NMR relaxation study of molecular motions between unequal potential wells in solid trans, trans- muconodinitrile

We report observations of extremely unusual proton NMR relaxation rates in solid trans, transmuconodinitrile (TMD, N=C-CH = CH-CH = CH-C = N). In particular we measured, over the temperature range 77-423 °K, proton dipolar relaxation times Tw and spin lattice relaxation times Tt (at 24 and 58 MHz). The relaxation pattern is characterized by the following features: (a) very long motional r, and F 1 0 even at their respective minima, (b) no detectable motional narrowing of the line even at the Tl c minimum, (c) unequal slopes at temperatures below and above the minimum of T, (and r , 0 ) vs 1/7". and (d) significant deviations from the usual linear dependence on resonance frequency of the values of the relaxation times at their respective minima. We extended an earlier NMR theory to the case of spin lattice relaxation due to molecular reorientations between the extremely unequal potential energy wells of TMD. We were able to explain all features of the above data in terms of this theory. By comparing our data to the results of several calculations of intermolecular potential energy which used different interatomic force parameters, we were able to rule out some of these, thereby determining the best choice for the parameters in this crystal. The detailed structure of this potential profile (i.e., relative depths of the wells and energy barriers hindering rotation) was then determined from the Tt and Tw data. We thus have observed and characterized in TMD a low concentration of orientational defects which occur when a molecule occupies a higher energy well. Our observations are probably the first of such extreme NMR relaxation effects due to motions between significantly inequivalent sites.

NMR relaxation study of molecular motions between unequal potential wells in solid trans,trans-muconodinitrilea) Micha Polakb) and David C. Ailion Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 17 May 1977) We report observations of extremely unusual proton NMR relaxation rates in solid trans,trans- muconodinitrile (TMD, N=C-CH = CH-CH = CH-C = N). In particular we measured, over the temperature riinge 77-423 °K, proton dipolar relaxation times Tw and spin lattice relaxation times T, (at 24 and 58 MHz). The relaxation pattern is characterized by the following features: (a) very long motional r, and Tlo even at their respective minima, (b) no detectable motional narrowing of the line even at the rlo minimum, (c) unequal slopes at temperatures beiow and above the minimum of T, (and T,D) vs \/T. and (d) significant deviations from the usual linear dependence on resonance frequency of the values of the relaxation times at their respective minima. We extended an earlier NMR theory to the case of spin lattice relaxation due to molecular reorientations between the extremely unequal potential energy wells of TMD. We were able to explain all features of the above data in terms of this theory. By comparing our data to the results of several calculations of intermolecular potential energy which used different interatomic force parameters, we were able to rule out some of these, thereby determining the best choice for the parameters in this crystal. The detailed structure of this potential profile (i.e., relative depths of the wells and energy barriers hindering rotation) was then determined from the Tj and Tw data. We thus have observed and characterized in TMD a low concentration of orientational defects which occur when a molecule occupies a higher energy well. Our observations are probably the first of such extreme NMR relaxation effects due to motions between significantly inequivalent sites. I. INTRODUCTION The study of nuclear magnetic resonance (NMR) re­laxation times has proven to be one of the most power­ful tools for investigation of atomic motions. In par­ticular, temperature and resonance frequency depen­dences of the spin lattice relaxation times of nuclei in solids can provide detailed information concerning char­acteristic times of nuclear motions and associated ac­tivation energies. Molecular crystals studied by NMR relaxation times typically have ordered structures and, for the case of molecular reorientations, the axis of hindered rotation usually coincides with a symmetry axis so that motion occurs between crystallographically (and therefore en­ergetically) equivalent sites. Common examples are methyl's rotation about its C3 axis and benzene's rota­tion about its C6 axis. The detailed NMR relaxation theories for such cases are well established both in the high field "weak-collision" region,1 where the internu­clear dipolar interaction is treated as a motionally modulated perturbation, and in the low field "strong- collision" region,2 in which the relaxation times are sensitive to much slower motions. Normally, a sym­metric minimum in a plot of the temperature depen­dence of the spin lattice relaxation time is observed when the characteristic frequency of motion is com­parable to the resonance frequency. Furthermore, high field spin lattice relaxation times (7\) for proton reso­nance at the minimum are typically of the order of 1­100 msec, depending on the internuclear distances and a,This research waa supported by the U.S. National Science Foundation under grant DMR-76-18966. b,Present address: Division of Chemistry and Chemical En­gineering, California Institute of Technology, Pasadena, CA 91125. the orientation of the internuclear vectors relative to the axis of motion. Shorter times are obtained for rotating frame relaxation (T'ip) and even shorter for dipolar re­laxation (T1d). The T1D minimum usually occurs at temperatures corresponding to the onset of motional narrowing. Furthermore, fluctuations of the entire dipolar Hamiltonian occurring at a frequency of order 1 /Tz cause the value of TlD at the minimum to be com­parable to Tz. These features are also typical of molecular reorien­tations in plastic crystals, even though jumps may then occur between crystallographically inequivalent sites. Nevertheless, as long as motions occur in the orienta­tionally disordered crystals between energetically equal or very similar potential wells, one may expect the above mentioned general behavior to characterize the relaxa­tion. A third class of crystals is characterized by what can