serge lacelle and luc tremblay- nmr multiple quantum dynamics in large spin networks

8/3/2019 Serge Lacelle and Luc Tremblay- NMR Multiple Quantum Dynamics in Large Spin Networks

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Vol. 18, No. 3/4 21 1NMR Multiple Quantum Dynamics in Large Spin Networks

Serge Lacelle and Luc TremblayDepartement de ChimieUniversite de SherbrookeSherbrooke, Quebec, Canada JlK 2R 1

ContentsI. Statistical Ph ysics of Large Spin NetworksII . Anisotropy of Growth of Multiple Spin CoherencesIII. Maximal Spanning Tree DynamicsIV. PerspectivesV. AcknowledgmentsV I. ReferencesI. Statistical Physics of Large

Spin NetworksExcitation of NMR multiple quantum (MQ) tran-sitions in a solid is possible due to the presence of anintricate dipolar couplings network amidst nuclearspins. In a magnetic field, a non-symmetric rigidN-spin 1/2 system, has

distinct dipolar coupling constants given byDij = 7 2 h2(3cos29 lJ - l ) / (2ry3 ) (1)

where the symbols represent the customary univer-sal constan ts and lattice p aramete rs. Radiative cou-pling among the 2N Zeeman levels, which are furtherspread by dipolar interactions, leads to

accessible MQ coherences of order n (n^O), where nis the difference in the magnetic quantum numbersof the coupled levels. Such MQ transitions a re de-tected indirectly through two-dimensional methods,some of which are specifically tailored for solids, e.g.,time-reversal excitation (1).

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In a solid, th e local dipolar field at a nuclear siteoriginates from all other magnetic moments in thesample. This mean field description is useful forunderstanding lineshapes and spin thermodynamicsarguments in some NMR experiments (2). The re-tarded time for the propagation of a dipolar field atthe speed of light on a length scale of a micron isabout 10""15 s. Hence, it could be argued that allaspects of the many-body problem need to be con-sidered in order to understand the dynamics of MQcoherences in experiments.However, a considerable simplification occurs byexamining the equation of motion of the density op-erator, p(r), for the spin system. While exposed toa typical MQ rf irradiation scheme, the spin sys-tem evolves under some non-secular average dipolarHamiltonian H according to ,

p(r) = p(0) + (i/h)r\p(0),n}(i/h)2(r2/2\)[[p(0),n], H] + .

(2 )

From this equation, one notes the appearanceof multiple spin coherences with increasing time.These coherences can only grow one spin at a timedue to the bilinear nature of H within this com-mutator algebra. Amalgamation of clusters of cor-related spins are forbidden under such selectionrule. In addition, the multiple spin coherences are


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212 Bulletin of Magnetic Resonanceweighted by dipolar coupling constants in the com-mutators (for two-spin coherences) and by prod-ucts of dipolar coupling constants arising from thenested commutators (for higher order spin coher-ences). Therefore, one can appreciate that while allspins are coupled to each other, the "propagation"of dipolar correlations depends on the values of thedipolar coupling constants and their products, inaddition to the evolution time r.

Another feature underlying eqn. 2 is that it onlydescribes the growth of multiple spin coherences.This is due to the nature of Schrodinger's equa-tion from which it is derived. The loss of coher-ence in the spin system can only be dealt with bytreating th e lattice and its interactions quantum me-chanically, thereby leading to decoherence. Alterna-tively, relaxation can be introduced phenomenolog-ically within the spin system evolution.Dipolar interactions are long-ranged due to ther~ 3 dependence (see eqn. 1). However, as seen fromthe discussion of eqn. 2, the time-resolved nature ofMQ NMR experiments effectively limits the rangeof dynamical coupling among spins. For finite timeevolution, only the m ultiple spin coherences with thelargest dipolar coupling constants and their prod-ucts should contribute significantly to MQ spectra,i.e., Tr[Iz p(t)]. As a consequence, under such con-ditions, a macroscopic solid can then be viewed asan ensemble of loosely coupled subsystems. The dy-

