Manipulating and probing the polarisation of a methyl tunnelling system by field-cycling NMR

In NMR the polarisation of the Zeeman system may be routinely probed and manipulated by applying resonant rf pulses. As with spin-1⁄2 nuclei, at low temperature the quantum tunnelling states of a methyl rotor are characterised by two energy levels and it is interesting to consider how these tunnelling states might be probed and manipulated in an analogous way to nuclear spins in NMR. In this paper experimental procedures based on magnetic field-cycling NMR are described where, by irradiating methyl tunnelling sidebands, the polarisations of the methyl tunnelling systems are measured and manipulated in a prescribed fashion. At the heart of the technique is a phenomenon that is closely analogous to dynamic nuclear polarisation and the solid effect where forbidden transitions mediate polarisation transfer between H Zeeman and methyl tunnelling systems. Depending on the irradiated sideband, both positive and negative polarisations of the tunnelling system are achieved, the latter corresponding to population inversion and negative tunnelling temperatures. The transition mechanics are investigated through a series of experiments and a theoretical model is presented that provides good quantitative agreement. # Now at: Laboratoire L.E.O.G, Ecole Normale Supérieure, B.P. 227, Mostaganem 27000, Algeria $ Now at: Department of Laser Physics, College of Science for Women, University of Babylon, Ministry of Higher Education and Scientific Research, Hilla, P.O.00964, Iraq


Introduction
Interesting quantum effects that have no classical analogue arise when spatial and nuclear spin degrees of freedom are combined in the description of symmetrical quantum rotors.At their heart is the antisymmetry principle which underpins the Pauli Exclusion Principle (PEP), requiring the complete wavefunction describing the rotor to be antisymmetric upon exchange of indistinguishable fermions.The wavelike characteristics of the rotor are naturally evident in the quantisation of energy and angular momentum, however, the PEP additionally leads to a partitioning of the eigenstates into discrete sets of nuclear spin isomers, alternatively described as nuclear spin-symmetry species.The most familiar example is molecular hydrogen which is characterised by two spin-isomers, ortho-H 2 and para-H 2 .The discrete combination of spatial and nuclear spin states is evident in that ortho-H 2 has total nuclear spin 1 I  and odd values of the rotational quantum number 1, 3, ... J  , whereas para-H 2 has 0 I  and even values of J.
Particularly at low temperature, the two sets of nuclear spin isomers often co-exist as independent species, remaining decoupled from each other for substantial periods of time.This is evident for the endofullerene H 2 @C 60 [1][2][3] where, in the absence of spin conversion catalysts, interconversion between ortho-and para-hydrogen cannot be detected in experiments conducted over several days.[6] Methyl groups, CH 3 , are quantum rotors with a three-fold symmetry axis.Rotation about the C 3 axis permutes the indistinguishable 1 H nuclei belonging to the three hydrogen atoms.As a result we identify two discrete sets of spin-symmetry species labelled A and E, classified by the irreducible representations of the C 3 symmetry group 7,8 .The Aspecies is a quartet with total nuclear spin 3 2 I  whereas there are two degenerate Especies, labelled E a and E b , which both have spin 1 2 I  .In a molecular environment there is a potential barrier to rotation and coherent quantum tunnelling gives rise to a splitting of the A-and E-species which are separated in energy by t h , where t  is the tunnelling frequency.
In accordance with the PEP, the spin-symmetry species of CH 3 are characterised by certain allowable combinations of space and spin wave-functions 9 .Transitions between A and E species involve simultaneous changes in spatial and spin eigenstates.As a result, at low temperature the interconversion is spin-restricted and relatively slow so we can identify a long-lived 'tunnelling reservoir' characterised by the energy t h and the polarisation of the A-and E-states.To investigate the tunnelling dynamics demands an interaction that has the capacity to bring about simultaneous changes in space and spin states and hence induce transitions between A-and E-species.One such interaction is the 1 H-1 H dipole-dipole interaction, which has played an important role in elucidating the quantum dynamics of this quantum rotor 8,10 .
Having discrete combinations of space and nuclear spin eigenfunctions, spin-isomers offer opportunities to act as vehicles for creating strongly non-equilibrium nuclear spin polarisations.This is well-established in the technique of PHIP (parahydrogen induced polarisation) and related techniques where para-H 2 is used as a hyperpolarisation agent 11 .It may be envisaged that given appropriately designed experimental protocols, the spin-symmetry species characterising other quantum rotors may be analogously exploited.In this respect, methyl groups have a potential advantage over molecular hydrogen because the splitting between states of different spin-symmetry is a variable of the height and shape of a potential barrier, and hence determined by the local molecular environment.Therefore, the A-E energy splitting may be optimally 'selected' by appropriate choice of host molecule.Indeed, progress has been made in utilising the Haupt effect, a phenomenon that applies to weakly hindered CH 3 groups 12,13 , in nuclear polarisation schemes.Additionally, the group of Levitt has explained and exploited remarkable hyperpolarisation effects in solution NMR attributed to quantum rotor effects of CH 3 5, 14, 15 which were first observed by Icker and Berger 16,17 .
To create non-equilibrium spin polarisation in methyl group containing materials, the protocols demand A-E transitions to be driven directly.Therefore, it is interesting to explore mechanisms and experimental protocols that will enable the polarisation of the tunnelling reservoir to be manipulated by design, somewhat analogous to the ubiquitous manipulation of spin states using pulsed NMR.This is the motivation for this paper.
Although the polarisations achieved here are relatively modest, largely because the tunnelling frequencies are relatively small, field-cycling NMR sequences have been devised to purposely drive positive and negative changes in both tunnelling and 1 Hpolarisations, allowing an exploration of the fundamental principles that may later be developed to form the foundations of protocols achieving more substantial spin polarisations.
In an earlier paper 18 , we described experiments that enabled the polarisation of the tunnelling reservoir to be manipulated and interrogated directly.These experiments demanded rf irradiation at low magnetic field and excursions to level-(anti)crossings that arise between A and E states at particular magnetic fields.Therefore, magnetic fieldcycling was incorporated in the NMR pulse sequence and the phenomenon was described as Dynamic Tunnelling Polarisation (DTP) because of its close analogy to dynamic nuclear polarisation (DNP) and the solid effect [18][19][20] .In this new study, DTP experiments on acetophenone, 2-butanone and thioanisole are described.These materials are characterised by tunnelling frequencies in the range 490 kHz -1.5 MHz.Field-cycling NMR experiments have been employed to quantify the positive and negative changes in 1 H polarisation that come about when 'tunnelling sidebands' are irradiated using rf.The mechanisms are investigated, including the measurement of the lifetime of the tunnelling reservoir and the role of the level-crossings in inducing resonant A-E conversion.Finally, the experimental results are successfully simulated using a thermodynamic model.