be called orientational defects. For these crystals a fraction of the molecules can occupy metastable orienta­tions corresponding to energies which are higher than that of the main minimum corresponding to the equi­librium orientation. When the energy difference be­tween inequivalent orientations is high compared to the thermal energy of the molecules, the population in a sub­minimum is relatively small and therefore undetectable by diffraction techniques. NMR, on the other hand, measures relaxation rates which are sensitive to the dynamics of the nuclei and, accordingly, may detect molecular jumps between such unequal potential wells. In a preliminary paper3 we noted that such motions may lead to anomalous relaxation rates which differ from those observed in the more "normal" cases of motions between equivalent wells. As predicted theoretical­ly, 4,5 NMR relaxation in unequal wells is relatively inef­ficient. This inefficiency arises from incomplete di- The Journal of Chemical Physics, Vol. 67, No. 7, 1 October 1977 Copyright © 1977 American Institute of Physics 3029 3030 M. P olak and D. C. A ilio n : NM R o f s o lid tra ns ,fra n s -m u c o n o d in itrile polar averaging resulting from the relatively short time the molecules spend in a subminimum compared to their longer stay at the low-energy orientation. Only few NMR relaxation studies known to us6'9 have been devoted to crystals of this type. Moreover, in the ma­jority of these studies the energy difference between wells was typically of the order of the average thermal energy. In this case the relaxation rates show nearly normal behavior. At the other extreme, where the well difference is far greater than kT, the population of the subminima may be too small for this mechanism to dominate the relaxation. The study reported here deals with trans, trans-m\i- conodinitrile (TMD) I I H H which, according to our results, is intermediate be­tween the above two cases. Specifically, in this crystal reorientations can occur between unequal potential wells whose differences in energy at the respective minima are considerably larger than kT but still not too big to provide a completely inefficient NMR relaxation mech­anism. Thus, the relaxation effects observed for the protons of TMD show that at room temperature about half a percent of the molecules occupy metastable ori­entations. The observed behavior exhibits the following unique features: (1) We observed extremely long spin lattic relaxation times (7\> Tx„, r1D) approximately three orders of mag­nitude longer than in the usual equal wells case. (2) No motional narrowing was observed in T2 mea­surements even at temperatures above the TVD minimum. Moreover, the value of ^1D at the minimum is about four orders of magnitude larger than the corresponding r2. This behavior is in contrast to the situation char­acterized by fluctuations of the entire dipolar energy resulting in T1D at the minimum being of order Tz. (3) We observed considerably different slopes above and below the minimum in the vs reciprocal tem­perature plot. (4) The frequency dependence of the relaxation time minima is weaker than the usual linear relationship. In Sec. V we analyze these peculiar data on the basis of a model of molecular jumps between unequal wells using an NMR relaxation theory developed for slightly different cases by Look and Lowe4 and by Anderson.5 Since their treatments are confined to equal barriers, we extended their equations to somewhat more general potential energy surfaces, such as those which occur for TMD's molecular rotations in the crystal. An independent calculation10 of the potential energy for reorientations of TMD using a van der Waals atom- atom potential function indicates that, when a TMD molecule reorients about its long axis, there exist in the crystal two subminima between three energy bar­riers of different heights. These theoretical results are only qualitative, primarily because the different van der Waals parameters introduced in the literature give somewhat different well depths and energy bar­riers. The analysis of the NMR relaxation data (Sec. V) provides a detailed quantitative description of the energy surface for molecular reorientations in solid TMD and thus allows one to determine the best set of van der Waals parameters for this case. II. THEORY A. NMR relaxation due to motions between unequal wells Motions between unequal wells constitute a distinct and unique NMR relaxation mechanism because such mo­tions may bring about a partial and correspondingly smaller averaging of the local dipolar fields than do mo­tions between symmetric minima. The limited fluctua­tion in local fields is directly related to the relative populations at the potential minima and is therefore temperature dependent. Thus, in addition to the usual effects of temperature on the correlation times char­acterizing the motions, temperature further affects the relaxation through its gradual effect on the degree of dipolar averaging. Theoretical studies of the problem were done by Look and Lowe4 for the case of motions in two unequal poten­tial wells and by Anderson5 for a three well potential. In particular, Anderson examined effects of reorienta­tions for a potential in which two of the minima have equal depth higher in energy than the equilibrium mini- ' mum (Fig. 1). For thermally activated motions the jump probabilities are given by fej =/f exp[-(i/+ A)/RT] (1) and fe2=/fexp(-H/RT). As we discuss in Sec. IV, the energy surface in an actual crystal may differ considerably from this relatively symmetric case. TMD exhibits a much more asym­metric potential and this was taken into account when using the expressions derived. Nevertheless, the basic assumptions and approach are still the same, namely, that the reorientation process consists entirely of in­dividual jumps to adjacent sites. Also, we assume that the energy barriers are the same for all molecules and further assume that there is no correlation between mo­tions of different molecules. FIG. 1. Three well potential energy profile for molecular re­orientation. In this model the jump probability fe2 is greater than k (. J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 M. P o la k and D. C. A ilio n : N M R o f s o lid fra n s ,fra /7 s -m u c o n o d in itrile 3031 Even though we are studying a dipolar minimum the low temperature side of