J . Chem. SOC., Faraday Trans. I , 1988, 84(11), 3713-3730 Spectral Spin Diffusion in Polycrystalline Solids under Magic-angle Spinning Atsushi Kubo and Charles A. McDowell" 2036 Main Mall, Department of Chemistry, University of British Columbia, Vancouver, B.C. V6T I Y6, Canada Spectral spin diffusion in 13C NMR of double 13C-labelled sodium acetate trihydrate (SAC), and in 31P NMR of zinc(I1) bis(0,O'-diethyldithio- phosphate) (ZNP) has been studied under magic-angle spinning conditions. Spin-diffusion time constants, qD, were determined from the intensities of the spinning sidebands in experiments using rotation-synchronized DANTE pulse sequences, at several different spinning frequencies. The theory of Suter and Ernst, developed for spectral spin diffusion in single crystals, was extended to the case of polycrystalline samples rotating under magic-angle spinning conditions.We considered two mechanisms for the spin diffusion, i.e. dipolar interaction and J-coupling. The spin-diffusion time constants, qT), were related to the zero-quantum lineshape functions in a manner similar to the theory of Suter and Ernst. The zero-quantum lineshape functions were estimated from the observed single-quantum lineshape functions. In the present studies the dependence of the experimental values for T,, on the rotational frequency v, are in good agreement with those calculated from the theory based on the dipolar interaction mechanism. The values of T,, for SAC showed a deep minimum at Am z 2m,, and a shallow minimum at Am z 3m,. This phenomenon is rotational relaxation resonance.1. Introduction A major development in solid-state nuclear magnetic resonance spectroscopy occurred with the introduction of magic-angle spinning (MAS) of samples.'. Because MAS imposes a time dependence on all nuclear anisotropic interactions, this causes the averaging of inhomogeneous interactions and yields well-resolved NMR spectra of amorphous and polycrystalline solids. Later, when combined with cross-p~larization,~ and high-power proton decoupling,** it led to modern high-resolution solid-state NMR spectroscopy.6 It was early observed that in addition to the narrow central line there appeared satellite lines, or sidebands, on either side at integral multiples of the rotation frequency.'. Subsequently it was shown that when the spinning frequency is comparable to, or less than, the chemical-shift anisotropy, information on the anisotropy and orientations of chemical shift tensor can be obtained by appropriate mathematical analysis.'-' The combination of two-dimensional methods with the MAS technique has recently led to important developments whereby accurate and detailed information can be obtained not only about the anisotropies of chemical shifts, but also direct and indirect dipolar couplings, and slow molecular orientations, from the appropriate analysis of the rotational sideband patterns. Spin diffusion between nuclei with different chemical shifts (spectral spin diffusion) has recently been studied by the combination of two-dimensional NMR exchange techniques,l0' 22 or by the application of selective pulse excitation Since spin diffusion is largely induced by homonuclear dipolar interaction, it is possible to determine the strength of the dipolar interaction, or the internuclear distance, from the 37133714 Spin Digusion in Solids under MAS observed spin-diffusion time constants.Several applications of spectral spin-diffusion measurements based on these ideas have recently been reported. 26-32 Theoretical analyses of spectral spin diffusion have recently been reported by Suter and Ernst22 and by Henrichs et al.,33 both using different approaches to analyse spin- diffusion rates. We have reported our detailed studies of 31P spectral spin diffusion in single crystals of dipotassium a-D-glucopyranose- 1 -phosphate dihydrate, and also triphenyl phosphine, and have assessed the above theories as to their applicability to the 31P NMR spectral spin-diffusion rates in single In this publication we report the results of an MAS study of 13C doubly labelled sodium acetate (SAC), and zinc(rr) bis(0,O’-diethyldithiophosphate) (ZNP) under MAS conditions.One aspect of this paper is the study of the effect of spinning frequency on the spin-diffusion time constant ( qD). There are few studies on spin diffusion which have paid attention to the effect of the spinning f r e q ~ e n c y . ~ ~ . ~ ~ Another important matter is the contribution of the homonuclear J coupling to spectral spin diffusion. The Hamiltonian of the homonuclear J coupling also includes the so-called flip-flop term, although it is often much smaller than the homonuclear dipolar coupling.Under MAS conditions, the J coupling Hamiltonian included the static part, while the dipolar interaction is modulated by co, and 2cou,, where u), is the spinning angular frequency. In favourable cases it may be possible to measure these different contributions to the spin diffusion process. 