I. URSU et al.: Spin Relaxation in the Presence of Coherent Sequences 309

phys. stat. sol. (b) 136, 309 (1986)

Subject classification: 76.20

Central Institute of Physics, Bucharest (a), Institute for Isotopic and Molecular Technology, Cluj-Napoca’) ( b ) , and Polytechnical Institute, Department of Physics, Cluj-Napoca ( G )

Spin Relaxation in the Presence of Coherent Averaging Pulse Sequences

BY I. URSU (a), F. BALIBANU (b), D. E. DEMCO (c), and M. BOGDAN (b)

Theoretical calculations of the effective transversal relaxation time for homo- and heteronuclea dipole-dipole interaction in the presence of the MW, WHH, and MREV sequences are performed in the theoretical frame of Griinder, Schmiedel, and Freude. The relaxation times are calculated as functions of the correlation time of the dipole-dipole interactions, the time period, and the pulse length. The possibility of selective cancellation of the dipolar relaxation mechanisms by using multiple pulse sequences is discussed.

Les calculs theoriques du temps effectif de relaxation transversale, pour les interactions dipblaires homo-et h6teronucl6aires en prhence des sequences MW, WHH et MREV ont 6t6 realises dans le cadre theorique de Griinder, Schmiedel et Freude. Les temps de relaxation ont 6tB calculees comme fonctions du temps de correlation des interactions dipble-dipble, de la phiode temporelle de la sequence, et de la durke des pulses. Nous discutons la possibilite d’annuler les mechanismes de relaxation dipblaire en ntilisant des sequences de pulses multiples.

1. Introduction Certain relaxation mechanisms may provide valuable information concerning the molecular interactions and dynamics or the isotopic content of the sample [I]. Fre- quently, a stronger relaxation mechanism covers a weaker and perhaps more “inter- esting” one. A large number of multiple pulse sequences have been designed in order t o average out the line-broadening interactions in solid state NMR. Thus weaker interactions have been revealed, giving useful information about molecular properties [2, 31. In a similar way, a manipulation of the relaxation mechanisms is achievable, permitting the observation of the less efficient ones, thus providing complementary insight into molecular dynamics. Various paths have been adopted in order to cal- culate the nuclear magnetic relaxation duiing multipulse sequences [4 t o 131.

A rigorous treatment has been attempted by Griinder et al. [5 t o 71 by calculating directly the time development of the magnetization in a transversal direction. This theory is based on the weak-collision assumption and on an exponential correlation function. The formalism has been applied on the homodipolar relaxation during MW2 and WHH4 sequences in the case of %pulses [6] and during MW4 and MW4x sequences with finite-length pulses. Heteronuclear dipole-dipole relaxation has been taken into account for MW4 and MW4x sequences in the case of %pulses 171.

In the present paper we extend these calculations to homo- and heteronuclear dipole-dipole relaxation during WHH, MREV, and MW sequences, taking into account the nonvanishing pulse length and discuss the selectivity of these sequences towards different relaxation mechanisms. A short review of the GSF theory [5 t o 7 J

l) POB 700, 3400 Cluj-Napoca 5, Romania.

310 I. URSU, P. BALIBANU, D. E. DEMCO, and M. BOGDAX

is presented in Section 2 . Sections 3 and 4 present our results of the theoretical ex- pression of the effective transversal relaxation rate for homo- and heterodipolar relaxation. In the last section we discuss the usefulness of different pulse sequences in manipulating dipolar relaxation mechanisms. We also give a comparison with ex- perimental data from [lo].

2. Theory

Using the density matrix operator e( t ) the expectation value of the magnetization along the z-axis may be written

(1 ) -~

(I&)) = Tr {eW I%> Y

where I , is the spin operator and the bar over the trace means an average performed over the spin system. The density matrix at a moment t may be written

U ( t ) = T exp [ - f 1 dt, H(t , ) ] (3) 0

is the propagator of the system under the action of the total Hamiltonian H ( t ) and T is Dyson’s time ordering operator.

