Accepted Manuscript Finding the True Spin-Lattice Relaxation Time for Half-Integral Nuclei with Non-Zero Quadrupole Couplings James P. Yesinowski PII: DOI: Reference:

S1090-7807(14)00360-7 http://dx.doi.org/10.1016/j.jmr.2014.12.012 YJMRE 5575

To appear in:

Journal of Magnetic Resonance

Received Date: Revised Date:

18 September 2014 17 December 2014

Please cite this article as: J.P. Yesinowski, Finding the True Spin-Lattice Relaxation Time for Half-Integral Nuclei with Non-Zero Quadrupole Couplings, Journal of Magnetic Resonance (2015), doi: http://dx.doi.org/10.1016/j.jmr. 2014.12.012

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Finding the True Spin-Lattice Relaxation Time for Half-Integral Nuclei with Non-Zero Quadrupole Couplings

James P. Yesinowski Chemistry Division, Naval Research Laboratory Washington, DC 20375-5342 [email protected]

Submitted to Journal of Magnetic Resonance, Revised December 17

1

TOC C Graphiccall Absttraact

2

Highlights •

71

•

Methods for measuring T1 of half-integral nuclei with non-zero NQCCs are compared.

•

Spin exchange during ST/CT zero-crossings under MAS invalidates some T1 methods.

•

Full saturation of STs and CT in MAS by an asynchronous pulse train yields true T1.

Ga NMR (I=3/2) of h-GaN:Ge provides good test case for static and MAS T1 methods.

3

Abstract Measuring true spin-lattice relaxation times T1 of half-integral quadrupolar nuclei having nonzero nuclear quadrupole coupling constants (NQCCs) presents challenges due to the presence of satellite-transitions (STs) that may lie outside the excitation bandwidth of the central transition (CT). This leads to complications in establishing well-defined initial conditions for the population differences in these multi-level systems. In addition, experiments involving magic-angle spinning (MAS) can introduce spin exchange due to zero-crossings of the ST and CT (or possibly rotational resonance recoupling in the case of multiple sites) and greatly altered initial conditions as well. An extensive comparison of pulse sequences that have been previously used to measure T1 in such systems is reported, using the 71Ga (I=3/2) NMR of a Ge-doped h-GaN n-type semiconductor sample as the test case. The T1 values were measured at the peak maximum of the Knight shift distribution. Analytical expressions for magnetization-recovery of the CT appropriate to the pulse sequences tested were used, involving contributions from both a magnetic relaxation mechanism (rate constant W) and a quadrupolar one (rate constants W1 and W2, approximately equal in this case).

An

asynchronous train of high-power saturating pulses under MAS that is able to completely saturate both CT and STs is found to be the most reliable and accurate method for obtaining the “true T1”, defined here as (2W + 2W1,2)-1. All other methods studied yielded poor agreement with this “true T1” value or even resulted in gross errors, for reasons that are analyzed in detail. These methods involved a synchronous train of saturating pulses under MAS, an inversion-recovery sequence under MAS or static conditions, and a saturating comb of pulses on a static sample. Although the present results were obtained on a sample where the magnetic relaxation mechanism dominated the quadrupolar one, the asynchronous saturating pulse train approach is not limited to this situation. The extent to which W1 and W2 are unequal does affect the interpretability of the experiment however, particularly when the quadrupolar mechanism dominates. A numerically approximate solution for the I=3/2 recovery case reveals the quantitative effects of any such inequality.

4

Introduction There is continually increasing interest in the high-resolution MAS-NMR of half-integral quadrupolar nuclei, which span the Periodic Table.[1] While the detailed study of spin-lattice relaxation processes in such nuclei has been important in studying the physics of materials such as superconductors, semiconductors and ionic conductors, the vast majority of such studies have been carried out on static samples. The use of spin-lattice relaxation measurements in samples undergoing MAS has the great advantage over static measurements of being able to observe relaxation at individual atomic sites resolved by differences in their chemical shifts, paramagnetic hyperfine shifts or Knight shifts. Detailed structural and dynamical information can therefore become available in MAS-NMR studies. Structural information is obtainable from the effects of localized or delocalized unpaired electron spins upon resolved nuclear sites. This includes both distance information in paramagnetic compounds or proteins and electronic structure information in semiconductors and metals. Dynamical information includes diffusion in ionic conductors or defect diffusion, motions of molecules in porous or mesoporous solids, lattice dynamics, and anion or cation dynamics in biological or electrochemical systems. However, the experimental means both to extract and to interpret T1 values from MAS-NMR of half-integral quadrupolar nuclei are considerably more complicated than for spin-1/2 nuclei, an aspect that may have limited the more widespread application of such measurements to chemical problems. In samples having large nuclear quadrupole coupling constants NQCC and where the satellite transitions (ST) are outside the bandwidth of excitation of the central ½ ↔ - ½ transition (CT), the time-dependences introduced by the magic-angle spinning process itself introduce a number of complications into the measurement of T1 that if ignored can lead to an order of magnitude or more error in the T1 values obtained, as will be shown here. The two major sources of complications involve: 1) establishing a well-defined set of initial conditions for the populations of the 2I+1 quadrupole-perturbed Zeeman levels; 2) the occurrence of zero-crossing (or rotational resonance) conditions between ST and CT during the rotation periods that alter the populations of levels via spin exchange. Kinetic equations describing changes in the populations of Zeeman levels of quadrupoleperturbed spin systems induced by magnetic or quadrupolar fluctuations[2] have been developed for static samples.[3],[4],[5],[6],[7] The effects of magic-angle spinning upon the establishment of well5

defined initial conditions by means of rf pulses was first discussed by Woessner and Timken.[8] The occurrence of zero-crossing conditions of the ST leading to spin exchange[8] (or rotational resonance in the case of differences in chemical and second-order quadrupolar shifts[9]) between ST and CT were observed to markedly influence the return of the spin system to equilibrium. However, neither of these two studies concerned themselves with the quantitative extraction of fundamental relaxation rates.