namics in any MQ NMR experiments correspond tothe average behavior over this ensemble.A few subtle points about this ensemble approx-imation should be clear. The correlation length ofany multiple spin coherences must be smaller thana subsystem size. Interaction energies between sub-systems are assumed weak in comparison to inter-action energies within subsystem s. This amountsto saying that surface effects between subsystemsare neglected for MQ dynamics during an experi-ment; however, in the end, they are necessary to

achieve equilibrium among subsystems in the en-semble, e.g. a uniform spin tem pera ture . Under-lying this ensemble construction is the property ofstatis tical homogeneity. On a subsystem scale, thestatistical properties are stationary in space, i.e., asubsystem is statistically translationally invariant inspace. A physical property of the macroscopic solidis equal to this property averaged over the subsys-tems, i.e., spatial ergodicity exists.

In a sense, this ensemble approach applied tothe non-equilibrium rf-driven spin system permitsthe growth process of multiple spin coherences tobe treated from a statistical mechanics perspective.For example, if all the possible coherences in anN-spin system are excited across such an ensem-ble of subsystems, and all coherences have the sameweight, as would be the case for an infinite excita-tion period, then the intensities in a MQ spectrumcould possibly exhibit a Gaussian profile, i.e.,

exp[-n2/N\ (3)Traditionally, this approximation has been used tointerpret MQ NMR spectra of solids even for finitetime excitation periods (1) when coherences cannotbe treated equally (3). Averaging over products ofdipolar coupling constants in the ensemble (at finitetimes) does not lead to Gaussian behavior (3,4). Inaddition, a thorough examination of the Gaussianapproximation and its implications reveals severalaspects of non-Gaussian behavior in XH MQ NMRspectra of adamantane and hexamethylbenzene (5).Actually, it appears that exponential behavior is amore appropriate description of MQ NMR spectralintensity profiles and their interpretation (5).

The growth dynamics of individual coherenceswithin a subsystem is dictated by eqn. 2. It is possi-ble to map, in an isomorphic fashion, this equationof motion onto a graph model (3). In this graphtheoretical method, the spins correspond to vertices.Colors of the vertices stand for the states of the spinoperators, e.g., Ix, Iy , and Iz. Pairwise dipolar cor-relations are delineated by edges whose weights areproportional to the absolute values of dipolar cou-pling constan ts. The sign of the dipolar couplingconsta nts appears as an irrelevant variable when oneconsiders MQ NMR experiments with time-reversalexcitation; the signal is proportional to the squareof the modulus of the density operator (see eqn.2). Hence, the analytical solution of eqn. 2 canbe examined with a graph ensemble. As an N-spincorrelation is characterized by a product of N spinoperators with N-l dipolar coupling constants (seeeqn. 2), the only permissible graphs in the ensem-ble are trees, i.e., graphs without loops (N verticesconnected with N-l edges). A tree of N vertices hasa weight proportional to the absolute value of theproduct of N-l dipolar coupling constants (edges).This visualization approach provides insight into th e


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Vol. 18, No. 3/4 21 3

0 O1 0 0Figure 1: Flow diagram for products of the angular part [|3cos2 - 1 |< 1. With9 in the angular ranges, 0 < 9 < 35.3 or 144.7< 0 < 180, |3cos2fl - 1 |> 1, hence, the productdiverges to infinity with increasing N. Finally, if theangles in the product vary between these three an-gular ranges, the pro duct will either converge to zeroor diverge to infinity depending on the relative pro-portions of the different ranges and the values of 9sin the product. One should note that for all threeclasses, the product of the angular factors with theuniversal co nstants and length scale factors alwaysconverges to zero in the limit of large N (3,4).The behavior of the products of the angular fac-tors can be summarized with a flow diagram (Fig-ure 1) for the absolute value of the product. Thedynamics of the products are determined by threefixed points as N increases in the product. The sta-ble fixed point 0 corresponds to 35.3 < 9 < 144.7,while the other stable fixed point, oo, attrac ts prod-ucts within the angular ranges of 0 < 9 < 35.3 or