Theoretical background
In a condensed phase, CH 3 rotors are sub-groups of a molecule, with the result that inter-and intra-molecular interactions give rise to a potential barrier to rotation about the three-fold axis.This barrier has 3-fold symmetry (or higher harmonics) and rotation of the molecule permutes the hydrogen atoms between energetically equivalent sites.
Quantum mechanically, exchange interactions associated with tunnelling through the potential barrier give rise to a splitting of the ground torsional state and A-and E-species are separated in energy by t h , where the tunnelling frequency t  is an exponential function of the barrier properties 21 .
The NMR properties of the methyl rotor may be described by the spin Hamiltonian 10 , where is the Zeeman interaction of the three 1 H spins labelled 1, 2, 3 j  with the applied field z B oriented along the z -axis.ˆdd H is the 1 H- we may write, The dipolar interaction is between the three pairs of 1 H spins comprising the CH 3 rotor, which has the important association of a spatial factor   q jk F with a factor   ˆq jk A comprising nuclear spin operators characterised by 0, 1, 2 q    spin flip terms.In coupling space and spin, ˆdd H is significant to quantum rotor physics, providing a conduit for interactions between A and E spin-symmetry species.Explicit expressions for   q jk F and   ˆq jk A are available in the NMR literature and have been defined in the context of methyl rotors in reference 22 , showing how ˆdd H may be written in symmetry-adapted form, explicitly revealing its capacity to drive A-E transitions.
Solving the spin Hamiltonian Eq.( 1) in a basis of nuclear spin eigenfunctions  n   for E-species.This provides an equivalent formalism for defining the thermodynamic properties of the tunnelling-Zeeman system.In the high temperature approximation we may define the populations of the tunnelling-Zeeman levels as , where any constants have been ignored since only population differences are important.
In this way the populations can be defined in terms of the inverse tunnelling and Zeeman temperatures.Alternatively, the inverse temperatures can be defined in terms of the populations using, Given Subsequently, by switching the field to a level-crossing, lc1 or lc2, the tunnelling and Zeeman systems will rapidly equilibrate and any tunnelling polarisation will be transferred to the previously saturated 1 H Zeeman reservoir.In the pulse sequence, Fig.

