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Solid echoes in the slow-motion region
JOURNAL
OF MAGNETIC
42, 381-389
RESONANCE
(1981)
Solid Echoes in the Slow-Motion H. Institut
fiir
Physikalische
Received
w.
SPIESS
AND
H.
SILLESCU
Chemie der Universitiit Mainz, D-6500 Mainz, Germany June
16, 1980;
revised
Region
September
Jakob-Welder-Weg
IS,
8, 1980
NMR lineshapes obtained by Fourier transform of the solid-echo decay are compared with those from the free-induction decay in the slow-motion region where the motional correlation times are of the same order as the reciprocal anisotropic spin interactions. The theory has been worked out explicitly for the example of twosite exchange in an Z = 1 spin system that applies, e.g., to ‘H or *D NMR of water molecules flipping by 180” about the twofold axis. Large differences are found in the region where the time between the two 90” pulses of the solid-echo sequence is comparable with the motional correlation time. The intensity of the solid echo is reduced by a factor R 2 0.5 relative to the free-induction decay for the example of anisotropic motion considered. Much smaller reduction factors are obtained for isotropic reorientation by small- or large-angle rotational jumps.
Application of Fourier transform techniques to NMR lineshape investigations in solids is limited by “dead time” effects due to pulse length, recovery times, magnetoacoustic ringing, etc. In order to circumvent dead time problems, Powles technique where two 7r/2 pulses and Strange (I) have proposed a “solid-echo” separated by a time distance T and shifted in phase by 7r/2 are applied. The solidecho decay at times t 2 27 is recorded which is identical to the FID for solid I = 1 systems experiencing a time-independent coupling. This technique is also applicable to 2D NMR in liquid crystals if rapid anisotropic motions give rise to a quasi-rigid spectrum given by the time-averaged Hamiltonian of the 2D quadrupolar coupling (2). Clearly, no solid echo is observable in real liquids where in the absence of magnetic field inhomogeneities the FID is irreversible, and the decay time T2 depends upon correlation times of molecular motion. In the slow-motion region with correlation times TVcomparable with the pulse separation 7, the solidecho decay can no longer be expected to be identical with the FID. For isotropic rotational diffusion, Woessner and collaborators (3) have calculated solid echoes following the stochastic Liouville approach (4) in the formulation first given by Korst and Khazanovich (5). In the present paper, we follow rather closely Abragam’s (6) treatment of NMR in the slow-motion region. The theory is applied explicitly to the problem of two-site exchange in 2D NMR which is relevant to recent investigations of lipid systems of biological interest (7, 8). Our discussion includes some comments on rotational diffusion (3) and short time expansions (9, 10). Our main goal is to investigate how NMR lineshapes are changed when they are obtained as Fourier transforms of the solid-echo decay rather than the FID itself. 381 0022-2364/81/030381-09$02.00/0 Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
382
SPIESS
AND
SILLESCU
THEORY
We restrict our treatment to adiabatic perturbations of the Zeeman coupling by dipolar or quadrupolar couplings both given in the rotating frame by X(t) = A(t)[Z:
- (l/3)1(1 + l)],
A(r) = Ao[3 co? e(t) - 11,
ill PI
where Z = 1. For dipolar coupling of two spins l/2 at a distance r A, = (3/4)y%r-3;
for quadrupolar
[31
coupling of one spin Z = 1, A0 = (3/8)e2Qqh.
[41
In Eq. [2], 8 is the angle the external field makes with the vector r for dipole coupling, and with the unique axis of the field gradient q for quadrupolar coupling, the latter assumed to be axially symmetric. Furthermore, we assume that the time dependence of 8 is given by a Markov process. The FID can be formulated for the Z = 1 system defined above by (6) s,(t)
= 2(cos
where
4(0,0),
[51
A(t’)dt’.
[61
Y2 4(f*,tz> =
I t1
The angle brackets denote averaging over the Markov process. As can be shown by standard procedures (6), the solid-echo decay is given by Sl(t,d = %cos [4(O,d - 447,t)l). [71 In the rigid limit, A is constant, and +(fl,f2) = (f$ - ?,)A. Thus, $(O,r) - +(r,t) = (t - 27)A and S,(~,T) = 2 cos [(t - 2~)A,(3 co? 0 - l)] PI is the NMR signal for a single crystal. The solid echo of a polycrystalline is obtained when Eq. [8] is averaged over all orientations (I). Let us formulate the average in Eq. [7] as (c“s [+(o,T) - ddT,t)l)
= (1/2)[K(t,T)
+ K(t,T)*],
powder
191
where The evaluation
of
K(t,T)
K(l,T) = (exP{i[+(o,T) - +(T,f)l)). follows closely that of the FID
[lOI
G(t) = (exp{W401) [Ill given on pages 448 ff. of Ref. (6). Using a matrix notation for exchange between a finite set of frequencies, a master equation z
G = G(iw + n)
is derived for a vector G(t), the components defined in Eq. [ 111;
[121
of which add up to the function G(t)
SOLID
ECHOES
IN
SLOW-MOTION
383
REGION
G(t) = W*exp[(io + ~)t].l, [I31 where o is a diagonal matrix with components that are the possible values of A in our case, rr is the transition rate matrix for transitions between these values, W = G(0) is a vector consisting of the time-independent probabilities of finding a system in one of the possible A values, and 1 is a vector with all components equal to unity. Along the same lines, one can derive the expression K(~,T) = W.exp{(io
+ 11)~) .exp{(-iw
+ ?r)(t - T)} -1.
