anti-ferromagnetic arrangemento f spins, and spin non-collinearity is distributeda roundt he Cr ion segmento ft he ring. It is calculated that in an applied magnetic field the Mn II ion moment partially aligns with the field vector creatinganode localised at the opposite side of the ring. Ac lassical calculation yields ap lanar configuration of minimum energy,w hile the quantum ground state dynamics is found to fluctuate betweenn on-planar states with opposite scalar chirality. mSR measurements performed as af unctiono f the magnetic field suggesta nS = 1/2!S = 3/2 ground state avoidedc rossing centred at 3.73(5) T. The crossingf ield is consistent with the model developedf or the description of the INS results; however, as mall DMI has to be included in the Hamiltonian to create an avoided crossing there. The INS simulations are insensitivet oas mall DMI term, whereas mSR spectroscopy in the region of the crossover field shows its potential as ap robe of electronic transitions at level crossings in molecular nanomagnets.
Even-membered antiferromagneticr ings have been proposed as candidates for quantum information processing [27] and synthetically linking rings has been realised for the design of prototype qubits. [28,29] In such systems, magnetic fields are necessary to control entanglement andi ng eneral to implement quantumg ates. Conversely,e xternal electric fieldsc an provide am ore localised method to control spin states with opposite chirality, and such am ethodology has been proposed and evaluated for the case of spin frustrated Cu 3 triangles. [30] The prediction of af luctuating scalar chirality within the spin frustrated ground state of Cr 8 Mn highlights the necessity for furtheri nvestigations of spin chirality within other odd-membered odd-electronm olecular nanomagnets.

Materials and general methods
Unless stated otherwise, all reagents and solvents were purchased from Sigma-Aldrich and used without further purification. Manganese(II) carbonate, Puratronic, 99.985 %( metals basis) was from Alfa Aesar.T he Erlenmeyer Te flon FEP flasks were supplied by Fisher.Analytical data were obtained by the microanalysis laboratory at the University of Manchester-carbon, hydrogen, nitrogen analysis (CHN) by aF lash 2000 elemental analyser and metals analysis by Thermo iCap 6300 inductively coupled plasma optical emission spectroscopy (ICP-OES). Electrospray ionisation mass spectrometry (ESI-MS) were recorded on Micromass "QTOF Micro" quadrupole time of flight mass spectrometer.
[H 2 N i (C 3 H 7 ) 2 ][Cr 8 MnF 9 (O 2 CtBu) 18 ]( 1):P ivalic acid (30.0 g, 293 mmol), diisopropylamine (1.25 g, 12.4 mmol), and chromium(III) fluoride tetrahydrate (5.0 g, 28 mmol) were stirred for 4h in an Erlenmeyer Teflon FEP flask in an oil bath at 160 8C. To this solution manganese(II) carbonate (0.75 g, 6.5 mmol) was added in small portions over ap eriod of about 10 min and the reaction mixture was stirred for af urther 66 ha t1 60 8Ci nas low flow of N 2 . The flask was then allowed to cool to room temperature, acetone (50 mL) was added and the resulting mixture stirred for 2h.T he precipitate was collected by filtration, washed with acetone (ca. 300 mL) and dried in air.T hen it was dissolved in pentane (100 mL) and filtered. The filtrate was diluted with toluene (50 mL) and the solution was concentrated by distillation to half of its initial volume, and allowed to cool to ambient temperature in ap artially open flask. After concentration by slow evaporation during two weeks dark green hexagonal shaped crystals of 1 formed as at oluene solvate 1·2C 6 H 5 CH 3 (1a). The crystals were collected by filtration, washed with toluene (3 10mL) and dried in air.Y ield:4 .0 g (42 %b ased on Cr). Elemental analysis calcd (%) for C 110  Crystallographic data collection:F or compound 1a crystallographic data was recorded on aX 8p rospector Bruker SMART CCD diffractometer with Cu Ka radiation (l = 1.54184 ) at at emperature of 100 K, equipped with an Oxford Cryosystems Cobra nitrogen flow gas system. X-ray data for 1b were collected at Diamond Light Source beamline I19 (l = 0.6889 ), [31] at at emperature of 30 K. Data were measured using CrystalClear-SM Expert 2.0 r5 suite of programs.
Crystal structure determinations and refinements:X -ray data were processed and reduced using the CrysAlisPro suite of programs. An absorption correction was performed using empirical methods based upon symmetry-equivalent reflections combined with measurements at different azimuthal angles., [32,33] The crystal structure was solved and refined against all F 2 values using the SHELXTL suite of programs. [34] For compound 1a,o nly the metal sites and fluoride atoms could be found due to the extremely high disorder found. No attempt was made to refine the structure to convergence. Crystal data for 1a:f ormula C 110 H 194 Cr 8 F 9 MnNO 36 , hexagonal, space group P6 3 /m, T = 100(2) K, a = b = 19.552(2), c = 24.489(3) .
The crystallographic data and experimental details of the structural refinement for the X-ray crystal structure of 1b are given in the Supporting Information. CCDC 1402276 contains the supplementary crystallographic data for this paper.T hese data are provided free of charge by The Cambridge Crystallographic Data Centre.
Magnetic measurements:T he magnetic properties of polycrystalline samples of 1a and 1b were measured with aQ uantum Design MPMS-XL7 SQUID. The samples were ground, placed in agel capsule and fixed with as mall amount of eicosane to avoid movement during the measurement. The data were corrected for the diamagnetism from the gel capsule and the eicosane with the diamagnetic contribution from the complexes calculated from Pascal constants.
Inelastic neutron scattering (INS) studies:T ime-of-flight INS measurements on non-deuterated polycrystalline samples of 1a were performed at the IN5 spectrometer at Institute Laue-Langevin, [35] Grenoble (France) and the FOCUS spectrometer at the Swiss spallation neutron source SINQ, Paul Scherrer Institute [36] (Switzerland). The sample of 1a was sealed inside ah ollow aluminium cylinder for measurement. Measurements were performed at various temperatures within ar ange from 1.5-18 K. IN5 data were measured with several chopper settings, for which different speeds and ratios were used to select the optimum resolution, energyand momentum-transfer ranges.
INS energy spectra were obtained by integration of scattering intensity over all detector angles (À12 to 1358 and 10 to 1308 in horizontal primary scattering plane for IN5 and FOCUS, respectively).
Detector efficiencies were normalised to as tandard vanadium measurement.

Muon spin relaxation (mSR) spectroscopy
Muon spin relaxation measurements were performed on 1b using the HiFi spectrometer at the ISIS facility,Rutherford Appleton Laboratory (UK). [25a] Twou naligned crystals of Cr 8 Mn were mounted on the cold-finger of ad ilution refrigerator.A ll measurements were performed at 100 mK with al ongitudinal magnetic field applied parallel to the initial muon spin polarisation. stitute for neutroni nstrument time which wass upported by the European Commission under the 7th Framework Programme through the "Research Infrastructures" action of the "Capacities" Programme, Contract No:C P-CSAI NFRA-2008-1.1.1Number 226507-NMI3. This work has been financially supportedb yF IRB Project No. RBFR12RPD1 of the Italian Ministry of Educationa nd Research (MIUR), by the EPSRC (UK) (EP/ J009377/1) and EPSRC fellowship (EP/G003092/2). REPW also thankst he Royal Society for aW olfson Merit Award.