2. Theoretical 2.1. Nuclear Relaxation under Modulation of the Hamiltonian Under MAS conditions, many terms in the Hamiltonian become time-dependent. Modulation of the Hamiltonian also can be introduced by applying a radio-frequency (r.f.) magnetic field, or by the use of various multiple-pulse experiments. Haeberlen and W a ~ g h ~ ~ discussed spin-lattice relaxation phenomena in periodically perturbed systems.They started from the following equation of motion of the density matrix p given by Abragam : 38 dp = - dT(W*(t), W * ( t - r), p 1 L . (1) - dt s:; In this paper we use a slightly different method to calculate the relaxation times; our method is probably easier to understand. The time-dependent Hamiltonian is generally written as %(I) = a#) T,+a2(t) q+ ... +a,(t) T, (2) where the a,(t) coefficients are time-dependent. The q terms are nuclear spin operators. The time-evolution operator, U(t), is given by U(t) = Texp( -i [ dtl.hY(tl)) where T is Dyson’s time-ordering operator. We assume the observable B does not commute with & but commutes with q, k # 1, because we can often find such cases. We divide the Hamiltonian into two parts, .hY1(t) and S 0 ( t ) , as defined by the following equations:A .Kubo and C. A . McDowell The time-evolution operator Uo(t) of the Hamiltonian So(?) is given by the equation 3715 Uo(t) = Texp -i dt, Zo(tl) . ( s , 1 Since Uo(t) commutes with B it does not affect the motion of the quantity B. following equation : By differentiating both sides of eqn (6) we obtain The time-evolution operator U*( t ) in the interaction representation is given by the (6) U*(t) = U&-’ U(t). (7 4 where Zl*(t) = Uo(t)-l A?#) q)(t). (7 b) (8) d - U*(t) = -iX,*(t) U*(t) dt Integrating this equation we obtain We define the decay function of the quantity B by the following equation: Tr{BU(t)-l BU(t)} Tr { B2} JB(t> = (9) On expanding the exponential in eqn (9) we obtain the result that The first-order term in eqn (10) is zero, because Tr{ABC) = Tr{CAB): The second-order term includes the two-times correlation function GBt(t), of the quantity B = i/d[T,, B], namely Tr { B’ &( t)-l B’ &( t)} Tr {(B’)21 G,,(t) = The factor, d, is defined as Tr(S2} = Tr{B2} and this relation is fulfilled, so that d is given by Thus second-order term in eqn (10) may now be written as The higher-order terms in eqn (10) consist of the more than two-times correlation function of the quantity c(t) = U;’(t) &(t).When X,*(t) is a random function of t, and the time-correlation functions of X:(t) have much shorter correlation times than the relaxation time &(I), it is often found experimentally that jB(t) is an exponential ... . (14) j B ( t ) = exp(-t/T,) = I--+-- function of t, i.e.t t2 TB 2! T i The relaxation time T, can be obtained when we compare eqn (12) or (1 3) with eqn (14) under the assumption that G,,(t) has a shorter correlation time than TB.3716 Spin Difiusion in Solids under MAS We write al(tl) as the sum of its Fourier components: al(t) = C A(w) exp(iwt). W Eqn (13) can thus be rewritten as - d2 2 A(w) A(w') f dt, r' dz exp [i(u + w') T3] GBf(z) exp (- iw'z) (16) w , 0, J o J o since G,,(t) vanishes in a much shorter time than TB, and because we are interested in the behaviour of j,(t) on the timescale of TB the upper limit of the second integral can be replaced by infinity. The terms w+w' = 0 give a contribution which is proportional to t. The other terms give rise to an oscillating term with frequency w + w'.The relaxation time TB is therefore given by (17) 0 1 - = d2 Ts A(w) A( - w) dz G,,(z) exp (- iwz). This equation can also be derived from eqn (1) by replacing Z * ( t ) by H,*(t) defined by eqn (7). Note that the time dependence of the terms of the Hamiltonian other than the perturbation term Zl(t) are included in the time-evolution operator Uo(t) given by eqn (5). The conditions which are necessary for eqn (17) to apply are that TB % 27r/w,, zB,, where w, is the frequency at which the Hamiltonian is modulated, and zB, is the correlation time of G,,(t). The following conditions have been fulfilled for the second condition TB % zBf to be satisfied; (1) 0, zBr -g 1 ; d << l/zB,. (2) If w, zB, 2 1, d << urn. 2.2. Hamiltonian under MAS Under MAS conditions, the secular Hamiltonian in the high magnetic-field region may be written as where Ttm are irreducible spherical tensor components of the nuclear spin operators.A represents the various types of interactions ; 2 (Zeeman interaction), Cx (chemical-shift interaction of the X spin), D,, (dipolar interaction between the X and Y spins) and J (J-coupling between the S , and S, spins). The list of the coupling parameters C', R', d' and T,I' is given by Spie~s,~' and we reproduce in table 1 the spin operators TAo, Tio and Tio. The second term, which represents the antisymmetric part of the interaction, differs from zero, only for the case involving J-coupling. The time-dependent factors cJ(t) are given by 1 Zy = C'( TAo R' + T;, E'( t ) + Tio d$ d' ti( t)) (18) ~ ~ ( t ) = C DL exp (imco, t ) (194 (19@ m--1 1 05 = - [P'l, %30(W + P A %(Q') + P L 9:1o(Q')l d3 where and pfo = -iy'2p&, pfil = p:zfip& and 9k,(n) are components of the Wigner rotation matrix.a' represents the Euler angles, which define the transformation from the principal-axes system of the interaction 3, to the rotor-fixed axes system. We used the same definition of the Euler angles as Spie~s.