[I41
It should be noted that K(t,O) = G(t)*, K(T,T) = G(T), and K(~,T) becomes G(t - 2T)* in the rigid limit rr + 0 where the solid-echo decay is identical with the FID except for a time shift by 27 (see Eq. [8]). The numerical evaluation of the expressions of Eqs. [13] and [14] is achieved by determining the eigenvalues of the complex symmetric matrix io + 1~. An appropriate computer program is described by Gordon and Messenger in Ref. (4). The simplest case of a 2 x 2 matrix is discussed in the following section. If A(t) depends upon time through a continuous Markov process a(t), the summations in Eqs. [13] and [14] are replaced by integrations over the fl space: G(t) = K(~,T) =
Here, G(R,&,t)
I
I
d%W(C$J
dfloW(fio)
I
I
[151
~fWWo,fLt),
dfl,G(%,R,,~)
I
d&G(Cb,,
iit, I - T)*,
[I61
is the solution of r171
subject to the initial condition GWo,Q,O) where 6(Q - Q,) is the 6 function
= W-4 - fro),
[181
and
is the rate for transitions R’ -+ sl,. The quantity P(R’lQ,t)di& is the probability of finding a system between fit and Q + dR, at time t subject to the condition that it was Cl’ fort = 0. For small-step rotational diffusion, the integral operation in Eq. [17] is replaced by the Laplacian differential
- mg-(n,) I dn’f(sl’)II(n’,n,) and P(fIo 1Cn,,t) becomes the solution P(Ro 1f-4,0)= &at - 00). LINESHAPE
of the rotational
WI diffusion
equation
for
CALCULATION
In this section we treat explicitly the simple case in which the slow motion merely consists of an exchange between two sites with frequencies
384 01
SPIESS
AND
=
6;
ij
+
SILLESCU
[211 Then the spectrum can be calculated analytically. The response to a single pulse, G(t), is obtained from Eq. [13] as described, e.g., in Refs. (II, 12): cos At + -
w2
=
sin At
0
-
6.
(A = (82 - wy,
G(t) = @~$-nt.2.
cl2 < CY),
(R2 = S”), (p = (W
WI
- fY)“2,
R2 > p’).
Here as in subsequent equations the first, second, and third rows apply for slow, intermediate, and fast exchanges, respectively, and the exchange parameter CR gives the probability of interchanging the two nuclei (6). The response to the solid-echo sequence, K(t,+ is obtained similarly from Eq. [14]: K(t,7) = e- iPt--P7)e--Rt.2 cos At + t
sin At + $
sin A7 sin A(t - 7)
1 + Rt + 2WT(f
X
- 7)
. 1231
cash pt + fl sinh pt + 262 sinh pi sinh p(t - T) P
P2
The spectrum Z(w) is obtained through Eq. [22], starting at time t = 0:
Fourier transform
462R Aw4 + 2Aw2(2R2 - P) + a4
Z(w) =
cf. also Ref. (6). Similarly FT of K( t ,T) starting .Z(W,T) as determined through the solid-echo J(w,r) it is convenient to bring K(t,T) into the t’ = t - 27. From Eq. [23] we then obtain, for fqt,7)
=
e-iGt’e-W
given in
(Aw = w - 0);
at time t = 27 gives the spectrum technique. In order to calculate same form as G(t) by introducing fast and slow exchange,
.3A
w
R
62
- -;;;- cos 2x7 + h sin 2Ar + s [
1
cos At’
w n [ 1 cl’ R 62 1 R 1 +
X
(FT) of G(t),
h2 sin 2x7 + x cos 2Ar
-;- cash 2pr + P
sinh 2~7 - 1
P
2
sin At’
cash pt’
P
sinh 2pr + -
P
cash 2~7
sin’h pt’
eeZRT. [25]
SOLID
FIG. 1. Calculated two sites for various row: distorted spectra
ECHOES
IN
SLOW-MOTION
385
REGION
powder spectra for a spin I = 1 system spectra jump rates R. Left row: “true” obtained through FT of solid echo with
undergoing rotational jumps between obtained through FT of FID. Right pulse separation 7 = 8.
Now FT starting at time t’ = 0 can easily be performed
to yield
clsin AT + -A62 sin2 hr A 1 A2 (1 + CW2 + Ao2r2 R A& + + - sinh pr sinh2 pr 2
(
cos A7 + -
.z(W,T) = Z(o).