which would normally be ana­lyzed by a strong collision theory, 2 we used a weak col­lision perturbation approach in evaluating the relaxation rates for the entire relaxation curves. This approach seems to be justified, since a molecule which jumps from a low energy minimum to a subminimum at a rate, say, r"1 ~ Tl1, will jump out of the subminimum in a much shorter time and thus cannot be described by a spin temperature prior to this second jump. In a later sec­tion of this paper we present experimental results which demonstrate the inapplicability of the strong col­lision theory. The dipolar and spin lattice relaxation effects due to the small fraction of the dipolar interactions which is modulated by the motion can be treated by the usual per­turbation theory. The correlation function for the modu­lation of the dipole-dipole interactions is evaluated by solving the differential equations characterizing the dy­namics of nuclei in the potential surface. Their solu­tion gives the time dependence of the probabilities of finding a molecule at each orientational site. Accord­ing to Anderson, the correlation function for the case of three equally spaced orientations (Fig. 1) consists of a linear combination of two exponentials with decays char­acterized by different correlation times. After powder averaging, the high field and the dipolar relaxation rates are given, respectively, by TV = yz{Mz)moi 3QiQz (t v 4ti 0' 1 t) 4 T, + %(l+4^071+l+4«grl )] (2a) and T = 1 ID 1 37 (M2)mod 2 (2b) In Eqs. (2) and Qz are the steady-state probabil­ities of finding a molecule at the equilibrium and meta­stable orientations, respectively, and are given in terms of the jump probabilities kx and kz by Qi kz + 2kx Qz '■ (3) The correlation times and t2 are given in terms of the same jump probabilities rl1=kz + 2kl and T?=3kz. (A^mod is the total motional change in the second mo­ment when the wells are equal5 (i. e., A = 0), a>0 is the high field Larmor frequency, and coD is the precession frequency in the local field HD and is given by uD=y [if the three well depths are equal, k1=kz=i3 rl1 = i rj1 and Q1=Qz = l/3, so Eqs. (2a) and (2b) reduce to the usual spin lattice relaxation rate ex­pressions. ] In case the energy barriers or the subminima depths are not equal, Eqs. (2a) and (2b) must be modified some­what, as we shall see in Sec. V. Nevertheless, the same general features characterize the expected re­laxation behavior: (1) Spin lattice and dipolar relaxa­tion times are longer than in the usual case of motions between equal potential wells; furthermore, larger en­ergy differences A between wells will result in longer relaxation times. (2) Motional narrowing is inhibited and, if the energy difference between the wells is far greater than kT, may be too small to be observed. As explained before, the origin of these phenomena is the decrease in rms fluctuations of the local fields with in­creasing well difference A, since a larger energy dif­ference causes a larger proportion of the molecules to occupy the low energy equilibrium position. B. Application to motions between extremely unequal wells (A 2>RT) Our potential energy calculations10 for TMD indicate that, at all temperatures up to the melting point, the condition A » RT is satisfied. As we shall now show, this feature results in rather dramatic effects on the relaxation. In the limit A » RT, kz is much larger than with the result that Qi = 1, - exp(- a/RT). (4) Then Eqs. (2) reduce to ,2/ n-l r Wmod 1 2 (t exp(-A/RT) Jz___ . 4 t2 _3(1 + U)Z0T j + 1 +4WqTi ) + + 1 + 4w2 *)] (5a) and -1 3y (Ma)mnd 2 T = 1 ID exp(-A/ijr)f---j-+---^5-j) . Vi+4u£tT 1+4o6ldj\) (5b) Furthermore, when A» RT, t1=3t2 = t0exp(H/RT). We can see several striking results of these equations: (1) Relaxation times become longer by the inverse Boltzmann factor exp(V.RT) compared to those for mo­tions between equal wells. (2) The slopes of a plot of the temperature depen­dence of each relaxation time are different above and below the minima. At temperatures above the minimum wt« 1 and Tu T1d °c exp[(A - H)/RT\, (6a) whereas at temperatures below the minimum wr » 1 and Tu TlD cc exp[(A + H)/RT] . (6b) Thus, a measurement of the high and low temperature slopes of the relaxation time vs temperature enables one to determine A and H separately. (3) The factor exp(-A/i?r) shifts the position of the minima. In Fig. 2 we calculated the value of (ur)mil as a function of A/H for both the 7\ and TlD minima. The mathematical form of (w0r)mln vs A/tf for a 7\ mini­mum is rather complicated and was calculated numeri­cally. On the other hand, for a TlD minimum (w0r)mln is given by J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 3032 M. P o la k and D. C. A ilio n : NM R o f s o lid fra /7 s ,fra n s -m u c o n o d in itrile PULSE SEQUENCE FOR T, MEASUREMENT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I A/H FIG. 2. Theoretical position of the relaxation minima vs A/H. The position is characterized here by (^lOnaa [or (^cr)mlJ for T, (or TlD). (UdT)„ (7) As we can see for both cases, the effect on unequal wells is to shift the relaxation time minima towards higher temperatures compared to their position when A =0. (4) A final interesting feature is that the factor exp(-A/RT), which causes an upward shift in the mag­nitude of all relaxation times, results in reduced fre­quency dependence of the value of T, (or at their minima. Since 1 / 7\ cc [exp - (A/RT)]/w0 and 1/ T1D cc[exp(-&/RT)\/ud at their respective minima, it is easy to see that for any relaxation time at its minimum, we have that l-< A/H) and (T\i))min °- (WZ)) ,l-< A/H) (8a) (8b) For unequal wells we thus have a reduced frequency de­pendence of the value of Tx (or T1D) at the minimum compared to the linear frequency dependence which holds for motions between equal wells (A/tf=0). This can result in a much shallower minimum for TlD than is usually the case (see Fig. 3). In Sec. IV of this paper we shall report experimental 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I A/H FIG. 3. Theoretical ratio of relaxation times at the minima (T1)inlll(24 MHz)/(T1B)mlD vs A/H. Note that, in the limit A-H, (T10)mIn approaches (T1)mln and thus the dependence on rf fre­quency disappears. (a) Saturating Pulses U 9CfPuise PULSE SEQUENCE FOR T, (or Tm ) MEASUREMENT -0.7 T,- 11D ' H f*- Variable Time (b) ir . FID Saturating Pjlses 90°Pulse 