Results
All NMR spectra in this investigation concern 1  The peak pair a  & a  are the two members of the 1 m   spectrum and are analogous with the DNP features observed in the solid effect 19,23,24 .However, whereas in DNP the pair of asymmetric peaks identify forbidden sideband transitions within the manifold of can only be explained by invoking mechanical coupling between pairs of CH 3 groups 25,26 and is not accommodated in the simple energy level scheme of Fig. 1a.The simplified energy level scheme for a pair of coupled CH 3 rotors shown in Fig. 1b provides a basis for interpreting these spectral features, where such sidebands may be identified.dipole interactions, so we infer these antisymmetric features reveal 1 H dipolar finestructure that is not elucidated in the abridged energy level scheme of Fig. 1.In an observation that is possibly related, we also tentatively identify an indication of some antisymmetric structure simultaneously appearing in the a  and b  peaks of the 28.8 mT spectrum.In general the aforementioned peak shapes may indicate the influence of dipolar order, however, these are secondary effects and beyond the scope of the present study.
Acetophenone has a tunnelling frequency 27 that is approximately three times that of thioanisole.DTP spectra have been recorded with     t kHz) have been reported earlier 18 .

b) Mapping out the level-crossing region, LC B
To investigate the level-crossings between 1  In both Figs 5a and 5b, to access the lower field values, it is an unavoidable consequence of the field-cycling sequence that the system first passes through levelcrossings that occur at higher field.Therefore, the DTP effect is an integral function of incremental contacts that are made between the two reservoirs as the field switches down to its stable value of LC B .Given plots of DTP signal vs. LC B have a rectangular appearance, as opposed to discrete peaks, we can infer the cross-relaxation process is rapid and highly efficient.


The DTP effect is determined by a polarisation transfer from the 1 H Zeeman system to the CH 3 tunnelling system.A population gradient is created within the Zeeman system during the polarisation step.This is subsequently transferred to the tunnelling system by pumping forbidden transitions within the manifold of tunnelling-Zeeman states.To

e) Tunnelling lifetime measurements
Applying the protocol depicted in Fig. 2b, where a relaxation step has been introduced into the DTP pulse sequence, the lifetime of the CH 3 tunnelling states has been measured.This is reported in the form of the tunnelling relaxation time and is studied by mT.Here, particularly at the higher fields, the data appear to be characterised by multi-exponential functions.Nevertheless to facilitate a quantitative analysis of the magnetic field dependence, the data have been fitted with a single exponential decay.
The magnetic field dependence of tun T is plotted in Fig. 8b

f) The heat capacity ratio of the Zeeman and tunnelling systems
The DTP pulse sequence involves field-cycling excursions to level-crossings in the tunnelling-Zeeman manifold of levels.This is a practical device to enable the deviations in tunnelling polarisation to be made visible by cross-relaxation between tunnelling and 1 H Zeeman reservoirs.The experiments provide limited insight into the detailed crossrelaxation mechanism, but they do reveal this is an efficient process.The extent of thermal mixing between the two reservoirs is determined by their relative heat capacities and experiments have been conducted to investigate this ratio.
We define Z C and t C to be the heat capacities of the Zeeman and tunnelling systems respectively.Since the 1 H Zeeman splitting is linearly dependent on the applied magnetic field, whereas the tunnelling splitting is independent of field, the heat capacity ratio . However, at each cycle the Zeeman system is saturated, so the decline of the DTP signal conforms with the function   . Already by the second contact the DTP signal has declined to 16% of its value at the first contact, 1 P .By the third contact it is smaller than experimental uncertainties.In these circumstances we cannot extract an accurate value of the heat capacity ratio, but by fitting with the function   , the data in Fig. 9 do permit upper and lower limits to be placed on the heat capacity ratio.We find, 5 8