~' Another time-dependent factor <'( t), is given by c'(t) = At exp (iw, t ) + dl exp ( - iw, t ) + A; exp (2iw, t ) + A!, exp ( - 2iw, t ) (20a)A . Kubo and C. A . McDowell 3717 Table 1. Spin-dependent operators Ti,,, in laboratory-axes systema ~~ ~ a This table is based on information in ref. (39). 2.3. Spin Diffusion under MAS We consider the case of spectral spin diffusion between two spins S = + in the presence of abundant I spins.The commutation relations of the Hamiltonian are discussed in detail by Suter and Ernst.22 The observable of the spin diffusion process B = Sl2--S2,. The operator corresponds to the z component of the zero-quantum transition We introduce the following zero-quantum transition operators : R, = %SI+ s2+ + SI- s2+> (21 4 Rz = ;(slz-s2z). (21 c ) The Hamiltonian of the dipolar interaction between the S, and S, spins is expressed (22) The term Sl,S2, commutes with the observable of the spin diffusion R,, and does not affect the spin diffusion between the S , and S, spins. We can choose r: fir;: R, yDs,sz(t) as Xl(t). The spin-diffusion time constant due to the dipolar interaction T& is given by the following equation : by using the zero-quantum transition operators as shown in the following equation: H;((t) = -7; Ar1,3(2S1, S2, - R,) yDsisz(t).C A S l s z A”;TrSz dz GRY(z) exp (- imwr 7). (23) som - = d2 1 T:D m-*1, k2 We introduce the zero-quantum lineshape function KR,(w), defined by KRY(w) = som dz G,$z) exp (- iwz). Eqn (23) can now be rewritten as 1 -- - d2{A~slszA”~~P[KRy(OT) + K R y ( - ~ , ) ] + A f s , s , A_D,s1s2[K420r)+ KRI(-2mr)]}. (25) GI, For powder samples, a distribution of T& can exist, if A f s , s , ADf,s, or Afs,s2 ADglse have strong angular dependences. However, if spin diffusion between polycrystallites3718 Spin Digusion in Solids under MAS is fast enough to maintain a uniform spin temperature over the sample, the sample can still have a single spin-diffusion time constant, which is calculated by integrating eqn (25) over OD.We thus obtain the equation We ignore the orientation dependence of the zero-quantum lineshape function KRy(co). The Hamiltonian for the case with J-coupling is given by The spin-diffusion time constant due to J-coupling, cD, is By taking the powder average, c,, is calculated as It is noteworthy that eqn (29) includes the KRy(0), co = 0, component of the zero-quantum lineshape function, while that term is absent from eqn (26). Eqn (26) and (29) are similar to the equation which Suter and Ernst,22 Henrichs et al.33 and Kubo and M ~ D o w e l l ~ ~ derived for qD, the spin-diffusion rate in single crystals. The corresponding equation for the single-crystal case only contains the KRy(0) term.If we wish to obtain information about the strength of the dipolar interaction, or of the J-coupling, we need to know the zero-quantum lineshape function.' 2.4. The Estimation of the Zero-quantum Lineshape Function from the Single-quantum Lineshape Functions Henrichs et al.33 proposed a method to estimate the zero-quantum lineshape function, KR (co), from the single-quantum lineshape functions measured without the application of 1 spin decoupling. In another p ~ b l i c a t i o n ~ ~ we have discussed in detail the limitations of this method. At first we assume that the decay function of the zero-quantum coherence is given (30) where Il is the intensity of the lth sideband of the zero-quantum line, TR is the linewidth of each zero-quantum line.The Fourier component is given by by GRY(t) = exp (- t / TR) f I,{[exp - i(co, I + Aco) t ] + exp [i(co, I + Am) t]> 1 KRy(?i'lcU,) = d t GRV( t ) eXp ( - imo, t ) T R ). (31) TR JI = ' I' (1 +i[co,(l+m)+Aco] TR+ 1 -i[co,(l-m)+dco] TRA . Kubo and C. A . McDowell 3719 If two spins S, and S , are far from each other, and the correlation of the dipolar fields at the S , nucleus and the S, nucleus produced by I spins can be ignored, &(t) can be written as the product of the two evolution operators, U f ) ( t ) Uf)(t). Note that Ut)(t) includes the Siz and I spin operators, referring to nuclei which are close to the Si spin. The following commutations for interaction relations are satisfied : [ UP)(t), Uf’(t)] = [ UF)(t), S,,] = [Uf’(t) S,,] = 0.(32) By using these operators the zero-quantum lineshape function can be rewritten as We now introduce the decay functions of the single quantum coherence, L By using eqn (34), we can write eqn (33) as GR(t) = $~y)(t)~?’(t) +j?’