A comparison of these rates derived from both static single-crystal and MAS-NMR

(polycrystalline powder) of

71

Ga (I=3/2) in the wide bandgap semiconductor h-GaN showed good

agreement when appropriate experimental conditions were used for the MAS-NMR experiments.[10] A Ge-doped polycrystalline sample of h-GaN was chosen for this 71Ga NMR comparison of various methods for measuring T1. This was done because of its good sensitivity and reasonably short T1 due to Korringa relaxation, as well as the detailed understanding of the anisotropic quadrupolar relaxation behavior obtained from the single-crystal study.[10] This work will show how to establish optimum initial conditions, and illustrate the potentially misleading effects arising when such conditions are not met. Magnetization recovery experiments in the same sample that are obtained with different sets of initial conditions and under both static and MAS conditions will be compared. A similar type of comparison has been made previously, but only more qualitatively in terms of an effective T1* rather than a true T1.[8] Both those results and the present ones point to some of the difficulties in obtaining accurate T1 values for half-integral spins with I ≥ 3/2. The theoretical framework for interpreting all of the relaxation results is described in Section 2 of the Results and Discussion. A procedure is described (in Section 3 of Results and Discussion) for making the most reliable T1 determination under MAS conditions, which can be accomplished using an asynchronous train of saturation pulses. Sections 4-7 describe alternative methods that in general are shown to be much less satisfactory (except for the static inversion-recovery method in Sections 4 and 6). Section 8 deals solely with aspects of Korringa relaxation relevant to metals or doped semiconductors.

Results and Discussion 1. Sample (h-GaN:Ge) and 71Ga spectrum

6

The saampple chhoseen ffor study is hh-G GaN N doopeed w withh 0.13% %G Ge (h-GaaN:G Ge)), w whose 771Ga G M MA ASN NMR R sspecctruum (seee F Figguree 1)) exxhibitss a brroadd distrribuution oof K Kniightt shhiftts ffrom m thhe connduuction ellecttronns, witth T1 valluess bbeinng sshorterr foor tthe larrgerr K Knigght shhiftss inn thhe m mannneer ppreddictted byy a K Korrringga rrelaaxattionn mech m hanism m.[111],[122] T Thee peeakk maaxim mum m ((whhosee K Knigght shiift iis 107.8 pppm m) hhas a T1 vaaluee att 2997 K andd 11.7 T of ca. 0.5 ss, w whiich is connsidderaablyy shorrterr thhan thee 3.51 s vallue mplle aandd atttribbuteed iin tthatt caase to a quaadruupoolarr reelaxxation obbseerveed aat 33044 K foor aan uunddopeed sam m mechhannism m due to llattiice

F Figu ure 1.

771

Gaa ssatuuratiionn-reccovveryy M MAS-N NM MR speectrra oof hh-G GaN N:G Ge oobtaaineed with an opptim mal

assynnchrronouss puulsee traain (seee ttextt), aat 11.77 T in a 4 mm mV Varriann rottor spiinniing at 5.00 kH Hz. Thhe small peeakk at rigght corrressponndss to a rregiion of thee saampple havvingg noo K Knigght shiift, andd w was theerefforee ussed as a 0 pppm mK Knigght shiift rrefeerenncee. T Thee solid blaack speectrrum m, oobtaaineed w withh a satturaatioon rrecooveery dellay m exxhibbitiing a w widde ddistrribuutioon oof K Kniightt shhiftss.[111] off 4..0 ss, correesppondds tto a fuullyy-reelaxxed speectrrum T The T1 meeasuurem mennts in thiis ppapeer w werre aall m madde aat 22944 K at thee poosittionn off the ppeakk m maxiimuum in thhis speectrrum m, aat thhe iindiicatted Knnighht sshifft oof 1107.8 pppm m. T Thee daashhed bluue sspeectruum m waas oobttainned w withh a ssatuurattionn reecovverry ddelaay oof 00.1 s bbut nnorrmaalizeed tto nneaarly thee saamee peeakk heeighht aas thhe ffulllyreelaxxedd speectrrum m; it shhow ws tthatt thee reegioon oof tthe speectrrum m haavinng llargger Knnighht shhiftts allso hass shhortter T1 vaaluees dduee to inccreaasedd Korr K ringga rrelaaxattionn, ass quuanntitaativvelyy annalyyzedd prreviiouuslyy.[11] m motiionss prrodduciing fluuctuuatiing eleectrric ffielld ggraddiennts..[111] It iis thhuss a goood appproxxim matiion thaat tthe reelaxxatiion beehavvior oof tthe peeak maxi m imuum in h--GaaN:G Ge is duue onlly to a m maggneetic reelaxxation

