Relaxation studies using saturated bandshapes in high-resolution NMR. I. Symmetric three spin-12 systems

JOURNAL

OF MAGNETIC

RESONANCE

22,491-508

(1976)

Rellaxation Studies Using Saturated Bandshapesin High-Resolution NlWR. I. Symmetric Three Spin-t Systems B. W. GOODWIN*

AND R. WALLACE

Department of Chemistry, University of Manitoba, AND

Winnipeg, Canada RT3 2N2

R. K. HARRIS School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, England Received December 5, 1975 Bandshapes of spectra of the types AB2 and AX2 have been calculated at various rf field strengths, for relaxation by the intramolecular dipole-dipole and random field mechanisms. In general, different relaxation processes give rise to different bandshapes, and for the symmetric three spin-+ cases studied here the two mechanisms are readily distinguishable if one or the other dominates. Spectra of an AB2 spin system, for example, show that the behavior of the asymmetric transition is very sensitive to the relaxation processes, with the random field mechanism producing a relatively sharp transition which tends to saturate at a low rf amplitude, in contrast to the dipole-dipole mechanism. The programs written for the calculation of these bandshapes are described in some detail, and they may be readily generalized to more complex spin systems and to other relaxation mechanisms in the extreme narrowing limit. These features follow because all required relaxation terms are calculable by the programs when the appropriate relaxation Hamiltonians are put into suitable forms. INTRODUCTION

Information regarding molecular motion in liquids may be obtained through a deta.iled study of the relaxation processes occurring in high-resolution NMR systems. Experiments available include selective pulsed NMR (I), double-resonance (2) and the single-resonance methods involving saturated bandshapes (34 or variations in the nonsaturated linewidths (9-12). In this paper, the latter method has been extended to a study of symmetric three spin-& systems, the object being to determine the sensitivity of these bandshapes to different relaxation processes. The relaxation mechanisms considered are the intramolecular dipole-dipole (DD) mechanism and the random field (RF) mechanism (often used to approximate other mechanisms (13)). Computed spectra reproduced here indicate that these mechanisms yield sufficiently distinct bandshapes so that, should one or the other mechanism be dominant, the relaxation parameters describing that mechanism may be experimentally accessible using the saturation method. Variations in linewidths may be detectable in the unsaturated limit, provided that inhomogeneity broadening is not predominant. As the * l?ostdoctorate Fellow, National Research Council of Canada. 491

Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

492

GOODWIN, WALLACE AND HARRIS

rf field amplitude becomes large enough to compete effectively with relaxation, however, differences between the two mechanisms become more apparent, especially with respect to the relative widths of the transitions and the relative order in which the transitions begin to show saturation effects. These effects may be more readily discerned from plots of a linewidth factor and a saturation factor for both mechanisms and for the major transitions of the symmetric three spin-3 system. Some comments on the sensitivity of the saturation factor to the relaxation parameters are also made. THEORY

In the rotating frame, the equation of motion of the spin density matrix p’ may be expressed as

&‘Pj = w, X0’ + Jf”,‘l + (+rlwrelax, where X0’ is the transformed high-resolution spin Hamiltonian tonian describing the coupling of the rf field with the spins:

111

and 3Ypr is the Hamil-

Pal PI using the usual notations, and with the rf field given by -2H,cos(ot) i (in the derivation of Eq. [2], that part of the rf field rotating in the opposite sense to that of the Larmor precession is neglected). The last term in Eq. [I] accounts for the interaction of the spin system with its surroundings, and provides the process by which the spins are relaxed toward thermal equilibrium in response to a perturbation by the rf field. Using the theory advanced by Redfield (Z4), the relaxation term in the extreme narrowing limit has the form with rij(P’ - P”) = T T R,JU * (~/cl’- ~/cl’)

WI

in terms of the Redfield relaxation matrix R and the thermal equilibrium density matrix p”. The Hamiltonian associated with the interaction of the spins with their surroundings is conveniently written in spherical tensor form : S”,(j) = 1 L: (-1Y F-qpL(O ZqcL, 141 P4 where the summation p is over all mechanisms, each with 2L + 1 components q, with L being the rank of the mechanism. The time dependence of the interaction is carried in the stationary random function F, while the function Z contains operators acting on the spins. The Redfield theory gives the result’

1 Equation [5] often appears expanded in terms of spectral densities; however, the form of Eq. [5] has been found more convenient for conversion to a program. Likewise the Liouville representation (see, for example, Ref. (3)), although more compact, is not employed, as the density matrix equations are somewhat more manageable.

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where the bar indicates a (semiclassical) ensemble average. Substitution into this last expression yields an expression of the form r(p’ - p”) = 2 2 [[p’ - p”, ZqpLJ,Z,,,,“‘] P.FJ'4.d

1% GqqrpprLL’(~),

6

493 of Eq. [4]

WI

where Gqq’pp’LL’(T)= FdqpL(t) F-q’p’L’(t + T) = 6,,, 6-,,. Goopp~(r)~(-l)q.