(t)~y’(t)]. We approximate the single-quantum lineshape function by (35) J‘,“)( - t ) = C 4”) exp [ T i(w, I + w(”) t] exp ( - t/ F,”)) (36) 1 where 4k) is intensity of the lth sideband of the spectrum of the Sk spin, a(”) is the angular frequency of the isotropic peak of the spectrum of the Sk spin. The term l/Tik) is the width of each sideband of the spectrum of the Sk spin. Eqn (35) can be rewritten by using eqn (36) to yield the expression x (exp {i[w,(l- m) + dl) - d2)] t } + exp { - i[w,(l- m) + - d2)] t}). (37) By comparing eqn (37) and (30) we obtain the relations 1 1 1 - _ --+- TR Tf) T p Am = ~ ( 1 ) -w(2) For powder samples we must take the average over all possible orientations of the crystallites; thus we obtain the result3720 Spin Difusion in Solids under MAS where ( }powder means the powder average.However, from the single-quantum spectra, we can only observe (j'~)(t)}powde. - and (~'~)(t)),owder. - We approximate eqn (39) by writing (GRy(t))powder i[(j:)('))powder (jy2)(t))powder + <jy)(t))powder (j:2)(t))powderl' (40) Eqn (38) can be applied to powder samples by using this approximation. 2.5. Calculation of the Zero-quantum Lineshape Function The calculation of the zero-quantum lineshape function for single-crystal samples is discussed in detail in our other publication^.^' We now outline the theory needed to calculate the zero-quantum lineshape function for rotating samples under the MAS condition.We shall try to show which factors contribute to the sideband intensities or the widths of the zero-quantum lines. To derive the time dependence of the zero-quantum coherence, the relevant terms of the Hamiltonian ' Z 0 ( t ) needed are given by the equation rA?o(t) = 7, B,,{o:;: - 0:;: + f[Ad"~"~i(t) - AcT'"~"s~(~)]> R, where b:: = ys y I h ( ~ - t ; ) - ~ . (41 b) r:: is the internuclear distance between the Si and the Ik spins. This Hamiltonian can be related to the relevant Hamiltonian of the single-quantum coherence Sxi if we replace the chemical-shift term oi:;+$Adi) ecsi(t) in eqn (1 1) by the difference of the chemical shift: in eqn (41a), and the dipolar coupling b,S,'cDsiik(t) by the difference of the dipolar couplings bs,' cDslzk(f) - bti <Ds2z,(t).Thus we see that the difference of chemical-shift anisotropies, and the difference of the dipolar interactions, contribute to the rotational sidebands of the zero-quantum spectra. Since <"s,(t) and eCs2(t) are functions of the two different Euler angles, Q"s,(t) and QCs2(t), the difference of the chemical-shift anisotropies $[Ad') cCsl(t) - A d 2 ) 5cs2(t)] is not zero, when the chemical shielding tensors of S, and S, spins have the same principal values, but different orientations. The third term in eqn (41), & , l Z k r I (t), is the dipolar interaction between the I spins. This term commutes with the obskivable R,, and the first term in eqn (41a), but does not commute with the second term, -a:,": + $[Ao"' cCsl(t) - Ad2) ccs2(t)] y, Bo{oi;: - CT~;; + $[Ad1) ccsl(t) - Ad2) cCs2(t)]> R, - 2 Ikz[bf: c D s i r k ( t ) - bt: cDsZrk(t)] R,.k This dependence is expected to broaden the zero-quantum lines, and changes the sideband intensities in a similar way as slow molecular motions change the shape of the single-quantum MAS To simplify the problem, we ignore the third term, and calculate the sideband intensities due to the difference of the chemical-shift anisotropies, and the difference in the static heteronuclear dipole interactions. We also assume that there is only one kind of I spin. If we introduce the operators R+, defined by the decay function of the zero-quantum coherence can be written as - R, - = Rx+iR, (42)A .Kubo and C. A . McDowell where Uo(t) is given by Uo(t) = exp - i 'X;(tl) dt, ( s : 3721 (44) and we have 'X;(t,) = 7, B0{ai;; - 0;;; + :[Ad1) r c s i ( t ) - A d 2 ) c"sz(t)]} R, -21,[bf' cDsir(t) - biI c " s z r ( t ) ] R,. (45) By calculating the trace over R and Z, eqn (43) can be rewritten as where ACO = y s B0(&; - CT~:;) (47) c'f ( t ) = ;[Ad1' tcsi( t ) - Ad*) 5"sz(t)] y s Bo f [bf' t"si I ( t ) - bfI <"sir( t)] (48) cT - ( t ) = Cf+ cos w, t + Sf* sin w, t + Cf* cos ~ C O , t + Sf* sin 2w, t. <",t) can be changed into the same form given by Maricq and Waughs or Herzfeld and Berger,g namely (49) exp ( + i [ <:(tl)dtl) can be expressed as the sum of the sideband terms: exp (i S, t:(tl) dt1) + exp (i S, dt1) = c I m exp (imw, t ) (50) where Im is the intensity of the mth sideband.We do not calculate the explicit forms of I m . Eqn (46) can now be written as (51) Multiplying eqn ( 5 1 ) by the relaxation factor exp (- t / q ) there results an expression which corresponds to eqn (30). As discussed el~ewhere,~~ the heteronuclear dipolar interactions Xtsirk are sometimes comparable to the homonuclear dipolar interactions X t r r l . In this situation the relaxation theory described in subsection 2.1 can be applied to calculate the linewidth, if the spinning speed is high enough to satisfy the condition w, % lXts I. We now calculate the width of the zero-quantum lines with this condition, i.e. at 6iih rotation speeds. Xl(t) and Xo(t) in eqn (4a) and (4b) can be chosen as m GRY(t) = C Zm {exp [i(mm, + A o ) t] + exp [ - i(mw, + Am) t]}.m The linewidth, l/TR, is calculated to be 1 - = C (b:: Azirx - bil A2zIk) TR k ; l ; m - * , f Z x (b::fA"$rk-bi: A!