90°Phase Shift FIG. 4. Pulse sequences used in measuring relaxation times. observations on TMD which exhibit each of these in­teresting predictions. III. EXPERIMENTAL PROCEDURE A. NMR pulse sequence The spin lattice relaxation times were measured at rf frequencies of 24 and 58 MHz by applying a train of a few close-spaced 90° saturating pulses followed at a variable time interval by a single probing 90° pulse [Fig. 4(a)]. The train of closely spaced pulses destroys all residual magnetization in a relatively short time, thereby eliminating the need for waiting several 7\'s between measurements as in the conventional 180°-90° or 90°-90° sequences. This technique is particularly advantageous when 7\ is long, as in the case of TMD for which the protons' at 24 MHz is 40 min at 77°K! The same method of preparative saturation was used in measuring the dipolar relaxation time Tio and the ro­tating frame spin lattice relaxation time in order to avoid a long delay between cycles. The Tlp (or TiD) measurements were then made by waiting a constant time of about 0. 7Tj after the saturating pulse train and then applying a"spin-lock" sequence (90° pulse-90° phase shift) followed by adiabatic demagnetization (ADRF) to a low (or zero) value. After waiting a variable time, Tu (or Tw) is measured by adiabatic re­magnetization along H^ and then observation of the mag­nitude of the initial signal following a sharp turnoff of Hx as a function of time in the demagnetized state (see Fig. 4(b)]. In some preliminary Tw measurements we used adiabatic frequency modulation11 instead of the spin- lock method in order to tilt the magnetization into the direction. This frequency modulation was followed as before by ADRF of Ht to zero field and subsequent remagnetization. Exact resonance is critically im­portant in measuring TlD and is achieved by adjusting H0 to give no signal for zero-field times much greater than T1d. J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 M. P o la k an d D. C. A il io n : NM R o f s o lid tra ns jra ns-mu conod 'm'WrWe 3033 FIG. 5. Measured relaxation times vs inverse temperature in polycrystalline TMD. Open and black circles refer to T]D of sample I before and after exposure to elevated (~ 150 °C) tem­peratures, respectively. The crosses are Tw values for sam­ple II, which was not exposed to elevated temperatures. Squares and triangles refer to Ti of sample I at 24 and 58 MHz, respectively. B. Characteristics of TMD sample The powder sample used in the experiments de­scribed in this paper was prepared at the University of Tel-Aviv by a procedure described in detail else­where. 12 The muconodinitrile Was synthesized in a two step gas-solid reaction. The trans, trans isomer was then separated from the other isomers by recrys­tallization from ethanol and identified by its melting point (159 °C). The extremely long 7\ values which we observed at low temperatures indicate that the sample had a rela­tively low impurity content. Upon excessive heating the white powder turns gradually darker, probably due to polymerization. IV. EXPERIMENTAL RESULTS In order to obtain information about the detailed shape of the potential energy surface, we measured proton spin lattice relaxation times of polycrystalline TMD over wide ranges of resonance frequencies and temperatures. The temperature dependence of T1D and of Ti at 24 and 58 MHz are plotted in Fig. 5 and exhibit a number of in­teresting features, which we summarize below. In the next section of this paper we will discuss the results in terms of the theory presented in Sec. II. (1) The observed relaxation rates are extremely low even at the and Tw minima. Thus, the minimal Tw value is 0.16 ±0.01 sec, compared to more typical val­ues of 0.1-1 msec. (2) Within the accuracy of the FID measurements Tz appears to be temperature independent (13±2 jjsec). Thus, using this technique, we didn't observe any de­tectable motional narrowing even at temperatures cor­responding to the T\d minimum, which is extremely long compared to Tz. The second moment of the ab­sorption line at the temperature of the Tlp minimum (approximately room temperature) was measured earlier by cw and pulse techniques.12,13 Only a very small re­duction (~8%) from the calculated rigid-lattice value was observed. It was attributed to librational effects. (3) The slope of vs 1/T above the minimum is considerably smaller than the slope at temperatures below the minimum, as can be seen in Fig. 6. The same effect may possibly be seen when comparing the slope of Tid vs 1/T below its minimum to the slope right above the minimum; however, it is difficult to estimate an accurate value for this high temperature slope since another mechanism, which becomes domi­nant at about 385 °K, causes T1D to drop sharply with temperature. At temperatures below ~200 °K there is a considerable decrease in the slopes of both 7\ and Tw, which indicates that a different mechanism dominates the spin lattice relaxation over the range from this tem­perature down to the lowest temperature measured in this study (liquid N2). Therefore, in order to eliminate the contributions of this mechanism to the higher tem- 1000 100 C 0 1 10 2.3 2.5 2.7 2.9 3.1 3.3 3.5 io3/t FIG. 6. Motional contribution to T, vs inverse temperature (expanded scale). The discontinuity in the Tt's at IQ3/T = 3.190K*' is indicated by the vertical dashed lines. Note the differences in slope below and above the minima. TMD Spin Lattice Relaxation Times ♦ * $ 24 MHz i ♦ 58 MHz : } !* I $ Sj £ $ ♦ i i____i____iii i J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 3034 M. Polak an d D. C. A ilio n : NM R o f s o lid tra ns ,tran s-m u c o n o d in itr ile l.0< 0.8 0.6 3 0.4 5 \ I 0.2 0.1 * Tn MAGNETIZATION DECAY 'ID f r AT 152° K 10 15 20 t (sec) 25 30 (J_) '-*1,10 'motion 1 1,1D \ *1,1 (9) 1L 'extrap Thus, for data analysis we used relaxation times (Ti,Emotion obtained from Eq. (9) rather than the ob­served values, [in Fig. 5 we plotted observed values, whereas in Fig. 6 we used (T^ouon values corrected according to Eq. (9). ] The slopes were evaluated by assuming thermally activated processes (Table I). Be­cause of the differences in slopes between Tx and T1D, it appears that they probably reflect different mecha­nisms. Furthermore, the 7\ and TlD minima occur at surprisingly close temperatures for them to reflect the same mechanism, specifically, 305 °K vs 345 °K for (T'loUn and (Ti)mln (24 MHz), respectively. (4) The frequency dependence of the spin lattice re­laxation times at the respective minima is smaller than that usually observed MHz) (r1)mln(24 MHz) = 1.13 ± 0. 