2
cash pi
P
P2
The spectrum J(w,T) as obtained from the FT of the solid echo thus is given by the “true” spectrum Z(w) multiplied by a reduction factor depending on the pulse separation T. RESULTS
AND
DISCUSSION
As an example of practical importance we have calculated powder spectra for deuterons exchanging between two sites, where the unique axes of the two axially symmetric field gradient tensors form the tetrahedral angle. Spectra of this kind are expected, e.g., for a water molecule flipping by 180” about its twofold axis. Recent experimental examples are provided by motionally narrowed deuteron spectra of lipids and collagen (7,s) and of the amorphous regions of polyethylene
386
SPIESS
AND
SILLESCU
FIG. 2. Calculated powder spectra for a spin I = 1 system undergoing rotational jumps between sites for various jump rates a. Left row: “true” spectra obtained through FT of FID. Right distorted spectra obtained through FT of solid echo with pulse separation 7 = 8.
(13, 14). The powder spectra are obtained by superposition in Eq. [26] with frequencies given by Eq. [2]: wk = &/A,, = 3 co.!? 8k - 1,
k = 1,2.
two row:
of the spectra given
1271 The numerical calculation corresponds to that of “true” absorption spectra described in Ref. (16). In Figs. 1 and 2 the powder spectra as obtained through the solid-echo technique are plotted for various exchange rates fi. The values of fl and r given are normalized by division by AO; cf. Eq. [27]. In order to obtain absolute values for a given system the quadrupolar coupling constant has to be incorporated. For deuterons in aliphatic C-D bonds the conversion factors are approximately A,, = 4 x lo5 see-* and Ai1 = 2.5 +ec for R and T, respectively (16). The spectra presented in Figs. 1 and 2 thus correspond to 7 = 20 psec. For easy comparison the “true” spectra are also plotted; cf. also Ref. (11). Whereas both lineshapes Z(w) and J(w,T) are identical for very slow (0 =+ 1) and very fast (a % 1) exchange rates, severe distortions are introduced by using the solid-echo technique in the transition region (0.1 5 1R 5 10). Therefore, one is not justified in trying to determine jump rates 0 by comparison of experimental spectra obtained via the solid echo with theoretical ones, calculated from the FID. Moreover, in an attempt to analyze such spectra quantitatively, not realizing that they are distorted because of finite T, one can easily be led to incorrect conclusions, e.g., “explaining” the lineshapes observed as superpositions of motionally narrowed spectra of different exchange rates. In Fig. 3 the T dependence of the lineshape is shown for intermediate exchange (a = 1). It is clear that significant distortions occur even for small values of T, e.g., 7 = 2 3 5 psec. With increasing T, on the other hand, the lineshape almost
SOLID
R=0.59
ECHOES
IN SLOW-MOTION
REGION
387
:
FIG. 3. Calculated powder spectra obtained going rotational jumps between two sites. separation 7 is varied.
through FT of solid echo for a spin I = 1 system underThe jump rate I2 has a fixed value I2 = 1; the pulse
converges; note the minor change of the lineshape on increasing 7 from 8 4 20 psec to 16 !! 40 psec compared with the differences in the lineshape obtained from the FID (T = 0). The reason for this behavior can be visualized considering the expression for J(w,T) as given in Eq. [26]. The intensity of the spectra contributing to the powder lineshape for which the condition of slow exchange, R2 =CS2, is fulfilled is damped by the factor exp(-2&). For the same value of a, however, there are other spectra that contribute to the powder lineshape, for which the condition of fast exchange (W > a*) is met. These are damped by the factor exp[-2(LR - p)~] only, and, therefore, will dominate the lineshape obtained via the solid echo for finite 7. In Figs. 1 to 3 reduction factors R are also included, giving the ratio of the highest points in the “stick spectra” for a given T and T = 0, respectively. These reduction factors are above 0.48 for all the spectra presented here, showing that considerable spectral intensity can be detected via the solid echo throughout the transition region from slow to fast exchange for moderate values of 7. In reality, of course, the solid echo will further be damped by transverse relaxation, taken into account here only by convoluting the final powder lineshape with a Gaussian exp(-A&/2u2) with u = 27r. 1.27 kHz, corresponding to the dipolar coupling in perdeuterated samples (14, 16). In partially deuterated compounds there is an additional contribution from dipolar deuteron proton interaction (14). Our present lineshape calculation makes clear that in the case of a limited motional narrowing as provided by the exchange between two sites only, the solid-echo technique will be able to provide the possibility of obtaining FT spectra of deuterons not only in the limits of very slow and very fast exchange but in
388
SPIESS
AND
SILLESCU
the intermediate region as well. The lineshapes obtained via the solid echo, however, are severely distorted compared with the “true” spectra that could in principle be obtained from FT of the FID. For proper analysis of experimental lineshapes, therefore, theoretical spectra also have to be calculated by taking the FT of the response to the solid-echo sequence as described here. Solid echoes with acceptable reduction factors R 2 0.5 can only be expected for anisotropic reorientation where the rapid motion limit leads to a “quasi-rigid” spectrum, the total width of which is comparable with that of the rigid spectrum; cf. Fig. 2. For isotropic reorientation, the solid echo is drastically reduced if the pulse separation r becomes longer than the rotational correlation time 7C unless A,,T~ + 1. This has precluded the application of the solid-echo technique to lineshape investigations of tetrahedral jumps, e.g., in deuterated hexamethylenetetramine (16, 17). Isotropic small-step rotational diffusion causes similar reductions, as is apparent from the calculation of Woessner et al. (3). An interesting aspect of the lineshape distortions produced by the second 90” pulse can be seen in a perturbation expansion of the master equation for isotropic rotational diffusion. For relatively rapid motion A,,rC 5 0.5 the FID and the solidecho decay (see Eqs. [ 151 and [ 161) can be approximated by (18,19) ~(t,~)*
G(t) = (1 + 3a + p)eeAot + (p* + a)eeAZt, = ~(1 + 3,)e-c~~~+Au’h+ pe-ALle-hot’ + [~*~-A27 + ae-ch+&)~]e-hf’,
[281 [291
a = (415)A;r: = ?,lTz,
[301
/3 = -2a
1311
+ i*1.280312,
A, = T;‘(l
+ 0.37~~ + i-0.64d2),
[321
A2 = ~;l(l
- 0.69a - i .0.640?‘~).