%'r) dtGIkIi(t) exp(-imco,.t) (54) 1: where GI, is the two-spin correlation function, given by I , ( ? ) = 4(Tr { 'I)-' Tr {rk Uo(t)-l rl Uo(t))* ( 5 5 )3722 Spin Diffusion in Solids under MAS Fig. 1. A rotationally synchronized DANTE sequence. This pulse sequence is similar to that used by Caravatti et al.,25 except that the sign of the last 13C or 31P 90" pulse is altered, instead of the first 'H 90" pulse. We approximate this function by writing where z, is the correlation time of the flip-flop transitions of the I spins. Eqn (54) can now be rewritten as (57) 2% - = 2 2 1 TR m-1,2 k 1 + (mu, zJ2 * Eqn (57) can be applied only when w, % IJFOO,,, I.Under the slow-spinning condition co, % lX$s,Ikl the lineshape of the zero-quantug line can be calculated from Floquet Hamiltonih t h e ~ r y , ~ ~ , ~ ~ which has been applied to describe the effect of the molecular motion on lineshapes in MAS 54, 55 3. Experimental 13C doubly labelled sodium acetate (SAC) was purchased from MSD Isotopes, Montreal, Canada, and used without further purification. Zinc(I1) bis(0,O'-diethyl- dithiophosphate), Zn[S,P(OC,H,),], (ZNP) was ~ynthesized~~ by mixing concentrated aqueous solutions of zinc sulphate (BDH Chemicals) and ammonium diethyldithio- phosphate (Aldrich) in a molar ratio of 1 : 2. The precipitate was purified by repeated recrystallization from acetone solutions.The chemical analysis results were : C, found 22.2Y0, calculated 22.1 YO; H, found 4.7'/0, calculated 4.6%. All the NMR experiments were carried out by using a Bruker CXP200 FTNMR spectrometer with operating resonance frequencies of 50.30 MHz for 13C, 80.98 MHz for 31P and 200.0MHz for lH, respectively. The MAS double air-bearing probe was purchased from Doty Scientific Inc., Columbia, South Carolina. The spinning frequency was controlled within f 50 Hz during each experiment by supplying a constant N, gas flow. The magic angle was adjusted using the 79Br resonance from solid KBr.49 (fig. 1) was used to determine the spin-diffusion time constant GD. However, we reversed the signs of the last 13C, 31P 90" pulse, instead of the first 'H 90" pulse. 2-6 transients were collected by using 6 ,us ;n/2 pulses, 1 ms contact time and 8-20 s recycling times.The carrier frequency was set to one of the isotropic peaks. The time duration between the short pulses, t,, was set equal to l / v r , where v, is the rotor spinning frequency. The rotationally synchronized DANTEA . Kubo and C. A . McDowell 3723 For slow spinning speeds, t, becomes long. Loss of magnetization and 'H r.f. power may occur during this time period and these cannot be ignored. The number of short pulses, N , and the lengths of these pulses, t,, were adjusted to obtain the maximum intensity of one of the sideband groups, while keeping the 7r/2 pulse condition Nt, z 6ps. Examples of the values of N and t, used are: (1) 12, 0 .5 ~ ~ ; (2) 10, 1.Ops and (3) 4, The 13C NMR spectra of SAC without lH decoupling were recorded by using a standard single contact lH-13C cross-polarization pulse sequence. Since the two 31P resonance frequencies of ZNP are very close together, the 31P NMR spectra without 'H decoupling for ZNP were observed after eliminating one group of the sidebands by employing the rotationally synchronized DANTE sequence. More than 30 transients were accumulated. 4. Results and Discussion The 13C CP-MAS spectra of SAC resembled those reported by Raleigh et aL50 We also observed the 23 Hz splittings of both the methyl and carboxy isotropic peaks which are due to the 13C-13C J-coupling. The 31P CP-MAS spectra of ZNP consist of two isotropic peaks at < = 95.5 ppm and a! = 99.5 ppm from the corresponding resonance for 85 % H3P0,, which was used as a standard.The same procedure as described by Conner et aL5' was used to determine the spin- diffusion time constant &,. The ratios r' of the intensities of the two isotropic peaks were measured as a function of the mixing time t,, and were normalized by the equilibrium value r& measured under the condition that t , + GD; r(t,) = r'(tm)/rLq, where r(t,) is the normalized value. The values of In [ 1 + r(t,)/ 1 - r(t,)] were plotted against t,. If the sample has a single-spin diffusion time constant &,, this plot becomes a straight line described by the equation I .5 p s . &, is obtained from the gradient of this line. Fig. 2 shows the v, dependence of TsD for SAC.The spin-diffusion process is well described by a single time constant T,,, except when v, > 4.9 kHz. For these regions of u,, significant deviations from eqn (58) were observed (fig. 3). The average value 7'& was determined by using the equation Fig. 2 displays the data