The spin Hamiltonian is a sum of a Zeeman term H Z = wORIZ, a radiofrequency term H f i ( t ) = -2w,lz[fz(t) I, + f 2 / ( t ) 12/] cos wet, and a spin-spin interaction term H,(t). j z ( t ) and fY( t ) are periodic rectangular functions which take the value 0 or *l and describe the particular multipulse sequence.

In the interaction representation of the Zeeman term (the rotating frame) the Hamiltonian is

Hr(t) = -mlh[ fz( t ) 1% + f y ( t ) I2/1 + Hir( t ) * (4) The rapidly oscillating terms of Hr(t) have been omitted in (4). If we take into account only the secular part of H,(t) , then Hir(t) = Hi(t).

In order to calculate (I,(t)) we have to perform a second transformation using a “toggling” reference frame [ Z ] . The toggling reference frame coincides with the rotating frame for t = nt,, where t , is the time period of the radiofrequency Hamil- tonian in the rotating frame. We perform this transformation to the radiofrequency interaction representation using the propagator

Ur(t) = ~ X P {- iwl [ fz( t ) -I, + f d t ) ‘,I>

H,(t) = U,(t) Hi$) U,l(t) = Hi$) .

(5 )

(6)

and the spin Hamiltonian becomes

The evolution of the magnetization in the toggling frame (as well as in the rotating frame for t =- ntc) is obtained from the above equations

( I & ) ) = Tr { U , ( t ) e(0) U,l ( t ) I z } * (7 1 In the high temperature-high field approximation the contribution of the internal

Hamiltonian to the equilibrium density matrix may be neglected because it is much smaller than the Zeeman contribution. After a first n12 preparation pulse, the density matrix is proportional to 1% [ 8 ] .

Spin Relaxation in the Presence of Coherent Averaging Pulse Sequences 31 1

To get rid of the nonanalytical form (3) of the time evolution operator we use the Magnus expansion 1141,

Expanding the exponential in power a series and using it in (7) with high temperature approximation for e (O) , we obtain [7]

where

2 = T r { e x p ( - g ) } .

The expression is valid in the weak collision limit for an arbitrary interaction Hi(t) .

3. Relaxation by Homonuclear Dipole-Dipole Coupling

We take into account only the secular part of the dipole-dipole interaction Hamil- tonian [151,

The nonsecular part has been taken into account by Blicharski [l l] for the MW4, 8-pulse sequence and his results show that its contribution is not affected by the radio- frequency pulses. Our calculations have been performed for five different pulse se- quences described by various fz ( t ) and fy(t) functions. For convenience we show them in Fig. 1.

Pulses are of a length t , so that coltw = zl2, except for the MW4x sequence, where q t , = z. Owing to the pulse sequence, H,(t) becomes, in the toggling reference frame, Hit(t), with the general form

Hit(t) = C Bij(t) [gzz(t) ( I z i I z j - f ZiZ,) + gyy(t) ( I v i I u j - f ZiZ,) + i <j

Prom (9) and (12) we obtain

31 2 I. URSU, F. BALIBANU, D. E. DEMCO, and M. BOGDAN

IT: <- 0

-7

I I - Fig. 1. The fi(t) and f u ( t ) functions for MW2 - various multiple pulse sequences n

~

- I I

-

I 1 I 1 1

0 2 4 6 8

1 I I 1

0 2 4 6 8

0 2 4 6 8

I I I I I 1 1 70 72 0 2 4 6 8

MpZV8

-1

70 72 0 2 4 6 8 f l z --

Using an exponential correlation function (i.e. random isotropic moleeular motions) and neglecting the correlations between different pairs of spins, i.e.

we obtain quasiesponential relaxation of the following type :

G&) == exp [-M,t,tR,(t,, t, i,) + M2z:RZ(zcr z, t,) (1 - e-t’*c)]

Spin Relaxation in the Presence of Coherent Averaging Pulse Sequences 31 3

T a b l e 1 The R,(t,, T, t,) functions for homonuclear relaxation during different pulse sequences. *)

multipulse sequence S(a, B, h)

