High pressure electron spin resonance of the endohedral fullerene 15 N@C 60
R. T. Harding, A. Folli, J. Zhou, G. A. D. Briggs, K. Porfyrakis, E. A. Laird
aa r X i v : . [ phy s i c s . c h e m - ph ] D ec High pressure electron spin resonance of the endohedral fullerene N@C R.T. Harding, A. Folli, J. Zhou, G. A. D. Briggs, K. Porfyrakis, and E.A. Laird Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH,United Kingdom School of Chemistry, Cardiff University, Cardiff CF10 3AT, United Kingdom (Dated: 5 November 2018)
We measure the electron spin resonance spectrum of the endohedral fullerene molecule N@C at pressuresranging from atmospheric pressure to 0.25 GPa, and find that the hyperfine coupling increases linearly withpressure. We present a model based on van der Waals interactions, which accounts for this increase viacompression of the fullerene cage and consequent admixture of orbitals with a larger hyperfine coupling.Combining this model with theoretical estimates of the bulk modulus, we predict the pressure shift andcompare it to our experimental results, finding fair agreement given the spread in estimates of the bulkmodulus. The spin resonance linewidth is also found to depend on pressure. This is explained by consideringthe pressure-dependent viscosity of the solvent, which modifies the effect of dipolar coupling between spinswithin fullerene clusters. I. INTRODUCTION
In the central-field approximation, atomic nitrogen hasno hyperfine coupling because the combined spin densityof the three unpaired p orbitals vanishes at the nucleusand is spherically symmetric outside it. In reality, in-teractions beyond this approximation lead to a non-zerohyperfine coupling for the free atom. In the presenceof other atoms or molecules, distortion of the electronorbitals further modifies the hyperfine coupling, whichtherefore offers an insight into the nature of the inter-atomic interactions. The endohedral fullerene N@C offers a nearly uniquesystem in which the fullerene cage stabilises the encap-sulated nitrogen such that it behaves like a free atom. However, the hyperfine coupling of N@C is enhancedby approximately 50% relative to the free atomic value,which reflects the effect of confinement of the nitrogen or-bitals by the cage. This increase has previously been at-tributed to nitrogen–cage interactions that mix in excitedstates with larger hyperfine couplings, although withoutproposing a microscopic model. Understanding the hyperfine coupling of this moleculeis important due to its potential use as a molecular spinqubit or frequency reference in an atomic clock. In par-ticular, N@C is suitable as a frequency reference dueto its sharp resonances and the existence of a clock tran-sition in its low-field spectrum. Since the frequency ofthis clock transition depends solely on the isotropic hy-perfine coupling constant A , it is important to charac-terise the mechanisms that affect it. For example, thehyperfine coupling is known to depend on temperature, which could reduce the long term stability of a fullereneclock against environmental temperature fluctuations.Here, we measure the hyperfine coupling strength andthe electron spin resonance (ESR) linewidth over a pres-sure range up to 0.25 GPa. The hyperfine coupling in-creases linearly with pressure, which we explain usinga microscopic model of electron wavefunction distortionmediated by van der Waals interactions between the ni- trogen atom and the cage. Using this model and theo-retical estimates of the bulk modulus of isolated fullerenemolecules, we then estimate the pressure shift andcompare it with our experimental results. The predictedshift is smaller than the observed shift, which may indi-cate contributions beyond the van der Waals interactionas well as uncertainty in the the theoretical bulk mod-ulus. To the best of our knowledge, the value deducedfrom our model is the first experimental estimate of bulkmodulus for an individual C molecule under conditionsof hydrostatic pressure.We find that the measured linewidth increases non-linearly with pressure. This is explained by the pressure-dependent viscosity of the solvent and its effect on ro-tational diffusion of fullerene clusters. At low pres-sure, the solvent is sufficiently non-viscous that dipole-dipole coupling between spins is averaged out by rotationof the cluster. At high pressure, the solvent viscosity in-creases, leading to an increase in the linewidth towardsthe rigid lattice limit. II. HYPERFINE COUPLINGA. Experiment A N@C sample was prepared by ion implantation, dissolved in toluene, and purified by high-performanceliquid chromatography to a purity (N@C :C ) of ∼ The sample used for high-pressure measure-ments was then concentrated by bubbling nitrogen gasthrough it to evaporate some solvent. This is because thesmall sample volume of the pressure cell leads to a lowresonator filling factor, which reduces the signal-to-noiseratio (SNR) of the measurement. This sample was theninjected into an yttria-stabilised zirconia cell attachedto a barocycler and