with error bars which express the distribution of the relaxation times. &, exhibits a deep minimum around v, = 3.9 kHz, and a shallow minimum around v, = 2.7 kHz. These frequencies correspond to and 5 of the frequency difference between the carboxy and methyl resonance lines. In fig. 4 we show the v, dependence of the T,, values for ZNP. The plots of In {[ 1 + r(tm)]/[ 1 - r(t,)]} us. t, are explained well by eqn (58) over the whole range of v,. The q,, values for ZNP decrease monotonically as v, increases.Fig. 5 shows the 13C NMR spectrum of SAC and the 31P NMR spectrum of ZNP without 'H decoupling. Each sideband resembles a Lorentzian shape. The widths at half- height, Av;, of the isotropic peaks and the sideband intensities were read from the spectra. The widths Av; decrease monotonically when v, increases (fig. 6). The v, dependences of T,, were analysed in terms of the theory proposed in section 2. We consider only the dipolar mechanism here, because the J-coupling is expected to be much smaller than the dipolar interaction for both the compounds studied. Since3724 Spin Diflusion in Solids under MAS 10 1 0.5 1.0 3.0 5.0 vr /kHz Fig. 2. The rotational frequency v, dependence of the spin-diffusion time constant T,, of SAC.The closed and open circles show the experimental and the calculated values, respectively. The exchange of the magnetization cannot be described by a single spin-diffusion time constant T,, above 4.9 kHz. The average values T,, were determined by the method described in the text and are shown with error bars. 0 tm 5 - 2.0 n I 4 v \ n + d v 4 I .o 0 10 20 30 cm /ms Fig. 3. The mixing time I,, dependence of In (I+ r ) / ( I - r), where r represents the normalized ratios of the intensities of the two isotropic peaks. The data for v, = 5480+20 Hz show deviation from the linear dependence given by eqn (58).A . Kubo and C. A . McDowell (4 I 3725 ZNP vr = 4120k20Hz O O k 50 I I I I I ZNP 0 80 0 0 eo I .o 5.0 v, IkHz Fig. 4. The rotational frequency v, dependence of the spin-diffusion time constant T,, of ZNP.The closed and open circles show the experimental and calculated values, respectively. I I 1 -5 0 5 vr = 4720220Hz 1 . -10 0 10 V k H Z Fig. 5. The 13C NMR spectrum of SAC (a) and 31P NMR spectrum of ZNP (b) without 'H decoupling condition. there is more than one pair of S spins interacting with each other in the crystals, the strength of the dipolar interaction d2 in eqn (23) should be replaced by the lattice sum k k where rik is the internuclear distance between ith S , spin and the kth S, spin. The values of x k dik were calculated from the known crystal-structural data.*** 51 The +or and f 2 0 r components of the zero-quantum spectra, KR*(+or) and3726 1 .o Spin Diffusion in Solids under MAS 2.0 5.0 v, /kHz Fig.6. The rotational frequency v, dependence of the widths at half-height Av; for each of the two resonance lines, in 13C NMR spectrum of SAC and in 31P NMR spectrum of ZNP. The solid circles (0) and the open circles (0) are the widths of the carboxyl and the methyl isotropic peaks of the 13C NMR spectrum of SAC. The solid triangles (A) and the open triangles (A) are the widths of the 95.5 and 99.5 ppm peaks of the 31P NMR spectrum of ZNP, respectively. KR,( & 2wr), were calculated from the observed single-quantum spectra using eqn (38). The calculated rotational frequency v, dependences of T,, are shown in fig. 2 and 4 by open circles. The calculated T,, values for SAC show a similar v, dependence as the experimental T,, values. The calculated T,D values for ZNP are in good agreement with the experimental values.When the spinning frequency is increased, the width of the zero-quantum line, 27r/TR, is expected to decrease in the same manner as the width of the single quantum line measured without 'H decoupling. If 2n/T, becomes much smaller than or, only specific sidebands of the zero-quantum line can contribute to spin diffusion. We discuss the two cases; (1) A o z 0, w, % 27c/TR and ( 2 ) A o = no,, or % 27r/TR. ZNP is an example of the first case, while SAC corresponds to the second one. In the first case Am z 0, the spin-diffusion time constant due to the dipolar coupling is given by (60) 1 d2 TR - = - (Il + I-1 + ;I2 + ;Ip2) TFD 15 1 +Aw2Tk' When the spinning speed is increased, all the sideband intensities In where n # 0, and the widths 2n/TR of the zero-quantum lines, decrease.If A o # 0, the condition A o > 27r/TR is satisfied at high spinning speeds. Eqn (60) can be rewritten as 1 d2 1 - = - ( I l + L 1 + ; I 2 + L I )- T:,, 15 -' Ao2TR' In this equation the two effects, the decrease of the sideband intensities and the decrease of the width, act in the same direction. Spin diffusion between spins with Ao # 0 can be suppressed at the high spinning speed limit. If, however, A o = 0, eqn (60) becomesA . Kubo and C. A . McDowell 3727 In this case the two effects consequent on increasing the spinning speed may cancel each other. It may, however, be possible to observe spin diffusion even in the high spinning- speed limit. We also have to distinguish between two possibilities for the spin pair with the same principal values of the chemical shielding tensors, because they may or may not have the same orientations.In the former case only, the dipolar interactions can contribute to the spin diffusion. It would be interesting to see if there is a difference in the w, dependence of T,, in both cases. When Aco % 0, LO, 9 2;n/TR, the spin-diffusion time constant due to J-coupling, G,, is given by 1 (zl+I-l++12++I-2) ) 1 +:E2Tk' (63) From comparison with eqn (60), the symmetric part of the J-coupling can contribute spin diffusion only when A J 2 - (1 +$) - d 2 . 