7

m mechhannism m oof thhe Koorrinngaa tyype.. H How wevver,, most m t off thhe aanaalyses bellow w doo nnot reqquirre aany suuch appprroxiimaation. 2. Thheoretiicall baackg kgroound too sppin--lattticee reelaxxatiion forr I= =3/22 The thhreee fuunddam mental ratte cconsstannts forr traansitioons bettweeen ennerggy lleveels of a qquaddruupollepeertuurbeed I = =3/22 sppin aree deepicctedd inn Fiigurre 22. Theesee traansiitioons maay bbe iinduuceed bby fflucctuaatioons of thhe eleectrric fieeld grradiientt ((quaadruupoolarr m mecchannism m) orr bby maagnneticc iinteeracctioons (m maggnettic m mechhannism m), eithherr dippollar iinteeracctioons witth ootheer nnucclei or,, inn thee presentt caase, contaact iinteeracctioons w withh coonduuctionn eleectrronns (Korrinngaa rellaxatioon). T Thee quuadrruppolaar m mecchannism m rratees W1 andd W2 aare noot nnecessarilly eequal aandd aree orrienntattionn-deepenndeent as w welll, w whiich cann leead to uunuusuaal m maggneetizaatioonreecoverry bbehavior.[[100] H How wevver, a dettailled

71

G Ga sstuddy of unddopped h-G GaN N iindiicattes thaat aassuuminng

W1 = W2 foor a poolyccrysstallline saampple is a veery goood appprooxim matiionn in this caase.[100]

F Figu ure 2. Sppin––laatticce rrelaaxattionn prroceessees ffor quuadrrupoole-peerturrbeed I = 3/22 nuucleeuss. Q Quaddruupollar reelaxxatiion is byy thhe sinnglee-quuanttum m pproccesss W1 and tthe dooublle quaantuum prroceess W2, whhereeas m magnnettic rrelaaxattionn innvollvess onnly thee sinnglle-qquanntum m pproccess W W. R m ref. [10]]. Repprinntedd frrom

Vaariouus ttheoreeticaal trreatmeentss off reelaxxatioon in qquaadruupoolar sysstem ms in tterm ms of thee kinnetiics of chhanngess inn poopulatiionss off ennerggy lleveels aree citted in thee inttrodducctionn. Thee exxacct sooluttionns oof reeferennce [66] ffor II=33/2 w willl bee reeprooduucedd heere,, inn ordderr to anaalyzze w withh a connsisstennt nnotaatioon thhe vvarriouus reesuults frrom m diiffeerennt m methhodds oobtaaineed herre. Thhose aauthhorss allso givve eexppresssioons forr thhe trrannsitiion rattes 8

between levels for I > 3/2 under the assumption of a short correlation time for fluctuations, in which case only the same three rate constants (but with different numerical coefficients depending upon the levels involved) are needed.[6] The time dependence of the vector N that corresponds to the deviation of the population difference between two adjacent Zeeman levels in a multilevel quadrupole-perturbed system is given by:  

N  RN

Eq. (1)

where R is the reduced relaxation coefficient matrix involving W, W1 and W2, which has eigenvalues λi and eigenvectors E.[6],[10] The specific solution to the set of homogeneous linear differential equations of Eq. (1) depends upon the vector N(0). This (normalized) initial condition vector represents the deviations from their equilibrium values of the population differences corresponding to the three transitions of an I=3/2 spin (see Figure 3). The solution results in a time-dependent magnetization Mj(t) for the observed transition j given by: M t  M∞1 ∑  exp  

Eq. (2)

with the coefficients ai of the eigenrates given by [10]:  

 బ

   .

Although Eqs. (1) and (2) are general for half-integral spins I ≥ 3/2, we give the exact results for the eigenrate vector λ of the relaxation matrix and the corresponding coefficients a1c, a2c, a3c for the central transition that have been previously derived for the I = 3/2 case for the case where all transitions (ST and CT) have been saturated and the normalized initial condition vector N(0) = [-1 -1 -1], as shown on the right side of Figure 3 [6]:

7           6  2    

7        

 

Eq. (3a)

    5     

  0  

Eq. (3b) Eq. (3c)

 

     5    

9

Eq. (3d)

w wherre β = [ (W W1− −W W2)2 + 66W((W1 −W W2) + 25W W2]1/2 andd thhe ssubsscriiptss c oon tthe coeffiicieentss indiccatees thhat itt iss thhe ceentraal traansitionn m maggneetizaatioon beiingg ddesccribbed.

Ass m menntionedd aabove, thhe good

appprroxiimaation (for unndoppedd h--GaaN aandd asssum medd foor hh-G GaN N:Gee as w welll) thhat W1 = W2 ressultts inn β = 5W W. T Thiss reesullts iin a1c = 1.00, a3c = =0, annd ppreddictts a siinglle-eexponeentiial reccoveery wiith thee “sslow w” eiigennraate oof − −(22W + W1 + W2) ≡ −2(W W + W1,2)). T Thuus, tthe eiggennratee hhas equuivaalennt tterm ms ffrom m tthe m magnnettic aandd thee quuaddruppolaar

F Figu ure 3.