Discussion and numerical simulations
The field-cycling sequence has been shown to successfully manipulate the tunnelling By selecting the appropriate tunnelling sideband, both positive and negative tunnelling polarisations are achieved.So the tunnelling system may be cooled, or alternatively heated to the extent that a population inversion can be induced.This facility enhances the degree to which the tunnelling system can be manipulated in a prescribed manner.
To gain further insight into the mechanisms, a thermodynamic model has been developed which provides the theoretical basis for a simulation of the DTP spectrum.
This model is a development of earlier publications by the Nottingham group, to which the reader is referred for further details 22,[28][29][30] .Forming the basis of the simulation, for a given applied magnetic field and tunnelling frequency the matrix of the spin Hamiltonian ˆs H , Eqn. ( 1), may be written directly in the basis of spin functions alternatively in symmetry-adapted form 10,22 .Then following ref.[22], the eigenvalues and eigenfunctions of the CH 3 system are calculated by numerical diagonalization of ˆs H .This determines the extent to which there is a mixing of the A and E states by the nonsecular terms in the dipolar interaction.The thermodynamic parameters, namely the populations, polarisations and inverse temperatures of the 1 H Zeeman and tunnelling systems are defined according to the framework described in section 2, Eqns.( 4) -( 6).
Then those thermodynamic parameters are evolved in a step-wise fashion imitating each stage of the field-cycling pulse sequence, Fig. 2a.Here, to simulate the effects of rf irradiation at low field, the changes in population of states labelled i and j that participate in a weakly allowed tunnelling sideband transition may be written as 22 , where using the eigenvectors calculated from the diagonalization of ˆs H , the transition rates per unit time, i j T  , are determined by Fermi's golden rule with the time- dependent Zeeman Hamiltonian Evolution also occurs in response to the switches in magnetic field between that arise in the field-cycling sequence.These are adiabatic, resulting in adiabatic magnetisation/demagnetisation of the system depending on the direction of the field switch.Accordingly the change in inverse 1 H Zeeman temperature that accompanies a field switch is calculated using, Additionally, at a level-crossing the cross-relaxation between the respective systems induces changes in inverse temperature of the two systems described by the pair of differential equations 22 , where W is the cross-relaxation rate.
The simulated DTP spectra for thioanisole are presented in Fig. 3 with solid lines.There is generally good quantitative agreement with experiment for the features associated with sidebands a  , a  , b  and b  , confirming the proposed mechanisms that form the basis of the thermodynamic model.The model does not address mechanical coupling so the double sideband feature 2b  does not appear in the simulation.The experiments in section 4f reveal the heat capacity ratio is somewhat larger than the value 3 Z t C C  derived from a statistical analysis of the manifold of tunnelling-Zeeman energy levels 22,28 .Firstly it should be noted there are additional non-methyl protons present in the molecule which, as a result of efficient spin diffusion processes, contribute to the heat capacity of the 1 H Zeeman reservoir.Secondly, the theoretical value 3 is only valid for isolated groups at the field corresponding to lc1.However mechanical coupling means the manifold of coupled rotor states is more complex than provided for in a statistical thermodynamic analysis of the energy levels of Fig. 1a.Moreover the fieldcycling switches that characterise the pulse sequence transit through the level crossing lcc which occurs at higher field than lc1.Therefore, the situation is complex but taking all these factors into account it is to be expected the experimental value observations make it clear that cross-relaxation is fast at the level-crossings and the value employed in the simulations for thioanisole, 0 10 W  s -1 is consistent with experiment.

Concluding remarks
These investigations were devised to develop techniques that enable the polarisation of a methyl tunnelling system to be probed and manipulated in a prescribed fashion.Using The experiments provide insights into the physics of space-spin entangled systems such as those which typify quantum rotors.This is interesting to broader studies seeking to employ quantum rotors in nuclear spin polarisation schemes.Informed by parameters determined from the eigenstates of the system's Hamiltonian, a simple thermodynamic model has been described which provides quantitative agreement with experiment.One finding that reveals itself repeatedly in different ways is that mechanical coupling between pairs of methyl groups is surprisingly prevalent.This means the simple model of an isolated methyl group, while generally adequate to account for many methyl-related phenomena, requires improvement to account for the full range of observations.
provides the manifold of tunnelling-Zeeman eigenstates that characterise the CH 3 rotor (