07, (T1)mlll(24 MHz) = 94 ±9. (5) The T1d values measured at the lowest tempera­ture range (below 200 °K) seem to depend on the sam­ple used and whether it was previously exposed to ele­vated temperatures (~ 150 °C). Moreover, unlike the results at higher temperatures, the magnetization in this region didn't go to zero exponentially as the demag­netization time was increased in the T1D measurement (see Fig. 7). (6) A break in the slope of appears at around 40 °C (10VT equals 3.19 °K"1) in both the 24 and 58 MHz data. In Fig. 6 7\ values corrected for the low- temperature mechanism are plotted with an expanded 1/T scale in order to display the observed small (~20%) discontinuity. In the next section we will present pos­sible explanations for this effect and the one mentioned TABLE I. Experimental values3 of H - A and H t A. H - A (kcal/mole) H + A (kcal/mole) Relaxation (temperatures (temperatures time above minimum! below minimum) T\d 1. 6 ±0.2 3± 1 7,7 ±0.5 llil FIG. 7. Magnetization in TMD vs time in the demagnetized state following the spin-lock ADRF pulse sequence of Fig. 4(b). perature relaxation, we had to subtract from the observed rates the extrapolated relaxation rates due to that mech­anism "These were obtained for TMD from experimental slopes at temperatures above and below the minima of logX) (or T1D) vs 1/T. in (5), neither of which are related to the main relaxa­tion mechanism. In order to investigate further the mechanism re­sponsible for the TlD minimum, we measured the rotat­ing frame spin lattice relaxation time for different values of the rf field Hx at two different temperatures, below and at the T,, collision theory2, dence on H Hi According to the strong should have the following depen- (10) Normally, this theory would apply only to the region be­low the T'ip minimum. As can be seen in Fig. 8, Tlp fol­lows the H\ law at both temperatures. However, for the temperature below the TlD minimum (250 °K) we see that the horizontal intercept occurs at a field signifi­cantly higher than the local field HD. The square of the local field was estimated to be 1/3 of the experimental second moment of the TMD protons' absorption line12 (3. 3± 0.1 g2) despite the fact that this value was mea­sured at 300 °K. We felt justified in neglecting changes due to temperature in the effects of librations on the second moment,13 since they do not depend strongly on temperature. Accordingly, we used at 250 °K the value 1.1 g2 for H%. (j 3 - <11 H( Dependence of f| at 250 ° K • at 295 0 K / / / /• 4* • 11 ■ 11111111111111111111111111111111111 10 15 20 25 30 35 40 45 Hf (GZ) Rotating-frame relaxation time T1p vs H \ (square of D FIG. _ rf field amplitude). Hq is the square of the dipolar local field measured in Ref. 12. The data at 250° (open circles) do not have a horizontal intercept at -H2D, thereby indicating the in­applicability of the strong collision theory. J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 M. P o la k a nd D. C. A il io n : NM R o f s o lid tra n s ,tra n s-m u c o n o d in itr ile 3035 V. DISCUSSION A. NMR determination of TMD well differences and energy barriers The anomalous NMR relaxation pattern observed for TMD's protons above ~200°K is indicative of nuclear motion between unequal potential wells as discussed in Sec. II. This experimental evidence is consistent with results calculated for the potential energy surface which in turn suggest the existence of such wells when a TMD molecule is reorienting around its long axis.10 As pre­dicted earlier by Look and Lowe4 and by Anderson,5 such a relaxation mechanism is less efficient than motions between equal wells because in the unequal case a smaller fraction of the dipolar local field is modulated by the motions. This leads to longer spin lattice re­laxation times and to inhibition of the apparent motional narrowing. A rough estimate of the fraction of the di­polar interaction which is averaged out by TMD's mo­lecular motions can be obtained from the experimental ratio (T1D)min/(Tz)Rh. According to Eq. (2b) rjj, at the minimum is given by . {M'z\ . ( (hd) D 'mod (ID RL where {HD)RL and (HzD)moi are, respectively, the rigid- lattice dipolar local field and the change in the square of the local field due to motions. [Note that (Af|)mod is appropriate to wells which may be unequal and would then be smaller than the (A/2)mod defined earlier. For A»RT, (M^)mods(M2)modexp(-A/«r). we have that Since (r2)j (^ln)mln ( ^)hl (12) TMD's potential surface. The calculated energy profile indicates that the barrier corresponding to a relative orientation of 90° is considerably higher in energy than the other two barriers. Within the uncertainties of the calculation we are left with the possibilities that the central barrier is either higher (as in Fig. 9) or lower than the barrier at 260°. (The experimental evidence for the existence of two distinct relaxation mechanisms rules out the possibility of a two well potential in TMD, even though such a result could be obtained with one of the possible van der Waals parameter sets.15) Consis­tent with the evidence for large Aj(Aj »RT) we assumed that the jump probability k5 is significantly larger than fe6 (see Fig. 9). Since the barrier at 90° appears to be quite large, we further assumed that the jump probabil­ities ki and k2 are considerably smaller than the rest. With regard to the relative heights of the barriers at 150° and 260° we found that only the potential surface consisting of a higher central barrier could fit the NMR data consistently. Figure 9 shows our model for the potential profile for reorientation of a single TMD molecule. Since the smallest barrier is at 260°, we found that it was reasonable to assume that k5» k3, kA. Using this model for the potential profile, we then pre­dict the existence of two distinct correlation times for TMD and thus, in principle, the occurrence of two mini­ma each in 7\ and in Tw. In particular, the two rele­vant correlation times for these unequal wells turn out to be ik;1 =T0exp(Hl/RT) and r2 = k 3_1 = r £ exp(Hz/R T), (13a) (13b) For the usual situation in which there is appreciable mo­tional narrowing (H2D)moi~ (HDfBh and (TiD)mln~ (T2)RL. We observed for TMD a ratio of 1. 