D Desscriptioon of thee thhreee poopuulatiion diffferrencess beetweenn thhe ffourr quuaddruppolee-ppertuurbbed

Z Zeem mann leevells oof aan II=3//2 nnucleus foor tthreee ddiffe ferent nnorrmaalizeed iinittial connditionn veectoors N((0). T The B Boltzm mannn (eequiilibbrium m) poppullatioon ddifffereencee iss n0. reelaxxatiion meechaanismss inn thhis ccasee. IIn tthe preesennt ssam mplee, thhe m maggneetic ratte W dom minaates beecauuse of K Korrringga rrelaaxattionn. H How wevver, thhis iis nnot a reequuireemeent of tthe meethood, as lonng aas tthe efffectts oof anny innequuality bettweeen W1 annd W2 aare takken intoo acccoountt, as diiscuusseed iin tthe folllow wingg paaraggraaph.. meaasuurem mennts ffor thee fiirst tim me oon som me sam mplle, uunllesss W1 annd In maakinng aandd annalyyzinng T1 m W2 hhavve bbeenn m measurred by some meeanns, oonee haas tto cconnsidder whhethher assum mingg W1 ≈ W2 m migght be juustifiedd, aand whhat tthe conseequencces of anyy innequualiity m migght be.. T Thee rattio W2/W W1 haas bbeenn m meaasurred inn onlyy a few w iinsttancces (reeferrennce [100] andd reeferrencess thhereein)), aand for som me iioniic ccom mpoounnds coontainingg mole m ecullar grooupps a raatioo of 00.2± ±0.006 [5] inn onne casse aandd off ~5 iin aanootheer [13]] was w deeterrmiinedd. Aggreeemeent wiith ttheoreeticaal ppoinnt cchaargee mode m els of fluuctuuatinng eleectriic ffield ggraddiennts prrodduceed bby nuccleaar ddispplaccem ments iin thhe ionnic lattticee duue tto vvibrratioonss was ppooor. Foortuunattelyy, oone 100

can gain insight into the consequences of any inequality between W1 and W2 by employing the following approximated expressions for the exact expressions in Eq. 3: λ1 = -[2W + W1 + W2 – 0.1(W1 - W2)]

Eq. (4a)

λ3 = -[12W + W1 + W2 + 0.1(W1 - W2)]

Eq. (4b)

a1c = 1 + 0.16 (W1 - W2)/W

Eq. (4c)

a3c = -0.16 (W1 - W2)/W

.

Eq. (4d)

Numerical comparisons of the exact and approximated rates and coefficients for a wide variety of assumed relaxation rates reveals that values obtained from Eqs. 4 are very close (~1-2%) to exact values as long as | W1 - W2 |/W ≤ 5. Agreement is still reasonable for | W1 - W2 |/W ≤ 10. This approximate solution shows clearly that a significant difference between W1 and W2 can still yield a nearly-single-exponential magnetization recovery where | a3c |

a1c and the (nearly) single rate is

given by Eq. (4a). It is apparent from this approximate solution that when the magnetic mechanism dominates the quadrupolar mechanism, the inequality between W1 and W2 can be very large without significantly affecting the results. However, when the quadrupolar mechanism dominates, this is no longer true, and a small inequality can result in significant contributions from the a3c term. This term may have a negative sign, leading to recovery curve shapes quite different from typical biexponential recoveries having identical signs of both terms. In the extreme limit of pure quadrupolar relaxation, it can lead to “overshoots” in the recovery, or to “false” apparent single exponential recovery curves.[10] Observing the former behavior might allow one to obtain the relevant rates, which would be difficult in the latter case. In general, sorting out the relative contributions of quadrupolar and magnetic relaxation mechanisms is very difficult in the “mixed” case, and has been analyzed in detail.[6] Measurements using favorable nuclear pairs such as 71Ga/69Ga whose ratios of quadrupole moment to magnetogyric ratio are very different is one solution (e.g. ref. [10]), as is obtaining results at lower temperatures where the quadrupolar mechanism typically drops off very steeply. Of course, in some cases such as doped semiconductors or compounds or proteins containing paramagnetic ions, the dominance of the magnetic mechanism can either be assumed or established by comparison with relaxation in the undoped systems.

11

The analysis above shows that saturating all of the transitions and measuring a singleexponential recovery of the CT provides the simplest method for measuring what is defined as the “true T1” in this sample, viz. [2(W + W1,2)]-1, whose value is reported in the next section. The CT magnetization recovery curves resulting from other experiments with two different initial conditions are also analyzed and compared to exact theoretical predictions.[6] In both cases this recovery is predicted to have only two different exponential rate constants λ1 and λ3 with differing coefficients. Thus, we are able to use a generalized biexponential magnetization recovery curve to fit all of the experimental data to theoretically-anticipated forms for I=3/2 relaxation by combined quadrupolar and magnetic mechanisms: M(t) = M(0) · (1 – B · [FA · exp(−t/TA) + (1 − FA) · exp(−t/TB) ] )

Eq. (5)