loaded into a dielectric resonator(Bruker ER4123D). Measurements were performed at the X band (frequency f ∼ . N@C spin system are de-scribed by the Hamiltonian H = g e µ B S z B − g N µ N I z B + A ˆ S · ˆ I , (1)where the first two terms parameterise the Zeeman in-teraction of the electron spin S = 3 / I = 1 / B . The electronicand nuclear g factors are denoted by g e and g N , respec-tively. The final term describes the hyperfine interaction,with coupling strength A . At high fields, where the Zee-man interaction dominates, the energy levels are labelledby the quantum numbers m S and m I , which give theprojection of the relevant spin onto the axis defined bythe direction of B . The ESR signals arise from mag-netic dipole transitions between different m S levels, withselection rules ∆ m S = ± N nuclear spin, as shown inthe upper inset of Fig. 1. The resonances are separated inthe field domain by the hyperfine splitting ∆ B . We ex-tract ∆ B by fitting the spectrum to identify the extremaof each resonance observed in the first-derivative spec-tra. This splitting is converted to a hyperfine couplingusing the relationship A = gµ B ∆ B , where the g factor g = hf /µ B B av . . Here, B av . is the centre field of the spec-trum and f is the frequency of the applied microwave ra-diation. This frequency depends weakly on pressure viathe changing dielectric constant of toluene, which al-ters the cavity frequency.Data for the hyperfine coupling constant as a functionof pressure are plotted in Fig. 1. Fitting the data by alinear relationship, such that A ( P ) = A + A P , gives A = 22 . A = 1 . × − Hz Pa − .The bracketed number gives the statistical error in thelast digit of the quoted value. B. Theory
We will now explain the pressure-dependent hyperfinecoupling displayed in Fig. 1. A similar effect has beenmeasured previously for atomic nitrogen generated usinga radio-frequency discharge plasma, where the hyperfinecoupling is proportional to the buffer gas pressure.
In contrast to the pressure-dependent coupling observedfor atoms with unpaired s electrons, which can eitherbe positive or negative, the magnitude of the nitrogenhyperfine coupling always increases with pressure. This observation has been explained by treating vander Waals interactions between the nitrogen atom andthe buffer gas atoms as the dominant contribution to theincreased hyperfine coupling. These interactions lead tospin-dependent excitation of the s electrons, which causesspin polarization of the s orbital at the nucleus and an Pressure (GPa) | A | ( M H z ) B (mT) S i gna l ( a r b . un i t s ) ΔBB av. δB m I = / − m I = + / FIG. 1. Hyperfine coupling constant A as function of pressure.The data (dark green circles with error bars) are fitted by alinear function (solid black line). The data points are themean value of A at each pressure, and the error bars arethe standard deviation of this measurement. Upper inset:ESR signal as function of magnetic field at frequency f ≈ .
334 GHz, demonstrating the hyperfine splitting ∆ B , peak-to-peak linewidth δB , and the centre field B av . Lower inset:schematic of high-field energy levels showing the two sets oftransition frequencies corresponding to the two values of m I . increased hyperfine coupling. Following an analogousmodel, we hypothesise that the increase in hyperfine cou-pling for the nitrogen endohedral fullerene is due to vander Waals interactions between the incarcerated nitrogenand the fullerene cage. There should also be a pressure-dependent exchange coupling between the cage and the p orbitals. However, because of spherical symmetry thisdoes not mix p and s orbitals and is therefore expected tomake a weaker contribution to the hyperfine coupling. We therefore model the shift as arising entirely from vander Waals interactions.To apply this model to the endohedral fullerene sys-tem, we consider the increase in hyperfine coupling ∆ A = A − A free , where A free is the value for free atomic nitro-gen. A perturbative calculation of the van der Waalsinteraction between atomic nitrogen and a nearby parti-cle predicts the relationship (Eq. (15) of Ref. 3):∆ A = kR , (2)where R is the distance between the nitrogen andfullerene charge distributions. Since the nitrogen atomis located at the centre of the cage, we set this distanceequal to the fullerene radius. The constant k parame-terises the strength of the interaction between the twocharge clouds. Performing an ab initio calculation of thevalue of k is beyond the scope of this work; instead, wetreat it as an experimentally determined parameter. Solvent ǫ r µ (D) | A | (MHz)Toluene 2.39 0.375 22.247(2)Carbon disulfide 2.63 0 22.240(7)Chlorobenzene 5.69 1.69 22.253(5)TABLE I. Hyperfine coupling at room temperature and pres-sure in solvents with different dielectric constants ǫ r anddipole moments µ . Values of the dielectric constant are givenfor atmospheric pressure and T = 293 . The bracketednumber gives the one standard deviation error calculated bytaking the sample standard deviation of six measurements.