WI

El

The last result in Eq. [6c] follows from the results of Hubbard (15) and Pyper (Z6), with the consequence that the required relaxation matrix may be written as a function of a minimal number of phenomenological parameters multiplied by the appropriate contributions from the spin operators. Two major types of ensemble averages will appear from Eq. [6]: those involving GOOy,p(~), which is an ensemble average within a given mechanism, and those involving Gool,p(r), which averages the functions F between two different mechanisms. Inasmuch as the mechanisms encountered here have different rank, only the first type of average will occur; if the anisotropic chemical shift mechanism were also to be considered, then the isecond type of average would occur between it and the DD mechanism. Definition qf the Relaxation Parameters The set of phenomenological relaxation parameters introduced above may now be defined for the two mechanisms of interest here. For the RF mechanism the required Hamiltonian, of rank L = 1, is of the form XrKFW = q;lg,, (-114F-q(t)lq,

where IO = Z, and Zi-i = TZJV?.

PaI

It is advantageous to partition Eq. [7a] as follows:

where the summation n runs over the spinning nuclei (or more accurately, the spinning nuclei which are relaxed by the RF mechanism), and where Jon= I,,, etc. The GOOPp(r) averages then yield, for an m spin-4 system, m “interaction” parameters and m(m - 1)/2 “correlation” parameters for the RF mechanism. For a three spin-f system, labeling the nuclei as A, B, and C, the three interaction parameters FA2, FB2,and Fc2 are defined through expressions of the form FA2 = md~ F,*(t) F,*(t + T) s 0

@aI

and the three correlation parameters Car,, CAC, and CBC,using CA, = &W,,*(t)FoB(t

+ r)/ IF*&,\.

0

For a symmetric three spin-+ system of nuclei A, B, and B’ (the prime not denoting 18

494

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magnetic inequivalence), symmetry arguments reduce the number of independent parameters to four (cf. Ref. (26)), namely: FB==FB'2;

FAN;

CAB = CAB,;

and

C,,..

Note that in some cases the use of the RF mechanism to approximate the spin-rotation mechanism may result in the further simplification that all correlations are equal to unity (23). The intramolecular dipole-dipole mechanism is described by the Hamiltonian of rank L = 2,

with

*rDD@)= ;<; q=.,s1*i2 C-1)’F-,““(t) 4”‘”

[W

IO,,= xn, I”,- v,, A-+Al,-L+P, 121 mn= T(&/~)(Z,,,,Z,, + I,,,? I,,,), I 042)&+In+_,

and

PI

+2 mn =

where the Hamiltonian has been partitioned to run over all pairs of interactions between the spinning nuclei m and n. For an m spin-3 system, the ensemble averages in GOOpp(t) then yield m(m - 1)/2 DD interaction parameters, and m(m - 2)(m2 - 1)/8 correlation parameters. For a three spin-3 system, there are once again six independent parameters for this mechanism: three interactions DAB', DACZ and DBc2, which are defined using expressions of the form m

D AB 2,

UW

dzFoAB(t)F,,AB(f + z)

s

0

and three correlations CABAc,CAaBc, and CAcBcdefined using CABAC= 6

1md~FoAB(t)FoAC(f+

For symmetric three spin-+ systems, the number of independent reduces to four (16): DAB'= DABr2;

D BBp2;

C ABAB';

[lob1

7) I IDABDAcI-

and

DD parameters

CAB,,, = CAB*,,,.

It is also possible to reduce the number of these DD mechanism parameters further by allowing the assumption of rigid molecules undergoing isotropic rotational diffusion (27, 18); i.e., all molecular orientations are assumed to be equally probable. This simplified model effectively decreases the number of parameters to one independent variable, with the interactions taking the form D AB 2

=

11Ial

yA2 yB2rABm6'K,

with the same proportionality constant K (itself proportional to the correlation time for molecular reorientation, which is considered to be the adjustable variable), and with the correlations taking the form C ABAB'=(3cos2&.,B,

- 1)/2.

[llbl

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SYSTEMS

The latter equations exhibit the usual dependence of the DD mechanism on the gyromagnetic ratios y,,,, internuclear distances P,, and internuclear angles &,, between rmn and v,,~;they may be considered to be implicitly contained within Eq. [lo] which can also describe more general DD relaxation models. Other mechanisms may be introduced into the programs provided that the extreme narrowing approximation applies and that the required Hamiltonian is expressed in a partitioned form as in Eq. [7a] or [9a], as the evaluation of the Redfield matrix may then be generated in a straightforward manner from the double commutator expression in Eq. [6a]; for example, a particular Redfield element Rijkl may be found from Eq. [6a] by adding those coefficients containing (p’ - p”)kl in [[p’ - P~,Z~~~],Z~‘~,L’]ij. The scalar type II mechanism, for example, has been included in the programs but is not discussed further here. Inclusion of other mechanisms, such as the anisotropic chemical shift relaxation mechanism, will introduce new complexities (the Go,,,, terms), but evaluation of the Redfield matrix still remains programmable and calculable. Program Details

Computation of a cw bandshape involves an evaluation, over some range of frequency o, of the expression for the signal :