However, the results from such experiments will be shown to suffer from major inaccuracies that result from two separate factors: (a) difficulty in establishing accurately known initial condition vectors in both static and MAS experiments; (b) presence of spin exchange or rotational resonance between ST and CT in the case of MAS. In contrast, the asynchronous train of saturating pulses described in the next section circumvents these difficulties. A limiting value for T1 is obtained from the single exponential recovery as the desired initial condition vector is more closely approached, strongly supporting the accuracy of the measurement compared to other approaches. Further support for the accuracy of the T1 as obtained by the asynchronous train of saturating pulses is its agreement with the value calculated for Korringa relaxation using the measured Knight shift (reference [11] and further results being prepared for publication). Thus, the limiting value obtained in the next section will be used as the “true T1” for comparison with values obtained in subsequent sections. 3. Saturation-recovery with asynchronous and synchronous trains of pulses under MAS When applied to half-integral quadrupolar spins with non-zero NQCC values, the widelyused technique for measuring T1 of saturation-recovery requires all transitions be saturated, not only the CT.[10] Thus, to establish the desired initial condition vector of [−1 −1 −1] for the I = 3/2 case requires that both STs be saturated as well as the CT. For higher half-integral spins, more generally an initial condition vector N(0) having all values equal to −1 must be created. For static samples this can be difficult to achieve, although changing the transmitter frequency during the saturation pulse 12

train proved an effective strategy for an h-GaN single-crystal film sample.[10] For quadrupolar nuclei having sufficiently long T1 (e.g. at cryogenic temperatures) hoisting the sample in and out of the B0 field enabled an initial condition of all transitions saturated (i.e. equilibrated in near-zerofield).[14] The very act of magic-angle spinning provides the convenient means with which to saturate the STs without changing the transmitter frequency: the instantaneous ST frequency crosses the CT frequency at least twice a rotor cycle, and thus can be affected by a saturation pulse at or near this crossing. In order that the ST arising from all crystallite orientations in the rotor be affected by at least one of the saturating pulses, the spacing tXsat should not equal any multiple of the rotor period tr (which would yield a “synchronous train”). Rather, the spacing of the n pulses should satisfy the condition n*tXsat/tr ≠ an integer for all n less than some sufficiently large value N, which then can be chosen as the number of saturating pulses to be used. One example of a proper choice of spacing for the 5.0 kHz MAS spinning speed would be 1008 µs. Since 5 rotor periods is 1000 µs, the first pulse is at 5.04 rotor periods, the second at 10.08, the third at 15.12, and so on. The net effect is to have pulses at some 25 different rotor orientations equally spanning a full rotor cycle. Another proper choice of spacing, used in two experiments in Table S1, is 1097 µs, corresponding to 5.485, 10.97, 16.455, and 21.94 rotor periods for the first four saturating pulses. After about 17 pulses the same number of different rotor orientations have been subjected to a pulse, but in this case subsequent pairs of pulses affect rotor orientations 180° from each other. This choice could conceivably offer an advantage in obtaining more complete saturation of the ST if T1 relaxation occurring during the saturating train is at all significant. The pulse sequence used in the asynchronous and synchronous trains in Table S1 consists of a saturating train of qXsat number of pulses with a spacing of tXsat greater than T2, followed by a variable recovery interval and a single detection pulse. The pulse lengths were chosen to correspond to a selective 90° pulse for the CT (measured from nutation data as one-half of a 180º pulse). Increasing the pulse length slightly to that calculated for the ST (for I=3/2 the CT 90° pulse length is 0.500 times that in solution, the factor for the ST is 0.578[15]) did not affect the results. For the ST of higher half-integral spins, the greater difference between the 90° pulse lengths of the higher ST and the CT might warrant using an increased pulse length in the saturating train (but see below).[15]

13

As justified in section 2, the CT magnetization recovery was fit to a single exponential T1 time constant corresponding to the eigenrate of −2(W + W1,2). Two different rf field strengths were used for the data given in Table S1, each with 90 saturating pulses. It is apparent that a higher value for T1 (491.4 ms) is obtained only with the higher rf field strength (1.7 µs 90° pulse), which is certainly not selective for the CT only. In other experiments not shown a qXsat of 60 gave close to this higher value for T1, but a qXsat=30 gave a significantly shorter value for T1. It is in this sense, i.e. T1 vs. the number of saturating pulses qXsat, that we treat T1=491.4 ms as the limiting value of T1. The T1 values obtained using a lower rf field strength (5.6 µs 90° pulse) are significantly shorter (411.5 ms for qXsat=90), and steadily decrease as qXsat decreases from 90 to 30 (results not shown). This trend is entirely plausible for the present situation where magnetic relaxation dominates, since any deviation from the ideal initial condition vector would result in a non-zero contribution from the coefficient a3c in Eq. (3d) characterizing the 6-fold faster eigenrate λ3 from Eq. (3a).