2  1 T 1 T
2a, this field is labelled LC B .Finally, a -pulse at the field nmr B enables that 1 H spin polarisation to be measured, from which we can infer the tunnelling polarisation that existed at the end of the rf irradiation.The variant of the field-cycling pulse sequence, Fig. 2b, contains an additional 'relaxation' step, inserted immediately after the rf irradiation.During this period the tunnelling reservoir may relax towards equilibrium with the lattice and by plotting the final tunnelling polarisation as a function of rel  , the tunnelling relaxation time may be determined.The field at which the relaxation is conducted, rel B , is a variable of the experiment enabling the relaxation to be investigated close by or remote from the level crossing fields.All experiments were conducted with a sample temperature of 4.2 K.The samples of thioanisole (C 6 H 5 SCH 3 ), acetophenone (C 6 H 5 C(O)CH 3 ) and 2-butanone (CH 3 C(O)CH 2 CH 3 ) were obtained from Aldrich and used as supplied.All three are liquids at room temperature.Quench cooling invariably rendered a solid sample with fast 1T , probably indicating a more disordered state but possibly also reflecting the presence of paramagnetic oxygen.However, by annealing the samples at a temperature approximately 5 K below their melting point for a period of order 1 hour, a more ordered material was generally obtained, characterised by a significantly longer spin-lattice relaxation time.The step possibly also enabled some dilution of oxygen in the sample due to the immersion in the helium exchange gas.In practise, aiming to achieve material with very long was not desirable since it is necessary to practically achieve high levels of 1 H polarisation at the beginning of the pulse sequence.Therefore the annealing stage was optimised experimentally to provide material with intermediate values of .Typically, in these experiments the spin-lattice relaxation times were in the range at B  1T and T  4.2 K.
H nuclei in the solid state at low temperature.a) DTP spectroscopy Using pulse sequence Fig 2a, DTP spectra of thioanisole have been recorded with three low field values lf B  31.7,29.7 and 28.8 mT and are presented in Fig. to a 1 H Larmor frequency nmr   37 MHz.The DTP spectra were systematically scanned by incrementing the secondary rf irradiation frequency at each cycle in the sequence in steps of 10 kHz in the range 500 3600 rf s.The tunnelling polarisation (vertical axis) is presented as the inverse tunnelling temperature,  t , recorded after equilibration with the saturated Zeeman reservoir.This polarisation data was calibrated with reference to a measurement of the equilibrium 1 H Zeeman polarisation recorded at a field of 100 mT.Peaks with positive polarity are observed for sideband transitions where rf H t a  and b  respectively.Inverted peaks with negative polarity are observed where rf H t a  and b  respectively.Fitting the peaks in the DTP spectra with Gaussians, we determine the tunnelling frequency,

2 b
electron-nuclear states, in the DTP experiment recorded here the antisymmetric pair relate to sideband transitions within the manifold of tunnelling-Zeeman states; the role of the electron in DNP has been replaced by the tunnelling CH 3 groups in DTP.The antisymmetric pair of peaks b  & b  have a similar origin, except these are the two members of the 2 m   spectrum.Some further insights into these tunnelling polarisation effects will be addressed in Section 5.In the lf B  31.7 mT spectrum an additional positive peak appears at 1565 rf  , being a sideband peak centred on by twice the tunnelling frequency from the centre of the 2 m   spectrum.The origin of this peak Further scrutiny of the spectra as lf B is decreased identifies additional features appearing at H  and 2 H  , growing in amplitude as the condition 2 where a  and b  begin to overlap.Appearing most clearly in the lf B  28.8 mT spectrum, these NMR peaks are both antisymmetric in shape, reminiscent of the derivative of an absorption peak,.The centres of these 'derivative' shaped peaks accurately identify H  and 2 H  respectively.The NMR linewidth is governed by the 1 H-1 H   5 s and are presented in Fig. 4. Spectra with polarisation times in the range 120 2400 pol    s are shown.We observe the a  , a  and b  DTP peaks, however, the frequency of the b  peak falls beyond the bandwidth of secondary rf irradiation so could not be accessed with our experimental set up. Analysis of the peak positions provides the tunnelling frequency