[331
For rapid motion (Y << 1 and t 9 r, we have G(t)
= K(t,T)*
[341
= e-t’T2.
The most important corrections to this exponential arise from the imaginary terms in the expression Eq. [28] for the FID, whereas just these terms are reduced by factors e-“‘c in the “solid echo,” Eq. [29]. Since the imaginary terms describe the onset of solid-like behavior, one should be aware of their absence if rotational correlation times are determined from exponential solid-echo decays. A further comment relates to a short time expansion that has been used in the analysis of solid echoes in polymer solutions and melts (9, 20). Here, molecular reorientation is constrained through entanglements, and a “nonzero average” dipolar coupling A that amounts to about 1% of the original interaction contributes to a “pseudosolid echo.” It has been shown (9, 10) that the normalized signal r(t,T)
= i
Re
K(t,d
- G(t) G(O) I
used in the analysis of the experiments can be described approximation r(t,7) = 7(t - 7>(Az)e-B’,
[351 adequately
by the t361
SOLID
ECHOES
IN
SLOW-MOTION
389
REGION
where the angle brackets denote an ensemble average, and B is an empirical parameter. We can readily reproduce Eq. [36] for our example of two-site exchange if we let ti = 0, S2 = (AZ), and R = B in Eqs. [22] and [23]. Equation [36] is then valid for all times if R2 = S2 and it is obtained for small times t, subject to the condition hr -+ 1 if R2 < S2, or pr -G 1 if fin2 > a2, respectively. Equation [36] is also obtained for isotropic rotational diffusion if we expand Eqs. [28] and [29] for small times, and let Af = (A’) and (2~~)~~ = B. Thus, the parameter B can be related to the decay of the anisotropic spin interactions. Applications to 2D NMR in deuterated polyethylene melts will be published elsewhere (20). CONCLUSIONS
Fourier transform of the solid-echo decay leads to NMR lineshapes that differ from the true absorption spectrum in the slow-motion region 0.1 5 0 5 10. For anisotropic motion leading in the fast exchange limit to a “quasi-rigid” spectrum comparable in width with that of the rigid one the signal intensity of a polycrystalline sample for intermediate exchange likewise is comparable with that for slow or rapid exchange. The distortions of the lineshape introduced by using the solid-echo technique, however, have to be taken into account if such lineshapes are analyzed to give information about type and time scale of the motion. For isotropic reorientation, the solid-echo technique produces drastically reduced intensities for T > T, where the correlation time 7C is a few microseconds only in the slow motion region of typical deuterated compounds. ACKNOWLEDGMENT Financial
support
by the
Deutsche
Forschungsgemeinschaft
is gratefully
acknowledged.
REFERENCES AND J. H. STRANGE, Proc. Phys. Sot. 82, 6 (1963). K. R. JEFFREY, M. BLOOM, M. I. VALIC, AND T. P. HIGGS, Chem. Phys. Letf. 42, 390 (1976). 3. D. E. WOESSNER, B. S. SNOWDEN, JR., AND G. H. MEYER, .I. Chem. Phys. 51, 2968 (1969). 4. L. T. Muus AND P. W. ATKINS (Eds.), “Electron Spin Relaxation in Liquids,” Plenum, New York, 1972. 5. N. N. KORST AND T. N. KHAZANOVICH, Sm. Phys. JETP 18, 1049 (1964); Zh. Eksp. Teor. Fiz. 45, 1523 (1963). 6. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford Univ. Press, Oxford, 1961. 7. R. G. GRIFFIN, Abstracts of the 21st ENC, Tallahassee, Florida, 1980. 8. L. W. JELINSKI, C. E. SULLIVAN, L. S. BATCHELDER, AND D. A. TORCHIA, Biophys. J., in 1. J. G. POWLES 2. J. H. DAVIS,
press. 9. J. P. COHEN-ADDAD AND R. VOGIN, Phys. Rev. Lett. 33, 940 (1974). IO. J. P. COHEN-ADDAD, J. Chem. Phys. 63, 4880 (1975). II. M. MEHRING, in “NMR, Basic Principles and Progress” (P. Diehl, 12. 13.