This

contribution would produce a shorter apparent T1 in a single-exponential fit. This effect is even more pronounced for the data obtained with a synchronous train, which yields T1 = 253.4 ms, due to far from ideal saturation of the ST. The overall conclusion is that a limiting (and maximal) value for T1 can be obtained from a saturation recovery experiment with an asynchronous train of pulses, as long as these are sufficient in number and rf power. The limiting value measured, T1 = 491.4±1.3 ms, represents what is termed here the “true T1”, i.e. the most accurate value obtainable, by the arguments given at the end of section 2. The required condition that the STs be saturated as well as the CT cannot be ascertained merely by observing a small or near-zero CT intensity for short recovery times, since the ST peaks are not being monitored. However, one way to establish that the STs are indeed being saturated is to monitor the CT intensity at a short recovery time for several recycle delay values and many acquisitions. If the CT intensity increases as the recycle delay goes from T1 , this indicates that spin exchange from STs that have relaxed towards equilibrium during the recycle delay and are not being fully saturated by the pulse train acts to enhance the CT intensity (see next section). One should note that the choice of a selective 90° pulse length for saturation, as discussed above, is not necessarily optimal. This is so because of the complicated interplay between the rf field strength, the NQCC, the pulse length (assumed short compared to the rotor cycle), the number of pulses qXsat, and the separation between an ST doublet and the CT of a given crystallite at each time 14

a pulse is applied. Crystallites oriented such that their ST doublets remain at all times much closer to the CT than the strength of the rf field will be most effectively saturated by a non-selective 90° pulse length, i.e. one twice as long as used. However, they generally represent a small fraction of the sample, i.e. those whose frequency variation over a rotor cycle remains entirely within a narrow bandwidth. The majority of crystallites will have ST doublet peaks at some different distance from the CT for each pulse, and thus will be excited by pulses that are not always completely selective. Although increasing the pulse length above that of the selective 90° would seem to help compensate, there will be an undesirable decrease in the larger effective bandwidth of excitation obtainable from shorter pulses. Fortunately, from an experimental standpoint it was possible in this study to establish that the desired initial condition of complete saturation of all transitions could be achieved, based upon the simple criterion of a maximal measured value for T1 as the number of saturating pulses was increased. In other applications, it may be worthwhile to treat the saturating pulse length(s) as an additional experimental variable to achieve the desired initial condition. 4. Inversion-recovery experiments: Static vs. MAS The static and

71

Ga MAS-NMR magnetization-recovery behaviors in the inversion-recovery

experiments are shown in Figure 4, and further information including fitting parameters is in Table S1. The 90° pulse length on the CT was 6.6 µs, which given the large axially-symmetric NQCC (e2qQ/ħ) of 1.72 MHz [11, 16]

implies a highly-selective pulse on the CT.

The general

biexponential fitting function of Eq. 5 used to describe M(t), the magnetization recovery of the CT at time t in terms of the equilibrium magnetization M(0) has B as an adjustable parameter that equals 2.0 in the case of ideal inversion efficiency, a parameter FA representing the fraction of signal having a recovery time constant of TA, and the time constant TB applying to the remaining fraction (1 − FA). For the static sample, FA was fixed at 0.9, since for a magnetic relaxation mechanism 90% of the recovery occurs at a rate 6 times faster than the remaining 10% (i.e. TA = TB /6).[17, 18] As Table S1 shows, a good fit is obtained with the two time constants TA = 119.9 ms and TB = 774.5 ms, with a TB/TA ratio of 6.46 ± 0.6 that agrees with the theoretical value of 6.0. However, this value for TB is much larger than the most reliable T1 value of 491.4 ms obtained and discussed in the previous section. A fit using the constraint TB/TA=6.0 offers little improvement, yielding TB = 690.4 ms. Figure 5 shows graphically the range of T1 values obtained in this experiment in comparison to the values obtained by the other methods. 15

The abbovve cconnstraaintts oon FA aandd TB/TA appply too reecoverry aafter anny seleectiive pulse onn thee C CT, w whettherr a 1880° puulsee inn thhe casse oof invverssionn reecovverry oor a 990° puulsee inn thhe ccase oof sseleectiive saaturration.. F For tthe invversionn reecooverry ccasee, thhe vvecctorr N((0) is [+11 −22 +1], whhereeas forr the 990° pullse caase it iis [ +00.5 −1 +00.5]. B Botth innitiial cconndittionns reesuult iin thhe sam me cconnstrrainnts oon tthe bieexpponeential reecoverry ccurvves..

F Figu ure 4. Fiit oof

71

Gaa innveersioon-reccoveery peeak inttenssityy off Knnigght-shifftedd m maxximuum off h-GaaN:G Ge

sppecctrum m uundder stattic andd MAS M S coondditioons to Eqquattionn 5 witth pparaameeterrs ggiveen iin T Table S S1. Thhe leeft siide shoowss ann exxpaandeed rregiionn of thee daata whhosee fuull rregiion is sshoownn onn the riightt, too deemoonsstratte tthe hiighh quualiity of botth ffits. N Notte tthe maarkkedlly iincrreassed rellaxatioon ratee foor tthe daata obttainned wiith M MAS S, aas a coonseequuencce oof zzeroo-crrosssinggs aand spiin eexchhannge bettweeen CT T annd S ST.. H How weveer, thee efffecct off magi m ic-aangle sspinnninng upoon tthe invverssionn reecoverry bbehaavior iis qquitte ddram mattic, ass Fiiguure 4 shhow ws. Thhe reccoveery curvee is bieexponeentiial as iin tthe staatic casse, butt clearrly ddoees nnot fit w well to thee fiixedd FA vvaluue oof 00.9 thaat ddesccribbed thee sttaticc reesults weell. Insteaad, a ggoood ffit ccouuld be obbtaiined w withh FA = 0.773 aassoociaatedd w with a vveryy shhortt TA = 31.5 m ms,, annd a muuchh loongeer TB = 2883.7 m ms. T The reaasonns forr this ddram maticaallyy diifferennt bbehaavioor ffrom m tthe staaticc innverrsioon-rrecoovery expperiimeent caan bbe uunddersstoood iin teerm ms oof thhe zzero-ccrosssinng spinn exxchaangge eeffect.[[8],[9] T The freequeenccy oof a givvenn ST T (ee.g.. +11/2 ↔ +33/2)) deepenndss uppon thee orrienntattionn off eaach crrysttallite’’s aaxiaallyy syymm mettric NQ QC CC ttenssor’s pprinncippal axis w withh reesppectt too thhe eexteernaal m maggnettic fiieldd. A Anyy ggiveen S ST frequeency oof a neeighhboorinng