We test this model by using Eq. (2) to predict thepressure shift given the bulk modulus B . By treatingthe cage as a sphere with volume V and bulk modulus B = − V ∂P/∂V , we obtain the expression (cid:18) ∂ ∆ A∂P (cid:19) = 2∆
AB . (3)For N@C , we take A free = − .
65 MHz and A = − .
35 MHz and hence ∆ A = 7 .
70 MHz. Thebulk modulus of crystalline C samples has been mea-sured experimentally, but the value for an isolatedC molecule under conditions of hydrostatic pressureis only known via simulations. The simulated val-ues range from 300 to 1200 GPa, with a grouping in therange 700 to 900 GPa.
Taking B ∼
800 GPa, Eq. 3predicts ∂ ∆ A/∂P ∼ × − . This value is approx-imately six times smaller than the value ∂ ∆ A/∂P =1 . × − Hz Pa − determined from the fit in Fig. 1.The discrepancy could have at least two causes. Itmay indicate that exchange interaction between the ni-trogen and the cage, not captured by our van der Waalsmodel, in fact contributes significantly to the nitro-gen hyperfine coupling. Alternatively, the discrepancycould indicate that previous predictions have overesti-mated the fullerene’s bulk modulus. To explain the mea-sured pressure shift using Eq. 3 would require a value B = 140 ±
13, which is outside the range of theoret-ical predictions. However, these predictions themselvescover a wide range, which may indicate a need for furthermodelling. One source of uncertainty in the modelling isthe effect of the solvent, which is expected to modify themolecule’s bulk modulus. In particular, the effect oftoluene solvent has not yet been modelled.
C. Solvent effects
The most direct interpretation of the pressure de-pendence is that the nitrogen wavefunction is modifiedby compression of the cage. However, for some dis-solved radicals, the hyperfine coupling constant dependson the solvent due to interactions between the solventmolecules and the unpaired electrons of the radical.
For N@C , such an effect should be small, since the fullerene cage effectively isolates the atomic nitrogenfrom the environment. However, changes in the elec-tronic properties of the solvent could plausibly alter theelectronic properties of the fullerene cage; consideringEq. 2, this would correspond to altering k . Therefore, wemust exclude the possibility that the pressure-dependenthyperfine coupling reflects changes in solvent properties,such as dielectric constant ǫ r or molecular electric dipolemoment µ , that are known to vary with pressure. At room temperature and atmospheric pressure,toluene has dielectric constant ǫ r = 2 .
39 and dipole mo-ment µ = 0 .
36 D. At 0.25 GPa, ǫ r increases to ∼ µ is likely to vary by less than 2%. To measurethe effect of changes in ǫ r and µ , we therefore comparethe spectrum of N@C dissolved in toluene with thoseof N@C dissolved in chlorobenzene ( ǫ r = 5 .
69 and µ = 1 .
69 D) and carbon disulfide ( ǫ r = 2 .
63 and µ = 0 D).The ESR spectra of N@C dissolved in these solventswere measured at room temperature and atmosphericpressure at the X band using a Bruker ER4122-SHQE-W1 resonator and EMXmicro spectrometer. Data fromthese experiments are presented in Table I.The data show that the effect of altering the dielec-tric properties of the solvent is negligible to within ex-perimental error. Moreover, the effect of using differentsolvents is less than the observed change in hyperfinecoupling as a function of pressure, despite varying thedielectric constant and dipole moment by considerablymore than the variation achieved by pressurising toluene.The data were obtained using a different spectrometer tothe data presented in Fig. 1, and the small differencein A between comparable values presented in Fig. 1 andTable I presumably reflects differences in magnet calibra-tion. Such a systematic error does not affect the valid-ity of the comparison between different solvents shownhere. Therefore, we conclude that the increased hyper-fine coupling at high pressures is not caused by pressure-dependent solvent properties, and that the cage compres-sion model is correct. III. LINEWIDTH
In addition to the pressure-dependent hyperfine cou-pling, we also observe a pressure-dependent linewidth.The field domain peak-to-peak linewidth δB , as shownin the upper inset of Fig. 1, was determined by measuringthe splitting between the extrema of each resonance. Asshown in Fig. 2, the linewidth increases with pressure.Pressure-dependent linewidths are typically gov-erned by spin-exchange processes. However, forN@C exchange interactions between the incarceratednitrogen atoms are suppressed by the cage, which pre-vents overlap of the electronic wavefunctions. We there-fore use a model that instead considers the dipole–dipole interactions between N@C spins in fullereneclusters. Within each cluster, N@C spins inter-act via their magnetic moments, which leads to dipolar Pressure (GPa) L i ne w i d t h ( μ T ) V i sc o s i t y ( P a · s ) × - FIG. 2. Peak-to-peak linewidth and toluene viscosity as afunction of pressure. The linewidth data (red circles witherror bars) are fitted (black dashed line) using a model basedon the rotation of fullerene clusters. The viscosity of toluene(solid orange line) over this pressure range is plotted from themeasured equation of state given in reference 15. broadening. At low pressures, the viscosity of the toluenesolvent is low.