3x 104, so only about 10"4 of the square of the local dipolar field (or second moment) is averaged out at room temperature by the motions! In contrast, we estimated a 30% reduction in the second moment of polycrystalline TMD in case the motions were between three equal wells along the same axis. The large difference between this result and our observations indicates the existence of a large energy difference between the main minimum and the higher subminima of the potential curve. As discussed in the previous section, the data suggest that the Tt and TlD minima reflect different mechanisms. Furthermore, calculated potential energies10 suggest that the three energy barriers (and subminima depths) in TMD may be different in size. Accordingly, these differences should be taken into account when applying the relaxation equations to the present case. Fortu­nately, the spacing of the wells' minima in TMD occurs fairly close to 120° and 240° as in Anderson's theory. The general case of the three well potential was treated theoretically by Hoffman14 in his study of the expected effects of reorientations between unequal wells on di­electric relaxation. His general expressions for the time dependence of the wells' populations should, how­ever, be transformed to fit the particular shape of where t2 is significantly longer than rt. (It should be noted that we observed only one 7i minimum and one TlD minimum, for reasons to be discussed shortly.) We then generalize Eqs. (5a) and (5b), respectively, to give for this case -1 _ Y (^2)mod 1 2 + exp(- A 2/RT) (t - a-?+ + <rf 1 + 4w0t 2 )] (14a) FIG. 9. Proposed three well model for potential energy vs orientation about its long axis of a single molecule in crystal­line TMD. Jump probabilities between different wells obey the following relations: kl,k2«k:s,ki,ks,ke and k5»k3,ki,k6. J, Chem. Phys., Vol. 67, No. 7, 1 October 1977 3036 M. P o la k and D. C. A ilio n : NM R o f s o lid tra ns ,tran s -m u c o n o d in itr ile TMD Molecule trans, Irans - Muconodinitrile FIG. 10. Structure of a TMD molecule in the solid. Indicated are intramolecular interprotonic distances as well as the direc­tion L of the inertia axis corresponding to the axis of molecular reorientation. The proton-proton distances were evaluated from the x-ray determined atomic positions of Filippakis et al. 24 and 3r2(M2)„ + exp(- A, exp(-Aj/i?r) (--^V-s) V1 + 4Ti * (14b) where and r2 are given by Eqs. (13). Accordingly, the Tx minimum seems to correspond to jumps over the low barrier Hu whereas the TlD minimum would reflect motions over the higher central barrier Hz. Hence, the four quantities of interest A1} and A2, Hz can be evaluated from the 1\ and T1D data, respectively. Equa­tion (6a) would apply to Tt relaxation at temperatures above the minimum 7\ cc exp[(At - HJ/RT], while for temperatures below the minimum Eq. (6b) gives Ty cc exp[(At +Hl)/RT\. Similarly, the high and low tem­perature slopes of T\d correspond to (A2 - Hz) and (A2 + HZ), respectively. So, both the energy difference A between the equilibrium and a metastable state and the activation energy .ff fora molecular jump from the me­tastable state are obtained individually from the sum and difference of slopes below and above the minimum. The results evaluated in this way (Table II) reveal that in­deed Aj + A2 and Ht * Hz. As we have seen in Sec. II the wells' relative depths Aj and A2 can be evaluated also from the experimental Tx and T1D minimum values, respectively. Using the graphs in Fig. 2 together with the experimental A/H ratios, we found that for both relaxation times in TMD a minimum should occur when (^r)min~ 0. 2. In order to evaluate A from the value of the relaxation time at the minimum using Eqs. (14), it is also necessary to calcu­late the expected total change in second moment (A/2)moci due to motion around the same axis between equal wells. Using a well known formula,16 we calculated the change in the intramolecular contribution due to such motions to be only 0.65 g2 (a 30% reduction). The smallness of the reduction is due mainly to the fact that the most sig­nificant internuclear vectors (i. e,, the short vectors be­tween the cis protons in Fig. 10) are almost parallel to the axis of motion and, accordingly, their orientations are not changed much by the motions. Assuming a simi­lar reduction in the intermolecular contribution we get that (M2)m0(j equals 1 g2. The values of (T1)miI1 and (T1D)min obtained from experiment together with the above (M2)mod and (wr)mln give a value for At of 3. 2 ± 0. 2 kcal/mole and for A2 a value of 4. 2 ± 0. 2 kcal/ mole, respectively, which agree very well with the results of the slopes' analysis (see Table II). Another interesting aspect of the nuclear relaxation induced by motions between unequal potential wells which we predicted in Sec. II is the weaker dependence of the relaxation time minimum on the resonance fre­quency [Eq. (8)]. Again, the dominant factor is A/H and, for high field relaxation times such as at 24 and 58 MHz, the ratio of their respective minimum values should obey _ (15) (T1)mln(24 MHz) As can be seen in Fig. 6 the experimental values for (?i)min differ by only 13%(compared to an expected 240% if A = 0)! This result leads to A/H = 0. 86 ± 0. 07, which is within the limits of the experimental error of the val­ue obtained from the slopes' analysis (Table II). Such a ratio may explain the reason for our not having ob­served at lower temperatures a minimum in Tw corre­sponding to the same mechanism as that responsible for the observed 7\ minimum. The calculated ratio ( ^1)min ^ (r1D)ml„ is plotted as a function of AjH in Fig. 3. Note that (r1D)ml„ becomes comparable to (Tt)mia as A becomes TABLE II. Orientational structure parameters1 of potential wells in TMD obtained from NMR mea­surements of relaxation times. Method Ai Hi ,i'2 A]/-^| Ag/-^^ Tt(24 MHz) vs 1/T (slopes) 3.1±0. 3 4.6 ±0.3 0. 7±0.1 (T^ (24 MHz) 3.2 ± 0.2 Txd vs 1/T (slopes) 4±1 7 ± 1 0. 6 ± 0. 2 4. 2 ± 0. 2 5 ± 1 (T1)mln(24 MHz)/(T,)mlB (58 MHz) 0.86 ± 0. 