771

71

G Ga spiin w willl haapppen to coiincidee wiith thee CT ffreqquenncyy off a

G sspinn too whicch it iss diipollar--couupleed at ssevveraal ppoinnts ddurringg a rottor cyccle.[8] Thhis Ga 166

m tthe faact thaat the first-oordeer quaadruupoole perturrbaationn inn tthe obbseervaatioon ccann bee eeasilly seeen ffrom H Ham miltooniaan aveeraggess too zeero foor aall cryystaallitee oorieentaationns in thee M MAS rrotoor rregaarddless oof tthe assym mmeetryy paraameeter,, annd tthuus m musst nneceessaarilyy ccrosss zzeroo att leaast tw wicee duurinng a rootorr cyyclee.[119] A At ssuchh ttimees of rottatioonaal rresoonaancee condditioonss, ccrosss-rrelaaxattionn (sspinn eexchhannge)) betw weenn tthe diiffeerennt trransitionss (C CT andd ST T) of tthe dippolaar-ccouupleed hhom monnucllearr paair of sspinns ccann takke pplace bby m meanns of thee fliip-fflopp (II1+I2− + I1−I2+) term ms in thee ddipoolar Haamiiltooniaan tthatt coonservve the Zeem man ennergy. T Thiss prroceess will teendd too eqqualizee thhe ppoppulaationn ddiffeerenncees bbetw weeen thhe twoo leevells oof tthe ST aand thoose of thee CT T.

F Figu ure 5. Com C mparrisoon oof 771Gaa T1 vaalues m meaasuuredd byy diiffeerennt teechnniqquess inn h-GaN N:G Ge at a Knig K ght shhiftt poositiionn off 1007.88 pppm (seee F Fig. 1)). R Ressultts aare ffor booth sstattic andd MAS M S (sspinn raate = 55.0 kH Hz) exxpeerim mennts. Thhe rrighhtm mostt exxpeerim ment yiieldds tthe moost rreliiablle, ii.e. “trrue T1 vallue, whic w ch iis shhow wn byy thhe rred dassheed liine.. Wee caan aachiieve a moore quaantiitattivee deescrriptiionn byy coonsiiderringg a pprooof givven by Sliichtter iin hhis diiscuussiionn of sspinn ttem mperratuure.[200]

H He dem monnstrratees froom sim mplle microoscoopicc rreveersiibiliity

arrguumeentss thaat aan II=3//2 sspinn syysteem initiallly awaay from m eequuilibbriuum (annd hhence nnott deescrribaablee byy a sppin tem mpeeratturee) w willl byy viirtuue oof thhe dippolaar fl flip--floop tterm mm menntionnedd abbovve aachiievee a poopullation 177

distribution among the four levels governed by a Boltzmann distribution (i.e. be described by a spin temperature). This argument assumes that the spin exchange process is much faster than relaxation to the lattice, which is often the case for abundant spins that have many strong homonuclear dipolar couplings and hence fast spin exchange. However, even though the system achieves a common spin temperature after internal spin equilibration, this temperature will not in general equal the lattice temperature. We can see this most clearly by considering the total Zeeman energy of an ensemble of I=3/2 spin systems: /

 total 

$ % 

Eq. 6

/

where pn is the population of the n-th level, whose Zeeman energy is given by En = − γ ħ B0 mn, with mn = 3/2, 1/2, −1/2, −3/2. We can with complete generality (for all but exceedingly low temperatures or low B0 fields) represent the level populations at some equilibrium lattice temperature as p+3/2 = N + 8δ, p+1/2 = N + 6δ, p−1/2 = N + 4δ, and p−3/2 = N + 2δ, since the population difference between adjacent levels is equal (and equal to 2δ). We then define a unit of Zeeman energy b as b = γħB0. The EZ(total) at equilibrium is then −10·δ·b, since the summation in Eq. 6 over the terms containing N is zero. An ideal selective 180° inversion pulse on the CT interconverts the equilibrium values of p+1/2 and p−1/2 and raises the total Zeeman energy to −8·δ·b. After spin exchange or rotational resonance yields an internal equilibrium in the spin system, the spins are described by a higher spin temperature (1/0.8=1.25 times the lattice temperature). The corresponding populations are p+3/2 = N + 7.4δ, p+1/2 = N + 5.8δ, p−1/2 = N + 4.2δ, and p−3/2 = N + 2.6δ. For comparison, an ideal selective 90° saturationpulse on the CT results in p+1/2 = N + 5δ, p−1/2 = N + 5δ, with the other two pn values unaffected, leading to a total Zeeman energy of −9·δ·b. In the case of complete saturation of both CT and ST by rf pulses, all pn values become N + 5δ and the total Zeeman energy is then zero. This situation corresponds to an infinite spin temperature, and does not require spin exchange to take place. We see immediately that the NMR signal strength of the CT in a sample at equilibrium at the lattice temperature is proportional to 2δ, that the signal strength immediately after an ideal selective 180° inversion-pulse on the CT is proportional to −2δ, and that after rapid spin exchange but before any significant spin-lattice relaxation the signal strength is proportional to 1.6δ. This implies that ideally 90% of the recovery of the inverted signal is due to the zero-crossing spin exchange process, and if this process can be represented by a single short relaxation time TA in Eq. 5 then FA = 0.9, and 18