WI

S(w) a trWZ+),

where the real and imaginary parts of the indicated trace (tr) correspond to the dispersion and absorption modes of detection, respectively. For an m spin-3 system of nuclei, Eq. [ 121 contains m *2”‘-’ off-diagonal elements pijr, each of which may be associated with one of the observable transitions in a spectrum of the first-order unsaturated limit. Evaluation of these elements is achieved by solution of the set of linear equations obtained from Eq. [l] after application of the steady-state approximation which allows that dp’ldt = 0. In the event that a spectrum consists of nondegenerate and nonoverlapping transitions, the saturated bandshape may be described by a sum of simple lines similar in form to that obtained from a solution of the Bloch equations (9). In this case, the absorption mode signal near a transition between eigenstates Ii> and Ij> is described by the expression

m&dwl cc

Cz+)ijzCT2)ij l +

(O*j

-

o)2

tT2)ij2

+

Y2 Hp2

Cz+)ij2

tTl)ij

CT2)ij

’

in which the (7’2)jj and (T&, refer to the effective spin-spin and spin-lattice relaxation times for that transition (at o = ol, with intensity (Z+)ij2). In terms of the Redfield elements, r.(T2)ij = Rijij-l, where r = 1 for single quantum transitions, 2 for double quantum transitions, etc., and (T,), is a solution of a set of equations involving the elements Rkkll (9). For overlapping or degenerate transitions, Eq. [13] becomes inapplicable and, except in a few special cases (Z8), solution of the general saturated bandshape in Eq. [12] involves a computation of all the elements contained in Eq. [l]. For an m spin-$ system, there are rz2= 22m e1ements, of which only the n(n + 1)/2 uppe:r or lower triangular elements need be considered because of the Hermicity of p. Separating the complex equations into real and imaginary parts, and noting that the

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equations for the diagonal elements involve only a real part. the resultant system of equations is of the order n2. In standard matrix notation, A(w)X = x0,

u41

where X is a column vector of length n2 containing the required real and imaginary parts of the elements of p in some order, A(o) is the matrix of coefficients as determined from Eq. [1], and X, is a column vector containing p” times its coefficients.2 This set of inhomogeneous equations is rendered linearly independent using the projection method of Hoffman (3), wherein the factor P*tr(@) = P.tr(pO)(=P) is added to Eq. [14], with P chosen to equal the Redfield element Rllll. Calculation of the signal quickly becomes a very formidable problem for more complex spin systems, since a large order of linear inhomogeneous (often ill-conditioned) equations must be solved at each point in the frequency spectrum. Further simplification for the general case is possible, however, since not all of the elements contained in the column vector X are frequency dependent. For a three spin-+ system, the eight diagonal elements and six complex off-diagonal elements corresponding to zero quantum transitions may be removed by a preliminary application of the Gaussian elimination algorithm, thereby reducing the system of equations from the order 64 to the order 44, this reduced system being used for the evaluation of the signal as a function of the frequency. Further reduction of the order of the system of equations is also possible if the spin system allows the “X” approximation. For an ABX spectrum, Eq. [14] may be reduced to a system of order 10 for the signal in the X-region, and to a system of order 20 for the AB-region. Savings in terms of computer execution time are considerable, inasmuch as the number of operations required for a direct method of solution varies as the cube of the order of the system. Careful attention must be paid to the solution of these large-order systems of equations, especially when the simple product representation is chosen as a basis set. A number of cases have been encountered for symmetric three spin-+ systems where, for particular combinations of the relaxation and high-resolution parameters, the equations are so ill-conditioned that round-off errors obscure the solution. In every such case, however, the use of an eigenbasis representation has resulted in spectra for which round-off errors do not have an observable effect. In the programs described here, all Hamiltonians are initially evaluated in the basis product representation, and then transformed to an eigenbasis representation (which is also calculated) before the pertinent equations from [l] are set up as required by Eq. [14]. After elimination of the frequency-independent terms, subsequent solution for the signal is achieved by means of a modified Gaussian elimination scheme with row pivoting. Programs have been written in FORTRAN IV which calculate ABC and ABX spectra (or simpler spectra such as AMX, A2X, etc.) for relaxation by the DD and RF mechanisms (scalar type II relaxation is also included). Aside from the usual highresolution parameters for the Larmor frequencies, coupling constants, and rf field ’ The Liouville representation (see, for example, Ref. (3)) results in a more suggestive matrix inversion formula; however, the solution of a set of linear inhomogeneous equations is subject to fewer round-off errors and is computationally faster than inversion of a matrix of the same order; such considerations are of importance for huge-order systems of equations. On the other hand it is not necessary to set up the equations for the Liouville representation as a matrix inversion problem.

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497

amplitudes, input also includes the relaxation parameters as defined above. An additiona’l input is an inhomogeneity parameter do which is incorporated empirically into the equations by modifying all linewidth terms Rijij to Ri,rj + do. 1AF,l, where AFz is the difference (Zz)it - (Z,)jj for the basis functions Ii) and /j). This procedure will yield sufficiently accurate bandshapes when the rf field is so weak as to preclude saturation effects (see, for example, Ref. (9)) but in general a standard convolution is more satisfactory (19,20). Examples of the required execution times on an IBM 370 are approximately 1.5 points/set for the ABC program, and an average of 15 points/set for the ABX program (an average for the AB and X regions); all calculations are performed in doublepreciision arithmetic. Copies of the program are available from the Atlas NMR Programme Library (21). COMPUTED