07 "In particular, A( and A, refer to energy differences between the equilibrium and metastable states and //[ and H2 correspond to barrier heights measured from these metastable states (see Figs. 9 and 11). All energies are in kcal/mole. J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 M. P o la k a n d D. C. A il io n : N M R o f s o lid fra n s ^ ra /is -m u c o n o d in itr ile 3037 FIG. II, IntermoLecular potential energy in solid TMD as a function of the orientation of a molecule around its long axis. The solid line is the profile calculated in Ref. 10 using atom- atom "6-exp" potential parameters from Refs. 17 and 19, the dashed curve is based on parameters from Refs. 17 and 18, and the dash-dotted line is based on parameters from Refs. 18 and 20. The dotted curve between experimental points repre­sents schematically the energy profile obtained from our NMR data and is similar to the curve in Fig. 9. It should be noted, however, that no independent measurements were made which determined the horizontal positions of these points; the sug­gested positions indicated in this figure give maximum consis­tency with theory. closer in magnitude to H. (Actually, the model used here predicts the disappearance of a relaxation mini­mum when A>HI Thus, the existence of a minimum in our Tx and T\d results means that, for both mech­anisms, A<H. ) Since (7'1)mtn(24 MHz) is only 15 sec, it is possible that the mechanism which dominates 7\ at lower temperatures (<200°K) also dominates and thus prevents us from observing a similarly long or, at best, leaves a very shallow, almost un­observable minimum (Fig. 5). Finally, A2 can also be estimated from the experi­mental ( TlD )min/ Tz ratio using Eq. (12). Thus, a rough estimate of A2= 5± 1 kcal/mole is obtained, in approxi­mate agreement with values for A2 obtained by the other procedures. Both At and A2 should be used to estimate the expected high-temperature (i. e., 7U r2« T2) ef­fects of the molecular reorientations on the protons' second moment. At 20 °C, however, only motions in­volving Aj are effective. We calculated, for motions between the unequal wells of TMD, reductions in the second moment of about 0. 5% at 20 °C and 2. 5% at 150 °C, which are small compared to the 6% predicted reduction due to lattice vibrations at room tempera­ture. 12 The larger reduction at 150 °C is due to the in­creased populations of both metastable states at the higher temperature. The fact that these effects on Tz should be even smaller explains the reason that no such motional narrowing was detected. Nevertheless, it is possible that careful measurements of the second mo­ment as a function of temperature up to the melting point may reveal the combined effect of lattice vibra­tions and molecular reorientations between TMD's un­equal potential wells. B. Comparison of NMR results with potential energy calculations In Fig, 11 we present potential energy profiles cal­culated10 using different possible van der Waals param­eters17- 20 for the 6-exp potential function along with our experimentally measured values for At, A2, Hu and Hg,. Our measurements resulted in absolute determina­tions of these quantities without adjustable parameters; only the reference energy at 0° orientation was made to fit the profile obtained with parameters from Refs. 17 and 19. The NMR results appear to be somewhat in­termediate between the three calculated profiles, demon­strating that none of the sets of van der Waals param­eters is completely suitable for TMD reorientations. The calculations indicate that the barriers at 90° and 260° are due primarily to C • • • H interactions, where­as the central barrier is more sensitive to N • ° - H in­teractions. Since there is greater uncertainty in the literature15'17'20 regarding N • • • H van der Waals pa­rameters, it is not surprising that the calculations do not show agreement in this central region. Further­more, the 6-exp potential function may be a poor ap­proximation here since the nitrogen and hydrogen atoms on neighboring molecules may overlap appreciably for these intermediate orientations. As can be seen, the ex­perimental energy differences between the low-lying equilibrium and the metastable states agree well with the corresponding results of potential energy calcula­tions using van der Waals parameters from Refs. 17­19. In view of the approximations involved in such semiempirical calculations, we view the general agree­ment to be quite satisfactory. It is interesting to note that our experiments enable us to rule out for TMD a profile calculated10 using ni­trogen parameters of Kuan et al.,15 since it predicts well-differences of 12-13 kcal/mole, which are much higher than our experimental values of 3-4 kcal/mole. Furthermore, with such high energy subminima as those calculated with these parameters, no NMR effects would have been observed due to motions between the wells, since the low-temperature mechanism would then have dominated the relaxation up to the melting point. It seems that the N • • -N parameters, no NMR effects unrealistically high repulsive interactions between hy­drogen and nitrogen atoms in TMD. The failure of the use of these parameters in TMD may be due to the fact that they were obtained from data of only one crystal, a-N2, whereas the carbon and hydrogen parameters of Refs. 18 and 19 were evaluated from data of many hy­drocarbons. C. Other features of NMR relaxation in TMD There are in the relaxation pattern a few features which are not necessarily associated directly with mo­J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 3038 WI. P o la k an d D. C. A ilio n : NM R o f s o lid fra n s ,fra n s -m u c o n o d in itr ile tion between inequivalent sites. These are the low-tem- perature relaxation (T<200°K), the small drop in Tx at 312-314 °K, and the sharp decrease in Txd from about 385 K up to the melting point. The relaxation pattern observed in the low temperature region suggests that the nuclear spins are probably relaxed via electron paramagnetic impurities. As can be seen in Fig. 7, the magnetization's spin lattice decay in zero field is nonexponential at these temperatures, with a fast initial decay of a small fraction of the dipolar order followed by a slower decay of the residual larger magnetization. Similar Tw effects were observed in CaF2 doped with paramagnetic impurities.21 Support for our assumption that the dominant mechanism for proton relaxation in this temperature region is due to paramagnetic elec­trons comes from the apparent dependence of TlD at low temperatures on the sample used and on whether it was exposed to elevated temperatures (see