Eds.), Vol. 11, Springer-Verlag, Berlin, 1976. H. W. SPIESS, in “NMR Basic Principles and Progress,” K. ROSENKE, H. SILLESCU, AND H. W. SPIESS, Polymer,
14. 15. 16.
D. D. U.
17. 18.
M. LAUSCH H. SILLESCU
19. 20.
H. SILLESCU, J. Chem. Phys. J. COLLIGNON, H. SILLESCU, AND Deutsche Physikalische Gesellschaft
Vol. 15, Springer-Verlag, 21, 757 (1980).
HENTSCHEL, H. SILLESCU, AND H. W. SPIESS, Makromol. Chem. HENTSCHEL, H. SILLESCU, AND H. W. SPIESS, to be published. PSCHORN AND H. W. SPIESS, J. Magn. Reson. 39, 217 (1980). AND H. W. SPIESS, AND D. KIVELSON,
E. Fluck,
180,
and
R.
Berlin,
Kosfeld, 1978.
241 (1979).
Chem. Phys. Lett. 71, 182 (1980). J. Chem. Phys. 48, 3493 (1968). 54, 2110 (1970). H.
W. SPIESS, presented at the 1980 spring meeting in Leoben, Austria; Progr. Polym. Sci., in press.
of
OF MAGNETIC
42, 381-389
RESONANCE
(1981)
Solid Echoes in the Slow-Motion H. Institut
fiir
Physikalische
Received
w.
SPIESS
AND
H.
SILLESCU
Chemie der Universitiit Mainz, D-6500 Mainz, Germany June
16, 1980;
revised
Region
September
Jakob-Welder-Weg
IS,
8, 1980
NMR lineshapes obtained by Fourier transform of the solid-echo decay are compared with those from the free-induction decay in the slow-motion region where the motional correlation times are of the same order as the reciprocal anisotropic spin interactions. The theory has been worked out explicitly for the example of twosite exchange in an Z = 1 spin system that applies, e.g., to ‘H or *D NMR of water molecules flipping by 180” about the twofold axis. Large differences are found in the region where the time between the two 90” pulses of the solid-echo sequence is comparable with the motional correlation time. The intensity of the solid echo is reduced by a factor R 2 0.5 relative to the free-induction decay for the example of anisotropic motion considered. Much smaller reduction factors are obtained for isotropic reorientation by small- or large-angle rotational jumps.
Application of Fourier transform techniques to NMR lineshape investigations in solids is limited by “dead time” effects due to pulse length, recovery times, magnetoacoustic ringing, etc. In order to circumvent dead time problems, Powles technique where two 7r/2 pulses and Strange (I) have proposed a “solid-echo” separated by a time distance T and shifted in phase by 7r/2 are applied. The solidecho decay at times t 2 27 is recorded which is identical to the FID for solid I = 1 systems experiencing a time-independent coupling. This technique is also applicable to 2D NMR in liquid crystals if rapid anisotropic motions give rise to a quasi-rigid spectrum given by the time-averaged Hamiltonian of the 2D quadrupolar coupling (2). Clearly, no solid echo is observable in real liquids where in the absence of magnetic field inhomogeneities the FID is irreversible, and the decay time T2 depends upon correlation times of molecular motion. In the slow-motion region with correlation times TVcomparable with the pulse separation 7, the solidecho decay can no longer be expected to be identical with the FID. For isotropic rotational diffusion, Woessner and collaborators (3) have calculated solid echoes following the stochastic Liouville approach (4) in the formulation first given by Korst and Khazanovich (5). In the present paper, we follow rather closely Abragam’s (6) treatment of NMR in the slow-motion region. The theory is applied explicitly to the problem of two-site exchange in 2D NMR which is relevant to recent investigations of lipid systems of biological interest (7, 8). Our discussion includes some comments on rotational diffusion (3) and short time expansions (9, 10). Our main goal is to investigate how NMR lineshapes are changed when they are obtained as Fourier transforms of the solid-echo decay rather than the FID itself. 381 0022-2364/81/030381-09$02.00/0 Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
382
SPIESS
AND
SILLESCU
THEORY
We restrict our treatment to adiabatic perturbations of the Zeeman coupling by dipolar or quadrupolar couplings both given in the rotating frame by X(t) = A(t)[Z:
- (l/3)1(1 + l)],
A(r) = Ao[3 co? e(t) - 11,
ill PI
where Z = 1. For dipolar coupling of two spins l/2 at a distance r A, = (3/4)y%r-3;
for quadrupolar
[31
coupling of one spin Z = 1, A0 = (3/8)e2Qqh.
[41
In Eq. [2], 8 is the angle the external field makes with the vector r for dipole coupling, and with the unique axis of the field gradient q for quadrupolar coupling, the latter assumed to be axially symmetric. Furthermore, we assume that the time dependence of 8 is given by a Markov process. The FID can be formulated for the Z = 1 system defined above by (6) s,(t)
= 2(cos
where
4(0,0),
[51
A(t’)dt’.