SPECTRA

AND

DISCUSSION

The transition frequencies of interest for an AB, system of spin-: nuclei are given in Table 1, along with the numbers used to identify the transitions in the accompanying plots. Also listed are the Redfield elements Rijij for the DD and RF mechanisms in the AX, limit. Due to the inherent degeneracies of such a system, only transitions 1 and 4 may be assigned effective spin-spin relaxation times, with the provision that they do not overlap with other transitions. In general, the degenerate transitions do not have a simple Lorentzian shape and require, in addition to their linewidth factors Rijij, a number of cross terms such as R 2645for transitions 2 and 3 to describe their (unsaturated) lineslnapes. Nevertheless, the listed elements, which are probably the major contributors to the width of a transition, suggest that for either mechanism the spectra in the AX, limit are symmetric about oA in the A region and about wx in the X region. Furthermore, only the asymmetric transition 3 has a linewidth factor which can become vanishingly

RF 2.5

1.0

3

I -0.3

I -I .o

log IJ/6(

FIG. 1. Plot of the linewidth factor RI,,,-’ vs log(J//d) for symmetric three spin-+ systems, where J = JllB and 6 = 10~ - ccBJrand for relaxation by the RF mechanism using the parameters listed in the text. ‘The eight major transitions are identified in Table 1. The break in the log axis is used to indicate that the points on the extreme r.h.s. are the RIJI,-’ in the AX2 limit.

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1

THE MAJOR TRANWION FREQUENCIES” AND MULTIPLE QUANTUM TRANSITION FREQUENCIES FOR AN A& SYSTEM OF SPIN-+ NUCLEI, AND CORRESPONDING LINEWIDTH FACTORS R,jij FORTHEDD ANDRF RELAXATION

Transition W + Ii>)

MECHANISMS

IN THE AX2

Frequency

LIMITS

Linewidth

factor R,,ij

-______

2 16) -+ 12)

1. 13) -+ II>

(WA + c&d/2 + 3Jl4 + A+ w.+A++A-

17/2D,~~ + 9/2Dw2 + 1/2C~,~~r + 2Fa2 + 2F,= 15/2D.m= + 3Dd + 5/2Cm.m, + 2FA2 + 3Fis2 + Cm, 1 5/2Dmz - 15/2G,.~ + 2FA2 + 3FB= - 3 CBB,

3. 15) + 14) 4. IS> -+ 17) 5. 17) -+ 12)

(WA + w.),? 3J/4 + Aag+A+-A-

6. 1.2)--f II>

(WA+ e3)/2 + 3Jl4 - A+

7. 16) --f 13)

8. IS>-+ 16) DQI. 16) --f 11)

cog-A++A(WA + o&2 - 3J/4 - A(WA + 3m + 3J/2 + 2A-)/4

DQz. 18) + 12) DQ3. 17) -+ II>

(WA+ 3wi, - 3J/2 + 2A+)/4 (WA+ 3% + 3J/2 - 2Ae)/4

DQ4. 18) + 13) T. 18) + 11)

(WA+ 3wg - 3J/2 - 2A+)/4 =R,717 (wA+ 2wdi3 9D.m’ + 9/2D~,g~ + 3/2C,mu + 3Cm,,, + 2Fa= + ~FB= i- 4Cm + 2C&

=R1313

25/4D.m2 + 6DBd 35/4Dm2 + 6De2

- 912cABBBr + WCABAB~ f Fj,= + 3F,= + Cm,’ + 9/2Cmm~ + 5/2CmB, + FA= + 3FB2 + Cm.

=R2727

=R121z 35/4D~~’

+

6D&

+

9/2C~,~~, + 5/2CasnBr -i2Fa2 + 3Fs2 + 2C,. + Cr,,,,

=Rm6 17/2D~a2 + 9/2Dm,,’ + 7/2C,,,,p

+ Fa2 + 4FB2 + 2Cw

’ A weak “mixed” transition has not been included here, its intensity being much less than the intesities of the other transitions in the plots shown. * The relaxation parameters are as defined in the text, but with the correlations abbreviated using, for example, CABBB, in place of C,aeB,. /DAB Dw I. The eigenbasis referred to is as follows : 11) = laaa> ; 14) = 12) = (c+l~‘%la~B> + laBa>)+ s+lBaa>; 13) = -b+l~~W$> + la/la>) + c+lBaa>;

(1/~2)(laaB> - I@>);

IS>=

Ul~i)(lf?a8>

-

IBBa>);

16)= W~2)WB>

+ lBBa>)- s-la/V>;

where c* = cos&, S* = sins* 17) = (s-P%(IBaB> + [BBx>) + c-la@>; and 18) = I@/O, tan20, = d\/WAB/(AcoABand I!I J&2) and Aw AB = (WA - w,). Frequencies are expressed in terms of the Larmor frequencies w,,,, the coupling constant J= J AB, and A* = {(dad2 k JAs/4)’ + Jas2/2}1’2. The numbers opposite each transition and the labels DQ1, etc., identify the transitions in the plots. In the AX2 limit, transitions 1 to 4 are “A” transitions and transitions 5 to 8 belong to the “X’‘-region.

small relative to the other linewidth factors, which suggests that this transition might prove to be a reliable indicator of the dominating relaxation mechanism. More information concerning the linewidth factor is contained in Figs. 1 and 2, for which RiJiJ-l for the eight single-quantum transitions listed in Table 1 are plotted against log(J AB/ 1WA - or, I), for the relaxation parameters listed below. A striking feature of the RF mechanism is that the asymmetric transition 3 is always much sharper than the other transitions (cf. Ref. (9)), while the nearby “symmetric” transition 2 is (usually) considerably broader. Figure 1 also indicates that transitions 1 and 4 and transitions 6 and 8 always have identical linewidths (the equality of the linewidths for 1 and 4 with 6 and 8 is an accidental result deriving from the choice FA2 = 2F,*; further, the linewidth