MW2

h(2h - 1) B s h ( a - 8) chB

h(2h - 1) B s h ( a - B ) shB

MW4 1 - - h - a a ch a

MW4n 1 - - h - a a s h a

2 5 (2h - 1) sh 28 - 2/? - ch2 8[4 sh2 a(5 ch a-- 1) h2] WHH4 - { l + % 3 a 2a sh 3a +

5 ch2 /? ch a(ch 2a - h) h 2a sh 3a +

- 2 sh2 a[5(2h - 1) ch - 2h] a sh 3a

5 (2h - 1) sh 28 - 28 h - C h 2 /? MREV8 - { 1 + % 3 a

5 c h a c h 3 a - c h 2 2 a a sh 6a

-

*) R, = T52/M2tc; a = t/te; /3 = t,/2tc; h = o:s:/(l + w?tz); M2 is the second moment of the normalised resonance line; te is the correlation time; t is half the maximum spacing between pulses, t , is the pulse length; w1 is the r.f. magnetic field.

with an effective relaxation rate T-1 -

2e - Jfz~eRi(zc, z, t ,) where M2 is the second moment of the resonance line. From (13) and (14) we can com- pute the specific Rl(z,, t, t , ) functions for each sequence in Fig. 1. The results are listed in Table 1.

The results for MW4 and MW4x sequences with finite-length pulses have been previously calculated by Gruiider [7]. In the &pulse limit (i.e. t , = 0 and h = 1) our results from Table 1 coincide with those obtained in [6] for MW2 and WHH4 se- quences.

Using the results from Table 1 we present the dependence of In Tze on the reciprocal temperature (Fig. 2) and on the spacing between pulses (Fig. 3).

In the limit of fast-pulse sequences (z < z,) we have found some asymptotic rela- tions :

1 t c t , T&(MW2) = - M,- , z2 z

314 I. URSU, F. BALIBANU, D. E. DEMCO, and M. BOGDAN

1, h‘ s

I, . ?IT-

Fig. 2 Fig. 3 Fig. 2. The variation of In Tze versus the reciprocal temperature for WHH4 (- - -) and MREV8 (- . -) pulse sequences in the case of homodipolar relaxation

Fig. 3. The dependence of In Tze on In t (where 2t is the maximum spacing between pulses) for MW2 and MW4 pulse sequences in the case of homodipolar relaxation. ~ MW2 and MW4, - _ _ MW4, - - - M W 2 , - - . . - M W 2 , - - - - - M W 2 , - - . . - M W 2

4. Relaxation by Heteronuclear Dipole-Dipole Coupling

Using a heteronuclear dipole-dipole secular Hamiltonian as H,( t ) and following a cal- culation procedure similar t o that employed in the previous section we obtain a relaxa-

Tab le 2 The R,(t,, t, tw) functions for heterodipolar relaxation during different pulse sequences*)

multipulse sequence R,(a, B. h9 9 )

/? 1 - - h - a

h(2h - 1) ch 2a - h ch 28 + 2gh sh (2a - 2/?) 2 a s h 2 a

MW2

B a 2a ch 2a

h(2h - 1) (sh 2a - sh 2B) - 4gh sh2 (a - B ) MW4 I - - h -

h(2h - 1) 1 - - - h - B s h ( a - B) chB MW4x a a c h a

- 3h(2h - 1) ch 3a - 3h ch a ch 2B - h(7 - h) sh a sh 28

4a sh 3a WHH4

3 4[h sh (a + /I) - g ch (a - /?)I2 + 2gh[3 sh (3a - 2B) + sh a] -

4a sh 3a ’

3(8 - g ch 2B) - - 3h(2h - 1 + 29 sh 28) sh 3a

h sh a ch 28(4 ch a + 1 ) + 2h[(2h - 1) sh 28 - g] (2 ch 2a + ch a) 4a ch 3a

MREV8 3 2a 4a ch 3a

Spin Relaxation in the Presence of Coherent Averaging Pulse Sequences 315

711 --- I i I I

001 07 70 10 - Fig. 4 Fig. 5

Fig. 4. The dependence of In Tee on reciprocal temperature for WHH4 pulse sequence in the case of heterodipolar relaxation mechanism