30 LCB
H Zeeman and tunnelling CH 3 groups in more detail, the intensity of the DTP spectrum of thioanisole was studied as the level-crossing was approached by varying LC B .In this series of experiments the intensity of the a  DTP peak was monitored as a function of LC B , thereby mapping out the level-crossing region, see Fig. 5a.With  mT there is no DTP effect, as the magnetic field is too high to bring the Zeeman and tunnelling systems into contact.As LC B is systematically reduced below 30 mT the DTP signal grows as the tunnelling polarisation is enabled to transfer to the previously saturated Zeeman reservoir in proximity to the level-crossing.at 13.1 mT and lc2 appears at half this field, 6.55 mT.However, in Fig. 5 the DTP effect first appears at twice the field of lc1, namely in the vicinity of 26 mT where 2 H t    .Corroborating the earlier observation, this provides further evidence for mechanical coupling of pairs of CH 3 groups.The level-crossing that facilitates this contact between tunnelling and Zeeman reservoirs, labelled lcc, is shown in the energy level scheme for a pair of coupled CH 3 rotors, Fig. 1b.Similar behaviour is observed for 2-butanone where the DTP signal peaks at the field LC B .5b, so we can infer the coupling of CH 3 pairs may not be uncommon.

1 polB
Fig. 7a in the range 1 8 lf    s with lf B  29.7 mT.It is evident A E  transitions become saturated after approximately 4 s, whereby the connected states have achieved the same population.Therefore the transition rate characterising the process is faster than 1 s -1 in this material.Data for 2-butanone is presented in Fig. 7b.The intensities of a  ,  a ,  b and b  DTP peaks have all been monitored as a function of lf  at the field lf B  27.8 mT.The growth monitoring the intensity of a DTP peak as a function of magnetic field, rel B .Experiments have been conducted on 2-butanone and thioanisole.For 2-butanone, the intensity of the DTP peak b  is plotted as a function of relaxation time, rel  , for several magnetic field values in the range 23.Fig.8a.Characteristic tunnelling relaxation times, tun T , have been determined by fitting with single exponential decays which generally represent the data well.In Fig. 8b the relaxation data is presented for thioanisole in the range 27.

Fig. 9 .
Fig.9.The data is presented as the ratio of the DTP signal after the n'th contact relative to the first,  1 n P P .For this experiment the level-crossing contact was made at the field

1
polarisations.Changes in tunnelling state can be driven in a prescribed manner by rf irradiation, pumping selective tunnelling sideband transitions.Additionally, as part of this sequence, the contact between the two systems established at a level-crossing enables the tunnelling state to be interrogated via a measurement of the 1 H Zeeman magnetisation.Tunnelling polarisations are achieved with magnitudes that are comparable with the equilibrium 1 H Zeeman polarisations.The mechanisms involved are analogous to DNP and the solid effect, except the role of the electron in DNP is replaced by the CH 3 tunnelling system in DTP.