SATURATION IN THREE SPIN SYSTEMS

499

factors for 1 and 4 and for 6 and 8 may differ slightly in the AB2 limit for another choice of RF parameters). In the AX, limit, at the extreme r.h.s. of the plot, there are only three distinct values for the Riiij, and such spectra would be relatively uninteresting from an experimental viewpoint, compared to spectra of second-order systems of nuclei. Figure 2, which contains the dependence of R,jij-’ upon log(J,J)o, - ~~1) for the DD mechanism, displays a considerably different and more complex pattern, although it may be noted that transitions 1 and 4 have the same limiting values in the AX, 50.

-\,,

Rifij (SK)

FIG. 2. As for Fig. 1, but for relaxation by the DD mechanismusing the parameterslisted in the text. For the transition labeled “7*,” the scale should be twice that shown; it has the same limiting AXI value as transition 5 (and as shown by the partial curve labeled “7”).

limit due to the choice of the DD relaxation parameters. For the parameters chosen, transition 3 is no longer unique; rather, it is transition 7 which is always much sharper than the other transitions. The difference between the two mechanisms for the asymmetric transition clearly derives from symmetry reasons, with the DD mechanism, whosle Hamiltonian contains operators acting simultaneously on two spins, being more effective in coupling asymmetric to symmetric spin states. The behavior of transition 7 is less transparent, but apparently it is largely due to symmetry (the parameter choice also has some influence), especially on the 1.h.s. of the plot, where, for example, 13) approaches the symmetrized basis function l/1/6( Icrcrj> + [@a) - 2\@a)). Convergence of the Ri,ij to the AX, limit is much slower for the DD mechanism, and there is a pairwise convergence to identical values for transitions 1 and 4, 5 and 7, and 6 and 8 (cf. Table 1). Another useful feature describing the saturation of single resonance spectra is the relative order in which the transitions begin to show saturation effects as the rf amplitude is increased. For the set of relaxation parameters listed below, a saturation factor Sij, defined using

(cf. Ref. (7)) for the transition Ij) -+ Ii), is plotted against log(J,,/Io, - 0~1) in Figs. 3 and 4 (evaluation of the effective spin-spin and spin-lattice relaxation times is made assuming that no degeneracies exist, and whereas this can never be true for AX2 spectra, the 6:ij still prove useful in interpreting the relative orders of saturation of the following

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10.0

RF

FIG. 3. Plot of the saturation factor (defined in the text) in units of se.condsZvslog(J/6) for symmetric three spin-+ systems, and for relaxation by the RF mechanism. Other details as in Fig. 1.

\

DD

FIG. 4. As for Fig. 3, but for relaxation by the DD mechanism. The scale for the transitions labeled “5*” and “7*” should be ten times the scale shown; drawn to the same scale as the other transitions, they are always above the other curves and have a limiting AX2 value of about 13.21 se?.

plots). For the RF mechanism, the asymmetric transition 3 will begin to saturate before the other transitions 1, 2, and 4 when log(J AB/I mA - 0~1) is less than (approximately) -0.2, with transition 2 always saturating last. In the “W-region, transitions 5 and 6 saturate before transitions 7 and 8, except in the AX2 limit where Fig. 3 indicates that the X transitions tend to begin saturating at the same rf amplitude, this occurring well before similar effects are detectable for transitions 1, 2, and 4,

501

SATURATION IN THREE SPIN SYSTEMS

As before, the pattern for the DD mechanism, presented in Fig. 4, is decidedly different from that for the RF mechanism; here transition 7 will always show saturation effects before the other transitions (note the change in scale for transitions 5 and 7), while transition 3 does not display such a unique behavior. In the AX2 limit, the saturation factors are identical for transitions 1 and 4, 5 and 7, and 6 and 8, suggesting that, in accordance with the RF mechanism, such spectra will possess a symmetry about oA and wx (subject to the equality of relevant cross terms such as Rlz2, and Rsee8, which is the case for AX, spectra). RF D2

- 8.0

28.0 Hz

FIG. 5. Computed absorption mode bandshapes for a strongly coupled AB, system of spin-) nuclei withJ,, = 7.95 Hz, wJ2n = 15.90 Hz, and wg = 0.0 Hz, for relaxation by the RF mechanism using the parameters listed in the text. The numbers identifying the transitions are listed in Table 1 and the calculations include inhomogeneity broadening with a linewidth of ca. 0.05 Hz (see text). The spectra labeled. (a) to (d) are at rf amplitudes of 0.01,8.0,32.0, and 64.0 nT, respectively.