144.7 < 6 < 180. The unstable fixed point, 1, issatisfied only in products where 0 = 35.3 or 144.7.The stable fixed point 0 can be arrived at directlyif any of the angles in the products is equal to 54.7or 125.3. On th e other ha nd, the stable fixed pointoo can only be reached asymptotically as N tends toinfinity.Therefore, anisotropy effects should influence themultiple spin growth processes. On the basis ofthe properties of products of angular factors, onewould expect that the individual correlated clustersof spins or trees to show some anisotropy. Indeed,it would appear that in a magnetic field the growthof large correlated clusters should be dominated bycorrelations within the angular ranges of 0 < 9


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Figure 2: Schematic growth of multiple spin coher-ences according to a maximal spanning tree (MST)algorithm as described in the tex t. The vertices rep-resent spins, while the edges correspond to estab-lished dipolar correlations according to eqn. 2. Twodifferent initial conditions, on the same spin system,are shown on both sides. Time steps: A) 1, B) 5, C)10, D) 15, E) 20, F) 24. Note that both realizationsconverge to an identical cluster of correlated spins(tree) at time steps E and F. The Zeeman field isoriented vertically and parallel to the plane.

size, the number of elements in the distribution, andthe realization of these elements in the sample (3,4).In large networks, it turns out that the most proba-ble value is dominated by the smallest and the mostnumerous dipolar coupling constants and their prod-ucts. On th e other hand, the average value is domi-nated by the largest and the least abundant nearestneighbor dipolar coupling constants and their prod-ucts.

Thus, for any finite excitation time, it is expectedthat the dynamics of the multiple spin coherenceswill be dom inated by nearest neighbor coupling con-stants for pairs of spins, and by the largest prod-ucts of coupling constants for correlated clusters of

Bulletin of Magnetic Resonancethree or more spins. An algorithm can therefore beconstructed, such that a correlation or a tree canonly grow by adding one spin at a time in the direc-tion with the largest coupling consta nt. As a conse-quence, the weight of this tree is always the largestpossible. This algorithm determines the maximalspanning tree (MST). The MST has the maximalweight and spans the entire spin network after someperiod of growth.Figure 2 shows the growth of a MST startingfrom different initial cond itions in a 25-spin network.The choice of a lattice with topological disorder isneeded in order to prevent any degeneracy for theMST. The same generic growth behavior would beobserved in a regular lattice, however, the degener-acy of the MST would obscure the present discus-sion. One should also appreciate th at in this net-work, there is only one 25-quantum coherence pos-sible (see eqn. 2). Yet, th is coherence is weightedby a product of 24 dipolar coupling constan ts chosenamong

hence, there are roughly 1032 possible such productsin this network. The M ST is the one with the largestproduct of coupling constants among all of thesetrees. That the MST is associated with the observ-able 25-quantum coherence in an experiment mightseem surprising at first sight in view of the possible1032 products. Nevertheless, as discussed above andelsewhere (3,4), the statistical properties of multi-plicative processes are dominated by rare events indistribut ions. This consequence is necessary for alogically consistent inte rpretation of MQNMR mea-surements over an ensemble of. loosely coupled sub-systems. In general, for an N-spin system coupledwith N-l dipolar coupling constants chosen among

there are ~NN 2 distinct trees.The multiple spin coherence growth depicted inthe simulations of Figure 2 shows that different ini-tial conditions lead to distinct clusters of correlatedspins at intermediate time steps. The trees of thisgrowth algorithm converge to a unique tree at laterstages (time step 20, Figure 2E) before reaching theMST at the final time s tep. The MST is the fixed


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Vol. 18, No. 3/4 215point attractor for the algorithm. Interestingly, thisgrowth process, at intermediate time steps, can beviewed as a prop agatio n of multiple spin correlationsdirectly on the MST. An ensemble of subsystems,with the growth limited onto the MST, would showa distribution of products at intermediate steps be-fore progressing towards a unique tree at later stagesof growth. In a sense, while such an ensemble looksrandom at early and intermediate times, there issome definite u nderlying order present in the growthprocess. As there exists exactly a unique path be-tween any two vertices on the MST, one could imag-ine simulating "spin diffusion" as a deterministicprocess on the MST of large spin networks.

Another appealing feature of the results in Fig-ur e 2 is the angular distribution of 0s in the MST.Out of 24 polar angles (9), 8 are found in the range35.3 < 9 < 144.7, while the rest are within 0

serge lacelle and luc tremblay- nmr multiple quantum dynamics in large spin networks

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