The fullerene cluster therefore rotatessufficiently rapidly to average out the dipolar couplingbetween the spins, leading to a narrow linewidth. Athigh pressures, however, the viscosity increases, whichreduces the rate of rotation. The linewidth then tendstoward the rigid lattice limit imposed by dipole–dipolecoupling between the spins in the cluster. In this model, the dephasing time T ∗ obeys the implicitequation ( T ∗ ) − = (2 /π ) C tan − (2 τ c /T ∗ ) , (4)where τ c is the rotational correlation time and C is thelinewidth due to dipole-dipole broadening as τ c → ∞ ,i.e. the rigid lattice limit. The rotational correlationtime τ c = 4 πηa / k B T, where η is the viscosity of thesolvent, a is the effective hydrodynamic radius of the ro-tating cluster, k B is the Boltzmann constant, and T isthe temperature. We fit the field domain linewidths δB presented in Fig. 2 as a function of solvent viscosity,which is known from the previously measured equationof state. From the fit, we extract C = 590 ±
100 kHzand a = 15 . ± . By considering the interaction energy of two spins,we determine the average spin–spin separation r ≈ (cid:0) µ S ( S + 1) g e µ B /hC (cid:1) / = 16 . ± . a is much greater than the radius of anindividual C molecule, while the value of r is much lessthan the expected separation for unclustered fullerenesgiven the spin density. Both these facts are evidence forthe clustering of fullerenes in our sample. However, theobserved spin–spin separation is larger than the separa-tion expected if the cluster were formed of close packedfullerenes. Such a cluster would have a spin–spin separa-tion of approximately 5 nm given the purity of the sam-ple and the diameter of an individual fullerene. However,the values are consistent with a porous structure for thefullerene clusters, which would reduce the effective spindensity in the cluster. The structure of fullerene clus-ters depends on the formation process. Slow aggrega- tion leads to densely packed clusters, whereas mechanicalagitation and exposure to light lead to the formationof fractal clusters. It is therefore possible that the con-centration procedure, during which nitrogen was bubbledthrough the solution under ambient light conditions, pro-moted the formation of porous clusters. Furthermore,the N@C sample used in this work was stored underambient conditions for approximately one year after theinitial purification procedure. The fullerene moleculesmay have oxidised during this storage period, which fur-ther promotes the formation of large clusters by formingepoxide bonds between individual fullerenes. This rotating cluster model explains why the observedlinewidth is much greater than the minimum achievablelinewidth set by relaxation mechanisms inherent to themolecule itself, such as the Orbach process. The nar-rowest linewidths measured previously required carefulsample preparation to inhibit clustering, which occurs atconcentrations above 0.06 mg/mL, and minimise para-magnetic impurities such as dissolved oxygen. However,using such a low concentration in our experiments wasnot feasible given the small sample volume of the pres-sure cell and the available sample purity.The field-independent clock transition in the low-field spectrum of N@C should suppress the dipolarbroadening. However, the clock transition does notprotect against all dipolar decoherence mechanisms. The maximum achievable stability of a fullerene clockcould therefore be constrained by the need to compro-mise between increasing spin density to increase signalintensity and the need to minimise linewidth broadeningcaused by dipole–dipole interactions. IV. CONCLUSIONS
The proposed model, based on van der Waals interac-tions between the nitrogen atom and the cage, explainsthe pressure dependence of the N@C hyperfine cou-pling. The model predicts a smaller pressure shift thanis observed experimentally, which may indicate that con-sidering only van der Waals interactions between the ni-trogen atom and the cage is insufficient. However, themodel provides reasonable agreement with the data giventhe spread in predictions of the bulk modulus. Whilethe small magnitude of the pressure shift likely precludesusing it to offset the temperature shift of the clock fre-quency, it ensures that a N@C based frequency refer-ence is insensitive to atmospheric pressure fluctuations.The pressure-dependent linewidth is well fitted bya model based on dipole–dipole interactions between N@C spins embedded in fullerene clusters. Dipo-lar broadening of the spin resonance depends on pres-sure via the viscosity of the solvent, which modifies therotational correlation time of the cluster. This modelprovides insight into spin relaxation processes in concen-trated N@C solutions that may limit the stability of afullerene clock. ACKNOWLEDGEMENTS
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