It might be noted here that the pattern of linewidth and saturation factors in Figs. 1 to 4 (and the spectra described below) show no dependence on the sign of JAB, nor is there any dependence on the sign or magnitude of JBB,. This behavior is not unexpected for the DD and RF relaxation mechanisms. The: spectra displayed in Figs. 5 to 10 show computed bandshapes for a number of symmetric three spin-q cases, for relaxation by the DD and RF mechanisms and for different amplitudes of the rf field. The relaxation parameters in each case have been chosen to correspond to those reported by Kumar and Rao (17) from their doubleresonance study of 2,6-dibromoaniline, i.e., Fa2 = 0.200 rad/sec;

c,, = 0.000;

and

FB2 = 0.100 rad/sec;

C,,, = 1.000

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for the RF mechanism, and

DAB2= 0.0729 rad/sec; CA,,,, = -0.125;

and

DBBj2= 0.00270 rad/sec; C,,,,,

= 0.625

for the DD mechanism. The calculated single-resonance bandshapes of the ring protons of 2,6-dibromoaniline, using the high-resolution spin parameters reported by Kumar and Rao (Z7), are reproduced in Figs. 7 and 8. DD

LIZ

(d)

(b)

(a) 2Ei.0 Hz

-0:o

Fig. 6. As for Fig. 5 but for relaxation by the DD mechanism using the parameters listed in the text.

The parameters chosen for the DD mechanism are based on the model of rigid molecules undergoing isotropic reorientation, and as such there is only one independent variable (vide supra). The ratio of DABto DBBsand the values for the correlation parameters have been estimated assuming that the molecular geometry of 2,6-dibromoaniline is similar to that of benzene (17). In the spectra reproduced here, the absolute value of the interactions was adjusted so that the elements R,jij for transitions 1 and 4 are identical for relaxation by either of the RF or DD mechanisms in the AX, limit (see Table 1). The lirst pair of plots in Figs, 5 and 6 shows the bandshapes for a strongly coupled AB, system over a range of rf amplitudes for which the multiple quantum transitions ~eventually appear; all spectra included here have been normalized to give identical heights for the largest peak. The inhomogeneity window of 0.05 Hz is somewhat less than the natural width of the transitions, and consequently the two different relaxation mechanisms yield noticeably different lineshapes even under the nonsaturating conditions of Figs. 5a and 6a, especially with respect to the linewidths, or more obviously the peak heights, of transitions 3 and 7. For the RF mechanism, transition 3 is much

SATURATION

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503

SYSTEMS

(d)

(a) loO.0 Hz

80.0

8.0

-8.0

FIG. ‘7. Computed absorption mode bandshapes for a weakly coupled ABz system of spin-f nuclei with Jai, = 7.95 Hz, o,,/2n = 89.95 Hz, and cu,,= 0.0 Hz, and for relaxation by the RF mechanism. The spectra labeled (a) to (d) are at rf amplitudes of 0.01, 8.0, 16.0 and 64 nT, respectively; other details as in Fig. 5.

---

-

100.0 Hz

80.0

0.0

1

-8.0

FIG. 8. As for Fig. 7 but for relaxation by the DD mechanism.

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sharper than, for example, transition 2; while for the DD mechanism it is transition 7 which is much sharper than transition 2. Although linewidth variations would be less apparent should inhomogeneity be of more importance, moderately saturated spectra, whicharelessdependent oninhomogeneityeffects, willstillexhibitsubstantialdeviations; compare, for example, Figs. 5c and 6c, particularly with respect to the heights of transitions 3 and 7 relative to transition 2. The relative linewidths and order in which saturation effects begin to appear are more conveniently unraveled for these spectra by referring back to Figs. 1 to 4. The intersections of a vertical line drawn through the log axis at (approximately) -0.30 reveals, for example, that, in general, the DD mechanism tends to sharpen the outer four transitions at the expense of the inner four transitions, and the RF mechanism tends to give alternating variations in the linewidths. Figure 3 also indicates that the RF mechanism tends to produce saturation effects first detectable in the inner four transitions, whereas the DD mechanism tends to saturate the “B” transitions 5 to 8 first. Normalization of the spectral heights has partially obscured some differences, particularly for Figs. 5d and 6d ; in fact, the spectra simulated using the DD mechanism should be enhanced by a factor of (approximately) 2, in which case the first three double quantum transitions and the triple quantum transitions are considerably more intense for the DD relaxed spectra (6,8), with DQ4 of comparable intensity for both mechanisms. Comparison of unnormalized spectra also show that the DD spectra tend to saturate slightly more slowly, although the influence of the cross terms may result in a greater degree of overlap and therefore an apparent greater extent of saturation. Absolute maxima in spectral heights were obtained for the RF mechanism near an rl amplitude of 4 nT, compared with 16 nT for the DD mechanism. Figures 7 and 8 show the saturated bandshapes for a weakly coupled AB, spin system, and once again the two mechanisms are distinguishable even in the unsaturated limit. From Fig. 1, at log(J AB/ 1wA - 0~1) N I .05, the disparity between the relative heights of transitions 2 and 3 in Fig. 7a may be seen to be a consequence of the difference between the linewidth factors which approach a ratio of 2 : 1, which is the reciprocal of the ratio approached by the peak heights (theoretically the intensities for transitions 2 and 3 should be nearly equal; also compare with Ref. (9)). The RF mechanism also has that transitions 5 to 8 have almost identical linewidths, and as a result the heights of transitions 5 and 6 and 7 and 8 are approximately equal. This is in contrast with the DD mechanism, for which transitions 5 and 7 are somewhat sharper than are 6 and 8, while the “A” transitions have linewidths which are not detectably different. The saturation pattern of these spectra is less readily deduced from Figs. 3 and 4 since many of the transitions exhibit substantial overlap. Neglecting such effects, however, the RF spectra and plots of the RF saturation factors agree in that transitions 5 and 6 saturate before transitions 7 and 8, so that at an rf amplitude of 64 nT the lineshapes of the two doublets are almost symmetric; this is in contrast to Fig. 4 and the spectra for the DD mechanism, for which the outer doublet saturates before the inner doublet. For the transitions in the A region, the RF mechanism has that transition 3 saturates well before transition 2, and transitions 1 and 4 saturate at an intermediate rf amplitude; while for the DD mechanism there is little variation between the saturation order of transitions 1 to 4. In terms of absolute peak heights, both mechanisms have absolute spectra maxima at approximately the same value of the rf field, and the