[61
Y2 4(f*,tz> =
I t1
The angle brackets denote averaging over the Markov process. As can be shown by standard procedures (6), the solid-echo decay is given by Sl(t,d = %cos [4(O,d - 447,t)l). [71 In the rigid limit, A is constant, and +(fl,f2) = (f$ - ?,)A. Thus, $(O,r) - +(r,t) = (t - 27)A and S,(~,T) = 2 cos [(t - 2~)A,(3 co? 0 - l)] PI is the NMR signal for a single crystal. The solid echo of a polycrystalline is obtained when Eq. [8] is averaged over all orientations (I). Let us formulate the average in Eq. [7] as (c“s [+(o,T) - ddT,t)l)
= (1/2)[K(t,T)
+ K(t,T)*],
powder
191
where The evaluation
of
K(t,T)
K(l,T) = (exP{i[+(o,T) - +(T,f)l)). follows closely that of the FID
[lOI
G(t) = (exp{W401) [Ill given on pages 448 ff. of Ref. (6). Using a matrix notation for exchange between a finite set of frequencies, a master equation z
G = G(iw + n)
is derived for a vector G(t), the components defined in Eq. [ 111;
[121
of which add up to the function G(t)
SOLID
ECHOES
IN
SLOW-MOTION
383
REGION
G(t) = W*exp[(io + ~)t].l, [I31 where o is a diagonal matrix with components that are the possible values of A in our case, rr is the transition rate matrix for transitions between these values, W = G(0) is a vector consisting of the time-independent probabilities of finding a system in one of the possible A values, and 1 is a vector with all components equal to unity. Along the same lines, one can derive the expression K(~,T) = W.exp{(io
+ 11)~) .exp{(-iw
+ ?r)(t - T)} -1.
[I41
It should be noted that K(t,O) = G(t)*, K(T,T) = G(T), and K(~,T) becomes G(t - 2T)* in the rigid limit rr + 0 where the solid-echo decay is identical with the FID except for a time shift by 27 (see Eq. [8]). The numerical evaluation of the expressions of Eqs. [13] and [14] is achieved by determining the eigenvalues of the complex symmetric matrix io + 1~. An appropriate computer program is described by Gordon and Messenger in Ref. (4). The simplest case of a 2 x 2 matrix is discussed in the following section. If A(t) depends upon time through a continuous Markov process a(t), the summations in Eqs. [13] and [14] are replaced by integrations over the fl space: G(t) = K(~,T) =
Here, G(R,&,t)
I
I
d%W(C$J
dfloW(fio)
I
I
[151
~fWWo,fLt),
dfl,G(%,R,,~)
I
d&G(Cb,,
iit, I - T)*,
[I61
is the solution of r171
subject to the initial condition GWo,Q,O) where 6(Q - Q,) is the 6 function
= W-4 - fro),
[181
and
is the rate for transitions R’ -+ sl,. The quantity P(R’lQ,t)di& is the probability of finding a system between fit and Q + dR, at time t subject to the condition that it was Cl’ fort = 0. For small-step rotational diffusion, the integral operation in Eq. [17] is replaced by the Laplacian differential
- mg-(n,) I dn’f(sl’)II(n’,n,) and P(fIo 1Cn,,t) becomes the solution P(Ro 1f-4,0)= &at - 00). LINESHAPE
of the rotational
WI diffusion
equation
for
CALCULATION
In this section we treat explicitly the simple case in which the slow motion merely consists of an exchange between two sites with frequencies
384 01
SPIESS
AND
=
6;
ij
+
SILLESCU
[211 Then the spectrum can be calculated analytically. The response to a single pulse, G(t), is obtained from Eq. [13] as described, e.g., in Refs. (II, 12): cos At + -
w2
=
sin At
0
-
6.
(A = (82 - wy,
G(t) = @~$-nt.2.
cl2 < CY),
(R2 = S”), (p = (W
WI
- fY)“2,
R2 > p’).
Here as in subsequent equations the first, second, and third rows apply for slow, intermediate, and fast exchanges, respectively, and the exchange parameter CR gives the probability of interchanging the two nuclei (6). The response to the solid-echo sequence, K(t,+ is obtained similarly from Eq. [14]: K(t,7) = e- iPt--P7)e--Rt.2 cos At + t
sin At + $
sin A7 sin A(t - 7)
1 + Rt + 2WT(f
X
- 7)
. 1231
cash pt + fl sinh pt + 262 sinh pi sinh p(t - T) P
P2
The spectrum Z(w) is obtained through Eq. [22], starting at time t = 0:
Fourier transform
462R Aw4 + 2Aw2(2R2 - P) + a4
Z(w) =
cf. also Ref. (6). Similarly FT of K( t ,T) starting .Z(W,T) as determined through the solid-echo J(w,r) it is convenient to bring K(t,T) into the t’ = t - 27. From Eq. [23] we then obtain, for fqt,7)
=
e-iGt’e-W
given in
(Aw = w - 0);
at time t = 27 gives the spectrum technique. In order to calculate same form as G(t) by introducing fast and slow exchange,
.3A
w
R
62
- -;;;- cos 2x7 + h sin 2Ar + s [
1
cos At’
w n [ 1 cl’ R 62 1 R 1 +
X
(FT) of G(t),
h2 sin 2x7 + x cos 2Ar
-;- cash 2pr + P
sinh 2~7 - 1
P
2
sin At’
cash pt’
P
sinh 2pr + -
P
cash 2~7
sin’h pt’
eeZRT. [25]
SOLID
FIG. 1. Calculated two sites for various row: distorted spectra
ECHOES
IN
SLOW-MOTION
385
REGION
powder spectra for a spin I = 1 system spectra jump rates R. Left row: “true” obtained through FT of solid echo with
undergoing rotational jumps between obtained through FT of FID. Right pulse separation 7 = 8.