1 B
is the amplitude of the rf field.

1 TFig. 3 . 3 
Fig. 3.The cross-relaxation rate at the level-crossings was simulated by a Gaussian function, agree within a factor two and the transition rates for the 1   m sidebands are systematically faster in both experimental and calculated data sets.This represents a favourable comparison given the uncertainties in modelled and experimental parameters, for example 1 B .It is pertinent to address here in further detail the mechanisms driving the tunnelling sidebands and which give rise to the prescribed manipulations of the tunnelling polarisation.In conventional DNP, 'forbidden' transitions among the manifold of electron -nuclear spin states are rendered weakly allowed by non-secular terms in the electronnuclear interaction which mix the pure basis states.The transitions in question involve simultaneous changes in electron spin and nuclear spin states.As described by Abragam 31 , pumping such forbidden transitions which occur in proximity to the electron spin resonance frequency, e  , gives rise to strong positive and negative deviations in nuclear spin polarisation, depending on the sign of the offset, e n    , forming the basis of the solid effect.In the DTP experiment with CH 3 quantum rotors, we identify 'forbidden' transitions among the manifold of tunnelling-Zeeman states.These involve A  E changes in nuclear spin-symmetry, representing simultaneous changes in tunnelling and nuclear spin states.These are the methyl tunnelling sidebands in NMR, offset from the nuclear spin resonance frequency by t   .These 'forbidden' transitions are rendered weakly allowed due to mixing of A and E states by the non-secular terms in the 1 H-1 H dipole-dipole interaction.Pumping these forbidden transitions drives the strong positive and negative deviations in tunnelling polarisation, depending on the sign of the offset.The non-secular terms in ˆdd H involve single spin-flip operators such as ˆẑ I I  which are responsible for 1   m sideband transitions, but additionally there are double spin-flip operators such as ˆÎ I forbidden at conventional magnetic fields employed in NMR, but as the dipolar energies become more comparable to the nuclear Zeeman energy at low field, both single and double spin-flip components become more strongly allowed.Therefore, analogous to DNP and the solid effect, the DTP spectrum comprises positive and negative peaks in tunnelling polarisation symmetrically displaced about the nuclear spin resonance frequency.However, because of the double spin-flip operators, a 2 m   spectrum is also observed, mirroring that of 1 m   .The field-cycling methods facilitate field-dependence measurements of the tunnelling relaxation time, Fig. 8.With field offsets of just 5 or 10 mT from a level-crossing, values of tun T are already very long, confirming the 1 H Zeeman and tunnelling systems are normally strongly decoupled at low temperatures.However, there is a steep decrease in tun T as the level-crossings are approached and cross-relaxation pathways become efficient.The field-offsets in question, of order 5 or 10 mT, are noteworthy, being of the order of the NMR linewidths.The latter are governed by 1 H-1 H dipolar interactions, so given ˆdd H contains the requisite space-spin coupled terms that can drive transitions between A and E states, we find confirmatory evidence for the important role that the dipolar Hamiltonian plays in tunnelling relaxation specifically and quantum rotor physics more generally.Although the field-switching timescales do not enable a closer approach to the level-crossing in these tunnelling relaxation experiments, the study does probe the same cross-relaxation process that is responsible for bringing the tunnelling and Zeeman reservoirs into contact at a level-crossing.Extrapolating the curve of tun T vs. rel B in Fig. 8c, the data seems consistent with the cross-relaxation rate value 0 10 W  s -1 employed in the simulations.
magnetic field-cycling NMR pulse sequences and irradiating forbidden transitions within the manifold of 1 H tunnelling-Zeeman energy levels, this has been achieved.It has proved possible to invert the tunnelling populations as well as create tunnelling states characterised by relatively low temperatures.The mechanisms are analogous to DNP and the solid effect.Comparing with those established techniques, the role of the electron spin in DNP is replaced by the tunnelling methyl system in these DTP phenomena.The effect relies on a polarisation transfer between 1 H Zeeman and tunnelling systems mediated by rf irradiation at the various tunnelling sidebands.Therefore, the tunnelling polarisations achievable are consistent with those characterising a nuclear spin system.
simulated DTP spectra of Fig.3.The rf field amplitude is 1

Fig. 1
Fig. 1 The manifold of tunnelling-Zeeman energy levels, showing the level-crossings lc1, lc2 and lcc (see main text for details): a) single CH 3 group, indicating tunnelling sideband transitions; b) mechanically coupled pairs of CH 3 groups (simplified with ˆdd H omitted).The arrows denote the forbidden transitions connecting A and E states that are responsible for the respective tunnelling sideband peaks in the DTP spectra.

Fig. 2
Fig. 2 The field-cycling NMR pulse sequences employed in the DTP experiments: a) DTP pulse sequence, b) DTP pulse sequence including a tunnelling relaxation step.

Fig. 6 1 polB
Fig. 6 The DTP peak intensity as a function of polarisation time, pol  : a) thioanisole, a + and b -peaks; b) b -peak intensities for acetophenone (filled squares) and 2-butanone (filled circles), superimposing the polarisation growth vs. time curve for 2-butanone recorded at the same field,

Fig. 7
Fig. 7 The DTP peak intensity as a function of irradiation time, lf  : a) thioanisole, a 

Fig. 8
Fig.8The relaxation of the tunnelling reservoir recorded for several relaxation fields,

Fig. 9
Fig. 9 Determining the heat capacity ratio, Z t C C in 2-butanone by monitoring the b - peak intensity while making repeated contacts between 1 H Zeeman and tunnelling reservoirs.The solid line is the function     1

Table 1 .
Neither in the simulation do the 'derivative' lineshape features appear at H -1 .This is qualitatively consistent with the growth of the DTP peaks as a function of irradiation time, lf  , shown in Fig.7a, providing further validation for the simulated DTP spectra.Since thioanisole and 2-butanone have similar tunnelling frequencies and therefore similar ˆs H , a more quantitative comparison is available from the data for 2-butanone, Fig.7band the calculated transition rates for thioanisole in Experimental and theoretical values for the s