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scale for spectra at the same field in Figs. 7 and 8 are approximately the same for the A-region, while the B-region is slightly attenuated for the spectra relaxed by the DD mechanism.

(c) (b)

(0) V, +lZ.OHz

v, -12.0

10.0

-10.0

FIG. 9. Computed absorption mode bandshapes for an AXI system of spin-4 nuclei with JAB = 7.95 Hz, wJ27c = 10’ Hz, and wg = 0.0 Hz, for relaxation by the RF mechanism. The spectra labeled (a) to (c) are at rf amplitudes of 0.01, 8.0 and 64.0 nT, respectively; other details as in Fig. 5.

DD

516 4I I L

(b)

(a) V, +12.0Hz

FIG.

1 -10.0

10. As for Fig. 9 but for relaxation by the DD mechanism.

The last pair of spectra, in Figs. 9 and 10, are of an AX2 system of nuclei, and the choice of the DD parameters is such that the differences in the linewidths are not as conspicuous as for the previous cases. As noted, transitions 1 and 4 have identical linewidths for both mechanisms; however the peak heights of these transitions are slightly greater for the DD mechanism than for the RF mechanism as a result of slightly

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varying scaling factors for the spectra, based on the height of the doublet of transitions 2 and 3. This is in agreement with Figs. 1 and 2 if the averages of Rijij-l for transitions 2 and 3, taken from the extreme r.h.s. of the plots, are compared for the two mechanisms (cross terms, not necessarily producing similar effects for the two mechanisms, are ignored). Similar averages for the doublets 5 and 6 and 7 and 8 have that the DD relaxed spectra should be slightly sharper; and although variations in the linewidths are not detectable, the absolute peak heights are slightly greater for the X transitions of the DD spectrum. For the saturated AX, spectra in Figs. 9c and lOc, transitions 1 and 4 saturate before the doublet 2 and 3 for the DD mechanism compared with the pattern for the RF mechanism, this feature being in accord with the results indicated using Figs. 3 and 4. Comparison of the relative ratios of the absolute heights of the A doublet to the X doublets also has that the X transitions saturate faster than the A transitions for the DD mechanism compared with the RF mechanism. It was previously noted that the model chosen for the DD mechanism is relatively simple, and effectively there is only one parameter which is independently adjustable, and the question arises as to whether or not the use of more sophisticated DD models might not give spectra which resemble more closely equivalent spectra relaxed by the RF mechanism. An attempt was made to obtain a closer fit to the RF spectra in Fig. 5 by allowing that, first of all, the two interaction parameters DAB and DBBtwere independently variable and, secondly, that all four DD parameters were independently variable. A program, based on the simple line approximation and to be reported on at a later date, was used in an endeavor to match effective spin-spin and spin-lattice relaxation times obtained for the RF mechanism to those obtained using the DD mechanism by varying the indicated parameters. Plots of the best fits thereby obtained indicate that spectra of the sort presented in Fig. 5 cannot be matched by the DD mechanism, with the major variation being with respect to the behavior of the asymmetric transition (a set of DD parameters could be found which gave a similar linewidth and saturation behavior for transition 3, but the remainder of the bandshapes were even more divergent from the RF generated bandshapes; similarly, a set of parameters could be found for which transitions 5 to 8 behaved much as for the RF mechanism, but then transitions 1 to 4 were considerably different). Parallel findings apply to the spectra in Fig. 7, although the results may be less conclusive inasmuch as, for those spectra, the fitting criteria (the effective relaxation times) cannot be rigorously defined. Finally, for the spectra in Fig. 9, adjustment of the DD parameters, based only on the analytical expressions for the linewidth factors Rijij in Table 1, could give DD spectra which closely resemble the RF spectra in the unsaturated limit; however, in general, the saturation behavior of the two different mechanisms showed detectable variation, as did the amount of overlap between neighboring transitions. A final consideration to be discussed here is based on the sensitivity of the bandshapes to the various relaxation parameters. Ideally, one would like some set of experimentally accessible data, such as relative peak heights for some pair of transitions, to be dependent on only one of the relaxation parameters; however, these spectra do not readily furnish such orthogonal characteristics. A few generalizations are made possible, however, by generating plots of the change in the linewidth and saturation factors for a given change in a relaxation parameter. From such plots it may be possible to deduce