Now FT starting at time t’ = 0 can easily be performed
to yield
clsin AT + -A62 sin2 hr A 1 A2 (1 + CW2 + Ao2r2 R A& + + - sinh pr sinh2 pr 2
(
cos A7 + -
.z(W,T) = Z(o).
2
cash pi
P
P2
The spectrum J(w,T) as obtained from the FT of the solid echo thus is given by the “true” spectrum Z(w) multiplied by a reduction factor depending on the pulse separation T. RESULTS
AND
DISCUSSION
As an example of practical importance we have calculated powder spectra for deuterons exchanging between two sites, where the unique axes of the two axially symmetric field gradient tensors form the tetrahedral angle. Spectra of this kind are expected, e.g., for a water molecule flipping by 180” about its twofold axis. Recent experimental examples are provided by motionally narrowed deuteron spectra of lipids and collagen (7,s) and of the amorphous regions of polyethylene
386
SPIESS
AND
SILLESCU
FIG. 2. Calculated powder spectra for a spin I = 1 system undergoing rotational jumps between sites for various jump rates a. Left row: “true” spectra obtained through FT of FID. Right distorted spectra obtained through FT of solid echo with pulse separation 7 = 8.
(13, 14). The powder spectra are obtained by superposition in Eq. [26] with frequencies given by Eq. [2]: wk = &/A,, = 3 co.!? 8k - 1,
k = 1,2.
two row:
of the spectra given
1271 The numerical calculation corresponds to that of “true” absorption spectra described in Ref. (16). In Figs. 1 and 2 the powder spectra as obtained through the solid-echo technique are plotted for various exchange rates fi. The values of fl and r given are normalized by division by AO; cf. Eq. [27]. In order to obtain absolute values for a given system the quadrupolar coupling constant has to be incorporated. For deuterons in aliphatic C-D bonds the conversion factors are approximately A,, = 4 x lo5 see-* and Ai1 = 2.5 +ec for R and T, respectively (16). The spectra presented in Figs. 1 and 2 thus correspond to 7 = 20 psec. For easy comparison the “true” spectra are also plotted; cf. also Ref. (11). Whereas both lineshapes Z(w) and J(w,T) are identical for very slow (0 =+ 1) and very fast (a % 1) exchange rates, severe distortions are introduced by using the solid-echo technique in the transition region (0.1 5 1R 5 10). Therefore, one is not justified in trying to determine jump rates 0 by comparison of experimental spectra obtained via the solid echo with theoretical ones, calculated from the FID. Moreover, in an attempt to analyze such spectra quantitatively, not realizing that they are distorted because of finite T, one can easily be led to incorrect conclusions, e.g., “explaining” the lineshapes observed as superpositions of motionally narrowed spectra of different exchange rates. In Fig. 3 the T dependence of the lineshape is shown for intermediate exchange (a = 1). It is clear that significant distortions occur even for small values of T, e.g., 7 = 2 3 5 psec. With increasing T, on the other hand, the lineshape almost
SOLID
R=0.59
ECHOES
IN SLOW-MOTION
REGION
387
:
FIG. 3. Calculated powder spectra obtained going rotational jumps between two sites. separation 7 is varied.