Fig. 5. The dependence of In Tze on the spacing between pulses in the case of heteronuclear relaxation for MREV8 pulse sequence

tion function for the I spins,

where gZ/(t) and g Z ( t ) are periodic functions which describe the evolution of the Hamil- tonian in the toggling frame, as gba(t) functions did in Section 3. The relaxation is also quasiexponential, and the Rl(t,, t, t , ) functions for the heteronuclear coupling are listed in Table 2.

In the case of 8-pulses (i.e. t , = 0, h = 1, g = 0) our results for MW4 and MW4x pulse sequences are identical to those given in ti']. The dependence of In Tze on the reciprocal temperature and on the spacing between pulses is presented in Fig. 4 and 5.

I n the limit of fast-pulse sequences (t Qz,) the asymptotic relation for the MREV8 pulse cycle is

5. Discussions and Conclusions

The results presented in Sections 3 and 4 predict the dependence of the effective trans- versal relaxation time during the coherent averaging pulse sequences on temperature, pulse length, and on the time interval between the pulses.

In the pulse limit the pairs MW2-MW4 and WHH4-MREV8 have the same effect on the homonuclear dipole-dipole relaxation time, while the MW4x pulse sequence has no effect a t all. I n the long-pulse limit, i.e. t , = 22 for the MW pulse sequences

316 I. URSU, F. BALIBANU, D. E. DEMCO, and M. BOGDAN

I

I -___- .

I ?l Fig. 6. The experimental data of Erofeev et al. [lo] plotted against our theoretical results for homonu- clear relaxation during WHH4 sequence with t , = O . l t h

and t , = z for WHH4 and MREV8 sequences, the behaviour of TBe is quite different. The MW4 and MW4x pulse sequences, which are composed of identical, in-phase, equidistant pulses lead to a T1, limit. The other sequences (MW2, WHH4, and MREV8), containing phase-alternated pulses, lose much of their efficiency in pro- longing the nuclear magnetic relaxation. However, the MREV8 pulse sequence is still more efficient than WHH4 within a 514 factor. l t is interesting to observe that, within the framework of this theory, for the alternating pulse sequences the reduction of the spacing between pulses prolongs the relaxation to z = (twz,)l/z. Going further, with z < (twt,)l/e the relaxation is accelerated again and thus In TZe versus In t presents a maximum (Fig. 3).

The MW2 and WHH4 pulse sequences do not prolong much the heteronuclear di- polar relaxation, though they prolong the homonuclear one. The MW4n pulse se- quence has an opposite effect, prolonging only the heteronuclear relaxation. These pulse sequences can be used in order t o eliminate one of the relaxation mechanisms, thus permitting the observation of the other. The use of MW4 or MREV8 pulse sequences prolongs both the homonuclear and the heteronuclear relaxation times.

The analysis of the variation of TBe versus temperature and versus the spacing between pulses shows that these pulse sequences have practically the same efficiency and the same measurement domain in the determination of the correlation time.

The experimental results of Erofeev et al. [lo] for Tze at different temperatures obtained with a WHH4 pulse sequence are compared in Fig. 6 with our theoretical results, given in Table 1. The experimental plot shows two minima, the second is produced by the resonance offset of the r.f. pulses [lo]. Our theoretical results predict one minimum, as our calculations have been performed considering resonant pulses. They do not predict the temperature independent Tee obtained at low temperatures, owing to the fact that exponential correlation functions and weak collision approxima- tions are not valid in the rigid lattice limit.

In conclusion we may say that these results are useful in selecting a certain multi- pulse sequence in a specific experiment, prolonging (and hence eliminating) either the heteronuclear or the homonuclear relaxation process.

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Spin Relaxation in the Presence of Coherent Averaging Pulse Sequences 317

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443 (1982).

(Received January 6 , 1986)