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507

an approach to the practical problem of analyzing experimental spectra in order to obta’in their particular set of relaxation parameters. By way of example, the following observations on the saturation factor were made for the weakly coupled spectra in Figs. 7 and 8: (a) For the RF mechanism, the A transitions I to 4 are strongly dependent on FA, their values for Sij decreasing by an average of about 26 % for a 10 % increase in FA, compared to a decrease of about 10% for the transitions 5 to 8; while the relative strengths of dependence is reversed for FB. Increasing CAB to a value of 0.10 increases Sij for the outer four transitions by an average of about 2 %, and decreases Sij for the inner transitions by about the same amount; while decreasing C,,, by 10 % results in an increase in Sij for transitions 5 to 8 by about 4 ‘A and has a mixed but weak effect on the A transitions. The asymmetric transition exhibits a strong dependence (ca -32%) on FA but no apparent effect on the other RF parameters, partially due to the choice of CBB,= 1.O(cf. Table 1). The stated dependence on FA and FB is enhanced in the X limit; furthermore AX2 spectra do not depend on the parameter C,, (cf. Ref. (17)). (b) For the DD mechanism, an increase of 10% in DAB, the largest contributor to relaxation by this mechanism, results in a large decrease of about 32% in Sij for all transitions, while a similar increase in D BB, yields a small increase for transition 7, no change for transition 3, and small decreases for the remaining transitions. Dependence on the two DD correlation parameters is weak, with transition 3 decreasing by about 2 % and the other transitions increasing by about 1.5 % for C,,,,, decreased by 10%; and with transitions 5 and 7 increasing by about 2.5 “/L, transitions 6 and 8 decreasing by about 2 %, and with small and mixed changes for transitions I to 4 when c ABBB’ is increased by 10%. These observations indicate that, at least for a set of relaxation parameters near those chosen, the bandshapes depend mostly on the three parameters FA, FB, and DAB, with the other parameters producing small but quite different effects on the spectra. Analysis of experimental spectra might be facilitated through the use of such a set of empirical observations, although the most satisfactory (but computationally formidable) approach would be a numerical fit of theoretical to experimental bandshapes (cf. Ref. (6)). In a future report, a program currently in development will be described which may simplify experimental analyses in many cases. CONCLUSIONS

The computed single-resonance bandshapes of symmetric three spin-3 systems are, in general, distinguishable for relaxation by the RF or DD relaxation mechanisms. Differences are detectable, especially for second-order spin systems, for unsaturated spectra when natural linewidths are not dominated by inhomogeneity broadening. These differences are more apparent for spectra in which the rf amplitude is strong enough to produce moderate saturation effects, and a single experimental spectrum of a strongly coupled spin system under such conditions may be expected to yield qualitative information as to which (if either) of the two mechanisms dominates the relaxation proczss. A series of experimental spectra at various rf amplitudes (and possibly in different static magnetic fields or at different temperatures) might provide sufficient data for a quantitative determination of the relevant relaxation parameters, this being particularly true for spectra of systems of strongly coupled nuclei.

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ACKNOWLEDGMENTS One of the authors (B.W.G.) would like to thank Dr. R. K. Harris for his kind hospitality while he was at the University of East Anglia; likewise, the many fruitful discussions with Dr. K. M. Worvill and W. S. Bitter should be mentioned. Parts of this work were also carried out at Simon Fraser University, where Dr. E. J. Wells helped elucidate some of the research. REFERENCES R. FREEMAN, S. WITTEKOEK, AND R. R. ERNST, J. Chem. Phys. 52,1529 (1970). 2. B. D. N. RAO, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 271, Pergamon Press, Oxford, 1970. 3. R. A. HOFFMAN, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 87, Pergamon Press, Oxford, 1970. 4. B. GESTBLOM AND 0. HARTMAN, J. Magn. Resonance 8,230 (1972). 1.

5. 6. 7. 8. 9. 10. Il. 12. 13. 14.

R. R. R. K. A. R. R. D.

K. HARRIS AND K. M. WORVILL, Chem. Phys. Left. 14,598 (1972). K. HARRIS AND K. M. WORVILL, J. Magn. Resonance 9,394 (1973). K. HARRIS, N. C. PYPER, AND K. M. WORVILL, J. Magn. Resonance 18, 139 (1975). M. WORVILL, J. Magn. Resonance 18,217 (1975). ALIRAGAM, “Principles of Nuclear Magnetism”, Oxford University Press, London, 1961. M. LYNDEN-BELL, Proc. Roy. Sot. A%, 337 (1965). K. HARRIS AND K. J. PACKER, J. Chem. Sot. (London), 4736 (1961). G. HUGHES, M. R. SMITH, AND D. H. M. SWITZER, J. Chem. Phys. 60,489O (1974). A. KUMAR AND B. D. N. RAO, J. Magn. Resonance 8,l (1972). A. G. REDFIELD, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 1, p. 1, Pergamon

Press, Oxford, 1965. 15. P. S. HUBBARD, Phys. Rev. 180,319(1969). 16. N. C. PYPER, Mol. Phys. 21, 1 (1971). 17. A. KUMAR AND B. D. N. RAO, Mol. Phys. 15,377 (1968). 18. K. F. KUHLMANN AND J. D. BALDEXHWIELER, J. Chem. Phys. 19. B. W. GOODWIN AND R. WALLACE, J. Magn. Resonance 12,60 20. B. W. GOODWIN AND R. WALLACE, J. Magn. Resonance 9,280

43,572 (1973).

(1965).

(1973). 21. Science Research Council ATLAS Computer Laboratory, NMR Program Library, c/o P. Anstey and R. K. Harris, School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, England.