9 This mechanism is limited to heavy atoms. The isotropic part, or the antisymmetric part, can be observed if d 2 15 Ji",, I, - - (Il + I-, + + iI-2) or Since the ratio (Il+Z-J/I0 decreases as the spinning frequency is increased, it is not impossible to observe these mechanisms in the high-speed spinning experiments.It is also possible to observe these mechanisms selectively if a train of the ;n-pulses is applied to the S spin system during the mixing time to eliminate the sidebands from the zero- quantum spectra. In the second case when Aw = n q , the spin diffusion time constant cD due to the dipolar interaction is given by If' n = 1 or k2, spin diffusion is expected to be greatly enhanced, because eqn (64) includes the isotropic peak intensity I,. The experimental values of T,, for SAC (fig. 2) show a deep minimum at Aco 2co, and a shallow minimum at ALO % 3~0,. The latter minimum is caused by the first and second sidebands of the zero-quantum line.A similar observation and explanation have been reported by Andrew et a1.46747 for 31P spin diffusion in a rotating sample of polycrystalline phosphorus pentachloride ; this phenomenon is rotational relaxation r e s ~ n a n c e . ~ ~ Another interpretation of this phenomenon is also possible. From eqn (10) T,, may be written as Eqn (65) consists of the time correlation function of XT(t), i.e. the flip-flop term of the3728 Spin Diffusion in Solids under MAS dipolar or the J-coupling Hamiltonian expressed in the interaction representation. This equation corresponds to the o = 0 Fourier component of the correlation function The quasi-static term of X,*(t) determines the spin-diffusion time constant qD. This concept may be useful for the interpretation of other phenomena observed with rotating samples under MAS conditions, such as the dipolar lineshapes** 4 7 9 50 and the excitation of the multiquantum coherence under MAS conditions. We consider the case where there are dipolar SD? s2 and chemical-shift interactions SC,,, S c S 2 .The time dependence of the dipolar Hamiltonian expressed in the interaction representation is given by 53 2S,, S,, - isl+ S,- exp - is,- S,, exp [ - i (Aot + [ tR(tl) dt,)]} tDsls2(t) = - d ( 2S1, S,, A$,s2 exp (imm, t ) m - + l , + 2 - _ 2 c S,, S,- A s l s 2 Il exp {i[(m + I ) or + Am] t> m - * l , * 2 1 S,-S,+ A$ls2Z, exp{i[(m-I)w,-Ao] t } m - r t l , + Z 1 The Hamiltonian can be truncated in the same way as we truncate the dipolar or the quadrupolar Hamiltonian, when we need to calculate the lineshape in a high magnetic field.If n o , + A o z 0, the truncated Hamiltonian tX* ( t ) is given by D s l S 2 c S,, Sz- A$,s2 In-m exp [i(no, + Am) t] d “X;, ( t ) = -- 2 m - + l , + 2 1 2 2 S,- S,, A$,s2 Zm+n exp [ - i(no, + Ao)t]. (68) d -- 2 m - + l , + 2 The time dependence of the operator B can be calculated from the expression Tr {BtU0(t)-lp(O) Wo(t)). Here Uo(t) is the time-evolution operator of the truncated Hamiltonian : and p(0) is the initial density matrix. From this equation it is clear that there are still quasi-static terms of the dipolar interactions even under MAS conditions. For example, when Am = 0, the truncated Hamiltonian becomes C S1- S,, A21s2Zm. (70) d d ( t ) = -- C S,, Sz- A>isz Im -- 1 2 2 m-+1, + 2 2 m - - + l , + 2A .Kubo and C. A . McDowell 3729 The second moment of the S, resonance line can be calculated from the expression This concept may also be useful in the following new experiment. Further modulation , of the spin Hamiltonian can be introduced by applying special r.f. pulses or multiple pulses. It is then possible to create the quasistatic Hamiltonian from the non-secular terms. For example, the flip-flop term S , I- can become the quasistatic Hamiltonian if we apply the r.f. field from the z direction with the frequency cu,-cus. The time dependence of the term S+I- in the interaction representation is given by Uo(t)-l S+ I- Uo(t) = S+ I- exp i (cus -coo,) t + (cu,, - q,) cos (a, -us) t , df,)] (72) where cu,, = ys H,,, and cu,, = y, Hl,.H,, is the amplitude of the r.f. magnetic field applied from the z direction. We can rewrite eqn (72) by using the Bessel functions of the mth kind, Jm(x), as [ ( s: The term m = - 1 in eqn (73) is the static term. Since S, and I, do not commute with S+ I-, the interaction S+ I- may be observable by monitoring the change of S, or I, after applying r.f. irradiation from the z direction. We thank the Natural Sciences and Engineering Research Foundation of Canada for grants to C.A.McD. Also A.K. thanks the Killam Foundation for the award of a postdoctoral fellowship. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 E. R. Andrew, A. Bradbury and R. G. Eades, Nature (London), 1959, 162, 1659. I. J. Lowe, Phys. Rev. Lett., 1959, 2, 285. S. R. Hartmann and E.L. Hahn, Phys. Rev., 1962, 128, 2042. A. Pines, M. G. Gibby and J. S. Waugh, J. Chem. Phys., 1973, 59, 569. J. Schaefer and E. 0. Stejskal, J. Am. Chem. SOC., 1976, 100, 1031. C. A. Fyfe, Solid State NMR for Chemists, (C.F.C. Press, Guelph, 1983). E. Lippmaa, M. Alla and T. Turherm, Proc. Of 19th Congr. Ampere (Groupement Ampere, Heidelberg, 1976), p. 113. M. M. Maricq and J. S. Waugh, J. Chem. Phys., 1979, 70, 3300. J. Herzfeld and A. E. Berger, J. Chem. Phys., 1980, 73, 6201. J. Jenner, B. H. Meier, P. Bachman and R. R. Ernst, J. Chem. Phys., 1979, 71, 4546. M. G. Munowitz, R. G. Griffith, G. Bodenhausen and T. W. Hwang, J. Am. Chem. SOC., 1981, 103, 2529. M. Munowitz and R. R. Griffith, J. Chem. Phys., 1982, 76, 2848. M. Munowitz, W. P. Aue and R. G.Griffith, J. Chem. Phys., 1982, 77, 1686. W. P. Aue, D. J. Ruben and R. G. Griffin, J. Chem. Phys., 1984, 80, 1729. G. R. Harbison and H. W. Spiess, Chem. Phys. Lett., 1986, 124, 128. E. M. Menger, S. Vega and R. G. Griffith, J. Am. Chem. SOC., 1986, 108, 2215. J. Herzfeld, J. E. Roberts and R. G. Griffin, J. Chem. Phys., 1987, 86, 597. G. S. Harbison, V. D. Vogt and H. W. Spiess, J. Chem. Phys., 1987, 86, 1206. Y. Yang, M. Schuster, B. Blumich and H. W. Spiess, Chem. Phys. Lett., 1987, 139, 239. A. E. DeJong, A. P. M. Kentgens and V. S. Veeman, Chem. Phys. Lett., 1984, 109, 337. H. Miura, T. Terao and A. Saika, J. Chem. Phys., 1986, 85, 2458.3730 Spin Diflusion in Solids under MAS 22 (a) D. Suter and R. R. Ernst, Phys. Rev. B, 1982, 25, 6038; (b) D. Suter and R.R 23 G. Bodenhausen, R. Freeman and G. A. Morris, J . Magn. Heson., 1976, 23, 17 24 P. Caravatti, G. Bodenhausen and R. R. Ernst, J . Magn. Reson., 1983. 55, 88. 25 P. Caravatti, M. H. Levitt and R. R. Ernst, J . Magn. Reson., 1986, 68, 323. 32, 5608. R. Freeman, J. Magn. Reson., 1978, 29, 433. Ernst, Phys. Rev. B, ; G. A. Morris and 26 N. M. Szeverenyi, M. J. Sullivan and G. E. Maciel, J. Magn. Reson., 1982, 47, 462. 27 T. A. Cross, M. H. Frey and S. J. Opella, J . Am. Chem. Soc., 1983, 105, 7471. 28 H. T. Edzes and J. P. C. Bernards, J. Am. Chem. Soc., 1984, 106, 1515. 29 P. Caravatti, P. Neuenschwander and R. R. Ernst, Macromolecules, 1986, 19, 1889. 30 P. Caravatti, J.A. Deli, G. Bodenhausen and R. R. Ernst, J . Am. Chem. SOC., 1982, 104, 5506. 31 M. Linder, P. M. Henrichs, J. M. Hewitt and D. J. Massa, J . Chem. Phys., 1985, 82, 1585. 32 K. Takegoshi and C. A. McDowell, J. Chem. Phys., 1986, 84, 2084. 33 P. M. Henrichs, M. Linder and J. M. Hewitt, J . Chem. Phys., 1986, 85, 7077. 34 A. Kubo and C. A. McDowell, J . Chem. Phys., 1988, 89, 63. 35 N. J. Clayden, J . Magn. Reson., 1986, 68, 360. 36 H. Kessemeier and R. E. Norberg, Phys. Rev., 1967, 155, 321. 37 V. Haeberlen and J. S. Waugh, Phys. Rev., 1969, 185, 420. 38 A. Abragam, The Principles of Nuclear Magnetism (Oxford University Press, New York, 1961) chap. 39 H. W. Spiess, in NMR, Basic Principles and Progress (Springer, Berlin, 1978), vol. 15. 40 0. W. Sorensen, G. W. Eich, M. H. Levitt, G. Bodenhausen and R. R. Ernst, Progr. NMR Spectrosc, 41 W. P. Rothwell and J. S. Waugh, J . Chem. Phys., 1981, 74, 2721. 42 D. Suwelack, W. P. Rothwell and J. S. Waugh, J. Chem. Phys., 1981, 73, 2559. 43 A. Schmidt, S. 0. Smith, D. P. Raleigh, J. E. Roberts, R. G. Griffin and S. Vega, J . Chem. Phys., 1986, 44 J. H. Shirley, Phys. Rev. B, 1965, 138, 979. 45 D. R. Dion and J. 0. Hirschfelder, in Adv. Chem. Phys. ed. I. Prigogine and S. Rice (Wiley, New York, 46 E. R. Andrew, A. Bradbury, R. G. Eades and V. T. Wynn, Phys. Lett., 1963, 4, 99. 47 E. R. Andrew, S. Clough, L. F. Farnell, T. D. Gledhill and I. Roberts, Phys. Lett., 1966, 21, 505. 48 T. Ito, T. Igarashi and H. Hagihara, Acta. Crystallogr., Sect. B, 1969, 25, 2303. 49 J. S. Frye and G. E. Maciel, J . Magn. Reson., 1982, 48, 125. 50 D. P. Raleigh, G. S. Harbison, T. G. Neiss, J. E. Roberts and R. G. Griffin, Chem. Phys. Lett., 1987, 51 T. S. Cameron, Kh. M. Mannan and Md. Obaidur Rahman, Acta. Crystallogr., Sect. B, 1976, 32, 52 C. Connor, A. Naito, K. Takegoshi and C. A. McDowell, Chem. Phys. Lett., 1985, 113, 123. 53 B. H. Meier and W. L. Earl, J . Chem. Phys., 1986, 85, 4905. 54 S. Vega, E. T. Olejniczak and R. G. Griffin, J . Chem. Phys., 1984, 80, 4832. 55 A. Schmidt and S. Vega, J . Chem. Phys., 1987, 87, 6875. VIII, eqn (33). 1983, 16, 163. 85, 4248. 1976), vol. 35. 138, 285. 87. Paper 8/0055 1 F ; Received 9th February, 1988