through FT of solid echo for a spin I = 1 system underThe jump rate I2 has a fixed value I2 = 1; the pulse
converges; note the minor change of the lineshape on increasing 7 from 8 4 20 psec to 16 !! 40 psec compared with the differences in the lineshape obtained from the FID (T = 0). The reason for this behavior can be visualized considering the expression for J(w,T) as given in Eq. [26]. The intensity of the spectra contributing to the powder lineshape for which the condition of slow exchange, R2 =CS2, is fulfilled is damped by the factor exp(-2&). For the same value of a, however, there are other spectra that contribute to the powder lineshape, for which the condition of fast exchange (W > a*) is met. These are damped by the factor exp[-2(LR - p)~] only, and, therefore, will dominate the lineshape obtained via the solid echo for finite 7. In Figs. 1 to 3 reduction factors R are also included, giving the ratio of the highest points in the “stick spectra” for a given T and T = 0, respectively. These reduction factors are above 0.48 for all the spectra presented here, showing that considerable spectral intensity can be detected via the solid echo throughout the transition region from slow to fast exchange for moderate values of 7. In reality, of course, the solid echo will further be damped by transverse relaxation, taken into account here only by convoluting the final powder lineshape with a Gaussian exp(-A&/2u2) with u = 27r. 1.27 kHz, corresponding to the dipolar coupling in perdeuterated samples (14, 16). In partially deuterated compounds there is an additional contribution from dipolar deuteron proton interaction (14). Our present lineshape calculation makes clear that in the case of a limited motional narrowing as provided by the exchange between two sites only, the solid-echo technique will be able to provide the possibility of obtaining FT spectra of deuterons not only in the limits of very slow and very fast exchange but in
388
SPIESS
AND
SILLESCU
the intermediate region as well. The lineshapes obtained via the solid echo, however, are severely distorted compared with the “true” spectra that could in principle be obtained from FT of the FID. For proper analysis of experimental lineshapes, therefore, theoretical spectra also have to be calculated by taking the FT of the response to the solid-echo sequence as described here. Solid echoes with acceptable reduction factors R 2 0.5 can only be expected for anisotropic reorientation where the rapid motion limit leads to a “quasi-rigid” spectrum, the total width of which is comparable with that of the rigid spectrum; cf. Fig. 2. For isotropic reorientation, the solid echo is drastically reduced if the pulse separation r becomes longer than the rotational correlation time 7C unless A,,T~ + 1. This has precluded the application of the solid-echo technique to lineshape investigations of tetrahedral jumps, e.g., in deuterated hexamethylenetetramine (16, 17). Isotropic small-step rotational diffusion causes similar reductions, as is apparent from the calculation of Woessner et al. (3). An interesting aspect of the lineshape distortions produced by the second 90” pulse can be seen in a perturbation expansion of the master equation for isotropic rotational diffusion. For relatively rapid motion A,,rC 5 0.5 the FID and the solidecho decay (see Eqs. [ 151 and [ 161) can be approximated by (18,19) ~(t,~)*
G(t) = (1 + 3a + p)eeAot + (p* + a)eeAZt, = ~(1 + 3,)e-c~~~+Au’h+ pe-ALle-hot’ + [~*~-A27 + ae-ch+&)~]e-hf’,
[281 [291
a = (415)A;r: = ?,lTz,
[301
/3 = -2a
1311
+ i*1.280312,
A, = T;‘(l
+ 0.37~~ + i-0.64d2),
[321
A2 = ~;l(l
- 0.69a - i .0.640?‘~).
[331
For rapid motion (Y << 1 and t 9 r, we have G(t)
= K(t,T)*
[341
= e-t’T2.
The most important corrections to this exponential arise from the imaginary terms in the expression Eq. [28] for the FID, whereas just these terms are reduced by factors e-“‘c in the “solid echo,” Eq. [29]. Since the imaginary terms describe the onset of solid-like behavior, one should be aware of their absence if rotational correlation times are determined from exponential solid-echo decays. A further comment relates to a short time expansion that has been used in the analysis of solid echoes in polymer solutions and melts (9, 20). Here, molecular reorientation is constrained through entanglements, and a “nonzero average” dipolar coupling A that amounts to about 1% of the original interaction contributes to a “pseudosolid echo.” It has been shown (9, 10) that the normalized signal r(t,T)
= i
Re
K(t,d
- G(t) G(O) I
used in the analysis of the experiments can be described approximation r(t,7) = 7(t - 7>(Az)e-B’,
[351 adequately
by the t361
SOLID
ECHOES
IN
SLOW-MOTION
389
REGION
where the angle brackets denote an ensemble average, and B is an empirical parameter. We can readily reproduce Eq. [36] for our example of two-site exchange if we let ti = 0, S2 = (AZ), and R = B in Eqs. [22] and [23]. Equation [36] is then valid for all times if R2 = S2 and it is obtained for small times t, subject to the condition hr -+ 1 if R2 < S2, or pr -G 1 if fin2 > a2, respectively. Equation [36] is also obtained for isotropic rotational diffusion if we expand Eqs. [28] and [29] for small times, and let Af = (A’) and (2~~)~~ = B. Thus, the parameter B can be related to the decay of the anisotropic spin interactions. Applications to 2D NMR in deuterated polyethylene melts will be published elsewhere (20). CONCLUSIONS
Fourier transform of the solid-echo decay leads to NMR lineshapes that differ from the true absorption spectrum in the slow-motion region 0.1 5 0 5 10. For anisotropic motion leading in the fast exchange limit to a “quasi-rigid” spectrum comparable in width with that of the rigid one the signal intensity of a polycrystalline sample for intermediate exchange likewise is comparable with that for slow or rapid exchange. The distortions of the lineshape introduced by using the solid-echo technique, however, have to be taken into account if such lineshapes are analyzed to give information about type and time scale of the motion. For isotropic reorientation, the solid-echo technique produces drastically reduced intensities for T > T, where the correlation time 7C is a few microseconds only in the slow motion region of typical deuterated compounds. ACKNOWLEDGMENT Financial
support
by the
Deutsche
Forschungsgemeinschaft
is gratefully
acknowledged.
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of