TB should then represent the true T1 of the spin system. We note that even with a non-ideal inversion pulse, e.g. with B in Eq. 5 equal to 1.58, or for a 90° CT-selective saturation pulse, the fraction FA = 0.9 under these assumptions of a common spin temperature due to spin exchange. It is important to note that the prediction of FA = 0.9 using this model of very rapid spin exchange due to zero-crossings under MAS arises from completely different reasons than in the static case analyzed earlier, which was for a magnetic relaxation mechanism without any spin exchange. In this static case, the transition rates between levels involving only the magnetic mechanism W (see Figure 2) yield kinetic equations for the population differences; the corresponding solutions involve two eigencoefficients (0.9 and 0.1, thus yielding FA = 0.9 in Eq. 5) and two eigenrates. However, unlike the situation in the MAS case, the two eigenrates here are necessarily related, with the time constant TA equal to TB/6. For the MAS inversion-recovery data in Fig. 4, however, constraining only the parameter FA in Eq. 5 to equal 0.9 resulted in a noticeably poor fit showing systematic deviations from the experimental points and a time constant TB significantly larger than the true T1 (see Table S1). Allowing FA to vary resulted in a much better fit (with FA = 0.73), but the time constant TB was now significantly smaller than the true T1 value in this case. The reason for the experimentally derived FA being less than the theoretical value is unclear. Possible effects of higher spinning speeds upon the spin-exchange that takes place during inversion-recovery under MAS have not been investigated here due to a desire to minimize the heating effects of induced eddy currents in the electrically conductive sample, which depend quadratically upon spinning speed.[21] The simplest view is that at a higher spinning speed the shorter period in which spin exchange can occur is exactly counterbalanced by the greater number of such periods, leading to the same fast-relaxing time constant TA. However, the ratio each saturating pulse width to the rotor cycle time would also increase, with resultant effects upon the saturation behavior of the train of pulses. As long as a completely saturated initial condition is obtained, this effect would not matter. A second possible complication is spinning-speed dependences of the linewidths due to second-order cross-terms between e.g. the quadrupole coupling and csa.[22] The possible effects of higher spinning speeds upon rotational resonance involving multiple sites are discussed in the next section.

19

5. Implications of zero-crossing spin exchange and rotational resonance recoupling effects for inversion-recovery experiments, including spins I > 3/2 and multiple sites The above results and discussion indicate that the population transfers between CT and ST effected by spin exchange under MAS conditions produce a population difference for the CT, after the 180° pulse and subsequent rapid spin exchange, much closer to the thermal equilibrium situation. Thus, only a small fraction of the magnetization recovery will be due to T1. For I=3/2 this fraction is 0.1, and it will be even smaller for higher spins, as an extension of the spin-temperature arguments shows. Furthermore, the assumption that spin exchange is much more rapid than T1 will not always apply, particularly for samples where the homonuclear dipolar spin couplings are weak. In such situations, a meaningful analysis of magnetization recovery becomes intractable. The preceding discussion of spin exchange has involved a situation where the pairs of dipolarcoupled nuclei are assumed to have identical chemical shifts, a moderate NQCC and identical csa and quadrupolar tensor orientations. This is a reasonable assumption for h-GaN:Ge. However, in the more general case, where multiple sites may be present, the description of spin exchange becomes very complicated, as discussed previously[23],[9],[24], and as can be surmised from the many single and double quantum transitions possible and depicted in the 16-level diagram of a homonuclear dipolar-coupled spin-3/2 pair (Figure 1 in reference [24]). Theoretical discussions of spontaneous recoupling and rotational resonance between ST and CT of homonuclear dipolar coupled pairs can be found in these papers and their references, and illustrate the importance of second-order quadrupolar shifts as well as the effects of spinning speed and deviations from the magic-angle. Because of this complexity of the recoupling behavior between half-integral quadrupolar spins, it is likely to prove difficult in any particular case to establish the rate at which any possible ST/CT spin exchange induced by recoupling between homonuclear dipolar-coupled pairs is taking place. Therefore, one conclusion of this work is that for MAS-NMR experiments, inversion-recovery or saturation-recovery experiments that operate selectively on the CT are not a useful way to measure T1, unless the spins are well-isolated from each other, either chemically, isotopically, or by motion of ions. 6. Static inversion-recovery with recycle delay ≈ T1

20

One might ask whether one could shorten the total experiment time for a static selective inversion-recovery experiment using a recycle delay