Fig. 5). Ex­cessive heating caused sample deterioration which was observed through the appearance of a yellow color in the normally white material. If we assume that the paramagnetic species which causes nuclear relaxation in TMD is free radicals and further assume that the observed deterioration is due to polymerization, we can explain the observed increase in low temperature re­laxation times for samples which had been heated. Since polymerization involves a reduction in the free radi­cals' concentration, such a deterioration of the material will cause an increase in the relaxation times. We de­tected such a trend at liquid N2 temperature, for ex­ample, where T1D was observed to increase from 24 ±4 to 66 ± 10 sec after exposure to temperatures of about 150 °C. The extremely long relaxation times ob­served at the lower temperatures indicate that the con­centration of paramagnetic impurities (e.g., free radi­cals) in our solid TMD sample is rather low. Figure 6 exhibits a sudden drop in 7\ at 312-314 °K at both 24 and 58 MHz. The fact that the temperature of the 7\ discontinuity appears to be independent of fre­quency suggests that the behavior may be due to a phase transition. However, we are not aware of any x-ray or heat-capacity data at the relevant temperatures which could verify this possibility. At any rate in both the 24 and 58 MHz data the observed effect is rather small (less than 20%) and is not reflected at all in the T1D minimum. Also, the slopes of Tx (58 MHz) vs 1/T be­fore and after the "break" seem to be identical. These results indicate that the previously discussed motions of TMD molecules probably do not participate to any ap­preciable extent in the possible occurrence of a phase transition in this crystal. Moreover, the invariance of the slopes require that the two structures would prob­ably have to be characterized by very similar well depths and energy barriers for reorientation. From the 7\ (24 MHz) data below the minimum and the experi­mental Aj we could estimate correlation times for re­orientation from the 230° well (see Fig. 12). r0 for the room temperature structure was evaluated to be (2.6 ±0. 5)xl0‘12 sec by extrapolation of the corresponding r's and is similar to the value (1.6±0. 5)xl0"12 sec ob­tained for the high temperature structure from the posi­tion of the Tx minimum, assuming the same activation energy. The r0 value evaluated for the 120v well from the (T1D)mln position is longer, (6±3)xl0‘u sec. The sharp decrease in TlD starting at about 385 K up to the melting point (432 °K) is characterized by a slope of 26 ±3 kcal/mole. Similar large activation energies are associated typically with translational diffusion of entire molecules in the crystal. 22,23 In many systems diffusional activation energies and lattice sublimation energies are found22 to be typically in the ratio of about 2. 0-2. 5. A comparison of our measured high-tempera- ture activation energy (26 ± 3 kcal/mole) with the cal­culated10 lattice energy (9-10 kcal/mole) gives a similar ratio for TMD, thus supporting the idea that the high temperature mechanism is translational diffusion of the TMD molecule. Such motions usually average out large portions of the dipolar local fields and result in T1D minimum values of the order of T2. Due to the crys­tal's melting temperature being at 159 : C, the mini­mum in T1d corresponding to the above process could not be observed. At any rate the relatively short TiD values observed prior to the melting of TMD suggest a strong-collision diffusion process between equal poten­tial wells. VI. CONCLUSIONS This paper demonstrates that NMR relaxation may provide a powerful tool for studying asymmetric poten­tial surfaces in solids. By modifying and applying the basic NMR theory4'5 to each particular system, it is pos­sible to evaluate from the relaxation data such quanti­ties as energy barriers for molecular reorientations and relative depths of potential wells. It should be noted that, although motions between unequal wells lead to a unique and distinct relaxation pattern, they do not constitute a very efficient relaxation mechanism, especially when the wells differ considerably in depth so that only a very small fraction of the dipolar field is modulated. Consequently, this mechanism will often be masked by other mechanisms such as motions be­tween equal wells or paramagnetic relaxation. There­fore, in order to study orientational defects by NMR, it is generally desirable to have very pure crystals com­posed of molecules which do not have easily rotating symmetric groups. >.£ a.- I03/T (° K'1) FIG. 12. Correlation times t, vs inverse temperature for re­orientations from the 230° well. The values for were ob­tained from the T,(24 MHz) data using Eq. (14a). J. Chem. Phys., Vol. 67, No. 7, 1 October 1977 M. P o la k a nd D. C. A ilio n : N M R o f s o lid tra ns ,tran s -m u c o n o d in itr ile 3039 Calculations of potential energy surfaces, such as were done for TMD,10 are desirable either in predicting possible motions in solids or in identifying the actual mechanism of experimentally detected motion. How­ever, it should be realized that such semiempirical calculations in their present state of art can yield only qualitative information, especially when close contacts and strong repulsion between atoms are involved. Nevertheless, as we have demonstrated for TMD, the combination of NMR relaxation measurements of re­orienting molecules and complementary potential en­ergy calculations can detect orientational defects in crystals and lead to a better understanding of their de­tailed structure. Furthermore, as we have shown, such studies may allow different models for interatomic forces to be distinguished in a particular molecular crystal. ACKNOWLEDGMENTS We wish to thank Professor M. Mehring, Dr. D. Wolf, and Dr. D. Paquette for useful discussions. We are grateful to Dr. S. Wei and particularly to Dr. H. Stokes for their generous assistance in the experiments. !N. Bloembergen, E. M Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948); D. C. Look and I. J. Lowe, J. Chem. Phys. 44, 2995 (1966). 2C. P. Slichter and D. C. Ailion, Phys. Rev. A 135, A1099 (1964); D. C. Ailion, Adv. Magn. Reson. 5, 177 (1971). 3M. Polak and D. C. Ailion. J. Magn. Reson. 26, 179 (1977). 4D. C. Look and I. J. Lowe, J. Chem. 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